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The Erwin Schrodinger International Pasteurgasse

ESI Institute for A Wien Austria

The Top ological GG WZW Mo del in the

Generalized Momentum Representation

Anton Yu Alekseev

Peter Schaller

Thomas Strobl

Vienna Preprint ESI May

Supp orted by Federal Ministry of Science and Research Austria

Available via httpwwwesiacat

ETHTH TUW

PITHA ESI

hepth

The Top ologi cal GG WZW Mo del in the

Generalized Momentum Representation

Anton Yu Alekseev

Institut fur Theoretische Physik ETHHonggerb erg

CH Zuric h Switzerland

Peter Schaller

Institut fur Theoretische Physik TU Wien

Wiedner Hauptstr A Vienna Austria

y

Thomas Strobl

Institut fur Theoretische Physik RWTH Aachen

Sommerfeldstr D Aachen Germany

May

Abstract

We consider the top ological gauged WZW mo del in the general

ized momentum representation The chiral eld g is interpreted as

a counterpart of the electric eld E of conventional gauge theories

The gauge dep endence of wave functionals g is governed by a new

gauge co cycle We evaluate this co cycle explicitly using the

GW Z W

machinery of Poisson mo dels In this approach the GWZW mo del

is reformulated as a Schwarz typ e top ological theory so that the ac

tion do es not dep end on the worldsheet metric The equivalence of

this new formulation to the original one is proved for genus one and

conjectured for an arbitrary genus Riemann surface As a bypro duct

we discover a new way to explain the app earence of Quantum Groups

in the WZW mo del

On leave of absence from Steklov Mathematical Institute Fontanka StPetersburg

Russia email alekseevitpphysethzch

y

email tstroblplutophysikrwthaachende

Intro duction

In this pap er we investigate the quantization of the gauged GG WZW

mo del in the generalized momentum representation The consideration is

inspired by the study of twodimensional YangMills and BFtheories in

the momentum representation

The problem of quantization of gauge theories in the momentum repre

1

sentation has b een attracting attention for a long time While in the

connection representation the idea of gauge invariance may b e implemented

in a simple way

g

A A

we get a nontrivial b ehaviour of the quantum wave functions under gauge

transformations in the momentum representation Indeed one can apply the

following simple argument The wave functional in the momentum represen

tation may b e thought of as a functional Fourier transformation of the wave

functional in the connection representation

Z Z

A E D A exp i tr E A

i i

Taking into account the b ehaviour of A and E under gauge transformations

g

1 1

g A g g g A

i i

i

g

1

E g E g

i

i

we derive

R

1

g i tr E g g

i i

E E e

We conclude that the wave functional in the momentum representation is not

invariant with resp ect to gauge transformations Instead it gains a simple

phase factor E g which is of the form

Z

1

E g tr E g g

i i

The innitesimal version of the same phase factor

Z

E tr E

i i

1

The authors are grateful to Prof RJackiw for drawing their attention to this pap er

and for making them know ab out the scientic content of his letter to Prof D Amati

corresp onds to the action of the gauge algebra

It is easy to verify that satises the following equation

g

E g h E g E h

This prop erty assures that the comp osition of two gauge transformations

with gauge parameters g and h is the same as a gauge transformation with

a parameter g h Equation is usually referred to as a co cycle condition

It establishes the fact that is a oneco cycle of the innite dimensional

gauge A oneco cycle is said to b e trivial if

g

E g E E

for some In this case the gauge invariance of the may b e

restored by the redenition

~

i(E )

E e E

An innitesimal co cycle E is trivial if it can b e represented as a

function of the commutator E

E E

It is easy to see that the co cycle is nontrivial Indeed let us cho ose b oth

a a

E and having only one nonzero comp onent E and in the

Then the commutator in is always equal to zero whereas the expression

is still nontrivial As a consequence also the gauge group co cycle is

nontrivial

On the other hand on some restricted space of values of the eld E the

co cycle may b ecome trivial generically if we admit nonlo cal expressions for

This is imp ortant to mention as one may rewrite the integrated Gauss

law as a triviality condition on the co cycle Let us parametrize as

E exp iE which is p ossible whenever and insert this ex

R

1 2

pression into The result is precisely with tr E g g In fact

i i

eg in two dimensions the wave functions of the momentum representation

2

More accurately one obtains only mo d But anyway this mo dication of

is quite natural in view of the origin of the co cycle within Alternatively one might

regard also a multiplicative co cycle exp i right from the outset cf App endix A

1

are supp orted on some conjugacy classes E x g x E g x with sp ecic

0

values of E But away from these sp ecic conjugacy classes and in partic

0

ular in the original unrestricted space of values for E the general argument

of the co cycle b eing nontrivial applies More details on this issue may b e

found in App endix A

It is worth mentioning that in the ChernSimons theory a co cycle app ears

in the connection representation as well

g i(Ag )

