Heat kernel expansion: user's manual

Heat kernel expansion: user's manual
Heat kernel expansion: user’s manual
D.V. Vassilevich a,b
a Institut
b V.A.
f¨
ur Theoretische Physik, Universit¨
at Leipzig, Augustusplatz 10, D-04109
Leipzig, Germany
Fock Insitute of Physics, St. Petersburg University, 198904 St. Petersburg,
Russia
Abstract
The heat kernel expansion is a very convenient tool for studying one-loop divergences, anomalies and various asymptotics of the effective action. The aim of this
report is to collect useful information on the heat kernel coefficients scattered in
mathematical and physical literature. We present explicit expressions for these coefficients on manifolds with and without boundaries, subject to local and non-local
boundary conditions, in the presence of various types of singularities (e.g., domain
walls). In each case the heat kernel coefficients are given in terms of several geometric
invariants. These invariants are derived for scalar and spinor theories with various
interactions, Yang-Mills fields, gravity, and open bosonic strings. We discuss the relations between the heat kernel coefficients and quantum anomalies, corresponding
anomalous actions, and covariant perturbation expansions of the effective action
(both ”low-” and ”high-energy” ones).
Key words: heat kernel, functional determinants, effective action, boundary
conditions, anomalies
PACS: 04.62.+v, 11.10.-z, 02.40.-k
Email address: [email protected] (D.V. Vassilevich).
Preprint submitted to Elsevier Science
9 September 2003
Contents
1 Introduction
3
2 Spectral functions: heat kernel, zeta function, resolvent
2.1 Differential geometry and the operators of Laplace type . . . . . . .
2.2 Spectral functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Lorentzian signature . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 Relevant operators and boundary
3.1 Scalar fields . . . . . . . . . . . .
3.2 Bosonic string . . . . . . . . . . .
3.3 Spinor fields . . . . . . . . . . . .
3.4 Vector fields . . . . . . . . . . . .
3.5 Graviton . . . . . . . . . . . . . .
conditions
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4 Heat kernel expansion on manifolds without
4.1 General formulae . . . . . . . . . . . . . . . .
4.2 Examples . . . . . . . . . . . . . . . . . . . .
4.2.1 Yang–Mills theory in flat space . . . .
4.2.2 Free fields in curved space . . . . . . .
4.3 DeWitt iterative procedure . . . . . . . . . .
4.4 Non-minimal operators . . . . . . . . . . . . .
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boundary
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5 Heat kernel expansion on manifolds with boundaries
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5.1 Two particular cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.2 Dirichlet and Neumann boundary conditions . . . . . . . . . . . . . 46
5.3 Mixed boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 49
5.4 Other boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 51
5.4.1 Boundary conditions with tangential derivatives and
Born–Infeld action from open strings . . . . . . . . . . . . . . 51
5.4.2 Spectral or Atiyah–Patodi–Singer (APS) boundary conditions 53
6 Manifolds with singularities
6.1 Non-integrable potentials . . .
6.2 Conical singularities . . . . . .
6.3 Domain walls and brane world
6.4 Non-smooth boundaries . . . .
6.5 Dielectric bodies . . . . . . . .
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7 Anomalies
7.1 Conformal anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Chiral anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Remarks on the Index Theorem . . . . . . . . . . . . . . . . . . . . .
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8 Resummation of the heat kernel
8.1 Modified large mass expansion
8.2 Covariant perturbation theory .
8.3 “Low energy” expansion . . . .
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expansion
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2
8.4
Heat kernel on homogeneous spaces . . . . . . . . . . . . . . . . . . .
79
9 Exact results for the effective action
9.1 The Polyakov action . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2 Duality symmetry of the effective action . . . . . . . . . . . . . . . .
83
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10 Conclusions
90
3
1
Introduction
It was noted by Fock [?] in 1937 that it is convenient to represent Green
functions as integrals over an auxiliary coordinate (the so-called “proper time”)
of a kernel satisfying the heat equation. Later on Schwinger [?] recognised that
this representations makes many issues related to renormalization and gauge
invariance in external fields more transparent. These two works introduced the
heat kernel to quantum theory. DeWitt made the heat kernel one of the main
tools of his manifestly covariant approach [?, ?, ?, ?] to quantum field theory
and quantum gravity which became dominant for many years.
Heat kernel is a classical subject in mathematics [?] 1 . Asymptotics of
the heat kernel are closely related to the eigenvalue asymptotics found by
H. Weyl [?, ?] and studied further in [?, ?]. The problem, as it was formulated
by Kac [?], reads: “Can one hear the shape of a drum?”. In other words, this
is the problem of recovering geometry of a manifold from the spectrum of a
natural differential operator. Heat kernel coefficients proved very useful in this
context. On the other hand, the heat kernel is also an adequate tool to study
the index theorem of Atiyah and Singer [?, ?, ?].
By about 1990 the heat kernel expansion on manifolds without boundaries or with boundaries and simplest local boundary conditions on them was
well understood. Also, the heat kernel became a standard tool in calculations of the vacuum polarisation, the Casimir effect, and in study of quantum
anomalies. Later on, progress in theoretical physics, especially in string theory and related areas, and parallel developments in mathematics made this
field highly specialised. New results on non-standard boundary conditions (as,
e.g., containing tangential derivatives on the boundary or non-localities) or on
non-standard geometries (domain walls) were scattered in large amounts of
physical and mathematical literature. The aim of this report is to present a
unifying approach to the heat kernel expansion and to supply the reader with
a “user-friendly” guide to the field. The main idea which we shall pursue is
the universality of the heat kernel. A single calculation (though, sometimes,
quite involved) may help in a large variety of applications regardless of such
details as spin, gauge group, etc. As well, just a single universal object in fact
describes counterterms, anomalies, some asymptotics of the effective action,
and much more.
To illustrate the use of the heat kernel in quantum field theory let us
consider the generating functional for the Green functions of the field φ in the
path integral representation
Z
Z[J] =
Dφ exp(−L(φ, J)) .
(1.1)
The heat kernel methods are almost exclusively used for the one-loop calcula1
A historical survey of the mathematical literature on the heat kernel expansion
can be found in [?].
4
tions. In this approximation it is enough to expand the action L to up to the
quadratic order in quantum fluctuations φ.
L = Lcl + hφ, Ji + hφ, Dφi ,
(1.2)
where Lcl is the action on a classical background, h., .i denotes an inner product
on the space of quantum fields. Usually, this inner product is just an integral
over the underlying space. For real one-component scalar fields it reads
hφ1 , φ2 i =
Z
√
dn x gφ1 (x)φ2 (x) .
(1.3)
The linear term in (1.2) contains in general contributions from the external
sources of the field φ and from the first variation of the classical action. If
the background satisfies classical equation of motion for the field φ, the latter
part of the linear term vanishes, though the former one (external sources)
should be kept arbitrary if one wishes to study correlation functions of φ. We
stress that the background and quantum fields may be of completely different
nature. For example, it is a meaningful problem to consider pure quantum
scalar fields on the background of pure classical gravity. D is a differential
operator. After a suitable number of integrations by part it is always possible
to convert the quadratic part of the action to the form given in (1.2). We
postpone discussion of possible boundary terms to the next sections. In the
simplest case of quantum scalar field on the background of a classical geometry
D is a Laplacian with a mass term:
D = D0 := −∇µ ∇µ + m2 .
(1.4)
Just this simple example is enough to illustrate the material of this section.
Note, that in this case J has no contribution from the first variation of the
classical action since gravity is not quantised.
The path integral measure is defined by:
Z
1=
Dφ exp(−hφ, φi) .
(1.5)
Strictly speaking the right hand side of (1.5) is divergent. The essence of the
condition (1.5) is that this divergence does not depend on external sources and
on the background geometry and, therefore, may be absorbed in an irrelevant
normalisation constant. The Gaussian integral (1.1) can be evaluated giving
1
Z[J] = e−Lcl det− 2 (D) exp
1
JD −1 J
4
.
(1.6)
We stress that the equation (1.6) is true only if the operator D is selfadjoint. This means that D is symmetric or formally self-adjoint with respect
to the scalar product h., .i:
hφ1 , Dφ2 i = hDφ1 , φ2 i
5
(1.7)
for any φ1 , φ2 , and that the domains of definition of D and its adjoint coincide.
We will not care about the second requirement since it involves mathematical
machinery [?] going beyond the scope of the present report. The first requirement (1.7) poses important restrictions on admissible boundary conditions.
To become convinced that (1.7) is really necessary, one can calculate a
finite-dimensional Gaussian integral with


a b

D=
(1.8)
cd
and a, b, c and d being real constants first by completing the squares in
the exponent, and then compare the result with det D (of course, one should
remember a factor of π). The two results coincide if b = c.
Let us return to the generating functional (1.6). To analyse the two multiplies on the right hand side of (1.6) which depend on the operator D it is
convenient to introduce the heat kernel
K(t; x, y; D) = hx| exp(−tD)|yi .
(1.9)
This somewhat formal expression means that K(t; x, y; D) should satisfy the
heat conduction equation
(∂t + Dx )K(t; x, y; D) = 0
(1.10)
K(0; x, y; D) = δ(x, y) .
(1.11)
with the initial condition
For D = D0 (1.4) on a flat manifold M = Rn the heat kernel reads:
(x − y)2
K(t; x, y; D0) = (4πt)−n/2 exp −
− tm2
4t
!
.
(1.12)
The equations (1.10) and (1.11) can be checked straightforwardly. Let us consider a more general operator D which contains also a potential term or a gauge
field. Then (1.12) still describes the leading singularity in the heat kernel as
t → 0. The subleading terms have a form of the power-law corrections 2 :
K(t; x, y; D) = K(t; x, y; D0 ) 1 + tb2 (x, y) + t2 b4 (x, y) + . . . .
(1.13)
The coefficients bk (x, y) are regular in the limit y → x. They are called the heat
2
On a curved space even the leading term must be modified (cf sec. 4.3). On
manifolds with boundaries also half-integer powers of t appear in the expansion,
and, consequently, b2j+1 6= 0.
6
kernel coefficients 3 . At coinciding arguments bk (x, x) are local polynomials of
background fields and their derivatives.
The propagator D −1 (x, y) can be defined through the heat kernel by the
integral representation
D −1 (x, y) =
Z
∞
dt K(t; x, y; D) ,
(1.14)
0
which follows from (1.9) if we suppose that the heat kernel vanishes sufficiently
fast as t → ∞. We can formally integrate the expansion (1.13) to obtain
−1
D (x, y) ' 2(4π)
−n/2
X
j=0
|x − y|
2m
!− 1 n+j+1
2
K− 1 n+j+1(|x − y|m)b2j (x, y) ,
2
(1.15)
where b0 = 1. By examining the behaviour of the Bessel function Kν (z) for
small argument z [?] we conclude that the singularities in the propagator at
coinciding points are described by the first several heat kernel coefficients bk .
Let us consider the part of the generating functional (1.6) which contains
det(D). The functional
1
ln det(D)
2
W =
(1.16)
is called the one-loop effective action. It describes the quantum effects due to
the background fields in the one-loop approximation of quantum fields theory.
To relate W to the heat kernel we shall use the arguments of [?].
For each positive eigenvalue λ of the operator D we may write an identity
ln λ = −
Z
∞
0
dt −tλ
e .
t
(1.17)
This identity is “correct” up to an infinite constant, which does not depend on
λ and, therefore, may be ignored in what follows 4 . Now we use ln det(D) =
Tr ln(D) and extend (1.17) to the whole operator D to obtain
W =−
1
2
Z
∞
0
dt
K(t, D) ,
t
(1.18)
where
−tD
K(t, D) = Tr e
Z
=
3
√
dn x gK(t; x, x; D) .
(1.19)
We shall mostly use the coefficients ak which differ from bk by a normalisation
factor.
4 To “prove” this statement one has to differentiate both sides of eq. (1.17) with
respect to λ.
7
Here we have only presented some heuristic arguments in favour of eq.
(1.18). A more rigorous treatment of functional determinants can be found in
sec. 2.2.
The integral in (1.18) may be divergent at both limits. Divergences at
t = ∞ are caused by zero or negative eigenvalues of D. These are the infra
red divergences. They will not be discussed in this section. We simply suppose
that the mass m is sufficiently large to make the integral (1.18) convergent at
the upper limit. Divergences at the lower limit cannot be removed in such a
way. Let us introduce a cut off at t = Λ−2 .
WΛ = −
1
2
Z
∞
Λ−2
dt
K(t, D) .
t
(1.20)
It is now easy to calculate the part of WΛ which diverges in the limit Λ → ∞:
WΛdiv
= −(4π)
−n/2
Z

X
√
d x g
n
2(j+l)<n
+
X
2 l
Λ
n−2j−2l
(−m2 )l l!
b2j (x, x)
n − 2j − 2l

ln(Λ)(−m ) l! b2j (x, x) + O(Λ ) .
0
(1.21)
2(j+l)=n
We see that the ultra violet divergences in the one-loop effective action are
defined by the heat kernel coefficients bk (x, x) with k ≤ n.
On non-compact manifolds the integral of b0 (x, x) is divergent. This divergence is removed by subtracting a “reference” heat kernel (see sec. 6.1).
Contributions from higher heat kernel coefficients bk , k > n to the effective
action are not divergent and can be easily calculated yielding in the limit
Λ→∞
1
− (4π)−n/2 mn
2
Z
√ X b2j (x, x)
dn x g
Γ(2j − n) .
m2j
2j>n
(1.22)
This is nothing else than the large mass expansion of the effective action. This
expansion is valid for relatively weak and slowly varying background fields.
We have seen that the heat kernel expansion describes
• short-distance behaviour of the propagator;
• one-loop divergences and counterterms;
• 1/m expansion of the effective action.
We shall see below that heat kernel provides a natural framework for studying
• quantum anomalies (sec. 7);
• various perturbative expansions of the effective action (sec. 8);
• selected non-perturbative relations for the effective action (sec. 9).
Of course, in all these applications the heat kernel methods have to compete with other techniques. The main advantage of the heat kernel is that it
delivers necessary information in terms of just few geometric invariants. This
method does not make distinctions between different spins, gauge groups, etc.
8
Even dependence of the space-time dimensionality is in most cases trivial.
Therefore, on one hand, just a single calculation serves then in many applications. On the other hand, calculations in simple particular cases give valuable
information on the general structure of the heat kernel. This property is especially useful when one deals with complicated geometries (like in the presence
of boundaries or singularities). During the last decade, many models which
lead to such complicated geometries were very actively studied in theoretical physics. The Dirichlet branes and the brane world scenario are the most
popular but not the only examples.
We have to mention also the limitations of the heat kernel formalism. It
works less effectively in the presence of spinorial background fields, i.e. when
there is mixing between bosonic and fermionic quantum fields. This problem
is probably of the technical nature, so that the corresponding formalism may
be developed some time in the future. A more serious drawback is that the
heat kernel expansion is not applicable beyond the one-loop approximation. It
is not clear whether necessary generalisations to higher loop could be achieved
at all.
It is not possible to write a review paper on heat kernel which would be
complete in all respects, especially in the bibliography. A more comprehensive
treatment of many mathematical problems related to the heat kernel expansion can be found in [?,?]. The book by Kirsten [?] considers also specific physical applications as the Casimir energy and the Bose–Einstein condensation.
The recent review paper [?] is devoted to the Casimir effect (see also [?]). The
monographs by Birrell and Davies [?] and Fulling [?] remain standard sources
on quantum field theory in curved space. Quantization of gauge theories is
explained in [?]. In [?, ?, ?, ?, ?] the heat kernel expansion is treated from the
point of view of quantum gravity and quantum cosmology. Useful information
about properties of the zeta function can be found in [?]. One may also consult [?]. The DeWitt approach to the heat kernel and its’ generalisations are
described in [?]. The path integral point of view on the heat kernel can be
found in [?].
Our primary goal is local formulae for the heat kernel coefficients. Therefore, in some cases global aspects will be somewhat neglected.
This report is organised as follows. The next section contains necessary
preliminary information on spectral geometry and differential geometry. There
we define main geometric characteristics of the manifold and of the boundary. We discuss the zeta function (which defines the effective action) and the
resolvent (which is a generalisation of the propagator). We relate the asymptotics of these functions to the heat kernel coefficients. Through this report
we work on Euclidean manifolds. A short remark on the analytical continuation to the Lorentzian signature is given at the end of section 2. In section 3
we consider the most widely used models of quantum field theory and open
bosonic strings. The one-loop dynamics in each of these models is defined by
a second order differential operator which depends on an effective connection
and on a matrix-valued potential. The connection and the potential serve as a
9
basis of an invariant description of all that models in the language of spectral
geometry. These quantities are written down explicitly for each model. We
also define suitable boundary conditions. In section 4 we consider the heat
kernel expansion on manifolds without boundary. We introduce a simple and
very powerful method and illustrate it by calculating several leading terms in
the heat kernel expansion. We also briefly discuss some other methods and
non-minimal operators. Section 5 is devoted to manifolds with boundary. The
heat kernel expansion for standard Dirichlet, Neumann, and mixed boundary
conditions is considered in some detail. We also describe less known oblique
and spectral boundary conditions (these are the ones which contain tangential
derivatives on the boundary or non-local projectors). We discuss loss of the
so-called strong ellipticity for oblique boundary conditions which corresponds
to the critical value of the electric field in string physics. With the example of
spectral boundary conditions we illustrate appearance of non-standard (ln t)
asymptotics in the heat kernel. The results of sections 4 and 5 are valid on
smooth manifolds with smooth potentials and gauge field. In section 6 we
consider the case when either the background fields or the manifold itself have
singularities. In particular, conical and domain wall singularities are considered.
Sections 7 - 9 are devoted to applications. In section 7 we relate conformal and chiral anomalies to certain heat kernel coefficients and re-derive the
anomalies in several particular models. This is a very spectacular but also
rather well known application of the heat kernel expansion. In section 8 we
go beyond the power series in the proper time t. We consider mainly two particular cases. The first one is the case when derivatives are more important
than the potentials (so that we sum up leading non-localities). The second one
is the so-called low energy expansion which neglects derivatives of the background fields starting with certain order (but treats all powers of background
potentials and curvatures exactly). In that section we also review some results
on the heat kernel on homogeneous spaces where spectrum of relevant operators may be found exactly. In section 9 we consider two examples when the
heat kernel can be used to obtain exact results for the effective action. The
first one is the famous Polyakov action. The second example is exact relations
between the effective actions in dual theories. In this section (in contrast to
section 8) we don’t have to neglect derivatives or powers of the background
fields. Section 10 contains concluding remarks.
10
2
Spectral functions: heat kernel, zeta function, resolvent
2.1 Differential geometry and the operators of Laplace type
Let M be a smooth compact Riemannian manifold 5 of dimension n with
smooth boundary ∂M. We shall also consider the case when the boundary
∂M is empty. Let V be a vector bundle 6 over M. This means that there is a
vector space attached to each point of the manifold. For example, this could
be a representation space of a gauge group or of the space-time symmetry
group. Sections of V are smooth functions bearing a discrete index which
could correspond to internal or spin degrees of freedom.
We study differential operators on V . We restrict ourselves to second order
operators of the Laplace type. Locally such operators can be represented as:
D = −(g µν ∂µ ∂ν + aσ ∂σ + b)
(2.1)
where g µν is the inverse metric tensor on M; aσ and b are matrix valued functions on M. There is a unique connection on V and a unique endomorphism
E of V (another matrix valued function) so that
D = −(g µν ∇µ ∇ν + E) ,
(2.2)
where the covariant derivative ∇ = ∇[R] + ω contains both Riemann ∇[R] and
“gauge” (bundle) ω parts. We may express:
1
ωδ = gνδ (aν + g µσ Γµσ ν IV ) and
2
E = b − g νµ (∂µ ων + ων ωµ − ωσ Γνµ σ ) ,
(2.3)
(2.4)
where
1
Γνµ σ = g σρ (∂µ gνρ + ∂ν gµρ − ∂ρ gµν )
2
(2.5)
is the Christoffel symbol. IV is the unit operator on V .
5
This simply means that we assume that there is a positive definite metric tensor
gµν on M .
6 Many physicists strongly dislike vector bundles. Nevertheless, there are two good
reasons for using the fibre bundles in this paper (in parallel with more familiar notations of matrix-valued functions, gauge fields, etc). First, our simplifying comments
and examples may help the reader to understand mathematical literature on the
subject. Second, one of the main ideas of this report is to reveal some universal
structures behind the heat kernel expansion. In particular, we shall see that there
is not much difference between different spins and symmetry groups. The vector
bundle language seems to be the most adequate language to achieve this goal. The
reader may consult the excellent review paper by Eguchi, Gilkey and Hanson [?].
11
It is important to understand how general the notion of the Laplace type
operator is. The most obvious restriction on D is that it is of second order, i.e.
it contains second derivatives, but does not contain higher derivative parts. In
this paper we shall also consider first order operators (Dirac operators, for example). Second, the operator D is a partial differential operator. This excludes
negative or fractional powers of the derivatives. The operators containing such
structures are called pseudo-differential operators. More information on spectral theory for these operators can be found in [?, ?, ?]. Third, the operator
(2.1) has a scalar principal part. This means that the second derivatives in
(2.1) are contracted with the metric, and the internal index structure of the
second derivative term is trivial. Such operators are also called minimal. Nonminimal operators will be briefly considered in sec 4.4.
We can define local invariants associated with ω and g. Let
Rµ νρσ = ∂σ Γµνρ − ∂ρ Γµνσ + Γλνρ Γµλσ − Γλνσ Γµλρ
(2.6)
be the Riemann curvature tensor, let Rµν := Rσ µνσ be the Ricci tensor, and let
R := Rµµ be the scalar curvature. With our sign conventions, R = 2 on the unit
sphere S 2 in the Euclidean space. Let Roman indices i, j, k, and l range from 1
through the dimension n of the manifold and index a local orthonormal frame
(vielbein) {e1 , ..., en } for the tangent space of the manifold. In components
we have: eµj eνk gµν = δjk , eµj eνk δ jk = g µν . The inverse vielbein is defined by the
relation ejµ eµk = δkj . These two objects, ekµ and eνj , will be used to transform
“curved” indices (µ, ν, ρ) to “flat” ones (i, j, k) and back. In Euclidean space
there is no distinction between upper and lower flat indices.
As usual, the Riemann part of the covariant derivatives contains the
Christoffel connection so that
∇µ[R] vν = ∂µ vν − Γρµν vρ
(2.7)
for an arbitrary vector vν . To extend this derivative to the objects with flat
indices one has to use the spin-connection σµ :
∇µ v j = ∂µ v j + σµjk vk .
(2.8)
The condition ∇µ ekν = 0 yields
σµkl = eνl Γρµν ekρ − eνl ∂µ ekν .
(2.9)
Let Ωµν be the field strength of the connection ω:
Ωµν = ∂µ ων − ∂ν ωµ + ωµ ων − ων ωµ .
(2.10)
The covariant derivative ∇ acts on both space-time and internal indices. For
example,
∇ρ Ωµν = ∂ρ Ωµν − Γσρµ Ωσν − Γσρν Ωµσ + [ωρ , Ωµν ] .
12
(2.11)
If the boundary ∂M is non-empty we have more invariants. Let en be
inward pointing unit vector field. Let Roman indices a, b, c and d range from
1 to n − 1 and index a local orthonormal frame for the tangent bundle of ∂M.
Let Lab := Γnab be the second fundamental form (extrinsic curvature) of the
boundary. We use the Levi-Civita (spin) connections and the connection ω
to covariantly differentiate tensors of all types. Let ‘;’ denote multiple covariant differentiation with respect to the Levi-Civita connection of M and let
‘:’ denote multiple tangential covariant differentiation on the boundary with
respect to the Levi-Civita connection of the boundary; the difference between
‘;’ and ‘:’ is measured by the second fundamental form. Thus, for example,
E;a = E:a since there are no tangential indices in E to be differentiated. On
the other hand, E;ab 6= E:ab since the index a is also being differentiated. More
precisely E;ab = E:ab − Lab E;n . Since L is only defined on the boundary, this
tensor can only be differentiated tangentially.
