Virasoro algebra and dynamics in the space of univalent functions

Virasoro algebra and dynamics in the space of univalent functions
Contemporary Mathematics
Virasoro algebra and dynamics in the space of
univalent functions
Irina Markina and Alexander Vasil’ev
Abstract. One of the important aspects in recent trends in complex analysis has
been the increasing degree of cross-fertilization between the latter and mathematical
physics with great benefits to both subjects. Contour dynamics in the complex plane
turned to be a meeting point for complex analysts, specialists in stochastic processes,
and mathematical physicists. This was stimulated, first of all, by recent progress in
understanding structures of evolution processes in the space of univalent functions
provided by the classical and stochastic Löwner evolutions, and by the Laplacian
growth. The Virasoro algebra represents a basic algebraic object in conformal field
theory (CFT) so it was not surprising that it turned to play an important role of
a structural skeleton for contour dynamics. The present paper is a survey of recent
progress in the study of the CFT viewpoint on contour dynamics, in particular, we
show how the Witt and Virasoro algebras are related to the stochastic Löwner and
classical Löwner-Kufarev equations.
1. Introduction
Conformal field theory (CFT) in two dimensions has a deep intrinsic connection to
the representation of infinite dimensional algebras. The Virasoro algebra is a vertex
algebra, which appeared in early 1970’s physics papers (see, e.g. [47]) on string
theory. Earlier in 1968 it was introduced by Gelfand and Fuchs [15] as a unique
(up to isomorphisms) central extension of the algebra of vector fields on the unit
circle. Later in 1980’s, it became clear that the Virasoro algebra turned out to be
a universal symmetry algebra in two-dimensional CFT. The infinitesimal conformal
transformations in the classical setup lead to an infinite dimensional algebra, called
the Witt algebra. Turning to quantum field theories, the conformal anomaly, or Weyl
Key words and phrases. Virasoro algebra, Löwner equation, SLE.
This work was completed with the support of the grants of the Norwegian Research Council
#177355/V30, of the European Science Foundation Research Networking Programme HCAA, and
of the NordForsk Network ‘Analysis and Applications’.
c
!0000
(copyright holder)
1
2
IRINA MARKINA AND ALEXANDER VASIL’EV
anomaly, leads to the appearance of a nontrivial central charge. So the Witt algebra
is modified by central extension to the Virasoro algebra. Infinite-dimensional algebras
have also been used recently in the theory of exactly solvable models. For example,
the Virasoro algebra plays a central role in the study of integrable systems, such
as those associated to the KdV and other soliton hierarchies. The Virasoro algebra
is intrinsically related to the KdV canonical structure where the Virasoro brackets
become just the Magri brackets for the Miura transformations of elements of the
phase space of the KdV hierarchy (see, e.g., [10, 16]). The lattice Virasoro algebra
appears in the study of the Toda field theory and Toda integrable systems [11, 21].
On the other hand, contour dynamics is a classical subject in complex analysis.
One of the typical dynamics started from classical Hele-Shaw experiments in 1897.
This leads to a sample free boundary problem, known also as the Laplacian growth
in two dimensions (see [20, 46]). Another one is the Löwner evolution, see [27, 38].
In the last decade, the progress in the study of Laplacian growth has resulted in its
integrable structure in terms of Richardson’s moments, namely these moments form
a dispersionless Toda hierarchy [32]. Richardson’s moments are conserved quantities
of the Laplacian growth. Recently, it was proved [31] that the Virasoro generators
in their co-vector representation are conserved quantities of the Löwner-Kufarev evolution. Friedrich and Werner [13], and independently Bauer and Bernard [6] found
relations between SLE (stochastic or Schramm-Löwner evolution) and the highest
weight representation of the Virasoro algebra. All these results encouraged us to
conclude that the Virasoro algebra is a common structural basis for these and possibly other types of contour dynamics and we decided to present our overview of the
development in this direction.
The survey is designed for a comprehensive reading of sufficiently wide mathematical audience, in particular graduate students. Therefore, we decided to include
several parts well-known for specialists in mathematical physics but less known for
analysts.
We start to show how the Virasoro algebra appears in CFT and in integrable
models. Then we proceed with the classical Löwner-Kufarev equation where the
central extension does not appear and we work only with the Witt algebra. The
central extension appears for the stochastic version of the Löwner equation. Finally,
we briefly describe connections between SLE and CFT.
This survey is partially based on a lecture given by the second author at the
II Winter School in Complex Analysis and Operator Theory, held in Seville (Spain),
February 5-9, 2008. He is thankful to the organizers for their hospitality. The authors
want to express their gratitude to Hélène Airault, Roland Friedrich, Paul Malliavin,
Yurii Neretin, and Dmitri Prokhorov for many fruitful discussions on the subject
during last years.
VIRASORO ALGEBRA AND DYNAMICS IN THE SPACE OF UNIVALENT FUNCTIONS
3
2. CFT and Virasoro algebra
There is a vast amount of standard references on basics of CFT, and classical and
quantum strings, see e.g., [19, 35], but since the topic is presented in a lecture form
for mathematics students, we take a risk to look briefly through a simplest example
of a closed bosonic string with periodic boundary conditions in order to introduce the
Virasoro algebra.
2.1. Classical bosonic string. We start with some basic definitions. A p-brane
is the spatial evolution of a p-dimensional object, which is reduced to a point particle
if p = 0, a string if p = 1, a membrane if p = 2, etc., in some D-dimensional ambient
spacetime M. The result of such evolution is called the worldline (p = 0), worldsheet
(p = 1), or worldvolume otherwise. We suppose M to be a D-dimensional vector
space endowed with the Lorentzian metric dsM of signature (1, D − 1); that is


−1 0 . . . 0
 0
1 ... 0 

ds2M = ηµν dxµ dxν , ηµν = 
 ... ... ... ... .
0
0 ... 1
We consider the toy model of a closed bosonic string. In order to see its dynamics we
construct dynamical variables xµ (σ α ) where µ = 0, 1, . . . , D − 1 and σ 0 = τ , σ 1 = σ
are coordinates on the canonical cylinder C = {τ ∈ (−∞, ∞), σ ∈ [0, 2π)}. The
intrinsic metric dsC on C is given by
'
(
−1 0
2
α
β
dsC = ηαβ dσ dσ , ηαβ =
.
0 1
We reserve the notation ηµν for the metric in D-dimensional space and ηαβ on the
cylinder C. The dynamical variables describe a C ∞ -smooth embedding of C in M,
and this allows us to consider the worldsheet (ws) as an embedded manifold in M.
The induced metric dsws on the worldsheet is given by
∂xµ ∂xν α β
= ηµν α β dσ dσ = gαβ dσ α dσ β .
∂σ ∂σ
µ
The dynamical variables x (τ, σ) satisfy the equations of motion derived by making
use of the least action principle. In Lagrangian mechanics, a system with a configuration space N is characterized by its Lagrangian L, a smooth real-valued functional
on the direct product of the tangent bundle T N and the real axis. Let P N stand
for the path space, that is the space of all paths having the fixed initial point (‘ini’)
and the end point (‘end’). This space has a structure of an infinite-dimensional
Fréchet manifold. Then, the action functional S is a real-valued integral functional
) fin
defined by S = ini Ldt, where t is a real curve parameter. Turning to the spacetime
configuration
space M and a worldsheet in M we work with the action functional
)
)
S = ws Φ(x)dx where the Lagrangian L is included into this integral L = Φ(x)dx̃,
ds2ws
4
IRINA MARKINA AND ALEXANDER VASIL’EV
where the integration is performed only along the spatial component x̃ of x. So
defined action has some physical and geometrical relevance. In classical mechanics,
action, e.g., is the difference between kinetic and potential energies. The minimizing
curve for the length functional and for the action given by the Lagrangian, which is
the kinetic energy, coincide. There is no direct analogue to energy in the relativistic mechanics. Therefore, Nambu in 1970 [33] and Gotô in 1971 [17] proposed to
choose the area of the worldsheet as the simplest action. This action, known as the
Nambu-Goto action, admits the form
*
*
+
2
SN G = −T dx = −T dσ | det gαβ |,
ws
C
where C is the canonical cylinder and the quantity T , the string tension, has dimension
mass per unit length. Observe that (-) in front of the integral is due to the Lorentzian
metric. The string possesses geodesic motion, i.e., the dynamical variables satisfy the
Euler-Lagrange equation
'+
(
µ
∂
1
αβ ∂x
,
= 0, µ = 0, . . . , D − 1,
| det gαβ | g
(2.1)
∂σ β
| det gαβ | ∂σ α
where g αβ and gαβ are obtained by rising and lowering of indices with respect to
the Lorentzian metric. The metric gαβ is defined on the worldsheet embedded into
M. It depends on the variables xµ , and the equation (2.1) for xµ is highly nonlinear. Moreover, the square root within the integral results in difficulties in further
quantization.
To overcome these problems Polyakov proposed to introduce an analogue of Lagrange multipliers in 1981 [36]. In fact, he considered a variable metric ds of index
1 on the worldsheet as on a 2-D manifold, such that ds2 = hαβ (τ, σ) dσ αdσ β , and the
Dirichlet integral
*
+
µ
ν
T
αβ ∂x ∂x
2
SP = −
dσ | det hαβ | h
2
∂σ α ∂σ β
C
as an action. The Euler-Lagrange equation (regarding to the variation of the dynamical variables) for the Polyakov action SP is formally the same as for the Nambu-Goto
action SN G
'+
(
µ
∂
1
αβ ∂x
,
= 0,
| det hαβ | h
(2.2)
∂σ β
| det hαβ | ∂σ α
but h does not depend on x any longer. So (2.2) are linear equations for xµ (τ, σ).
Moreover, the Polyakov action is quantized easier due to its linearity.
There are three degrees of freedom in h because it is symmetric. They can be
removed by using the equation of motion for h
δSP
(2.3)
= 0,
δhαβ
VIRASORO ALGEBRA AND DYNAMICS IN THE SPACE OF UNIVALENT FUNCTIONS
5
where the left-hand side is a functional derivative. The equations (2.2–2.3) perform
the necessary conditions for minimizing the action. The two-dimensional energymomentum tensor is defined as
δSP
−2
1
,
Tαβ =
,
T
| det hαβ | δhαβ
and the equation of motion implies Tαβ = 0. Moreover, SP = SN G under Tαβ = 0,
whereas in general, SP ≥ SN G .
A guiding principle in physics is that symmetries in the geometry of spacetime
define the standard conservation laws. There is no single theorem describing such a
connection, however this principle is sometimes referred to as the ‘Noether theorem’,
although the Noether 1918 theorem [34] itself contains only a very partial statement
of it.
By symmetries for the Polyakov action SP we mean transformations keeping the
action invariant. They are:
• Global symmetries
– Poincaré invariance
xµ → xµ + bµ ;
xµ → xµ + ωνµ xν ;
where ωνµ = −ωµν are infinitesimal Lorentzian transformations.
• Local symmetries
– reparametrization invariance by diffeomorphisms in 2-D: σ α → σ̃ α (τ, σ)
implies the invariance of the area element
+
,
2
dσ̃ | det h̃| = dσ 2 | det h|.