A e A

The co cycle A g is ususally called WessZumino action It is intimately

related to the theory of anomalies

Recently a co cycle of typ e has b een observed in twodimensional BF

and YMtheories In this pap er we consider the somewhat more complicated

example of the gauged WZW GWZW mo del for a semisimple

Like the BF theory it is a twodimensional top ological eld theory for a

detailed account see It has a connection oneform gauge eld as one of

its dynamical variables and p ossesses the usual gauge symmetry However

there is a complication which makes the analysis dierent from the pattern

In the GWZW mo del the variable which is conjugate to the gauge eld

and which shall b e denoted by g in the following takes values in a Lie group G

instead of a linear space So we get a sort of curved momentum space We

calculate the co cyle which governs the gauge dep endence of wave

GW Z W

functions in a g representation and nd that it diers from the standard

expression We argue that while the co cycle corresp onds to a Lie

group G our co cycle is related to its quantum deformation G In the course

q

of the analysis we nd that the GWZW mo del b elongs to the class of Poisson

mo dels recently discovered in This theory provides a technical to ol for

the evaluation of the co cycle

GW Z W

Let us briey characterize the content of each section In Section we

develop the Hamiltonian formulation of the GWZW mo del nd canonically

conjugate variables and write down the gauge invariance equation for the

wave functional in the g representation

Section is devoted to the description of Poisson mo dels A two

dimensional top ological mo del of this class is dened by xing a Poisson

bracket on the target space Using the Hamiltonian formulation the top ology

of the spacetime b eing a torus or cylinder we prove that the GWZW mo del

is equivalent to a certain Poisson mo del coupled to a top ological term S

that has supp ort of measure zero on the target space of the eld theory The

target space of the coupled Poisson mo del is the Lie group G We start

from the GWZW action evaluate the Poisson structure on G and discover

its relation to the theory of Quantum Groups The origin of the term S is

considered in details in App endix B

In Section we solve the gauge invariance equation and nd the gauge

dep endence of the GWZW wave functional in the g representation This

provides a new co cycle Calculations are p erformed for the Poisson

GW Z W

mo del without the singular term In App endix C we reconsider the problem

in the presence of the top ological term It is shown that in the case of

G SU at most one is aected We compare the results

with other approaches

In some nal remarks we conjecture that the Poisson mo del coupled to

S gives an alternative formulation of the GWZW mo del valid for a Riemann

surface of arbitrary genus We comment on the new relation b etween the

WZW mo del and Quantum Groups which emerges as a bypro duct of our

consideration

Hamiltonian Formulation of GWZW Mo del

The WZW theory is dened by the action

Z Z

k k

1 1 2 1 1 3

tr g g g g d x tr d dg g WZW g

where the elds g take values in some semisimple Lie group G and indices

are raised with the standard Minkowski metric The case of a Euclidean

metric may b e treated in the same fashion Some remarks concerning the

second term in may b e found in App enidx B

The simplest way to gauge the global symmetry transformations g

1

l g l is to intro duce a gauge eld h taking its values in the gauge group the

action

1

GW Z W h g WZW hg h

1 1

is then invariant under the lo cal transformations g l g l h hl With

1

the celebrated PolyakovWigmann formula and a h h where

GWZW can b e brought into the standard form

o 1

GW Z W g a a WZW g

+

R

k

1 1 1 2

tr a g g a g g a g a g a a d x

+ + + +

4

+

In the course of our construction of GWZW a a dx a dx has b een

+

2

sub ject to the zero curvature condition da a This condition results

also from the equations of motion arising from So further on we treat

a as unconstrained elds taking their values in the Lie algebra of the chosen

gauge group

In order to nd a Hamiltonian formulation of the GWZW mo del we rst

bring into rst order form To this end we intro duce an auxiliary eld

px into the action by the replacement g g

0

k

1 1 2 1 1 2

g g a g a g pg g a g a g p

+ +

k

As p enters the action quadratically it may b e eliminated always by means

of its equations of motion so as to repro duce the original action In the

functional integral approach this corresp onds to p erforming the Gaussian

integral over p

Now the action may b e seen to take the form with g g

1

Z Z

k

1 1 3 2 1

GW Z W g p a tr d dg g d x tr pg g

k

1 1 1

g g g g g pg p a

k

1 2 1

a a p g g p g g

+

k

This is already linear in time derivatives After the simple shift of variables

1

a a a a p g g

+ + +

k

the last term is seen to completely decouple from the rest of the action

Therefore one can exclude it from the action without loss of information

We can again employ the argument ab out integration overa or also a in

+ +

So we have intro duced one extra variable p and now one variable is

found to drop out from the formalism

After a is excluded the rest of formula provides the Hamiltonian

+

formulation of the mo del The rst two terms play the role of a symplectic

3

p otential giving rise to the symplectic form

I

2

k

1 1 1 1 f ield

g g dx dg g dpdg g p tr

Here d is interpreted as an exterior derivative on the phase space It is

interesting to note that the nonlo cal term in gives a lo cal contribution

to the symplectic form on the phase space The third term which includes

a represents a constraint

k

1 1 1

g g g g g pg p

The variable a is a Lagrange multiplier and the constraint is nothing but

the Gauss law of the GWZW mo del It is a nice exercise to check with the

help of that the constraints are rst class and that they generate

2

k

1 1

the gauge transformations Equation implies tr g pg g g

4

k

1 2

g g and hence tr p

4

1

tr p g g

This p ermits to eliminate the Hamiltonian in in agreement with the fact

that the mo del is top ological

Being a Hamiltonian formulation of the GWZW mo del the form is

not quite satisfactory if one wants to solve the Gauss law equation We

therefore apply here some trick usually referred to as b osonization The

main idea is to substitute the Gauss decomp osition for the matrix g into the

GWZW action

1

g g g

where g is lower triangular g is upp er triangular and b oth of them are

elements of the complexication of G Note however that we do not com

plexify the target space G here but only use complex co ordiantes g g on

3

Cf also App endix C

it If the diagonal parts of g and g are taken to b e inverse to each other

this splitting is unique up to sign ambiguities in the evaluation of square

ro ots Analogously any element of the Lie algebra G corresp onding to G may

b e split into upp er and lower triangular parts according to

Y Y Y Y Y Y

d

d d

where a subscript d is used to denote the diagonal parts of the corresp onding

matrices

1 3

Observe that the threeform tr dg g may b e rewritten in terms of g

and g as follows

k k

1 3 1 1

tr dg g d tr dg g dg g

Here is a threeform on G supp orted at the lower dimensional subset of

G which do es not admit the Gauss decomp osition Now we can rewrite the

top ological WessZumino term as

Z Z Z

k k k

1 1 1 1 3 1

WZ g dg g d tr d dg g tr dg g

1

In this way we removed the symb ol d in the rst term of the right hand

side The top ological term

Z

k

1

d S

is considered in details in App endix B In contrast with the conventional WZ

term the new top ological term inuences the equations of motion only

on some lower dimensional subset of the target space

Let us return to the action of the GWZW mo del We make the substitu

tion and intro duce a new momentum variable

k

1 1 1

g pg g g g g

k

a we now may rewrite the GWZW action Rescaling a according to

2

in the form

GW Z W g S g S g

P

where S has b een intro duced in and S is given by

P

n

R

1 1

2

S g d x tr g g g g

P 0 0

h io

1 1 1 1

2

g g g g g g g g

1 1

k

In the further consideration we systematically disregard the top ological term

S In App endix C we prove that if we take into account the results

