1. Introduction

GAUDIN MODEL BETHE ANSATZ AND CRITICAL LEVEL

BORIS FEIGIN EDWARD FRENKEL AND NIKOLAI RESHETIKHIN

January

Introduction

Gaudins mo del describ es a completely integrable quantum spin chain Originally

it was formulated as a spin mo del related to the Lie algebra sl Later it was

realized cf Sect and that one can asso ciate such a mo del to any

semisimple complex Lie algebra g and a solution of the corresp onding classical Yang

Baxter equation In this work we will fo cus on the mo dels corresp onding to

the rational solutions

Denote by V the nitedimensional irreducible representation of g of dominant

integral highest weight Let b e a set of dominant integral

V and asso ciate with weights of g Consider the tensor pro duct V V

of this tensor pro duct a complex numb er z The hamiltonians of each factor V

Gaudins mo del are mutually commuting op erators z z i N

i i N

acting on the space V

Denote by h i the invariant scalar pro duct on g normalized as in Let fI g a

d dim g b e a basis of g and fI g b e the dual basis For any element A g

denote by A the op erator A which acts as A on the ith factor

of V and as the identity on all other factors Then

i aj

X X

I I

z z

i j

j i

One of the main problems in Gaudins mo del is to nd the eigenvectors and the

eigenvalues of these op erators

Bethe ansatz metho d is p erhaps the most eective metho d for solving this problem

for g sl As general references on Bethe ansatz cf Applied to

Gaudins mo del this metho d consists of the following There is an obvious eigenvector

s One constructs in V the tensor pro duct of the highest weight vectors of the V

other eigenvectors by acting on this vector by certain elementary op erators dep ending

on auxiliary parameters w w The vectors obtained this way are called Bethe

vectors Such a vector is an eigenvector of the Gaudin hamiltonians if a certain

BORIS FEIGIN EDWARD FRENKEL AND NIKOLAI RESHETIKHIN

system of algebraic equations involving the parameters z s and w s and the highest

i j

weights s is satised These equations are called Bethe ansatz equations

One usually constructs eigenvectors in statistical mo dels asso ciated to a general

simple Lie algebra g by cho osing a sequence of emb eddings of Lie algebras of lower

rank into g and inductively solving diagonalization problems for these subalgebras

This leads to a rather complicated combinatorial algorithm which very

much dep ends on the structure of g

Another metho d was recently prop osed by Babujian and Flume They con

sider as analogues of Bethe vectors the rational V valued functions which enter

SchechtmanVarchenko solutions of the KnizhnikZamolo dchikov KZ equa

tion asso ciated to g cf also Varchenko and one of us show elsewhere

that such a vector can b e obtained as a quasiclassical asymptotics of a solution

of the KZ equation This establishes a remarkable connection b etween a mo del of

statistical mechanics and correlation functions of a conformal eld theory

In this work we prop ose a new metho d of diagonalization of Gaudins hamiltonians

which is based on Wakimoto mo dules over the ane algebra g at the critical level

and the concept of invariant functionals correlation functions on tensor pro ducts

of gmo dules This will allow us to treat the diagonalization problem and the KZ

equation on equal fo oting and thus explain the connection b etween them

The critical level is the level k h where h is the dual Coxeter numb er of

g The p eculiarity of this value of level is that it is only for k h that the lo cal

b b

completion U g of the universal enveloping algebra of g contains central elements

k lo c

g was describ ed in cf also Elements of The center of U

lo c h

the center act on the space of functionals on tensor pro ducts of representations of g

at the critical level which are invariant with resp ect to a certain Lie algebra One

can interpret the hamiltonian as the action of a quadratic central element from

g on the ith factor of invariant functional Other central elements give rise U

lo c h

to other op erators which commute with each other and with Gaudins hamiltonians

We call them higher Gaudins hamiltonians

Wakimoto mo dules at the critical level are sp ecial b osonic representations

of g which are essentially parametrized by functions on the circle with values in

the dual space of the Cartan subalgebra h of g We will construct eigenvectors of

Gaudins hamiltonians by restricting certain invariant functionals on tensor pro ducts

of Wakimoto mo dules In conformal eld theory language those are certain b osonic

correlation functions Analogues of Bethe ansatz equations will naturally app ear

as KacKazhdan typ e equations on the existence of certain singular vectors in

Wakimoto mo dules The existence of these singular vectors will ensure that the

vector constructed this way is an eigenvector of the Gaudin hamiltonians and their

generalizations

We will express the eigenvalues of these hamiltonians in terms of the Miura trans

formation and show that Bethe ansatz equations app ear as certain analytic conditions

GAUDIN MODEL BETHE ANSATZ AND CRITICAL LEVEL

on the eigenvalues The formulas we obtain for the eigenvalues generalize Sklyanins

formula for g sl The app earance of the Miura transformation in this con

g is isomorphic to the classical text is not surprising b ecause the center of U

lo c h

W algebra asso ciated to g the Langlands dual Lie algebra to g

The space of invariant functionals makes sense for an arbitrary level k For non

critical values of k it coincides with the space of genus correlation functions or con

formal blo cks of the corresp onding WessZuminoNovikovWitten WZNW mo del

This denition is equivalent to the more common denition of

correlation functions as matrix elements of certain vertex op erators acting b etween

representations of g It is known that for noncritical k these functions satisfy the

KZ equation

One can obtain solutions of the KZ equation by restricting certain invariant func

tionals on tensor pro ducts of Wakimoto mo dules but those of noncritical level The

rational function which enters such a solution cf also then

app ears as a b osonic correlation function cf which coincides with the formula

for an eigevector obtained from Wakimoto mo dules at the critical level This gives

an explanation of the connection b etween the eigenvectors of the Gaudin mo del and

solutions of the KZ equation asso ciated to g which was observed in

Gaudins hamiltonians can b e obtained from the transfermatrix of a quantum

spin chain related to ane quantum groups U g in a certain limit when q

A generalization of the results presented here to the case of U g will b e published

separately

We also want to remark that our construction of eigenvectors naturally ts into a

program of geometric Langlands corresp ondence prop osed by Drinfeld This corre

sp ondence relates equivalence classes of G bundles over a complex algebraic curve

