Tesi Latini Emanuele

Tesi Latini Emanuele
Alma Mater Studiorum - Universitá di Bologna
DOTTORATO DI RICERCA
Fisica
Ciclo XX
Settore scientifico disciplinare di afferenza: FIS/02
TITOLO TESI
WORLDLINE APPROACH
TO
HIGHER SPIN FIELDS
Presentata da: Dott. Emanuele Latini
Coordinatore Dottorato
Prof. Fabio Ortolani
Relatore
Prof. Fiorenzo Bastianelli
Esame finale anno 2008
Introduction and motivations
The main object of this thesis is the analysis and the quantization of spinning particle models
which employ extended ”one dimensional supergravity” on the worldline, and their relation
to the theory of higher spin fields (HS).
In the first part of this work we will describe the classical theory of massless spinning
particles with an SO(N ) extended supergravity multiplet on the worldline, in flat and more
generally in maximally symmetric backgrounds. These (non)linear sigma models describe,
upon quantization, the dynamics of particles with spin N2 [1] [2] (see also [3] [4] [5] for further
analysis on the N = 2 case).
Then we will analyze carefully the quantization of spinning particles with SO(N ) extended supergravity on the worldline, for every N and in every dimension D. The physical
sector of the Hilbert space reveals an interesting geometrical structure: the generalized higher
spin curvature (HSC). We will see, in particular, that these models of spinning particles describe a subclass of HS fields whose equations of motions are conformally invariant at the
free level [6] [7] [8] [9] [10]; in D = 4 this subclass describes all massless representations of
the Poincaré group.
In the third part of this work we will consider the one-loop quantization of SO(N ) spinning particle models by studying the corresponding partition function on the circle. After
gauge fixing the supergravity multiplet, the partition function reduces to an integral over the
corresponding moduli space which will be computed using orthogonal polynomial techniques.
Finally we will extend our canonical analysis, described previously for flat space, to maximally symmetric target spaces (i.e. (A)dS background). The quantization of these models
produce (A)dS HSC as the physical states of the Hilbert space; we will then use an iterative
procedure and Pochhammer functions to solve the differential Bianchi identity in maximally
symmetric spaces.
In the last part of this work we will construct spinning particle models with sp(2) R
symmetry, coupled to Hyper Kähler and Quaternionic Kähler (QK) backgrounds1 .
1
Spinning particle models with U (N ) R symmetry in Kähler background have been considered in [11].
ii
Introduction and motivations
Motivated by the correspondence between SO(N ) spinning particle models and HS gauge
theory, and by the notorious difficulty one finds in constructing an interacting theory for
fields with spin greater than two [12], we intend to employ these one dimensional supergravity models to study and extract informations on HS.
One advantage of using the ”worldline” point of view, for studying HS fields, is the fact that
this approach gives a quite interesting representation of the one-loop quantum field theory
(qft) effective action in terms of a quantum mechanical model (qm) (see [13] [14] [15] and
references therein). Let us briefly review the relation of first quantized particle to the formalism of qft.
The worldline approach to qft is a powerful technique to calculate effective actions in
quantum field theories and study anomalies, amplitudes etc... . The main idea can
be introduced as follows: consider a qft action principle and use a path integral on
the fields φ(x) to construct the effective action Γqf t (second quantization); after that,
one can convert Γqf t into a path integral over space-time coordinate x (first quantized
theory, or equivalently particle theory). These two kinds of approaches are equivalent,
at least at the perturbative level.
As an example one may consider the simple case of a free scalar field φ
Z
Z
1
4 1 µ
Sqf t [φ] = d x ∂ φ∂µ φ = d4 x φ(−2)φ .
2
2
(1)
The partition function reads
−Γqf t
Zqf t ≡ e
Z
=
1 − 12
det(− 2)
2
1
1 = exp − tr(ln(− 2) .
2
2
be re-expressed as
1 1
tr ln(− 2)
2
2
Z ∞
1
1
dβ
−
tr e−β − 2 2
2 0 β
Z
Z
1 ∞ dβ
−
e−Sqm [x]
2 0 β P BC
Dφe−Sqf t [φ] =
Then the effective action Γqf t can
Γqf t =
=
∼
(2)
(3)
where β, the Schwinger parameter, will be recognized as the proper time of a relativistic
particle and P BC means that we take periodic boundary conditions x(β) = x(0). In
the second line of (3) there appears the operator (− 12 2) which may be interpreted as
the hamiltonian operator of a suitable qm model. The trace is then computed by using
the path integral for the qm action Sqm given by
Z β
1
Sqm [x] =
dτ ẋ2 .
(4)
2
0
Introduction and motivations
iii
Originally this qm model was considered to be ”fictitious”; however, later, it has been
realized that it arises from the quantization of the bosonic particle model
Z 1
ẋ2
Sqm [x, e] =
dτ
(5)
2e
0
with a suitable gauge fixing of the one dimensional einbein e2 . The partition function
on the circle (P BC), for the model (5) reads formally as
Z
DxDe
Zqm =
e−Sqm [x,e] ;
(6)
Vol(Gauge)
P BC
the gauge fixing of the einbein to the proper time (e = β) produces an integral over
the modulus β [16]. The gauge fixed partition function is:
Z ∞
Z
dβ
Zqm ≡
Dxe−Sqm [x,e=β]
β P BC
0
Z ∞
dβ
Zqm (β)
=
(7)
β
0
and reproduces (3). Concluding one has:
Z
DxDe
Γqf t ∼
e−Sqm [x,e] .
Vol(Gauge)
P BC
(8)
Thus we see that a qft object (in particular Γqf t ) can be calculated by using a qm
model.
After having emphasized the relations between particle and fields (first and second
quantization) let us focus our attention on qm path integrals.
Consider the action principle
Z
i
1 1 h1 α β
2
(9)
Sqm =
dτ ẋ ẋ δαβ + β V (x)
β 0
2
where V (x) is a scalar potential. To compute the path integral one can extracts the
dependence on the zero modes xα0
xα (τ ) = xα0 + y α ;
(10)
y α is the quantum fluctuation and we impose Dirichlet boundary conditions y α (0) =
y α (1) = 0. This describes a loop with a fixed point.
2
Note that a canonical analysis of (5) produces the massless Klein-Gordon equation of motion as a
constraint of the physical sector of the Hilbert space.
iv
Introduction and motivations
At this point the computation is straightforward and the qm path integral can be
evaluated in perturbative expansion:
Z
Z
R
1
− β1 01 dτ [ 12 ẋ2 +β 2 V (x)]
D
2
Dxe
(11)
=
d x0 a0 + a1 β + a2 β + · · ·
{z
}
|
(2πβ)2
P BC
=0 in the free case
where the coefficients ai are the so called Seeley-DeWitt coefficients [17] in the coincidence limit; they characterize the quantum theory, for example we can use them to
identify the counterterms needed to renormalize the full one loop effective action. In
the free case limit one finds that ai = 0 ∀i 6= 0 whiel a0 counts the degrees of freedom
(Dof ) propagated into the loop. In the example discussed above one finds, in fact,
a0 = 1, that coincide with the degrees of freedom associated to a scalar field.
At this point one can introduce more general backgrounds, for example electromagnetic background (V (x, ẋ) = −igA(x) · ẋ) or gravitational background, (δαβ → gµν (x)
and V (x) = ξR(x)). In this cases, in order to define properly the path integrals, one
requires the introduction of a regularization scheme to make sense on the integration
over paths, and the fixing of certain renormalization conditions, which makes sure that
different regularization schemes will produce the same results. We are not going to
describe all the details, since regularization is not the main object of our work (more
details on path integral and anomalies could be found in [18]).
The previous considerations can be extended to the spinning case by introducing extra
fermionic degrees of freedom ψ α , the supersymmetric partner of xα , on the worldline [19]
[20] [21].
Note that supersymmetry on the worldline does not imply supersymmetry on space-time. In
fact all the spinning particle models we will deal with describe the propagation of multiplets
of the same statistical nature (either fermionic or bosonic).
In fact it is well known that the dynamics of a spin 1/2 particle in D = 4 Minkowsky
background can be described by using a quantum mechanical model with N = 1 local
supersymmetry on the worldline:
Z
1
i
(1)
(12)
S = dt pα ẋα + ψ α ψ̇ β ηαβ − e pα pα − iχ(ψ α pα ) .
2
2
A canonical analysis of this model produces the massless Dirac equation as constraints for
the physical sector of the Hilbert space. In (2.30) e and χ are lagrangian multipliers introduced to implement into the action the first class constraints H = p2 (i.e. the hamiltonian,
the generator of worldline diffeomorphism) and Q = ψ α pα (i.e. supercharge, the generator
of supersymmetry).
More in general, in this work, we will study spinning particle models enjoying N local worldline supersymmetries, SO(N ) gauge symmetry and worldline diffeomorphism: the SO(N )
spinning particle models.
The SO(N ) spinning particle models have been extensively studied in [1] [2] [3] [8] [22]
[23] [24] [25] [26].
Introduction and motivations
v
In Chapter 1 we will start describing the model in flat Minkowsky target space and
we will analyze in details its symmetries and the first class constraints algebra, namely
the SO(N ) extended supersymmetry algebra. After that we will focus our attention on
the possible coupling with gravitational background preserving one dimensional local
supersymmetry. Howe et al. have analyzed this problem in [2] and have concluded
that, with a given generalization of the worldline local supersymmetry transformation,
SO(N ) spinning particle models can be consistently coupled to a gravitational background only when N = 0, 1, 2; however Kuzenko and Yarevskaya have relaxed this
no go theorem introducing in [27] a coupling with maximally symmetric space (i.e.
(A)dS) for every N , by extending in a clever way the local worldline supersymmetry
transformation rules. In Chapter 1 we will propose also an alternative approach to
this problem; we will work in first order formalism and we will study the first class
constraints algebra in curved space showing explicitly that in maximally symmetric
space, turns out to be closed, yet non linear. Finally we will compare our results with
the Kuzenko and Yarewskaya ones. In Appendix B we will also discuss the construction of the BRST charge associated to this non linear SO(N ) extended supersymmetry
algebra.
The SO(N ) spinning particles, from our point of view, are very interesting because in D = 4
Minkowskian background, one finds, via canonical analysis, the well known Bargman Wigner
equation of motion [28] describing the dynamics of spin s = N2 fields (i.e. Higher Spin Field).
Higher Spin Fields and in particular HS gauge theory have attracted a great deal of
attention in the search for generalizations of the known gauge theories of fields of spin
1 (Maxwell and Yang-Mills theory), spin 2 (Einstein general relativity) and spin 32
(supergravity); see [29] [30] [31] [32] [33] and references therein.
Further motivations in studying HS arise from Superstring Theory; superstring theory
is, in fact, the most important candidate for a unified theory of interactions. The key
idea of string theory is very simple: extension of point particles (i.e. 0 dimensional
object) to one dimensional object (i.e. string) with length ls ∼ 10−33 cm. The Regge
slope α0 and the string tension T are defined as
ls =
√
1
2α0 = √
.
πT
(13)
Vibrational string states generate an infinite number of states with mass ms and spin
s given by:
1
m2s ∼ 0 (s − 1) ∼ T (s − 1) .
(14)
α
String theory naturally describes an infinite number of excited massive states, with
increasing mass and spin. String tension is usually taken to be of the same order of the
Plank mass (Mpl ∼ 1019 Gev), for this reason at low energy just the massless modes
are exited (with s 6 2) and HS states are too heavy and cannot be not observed at
low energy. In the high energy limit, all the HS string states may effectively become
massless, so that one may be left with an infinite number of massless HS states, i.e. a
vi
Introduction and motivations
HS gauge theory [35].
For all this reason a better understanding of the dynamics of HS states is important
for the analysis of quantum properties of String Theory.
The construction of an interacting HS theory is a main long standing problem (see
[36] for recent developments in the spin 3 case); the general Coleman-Mandula and
Haag-Lopuszanski-Sohnius theorem of the possible symmetries of the unitary S-matrix
of the quantum field theory in D = 4 Minkowsky space [37] does not allow conserved
currents associated with the symmetries of fields with spin greater than two to contribute to the S-matrix (see also [38]). This no-go theorem might be overcome if the
higher spin symmetries would be spontaneously broken. There exists also another way
out: one could construct the interacting HS field theory in a vacuum background with
a non-zero cosmological constant (i.e. (A)dS), in which the S-matrix theorem does not
apply. Positive results along these lines have been achieved and the most notorious is
perhaps the Vasilievs interacting field equations, which involve an infinite number of
fields with higher spin [12], but an action principle for them is still lacking; further and
interesting analysis on HS fields in (A)dS background can be found in [39] and [40] [41].
Relations between HS field theory and Superstring Theory suggest us that one has
to study the HS dynamics in D 6= 4 and in particular in D = 10, that is the superstring critical dimension. Note now that in D = 4 symmetric tensors of rank s (that
we denote with φα1 ...αs ) describe all possible higher spin representation of the Poincaré
group; in this case, in fact, all the irreducible massless representations of the Poincaré
group are classified by using the group SO(2) whose Young tableaux are single rows;
this implies that massless HS fields could be represented by totally symmetric tensors.
Something news happens in D 6= 4; in higher dimension, symmetric tensors do not
describe all possible HS fields. For example in D = 10 the compact subgroup of the
little group is SO(8); the representations of SO(8) are, in general, Young tableaux
with mixed symmetry; see [42] for a recent review on this topics.
In Appendix A we will discuss different approaches to HS gauge theory. In particular we
will focus our attention on the geometric approach to HS and conformal HS gauge theory.
Conformal HS is an interesting and important subclass of HS fields whose (at least linearized)
equations of motion are conformally invariant; it’s important to observe that in this case,
in every even D > 4 dimensions, conformal HS fields are not completely symmetric tensors,
and the corresponding Young tableaux are rectangles with s columns and D2 − 1 = n − 1 rows
(i.e. s ⊗ (n − 1) Young tableaux); fields with this kind of symmetries will be denoted with
φ[n−1]1 ···[n−1]s (further analysis and references on conformal HS gauge theory can be found in
[43]).
In general, integer HS fields dynamics, both for completely symmetric or with the symmetries
of a rectangular Young tableaux, is described by the Frosndal (for completely symmetric
tensor) and Fronsdal-Labastida (that is the generalization for tensors with mixed symmetry)
equation of motion:
Gφα1 ···αs = 0
and
Gφ[n−1]1 ···[n−1]s = 0
(15)
Introduction and motivations
vii
where G is the Fronsdal-Labastidal kinetic operator. It is important to emphasize that this
operator is a second order differential operator which guarantees the unitary of the theory.
Generalized Higher Spin Curvatures (HSC), or equivalently HS field strengths, are the
key ingredients of the geometrical approach to HS gauge theory.
More in general gauge theories are usually presented in terms of a local symmetry of
an action principle. Other ingredients, however, are gauge parameters, gauge fields
equation of motion and Bianchi identities.
Let now Yp be a Young diagram such that the number of cells of Yp is p. We define
Ωp(Y ) (RD ) to be the vector space of tensor fields of rank p on RD which have the Young
symmetry type Yp . We define now the differential operator
D
d = (−)p Yp+1 ∂ : Ωp(Y ) (RD ) → Ωp+1
(Y ) (R ).
(16)
This operator acts as partial derivative of the tensor T ∈ Ωp(Y ) (RD ), then we have to
apply the Young symmetrizer Yp+1 to obtain a tensor in Ωp+1
(Y ) ; note that in general
2
d 6= 0.
Let us restrict now ourselves to tensor fields in RD with the symmetries of a Young
tableaux with number of columns strictly smaller than s + 1, and we consider gauge
potential with the symmetry of a rectangular Young tableaux with s columns3 . Gauge
theory for arbitrary spin s in RD , by using the differential operator (16), can be packaged into a complex, that schematically one can writes as:
ds
d
|
{z
} →|
Gauge parameter
{z
Potential
}→|
d
{z
Curvature
}→ |
{z
} ;
(17)
Diff. Bianchi id.
the fact that (17) is a complex follows form the property ds+1 = 0 [44]; moreover the
generalized Poincaré Lemma implies that the complex (17) is also an exact sequence.
This assures that the the differential Bianchi identity can be solved by introducing the
gauge potential, and this solution is unique (at least in RD ); moreover exactness of
(17) implies that the curvature is gauge invariant only with respect the transformation
δ(potential) = d(gauge parameter).
In particular, let now di be the usual exterior derivative acting on the ith column so
that:
s
Y
di
dj = 0
∀ i = 1, ..s .
(18)
j=1
In virtue of the generalized Poincaré Lemma one has
di R = 0 ∀ i = 1, 2, ..., s ⇒ R =
s
Y
dj φ
(19)
j=1
3
Let us recall that in D = 4 gauge fields have the symmetry of a Young tableaux s ⊗ 1 while in D 6= 4
and in particular in even dimension D = 2n we can construct also representation of the Poincaré group with
the symmetry of the Young tableaux s ⊗ (n − 1).
viii
Introduction and motivations
where R (i.e. curvature) and φ (i.e. potential) are multiforms which components are
irreducible tensor fields with the symmetry of the Young tableaux s⊗(m+1) and s⊗m
respectively, for some m. More details and generalization to tensor with different kind
of symmetries can be found in [45] [46].
Higher spin curvatures are constructed as the natural generalization of the spin 1
Maxwell field strength and of the linearized Riemann tensor in the case of spin 2. HS
gauge potential can be introduced by solving the differential Bianchi identity (19). The
main advantage one has by using this geometrical approach, is the fact that the theory
is automatically gauge invariant; otherwise HS potential equation of motion (obtained
from the traceless condition on the HSC) suffers for higher derivative problem, and
this implies that the theory is not unitary. This problem can be cured as follows: in
virtue of the generalized Poincaré Lemma one can introduce the so called compensator
field and after having gauge fixed it to zero the HS equation of motion reduces to (15)
(more details can be found in Appendix A).
In Chapter 2 we will analyze the physical contents of spinning particle models with
SO(N ) extended supergravity multiplet on the world line, in flat target space, and for
every dimension D. We will focus our attention on the integer spin case (even N ), and
we will quantize the model ”á la Dirac”; the phase space coordinates will be turned into
operators and the Hamiltonian, supersymmetry and SO(N ) constraints are imposed as
operatorial constraints on the Hilbert space states. We will compare our results with
the conformal HS gauge theory and we will show that a canonical analysis produce
”conformal” HSC as physical states of the Hilbert space.
After having analyzed the spectrum of the SO(N ) spinning particle model, and after having
understood its physical contents we proceed computing the path integral.
One loop quantization of spinning particles is the main objective of Chapter 3. First
of all we will gauge fix the one dimensional supergravity multiplet on the circle and
we will introduce the Faddeev-Popov determinant to extract the volume of the gauge
group. We will calculate the path integral on the one-dimensional torus in order
to obtain a compact formula which gives the number of physical degrees of freedom
of the spinning particles for all N in every dimensions D. Let us emphasize that
the computation of the path integral allows us to obtain the correct measure on the
moduli space of the supergravity multiplet on the circle; this is going to be extremely
useful also to compute more general quantum corrections arising when couplings to
background fields are introduced.
Spinning particles with SO(N ) supergravity multiplet on the worldline, can also propagate, in fact, in maximally symmetric space (i.e. (A)dS). In this work we intend to analyze
the physical content of the spinning particle model with SO(N ) supergravity multiplet on
the worldline, coupled to maximally symmetric space. In the literature the dynamics of HS
in (A)dS background has been extensively studied; in (A)dS4 , for example, integer HS gauge
potential φα1 ···αs equation of motion reads
G(A)dS φα1 ···αs = 0;
(20)
Introduction and motivations
ix
where G(A)dS is the Fronsdal-Labastida kinetic operator obtained starting from G by minimal
coupling plus terms introduced to restore gauge invariance. However it is not yet clear how
to obtain this operator starting from a pure geometrical object (i.e. (A)dS HSC). A possible
explicative scheme is the following:
HS
O
O
r2 HSC V V V
r2 r2 r2 r2
V V
2
r
2
r
2
r
V V
2r 2r 2r
r
2
V V
r
2
r
2
2r 2r
V V+
r
2
r
2
r
2
r
2
r
2
r
2
r
(A)dS HSC
?
O
Fronsdal-Labastida Xe.o.m.
X
X X X
X X X
X X X
X,
h HS
h
h
h
hhh
?hhhhhhh
h
h
hh
hs hhh
Fronsdal-Labastida e.o.m in (A)dS
where:
A o/ /o /o / B
• means that we go from A to B by using the generalized Poincaré Lemma
A _ _ _/ B
• means that B is obtained by minimal covariantization of A.
A
?
/B
• means that one should obtain B, starting from A by using the generalized Poincaré
Lemma; the interrogation point on the arrow emphasizes that an extension of this
Lemma in (A)dS space is still lacking, and we cannot use it as a guide line.
Recently some progresses in constructing (A)dS HSC have been obtained by Enquist and
Hohm in [47] where they have studied the dynamic of HS in (A)dS in frame-like formulation4 , and by Manvelyan and Ruhl in [49].
Conformal HSC in (A)dS2n will be extensively studied in Chapter 4. A canonical analysis of SO(N ) spinning particles produces (A)dS HSC as the physical sector of the
Hilbert space. In order to find the relation between HS field strength and the gauge
potential, and in particular construct the Fronsdal-Labastida operator in (A)dS, we
cannot reproduce the procedure described for the flat case because the generalized
Poincaré Lemma is the key ingredient one needs to extract the potential from the curvature (equivalently solve differential Bianchi identity) and construct gauge theories.
4
Gravity (i.e. spin 2 theory) could be described using metric and curvature, or vielbein and spin connection. Frame-like approach is based on the generalization of vielbein and spin connection, instead of metric
and curvature, to the HS case (see [48] for a pedagogical review and references therein).
x
For this reason we will propose a different approach: we start from the results obtained
in the flat case limit, we will add suitable (A)dS corrections and we will use supercharges constraints (i.e. differential Bianchi identity) to fix extra terms. In particular
we will propose an iteration procedure that let us to solve differential Bianchi identity
for every even dimension D and every integer spin s. The analysis is technically complicated, for this reason we have decided to postpone part of it, regarding Pochhammer
function, in Appendix C.
Recently spinning particle models with Hyper Kähler (HK) and Quaternionic-Kähler (QK)
background have attracted a great deal of attention in the context of studying radial quantization of BPS black-hole [50].
HK and QK N = 4 one dimensional supergravity will be analyzed in Chapter 5.
HK and QK geometries in dimension 4n and signature (2n, 2n) enjoy sp(2n) and
sp(2) ⊗ sp(2n) holonomy, respectively. Thus we will decompose SO(2n, 2n) tangent
space indices with respect to the sp(2) ⊗ sp(2n) subgroup and we will construct spinning particle models with N = 4 supersymmetry and sp(2) ”internal” symmetry: the
sp(2) spinning particle model.
In HK this model enjoys rigid worldline translation, sp(2) symmetry and N = 4 supersymmetry. We will thus gauge these symmetries and we will study the first class
constraints algebra.
Then we will analyze the model with QK background and we will show that in QK
target space is no longer possible to maintain rigid supersymmetry, just a model with
local supersymmetry is allowed. We will analyze the first class constraints algebra, that
in this case becomes a non Lie algebra, and we will show that two possible interesting
gauged model should be studied with:
- Rigid sp(2) symmetry, local supersymmetry and worldline diffeomorphism.
- Local sp(2) symmetry, local supersymmetry and worldline diffeomorphism.
We will focus our attention on the first one and we will construct the BRST charge.
Moreover we will gauge fix the one dimensional supergravity multiplet on the circle
and we will construct also the gauge fixed action.
Contents
1 SO(N ) spinning particle model
1.1 Kuzenko and Yarevskaya construction . . . . . . . . . . . . . . . . . . . . . .
1.2 Hamiltonian analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 From spinning particle to HS
2.1 Bargman Wigner equation of motion . . . .
2.2 Gravitino . . . . . . . . . . . . . . . . . . .
2.3 Graviton . . . . . . . . . . . . . . . . . . . .
2.4 From spinning particle to integer higher spin
2.5 HS gauge potential . . . . . . . . . . . . . .
1
4
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8
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gauge theory
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11
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3 One loop quantization and counting degrees of freedom
3.1 Gauge fixing on the torus . . . . . . . . . . . . . . . . . .
3.2 Computing the partition function . . . . . . . . . . . . . .
3.2.1 Even case: N = 2r . . . . . . . . . . . . . . . . . .
3.2.2 Odd case: N = 2r + 1 . . . . . . . . . . . . . . . .
3.3 Orthogonal Polynomials method . . . . . . . . . . . . . . .
3.4 Counting degrees of freedom . . . . . . . . . . . . . . . . .
3.5 Rectangular SO(D − 2) Young tableaux . . . . . . . . . .
3.6 The case of N = 4 and the Pashnev–Sorokin model . . . .