change only for wave functions having supp ort on those adjoint orbits in G

one in the case of G SU on which the Gauss decomp osition breaks

down

For the formulation of a quantum theory in the g representation the

momentum should b e replaced by some derivative op erator on the group

The rst term in represents the symplectic p otential on the phase space

and suggests the ansatz

g g g ig

g g

At this p oint some remark on the notational convention is in order On

GLN co ordinates are given by the entries g of the matrix representing an

ij

element g GLN The corresp onding basis in the tangent space may b e

arranged into matrix form via

g g

j i

ij

With this convention the entries of g are seen to b e the right translation

g

invariant vector elds on GLN Given a G of GLN the trace

can b e used to pro ject the translation invariant derivatives from GLN to

G In more explicit terms given an element Y of the Lie algebra of G

a right translation invariant derivative on G is dened by tr Y g The

g

matrix valued derivatives in this pap er are to b e understo o d in this sense

In particular means that the quantum op erator asso ciated to tr Y is

given by

tr Y i tr Y g Y g

g g

With this interpretation it is straightforward to prove that commutators of

the quantum op erators dened in repro duce the of the

corresp onding classical observables as dened by the symplectic p otential

term in

Let us lo ok for the wave functionals of the GWZW mo del in the g

representation This means that we must solve the equation

1 1

g g g g g g

1 1

i

1 1

g g g g g g g g g g

k g g g g

for b eing a wave functional the functional derivatives are understo o d to

act on only but not on everything to their right The problem is clearly

formulated but at rst sight it is not evident how to solve equation To

simplify it we intro duce another parametrization of the matrix g

1

g h g h

0

Here g is diagonal and h is dened up to an arbitrary diagonal matrix which

0

may b e multiplied from the left The part of the op erator which includes

functional derivatives simplies dramatically in terms of h One can rewrite

equation as

i

1 1

h g h g g g g

0 1 1

k h

where g g are determined implicitely as functions of h and g via

0

1 1

g h g h g

0

We discuss the interpretation of equations and in Section and

solve them eciently in Section

Gauged WZW as a Poisson  Mo del

The Gauss law equations of the previous section may b e naturally acquired

in the theory of Poisson mo dels We start with a short description of this

typ e of top ological mo del

The name Poisson mo del originates from the fact that its target space

N is a Poisson manifold ie N carries a Poisson structure P We denote

co ordinates on the twodimensional worldsheet M by x and co

i

ordinates on the target space N by X i n A

f g on N is dened by sp ecifying its value for some co ordinate functions

i j ij

fX X g P X Equivalently the Poisson structure may b e represented

by a bivector

ij

P X P

i j

X X

In terms of this tensor the Jacobi identity for f g b ecomes

j k ij k i

P P P

li lk lj

P P P

l l l

X X X

For nondegenerate P the notion of a Poisson manifold coincides with that of

a symplectic manifold In general however P need not b e nondegenerate

i

In the worldsheet picture our dynamical variables are the X s and a

eld A which is a oneform in b oth worldsheet and target space In lo cal

co ordinates A may b e represented as

i

A A dX dx

i

The top ological action of the Poisson mo del consists of two terms which

we write in co ordinates

Z

i

X

ij

S X A A dx dx P A A

P i i j

x

M

Here A and X are understo o d as functions on the worldsheet Both terms

in are twoforms with resp ect to the worldsheet Thus they may b e

integrated over M The action is obviously invariant with resp ect to dif

feomorphisms of the worldsheet It is also invariant under dieomorphisms

of the target space which preserve the Poisson tensor Equations of motion

for the elds X and A are

i ij

X P A

j

j k

P

A A A A

i i j k

i

X

Here is the derivative with resp ect to x on the worldsheet

Let us remark that the twodimensional BF theory may b e interpreted

as a Poisson mo del Indeed if one cho oses a Lie algebra with structure

ij

constants f as the target space N and uses the natural Poisson bracket

k

i j ij k

fX X g f X

k

one repro duces the action of the BF theory

Z

2

BF X A tr X dA A

M

2

In the traditional notation X is replaced by B and the curvature dA A

of the gauge eld is denoted by F The class of Poisson mo dels includes

also nontrivial examples of twodimensional theories of gravity for details

see

We argue that the gauged WZW mo del is equivalent to a Poisson mo del

coupled to the term The target space is the group G parametrized by

g and g The form A is identied readily from

0 1 1 1 1 1

dx dg dg g dx g dg g A dg g

Here we have interpreted the terms linear in and

Then the part of the action quadratic in and directly determines the

Poisson structure In our formulation of the general Poisson mo del

i

the indices i of A corresp ond to a co ordinate basis dX in T N and dx

in T M In such a formulation we simply have to replace A by in

i

i

X

the quadratic part of the action to obtain the Poisson bivector as the

co ecient of the volumeform dx dx Each of the matrixvalued one

1 1 1 1

dg in the present expression for dg g and g dg g forms dg g

A however represents a nonholonomic basis in the cotangent bundle of the

target space G In such a case the corresp onding comp onents of A ie

and in our notation have to b e replaced by the resp ective dual derivative

matrices Applying this simple recip e to the quadratic part of we nd

the Poisson bivector on G

g g g g

g g g g

" # " #

4

g g tr P

k g g

" #

1 1

g g g g g g g g

g g g g

# " # "

Using the parametrization this expression can b e formally simplied

to

tr h g g P

k g g h

For means of completeness we should check now that this bivector fullls the

Jacobi identity In our context the simplest way to do so is to recall that

the constraints of the GWZW mo del are rst class this suces b ecause one

can show that the constraints of any action of the form are rst class

ij

exactly i P ob eys Certainly one can verify the Jacobi identity also by

some direct calculation and in fact this is done implicitely when establishing

and b elow

The ab ove Poisson bracket on G requires further comment For this

purp ose it is useful to intro duce some new ob ject We always assume that

the group G is realized as a subgroup in the group of n by n matrices Then

n n

the following matrix r acting in C C is imp ortant for us

X X

i

r h h t t

i

i

i

Here h and h are generators of dual bases in the Cartan subalgebra t and

i

t are p ositive and negative ro ots resp ectively The matrix r is usually

called classical r matrix It satises the classical YangBaxter equation in

the triple tensor pro duct which reads

r r r r r r

12 13 12 23 13 23

Here r r is emb edded in the pro duct of the rst two spaces and so

12

on An imp ortant prop erty of the r matrix is the following

1 2

tr r A B tr A B tr A B

12 d d

where the trace in the left hand side is evaluated in the tensor pro duct of

two spaces and

1 2

A A B B

Now we are ready to represent the bracket in a more manageble

way As the most natural co ordinates on the group are matrix elements we

are interested in Poisson brackets of entries of g and g Using shorthand

notations and the denition of the r matrix we arrive at the following

elegant result

1 2 1 2

r g g fg g g

k

1 2 1 2

fg g g r g g

k

1 2 1 2

r g g fg g g

k

We omit the calculation which leads to as it is rather lengthy but

straightforward Each equation in provides a Poisson bracket b etween

any matrix element of the matrix with sup erscript with any matrix element

of the matrix with sup erscript In order to clarify this statement we rewrite

the rst equation in comp onents

~ ~ ~

~ ~

~

j j ij ij ij

k l ii k l k l

r g g r g g g fg g

~

~

k k

ll

k

Here summation over the indices with tilde in the right hand side is under

sto o d The remaining equations of can b e rewritten in the same fashion

The formulae dene the Poisson bracket only on the subset of the group