– Weyl rescaling
hαβ dσ α dσ β → eρ(τ,σ) hαβ dσ α dσ β .
,
Weyl rescaling leaves | det h| hαβ invariant.
The Poincaré and reparametrization invariance of SP allow us to choose a gauge
in which the three independent components of h are expressed with just one function,
typically the conformal flat gauge hαβ = eρ(τ,σ) ηαβ . Substituting this in the Polyakov
action we obtain
*
∂xµ ∂xν
T
2 αβ
dσ η ηµν α β ,
SP = −
2
∂σ ∂σ
C
so the gauge fixed action is a quadratic functional with respect to x. Varying it with
respect to x we arrive at a free wave equation of motion
(2.4)
x¨µ − (xµ )"" = 0,
where ẋ is the τ - and x" is the σ- derivative.
6
IRINA MARKINA AND ALEXANDER VASIL’EV
Weyl invariance implies that SP does not depend on ρ(τ, σ). Therefore, the variation δSP /δρ = hαβ Tαβ = Tαα vanishes that makes Tαβ traceless.
Keeping in mind that Tαβ = 0 one can derive the constraints T01 = T10 = ẋµ (xµ )" =
0 and
1
(2.5)
T00 = T11 = (ẋµ ẋµ + x"µ (xµ )" ) = 0.
2
" 2
This yields the equations (ẋ ± x ) = 0, which are called the Virasoro constraints.
The equation (2.4) with the constraints (2.5), subject to some boundary conditions
describes the motion of a bosonic string.
Let us use periodic boundary conditions xµ (τ, σ) = xµ (τ, σ + 2π). The general
solution to the wave equation of motion is
xµ (τ, σ) = xµR (τ − σ) + xµL (τ + σ).
Let us show that the center of mass of the worldsheet moves as a free particle. Satisfying the boundary condition and separating the linear part, we use the Fourier series
representation of the solution as
1
1
i - 1 µ −in(τ −σ)
xµR (τ − σ) = xµ0 +
(τ − σ)pµ + √
αn e
,
2
2πT
2πT n#=0 n
1
1
i - 1 µ −in(τ +σ)
xµL (τ + σ) = xµ0 +
(τ + σ)pµ + √
βn e
,
2
2πT
2πT n#=0 n
where xµ0 is the center of mass and pµ is the momentum. The functions xµR and xµL
µ
µ
are real, and hence, ᾱnµ = α−n
and β̄nµ = β−n
. These coefficients are known under the
name of oscillators in physics.
The position of the center of mass of the string is calculated as
* 2π
1
pµ
µ
X =
xµ (τ, σ)dσ = xµ0 +
τ,
2π 0
πT
that shows that the center of mass moves as a free particle starting from xµ0 . The
momentum of the center of mass is written
* 2π
*
T 2π µ
T
µ
µ
P =
ẋ dσ = pµ , Πµ = ẋµ .
Π dσ =
2 0
2
0
In classical) Hamiltonian mechanics the action S is a time-integral of the Lat
grangian S = t01 L(x, ẋ)dt. In our case the Lagrangian becomes
*
∂xµ ∂xν
T 2π αβ
η ηµν α β dσ.
L=−
2 0
∂σ ∂σ
The Hamiltonian function is given by
* 2π
*
T 2π µ
µ
H=
(ẋ Πµ )dσ − L =
(ẋ ẋµ + (xµ )" x"µ )dσ.
2
0
0
VIRASORO ALGEBRA AND DYNAMICS IN THE SPACE OF UNIVALENT FUNCTIONS
7
Using an identity for the Kronecker delta
* 2π
1
ei(n−m)σ dσ = δn,m ,
2π 0
we obtain the Hamiltonian function in terms of oscillators as
1H=
(α−n αn + β−n βn ),
2 n∈Z
1
where we set α0µ = β0µ = 2πT
pµ .
The standard Heisenberg-Poisson-Dirac bracket on phase coordinates looks like
1
{xµ (τ, σ), ẋν (τ, σ " )} = η µ,ν δ(σ − σ " ),
T
µ
ν
"
µ
{x (τ, σ), x (τ, σ )} = {ẋ (τ, σ), ẋν (τ, σ " )} = 0.
It is convenient to turn to light-cone coordinates on C assuming light speed to
be 1. They are ζ± = σ ± τ and the flat metric ds2C becomes ds2C = dζ+ dζ− . The
metric components of ηαβ in the light-cone coordinates are η++ = η−− = 0 and
η+− = η−+ = 21 . The differential operators become ∂± = 21 (∂σ ∓ ∂τ ).
The Virasoro generators are defined by
*
1 2π
1Lm =
T++ eim(τ −σ) dσ =
αm−n αn ,
2 0
2 n∈Z
*
11 2π
T−− eim(τ +σ) dσ =
βm−n βn ,
L̃m =
2 0
2 n∈Z
where T++ and T−− are the principal diagonal components of the energy-momentum
1
pµ . The commutator
tensor in the real light-cone coordinates and α0µ = β0µ = 2πT
relation for Ln given by the Poisson structure is
{Lm , Ln } = i(n − m)Ln+m ,
{L̃m , L̃n } = i(n − m)L̃n+m ,
{Lm , L̃n } = 0.
The next useful operation is so-called Wick rotation when the non-physical time
τ is replaced by the imaginary time iτ . Then the light-cone coordinates change to
ζ+ → z, ζ− → z̄, where z = σ + iτ . The differential operators become ∂+ → ∂z ,
∂− → ∂z̄ . Then the Polyakov action in the conformal gauge reads as
*
dz ∧ dz̄
SP = −2T
ηµν ∂z xµ ∂z̄ xν ,
2i
C
and the equation of motion is Laplacian ∂z2z̄ xµ = 0. The energy-momentum tensor
has the following components
1
Tzz = ηµν ((xµ )" − iẋµ )((xν )" − iẋν ) = T00 + 2iT10 ,
2
1
Tz̄ z̄ = ηµν ((xµ )" + iẋµ )((xν )" + iẋν ) = T00 − 2iT10 ,
2
8
IRINA MARKINA AND ALEXANDER VASIL’EV
and Tz z̄ = Tz̄z = 0. The Virasoro constraints in these coordinates are written as
Tzz = Tz̄ z̄ = 0.
The invariance of SP with respect to an infinitesimal reparametrization σ α →
α
σ̃ + εα (σ) implies the conservation law for the energy-momentum tensor ∇α Tαβ = 0
which in z-coordinates becomes
∂z̄ Tzz + ∂z Tz̄z = ∂z Tz̄ z̄ + ∂z̄ Tz z̄ = 0,
and making use of Tz z̄ = Tz̄z = 0, we conclude that Tzz is analytic and Tz̄ z̄ is antianalytic components of T .
Then in complex coordinates
1 - µ −inz
∂z xµ = √
βn e
,
4πT n∈Z
∂z̄ xµ = √
1 - µ −inz̄
αn e
.
4πT n∈Z
The Wick rotation leads to the complex coordinates and we conclude that in
complex coordinates the Virasoro generators Ln are the coefficients of the formal
Laurent series
- Ln
Tzz =
,
z n+2
n∈Z
and the commutator relations for Ln in these coordinates are
{Lm , Ln } = i(n − m)Ln+m ,
so span{Ln } forms the Witt algebra.
2.2. Canonical quantization. In classical mechanics the motion is completely
defined by the position x and momentum p, which are the elements of a symplectic
manifold. They specify the state of the classical mechanical system. There is a specific
function, the Hamiltonian function H, that defines the motion by the equations ẋ =
∇p H, ṗ = −∇x H. The symplectic structure (also known as the canonical structure)
defines the Poisson bracket by {x, p} = 1.
The main goal of the classical mechanics is to find the trajectories of motion of
particles, which are the solutions to a Hamiltonian system. The task of quantum mechanics differs because we have no trajectories any longer and a particle is found at a
point of spacetime with some probability (complex in general). The entire information
about the quantum system is given by the spectrum and the corresponding collection
of eigenfunctions for a quantum operator. The initial task of the first (canonical)
quantization is to find operators X and P corresponding to the dynamical variables
x and p, which act over an infinite dimensional Hilbert space of smooth functions,
quantum states. The next step is to find quantum analogue of the Poisson bracket
and the operator, the Hamiltonian, which becomes now a self-adjoint operator and
VIRASORO ALGEBRA AND DYNAMICS IN THE SPACE OF UNIVALENT FUNCTIONS
9
defines the total energy of a system, or in other words, an analogue of the Hamiltonian function. Finally, the states of the quantum system can be labelled by the
eigenvalues of the Hamiltonian.
Let us start with the definition of the operators X and P . In quantum mechanics we replace x → ·x ≡ X, p → −i!∇x ≡ P . Acting on quantum states they
result in Xψ = xψ, P ψ = −i!∇x ψ. The operator X acts on smooth functions
as a multiplication by independent variable x and P is an unbounded operator of
differentiation. There exist many reasons of this replacement. Let us give a toy reason related to the wave function, which appears in the harmonic oscillator with the
phase ϕ = kx − ωt, where ω is the frequency and k is the coefficient of displacement
(Hooke’s law). The Planck constant is the proportionality constant between energy
(E) of a photon and the frequency of its associated electromagnetic wave: E = !ω,
! = 6.62606896(33) × 10−34 /2π J s (Joule×seconds). The value !k is the momentum
of the displacement in a given direction x. Then the expression !dϕ = !kdx − !ωdt
can be interpreted as !dϕ = pdx − Hdt = dS, where S is the action. The real wave
function is cos ϕ, and the complex wave function is ψ = exp( !i S) with the amplitude 1.
We consider the simplest case in which the amplitude is constant=1. Generally, it
can be some non-negative function.
Let us consider the action S, i.e., the time integral of the Lagrangian, on the
extremal trajectories, the solutions to the Euler-Lagrange equation. Then the result
is a function S(t0 , t, x0 , x), that depends on the initial and final time t0 and t, and
on the initial and final point of the trajectory x0 and x. Differentiating with respect
to the final point x gives p = ∇x S, where the momentum p is the solution of the
corresponding Hamiltonian system ẋ = ∇p H, ṗ = −∇x H taken at the final time.
So we immediately obtain −i!∇Ψ = pΨ and the momenta represent eigenvalues of
the operator −i!∇. Thus, the quantum commutator is [X, P ]ψ = (XP − P X)ψ =
(x(i!)∇ − i!∇x)ψ = i!ψ, or [X, P ] = i!. The quantum commutator is modified in
order to preserve the classical properties in which the Poisson brackets {x, p} = 1,
therefore we come to the conclusion
[X, P ]quantum
{X, P }PB =
.
i!
Moreover, in the canonical quantization the fields are replaced by Hermitian operators. Rescaling our bosonic string, such that ! → 1, T → 1, we have
{xµ (τ, σ), ẋν (τ, σ " )} = −iη µ,ν δ(σ − σ " ),
{xµ (τ, σ), xν (τ, σ " )} = {ẋµ (τ, σ), ẋν (τ, σ " )} = 0.