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39
4 Higher spin generalized curvature in (A)dS2n
4.1 (A)dS quantum Algebra . . . . . . . . . . . . . . . . .
4.1.1 The special even N case . . . . . . . . . . . . .
4.2 Canonical analysis and the generalized Poincaré Lemma
4.3 Warm up examples in D=2 . . . . . . . . . . . . . . . .
4.4 Fixing the strategy . . . . . . . . . . . . . . . . . . . .
4.4.1 Spin 2 in (A)dS2n . . . . . . . . . . . . . . . . .
4.4.2 Spin 3 in (A)dS2n . . . . . . . . . . . . . . . .
4.4.3 Spin 4 in (A)dS2n . . . . . . . . . . . . . . . . .
4.4.4 Conjecture . . . . . . . . . . . . . . . . . . . . .
4.5 HSC in (A)dS2n . . . . . . . . . . . . . . . . . . . . . .
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41
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xii
5 HK
5.1
5.2
5.3
5.4
CONTENTS
and QK N = 4 one dimensional SUGRA
Special Geometry . . . . . . . . . . . . . . . .
HK Sigma Model . . . . . . . . . . . . . . . .
QK N = 4 d = 1 SUGRA . . . . . . . . . . .
BRST charge and gauge fixed action . . . . .
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59
59
61
62
64
A Tour on higher spin gauge theory
A.1 Conformal HS theory in various dimension . . . . . . . . . . . . . . . . . . .
A.2 Integer HS in (A)dS4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
72
74
B Koszul-Tate alghoritm and gauge fixed action
B.1 Koszul-Tate differential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.2 Non linear supersymmetry algebra . . . . . . . . . . . . . . . . . . . . . . . .
B.3 Spinning particle in (A)ds: gauge fixing . . . . . . . . . . . . . . . . . . . . .
77
77
78
79
C Summing with Mr. Pochhammer
C.1 Pochhammer function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.2 Solution by iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
83
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Notations
During this work we will use the following notation:
Notation
Meaning
D
Space time dimension
D = 2n
Even space time dimension
α, β, ...
Flat space time indices
ηαβ ∼ (−, +, ..., +)
Minkowskyan metric
δαβ ∼ (+, +, ..., +)
Euclidean metric
µ, ν, ...
Curved space time indices
gµν
Curved metric
N
Number of one dimensional supercharges
i, j, .. = 1, 2, ..., N
SO(N ) vector indices
I, J, ... = 1, 2, ..., N/2
SO(N ) vector indices in complex base
·
Contraction over space time indices
◦
Contraction over SO(N ) vector indices
{ ,
}pb
Poisson and Dirac bracket
]
Commutators or anticommutators
[ ,
[xy] = 12 (xy − yx)
Antisimmetrization
[m]
m indices totally antisimmetrized
{xy} = 12 (xy + yx)
symmetrization
HS
Higher Spin field
HSC
Generalized Higher Spin Curvature
R
Background curvature
R
HSC
φ
Integer HS field
Chapter 1
SO(N ) spinning particle model
In this Chapter we will describe in details the main object of our study: the spinning
particle model with SO(N ) extended supergravity multiplet on the worldline. This model
has attracted a great deal of attention in the context of studying the dynamics of higher spin
field (HS). As was shown by Gershun and Tkach in [1], and later by Howe et al. in [2], the
mechanics action of spinning particle with a gauged N -extended worldline supersymmetry
and a local SO(N ) invariance, in D = 4 Minkowskian background, describes the dynamics of
free massless HS. In Ref [1] the massive case was completely treated too. For even dimension
it was thoroughly shown in [2] that upon quantization, the physical wave functions are
subject to a relativistic conformally invariant equation for pure spin N2 . Let us postpone the
canonical analysis to the next Chapter; in the following we shall focus our attention on the
symmetries of the model and on the interaction with gravitational background preserving
one dimensional supersymmetry.
These one dimensional supergravity models were thought for a while to be consistently
extended to include coupling with an arbitrary gravitational background only for N = 0, 1, 2.
When N is bigger than 2, it was originally concluded in [2] that the only space compatible
with ”standard ” local worldline supersymmetry transformations rules, is the flat one.
Nevertheless Kuzenko and Yarevskaya have shown in [27] that, with a suitable generalization
of worldline supersymmetry transformation rules, this statement can be generalized, for every
N , to space-time with constant non-zero curvature (i.e. maximally symmetric space):
Rµνρσ = b(gµρ gνσ − gµσ gνρ )
→
R = b d(d − 1)
(1.1)
where b is a parameter related to the cosmological Λ by the relation
Λ=b
(d − 1)(d − 2)
.
2
(1.2)
The case b > 0 corresponds to dS space, b < 0 to AdS.
In the following we intend to discuss and analyze the results obtained by Howe et al. in [2]
and Kuzenko and Yeravskaya in [27]; we will rederive them by using Hamiltonian analysis
that reveals an interesting non linear structure of the SO(N ) extended superalgebra.
The SO(N ) spinning particle action principle is characterized by a N extended supergravity
2
SO(N ) spinning particle model
multiplet on the worldline. The gauge fields (e, χi , aij ) of the N supergravity contain in
particular the einbein e which gauges worldline translation, N gravitinos χi which gauges
the N worldline supersymmetry, and the standard SO(N ) gauge fields. The einbein and
the gravitinos correspond to constraints that eliminate negative norm state and make the
particle model consistent with unitarity [1]. Let us start discussing the ”rigid” model. The
coordinate space is spanned by the real bosonic variable xα (τ ), that represents a map from
the worldline to the D-dimensional Minkowsky space time, and its supersymmetric partners
ψiβ ; α, β = 1, .., D are Lorentz indices, while i, j = 1, ..., N are SO(N ) vector indices.
The dynamics is governed by the simple action principle
Z
1
(1.3)
S = dt (ẋα ẋβ + iψiα ψiβ )ηαβ .
2
This model is invariant with respect rigid time translation, SO(N ) and rigid supersymmetry;
the corresponding Noether charges are
1 α
ẋ ẋα
2
= ẋα ψiα
H =
Qi
Jij = iψ[iα ψj]β ηαβ .
We extend now the model by gauging the symmetry generated by H, Qi and Jij :
Z h
i
i
i
1 −1
β
α
α
β
α β
α β
S = dt e ηαβ (ẋ − iχi ψi )(ẋ − iχi ψi ) + ηαβ ψi ψ̇i − aij ψi ψj ηαβ
2
2
2
(1.4)
where (e, χi , aij ) is the one dimensional SO(N ) extended supergravity multiplet; thus (1.4)
enjoys local symmetries
Supersymmetry
δxβ
δψiβ
δe
δχi
δaij
=
=
=
=
=
ii ψiβ
−i e−1 (ẋβ − iχj ψjβ )
2iχi i
˙i − aij j
0
(1.5)
=
=
=
=
=
(1.6)
Local SO(N )
δxβ
δψiβ
δe
δχi
δaij
0
αij ψjβ
0
αij χj
α̇ij + αik akj − αjk aki
3
Worldline diffeomorphism
δxβ
δψiβ
δe
δχi
δaij
=
=
=
=
=
ξ ẋβ
ξ ψ̇iβ
∂τ (ξe)
∂τ (ξχi )
∂τ (ξaij ).
(1.7)
We introduce now pα , the conjugated momentum to xα ; in first order formalism the action
(1.4) becomes:
Z h
i
1
i
i
(1)
(1.8)
S = dt pα ẋα + ψiα ψ̇iβ ηαβ − e( pα pα ) − iχi (ψiα pα ) − aij (ψiα ψiβ ηαβ ) .
2
2
2
In the previous expression we recognize the constraints
Qi = pα ψiα
1
H =
ηαβ pα pβ
2
Jij = iψ[iα ψj]β ηαβ .
Note that these constraints are first class and the algebra reads:
{Qi , Qj }pb = −2iδij H,
{Jij , Qk }pb = −2iδk[i Qj] ,
[k
l]
{Jij , J kl }pb = −iδ[i Jj]
(1.9)
Let us recall to the reader that in the fermionic sector we have eliminated the second-class
constraint
i
pαψ + ψ α ≈ 0
(1.10)
2
and we use Dirac bracket
(1.11)
{ψiα , ψjβ }pb = −iδij η αβ .
We intend now to extend the analysis introducing a coupling with gravitational background.
Our convention is the following: µ, ν, ... are curved indices, we use the flat ones α, β, .. for
worldline fermions and we introduce the vielbein eµα to raise and lower these indices:
eαµ eβν ηαβ = gµν
ψiα = ψiµ eαµ .
(1.12)
The action proposed in [2], as the natural extension of (1.4) to the curved background case,
reads:
Z
S=
dt
h1
i
gµν (ẋµ − iχi ψiα eµα )(ẋν − iχi ψiα eνα ) + ψiα (ψ̇iα − aij ψjα + ẋµ ωµαβ ψiβ )
2e
2
i
1
+ eψiα ψiβ ψjγ ψjδ Rαβγδ
(1.13)
8
4
SO(N ) spinning particle model
where ωµαβ is the spin connection field, and Rαβγδ is the curvature tensor.
A suitable generalization of (1.5) to the curved target space case, reads1 :
δxµ
δψiα
δe
δχi
δaij
=
=
=
=
=
ii ψiµ
−i e−1 (ẋµ eαµ − iχj ψjα ) + ij ψjβ ψiγ eµβ ωµγ α
2iχi i
˙i − aij j
0.
The Action (1.13) transforms under (1.14) as
Z
i
1
i
δS = dt (i χk − δik l χl )ψkα ψiβ ψjγ ψjδ Rαβγδ + ek ψkµ ψiα ψiβ ψjγ ψjδ ∇µ Rαβγδ .
2
2
8
(1.14)
(1.15)
Howe, Penati, Pernici and Towsend have shown in [2] that this term vanishes only for N = 0, 1
or 2:
• N = 1 → ψ α ψ β ψ γ ψ δ Rαβγδ vanishes because of the cyclic symmetry of Riemann tensor;
• N = 2 → ψkµ ψiα ψiβ ψjγ ψjδ ∇µ Rαβγδ vanishes because of Bianchi identity, while the first
one is zero because in this case the contraction of Riemann tensor with four Grassmann
variables has to be proportional to δik .
Thus for N = 0, 1,2 there are no constraints on the background; if N ≥ 3 equation (1.15)
vanishes only if the space time is flat: Rαβγδ = 0. This suggests that worldline supersymmetry
(1.14) is not compatible with gravitational background.
Howe, Penati, Pernici and Towsend concluded that this result seems to be very ”natural”
and strictly related to the old problem of constructing interaction between massless higher
spin field and gravitational background.
However there is still an opportunity to construct SO(N ) spinning particle models coupled
to gravitational background: modify (1.14). We will discuss it in the next section.
1.1
Kuzenko and Yarevskaya construction
Kuzenko and Yeravskaya in [27] have shown how to introduce a coupling to constant non zero
curvature space. The starting point is the ansatz proposed by Siegel in [51] in which he has
constructed the D-dimensional action principle (1.4) starting from an explicitly conformal
O(D, 2) invariant mechanics action in D space and 2 time dimensions; Siegel has also shown
in [52] that all conformal wave equations, in all dimensions, can be derived in this way.
In the following will not focus our attention on the technique proposed in [27] to introduce a
coupling with maximally symmetric space; more details can be found in the original paper.
1
Note that this generalization in phase space coordinate corresponds with the minimal substitution of
the momentum p with covariant momentum π.
1.2 Hamiltonian analysis
5
We limit ourselves in describing the results, and we will compare them with our Hamiltonian
analysis in the next section.
The action principle proposed by Kuzenko and Yarevskaya reads:
Z h
i
i
1
KY
gµν (ẋµ − iχi ψiα eµα )(ẋν − iχi ψiα eνα ) + ψ̇iα (ψiα − ãij ψjα + ẋµ ωµαβ ψiβ ) , (1.16)
S
= dt
2e
2
where gµν is the (A)dS metric.
This action enjoys local supersymmetry:
δxµ
δψiα
δe
δχi
δãij
=
=
=
=
=
ii ψiµ
−i e−1 (ẋµ eαµ − iχj ψjα ) + ij ψjβ ψiγ eµβ ωµγ α
2iχi i
˙i − aij j
−ib[i ψj]α ẋµ eαµ .
(1.17)
Note now that in maximally symmetric space (i.e. Rαβγδ = b(ηαγ ηβδ − ηαδ ηβγ )) the action
principle (1.16) coincides with (1.13) except for the following redefinition of the SO(N ) gauge
field.
i
(1.18)
ãij = aij − ebψiα ψjα .
2
Thus we can conclude that spinning particle models with SO(N ) extended supergravity on
the worldline, can be coupled to (A)dS background. In order to preserve local worldline
supersymmetry, the SO(N ) gauge field has to transform under supersymmertry. In particular, from (1.17) and (1.18), one finds that (1.13) is invariant with respect to the following
supersymmetry transformation rules:
δxµ
δψiα
δe
δχi
δaij
1.2
=
=
=
=
=
ii ψiµ
−i e−1 (ẋµ eαµ − iχj ψjα ) + ij ψjβ ψiγ eµβ ωµγ α
2iχi i
˙i − aij j
−bχk (k ψiα ψjα + ψkα [i ψj]α ).
(1.19)
Hamiltonian analysis
We would like now to reanalyze, and extend just a little bit, the results discussed above by
using Hamiltonian approach.
First of all let us recall some useful formula [53].
Le X be matter fields both fermonic and bosonic and PX their conjugated momentum. We
consider the action principle, written in first order formalism
Z
(1)
S = dt[ẊPX − H0 ] .
(1.20)
6
SO(N ) spinning particle model
We introduce now a set of gauge fields Gp to enforce into the action a set of constraints
Vp ≈ 0
Z
(1)
S = dt[ẊPX − H0 − Gp V p ] .
(1.21)
These constraints are first class and the algebra reads
t
Vt
{Vp , Vq }pb = Cpq
{H0 , Vp }pb = Kpq Vq
(1.22)
t
and Kpq are, in general, functions of the phase space coordinates.
where Cpq
Action (1.21) is invariant with respect the following transformation:
δ X = {X, p Gp }pb
q
δ Gq = ˙q + Gt p Cpt
− p Kpq
(1.23)
where in an obvious notation q is the gauge parameter associated to the gauge generator
(or equivalentely first class constraint) V q .
We use now relations (1.22) and (1.23) to analyze spinning particle model in (A)dS. Let us
recall that the SO(N ) extended algebra in flat background reads
{Qi , Qj }pb = −2iδij H,
{Jij , Qk }pb = −2iδk[i Qj] ,
[k
l]
{Jij , J kl }pb = −iδ[i Jj] .
In a generic curved space one may attempt to covariantize it. Using tangent space flat indices
instead of curved ones it is straightforward to generalize the SO(N ) generators, and using
the vielbein eµα , one obtains the covariantization of susy generators that thus reads
Qi = ψiα eµα πµ
(1.24)
where πµ is the “covariant” momentum πµ = pµ − 2i ωµαβ ψ α ◦ψ β ,and ψ α ◦ψ β /2i the generators
of the SO(D) Lorentz group. In our notation ◦ means contraction over SO(N ) vector indices.
The Hamiltonian constraint in (A)dS background becomes
1
1
H = π µ π ν gµν − ψ a ◦ ψ b ψ c ◦ ψ d Rabcd .
|2 {z } |8
{z
}
H0
(1.25)
HR
After a straightforward computation one obtains
i
α β
{Qi , Qj }pb = −ig µν πµ πν δij + Rαβγδ ψ{i
ψj} ψ γ ◦ ψ δ
2
1
σ α
{H, Qi }pb = − ∇σ Rαβγδ ψi ψ ◦ ψ β ψ γ ◦ ψ δ .
8
(1.26)
(1.27)
The latter vanishes for locally (Riemannian) symmetric spaces. However, even in such a case
the closure of (1.26) seems not to be guaranteed, at least not in a trivail way. Let us now
restrict ourselves to manifolds equipped with the Riemann tensor
Rαβγδ = f (x)(ηαγ ηβδ − ηαδ ηβγ ).
(1.28)
1.2 Hamiltonian analysis
7
The algebra now reads
{Qi , Qj }pb = −2iH0 δij + if (x)Jik Jjk
= −2iHδij + if (x)Jik Jjk − i
f (x)
Jlk Jlk δij
2
1
{Qi , H}pb = ψiµ ∂µ f (x)Jjk J jk
4
{Qi , Jjk }pb = 2δk[i Qj]
{Jij , Jlm }pb = SO(N ) algebra .
(1.29)
from which one can compute the transformation rules for the SO(N ) gauge fields:
f (x) α
f (x) α
ψj ψkα (χi k + χk i ) −
ψ ψkα (χj k + χk j )
2
2 i
X i µ
+f (x)ψiα ψjα χk k +
ψk ψiα ψjα ∂µ f (x)ek
2
k6=i,j
X iξ µ
= ∂τ (ξaij ) +
ψk ψiα ψjα ∂µ f (x)χk ;
2
k6=i,j
δ susy aij =
δ dif f aij
(1.30)
(1.31)
when f (x) = b one recovers the transformation rules (1.23).
Equation (1.23) and (1.30) are not exactly the same, but they differ for a trivial symmetry
δ new , which vanishes on shell. In particular
i
i
i
i
i
1
δ new aij = ψ[i ψk (χ[k j]] ) = − ψp ψq ( δjp χ[i q] + δiq χ[j p] + δip χ[q j] + δjq χ[p i] ) (1.32)
2
2
2
2
2
2
that looks like δaδSpq Cijpq with Cijpq = −Cpqij ; note that the variation of the action with
respect (1.32) vanishes, in fact, only for symmetry reason.
We will analyze now, for every dimension D, the space-time with non constant curvature we
have used above (1.28):
Rαβγδ = f (x)(ηαγ ηβδ − ηαδ ηβγ )
Rβδ = Rαβαδ = (D − 1)f (x)ηβδ
R = Rαα = D(D − 1)f (x).
(1.33)
(1.34)
(1.35)
We impose the Bianchi identity:
∇σ Rαβγδ + ∇c Rαβδσ + ∇d Rαβσγ = 0.
(1.36)
Let us now contract the previous formula with η αγ η βδ
1
∇α (Rαβ − δβα R) = 0.
2
(1.37)
8
SO(N ) spinning particle model
When we substitute (1.34) (1.35) in (1.37) we obtain
(D − 1)(D − 2)
∂β f (x) = 0.
2
(1.38)
This condition is satisfied in every dimension when f (x) is a constant ((A)dS and Minkowsky)
and just in dimension two for every f (x).
Finally the SO(N ) extended algebra (1.29) in (A)dS background becomes:
{Qi , Qj }pb
= −2iHδij + ibJik Jjk − i 2b Jlk Jlk δij
{Qi , Jjk }pb
=
2δk[i Qj]
=
[k l]
−iδ[i Jj]
{Jij , J kl }pb
(1.39)
In Appendix B we will use Koszul-Tate algorithm to the construct the BRST charge associated to this quadratic superalgebra.
1.3
Comments
In this Chapter we have constructed and analyzed spinning particles with SO(N ) extended
supergravity on the worldline. In particular we have rederived the model coupled to maximally symmetric background, with a canonical analysis at the classical level. This approach
has produced as a byproduct more general coupling in D = 2, and has revealed an interesting
non linear structure of the first class constraints algebra. The action principle we started
with is the one proposed by Kuzenko and Yarevskaya in [27]
Z
1
i
KY
S
= dt[ gµν (ẋµ − iχi ψiα eµα )(ẋν − iχi ψiα eνα ) + ψ̇iα (ψiα − ãij ψjα + ẋµ ωµαβ ψiβ )].
2e
2
Let us redefine now the SO(N ) gauge fields ãij as (this is obviously legal):
ãij = aij + ibk e ψiα ψjα
(1.40)
where k is an arbitrary constant. The first class constraints become
Qi = ψiµ πµ
1 2 k a b c d
π + ψi ψi ψj ψj Rabcd
2
4
= iψ[ia ψj]a
H =
Jij
(1.41)
and the algebra reads:
1
{Qi , Qj }pb = −2iHδij + ib( − k)Jik Jjk + ikbJlk Jlk δij
2
1
{Qi , H}pb = = 2ib( + k)Qj Jij .
2
(1.42)
1.3 Comments
9
During this work we prefer to use k = − 12 because with this choice the gauge fixing procedure
seems to be easier. In Appendix B we will discuss also the gauge fixing of the SO(N ) spinning
particles in (A)dS background and we will construct the gauge fixed action.
We resume now, for commodity, the results obtained in this chapter, regarding spinning
particles models with SO(N ) extended supergravity multiplet on the worldline with (A)dS
background:
• Action:
Z
S=
h1
i
dt
gµν (ẋµ − iχi ψiα eµα )(ẋν − iχi ψiα eνα ) + ψ̇iα (ψiα − aij ψjα + ẋµ ωµαβ ψiβ )
2e
2
i
1
+ eψiα ψiβ ψjγ ψjδ Rαβγδ .
8
• Symmetries:
Supersymmetry
δxµ
δψiα
δe
δχi
δaij
=
=
=
=
=
ii ψiµ
−i e−1 (ẋµ eαµ − iχj ψjα ) + ij ψjβ ψiγ eµβ ωµγ α
2iχi i
˙i − aij j
−bχk (k ψiα ψjα + ψkα [i ψj]α ).
Local SO(N )
δxµ
δψiβ
δe
δχi
δaij
=
=
=
=
=
0
αij ψjβ
0
αij χj
α̇ij + αik akj − αjk aki
Worldline diffeomorphism
δxµ
δψiβ
δe
δχi
δaij
=
=
=
=
=
ξ ẋµ
ξ ψ̇iβ
∂τ (ξe)
∂τ (ξχi )
∂τ (ξaij ).
10
SO(N ) spinning particle model
• Algebra:
b
{Qi , Qj }pb = −2iHδij + ibJik Jjk − i Jlk Jlk δij
2
{Qi , Jjk }pb = 2δk[i Qj]
[k
l]
{Jij , J kl }pb = −iδ[i Jj]
Chapter 2
From spinning particle to HS
In this Chapter we give a detailed analysis of the quantization of the SO(N ) spinning particle models with flat background. We will show that spinning particles with SO(N ) extended
local supersymmetries on the worldline, constructed and analyzed in the previous Chapter,
describe the propagation of particles of spin N/2 in four dimension. A canonical analysis
produces, in fact, the massless Bargmann-Wigner equations [28] as constraints for the physical sector of the Hilbert space, and these equations are known to describe massless free
particles of arbitrary spin.
We proceed analyzing in details two very interesting cases, N = 3 and 4, corresponding to
spin 23 and 2, in order to derive explicitly the well known massless Rarita-Schwinger and
graviton equation of motion in D = 4.
More generally, SO(N ) spinning particles are conformally invariant and describe all possible conformal free particles in arbitrary dimensions, as shown by Siegel in [7]. In the second
part of this Chapter we will analyze carefully the even N case (i.e. integer spin) in every
even dimension D = 2n. In particular we will show that SO(N ) spinning particle models
produce, upon quantization, conformal higher spin curvature (HSC) as physical states of the
Hilbert space; more details about HSC and the geometrical approach to HS fields can be
found in Appendix A.
Finally we will solve the differential Bianchi identity and we will derive the FronsdalLabastida kinetic operator, describing the dynamics of free HS fields with mixed symmetry.
2.1
Bargman Wigner equation of motion
It is well known that the Klein-Gordon equation for a spin 0 particle can be obtained by
quantization of a relativistic particle model. Moreover a canonical analysis of spinning particles with N = 1 worldline supersymmetry produce the massless Dirac equation of motion.
More in general the wave equations for arbitrary spin can be obtained by quantization of
spinning particle models with SO(N ) extended local worldline supersymmetry [2]. Precisely,
in D = 4, SO(N ) spinning particle action describes, upon quantization, a massless particle
of spin N2 .
12
From spinning particle to HS
Let us rewrite for commodity, the one dimensional supergravity model we would like to
analyze from a first quantized point of view:
S
(1)
Z
=
i
i
dt[pα ẋα + ψiα ψ̇iβ ηαβ − eH − iχi Qi − aij Jij ]
2
2
(2.1)
where (e, χi , aij ) is the one dimensional supergravity multiplet we introduce to enforce into
the action the constraints
1
H = p2 ≈ 0
2
(2.2)
Qi = ψiα pα ≈ 0
(2.3)
Jij = iψiα ψjα ≈ 0
.
(2.4)
When one quantizes ”a la Dirac” these models, the constraints H, Qi and Jij have to be
imposed as operatorial conditions on the states on the Hilbert space. In principle these states
are complex, but in what follows we shall suppose that they satisfy a reality condition. This
is possible for arbitrary even D when N is even. The required reality condition takes the
form of identification under worldline time reversal and was discussed in [54] for the spin 12
particle. We shall now show that the equations obtained in this way are relativistic wave
equations for spin N2 . For the sake of concreteness we shall discuss the D = 4 case, extension
to D 6= 4 will be analyzed later.
To perform the quantization let us turn phase space variables into operators
O→O;
(2.5)
Poisson and Dirac brackets are promoted to (anti)commutators (we use ~ = 1)
{·, ·}pb → i[·, ·] .
(2.6)
Phase space coordinates, both bosonic and fermionic, satisfy:
[xα , pβ ] = iδβα
[ψ αi , ψ βj ] = δij η αβ ;
(2.7)
and the SO(N ) extended supersymmetry algebra (1.9) becomes:
[Qi , Qj ] = 2δij H
[J ij , Qk ] = 2δk[i Qj]
[k
l]
[J ij , J kl ] = δ[i J j] .
(2.8)
2.1 Bargman Wigner equation of motion
13
From the previous algebra one can observe that, after having imposed the condition Qi |φi =
J ij |φi = 0 the constraint H|φi = 0 is automatically satisfied.
The wave function is a multispinor φ(x)a1 .....aN , where ai are spinorial indices.
The Grassmann operators ψ αi are the generators of Clifford algebra; they can be represented
in terms of Dirac γ-matrices as follows1
1
... ⊗ 1} .
ψ αi = √ γ∗ ⊗ ..... ⊗ γ∗ ⊗γ α ⊗ |1 ⊗ {z
{z
}
2|
(2.9)
N −i
i−1
In the bosonic sector the x, p commutation relation is realized in the usual way by settings
xα = x α
pα = −i
∂
.
∂xα
We start imposing the constraint (2.3); this implies that, with respect each spinorial index,
φa1 .....aN satisfies a Dirac-type equation:
Qi |φi = 0
⇒
(/
∂ )bai φa1 ...ai ...aN = 0 .
(2.10)
Let us now use the operator Jij . It could be written in gamma matrices basis as
i
α
⊗ γ ⊗γ ⊗ 1 ⊗ ... ⊗ 1 .
Jij = − 1| ⊗ .....
{z ⊗ 1} ⊗γ∗ γ ⊗ γ
| ∗ {z }∗ α | {z }
2
i−1
j−i−1
(2.11)
N −j
The relation Jij |φi = 0 yields
(γ α )ai ãi (γ α )aj ãj φa1 ...ãi ...ãj ...aN = 0.
(2.12)
We contract the previous expression with any element Γ(m) of the four-dimensional Clifford
algebra basis
Γ(m) = (1, γ α , γ αβ , γ ∗ γ α , γ ∗ )
(2.13)
and we obtain a set of relations:
(γ α Γ(m) γα )ãi ãj φa1 ...ãi ...ãj ...aN = 0 .
(2.14)
Note that we use the charge conjugation matrix to lower and riser spinorial indices
(Γ(m) )ab = (Γ(m) )ab̃ Cb̃b .
(2.15)
(γ α Γ(m) γα ) = (−1)n (D − 2m)Γ(m)
(2.16)
We use now the identity
1
We work, now, in four dimension; this construction could be generalized to every even D. Let us
comment that γ ∗ plays a fundamental role to realize the anticommutation relation (2.7), this is one of the
most relevant obstacle in generalizing this construction for odd D where γ ∗ = 1.
14
From spinning particle to HS
and in D = 4 equation (2.14) reduces to
(Γ(m) )ãi ãj φa1 ...ãi ...ãj ...aN = 0
with
Γ(m) 6= γ αβ .
(2.17)
Since in D = 4 γ αβ is symmetric in spinor indices, the previous relation implies that
φa1 ...ãi ...ãj ...aN is a completely symmetric multispinor. This could be easily understood by expanding φ on the Clifford algebra basis; relation (2.17) implies that just terms proportional
to γ αβ survive. This conclude the proof since from (2.17) and (2.10) we obtain explicitly the
Bargman-Wigner e.o.m.
(/
∂ )bai φa1 ...ai ...aN = 0
where
φa1 .....aN
is a completely symmetric multispinor
(2.18)
2.2
Gravitino
In this section we will show that, by solving explicitly the equation (2.17) for N = 3 in D = 4,
one obtains the massless Rarita-Schwinger equation of motion describing the dynamics of a
spin 32 field2 :
(γ αβδ )ac ∂β Ψδ|c ≡ γ αβδ ∂β Ψδ = 0
(2.19)
with
γ αβδ = γ α γ β γ δ − η αβ γ δ − γ βδ γ α + η αδ γ β .
It is very useful to rewrite (2.19) by contracting it with γα and ∂α :
γα γ αβδ ∂β Ψδ = 0
⇒
∂
/ (γ · Ψ) − ∂ · Ψ = 0
(2.20)
∂α γ αβδ ∂β Ψδ = 0
⇒
∂
/∂
/ (γ · Ψ) − ∂ 2 (γ · Ψ) = 0 .
(2.21)
We expand now the wave function on the gamma matrix basis, in this way:
φabc = (Γ(n) )ab χ(n)|c .
As was just discussed in the previous section, Jij |φi = 0, in four dimension implies:
2
Where we don’t write spinor indices a, b, c, .. it means that they are contracted.
(2.22)
2.3 Graviton
15
(1)ai aj φa1 a2 a3 = 0
(2.23)
(γ α )ai aj φa1 a2 a3 = 0.
(2.24)
From (2.22), (2.23) and (2.24) one finds that the wave function can be written as
φabc = (γ αβ )ab χαβ|c
(2.25)
(γ α )ac χαβ|c = 0 .
(2.26)
and χαβ|c has to satisfy the relation
Now we use the supercharges constraint (2.3) and we obtain:
(γ ρ )ãi ai ∂ρ φa1 a2 a3 = 0
i = 1, 2, 3 .
(2.27)
This relation plays a key role; it implies that χαβ|c is a closed two form, so it could be written
as
χαβ|c = ∂α Ψβ|c − ∂β Ψα|c .
(2.28)
We substitute now the previous expression in (2.26) and we obtain:
∂
/ Ψα − ∂α (γ · Ψ) = 0
(2.29)
Contraction of the previous relation with γ α and γ α ∂
/ produces
γ α (/∂ Ψα − ∂α (γ · Ψ)) = 0
⇒
∂ ·Ψ−∂
/ (γ · Ψ) = 0
(2.30)
γ α∂
/ (/∂ Ψα − ∂α (γ · Ψ)) = 0
⇒
∂
/∂
/ (γ · Ψ) − ∂ 2 (γ · Ψ) = 0 .
(2.31)
Note now that (2.30) and (2.31) coincide with (2.20) and (2.21) and this conclude our proof.
2.3
Graviton
In this section we will discuss the case N = 4 (i.e. spin 2) in D = 4. This example reveals to
be very useful for future analysis; we will use a different way, with respect the one we have
discussed in the previous section, to realize the algebra.
In particular we use complex combination of SO(N ) vector indices
(ψ1α , ψ2α , ψ3α , ψ4α )
⇒
(λα1 , λ̄α1 , λα2 , λ̄α2 )
(2.32)
16
From spinning particle to HS
where
λα1 =
ψ1α + iψ2α
2
λ̄α1 =
ψ1α − iψ2α
2
λα2 =
ψ3α + iψ4α
2
λ̄α2 =
ψ3α − iψ4α
.
2
Equations (2.2) (2.3) and (2.4) in this complex base become.
H =
1 2
p
2
α
QI = λαI pα
QI = λ̄I pα
J IJ = iλI λJ
J IJ = iλ̄I λ̄J
JIJ =
i
[λI , λ̄J ].
2
where I, J = 1, 2. In our notation raised indices are complex indices
J I J¯ ≡ J I J
J I¯J¯ ≡ J IJ .
In the following we prefer, sometimes, consider the operators J I J with I 6= J and I = J
separately; to this aim we introduce the notation