G which admits the Gauss decomp osition One can easily recover the

Poisson brackets of matrix elements of the original matrix g We leave this

as an exercise to the reader The result may b e presented in tensor notation

1 2 2 1 1 2 1 2 1 2

g r g g r g r g g g g r fg g g

k

Here r is obtained from r by exchanging the two copies of the Lie algebra

X X

i

r t t h h

i

i

The Poisson bracket is quadratic in matrix elements of g and obviously

smo oth This means that the bracket which has b een dened only on the

part of the group G where the Gauss decomp osition is applicable may now

b e continued smo othly to the whole group Eg for the case of G SU it

is straightforward to establish that the righthand side of and thus also

the smo othly continued Poisson tensor P vanishes at antidiagonal matrices

g SU The latter represent precisely the onedimensional submanifold of

SU where a decomp osition for g do es not exist It is worth mentioning

that the bracket app eared rst in within the framework of the theory

of PoissonLie groups

The group G equipp ed with the Poisson bracket may b e used as

a target space of the Poisson mo del We have just proved that in the

Hamiltonian formulation geometry of the worldsheet is torus or cylinder

this Poisson mo del coupled to the top ological term coincides with the

gauged WZW mo del

Solving the Gauss Law Equation

This section is devoted to the quantization of Poisson mo dels More ex

actly we are interested in the Hilb ert space of such a mo del in the Hamilto

nian picture This implies that we need a distinguished time direction on the

worldsheet and thus we are dealing with a cylinder The remarkable prop

erty of Poisson mo dels is that the problem of nding the Hilb ert space

in this twodimensional eld theory may b e actually reduced to a quantum

mechanical problem This has b een realized in and here we give only a

4

short account of the argument

i

It follows form that in the Hamiltonian formulation the variables X

and A are canonically conjugate to each other this changes slightly when

i1

the Poisson mo del is coupled to the term S see App endix C In the X

i

representation of the quantum theory the X act as multiplicative op erators

and the A act as functional derivatives

i1

A i

i1

i

X

The comp onents A enter the action linearly They are naturally inter

i0

preted as Lagrange multipliers The corresp onding constraints lo ok as

i i ij

G X P X A

1 j 1

Combining and one obtains an equation for the wave functional in

4

But cf also Appp endix C

the X representation

i ij

X X iP X

1

j

X

Equations and are particular cases of this equation In order to

solve we rst turn to a family of nite dimensional systems on the

target space N dened by the Poisson structure P

As the target space of a Poisson mo del carries a Poisson bracket it

may b e considered as the starting p oint of a quantization problem Namely

one can consider the target space as the phase space of a nite dimensional

Hamiltonian system which one may try to quantize The main obstruction

on this way is that the Poisson bracket P may b e degenerate This means

that if we select some p oint in the target space and then move it by means of

all p ossible Hamiltonians we still do not cover the whole target space with

tra jectories but rather stay on some surface S N The simplest example

3

of such a situation is a threedimensional space N IR with the Poisson

bracket

i j k

fX X g X

ij k

This Poisson bracket describ es a threedimensional angular momentum and

it is wellknown that the square of the length

X

i 2 2

X R

i

i 2

commutes with each of the X So R cannot b e changed by means of

Hamiltonian ows and the surfaces S are twodimensional spheres

If the Poisson bracket P is degenerate we cannot use N as a phase

space However if we restrict to some surface S these surfaces are also

called symplectic leaves the Poisson bracket b ecomes nondegenerate and

one can try to carry out the quantization program In the functional integral

approach we are interested in the exp onent expiA of the classical action A

b eing the main ingredient of the quantization scheme In order to construct

the classical action A we invert the matrix of Poisson brackets restricted to

some particular surface S and obtain a symplectic twoform

i j

dX dX

ij

X

j

j k

P

ik

i

k

As a consequence of the Jacobi identity the form is closed

d

and we can lo ok for a oneform which solves the equation

d

If b elongs to some nontrivial cohomology class pdq do es not exist

globally Still the expression

Z

1

X exp i d

makes sense if the cohomology class of is integral ie if

I

n n Z

R

1

for all twocycles S in this case A d is dened mo d and

is onevalued cf also b elow Alternatively to the functional integral

approach we might use the machinery of geometric quantization to ob

tain condition Within the approach of geometric quantization it is a

wellknown fact that a Hamiltonian system S may b e quantized consis

tently only if the symplectic form b elongs to an integral cohomolgy class

of S In the example of twodimensional spheres in the threedimensional

target space considered ab ove the requirement of the symplectic leaf to b e

quantizable obtained in any of the two approaches suggested ab ove implies

that the radius R of the sphere is either or halfinteger for more

details confer This is a manifestation of the elementary fact that a

threedimensional spin has to b e either integer or halfinteger

After this excursion into we return to equation

It is p ossible to show that formula provides a solution of equation

Moreover any solution of can b e represented as a linear combina

tion of expressions corresp onding to dierent integral symplectic leaves

Let us explain this in more detail The wave functional X of the eld

i

theory dep ends on n functions X on the circle They dene a parametrized

closed tra jectory lo op in the target space N Now it is a more or less

immediate consequence of that the quantum constraints of the eld

1

theory restrict the supp ort of to tra jectories lo ops X x which lie com

i

pletely within a symplectic leaf S just use co ordinates X in the target space

adapted to the foliation of N into symplectic leaves A further analysis re

capitulated in part in App endix C within the more general framework of a

Poisson mo del coupled to a top ological term shows that these leaves have

to b e quantizable and that admissible quantum states are indeed all of the

form or a sup erp osition of such functionals In the case that S is simply

connected may b e rewritten more explicitely as

Z

mo d X expiAX AX



1

where the twodimensional surface is b ounded by the closed path X x

5

lying in some quantizable leaf S As b elongs to an integral cohomology

1

class by the choice of S is a globally welldened functional of X x

As stated already b efore any such a functional solves the quantum constraints

and vice versa any solution to the latter has to b e a sup erp osition of

states On the other hand or may b e also reinterpreted as

1

exp onentiated p oint particle action x then is the timeparameter of the

1

tra jectory X x which one requires to b e p erio dic in time

So we obtain the following picture for the relation b etween the Poisson

mo del and nite dimensional In order to obtain the

Hilb ert space of the mo del on the cylinder one may regard the target

space as a phase space of a dynamical system This space splits into a set

of surfaces on which the Poisson bracket is nondegenerate creating a family

of nite dimensional systems Some of these systems are quantizable in the

sense that the cohomology class of the symplectic form is integral To each

quantum system generated in this way corresp onds a linearly indep endent

vector in the Hilb ert space H of the mo del In the case that the resp ective

quantizable symplectic leaf S is not simply connected however there is a

linearly indep endent vector in H for any element of S This idea may

1

b e successfully checked for BF theories in two dimensions for more details

confer

Now we apply the machinery of this section to the GWZW mo del First

we should lo ok at the surfaces S in the group G where the restriction of the

5

In the language of Ap endix C the denition corresp onds to the choice of a constant