Consider an example of the circle S 1 in the plane R2 . The line element
has the form
(ds)2 = (dr)2 + r 2 (dθ)2 ,
(2.12)
where 0 ≤ r < ∞, 0 ≤ θ < 2π. Then gµν = diag(1, r 2). S 1 is defined by the
condition r = r0 . We may choose e1θ = r, e2r = −1 (the minus sign appears
because e2 is an inward pointing unit vector for the disk with the boundary
S 1 ). Then the second fundamental form of S 1 is
L11 = e2r eθ1 eθ1 Γrθθ r=r0
=
1
.
r0
(2.13)
In general, on S n−1 considered as a boundary of the ball in Rn the extrinsic
curvature is Lab = r10 δab .
If the boundary ∂M is non-empty, one has to define boundary conditions
for the field φ. A convenient way to write them down is:
Bφ = 0
(2.14)
where B is called the boundary operator. In general, the operator B calculates
a linear combination of the boundary data for any given function φ. If D is of
Laplace type, the boundary data include value of the function at the boundary
and value of it’s first normal derivative. The most frequently used choices are
the Dirichlet and Neumann boundary operators which we denote B− and B+
respectively:
B− φ = φ|∂M ,
B+ φ = (φ;n + Sφ)|∂M ,
(2.15)
(2.16)
where S is a matrix valued function defined on ∂M. The boundary conditions
(2.16) are also called Robin or generalised Neumann. In some literature they
13
are called mixed boundary conditions. We shall not use this latter terminology.
The name “mixed” is reserved for another type of boundary conditions. Let
Π− and Π+ be two complementary projectors defined on V |∂M , (Π± )2 = Π± ,
Π+ + Π− = I. There is a decomposition V |∂M = VN ⊕ VD , where VN,D =
Π± V |∂M . Decompose also φ = φN ⊕ φD and set
N
Bφ := φD ⊕ (φN
;n + Sφ )|∂M .
(2.17)
The matrix valued function (endomorphism) S acts on VN only, S = Π+ S =
SΠ+ . In other words, we define Dirichlet boundary conditions on VD and generalised Neumann boundary conditions on VN . For obvious reason the boundary conditions (2.17) will be called mixed. In sec. 3 we shall see that natural
boundary conditions for spinor and vector fields are of this type.
2.2 Spectral functions
For the boundary conditions considered in this section as well as on manifolds without a boundary the operator exp(−tD) with positive t is trace class
on the space of square integrable functions L2 (V ). This means that for an
auxiliary smooth function f on M
K(t, f, D) = TrL2 (f exp(−tD))
(2.18)
is well defined. We also write
Z
K(t, f, D) =
M
√
dn x gtrV K(t; x, x; D)f (x) ,
(2.19)
where K(t; x, x; D) is an y → x limit of the fundamental solution K(t; x, y; D)
of the heat equation (1.10) with the initial condition (1.11). If there is a
boundary, the kernel K(t; x, y; D) should also satisfy some boundary conditions Bx K(t; x, y; D) = 0 in one of the arguments. We stress that K(t; x, y; D)
is a matrix in the internal indices. trV denotes the trace over these indices.
Let D be self-adjoint. This implies that in a suitable basis in the internal
space the matrix ωµ is anti-hermitian and E is hermitian. Let {φλ } be a
complete basis of orthonormal eigenfunctions of the operator D corresponding
to the eigenvalues {λ}. Then
K(t; x, y; D) =
X
φ†λ (x)φλ (y)e−tλ .
(2.20)
λ
We shall almost exclusively work either on manifolds without boundaries,
or on manifolds with boundaries with the fields subject to local boundary
conditions (2.15), (2.16), or (2.17). In all these cases there is an asymptotic
14
expansion as t ↓ 0 [?, ?, ?] 7 :
TrL2 (f exp(−tD)) ∼
=
X
t(k−n)/2 ak (f, D) .
(2.21)
k≥0
This expansion is valid for almost all boundary conditions appearing in applications to physics. There are, however, some exceptions which will be discussed
in sec. 5.4.
The coefficients ak and the coefficients bk introduced in the previous section (cf. (1.13)) are related by the equation (m = 0 for the simplicity):
ak (f, D) = (4π)−n/2
Z
M
√
dn x gbk (x, x)f (x) .
(2.22)
Note, that the definition (1.13) is valid on flat manifolds without boundary
only (although generalisations to other cases are possible).
The key property of the heat kernel coefficients ak is that they are locally
computable in most of the cases. This means that they can be expressed in
term of the volume and boundary integrals of local invariants.
For a positive operator D one can define the zeta function [?, ?] by the
equation:
ζ(s, f, D) = TrL2 (f D −s ) .
(2.23)
The zeta function is related to the heat kernel by the integral transformation
ζ(s, f, D) = Γ(s)
−1
Z∞
dt ts−1 K(t, f, D) .
(2.24)
ds t−s Γ(s)ζ(s, f, D) ,
(2.25)
0
This relation can be inverted,
1
K(t, f, D) =
2πi
I
where the integration contour encircles all poles of the integrand. Residues at
the poles can be related to the heat kernel coefficients:
ak (f, D) = Ress=(n−k)/2 (Γ(s)ζ(s, f, D)) .
(2.26)
an (f, D) = ζ(0, f, D) .
(2.27)
In particular,
Zeta-functions can be used to regularize the effective action [?, ?]. The
7
These papers contain also a method allowing to calculate the coefficients of the
expansion. The method is, however, too complicated to use it on practice.
15
regularization is achieved by shifting the power of t in (1.18):
1 2s
˜
Ws = − µ
2
Z
∞
0
dt
t1−s
K(t, D) ,
(2.28)
where µ
˜ is a constant of the dimension of mass introduced to keep proper
dimension of the effective action. The regularization is removed in the limit
s → 0. Eq. (2.28) can be considered as a definition of the regularized effective
action without any reference to (1.18). One can also rewrite (2.28) in terms of
the zeta function:
1 2s
Ws = − µ
˜ Γ(s)ζ(s, D) ,
2
(2.29)
where ζ(s, D) := ζ(s, 1, D).
The gamma function has a simple pole at s = 0:
Γ(s) =
1
− γE + O(s) ,
s
(2.30)
where γE is the Euler constant. The regularized effective action (2.29) has also
a pole at s = 0:
1 1
1
Ws = −
˜2 ζ(0, D) − ζ 0(0, D) .
− γE + ln µ
2 s
2
(2.31)
According to (2.27) the divergent term in the zeta function regularization is
proportional to an (D) (cf. (1.21) for another regularization scheme). The pole
term in (2.31) has to be removed by the renormalization. The remaining part
of Ws at s = 0 is the renormalised effective action:
1
1
W ren = − ζ 0 (0, D) − ln(µ2 )ζ(0, D) ,
2
2
(2.32)
where we have introduced a rescaled parameter µ2 = e−γE µ
˜2 . In this approach
µ2 describes the renormalization ambiguity which must be fixed by a suitable
normalisation condition. Let us remind that here we are working on a compact manifold. On non-compact manifolds ζ(s, D) may have divergent contributions proportional to the volume. Such divergences are usually removed by
the subtraction of a “reference” heat kernel (see sec. 6.1).
Together with (1.16) equation (2.32) yields a definition of the functional
determinant for a positive elliptic second order operator which is frequently
used in mathematics:
ln det(D) = −ζ 0 (0, D) − ln(µ2 )ζ(0, D) .
(2.33)
Note, that the definitions (2.23), (2.33) are valid for positive operators
only. Elliptic 2nd order differential operators have at most finite number of
16
zero and negative modes 8 which must be treated separately. However, one can
extend the definition of the zeta function to operators with negative modes:
ζ(s, D) =
X
|λ|−s ,
(2.34)
where the sum extends over all non-zero eigenvalues λ. One can also define
another spectral function in a similar way:
η(s, D) =
X
sign (λ)|λ|−s .
(2.35)
This function is especially useful in spectral theory of Dirac type operators
where η(0, D)
/ measures asymmetry of the spectrum.
Another function which is frequently used especially in the mathematical
literature is the resolvent (or, more precisely, its’ powers):
Rl (z) := (D + z 2 )−l .
(2.36)
If D is on operator of Laplace type subject to “good” boundary conditions 9
and if l sufficiently large (l > n/2) there is a full asymptotic expansion
Tr(Rl (z)) =
XΓ l+
k
k−n
2
Γ(l)
ak (D)z −2l+n−k
(2.37)
as z → ∞. The coefficients ak are the same as in the heat kernel expansion
(2.21).
2.3 Lorentzian signature
Locality of the heat kernel coefficients in the Euclidean domain can be
easily understood by examining the free heat kernel (1.12). For small t the
first term in the exponential strongly suppresses non-local contributions. For
Lorentzian metrics the squared distance function (x − y)2 is no longer positive
definite. Therefore, the simple arguments given above do not work. A partial
solution to this problem is to consider the “Schr¨odinger” equation
˜ ; x, y; D) = 0
(i∂τ − D) K(τ
(2.38)
˜ instead of the heat conduction equation (1.10). Then K
˜
for the kernel K
oscillates at large distances. However, even though non-local contributions
˜ oscillate furiously as τ → 0 they are not small. Consequently, local
to K
asymptotic series do not exist in many cases. A discussion on this point can
be found in chapter 9 of [?].
8
On manifolds with boundaries one has to require strong ellipticity of the boundary
value problem (cf. sec. 5.4).
9 A more precise meaning of this restriction will be explained in sec. 5.4.
17
Of course, the heat kernel expansion can be used also on Lorentzian manifolds at least for the renormalization theory where non-local terms are of less
importance. Counterterms are still defined by the same heat kernel coefficients
with the same functional dependence on the Lorentzian metric. Explicit definitions with the imaginary “proper time” τ can be found in [?,?]. One should
note that some background fields receive an imaginary phase when being continued to the Euclidean domain (cf. secs. 3.2 and 3.3).
18
3
Relevant operators and boundary conditions
The operator of Laplace type is not necessarily the scalar Laplacian. In
fact, in almost all models of quantum field theory the one loop effective action
is defined by an operator of this type. This can be demonstrated by bringing
relevant operators to the canonical form (2.2). In this section we give explicit construction of the connection ω and the endomorphism (matrix-valued
potential) E for scalar, spinor, vector and graviton fields. We also describe
appropriate boundary conditions.
3.1 Scalar fields
Consider first the example of the multi-component real scalar field ΦA in
n dimensions. The action reads
Z
√
L=
dn x g(g µν ∇µ ΦA ∇ν ΦA + U(Φ) + ξRΦA ΦA ) ,
(3.1)
M
where ξ is the conformal coupling parameter, U is a potential. The covariant
A
derivative ∇µ contains the background gauge field GAB
= ∂µ ΦA +
µ : ∇µ Φ
B
GAB
µ Φ . Gµ is antisymmetric in internal indices A, B. To evaluate the one¯
loop effective action one should expand (3.1) around a background field Φ,
¯ + φ, and keep the terms quadratic in fluctuations:
Φ=Φ
Z
√
1
¯ 00 )AB + ξRδ AB φB
dn x gφA −(∇µ ∇µ )AB + (U(Φ)
2
M
Z
√
−
dn−1 x hφA ∇n φA ,
L2 =
∂M
(3.2)
h is the determinant of the induced metric on the boundary. The inner product
for quantum fields φ reads
Z
< φ1 , φ2 >=
M
√
A
dn x gφA
1 φ2 .
(3.3)
The operator D is defined by the bulk part of the action (3.2):
1
¯ 00 )AB + ξRδ AB .
DAB := −(∇µ ∇µ )AB + (U(Φ)
2
(3.4)
For a special choice of the parameter ξ:
ξ=
n−2
4(n − 1)
(3.5)
¯ 00 = 0). To bring
the operator D (3.4) is conformally covariant (if also U(Φ)
the operator (3.4) to the canonical form (2.2) we introduce
ωµAB = GAB
µ ,
1
¯ 00 )AB − ξRδ AB .
E AB = − (U(Φ)
2
19
(3.6)
For this case, Ωµν is just the ordinary Yang-Mills field strength.
The operator (3.4) is symmetric with respect to the inner product (3.3)
if the surface integral
Z
∂M
√
A
A
A
dn−1 x h(φA
1 ∇n φ 2 − φ 2 ∇n φ 1 )
(3.7)
vanishes for arbitrary φ1 and φ2 belonging to its’ domain of definition. This
may be achieved if one imposes either Dirichlet
φA |∂M = 0
(3.8)
(∇n φA + S AB φB )|∂M = 0
(3.9)
or modified Neumann
boundary conditions. S AB is an arbitrary symmetric matrix. Note, that the
integral (3.7) vanishes also if S AB is an arbitrary symmetric (differential) operator on the boundary.
For the Dirichlet conditions (3.8) the boundary term in (3.2) vanishes
automatically. To ensure absence of the surface term for the modified Neumann
conditions (3.9) one should add to (3.1) an appropriate surface action.
3.2 Bosonic string
Our next example is the non-linear sigma model in two dimensions described by the action
L
Z
[σ]
=
M
d2 x
+
√
Z
∂M
gGAB (X)g µν ∂µ X A ∂ν X B + µν BAB (X)∂µ X A ∂ν X B
AB dX B .
(3.10)
From the point of view of two-dimensional world sheet the fields X A (x) are
scalars. In string theory they are interpreted as coordinates on a d-dimensional
target manifold with the metric GAB (X). µν is the Levi-Civita tensor density,
12 = −21 = 1. BAB (X) is an antisymmetric tensor field on the target space.
AB (X) is the electromagnetic vector potential. The action (3.10) describes
charged open stings. For simplicity, we absorb the inverse string tension α0
into a field redefinition. We do not include tachyon and dilaton couplings in
the bulk or on the boundary.
Usually the term with the B-field gets an imaginary coefficient in the
Euclidean space. Since the physical space-time has Minkowski signature it is
not especially significant which way of continuation to the Euclidean space has
been chosen provided the results are properly continued back to Minkowski
space after the calculations. As we will see below, real coefficient in front of
the B-field leads to a well-defined spectral problem. This situation is in close
20
analogy with the continuation rules for the axial vector field in the spinor
determinant [?, ?]. The other way to deal with the field B is to keep the
coefficient of the B-term in (3.10) imaginary at the expense of introducing a
more sophisticated conjugation operation [?] containing the sign-reversion of
the B-field. The same refers to the electromagnetic potential AB .
The field X enters the action (3.10) at many places making the background field expansion a quite cumbersome procedure. The most economic
way to arrange such an expansion and to calculate higher derivatives of the
action (3.10) is to introduce the geodesic coordinates in the target space. A
detailed explanation of the method as well as further references can be found
in [?]. Consider the target space geodesics defined independently at each point
of the two-dimensional world surface and parametrised by the arc length s in
the target space. They satisfy the usual geodesic equation
d2 A
d
d
A
X (x, s) + γBC
(X) X B (x, s) X C (x, s) = 0 ,
2
ds
ds
ds
(3.11)
A
where γBC
(X) is the Christoffel connection corresponding to the target space
metric GAB . Let us supplement the equation (3.11) by the initial conditions
d A
X (x, 0) = ξ A (x) ,
ds
¯ A (x) ,
X A (x, 0) = X
(3.12)
¯ is the background field. ξ A parametrises deviations from X
¯ and,
where X
therefore, can be identified with quantum fluctuations. The k-th order term
¯ is given by
of the expansion of the action (3.10) around the background X
Lk =
1 dk
L(X(s))|s=0 .
k! dsk
(3.13)
Higher order derivatives of X(x, s) with respect to s can be traded for the first
derivatives by means of the geodesic equation (3.11) and then replaced by ξ
at s = 0 with the help of the initial conditions (3.12).
It is convenient to introduce the Riemann curvature of the target space
metric GAB , RABCD , and a 3-index field strength
HABC = ∂A BBC + ∂B BCA + ∂C BBA .
(3.14)
The covariant derivative ∇ is
A
¯ C )ξ B + 1 µ ν (∂ν X
¯ B )H A BC ξ C .
¯ µX
∇µ ξ A = ∂µ ξ A + γBC
(X)(∂
2
The quadratic part of the action (3.10) reads:
21
(3.15)
L2 =
Z
M
√ ¯ A )(∂ µ X
¯ D )ξ B ξ C
¯ µ ξ A ∇µ ξ B − RABCD (∂µ X
d2 x h GAB (X)∇
1
¯ C )(∂ν X
¯ D )µν ξ A ξ B DA HBCD
− (∂µ X
2
1
B
µ ¯D
C
A E
¯
+ (∂µ X )(∂ X )HABC H DE ξ ξ
4
Z
+
∂M
(3.16)
¯ B )DA (BBC − FBC )ξ A ξ C ,
dτ Dτ ξ A ξ B (BAB − FAB ) + (∂τ X
where τ is the arc length along the boundary. FBC = ∂B AC − ∂C AB . The
covariant derivatives Dτ and DA contain the Christoffel connection on the
target space (but not HABC as the full covariant derivative ∇, eq. (3.15)).
The natural inner product in the space of fluctuations ξ reads
Z
< ξ(1) , ξ(2) >=
M
√
A
B
¯
d2 x gGAB (X(x))ξ
(1) (x)ξ(2) (x) .
(3.17)
The volume part of the action (3.16) has now the canonical form hξ, Dξi with
the operator D (2.2) which is obviously of Laplace type. The connection ω is
defined in (3.15) and the endomorphism E can be easily extracted from (3.16).
The operator D is formally self-adjoint with respect to (3.17) if we impose the
boundary conditions of Neumann type
Bξ = (∇n ξ A + SBA ξ B )|∂M = 0
(3.18)
with arbitrary operator S which should be symmetric with respect to the
restriction of (3.17) to the boundary:
Z
∂M
A
A
dτ (ξ(1)
SAB ξ(2)B − ξ(2)
SAB ξ(1)B ) = 0
(3.19)
There is a preferable choice of the boundary operator. Let us vary the action
(3.16) with respect to the fluctuation field ξ:
Z
Z
√
1
2
δS2 =
d x h(δξ)Dξ −
dτ (δξ)Bξ .
2
M
∂M
(3.20)
Now we require that the boundary integral in (3.20) vanishes for arbitrary δξ.
Hence we arrive at the boundary conditions (3.18) with the operator S given
by
1
B
B
B
SAB = (Γ∇τ + ∇τ Γ)B
ΓB
(3.21)
A + SA ,
A = BA − FA ,
2
h
i
1
¯ C ) H DA C (BDB − FDB ) + H D BC (BD A − FD A )
SAB = (∂n X
4
h
i
1
¯ C ) D A (BBC − FBC ) + DB (B A C − F A C ) .
+ (∂τ X
2
Note that the operator in (3.21) is not of the ordinary Neumann (or Robin)
type since it contains tangential derivatives on the boundary (cf. sec. 5.4.1).
22
The variation (3.20) vanishes also if we choose Dirichlet boundary conditions for some of the coordinates of the string endpoints. Namely, we can
b
A
B
take a projector Π+ and impose (3.18) on (Π+ )A
B ξ and (1 − Π+ )B δξ |∂M = 0.
Physically this means that the endpoints of the bosonic string are confined
to a submanifold in the target space. Such configurations [?] are called the
Dirichlet branes.
3.3 Spinor fields
The action for the spinor fields ψ
L=
Z
M
√
dn x g ψ¯D
/ψ
(3.22)
contains a first order operator D
/ of Dirac type. In Euclidean space the conjugate spinor ψ¯ is just the hermitian conjugate of ψ: ψ¯ = ψ † . By definition, an
operator D
/ is of Dirac type if its square D = D
/ 2 is of Laplace type. Spectral
theory of general operators of Dirac type both on manifolds without boundaries and with local boundary conditions on manifolds with boundaries can
be found in [?,?]. Here we consider some physically motivated particular cases
only. Let us introduce the Euclidean Dirac γ-matrices which satisfy the Clifford commutation relations:
γµ γν + γν γν = 2gµν .
(3.23)
The γ-matrices defined in this way are hermitian, 㵆 = γµ . We also need the
chirality matrix which will be denoted γ 5 independently of the dimension. It
satisfies
(γ 5 )† = γ 5 ,
γ 5 γµ = −γµ γ 5 .
(3.24)
From now on we suppose that the dimension n is even. We fix the sign in γ 5
by choosing
5
γ =
i
n(n−1)
2
n!
µ...ν γµ . . . γν .
(3.25)
Let D
/ be the standard Dirac operator in curved space with gauge and axial
gauge connection
1
D
/ = iγ µ ∂µ + [γν , γρ]σµνρ + Aµ + iA5µ γ 5 .
8
(3.26)
Here σµνρ is the spin-connection (2.9). Aµ and A5µ are vector and axial vector
fields respectively taken in some representation of the gauge group. Both Aµ
and A5µ are antihermitian in the gauge indices. The operator D
/ (3.26) is for2
mally self-adjoint in the bulk. The operator D = D
/ is of Laplace type (2.2)
23
with 10
1
i
ωµ = [γν , γρ ]σµνρ + Aµ + [γµ , γν ]A5ν γ 5 ,
8
2
1
1 µ ν
E = − R + [γ , γ ]Fµν + iγ 5 D µ A5µ − (n − 2)A5µ A5µ
4
4
1
− (n − 3)[γ µ , γ ν ][A5µ , A5ν ]
4
(3.27)
with obvious notations Fµν = ∂µ Aν −∂ν Aµ +[Aµ , Aν ], Dµ A5ν = ∂µ A5ν −Γρµν A5ρ +
[Aµ , A5ν ]. The expression for Ωµν is a little bit lengthy:
1
Ωµν = Fµν − [A5µ , A5ν ] − γ σ γ ρ Rσρµν − iγ 5 γ ρ (γν Dµ A5ρ − γµ Dν A5ρ )
4
5 5
5
+iγ Aµν + [Aµ , A5ρ ]γ ρ γν − [A5ν , A5ρ ]γ ρ γµ
−γ ρ A5ρ γµ γ σ A5σ γν + γ ρ A5ρ γν γ σ A5σ γµ ,
(3.28)
where
A5µν = ∂µ A5ν − ∂ν A5µ + [Aµ , A5ν ] − [Aν , A5µ ]
(3.29)
Consider now manifolds with boundary. The specific feature of the action
(3.22) is that it contains first order derivatives only. Consequently, boundary
conditions should be imposed on a half of the spinor components. Let these
be Dirichlet boundary conditions
Π− ψ|∂M = 0
(3.30)
where Π− is a hermitian projector, Π2− = Π− , Π†− = Π− . Due to the hermiticity
¯ − |∂M = 0
ψΠ
(3.31)
Following Luckock [?] let us consider a family of the projectors
Π− =
1
1 + γ n exp(iqγ 5 ) ,
2
(3.32)
where q is a scalar which can depend on the coordinate on the boundary.
To make the operator D
/ formally self-adjoint including the boundary we
must require that
Z
∂M
√
dn−1 x hψ¯1 γn ψ2 = 0
10
(3.33)
The present author is grateful to Valery Marachevsky for his help in deriving and
checking eqs. (3.27), (3.28) and (3.37), see also [?].
24
for all ψ1,2 satisfying the boundary conditions (3.30). Since ψ1 and ψ2 are
arbitrary, the projector Π− should satisfy
(1 − Π− )γ n (1 − Π− ) = 0 .
(3.34)
This condition yields q = ±π/2. The projector (3.32) takes the form
Π− =
1
1 ± iγ n γ 5 .
2
(3.35)
To formulate the spectral problem for the second order operator D = D
/2
we need boundary conditions for the second half of the spinor components.