when I 6= J
 J˜I J
J
JI =
;
(2.33)

I
JI
when I = J
note that this notation doesn’t imply in J I I , where is not explicitly indicated, a sum over
complex SO(N ) vector indices.
In this new base Grassmann variables satisfy the anticommutation relation
α
[λ̄I , λβJ ] = δIJ η αβ
(2.34)
and the algebra is realized by setting
λαI = λαI
α
λ̄I =
∂
.
∂λαI
(2.35)
Note that in writing JI I we have fixed the following ordering
i α ∂
∂
∂
JII ≡
λI α − α λαI = iλαI α − 2i .
2
∂λI
∂λI
∂λI
States of the full Hilbert space can be described as functions of the coordinates xα and λαI .
We denote, as usual, with xα the eigenvalues of the operator xα , while for the fermionic
variables we use bra coherent states defined by
hλαI |λαI = hλαI |λαI .
2.3 Graviton
17
Any state |φi can then be described by the wave function
φ(x, λI ) = (hx| ⊗ hλ1 | ⊗ hλ2 |)|φi.
(2.36)
Since λI are anticommuting variables, the wave function has a finite expansion
1
α 1 β1
α1 α2
IJ
φ(x, λI ) = A(x) + A(x)Iα1 λαI + A(x)II
α1 α2 λI λI + A(x)α1 β1 λI λJ +
α 1 α 2 α3
α 1 α 2 β1
IIJ
+A(x)III
α1 α2 α3 λI λI λI + A(x)α1 α2 β1 λI λI λJ + ....
(2.37)
We start solving the constraint J I I . In particular we use the relation J I I |φi = 0 to select
only one tensorial structure on the expression (2.37):
J I I φ(x, λI ) = 0
⇒
φ(x, λI ) = A(x)α1 α2 β1 β2 λα1 1 λα1 2 λβ2 1 λβ2 2 .
(2.38)
From the other SO(4) operator we learn that:
J 12 φ = 0
⇒
η α1 β1 A(x)α1 α2 β1 β2 = 0
(2.39)
J 12φ = 0
⇒
A(x)[α1 α2 β1 ]β2 = 0.
(2.40)
It is not hard to show that at this point the constraint J 12 |Ri = 0 is automatically satisfied.
Let us now resume the informations we have obtained above:
• The wave function is proportional to a 4-rank tensor φ(x) v A(x)α1 α2 β1 β2 .
• From (2.38) one can easily reads the symmetry property of A:
A(x)α1 α2 β1 β2 = −A(x)α2 α1 β1 β2 = A(x)β1 β2 α1 α2 = −A(x)α1 α2 β2 β1
• The 4-rank tensor A(x)α1 α2 β1 β2 satisfies the algebraic Bianchi identity (2.40);
• Note also that Aα1 α2 β1 β2 has to be traceless (2.57).
We use now supercharges and we obtain:
QI φ = 0
⇒
∂[δ A(x)α1 α2 ]β1 β2 = 0
A closed
(2.41)
and
QI φ = 0
⇒
∂ δ A(x)δα2 β1 β2 = 0
A co-closed.
(2.42)
We proceed now analyzing carefully relations (2.39)-(2.42).
Note that equation (2.41) implies that A(x) is a closed multiform, equivalently satisfies
differential Bianchi identity with respect the first couple of indices α1 , α2 or the second
couple β1 , β2 . Thus we can use the Poincaré Lemma to introduce a rank-3 tensor Kα1 β1 β2
defined by:
A(x)α1 α2 β1 β2 = ∂α1 Kα2 β1 β2 − ∂α2 Kα1 β1 β2 .
(2.43)
18
From spinning particle to HS
We substitute now (2.43) in (2.40), and we solve it as a function of K; explicitly we obtain:
⇒
∂[α1 K̃α2 β1 ]β2 = 0
K̃α2 β1 β2 = ∂α2 χ̃β1 β2 − ∂β1 χ̃α2 β2
(2.44)
where K̃α2 β1 β2 = K[α2 β1 ]β2 ; this tensor with mixed symmetry is strictly related to the Fierzfield that has been extensively studied by Novello and Neves in [55] as an alternative approach
to describe gravity (known in literature as Teleparallel gravity [56]).
An iterative procedure let us write (2.44) in terms of K and in particular one finds:
2Kα2 β1 β2 = ∂α2 χ̃[β1 β2 ] − ∂β1 χα2 β2 + ∂β2 χα2 β1
(2.45)
where we have defined χαβ = χ̃{αβ} .
We are ready now to write the rank-4 tensor A(x) as a function of a rank-2 tensor χ:
A(x)α1 α2 β1 β2 = ∂β1 ∂[α2 χα1 ]β2 − ∂β2 ∂[α2 χα1 ]β1 .
(2.46)
Note that, at this point, condition (2.42) is automatically satisfied.
We substitute now (2.46) in (2.57), and we obtain the graviton e.o.m.
∂α ∂β χαδ − ∂α ∂δ χαβ − 2χδβ − ∂β ∂δ χ = 0
(2.47)
where χ = η αβ χαβ . Finally it’s not hard to prove that (2.47) enjoys the gauge symmetry:
δχαβ = ∂{α ξβ} .
2.4
(2.48)
From spinning particle to integer higher spin gauge
theory
We use now the idea inherited from the N = 4 case. We consider even N = 2s and we work
in even dimension D = 2n. Let us forget, in the following, about the bold font, as its obvious
that we are now dealing with quantum operators.
We use complex combinations of the SO(N ) vector indices
ψiα
with i = 1, . . . , N = 2s ,
⇒
(λαI , λ̄αI ) with I = 1, .., s
(2.49)
so that
[λαI , λ̄βJ ] = η αβ δI J
with
λ̄βI =
∂
.
∂λβI
(2.50)
(2.51)
Explicitly the SO(N ) generators are:
JIJ ≡ iλαI λJα
∂
JI J ≡ iλαI α − inδIJ
∂λJ
∂
∂
J IJ ≡ i α
∂λI ∂λJα
(2.52)
(2.53)
(2.54)
2.4 From spinning particle to integer higher spin gauge theory
19
and the algebra is
[QI , QJ ] = 2δI J H
[JIJ , QK ] = iδ[J K QI]
[JI J , QK ] = iδK J QI
[J IJ , QK ] = iδK [J QI]
[JI J , QK ] = −iδI K QJ .
The wave function, R(xα , λαI ) = (hx| ⊗ hλI |)|Ri, depends on the coordinate xα and on the
fermionic degrees of freedom λαI ; in analogy with the spin 2 case, it can be Taylor expanded
in terms of λ.
We start imposing the constraint JI I |Ri = 0 and we find that:
JI I |Ri = 0 ⇒ R(xα , λαI ) = Rα1 ..αn |...|β1 ..βn (x) λα1 1 ..λα1 n ...λβs 1 ..λβs n
.
(2.55)
Note that this implies that in the wave function survives only a tensor structure that has
s blocks of n antisymmetric indices. For commodity in the following we prefer to use the
[n]
[n]
shortcut notation R[n]1 ...[n]s λ1 1 · · · λs s .
We proceed now imposing the other constraints on the wave function (2.55).
The constraint J˜I J remove a λσJ from the J th block and add a λσI in the I th one; it
produces the condition
R[n]1 ···[n+1]I ···[n−1]J ...[n]s = 0 .
(2.56)
This is a constraints that guaranties the algebraic Bianchi identity.
We use now J IJ . It removes two Grassmann variables, one from the I th and one from
the J th block; the corresponding vector indices are contracted and the condition J IJ |Ri = 0
yields:
η σγ R[n]1 ,··· ,σ[n−1]I ,··· ,γ[n−1]J ,··· ,[n]s = 0 .
(2.57)
Thus this constraint remove all the traces into R.
One can notes that at this point the condition JIJ |Ri = 0 is automatically satisfied. It
does not arise as a consequence of the algebra, but it is related to the hermiticity properties.
One could view it as a consequence of a duality symmetry which takes the dual in the I th
block of indices by the operation
λI ↔ λ̄I
(2.58)
which can be obtained with a discrete O(N ) transformation and in particular a reflection on
one real coordinate. In particular JIJ |Ri = 0 reduces to a sum of traceless constraints that
20
From spinning particle to HS
are just implemented into the condition J IJ |Ri = 0. Explicitly one has that the operator
JIJ adds two Grassmann variables into the I th and J th block:
[n]
[n]
[n]
R[n]1 ···[n]s λ1 1 ..λI I λσI ..λJ J λIσ ..λs[n]s = 0.
(2.59)
Note now that when vector indices into [n]I and [n]J assume the value σ the previous relation
vanishes identically because of the property of the Grassmann variables. Let us recall that
2n = D, for this reason
[n]
[n]
[n]
R[n]1 ···[n]s λ1 1 ..λI I λcI ..λJ J λIc ..λs[n]s
does not vanish identically only if at least one index into the I th block is contracted with
one index into the J th block; it implies that the constraint JIJ |Ri = 0 produces a sum of
traceless, double traceless and multiple traceless conditions
n
X
ck tr k R[n]1 ···[n]s = 0
(2.60)
k=1
for some coefficients ck .
We are ready now to attack the supercharges constraints. QI adds a Grassmann variable
into the I th block and its vector index is contracted with a partial derivative
[n]1
∂σ R[n]1 ···[n]s λ1
[n]
· · · λI I λσI · · · λs[n]s = 0;
(2.61)
thus QI produces the differential Bianchi identity (i.e. closing condition) as a constraint on
the wave function.
In addition QJ removes a Grassmann variable from the J th block and substitute it with a
partial derivative:
[n]1
∂ σ R[n]1 ··· ,[n−1]I σ,··· ,[n]s λ1
[n−1]I
· · · λI
· · · λs[n]s = 0,
(2.62)
that is the co-closing condition on R, that at this point it is automatically satisfied as a
consequence of the algebra ([J IJ , QK ] = iδK J QI − iδK I QJ ).
Let us recall that [QI , QI ] = 2H; it implies that, at this point, also the constraint H is
automatically satisfied.
Let us summarize and explain further what we have obtained before:
• The constraints JI J |Ri = 0 correspond to the subgroup U (s) ⊂ SO(2s), which is
manifestly realized in the complex basis. The curvature R that solves these U (s)
constraints has “s” symmetric blocks of “n” antisymmetric indices each, and satisfies
the algebraic Bianchi identities . Antisymmetry in each block is manifest.
• Symmetry between blocks can be shown by using finite SO(s) ⊂ U (s) rotations. For
example, consider the rotation that exchanges λI → λJ and λJ → −λI for fixed I and
J. This proves symmetry under exchange of the block relative to the fermions λI with
the one relative to the fermion λJ .
2.4 From spinning particle to integer higher spin gauge theory
21
• Note that the fermionic Fock vacuum |V i ∼ V (x) is not invariant under the subgroup
0
[U (1)]s ⊂ U (s), as the generators JI I for I = 1, .., s transform it by an infinitesimal
0
phase (JI I |V i = −i D2 |V i). It is the vector |Ri in eq. (2.55) that is left invariant.
This is collected in the following table:
JI I |Ri = 0
⇒
|Ri ∼ R[n]1 ···[n]s
J˜I J |Ri = 0
⇒
R[n+1]1 [n−1]2 ...[n]s = 0
Algebraic Bianchi identity
QI |Ri = 0
⇒
∂[β Rα1 ..αn ][n]2 ...[n]s = 0
R closed
J IJ |Ri = 0
⇒
Rσ[n−1]1 ,σ[n−1]2 ...[n]s = 0
R traceless
JIJ |Ri = 0
⇒
QI |Ri = 0
⇒
P
HSC
ck tr k R[n]1 ···[n]s = 0 = 0 Sum of traceless conditions
∂ α1 Rα1 ..αn [n]2 ...[n]s = 0
R co-closed
These coincides with the geometrical equations of de Wit-Freedman [31] but in a different basis [29] [9].
Thus we can conclude that spinning particle models with SO(N ) extended supergravity
on the worldline, produce, upon quantization, conformal HSC as the physical sector of the
Hilbert space (more details could be found in Appendix A).
In Young tableaux language we have:
|Ri ∼ Ra11 ···a1n |...|as1 ···asn ≡
a11
.
.
.
a1n
.
.
.
.
.
. as1
. .
. .
. .
. asn
(2.63)
.
.
.
.
.
. as1
. .
. .
=0
. .
. asn
(2.64)
and
∂[a Ra11 ···a1n ]|...|as1 ···asn ≡
a11
.
.
.
a1n
∂
22
From spinning particle to HS
2.5
HS gauge potential
We would like now to solve the differential Bianchi identity (2.40) and introduce the gauge
potential (i.e. HS field).
We define the operator
qs ≡ Q1 Q2 ..Qs
(2.65)
which satisfies QI qs = qs QI = 0 for any I. In fact, powers of the QI may be non vanishing up
to the s-th power, since then any additional application of any of the QI ’s makes it vanish,
because of the algebra [QI , QI ] = 0 which in particular implies Q2I = 0 at fixed I. Then the
differential Bianchi identity (i.e. QI |Ri = 0) can be solved by using the generalized Poincaré
Lemma 3 [45] [46] as
|Ri = qs |φi .
(2.66)
We focus now our attention on the U (S) constraints; we compute
JI J qs |φi = ([JI J , qs ] + qs JI J )|φi = qs (iδI J + JI J )|φi = 0
(2.67)
which can be solved by requiring that
JI J |φi = −iδI J |φi
(2.68)
α
(2.69)
which says that |φi has the form
|φi ∼ φα1 ..αn−1 |...|α1 ..αn−1 (x) λα1 1 ..λ1 n−1 ...λαs 1 ..λαs n−1
and satisfies the algebraic Bianchi identities. In particular, the tensor φ is symmetric under
block exchange. Thus equation (2.55), the algebraic and differential Bianchi identity are
solved.
Note now that φ and is represented by the Young tableaux:
α11
.
φα1 ..αn−1 |...|α1 ..αn−1 (x) = .
1
αn−1
.
.
.
.
. α1s
. .
. .
s
. αn−1
(2.70)
One may now implement the traceless condition and then all the other constraints are solved
automatically. We force now the generalized curvature to be traceless (in our mind this step
3
This Lemma assures also that the solution we are going to propose is unique.
2.5 HS gauge potential
23
coincides with the computation of the e.o.m.):
J 12 qs |φi = J 12 Q1 Q2 Q3 ...Qs |φi = Q3 ...Qs J 12 Q1 Q2 |φi
| {z }
qextra
i
h
12
12
12
= qextra [J , Q1 ]Q2 + Q1 [J , Q2 ] + Q1 Q2 J |φi
h
i
= qextra − iQ2 Q2 + iQ1 Q1 + Q1 Q2 J 12 |φi
h
i
= qextra − 2iH + i(Q1 Q1 + Q2 Q2 ) + Q1 Q2 J 12 |φi
h
i
1
= qextra − 2iH + iQI QI + QI QJ J IJ |φi
2
= qextra iG|φi
(2.71)
where we have defined the Fronsdal-Labastida operator
i
G = −2H + QI QI + QI QJ J JI
2
(2.72)
which is manifestly U (s) invariant. In fact, one may compute [JI J , G] = 0. A similar
expression holds for J 12 → J IJ so that imposing the traceless condition one obtains (in an
obvious notation)
qextra IJ iG|φi = 0 .
(2.73)
Before describing a solution to this equation let us discuss the gauge symmetries. Using an
arbitrary vector field V µ (x) we may define
V̄ I = V µ λ̄Iµ
(2.74)
and use it to describe the following gauge transformation
δ|φi = QK V̄ K |ξi
(2.75)
which is a gauge symmetry of |Ri = qs |φi. Since [JI J , QK V̄ K ] = 0, one requires that
JI J |ξi = −iδI J |ξi to guarantee that |φi and δ|φi describe tensors with the same Young
tableaux. Having familiarized with these techniques, let us describe the solution to eq.
(2.73). Recalling that the product of s + 1 QI ’s must vanish, one obtains the following
solution
(2.76)
G|φi = QI QJ QK W̄ K W̄ J W̄ I |ρi
which depends on an arbitrary vector field contained in W̄ I = W µ λ̄Iµ and on |ρi which
satisfies JI J |ρi = −iδI J |ρi (so that it belongs to the same space of |φi and |ξi, i.e. it has the
same Young tableaux). These are the equation of motion for HS fields written using the so
called compensator fields described by W̄ K W̄ J W̄ I |ρi (see Appendix A, [29] and references
therein). To study how gauge symmetries act on these equations, one may compute the
gauge variation of G|φi using (2.75)
i
Gδ|φi = QI QJ QK V̄ K J¯JI |ξi .
2
(2.77)
24
From spinning particle to HS
This implies that the compensator field transforms as:
i
δ(W̄ K W̄ J W̄ I |ρi) = V̄ [K J¯JI] |ξi
2
(2.78)
With the choice V µ = W µ and J JI |ξ >= −W J W I |ρ > we can gauge fix the compensator to
zero.
Now the HS potential e.o.m. reads
G|φ >= 0 .
(2.79)
The previous e.o.m. is gauge invariant
i
Gδ|φ >= QI QJ QK J [JI V K] |ξ >
2
(2.80)
only if we force the gauge parameter to be traceless. Thus we have obtained the FronsdalLabastida equation of motion, with the correct traceless condition on the gauge parameter.
We apply now the operator
i
QI + QJ J JI
(2.81)
2
on (A.38) one obtains the relation
1
QJ QK QM J KM J JI |φi = QJ QK QM (−iQP − iQN J N P )W J W K W M + iQM W J W K W P )|ρi
4
(2.82)
which implies that the gauge fixing of the compensator to zero forces the HS spin field to be
double traceless.
This conclude our analysis; we have in fact solved explicitly (2.2) (2.3) and (2.4) for every
even N (i.e. integer spin) in every dimension D = 2n. In particular we have reobtained,
by using our language and notations, the well known Fornsdal-Labastida e.o.m. describing
the dynamics of free higher spin field with mixed symmetry. One advantage one finds by
using the approach described above is the fact that equations are quite compact and do not
depend on the space time dimension.
Chapter 3
One loop quantization and counting
degrees of freedom
In this Chapter we study the one-loop quantization of the spinning particle model with
a SO(N ) extended local supersymmetry on the worldline. We restrict our analysis to flat
space, and we will calculate the path integral on the one-dimensional torus to obtain compact
formulas which give the number of physical degrees of freedom of the spinning particles for
all N in every dimensions. We will obtain also the correct measure on the moduli space
of the supergravity multiplet on the one-dimensional torus; this should be useful also to
construct the quantum field theory effective action and to compute more general quantum
corrections arising when couplings to background fields are introduced.
Our starting point is the Minkowskian action (1.4) or equivalently (1.8). In the following we
prefer to use euclidean conventions, and perform a Wick rotation to euclidean time t → −iτ ,
accompanied by the Wick rotations of the SO(N ) gauge fields aij → iaij , just as done in [4]
for the N = 2 model. We obtain the euclidean action
Z 1
1 α
1 −1 α
α 2
e (ẋ − χi ψi ) + ψi (δij ∂τ − aij )ψjα
(3.1)
S[X, G] =
dτ
2
2
0
where X = (xα , ψiα ) collectively describes the coordinates xα and the extra fermionic degrees
of freedom ψiα of the spinning particle, and G = (e, χi , aij ) represents the set of gauge fields
of the SO(N ) extended worldline supergravity. The euclidean model (3.1) enjoys the gauge
symmetries on the supergravity multiplet given by
δe = ξ˙ + 2χi i
δχi = ˙i − aij j + αij χj
δaij = α̇ij + αim amj + αjm aim
(3.2)
where (ξ, i , αij ) are diffeomorphism, supersymmetry and SO(N ) gauge parameters; in the
previous formula we have also Wick rotated the gauge parameters i → −ii , ξ → −iξ.
In this chapter, in particular, we would like to study the partition function on the onedimensional torus T 1
26
One loop quantization and counting degrees of freedom
Z
Z∼
T1
DXDG
e−S[X,G] .
Vol (Gauge)
(3.3)
Bosonic coordinates have periodic boundary conditions (PBC), while we will take fermions
and gravitini with antiperiodic boundary conditions (ABC) on the one dimensional torus.
This choice will reveals to be very useful soon.
Our strategy reads:
• Gauge fix the local symmetries (3.2). In particular we will choose a gauge which fixes
completely the supergravity multiplet up to some moduli.
• Use the Faddeev-Popov method to extract the volume of the gauge group.
• Calculate the path integral.
3.1
Gauge fixing on the torus
We start now fixing the gauge freedom. In particular using (3.2) we choose the convenient
gauge configuration
(e, χi , aij ) = (β, 0, âij )
(3.4)
where β and âij are constants. Let us now discuss in more details this gauge choice.
EINBEIN:
The gauge choice of the einbein is rather standard, and produces an integral over
the proper time β [16].
GRAVITINI:
The fermions and the gravitinis are taken with antiperiodic boundary conditions (ABC).
This implies that the gravitino can be completely gauged away as there are no zero
modes for the differential operator that relates the gauge parameters i to the gravitinos, see eq. (3.2).
SO(N ) GAUGE FIELDS:
For the SO(N ) gauge fields, the gauge conditions aij = âij (θk ) can be chosen to depend
on a set of constant angles θk , with k = 1, ..., r, where r is the rank of group SO(N ),
taking values on the Cartan torus of the Lie algebra of SO(N ). These angles are the
moduli of the gauge fields on the torus and must be integrated over a fundamental
region. Let us now show explicitly how to reach this gauge configuration.
We parametrize the one-dimensional torus of the worldline by τ ∈ [0, 1] with periodic
boundary conditions on τ .
3.1 Gauge fixing on the torus
27
Let us start with the simpler SO(2) = U (1) group. For this case the finite version of
the gauge transformations (3.2) looks similar to the infinitesimal one
a0 = a + α̇
1
= a + g −1 ġ ,
i
g = eiα ∈ U (1) .
One could try to fix the gauge field to zero by solving
Z
a + α̇ = 0
⇒
α(τ ) = −
(3.5)
τ
dt a(t) ,
(3.6)
0
but this would not be correct as the gauge transformation
g̃(τ ) ≡ e−i
Rτ
0
dt a(t)
(3.7)
is not periodic on the torus, g̃(0) 6= g̃(1). In general this gauge transformation is not
admissible as it modifies the boundary conditions of the fermions. Thus one introduces
the constant
Z
1
dt a(t)
θ=
(3.8)
0
and uses it to construct a periodic gauge transformation connected to the identity
(“small” gauge transformation)
g(τ ) ≡ e−i
Rτ
0
dt a(t) iθτ
e
.
(3.9)
This transformation brings the gauge field to a constant value on the torus
a0 (τ ) = θ .
(3.10)
Now “large” gauge transformations eiα(τ ) with α(τ ) = 2πnτ are periodic and allow to
identify
θ ∼ θ + 2πn ,
n integer .
(3.11)
Therefore θ is an angle, and one can take θ ∈ [0, 2π] as the fundamental region of the
moduli space for the SO(2) gauge fields on the one-dimensional torus.
The general case of SO(N ) can be treated similarly, using path ordering prescriptions to take into account the non-commutative character of the group. Finite gauge
transformations can be written as
1
a0 = g −1 ag + g −1 ġ ,
i
g = eiα ,
α ∈ Lie(SO(N )) .
(3.12)
One can define the gauge transformation
g̃(τ ) = Pe−i
Rτ
0
dt a(t)
(3.13)
where “P” stands for path ordering. This path ordered expression solves the equation
∂τ g̃(τ ) = −ia(τ )g̃(τ )
(3.14)
28
One loop quantization and counting degrees of freedom
and could be used to set a0 to zero, but it is not periodic on the torus, g̃(0) 6= g̃(1),
and thus is not admissible. Therefore one identifies the Lie algebra valued constant A
by
R1
e−iA = Pe−i 0 dt a(t)
(3.15)
so that the gauge transformation given by
g(τ ) ≡ Pe−i
Rτ
0
dt a(t) iAτ
e
(3.16)
is periodic and brings the gauge potential equal to a constant
a0 (τ ) = A .
(3.17)
Since the constant A is Lie algebra valued, it is given in the vector representation
by an antisymmetric N × N matrix, which can always be skew diagonalized by an
orthogonal transformation. One can recognize that the parameters θi contained in
the latter equations are angles, since one can use “large” U (1) gauge transformation
contained in SO(N ) to identify
θi ∼ θi + 2πni ,
ni integer .
(3.18)
The range of these angles can be taken as θi ∈ [0, 2π] for i = 1, . . . , r, with r the rank
of the group. Further identifications restricting the range to a fundamental region are
discussed in the next sections.
3.2
Computing the partition function
We are ready now to write the gauge fixed partition function
Z
Z
1 ∞ dβ
dD x
Z = −
D
2 0 β
(2πβ) 2
Y
D2 −1
r Z 2π
dθk
×
K
Det (∂τ − âvec )ABC
Det0 (∂τ − âadj )P BC
| {zN }
|
{z
}
2π
0
{z
}
(1)
(4)
}|
| k=1 {z
(2)
(3.19)
(3)
The previous formula contains the well-known proper time integral with the appropriate
measure for one-loop amplitudes, and the spacetime volume integral with the standard free
D
particle measure ((2πβ)− 2 ). In addition we have:
(1) KN is a normalization factor that implements the reduction to a fundamental region of
moduli space and will be discussed in the next section separately for even and odd N .
(2) This contribution contains the integrals over the SO(N ) moduli θk and the determinants
of the ghosts and of the remaining fermion fields.
(3) This is the determinants of the susy ghosts and of the Majorana fermions ψiα which all
have antiperiodic boundary conditions (ABC) and transform in the vector representation of SO(N ).
3.2 Computing the partition function
29
(4) This determinant is due to the ghosts for the SO(N ) gauge symmetry. They transform
in the adjoint representation and have periodic boundary conditions (PBC), so they
have zero modes (corresponding to the moduli directions) which are excluded from the
determinant (this is indicated by the prime on Det0 ).
The whole second line (1) + (2) + (3) + (4) computes the number of physical degrees of
freedom, normalized to one for a real scalar field,
Dof (D, N ) = KN
Y
r Z
k=1
2π
0
dθk
2π
D2 −1
Det (∂τ − âvec )ABC
Det0 (∂τ − âadj )P BC
(3.20)
In fact, for N = 0 there are neither gravitinos nor gauge fields, K0 = 1, and all other terms
in the formula are absent [19], so that
Dof (D, 0) = 1
(3.21)
as it should, since the N = 0 model describes a real scalar field in target spacetime. We now
present separate discussions for even N and odd N , as typical for the orthogonal groups,
and explicitate further the previous general formula.
3.2.1
Even case: N = 2r
To get a flavor of the general formula let us briefly review the N = 2 case treated in [4]. We
have a SO(2) = U (1) gauge field aij which can be gauge fixed to the constant value
0 θ
âij =
(3.22)
−θ 0
where θ is an angle that corresponds to the SO(2) modulus. A fundamental region of
gauge inequivalent configurations is given by θ ∈ [0, 2π] with identified boundary values: it
corresponds to a one-dimensional torus. The factor K2 = 1 because there are no further
identifications on moduli space, and the formula reads
Z
Dof (D, 2) =
0
2π
dθ
2π
D2 −1
Det (∂τ − âvec )ABC
|
{z
}
(2 cos θ2 )2
(
=
(D−2)!
[( D
−1)!]2
2
even D
0
odd D
.
Det0 (∂τ )P BC
|
{z
}
1
(3.23)
This formula correctly reproduces the number of physical degrees of freedoms of a gauge
( D2 − 1)–form in even dimensions D. Instead, for odd D, the above integral vanishes and one
has no degrees of freedom left. This may be interpreted as due to the anomalous behavior
of an odd number of Majorana fermions under large gauge transformations [57]. In this
formula the first determinant is due to the D Majorana fermions, responsible for a power D2
30
One loop quantization and counting degrees of freedom
of the first determinant, and to the bosonic susy ghosts, i.e. the Faddeev–Popov determinant
for local susy, responsible for the power −1 of the first determinant. This determinant is
more easily computed using the U (1) basis which diagonalizes the gauge field in (3.22). The
second determinant is due the SO(2) ghosts which of course do not couple to the gauge
field in the abelian case. A zero mode is present since these ghosts have periodic boundary
conditions and is excluded from the determinant. This last determinant does not contribute
to the SO(2) modular measure.
In the general case, the rank of SO(N ) is r = N2 for even N , and by constant gauge
transformations one can always put a constant field aij in a skew diagonal form