p ointlike lo op of reference for

0

Poisson bracket is nondegenerate For generic leaves this problem has

b een solved in In order to make P nondegenerate one should restrict

to some conjugacy class in the group

1

g h g h

0

Each conjugacy class may b e used as the phase space of a Hamiltonian sys

tem However in the case of G SU we found that the Poisson bracket

vanishes on the subset of antidiagonal matrices Hence any antidiagonal ma

trix represents a zerodimensional symplectic leaf in G SU So some

exceptional conjugacy classes may further split into families of symplectic

leaves This o ccurs precisely where the Gauss decomp osition do es not hold

The form on a generic orbit characterized by g has b een recently

0

evaluated in a presentation more adapted to the physical audience can

b e found in and has the form

i h

k

1 1 1

dg dg g tr h dh g

where g g and h are related through The corresp onding p oint particle

action or phase factor of the quantum states resp ectively is

Z

h i

k

1 1

1 1

d tr h dh g dg g dg A g

GW Z W

As outlined ab ove quantum states are assigned only to integral symplec

tic leaves In App endix C the corresp onding integrality condition is

evalutated explicitly for the example of G SU

The exceptional conjugacy classes require some sp ecial attention From

the p oint of view of the pure Poisson mo del there corresp onds a quantum

state to any integral symplectic leaf which the resp ective conjugacy class

may contain For the case of SU eg there is one exceptional two

dimensional conjugacy class characterized by tr g It contains

the onedimensional submanifold C of antidiagonal matrices in SU Any

p oint of C is a zerodimensional symplectic leaf and b ecause zerodimensional

leaves are always quantizable one would b e left with a whole bunch of states

corresp onding to this exceptional conjugacy class

However we know that in order to describ e the GWZW mo del in full

generality we need to add the top ological term S to the pure Poisson

part of the action Also appropriate b oundary conditions of A have to b e

taken into account at the part of G where the Gauss decomp osition breaks

down Whereas S and these b oundary conditions may b e seen to b e irrele

vant for the quantum states corresp onding to generic conjugacy classes they

decisively change the picture at the exceptional ones Eg for G SU

the net result is that there corresp onds only one or even no quantum state

to the exceptional conjugacy class dep ending on whether k is even or o dd

resp ectively Further details on this may b e found in App endix C

From it is straightforward to evaluate the co cycle which

GW Z W

controls the b ehaviour of the wave functional with resp ect to gauge transfor

mations Eg for the case of innitesimal transformations

g g h h

the new gauge co cycle lo oks as

Z

k

1 1

tr g g dg g dg

GW Z W

An integrand of this typ e has b een studied in the framework of PoissonLie

However the gauge algebra interpretation is new

In order to check that the co cycle is nontrivial it is convenient

GW Z W

to use the same trick as we applied in Intro duction Indeed cho ose b oth g

and to b e diagonal Then any trivial co cycle vanishes but is not equal

to zero for generic diagonal g and

Discussion

Let us briey recollect and discuss the results of the pap er Using the Hamil

tonian formulation we have proved that the GWZW mo del is equivalent to

a Poisson mo del coupled to the top ological term S

GW Z W g A S g A S g

P

It is natural to conjecture that this equivalence holds true for a surface of

arbitrary genus Let us mention that originally the GWZW is formulated as a

Witten typ e top ological eld theory This means that the action includes the

kinetic term and explicitly dep ends on the worldsheet metric Then one can

use some sup ersymmetry to prove that in fact the terms including the world

sheet metric do not inuence physical correlators The Poisson mo del

provides a Schwarz typ e formulation of the same theory The righthand side

of is expressed in terms of dierential forms exclusively and do es not

include any metric from the very b eginning

At the moment the GWZW mo del is solved in many ways whereas the

general Poisson mo del has not b een investigated much Applying various

metho ds which work for the GWZW to Poisson mo dels one can hop e to

achieve two goals First one can select the metho ds which work in a more

general framework and hence which are more reliable This is esp ecially

imp ortant when one deals with functional integrals The other ambitious

program is to solve an arbitrary Poisson mo del coupled to a top ological

term explicitly Solution should include an evaluation of the partition func

tion and top ological correlators in terms of the data of the target space In

this resp ect an exp erience of the GWZW mo del may b e very useful

Another issue which deserves some comment is the relation b etween

Quantum Groups and WZW mo dels This issue has b een much studied in the

literature The picture of the quantum symmetry in WZW mo dels may

b e describ ed in short as follows Separating leftmoving and rightmoving

sectors of the mo del we add some nite numb er of degrees of freedom to

the system The Quantum Group symmetry is a gauge symmetry acting on

the left and rightmovers The physical elds are invariants of the Quan

tum Group action Usually one can cho ose some sp ecial b oundary conditions

when separating the sectors in order to make the Quantum Group symmetry

transparent

Let us compare this picture to the considerations of the present pap er

The gauged WZW mo del app ears to b e equivalent to some Poisson mo del

with gauge group G as target space We derive the Poisson bracket

directly from the GWZW action This bracket is quite remarkable Quan

tizing the bracket one gets the generating relations of the Quantum

Group We have found that the gauge dep endence of the wave function

als of the GWZW mo del is describ ed by the classical action dened on the

symplectic leaves This typ e of action for the bracket has b een consid

ered in It is proved there that such an action p ossesses a symmetry

with resp ect to the Quantum Group So conrming our exp ectations the

Quantum Group governs the nonphysical gauge degrees of freedom of the

GWZW mo del The new element of the picture is that we do not have to

intro duce any new variables or cho ose sp ecic b oundary conditions in order

to discover the Quantum Group structure Let us remark that the treatment

may lo ok somewhat more natural for GWZW than for the original WZW

mo del The reason is that GWZW may b e viewed as a chiral theory from

6

the very b eginning The only choice which we make is the way how we

b osonize the WZW action We conclude that the Quantum Group degrees of

freedom are intro duced by b osonization It would b e interesting to explore

this idea from a more mathematical p oint of view

App endices

A Gauge Co cycles and Integral Coadjoint Orbits

Here we study in details the oneco cycle

I

1

E g tr E g g dx A

of the lo op group LG which plays the role of the gauge group on the circle

Along with the additive co cycle we consider a multiplicative co cycle

E g expiE g A

The counterparts of the co cycle and cob oundary conditions in the multiplica

tive setting are

g

1

E g g E g E g A

1 2 2 1

g 1

E g E E A

Let us observe that one can consistently restrict the region of denition

of E from the lo op algebra l G to any subspace invariant with resp ect to the

action of the gauge group by conjugations Let us cho ose such a subset in

the form

1

E hx E hx A

0

for E b eing a constant diagonal matrix For xed x equation A denes

0

a conjugacy class in the algebra G coadjoint orbit

6

We are grateful to KGawedzki for this remark

The diagonal matrix E may b e decomp osed using a basis of fundamental

0

weights w in the Cartan subalgebra

i

X

i

E w A E

i 0

0

i

In the case of compact groups the co cycle is trivial if and only if all co e

i

cients E are integer To demonstrate this let us present the explicit solution

0

for It is given by

I

1

exp i tr E hh dx A

0

It is easy to check that A provides a solution of the cob oundary problem

It is less evident that A is welldened The group element hx is dened

by E x only up to an arbitrary diagonal left multiplier When co ecients

in A are integral this multiplier do es not inuence A

For noncompact groups though A may turn out to b e welldened

even for continuously varying choices of E

0

To establish contact with the presentation in the main text one may

observe that the additive cob oundary generically not welldened

I

1

tr E hh dx A