E with at connections and D mo dules on the mo duli space of Gbundles over E

The pap er is organized as follows In Sect we recall the denition of Gaudins

mo del and Bethe ansatz pro cedure in the case g sl Sect contains an inter

pretation of the mo del in terms of invariant functionals at the critical level We

show how singular vectors of imaginary weight give rise to Gaudins hamiltonians

and their generalizations In Sect we recall the denition of Wakimoto mo dules

g We use these results in Sect and discuss the structure of the center of U

lo c h

to construct eigenvectors of Gaudins hamiltonians and to compute their eigenval

ues In Sect we derive SchechtmanVarchenko solutions of the KZ equation using

Wakimoto mo dules We show that the rational function entering a solution coincides

with the formula for an eigenvector In the App endix we prove some technical results

concerning Wakimoto mo dules which are used in the main text

Gaudins model

Let g b e a simple Lie algebra over C of rank l and dimension d and let U g

b e its universal enveloping algebra For a dominant integral weight denote by V

BORIS FEIGIN EDWARD FRENKEL AND NIKOLAI RESHETIKHIN

the irreducible representation of g of highest weight Denote by h i the invariant

scalar pro duct on g normalized as in Let I a d b e a ro ot basis of

g and I b e the dual basis Denote by the quadratic Casimir op erator from the

center of U g

I I

The op erator acts on V by multiplication by a numb er which we will denote by

Now x a p ositive integer N and a set of dominant highest

weights Denote by V the tensor pro duct V V

Let u z z b e a set of distinct complex numb ers Intro duce linear op erator

S u which we will call the Gaudin hamiltonian on V by the formula

N N

X X

i i

S u

u z u z

i i

where is given by formula

One can check directly that the op erators commute with each

other Therefore the op erators S u commute with each other for dierent values of

u In the next section we will give a new pro of of this fact

One of the main problems in Gaudins mo del is to diagonalize op erators S u This

is equivalent to simultaneous diagonalization of the op erators

For g sl this problem can b e solved by algebraic Bethe ansatz

Denote by v the highest weight vector of the mo dule V The tensor pro duct of

the highest weight vectors

ji v v V

is an eigenvector of the op erators Indeed if I and I are not elements of the

i a

i aj

Cartan subalgebra of g then the op erator I I acts by on ji If they are then

this op erator multiplies ji by a certain numb er

Therefore ji is an eigenvector of S u

The idea of the Bethe ansatz metho d is to pro duce new eigenvectors by applying

certain elementary op erators to the vacuum ji

Let fE H F g b e the standard basis of sl Intro duce op erators F w on the space

V by the formula

F w

w z

where F is the op erator which acts as the generator F of sl on the ith factor of

V and as the identity on all other factors Here w is a complex numb er which is

not equal to z z

GAUDIN MODEL BETHE ANSATZ AND CRITICAL LEVEL

Now consider Bethes vector

jw w i F w F w ji

m m

in V

Explicit computation shows that

S ujw w i s ujw w i

m m m

jw w u w w i

j j m

u w

where s u is a function in u and

X X

w z w w

j i j s

s j

If all f s vanish then jw w i is an eigenvector of S u The corresp onding

j m

equations

X X

j m

w z w w

j i j s

s j

are called Bethe ansatz equations

The eigenvalue of S u on the vector jw w i is given by

u u s u

m u m m

where

m N

X X

u z u w

i j

j i

There are several approaches to constructing eigenvectors of Gaudins hamiltonians

for general g

One can construct eigenvectors by cho osing a sequence of emb eddings of Lie al

gebras of lower rank into g and inductively solving the diagonalization problems

for these subalgebras This leads to a rather complicated combinatorial

algorithm which very much dep ends on the typ e of g

Another metho d was prop osed by Babujian and Flume in The eigenvectors

should b e constructed by applying to the vacuum ji the op erators

F w j l

w z

However since such op erators no longer commute with each other one should also

add some extra terms which account for the commutators The right formula can

BORIS FEIGIN EDWARD FRENKEL AND NIKOLAI RESHETIKHIN

b e extracted from solutions of the KZ equation and in fact can b e obtained

as quasiclassical asymptotics of such solutions

j j j

F F F

j j j

X Y

i i i

2 1

i i

i jw w ji

w j w j w j w j w j z

1 N

j i i i i i

pI I

1 2 2 3

Here the summation is taken over all ordered partitions I I I of the set

j j

f mg where I fi i i g Note that one can consider vector as

an element of the tensor pro duct of Verma mo dules M M with arbitrary

highest weights

i satises an analogue of the It was claimed in that the vector jw w

equation Namely

i i i i

m m

1 1

X i s i w S ujw w ujw

j i i

1 m m

u w

where s u is a rational function in u

i i

X X

i i i i

j j

w z w w

j i j s

s j

and X is a certain vector in V Thus if the equations f are satised for all

i is an eigenvector of Gaudins hamiltonians j m then vector jw w

i a It is natural to call these equations Bethe equations and the vector jw w

Bethe vector

The direct pro of of formula seems rather complicated and it is not clear

how to generalize it to higher Gaudins hamiltonians which we intro duce in the next

section

In this work we prop ose an alternative approach to the diagonalization of Gaudins

hamiltonians which is based on a concept of invariant functionals correlation func

tions on tensor pro ducts of representations of ane algebras at the critical level

In the next section we will interpret Gaudins mo del in terms of such functionals

We will show that Gaudins hamiltonians are memb ers of a family of commuting

linear op erators on the invariant functionals which come from singular vectors of

imaginary weight in the vacuum representation of g at the critical level

In Sect we will construct eigenvectors of Gaudins hamiltonians by restricting

certain invariant functionals on tensor pro ducts of Wakimoto mo dules This con

struction will allow us to prove that if Bethe equations are satised then vector

i is an eigenvector of the op erators S u and their generalizations de jw w

ned in the next section We will also compute the eigenvalues of these op erators

This construction will then b e used in Sect to give an explanation of the app earance

of Bethe vectors in solutions of the KZ equation

GAUDIN MODEL BETHE ANSATZ AND CRITICAL LEVEL

Correlation functions

Let g b e the ane algebra corresp onding to g It is the extension of the Lie

algebra g C t by onedimensional center C K Denote by g the Lie subalgebra

g C t C K of g For any nitedimensional representation V of g denote by V the

representation of g on which g tC t acts trivially and K acts by multiplication

by k C Denote by V the induced representation of g of level k

k k

V U g V

U bg

Consider the pro jective line CP with a global co ordinate t and N distinct nite

p oints z z CP In the neighb ourho o d of each p oint z we have the lo cal

N i

e b

co ordinate t z Denote gz g C t z Let g b e the extension of the

i i i N

gz by onedimensional center C K such that its restriction to any Lie algebra

summand gz coincides with the standard extension For f f t z f t

i N

gz the co cycle dening z and g g t z g t z from

i N N N N

this central extension is given by

f g K hf dg i

N i i

stands for the integral over a small contour around the p oint z where

k k k

in V The Lie algebra g naturally acts on the tensor pro duct V V

particular K acts by multiplication by k We will say that we have assigned mo dules

k k

to the p oints z z V V

Let g g b e the Lie algebra of gvalued regular functions on CP nfz

z z z

z g ie rational functions on CP which may have p oles only at the p oints

z z which vanish at innity Clearly such a function can b e expanded into

a Laurent p ower series in the lo cal co ordinate t z at each p oint z This gives an

i i

e e

gz element of gz Thus we obtain an emb edding of g into the direct sum

i i z

The restriction of the central extension to the image of this emb edding is trivial