0 θ1 0
0 . 0
0
 −θ1 0
0
0 . 0
0 


 0
0
0 θ2 . 0
0 


 .
0
0
−θ
0
.
0
0
âij = 
(3.24)
2


 .

.
.
.
.
.
.


 0
0
0
0 . 0 θr 
0
0
0
0 . −θr 0
The θk are angles since large gauge transformations can be used to identify θk ∼ θk + 2πn
with integer n. The determinants are easily computed pairing up coordinates into complex
variables that diagonalize the matrix (B.15). Then
Det (∂τ − âvec ) =
r
Y
Det (∂τ + iθr ) Det (∂τ − iθr )
(3.25)
k=1
and thus
D2 −1
Det (∂τ − âvec )ABC
=
r Y
k=1
2 cos
θk D−2
.
2
(3.26)
Similarly
0
Det (∂τ − âadj )P BC =
r
Y
Det0 (∂τ )
k=1
×
Y
×
Y
=
Y
Det (∂τ + i(θk + θl )) Det (∂τ − i(θk + θl ))
k<l
Det (∂τ + i(θk − θl )) Det (∂τ − i(θk − θl ))
k<l
k<l
2 sin
θ k + θ l 2 θk − θl 2
2 sin
.
2
2
Thus, with the normalization factor KN = 2r2r! one obtains the final formula
r Z
2 Y 2π dθk θk D−2
Dof (D, N ) = r
2 cos
2 r! k=1 0 2π
2
Y
θk + θ l 2 θ k − θl 2
×
2 sin
2 sin
.
2
2
k<l
(3.27)
(3.28)
3.2 Computing the partition function
31
The normalization KN = 2r2r! can be understood as follows. A factor r!1 is due to the fact that
with a SO(N ) constant gauge transformation one can permute the angles θk and there are r
angles in total. The remaining factor 22r can be understood as follows. One could change any
angle θk to −θk if parity would be allowed (i.e. reflections of a single coordinate) and this
would give the factor 21r . Thus we introduce parity transformations, which is an invariance of
(3.28), by enlarging the gauge group by a Z2 factor and obtain the group O(N ). This justifies
the identification of θk with −θk and explains the remaining factor 2; equivalently, within
SO(N ) gauge transformations one can only change signs to pairs of angles simultaneously.
It is perhaps more convenient to use some trigonometric identities and write the number of
degrees of freedom as
r Z
θk D−2
2 Y 2π dθk 2 cos
Dof (D, N ) = r
2 r! k=1 0 2π
2
2
Y θ k 2 θ l 2
×
2 cos
− 2 cos
.
2
2
k<l
3.2.2
(3.29)
Odd case: N = 2r + 1
The case of odd N describes a fermionic system in target space. In fact, the simplest example
is for N = 1, which gives a spin 1/2 fermion. It has been treated in [20] on a general curved
background, but there are no worldline gauge fields in this case. For odd N > 1 the rank of
the gauge group is r = N 2−1 and the gauge field in the vector representation aij can be gauge
fixed to a constant matrix of the form






âij = 





0 θ1 0
0
−θ1 0
0
0
0
0
0 θ2
0
0 −θ2 0
.
.
.
.
0
0
0
0
0
0
0
0
0
0
0
0
. 0
0
. 0
0
. 0
0
. 0
0
.
.
.
. 0 θr
. −θr 0
. 0
0
0
0
0
0
.
0
0
0






 .





(3.30)
Then, in a way somewhat similar to the even case, one gets
Det (∂τ − âvec ) = Det (∂τ )
r
Y
Det (∂τ + iθk ) Det (∂τ − iθk )
(3.31)
k=1
and thus
D2 −1
Det (∂τ − âvec )ABC
=2
D
−1
2
r Y
k=1
2 cos
θk D−2
.
2
(3.32)
32
One loop quantization and counting degrees of freedom
Similarly for the determinant in the adjoint representation
0
Det (∂τ − âadj )P BC =
r
Y
Det0 (∂τ ) Det (∂τ + iθk ) Det (∂τ − iθk )
k=1
×
Y
Det (∂τ + i(θk + θl )) Det (∂τ − i(θk + θl ))
k<l
×
Y
Det (∂τ + i(θk − θl )) Det (∂τ − i(θk − θl ))
(3.33)
k<l
which gives
0
Det (∂τ − âadj )P BC =
r Y
k=1
×
Y
k<l
θ k 2
2 sin
2
θk + θl 2 θk − θl 2
2 sin
2 sin
.
2
2
(3.34)
Thus, with a factor
KN =
1
2r r!
(3.35)
one gets the formula
D
r Z
2 2 −1 Y 2π dθk θk D−2 θk 2
Dof (D, N ) =
2 cos
2 sin
2r r! k=1 0 2π
2
2
Y
θk + θl 2
θk − θl 2
2 sin
×
2 sin
.
2
2
k<l
(3.36)
In the expression for KN the factor 2 that appeared in the even case is now not included,
since in the gauge (3.30) one can always reflect the last coordinate to obtain a SO(N )
transformation that changes θk into −θk .
For explicit computations it is perhaps more convenient to write the number of degrees
of freedom as
D
r Z
θk D−2 θk 2
2 2 −1 Y 2π dθk 2
cos
2
sin
Dof (D, N ) =
2r r! k=1 0 2π
2
2
2
Y θk 2 θl 2
×
2 cos
− 2 cos
.
2
2
k<l
Let us resume, for commodity, the results we have obtained in this section
(3.37)
3.3 Orthogonal Polynomials method
r
2 Y
Dof (D, 2r) = r
2 r! k=1
33
Z
2π
dθk θk D−2
2 cos
2π
2
0
2
Y
θl 2 θ k 2
×
2 cos
− 2 cos
2
2
1≤k<l≤r
D
r Z
2 2 −1 Y 2π dθk θk D−2 θk 2
Dof (D, 2r + 1) = r
2 cos
2 sin
2 r! k=1 0 2π
2
2
2
Y
θl 2 θ k 2
×
2 cos
− 2 cos
2
2
1≤k<l≤r
(3.38)
(3.39)
with N = 2r and N = 2r + 1, respectively. It is obvious that Dof (D, N ) vanishes for an
odd number of dimensions
Dof (2d + 1, N ) = 0,
∀N > 1
(3.40)
as in such case the integrands are odd under the Z2 symmetry 2θ → π − 2θ . Only for N = 0, 1
these models have a non-vanishing number of degrees of freedom propagating in an odddimensional spacetime, as in such cases there are no constraints coming from the vector
gauge fields. Also for N = 2 these models can have degrees of freedom propagating in odddimensional target spaces, provided a suitable Chern-Simons term is added to the worldline
action [2]. However, Chern-Simons couplings are not possible for N > 2.
To compute (3.38) and (3.39) for an even-dimensional target space and for every N we
need to introduce a strong mathematical technique. In the next sections we will describe the
orthogonal polynomials method and we will use it to solve explicitly the integral (3.38) and
(3.39).
3.3
Orthogonal Polynomials method
Let us now briefly review some properties of the Van der Monde determinant and the orthogonal polynomials method. Further details and applications of the method can be found
in Mehta’s book on random matrices [58]. The Van der Monde determinant is defined by
x1 0 · · · xr 0 x1 1 · · · xr 1 Y
(3.41)
∆(xi ) =
(xl − xk ) = :
:
·
·
1≤k<l≤r
r−1
x1
· · · xr r−1 where the second identity can be easily proved by induction; one can observe that:
34
One loop quantization and counting degrees of freedom
1. the determinant on the right hand side vanishes if xr = xi , i = 1, . . . , r − 1
2. the coefficient of xr r−1 is the determinant of order r − 1.
Furthermore, using basic theorems of linear algebra the Van der Monde determinant can be
written as
p0 (x1 ) · · · p0 (xr ) p1 (x1 ) · · · p1 (xr ) ∆(xi ) = (3.42)
:
:
·
·
pr−1 (x1 ) · · · pr−1 (xr ) where pk (x) is an arbitrary, order−k polynomial in the variable x, with the only constraint
of being monic, that is pk (x) = xk + ak−1 xk−1 + · · · .
Interesting properties are associated with the square of the Van der Monde determinant,
which can be written as



p0 (x1 ) · · · pr−1 (x1 )
p0 (x1 ) · · · p0 (xr )
 p0 (x2 ) · · · pr−1 (x2 )   p1 (x1 ) · · · p1 (xr ) 



∆2 (xi ) = det 


:
:
:
:



·
·
·
·
p0 (xr ) · · · pr−1 (xr )
pr−1 (x1 ) · · · pr−1 (xr )
(3.43)
= det K(xi , xj )
where the kernel matrix K reads as
K(xi , xj ) =
r−1
X
pk (xi )pk (xj ) .
(3.44)
k=0
The above polynomials can be chosen to be orthogonal with respect to a certain positive
weight w(x) in a domain D
Z
(3.45)
dx w(x)pn (x)pm (x) = hn δn,m .
D
However, monic polynomials cannot in general be chosen to be orthonormal. Of course, one
can relate them to a set of orthonormal polynomials p̃n (x)
p
(3.46)
pn (x) = hn p̃n (x)
and the square of the Van der Monde determinant can be written in terms of a rescaled
kernel
2
∆ (xi ) =
r−1
Y
hk det K̃(xi , xj )
(3.47)
k=0
with an obvious definition of the latter kernel in terms of the orthonormal polynomials.
Thanks to the orthonormality condition, the rescaled kernel can be shown to satisfy the
property
Z
dz w(z)K̃(x, z)K̃(z, y) = K̃(x, y) ,
(3.48)
D
3.4 Counting degrees of freedom
35
that can be applied to prove (once again by induction) the following identity
Z
Z
Z
dxr w(xr )
dxr−1 w(xr−1 ) · · ·
dxh+1 w(xh+1 ) det K̃(xi , xj )
D
D
D
= (r − h)! det K̃
(h)
(xi , xj )
where K̃ (h) (xi , xj ) is the order−h minor obtained by removing from the kernel the last r − h
rows and columns. In particular
Z
Z
dxr w(xr ) · · ·
dx1 w(x1 ) det K̃(xi , xj )
D
D Z
dx1 w(x1 )K̃(x1 , x1 ) = r!
(3.49)
= (r − 1)!
D
and
1
r!
3.4
Z
Z
dxr w(xr ) · · ·
D
2
dx1 w(x1 ) ∆ (xi ) =
D
r−1
Y
hk .
(3.50)
k=0
Counting degrees of freedom
We are ready now to solve explicitly (3.38) and (3.39). We first observe that the integrands
are even under the aforementioned Z2 symmetry, and thus we can restrict the range of
integration
r Z
2 Y π dθk θk D−2
Dof (D, 2r) =
2 cos
r! k=1 0 2π
2
2
Y
θl 2 θ k 2
2 cos
(3.51)
×
− 2 cos
,
2
2
1≤k<l≤r
D
r Z
θk D−2 θk 2
2 2 −1 Y π dθk 2 cos
2 sin
Dof (D, 2r + 1) =
r! k=1 0 2π
2
2
2
Y
θl 2 θ k 2
2 cos
×
− 2 cos
.
2
2
1≤k<l≤r
Now, upon performing the transformations xk = sin2
θk
,
2
(3.52)
we get
22(d−1)r+(r−1)(2r−1)
π r r!
Z
r
1
Y
Y
−1/2
×
dxk xk (1 − xk )d−3/2 (xl − xk )2 ,
Dof (2d, 2r) =
k=1
0
(3.53)
k<l
2(d−1)+r(2r+2d−3)
π r r!
Z
r
Y 1
Y
1/2
×
dxk xk (1 − xk )d−3/2 (xl − xk )2 .
Dof (2d, 2r + 1) =
k=1
0
k<l
(3.54)
36
One loop quantization and counting degrees of freedom
We have made explicit in the integrands the square of the Van der Monde determinant: it is
then possible to use the orthogonal polynomials method to perform the multiple integrals.
Note in fact that in (3.53) and (3.54) the prefactors of the Van der Monde determinant have
(p,q)
the correct form to be weights w(p,q) (x) = xq−1 (1 − x)p−q for the Jacobi polynomials Gk
with (p, q) = (d − 1, 1/2) and (p, q) = (d, 3/2), respectively. The integration domain is also
the correct one to set the orthogonality conditions
Z
1
(3.55)
dx w(x)Gk (x)Gl (x) = hk (p, q) δkl
0
with the normalizations given by
hk (p, q) =
k! Γ(k + q)Γ(k + p)Γ(k + p − q + 1)
,
(2k + p)Γ2 (2k + p)
(3.56)
see [59] for details about the known orthogonal polynomials. Since the Jacobi polynomials
(p,q)
Gk are all monic, the even−N formula reduces to
r−1
22(d−1)r+(r−1)(2r−1) Y
hk (d − 1, 1/2)
Dof (2d, 2r) =
πr
k=0
Γ(2d − 1) 1
2(r−1)(2r+2d−3)
Γ2 (d) π r−1
=
r−1
Y
hk (d − 1, 1/2)
(3.57)
k=1
where in the second identity we have factored out the normalization of the zero-th order
polynomial. It is straightforward algebra to get rid of all the irrational terms and reach the
final expression
Dof (2d, 2r) = 2r−1
r−1
(2d − 2)! Y k (2k − 1)! (2k + 2d − 3)!
.
[(d − 1)!]2 k=1 (2k + d − 2)! (2k + d − 1)!
(3.58)
For odd N we have instead
Dof (2d, 2r + 1) =
=
r−1
2(d−1)+r(2r+2d−3) Y
hk (d, 3/2)
πr
k=0
r−1
2(2−d)+r(2r+2d−3) Γ(2d − 1) 1 Y
hk (d, 3/2)
d
Γ2 (d) π r−1 k=1
(3.59)
which can be reduced to
r−1
2d−2+r (2d − 2)! Y (k + d − 1) (2k + 1)! (2k + 2d − 3)!
Dof (2d, 2r + 1) =
d [(d − 1)!]2 k=1
(2k + d − 1)! (2k + d)!
(3.60)
3.5 Rectangular SO(D − 2) Young tableaux
37
From these final expressions we can single out a few interesting special cases
(i) Dof (2, N ) = 1,
∀N
(3.61)
(ii) Dof (4, N ) = 2,
∀N
(3.62)
(iii) Dof (2d, 2) =
(2d − 2)!
[(d − 1)!]2
(3.63)
(iv) Dof (2d, 3) =
2d−1 (2d − 2)!
d [(d − 1)!]2
(3.64)
1
(v) Dof (2d, 4) =
(2d − 1)(2d + 2)
(2d)!
[d!]2
2
(3.65)
2
(2d + 2)!
[(d + 1)!]2
2
12
(2d + 2)!
(vii) Dof (2d, 6) =
.
(2d − 1)(2d + 6)(2d + 1)2 (2d + 4)2 [(d + 1)!]2
3 · 2d−2
(vi) Dof (2d, 5) =
(2d − 1)(2d + 4)(2d + 1)2
(3.66)
(3.67)
In particular, in D = 4 one recognizes the two polarizations of massless particles of spin N/2.
The cases of N = 3 and N = 4 correspond to free gravitino and graviton, respectively, but
this is true only in D = 4. In other dimensions one has a different field content compatible
with conformal invariance.
Let us resume our results:
r−1
Dof (2d, 2r) = 2
r−1
(2d − 2)! Y k (2k − 1)! (2k + 2d − 3)!
[(d − 1)!]2 k=1 (2k + d − 2)! (2k + d − 1)!
(3.68)
and
r−1
2d−2+r (2d − 2)! Y (k + d − 1) (2k + 1)! (2k + 2d − 3)!
Dof (2d, 2r + 1) =
d [(d − 1)!]2 k=1
(2k + d − 1)! (2k + d)!
(3.69)
3.5
Rectangular SO(D − 2) Young tableaux
In the following we would like to prove that (3.68) coincide with the dimensions of the
rectangular SO(D − 2) Young tableaux with (D − 2)/2 rows and N/2 columns 1 .
1
Equivalently one can show that (3.69) coincide with the dimension of a spinorial rectangular SO(D − 2)
Young tableaux with (D − 2)/2 rows and (N − 1)/2 columns; extension to the odd N case is straightforward
and will not be discussed in this section.
38
One loop quantization and counting degrees of freedom
Let us start fixing the notation; our convention is to label a rectangular young tableaux with
n row and λ column with
[λ, λ, ..., λ] ≡ [λq ] ≡
| {z }
(3.70)
q times
We indicate with d[λq ] (SO(D − 2)) the dimension of the representation [λq ] of the SO(D − 2)
Lie group. In order to explicitly compute d[rn−1 ] (SO(D − 2)) we use the hook rule algorithm
(more details about this technique could be found in [60]). In particular after a straightforward and boring computation one can note some recursive relation that let us to write:
d[rn−1 ] (SO(D − 2)) =
Yt
Yh
(3.71)
where
Yt =
n−1
Y
i=1
(r + i − 2)!(2r + 2i − 2)!
(2i − 2)!(2r + i − 2)!
(3.72)
and
Yh =
n−1
Y
i=1
(r + i − 1)!
(i − 1)!
(3.73)
We would like now to show that (3.71) reproduce (3.68) for every r and n; to this aim we
use a very simple trick. First of all it’s easy to verify that (3.71)=(3.68) for the first few
simple case.
Now we fix r and if we calculate how d[rn ] (SO(2n − 2)) and Dof (2n, 2r) scale when one
increases the number of the dimensions; in particular we obtain:
d[rn−1 ] SO(2n − 2)
Dof (2n, 2r)
2(n − 1)(2r + 2n − 3)
=
=
d[rn ] SO(2n)
Dof (2n + 2, 2r)
(2r + n − 2)!(2n − 2)!
(3.74)
We reverse now the point of view. We fix D and we calculate the scale factor when one r
increases:
d[(r+1)n−1 ] SO(2n)
Dof (2n, 2r + 2)
2r(2r − 1)!(2r + 2n − 3)
=
=
d[rn ] SO(2n)
Dof (2n, 2r)
(2r + n − 2)!(2r + n − 1)!
(3.75)
This conclude our proof since d[rn ] (SO(2n − 2)) = Dof (2n, 2r) fot n = 1 and r = 1 and they
scale in the same way with respect r and n.
3.6 The case of N = 4 and the Pashnev–Sorokin model
3.6
39
The case of N = 4 and the Pashnev–Sorokin model
We would like now to analyze the so called N = 4 Pashnev and Sorokin model (PS) constructed in [22]; possible couplings to curved backgrounds have been studied in [61]. This
model is constructed writing the R symmetry group as SO(4) = SU (2)local × SU (2)global .
In the analysis of Pashnev and Sorokin the model corresponds to a conformal gravitational
multiplet, and it was left undecided if the field content in D = 4 is that of a graviton plus
three scalars (five degrees of freedom) or that of a graviton plus two scalars (four degrees of
freedom). Thus, we apply the techniques discussed in the previous chapter, to compute the
number of physical degrees of freedom to clarify the field content of the PS model.
In order to count the physical degrees of freedom let us consider the change of variables
(3.76)
ψ i = ψ aȧ σ i aȧ
where
σ̄ i
ȧa
= (−i1, σ)ȧa ,
σi
aȧ
The transformation (3.76) can be inverted as
= (i1, σ)aȧ = −ab ȧḃ σ̄ i
ḃb
.
(3.77)
2
1
ψ aȧ = ψ i (σ̄i )ȧa .
2
(3.78)
The reality condition on ψ i , along with the expressions (3.77), allows to write it also in the
form
ψ i = ψ̄aȧ (σ̄i )ȧa
(3.79)
with
ψ̄aȧ = −ab ȧḃ ψ bḃ .
(3.80)
Thus, the fermion part of the lagrangian can be written as
1 i
ψ (δij ∂τ − aij )ψ j = ψ̄aȧ δ a b δ ȧ ḃ ∂τ − Aa b ȧ ḃ ψ bḃ
2
(3.81)
ȧa j 1
Aa b ȧ ḃ = aij σ̄ i
σ bḃ
2
(3.82)
1
(σi )aȧ (σ̄j )ḃb Aa b ȧ ḃ .
2
(3.83)
where
and
aij =
The SU (2) × SU (2) gauge invariance of the action is now manifest. To gauge only a SU (2)
subgroup one may choose
Aa b ȧ ḃ = δ a b B ȧ ḃ
2
⇒
1
aij = tr (σi B σ̄j )
2
Here we make use of the well-known properties
ḃb
2δ ij δ ȧ ḃ , σ i aȧ (σ̄i ) = 2δa b δȧ ḃ .
σ i σ̄ j + σ j σ̄ i
b
a
= 2δ ij δa b ,
(3.84)
σ̄ i σ j + σ̄ j σ i
ȧ
ḃ
=
40
One loop quantization and counting degrees of freedom
and gauge fix B to
i
B ȧ ḃ = 2θ ( σ 3 )ȧ ḃ = iθ (σ 3 )ȧ ḃ
2
(3.85)
which gives