When the canonical quantization procedure is applied in the frames of a given
quantum field theory, the classical field variable becomes a quantum operator, which
acts on a quantum state to increase or decrease the number of particles by one. For
a boson there are two operators: the boson’s creation operator b† and the boson’s
annihilation operator b (commonly known as ‘ladder operators’) for the system with
10
IRINA MARKINA AND ALEXANDER VASIL’EV
one degree of freedom, or b†k and bk for fields, k ∈ Z. Each operator creates or annihilates particles in a certain state ψ. The next step in quantization is establishing the
normal ordering operation between creators and annihilators following the Heisenberg
uncertainty principle expressed in above Poisson bracket. The commutation relations
for creators and annihhilators are {b† , b† } = 0, {b, b} = 0, and {b, b† } = 1. The latter
can be rewritten as bb† = b† b + 1. A product of creation and annihilation operators
is said to be in the normal order when all creation operators are to the left of all
annihilation operators in the product. The process of putting a product into normal
order is called normal ordering. The normal ordering operation is denoted by : ab :.
In the case of two boson’s operators : b† b := b† b but : bb† := b† b and bb† =: bb† : +1.
We define the Virasoro generators in the quantum system by introducing normal
ordering for positively indexed oscillators αn , βn as annihilators and for the negatively
indexed ones as creators:
1Lm =
: αm−n αn :
2 n∈Z
L̃m =
1: βm−n βn :
2 n∈Z
where m ,= 0. The only operator L0 produces an anomaly because it is already in
normal order,
1
11
1L0 = α0 α0 +
: α−n αn := α0 α0 +
α−n αn .
2
2
2
2
"
"
n∈Z
n∈Z
In order to keep the commutation law for oscillators one must subtract from L0 some
constant c when taking commutator of symmetric operators Ln and L−n . Taking into
account quantum brackets one arrives at the Virasoro commutation relation
c
(2.6)
[Lm , Ln ]Vir = (n − m)Ln+m + n(n2 − 1)δn,−m ,
12
where the constant c is called the central charge and in this particular case it coincides
with the dimension D.
The Virasoro constraints can not be written as Ln |ψ- = 0 for all n, because
.ψ|[Ln , L−n ]|ψ- = 2n.ψ|L0 |ψ- +
c
n(n2 − 1).ψ|ψ- =
, 0.
12
So the correct Virasoro constraints are imposed by Ln |ψ- = 0 for n > 0 and (L0 −
c)|ψ- = 0. Here .ψ| and |ψ- are the standard Dirac notations of bra and ket vectors
of states. Our system is in the physical state |ψ- and c = D.
Early string theory proposed by Yoichiro Nambu and others in 1970 was only
bosonic. Pierre Ramond, André Neveu, and John Schwarz completed the theory by
inventing fermionic strings to accompany the bosonic ones.
VIRASORO ALGEBRA AND DYNAMICS IN THE SPACE OF UNIVALENT FUNCTIONS
11
3. KdV and Virasoro algebra
The KdV equation appeared in a paper by Diederik Johannes Korteweg and his
student Gustav de Vries [25] in Philosophical Magazine, 1895, and originally described
the solitary wave discovered by the Scottish engineer John Scott Russell about half a
century earlier. Later on, it became the condition on the potential of the spectral stability of the Schrödinger operator, eigenvalues of which represent energy quantization
for bound systems. The existence of quantized energy levels is verified experimentally
by observation of the energy emitted or absorbed when the system makes a transition
from one level to another.
3.1. Hamiltonian dynamics and integrability. In order to speak on integrability of KdV let us introduce this notion briefly for finite dimensional Hamiltonian
systems which will be used in the sequel. There exists a vast amount of modern
literature dedicated to different approaches and definitions of integrable systems (see,
e.g., [3, 4, 7, 49]).
The classical definition of an integrable system in the sense of Liouville applied to
a Hamiltonian system says, that if we can find independent conserved integrals which
are pairwise involutory (have vanishing Poisson brackets), this system is partially or
completely integrable depending on the number of pairwise involutory integrals (see
e.g., [3, 4, 7]). That is each first integral allows us to reduce the order of the system
not just by one, but by two. We give now the precise definitions.
Let (N, ω) be a symplectic manifold with the symplectic 2-form ω and H be a
−
→
∞
C -function defined on N. Then we write H to denote the Hamiltonian vector field
−
→
associated to H. By definition H is a vector field on N, such that
−
→ −
→
−
→
ω( V , H ) = dH( V ),
−
→
for every smooth vector field V on N. In this case the function H is called the
Hamiltonian function. If H, K ∈ C ∞ (N), then the Poisson bracket {H, K} is the
−
→
directional derivative of K in the direction of H , i. e.
−
→
−
→ −
→
{H, K} = dK( H ) = ω( H , K ).
−
→
The Jacoby identity and the Leibniz property show that the map H → H is a Lie
algebra homomorphism
(C ∞ (N), {·, ·}) to (V ∞ (N), [·, ·]),
where V ∞ (N) denotes the set of C ∞ -vector fields on N. An equation of the form
−
→
(3.1)
ẋ(t) = H (x(t)),
is called the Hamiltonian system.
Let us consider the particular case of n-dimensional complex vector space Cn .
In this case the cotangent bundle T ∗ (Cn ) for Cn , that is, isomorphic to C2n , has a
12
IRINA MARKINA AND ALEXANDER VASIL’EV
natural symplectic structure. Relatively to the coordinate chart
T ∗ (Cn ) = (z1 , . . . , zn , ψ̄1 , . . . , ψn ),
we have the formulas
ω=
n
k=1
−
→
H =
n
k=1
∂H ∂
∂H ∂
,
−
∂z k ∂ ψ̄k ∂ ψ̄k ∂z k
dz k ∧ dψ̄k ,
n
∂H ∂K
∂H ∂K
{H, K} =
.
−
∂z k ∂ ψ̄k ∂ ψ̄k ∂z k
k=1
Then the system (3.1) can be written in the form
∂H
∂H
(3.2)
ż k (t) =
, ψ˙ k (t) = − k , k = 1, 2 . . . , n.
∂z
∂ψ k
and, using the Poisson brackets, we get
(3.3)
ż k (t) = {z k , H, }, ψ˙ k (t) = {ψ k , H}, k = 1, 2 . . . , n.
The latter system has n degrees of freedom. A smooth function Φ satisfying {H, Φ} =
0 is called the first integral of the Hamiltonian system. In particular, {H, H} = 0,
and the Hamiltonian function H is a first integral of the system (3.1). If the system
(3.3) has n functionally independent first integrals Φ1 , . . . , Φn , which are pairwise
involutory, that is {Φk , Φj } = 0, k, j = 1, . . . , n, then the system is called completely
integrable in the sense of Liouville. The function H is included in the set of the first
integrals. The classical theorem of Liouville and Arnold [3] gives a description of
the motion generated by the completely integrable system (3.3). It states that such
a system admits action-angle coordinates on a connected regular compact invariant
manifold.
3.2. From KdV to Virasoro. Let us consider the phase coordinates (field variables) defined on the spacetime S 1 × R, which are from the Sobolev space H α (S 1 ),
α ≥ −1, with respect to x ∈ S 1 for each fixed t ∈ R, i.e., real valued distributions on
the unit circle u(eix , t) . Let us simplify u → u(x, t), where the new u becomes a 2π
periodic smooth function in x. Observables are C 1 functionals on H α (S 1 ).
The famous KdV equation is ut = 6uu" +u""" on the unit circle and can be viewed as
an integrable Hamiltonian system of infinite dimensions. Several Poisson structures
can be defined on the space of observables. One was proposed by Gardner [14], and
independently, by Zaharov and Faddeev [50], see others in, e.g., [9, 18, 29].
Let us consider two functionals F (u) and G(u), where u ∈ H α (S 1 ) with the
vanishing mean value. Expanding u into the Fourier series we obtain
u(x) =
un einx ,
n∈Z"
"
where u−n = ūn , and Z = Z \ {0}. Then let us consider the functionals F and G as
functions F = F (q1 , q2 , . . . , p1 , p2 . . . ) and G = G(q1 , q2 , . . . , p1 , p2 . . . ) with respect
VIRASORO ALGEBRA AND DYNAMICS IN THE SPACE OF UNIVALENT FUNCTIONS
13
qn = un /n and pn = ū−n , n ∈ Z" . The canonical symplectic form dq ∧ dp̄ implies the
Poisson structure
(
∞ '
1 - ∂F ∂G
∂F ∂G
1 - ∂F ∂G
{F, G} =
−
=
n
.
2π n=1 ∂qn ∂ p̄n ∂ p̄n ∂qn
2π
∂un ∂u−n
"
n∈Z
Observe that u0 can be included now in the latter sum.
Take into account a useful formula for variational derivatives. If
* 2π
F =
f (u, u", u"" , . . . )dx,
0
then
∂f
d ∂f
d2 ∂f
δF
=
−
+
−...
δu
∂u dx ∂u" dx2 ∂u""
Hence, if the function u depends on a parameter α, then integrating by parts we
conclude that
* 2π
dF
δF ∂ u
=
dx.
dα
δu ∂ α
0
In particular,
* 2π
∂F
δF inx
δF
1 - ∂F inx
1 - ∂F −inx
=
e dx,
=
e =
e
.
∂un
δu
δu
2π
∂un
2π
∂u−n
0
"
"
n∈Z
n∈Z
Substituting this in the formula for the Poisson bracket we obtain
*2π *2π 1
∂F ∂G
1
∂F inx ∂G
{F, G} = 2
n
dx = 2
e
ne−inx dx
4π
∂u
∂u
4π
∂u
∂u
n
−n
n
−n
"
"
0
1
= 2
4π
n∈Z
*2π 0
n∈Z"
0
n∈Z
∂F inx - ∂G −ikx
e
ke
dx = i
∂un
∂u−k
"
k∈Z
*2π
δF d δG
dx,
δu dx δu
0
which is true for all functionals of the above integral form.
) 2π
If we choose the Hamiltonian function in the form H = −i 0 ( 12 (u"2 ) + u3 )dx,
then the evolution equation u̇ = {u, H} admits the form
d δH
= 6uu" + u""" .
u̇ = i
dx δu
One of the most important features of this equation is that it possesses an infinite
number of conserved quantities (first integrals) Ik [u], e.g.,
* 2π
* 2π
* 2π
1
2
I−1 =
udx, I0 =
u dx, I1 =
( (u"2 ) + u3 )dx, . . .
2
0
0
0
* 2π
d
...,I =
polynomial ( dx
, ·u)dx.
0
14
IRINA MARKINA AND ALEXANDER VASIL’EV
which are all in involution. One observes the relation of this fact to the Liouville
integrability for Hamiltonian systems. However, the proof of complete integrability
is rather difficult task, which was fulfilled in [14, 50].
If we consider the conserved quantities as Hamiltonians, then we obtain a hierarchy
constructed as
d δIn
u̇ = {u, −iIn } ≡
,
dx δu
which is called the KdV hierarchy.