Indeed according to formula this restriction is given by the sum of all residues

of a certain oneform on CP hence it should vanish Therefore we can lift the

e b

emb edding g gz to an emb edding g g

z i z N

k k

Denote by H the space of linear functionals on V which are invariant with

resp ect to the action of the Lie algebra g Such a functional is a linear map

C which satises V

g y g g y V

By construction we have a canonical emb edding of a nitedimensional represen

tation V into the mo dule V

x V x V

BORIS FEIGIN EDWARD FRENKEL AND NIKOLAI RESHETIKHIN

which commutes with the action of g on b oth spaces recall that g is emb edded into

g as the constant subalgebra Thus we have an emb edding of V V V

into V We will keep the same notation V for the image of this emb edding

k k

The restriction of any invariant functional H to the subspace V V

k k

denes a linear functional on V Thus we obtain a map H V

k k

Lemma The map H V is an isomorphism

e e

Proof Denote g z g C t z gz We have a direct sum decomp osition

i i i

N N

e e

gz g z g

i i z

i i

Indeed for g t g C t denote by g t and g t its regular and singular

gz can b e parts resp ectively Any element f f t z f t z of

i N N

uniquely represented as the sum of

f t z g f

i z

and

g z t z f t z t z f t z f f

i N N N

i N

where f t z denotes the regular part of the expansion of the function f at

i i

N k

e b

g z C K on is induced from the mo dule V over The g mo dule V

i N

which t z g z acts trivially and K acts by multiplication by k Therefore

i i

is isomorphic to the free mo dule generated by V Hence the as a g mo dule V

is isomorphic to V space of g invariant functionals on V

Lemma shows that an invariant functional is uniquely dened by its restriction

to the subspace V V Thus for any V there exists a g invariant

k k k

e e

C such that the restriction of to V V V functional

coincides with

Consider the Lie algebra g g g g where g is the constant diagonal

z N

subalgebra of g Note that g do es not lie in g b ecause by denition elements of

N z

invariant functionals g must vanish at innity One can consider the space of g

on V This space is isomorphic to the space of ginvariants of V In conformal

invariant functional on V is called a correlation function more eld theory a g

precisely conformal blo ck and the equation is called Wards identity

Remark Sometimes it is convenient to consider instead of the space of g invariant

functionals on V the space of coinvariants of V with resp ect to g These spaces

are dual to each other For a detailed study of the functor of coinvariants cf

GAUDIN MODEL BETHE ANSATZ AND CRITICAL LEVEL

Let M b e a gmo dule and let M b e the restricted dual linear space to M We

can dene a structure of gmo dule on M as follows

g f m f g Y f M g g m M

where stands for the Cartan antiinvolution on g It is dened on the generators of

g as follows

E F F E H H

i i i i i i

This gives a structure of gmo dule on M which is called contragradient to M It

is known that if the mo dule M b elongs to the category O of gmo dules so do es

M In particular the mo dule contragradient to an irreducible representation V

with highest weight is isomorphic to itself Hence as a contragradient mo dule V

is isomorphic to V Therefore H V

b e the representation of g which corresp onds to the onedimensional trivial Let V

gmo dule V We will call it the vacuum module Denote by v the generating vector

of V We assign the vacuum mo dule to a nite p oint u CP which is dierent

k k k

V the space of g invariant functionals on V from z z Denote by H

zu N

with resp ect to the Lie algebra g Lemma tells us that H V

consider the corresp onding g invariant For any V Let X b e a vector in V

e e

functional H Its restriction X to the subspace V X V denes

another linear functional on V Thus we obtain a linear op erator dep ending

on u X u V V which sends to Since V V we can consider

X u as an op erator acting on V

For A g and m Z denote by Am the element A t g Now intro duce

the following vector in V

S I I v

This vector denes a linear op erator

S u V V

Prop osition The operator S u coincides with the Gaudin hamiltonian

Proof For any A g consider the element

g g

t u

The expansion of g at z is equal to

z u

uz i

BORIS FEIGIN EDWARD FRENKEL AND NIKOLAI RESHETIKHIN

Therefore the action of g on the ith factor of the tensor pro duct of V is given by

i n

n m

n u u z

Thus from we obtain for any Y V

i n

X X

e e

Y X Y AnX

n m

n u u z

m i

This identity allows to swap the op erator An from the mo dule assigned to the

p oint u to the mo dules assigned to the p oints z z

Put n in Since any element of g t z C t z acts trivially on the

i i

we obtain for any V subspace V of V

e e

X X V AX

u z

we obtain Applying this identity twice to the element S V

d d N N

j ai

X X X X

I I

e e

v S

u z u z

i j

a a

j i

d N d N

a i j

X X X X

I I

u z u z

j i

a i a j

by formula Hence

d d N

aj i ai i

X X X X

I I I I

a a

S u

u z u z u z

i i j

a ij a i

But this coincides with formula b ecause

u z u z z z u z u z

i j i j i j

We can now prove that the op erators S u commute for dierent values of u In

fact we will prove a more general result

Let us sp ecialize to the value k h where h is the dual Coxeter numb er

This value of level is called critical For simplicity in what follows we will omit the

etc sup erscript k when k h Thus we will write V instead of V

This value is sp ecial b ecause it is only at the critical level that S V is a singular