0
 0
aij = θ 
 0
1
so that
Z
1
Dψ exp −
2
Z
ψαi (∂τ δij
−
aij )ψαj

0 0 −1
0 −1 0 

1 0
0 
0 0
0
(3.86)
= DetD (∂τ + iθ)ABC DetD (∂τ − iθ)ABC
=
θ
2 cos
2
2D
.
(3.87)
The Faddeev-Popov determinant associated to the gauge-fixing of the SU (2) gauge group
reads
Det (∂τ 1adj − Badj )P BC = (2 sin θ)2
since eq. (3.85) in the adjoint representation becomes


0 −1 0
Badj = 2θ  1 0 0  .
0 0 0
(3.88)
(3.89)
Finally, the Faddeev-Popov determinant associated to gauge-fixing the local supersymmetry
reads
−4
θ
−1
(3.90)
Det (∂τ δij − aij )ABC = 2 cos
.
2
Assembling all determinants one gets (3.91), where the factor 1/2 is due to the parity transformation θ → −θ.
We are ready now to write the counting Dof formula that reads:
Z
2
θ 2(D−2) 1 2π dθ 2 cos
2 sin θ .
(3.91)
Dof (D, PS) =
2 0 2π
2
This can be cast in a form similar to those obtained in the previous section in order to
extract the Van der Monde determinant
Z
22D 1
dx (1 − x)D−3/2 x1/2 .
(3.92)
Dof (D, PS) =
2π 0
Expilcitly computation gives
(2D − 3)!!
Dof (D, PS) = 2D−1
(3.93)
D!
producing Dof (D, PS) = (1, 2, 5, 14, 42, 132, 429, . . . ) for D = (2, 3, 4, 5, 6, 7, 8, . . . ). Thus
in D = 4 one gets 5 degrees of freedom, which must correspond to a graviton plus three
scalars. Notice that the Pashnev–Sorokin model contains physical degrees of freedom also
in spacetimes of odd dimensions.
Chapter 4
Higher spin generalized curvature in
(A)dS2n
In Chapter 1 we have studied spinning particle models with SO(N ) supergravity multiplet
on the worldline, coupled with gravitational background; we have shown explicitly that only
maximally symmetric spaces (i.e. (A)dS space) preserve local worldline supersymmetry; we
would like now to analyze the physical content of this model. The flat case was extensively
analyzed in Chapter 2, where, in particular, we have shown that for every even dimension
D = 2n and for every integer spin (even N ) a canonical analysis produces conformal HSC
as physical sector of the Hilbert space; we have then used the generalized Poincaré Lemma
to solve the differential Bianchi identity and introduce the HS gauge potential.
In the following we intend to quantize á la Dirac SO(N ) spinning particle model, with
N extended supergravity multiplet on the worldline, coupled with (A)dS background and
analyze carefully higher spin curvature in maximally symmetric background.
Let us recall, for commodity, the non linear sigma model we are going to study
Z
i
i
(1)
S = dτ [pµ ẋµ + ψiα ψ̇iβ ηαβ − eH − iχi Qi − aij Jij ] .
2
2
(4.1)
In the previous action principle H, Qi and Jij are the constraints we have to impose as
operatorial conditions on the Hilbert space states
1
1
H ≡ π 2 − ψiα ψiβ ψjγ ψjδ Rαβγδ ≈ 0
2
|8
{z
}
(4.2)
Qi ≡ ψiµ pµ ≈ 0
(4.3)
∼J 2
42
Higher spin generalized curvature in (A)dS2n
Jij ≡ ψiα ψjα ≈ 0
(4.4)
where µ, ν, ... are curved indices, α, β, ... flat ones, Rαβγδ = b(ηαγ ηβδ − ηαδ ηβγ ) is the (A)dS
curvature and πµ is the covariant momentum. It’s very important to note that in maximally
symmetric space the term proportional to the curvature in the r.h.s. of (4.2) it’s proportional
to the product of two SO(N ) generators.
4.1
(A)dS quantum Algebra
First of all we need to discuss the quantization of the supersymmetry algebra (1.39) described
in Chapter 1. We will thus replace as usual, coordinate with operator and Poisson brackets
with (anti)-commutators:
[xµ , pν ] = iδνµ
[ψiµ , ψjν ] = g µν δij .
(4.5)
The Grassmann variables are respresented as Gamma matrices of a multiple Clifford algebra
√
(more details could be found in Chapter 2); we recall to the reader that the operator 2ψiα
is the Gamma matrix (γ α )a0 i ai in the basis |a1 , · · · , aN i. The SO(N ) and susy generators
become
i
Jij = (ψi · ψj − ψj · ψi )
2
(4.6)
Qi = ψiα eµα πµ .
(4.7)
The covariant momentum acts on the state of the Hilbert space as the covariant derivative
i
πµ = −i∂µ − ωµαβ ψ α ◦ ψ β = −i∇µ (ω)
2
(4.8)
where the symbol ◦ means contraction over SO(N ) vector indices. The ordering in (4.7) is
the one suggested by Einstein covariance, in other words the operators written in such a form
will give rise to Einstein covariant (anti)-commutators. At this point one may start checking
the algebra and identify a suitable quantum hamiltonian operator. We get the preliminary
result
[Jij , Jkl ] = iδjk Jil − iδik Jjl − iδjl Jik + iδil Jjk
[Jij , Qk ] = iδjk Qi − iδik Qj
1
[Qi , Qj ] = 2δij H0 − ψiα ψjβ Rαβγδ M γδ
2
(4.9)
where M γδ is the Lorenz generator defined as M γδ = 12 [ψiγ , ψiδ ] and H0 is
H0 =
1 α
π πα − iω α αβ π β ;
2
(4.10)
4.1 (A)dS quantum Algebra
43
note that this is the correct quantum ordering one must choose in order to reproduce the
laplacian operator in curved space.
We restrict now ourselves to maximally symmetric spaces in which one can rewrite the term
appearing in (4.9) as
i
h
D D2 δij + Jik Jjk + Jjk Jik .
(4.11)
ψiα ψjβ Rαβγδ M γδ = b (2 − N ) −
2
2
Let us write now the complete hamiltonian as H = H0 + ∆H; we have now to fix ∆H in
order to close the algebra quadratically. We start computing
b
[H0 , Qi ] = i (Qk Jki − Jik Qk ) .
2
(4.12)
The clever choice
b
bC
∆H = − Jij Jij −
,
4
4
C = (2 − N )
D D2
−
2
2
(4.13)
produce the desired commutator
[H, Qi ] = 0 .
(4.14)
Thus the algebra is closed. The constant C in (4.13) is needed to have a consistent first class
algebra of constraints.
In summary, we have the quantum constraints
i α
[ψ , ψjα ]
2 i
i βγ β γ α µ
Qi = ψi eα pµ − ωµ ψj ψj ≡ ψiα πα
2
b
1 α
bC
α
β
H =
π πα − iω αβ π − Jij Jij −
2
4
4
Jij =
(4.15)
satisfying the following quadratic algebra
[Jij , Jkl ] = iδjk Jil − iδik Jjl − iδjl Jik + iδil Jjk
[Jij , Qk ] = iδjk Qi − iδik Qj
b
[Qi , Qj ] = 2δij H − (Jik Jjk + Jjk Jik − δij Jkl Jkl ) .
2
(4.16)
One may check that this algebra coincides with the zero mode restriction in the Ramond
sector of the nonlinear superconformal algebras discovered by Knizhnik and Bershadsky in
two dimensions [62], [63].
4.1.1
The special even N case
Reveals to be very useful for future analysis write the quantum algebra, discussed above, for
even N , by using complex combinations of the SO(N ) indices.
ψi
→
(λI , λ̄I )
i = 1, 2, ..., 2s
I = 1, 2, ..., s
44
Higher spin generalized curvature in (A)dS2n
so that1
[λαI , λ̄βj ] = δIJ η αβ
(4.17)
and the supercharges are
QI =
−iλγI
QI = −i
eµγ
∂
∂λIγ
ωµαβ λαJ
∂µ +
µ
eγ ∂µ + ωµαβ
∂
∂λJβ
α ∂
λJ
.
∂λJβ
(4.18)
(4.19)
The most interesting and useful (anti)commutators in this base are
[JI K , JJ P ] = iδJ K JI P − iδI P JJ K
[JI K JJP ] = iδJ K JIP
I JK
J
[QI , QJ ] = b(JK
J + J IK JK
)
[QI , QJ ] = b(JKI JJ K + JKJ JI K )
[QI , QJ ] = b(JIK J KJ + JI K JK J )
[QI , QI ] = 2H0 + b(JK I JI K
with I 6= J
X
JK K + Ãs (D)
− JIK J IK ) + ibJI I − ib
(4.20)
K
where Ãs (D) = b D2 (s + D2 − 1); note also that in the last anticommutatorPI is a fixed index
while K runs from 1 to s and just in some particular case one can write K JK K = sJI I .
4.2
Canonical analysis and the generalized Poincaré
Lemma
We are now ready to analyze the physical states of the Hilbert space. We focus our attention
on the even N (i.e. integer spin) case and even dimension D = 2n. We proceed in analogy
with the flat case and we start imposing the condition JI I |Ri = 0 and we obtain
JI I |Ri = 0
⇒
R(x, λ) = Rα1 ..αn |...|β1 ..βn (x) λα1 1 ..λα1 n ...λβs 1 ..λβs n .
From the other constraints constraints we learn:
1
As in the flat case we realize the algebra by setting λ̄α
I =
∂
∂λIα .
(4.21)
4.2 Canonical analysis and the generalized Poincaré Lemma
J˜I J |Ri = 0
⇒
R[n+1]1 [n−1]2 ...[n]s = 0
QI |Ri = 0
⇒
∇[µ Rµ1 ..µn ][n]2 ...[n]s = 0
R ∇-closed
J IJ |Ri = 0
⇒ Rµ[n−1]1 ,σ[n−1]2 ...[n]s = 0
R traceless
JIJ |Ri = 0
⇒
P
QI |Ri = 0
⇒
∇µ1 Rµ1 ..µn [n]2 ...[n]s = 0
45
Algebraic Bianchi identity
ck tr k R[n]1 ···[n]s = 0 Sum of traceless conditions
R ∇-co-closed
Using the Young tableaux language we have:
|Ri ∼ Rα11 ···α1n |...|αs1 ···αsn ≡
α11
.
.
.
αn1
.
.
.
.
.
. α1s
. .
. .
. .
. αns
(4.22)
.
.
.
.
.
. α1s
. .
. .
=0
. .
. αns
(4.23)
such that
∇[α Rα11 ···α1n ]|...|αs1 ···αsn ≡
α11
.
.
.
αn1
∇
Note that the relations we have obtained above are the minimal coupling of the flat one.
In the following we will give a detailed analysis of the differential Bianchi identity in maximally symmetric space (∇-closed condition).
In the flat case we have solved the differential Bianchi identity by using the the generalized
Poincaré Lemma. Unfortunately in (A)dS we can’t use this Lemma as a guide line. Let us
emphasize that in literature it is well known how to make covariant and gauge invariant the
Fronsdal-kinetic operator (see Appendix A fot more details and references), but it is not yet
clear how to derive it starting from (A)dS higher spin curvature.
In the following we will look for a solution of the differential Bianchi identity in curved
space, but note that this could be just a particular solution and probably not the most
46
Higher spin generalized curvature in (A)dS2n
generical one.
4.3
Warm up examples in D=2
We start considering some examples in D = 2.
Our strategy reads:
1. write the (A)dS HSC as
R(A)dS = Rf lat + (A)dS corrections;
(4.24)
note that in the r.h.s. of the previous relation we put all possible correction proportional
to the (A)dS curvature, preserving the symmetry of R.
2. Impose closing condition (i.e. differentialBianchi identity) to fix the extra terms.
3. Use the traceless condition to find the HS gauge potential e.o.m..
Spin 1 in two dimension:
This is rather trivial. We start with Rµ . Then
∇[µ Rν] = 0
→
Rµ = ∂µ φ
(4.25)
∇µ Rµ = 0
→
∇2 φ = 0
(4.26)
so that
Spin 2 in two dimension:
The HSC is now the symmetric tensor Rµν . We relax the trace constraint and try
to solve Bianchi by using the ansatz
Rµν = ∇µ ∇ν φ + αgµν φ .
(4.27)
The differential Bianchi constraints fix α = 1, i.e.
∇[µ Rν]λ = 0
→
α=1.
(4.28)
Imposing the trace constraint gives an e.o.m. on the potential
Rµ µ = 0
→
(∇2 + 2)φ = 0
(4.29)
which corresponds to a nonminimally coupled scalar. Co-closing condition is consistently satisfied, as one can compute
∇µ Rµν = 0
→
∇ν (∇2 + 2)φ = 0
(4.30)
4.4 Fixing the strategy
47
Spin 3 in two dimension:
In this case we use the following ansatz
Rµνλ = (∇[ µ∇ν ∇λ] φ + αg[µν ∇λ] φ)
(4.31)
Imposing Bianchi identities we find
∇[α Rµ]νλ = 0
→
α=4.
(4.32)
Finally we compute e.o.m. from the traceless condition
Rµ µλ = 0
→
3∇λ (∇2 + 6)φ = 0
(4.33)
that is also consistent with co-closing condition.
4.4
Fixing the strategy
In (A)dS target space, in order to solve the differential Bianchi identity, we use as a
guide line the results obtained in the flat limit case, resumed in Chapter 2. Let us
emphasize that by using the algebra, the results we obtain do not depend on the target
space dimension.
In particular our strategy for future analysis reads:
1. Write
|Ri = (Q[1 ....Qs] + (A)dS corrections)|φi ;
(4.34)
The HS potential |φi satisfies the condition
JI J |φi = −iδIJ |φi
(4.35)
that
- for I 6= J implies that the potential satisfies the algebraic Bianchi identity
m
n−1
1
- for I = J one has that |φi = φ[n−1]1 ···[n−1]s λm
· · · λns 1 · · · λns n−1 .
1 · · · λ1
Note that the r.h.s. of (4.34) has to be U (s) invariant and we have to take into
account it in adding (A)dS corrections, in order to preserve all the symmetry of
the HSC.
2. Impose differential Bianchi identity (i.e. QI |Ri = 0 ∀ I = 1, ....s) to fix the
extra terms
and for the first few N case
3. Analyze the gauge symmetry
48
Higher spin generalized curvature in (A)dS2n
4. Extract the higher derivative e.o.m.
5. Introduce the compensator field
6. Extract the linearized e.o.m.
In order to fix the notation and the idea we start considering the first easier example,
N = 4, 6, 8 (i.e. spin 2, 3, 4). Note that in (A)dS supercharges do not anticommute; for
this reason we prefer to start with an explicitly U (s) invariant ansatz. The calculation
become more complicated but also more controlable.
4.4.1
Spin 2 in (A)dS2n
Curvature: The starting point is the manifest U (2)-invariant expression
"
#
1 I1 I2
QI1 QI2 + qJI1 I2 |φi .
|Ri = 2!
(4.36)
We impose now the differential Bianchi identity on the latter curvature
QI |Ri = 0 .
(4.37)
This will suffice to fix the constant q. In particular, thanks to the symmetry
property of (4.36) it will be enough to require Q1 |Ri = 0. Let us rewrite (4.36)
in a less elegant yet more convenient form. By making use of the commutators
(4.38)
[QI , QJ ] = −b JIL JJ L + JJL JI L
J
QI
[JI J , QK ] = iδK
(4.39)
one can easily get
#
"
|Ri = Q1 Q2 + q J12 |φi
(4.40)
and Q1 |Ri = 0 uniquely fixes
q = ib
(4.41)
so that
"
#
1 I1 I2
|Ri =
QI1 QI2 + ib JI1 I2 |φi.
2!
(4.42)
We conclude that the final form of the spin-2 curvature is
"
#
|Ri = Q1 Q2 + ib J12 |φi
(4.43)
4.4 Fixing the strategy
49
Gauge invariance: Let us consider the transformation
δ|φi = QK V̄ K |ξi
(4.44)
where V̄ K = V m (x) λ̄K
m ; note that |ξi and |φi have the same Young tableaux. In
particular it’s easy to verify that:
[JI J , V̄ K ] = −iδIK V̄ J
(4.45)
and QK V̄ K is then a U (2)-scalar
[JI J , QK V̄ K ] = 0 .
(4.46)
By using the previous relations it’s thus easy to obtain:
δ Q1 Q2 |φi = −ib J12 QK V̄ K
=⇒
δ|Ri = 0 .
This prove that the spin 2 curvature is invariant with respect the gauge transformation (4.44).
Equation of motion: We derive now the equation of motion satisfied by the spin2 potential. The curvature |Ri constructed previously satisfies all the required
constraints except the trace J IJ |Ri = 0. Imposing the latter condition we will
obtain the equation of motion satisfied by the potential; using the susy algebra
we obtain
J 12 |Ri = 0
⇓
#
"
i −2H0 + QI QI − 2i QI QJ J IJ + bJIJ J IJ + bA2 (D) |φi = 0
(4.47)
A2 (D) = 4 − Ã2 (D) .
(4.48)
where
Into the bracket we recognize the spin 2 Fronsdal-Labastida kinetic operator
(A)dS
G2
i
= −2H0 + QI QI − QI QJ J IJ +bJIJ J IJ + bA2 (D)
|
{z 2
}
Gf lat
(4.49)
50
Higher spin generalized curvature in (A)dS2n
Graviton in (A)dSD : We would like now to show that this is the correct result by
rederiving it starting from Einstein general relativity principles.
We consider a small perturbation to the background metric
g̃αβ = gαβ + hαβ
(4.50)
Our convention for the connection and curvature reads:
Γ̃δαβ (g + h) = Γδαβ + δΓδαβ
R̃αβδσ (g + h) = Rαβδσ + δRαβδσ + ...
The linearised contribution to the Christoffel connection can be written as
1 λσ
λ
−∇σ hαβ + ∇β hσα + ∇α hβσ
(4.51)
δΓαβ = g
2
and, since the Riemann tensor is given by
Rαβ λ σ = ∇α Γλβ|σ| − ∇β Γλα|σ|
(4.52)
where the notation | · · · | means that the covariant derivative does not act on such
index, we have
δRαβ λ σ = ∇α δΓλβσ − ∇β δΓλασ
(4.53)
that, using (4.51), reduces to
δRαβ
λ
σ
1 λρ
=
g
−∇α ∇ρ hβσ + ∇α ∇σ hρβ + ∇α ∇β hσρ
2
+∇β ∇ρ hασ − ∇β ∇σ hρα − ∇β ∇α hσρ
(4.54)
from which it’s easy to compute the linearized Ricci tensor:
1
2
(4.55)
δRαβ = − ∇ hαβ + ∇α ∇β h + ∇λ ∇{α hβ}λ
2
1
2
= − ∇ hαβ + ∇α ∇β h + ∇{α ∇λ hβ}λ + Rρ αβ λ hρλ + Rλ {α hβ}λ(4.56)
.
2
In (A)dSD background the previous formula reduce to
1
2
δRαβ = − ∇ hαβ + ∇α ∇β h + ∇{α ∇λ hβ}λ + b (Dhαβ − gαβ h) .
2
(4.57)
We are ready now to compute the the Einstein equation in (A)dS that at the
linearised level reads:
δRαβ = (D − 1)bhαβ
(4.58)
We substitute now (4.50) and (4.57) in (4.58) and we find:
1
2
− ∇ hαβ + ∇α ∇β h + ∇(α ∇λ hβ)λ + b (hαβ − gαβ h) = 0
2
(4.59)
that is the equation of motion for a graviton in (A)dSD space. Note that in D = 4
(4.59) coincides with (4.47) and it is just written in a different language.
4.4 Fixing the strategy
4.4.2
51
Spin 3 in (A)dS2n
Curvature: Similarly to the previous section we start from the most generic U (3)invariant expression
#
"
1 I1 I2 I3
QI1 QI2 QI3 + pJI1 I2 QI3 |φi .
|Ri = (4.60)
3!
We solve now the constraint Q1 |Ri = 0 that uniquely fixes
(4.61)
p = 4ib
so that one can write
#
"
1 I1 I2 I3
QI1 QI2 QI3 + i4b JI1 I2 QI3 |φi
|Ri = 3!
(4.62)
or equivalently
"
|Ri =
Q1 Q2 Q3 + ib
#
J12 Q3 + J31 .Q2 +i2b J23 Q1 |φi
(4.63)
We use the trace operator to extract the gauge potential e.o.m.. To this aim the
following algebraic identities reveal to be very useful:
2iJ12 Q3 J 23 ∼ Q1 Q1 QK J 1K + iQ1 J1K J 1K + ”algebraic Bianchi identity”
iJ1K QK ∼ Q1 Q1 Q1 + ”algebraic Bianchi identity” .
By using the previous relations we obtain a very elegant and explicitly U (3) result:
J 23 |Ri = 0
⇓
"
#
iQ1 −2H0 + QI QI − 2i QI QJ J IJ + bJIJ J IJ + bA3 (D) |φi = 0
(4.64)
A3 (D) = 9 − Ã3 (D)
(4.65)
where
In the formula (4.64) we can read inside the square bracket the spin the spin 3
Fronsdal-Labastida kinetic operator
i
(A)dS
G3
= −2H0 + QI QI − QI QJ J IJ +bJIJ J IJ + bA3 (d)
(4.66)
|
{z 2
}
Gf lat
52
Higher spin generalized curvature in (A)dS2n
Gauge invariance: Using the experience inherited from the flat case, we would like
now to study the gauge invariance; we introduce also the compensator field
W K W J W I |ρi
(4.67)
in order to linearize the e.o.m.. First of all, by using (4.63) and (4.64) we can
write
(A)dS
G3
= (QI QJ QK + i4bJIJ QK )W K W J W I |ρi
(4.68)
The gauge transformation:
δ|φi = QK V̄ K |ξi
(4.69)
leaves (4.63) invariant, while (4.66) trasforms as:
i
(4.70)
δ|φi = (QI QJ QK + i4bJIJ QK )J {[IJ V̄ k]} |ξi .
2
From the previous expression is manifest that the Fronsdal-Labastida spin 3 kinetic operator isn’t gauge invariant.
(A)dS
G3
Equation of motion: Let us analyze how the compensator field transform under
(4.69). After some straightforward computation it’s easy to obtain:
i
(4.71)
δ(W K W J W I |ρi) = J [IJ V̄ k] |ξi .
2
The previous expression plays a key role. We use it to gauge fix the compensator
to zero; this gauge choice let us write
|ρi = 0
⇒
(A)dS
G3
|φi = 0
(4.72)
that is the ”linearized” spin 3 e.o.m. in (A)dS. Let us emphasize that this relation
is gauge invariant only if we force the gauge parameter to be traceless.
4.4.3
Spin 4 in (A)dS2n
Curvature: We start from the manifestly U (4)-invariant expression
"
#
1 I1 I2 I3 I4
|Ri = QI1 QI2 QI3 QI4 + 3! p KI1 I2 QI3 QI4 + 3q KI1 I2 KI3 I4 |φi (4.73)
4!
and use the commutation rules to bring Q1 in front of everything. After some
straightforward yet tedious algebra one gets
"
2i
|Ri = Q1 Q2 Q3 Q4 + p − b
K12 Q3 Q4 + K31 Q4 Q2 + K14 Q2 Q3 +
3
i
4i
K42 Q1 Q3 + K23 Q1 Q4 +
p + b K34 Q1 Q2 + p + b
3
3
#
q K12 K34 + K31 K24 + K14 K23
|φi
(4.74)
4.4 Fixing the strategy
53
so that requiring Q1 |Ri = 0, it yields p =
"
|Ri =
5i
b
3
and q = −3b2 . Hence one has
Q1 Q2 Q3 Q4 + ib K12 Q3 Q4 + K31 Q4 Q2 + K14 Q2 Q3
i2b
+ i3b K34 Q1 Q2 +
K42 Q1 Q3 + K23 Q1 Q4
−3b2
#
K12 K34 + K31 K24 + K14 K23
|φi
(4.75)
or equivalently
#
"
1 I1 I2 I3 I4
QI1 QI2 QI3 QI4 + i10b KI1 I2 QI3 QI4 − 9b2 KI1 I2 KI3 I4 |φi
|Ri = 4!
(4.76)
that is the final form of the spin-4 curvature.
Equation of motion: The traceless condition produces
"
#
i
(iQa Qb − Jab ) −2H0 + QI QI − QI QJ J IJ + bJIJ J IJ + bA4 (D) |φi = 0 (4.77)
2
where
A4 (D) = 16 − Ã4 (D)
(4.78)
In analogy with the other case we define
(A)dS
G4
i
= −2H0 + QI QI − QI QJ J IJ +bJIJ J IJ + bA4 (D)
|
{z 2
}
(4.79)
Gf lat
4.4.4
Conjecture
The results we have obtained above suggest us that, for every spin and for every even
dimension D, the Fronsdal-Labastida kinetic operator in (A)dS becomes:
#
i
= −2H0 + QI QI − QI QJ J IJ + bJIJ J IJ + bAs (D)
2
"
G(A)dS
s
where
As (D) = s2 −
D
D
(s + − 1).
2
2
(4.80)
(4.81)
54
Higher spin generalized curvature in (A)dS2n
Let us emphasize that in D = 4 this operator reproduces the extension of the Fronsdal
one in (A)dS spaces: moreover if we force the HS field (|φi) to be double traceless
and the gauge parameter (|ξi) to be traceless it is invariant with respect the gauge
tranformations:
δ|φi = QK V̄ K |ξi .
4.5
HSC in (A)dS2n
Before starting, for matter of convenience, we change our notation and we indicate