The above Poisson structure considered on the phase space formally can be rewritten (modulo a constant factor) as
[u(x), u(y)] = γδ " (x − y),
γ > 0.
It is degenerate because the center is one-dimensional and the admissible element
* 2π
P =
u(x)dx.
0
commutes with all observables. Fixing P we get a submanifold which is symplectic.
The Poisson structure on the phase space gives the Lie structure on the space of
observables.
The Hamiltonian itself is an observable, and for example, consideration of
* 2π
1
H=
u2 (x, t)dx
2γ 0
(corresponding to I0 ) gives a trivial evolution equation of motion u̇ = [H, u] = −u" ,
u = u(x − t). The infinite number of evolution equations generated by all integrals is
the above KdV hierarchy. The Miura transformation s = u2 + u" reduces the modified
KdV equation u̇ = u""" + u2 u" to the usual KdV and leads to the Hamiltonian
* 2π
1
(u2 + u")dx.
H=
2γ 0
The Poisson brackets for s satisfy the relation
[s(x), s(y)] = γ(2(s(x) + s(y)) + δ " (x − y) + δ """ (x − y),
which is called Magri brackets [29].
Fourier coefficients of s(x) are given as
* 2π
1
Ln (s) =
einx s(x)dx.
2π 0
The corresponding Lie brackets are [Lm , Ln ]Vir where c is the central charge, c = 6π
γ
(J.-L.Gervais [16]), or taking into account quantum effects c = 1 + 6( πγ + πγ + 2), see
[10] (one can observe here the famous gap in CFT for real c ∈ (1, 25)).
VIRASORO ALGEBRA AND DYNAMICS IN THE SPACE OF UNIVALENT FUNCTIONS
15
3.3. From Virasoro to KdV. Let us define
1
6Ln e−inx −
u=
c n∈Z
4
.
1
inx
Then, using δ(x) = 2π
and the Virasoro commutation relation (2.6), we
n∈Z e
obtain
6π
[u(x), u(y)] =
(−δ """ (x − y) + 4u(x)δ " (x − y) + 2u" δ(x − y)).
c
) 2π
Taking I0 = 12 0 u2 dx, we obtain
c
[u, I0] = u""" + 6uu".
u̇ =
6π
KdV as a non-linear PDE is related to the classical and quantum field theories in
which the infinite number of degrees of freedom follows from the infinite number of
degrees of freedom for the initial conditions. So it is not surprising to see relations
between the Virasoro algebra and KdV. As we shall show in forthcoming sections,
problems of completely different nature (the Löwner-Kufarev evolution) but also of
the infinite number of degrees of freedom, lead to a rigid algebraic structure given by
the Virasoro algebra.
4. Realization on the unit circle
Mathematically, the Virasoro algebra appeared first as a central extension of the
d
Lie algebra of smooth vector fields φ dθ
on the unit circle S 1 (see [15]). Let us denote
the Lie group of C ∞ sense preserving diffeomorphisms of the unit circle S 1 by Diff S 1 .
Each element of Diff S 1 is represented as z = eiα(θ) with a monotone increasing C ∞
real-valued function α(θ), such that α(θ + 2π) = α(θ) + 2π. The space of smooth
vector fields on a differentiable manifold S 1 forms a Lie algebra, where the Lie bracket
is defined to be the commutator of vector fields. The relation of this Lie algebra to
Diff S 1 is subtile. The Lie algebra to Diff S 1 can be associated with the left-invariant
vector fields Vect S 1 . But the exponential map, which is the same as the exponential
map from the tangent space at the origin, is not even locally a homeomorphism. The
infinitesimal action of Vect S 1 is θ → θ + εφ(θ). To φ we associate the vector field
d
φ dθ
, and the Lie brackets are given by
(4.1)
[φ1 , φ2 ] = φ1 φ"2 − φ2 φ"1 .
The Virasoro algebra is the unique (up to isomorphism) non-trivial central extension
of Vect S 1 by R given by the Gelfand-Fuchs cocycle [15].
4.1. Canonical identification. As an infinite dimensional Lie-Fréchet group,
Diff S 1 undergoes certain irregular behaviour, in particular, the exponential map from
Vect S 1 is not a local homeomorphism. The entire necessary background of unitary
representations of Diff S 1 is found in the study of Kirillov’s homogeneous Kählerian
manifold Diff S 1 /S 1 .
We deal with the analytic representation of
16
IRINA MARKINA AND ALEXANDER VASIL’EV
Diff S 1 /S 1 . Let S stands for the whole class of univalent functions f in the unit
.
n
disk U normalized by f (z) = z(1 + ∞
n=1 cn z ) about the origin. By S̃ we denote the
class of functions from S smooth (C ∞ ) on the boundary S 1 of U. Given a map f ∈ S̃
we construct the adjoint univalent meromorphic map
d−1
+ ...,
g(z) = d1 z + d0 +
z
defined in the exterior U ∗ = {z : |z| > 1} of U, and such that Ĉ\f (U) = g(U ∗ ). Both
functions are extendable onto S 1 . This conformal welding gives the identification of
the homogeneous manifold Diff S 1 /S 1 with the space S̃: S̃ / f ↔ f −1 ◦ g|S 1 ∈
Diff S 1 /S 1 , or with the smooth contours Γ = f (S 1 ) that enclose univalent domains
Ω of conformal radius 1 with respect to the origin and such that ∞ ,∈ Ω, 0 ∈ Ω,
see [1], [23]. Being quasicircles, the smooth contours allow us to embed Diff S 1 /S 1
into the universal Teichmüller space making use of the above conformal welding, see
[44]. Coefficients of the univalent functions from S̃ are the natural coordinates on
the Teichmüller space. So one can construct complexification of Vect S 1 and further
projection of the holomorphic part to the set M ⊂ CN which is the projective limit
of the coefficient bodies M = limn←∞ Mn , where
(4.2)
Mn = {(c1 , . . . , cn ) : f ∈ S̃}.
This construction relates the Kähler structure of both manifolds. The holomorphic
Virasoro generators can then be realized by the first order differential operators
∞
Lj = ∂j +
(k + 1)ck ∂j+k , j ∈ N,
k=1
in terms of the affine coordinates of M, acting over the set of holomorphic functions,
where ∂k = ∂/∂ck . We explain the details in the next subsection.
4.2. Complexification. Let us introduce local coordinates on the manifold
M = Diff S 1 /S 1 in the concordance with the local coordinates on the space S̃ of
univalent functions smooth on the boundary. Observe that M is a real infinitedimensional manifold, whereas S̃ is a complex manifold. We are aimed at a complexification of T M which admits a holomorphic projection to T S̃, where as usual,
Vect 0 S 1 = Vect S 1 /const is a module over the ring of smooth functions, which is
associated with the tangent bundle T M.
Two operations are to be considered: complexification, conjugation, and almost
complex structure. Given a real vector space V the complexification VC is defined
as the tensor product with the complex numbers V ⊗R C, that often is written as
VC = V ⊕iV . The subscript R indicates that we take the real tensor product, we omit
it in the sequel. Elements of VC are of the form v ⊗ z. In addition, the vector space VC
is a complex vector space that follows by defining multiplication by complex numbers,
α(v ⊗z) = v ⊗αz for complex α and z and v ∈ V . The space V is naturally embedded
VIRASORO ALGEBRA AND DYNAMICS IN THE SPACE OF UNIVALENT FUNCTIONS
17
into V ⊗ C by identifying V with V ⊗ 1. Conjugation is defined by introducing a
canonical conjugation map on VC as v ⊗ z = v ⊗ z̄.
An almost complex structure J on V is a linear transformation J : V → V such
that J 2 = −I. It can be extended by linearity to the complex structure J on VC by
J(v ⊗ z) = J(v) ⊗ z. Observe that
J(v ⊗ z) = Jv ⊗ z = Jv ⊗ z̄ = J(v ⊗ z̄) = J(v ⊗ z).
Eigenvalues of extended J are ±i, and there are two eigenspaces V (1,0) and V (0,1)
corresponding to them given by projecting 21 (1 ∓ iJ)v. VC is decomposed
into the
/
(1,0)
(0,1)
(1,0)
direct sum VC = V
, where V
= {v ⊗ 1 − J(v) ⊗ i/v ∈ V } and
/⊕ V
(0,1)
/
V
= {v ⊗ 1 + J(v) ⊗ i v ∈ V }. In the case of existence of such a complex structure
J, the vector spaces V (1,0) and V (0,1) give complex coordinates on V .
An almost complex structure on Vect 0 S 1 may be defined as follows (see [1]). We
identify Vect 0 S 1 with the functions with vanishing mean value over S 1 . It gives
∞
φ(θ) =
an cos nθ + bn sin nθ.
n=1
Let us define an almost complex structure by the operator
∞
J(φ)(θ) =
−an sin nθ + bn cos nθ.
n=1
2
1
Then J = −id. On Vect 0 S ⊗ C, the operator J diagonalizes and we have the
identification
∞
1
1
Vect 0 S / φ ↔ v := (φ − iJ(φ)) =
(an − ibn )einθ ∈ (Vect 0 S 1 ⊗ C)(1,0) ,
2
n=1
and the latter extends into the unit disk as a holomorphic function.
The Kirillov infinitesimal action [22] of Vect 0 S 1 on S̃ is given by a variational
formula due to Schaeffer and Spencer [42, page 32] which lifts the actions from the
Lie algebra Vect 0 S 1 onto S̃. Let f ∈ S̃ and let φ(eiθ ) := φ(θ) ∈ Vect 0 S 1 be a C ∞
real-valued function in θ ∈ (0, 2π]. The infinitesimal action θ 5→ θ + εφ(eiθ ) yields a
variation of the univalent function f ∗ (z) = f + ε δv f (z) + o(2), where
(2
* ' "
v(w)dw
wf (w)
f 2 (z)
,
(4.3)
δv f (z) =
2π
f (w)
w(f (w) − f (z))
S1
and φ ↔ v by the above identification. Kirillov and Yuriev [23], [24] (see also [1])
established that the variations δφ f (ζ) are closed with respect to the commutator (4.1),
and the induced Lie algebra is the same as Vect 0 S 1 . The Schaeffer-Spencer operator
is linear.
Treating T M as a real vector space, the operator δφ transfers the complex structure J from Vect 0 S 1 to T M by J(δφ ) := δJ(φ) . By abuse of notation, we denote
18
IRINA MARKINA AND ALEXANDER VASIL’EV
the new complex structure on T M by the same character J. Then it splits the
complexification T MC into two eigenspaces T MC = T M(1,0) ⊕ T M(0,1) . Therefore,
δv = δφ−iJ(φ) := δφ − iJ(δφ ) ∈ T M(1,0) . Observe that 2z∂z = −i∂θ on the unit
circle z = eiθ , and Lk = z k+1 ∂/∂z = − 21 ieikθ ∂/∂θ on S 1 . Let us take the basis of
Vect 0 S 1 ⊗C in the form νk = −ieikθ in order to keep the index of vector fields the same
as for Lk . Then, the commutator satisfies the Witt relation {νm , νn } = (n − m)νn+m .