The relevant functional space should be spanned by the eigenfunctions of the
Dirac operator D
/ . It is clear that on this space the functions Dψ
/ should satisfy
the same boundary conditions (3.30) as the ψ’s themselves:
Π−D
/ ψ|∂M = 0 .
(3.36)
/ in (3.36) we
Let us adopt the choice (3.35) for Π− . By commuting Π− with D
obtain
(∇n + S)Π+ ψ|∂M = 0,
1
S = − Laa Π+ ,
2
Π+ = 1 − Π− .
(3.37)
We remind that Lab is extrinsic curvature of the boundary. The boundary
conditions (3.30), (3.37) with (3.35) are mixed (cf. (2.17)). Spectral geometry
of the Dirac operator with these boundary conditions has been thoroughly
studied by Branson and Gilkey [?].
We can generalise the boundary conditions presented above by considering non-hermitian projectors, Π†− 6= Π− . Then the boundary condition for the
conjugated spinors reads:
¯ †− |∂M = 0 .
ψΠ
(3.38)
Instead of (3.34) we have the condition
(1 − Π†− )γ n (1 − Π− ) = 0 ,
(3.39)
which yields
Π− =
1
a 5
1 + iγ n γ 5 er(x )γ
2
(3.40)
with an arbitrary real function (or even with an arbitrary hermitian matrix
valued function) r(xa ).
In the Minkowski signature space the spinor conjugation includes γ 0 .
Therefore, the boundary conditions on the conjugate spinor (3.31) are changed.
Roughly speaking, to continue Π− to the Minkowski signature space time one
25
has to replace γ n γ 5 by γ n in (3.35) and (3.40) and to take into account powers
of i which appear in the Dirac gamma matrices.
In particle physics, interest to the boundary conditions defined by (3.35)
is due to the MIT bag model of hadrons proposed in [?, ?, ?]. This model was
modified later [?, ?, ?, ?, ?, ?] to include a chiral phase on the boundary in a
manner similar to (3.40) (for a review see [?]). Renormalization of quantum
field theory with the boundary conditions defined by (3.35) was considered by
Symanzik [?].
As we have already mentioned above, bag boundary conditions defined
by the projector (3.35) are a particular case of mixed boundary conditions
(2.17). They will be considered in detail in sec. 5.3 (see also sec. 5.1 for further
references to calculations in a ball). Chiral bag boundary conditions defined
by (3.40) with r 6= 0 are considerably more complicated because the equation
(3.36) contains a mixture of normal and tangential derivatives. We refer to
[?, ?, ?, ?, ?, ?, ?, ?, ?] for more calculations of spectral functions in this latter
case.
The boundary conditions considered in this section are local, i.e. they
treat the fields at each point of the boundary independently. One can also
define global boundary conditions for the Dirac operator (cf. sec. 5.4.2).
Due to the fermionic nature of the spinor field the path integral over ψ
and ψ¯ gives determinant of the operator D
/ to a positive power,
Z
Z=
¯ ψ)) = det D
D ψ¯ Dψ exp(−L(ψ,
/
(3.41)
¯ ψ are complex Dirac spinors.
where the action L is given by (3.22) and ψ,
3.4 Vector fields
Consider the Yang–Mills action for the gauge field Aαµ (Greek letters from
the beginning of the alphabet label generators of the gauge group):
1
L=
4
Z
M
√ α µνα
dn x gFµν
F ,
(3.42)
α
where, as usual, Fµν
= ∂µ Aαν − ∂ν Aαµ + cαβγ Aβµ Aγν . cαβγ are totally antisymmetric
structure constants of the gauge group. Let us introduce the background field
Bµα by the shift Aαµ → Bµα + Aαµ . From now on Aαµ plays the role of quantum
fluctuations. The quadratic part of the action reads:
1
L2 =
2
+
Z
1
2
M
√ h
dn x g −Aαν ∇µ ∇µ Aαν + Aαµ ∇µ ∇ν Aαν + Aαν Aαµ Rµν
Z
∂M
i
α
+2Fµν
(B)cαβγ Aβµ Aγν
√
dn−1 x h [Aαν ∇ν Aαn − Aαν ∇n Aαν ] .
26
(3.43)
The covariant derivative ∇ contains both metric and gauge parts: ∇ν Aαµ =
∂ν Aαµ + Bνβ cαβγ Aγν − Γρνµ Aαρ . One should impose a gauge condition on the fluctuation Aαµ . We choose
∇µ Aαµ = 0 .
(3.44)
In the gauge (3.44) the bulk term in (3.43) defines an elliptic operator of the
Laplace type with
γ α ρ
ρ α
(ωµ )αρ
νβ = Bµ cγβ δν − Γµν δβ ,
ρ α
γρ γ
(E)αρ
νβ = −Rν δβ + 2F (B) ν cβα .
(3.45)
(3.46)
The field strength corresponding of the connection (3.45) reads
σ
α
δ α σ
(Ωµν )ασ
ρβ = R ρµν δβ + F (B)µν cδβ δρ .
(3.47)
Note, that all the quantities above (3.45) - (3.47) are matrix functions with
both gauge and vector indices.
On a manifold without boundary the operator D with (3.45) and (3.46)
is symmetric with respect to the standard inner product in the space of the
vector fields
Z
< A(1) , A(2) >=
√
M
gdn x A(1)αµ A(2)α
.
µ
(3.48)
The ghost operator corresponding to the gauge (3.44) is just the ordinary
scalar Laplacian, D gh = −∇[gh]µ∇[gh]
µ , with the connection
(ωµ[gh])αγ = Bµβ cαβγ .
(3.49)
The one-loop path integral reads:
−1
1
Z(B) = det(D)⊥ 2 det(D [gh] ) 2
(3.50)
where the first determinant is restricted to the fields satisfying the gauge
condition (3.44). Note, that pure gauge fields A = ∇ξ are zero modes of the
total operator D + ∇∇ of the bulk action (3.43) only on shell, i.e. when the
background field B satisfies the classical equation of motion ∇µ F (B)αµν = 0.
Therefore, the path integral is gauge-independent on shell, but it depends on
the gauge choice off shell. For example, the Feynman gauge path integral
1
ZF (B) = det(D)− 2 det(D [gh] ) ,
(3.51)
where the first determinant is calculated on the space of all vector field, is equal
to the Z(B) defined in (3.50) only on shell. However, physical predictions of
the two path integrals are, of course, equivalent.
27
The path integral (3.51) can be obtained by adding the gauge fixing term
1
Lgf =
2κ
Z
M
√
dn x g(∇µ Aµ )2
(3.52)
to (3.43) with κ = 1. The case κ 6= 1 yields a non-minimal operator on the
gauge field fluctuations (cf sec. 4.4).
One can define “total” heat kernel coefficients for the path integral in a
certain gauge χ(A)
χ
[gh]
atot
k = ak (D ) − 2ak (Dχ ) ,
(3.53)
where Dχ is defined by the action (3.43) with the gauge fixing term χ2 (A), and
the ghost operator is Dχ[gh] ξ = χ(∇ξ). Even on shell only the coefficient atot
n is
gauge-independent. Only this coefficient contains information on the one-loop
divergences in a gauge-invariant regularization like the ζ-function one.
On a manifold with boundary one should impose boundary conditions
on gauge fields and ghosts. The boundary conditions should be gauge invariant. Consider a more general set-up when we have some quantum fields Φ
and a linearised gauge transformation Φ → Φ + δξ Φ with local parameter ξ.
Boundary operator B defines gauge invariant boundary conditions
BΦ = 0
(3.54)
if there exist boundary conditions for the gauge parameter ξ
Bξ ξ = 0
(3.55)
Bδξ Φ = 0 .
(3.56)
such that
The equation (3.56) means that the functional space defined by the boundary
conditions (3.54) is invariant under the gauge transformations provided the
gauge parameter ξ satisfies (3.55). Upon quantisation (3.55) become boundary
conditions for the ghost fields. The condition (3.56) ensures validity of the
Faddeev-Popov trick on a manifold with boundary and guarantees gaugeindependence of the on-shell path integral [?].
Since the boundary term in (3.43) is diagonal in the gauge index α we can
consider the case of an abelian gauge group and drop α from the notations.
The general non-abelian case does not offer considerable complications. There
are two sets of local boundary conditions which satisfy the gauge-invariance
requirements above. The first set is called absolute boundary conditions. It
reads
An |∂M = 0 ,
∇n ξ|∂M = 0 .
∂n Aa |∂M = (∇n δab − Lab )Ab |∂M = 0 ,
(3.57)
28
The second set
(∇n − Laa )An |∂M = 0 ,
ξ|∂M = 0
Aa |∂M = 0 ,
(3.58)
is called relative boundary conditions. The projectors on the Dirichlet and
Neumann subspaces and the endomorphism S are (Π− )ij = δin δjn , (Π+ )ij =
δij − δin δjn , Sij = −Lab δia δjb and (Π− )ij = δij − δin δjn , (Π+ )ij = δin δjn , Sij =
−Laa δin δjn for the absolute and relative boundary conditions respectively.
It is straightforward to check gauge invariance of the absolute boundary
conditions (3.57). A bit more job is required to show that the condition (3.58)
for the normal component is also gauge invariant. The key observation is
that near the boundary the scalar Laplacian can be represented as D [gh] =
−(∇n −Laa )∇n +E, where E does not contain normal derivatives and, therefore,
Eξ|∂M = 0 under the boundary conditions (3.58) for the ξ. Consequently,
(∇n − Laa )δξ An |∂M = (∇n − Laa )∇n ξ|∂M = (−D [gh] ξ + Eξ)|∂M = 0 (3.59)
where D[gh] ξ|∂M = 0 on the eigenfunctions of the operator D [gh] which is
enough for our purposes.
The boundary term in the action (3.43) can be rewritten as 11
1
2
Z
∂M
√
dn−1 x h[Aa (An:a + Lab Ab − Aa;n )] .
(3.60)
This boundary action vanishes for both types of the boundary conditions
(3.57) and (3.58). Hence the operator D is symmetric for these boundary
conditions. Another remarkable fact is that the fields A⊥ satisfying the gauge
condition (3.44) are orthogonal to the gauge transformations. Indeed,
< A⊥ , ∇ξ >= −
Z
∂M
√
dn−1 x hA⊥
nξ = 0
(3.61)
for both (3.57) and (3.58).
We should also check whether the gauge condition (3.44) is indeed compatible with the boundary conditions (3.58) and (3.57). For an arbitrary A
there should exist unique ξ such that
∇µ (Aµ + ∇µ ξ) = 0.
(3.62)
∇µ Aµ = −∆ξ .
(3.63)
We rewrite (3.62) as
Let us start with relative boundary conditions (3.58). In this case the left hand
side of (3.63) satisfies Dirichlet boundary conditions. The scalar Laplacian ∆
11
The equation to follow contains two types of covariant derivatives. Definitions
can be found in sec. 2.1.
29
for these boundary conditions is invertible. Therefore, a solution for (3.63)
always exists, and it is unique. The case of absolute boundary conditions is a
bit more involved. One should take care of a zero mode in the ghost sector.
We leave this case as an exercise. A more extensive discussion of compatibility
of gauge and boundary conditions can be found in [?].
3.5 Graviton
We start this section with the Einstein-Hilbert action on a four dimensional Euclidean manifold without boundary:
Z
1
√
L=
d4 x g(R − 2Λ) ,
16πGN
(3.64)
where R is the scalar curvature, GN is the Newton constant, Λ is the cosmological constant. As usual, let us shift the metric gµν → gµν +hµν . From now on gµν
will denote the background metric, hµν will be the quantum fluctuations. We
can decompose the hµν further in trace, longitudinal and transverse-traceless
part:
1
hµν = hgµν + (Lξ)µν + h⊥
µν ,
4
(3.65)
ν ⊥
where g µν h⊥
µν = ∇ hµν = 0 and
1
(Lξ)µν = ∇µ ξν + ∇ν ξν − gµν ∇ρ ξρ .
2
(3.66)
The decomposition (3.65) is orthogonal with respect to the inner product:
0
Z
< h, h >=
√
d4 x gGµνρσ hµν h0ρσ ,
(3.67)
1
Gµνρσ = (g µρ g νσ + g µσ g νρ + Cg µν g ρσ ) .
2
Here C is a constant. For positivity of (3.67) C should be greater than − 12 .
Under the action of infinitesimal diffeomorphism generated by a vector µ the
components of (3.65) transform as
ξµ → ξµ + µ ,
h → h + 2∇µ µ ,
⊥
h⊥
µν → hµν
(3.68)
One can fix the gauge freedom (3.68) by the condition
ξµ = 0 .
(3.69)
If the background admits conformal Killing vectors (these are the vectors
which are annihilated by the operator L (3.66)) the condition (3.69) is not
enough and one should impose one more gauge condition on the trace part
(see e.g. [?]). We suppose that conformal Killing vectors are absent.
30
The Jacobian factor induced by the change of variables (3.65) hµν →
(h, ξµ , h⊥
µν ) is
1
J = det(L† L) 2 ,
(3.70)
V
where the determinant is calculated on the space of the vector fields (excluding the conformal Killing vectors which we do not take into account). It is
convenient to shift the scalar part of the metric fluctuations by ∇µ ξµ so that
the decomposition (3.65) becomes
1
hµν = (σ + 2∇ρ ξρ )gµν + (Lξ)µν + h⊥
µν .
4
(3.71)
Since the change of the variables h → σ does not introduce any Jacobian
factor we conclude that the path integral measure is
1
Dhµν = det(L† L) 2 DσDξDh⊥ .
(3.72)
V
To simplify the discussion we suppose that the background metric gµν
satisfies the Einstein equations
Rµν (g) = Λgµν .
(3.73)
The quadratic part of the action reads:
1
L2 =
16πG
Z
√ 1 ⊥µν
h (−∆gµρ gνσ + 2Rµρνσ )h⊥ρσ
dx g
4
3
R
− σ −∆ −
σ .
32
3
4
(3.74)
Due to the gauge invariance (3.74) does not contain ξ. Functional integration
over ξ produces an infinite constant equivalent to the volume of the diffeomorphism group which will be neglected.
The kinetic term for σ has a wrong (negative) sign. This represents the
well known conformal factor problem of quantum general relativity. Different
explanations of this phenomenon [?,?,?,?] suggest (roughly) the same remedy:
the conformal mode must be rotated to the imaginary axis, σ → iσ.
The path integral can be written in terms of functional determinants
corresponding to vector fields (indicated by the subscript V ), scalars (indicated
by S), and transverse traceless tensors (indicated by T ⊥):
R
Z = det(L L) det −∆ −
V
S
3
†
1
2
− 21
1
det(−∆gµρ gνσ + 2Rµρνσ )− 2 .
T⊥
(3.75)
This expression is, in principle, suitable for calculations on some homogeneous
spaces since the harmonic expansion on such spaces usually respects separation of tensors to transverse and longitudinal parts. However, we must warn
31
the reader again that we have neglected the presence of Killing vectors and
of conformal Killing vectors. The ways to treat these vectors created a long
discussion in the literature [?, ?, ?, ?, ?, ?, ?, ?, ?].
The representation (3.75) is not convenient on generic manifold since the
tensor operator is restricted to the transverse modes. To remove this restriction one can either multiply (3.75) by a compensating vector determinant,
or add a suitable gauge fixing term to the classical action (3.64) and repeat
the quantisation procedure right from the beginning. The result can be found
elsewhere in the literature (see, e.g., [?, ?, ?, ?]).
Boundary conditions for one-loop Euclidean quantum gravity must be
diffeomorphism invariant and must lead to “good” (i.e. symmetric and elliptic)
operators describing metric and ghost fluctuations. The problem of finding a
suitable set of boundary conditions in gravity appeared to be much harder
that for the lower spin fields. There exist several proposals on the market
[?,?,?,?,?,?,?,?,?,?]. However, neither of these proposals is fully satisfactory.
It is clear now that proper boundary conditions must depend on tangential
derivatives. Moreover, they probably must be non-local.
The material of this section can be generalised to the case when an independent torsion field is present. Explicit expressions for relevant operators
acting on fields with different spin in the Riemann–Cartan space can be found
in [?, ?, ?].
32
4
Heat kernel expansion on manifolds without boundary
4.1 General formulae
For physicists, the most familiar way to calculate the heat kernel coefficients is the DeWitt iterative procedure which will be briefly described in sec.
4.3. Here we take a different route following the method of Gilkey. Advantages
of this method are most clearly seen on manifolds with boundaries. However,
even if boundaries are absent we use the same method for the following reasons: (i) this will ensure a smooth transition to more complicated material of
the next section; (ii) we believe that calculations of the coefficients ak (f, D)
are a little bit easier even without a boundary. We are not going to give complete proofs of all statements. Instead, we concentrate on main ideas of the
method. For the details an interested reader can consult the original paper [?]
and the monographs [?, ?].
We start with very general properties of the heat kernel coefficients. Let
us consider a smooth compact Riemannian manifold M without boundary. To
be able to define functions on M which carry some discrete (spin or gauge)
indices we need a vector bundle V over M. Let D be an operator of Laplace
type on V , and let f be a smooth function on M. There is an asymptotic
expansion (2.21) and
1) Coefficients with odd index k vanish, a2j+1 (f, D) = 0;
2) Coefficients a2j (f, D) are locally computable in terms of geometric invariants.
Already the existence of the power-law asymptotic expansion (2.21) is a
non-trivial statement. We postpone the discussion of this property to sec. 5.4.
The second statement above is very important. It means that the heat
kernel coefficients can be expressed as integrals of local invariants:
Z
ak (f, D) = trV
M
X
√
dn x g{f (x)ak (x; D)} =
trV
I
Z
M
√
dn x g{f uI AIk (D)} ,
(4.1)
where AIk are all possible independent invariants of the dimension k constructed from E, Ω, Rµνρσ and their derivatives. We use usual assignments
of the dimensions when E has dimension two, any derivative has dimension
one, etc. uI are some constants. For example, if k = 2 only two independent
invariants exist. These are E and R. Note, that we can always integrate by
parts to remove all derivatives from f . The first statement, a2j+1 = 0, is clear
now. One cannot construct an odd-dimension invariant on a manifold without
boundary.
Let us study further relations between the heat kernel coefficients which
will turn out to generate relations between the constants uI . Consider now
the case when the manifold M is a direct product of two manifolds, M1 and
M2 , with coordinates x1 and x2 respectively, and the operator D is a sum of
33
two operators acting independently on M1 and M2 , D = D1 ⊗ 1 + 1 ⊗ D2 .
This means that the bundle indices are also independent. As an example, one
can consider the vector Laplacian on M1 × M2 . One can write symbolically
exp(−tD) = exp(−tD1 ) ⊗ exp(−tD2 ). Next we multiply both sides of this
equation by f (x1 , x2 ) = f1 (x1 )f2 (x2 ), take the functional trace and perform
the asymptotic expansion in t to get
X
ak (x; D) =
ap (x1 ; D1 )aq (x2 ; D2 ) .
(4.2)
p+q=k
The consequences of eq. (4.2) are very far reaching. In particular, eq.
(4.2) allows to fix the dependence of uI on the dimension of the manifold M.
Consider an even more specialised case when one of the manifolds is a onedimensional circle: M1 = S 1 , 0 < x1 ≤ 2π. Let us make the simplest choice for
D1 : D1 = −∂x21 . All geometric invariants associated with D are defined solely
by the D2 -part. Moreover, all invariants are independent of x1 . Therefore, by
the equation (4.1),
Z
ak (f (x2 ), D) =
S 1 ×M2
√ X
dn x g
trV {f (x2 )uI(n) AIk (D)}
I
Z
= 2π
d
M2
n−1
√ X
x g
trV {f (x2 )uI(n) AIk (D2 )} .
(4.3)
I
Here dependence of the constants uI(n) on the dimension n of the manifold is
shown explicitly. On the other hand, we can use (4.2). Spectrum of the operator
D1 is known. The eigenvalues are l2 , l ∈ Z. The heat kernel asymptotics for
D1 can be easily obtained by using the Poisson summation formula
K(t, D1 ) =
X
r
exp(−tl2 ) =
l∈Z
r
'
π
+ O e−1/t .
t
πX
exp(−π 2 l2 /t)
t l∈Z
(4.4)
Since exponentially small terms have no effect
√ on the heat kernel coefficients,
the only non-zero coefficient is a0 (1, D1 ) = π. Therefore, we obtain by eq.
(4.2)
ak (f (x2 ), D) =
√ Z
π
M2
√ X
dn−1x g
trV {f (x2 )uI(n−1) AIn (D2 )} .
(4.5)
I
There are no restrictions on the operator D2 or on the manifold M2 . By comparing the equations (4.3) and (4.5) we obtain
√
uI(n) = 4πuI(n+1) .
(4.6)
This proves that the constants uI depend on the dimension n only through
the overall normalisation factor (4π)−n/2 .
34
To calculate the heat kernel coefficients we need also the following variational equations (see [?], Lemma 4.1.15):
d
|=0 ak (1, e−2f D) = (n − k)ak (f, D) ,
d
d
|=0 ak (1, D − F ) = ak−2 (F, D) ,
d
d
|=0 an−2 (e−2f F, e−2f D) = 0 ,
d
(4.7)
(4.8)
(4.9)
where f and F are some smooth functions.
To prove the first property (4.7) we note that
d
d
|=0Tr(exp(−e−2f tD)) = Tr(2f tD exp(−tD)) = −2t Tr(f exp(−tD))
d
dt
and expand both sides of this equation in the power series in t. Eq. (4.8) can
be checked in a similar way. To prove (4.9) consider the operator
D(, δ) = e−2f (D − δF ) .
(4.10)
We use first (4.7) with k = n to show:
d
|=0 an (1, D(, δ)).
d
0=
(4.11)
Then we vary the equation above with respect to δ:
0=
d
d
d
d
|δ=0 |=0 an (1, D(, δ)) = |=0 |δ=0 an (1, D(, δ)) .
dδ
d
d
dδ
(4.12)
Finally, eq. (4.8) yields (4.9).
Eq. (4.8) restricts dependence of the heat kernel coefficients on the “potential” E while (4.7) and (4.9) describe properties of the heat kernel coefficients
under local scale transformations.
To calculate the heat kernel coefficients we adopt the following strategy.
First we write down a general expression for ak containing all invariants AIn of
dimension k with arbitrary coefficients uI . The constants uI are then calculated
by using the properties derived above. The first three coefficients read [?, ?]:
a0 (f, D) = (4π)
−n/2
a2 (f, D) = (4π)−n/2
Z
M
1
6
√
dn x gtrV {α0 f }
Z
M
√
dn x gtrV {f (α1 E + α2 R)},
(4.13)
(4.14)
Z
1
√
dn x gtrV {f (α3 E;kk + α4 RE + α5 E 2
360 M
+α6 R;kk + α7 R2 + α8 Rij Rij + α9 Rijkl Rijkl + α10 Ωij Ωij )}.(4.15)
a4 (f, D) = (4π)−n/2
Instead of the uI we use rescaled constants αI . By (4.6) the coefficients αI
are true constants, i.e. they do not depend on n. One can check that indeed
35
no more invariants exist. For example, Rij;ij is proportional to R;jj due to the
Bianchi identity.
The coefficient α0 follows immediately from the heat kernel expansion for
the “free” scalar Laplacian on S 1 (see eq. (4.4)). We obtain α0 = 1. Let us
now use (4.8). First take k = 2. Then
Z
1Z n √
√
d x gtrV {α1 F } =
dn x gtrV {F } .
6 M
M
(4.16)
This gives α1 = 6. Take k = 4 to see
1 Z n √
1Z n √
d x gtrV {α4 F R + 2α5 F E} =
d x gtrV {α1 F E + α2 F R} .
360 M
6 M
(4.17)
The equation (4.17) shows that α5 = 180, α4 = 60α2 .