with KIJ = JIJ and K IJ = J IJ .
In this section we will look for for a generic U (s)-invariant expression (N = 2s, s ∈ N )
of the form:
|Ri =
[s/2]
X
(ib)n rn (s)Rn (s) |φi ,
with r0 (s) ≡ 1
(4.82)
n=0
where in this case [s/2] means integer part and
1 I1 I2 ···Is
KI1 I2 · · · KI2n−1 I2n QI2n+1 · · · QIs .
s!
We fix the numerical coefficients rn (s) by requiring
Rn (s) ≡
QI |Ri = 0 .
(4.83)
(4.84)
In particular, thanks to the symmetry properties of (4.82) it will suffice to require
Q1 |Ri = 0. In order to achieve such a task we shall need a few recursive relations that
we now derive using the commutation relations
[QI , QJ ] = −b KIL JJ L + KJL JI L
(4.85)
J
[JI J , QK ] = iδK
QI
[KIJ , KKL ] = [KIJ , QK ] = 0
(4.86)
(4.87)
and the condition (4.35). Let us split now the s indices into a “time-like” index 1 and
s − 1 “space-like” indices
I = (1, i) ,
i = 2, . . . , s
(4.88)
and let us first define a shortcut notation that reveals to be extremely useful
i1 ···is−1 Qi1 · · · Qin Q1 Qin+1 · · · Qis−1 |φi ≡ Q[n] Q1 Q[s−1−n]
(4.89)
i1 ···is−1 K1i1 Qi3 · · · Qin Q1 Qin+1 · · · Qis−1 |φi ≡ K1i1 Q[n−2] Q1 Q[s−1−n] (4.90)
and whenever we encounter a Kij tensor we use the commutation rules above and the
antisymmetry provided by the tensor to bring them in front of everything and give
them the first indices of the set i1 , i2 , . . . .
After some algebraic manipulation, and by using the quantum algebra (4.20), it’s easy
to prove the following Lemma:
4.5 HSC in (A)dS2n
55
Lemma 1.
n
(−) Q[n] Q1 Q[s−1−n]
n(n − 1)
K1i1 Q[s−2]
= Q1 Q[s−1] − ib n(s − 2) −
2
n
s−3
X
X
+ibKi1 i2
(−)k Q[k] Q1 Q[s−k−3] .
(4.91)
m=1 k=m−1
Note now that the previous formula can be easily iterated by noting that the last term
is just equal to the l.h.s. if one performs the substitution s → s − 2. The iteration
process thus yields
s−1
X
(−)n Q[n] Q1 Q[s−1−n] = sQ1 Q[s−1]
n=0
−(ib)a2 (s) K1i1 Q[s−2] − Ki1 i2 Q1 Q[s−3]
2
−(ib) a4 (s) K1i1 Ki2 i3 Q[s−4] − Ki1 i2 Ki3 i4 Q1 Q[s−5]
.
.
.
−(ib)p a2p (s)K1i1 Ki2 i3 · · · Ki2(p−1) i2p−1 Q[s−2p]
p
+(ib)
k0
s−1 X
X
s−3
X
···
k0 =1 m1 =1 k1 =m1 −1
kp−1
s−2p−1
X
X
a2n (s) ≡
(4.92)
mp =1 kp =mp −1
with
k0
s−1 X
X
(−)kp Q[kp ] Q1 Q[2−2p−1−kp ]
s−3
X
k0 =1 m1 =1 k1 =m1 −1
ku−1
···
X
s−2n−1
X
1;
(4.93)
mu =1 kn =mn −1
the iterative expression (4.92) stops at the last-but-one entry if s = 2p whereas it stops
at the last if s = 2p + 1.
We will analyze carefully relation (4.93) In Appendix C, where, in particular, we will
show that a2n (s) ”factorize” as
a2n (s) = f (n) (s − 1)(s − 2) · · · (s − 2n)
(4.94)
for some function f (n) defined as
f (n) =
n−2
X
q̃=0
(−)n−1
(−)q̃
f (n − q̃ − 1) +
,
(2q̃ + 2)!
(2n − 1)!
with
f (0) = 0 .
(4.95)
The (A)dS quantum algebra could be used to derive other very useful relations. In
particular it’s not hard to prove that
56
Higher spin generalized curvature in (A)dS2n
Lemma 2.
Q21 Q[s−1]
= −ibK1i1
s−2
X
(−)n Q[n] Q1 Q[s−2−n] ;
(4.96)
n=0
that by using Lemma 1, could be reduced to
2
Q1 Q[s−1] = −ibK1i1 a0 (s − 1) Q1 Q[s−2] + ib a2 (s − 1) Ki2 i3 Q1 Q[s−4]
p−1
+ · · · (ib)
a2(p−1) (s − 1) Ki2 i3 · · · Ki2(p−1) i2p−1 Q1
(4.97)
noting that expressions containing K1i1 K1i2 are vanishing thanks to the implied antisymmetrization. In the latter we have defined a0 (s) ≡ s.
It is easy now to convince ourselves that the zero-th order operator R0 (s) can be
written as
s! R0 (s)|φi =
s−1
X
(−)n Q[n] Q1 Q[s−1−n]
(4.98)
n=0
so that making use of (4.92), and assuming for definiteness that s = 2p one gets
s! Q1 R0 (s)|φi = a0 (s)Q21 Q[s−1]
2
−(ib)a2 (s) K1i1 Q1 Q[s−2] − Ki1 i2 Q1 Q[s−3]
2
2
−(ib) a4 (s) K1i1 Ki2 i3 Q1 Q[s−4] − Ki1 i2 Ki3 i4 Q1 Q[s−5]
.
.
.
(4.99)
−(ib)p a2p (s)K1i1 Ki2 i3 · · · Ki2(p−1) i2p−1 Q1 .
One can now easily use Lemma 2 and get
s! Q1 R0 (s)|φi = −
s/2
X
(ib)n α2n (s)Q1 In (s)
(4.100)
n=1
where
In (s) ≡ K1i1 Ki2 i3 · · · Ki2(n−1) i2n−1 Q[s−2n]
(4.101)
4.5 HSC in (A)dS2n
57
and
α2n (s) ≡
n
X
a2k (s)a2(n−1−k) (s − 1 − 2k)
(4.102)
k=0
with a−2 (u) ≡ 1. This completes the first part of the analysis.
The next step will be to rewrite expression (4.100) in terms of SU (s)-covariant tensors.
The covariantization of the tensors In (s) is again an iterative process. Note in fact that
one can write
!
1
Jn (s) − Ki1 i2 · · · Ki2n−1 i2n (s − 2n)! R0 (s − 2n)
In (s) =
(4.103)
2n
that using (4.100) and noting that
Ki1 i2 · · · Ki2n−1 i2n Im (s − 2n) = In+m (s)
(4.104)
!
s/2
X
1
Jn (s) +
(ib)m−n α2(m−n) (s − 2n) Im (s) .
In (s) =
2n
m=n+1
(4.105)
allows to write
that finally yields
Q1
[s/2]
X
(ib)n rn (s)Rn (s)|φi = 0
(4.106)
n=0
with
n
rn (s) =
1 X
rn−k (s)α2k (s − 2(n − k)) .
2n k=1
(4.107)
Note that in (4.106) we have replaced s/2 with its integer part: it is in fact not difficult
to check that the latter holds for odd s as well, with that precise replacement.
In order to save the iteration process reveals to be very important the following Lemma:
Lemma 3.
α2n (s) = a2n (s + 1)
(4.108)
or equivalently
n−1
X
k=0
f (k + 1)f (n − k) = (2n + 1)f (n + 1)
(4.109)
58
Higher spin generalized curvature in (A)dS2n
The proof of the previous Lemma can be found in Appendix C.
From the definition of rn (s), and by using (4.108) we find
n
1 X
rn (s) =
rn−k (s) a2k (s − 2(n − k) + 1) .
2n k=1
(4.110)
We thus have
r0 (s) = 1
(s − 1)s(s + 1)
1
a2 (s + 1) =
r1 (s) =
2
6
1
1
a4 (s + 1) + a2 (s + 1)a2 (s − 1)
r2 (s) =
4
2
1
=
(5s + 7)(s + 1)s(s − 1)(s − 2)(s − 3)
5
· 23 · 32
1
1
1
r3 (s) =
a6 (s + 1) + a4 (s + 1)a2 (s − 3) + a2 (s + 1)a4 (s − 1)
6
4
2
1
+ a2 (s + 1)a2 (s − 1)a2 (s − 3)
8
1
=
(35s2 + 112s + 93)(s + 1)s(s − 1)(s − 2)(s − 3)(s − 4)(s − 5)
7 · 5 · 24 · 34
(4.111)
Note now that substituting (4.110) and (4.111) into (4.82) we produce the correct
result obtained in the previous sections (4.43), (4.63) and (4.76).
In addition we have the following predictions for s = 5, 6 and s = 7
"
#
1 I1 ···I5
|Ri =
QI1 · · · QI5 + 20ib KI1 I2 QI3 · · · QI5 − 64b2 KI1 I2 KI3 I4 QI5 |φi ,
5!
(4.112)
"
|Ri =
1 I1 ···I6
QI1 · · · QI6 + 35ib KI1 I2 QI3 · · · QI6 − 259b2 KI1 I2 KI3 I4 QI5 QI6
6!
#
−225ib3 KI1 I2 KI3 I4 KI5 I6 |φi ,
(4.113)
"
1 I1 ···I7
|Ri =
QI1 · · · QI7 + 56ib KI1 I2 QI3 · · · QI7 − 784b2 KI1 I2 KI3 I4 QI5 QI6 QI7
7!
#
−2304ib3 KI1 I2 KI3 I4 KI5 I6 QI7 |φi .
(4.114)
More in general, starting from the relations (4.82), (4.94), (4.95) and (4.110) one can construct HSC for every integer spin in every even dimension D = 2n.
Chapter 5
HK and QK N = 4 one dimensional
SUGRA
Recently supersymmetric non linear sigma model on Quaternionic-Kähler manifold (QK)
have attracted a great deal of attention in the context of studying radial quantization of
BPS black-hole [50].
In this chapter chapter we intend to construct and analyze N = 4, one dimensional supergravity model on Hyper Kähler (HK) and QK background. In particular we will analyze the
symmetries of these models and we will study the first class constraints algebra that in QK
case reveals an interesting ”non Lie” structure. In addition we construct the BRST charge
and the gauge fixed action on the circle.
5.1
Special Geometry
Let us consider HyperKähler and Quaternionic Kähler geometries in dimension 4n and signature (2n, 2n); they enjoy sp(2n) and sp(2) ⊗ sp(2n) holonomy, respectively; more details
on HK and QK geometry can be found in [64] [65] and references therein.
In this Chapter we will use the following notation
Curved indices
µ, ν = 1, 2, ..., 4n
Flat indices
m, n = 1, 2, ..., 4n
Fund. representation of sp(2n)
A, B = 1, 2, ..., 2n
Fund. representation of sp(2)
α, β = 1, 2
The symplectic special holonomies allow us to decompose SO(2n, 2n) tangent space indices
with respect to the sp(2) ⊗ sp(2n) subgroup.
m = Aα.
(5.1)
60
HK and QK N = 4 one dimensional SUGRA
The invariant SO(2n, 2n) metric decomposes as
ηmn = αβ AB
(5.2)
where αβ and AB are the sp(2) and sp(2n) invariant, antysimmetric tensors, that we use to
raise and lower sp(2) and sp(2n) indices by using the following rules
vA ≡ AB v B
vα ≡ αβ v β
v A ≡ vB BA
v α ≡ vβ βα
(5.3)
and in our notation we take αβ = δαβ = −β α . We denote the vielbein as Vµm
Vµm Vνm = gµν
Vµm Vnµ = ηmn
(5.4)
or equivalently
A α
VνA = −gµν ,
Vµα
A µβ
VB = −δAB δαβ .
Vµα
(5.5)
Special holonomy implies also that
1
β
A
V{µα
Vν}A
= − gµν δαβ ,
2
A
α
V{µα
Vν}
=−
1
gµν δAB .
2n
(5.6)
The einbein covariantly constant condition reads
∇V m ≡ dV m + Pn m ∧ V n = 0
(5.7)
where the spin connection Pn m decomposes as
Pn m = δBA ωβ α + δα β ΩA
B .
(5.8)
The one-forms ωβα and ΩA
B are symmetric in their sympectic indices. On HK manifold just
B
the sp(2n) connection ΩA is non vanishing while both are present for QK manifold. Let us
also emphasize that the tangent bundle decompose into rank 2 and rank 2n vector bundles
T M = H ⊗ E;
(5.9)
the connection acts on sections of H and E respecively as:
∇X α = dX α + ωβ α X β ,
∇X A = dX A + ωB A X B .
(5.10)
We need now another geometric ingredient, that is the Riemann tensor:
1
A
Rmn ≡ dP mn + P mr P rn = δβα RB
+ δBA Rβα = V r ∧ V s Rmnrs .
2
(5.11)
Rmnrs = λ(α|γ| β)δ AB CD + αβ γδ [λ(A|C| B)D + ΩABCD ] .
(5.12)
It decompose as
The sp(2n) curvature ΩABCD is totally symmetric while the Bianchi identities force the
sp(2) curvature to vanish. The terms proportional to the constant λ are present just in QK
manifold and vanisch in HK case. It is important also to note that these are not proportional
to ηr[m ηn]s , the constant curvature Riemann tensor, since constant curvature manifolds are
not QK.
5.2 HK Sigma Model
5.2
61
HK Sigma Model
Our starting point is the N = 4 susy algebra
{Qiα , Qjβ }pb = −ij αβ ;
(5.13)
the indices i, j = 1, 2 label the fundamental representation of an sp(2) R-symmetry with
invariant tensor ij . Let now (M, gµν ) be the 4n-dimensional HK target space. The field
content of the model consists of bosonic worldline embedding coordinate xµ (t), and fermionic
spinning degrees of freedom ψAi (t). The model is governed by the simple action
Z h
∇ψiA i
1
(5.14)
dt ẋµ ẋν gµν + ψAi
S=
2
dt
where we have defined the covariant derivative
the rigid symmetry
∇ψiA
dt
A C
ψi . This model enjoys
≡ ∂t ψiA + ẋµ ΩµC
Worldline translation:
δxµ = ξ ẋµ ,
δψAi = ξ ψ̇Ai
(5.15)
sp(2) symmetry:
δxµ = 0,
δψAi = λij ψAj
(5.16)
δxµ = VαµA ψAi αi ,
C i
α i
δψAi = −ẋµ VµA
α + VβµB ψBj βj ΩµA
ψC .
(5.17)
N = 4 supersymmetry:
The model (5.14) in first order form reads
Z h
i
1
1
(1)
S = dt pµ ẋµ + ψAi ψ̇iA − πµ πν g µν
2
2
(5.18)
with the canonical Poisson bracket
{pµ , xν }pb = −δµν ,
{ψAi , ψBj }pb = −ij AB .
(5.19)
In the equations above, πµ is the covariant momentum
i
πµ = pµ − Pµmn M mn
2
(5.20)
where M mn are quadratic in spinning degree of freedom and generate the Lorentz algebra
{M mn , M rs }pb = −iM [m|s| η n]r + iM [m|r| η n]s
(5.21)
62
HK and QK N = 4 one dimensional SUGRA
For special holonomy manifolds just a subgroup of the full orthogonal group need appear.
In particular, in our case, for HK manifolds we have
Pmn M mn = ΩAB T AB
(5.22)
{T AB , T CD } = −iC{A T B}D − iA{C T D}B .
(5.23)
where T AB generates sp(2n)
Explicitly one has
B}
T AB ≡ iψ i{A ψi
(5.24)
and
1
A B
ψi .
(5.25)
πµ = pµ − ψAi ΩµB
2
The diffeomorphism, sp(2) and susy conserved charges associated to the symmetry (5.15),
(5.16) and (5.17) are
1
H = π2
2
{i
Lij = −iψA ψ j}A
Qiα = ψAi V µA
α πµ .
(5.26)
It remains to compute the superalgebra. This computation is greatly simplified by using the
central relations
i
C
C
{πAα , πBβ }pb = {πAα , πBβ }pb = αβ ΩABCD T CD + ΩBβ|A πCα − ΩAα|B πCβ
2
µ
{πAα , ψjC }pb = VAα
ΩµM N ψjM N C
(5.27)
(5.28)
Let us emphasize that in the r.h.s. of the previous relations, only the sp(2n) component of
the connection and curvature appear.
After some straightforward computation one finds
{Qiα , Qjβ }pb = −ij αβ H
{Lij , Qkα }pb = 2k{i Qj}
α
{Lij , Lkl }pb = −iki f jl − ikj f il − ili f jk − ilj f ik
{H, Lij }pb = {H, Qiα }pb = 0 .
5.3
(5.29)
QK N = 4 d = 1 SUGRA
We replace the HK target space with a QK one. First of all one has to observe that is no
longer possible to maintain the N = 4 supersymmetry algebra (5.29); this is due to the fact
that the central relations (5.27) changes when one turns on the sp(2) connection:
i
C
C
{πAα , πBβ }pb = αβ ΩABCD T CD + ΩBβ|A πCα − ΩAα|B πCβ
2
γ
γ
+ ωBβ|α πAγ − ωAα|β πBγ
(5.30)
5.3 QK N = 4 d = 1 SUGRA
63
The superalgebra (5.29) becomes:
γ
γ
{Qiα , Qjβ }pb = (−ij αβ )H + (−ψjB ωBβ|α )Qiγ + (−ψiA ωAα|β )Qjγ
1
γ
{Qiα , H}pb = (π Bβ ωBβ|α )Qiγ + ( ψiA ψ jA )Qjα
2
where in the r.h.s. of (5.31) we have recognized the Hamiltonian
(5.31)
(5.32)
H = 21 πAα π Aα + λ4 ψ A ◦ ψ B ψA ◦ ψB .
where ψ A ◦ ψ B = ψ iA ψiB . The algebra, QK supersymmetry algebra (5.32), has some very
important consequences:
• In QK manifold is no longer possible to maintain rigid supersymmetry, just a model
with local susy is allowed.
• The algebra is no more a Lie algebra since structure functions appear in the r.h.s. of
(5.31) (5.32).
• In QK manifold one has to introduce a curvature coupling with fermions.
• One can also observe that in (5.32) we have choosen to use just the supercharges but
also the sp(2) generator is present in the r.h.s.. The reason is just matter of convenience
and will be clarified in 2 minutes.
In QK the bracket involving the operator Lij doesn’t change with respect the one we have
computed in HK.
Two possible interesting gauged model should be studied
Constraints
Gauge fields
H=0
one dimensional einbein N
Qαi = 0
one dimensional gravitini ψiα
H=0
one dimensional einbein N
Lij = 0
Yang-Mills f ij
Qαi = 0
one dimensional gravitini χαi
1.
2.
In the following we concentrate our attention on the first one because intimately related to
black hole supersymmetric quantum mechanics [50]. In particular we will study the first
order action with QK background
Z
1
(1)
(5.33)
S = dt[pµ ẋµ + ψAi ψ̇iA − N H − χαi Qiα ]
2
64
HK and QK N = 4 one dimensional SUGRA
and after Legendre transformation one obtains
Z
S=
dt[
1 i ∇ψiA λN i
1 µ
i α
ν
νA i α
(ẋ − V µA
ψ
χ
)(
ẋ
−
V
ψ
χ
)g
+
ψ
+
ψ ψiB ψ jA ψjB ] (5.34)
µν
α A i
α A i
2N
2 A dt
4 A
wich enjoys the symmeties
Local worldline reparametrizations:
δxµ
δψAi
δN
δχαi
=
=
=
=
ξ ẋµ ,
ξ ψ̇Ai
∂t (ξN )
∂t (ξχαi )
(5.35)
Local N = 4 supersymmetry:
i α
δxµ = V µA
α ψA i
1
j γ
B i
δψAi = − (ẋµ − VβµB ψBi χβi )VµAα iα + V µA
γ ψA j ΩµA ψB
N
δN = χiα αi
∇iα λN i A j
j γ
β i
+
ψα ψj α + V µA
δχiα =
γ ψA j ωµα χβ
dt
2
(5.36)
Rigid sp(2):
δψAi = λij ψAj
δχiα = λij χαj
5.4
(5.37)
BRST charge and gauge fixed action
We construct now the BRST charge associated to the algebra (5.31) and (5.32); note that this
algebra is a non Lie. We define the fermi (c, ρ) and bosonic (η iα , Pkα ) fields corresponding
to reparametrization and susy ghost, and ghost momenta respectively, equipped with the
Poisson structure bracket
{c, ρ} = {ρ, c} = −1
{η iα , Pkβ } = −{Pkβ , η iα } = δki δβα .
(5.38)
The BRST charge reads:
γ
γ
Qbrst = η iα Qiα + cH − η iα η jβ ψjC ωCβ|α Piγ − η iα cπ Cβ ωCβ|α Piγ
1
λ
− η iα η jβ ij αβ ρ − η iα cψiC ψ kC Pkα + higher order terms;
2
2
(5.39)
5.4 BRST charge and gauge fixed action
65
We have now to fix the higher order terms in ghost momenta. In order to do that we will
use a very funny and elegant trick; in the previous BRST expansion, infact, we recognize a
covariant structure underlining. We define the ghost generator
α
Pj}α
Lgh
= η{i
ij
{α
τ αβ = 2iηi P iβ}
(5.40)
The sp(2) generators Lgh
ij obey the sp(2) algebra; the holonomy generators similarly are
subject to
{τ αβ , τ γδ } = γα τ βδ + γβ τ αδ + δα τ βγ + δβ τ αγ .
(5.41)
Therefore we can construct a covariant momentum in the extended (i.e. sp(2) ⊗ sp(2n))
sense
i
i
(5.42)
Πµ = pµ − ΩµAB T AB − ωµαβ τ αβ .
2
2
Observing carefully (5.39), one can observe that by adding the higher order terms proportional to τ αβ τ δσ , everything could be written in terms of the extended hamiltonian and
supercharges
1 2 λ A
Π + ψ ◦ ψ B ψA ◦ ψB
2
4
A
= ψi Πµ V µα
A
H =
Qαi
Therefore we can immediately construct the covariant, in the extended sense, BRST charge
λ
1
Qbrst = η iα Qiα + cH − η iα ηiα ρ − η iα cψiC ψ kC Pkα
2
2
(5.43)
and it’s easy to verify that the higher order corrections we have added by using the argument
proposed above, play a fundamental role in order to make (5.43) nilpotent.
We intend now to gauge fix the supergravity multiplet on the circle and construct the
gauge fixed action.
The trasformation rule for gravitini and einbein are:
δe = ξ˙ − χiα jβ (ij αβ )
δ
δ
δχkδ = ˙kδ − χiα jβ (ψjC ωCβ|α δik + ψiC ωCα|β δjk )
λ
λ
δ
δ
−eiα (π Bβ ωBβ|α δik + ψiC ψ kC δαδ ) + χiα ξ(π Bβ ωBβ|α δik + ψiC ψ kC δαδ ) .
2
2
(5.44)
We choose antiperiodic boundary condition for fermions and gravitini and this let us to gauge
fix gravitini to zero. At this point the transformation rule for the einbein coincide with the
one we have discussed in the SO(N ) spinning particle models. Thus the einbein could be
66
HK and QK N = 4 one dimensional SUGRA
gauge fixed to the proper time β.
We introduce the gauge fixing fermion:
K = −βρ
(5.45)
that let us to implement the gauge choice
(e, χαi ) = (β, 0)
(5.46)
The gauge fixed action can be computed by
H GF = H + {K, Qbrst }
(5.47)
and explicitly we obtain the gauge fixed Hamiltonian
H GF =
β
i
i
(pµ − ΩµAB T AB − ωµαβ τ αβ )2
2
2
2
λ
λ
kC iα
−β ψiC ψ η Pkα + β TAB T AB .
2
4
(5.48)
Appendix A
Tour on higher spin gauge theory
In this appendix we intend to describe briefly some aspect of massless higher spin theory (HS
gauge theory). In the first part we will focus our attention on the ”linearized approach” to HS;
we will derive, in D = 4, the Fronsdal (for bosons) and Fang-Fronsdal (for fermions) e.o.m., as
the natural generalization of the KleinGordon/Einstein and Dirac/Rarita-Schwinger e.o.m.,
respectively.
We proceed analyzing an important subclass of HS field, conformal HS gauge theory, in
various dimensions, because strictly related to SO(N ) spinning particle. We will focus
our attention on the relation between geometrical approach to higher spin, that is higher
derivative, and the linearized one; we will study the flat case and we will show how to obtain
the e.o.m. and the Fronsdal-Labastida kinetic operator, which generalize those of Fronsdal
for mixed symmetry fields.
Finally we will discuss how to modify the Fronsdal operator in (A)dS4 background. Let
us emphasize that in literature is well known how to make covariant and gauge invariant
the Fronsdal kinetic operator, but is not clear how to derive it starting from geometrical
objects: HSC or equivalently HS field strength. For a review on related topics and additional
references see [33] [32] [29].
Let us start discussing the D = 4 case. In D = 4 symmetric tensor describe all possible
representetion of the Poincaré group; to describe integer HS we use the completely symmetric
tensor φα1 ···αs while physical states of half integer spin s are indicated with ψαa 1 ···α 1 .
s− 2
We resume, for commodity, quantum field theory for spin lower than 2, in the following
68
Tour on higher spin gauge theory
tableaux:
Spin
Equation of motion
Gauge symmetry
Scalar field
φ(x) (s = 0)
2φ = 0
No gauge symmetry
Spinor field
ψa (s = 12 )
(γ α )ab ψ b ∂α ψ b = 0
No gauge symmetry
Maxwell field
Aα (s = 1)
2Aα − ∂α ∂β Aβ = 0
δAα = ∂α ξ
Rarita-Schwinger
ψaα (s = 32 )
γαβγ ∂ β ψaγ = 0
δψaα = ∂ α ξa
Graviton
hαβ (s = 2)
∂α ∂β hαδ + ∂α ∂δ hαβ − 2hδβ − ∂β ∂δ h = 0
δhαβ = ∇α ξβ + ∇β ξα
Note now that except for the scalar and the spinor field, all other massless fields are gauge
fields. For these reason seems to be reasonable assuming that all massless higher spin fields
are also gauge fields.
HS gauge theory is constructed as the natural generalization of the well known massless
quantum field theory; in the following, we will discuss gauge symmetry and equations of
motion.