vector of imaginary weight This means that A X for any A g C t

GAUDIN MODEL BETHE ANSATZ AND CRITICAL LEVEL

Prop osition Let Z and Z be singular vectors of imaginary weight in V Then

for any pair of complex numbers u and v the corresponding linear operators Z u

and Z v on V commute In particular for Gaudins hamiltonians we have

S u S v

Proof Consider a generalization of the previous construction with two auxiliary

p oints u and v We can assign to each of these p oints the mo dule V The space

H is again isomorphic to V Therefore we can dene an action of a pair of

elements X X V on V Namely for any V we consider the corresp ond

e e

ing g invariant functional H Its restriction X X to the subspace

zuv

V X X V denes another linear functional on V Thus we

obtain a linear op erator dep ending on u and v X X u v V V which

sends to

We will show now that if X and X are singular vectors of imaginary weight then

X X u v X uX v and X X u v X v X u This will prove that

X u X v

Indeed X can b e written as a linear combination of monomials A n

A n v where A g n We can apply Wards identity to each

m m j j

summand and inductively swap op erators A n to the mo dules assigned to the

j j

p oints z z and v This amounts to acting on them by

v i n

X X

A m A

n n m

j j

n v z n u u v

j i j

m i

But the last term of such a sum vanishes if we apply it to X by denition of singular

vector of imaginary weight Therefore this only aects the mo dules at the p oints z

just as if we had v at the p oint v instead of X After that we can swap X

We obtain

X X X uX v

Hence X X u v X uX v In the same way we can show that X X u v

X v X u and Prop osition follows

A description of the space Z g of singular vectors of imaginary weight in V is

known It is related to the structure of the center of the lo cal completion

b b

g of the universal enveloping algebra of g at the critical level Let U

lo c h

us recall this description

Recall that as a mo dule over g g t C t V is isomorphic to its universal

b b

enveloping algebra U g Intro duce a Zgrading on U g and on V by putting

deg Am m deg v There is an op erator of derivative L of degree

on the mo dule V and hence on U g such that Am mAm and

Denote by d d the exp onents of g

BORIS FEIGIN EDWARD FRENKEL AND NIKOLAI RESHETIKHIN

Prop osition The space Z g of singular vectors of imaginary weight of V co

incides with the polynomial algebra in S i l n applied to vector

v where S i l are mutual ly commuting elements of U g of degrees

d i l

A more precise description of the center will b e given in the next section Note

that S is equal to S

Prop osition and Prop osition show that S u S u constitute a family of

mutually commuting linear op erators on the space V We call S u i higher

Gaudins hamiltonians It would b e interesting to nd explicit formulas for them

m m

Remark For Z Z g denote Z Res u z Z u These op erators

uz i

i i

S given by They all mutually commute generalize the op erators

Alternatively these op erators can b e dened as follows

Since V is a vertex algebra we can asso ciate a lo cal current X z to any

X V Denote by X the m st Fourier comp onent of this current This

g Therefore it acts on the ith comp onent of the is a central element of U

lo c h

V The dual op erator gives rise to an op erator on the tensor pro duct V

space of g invariant functionals H V Using the generalized Ward identity

from Sect we can show that the corresp onding op erator on V coincides with

Note that we can dene the space of g invariant functionals in a more general

situation To any gmo dule M we can asso ciate the induced gmo dule of level k

M U g M

U bg

Fix N representations M M of g and let M M b e the induced g

N N

mo dules We can dene the space of g invariant functionals on the tensor pro duct

denotes where M M M M This space is isomorphic to M

i N

the mo dule contragradient to M In the same way as ab ove we can asso ciate to a

singular vector of imaginary weight Z V a family of commuting linear op erators

on this space

It is clear from construction that the op erators Z u can b e viewed as elements of

the N th tensor p ower U g of the universal enveloping algebra of g dep ending on

z z and u Therefore they automatically act on any N fold tensor pro duct

of gmo dules

Denote by M the Verma mo dule over with highest weight h over g and by

M the contragradient mo dule

with arbitrary h i From now on we will cho ose as our mo dules M

l Then the space of g invariant functionals is isomorphic to the tensor pro d

uct of Verma mo dules M

GAUDIN MODEL BETHE ANSATZ AND CRITICAL LEVEL

If s are integral dominant weights then there is a surjective g homomorphism

N N

M V The op erators Z u Z Z g act on b oth spaces and commute

i i

with this homomorphism Thus the pro jection of any eigenvector of Z u from

N N

M on V is again an eigenvector

i i

Our construction of eigenvectors of Gaudins hamiltonians in the tensor pro duct

M M will b e based on Wakimoto mo dules

Wakimoto modules at the critical level

In this section we briey describ e imp ortant facts ab out Wakimoto mo dules

cf also

Let us rst intro duce the Heisenb erg Lie algebra g It has generators a m

a m m Z where is the set of p ositive ro ots of g and central element

They satisfy the relations

m n a m a n a m a a n a

Now let M b e the irreducible representation of g which is generated from the

vacuum vector v satisfying

a mv m a mv m

for all and v v

Let h b e the commutative Lie algebra h C z It has a linear basis h n

H t i l n Z Any oneform z dz h C z dz denes a one

dimensional representation of h

f z z f z dz f z h

where the integral is taken around the origin

Wakimoto mo dules constitute a family of representations of the ane algebra g of

the critical level on the space M

In order to dene these representations we should construct a homomorphism

from g to the lo cal completion of U g U h Here U g stands for the

Heisenb erg algebra which is the quotient of the universal enveloping algebra of g

by the ideal generated by We recall the construction of the homomorphism

First we have to cho ose co ordinates on the big cell U of the ag manifold B nG of

g This big cell is isomorphic to the nilp otent subgroup N of the Lie group G of g

Since N is isomorphic to its Lie algebra n via the exp onential map U is isomorphic

to a linear space with co ordinates x We assign to the co ordinate x the

degree The right action of G on the ag manifold gives an emb edding of the Lie

algebra g into the Lie algebra of vector elds on U This emb edding can b e lifted to

a family of emb eddings into the Lie algebra of the rst order dierential op erators