Taking elements νk = −iw k , |w| = 1 in the integrand of (4.3) we calculate the residue
in (4.3) and obtain so called Kirillov operators
Lj [f ](z) = δνj f (z) = z j+1 f " (z),
j = 1, 2, . . . ,
which are the holomorphic coordinates on T M(1,0) . In terms of the affine coordinates
in M we get the Kirillov operators as
Lj = ∂j +
∞
-
(k + 1)ck ∂j+k ,
k=1
where ∂k = ∂/∂ck . They satisfy the Witt commutation relation
[Lm , Ln ] = (n − m)Ln+m .
For k = 0 we obtain the operator L0 , which corresponds to the constant vectors
from Vect S 1 , L0 [f ](z) = zf " (z) − f (z). The elements of the Fourier basis −ie−iθk
with negative indices (corresponding to T M(0,1) ) are extended into U by −iz −k .
Substituting them in (4.3) we get very complex formulas for L−k , which functionally
depend on Lk (see [1], [22]) and might play the role of conjugates to Lk . The first
two operators are calculated as
L−1 [f ](z) = f " (z) − 2c1 f (z) − 1,
f " (z)
1
L−2 [f ](z) =
−
− 3c1 + (c21 − 4c2 )f (z),
z
f (z)
see [24].
This procedure gives a nice links between representations of the Virasoro algebra and the theory of univalent functions. The Löwner-Kufarev equations proved
to be a powerful tool to work with univalent functions (the famous Biberbach conjecture was proved [8] using Löwner method). In the following section we show how
Löwner-Kufarev equations can be used in a representation of the Virasoro algebra. In
particular, we identify T M(1,0) with T M, equipped with its natural complex structure
given by coefficients of univalent functions, by means the Löwner-Kufarev PDE.
5. Löwner-Kufarev Equations
A time-parameter family Ω(t) of simply connected hyperbolic univalent domains
forms a Löwner subordination chain in the complex plane C, for 0 ≤ t < τ (where τ
VIRASORO ALGEBRA AND DYNAMICS IN THE SPACE OF UNIVALENT FUNCTIONS
19
may be ∞), if Ω(t) " Ω(s), whenever t < s. We suppose that the origin is an interior
point of the Carathéodory kernel of {Ω(t)}τt=0 .
A Löwner subordination chain Ω(t) is described by a time-dependent family of
conformal maps z = f (ζ, t) from the unit disk U = {ζ : |ζ| < 1} onto Ω(t), normalized
by f (ζ, t) = a1 (t)ζ + a2 (t)ζ 2 + . . . , a1 (t) > 0, ȧ1 (t) > 0. After Löwner’s 1923 seminal
paper [27] a fundamental contribution to the theory of Löwner chains was made by
Pommerenke [37, 38] who described governing evolution equations in partial and
ordinary derivatives, known now as the Löwner-Kufarev equations due to Kufarev’s
work [26].
One can normalize the growth of evolution of a subordination chain by the conformal radius of Ω(t) with respect to the origin by a1 (t) = et .
Löwner [27] studied a time-parameter semigroup of conformal one-slit maps of
the unit disk U arriving then at an evolution equation called after him. His main
achievement was an infinitesimal description of the semi-flow of such maps by the
Schwarz kernel that led him to the Löwner equation. This crucial result was then
generalized in several ways (see [38] and the references therein).
We say that the function p is from the Carathéodory class if it is analytic in U,
normalized as p(ζ) = 1 + p1 ζ + p2 ζ 2 + . . . , ζ ∈ U, and such that Re p(ζ) > 0
in U. Pommerenke [37, 38] proved that given a subordination chain of domains Ω(t)
defined for t ∈ [0, τ ), there exists a function p(ζ, t), measurable in t ∈ [0, τ ) for any
fixed z ∈ U, and from the Carathéodory class for almost all t ∈ [0, τ ), such that the
conformal mapping f : U → Ω(t) solves the equation
(5.1)
∂f (ζ, t)
∂f (ζ, t)
=ζ
p(ζ, t),
∂t
∂ζ
for ζ ∈ U and for almost all t ∈ [0, τ ). The equation (5.1) is called the LöwnerKufarev equation due to two seminal papers: by Löwner [27] who considered the
case when
(5.2)
p(ζ, t) =
eiu(t) + ζ
,
eiu(t) − ζ
where u(t) is a continuous function regarding to t ∈ [0, τ ), and by Kufarev [26] who
proved differentiability of f in t for all ζ from the kernel of {Ω(t)} in the case of
general p in the Carathéodory class.
Let us consider a reverse process. We are given an initial domain Ω(0) ≡ Ω0
(and therefore, the initial mapping f (ζ, 0) ≡ f0 (ζ)), and a function p(ζ, t) of positive
real part normalized by p(ζ, t) = 1 + p1 ζ + . . . . Let us solve the equation (5.1) and
ask ourselves, whether the solution f (ζ, t) defines a subordination chain of simply
connected univalent domains f (U, t). The initial condition f (ζ, 0) = f0 (ζ) is not
given on the characteristics of the partial differential equation (5.1), hence the solution
exists and is unique but not necessarily univalent. Assuming s as a parameter along
20
IRINA MARKINA AND ALEXANDER VASIL’EV
the characteristics we have
dt
dζ
df
= 1,
= −ζp(ζ, t),
= 0,
ds
ds
ds
with the initial conditions t(0) = 0, ζ(0) = z, f (ζ, 0) = f0 (ζ), where z is in U.
Obviously, t = s. Observe that the domain of ζ is the entire unit disk. However, the
solutions to the second equation of the characteristic system range within the unit
disk but do not fill it. Therefore, introducing another letter w (in order to distinguish
the function w(z, t) from the variable ζ) we arrive at the Cauchy problem for the
Löwner-Kufarev equation in ordinary derivatives
dw
= −wp(w, t),
dt
for a function ζ = w(z, t) with the initial condition w(z, 0) = z. The equation (5.3) is a
non-trivial characteristic equation for (5.1). Unfortunately, this approach requires the
extension of f0 (w −1 (ζ, t)) into the whole U (w −1 means the inverse function) because
the solution to (5.1) is the function f (ζ, t) given as f0 (w −1 (ζ, t)), where ζ = w(z, s) is
a solution of the initial value problem for the characteristic equation (5.3) that maps
U into U. Therefore, the solution of the initial value problem for the equation (5.1)
may be non-univalent.
Solutions to the equation (5.3) are holomorphic univalent functions w(z, t) =
e−t z + a2 (t)z 2 + . . . in the unit disk that map U into itself. Every function f from
the class S can be represented by the limit
(5.3)
(5.4)
f (z) = lim et w(z, t),
t→∞
where w(z, t) is a solution to (5.3) with some function p(z, t) of positive real part for
almost all t ≥ 0 (see [38, pages 159–163]). Each function p(z, t) generates a unique
function from the class S. The reciprocal statement is not true. In general, a function
f ∈ S can be obtained using different functions p(·, t).
Now we are ready to formulate the condition of univalence of the solution to the
equation (5.1), which can be obtained by combination of known results of [38].
Theorem 1. [38, 39] Given a function p(ζ, t) of positive real part normalized
by p(ζ, t) = 1 + p1 ζ + . . . , the solution to the equation (5.1) is unique, analytic and
univalent with respect to ζ for almost all t ≥ 0, if and only if, the initial condition
f0 (ζ) is taken in the form (5.4), where the function w(ζ, t) is the solution to the
equation (5.3) with the same driving function p.
Recently, we started to look at Löwner-Kufarev equations from the point of view
of motion in the space of univalent functions where Hamiltonian and Lagrangian formalisms play a central role (see, [45]). Some connections with the Virasoro algebra
were also observed in [31, 45]. The present paper generalizes these attempts and
gives their closed form. The main conclusion is that the Löwner-Kufarev equations
VIRASORO ALGEBRA AND DYNAMICS IN THE SPACE OF UNIVALENT FUNCTIONS
21
are naturally linked to the holomorphic part of the Virasoro algebra. Taking holomorphic Virasoro generators Ln as a basis of the tangent space to the coefficient
body for univalent functions at a fixed point, we see that the driving function in the
Löwner-Kufarev theory generates generalized moments for motions within the space of
univalent functions. Its norm represents the energy of this motion. The holomorphic
Virasoro generators in their co-tangent form will become conserved quantities of the
Löwner-Kufarev ODE. The Löwner-Kufarev PDE becomes a transition formula from
the affine basis to Kirillov’s basis of the holomorphic part of the complexified tangent
space to M at any point. Finally, we propose to study an alternate Löwner-Kufarev
evolution instead of subordination.
6. Witt algebra and the classical Löwner-Kufarev equations
In the following subsections we reveal the structural role of the Witt algebra as
a background of the classical Löwner-Kufarev contour evolution. As we see further,
the conformal anomaly and the Virasoro algebra appear as a quantum or stochastic
effect in SLE.
6.1. Löwner-Kufarev ODE. Let us consider the functions
0
1
∞
−t
n
w(z, t) = e z 1 +
cn (t)z ,
n=1
satisfying the Löwner-Kufarev ODE
(6.1)
dw
= −wp(w, t),
dt
with the initial condition w(z, 0) = z, and with the function p(z, t) = 1 + p1 (t)z + . . .
which is holomorphic in U and measurable with respect to t ∈ [0, ∞), such that
Re p > 0 in U. The function w(z, t) is univalent and maps U into U.
Lemma 1. Let the function w(z, t) be a solution to the Cauchy problem for the
equation (6.1) with the initial condition w(z, 0) = z. If the driving function p(·, t),
being from the Carathéodory class for almost all t ≥ 0, is C ∞ smooth in the closure Û
of the unit disk U and summable with respect to t, then the boundaries of the domains
B(t) = w(U, t) ⊂ U are smooth for all t.
Proof. Observe that the continuous and differentiable dependence of the solution
of a differential equation ẋ = F (t, x) on the initial condition x(0) = x0 is a classical
problem. One can refer, e.g., to [48] in order to assure that summability of F (·, x)
regarding to t for each fixed x and continuous differentiability (C 1 with respect to
x for almost all t) imply that the solution x(t, x0 ) exists, is unique, and is C 1 with
respect to x0 . In our case, the solution to (6.1) exists, is unique and analytic in U,
22
IRINA MARKINA AND ALEXANDER VASIL’EV
and, moreover, C 1 on its boundary S 1 . Let us differentiate (6.1) inside the unit disk
U with respect to z and write
log w " = −
*t
(p(w(z, τ ), τ ) + w(z, τ )p" (w(z, τ ), τ ))dτ,
0
choosing the branch of the logarithm such as log w " (0, t) = −t. This equality is
extendable onto S 1 because the right-hand side is, and therefore, w " is C 1 and w is
C 2 on S 1 . We continue analogously and write the formula
*t
""
"
w = −w (2w "(z, τ )p" (w(z, τ ), τ ) + w(z, τ )w " (z, τ )p"" (w(z, τ ), τ ))dτ,
0
which guarantees that w is C 3 on S 1 . Finally, we come to the conclusion that w is
C ∞ on S 1 .