Gauge transformations are assumed to be
δφα1 ···αs = ∂{α1 ξa2 ···as }
δψαa 1 ···α
1
s− 2
= ∂{α1 ξαa 2 ···α
1
s− 2
(A.1)
}
.
(A.2)
Note that for s = 1, 32 , 2 the previous relations reproduce the well known transformations rule for Maxwell, Rarita-Schwinger and graviton field.
Free equations of motion of the HS are second order linear differential equations in the
case of the integer spins and the first order differential equations in the case of the
half integer spins. This is required by the unitary and ensures that the fields have
a positive-definite norm. The massless higher spin equations have been derived from
the massive higher spin equations [66] by Fronsdal for bosons [30] and by Fang and
69
Fronsdal for fermions [67], andPstudied in more detail in [31] .
In our convention the symbol
denote symmetrized sum with respect free indices. In
the bosonic case, the dynamics is governed by the so called Fronsdal e.o.m
X
X
∂α1 ∂α2 φββα3 ···αs (x) = 0
Gα1 ···αs ≡ 2φα1 ···αs (x) −
∂α1 ∂β φβα2 ···αs (x) +
≡ Gφα1 ···αs
(A.3)
where G is the Fronsdal-Labastida kinetic operator that in an obvious notation we
write as:
G ≡ 2 − grad div + grad2 tr .
(A.4)
In the fermionic sector the first order equations are a natural generalization of the
Dirac and Rarita–Schwinger equation
X
∂α1 (γ β ψ)aβα2 ···α 1 = 0 .
(A.5)
Gaα1 ···α 1 (x) ≡ (/∂ ψ)aα1 ···α 1 −
s− 2
s− 2
s− 2
Constraints on higher spin gauge parameters is the next arguments we intend to analyze.
Let us now analyze how equations of motion (A.3) and (A.5) transform under gauge
transformations (A.1) and (A.2). The direct computations gives
X
(A.6)
δGα1 ···αs = 3
∂α3 1 α2 α3 ξ ββα4 ···αs ,
X
∂α2 1 α2 γ βa b ξ bβα3 ···α 1 ,
δGαα1 ···α 1 = −2
(A.7)
s− 2
s− 2
where ∂α2 1 α2 = ∂α1 ∂α2 and ∂α3 1 α2 α3 = ∂α1 ∂α2 ∂α3 .
Note now that the previous relations vanish only if the parameters of the transformations (A.6), for s ≥ 3, are traceless, equivalently for (A.7) and s ≥ 5/2 are γ-traceless.
ξ ββα4 ···αs = 0
(γ β ξ)αβα3 ···α
s− 1
2
= 0.
(A.8)
(A.9)
Another key ingredient we have to take into account is the Bianchi identities that
imply that the traceless divergence on the left-hand-side of equations of motion must
vanish; equivalently the currents of the matter fields if coupled to the gauge fields are
conserved.
Let us consider Maxwell theory:
∂α (∂β F αβ ) ≡ 0,
(A.10)
where Fαβ is the field strength defined as Fαβ = ∂α φβ − ∂β φα . Electric current into the
Maxwell equations ∂α F αβ = J β is conserved ∂α J α = 0.
In the spin 2 case (theory of gravity) coupled to matter fields and described by the
Einstein equation
1
Rαβ − gαβ R = Tαβ
2
70
Tour on higher spin gauge theory
the energy-momentum conservation ∇α T αβ = 0 is related to the Bianchi identity
1
∇α Rαβ − ∇β Rαα ≡ 0 .
2
(A.11)
Generalization of (A.11) to the case of the bosonic higher spin fields reads:
1X
3X 3
∂β Gβα2 ···αs −
∂α2 Gββα3 ···αs = −
∂α2 α3 α4 φβγβγα5 ···αs .
(A.12)
2
2
Note now that the previous relation vanishes only if the HS fields for s ≥ 4 are doubletraceless
(A.13)
tr 2 φα1 ···αs ≡ φβγβγα5 ···αs = 0.
In analogy, in the fermionic sector we have
1X
1
∂β Gaβα2 ···α 1 −
∂α2 Gaββα3 ···α 1 − (/
∂ γ n G)aβα2 ···α 1
s− 2
s− 2
s− 2
2
2
X
=
∂α2 2 α3 (γ β ψ)aγβγα4 ···α 1
(A.14)
s− 2
that vanishes only if
(γ β γ γ γ σ ψ)a βγσα4 ···α
1
s− 2
≡ (γ β ψ)aγβγα4 ···α
s− 1
2
= 0.
(A.15)
We can conclude that in order to obtain a consistent gauge theory we have to force
the HS fields to be double traceless or gamma-triple traceless.
We would like now to show that the traceless condition on integer HS field is important
in order to obtain the correct number of independent polarization a massless integer
spin s field field propagates. We start imposing the de Donder gauge condition
1X
∂α2 φσσα3 ···αs = 0 .
(A.16)
Dα2 ···αs = ∂σ φσα2 ···αs −
2
Then the HS equation of motion reduces to the Klein Gordon one
2φα1 ···αs = 0.
(A.17)
Note now that under (A.1) de Donder gauge transforms as
1X
δ(∂σ φσα2 ···αs −
∂α2 φσσα3 ···αs ) = 2ξα1 ···αs−1
(A.18)
2
so we have further gauge freedom if 2ξα1 ···αs−1 = 0. We are ready now to count the
number of degree of freedom the HS field propagates; a totally symmetric rank tensor
in D = 4 has 2(s2 + 1) indipendent components. Note now that double traceless
condition on HS implies that de Donder gauge is traceless:
X
Dσ σα1 ···αs−4 ∼
∂α1 φρσ ρσα2 ···αs−4 = 0
(A.19)
and contains s2 independent constraints. Furthermore gauge fields components, not
till fixed, satisfies wave function e.o.m. and we can use residual gauge freedom to
eliminate further s2 independent components. Finally we have 2s2 + 2 − 2s2 = 2
degrees of freedom corresponding to the two polarization of a massless field.
71
The HS action are:
• Bosonic case
Z
1 α1 ···αs
1
βα3 ···αs
σ
D
G σα3 ···αs
SB = d x
φ
Gα1 ···αs − s(s − 1) φβ
2
8
(A.20)
This action enjoys local symmetry (A.1) only if the gauge parameter is traceless
and the gauge field double traceless.
• Fermionic case
R D 1 α1 ···αs− 1
α2 ···αs− 1 n
1
σ
2G
2 γ γ G
SF =
d x − 2 ψ̄
α1 ···αs− 1 + 4 s ψ̄
β
σβ2 ···αs− 1
2
2
βα3 ···αs− 1
2
Gσσα3 ···α 1 ,
(A.21)
+ 18 s(s − 1)ψ̄β
s− 2
This action enjoys local symmetry (A.2) only if the gauge parameter is gammatraceless the gauge field gamma-tripletraceless.
The presence of constraint on HS field and on gauge parameter suggest that the formulation discussed above is incomplete.
Different approach to analyze this problem have been suggested in literature. Let us analyze
them briefly:
- In order to remove trecelessness constraints has been proposed to add an appropriate
number of auxiliary fields, satisfying certain equation of motion. The main advantage
of this approach is the fact that HS equation of motion remain lagrangian [69] [39].
- Another interesting way out was proposed by Francia and Sagnotti in [32] [34]. The
key idea of this approach is the renounce of the locality of the theory. In particular
they have shown that equations of motion for unconstrained HS and the HS actions
can be implemented to an invariant form with respect unconstrained gauge parameter
if enlarged with non local terms.
- Another approach to remove constraints on gauge fields and gauge parameters is based
on the consideration that gauge theory can be easily constructed in terms of the field
strength (HSC). Geometric formulation of free HS have been considered for the first
time many years ago by Bargmann and Wigner [28] .The one advantage one has by
using geometrical approach is the fact that the theory is manifestly gauge invariant in
the unconstrained sense; however HS potential e.o.m. is higher derivative. Otherwise
this does not spoil the unitarity of the theory and it’s possible to show that, upon some
manipulations, HS e.o.m. reduces to the Fronsdal-Labastida one.
In the next section we will focus our attention on the geometrical approach to HS. In particular we will discuss conformal HS theory that is an important and interesting subclass of
HS field whose (at least linearized) equation of motion are conformally invariant.
72
Tour on higher spin gauge theory
A.1
Conformal HS theory in various dimension
In D = 4 symmetric tensors describe all possible higher spin representation of the Poincaré
group; in this case, in fact, all the irreducible massless representation of the Poicaré group
are classified by using the group SO(2) whose Young tableaux is a single rows; this implies
that massless integer HS fields could be represented by totally symmetric tensor.
φα1 ···αs ≡ α1 . . αs
(A.22)
In higher dimension, symmetric tensors do not describe all possible HS fields, and one
has to take into account also tensor with mixed symmetry.
In this section we would like to construct conformal HSC for integer spin, and we will show
how to obtain the linearized HS field e.o.m. (i.e. Fronsdal-Labastida e.o.m).
We will work in even D = 2n dimension and we will use the following notation:
[n] ≡ α[1 · · · αn] ,
(A.23)
stands for n antisymmetrized indices. In particular, in this appendix, two cumulative indices
which denote the same number of antisymmetric indices will be assumed to be symmetric
[n]1 [n]2 = [n]2 [n]1 .
(A.24)
We define the curvature (or the field strength) of a conformal integer spin s as
R[n]1 ···[n]s = Rα11 ···α1n ,··· , αs1 ··· αsn ;
(A.25)
note that its Young diagram is a rectangle with n rows and s columns. The curvature tensor,
by construction, is symmetric under exchange of any two blocks of antisymmetrized indices.
In addition
• it satisfies the algebraic Bianchi identity
R[n+1]1 [n−1]2 [n]3 ···[n]s = 0.
• The curvature is closed (i.e. satisfies the differential Bianchi identity)
∂[β Rα1 ···αn ][n]2 ···[n]s = 0 ,
(A.26)
∂ β Rβ[n−1]1 [n]2 ···[n]s = 0 .
(A.27)
• and is co-closed
Note now that that the Bianchi identity can be written as an exterior derivative acting on
one of the groups of antisymmetric indices of the multiform R[n]1 ···[n]s
∂1 R[n]1 ···[n]s = 0 ,
(A.28)
A.1 Conformal HS theory in various dimension
73
where in our notation ∂i means derivative and the index of the derivative is antisymmetrized
together with the ith group of antisymmetric indices (i.e. exterior derivative acting on the
ith column):
(A.29)
∂i R[n]1 ···[n]i ···[n]s ≡ ∂[αin+1 R[n]1 ···αi1 ···αin ],[n]i+1 ···[n]s ;
note now that
∂i ∂j = (1 − δij )∂j ∂i .
(A.30)
We introduce now the differential operator
∂≡
s
Y
∂i .
(A.31)
i=1
It’s not hard to convince ourselves, since ∂i2 = 0, that
∂ s+1 ≡ ∂i ∂ = 0
∀ i = 1, ..., s .
(A.32)
The generalized Poincaré Lemma [45], and Bianchi identity (A.26) implies that (at least
locally) the curvature can be written as the s-th derivative of a potential:
R[n]1 ···[n]s = ∂1 · · · ∂s φ[n−1]1 ···[n−1]s ,
(A.33)
where the field φ ≡ φ[n−1]1 ···[n−1]s is the conformal HS gauge potential of integer spin s and
is characterized by the rectangular Young diagram s ⊗ (n − 1) and satisfies the algebraic
Bianchi identity
φ[n] [n−2] [n−1] ···[n−1] = 0.
1
2
3
s
One can observe that (A.33) is the generalization to the higher spin s case of the well known
expression of the electromagnetic field strength in terms of the potential F = ∂ A.
HSC, due to (A.30), are invariant under the following gauge transformations of the gauge
potential [45]
δφ[n−1]1 ··· [n−1]s =
s
X
∂i ξ[n−1]1 ··· [n−2]i ··· [n−1]s ,
(A.34)
i=1
where ξ(x) is an unconstrained gauge function characterized by the Young tableaux (s, · · · , s, s−
1) ⊗ [n − 1].
Equation of motion are obtained by imposing tracelessness condition of the curvature tensor
tr R[n]1 ···[n]s = 0 .
(A.35)
Note that this is the field equation that generalizes the linearized Einstein equation Rαβ = 0.
In terms of the gauge field potential, equation (A.35) implies a generalization of the spin 3
Damour-Deser identity [68]
tr R[n]1 ···[n]s = ∂1 · · · ∂s−2 G[n−1]1 ···[n−1]s−2 [n−1]s−1 [n−1]s = 0
(A.36)
74
Tour on higher spin gauge theory
where G is the kinetic operator acting on the gauge field potential
G[n−1]1 ··· [n−1]s = φ[n−1]1 ··· [n−1]s −
s
X
∂i ∂ β φ[n−1]1 ··· , β[n−2]i ,··· [n−1]s
i=1
s
X
+
∂i ∂j η αβ φ[n−1]1 ··· ,α[n−2]i ,··· ,β[n−2]j ,··· [n−1]s ,
(A.37)
j>i=1
The left hand side of eq. (A.36) vanishes and this implies that G is ∂ s−2 -closed and the
generalized Poincaré lemma [45] implies that G is also ∂ 3 -exact
s
X
G[n−1]1 ··· [n−1]s =
∂i ∂j ∂k ρ[n−1]1 ··· ,[n−2]i ,··· ,[n−2]j ,··· [n−2]k ,··· [n−1]s ,
(A.38)
k>j>i=1
where we have introduced the so called ‘compensator’ field since its gauge transformation
compensates the non-invariance of the kinetic operator G(x) under the unconstrained local
variations (A.34) of the gauge field potential φ(x).
The gauge variation of G(x) is
δG[n−1]1 ··· [n−1]s =
s
X
∂i ∂j ∂k η αβ ξ[n−1]1 ··· ,[n−2]i ,··· ,[n−2]j α,··· [n−2]k β,··· [n−1]s
(A.39)
k>j>i=1
and it is compensated by the gauge shift of the field ρ(x) with the trace of the gauge
parameter
δρ[n−2]1 [n−2]2 [n−2]3 [n−1]4 ··· [n−1]s = η αβ ξ[n−2]1 α , [n−2]2 β, [n−2]3 [n−1]4 ··· [n−1]s .
(A.40)
Now we can gauge fix the compensator to zero. Then the equations of motion of the gauge
field φ(x) become the second order differential equations of Fronsdal-Labastida, which generalize those of Fronsdal for mixed symmetry fields
G[n−1]1 ··· [n−1]s = G φ[n−1]1 ··· [n−1]s = 0 .
(A.41)
where G is the Fronsdal-Labastida kinetic operator, that in D = 4 coincide with the Fronsdal
one (A.4). Note also that (A.41) is invariant under the gauge transformations (A.34) only if
ξ(x) is tracelss and the HS gauge field is double traceless.
A.2
Integer HS in (A)dS4
We intend now to analyze the dynamics of HS in (A)dS4 background. In literature this
problem has been extensively studied from the linearized point of view. In literature it is
not yet clear how to derive the Fronsdal e.o.m. in (A)dS4 from the (A)dS HSC since in
maximally symmetric space an extension of the generalized Poincaré Lemma is still lacking
and one cannot use it as a guide line.
A.2 Integer HS in (A)dS4
75
We will focus now our attention on integer spin in (A)dS4 . We assume also that the HS
gauge field is double traceless and the gauge parameter is traceless.
In order to become familiar with our notation let us recall that covariant derivatives don’t
commute and in (A)dS one finds that
Rµνρσ = b(gµρ gνσ − gµσ gνρ )
→
[∇µ , ∇ν ]Vρ = Rµνρσ Vσ = b(gµρ Vν − gνρ Vµ ) (A.42)
for some vector V µ ; we will replace now, as usual, ordinary derivative with the covariant one
∂α → ∇µ in (A.3). Non commutativity implies that the covariant Fronsdal kinetic operator
X
X
2
ρ
Gcov
=
∇
φ
−
∇
∇
φ
+
{∇µ1 , ∇µ2 }φρ ρµ3 ...µs
(A.43)
µ1 ...φs
µ1 ρ
µ1 ...µs
µ2 ···µs
is not invariant with a suitable covariantization of (A.1):
δφµ1 ···µs (x) = ∇{µ1 ξµ2 ···µs }
(A.44)
In order to solve this problem one has to add a new covariant term that cancels the ”extra
contributions”. Explicitly one obtains
X
cov
2
(A.45)
GAdS
=
G
+
b[s
−
2(s
+
1)]φ
+
2b
gµ1 µ2 φρ ρµ3 ...µs
µ1 ...µs
µ1 ...µs
µ1 ...µs
and the Fronsdal kinetic operator (A.4) in (A)dS4 becomes
G(A)dS ≡ (∇2 − div grad + div 2 tr + 2b g tr + bAs )
(A.46)
As = s2 − 2(s + 1).
(A.47)
where
and the operator g acts on a completely symmetric tensor Vµ3 ···µs as:
X
gVµ3 ···µs =
gµ1 µ2 Vµ3 ···µs .
(A.48)
Note now that in (A.46) a mass term associated to the curvature appears. Further analysis
about massless HS and partially massless HS in (A)dS can be found in [41].
76
Tour on higher spin gauge theory
Appendix B
Koszul-Tate alghoritm and gauge
fixed action
In this appendix we will describe a very useful technique to construct BRST charges, known
as Koszul-Tate alghoritm. In particular we will use it to study an interesting class of non lie
superalgebra, and to construct SO(N ) spinning particle in (A)dS background gauge fixed
action on the circle. More details about our analysis can be found in [53].
B.1
Koszul-Tate differential
We start considering Hamiltonian system with first class constraints
GA = 0 .
(B.1)
Let z F be a set of phase space variable (including bosons fermions and gauge fields); we
extend now this space by introducing as many ghost η A and ghost momenta PA as there are
constraints GA , and with opposite grading:
(GA ) = A
(PA ) = A + 1
(η A ) = A + 1 .
In this extended phase space the Koszul-Tate differential δ acts in the following way:
δz M = 0
δη A = 0
δPA = −GA ;
(B.2)
on an arbitrary polymonial in the PA ’s, δ acts as an odd-right derivative. Since δ 2 vanishes
on all the generators, δ is nilpotent. We use now this operator to construct the BRST
generator Qbrst .
To this aim let us write
X
Qbrst ≡
Ωp
(B.3)
p≥0
78
Koszul-Tate alghoritm and gauge fixed action
where Ω0 = η A GA and
Ωp = U a1 ...ap Pap .....Pa1
a ...ap
.
U a1 ...ap = η b1 ...η bp+1 Ub11...bp+1
a ...a
p
” are only functions of the original phase space. Now one can rewrite
The coefficents ”Ub11...bp+1
the nilpotency condition {Qbrst , Qbrst }pb = 0 as a set of equation for the unknown function
Ωp :
δΩp+1 = −Dp
(B.4)
where
p
2D =
p
X
k=0
k
{Ω , Ω
p−k
}pb +
p−1
X
{Ωk+1 , Ωp−k }ghost .
(B.5)
k=0
In the previous formula {, }pb refers to the poisson bracket in the original phase space (without
ghost), whereas {, }ghost refers to the poisson bracket acting only on the ghost and ghost
momenta:
{PA , η B }ghost = −(−)(B +1)(A +1) {η B , PA }ghost = −δAB
(B.6)
B.2
Non linear supersymmetry algebra
Let Li and Tα be some bosonic and fermionic generators respectively and the superalgebra
{Li , Lj }pb = Cijk Lk
β
{Li , Tα }pb = Ciα
Tβ
ij
i
{Tα , Tβ }pb = Cαβ
Li + Dαβ
L i Lj
(B.7)
β
ij
i
with Cijk Ciα
Cαβ
Dαβ
some structure constants.
We would like now to use the Koszul-Tate algorithm described in the previous section to
construct the BRST generator associated to this algebra.
First of all we use the generalized Jacobi identity in order to extract the following very useful
relations:
p
m p
m p
Cijm Cmk
+ (−)k ij Cki
Cmj + (−)i kj Cjk
Cmi = 0
β
β
σ
σ
σ
Cijβ Cβα
+ (−)α ij Cαi
Cβj
+ (−)i αj Cjα
Cβi
=0
j
σ
k
σ
k
Ciα
Cσβ
+ (−)β iα Cβi
Cσα
+ (−)i αβ Cαβ
Cjik = 0
i
σ
i
σ
i
+ (−)α δβ Cβδ
Ciα
=0
Cαβ
Ciδσ + (−)δ αβ Cδα
Ciβ
kj
j
kp
kj
+ (−)p (1+αβij )j αβ Cpi
Dαβ
C σiα Dσβ
+ (−)αβ +1 C σiβ Dσα
jp
k
+(−)p (1+αβik )+k αβ Cpi
Dαβ
=0
ij
ij
ij
σ
σ
σ
Dαβ
Cjδ
+ (−)δ αβ Dδα
Cjβ
+ (−)α βδ Dβδ
Cjα
=0
(B.8)
B.3 Spinning particle in (A)ds: gauge fixing
79
where i..j = i + .. + j .
Let us define A = (i, α) and GA = (Li , Tα ); our starting point is the linear term
Ω0 = η A G A .
(B.9)
We use now (B.4) and relations (B.8), and we obtain:
1
1
ij
D
PD − (−)α η α η β Dβα
Li Pj
Ω1 = − (−)A η A η B CBA
2
2
Ω2 = 0
1
ij
kl t
Dσδ
Cik Pt Pl Pj
Ω3 = − (−)αδl +i kl δσ ij η δ η σ η α η β Dβα
24
Ωp = 0
∀p > 3
We conclude that the Qbrst associated to the algebra (B.7) reads:
1
1
ij
D
Qbrst = − (−)A η A η B CBA
PD − (−)α η α η β Dβα
Li Pj
2
2
1
ij
kl t
− (−)αδl +i kl δσ ij η δ η σ η α η β Dβα
Dσδ
Cik Pt Pl Pj
24
B.3
(B.10)
Spinning particle in (A)ds: gauge fixing
We would like now to use the Koszul-Tate algorithm to analyze spinning particle model in
(A)dS space.1 We rewrite the action principle (1.43) using euclidean convention2
Z
1
1
S =
dτ [ gµν (ẋµ − χi ψiα eµα )(ẋν − χi ψiα eνα ) + ψiα (ψ̇iα − aij ψjα + ẋµ ωµαβ ψiβ )
2e
2
1
− eψiα ψiβ ψjγ ψjδ Rαβγδ ].
8
This action enjoys local supersymmetry (gauge parameter i ) local SO(N ) (gauge parameter αij ) and worldline diffeomorphism (gauge parameter ξ). The corresponding constraints
Qi , Jij , H are first class and the algebra reads
b
{Qi , Qj } = −2iHδij + ibJik Jjk − i Jlk Jlk δij
2
{Qi , Jjk } = 2δk[i Qj]
{Jij , Jlm } = SO(N ) algebra .
(B.11)
(B.12)
(B.13)
The gauge field transforms as:
δe = ξ˙ + 2χi i
δχi = ˙i − aij j + αij χj
δaij = α̇ij + αim amj − αjm ami
b
b
− ψjα ψkα (χi k + χk i ) + ψiα ψkα (χj k + χk j ) − bψiα ψjα χk k
2
2
1
(B.14)
We recall to the reader that in our convention (A)dS curvature is Rµνρσ = b(gµρ gνσ − gµσ gνρ ).
We Wick rotate the time as t → −iτ and aij → iaij ; note that we rotate also the gauge parameters
i → −ii and ξ → −iξ.
2
80
Koszul-Tate alghoritm and gauge fixed action
One can note that, except for the susy aij transformation rule (that we have emphasized
writing it in red), relations (B.14) coincides with the one we have discussed in the flat case
in Chapter 3 (3.2).
Let us restrict ourselves to periodic boundary conditions for bosons and antiperiodic
boundary conditions for fermions and this let us gauge fix gravtini to zero, as was extensively
discussed in Chapter 3. It is interesting to note that when gravitini vanish one leaves with
the same flat case transformation rules for the SO(N ) gauge fields and for the einbein e.
This simplify strongly the rest of the discussion; by using the same arguments we have used
in Chapter 3 one can bring the gauge configuration (e, χ, aij ) = (β, 0, âij ) where