To proceed further we need local scale transformations defined in (4.7)
and (4.9). These scale transformations look similar to the local Weyl transformations but are not exactly the same. The scale transformations in (4.7)
are designed in such a way that the operator D always transforms covariantly.
This is not the case of the Weyl transformations of an arbitrary operator of
Laplace type. For example, the scalar Laplacian (3.4) is conformally covariant
for a special value (3.5) of the conformal coupling only. Therefore, some of
the basic quantities are transformed in a somewhat unusual way. The metrics
transforms as gµν → e2f gµν thus defining standard conformal properties of the
Riemann tensor, the Ricci tensor and the scalar curvature. The functions aσ
and b in eq. (2.1) transform homogeneously. Transformation properties of ωµ
and E are then defined through (2.3) and (2.4). One can obtain the following
relations [?, ?]
d
√
√
|=0 g = nf g ,
d
d
|=0 Rijkl = −2f Rijkl + δjl f;ik + δ;ik f;jl − δil f;jk − δjk f;il ,
d
d
1
|=0 E = −2f E + (n − 2)f;ii ,
d
2
d
|=0 R = −2f R − 2(n − 1)f;ii ,
d
d
1
|=0 E;kk = −4f E;kk − 2f;kk E + (n − 2)f;iijj + (n − 6)f;k E;k ,
d
2
1
d
|=0 RE = −4f RE + (n − 2)f;ii R − 2(n − 1)f;ii E ,
d
2
d
|=0 E 2 = −4f E 2 + (n − 2)f;ii E ,
d
36
d
|=0 R;kk = −4f R;kk − 2f;kk R − 2(n − 1)f;iijj + (n − 6)f;iR;i ,
d
d
|=0 R2 = −4f R2 − 4(n − 1)f;iiR ,
d
d
|=0 Rij Rij = −4f Rij Rij − 2f;ii R − 2(n − 2)f;ij Rij ,
d
d
|=0 Rijkl Rijkl = −4f Rijkl Rijkl − 8f;ij Rij ,
d
d
|=0 Ω2 = −4f Ω2
d
(4.18)
Let us remind that the indices i, j, k, l are flat, so we can put them all down
and sum up over the repeated indices by contracting them with the Kronecker
δ.
Let us apply (4.9) to n = 4.
d
|=0a2 (e−2f F, e−2f D) = 0 .
d
R
√
By collecting the terms with trV M dn x g{F f;jj } we obtain α1 = 6α2 and
consequently α2 = 1 and α4 = 60α2 = 60. This completes calculation of a2 .
Let M = M1 × M2 with a product metric and let D = (−∆1 ) + (−∆2 )
where ∆1,2 are scalar Laplacians on M1 and M2 respectively 12 . Eq. (4.2) yields
a4 (1, −∆1 − ∆2 ) = a4 (1, −∆1 )a0 (1, −∆2 ) + a2 (1, −∆1 )a2 (1, −∆2 ) +
(4.19)
+a0 (1, −∆1 )a4 (1, −∆2 )
It is clear from (3.6) that E = 0 and Ω = 0. By collecting the terms with
R1 R2 (where R1 and R2 are scalar curvatures on M1 and M2 respectively) we
obtain
2
2
α2
α7 =
.
360
6
Consequently, α7 = 5.
Let us apply (4.9) to n = 6. We obtain with the help of the variational
equations (4.18)
Z
0 = trV
M
√
dn x g{F ((−2α3 − 10α4 + 4α5 )f;kk E
+(2α3 − 10α6 )f;iijj
+(2α4 − 2α6 − 20α7 − 2α8 )f;ii R
+(−8α8 − 8α9 )f;ij Rij )} .
(4.20)
The coefficients in front of independent invariants in (4.20) must be zero. We
determine α3 = 60, α6 = 12, α8 = −2 and α9 = 2.
12
More precisely, we assume that in (3.4) the potential U = 0, the conformal coupling is minimal ξ = 0, and there is no gauge coupling Gµ = 0 on both M1 and
M2 .
37
The most elegant way to calculate the remaining constant α10 is based on
the Gauss-Bonnet theorem (see [?]). We use here a more lengthy way [?] which
however works perfectly on flat manifolds without boundary and of trivial
topology of Rn . Note that Rn is non-compact. To make the heat kernel well
defined we should suppose certain fall-off conditions on the background fields
and on the smearing function f . The basis in the space of the square integrable
functions is given by the plane waves exp(ikx). Therefore, for M = Rn with
flat metric the heat kernel reads:
K(f ; t) = TrL2 (f exp(−tD))
Z
Z
dn k −ikx
n
= d x
e
trV {f (x) exp(−tD)eikx }
n
(2π)
Z
Z
dn k
n
trV {f (x) exp(t((∇µ + ik µ )2 + E))} .
= d x
n
(2π)
(4.21)
The following integrals will be useful:
Z
dn k −tk2
1
e
=
,
n
(2π)
(4tπ)n/2
Z
dn k −tk2 µ ν
1 µν
1
g ,
e
k k =
n
n/2
(2π)
(4tπ) 2t
Z
dn k −tk2 µ ν ρ σ
1 µν ρσ
1
e
k k k k =
(g g + g µρ g νσ + g µσ g νρ). (4.22)
n
n/2
(2π)
(4tπ) 4t2
Now we isolate exp(−tk 2 ) on the right hand side of (4.21) and expand the rest
of the exponent in a power series of t.
Z
K(f ; t) =
n
d x
Z
d4 k −tk2
e
trV
(2π)n
(
t2
f (x) 1 + t(∇ + E) − 4(k∇)
2
2
t2 2 2
(∇ ∇ + ∇2 E + E∇2 + E 2 )
2
4t3
((k∇)2 E + E(k∇)2 + (k∇)E(k∇))
−
6
4t3
−
((k∇)2 ∇2 + ∇2 (k∇)2 + (k∇)∇2 (k∇))
6
!)
16t4
4
+
(k∇) + . . .
24
+
We use the integrals (4.22) to obtain
38
(4.23)
Z
1
K(f ; t) =
dn xtrV {f (x) (1 + tE
(4πt)n/2
t2
+ (∇2 ∇2 + ∇2 E + E∇2 + E 2 )
2
t2
− (∇2 E + E∇2 + ∇µ E∇µ )
3
t2
− (2∇2 ∇2 + ∇µ ∇2 ∇µ )
3
!)
t2 µ ν
2 2
µ 2
3
+ (∇ ∇ ∇µ ∇ν + ∇ ∇ + ∇ ∇ ∇µ ) + O(t )
.
6
(4.24)
All derivatives combine into commutators. Finally we get:
K(f ; t) =
1
(4πt)n/2
Z
2
+t
dn xtrV {f (x) (1 + tE
1 2 1
1
E + E;µµ + Ωµν Ωµν + O(t3 )
2
6
12
.
(4.25)
From the equation (4.25) we conclude that α10 = 30. We have obtained also
an independent confirmation for the values of α0 , α1 , α3 and α5 .
The method we used above in eqs. (4.21) - (4.25) can be applied to more
general manifolds and operators as well. The key ingredient is a convenient
basis which should be used instead of the plane waves. For the case of a
box with (anti-) periodic boundary condition such a basis is rather obvious
(see [?,?]). On curved manifold M one has to use the so-called geodesic waves
[?, ?, ?, ?, ?] although calculations with ordinary plane waves are also possible
[?].
For calculation of the coefficient a6 we refer to [?]. The results for the
leading heat kernel coefficients are summarised in the following equations
39
a0 (f, D) = (4π)
−n/2
Z
M
a2 (f, D) = (4π)−n/2 6−1
√
dn x gtrV {f }.
Z
√
dn x gtrV {f (6E + R)}.
MZ
a4 (f, D) = (4π)−n/2 360−1
M
(4.26)
(4.27)
√
dn x gtrV {f (60E;kk + 60RE + 180E 2
+12R;kk + 5R − 2Rij Rij + 2Rijkl Rijkl + 30Ωij Ωij )}.
(4.28)
Z
nf
√
a6 (f, D) = (4π)−n/2
dn x gtrV
(18R;iijj + 17R;k R;k − 2Rij;k Rij;k
7!
M
−4Rjk;n Rjn;k + 9Rijkl;n Rijkl;n + 28RR;nn − 8Rjk Rjk;nn
+24Rjk Rjn;kn + 12Rijkl Rijkl;nn + 35/9R3 − 14/3RRij Rij
+14/3RRijklRijkl − 208/9Rjk Rjn Rkn − 64/3Rij Rkl Rikjl
−16/3Rjk Rjnli Rknli − 44/9RijknRijlp Rknlp
−80/9RijknRilkp Rjlnp ) + 360−1f (8Ωij;k Ωij;k + 2Ωij;j Ωik;k
+12Ωij;kk Ωij − 12Ωij Ωjk Ωki − 6Rijkn Ωij Ωkn − 4Rjk Ωjn Ωkn
+5RΩkn Ωkn + 6E;iijj + 60EE;ii + 30E;iE;i + 60E 3
+30EΩij Ωij + 10RE;kk + 4Rjk E;jk + 12R;k E;k + 30EER
+12ER;kk + 5ER2 − 2ERij Rij + 2ERijkl Rijkl )}.
(4.29)
2
Everyone who ever attempted calculations of the heat kernel coefficients
on curved background for arbitrary spin should appreciate that the method
presented here is a quite efficient one. One should also take into account that
some of the universal constant were indeed calculated twice.
The coefficients a0 – a4 are contained in [?, ?]. a6 was first computed by
Gilkey [?]. The next coefficient a8 has been calculated by Amsterdamski et
al [?] for the scalar Laplacian and by Avramidi [?, ?] for the general operator
of Laplace type. The coefficient a10 has been calculated by van de Ven [?].
Higher heat kernel coefficients in flat space were studied in [?, ?].
4.2 Examples
Here we consider several simple physical systems in four dimensions and
calculate the heat kernel coefficient a4 which defines the one-loop divergences
in the zeta function regularization.
4.2.1 Yang–Mills theory in flat space
Our first example is pure Yang–Mills theory in flat space. We are interested in the “total” heat kernel coefficient atot
4 defined by (3.53). Let us start
with the first term describing contribution from the vector fields. We choose
the gauge (3.44). The only non-vanishing invariants are E and Ω. The coefficient a4 is quadratic in these quantities. By using eqs. (3.46) and (3.47) we
obtain:
40
αρ βν
δ
γ
trV E 2 = Eνβ
Eρα = 4Fρν
Fρν
Kδγ ,
(4.30)
δ
γ
Fρν
Kδγ ,
trV (Ωij Ωij ) = −4Fρν
(4.31)
where
Kδγ = cδαβ cγαβ
(4.32)
is the Killing form of the gauge algebra.
Let us remind that the indices i, j, k in (4.28) refer to a local orthonormal
frame. In flat space they may be identified with the vector indices µ, ν, ρ. In the
case of vector fields the trace in (4.30) and (4.31) is taken over pairs consisting
of a gauge index α, β or γ and of a vector index µ, ν or ρ.
For the ghost operator we have to put R = U = 0 in (3.6) and identify the
connection with the background gauge field according to (3.49). Consequently,
E [gh] = 0 and
[gh]
[gh]
trV Ωij Ωij
δ
γ
= −Fρν
Fρν
Kδγ .
(4.33)
Next we substitute (4.30), (4.31) and (4.33) in (4.28) to obtain
[tot]
a4
[vec]
= a4
[gh]
− 2a4
=
11
96π 2
Z
M
√ δ γ
d4 x g Fρν
Fρν Kδγ .
(4.34)
The Yang–Mills gauge group G has usually a direct product structure,
G = G1 × G2 × . . . , so that on each of the irreducible components Gi the
Killing form K is proportional to the unit matrix. Therefore, the one-loop
divergence (4.34) reproduces the structure of the classical action with different
charges for each Gi . We also recover the coefficient 11/3 which is familiar from
computations of the Yang–Mills beta functions.
4.2.2 Free fields in curved space
Consider free quantum fields on a Riemannian manifold without boundaries. “Free” means that we neglect all interactions except for the one with
the background geometry. The heat kernel coefficients can be expressed then
in terms of local invariants of the metric. In particular,
1
1
a4 (x) =
aCµνρσ C µνρσ + b Rµν Rµν − R2 + cR;µ µ + dR2 , (4.35)
2
2880π
3
where Cµνρσ is the Weyl tensor,
1
Cµνρσ C µνρσ = Rµνρσ Rµνρσ − 2Rµν Rµν + R2 .
3
(4.36)
a, b, c and d are some constants depending on the spin. The first two structures
in (4.35) which appear together with a and b are conformally invariant in four
dimensions. This explains our choice of the basis in the space of invariants.
41
Table 1
a4 for various spins
Spin
a
b
c
d
0
1
1
30ξ − 6
90(ξ − 1/6)2
1/2
−7/2
−11
6
0
1
−13
62
18
0
2
212
0
0
717/4
The constants a, b, c and d can be evaluated by substituting particular
expressions for E and Ω obtained in sec. 3 in (4.28). Alternatively, one can use
the analysis of Christensen and Duff [?] who calculated a4 for arbitrary spin
fields (see also [?]). The results are collected in Table 1. Some comments are in
order. Spin 1/2 means 4-component Dirac spinors. For spin 1 and spin 2 fields
we took into account contributions from corresponding ghost fields. Note, that
vector Yang-Mills fields and vector ghosts for gravity interact differently with
the background geometry. The cosmological constant is taken to be zero.
As a physical application of the heat kernel expansion in curved space we
may mention, for example, the asymptotic conformal invariance phenomenon
which was studied by using this technique in [?, ?, ?, ?].
A similar analysis can be performed also in the presence of boundaries [?].
4.3 DeWitt iterative procedure
The iterative method by DeWitt uses separation of the heat kernel with
non-coinciding arguments into a leading part (which is non-analytic in t) and
the power-law corrections. For a flat manifold this separation has been described in sec. 1 (cf. eqs. (1.12) and (1.13)). Let us consider a massless scalar
field on a curved compact Riemannian manifold M. The DeWitt ansatz reads
in this case:
!
K(t; x, y; D) = (4πt)
−n/2
1/2
∆V V M (x, y) exp
σ(x, y)
−
Ξ(t; x, y; D) ,
2t
(4.37)
where σ(x, y) is one half the square of the length of the geodesic connecting
x and y. In Cartesian coordinates on a flat manifold σflat (x, y) = 12 (x − y)2 .
∆V V M is the so called Van Vleck–Morette determinant
∆V V M (x, y) =
det − ∂x∂µ ∂y∂ ν σ(x, y)
q
.
(4.38)
g(x)g(y)
As a consequence of the heat equation (1.10) the kernel Ξ should satisfy
−1/2
∂t + t−1 (∇µ σ)∇µ + ∆V V M D∆V V M Ξ = 0
42
1/2
(4.39)
with the initial condition
Ξ(0; x, y; D) = 1 .
(4.40)
The essence of the DeWitt method is to look for a solution of the equation
(4.39) in the following form
Ξ(t; x, y; D) =
∞
X
tj b2j (x, y; D) .
(4.41)
j=0
The initial condition (4.40) yields
b0 = 1 .
(4.42)
The recursion relation
−1/2
(j + (∇µ σ)∇µ ) b2j + ∆V V M D∆V V M b2(j−1) = 0 ,
1/2
(4.43)
which follows from (4.39) and (4.41), allows, in principle, to find the coincidence limits x = y of higher heat kernel coefficients b2j . An important ingredient of such calculations is the coincidence limits of symmetrized derivatives of
the geodesic interval σ(x, y) [?,?]. This method becomes very cumbersome beyond a4 . A refined nonrecursive procedure to solve the DeWitt equation (4.43)
was used by Avramidi [?] to calculate a8 (see also [?] for a short overview).
The method of DeWitt can be naturally extended to treat coincidence limits
of the derivatives of b2j (x, y; D).
The recursion relations can be generalised to the case of manifolds with
boundaries [?, ?]. However, for a practical use the functorial methods of sec.
5 seem to be more convenient.
4.4 Non-minimal operators
Quantisation of gauge theories oftenly leads to second order differential
operators which are not of the Laplace type. For example, by taking κ 6= 1 in
(3.52) one obtains the following operator acting on the gauge field fluctuations:
[nm]
Dµν
= −∆gµν
1
+ 1−
∇ν ∇µ ,
κ
(4.44)
where, for simplicity, we suppose that the manifold M is flat (Rµνρσ = 0) and
the gauge group is abelian (cγαβ = 0). The leading symbol of the operator (4.44)
(the part with the highest derivatives) has a non-trivial matrix structure. Such
operators are called non-minimal.
In some simple cases (see, e.g., [?, ?, ?]) the spectral problem for nonminimal operators can be reduced to the Laplacians. If the leading part of a
non-minimal operator has a form similar to (4.44) but the lower order part
43
is more or less arbitrary 13 , necessary generalisations of the DeWitt technique
were suggested by Barvinsky and Vilkovisky [?]. The technique was further
developed in [?, ?]. Complete calculation of a4 required extensive use of the
computer algebra [?]. Most general non-minimal operators were considered
in [?] where, because of great technical complexity, only the first two heat
kernel coefficients were analysed. Some calculations in various physical systems
with non-minimal operators can be found in [?, ?, ?, ?].
13
This case covers most of the physical applications.
44
5
Heat kernel expansion on manifolds with boundaries
5.1 Two particular cases
We start our analysis of manifolds with boundaries with two simple examples. First, let us consider a one-dimensional manifold M = [0, π]. Let
D = −∂x2 . We consider both Dirichlet (2.15) and Neumann (2.16) boundary
conditions (taking S = 0 in (2.16) to simplify the calculations). The eigenfunctions of D are:
Dirichlet: sin(lx), l = 1, 2, . . . ;
Neumann: cos(lx), l = 0, 1, 2, . . .
(5.1)
The eigenvalues are l2 in both cases, where l is a positive integer for Dirichlet
boundary conditions, and l is a nonnegative integer for Neumann boundary
conditions. The heat kernel asymptotics can be calculated with the help of the
Poisson summation formula (4.4):
r
π
1
− 1 + O e−1/t ,
=
exp(−tl ) =
2
t
l>0
r
X
π
1
2
+
2
+ 1 + O e−1/t ,
K(t, −∂x , B ) =
exp(−tl ) =
2
t
l≥0
K(t, −∂x2 , B− )
X
2
(5.2)
(5.3)
where for the later use we explicitly mention the boundary operators B− and
B+ which define Dirichlet (2.15) and Neumann (2.16) boundary conditions
respectively.
Let us modify a little bit the example above by allowing for a non-zero S
at one of the components of the boundary:
∂x φ|x=0 = 0,
(∂x − S)φ|x=π = 0
(5.4)
Note, that at x = π the derivative with respect to an inward pointing unit
normal is −∂x . The eigenfunctions
φk = cos(kx)
(5.5)
satisfy the boundary conditions (5.4) at x = 0. The spectrum is defined by
the condition at x = π which reads 14 :
−k sin(kπ) = S cos(kπ) .
(5.6)
In this example we restrict ourselves to the linear order in S. We suppose that
S is small and that the spectrum (5.1) is only slightly perturbed:
kl = l + zl .
14
For S > 0 also a negative mode cosh(kx) appears. Here we take S ≤ 0.
45
(5.7)
The equation (5.6) gives in this approximation
S
lπ
S
z02 = −
π
zl = −
for
l > 0,
for
l = 0.
(5.8)
We expand also the heat kernel in a power series in S and use (5.8) and (5.2)
to obtain:
K(t, −∂x2 , BS+ ) =
X
exp(−tkl2 )
l≥0
X
2St
St
+ O(S 2 )
=
exp(−tl ) 1 +
+1+
π
π
l>0
1
=
2
r
2
s
π
t
+1 +S
+ O(S 2 ) ,
t
π
(5.9)
where we also dropped exponentially small terms in t.
Our next example is the heat kernel expansion for scalar fields in a ball
with Dirichlet and Neumann boundary conditions [?, ?, ?, ?, ?]. The metric of
the unit ball in Rn reads:
ds2 = dr 2 + r 2 dΩ2 ,
0 ≤ r ≤ 1,
(5.10)
where dΩ2 is the metric on unit sphere S n−1 . The scalar Laplace operator has
the form
∆φ =
∂r2
n−1
∂r +(n−1) ∆ φ ,
+
r
(5.11)
where (n−1) ∆ is the Laplace operator on S n−1 . The eigenfunctions of the operator (5.11) are well known
φl,λ ∝ r (2−n)/2 J(n−2)/2+l (rλ1/2 )Y(l) (xa ).
(5.12)
Jp are the Bessel functions. The eigenvalues −λ are defined by boundary conditions. Y(l) (xi ) are n-dimensional scalar spherical harmonics. Their degeneracies
are
Nl =
(2l + n − 2)(l + n − 3)!
.
l!(n − 2)!
(5.13)
To proceed further it is convenient to consider the zeta function which
46
may be presented through a contour integral [?] 15 :
ζ(s, D) =
∞
X
l=0
Z
Nl
γ
dk −2s ∂
k
Φ(n−2)/2+l (k) ,
2πi
∂k
where Φν is a function which has zeros at the spectrum k =
boundary conditions this function reads:
Φν = k −ν Jν (k) .
√
(5.14)
λ. For Dirichlet
(5.15)
For Robin boundary condition Φν is given by a somewhat more complicated
combination of the Bessel functions. The contour γ runs counterclockwise and
encloses all the solutions of Φν = 0 on the positive real axis. Note the presence
of k −ν in (5.15) which is included to avoid unwanted contributions coming from
the origin k = 0.
The next step is to rotate the contour to the imaginary axis and to calculate residues of Γ(s)ζ(s, D) (as prescribed by (2.26)). It is interesting to note
that the heat kernel coefficients are defined by several leading terms in the
uniform asymptotic expansion of Φν (iνz) for large ν and fixed z. For further
details we refer to [?].
Euclidean ball was frequently used in calculations of the heat kernel coefficients and functional determinants. Apart from the papers already quoted
above also the computations for a scalar field [?, ?, ?, ?, ?], spinors [?, ?, ?, ?,
?, ?, ?, ?], abelian gauge fields [?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?] 16 should be mentioned. A similar technique works also for more complicated geometries as,
e.g., the spherical cap [?, ?, ?, ?, ?]. Note that here we consider local boundary
conditions without tangential derivatives only. For other types of boundary
operators see sec. 5.4.
5.2 Dirichlet and Neumann boundary conditions
Let us now find analytic expressions for the heat kernel coefficients in
terms of geometric invariants. Here we follow the method of Branson and
Gilkey [?, ?]. For both Dirichlet and modified Neumann (Robin) boundary
conditions the heat kernel coefficients are locally computable. This means
that ak may be represented as a sum of volume and boundary integrals of
15
A general discussion of the representation of the zeta function by contour integrals
can be found in [?].
16 Some of these works consider contributions of the so called physical modes only.
As explained in [?] complete answer for the effective action and for the scaling
behaviour must include contributions from ghosts and non-physical modes. This
applies also to the spherical cap case considered below.
47
some local invariants:
ak (f, D, B) =
Z
k−1
X
√
d x gf (x)ak (x, D) +
n
M
Z
j=0 ∂M
√
dn−1x hf (j) ak,j (x, D, B) ,
(5.16)
where f (j) denotes j-th normal derivative of the smearing function f . Note,
that we cannot now integrate by parts to remove normal derivatives from f .
This reflects the distributional nature of the heat kernel asymptotics.
The volume terms ak (x, D) are the same as in the previous section (see
(4.26) - (4.29)). Here we evaluate the boundary terms ak,j (x, D, B). Obviously,
the canonical mass dimension of ak,j is k − j − 1. In addition to the usual bulk
invariants E, R, etc, ak,j can also contain specific boundary quantities as the
extrinsic curvature Lab or S (for Robin boundary conditions). Note, that Lab
and S are defined on the boundary only and, therefore, can be differentiated
only tangentially 17 . Canonical mass dimension of Lab and S is +1.