BORIS FEIGIN EDWARD FRENKEL AND NIKOLAI RESHETIKHIN

dep ending on h Denote these emb eddings by Explicitly we have for the

generators E H F i l of g

i i i

E P

x x

where P is a certain p olynomial in x of degree

H H x H

i i i

and

F H x Q

i i

is a certain p olynomial in x of degree where Q

The homomorphism denes a structure of gmo dule on the space of algebraic

functions on the big cell C x This mo dule is isomorphic to the mo dule M

which is contragradient to the Verma mo dule over g of highest weight

We also have another map from the Lie algebra n to the Lie algebra of vector

elds on U N which comes from the left innitesimal action of n on its Lie

group The ith generator of n maps to the vector eld

G R

x x

where R is a certain p olynomial in x of degree These op erators generate

another action of n which commutes with the action of n dened by

Now we can construct the homomorphism Intro duce the notation

Az Anz A g

and

X X X

n n n

a z a nz a z a nz h z h nz

i i

nZ nZ nZ

We put

z E z a z a z P

H a z a z h z H z

i i i

F z Q z a z h z a z c a z

i i i z

i i

GAUDIN MODEL BETHE ANSATZ AND CRITICAL LEVEL

i i

where c is a constant Here P z and Q z stand for the expressions which are

i i

obtained from the p olynomials P and Q by insering a z instead of x and dots

denote normal ordering

We also dene op erators G n i l n Z by the generating functions

X X

n i

G z G nz a z R z a z

i i

Theorem There exist such c in the formula for F z i l that the

i i

map denes a homomorphism from the Lie algebra g to the local completion of

U g U h which sends the central element K g to h

This was proved in for g sl and in in general

Example We will write down the precise formulas in the case g sl We will

omit the unnecessary subscripts We have

E z az

H z az a z hz

F z az a z a z hz a z a z

and

Gz az

For explicit formulas in the case g sl cf

For any z dz h C z dz we can consider representation M of

g h According to Theorem the homomorphism denes on M a

structure of gmo dule of the critical level which we denote by W We call this

mo dule Wakimoto module with highest weight z

Remark The mo dules which were originally constructed in only dep ended

on h h dz z However the generalization of that construction to arbitrary

z is straightforward

Remark The Laurent p ower series z should b e considered as an element of the

space of h connections on a principal H bundle on the formal punctured disc

where H is a Lie group of h Lo cally such a connection can b e written as a rst order

dierential op erator h z dz where z is an element of h C z which

is isomorphic to h C z dz via the nondegenerate scalar pro duct on h We

can view such a connection as a linear functional on h which is equal to h on the

central element Here h is the Heisenb erg Lie algebra which is the central extension

of the Lie algebra of gauge transformations of this bundle

The space of connections is a torsor over the space of oneforms h C z dz If

we trivialize the bundle we can identify this torsor with the space of oneforms

BORIS FEIGIN EDWARD FRENKEL AND NIKOLAI RESHETIKHIN

The following result which is proved in the App endix will b e imp ortant in the

next section

Lemma Let z be a highest weight of the form

n n n

z z h

The vector G v W is a singular vector of imaginary weight if and only if

i z

We can also dene another representation of g at the critical level Let

C h n i l n b e the hmo dule on which h n n act by and

i i

h n n act as on the free mo dule with one generator Then M is a gmo dule

via the homomorphism We denote this mo dule by W

One can check directly that the generating vector v v of W is annihilated

by Am A g m Therefore the map denes a homomorphism V W

e e

which sends v V to v W One can show that is a emb edding Under this

homomorphism all singular vectors of imaginary weight from V get mapp ed into the

subspace W Thus we obtain an injective map Z g

In the image of this map was describ ed Let us recall this description

We can consider the algebra as the algebra of dierential p olynomials The

action of derivative on is dened by its action on the generators h n

nh n and by the Leibnitz rule The action of on Z g and on commutes

with the map Therefore the image of Z g is a dierential subalgebra of

On the other hand for any g one can dene a remarkable dierential subalgebra

of g which consists of densities of the corresp onding classical W algebra

dened by the DrinfeldSokolov reduction This subalgebra can b e charac

terized as the space of invariants of the nilp otent subalgebra n of g which acts on

cf Sect It can b e considered as a classical limit of the vertex op erator

algebra of the quantum W algebra We denote this subalgebra by W g

Now let g b e the Langlands dual Lie algebra of g Recall that the Cartan matrix

of g is the transp ose of the Cartan matrix of g Therefore there is an isomorphism

b etween the Cartan subalgebras of g and g which preserves the scalar pro ducts

This induces an isomorphism b etween g and g Therefore we can consider

W g as a dierential subalgebra of g

Theorem The image of Z g in g under the homomorphism co

incides with W g

Prop osition follows from this Theorem b ecause it is known that W g is a

P i l n where free commutative subalgebra of generated by

deg P d The generator P is the image of S Z g from Prop osition

i i i i

GAUDIN MODEL BETHE ANSATZ AND CRITICAL LEVEL

Example If g sl then C hn The image of Z g in consists of

p olynomials in P m where

P h h

This space coincides with W sl cf Sect

b b

g of the universal enveloping alge The center Zg of the lo cal completion U

lo c h

bra of g at the critical level was describ ed in It consists of lo cal expressions in

S S ie Fourier comp onents of all dierential p olynomials in S z S z

l l

where S z is the lo cal current asso ciated to S in the vertex algebra V There

i i

b b

fore Zg is isomorphic to the classical W algebra asso ciated to g by means of the

DrinfeldSokolov reduction cf This W algebra is the space of lo cal functionals

on a certain hamiltonian space H g

Now consider the space F of all Fourier comp onents of dierential p olynomials

in h z h z cf It is isomorphic to the space of lo cal functionals on the

space T g of h connections h z dz The map Z g gives us

z Z

an emb edding Zg F The corresp onding map of sp ectra T g H g

is nothing but the generalized Miura transformation

Elements of Zg act on Wakimoto mo dules by multiplication by constants One

can describ e the map in terms of Wakimoto mo dules as follows for X Zg

the value of X at h z dz is equal to the action of X on the Wakimoto

Z z

mo dule W This shows that the map is an ane analogue of the Harish

Chandra homomorphism

Recall that the HarishChandra homomorphism is a map from the center Zg

of the universal enveloping algebra U g of g to the algebra of p olynomials on the

dual space h of the Cartan subalgebra h of g Any element of Zg acts on Verma

mo dules by multiplication by a constant HarishChandra homomorphism maps a

central element X to the p olynomial on h whose value at is equal to the action

of X on M

Example If g sl then Zg consists of all lo cal expressions in the comp onents S

of the current

X X

a n

S z I z I z S z

a n

It is therefore sucient to describ e the values of on S Denote

Z n

sz s z S z

n Z

Then we have according to the previous example

hz hz sz

BORIS FEIGIN EDWARD FRENKEL AND NIKOLAI RESHETIKHIN

Therefore the central element s acts on the mo dule W by multiplication by

n m nm

m m

z where we put z

In the next section we will use Wakimoto mo dules and the description of the center

to construct eigenvectors of the Gaudin hamiltonians and to compute their sp ectrum