0

may b e reformulated in terms of the welldened Kirillov form on the coad

joint orbit

1 1 2

tr dE h dh A tr E dhh

0

as

Z

E x A



The ambiguity in the choice of do es not inuence the multiplicative co cycle

i the Kirillov form is integral ie i satises

For compact groups the integrality condition on coincides with

i

the b eforementioned condition on the E If is fullled with n not

0

only the multiplicative but also the additive co cycle b ecomes trivial This

o ccurs eg in the noncompact case G sl IR

It is worth mentioning that A is the action for a quantum mechanical

system with the phase space b eing a coadjoint orbit We consider a similar

system in Section There the quantum mechanical phase space is a conju

gacy class in the group and the analogue of the Kirillov form A is

the Kirillov form of the Quantum Group

We conclude that for certain restricted subspaces of the lo op algebra the

co cycle E g may b ecome trivial In two dimensions the wave functionals

in the momentum representation are supp orted on these sp ecial subspaces

The corresp onding cob oundary governs the gauge dep endence of the wave

functionals

A

0

for b eing a gauge indep endent distribution with supp ort on lo ops in in

0

tegral coadjoint orbits

Let us stress again that the triviality condition A is actually an inte

grated form of the Gauss law as shown in the intro duction Then A

provides a universal solution of the Gauss law In the example which we

considered in this App endix we observe a new phenomenon in the theory

of gauge co cycles A nontrivial co cycle may shrink its supp ort in order to

b ecome trivial and pro duce a physical wave functional This may lead as

in the example of D YM theory with compact gauge group to a discrete

sp ectrum in the momentum representation

B Top ological Term for G SU

The top ological WessZumino term in the WZW mo del is usually represented

in the form

Z

k

1 1 3

d dg g B tr WZ g



The integration is formally p erformed over the twodimensional surface

Here is the of the worldsheet M under the map g x from M G

1

The symb ol d is understo o d in the following way One cho oses a three

dimensional submanifold B in the group G so that B The integration

over is replaced by an integration over B

Z

k

1 3

WZ g tr dg g B

B

The denition B is ambiguous as B may b e chosen in many ways The

p ossible ambiguity in the denition of WZ g is an intergral over the union

of two p ossible B s

Z Z

k

1 3 1 3

tr dg g WZ g dg g

0 00

B B

Z

k

1 3

dg g B tr

0 00



B B

Here we denote by B the manifold B with opp osite orientation Let us

restrict our consideration to the case of G SU The only nontrivial

threedimensional cycle in SU is the group itself It implies that the

integral B is always prop ortional with some integer co ecient to the

normalization integral

Z

k

1 3

I tr dg g k B

G

Here we used the fact that the volume of the group SU with resp ect to the

1 3 2

form tr dg g is equal to This calculation explains why one should

cho ose integer values of the coupling constant k In this case the Wess

Zumino term WZ g is dened mo dulo and its exp onent expiW Z g is

welldened

Usually WZ g is referred to as a top ological term b ecause the dening

1 3

threeform tr dg g on the group G is closed and b elongs to a nontrivial

cohomology class This implies that the integral B do es not change when

we x and vary B in a smo oth way Cho osing the prop er co ecient

k k N we get a threeform which b elongs to an integer cohomology

class As we have seen this ensures that expiW Z g is preserved even by a

top ologically nontrivial change of B

So the fact that the threeform

k

1 3

tr dg g

is closed and b elongs to integer cohomology of G makes the action WZ g

welldened However it is not true that WZ g is dened already by the

cohomology class of If we cho ose some other representative in the same

class as eg in Eq B b elow we get a new top ological term which

is welldened for the same reason as WZ g In fact the new action will

dier from WZ g The reason is that the integral B is dened over the

manifold with a b oundary and hence it is not dened by the cohomology

class of the integrand It dep ends on the representative as well

Now we are prepared to intro duce a new top ological term for the WZW

mo del As it was explained in Section we use the Gauss decomp osition for

the group element g

1

g g g B

Observe that the Gauss decomp osition is not applicable for some elements

in SU The Gauss comp onents g g do not exist on the subset of antidi

agonal unitary matrices In a parametrization

p

i

z z e z

A

z C jz j g

p

i

z z e z

B

these elements are given by z They form a circle C parametrized by

We can apply the Gauss decomp osition on the rest of the group in order

1

to remove the symb ol d from the top ological term Indeed consider a

twoform

k

1 1

B dg g tr dg g

on the compliment of C It is easy to verify the relation

d B

Here we have used the fact that

3 3 1

tr dg g B tr dg g

1 m

which holds since the diagonal parts of dM M vanish for any triangular

matrix M if m

We established equation B on the part of the group G which admits

the Gauss decomp osition It is easy to see that this equation cannot hold

true on all of G Indeed the lefthand side is represented by the exact form

d whereas the righthand side b elongs to a nontrivial cohomology class In

order to improve B we intro duce a correction to it

B d

This equation is to b e understo o d in a distributional sense The threeform

is supp orted on C Moreover it is closed and b elongs to the same cohomology

class as

To determine for G SU we return to the parametrization B

In these co ordinates B takes the form

dz

d B i z dz z dz

z

Multiplying by test oneforms the resulting threeforms are integrable on

G So is a regular distribution and therefore the derivative d is also well

dened Using ddz z Rez Imz dz dz i C where C

has b een intro duced to denote the deltatwoform supp orted on the critical

circle C we obtain

C d B

Let us conclude that the top ological WessZumino term may b e replaced

by the sum of a lo cal term and a top ological term supp orted on the set C of

antidiagonal matrices

Z

k

1 1

WZ g tr dg g dg g S g



Z Z

k

S g C d B k

B B

The new top ological term S g dep ends exclusively on the p ositions of the

p oints where intersects C In particular it vanishes if b elongs to the

part of the group which admits the Gauss decomp osition

In Section we showed that the lo cal term of B ts nicely into the

formalism of Poisson mo dels Coupling of such a mo del to the top ological

term S is the sub ject of App endix C

C Poisson mo del coupled to a Top ological Term and

Quantum States for SU GWZW

Within this last App endix we pursue the following three goals First we

investigate the change of a Poisson mo del

Z

i

X

ij

dx dx C P A A S X A A

i j P i

x

M

under the addition of a top ological term

S X A S X A S X C

P top

Here S X is supp osed to b e given by some closed threeform

top top

Z

S X B Image M C

top top

B

of generically nontrivial cohomology on the target space N of the mo del cf

also App endix B To not sp oil the symmetries of C we further require

to b e invariant under any transformation generated by vector elds of

top

ij

the form P We will fo cus esp ecially on the change in the Hamiltonian

j

sturcture that is induced by C

Next we will sp ecify the considerations to the GWZW mo del In the

main text and the previous App endix we have shown that the Hamilto

nian GWZW action may b e rewritten identical ly in the form C

with However an additional complication arises due to the fact

top

that the matrixvalued oneform

1 i 1

dg C dg g dX g

i

which we used in the identication for A b ecomes singular at the part

of G where the Gauss decomp osition breaks down The singular b ehaviour

of A has to b e taken into account in the variation for the eld equations if

we want to describ e the GWZW mo del by means of C globally We will

show that the bulk of the quantum states obtained in the main text remains

unchanged by these mo dications The considerations change only for states

that have supp ort on lo ops lying on exceptional conjugacy classes in G

Finally we will make the considerations more explicit for G SU and

compare the resulting picture to the literature

In the classical Hamiltonian formulation the term C contributes only

into a change of the symplectic structure of the eld theory With

(top)

i j k

dX dX dX C

top

ij k

the symplectic structure takes the form

I

f ield

f ield 1 i 1 1

X A dA x dX x dx C

i1

top

1

S

with the extra piece

I

(top)