In this section we explicitly calculate the first three heat kernel coefficients
a0 , a1 and a2 . Basing on the considerations of the preceding paragraph we may
write:
Z
√
dn x gtrV (f ),
M Z
√
±
−(n−1)/2
a1 (f, D, B ) = (4π)
dn−1 x h b±
1 trV (f ),
±
a0 (f, D, B ) = (4π)
−n/2
Z
∂M
1
√
dn x gtrV (6f E + f R)
a2 (f, D, B ) = (4π)−n/2
6
M
Z
√
n−1
±
±
±
+
d x htrV (b2 f Laa + b3 f;n + b4 f S) ,
±
∂M
(5.17)
(5.18)
(5.19)
where B± denotes either Robin (B+ ) or Dirichlet (B− ) boundary operator. b±
are some constants. To keep uniform notations we formally included S also in
the expression for Dirichlet boundary conditions. It should be assumed that
S = 0 for that case.
Since the product formula (4.2) is still valid with obvious modifications
for the boundary contributions, we can again consider the case M = M1 × S 1
and repeat step by step the calculations (4.3) - (4.6) thus arriving to the same
conclusion that the constants b± do not depend on dimension of the manifold
(all explicit n-dependence of the heat kernel is contained in the power of 4π).
This property is crucial for our calculations.
17
See sec. 2.1 for more information on differential geometry of manifolds with
boundary.
48
The one dimensional examples 18 (5.2) and (5.3) immediately give
−
b+
1 = −b1 =
1
.
4
(5.20)
Our next example (5.9) controls linear terms in S and gives
b+
4 = 12 .
(5.21)
For Dirichlet boundary conditions we have set S = 0. Therefore, b−
4 plays no
role.
±
To define the constants b±
2 and b3 we use the conformal variation equation
(4.7). Variations of E and R are given by (4.18). Conformal properties of the
second fundamental form Lab are dictated by that of the metric:
d
|=0 Laa = −f Laa − (n − 1)f;n .
d
(5.22)
We wish to keep the boundary conditions invariant under the conformal transformations. This implies that the boundary operators B± must transform homogeneously. This property holds automatically for Dirichlet boundary conditions (2.15). In the case of modified Neumann boundary conditions conformal
transformation of S should cancel inhomogeneous term in the connection ωn .
This yields
d
1
|=0 S = −f S + (n − 2)f;n .
d
2
(5.23)
Next we substitute the variational formulae (4.18), (5.22) and (5.23) in (4.7)
with k = 2. We collect the terms with f;n on the boundary to obtain
−
−(n − 4) − (n − 1)b−
2 = (n − 2)b3
(5.24)
for Dirichlet conditions and
+
−(n − 4) − (n − 1)b+
2 + 6(n − 2) = (n − 2)b3
(5.25)
for generalised Neumann (Robin) ones. Since the constants b± do not depend
on n the two equations (5.24) and (5.25) are enough to define these constants:
−
b+
2 = b2 = 2 ,
−
b+
3 = −b3 = 3 .
(5.26)
This completes the calculation of a2 .
The coefficients a3 and a4 can be obtained as particular cases of more
general formulae of the next subsection.
18
For M = [0, π] the boundary integral reduces to a sum of two contributions from
x = 0 and x = π.
49
5.3 Mixed boundary conditions
Let us now turn to mixed boundary conditions (2.17) which, as we have
seen in sections 3.3 and 3.4, are natural boundary conditions for spinor and
vector fields. These boundary conditions depend on two complementary local
projectors Π− and Π+ = 1 − Π− which define subsets of components of the
field satisfying Dirichlet and Robin boundary conditions respectively 19 . More
precisely,
Π− φ|∂M = 0 ,
(∇n + S)Π+ φ|∂M = 0 .
(5.27)
Consequently, there is one more independent entity (as compared to Robin
case, for example) on which the heat kernel coefficients for mixed boundary
conditions can depend. This makes the calculations somewhat more complicated. It is convenient to define
χ = Π+ − Π− .
(5.28)
Calculation of the coefficients ak up to k = 4 can be found in [?] (see
also [?] for some corrections). The result reads:
19
Examples of such projectors are given by eq. (3.35) and below eq. (3.58).
50
Z
√
dn x gtrV (f ).
M
Z
√
1
−(n−1)/2
dn−1 x htrV (χf ).
a1 (f, D, B) = (4π)
4
∂M
Z
1
√
a2 (f, D, B) = (4π)−n/2
dn x gtrV (6f E + f R)
6
M
Z
√
n−1
+
d x htrV (2f Laa + 3χf;n + 12f S) .
a0 (f, D, B) = (4π)
−n/2
∂M
(5.29)
(5.30)
(5.31)
Z
n
√
1
(4π)−(n−1)/2
dn−1 x htrV f (96χE + 16χR
384
∂M
+8f χRanan + (13Π+ − 7Π− )Laa Lbb + (2Π+ + 10Π− )Lab Lab
+96SLaa + 192S 2 − 12χ:a χ:a ) + f;n ((6Π+ + 30Π− )Laa
+96S) + 24χf;nn }.
(5.32)
nZ
1
√
a4 (f, D, B) =
(4π)−n/2
dn x gtrV {f (60E;ii + 60RE + 180E 2
360
M
+30Ωij Ωij + 12R;ii + 5R2 − 2Rij Rij + 2Rijkl Rijkl )}
Z
n
√
+
dn−1 x htrV f {(240Π+ − 120Π− )E;n
a3 (f, D, B) =
∂M
+(42Π+ − 18Π− )R;n + 24Laa:bb + 0Lab:ab + 120ELaa
+20RLaa + 4Ranan Lbb − 12Ranbn Lab + 4Rabcb Lac
1
+ {(280Π+ + 40Π− )Laa Lbb Lcc + (168Π+
21
−264Π− )Lab Lab Lcc + (224Π+ + 320Π− )Lab Lbc Lac }
+720SE + 120SR + 0SRanan + 144SLaa Lbb + 48SLab Lab
+480S 2Laa + 480S 3 + 120S:aa + 60χχ:a Ωan − 12χ:a χ:a Lbb
−24χ:a χ:b Lab − 120χ:a χ:a S} + f;n (180χE + 30χR + 0Ranan
1
+ {(84Π+ − 180Π− )Laa Lbb + (84Π+ + 60Π− )Lab Lab }
7
+72SLaa + 240S 2 − 18χ:a χ:a ) + f;nn (24Laa + 120S)
+30χf;iin
oo
.
(5.33)
Dirichlet and modified Neumann (Robin) boundary conditions are recovered
when Π+ = 0 or Π− = 0 respectively.
The coefficients ak with k = 0, 1, 2, 3, 4 were calculated in [?, ?, ?, ?, ?, ?,
?, ?, ?]. A different algorithm was suggested in [?]. For “pure” (not mixed)
boundary conditions the coefficient a5 was calculated by Branson, Gilkey and
Vassilevich [?] for the special case of a domain in flat space or of a curved
domain with totally geodesic boundaries. Kirsten [?] generalised these results
for arbitrary manifolds and boundaries. Branson, Gilkey, Kirsten and Vassilevich [?] calculated the coefficient a5 for mixed boundary conditions.
51
5.4 Other boundary conditions
From the technical point of view boundary conditions for a Laplace type
operator are needed to exclude infinite number of negative and zero modes
and to ensure self-adjontness. In principle, any linear relation between the
boundary data φ|∂M and φ;n |∂M is admissible as long as it serves this purpose.
Here we consider two physically motivated examples of boundary conditions
which contain tangential derivatives on the boundary. These examples should
give the reader an idea of what can be expected for a more general boundary
value problem.
5.4.1 Boundary conditions with tangential derivatives and Born–Infeld action from open strings
The boundary condition
1
∇n + (∇a Γa + Γa ∇a ) + S φ = 0
2
∂M
(5.34)
is the simplest condition containing both normal and tangential derivatives.
Γa and S are some matrix valued functions defined on the boundary. Such or
similar structures appear in open strings (cf. sec. 3.2) and in chiral bags (cf.
sec. 3.3). They also describe photons with the Chern-Simons interaction term
concentrated on the boundary [?,?] and may be relevant for solid state physics
applications. The conditions (5.34) appeared in the mathematical literature
[?, ?].
Several heat kernel coefficients for the boundary conditions (5.34) (which
are called oblique) have been calculated by McAvity and Osborn [?] and by
Dowker and Kirsten [?, ?]. Avramidi and Esposito [?, ?, ?, ?] lifted some commutativity assumptions and proved a simple criterion of strong ellipticity (see
below).
Consider a differential operator Q, not necessarily of the second order. Let
is separate the part containing highest derivatives and replace ∂µ → ikµ , like
in doing the Fourier transformation. In this way we obtain an object AQ =
aµν...ρ (x)kµ kν . . . kρ which is called the leading symbol of Q. If the operator
Q acts on fields with indices (i.e. in a vector bundle), the index structure of
Q is inherited by AQ . In other words, for fixed k the leading symbol A is a
matrix valued function (an endomorphism of the vector bundle). If AQ (x) is
non-degenerate for all k 6= 0, the operator Q is called elliptic. For the Dirac
operator Q = D
/ the leading symbol is AQ = γ µ kµ. Therefore, D
/ is obviously
elliptic. The Laplace type operators are also elliptic since AD = k 2 . Ellipticity
means that “at large momenta” the operator Q is dominated by its’ highest
derivative part. This is the property which guarantees that on a compact
manifold without boundaries Laplace operators have at most finite number of
negative and zero eigenvalues and which ensures existence of the heat kernel.
On manifolds with boundaries just ellipticity is not enough to ensure nice
52
properties of the spectrum. There is an additional requirement, called strong
ellipticity, which should be satisfied by the boundary operator (see [?] for
details). Dirichlet and Robin boundary conditions are always strongly elliptic.
Oblique boundary conditions are strongly elliptic if and only if |Γ2 | < 1. If
this inequality is violated, infinite number of negative eigenmodes appears (a
simple example can be found in Appendix B of [?]). Note that the boundary
conditions (5.34) correspond to a symmetric D if the matrices Γa are antihermitian. Therefore, Γ2 = Γa Γa is typically negative.
Another complication stems from the fact that Γa is dimensionless. Consequently, arbitrary powers of Γa may enter ak , so that instead of the undetermined constants (cf. (5.17) - (5.19)) one has to deal with undetermined
functions of Γ. The problem becomes more tractable if we suppose that Γa ’s
commute among themselves, [Γa , Γb ] = 0. For this case, the coefficients a1
and a2 have been calculated by McAvity and Osborn [?] (a0 is still given by
(5.17)):
a1 (f, D) = (4π)−(n−1)/2
Z
∂M
√
dn−1 x htrV (f γ(Γ)) ,
(5.35)
Z
1
√
a2 (f, D) = (4π)−n/2
dn x gtrV (6f E + f R)
M
Z6
√
n−1
+
d x htrV (b0 (Γ)f Laa + b1 (Γ)f;n
∂M
+b2 (Γ)f S + f σ(Γ)Lab Γa Γb )} ,
(5.36)
where
"
#
1
2
√
γ=
−1 ,
4
1 + Γ2
#
"
√
1
1
b0 = 6
− √ 2 artanh −Γ2 + 2 ,
1 + Γ2
−Γ
√
6
b1 = √ 2 artanh −Γ2 − 3 ,
−Γ
12
,
b2 =
1 + Γ2
1
σ = 2 (2 − b0 ) .
Γ
(5.37)
We see, that if Γ2 approaches −1 the heat kernel blows up indicating violation
of the strong ellipticity condition.
As an example, let us consider the open string sigma model of sec. 3.2.
To simplify the subsequent calculations we put BAB = 0. In the zeta function
regularization divergent part of the effective action (2.31) reads (1/s)a2 (D) =
(1/s)a2 (1, D). On general grounds we expect that a2 repeats the structure of
53
the classical action (3.10):
Z
a2 =
∂M
[A]
¯ A + bulk terms,
dτ βA ∂τ X
(5.38)
[A]
where βA is a beta function 20 . We put Lab = 0 (otherwise we have had to
introduce a dilaton coupling on the boundary to achieve renormalizability). We
also suppose that the target space metric GAB is trivial. With these simplifying
assumptions the beta function can be easily calculated from (5.38), (5.36),
(5.37), (3.21):
[A]
βC = −
1
(∂A FBC )(1 + F 2 )−1
BA .
2π
(5.39)
By a lengthy but straightforward calculation one can demonstrate that the
[A]
condition βC = 0 is equivalent to the equations of motion following from the
Born-Infeld action on the target space:
LBI =
Z
q
¯ det(1 + iF ) =
dX
Z
¯ exp 1 tr ln(1 + F 2 ) ,
dX
4
(5.40)
where i appeared due to our rule of the Euclidean rotation for the gauge fields.
The Born-Infeld action has been derived from the beta functions of the
open string sigma model in [?, ?] confirming an earlier work [?] which used
different methods. The heat kernel analysis was performed by Osborn [?] and
then repeated in [?] for more general couplings.
5.4.2 Spectral or Atiyah–Patodi–Singer (APS) boundary conditions
Spectral boundary condition were introduced by Atiyah, Patodi and Singer
in their study of the Index Theorem [?, ?, ?]. These boundary conditions are
global, i.e. they cannot be defined by using local data only.
Consider a Dirac type operator
˜µ +E ,
D
/ = iγ µ ∇
(5.41)
˜ is a covariant derivative with a compatible connection:
where ∇
˜ µ γν = 0,
∇
(5.42)
i.e., the gamma matrices are covariantly constant. E is a zeroth order opera˜ is
tor (a matrix valued function). We also suppose that the connection in ∇
unitary. This means that the connection one-form is represented by an antihermitian matrix in a suitable basis. We restrict ourselves to the case when
E † = E, so that the operator D
/ is formally self-adjoint in the bulk. Note, that
20
In this case the subscript A is a target space vector index, while the superscript
[A] indicates the coupling. Here we do not consider other beta functions β [G] , β [B]
etc.
54
compatible unitary connection is not unique. Consider a first order differential
operator on the boundary:
˜a +
P = γn γ a ∇
i
(Eγn − γn E) + Θ(x) ,
2
(5.43)
where Θ(x) is a hermitian matrix valued function on ∂M. The operator P is a
self-adjoint operator of Dirac type on the boundary. All functions on ∂M can
be decomposed in positive, negative, and zero modes of P . Let us define Π−
as a projector on the space spanned by non-negative eigenspaces of P . Then
the equation
Π− φ|∂M = 0
(5.44)
defines the APS boundary conditions.
Of course, relative complexity and non-locality limits physical applications of spectral boundary conditions. However, they appeared in a number
of axial anomaly calculations [?, ?, ?, ?, ?, ?, ?, ?], quantum cosmology [?], and
in works on the Aharonov-Bohm effect [?,?]. In brane models spectral boundary conditions describe T-selfdual configurations which may be interpreted as
mixtures of D-branes and open strings [?].
General form of the heat kernel expansion for spectral boundary conditions was established by Grubb and Seeley [?, ?, ?]:
K(t, f, D) '
n−1
X
ak t(k−n)/2 +
∞ X
j=n
k=0
a0j ln t + a00j t(j−n)/2 .
(5.45)
In the contrast to all previous cases the expansion (5.45) contains logarithms of the proper time t. Although, such terms appear “typically” [?] for
pseudo-differential operators, they may lead to rather unpleasant physical consequences. As follows from (2.25), non-zero a0n means that the zeta function
has a pole at s = 0, and, therefore, the expression (2.32) for the renormalised
effective action does not make sense. Fortunately, for the APS boundary conditions a0n = 0 if f = 1 near the boundary. Consequently, the integrated zeta
function is regular at s = 0 and eq. (2.32) still can be used. However, all calculations involving localised heat kernel coefficients remain problematic. Many
logarithmic terms vanish if the manifold M has a product structure near ∂M.
In non-product cases strong criteria of “partial vanishing of logarithms” have
been found recently [?].
Another problem with the heat kernel expansion (5.45) is that the coefficients may have a more complicated dependence on n than just a power 21
21
Let us remind that to prove the simple dependence on n we used the product
formulae (4.3), (4.4). We assumed that the spectral problem can be “trivialised”
in one direction. In the present case all tangential coordinates enter the operator
(5.43) on equal footings. Therefore, the proof does not go through.
55
of (4π). After this long list of troubles it is not a surprise that only the coefficients ak with k < n are locally computable. The coefficient a0 is given by
(4.26). a1 and a2 have been calculated in [?]:
a1 = (4π)
(1−n)/2 1
4
(β(n) − 1)
Z
Z
∂M
√
dn−1x htrV (f ) ,
1
√
dn x gtrV f
a2 = (4π)−n/2
R+E
6
M
Z
√
1
3
+
dn−1 x htrV f E +
1 − πβ(n) Laa f
3 !# 4
∂M
n−1
1
−
1 − πβ(n) f;n
,
2(n − 2)
2
(5.46)
where
−1
n
1
β(n) = Γ
Γ
2
2
n+1
Γ
2
−1
.
(5.47)
The coefficient a3 can be found in [?].
Some string theory applications suggest [?,?] that spectral boundary conditions can be defined directly for a second order differential operator. Existence of the asymptotic expansion (5.45) and vanishing of leading logarithms
for such problems have been stated in [?].
56
6
Manifolds with singularities
All results on the heat kernel expansion formulated in the previous sections are valid on smooth manifolds only. If there are boundaries, they also
have to be smooth. As well, any singularities in the potential term or in the
field strength are strictly speaking forbidden. However, many physical models
deal with singular backgrounds. Even if such backgrounds may be represented
through certain limiting procedures from smooth configurations, the heat kernel coefficients are not given by limits of their “smooth” values. The most
visible manifestation of failure of the smooth field approximation is that ak
with sufficiently large k are divergent. Usually, the presence of singularities
changes even the structure of the heat kernel kernel expansion as compared
to the smooth case.
6.1 Non-integrable potentials
According to (1.21) divergences in the effective action are defined by
integrated heat kernel coefficients. Although the formulae (4.26) - (4.29) for
the localised heat kernel coefficients are valid on non-compact manifolds (provided the smearing function f falls off sufficiently fast), transition to the
integrated heat kernel is not that straightforward. Already the coefficient
a0 (D) = a0 (1, D), which is proportional to the volume, is divergent. This
divergence is usually removed by replacing det(D) in (1.6) by
det(D)/ det(D0 ) ,
(6.1)
where the operator D0 = −∂ 2 + m2 describes a free particle propagation in
an “empty” space. It is argued that since D0 does not depend on “essential”
variables division by det(D0 ) does not change physical predictions of the theory. In all subsequent formulae the heat kernel K(t; x, y; D) is then replaced
by the subtracted heat kernel
Ksub (t; x, y) = K(t; x, y; D) − K(t; x, y; D0 ) .
(6.2)
In flat space the coefficient a0 corresponding to Ksub is identically zero 22 . If
the field strength Ωµν and the (subtracted) potential E + m2 have a compact
support or decay sufficiently fast at the infinity, the small t asymptotic expansion of Ksub (t; x, x) is integrable on the whole M. If not, the very structure of
the global heat kernel may be changed.
As an example of non-integrable potentials consider the harmonic oscillator in one dimension. The Schr¨odinger operator reads
D = −∂x2 + ν 2 x2 .
22
(6.3)
On a curved manifold the subtraction procedure is more subtle. On has to define
a reference metric which differs from the physical one on a compact submanifold.
57
If we consider the problem on the whole real axis, M = R, the potential term
is not integrable. Already the expression (4.27) for a2 (1, D) diverges. Therefore, analytic expressions of sec. 4 cannot be used in this case. However, the
(integrated) heat kernel can be easily calculated. Eigenvalues of the operator
(6.3) are contained in almost any textbook on quantum mechanics:
λj = ν(2j + 1),
j = 0, 1, 2, . . .
(6.4)
The integrated heat kernel reads
K(t; D) =
∞
X
1
e−tν(2j+1) = [sinh(νt)]−1 .
2
j=0
(6.5)
As t → 0 it behaves like 1/t while for smooth rapidly decaying
√ potentials in
one dimension the leading singularity in the heat kernel is 1/ t. This statement may be generalised to higher dimensions. If D = −∂ 2 + Pµν xµ xν with
a non-degenerate matrix Pµν on M = Rn , the leading term in K(t; D) is
(2t)−n (det P )−1/2 [?].
6.2 Conical singularities
Conical space is defined as M = [0, 1]×N where N is an n−1-dimensional
manifold called the base. The metric of the cone has the form
(ds)2 = dr 2 + r 2 dΩ2 ,
(6.6)
where r ∈ [0, 1] and dΩ2 is the line element on the base N. This metric is,
in general, singular at r = 0. However, if we take the unit sphere S n−1 = N
with standard round metric, the singularity disappears and we obtain the
n-dimensional unit ball (5.10). If a manifold has singular points where the
metric can be approximated by (6.6) we say that this manifold has conical
singularities.
Conical singularities appear in many physical applications. First of all,
with N = S 1 the metric (6.6) is the Euclidean version of the Rindler metric.
Conical singularities appear in classical solutions of the Einstein equations [?,
?,?] and in the supermembrane theory [?]. Gravitational field of a point mass
in three dimensional gravity is a conical space [?, ?]. There are evidences [?]
that “conifolds” dominate the path integral for quantum gravity in topological
sectors.
Sommerfeld [?] was probably the first to consider the heat kernel in the
presence of conical singularities. The mathematical theory of the heat kernel
asymptotics with conical singularities was developed almost 100 years later
[?,?,?]. There two peculiar features of these asymptotics. First, the heat kernel
expansion contains in general both integer and half-integer powers of t even
without boundaries. Second, a non-standard ln t term may be contained in the
58
asymptotic series 23 .
On a manifold with conical singularities no closed analytical expression
for the heat kernel coefficients is available. However, usually it is possible
to disentangle contributions of the singularities from the smooth part. For
example, if N = S 1 , dΩ2 = dϕ2 with ϕ ∈ [0, α] only a2 receives a contribution
from the tip of the cone:
a2 (tip) =
4π 2 − α2
.
24πα
(6.7)
In many particular cases of conical singularities a very detailed analysis of
the heat kernel expansion has been performed [?,?,?,?,?,?,?,?,?,?,?,?,?,?].
One-loop computations on general orbifolds were considered recently in [?].
6.3 Domain walls and brane world
Delta function is an example of an extremely sharp background potential.
Let us consider a manifold M and a submanifold Σ of the dimension n − 1.
Let
D[v] = D + vδΣ .
(6.8)
D is an operator of Laplace type (2.2). Let h be the determinant of the induced
metric on Σ. Then δΣ is a delta function defined such that
Z
M
√
dx gδΣ f (x) =
Z
√
dx hf (x) .
(6.9)
Σ
The spectral problem for D[v] on M as it stands is ill-defined owing to
the discontinuities (or singularities) on Σ. It should be replaced by a pair of
spectral problems on the two sides M ± of Σ together with suitable matching
conditions on Σ. In order to find such matching conditions, we consider an
eigenfunction φλ of the operator (6.8):
D[v]φλ = λφλ .
(6.10)
Let us choose the coordinate system such that en is a unit normal to Σ and
xn = 0 on Σ. It is clear that φλ must be continuous on Σ:
φ|xn=+0 = φ|xn=−0 .
(6.11)
Otherwise, the second normal derivative of φλ would create a δ 0 singularity on
Σ which is absent on the right hand side of (6.10). Let us integrate (6.10) over
23
The present author is not aware of any simple example where the ln t terms
actually appear.
59
a small cylinder C = C n−1 × [−, +]
Z
C
h
i
√ d x g −∇2n φλ − ∇2a φλ + (E + λ)φλ +
n
Z
C
√
dn−1x hvφλ = 0 .