Bethe vectors from Wakimoto modules

We will construct eigenvectors using invariant functionals on tensor pro ducts of

Wakimoto mo dules We will restrict ourselves with mo dules W for which z

has the form

n n n

z z h

In other word we consider only the mo dules asso ciated to h connections on the

formal disc which are regular or have regular singularity at the origin We will call

such highest weight z regular

Slightly abusing notation we will write instead of Res z

Let us recall the geometric construction of the Heisenb erg algebra g cf

eg Consider the spaces F U C t and F U C tdt of functions

and oneforms on the formal punctured disc with values in the linear space U with

co ordinates x There is a natural nondegenerate pairing b etween them

F F C

hf t g tidt Here h i denotes the scalar which sends f t g tdt to i

pro duct on the linear space U with resp ect to which x s are orthonormal To

the scalar pro duct we can asso ciate in the standard way a central extension

of the commutative Lie algebra F F This is our Heisenb erg Lie algebra

m m

m corresp ond to x t F and x t dt F The generators a m a

resp ectively

Let x x x b e a set of distinct p oints on the pro jective line Consider

Wakimoto mo dules W with regular highest weights z i p

z i

Let us cho ose a global co ordinate t on CP and the corresp onding lo cal co ordinates

t x i p at our p oints Denote x U C t x U C t x dt

i i i i

Let b e the central extension of the Lie algebra x by a onedimensional

p i

center which coincides with our standard central extension on each of the summands

The Lie algebra acts on M in a natural way in particular the central element

acts as the identity

Consider the commutative Lie algebra F F F where F F is the

x x x x

space of U valued regular functions oneforms on CP nfx x g which vanish

GAUDIN MODEL BETHE ANSATZ AND CRITICAL LEVEL

have regular singularity at innity We have an emb edding of F into x

x i

obtained by expanding a function and a oneform at a given p oint The restriction

of the central extension to the image of this emb edding is trivial Therefore we can

lift it to an emb edding F

x p

We also intro duce the commutative Lie algebra h of hvalued regular functions

on CP nfx x g which vanish at innity We have an emb edding h h

p x

C t x Denote z z z where z h C z dz and let

i p

b e the onedimensional representation of h C t x on which h C t x

i i

acts according to its character z Thus h also acts on

i x

The Lie algebra H F h acts on

x x x

M W

z z

Denote by J the space of H invariant linear functionals on this tensor pro duct

Prop osition Suppose that

z z i p

If t x is the expansion of

i i

t x

at the point x for al l i p then the space J is onedimensional It is

generated by a functional whose value on the tensor product of the vacuum vectors

of W is equal to

Otherwise J

Proof The space J is isomorphic to the tensor pro duct of the dual space of the

space of coinvariants of M with resp ect to the action of F and the space of

h invariants of

x z

Since M is free over the Lie subalgebra of generated by a m m

and a m m we can use the same argument as in the pro of of Lemma

We conclude that the space of coinvariants of M with resp ect to the action of F

is onedimensional and that the pro jection of the tensor pro duct of the vacuum

vectors of W on this space is nontrivial Therefore there exists an H invariant

z x

functional on W whose value on the tensor pro duct of the vacuum vectors

is equal to

Let f t b e an element of h This is a meromorphic hvalued function on CP

which may have singularities only at x s and which vanishes at innity It acts on i

BORIS FEIGIN EDWARD FRENKEL AND NIKOLAI RESHETIKHIN

the onedimensional mo dule by multiplication by

t z f tdt

i i

stands for the integral over a small contour around x This sum vanishes where

for any f t if and only if t x is the expansion of

i i

t x

at the p oint x for all i p Therefore the space of h invariants of is

i x z

nonzero if and only if the condition of Prop osition is satised

Now x highest weights of g and a set of simple ro ots of g

i N i

Consider the function

m N

X X

t z t w

i j

j i

Denote by t z the expansions of t at the p oints z i N and by

i i i

t w the expansions of t at the p oints w j m We have

j j j

z z

i j

z z

where

X X

i i

w z w w

j i j s

s j

We put p N m and denote x z z z i N x

i i i i N j

z j m According to Prop osition the space J w z

j N j i

z z

N m

of H invariant functionals on the tensor pro duct W W is one

zw z z

i i j j

dimensional It is generated by a functional whose value on the tensor pro duct

N m

of the vacuum vectors is equal to This is a linear map

N m

W W C

N m

z z

i j

We will obtain eigenvectors of the op erators Z u Z Z g by restricting the func

tional to a certain subspace

N m

According to Lemma the vectors G v W j m are singular

of imaginary weight if and only if the following system of equations is satised

X X

i i i i

j j

j m

w z w w

j i j s

s j

These are precisely the Bethe ansatz equations

GAUDIN MODEL BETHE ANSATZ AND CRITICAL LEVEL

Denote by W the subspace of a Wakimoto mo dule W which is generated from

z z

the vector v by the op erators a This space is stable under the action of

the constant subalgebra g of g and is isomorphic to the mo dule M contragradient

to the Verma mo dule over g with highest weight Res z Indeed only the

th comp onents a and a will contribute to the action of g on W Our

formulas show that if we restrict the homomorphism to g and replace a by x

and a by x we will obtain the formulas for the homomorphism which

denes on C x W a structure of gmo dule isomorphic to M

The restriction of the homomorphism to the subspace

N m

W G v G v

i i

i 1

denes a linear functional

N i i

C M w w

m i

is graded by the weights of the Cartan subalgebra h of g The gmo dule M

so that all homogeneous comp onents are nitedimensional We will show in the

vanishes on all homogeneous pro of of Lemma that the functional w w

P P

m N

Therefore it corresp onds to comp onents except the one of weight

i i

j j i

i N

a vector w w in the tensor pro duct of the Verma mo dules over g M

m i

P P

m N

of weight

i i

j i

There is a dierential algebra homomorphism r from to the algebra of ra

N m

tional functions in u having singularities at z z w w on which the

N m

derivative acts as In order to dene it we have to sp ecify the values of this

homomorphism on the elements h of We put

N m

X X

s i

i s

r h h u

N m s s

u z u w

i j

This implies that

r h n h n h u

N m s s M s

M k

n u

Theorem If Bethe ansatz equations are satised then vector w w

is an eigenvector of the operator Z u for any Z Z g with the eigenvalue r Z

N m Z

In order to prove the Theorem we will need some prop erties of the vertex algebra