f ield

1 i 1 j 1 k 1 1

X x X x dX x dX x dx C

1

top

ij k

1

S

Note that as resp if is nontrivial in cohomology on the target space

top

f ield

b ecomes nontrivial as well ie globally there will not exist any sym

f ield f ield f ield

plectic p otential such that d

In the case N G and the symplectic forms C and

top

1 i 1

in the main text coincide Actually A x and X x are Darb oux co

i1

f ield f ield

ordinates of the symplectic form of the GWZW mo del As

has nontrivial cohomology such Darb oux co ordinates cannot exist globally

The situation may b e compared to the one of a sphere with standard sym

plectic form sin d d trying to extend the lo cal Darb oux co or

dinates cos and as far as p ossible one nds again in a distributional

2 2

sense d cos d southp ole northp ole Here we

P

2

used dd pol e resulting from the breakdown of d as a

poles

co ordinate dierential at the p oles while it still remains a regular oneform

in a distributional sense By the way one may infer eq also from

i

CC Just replace the co ordinate basis dX by the leftinvariant basis

1 1 1 1 2

dg g and note that dpdg g dpdg g pdg g has to b e substituted

i i

for dA dX dA dX

i1 i1

The classical Gauss law on the other hand remains unaltered by

i

the addition of a term C to the action Indeed the constraints G

emerge as the co ecient of A within the action S S S and S do es

i0 P

not dep end on A

Now let us turn to the quantum theory of the coupled mo del C Again

we go into an X representation In general wave functions may b e regarded

as sections of a line bundle the curvature of which is the symplectic form

In the case that this line bundle is trivial ie when the symplectic

f ield

form allows for a global symplectic p otential one may cho ose a global

nonvanishing section in the bundle The relative co ecient of any other

section with resp ect to the chosen one is then a function the wave function

X This pro cedure is called trivialization of the line bundle In the case

f ield

of prominent interest for us in which and thus b elong to some

top

nontrivial cohomology class the quantum line bundle over the lo op space

will b e nontrivial Sections may b e represented by functions X then

only within some lo cal charts

i

The X may still b e represented as multiplicative op erators However

the change in the symplectic structure implies that one cannot represent A

i1

f ield

as the derivative op erators any more Indeed the mo dication

top

i

preserves commutativity of the X as well as the commutation relations b e

i

tween the A and the X however the A do not commute among each

i1 i1

other any longer The net result of the change in the symplectic structure is

that we have to add some X dep endent piece to the op erator representation

of A

i1

f ield

X C A i

i1

i

i

X

f ield f ield

is a symplectic p otential to the nontrivial part The new quantity

top

i

of the symplectic form ie

I

f ield f ield

i 1 1

dX x dx C

top

i

is a solution to the equation

f ield f ield

d lo cally C

top top

f ield

is not unique and may b e chosen in many ways If b elongs to a

top

top

f ield

trivial cohomology class C may b e solved globally Any choice for

i

then corresp onds to the choice of a trivialization of this line bundle If on the

other hand b elongs to some nontrivial cohomology class we can sp eak

top

ab out a solution to C only lo cally Still any choice of a lo cal p otential

f ield

corresp onds to a lo cal trivialization of the quantum line bundle within

top

some chart Within the latter quantum states may b e represented again as

ordinary functions X on the lo op space and C gives the corresp onding

op erator reprensentation of A

i1

Let us nally write down the new quantum Gauss law Within a lo cal

chart on the lo op space it takes the form

f ield

ij i ij

P i X P C

j

j

X

f ield

these constraints yield a restriction to function For nonsingular forms

top

als with supp ort on lo ops lying entirely within some symplectic leaf again

f ield

This holds true also for a singular as long as its contraction with

j

ij

the Poisson tensor P vanishes To see this just use the rst k co ordi

i

nates X to parametrize the symplectic leaves in any considered region of

i

N Then C yields X for i k So strictly sp eaking the

physical wave functionals will b e distributions that restrict the lo ops to lie

entirely within symplectic leaves The remaining n k equations C then

determine the form of on each leaf

Let us show this for trivial cohomology of the dening threeform in C

ie for the sp ecial case that

d C

top top

(top)

f ield

1 k 1

X x X x globally on the phase globally on N Then

1

j

j k

space To nd the form of on a given symplectic leaf S we multiply C

for i k n by from the left cf Eq The resulting equation

li

can b e integrated easily to yield

Z

1

exp i d C

0 top

where is an integration constant which however may dep end on the cho

0

sen symplectic leaf and if S is not simply connected also on the homotopy

class of the argument lo op of may b e regarded as the evaluation of

0

on some reference lo op on S and the phase is determined by the integration

of the twoform over a twosurface that is enclosed b etween the ref

top

erence lo op and the argument lo op of Indep endence of the choice of the

chosen twosurface requires eg for a simply connected S

I

n n Z C

top

for all twocycles S This generalizes the integrality condition

which corresp onds to C is a wellformulated condition as the

top

invariance requirement for C under the symmetries of C may b e seen

to imply that the restriction of onto any symplectic leaf must b e a closed

top

twoform while certainly will not b e closed in general on all of N

top

For a truely topological term C equation C holds only lo cally

Still C provides the lo cal solution to the quantum constraints C in

the space of lo ops on S However as the form is not dened globally

top

on S in general the global integrability for C do es not have the simple

form C Instead the use of various charts in the line bundle over the

lo op space will b e unavoidable to determine integrability of C on a leaf

and thus the existence of a quantum state lo cated on that leaf We will not

study this problem in full generality here further Rather we will restrict our

attention to the GWZW mo del in the following

Everything that has b een written ab ove applies to the GWZW mo del to o

except for one small change Actually the correct Gauss law for GWZW is

i

not G but

i

G C

i

where the matrixvalued co ecients have b een dened in C To see

i

this we recall that the constraints of the GWZW mo del given rst in Eq

result from a variation for a within the action According to

A diers from the comp onents of by A tr So the correct

i0 i0 i

GWZW Gauss law may b e rewritten as C For lo ops inside the

Gaussdecomp osable region of G this is equivalent to the old form of the

i

constraints G since on that part of G the dierence corresp onds merely

to a change of basis in T G However as b ecomes singular at that lower

dimensional part of G where the Gauss decomp osition breaks down the

i

constraints C have somewhat dierent implications than G in that

region

This consideration applies also to the quantum constraints we have to

multiply C by from the left The result is

i

f ield

ij i ij

i P X P C

i i i

j

j

X

or equivalently

i

f ield

1 1 ij

h g h C g g g g P

0 1 1 i

j

k h

f ield

ij

The part P which may b e rewritten also as the insertion of the

i

j

f ield f ield

is the new con vector eld k hh into the oneform

top

tribution from S that has b een dropp ed in the derivation of

It is not dicult to see that for lo ops that lie at least partially outside

exceptional conjugacy classes critical region in G one may solve in

stead of C or C Indeed close to any part of the lo op outside the

critical region we may use C as the quantum constraint b ecause C

is invertible in that part of G But as argued ab ove this restricts the lo op to

lie entirely within a symplectic leaf outside the critical region in G For such

lo ops now we may always cho ose

f ield

C

i

f ield

as vanishes on that part of the phase space This justies that in the

main text we dropp ed the contributions from S as well as the multiplicative

factor and restricted our attention to the solution of Also we had

i

not to think of a nontrivial quantum line bundle in this way The main

part of the states could b e obtained within one lo cal trivialization of the line