!
Let f (z, t) denote et w(z, t). The limit limt→∞ f (z, t) is known [38] to be a representation of all univalent functions.
Let the driving term p(z, t) in the Löwner-Kufarev ODE be from the Carathéodory
class for almost all t ≥ 0, C ∞ smooth in Û, and summable with respect to t. Then
the domains Ω(t) = w(U, t) have smooth boundary ∂Ω(t). So the Löwner equation
can be extended onto the closed unit disk Û = U ∪ S 1 .
Consider the Hamiltonian given by
*
dz
(6.2)
H=
f (z, t)(1 − p(e−t f (z, t), t))ψ̄(z, t) ,
iz
z∈S 1
on the unit circle z ∈ S 1 , where ψ(z, t) is a formal series
ψ(z, t) =
∞
-
ψn z n ,
n=−k
defined about the unit circle S 1 for any k ≥ 0. The formula (6.2) is formal in the
sense it gives a short way to write infinite systems by variational derivatives. The
Poisson structure on the space (f, ψ̄) is given by the canonical brackets
{P, Q} =
δP δQ δP δQ
,
−
δf δ ψ̄
δ ψ̄ δf
or in coordinate form (only ψn for n ≥ 1 are independent co-vectors corresponding to
the tangent vectors ∂n with respect to the canonical Hermitean product for analytic
functions)
∞
∂p ∂q
∂p ∂q
{p, q} =
−
.
∂c
∂c
∂
ψ̄
∂
ψ̄
n
n
n
n
n=1
VIRASORO ALGEBRA AND DYNAMICS IN THE SPACE OF UNIVALENT FUNCTIONS
23
Here
P (t) =
*
dz
p(z, t) ,
iz
Q(t) =
z∈S 1
*
q(z, t)
dz
.
iz
z∈S 1
The Hamiltonian system becomes
df (z, t)
δH
= f (1 − p(e−t f, t)) =
= {f, H},
dt
δψ
(6.3)
for the position coordinates and
(6.4)
dψ̄
−δH
= −(1 − p(e−t f, t) − e−t f p" (e−t f, t))ψ̄ =
= {ψ, H},
dt
δf
δ
for the momenta, where δfδ and δψ
are the variational derivatives. So the phase
coordinates (f, ψ̄) play the role of the canonical Hamiltonian pair. The coefficients of
ψ(z, t) with negative indices will be defined later.
The coefficients cn are the complex local coordinates on M, so in these coordinates
we have
*
et
dz
ċn = cn −
w(z, t)p(w(z, t), t) n+2 ,
2πi
z
S1
* n
dz
1
e−kt (et w)k+1pk n+2 ,
= −
2πi
z
k=1
n ≥ 1.
S1
Let us fix some n and project the infinite dimensional Hamiltonian system on an ndimensional Mn .
Momenta in coordinates form an adjoint vector
T
ψ̄(t) = (ψ̄1 (t), . . . , ψ̄n (t)) , with complex-valued coordinates ψ1 , . . . , ψn for any fixed
n. The dynamical equations for momenta governed by the Hamiltonian function (6.2)
are
*
n
1 dz
˙
ψ̄j = −ψ̄j +
ψ̄k (p + wp") k−j+1 , j = 1, . . . , n − 1,
2πi k=1
z
S1
and
(6.5)
ψ̄˙ n = 0.
In particular,
ċ1 = −e−t p1 ,
ċ2 = −2e−t p1 c1 − e−2t p2 ,
ċ3 = −e−t p1 (2c2 + c21 ) − 3e−2t p2 c1 − e−3t p3 ,
...
...
24
IRINA MARKINA AND ALEXANDER VASIL’EV
for n = 3 we have
ψ̄˙ 1 = 2e−t p1 ψ̄2 + (2e−t p1 c1 + 3e−2t p2 )ψ̄3 ,
ψ̄˙ 2 = 2e−t p1 ψ̄3 ,
ψ̄˙ = 0.
3
Let us set the function L(z) := f " (z, t)ψ̄(z, t). Let (L(z))<0 mean the part of the
Laurent series for L(z) with negative powers of z,
∞
Lk
1
1
.
(L(z))<0 = (ψ̄1 + 2c1 ψ̄2 + 3c2 ψ̄3 + . . . ) + (ψ̄2 + 2c1 ψ̄3 + . . . ) 2 + · · · =
z
z
zk
k=1
Then, the functions L(z) and (L(z))<0 are time-independent for all z ∈ S 1 .
It is easily seen that, passing from the cotangent vectors ψ̄k to the tangent vectors
∂k , the coefficients Lk of (L(z))<0 defined on the tangent bundle T M(1,0) are exactly
the Kirillov vector fields Lk . Corresponding co-vector fields Lk are conserved by the
Löwner-Kufarev ODE because L̇k = {Lk , H} = 0. The above Poisson structure
coincides with that given by the Witt brackets introduced for Lk previously. For
finite-dimensional grades this result was obtained in [31].
Let us formulate the result as a theorem.
Theorem 2. Let the driving term p(z, t) in the Löwner-Kufarev ODE be from
the Carathéodory class for almost all t ≥ 0, C ∞ smooth in Û, and summable with
respect to t. Then the Kirillov co-vector fields are the conserved quantities for the
Hamiltonian system (6.3–6.4) generated by the Löwner-Kufarev ODE.
Remark 1. Another way to construct a Hamiltonian system could be based on
the symplectic structure given by the Kählerian form on Diff S 1 /S 1 . However, there
is no explicit expression for such form in terms of functions f ∈ S̃. Moreover, there
must be a Hamiltonian formulation in which the Löwner-Kufarev equation becomes
an evolution equation. This remains an open problem.
Remark 2. At a first glance the situation with an ODE with a parameter is quite
˙ eiθ ) = F (f (t, eiθ ), t), then fixing
simple. Indeed, if we solve an equation of type f(t,
θ we have an integral of motion C = I(f (t, ·), t) = const. Then, releasing θ, we
have C(eiθ ) = I(f (t, eiθ ), t). Expanding C(eiθ ) into the Fourier series, we obtain an
infinite number of conserved quantities, but they do not manifest an infinite number of
degrees of freedom that govern the motion as in the field theory where the governing
equations are PDE. In our case, we have not only one trajectory fixing the initial
condition but a pensil of trajectories because our equation has an infinite number
of control parameters, the Taylor coefficients of the function p(z, t), which form a
bounded non-linear set of admissible controls. Therefore, we operate with sections of
the tangent and co-tangent bundles to the inifinite dimensional manifold M instead
of vector fields along one trajectory as in usual ODE.
VIRASORO ALGEBRA AND DYNAMICS IN THE SPACE OF UNIVALENT FUNCTIONS
25
Remark 3. No linear combinations L∗k of L1 , . . . , Ln , . . . allows us to reduce the
system of {Lk } to a new system of involutory {L∗k } in order to claim the Liouville
integrability of our system. Observe that the coefficients in these linear combinations
must be constants to keep conservation laws.
6.2. Construction of L0 and L−n . Consider again the generating function
L(z) = f " (z, t)ψ̄(z, t) and the ‘non-negative’ part (L(z))≥0 of the Laurent series for
L(z),
(L(z))≥0 = (ψ̄0 + 2c1 ψ̄1 + 3c2 ψ̄2 + . . . ) + (ψ̄−1 + 2c1 ψ̄0 + 3c2 ψ̄1 + . . . )z + . . .
=
∞
k=0
L−k z k .
All L−k are conserved by the construction. Define ψ̄0∗ = −
L0 = L0 − (ψ̄0 − ψ̄0∗ ).
.∞
n=1 ck ψ̄k ,
and
The operator L0 acts on the class S by L0 [f ](z) = zf " (z) − f (z). Next define L−1 =
∗
∗
L−1 − (ψ̄−1 − ψ̄−1
) − 2c1 (ψ̄0 − ψ̄0∗ ), where ψ̄−1
= 0. Then,
Finally,
L−1 [f ](z) = f " (z) − 2c1 f (z) − 1
∗
∗
L−2 = L−2 − (ψ̄−2 − ψ̄−2
) − 2c1 (ψ̄−1 − ψ̄−1
) − 3c2 (ψ̄0 − ψ̄0∗ ).
∗
We choose ψ̄−2
= (c3 − 3c1 c2 + c31 )ψ̄1 + . . . , so that
1
1
∗
ψ̄−2
[f ](z) = −
− c1 − (c2 − c21 )f (z),
z f (z)
and
f " (z)
1
L−2 [f ](z) =
−
− 3c1 + (c21 − 4c2 )f (z).
z
f (z)
An important fact is that
L0 = c1 ψ̄1 + 2c2 ψ̄2 + . . . ,
L−1 = (3c2 − 2c21 )ψ̄1 + . . . ,
L−2 = (5c3 − 6c1 c2 + 2c31 )ψ̄1 + . . . ,
are linear with respect to ψ̄k , k ≥ 1, and therefore, are co-vectors. Equivalently
∂f
.
L0,−1,−2 [f ](z) = function(c1 , c2 , . . . )z 2 + . . . , z k =
∂ck−1
All other co-vectors we construct by our Poisson brackets as
1
1
L−n =
{L−n+1 , L−1 } =
{L−n+2 , L−2 }.
n−2
n−4
The form of the Poisson brackets guarantees us that all L−n are linear with respect
∗
to ψ̄1 , ψ̄2 , . . . and span the anti-holomorphic part of the co-tangent bundle T (0,1) M.
26
IRINA MARKINA AND ALEXANDER VASIL’EV
Let us summarize the above in the following conclusion. We considered a nonlinear contour dynamics given by the Löwner-Kufarev equation. It turned out to be
underlined by an algebraic structure, namely, by the Witt algebra spanned by the
Virasoro generators Ln , n ∈ Z.
• Ln , n = 1, 2, . . . are the holomorphic Virasoro generators. They span a
holomorphic vector bundle over the space of univalent functions S, smooth
on the boundary. In their co-vector form, Ln are conserved by the LöwnerKufarev evolution.
• L0 is the energy operator.
• L−n , n = 1, 2, . . . are the antiholomorphic Virasoro generators. They span
the antiholomorphic vector bundle. In their co-vector form, L−n contain a
conserved part and we give an iterative method to obtain all of them based
on the Poisson structure of the Löwner-Kufarev evolution.
6.3. Löwner-Kufarev PDE. The Löwner equation in partial derivatives is
ẇ(ζ, t) = ζw "(ζ, t)p(ζ, t),
Re p(ζ, t) > 0,
|ζ| < 1.
with some initial condition w(z, 0) = f0 (z).
the one-parameter family
.∞Let us consider
−t
n
of functions f (z, t) = e w(z, t) = z(1 + n=1 cn (t)z ), f (z, 0) = f0 (z) as a C 1 path
in S̃. At the identity id we have that Tid S̃ = Tid M(1,0) = Tid M. A path in the
coefficient body M in the neighbourhood of the identity is (c1 (t), . . . , cn (t), . . . ) with
the velocity vector ċ1 ∂1 + · · · + ċn ∂n + · · · ∈ Tid M.