W cf Each vector A W denes a formal p ower series Az whose

co ecients are linear op erators acting on representations of g h We call these

formal p ower series local currents note that they were called elds in They

satisfy all axioms of vertex op erator algebras except for the existence of the

Virasoro element that is why we call W a vertex algebra

BORIS FEIGIN EDWARD FRENKEL AND NIKOLAI RESHETIKHIN

The lo cal current corresp onding to a monomial basis element A n A n v

m m

W where A stands for a a or h is given by

i i

n dA n dA

m m

1 1

A z A z

z z

n dA n dA

m m

Here we put da dh da This current do es not dep end on the

order in which we apply normal ordering In general one should use normal ordering

nested from right to left

The space V is also a vertex algebra but not a vertex op erator algebra and

the homomorphism denes an emb edding of this vertex algebra into W

Let us assign representations L of g h to the p oints x i p of CP

i i

Denote by J L L the space of H invariant functionals on L

p x i

n dh n n Intro duce a Zgrading d on W by putting da n da

Let Y z b e the lo cal current corresp onding to an element Y W of degree dY We

ndY dY dY

will identify the dY dierential t x dt C t x dt

j i

dY j

L Consider the space F with the op erator Y n acting on the mo dule

x i

of regular dY dierentials on CP nfx x g which have zero of order at

least dY at innity Any element Y t of this space can b e expanded in p owers

of t x at each p oint x The corresp onding formal p ower series Y t x can then

i i i

b e viewed as an innite sum of op erators c Y m on the space L

m i

mdY

Denote

Y Y t x

Prop osition For any J L L Y W and X L Y X

p i

z or h z this In the case when Y z is one of the elementary currents a z a

follows from the denition of the space of invariant functionals The general result

follows by induction on the p ower of Y By linearity it is sucient to prove the

ndY dY

statement in the case when Y t t x dt n dY Prop osition

is equivalent in this case to the statement that

ndY

Y n X

ndY

n dY

X X

Y m

mdY

x x

j i

i j

mdY

This is a generalization of the Ward identity Note that this identity and

Prop osition hold for arbitrary vertex algebras

Since each of the mo dules L is automatically a gmo dule the Lie algebra g

i x

which was intro duced in Sect acts on the tensor pro duct L

GAUDIN MODEL BETHE ANSATZ AND CRITICAL LEVEL

Corollary For any J L L g g and X L g X

p x i

Proof We have to apply Prop osition in the case when Y z Az A g

Corollary means that any H invariant functional is g invariant This will allow

x x

us to prove Theorem

Proof of Theorem Let z and t b e the expansions of We assign

i j

the mo dules W to the p oints z i N the mo dules W to the p oints

z z

i j

w j m and the mo dule W to the p oint u of CP

The mo dule W is free over the Lie subalgebra of h generated by a m

m a m m h m i l m In the same way as in

the pro of of Lemma we can show that the space of H invariant functionals is

zw u

onedimensional and is generated by a functional whose value on the tensor pro duct

of the vacuum vectors is equal to It clearly has the prop erty X v X

N m

m N

W where v is the generating vector of W and W for any X

N m

z z

j i

is the generator of the space J cf

z z

b b

Recall that we have an emb edding V W of gmo dules Now let Z Z g

v G V and consider G W v Z where

i N m i

1 i

W This is a linear functional of

We can express this linear functional in two dierent ways On the one hand we

can represent Z as an element of U g ie in terms of Am A g m By

Corollary the functional is invariant with resp ect to the Lie algebra g So

zw u

v W j m are we can use formula to swap Z Since G

v for any m Therefore singular vectors of imaginary weight Am G

the action of the corresp onding elements of the Lie algebra g on these vectors is

zw u

equal to So we obtain in the same way as in the pro of of Prop osition

i i

v G G w v Z Z u w

i i

1 m

On the other hand we know that Z lies in W According to

G v G v h nX

N m i i i

r h n G v G v X

N m i N m i i

By applying this formula several times to Z we obtain

i i

e e

v G G v Z r w Z w

i N m i N m Z

From and we obtain

i i i i

m m

1 1

w Z u w r w Z w

N m Z

m m

and Theorem follows

BORIS FEIGIN EDWARD FRENKEL AND NIKOLAI RESHETIKHIN

Example For the element S given by we have

S h h

Z i

is the basis of h which is dual to the basis h h with resp ect to h Here h

the scalar pro duct on h h and h is such that Therefore

i i i

is equal to the eigenvalue of the element S u on the vector w w

h uh u u s u

i u i i

i 1

m m N N

X X X X

i i i

i i

j j j

u z u w u z u w

i j i j

j j i i

For g sl this formula coincides with It was written in cf formula

of the op er Using this formula we can nd the eigenvalues s w w

S u this Since Res ators given by on the vector w w

i uz i

i m

eigenvalue is equal to Res s u Thus we obtain

uz i i

i 1

X X

i j i i

i i

s w w

z z z w

i j i s

j i

coincides up to a sign with the Bethe vector Lemma The vector w w

i given by jw w

Proof We will nd an explicit formula for

i i

w w v G G v

i N m i

i and compare it with formula for jw w

This computation is essentially equivalent to the computation of a b osonic corre