âij = 




0 θ1 0
0
−θ1 0
0
0
0
0
0 θ2
0
0 −θ2 0
.
.
.
.
0
0
0
0
0
0
0
0
. 0
0
. 0
0
. 0
0
. 0
0
.
.
.
. 0 θr
. −θr 0





 .




(B.15)
for even N = 2r and






âij = 





0 θ1 0
0
−θ1 0
0
0
0
0
0 θ2
0
0 −θ2 0
.
.
.
.
0
0
0
0
0
0
0
0
0
0
0
0
. 0
0
. 0
0
. 0
0
. 0
0
.
.
.
. 0 θr
. −θr 0
. 0
0
0
0
0
0
.
0
0
0






 .





(B.16)
for odd N = 2r − 1.
We would like now to construct the gauge fixed action. The extended phase space contains
the original variables (xµ , ψiα ), the gauge field λA = (e, χi , aij ) and their momenta πA =
(πe , πχi , πaij ) and the ghost (η 1 , η 2i , η 3ij ), and ghost momenta (P1 , P2i , P3ij ) associated with
the first class constraints (H, Qi , Jij ).
From equations (B.4) and (B.8)we find explicitly
i 2i 2j 1
η η ( δij δpk δql − δik δpj δql )Jkl P3pq
2r2
2
2i 2
2i 3ij
im mj
−i(η ) P1 − η η P2j − η η P3ij
1 2i 2j 2t 2s 1
1
η
η
η
η
(
δ
δ
δ
−
δ
δ
δ
)(
δts δkr δzw − δsk δtr δzw )(δv[z δk][q δp]h )P3hv P3rw P3lm
+
ij
pl
qm
jp
qm
li
24r4
2
2
1
1
= η 1 H + η 2i Qi + η 3ij Jij + 2 η 2i η 2j ( δij δpk δql − δik δpj δql )Jkl P3pq
2r
2
−i(η 2i )2 P1 − η 2i η 3ij P2j − η im η mj P3ij
1 2i 2j 2t 2s
+
η η η η (δij δts P3hv P3mv P3lm − 3δts P3jv P3mv P3im )
(B.17)
24r4
Qbrst = η 1 H + η 2i Qi + η 3ij Jij +
B.3 Spinning particle in (A)ds: gauge fixing
81
In order to implement the ”temporal gauge fixing” (e, χi , aij ) = (β, 0, âij ), we choose the
gauge fixing fermion
K = −βP1 − âij P3ij
(B.18)
The gauge fixed Hamiltonian reads:
(B.19)
H GaugeF ixed ≡ H + [K, Qbrst ]
⇓
H GaugeF ixed = H + βH + âij Jij − âij η2i P2j
(B.20)
We are now ready to write the gauge fixed action and in particular we obtain
S
GaugeF ixed
F
A
Z
[z , e, χi , aij , η , PA ] =
h
i
dτ ẋµ pµ + iψ̇ia ψib ηab + η̇ A PA − H GaugeF ixed
= S[z F , e = β, χi = 0, aij = âij , PA ]
Z
∂
+ dτ P 2i (
− âij )η 2j .
∂τ
(B.21)
The last term is exactly the Faddeev-Popov determinant. This is not obvious at all because
the algebra is non linear; otherwise one can notes that the higher order terms vanish when
we gauge fix gravitini to zero, thus everything reduces to the usual Lie algebra case.
82
Koszul-Tate alghoritm and gauge fixed action
Appendix C
Summing with Mr. Pochhammer
In this appendix we will solve explicitly the series (4.93). The results we will obtain plays
a fundamental role in constructing higher spin curvature, and reveals also a funny and
interesting mathematical structure.
C.1
Pochhammer function
We start defining P ochhammer function. Let k, l, m, i ∈ N and x, y ∈ R. We define the
function P (x, k) as:
k
Y
P (x, k) =
(x − i)
(C.1)
i=0
that is related to the well known P ochhammer function P̃ (x, k) by the relation
P (x, k) = (−)k+1 P̃ (−x, k + 1);
(C.2)
for matter of convenience we use P instead of P̃ and, sometimes, we refer to it as ”modified
P occhammer function”. By definition one also has:
P (x, −1) = 1
∀x ∈ R;
(C.3)
by using (C.1) and (C.3) it’s not hard to prove that P (k, l) satisfies the following relations:
Property 1:
m
X
k=1
P (k, l) =
1
P (m + 1, l + 1)
l+2
(C.4)
Property 2:
P (S + 2k, 2k) = P (S + 2(k − l), 2(k − l))P (S + 2k, 2l − 1)
(C.5)
84
Summing with Mr. Pochhammer
C.2
Solution by iteration
We consider now the function a2n (s), defined in (4.94) as
a2n (s) ≡
k0
s−1 X
X
kn−2
s−3
X
···
=
kn−2
s−3
X
=
X
(s − 2n + 1 − mn )
mn −1=1 kn−1 =mn−1 −1 mn =1
kn−2
s−3
X
1
kn−1
s−2n−3
X
X
···
k0 =1 m1 =1 k1 =m1 −1
k0
s−1 X
X
s−2n−1
X
X
mn −1=1 kn−1 =mn−1 −1 mn =1 kn =mn −1
k0 =1 m1 =1 k1 =m1 −1
k0
s−1 X
X
kn−1
s−2n−3
X
X
···
k0 =1 m1 =1 k1 =m1 −1
s−2n−3
X
X
mn −1=1 kn−1 =mn−1
1
kn−1 (s − 2n) − kn−1 (kn−1 − 1) .
| {z }
{z
}
2|
−1
\
]
(C.6)
Note that in the last line of the previous equation we recognize the modified P ochhammer
functions P (kn , 0) (\) and P (kn , 1) (]). This observation plays a key role in our analysis.
Let us start fixing the notation. We introduce the step parameter z = 0, 1, ..., n − 1; in our
language doing a step means that we solve a couple of summatories :
kµ−1 s−2(µ)−1
···
X
X
···
µ = n − z = 0, 1, ..., n − 1;
(C.7)
mµ =1 kµ =mµ −1
|
{z
1 STEP
}
equivalently we define the z th step as the result we obtain after having solved z couple of
sommatories (i.e. after z step)
kn−z
s−2(n−z+1)−1
kn−2
X
X
X
···
s−2n−3
X
kn−1
s−2n−1
X
X
mn−z+1 =1 kn−z+1 =mn−z+1 −1
mn −1=1 kn−1 =mn−1 −1 mn =1 kn =mn −1
|
|
{z
z th couple
}
{z
2nd couple
}|
{z
1st couple
1.
(C.8)
}
Sometimes, in the following, we prefer to use the shortcut notation
P (kn−z , q) = [q],
(C.9)
and the step parameter (i.e. n − z) will be clear from the context.
In order to become familiar with this language let us reanalyze, from this point of view,
equation (4.82). In the first line of (4.82) we start summing ”1” that could be thought
as P (kn , −1). In the third line we have a linear combination of P (kn−1 , 0) and P (kn−1 , 1).
Formally we say that [−1] generates [0] and [1]1 ; this could be resumed in the following
1
In the first part of this analysis we don’t care about the coefficients in front of the P functions. we will
fix them later.
C.2 Solution by iteration
85
diagram:
[0]
{=
{
{{
{{
{{
[−1]
CC
CC
CC
CC
!
(C.10)
[1]
The above consideration could be generalized to every step z; by using the Property 1,
infact, we have:
kn−z−1
s−2(n−z)−1
X
X
P (kn−z , q) =
mn−z =1 kn−z =mn−z −1
1
P (s − 2(n − z), q + 1)P (kn−z−1 , 0)
q+2
−
(q + 1)!
P (kn−z−1 , q + 2).
(q + 3)!
(C.11)
The previous formula implies that, for every [q] and z, the modified P ochhammer function
[q] generates in the (z + 1)th step [0] and [q + 2]:
(C.12)
< [0]
yy
y
y
yy
yy
[q]
EE
EE
EE
EE
"
[q + 2]
Start to become clear that after z step we leave with a linear combination of [q] function;
this could be understood better looking at the diagramm (C.1):
At this point it’s not hard to convince ourselves that, after z steps, the parameter q can
take the values 2q̃ with q̃ = 0, 1, ..., z − 1, or 2z − 1. In particular at the z th step we obtain:
z−1
X
C(z, 2q̃)[2q̃] + C(z, 2z − 1)[2z − 1]
(C.13)
q̃=0
for some function C(·, ·) that we have to evaluate step by step. From the diagram (C.1) we
can observe that the ”odd” coefficient C(z, 2z − 1) is obtained by itering z times the second
line of equation (C.11). In particular one finds:
C(z, 2z − 1) =
(−)z
.
(2z)!
(C.14)
Computing the other coefficients seems to be too hard because at every step we double the
number of [q] functions. Fortunately we can simplify the computation by using the following
Lemma:
86
Summing with Mr. Pochhammer
[0] · · ·
z<
z
zz
zz
zz
[0]
G DD
DD
DD
DD
"
[2] · · ·
@ [0]00
00
00
00
[0] · · ·
00
z<
z
0
z
00
z
0 zzzz
[2]
DD
DD
DD
DD
"
[4] · · ·
[−1]
<<
<<
<<
<<
<<
[0] · · ·
<
<<
zz
<<
z
z
<<
zz
<<
zz
<<
[0]
G DD
<<
DD
<<
DD
<<
DD
<<
"
<<
[2] · · ·
<<
<<
<< [1]
00
00
00
00
[0] · · ·
00
<
zz
00
z
z
00
zz
zz
[3]
DD
DD
DD
DD
"
[5] · · ·
Figure C.1: Pochhammer family tree
C.2 Solution by iteration
87
Lemma 4.
At every step z one has
C(z, 0) = g(z)P (s − 2(n − z + 1), 2(z − 1))
∀z = 0, 1, ..., n − 1
(C.15)
for some function g(z).
Proof:
We prove this Lemma by induction.2
Using equation (C.11) and the Property 2 one can verify explicitly that for z = 0, 1, 2, 3
equation (C.15) is satisfied (base of induction porcedure).
Let us now suppose that (C.15) is satisfied till the (z − 1)th step (induction HP ). This assumption has an interesting consequence; it implies, infact, that the diagram (C.1) collaps
into (C.2).
In particular, the modified P ochhammer function P (kn−z , 0) is generated by P (kn−z−1 , 2q̃)
and P (kn−z−1 , 2z − 3). Observing carefully the diagram (C.2) we learn that the coefficients P (kn−z−1 , 2q̃) could be obtained just by applying q̃ times the second line of (C.11) to
C(z − 1 − q̃, 0). So we can conclude that
C(z − 1, 2q̃) = C(z − 1 − q̃, 0)
≡
(−)q̃
(2q̃ + 1)!
(−)q̃
g(z − 1 − q̃)P (s − 2(n − z + q̃ + 2), 2(z − q̃ − 2)) (C.16)
(2q̃ + 1)!
Using the first line of (C.11) and the Property 2 it’s easy to show that going from z −1 ⇒ z
one obtains (induction step)
C(z, 0) v P (s − 2(n − z + 1), 2q̃ + 1)P (s − 2(n − z + q̃ + 2), 2(z − q̃ − 2))
(C.17)
v P (s − 2(n − z + 1), 2(z − 1))
Let me emphasize that in the previous formula we don’t care about the contribution arising
from the odd modified P occhammer function [2z − 3]. Otherwise it is still proportional to
P (s − 2(n − z + 1), 2(z − 1)). This conclude the proof.
2
In particular we are going to use the so called ”extended” induction principle. Let P rop(n) be some
preposition depending on n ∈ N.
1. P rop(n0 ) is true for some n0 ∈ N (base of induction porcedure)
2. we suppose now that ∀k > no P rop(k) is true (induction HP )
3. if P rop(k) true ⇒ P rop(k + 1) true (induction step)
we can conclude that P rop(n) is true for every n ∈ N.
88
Summing with Mr. Pochhammer
[6]
}>
}
}}
}}
}}
[4]
[4]]
}>
}>
}
}
}
}}
}}
}}
}}
}}
[2]
[2]
[2]
}>
}>
}>
}
}
}
}
}
}}
}}
}}
}}
}}
}}
}}
/ [0]
/ [0]
/ [0]
/ [0]
AA
AA
AA
AA
AA
AA
AA
AA
[−1]
[1]
AA
AA
AA
A
[3]
AA
AA
AA
A
[5]
AA
AA
AA
AA
[7]
Figure C.2: Pochhammer family tree modified. Note that we have used the red square
bracket [0] to emphasize that the coefficient that multiply the function [0], at every step, is
just summed by using the hypothesis of the Lemma 4.
C.2 Solution by iteration
89
We resume now the results obtained from the Lemma4:
(−)z
C(z, 2z − 1) =
(2z)!
(C.18)
(−)q̃
(2q̃ + 1)!
C(z, 0) = g(z)P (s − 2(n − z + 1), 2(z − 1)) ;
C(z, 2q̃) = C(z − q̃, 0)
(C.19)
(C.20)
in particular we also find that:
g(z) =
z−2
X
q=0
(−)z−1
(−)q
g(z − 1 − q) +
(2q + 2)!
(2z − 1)!
with
g(0) = 0 .
(C.21)
We are ready now to calculate a2n (s). After n step equation (C.11) reduces to:
a2n (s) = g(n)P (s − 2n, 2(n − 1))
s−1
X
P (k0 , 0)
k0
.
.
+C(n, 2q̃)
s−1
X
P (k0 , 2q̃)
k0 =1
.
.
s−1
(−)n X
P (k0 , 2n − 1) .
(2n)! k =1
0
(C.22)
We use now Property 1 and Property 2, and (C.18)-(C.20), in order to write the previous
equation in a more compact and elegant form. In particular we can write the first n lines of
the previous equation as
s−1
X
(−)q̃
g(n − q̃)P (s − 2(q̃ + 1), 2(n − q̃ − 1))
P (k0 , 2q̃)
(2q̃ + 1)!
k
0
q̃
=
(−)
g(n − q̃) P (s − 2(q̃ + 1), 2(n − q̃ − 1))P (s, 2q̃ + 1)
|
{z
}
(2q̃ + 2)!
P (s,2n)
=
(−)q̃
g(n − q̃)P (s, 2n)
(2q̃ + 2)!
(C.23)
We substitute now (C.23) into (C.22) and we find:
a2n (s) = f (n)P (s, 2n)
(C.24)
90
Summing with Mr. Pochhammer
where
f (n) =
n−1
X
q̃=0
(−)n
(−)q̃
g(n − q̃) +
.
(2q̃ + 2)!
(2n + 1)!
(C.25)
It’s interesting to note that f (n) is defined in terms of linear combination of function g.
Everything become easier noting that f (n) = g(n + 1).
Finally we have:
a2n (s) = f (n)s(s − 1)(s − 2) · · · (s − 2n + 1)(s − 2n)
(C.26)
where f (n) is defined by recurrence as
f (n) =
n−2
X
q̃=0
(−)q̃
(−)n−1
f (n − q̃ − 1) +
,
(2q̃ + 2)!
(2n − 1)!
(C.27)
and this conclude the first part our analysis of equation (4.82).
In the second part of this appendix we would like to discuss an important property of
the function f (n). We introduce the shortcut notation:
f (n) =
n−2
X
A(q)f (n − q − 1) + C(n)
A(q) =
(−)q
(2q+2)!
C(n) =
(−)n−1
(2n−1)!
(C.28)
with
q=0
One can observes that for the first few case the function f (n) satisfies (base of induction
procedure):
Property 3:
n−1
X
f (k + 1)f (n − k) = (2n + 1)f (n + 1)
(C.29)
k=0
We would like to prove that this relation is valid for every n by using only (C.28) and the
initial condition f (0) = 0 (i.e. f (1) = 1). In order to fix the notation and the ideas we start
considering the case f (6). Let us suppose that (C.29) is satisfied for every n 6 4 (induction
HP ), or equivalentlly
n−2
X
f (k + 1)f (n − k − 1) = (2n − 1)f (n)
∀65
(C.30)
k=0
We would like to recognize and isolate terms proportional to (2q − 1)f (q). In particular one
C.2 Solution by iteration
91
has:
11f (6)
=
11
f (5)
2
− 11
f (4)
4!
+ 11
f (3)
6!
− 11
f (2)
8!
11
+ 10!
f (1)
=
f (5)f (1)
+C(2)f (4)
+C(3)f (3)
+C(4)f (2)
+C(5)f (1)
1
− 10!
+A(0)9f (5) +A(1)7f (4) +A(2)5f (3) +A(3)3f (2) +A(4)3f (1)
Note now that:
• Terms in red looks like f (6 − p)C(p)
• Terms in blue looks like f (p)(2p − 1)
We apply now the induction HP (C.30) on the blue terms and we can observe that
everything collapses into
f (1) (C(5) + A(0)f (4) + A(1)f (3) + A(2)f (2) + A(3)f (1))
|
{z
}
f (5)
+ f (2) (C(4) + A(0)f (3) + A(1)f (2) + A(2)f (1))
|
{z
}
f (4)
+ f (3) (C(3) + A(0)f (2) + A(1)f (1))
|
{z
}
f (3)
+ f (4) (C(2) + A(0)f (1))
|
{z
}
f (2)
+ f (5) (C(1)
| {z }
(C.31)
f (1)
Explicitly one finds the desired result
f (6) = (f (5)f (1) + f (4)f (2) + f (3)f (3) + f (2)f (4) + f (1)f (5))
(C.32)
We are ready now to attack the general n case. In analogy with the example discussed above
we suppose that
m−2
X
f (k + 1)f (m − k − 1) = (2m − 1)f (m)
∀ m 6 n − 1.
(C.33)
k=0
Another ingredient we need is the following relation:
(2n − 1)A(k) = (2(n − k − 1) − 1)A(k) + C(k + 1).
(C.34)
92
Summing with Mr. Pochhammer
We write now f (n) using the previous relation and its definition, in a clever way :
(2n − 1)f (n) =
n−3
X
f (n − k − 1)(2(n − k − 1) − 1)A(k) +
n−2
X
f (k + 1)C(n − k − 1). (C.35)
k=0
k=0
We apply now the induction HP on the blue part of the previous relation
n−3 n−k−3
X
X
k=0
A(k)f (n − k − q − 2)f (q + 1) +
q=0
n−2
X
f (k + 1)C(n − k − 1).
(C.36)
k=0
It’s very important to note that
n−3 n−k−3
X
X
k=0
A(k)f (n − k − q − 2)f (q + 1) =
q=0
n−3
X
f (k + 1)
n−k−3
X
A(q)f (n − k − q − 2) (C.37)
q=0
k=0
Substituting (C.37) and (C.36) into (C.35) we obtain:
(2n − 1)f (n) =
n−3
X
f (k + 1) C(n − k − 1) +
q=0
k=0
|
=
n−2
X
n−k−3
X
f (k + 1)f (n − k − 1)
k=0
and this conclude our proof.
A(q)f (n − k − q − 2) + f (n − 1) C(1)
| {z }
f ((1)
{z
}
f (n−k−1)
(C.38)
Conclusions and outlook
In this thesis we have studied spinning particles with an SO(N ) extended supergravity multiplet on the worldline. In particular we have analyzed the symmetries of these models with
flat and curved background and we have shown explicitly that in maximally symmetric space
the first class constraints algebra, for every N , turns out to be closed, yet non linear; this
approach has produced, as a byproduct, a more general coupling in D = 2 with certains
spaces with non constant curvatures. An action principle with gauge symmetries (i.e. local
symmetries), contains a 1st class constraints algebra, but it need not be necessary a Lie algebra but could be more complicated. It would be interesting, by using the idea we have used
in Chapter 1, to study SO(N ) spinning particles coupled to a more general background.
In order to preserve local symmetries one can try to close SO(N ) extended supersymmetry
algebra by admitting structure functions, instead of structure constant, or a more general
non linear structure of the algebra.
This problem might be overcome also by adding one more constraint (constructed for example with four Grassmann variables), but note that at this point, in general, the physical
meaning of the model could be modified and become less interesting.
In Chapter 2 we have shown that, in every even space time dimensions D = 2n and
for every even N , a canonical analysis of SO(N ) spinning particles with flat background,
produces as physical sector of the Hilbert space, tensors with mixed symmetry, and in particular with the symmetries of a rectangular Young tableaux with s columns and n rows
(i.e. s ⊗ n Young tableaux). These tensors have to be closed and co-closed and we have
recognized them as the conformal higher spin curvatures (HSC) describing the dynamic of
conformal higher spin fields (HS) whose Young tableaux are s ⊗ (n − 1) rectangles. We
have also solved, by using the generalized Poincaré Lemma, the differential Bianchi identity (i.e. closing condition) and we have shown that the traceless condition on the HSC
(that arise naturally from the quantization of spinning particles) produces an higher order
derivative equation of motion for the HS fields. These equations, in virtue of the generalized Poincaré Lemma and after having introduced and gauge fixed to zero the compensator
field, can be linearized; this analysis produces in particular, the Fronsdal-Labastida equation
of motion, describing the dynamics of massless HS with mixed symmetry, and the correct
traceless condition and double traceless conditions on the gauge parameters and gauge fields.
In Chapter 3 we have studied the one loop quantization of SO(N ) spinning particles
on the circle. We have considered propagation on a flat target space time and obtained the
measure on the moduli space of the SO(N ) supergravity on the circle. We have used it to
94
Conclusions and outlook
compute the propagating physical degrees of freedom for every N and every D, described
by the spinning particles. In particular we have shown explicitly that spinning particles do
not contain physical degrees of freedom in space times of odd dimensions. It would be very
interesting, in a future project, to study the one loop quantization of a bigger class of models,
orthosymplectic spinning particle models, describing tensors with mixed symmetry [70].
Let us also emphasize that the correct measure on the moduli space of the SO(N ) supergravity multiplet on the circle, we have obtained in this work, is necessary for computing more
general quantum corrections arising when couplings to background fields are introduced.
SO(N ) spinning particles, in fact, can consistently propagate in maximally symmetric space.
In Chapter 4 we have studied the physical contents of SO(N ) spinning particles on
(A)dS backgrounds, via a canonical analysis. We have focused our attention on the integer
spin case in every even dimension D = 2n and we have shown explicitly that the quantization
á la Dirac produces (A)dS HSC as the physical states of the Hilbert space.
The dynamics of massless integer HS in (A)dS background is described by the generalization
of the Fronsdal-Labastida equation of motion in maximally symmetric space. However in
the literature it is not yet clear how to derive these equations of motion starting from
pure geometrical object (i.e. HSC); in curved space, in order to solve the differential Bianchi
identity, one cannot use the generalized Poincaré Lemma as a guide line (covariant derivative
do not commute). For this reasons we have proposed a different approach; we have written
the (A)dS HSC |Ri as
|Ri =
[s/2]
X
(ib)m rm (s)Rm (s) |φi
m=0
where in this case [s/2] means integer part, Rm (s) is an operator containing s−2m covariant
derivatives and |φi is the HS gauge potential. We have then imposed the Bianchi identity to
fix the functions rm (s); these functions have been carefully evaluated by using the Pochhammer functions in Appendix C and we have found an elegant way to write them by using
recursive relation with themselves.
More in general during this work, we have proposed a solution of the ”∇-closed” condition
of multiforms with the symmetries of rectangular s ⊗ n Young tableaux, living in maximally
symmetric space; thus could be very interesting to study possible extensions to maximally
symmetric space of the generalized Poincaré Lemma, by using our results as a guideline.
In the cases of N = 4, 6, 8 we have solved explicitly the differential Bianchi identity
and we have shown that traceless condition on the HSC produces the Fronsdal-Labastida
equation of motion in (A)dS with the correct traceless and double traceless constraint on
the gauge parameter and HS gauge potential.
It would be interesting, for further analysis, impose the traceless condition on the (A)dS
HSC for every integer spin. We expect that the higher derivative equations of motion for HS
gauge fields reduce to a suitable generalization to curved space of the ”∂ s−2 ” closed condition
Conclusions and outlook
95
(A)dS
for the Fronsdal-Labastida operator Gs
that we write as
[s/2−1]
X
0
(ib)m rm
(s)Rm (s)G(A)dS
|φi = 0
s
m=0
0
(s); at this point one can try to introduce the compensator field and
for some functions rm
gauge fix it to zero and obtain the Fronsdal-Labastida equation of motion in (A)dS background
G(A)dS
|φi = 0 .
s
After having checked explicitly that SO(N ) spinning particle models describe, also in maximally symmetric background, the dynamics of conformal HS fields, it would be interesting
to calculate the one loop partition function on the circle. To this aim we have constructed
in Appendix B, by using the Koszul-Tate algorithm, the BRST charge associated to the
SO(N ) extended supersymmetry algebra, that, in maximally symmetric space is a quadratic
superalgebra; moreover we have gauge fixed the SO(N ) supergravity multiplet on the circle,
and we have used the BRST charge to construct the gauge fixed action.
Note that in D = 4 flat target space, massless HS fields propagate 2 degrees of freedom
associated to the two independent polarization, while massive fields with arbitrary spin s
propagate 2s + 1 independent polarizations. In AdS4 partially massless states appear. These
states propagate, for some particular values of the mass m and the cosmological constant λ,
2s + 1 − 2c degrees of freedom, for some c = 0, 1, 2, ..., s that is called step parameter [41].
For all the reasons discussed above another possible direction in which this work might
be developed is the analysis of the massive SO(N ) spinning particles in (A)dS target space.
The computation of the partition function and the calculation of the degrees of freedom
should be very interesting in order to study partially massless states in D = 4 and more in
general in every dimension D = 2n.
In Chapter 5 we have constructed N = 4 one dimensional supergravity models on Hyper Kähler (HK) and quaternionic Kähler (QK) manifolds. In particular we have analyzed
the symmetries of these models and in the QK background case, we have constructed the
BRST charge and the gauge fixed action on the circle. Thus, in the future, should be very
interesting calculate the one loop partition function.
HK manifold endowed with homothetic killing vector have been extensively studied in
literature (see for example [65] [71] and references therein); this is, in fact, an important
subclass of HK manifolds admitting conformal symmetry and we refer to them as Hyper
Kähler Cone (HKC). A map from a 4(n + 1) HKC to a 4n dimensional QK manifold for
bosonic non linear sigma models, have been constructed and analyzed carefully in [72]. This
procedure, known as superconformal reduction, is based on the gauging of the isometry of
the HKC.
96
Conclusions and outlook
In a forthcoming paper [73] we will consider, as a toy model, supersymmetric (non linear)
sigma model on flat and curved D + 1 dimensional cones; in particular we propose a map
for SO(N ) spinning particle and U (1) spinning particle models, from the cone to the D
dimensional base manifold.
It would be interesting to extend our analysis to sp(2) spinning particle models with (4n + 1)
dimensional HKC background, and by gauging conformal and sp(2) isometry, obtain sp(2)
spinning particle models propagating in 4n dimensional QK target space.
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Conclusions and outlook
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