(1,0)
Taking the Virasoro generators {Lk }, k ≥ 1, as an algebraic basis in T Mid we
wish the velocity vector written in this new basis to be
(6.6)
ċ1 ∂1 + · · · + ċn ∂n + · · · = u1 L1 + . . . un Ln + . . ..
We compare (6.6) with the Löwner-Kufarev equation
(6.7) f˙ = ċ1 ∂1 + · · · + ċn ∂n + · · · = zf " p(z, t) − f = L0 + u1 L1 + . . . un Ln + . . .,
where p(z, t) = 1 + u1z + · · · + un z n + . . . , and L0 f = zf " − f . In view of similarity
between these two expressions (6.6) and (6.7), we notice that
• a new term L0 appears in the Löwner-Kufarev equation;
• the function p(z, t) with positive real part corresponds to subordination,
whereas for generic trajectories it may have real part of arbitrary sign. We
call this an alternate Löwner-Kufarev evolution;
• the vector L0 corresponds exactly to the rotation:
eiε f (e−iε z) = f (z) − iε(zf " (z) − f (z)) + o(ε).
Let us consider the set S̃0 of non-normalized smooth univalent functions of the
form F (z, t) = a0 (t)z + a1 (t)z 2 + . . . , with a tangent vector ȧ0 ∂0 + · · · + ȧn ∂n + . . . ,
where ∂k = ∂/∂ak , k = 0, 1, 2, . . . . Our aim is to define two distributions of codimension 1 for the tangent bundle T S̃0 , that form the tangent bundle T S̃. This will
VIRASORO ALGEBRA AND DYNAMICS IN THE SPACE OF UNIVALENT FUNCTIONS
27
be realized by means of formulas (6.6) and (6.7). Notice that ∂k F = z k+1 . Setting
Lk (F ) := z k+1 F " we get
Ḟ = ȧ0 ∂0 + · · · + ȧn ∂n + · · · = zf " p(z, t) = u0 L0 + u1 L1 + . . . un Ln + . . . ,
where p(z, t) = u0 + u1 z + · · · + un z n + . . . . This alternate Löwner-Kufarev equation
represents recalculation of the tangent vector in the new basis
ȧ0 ∂0 + · · · + ȧn ∂n + · · · = u0 L0 + . . . un Ln + . . .,
where Lk = a0 ∂k + 2a1 ∂k+1 + . . . .
Let us present the distributions. We start with F ∈ S̃0 , then we define f ∈ S̃.
The necessary distribution is the map
S̃0 / F → Tf S̃ 4→ TF S̃0 .
The analytic form of the first distribution is the following factorization f1 (z, t) =
= z + aa01 z 2 + . . . , so that
1
F (z, t)
a0
ȧ0
f˙1 = zf1" p(z, t) − f1 ,
a0
(6.8)
where u0 =
ȧ0
.
a0
Then we obtain
ċ1 ∂1 + · · · + ċn ∂n + · · · = L̂0 + u1 L̂1 + · · · + un L̂n + . . .
where L̂0 f1 = u0 (zf1" − f1 ), L̂k f1 = z k+1 f1" , ck = aak0 , ∂k = ∂c∂k . In particular, a0 = et
implies the Löwner-Kufarev equation for arbitrary sign of Re p.
The analytic form of the second distribution becomes f2 (z, t) = F ( a10 z, t) = z +
a1 2
z + . . . , so that
a2
0
ȧ0
z
f˙2 = zf2" p( , t) − zf2" ,
a0
a0
(6.9)
where again u0 =
ȧ0
.
a0
In the coefficient form we get
ċ1 ∂1 + · · · + ċn ∂n + · · · = u1 L̃1 + · · · + un L̃n + . . .
where L̃k f2 = z k+1 f2" , ck =
ak
,
ak+1
0
∂k =
∂
.
∂ck
Observe that the equation (6.9) gives another identification of T M(1,0) with T M.
Finally, let us make an explicit calculation of L̂0 , which for a0 = et we continue
to denote by L0 . Using Kirilov’s basis L1 , L2 , . . . as a linear combination we write
L0 =
∞
-
m=1
Πm Lm .
28
IRINA MARKINA AND ALEXANDER VASIL’EV
The coefficients Πm are polynomials, which can be obtained using the following recurrent formulas
m−1
m
K1 = 0, Km = −
j(m − j + 1)cm−j cj ,
Πm = mcm +
Km−j+1 Pj−1,
j=1
j=1
where Pk are polynomials
(6.10)
P0 = 1,
P1 = −2c1 ,
P2 =
4c21
− 3c2 ,
Pk = −
k
-
(j + 1)cj Pk−j ,
j=1
Let us summarize the above considerations in the following theorem.
Theorem 3. The Löwner-Kufarev PDE (6.8) gives the distribution of co-dimension
1 inside the tangent bundle T S̃0 of non-normalized smooth univalent functions S̃0 , that
forms the tangent bundle T S̃.
The equation (6.9) gives another distribution, and moreover, it makes the explicit
correspondence between the natural complex structure of T S̃ and the complex structure
of T M(1,0) at each point f ∈ S̃.
One of the reason to consider the alternate Löwner-Kufarev PDE is the regularized
3
canonical Brownian motion on smooth Jordan curves. For all Sobolev metrics H 2 +ε ,
the classical theory of stochastic flows allows to construct Brownian motions on C 1
diffeomorphism group of S 1 . The case 3/ 2 is critical. Malliavin [30] constructed the
canonical Brownian motion on the Lie algebra Vect S 1 for the Sobolev norm H 3/2 .
Another construction was proposed in [12]. Airault and Ren [2] proved that the
infinitesimal version of the Brownian flow is Hölder continuous with any exponent
β < 1.
The regularized canonical Brownian motion on Diff S 1 is a stochastic flow on S 1
associated to the Itô stochastic differential equation
r
r
r
dgx,t
= dζx,t
(gx,t
),
r
ζx,t
(θ)
=
∞
n=1
√
rn
(x2n (t) cos nθ − x2n−1 (t) sin nθ),
n3 − n
where {xk } is a sequence of independent real-valued Brownian motions and r ∈
r
(0, 1) and the series for ζx,t
(θ) is a Gaussian trigonometric series. Kunita’s theory of
r
stochastic flows asserts that the mapping θ → gx,t
(θ) is a C ∞ diffeomorphism and the
r
limit lim− gx,t = gx,t exists uniformly in θ. The random homeomorphism gx,t is called
r→1
canonical Brownian motion on Diff S 1 , see [2, 12, 30, 40]. It was shown in [2, 12],
that this random homeomorphism is Hölder continuous.
The canonical Brownian motion can be defined not only on Diff S 1 , but also on
the space of C ∞ -smooth Jordan curves by conformal welding. This leads to dynamics
of random loops which are not subordinated.
VIRASORO ALGEBRA AND DYNAMICS IN THE SPACE OF UNIVALENT FUNCTIONS
29
7. Elliptic operators over the coefficient body
. The2 Kirillov first order differential operators Lk generate the elliptic operator
|Lk | . In this section we construct the geodesic equation and find geodesics with
constant velocity coordinates in the field of this operator. In particular, we shall
prove that the norm of the driving function in the Löwner-Kufarev theory gives the
minimal energy of the motion in this field.
7.1. Dynamics within the coefficient body. This subsection is auxiliary.
Let us recall the geometry of the coefficient body Mn for finite n with respect to
Kirillov’s basis Lk , k = 1, . . . , n of the tangent bundle T Mn . The affine coordinates
are introduced by projecting
∞
3
2
ck z k 5→ (c1 , . . . , cn ) ∈ Mn .
M/f =z 1+
k=1
The manifold Mn was studied actively in the middle of the last century, see e.g.,
[5, 42]. We compile some important properties of Mn below:
(i) Mn is homeomorphic to a (2n − 2)-dimensional ball and its boundary ∂Mn
is homeomorphic to a (2n − 3)-dimensional sphere;
(ii) every point x ∈ ∂Mn corresponds to exactly one function f ∈ S which is
called a boundary function for Mn ;
(iii) boundary functions map the unit disk U onto the complex plane C minus
piecewise analytic Jordan arcs forming a tree with a root at infinity and
having at most n tips,
(iv) with the exception for a set of smaller dimension, at every point x ∈ ∂Mn
there exists a normal vector satisfying the Lipschitz condition;
(v) there exists a connected open set X1 on ∂Mn , such that the boundary ∂Mn
is an analytic hypersurface at every point of X1 . The points of ∂Mn corresponding to the functions that give the extremum to a linear functional
belong to the closure of X1 .
Properties (ii) and (iii) imply that the functions from S̃ deliver interior points of
Mn . The Kirillov operators Lj restricted onto Mn give truncated vector fields
Lj = ∂j +
n−j
-
(k + 1)ck ∂j+k ,
k=1
which we, if it causes no confusion, continue denoting by Lj in this section. In [31]
based on the Löwner-Kufarev representation, we showed that these Lj can be obtained
from a partially integrable Hamiltonian system for the coefficients in which the first
integrals coincide
4 with Lj .
5
Let c(t) = c1 (t), . . . , cn (t) be a smooth trajectory in Mn ; that is a C 1 map
c : [0, 1] → Mn . Then the velocity vector ċ(t) written in the affine basis as ċ(t) =
30
IRINA MARKINA AND ALEXANDER VASIL’EV
ċ1 (t) ∂ 1 + . . . + ċn (t) ∂ n can be also represented in the basis of vector fields L1 , . . . , Ln
(compare with (6.9)) as
(7.1)
ċ(t)
= ċ1 (t) ∂ 1 + . . . + ċn (t) ∂ n
= u1 L1 + u2 L3 + . . . + un Ln ,
where the coefficients uk can be written in the recurrent form as
(7.2)
u1 = ċ1 ,
uk = ċk −
Expressing uk in terms of ck and ċk , we get
(7.3)
uk = ċk +
k−1
-
k−1
-
(j + 1)ċj uk−j .
j=1
Pj ċk−j .
j=1
One may notice that these polynomials are the first coefficients of the holomorphic
function 1/f "(z), where f ∈ S̃. In the infinite dimensional case this follows from
the Löwner-Kufarev equation (6.9) with a0 = et . Kirillov’s fields Lk act over these
polynomials as
Lk Pn = (n − 2k − 1)Pn−k
n≥k
and Lk Pn = 0 n < k.
Proposition 1. Let us give the conjugate to {L1 , . . . , Ln } basis of one-forms.
We define
ω1 = dc1 ,
ω2 = dc2 − 2c1 ω1 ,
... ... ...............,
n−1
ωn = dcn −
(j + 1)cj ωn−j .
(7.4)
j=1
Then
ωn (Ln ) = 1,
ωn (Lk ) = 0 if k ,= n.
Proof. If k > n, then the vector fields Lk do not contain ∂ n for k > n. Since
the form ωn depends only on dcj with j < n, then
ωn (Lk ) = ∂ n (Lk ) −
If n = k, then
ωn (Ln ) = ∂ n (Ln ) −
n−1
j=1
(j + 1)cj ωn−j (Lk ) = 0 for k > n > n − j.
n−1
j=1
(j + 1)cj ωn−j (Ln ) = 1 + 0 for n > n − j.