(6.12)
We now take the limit as → 0. Since the expression in the square brackets
in (6.12) is bounded, the contribution that this term makes vanishes in the
limit. We obtain
Z
√
0=
dn−1x h (−∇n φλ |xn =+0 + ∇n φλ |xn =−0 + vφλ) .
(6.13)
C
Since C and λ are arbitrary, we conclude that a proper matching condition
for the normal derivatives is
−∇n φ|xn=+0 + ∇n φ|xn=−0 + vφ = 0 .
(6.14)
Physically this problem corresponds to two domains separated by a penetrable membrane Σ (a domain wall). In many cases penetrable membranes
are better models of physical boundaries then just boundary conditions which
are imposed on each side of Σ independently and thus exclude any interaction
between the domains [?, ?]. δ-potentials are being used in quantum mechanics [?] where one studies the Schr¨odinger equation (which is nothing else than
the imaginary time heat equation). The Casimir energy calculations have been
performed e.g. in [?]. In the formal limit v → ∞ one obtains Dirichlet boundary conditions on Σ, although the heat kernel coefficients are divergent in this
limit (see below).
Further generalisations are suggested by the brane world scenario [?, ?]
which assumes that our world is a four dimensional membrane in a five dimensional space 24 . According to the Israel junction condition [?] the metric
in such models cannot be smooth on Σ. Typical form of the metric near Σ is
n
(ds)2 = (dxn )2 + e−α|x | (dsn−1 )2 ,
(6.15)
where α is a constant and where (dsn−1 ) is a line element on the (n − 1)dimensional hypersurface Σ. Due to the presence of the absolute value of the
n-th coordinate in (6.15), the normal derivative of the metric jumps on Σ. One
can think of two smooth manifolds M + and M − glued together along their
common boundary Σ. Neither Riemann tensor, nor matrix potential E must
−
be continuous on Σ. Also, the extrinsic curvatures L+
ab and Lab of Σ considered
+
−
as a submanifold in M and in M respectively are, in general, different. All
geometric quantities referring to M − (respectively, M − ) and their limiting
values on Σ will be supplied by a superscript “+” (respectively, “−”).
For the case at hand there is still an asymptotic expansion (2.21) for the
heat kernel. The heat kernel coefficients can be decomposed as
−
−
Σ
ak (f, D[v]) = a+
k (f, D) + ak (f, D) + ak (f, D, v) ,
24
A similar scenario was proposed earlier in [?], see [?] for a review.
60
(6.16)
±
where a±
(cf.
k (f, D) are known volume contributions corresponding to M
(4.26) - (4.29)). The coefficients aΣ
are
given
by
integrals
over
Σ
of
some
local
k
invariants. Note, that
−
ω
˜ a = ∇+
a − ∇a
(6.17)
being a difference of two connection is a (pseudo-) vector with respect to all
space-time and gauge symmetries. Therefore, ω
˜ a can be used for constructing
the surface invariants.
To make the formulae more symmetric we introduce two inward pointing
unit normals ν + and ν − to Σ in M + and M − respectively. We do not suppose
that the smearing function f is smooth on Σ (therefore, there is no relation
between f;ν + and f;ν − ), but we assume continuity of f : f + = f − = f on Σ.
±
±
The surface invariants can be constructed from L±
˜ a and
ab , Rijkl , E , v, ω
their derivatives. This gives much more invariants than we usually have for
a boundary value problem. There are, however, some properties of aΣ
k which
Σ
simplify the calculations considerably. First of all, ak must be invariant with
respect to interchanging the roles of M + and M − . Also, aΣ
k must vanish when
the singularity disappears. The first property excludes, for example, the term
f (E + − E − ) which changes sign under M + ↔ M − . The second requirement
excludes the invariant f (E + + E − ) because it survives even if there is no
±
singularity on Σ. These simple arguments show that aΣ
3 does not contain E
even though such terms are allowed on dimensional grounds. It is also very
helpful that in some particular cases the problem in question can be reduces
to a sum of Dirichlet and Robin boundary value problems [?, ?].
The coefficients aΣ
k , k = 0, 1, 2, 3, read
aΣ
0 (f, D, v) = 0.
aΣ
1 (f, D, v) = 0.
Z
√
−
dn−1 x htrV {2f (L+
(6.18)
aa + Laa ) − 6f v}.
Σ
Z
√
3
(1−n)/2 1
n−1
+
− −
f (L+
(f,
D,
v)
=
(4π)
d
x
htr
aΣ
V
3
aa Lbb + Laa Lbb
384 Σ
2
−
+ +
− −
+ −
+2L+
aa Lbb ) + 3f (Lab Lab + Lab Lab + 2Lab Lab )
−
+
−
2
+9(L+
˜ aω
˜a
aa + Laa )(f;ν + + f;ν − ) + 48f v + 24f ω
−n/2
aΣ
2 (f, D, v) = (4π)
1
6
o
−
+
−
−24f (L+
aa + Laa )v − 24(f;ν + + f;ν − )v .
Σ
The coefficients aΣ
4 and a5 are too long to be presented here in full generality.
+
−
Therefore, we restrict ourselves to the case of smooth geometry (Rijkl
= Rijkl
,
−
L+
=
−L
),
smooth
connection
(˜
ω
=
0),
and
smooth
smearing
function
a
ab
ab
(f;ν + = −f;ν − = f;n ). In other words, the only singularity comes from the
surface potential v.
61
aΣ
4 (f, D, v)
= (4π)
−n/2
Z
d
n−1
Σ
√
1
1
x htrV − f v 3 − f Rv − f Ev
6
6
1
1
1
− f v:aa + f;n vLaa − f;nn v
6
6
6
Z
√
1
1 4
−(n−1)/2
n−1
f v + f Rv 2
aΣ
(f,
D)
=
(4π)
d
x
htr
V
5
64
48
Σ
1
1
1
1
Rnn v 2 + f v 2 E −
f v 2Laa Lbb +
f v 2 Lab Lab
+
192
8
256
128
1
5
5
1
2
2
f v:a v:a −
f;n v Laa + f;nn v
+ f v:aa v +
24
192
384
64
(6.19)
The heat kernel coefficients for Σ = S n−1 ⊂ Rn and v = const. were calculated in [?]. Generic Σ with arbitrary v was considered in [?]. Moss [?]
added a non-smooth connection (see also [?]). Calculations on a particular
brane-world background can be found in [?]. The heat kernel coefficients in
the general setting described here were calculated in [?]. This latter paper
also considered renormalization of the brane-world scenario and predicted a
non-standard Higgs potential on the brane. Related calculations in wormhole
models were done in [?, ?]. The ζ function for brane-world geometries with
matching conditions (6.11), (6.14) was considered recently in [?,?] (where one
can find some further references).
It is very well known [?] that the conditions (6.11), (6.14) (which we
call transmittal) are not the most general matching conditions which can be
defined on a surface. In general, boundary values of a function and of its’
normal derivatives are related by a 2 × 2 transfer matrix:

+
 ∇ν +
0=

+S
S −+ ,
++
,
S
+−
∇−
ν− + S

φ 

 .
−−
−
+
φ
(6.20)
Σ
Note, that the transfer conditions (6.20) do not assume identification of φ+ and
φ− on Σ. In other words, there is no ad hoc relation between the restrictions of
the vector bundles V + |Σ and V − |Σ . We can even consider the situation when
we have dimV + 6= dimV − , i.e. the fields on M − and M − can have different
structures with respect to space-time and internal symmetries. S ±± are some
matrix valued functions on Σ (one can even consider the case when they are
differential operators). It is clear from the notations on which spaces they act.
For example, S +− : V − |Σ → V + |Σ .
The matching conditions (6.20) arise in heat transfer problems [?], some
problems of quantum mechanics [?], and in conformal field theory [?]. In a
formal limiting case S ++ − S −+ = S −− − S +− → ∞ while v = 2(S ++ + S +− )
is kept finite one arrives at transmittal boundary conditions (6.11), (6.14).
The heat kernel coefficients are divergent in this limit.
In a particular case of spherical Σ the heat kernel expansion with transfer
boundary conditions was evaluated in [?]. General expressions for ak with
k = 0, 1, 2, 3, 4 were obtained in [?]. Somewhat surprisingly, the calculations
62
for (6.20) are easier than for the singular particular case (6.11), (6.14).
We should note that not all singular limiting cases of (6.20) are described
by the transmittal conditions (6.11), (6.14). The heat kernel expansion for a
generalisation of transmittal condition is known in the spherically symmetric
case only [?]. Very little is known about the heat kernel if the transfer matrix
contains differential operators on Σ (conformal walls of ref. [?] belong to this
class of problems).
The case when the singular potential is located on a surface of co-dimension
larger than one (i.e. when dimΣ < n − 1) is rather complicated. Even a careful translation of this problem to the operator language was done only in
1960’s [?]. Direct calculations show [?, ?] that the heat kernel asymptotics
may contain very unusual (ln t)−1 terms. More references can be found in [?].
6.4 Non-smooth boundaries
Rectangular region in a plane is probably the simplest manifold with
boundaries as far as eigenvalues of the Laplacian are concerned. However, the
formulae (5.29) - (5.33) are not valid for this case because of the presence
of corners. The heat kernel expansion on manifolds with piecewise smooth
boundaries was considered by Kac in his famous paper [?]. He demonstrated
that for a region in R2 each corner with the inside facing angle α contributes
a2 (corner) =
π 2 − α2
24πα
(6.21)
to the coefficient a2 while a0 and a1 are still defined by their “smooth” expressions.
The formula (6.21) looks similar to the contribution of a conical singularity (6.7). The reason for this similarity is that the cone can be obtained
from the wedge by gluing the sides together and imposing the periodicity
conditions.
The study of boundary discontinuities was continued by Apps and Dowker
who calculated the coefficients a3 [?] and a4 [?] for piecewise smooth boundaries. We also refer to [?] where functional determinants on simplicial complexes were analysed, and to [?] where divergences in the Casimir energy found
in [?] were attributed to non-smoothness of the boundaries. A recent study [?]
should also be mentioned.
When the angle α goes to 0 we obtain a cusp. In this limit (6.21) is divergent. Presence of the cusp is an essential singularity which modifies powers
of the proper time t which appear in the asymptotic expansion of the heat
kernel [?].
6.5 Dielectric bodies
Calculations of the Casimir energy of a dielectric body have attracted
much attention and created many controversial results. A (rather large) liter63
ature on this subject is reviewed in [?,?]. Quantum field theory formulation of
this problem is known for a long time already (cf. [?, ?, ?]). However, the heat
kernel analysis of divergences in the Casimir energy in dielectric is a relatively
new subject.
Wave propagation with variable speed of light c(x) is described by the
operator D = −c(x)2 ∇2 + . . . , where we neglected the lower order terms. In a
dielectric medium c is expressed in terms of the dielectric permittivity ε and of
the magnetic permeability µ: c(x)2 = 1/(ε(x)µ(x)). For a smooth distribution
of ε and µ the operator D is a particular case of curved space Laplacian with an
effective metric defined by c(x). The heat kernel coefficients can be calculated
in the standard way [?].
Consider now a dielectric body bounded by Σ. Typically, c(x) (and the
effective metric!) jumps on Σ. This singularity is much stronger than the one
considered in sec. 6.3. Thus the geometric interpretation of this problem is very
difficult. Very little is known about the heat kernel expansion in a dielectric
body of an arbitrary shape. There are calculations for a dielectric ball [?] and
for a dielectric cylinder [?]. These calculations exhibit a puzzling property of
the heat kernel expansion in dielectrics in dilute approximation (ε ∼ 1): for a
dielectric body the Casimir energy in the ultra violet limit behaves better than
for the “smooth” case. The heat kernel expansion for a frequency dependent
ε was considered in refs. [?, ?].
64
7
Anomalies
The most immediate application of the technique developed in the previous sections is calculation of quantum anomalies which are defined as (nonzero) variations of quantum effective action with respect to symmetry transformations of the classical theory. In this section we consider two most important examples of quantum anomalies. These are conformal (scale) and chiral
anomalies. We also discuss briefly the Index Theorem.
There exists also a broader view on the anomalies which includes any qualitatively new phenomena of a quantum theory which are absent in its’ classical
counterpart. An example is the so-called dimensional reduction anomaly [?].
For two operators D1 and D2 the quantity det(D1 ) det(D2 )/ det(D1 D2 ) is
called the multiplicative anomaly [?] since det(D1 D2 ) 6= det(D1 ) det(D2 ) is
an “anomalous” property of infinite dimensional operators. It is interesting to
note that the heat kernel is also useful for treatment of these non-standard
anomalies.
To simplify the discussion in this section we work on manifolds without
boundaries.
7.1 Conformal anomaly
Conformal invariance is one of the symmetries which are usually broken by
quantisation. This phenomenon (called conformal or trace anomaly) is known
since mid 1970’s [?, ?, ?, ?, ?] (see [?, ?] for more extensive literature).
The vacuum polarisation induced by quantum effects is described by the
energy-momentum tensor
2 δW
Tµν = √
,
g δg µν
(7.1)
where W is the quantum effective action calculated on the background with
the metric tensor gµν . Consider the conformal transformation
gµν → e2ρ(x) gµν
(7.2)
for an infinitesimal value of the parameter: gµν → (1 + 2δρ)gµν , g µν → (1 −
2δρ)g µν . According to (7.1) variation of the effective action reads:
δW =
1
2
Z
M
√
dn x gTµν δg µν = −
Z
M
√
dn x gTµµ δρ .
(7.3)
It is clear from the equation above that the trace of the energy-momentum tensor measures conformal non-invariance of the theory. If the classical action L
is conformally invariant, classical energy-momentum tensor is traceless. However, even in this case conformal invariance is typically broken by quantum
effects. For this reason, quantum Tµµ is called trace or conformal anomaly.
65
If the classical action L is conformally invariant, the fluctuation operator
D is conformally covariant. This means that D transforms homogeneously
D → e−2ρ D under (7.2) 25 .
We restrict ourselves to the one-loop level and employ the zeta function
regularization in which the effective action is expressed through the zeta function of the operator D (see (2.32)). Hence, we have to study the conformal
properties of ζ(s, D). One can prove that the variation of the zeta function
with respect to variation of the operator D reads 26
δζ(s, D) = −sTr((δD)D −s−1) .
(7.4)
Next we use that under infinitesimal conformal transformations the operator D transforms as
δD = −2(δρ)D .
(7.5)
δζ(s, D) = 2sζ(s, δρ, D) .
(7.6)
This equation yields
For the operators which we consider in this section the zeta function is regular
at s = 0. Consequently, the variation of the effective action (2.32) reads:
δW = −ζ(0, δρ, D) = −an (δρ, D) ,
(7.7)
where we have used the relation (2.27) between the zeta function at zero
argument and the heat kernel coefficient an .
We compare (7.3) with (7.7) to see that
Tµµ (x) = an (x, D) .
(7.8)
Note, that ζ(0, D) (as well as the pole term in (2.31)) is conformally invariant.
Consequently, the conformal anomaly is not divergent, and (7.8) does not
contain the normalisation scale µ.
The anomaly (7.8) is defined by the same coefficient an as the divergent
part of the effective action in the zeta function regularization. Important difference is that the divergence is given by the integrated coefficient an (D), while
the conformal anomaly is defined by the localised coefficient an (x, D). The use
Strictly speaking this property holds up to a similarity transformation D →
eαρ De−αρ which does not change the functional determinant.
26 To derive this formula rigorously we have to use the method of [?, ?]. We first
assume that s is sufficiently large to keep us away from the singularities, then use
the Mellin transformation (2.24), perform the variation, then perform the transformation back, and then continue the result to s = 0. This is a perfectly standard
procedure which allows us to work with variations of positive integer powers of D;
see [?] for further detail.
25
66
of the integrated coefficient in (7.8) is insufficient to recover total derivatives
in the anomaly.
As an example, consider quantum scalar field φ in two dimensions 27
coupled to the background dilaton Φ:
L=
Z
√
d2 x ge−2Φ (∂µ φ)(∂ν φ)g µν .
(7.9)
On dimensional and symmetry grounds the inner product may also contain
an arbitrary function Ψ of the dilaton:
hφ1 , φ2 i =
Z
√
d2 x ge−2Ψ φ1 (x)φ2 (x) .
(7.10)
Such couplings and inner products appear, for example, after the spherical
reduction of higher dimensional theories to two dimension (see the review
paper [?] for more details and further references).
The rescaled field φ˜ = e−Ψ φ possesses the standard dilaton-independent
inner product (3.3). In terms of φ˜ the action (7.9) reads
L=
Z
√ ˜ ˜
d2 x g φD
φ,
(7.11)
D = −e2(Ψ−Φ) g µν (∇µ ∇ν + 2(Ψ,µ − Φ,µ )∂ν
+Ψ,µν + Ψ,µ Ψ,ν − 2Ψ,µ Φ,ν ),
(7.12)
where comma denotes covariant differentiation with ∇: Ψ,ν = ∇ν Ψ. We can
bring D to the standard form (2.2)
ˆ µ∇
ˆ ν + E)
D = −(ˆ
g µν ∇
(7.13)
by introducing the effective metric gˆµν = e2(Ψ−Φ) g µν and the covariant derivative
ˆ µ = ∂µ + Γ
ˆµ + ω
∇
ˆµ ,
ω
ˆ µ = Ψ,µ − Φ,µ ,
(7.14)
ˆ is the Christoffel connection for the metric gˆ. Here the potential E
where Γ
reads
E = gˆµν (−Φ,µ Φ,ν + Φ,µν ) .
(7.15)
Now we combine (7.8) with (4.27) and the definitions given above to obtain:
Tµµ =
1
(R − 6(∇Φ)2 + 4∇2 Φ + 2∇2 Ψ) .
24π
27
(7.16)
Much work on conformal anomalies on two-dimensional manifolds was done in
the context of string theory. Here we like to mention the papers [?, ?, ?, ?, ?].
67
In the case Ψ = Φ the expression (7.16) was first obtained in [?], and in the
general case in [?] 28 .
In four dimensions the conformal anomaly for different spins can be read
off from (4.35) with the numerical coefficients given in Table 1.
7.2 Chiral anomaly
Chiral anomaly was discovered in 1969 by Adler, Bell and Jackiw [?, ?]
and since that time plays a crucial role in understanding of the low energy
hadron physics. A detailed introduction to the field and extensive literature
can be found in [?, ?].
The spinor action (3.22) with the Dirac operator given in (3.26) is invariant under the gauge transformations
¯ ,
δλ ψ = −λψ ,
δλ ψ¯ = ψλ
δλ A5µ = [A5µ , λ] ,
δλ Aµ = ∂µ λ + [Aµ , λ]
(7.17)
and local (Euclidean) chiral transformations
¯ 5,
δ˜ϕ ψ = −iϕγ 5 ψ ,
δ˜ϕ ψ¯ = −iψϕγ
δ˜ϕ A5 = ∂µ ϕ + [Aµ , ϕ] ,
µ
δ˜ϕ Aµ = −[A5µ , ϕ]
(7.18)
with anti-hermitian local matrix parameters λ and ϕ. The Dirac operator
transforms as:
δλD
/ = [D,
/ λ] ,
δ˜ϕD
/ = i{ϕγ 5 , D}
/ .
(7.19)
The Lie algebra structure of the transformations (7.17) and (7.18) is encoded
in the following relations:
δλ1 δλ2 − δλ2 δλ1 = δ[λ1 ,λ2 ]
δλ δ˜ϕ − δ˜ϕ δλ = δ˜[λ,ϕ]
(7.20)
δ˜ϕ1 δ˜ϕ2 − δ˜ϕ2 δ˜ϕ1 = −δ[ϕ1 ,ϕ2 ] ,
(7.22)
(7.21)
where all transformation parameters are taken at the same space-time point. If
the matrices λ(x) and ϕ(x) belong to a finite-dimensional compact Lie algebra
of some Lie group G, the transformations (7.17) and (7.18) generate locally
the group G ×G with gauge transformations belonging the diagonal sub-group.
28
Different values for the numerical coefficients in the conformal anomaly (7.16)
were reported in [?, ?, ?, ?]. The reason for these discrepancies was clearly stated by
Dowker [?] who confirmed the result (7.16). More extensive literature can be found
in [?].
68
The gauge invariance can (and should) be retained in quantum theory
while the chiral invariance is typically broken by the quantisation. Let us
study these effects in the zeta function regularization. Define determinant
of the Dirac operator as a square root of the determinant of the associated
Laplace operator:
log det D
/=
1
1
log det D
/ 2 = log det D
2
2
(7.23)
The effective action W corresponding to the path integral (3.41) reads
1
1
1
W = − log Z = − log det D = ζ(0, D)0 + ln(µ2 )ζ(0, D) ,
2
2
2
(7.24)
where µ is a normalisation scale.
By virtue of (7.4)
δλ ζ(s, D) = − sTr([D, λ]D −s−1) = Tr([D −s , λ]) = 0 .
(7.25)
This proves gauge invariance of the effective action.
For the chiral transformation we have:
A(ϕ) := δ˜ϕ W = −2iTr(γ 5 ϕD −s )|s=0 .
(7.26)
A(ϕ) measures non-invariance of the effective action with respect to the chiral
transformations. It is called the chiral anomaly. One can express A through
the heat kernel coefficients:
A(ϕ) = −2ian (γ 5 ϕ, D) ,
(7.27)
where an (γ 5 ϕ, D) is defined as in (2.21) but with a matrix-valued smearing
function instead of the scalar one.
The chiral anomaly A(ϕ) should satisfy certain consistency conditions
following from the Lie algebra identities (7.20)-(7.22) and gauge invariance of
the effective action:
δλ A(ϕ) = A([λ, ϕ]) ,
δ˜ϕ A(ϕ2 ) − δ˜ϕ A(ϕ1 ) = 0 .
1
(7.28)
(7.29)
2
The relations (7.28) and (7.28) are called the Wess-Zumino consistency conditions [?]. One can check by a direct calculation that the anomaly defined in
this section indeed satisfies these conditions. For this reason A is called the
consistent anomaly (as opposed to the covariant anomaly which we do not
consider here).
Let us turn now to calculation of the chiral anomaly A. Since the smearing
function γ 5 ϕ in the heat kernel coefficient (7.27) is matrix-valued we need
more information than we possess at the moment. To recover the missing
terms one can adopt the strategy of the paper [?]. Consider Tr(Q exp(−tD))
69
with arbitrary matrix-valued function Q and arbitrary Laplace type operator
D. There is an asymptotic expansion
TrL2 (Q exp(−tD)) ∼
=
X
t(k−n)/2 ak (Q, D)
(7.30)
k≥0
where the coefficients ak are locally computable. This means that they can
be represented as integrals of local invariants constructed from Q and local
invariants of the operator D. These local invariants enter with some coefficients
(universal constants) which are to be defined. Since Q does not commute with
E and Ω there are more invariants than before. In the particular case Q = IV f
(where IV is the unit matrix and f is a function) we should recover the old
result (4.26) - (4.29). For k = 0, 2, 4 this last requirement is strong enough
to recover ak (Q, D) completely. Therefore, calculation of the chiral anomaly
in the dimensions n = 2 and n = 4 is quite simple: take the expression
(4.27) or (4.28) and substitute 2iγ 5 ϕ for f and (3.27), (3.28) for ωµ , E and
Ωµν . Computation of some γ-matrix traces is still required, but nevertheless
calculations are considerably easier than presented in many papers. In flat
space and with A5 = 0 the result is particularly simple:
n=2:
n=4:
1 Z 2
A(ϕ) = −
d xtr(ϕµν Fµν )
2π Z
i
A(ϕ) =
d4 xtr(ϕµνρσ Fµν Fρσ )
16π 2
(7.31)
(7.32)
where tr denotes trace over internal (flavour or colour) indices. We can easily
generalise this result for arbitrary even dimension n. First we observe that the
only way to construct a pseudoscalar density of appropriate dimension (for
A5 = 0) is to contract n/2 tensors Fµν with the Levi-Civita tensor µ1 µ2 ...µn .