bundle given by C

What has to b e considered separately only are p ossible states that have

supp ort on lo ops lying entirely within the critical region of G In this case

the full quantum constraints C have to b e taken into account It is also

in this region of G furthermore where the notion of symplectic leaves and

conjugacy classes do not coincide From C we learn that it is precisely

the mo dications of that restore the adjoint transformations as symme

tries on the quantum level diverges precisely where P vanishes so as to

give rise to the nite contibution h h in C As a result there will

corresp ond at most one quantum state to an exceptional conjugacy class

even if the resp ective orbit splits into several p ossibly in part integrable

symplectic leaves

Let us now sp ecify our considerations to G SU In particular we

want to determine all quantum states within our approach For this pur

3

p ose let us rst consider the splitting of SU S into conjugacy classes

Parametrizing conjugacy classes by cf also B

tr g Rez cos const C

we nd that top ologically sp eaking these orbits are twospheres for

and p oints for z Only one of the conjugacy classes

is exceptional it corresp onds to tr g Parametrizing this

2

critical S by p olar co ordinates and arccos Imz the part C of SU

on which the Gauss decomp osition is not applicable is identied with the

equator of this twosphere

3

So the picture we obtain is that N S is foliated into twospheres

except for its p oles z The equator of the threesphere itself an

2 1

S is what we called an exceptional conjugacy class The equator C S

2

of this S is precisely the subset of N G where the Gauss decomp osition

breaks down and corresp ondingly where the supp ort of lies The

top

exceptional conjugacy class splits into several symplectic leaves the Northern

2

part of the S its Southern part and the p oints of the equator C where P

vanishes According to our general considerations ab ove this splitting is

however irrelevant there will corresp ond at most one quantum state to the

exceptional conjugacy class

On the other hand there corresp onds precisely one quantum state to any

integral nonexceptional conjugacy class as all of these orbits are simply

connected So let us evaluate the integrality condition for the non

exceptional conjugacy classes in SU From we nd that in the co or

dinates B

ik dz

d C

z

In the parametrization C for the adjoint orbits this yields for the integral

of over the resp ective twospheres

Z

k

C

2

k

S

Here we have taken into account that the imaginary part Imz of z runs

only from sin to sin since jz j

For the critical orbit at the symplectic volume C b ecomes

illdened This comes as no surprise Here obviously the choice C

do es not apply for all lo ops on the critical conjugacy class Still the correct

integrability condition may b e guessed from a simple limiting pro cedure

From C we obtain

Z

lim k C

2

2

S

It is plausible to assume that the critical orbit will carry a quantum state

i again C is an integer multiple of cf Eq

In fact one can prove that this is indeed correct To do so one might

use two charts in the quantum line bundle First C which works for all

lo ops that do not intersect the equator C of the critical conjugacy class And

second

k k

f ield

i dz d i dz z d

1 1

i

z z

C

This second chart is applicable to all lo ops on the critical conjugacy class that

do not touch its p ole z i The solution to the full quantum constraints

C has again the form C within the resp ective domain of denition

of the two charts Now one might regard the value of the wave functional in

both charts for two small lo ops close to the p ole z i one of which with

winding numb er one around this p ole the other one with winding numb er

zero In the rst chart continuity of the wave function implies that the

wave functional will have basically the same value for b oth lo ops In the

second chart the two lo ops are separated from each other by a twosurface

2

that encloses basically all of the critical S since in this chart the rst lo op

may not b e transformed into the second one through the p ole z i but

instead one has to move through the other p ole z i this gives a relative

phase factor of the wave functions in this chart that may b e determined by

means of C The corresp onding phase need however not b e a multiple

of Instead the result of chart two has to coincide with the result of chart

one only after taking into account the transition fucntions b etween the two

charts Note that b oth lo ops lie in b oth charts In fact for the rst lo op

one picks up a nontrivial contribution to the integrality condition from there

Further details shall b e left to the reader In any case the result coincides

with the one obtained from the limit ab ove So one nds that there exists a

quantum state with supp ort on the critical orbit for even values of

k and no such a state for o dd values of k

Let us remark here that in the latter case al l physical quantum states

ie all states in the of the quantum constraints may b e describ ed

within just one chart of the quantum line bundle as eg by C So the

restriction to physical states may yield the originally nontrivial quantum

line bundle of a coupled mo del C to b ecome eectively trivial

Summing up the results for G SU we conclude that the integral

orbits ie the orbits allowing for nontrivial quantum states of the SU

GWZW mo del are given by n k n k

Now we want to compare this result with the current literature Aco

ording to there are two dierent pictures for the space of states of the

GWZW mo del In they consider partition functions of the WZW mo del

However these two issues may b e related using results of The rst pic

ture eventually coincides with our answer The second one suggests the nite

renormalization k k In this case the integral orbits are characterized

by n k n k However in this picture the singular

orbits with n and n k corresp onding to the central elements

I SU should b e excluded In it is proved that the two pictures

are equivalent However it would b e interesting to establish this equivalence

in the language of Poisson mo dels One motivation is to compare the re

sults with the similar formalism Also it seems to b e easier to handle the

sp ectrum of the mo del in the second picture

Acknowledgements

We are grateful to MBlau KGawedzki and RJackiw for the interest in

this work and useful comments AA thanks the Erwin Schrodinger Insti

tute Vienna Austria for hospitality during the p erio d when this work was

initiated

References

DAmati SElitzur and ERabinovici Nucl Phys B

DCangemi RJackiw Phys Rev D

SShabanov PSchaller and TStrobl unpubl collab oration Nov

see also reference b elow

JGoldstone and RJackiw Phys Lett B

PSchaller and TStrobl

Quantization of Theories Generalizing GravityYangMil ls Sys

tems on the Cylinder In LNP p Integrable Mo dels and

Strings eds AAlekseev et al Springer or grqc

Mod Phys Letts A

cf also Poisson models A generalization of to app ear in Pro c of

Conference on Integrable Systems Dubna or hepth

JWess and BZumino Phys Lett B

LDFaddeev and SLShatashvili Teor Math Phys

MSpiegelglas Phys Lett B

MBlau and GThompson Nucl Phys B

AGerasimov Localization in GWZW and Verlinde formula Uppsala

preprint and hepth

KGawedzki and AKupiainen Nucl Phys B

AAlekseev and SShatashvili Nucl Phys B

MBershadskii and HOoguri Comm Math Phys

AGerasimov AMarshakov AMorozov MOlshanetsky SShatashvili

Int J Mo d Phys A

TStrobl Phys Rev D

TStrobl and TKlo esch Al l Global Solutions of Dimensional Grav

ity preprint

MASemenovTianShansky Dressing transformations and PoissonLie

group actions in Publ RIMS Kyoto University

NMJWo o dhouse Geometric Quantization second Edition

Clarendon Press Oxford

HBNielsen and DRohrlich Nucl Phys B

AAlekseev LFaddeev and SShatashvili J Geom Phys

AYuAlekseev and AZMalkin Comm Math Phys

AAlekseev and ITo dorov Nucl Phys B

JHLu and AWeinstein J Di Geom

LDFaddeev NYuReshetikhin and LATakhtadjan Leningrad Math

J

BBlo ck Phys Lett B

AAlekseev and SShatashvili Comm Math Phys

FFalceto and KGawedzki On quantum group symmetries of conformal

eld theories in Pro c XXth Int Conf on Dierential geometric meth

o ds in New York

BursurYvette preprint IHESP

JBalog LDabrowski LFeher Phys Lett B

MChu PGo ddard IHalliday DOlive ASchwimmer Phys Lett B

KGawedzki Topological actions in twodimensional quantum eld the

ory in Nonp erturbative quantum eld theory pp eds

GtHo oft et al Plenum Press New York London

SAxelro d SDella Pietra EWitten J Di Geom