VIRASORO ALGEBRA AND DYNAMICS IN THE SPACE OF UNIVALENT FUNCTIONS
31
To prove the case k < n we apply the induction. Let us show for L1 . We have
ω2 (L1 ) = dc2 (L1 ) − 2c1 (L1 ) = 2c1 − 2c1 = 0.
We suppose that ωn (L1 ) = 0. Then
ωn+1 (L1 ) = dcn+1 (L1 ) −
n
j=1
(j + 1)cj ωn+1−j (L1 ) = (n + 1)cn − (n + 1)cn ω1 (L1 ) = 0.
The same arguments work for ωn (Lk ) with k < n.
!
In the affine basis the forms can be written making use of the polynomials Pn .
We observe that one-forms ωk are defined in a similar way as the coordinates uk with
respect to the Kirillov vector fields Lk . Thus, if we develop the recurrent relations (7.4)
and collect the terms with dcn we get
ωk = dck +
k−1
-
Pj dck−j . k = 1, . . . , n.
j=1
By the duality of tangent and co-tangent bundles the information about the motion is encoded by these one-forms.
7.2. Hamiltonian equations. There exists an Hermitian form on T Mn , such
that .
the system {L1 , . . . , Ln } is orthonormal with respect to this form. The operator
L=
|Lk |2 is elliptic, and we write the Hamilton function H(c, c̄, ψ, ψ̄) .
defined on
the co-tangent bundle, corresponding to the operator L as H(c, c̄, ψ, ψ̄) = nk=1 |lk |2 ,
where
n−k
lk = ψ̄k +
(j + 1)cj ψ̄k+j .
j=1
The corresponding Hamiltonian system admits the form
∂H
= ¯l1
∂ ψ̄1
... = ............
n−1
∂H
¯
ċn =
= ln +
(j + 1)cj ¯ln−j
∂ ψ̄n
j=1
ċ1 =
n−p
∂H
˙
= −(p + 1)
lk ψ̄k+p
ψ̄p = −
∂ cp
k=1
... = ............
∂H
ψ̄˙ n = −
= 0.
∂ cn
32
IRINA MARKINA AND ALEXANDER VASIL’EV
Let us observe that
l˙k =
(7.5)
n−k
j=1
(j − k)¯lj lj+k .
Expressing ¯lk from the first n Hamilton equations we get
¯lk = ċk +
(7.6)
k−1
-
Pj ċk−j ,
k = 1, . . . , n.
j=1
We can decouple the Hamiltonian system making use of (7.5) and (7.6) which leads
us to the following non-linear differential equations of the second order
c̈k = ¯l˙k +
k−1
-
(j + 1)cj ¯l˙k−l +
j=1
k−1
-
(j + 1)ċj ¯lk−l ,
j=1
where l˙k are expressed in terms of the product of ¯lj lj+k by (7.5), and the last products
depend on Pj , P̄j and ċ, c̄˙j for the corresponding indices j by (7.6). For example,
c̈1 = ¯l˙1 =
n−1
j=1
j−1
j
2
32
3
(j − 1) ċj +
Pp ċj−p ċj+1 +
Pq ċj+1−q .
p=1
q=1
Comparing (7.6) and (7.3), we conclude that ¯lk = uk and uk satisfy the differential
equations
(7.7)
u̇k =
n−k
j=1
(j − k)ūj uj+k ,
on the solution of the Hamiltonian system. Observe that any solution of (7.7) has a
velocity vector of constant length. It is easy to see from the following system
(7.8)
ū1 u̇1
ū2 u̇2
ū3 u̇3
ū4 u̇4
ū5 u̇5
ū6 u̇6
=
=
=
=
=
=
0ū1 ū1 u2 + ū1 ū2 u3 + 2ū1 ū3 u4 + 3ū1ū4 u5 + 4ū1 ū5 u6 + . . . ,
−1ū1 ū2 u3 + 0ū2 ū2 u4 + 1ū2 ū3 u5 + 2ū2 ū4 u6 + . . . ,
−2ū1 ū3 u4 − 1ū2 ū3 u5 + 0ū3ū3 u6 + . . . ,
−3ū1 ū4 u5 − 2ū2 ū4 u6 + . . . ,
−4ū1 ū5 u6 + . . . ,
...
Then,
n
d|u|2
(ūk u̇k + uk ū˙ k ) = 0,
=2
dt
k=1
VIRASORO ALGEBRA AND DYNAMICS IN THE SPACE OF UNIVALENT FUNCTIONS
33
for any n, thanks to the cut form of our vector fields and the skew symmetry of (7.8).
The simplest solution may be deduced for constant driving terms uk , k = 1, . . . , n.
The Hamiltonian system immediately gives the geodesic
c1 = ū1 (0)s + c1 (0),
c2 = ū21 (0)s2 + ū2 (0)s + c2 (0),
4 2 s3
5
4
5
s2
s2
c3 = 3ū1 (0) ū1 (0) + ū2 (0) + c2 (0) + 2ū2 (0) ū1 (0) + c1 (0)s + ū3 (0)s + c2 (0),
3
2
2
... = ..................
In general, cn becomes a polynomial of order n with coefficients that depend on the
initial data c(0) and on the initial velocities ū(0).
The Lagrangian L corresponding to the Hamiltonian function H can be defined
by the Legendre transform as
k−1
n 2
n
3 1¯
¯
lk ψ̄k + ψ̄k
(j + 1)cj lk−j −
L = (ċ, ψ̄) − H =
|lk |2 .
2
j=1
k=1
k=1
Taking into account that
ψ̄k ċk =
k−1
-
(j + 1)cj ψ̄k ¯lk−j + ψ̄k ¯lk .
j=1
Summing up over k, we obtain (ċ, ψ̄) =
¯ = .n ūk uk , that gives us
k=1
.n
L(c, ċ) =
k=1 lk lk
n
-
1
2
k=1
|uk |2 .
All these considerations can be generalized for n → ∞. Thus, we conclude that
the coefficients of the function p(z, t) in the Löwner-Kufarev PDE play the role of
generalized moments for the dynamics in Mn and M with respect to the Kirillov
basis on the tangent bundle. Moreover, the L2 -norm of the function p on the circle
S 1 is the energy of such motion.
8. SLE and CFT
In this section we review the connections between conformal field theory (CFT)
and Schramm-Löwner evolution (SLE) following, e.g., [6], [13]). SLE (being, e.g.,
a continuous limit of CFT’s archetypical Ising model at its critical point) gives an
approach to CFT which emphasizes CFT’s roots in statistical physics.
SLEκ is a κ-parameter family of conformally covariant processes describing the
evolution of random sets called the SLEκ hulls. For different values of κ these sets can
be either a simple fractal curve κ ∈ [0, 4], or a self-touching curve κ ∈ (4, 8), or a space
filling Peano curve κ ≥ 8. At this step we deal with the chordal version of SLE. The
complement to a SLEκ hull in the upper half-plane H is a simply connected domain
34
IRINA MARKINA AND ALEXANDER VASIL’EV
that is mapped conformally onto H by a holomorphic function g(z, t) satisfying the
equation
2
dg
=
, g(z, 0) = z,
(8.1)
dt
g(z, t) − ξt
√
where ξt = κBt , and Bt is a normalized Brownian motion with the diffusion constant
2t
+ . . . . The equation (8.1) is
κ. The function g(z, t) is expanded as g(z, t) = z +
z
called the Schramm-Löwner equation and was studied first in [28], [43], see also [41]
for basic properties of SLE. Special values of κ correspond to interesting special cases
of SLE, for example κ = 2 corresponds to the loop-erasing random walk and the
uniform spanning tree, κ = 4 corresponds to the harmonic explorer and the Gaussian
free field. Observe, that the equation (8.1) is not a stochastic differential equation
(SDE). To rewrite it in a stochastic way (following [6], [13]) let us set a function
kt (z) = g(z, t) − ξt , where kt (z) satisfies already the SDE
2
dkt (z) =
dt − dξt .
kt (z)
For a function F (z) defined in the upper half-plane one can derive the Itô differential
κ
(8.2)
dF (kt ) = −dξt L−1 F (kt ) + dt( L2−1 − 2L−2 )F (kt ),
2
d
d
and L−2 = − z1 dz
. These operators are the first two
with the operators L−1 = − dz
Virasoro generators in the ‘negative’ part of the Witt algebra spanned by the operators
d
acting on the appropriate representation space. For any state |ψ-, the state
−z n+1 dz
L−1 |ψ- measures the diffusion of |ψ- under SLE, and ( κ2 L2−1 − 2L−2 )|ψ- measures the
drift. The states of interest are drift-less and given by so-called null-vectors, i.e., the
second term in (8.2) vanishes. Such states are annihilated by κ2 L2−1 − 2L−2 , which
is true if we choose the state |ψ- as the highest weight vector in the highest weight
representation of the Virasoro algebra with the central charge c and the conformal
weight h given by
6−κ
(6 − κ)(3κ − 8)
, h=
,
c=
2κ
2κ
and the operators L−1 and L−2 are taken in the corresponding representation. It
was obtained in [6] and [13], that F (kt ) is a martingale if and only if ( κ2 L2−1 −
2L−2 )F (kt ) = 0. We define a CFT with a boundary in H such that the boundary
condition is changed by a boundary operator. The random curve in H defined by SLE
is growing so that it has states of one type to the left and of the other type to the
right (the simplest way to view this is the lattice Ising model with the states defined
as spin positions up or down). The mapping g satisfying (8.1) ‘unzips’ the boundary.
The primary operator that induces the boundary change with the conformal weight
h is drift-less, and therefore, its expectation value does not change in time under
the boundary unzipping. Hence all correlators computing with this operator remain
VIRASORO ALGEBRA AND DYNAMICS IN THE SPACE OF UNIVALENT FUNCTIONS
35
invariant. Analogous considerations one may provide for the ‘radial’ version of SLE
in the unit disk, slightly modifying the above statements.
Observe that in this formulation two Virasoro generators can generate the nontrivial ‘negative’ part of the Witt algebra by the commutation relation
[Lm , Ln ] = (n − m)Ln+m .
There are many forthcoming directions that can follow the described study of
the Löwnere-Kufarev equations. One of possible directions is to study the subRiemannian geometry naturally related to the bracket generating structure of the
Virasoro algebra and to the hypoellipticity of the drift operator for SLE. Another is
to consider analogues of SLE in the case of infinite degrees of freedom (stochastic
version of the Löewner-Kufarev equation). The alternate Löwner-Kufarev equation,
infinite dimensional controllable systems analogous to one considered here, are also
new objects to study. We hope this survey will encourage a new wave of interest to
this classical subject.
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University of Bergen, Johannes Brunsgate 12, N-5008 Bergen, Norway
E-mail address: [email protected]
University of Bergen, Johannes Brunsgate 12, N-5008 Bergen, Norway
E-mail address: [email protected]
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