Such tensor structure can be produced only by a trace of γ 5 and the maximal
number (n) of the gamma matrices. The only invariant having the required
form is
Z
dn xtr(QE n/2 ) .
(7.33)
This term does not contain derivatives. Under the trace Q and E commute.
Therefore, we can calculate the coefficient in front of (7.33) by considering the
particular case E ∼ IV × const.. For this simple case the dependence of the
heat kernel on E is given simply by etE . By picking up an appropriate term in
the expansion of the exponential we find that the coefficient in front of (7.33)
in an is (4π)−n/2 ((n/2)!)−1 . Next we substitute E = 12 γ µ γ ν Fµν and take trace
over the spinorial indices to obtain:
A(ϕ) = −
2i(−i)n(n−1)/2
(4π)n/2 (n/2)!
Z
dn xtr ϕµ1 µ2 ...µn Fµ1 µ2 . . . Fµn−1 µn
(7.34)
For global chiral transformations (ϕ = const.) this formula was obtained in [?].
70
As soon as we know the anomaly (7.34) for A5µ = 0, many more terms
(containing A5µ ) can be restored by using the Wess-Zumino condition (7.29).
Some comments on the Fujikawa approach [?, ?] to the chiral anomaly
are in order. Consider the path integral (3.41). Since the action is invariant
under the chiral rotations, the only source of the chiral anomaly could be noninvariance of the path integral measure. The Jacobian factor appearing due
to the change of the spinorial variables (7.18) can be formally represented as
J = det(1 − 2iϕγ 5 ) .
(7.35)
Then, to the first order of ϕ,
A(ϕ) = δ˜ϕ W = − log J −1 ' −2iTr(ϕγ 5 )
(7.36)
where J −1 has appeared due to the negative homogeneity of the fermionic
measure. The operator ϕγ 5 is not trace class on the space of square integrable
spinors (though formally its’ trace is zero at every point). The right hand side
of (7.36) is therefore ill defined. Fujikawa suggested to replace (7.36) by a
regularized expression
A(ϕ) = −2i lim Tr ϕγ 5 e−D/M
M →∞
2
.
(7.37)
The chiral anomaly (7.37) has now the heat-kernel form with the identification
t = M −2 . If we suppose that all positive powers of the regularization parameter
M in the small t (large M) asymptotic expansion are somehow absorbed in
the renormalization, we arrive at
A(ϕ) = −2ian (γ 5 ϕ, D)
(7.38)
that is just the expression (7.27) obtained above in the zeta function regularization. This method, based on calculations of the regularized Jacobians, can
be applied to conformal anomaly as well [?] 29 .
The most essential ingredient of the anomaly calculation presented in this
section is the homogeneous transformation low (7.19) for the Dirac operator.
This homogeneity allowed us to restore the power −s in the transformation law
(7.4) for the zeta function and to obtain a simple expression for the anomaly
(7.26). It is clear therefore that as long as the operators transform homogeneously we shall obtain relatively simple local expressions for corresponding
anomalies (understood as variations of the effective action with respect to
infinitesimal transformation of the background fields). This suggests to consider extensions of the chiral group. For example, one of such extensions [?]
identifies the group parameters with the diquark fields.
29
Applications of this method to topological anomalies can be found in [?].
71
7.3 Remarks on the Index Theorem
This report is mainly devoted to local aspects of the heat kernel expansion.
There is however one “global” application of the heat kernel which cannot be
ignored. This is the Index Theorem. In this section we briefly sketch formal
mathematical aspects of the index construction and its’ relation to the heat
kernel. For more details we refer to [?, ?, ?, ?, ?]. Physical applications of the
index theorem to gravity, gauge theories and strings are so numerous that we
cannot even mention them all.
The index theorem was first formulated by Atiyah and Singer [?], and
the heat kernel approach appeared later in [?,?]. Roughly speaking, their construction is as follows. Consider two vector bundles V1 and V2 over a manifold
M. Let the operator P map V1 to V2 , and let V1,2 have non-degenerate inner products ( , )1,2 so that one can define an adjoint P † by the equation
(φ2 , P φ1)2 = (P † φ2 , φ1 )1 . As an example, one can keep in mind V1 (respectively, V2 ) describing positive (respectively, negative) chirality spinors. In this
example P is a part of the Dirac operator (see eq. (7.44) below). Let us define
two operators D1 = P † P and D2 = P P † acting on (smooth sections of) V1
and V2 , and let us suppose that D1 and D2 are elliptic. In this case we deal
with an elliptic complex.
Since elliptic operators may have only a finite number of zero modes
(say, N1 and N2 for D1 and D2 respectively), we may define the index by the
following equations:
index(P ) = N1 − N2
= dim ker D1 − dim ker D2
= dim ker P − dim ker P † .
(7.39)
We also have the intertwining relations
D1 P † = P † D2 ,
P D1 = D2 P ,
(7.40)
which tell us that non-zero eigenvalues of D1 and D2 coincide. Consider now
the heat kernels,
K(t, D1 ) − K(t, D2 ) =
X
(λ1 )
e−tλ1 −
X
e−tλ2 = N1 − N2 = index(P )
(7.41)
(λ2 )
(where λ1,2 are eigenvalues of D1,2 ).
To be more specific, let us suppose that D1 and D2 are of Laplace type and
that boundary conditions and singularities (if any) are such that the powerlaw expansion (2.21) exists. Then, expanding both sides of (7.41) in a power
series of t one obtains:
ak (D1 ) − ak (D2 ) = 0
for k 6= n,
an (D1 ) − an (D2 ) = index(P ) .
72
(7.42)
The last equation provides a simple way to calculate the index from known
heat kernel expansion. It also allows to understand a very important property
of the index, namely, that it is a homotopy invariant under quite general assumptions. Indeed, suppose that P depends on a parameter α in such a way
that all geometric quantities (including the metric, connection, the matrix potential, the boundary conditions etc.) corresponding to D1 and D2 are smooth
functions of α. Clearly, no “essential” deformations, like changing the order of
the operator, or adding higher derivative terms to the boundary conditions, or
including new types of the singularities, are allowed. Under these smoothness
assumptions an (D1 ) and an (D2 ) are smooth functions of α as well. Therefore,
the index is also a smooth function of α. Since the index is an integer, it
can only be a constant. Hence, index(P ) is invariant under the deformations
described above.
The relations (7.42) are also useful for calculations of the heat kernel
coefficients as they give restrictions on the universal constants appearing in
front of independent invariants (see, e.g., [?]).
Let us consider an example of the spin complex. Let M be an even dimensional manifold admitting a spin structure (so that one can define spinor
fields on M). In a suitable basis the chirality matrix may be presented in the
diagonal form (at this point it may be useful to consult sec. 3.3):


I 0 
 .
γ5 = 
(7.43)
0 −I
Let V1 correspond the +1 eigenvalue of γ5 (positive chirality), and V2 – to
−1. The Dirac operator (3.26) anti-commutes with γ5 and, therefore, may be
represented in the form:


†
0 P 
D
/ =
 .
P 0
(7.44)
We also have:


†
P P
D
/2 = 
0 
PP†
0
 .
(7.45)
We conclude that index(P ), which now measures difference between dimensions of the null subspaces of D
/ with positive and negative chirality, is equal
to global chiral anomaly:
−tP † P
index(P ) = Tr e
−tP P †
− Tr e
2
= Tr γ5 e / = an (γ5 , D
/ 2 ) (7.46)
−tD
Therefore, the index can be calculated by the methods of the previous section.
In particular, the index of the Dirac operator in two and four dimensions on
73
flat manifolds without background axial vector fields (A5µ = 0) can be read off
from (7.31) and (7.32) with ϕ := i/2. We also see that chiral anomaly provides
very important information on topology of the background.
One can associate an index with any elliptic complex which must not
necessarily contain only two vector bundles. In general, one has a sequence
of bundles Vp and a family of operators Pp which map (smooth sections of)
Vp to (smooth sections of) Vp+1 . One requires that Pp Pp+1 = 0 and that the
†
are elliptic. Important
corresponding “Laplacians” Dp = Pp† Pp + Pp−1 Pp−1
examples include the de Rham complex (for which Vp consists of p-forms on
M and P is the exterior derivative), the Dolbeaux complex which deals with
the forms on a complex manifold, and the signature complex which treats
selfdual and anti-selfdual forms. More details can be found in [?]. Another very
important construction of this type is the Witten index [?] which constraints
the supersymmetry breaking.
74
8
Resummation of the heat kernel expansion
The heat kernel coefficients define the one-loop counterterms in the background field formalism. In many cases the heat kernel can also give a useful
information on the finite part of the effective action. Just one of the examples
is the large mass expansion (1.22) which is valid when all background fields
and their derivatives are small compared to the mass of the quantum field. In
order to get the effective action in other limiting cases one has to re-arrange
the heat kernel expansion 30 .
8.1 Modified large mass expansion
In many physical applications there is a quantity M2 which is large compared to the rest of the background fields and their derivatives. Therefore, it
is a well motivated problem to construct an expansion of the effective action
in a power series of M−1 . To do so, one has to re-express the heat kernel as
2
K(t; D) = e−tM
X
t(k−n)/2 a
˜k .
(8.1)
k
To obtain the effective action one has to integrate the heat kernel over t (cf
eq. (1.18)). To simplify the argumentation we assume the cut-off regularization (1.20), although in other regularization schemes the result will be essentially the same. The divergent and finite parts of the effective action are given
by (1.21) and (1.22) respectively up to the obvious replacements: m → M,
(4π)−n/2 bk → a
˜k . After a suitable renormalization, this procedure indeed gives
a large M expansion of the effective action.
Therefore, the problem is reduced to calculation of a
˜k . If M commutes
with D the coefficient a˜k are the heat kernel coefficients for the operator
D − M2 . This case returns us to the standard large mass expansion. If M2
does not commute with D then
2
2
e−tD 6= e−t(D−M ) e−tM .
(8.2)
To achieve an equality the right hand side of (8.2) must be corrected by
commutator terms. In this case calculation of a
˜k requires some amounts of
extra work.
Heavy particles of non-equal masses described by the mass matrix M2
is probably the most immediate example of a physical system to which the
modified large mass expansion should be applied. Corresponding technical
tools were developed recently [?, ?, ?].
The next example is a scalar field in curved space (cf. sec 3.1). Parker
and Toms suggested [?, ?] to use the modified large mass expansion with
30
A diagrammatic technique which can be used in resummations of the heat kernel
expansion is described in [?].
75
M2 = m2 + ξR which partially sums up contributions of the scalar curvature
R to the effective action. This formalism was developed further in [?, ?].
In some cases the term-by-term integration of the heat kernel expansion
gives good estimates of the vacuum energy even if no large parameter is explicitly present in the model (see, e.g. calculations of quantum corrections to
the mass of two dimensional solitons [?, ?]).
8.2 Covariant perturbation theory
Suppose that the matrix potential E, the Riemann curvature, and the
field strength Ω are small but rapidly varying. Can we get any information
on the heat kernel coefficients containing a fixed power of the quantities listed
above and arbitrary number of derivatives? For the linear and quadratic orders
on a manifold without boundaries the answer is positive and the results can
be obtained either by the functorial methods of sec. 4.1 [?, ?] or by solving
the DeWitt equation [?, ?, ?] (cf. sec. 4.3). Retaining only the leading terms
we have for k ≥ 1:
h
a2k (x, D) = (4π)−n/2 α1 (k)∆k−1 R + α2 (k)∆k−1 E
(8.3)
+higher order terms] ,
where [?, ?]
α1 (k) =
k!k
,
(2k + 1)!
α2 (k) =
2k!
.
(2k)!
(8.4)
One can sum up the expansion (8.3) and corresponding higher order terms
to obtain an information on the behaviour of the full heat kernel in the limit
described above. There is, however, a more straightforward method to control
non-localities which is called the covariant perturbation theory [?, ?, ?]. To
make the idea of this method most transparent we consider a simple example
of flat M and zero connection ω. The exponent exp(−tD) = exp(t(∆ + E))
with ∆ = ∂µ2 can be expanded in a power series in E:
−tD
e
t∆
=e
Zt
(t−s)∆
+
dse
0
s∆
Ee
Zs2
Zt
+
ds2
0
ds1 e(t−s2 )∆ Ee(s2 −s1 )∆ Ees1 ∆ + . . .
0
(8.5)
The heat trace can be also expanded,
K(t, D) = Tr e−tD =
∞
X
j=0
where Kj contains the jth power of E.
76
Kj (t),
(8.6)
Covariant perturbation theory approach prescribes to take the 0th order
heat kernel in the free space form (cf. eq. (1.12) for m = 0):
(x − y)2
K0 (x, y; t) = (4πt)−n/2 exp −
4t
!
,
(8.7)
which is an exact kernel on M = Rn only 31 . This formula neglects all global
contributions and, therefore, is valid only for sufficiently close x and y and for
small t. We have:
Z
K0 (t) =
√
d x gtrV K0 (x, x; t) = (4πt)−n/2
n
M
Z
M
√
dn x gtrV (IV ) ,
(8.8)
where trV IV simply counts discrete (spin and internal) indices of D. This
formula reproduces the a0 contribution to the heat kernel.
In the next order, we have:
Z
K1 (t) = Tr
t
(t−s)∆
dse
Z0
= t trV
M
s∆
Ee
Z
= Tr
√
dn x gK0 (x, x; t)E(x)
t
=
(4πt)n/2
Z
M
t
t∆
dse E
0
√
dn x gtrV E(x) ,
(8.9)
where we used the cyclic property of the trace and the expression (8.7) for
the 0th order heat kernel. This expression is consistent with the k = 1 term
of (8.3). The terms with k > 1 are total derivatives, and, therefore, they do
not contribute to the integrated heat kernel.
The quadratic order of the heat kernel reads

K2 (t) = Tr 
Zs2
Zt
ds2
0
ds1 e(t−s2 )∆ Ee(s2 −s1 )∆ Ees1 ∆ 
0
Z
= trV

M
dy
Zs2
Zt
Z
dz
M
ds2
0
ds1 K0 (z, y; t − s2 + s1 )E(y)
0
×K0 (y, z; s2 − s1 )E(z) .
(8.10)
Next we introduce the rescaled variables ξ = s/t and get rid of redundant
integrations to obtain:
t2
K2 (t) = trV
2
Z
Z
M
dy
M
Z1
dz
dξK0 (z, y; t(1 − ξ))E(z)
0
×K0 (y, z; tξ)E(z) .
31
(8.11)
The original paper [?] contained a curved space generalisation of (8.7). This,
however, does not improve the global issues discussed below.
77
Now we use the identity
K0 (z, y; t(1 − ξ))K0 (y, z; tξ) = (4πt)−n/2 K0 (z, y; tξ(1 − ξ))
(8.12)
and relate the heat kernel on the right hand side of (8.12) to a matrix element
of exp(tξ(1 − ξ)∆) to obtain the final result:
K2 (t) =
t2
(4πt)n/2
Z
M
√
dn x gEf (−t∆)E ,
(8.13)
where the non-local form-factor f reads:
q
1Z
1
√
f (q) =
dξe−qξ(1−ξ) = e−q/4 π/qErfi [ q/2] .
2
2
1
(8.14)
0
As we have already discussed above, applicability of this formula is limited by
our choice of K0 (x, y; t). Namely, the potential E(x) should have a compact
support and t should be reasonably small. Note, that the small t approximation
in the expansion (8.5) is self-consistent: if t is small, the integration variables
si are even smaller.
The main difficulty in constructing an expansion in powers the Riemann
curvature and of the field strength is to organise the procedure in a covariant
way. The details of the construction and higher order form-factors can be found
in [?,?,?,?,?]. From further developments of this method we mention the work
of Gusev and Zelnikov [?] who demonstrated that in two dimension one can
achieve considerable simplifications in the perturbation expansion by using the
dilaton parametrisation of the potential (cf. eq. (7.15)). Recently Barvinsky
and Mukhanov [?] suggested a new method for calculation of the non-local
part of the effective action based on the resummation of the perturbation
series for the heat kernel. This method was extended in [?] to include late
time asymptotics of the heat kernel in curved space.
Let us stress that the expansion (8.5) can be used also for singular potentials. For example, it is very effective for the calculation of the heat kernel
for δ-potentials concentrated on a co-dimension one subsurface [?, ?, ?] (cf.
sec. 6.3). If the δ-potential has its support on a submanifold of co-dimension
greater than one, the expansion diverges [?]. With a suitable choice of the
zeroth order heat kernel K0 (x, y; t) and of an operator to replace E in (8.5)
one can treat manifolds with boundaries [?] where the perturbative expansion
takes the form of the multiple reflection expansion [?, ?, ?].
8.3 “Low energy” expansion
Let us now turn to the opposite case when the derivatives are less important when the potential and the curvatures. In this case one has to collect the
terms which are of a fixed order in the derivatives, but contain all powers of
E, R and Ω. Since the derivatives are sometimes identified with the energy,
78
this approximation is being called the low energy expansion. This scheme goes
back to Schwinger’s calculations [?] in constant electromagnetic fields 32 .
Let us consider a simple example [?]. Let M = R2 . Consider a scalar
particle in constant electromagnetic field with the field strength F12 = −F21 =
B. As a potential we choose A1 = 0, A2 = Bx1 . Then the operator acting on
quantum fluctuations is
D = −∂12 − (∂2 − iBx1 )2 + m2 .
(8.15)
This operator commutes with ∂2 . Therefore, we can look for the eigenfunctions
˜ 1 ).
of D in the form φk (x) = exp(ikx2 )φ(x
Dφk (x) = −∂12 + B 2 (x1 − k/B)2 + m2 φk (x) .
(8.16)
In the x1 direction we have the one-dimensional harmonic oscillator potential
(cf. (6.3)). Therefore, the eigenvalues are λk,p = (2p + 1)|B| + m2 . These
eigenvalues do not depend on k. For this reason the heat kernel
K(t, D) =
X
exp(−tλk,p )
(8.17)
k,p
is ill defined. To overcome this difficulty it was suggested [?] to put the system
in a box (without specifying any boundary conditions, however) and replace
the sum over k by the degeneracy factor (|B|Vol)/(2π), where Vol is volume
of the box. The degeneracy factor is chosen in such a way that the resulting
heat kernel
K(t, D) =
BVol −m2 t
e
[sinh(Bt)]−1
4π
(8.18)
behaves as Vol/(4πt) in the limit B → 0.
Calculations of the effective action in covariantly constant background
gauge fields by the spectral theory methods have been performed by many
authors [?, ?, ?, ?, ?]. Such calculations were motivated, at least partially, by
various models of quark confinement (see [?, ?] and references therein).
There exists an algebraic method [?, ?] which allows to evaluate the low
energy heat kernel by using exclusively the commutator algebra. Let us briefly
formulate the results of [?, ?]. Consider a flat manifold (Rµνρσ = 0) with the
background fields satisfying the “low-energy conditions”:
∇µ Ωµν = 0,
∇ µ ∇ν ∇ρ E = 0
(8.19)
with usual definitions of E and Ω (see (2.1) - (2.10)). Moreover, let us suppose
that the background is “approximately abelian”, i.e. that Ωµν , E and all their
covariant derivatives are mutually commuting matrices.
32
We like to mention also the paper [?] which treated the effective action in external
electromagnetic field from a different point view.
79
With all these assumptions a closed expression for the heat kernel may
be obtained:
K(t; x, x; D) = (4πt)
−n/2
t3
exp tE + Φ(t) + E;µ Ψµν E;ν
4
!
,
(8.20)
where Φ and Ψ are complicated functions of t, E and Ω [?]. If ∇µ ∇ν E = 0,
!
1
sinh tΩ
Φ(t) = − ln det
,
2
tΩ
Ψ(t) = (tΩ)−2 (tΩ coth(tΩ) − 1) ,
(8.21)
where Ω has to be understood as a space-time matrix Ωµν , so that multiplication in (8.21) is the matrix multiplication, and det is the determinant of an
n × n matrix. These formulae generalise the equation (8.18) and justify the
choice of the degeneracy factor made to derive it.
If Ω = 0,
√ !
1
sinh(2t P
√
Φ(t) = − ln det
,
2
2t P
√ −3 √
√ Ψ(t) = − t P
tanh(t P ) − t P ) ,
(8.22)
where Pµν = −(1/2)∇µ ∇ν E. In the particular case of one-dimensional harmonic oscillator these formulae reproduce (6.5).
In curved space the best one can do in the framework of the low-energy
expansion is to consider locally symmetric manifolds, i.e. vanishing Ω, covariantly constant Riemann curvature and constant E. In this case formulae
similar to the ones presented above are available [?, ?]. Covariantly constant
curvature means that locally the manifold M is a symmetric space. Various
approaches to the heat kernel on such manifolds are described in detail in
monographs and survey articles [?, ?, ?]. In particular, very detailed information may be obtained for group manifolds (see, for example, [?]) and for
hyperbolic spaces [?].
8.4 Heat kernel on homogeneous spaces
In this section we briefly explain how one can find the spectrum of some
“natural” differential operators on homogeneous spaces by purely algebraic
methods. We start with some basic facts from differential geometry and harmonic analysis [?, ?]. Consider a homogeneous space G/H of two compact
finite-dimensional Lie groups G and H. The Lie algebra G of G can be decomposed as
G = H ⊕ M,
80
(8.23)
where H is the Lie algebra of H and M is the complement of H in G with
respect to some bi-invariant metric. We have:
[H, M] ⊂ M ,
(8.24)
where [ , ] is the Lie bracket on G. If, moreover, [M, M] ⊂ H, then G/H is a
symmetric space. We do not impose this restriction.
M can be identified with the tangent space to M = G/H at the origin
(i.e. at the point which represents the unit element of G). Eq. (8.24) tells us
that H acts on M by some (orthogonal) representation. This action defines
the embedding
H ⊂ so(n) .
(8.25)
All physical fields are classified according to certain representations of the Lie
algebra so(n). Restrictions of these representations to H define transformation
properties of the field with respect to H. From now on we work with each
irreducible representation of H separately.
The field ΦA belonging to an irreducible representation T (H) can be
expanded as (see, e.g., [?]):
− 12
ΦA (x) = (Vol)
X
s
j,ξ,q
dj (j) −1 (j)
T (g )φq,ξ ,
dT Aξ,q x
(8.26)
where Vol is the volume of G/H, dT = dimT (H). We sum over the representations T (j) of G which give T (H) after reduction to H. ξ labels multiple
components T (H) in the branching T (j) ↓ H, dj = dim T (j) . q runs from
1 through dj . The matrix elements of T (j) have the following orthogonality
property
Z
M
√ (j)†
(j 0 )
dn x gTAζ,q (gx−1)TAξ,p (gx−1 ) = (Vol)d−1
j dT δζξ δpq δjj 0
(8.27)
Therefore, to construct the harmonic expansion on G/H it is necessary to have
powerful methods for reduction of the representations from G to H. There are
several standard [?, ?] and less standard [?, ?] techniques which may be used
depending on the particular homogeneous space.
It is important that not only the harmonic expansion but also the spectrum of the invariant operators can be analysed by the group theoretical methods. The covariant derivative on G/H reads [?]:
[R]
∇µ = ∇[c]
µ + Γµ .
(8.28)
Here ∇[c] is the canonical covariant derivative on G/H. At the origin ∇[c]
can be identified with the tangent space generators from M taken in the
representation T (j) . The part Γµ[R] depends on the invariant metric on G/H
and on the structure constants of G restricted to M. On symmetric spaces such
81
structure constants are zero and, therefore, the Laplace-Beltrami operator has
a particularly simple form:
D ' ∇µ[c] ∇[c]
µ ' C2 (G) − C2 (H) ,
(8.29)
where C2 are quadratic Casimir operators of G and H which depend on the
representations T (i) and T (H) respectively. On general homogeneous spaces
the expressions are a bit more complicated (see, e.g., [?, ?] for explicit examples). In any case, eigenvalues of D are given by a second order polynomial
Q(m1 , . . . , mk ) of several natural numbers ml . These eigenvalues are in general
degenerate with multiplicities defined essentially by dimensions of the representations of G and H. They are also polynomials 33 in ml . The heat kernel
is then represented as an infinite sum
K(t, D) =
X
ml
N(m1 , . . . , mk ) exp(−tQ(m1 , . . . , mk )) .
(8.30)
There exist several tricks which can be used to evaluate the t → 0 asymptotics
of such sums. For example, one may use the Poisson summation formula (4.4).
By taking derivatives with respect to t one obtains
X
√
l2r exp −tl2 ' π2−r (2r − 1)!!t−(2r+1)/2 + O e−1/t
(8.31)
l∈Z
for r ∈ N.
More general polynomials of l may be treated by the Euler–Maclaurin
formula which reduces sums to the integrals. Let F (τ ) be a function defined
on 0 ≤ τ < ∞. If the 2m-th derivative F (2m) (τ ) is absolutely integrable on
(0, ∞)
k
X
i=0
F (i) −
Z
∞
0
1
F (τ )dτ = [F (0) + F (k)]
2
+
m−1
X
s=1
B2s (2s−1)
[F
(k) − F (2s−1) (0)] + Remm (n),
(2s)!
(8.32)
where the reminder Remm satisfies:
|Remm (n)| ≤ (2 − 2
1−2m
|B2m |
)
(2m)!
Z
0
n
|F (2m) (τ )|dτ.
(8.33)
Bs are the Bernoulli numbers. If F (i) is taken to be Ni exp(−tλi ), one can take
limit k → ∞ in (8.32) and restrict the summation to some finite m to obtain
asymptotic series for the heat kernel. For example, one can easily recover the
expansion (8.31).
33
The spectrum of differential operators on coset spaces can be calculated exactly
even if one adds a homogeneous gauge field (see [?] and references therein).
82
In many cases calculation can be done by means of the Mellin transform.
This method will be clear from the following example. Consider
K(t) =
∞
X
k exp(−tk 2 )
(8.34)
k=1
The Mellin transform of K(t) is
˜
K(s)
=
X
k
Z
k
∞
dx xs−1 exp(−xk 2 ) =
0
∞
X
k 1−2s Γ(s) = ζR (2s − 1)Γ(s),
k=1
(8.35)
where ζR is the Riemann zeta-function. Inverse Mellin transformation gives
1
K(t) =
2πi
Z
C
t−s ζR (2s − 1)Γ(s)ds.
(8.36)
The contour C covers all poles of the integrand at s = 1, 0, −1, −2, . . . . By
calculating the residues we obtain the desired expansion:
K(t) =
1
1
t
t2
t3
−
−
−
−
+ O(t4 ).
2t 12 120 504 1440
(8.37)
Actually, the techniques introduced above should only be used for complicated multi-parameter sums (see, e.g., [?]). For a simple one-parameter sum
it is easier to use combinations of known asymptotic series (cf. Appendix of
Ref. [?]).
In this method complexity of calculations of the heat kernel coefficients ak
is almost independent of k. Therefore, the algebraic techniques were applied
to higher dimensional theories where one needs higher heat kernel coefficients
to perform renormalization or to calculate the anomalies. Rather naturally,
the most simple toroidal spaces were considered first [?, ?, ?, ?, ?, ?], and then
computations on spheres were performed [?, ?, ?, ?, ?, ?, ?, ?, ?]. Other homogeneous spaces were considered for example in [?, ?]. In the same approach
non-minimal operators on homogeneous spaces were treated in [?,?]. We refer
to [?] for a more extensive literature survey.
83
9
Exact results for the effective action
In the previous section we have seen that the heat kernel can be calculated
exactly for all values of t if the background satisfies certain symmetry or
smoothness conditions. In this section it will be demonstrated that for some
classes of the operators the effective action can be calculated exactly without
imposing any restrictions on behaviour of the background fields. These are
the cases when the variation of ζ 0 (0) with respect to the background can be
reduced to the heat kernel coefficients which are locally computable. Then the
variation is integrated to the effective action.
The most immediate example of the variations admissible for this scheme
is quantum anomalies (cf. sec. 7). Historically, the first action obtained by
integration of an anomaly was the chiral Wess-Zumino action. We do not
consider this action here since such a consideration would require a lot of
additional technical devices. The reader can consult excellent original papers
[?, ?] 34 . The next subsection is devoted to another well know example, which
is the Polyakov action obtained through integration of the conformal anomaly
in two dimensions.
In sec. 9.2 below we consider a more complicated situation when relevant
variation of the effective action leads to a linear combination of the heat kernel
coefficients of several different operators. Typically in this case one obtains
exact relations between the effective actions rather than the effective actions
themselves.
9.1 The Polyakov action
Let us consider a two-dimensional Riemannian manifold M without boundary and a scalar field φ minimally coupled to the geometry. This means that
we choose Φ = 0 in the action (7.9) and Ψ = 0 in the inner product (7.10). In
two dimensions any metric is conformally flat, i.e. by a suitable choice of the
coordinates one may transform it to the conformal gauge
gµν = e2ρ(x) ηµν ,
ηµν = diag(1, 1).
(9.1)
In this gauge
√
gR = −2η µν ∂µ ∂ν ρ .
(9.2)
The effective action W depends on ρ only (since there is no other external
field in this problem). We substitute (7.16) with Φ = Ψ = 0 in (7.3) to obtain
1 Z 2
δW =
d x(δρ)η µν ∂µ ∂ν ρ .
12π M
34
(9.3)
Calculation of the effective action in two dimensional QED by integration of the
anomaly may be found in [?].
84
This equation can be integrated with the initial condition W (ρ = 0) = 0.
1
W =
24π
Z
M
d2 x ρη µν ∂µ ∂ν ρ .
(9.4)
By returning to the covariant notations we obtain the famous Polyakov action
[?] (see also [?]):
1
W =
96π
Z
1
√
d2 x gR R .
∆
M
(9.5)
Note, that on a compact manifold one has to take into account contributions from the zero modes which lead to an additional term in the Polyakov
action [?].
The problem of finding an action whose conformal variation reproduces
the conformal anomaly can be posed in higher dimension as well [?] and some
physical consequences of such actions can be studied [?, ?]. Complete expressions for the conformal anomaly induced effective action in four dimensions
were obtained in [?, ?]. Conformal action in the presence of boundaries was
constructed in [?]. One should keep in mind that unless the background is
conformally flat the conformal action represents only a part of the full effective action. Applicability and limitations of the conformal action approach are
discussed in [?, ?, ?].
9.2 Duality symmetry of the effective action
In this section we deal with variations of the effective action which can be
expressed through zeta functions of several operators (in contrast to genuine
anomalous transformations of sec. 7 which involve a single zeta function each).
A rather simple example can be found in the paper [?] which considered
two operators of Dirac type:
D
/ † = iγ µ e−Φ ∂µ eΨ
D
/ = iγ µ eΨ ∂µ e−Φ ,
(9.6)
on flat two-dimensional manifold without boundary depending on two scalar
functions Φ and Ψ. Let us now consider small variations of Φ and Ψ. According
to (7.4) the zeta function of corresponding Laplacian changes as
h
i
δζ(s, D
/D
/ † ) = −2sTr (D
/D
/ † )−s δΨ − (D
/ †D)
/ −s δΦ .
(9.7)
Variation of ζ 0 (0) reads:
h
δζ 0 (0, D
/D
/ † ) = −2 ζ(0, δΨ, D
/D
/ † ) − ζ(0, δΦ, D
/ †D)
/
h
i
= −2 a2 (δΨ, D
/D
/ † ) − a2 (δΦ, D
/ †D)
/ .
i
(9.8)
Here we used that D
/D
/ † and D
/ †D
/ are operators of Laplace type, and, therefore,
85
the corresponding zeta functions are regular at s = 0. In the last line we
replaced ζ(0) by the heat kernel coefficients by means of (2.27).
Next we have to rewrite D
/D
/ † and D
/ †D
/ in the canonical form (2.2) with
the help of (2.3), (2.4). For D
/D
/ † we have:
g µν = e2(Ψ−Φ) ,
ωµ = Ψ,µ − γ ν γµ Φ,ν ,
R = −2(Φ − Ψ),µ µ ,
E = Φ,µ µ ,
(9.9)
where comma denotes ordinary partial derivatives, all contractions are performed with the effective metric g µν . Note, that in two dimensions Γρµν g µν = 0
in the conformal coordinates. To obtain relevant geometric quantities for D
/ †D
/
one has to replace Φ by −Ψ and vice versa. Equation (4.27) gives:
δζ 0 (0, D
/D
/ †) = −
1
6π
Z
d2 x [δΨ(∆Ψ + 2∆Φ) + δΦ(∆Φ + 2∆Ψ)] ,
(9.10)
where ∆ is the flat space Laplacian. Let us recall, that ζ 0(0, D) defines the
functional determinant of D (see eq. (2.33)). The variation (9.10) can be integrated to give 35
1
ln det(D
/D
/ ) = −ζ (0, D
/D
/ )=
12π
†
0
†
Z
d2 x(Ψ∆Ψ + 4Ψ∆Φ + Φ∆Φ) .
(9.11)
We see from (9.11) that ln det(D
/D
/ † ) = ln det(D
/ †D)
/ which is obviously true
up to possible topological contributions from zero modes which have been
neglected in this calculation. The result (9.11) was confirmed in [?] by the
methods of covariant perturbation theory.
This example is clearly of (at least an academic) interest since it adds
up to very few cases discussed above in this section when a closed analytic
expression for the determinant may be obtained without any assumptions on
smoothness of the background fields. The operator D
/D
/ † shares some similarities with the kinetic operator for non-minimally coupled scalars in two
dimensions (7.12). There are good grounds to believe [?] the functional determinant (9.11) may describe the spherically symmetric part of the Hawking
flux in four dimensions.
Let us consider the dilatonic operator (7.12). We take Ψ = Φ for simplicity 36 . We shall be interested in properties of det D under reflection of the
dilaton, Φ → −Φ. The quantity ln det D(Φ)−ln det D(−Φ) defines the dilaton
shift under the T -duality transformations in string theory. In the string theory
context this problem was solved in [?] basing on earlier results of [?]. Here we
follow presentation of the paper [?].
The heat kernel coefficients a2 (x, D
/D
/ † ) and a2 (x, D
/ †D
/ ) are total derivatives.
Therefore, the second term on the right hand side of (2.33) does not contribute.
36 Variation of det D with respect to Ψ may be expressed through the scale anomaly.
Therefore, it is not a problem to find exact dependence of det D on Ψ by using the
methods of sec. 7.
35
86
Let us introduce two first order differential operators
Lµ = e−Φ ∂µ eΦ ,
L†µ = −eΦ ∂µ e−Φ
(9.12)
D− = Lµ L†µ .
(9.13)
and associated second order operators
D+ = L†µ Lµ ,
We restrict ourselves to flat two-dimensional manifolds. Therefore, position of
the indices (up or down) plays no role.
We calculate again variation of the ζ function with respect to variation
of Φ:
h
−s−1
−s−1 †
δζ(s, D+) = −2sTr (δΦ) L†µ Lµ D+
− Lµ D+
Lµ
i
.
(9.14)
−s
. The situation with the
The operators in the first term here recombine in D+
second term is more subtle. Strictly speaking, to treat this term one has to
perform the Mellin transform and use analytic continuation in s (see footnote
26 in sec. 7.1). However, the result of such manipulations is almost obvious:
h
i
h
i
−s−1 †
Tr (δΦ)Lµ D+
Lµ = Tr (δΦ)H −s ,
(9.15)
Hµν = Lµ L†ν
(9.16)
where
is a matrix operator acting on the space V+ of the vector fields which can
be represented as vν = Lν f with some scalar function f . The operator (9.16)
is not of Laplace type as it contains a complicated differential projector on
V+ . Therefore, even though the variation (9.14) looks very similar to (9.7), for
example, it cannot be used directly to evaluate ζ 0 (0, D+ ) since we cannot even
guarantee that (9.15) is regular at s = 0 (in fact, it is not regular, see [?]).
We also have:
h
−s
¯ −s
δζ(s, D− ) = 2sTr (δΦ) D−
−H
i
,
(9.17)
where the operator
¯ µν = µµ0 νν 0 L† 0 Lν 0
H
µ
(9.18)
acts on the space V− of the vector fields of the form v¯µ = µµ0 L†µ0 . All epsilon
tensors cancel after taking the trace in (9.17) and we arrive at the same formula
as for D+ but with the replacement Φ → −Φ. The spaces V+ and V− are
orthogonal, and V+ + V− = V is the space of all vector fields on M (we neglect
possible zero modes). Therefore,
h
i
¯ − ζ(s, δΦ, D+) − ζ(s, δΦ, D− ) .
δ [ζ(s, D+ ) − ζ(s, D− )] = 2s ζ(s, δΦ, H + H)
(9.19)
87
The operator
¯ µν = δµν D+ + Φ,µν
(H + H)
(9.20)
is of Laplace type. We can now act in the standard manner to obtain
h
i
¯ − a2 (δΦ, D+ ) − a2 (δΦ, D− ) .
δ [ζ 0(0, D+ ) − ζ 0 (s, D− )] = 2 a2 (δΦ, H + H)
(9.21)
The right hand side can be evaluated by using (4.27). We leave it is an exercise
to show that the variation (9.21) is zero (alternatively, one may look up in [?]).
This leads us to the conclusion that
ζ 0(0, D+ ) − ζ 0 (0, D− ) = − ln det D+ + ln det D− = 0.
(9.22)
In the one-dimensional case a similar relation may be obtained by methods of supersymmetric quantum mechanics. In this simplest case, and in higher
dimensions if Φ depends on one of the coordinates only, D+ and D− are
isospectral up to zero modes and (9.22) follows immediately. In two dimensions, D+ and D− are not isospectral in general. Although D+ and D− have
coinciding determinants, other spectral functions can be different.
Extensions of (9.22) to the case of matrix-valued Φ, manifolds with boundaries, and the dilaton-Maxwell theory in four dimensions can be found in [?].
A generalisation to arbitrary dimension has been achieved recently [?]. Let
Ap be a p-form field with the field strength F = dAp . Consider the classical
action
L=
Z
M
e−2Φ F ∧ ?F ,
(9.23)
where ? is the Hodge duality operator. Such actions appear, for example, in
extended supergravities [?] and bosonic M-theory [?]. Instead of the dilaton,
also a tachyon coupling may appear [?]. We are interested in the symmetry
properties of the effective action under the transformation p → n − p − 2,
Φ → −Φ. In higher dimensional supergravity theories this is a part of the
S-duality transformation.
It is convenient to define the twisted exterior derivative
dΦ := e−Φ deΦ
(9.24)
and the associated twisted coderivative and twisted Laplacian
δΦ := eΦ δe−Φ ,
∆Φ := (dΦ + δΦ )2 .
(9.25)
Since d2Φ = 0 we have an elliptic complex. The restriction of ∆Φ on the space
of p forms will be denoted by ∆pΦ . This twisted de Rham complex was introduced by Witten in the context of Morse theory and supersymmetric quantum
mechanics [?].
88
Any p-form can be decomposed as the sum of a twisted exact, twisted
co-exact, and twisted harmonic form:
Ap = dΦ Ap−1 + δΦ Ap+1 + γp ,
∆Φ γp = 0 .
(9.26)
The projections on the spaces of twisted exact and twisted co-exact forms will
be denoted by the subscripts k and ⊥ respectively.
We assume that the fields A˜p = e−Φ Ap have a standard Gaussian measure
and are to be considered as fundamental fields in the path integral. The action
given in eq. (9.23) is invariant under the gauge transformation which sends A˜p
to A˜p + dφ A˜p−1 . This means that the p-forms which are dφ exact have to be
excluded from the path integral, but that a Jacobian factor corresponding to
the ghost fields A˜p−1 has to be included in the path integral measure. Next we
note that dφ -exact (p − 1)-forms do not generate a non-trivial transformation
of A˜p . Hence, such fields must be excluded from the ghost sector. Then we
have to include “ghosts for ghosts”. This goes on until the zero forms have
been reached. By giving these arguments an exact meaning, one arrives at
the Faddeev–Popov approach to quantisation of the p-form actions [?, ?, ?, ?].
We note that the procedure of [?, ?] is valid also in the presence of a dilaton
interaction if one simply replaces the ordinary derivatives by the twisted ones.
As a result, we have the following expression for the effective action:
Wp (Φ) =
p
1X
(−1)p+k ln det(∆kΦ |⊥ ) + Wptop .
2 k=0
(9.27)
Wptop depends on certain topological characteristics of the manifold (the Betti
numbers). We shall neglect Wptop and some other topological contributions in
what follows.
By combinatorial arguments, similar to the presented above, one can show
that [?]:
δ (Wp (Φ) − Wn−p−2(−Φ)) =
m
X
(−1)p+k an (δΦ, ∆kΦ ) .
(9.28)
k=0
One should distinguish between δΦ (which is a variation of the dilaton) and
δΦ (which is a twisted co-derivative). Therefore, we have related the variation
of the effective actions with respect to Φ to a combination of the heat kernel
coefficients which is called the supertrace of the twisted de Rham complex. A
somewhat surprising feature of the supertrace is that it can be calculated for
any n, with or without boundaries [?]. For example, the volume term in (9.28)
does not depend on Φ and, therefore, is the standard Euler density which is
given by
En := (4π)−¯n 2n¯1n¯ ! i1 ...im j1 ...jn Ri1 i2 j2 j1 ...Rin−1 in jn jn−1
(with n
¯ = n/2) for n even. En = 0 for n odd.
89
(9.29)
It is known [?,?] that for Φ = 0 the dual theories are quantum equivalent:
Wp (0) − Wn−p−2 (0) = 0 .
(9.30)
By using this equation as an “initial condition” we can integrate the variation
(9.28) to obtain:
Wp (Φ) − Wn−p−2 (−Φ) = (−1)
P
p
Z
M
√
dn x gΦEn .
(9.31)
Further terms p (−1)p ak (x, ∆pΦ ) are of some interest in mathematics.
They can be calculated [?, ?] for k ≤ n + 2 and any n.
90
10
Conclusions
Here we present a short guide in this report. In other words, we are
going to answer the following question. What should one do if one likes to
calculate one-loop counterterms, anomalies, an expansion term in the effective
action, or something else which is defined by the heat kernel expansion? The
first step is to find the bulk part of the variation of the classical action (1.2)
and corresponding operator D. Next one has to bring this operator to the
canonical form (2.2) with the help of (2.1), (2.3) and (2.4). However, one can
first consult sec. 3. Probably, relevant expressions can be found there. If the
problem in question contains neither boundaries nor singularities, one can look
in sec. 4 for an expression for the heat kernel coefficient, or for a reference,
or for a method. In the case of the boundaries one has to proceed with sec.
5, in the case of singularities – with sec. 6. Relations between the heat kernel
coefficients and quantum anomalies can be found in sec. 7. The results going
beyond the standard heat kernel expansion are collected in sec. 8. In particular,
in this section we explain how one can extract leading non-localities from
the effective action, and what the heat kernel looks like if the background is
approximately covariantly constant (in this context we also consider invariant
operators on homogeneous spaces). Exact results for the effective action which
can be obtained with the help of the heat kernel expansion are reviewed in
sec. 9.
The Casimir energy is one of the most “classical” applications of the
heat kernel and zeta function technique (cf. [?]). It follows from the locality
of the heat kernel expansion that the divergences in the Casimir energy are
given by volume and surface integrals of some local invariants. Therefore, if
the boundaries are being moved in such a way that the boundary values of
the background fields remain unchanged, the boundary contributions to the
divergences also remain constant. This leads to the well known conclusion [?]
that there are no boundary divergences in the Casimir force (which is defined,
roughly speaking as a variation of the Casimir energy under infinitesimal translations of the boundary). Consequently, one can assign a unique value to the
Casimir force (see, e.g., [?]). This observation, however does not mean that the
quantum field theory on a manifold with boundary is finite. In general, some
surface counterterms are required (at one-loop order they may be read off
from sec. 5). Moreover, if the background field are non-trivial, the boundary
divergences will not be constant. Similar arguments created certain scepticism
towards reliability of the Casimir energy calculation [?]. This point has not
been settled so far for a “generic” theory. We may add that in supersymmetric
theories cancellations between divergences in the bosonic and fermionic sectors appear if the boundary terms are considered together with the volume
terms [?], therefore, separation of boundary and volume contributions is not
always natural for that theories. Renormalization of self-interacting theories
on manifolds with boundaries was considered in [?, ?, ?, ?, ?] where one can
find further references. Some aspects of the relationship between the Casimir
91
energy calculations and the heat kernel coefficients have been clarified recently
by Fulling [?].
Of course, not everything can be found in this report. Several topics are
very close to the subject of this review, but are not included.
(1) The heat kernel expansion can be successfully applied to quantum field
theory at finite temperature [?]. A new interesting development in this
field is related to the so called non-linear spectral problem [?] (see [?] for
an overview).
(2) The heat kernel expansion has interesting applications to integrable models and, in particular, to the KdV hierarchies (see [?] for an elementary
introduction).
(3) Recently some attention has been attracted [?, ?, ?, ?] to the so called
N/D or Zaremba problem which appears when one defines Neumann and
Dirichlet boundary conditions on two (intersecting) components of the
boundary. It is unclear whether this problem may have applications to
quantum theory.
(4) Instead of considering asymptotics of the heat trace Tr(f e−tD ) one can
also consider an asymptotic expansion for individual matrix elements of
the heat kernel hf1 , e−tD f2 i (which are called the heat content asymptotics
since they remind short time asymptotics of the total heat content in a
manifold with the specific heat f1 and the initial temperature distribution
f2 ). Such asymptotics do not contain negative powers of t. More details
can be found in [?].
(5) Many results on the heat kernel asymptotics can be extended to higherorder differential operators (see, e.g., [?, ?]) and to differential operators
in superspace [?].
Although quite a lot is already known about the heat kernel expansion,
many interesting problems still remain open. There are many opportunities to
extend and generalise the results presented in this report. This refers especially
to the material of sec. 5-9 where one could add new types of the operators,
boundary conditions, geometries, and singularities. There is a completely new
field of research related to the heat kernel expansion where very little has been
done so far. This is an extension to non-commutative geometry. This problem
is an especially intriguing one since one can expect very unusual properties
of the spectral functions because of very unusual properties of corresponding
field theories in the ultra violet asymptotics.
Acknowledgements
I am grateful to my collaborators S. Alexandrov, M. Bordag, T. Branson,
E. Elizalde, H. Falomir, P. Gilkey, D. Grumiller, K. Kirsten, W. Kummer,
H. Liebl, V. Lyakhovsky, V. Marachevsky, Yu. Novozhilov, M. Santangelo,
N. Shtykov, P. van Nieuwenhuizen and A. Zelnikov who contributed in many
ways to the material presented in this report. I have benefited from enlight92
ening discussions with I. Avramidi, A. Barvinsky, G. Esposito, D. Fursaev,
G. Grubb, A. Kamenshchik, V. Nesterenko and with my friends and colleagues
at St. Petersburg, Leipzig and Vienna.
I am grateful to all readers who suggested their comments on the previous
version of the manuscript. I also thank the referee for useful critical remarks.
This work has been supported by Project BO 1112/12-1 of the Deutsche
Forschungsgemeinschaft. I am grateful to E. Zeidler for his kind hospitality at
the Max Planck Institute for Mathematics in the Sciences where a part of this
work has been done.
93
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