denote by P the corresp onding p olynomial lation function from For X M

in x s Let P z b e the lo cal current which is obtained from P by replacing x

X X

with a z In conformal eld theory language

X X G v G v

N m N i i

is the b osonic correlation function where X M

m N

Y Y

G w P z

i j X i

j i

We will use the generalized Ward identity in the case when Y where

G is a ro ot generator of the left nilp otent subalgebra n of g

GAUDIN MODEL BETHE ANSATZ AND CRITICAL LEVEL

This identity reads

G v G v G v

N m

1 j

G G v v G v

N m

w z

j i

G v v G v G v

N m

s m

1 j

w w

j s

N N

where W M and c are the structure constants in the nilp otent

i i

subalgebra n of g

By successive use of this identity we obtain

v G G v v v

i N m i N m

M for some

C such that hF Y X i Recall that there is a linear pairing h i M M

C the pairing with the highest hY E X i i l cf Denote by M

weight vector x of M X hx X i

the Ward identity reads as follows When Y a

v v a

N m

Therefore

v v

N m

If is homogeneous of weight then is also homogeneous of weight

j j

But if the weight of is not equal to Therefore w w

m i

P P

m N

if the weight of is not equal to

i i

j j i

Following we obtain by induction the following formula

G j G j P

Y X

i i

w w X X

j j j

z w w w

1 N

i i i j

pI I

2 1

where we used and notation from Sect

But

G G P E E P

i i i i

m m

1 1

cf Lemma from Therefore we have

i i

w w X X

j j N

P E E

Y X

i i

j j j

w w w z

1 N

j i i i

pI I

1 2 j

BORIS FEIGIN EDWARD FRENKEL AND NIKOLAI RESHETIKHIN

On the other hand the pairing of a monomial basis element F F x and

i i

1 j

X M is equal to

hx E E X i E E P

i i i i X

j 1 j 1

N i

i M given by denes a linear Thus we see that the vector jw w

m i i

N m i

functional on M which coincides with the functional w w

m i

i i

i m i

1 1

m m

Therefore w w jw w i

m m

There is an interesting analytic interpretation of Bethe ansatz equations

Prop osition The Bethe ansatz equations are satised if and only if the

eigenvalue of any Z Z g on the vector w w is nonsingular at the

points w w

the orthogonal complement to the onedimensional subspace Proof Denote by h

of h generated by H h Polynomials in h n n form a subspace of

i i i

of We have n form a subspace Polynomials in hn h h

i i

The space is isomorphic to sl Denote by W the subspace corre

i i i

sp onding to W sl sl

Now x j m The image of Z g in is contained in the tensor pro duct

X Y where X Therefore Z can b e decomp osed as W

m m m Z i

j i

Then we have Y W

m i

r r X r Y Z

N m N m m N m m Z

From the denition of the homomorphism r it is clear that r X is non

N m N m m

singular at u w We will show now that r Y is nonsingular at u w for any

j N m j

if and only if the Bethe ansatz equation is satised Y W

is the space of dierential p olynomials in We know that W

h h P

i i i

j j j

Since r is a homomorphism of dierential algebras it is sucient to check that

N m

r P is nonsingular at u w if and only if the Bethe ansatz equation is

N m i j

satised Applying formula we obtain that r P is equal to

N m i

N m m N

X X X X

H H H H

i i i i i i i i

s s

j j j j

u z u w u z u w

i s i s

i s s i

The p ossible singular terms at u w are

H H

i i i i

j j j j

u w j

GAUDIN MODEL BETHE ANSATZ AND CRITICAL LEVEL

which vanishes b ecause H and

i i

j j

X X

i i i i

j j

u w u z u w

j i i s i

j j

s j

note that H The latter is nonsingular at u w if and

i i i i j

j j j j

only if the Bethe ansatz equation is satised

Remark Prop osition can b e considered as a guiding principle for nding eigen

values of Gaudins hamiltonians This analytic Bethe ansatz has b een successfully

applied to various mo dels of statistical mechanics cf eg

In conclusion of this section let us remark that the Gaudin mo del can b e general

ized to an arbitrary KacMo o dy Lie algebra g asso ciated to a symmetrizable Cartan

matrix

Indeed for such g we can cho ose the bases fI g and fI g which are orthogonal to

each other with resp ect to the invariant scalar pro duct cf Then the op erators

given by will b e welldened on the N fold tensor pro duct of gmo dules

from the category O For instance we can take the tensor pro duct of integrable

M V or Verma mo dules M representations V

1 1

N N

Moreover formula for the Bethe vector and Bethe ansatz equations make

p erfect sense in this general context So do es formula This leads us to the

following conjecture

Conjecture For an arbitrary symmetrizable KacMoody algebra g the vector

jw w given by is an eigenvector of the operators M i M

N m

given by if the Bethe i N with the eigenvalues s w w

i i

ansatz equations are satised

This conjecture can b e proved directly using metho ds of Sect note

that the pro of in works for an arbitrary KacMo o dy algebra One may also

hop e to prove this conjecture by developing theory of Wakimoto mo dules for general

KacMo o dy algebras

Connection with KZ equation

We x k h Let z z b e a function with values in V The

KZ equation is the system of partial dierential equations with regular singularities

k h i N

where is given by formula i

BORIS FEIGIN EDWARD FRENKEL AND NIKOLAI RESHETIKHIN

This equation can b e considered as the equation on the space H which was

dened in Sect for more details cf

Consider the lo cal current S z corresp onding to the element S of the vertex

op erator algebra V given by

X X

a n

S z I z I z k h L z

a n

It is known that the op erators L generate an action of the Virasoro algebra on any

representation of g of level k They have the following commutation relations with

generators of g

L Am mAn m A g

Let C b e the space C with co ordinates z z without the diagonals and

N N

b e the space C with co ordinates z z t without the diagonals Denote C

by B and B the algebras of regular functions on C and C resp ectively

k k

the dual space to V Denote by V

k k

b b

g B Intro duce the B mo dules V z V B g z g B and g

C C N N C

The Lie algebra g z naturally acts on V z The Lie algebra g emb eds into

b b

g z in the same way as the Lie algebra g emb eds into g cf Sect Hence g

N z N

k k

z is a B mo dule By z The space of invariants of this action H acts on V

B z with V Lemma we can identify H

Since z i N are derivations of B they act on the B mo dules g z

i N

and V z g

Denote by L the op erator on V z which acts as the op erator dual to L on

the ith factor and as the identity on all other factors

Lemma The operators

r L i N

acting on the space V z commute with each other and normalize the action of the

Lie algebra g

Proof We have

j i i j

r r L L L L

i j

z z

i j

The rst two terms in this commutator vanish b ecause the op erators L do not

i j

and L act dep end on z and the last term vanishes b ecause the op erators L

on dierent factors of V z

GAUDIN MODEL BETHE ANSATZ AND CRITICAL LEVEL

To prove the second statement consider

g A g

t z

We have to show that the commutator of this element with r i N is again

an element of g This will prove the Lemma b ecause such elements generate g

z z

The action of the element on V z is given by

s n

X X

j j

A An

n z z

j s

s i

cf Straightforward computation using gives

j j

A A L nA

n n

z we obtain Then for the dual op erators acting on V

nA L A

and Lemma follows

Lemma means that the op erators r dene a at connection on the trivial bundle

the contragradient Recall that we consider on V V over C with the b er H

structure of gmo dule with resp ect to which it is isomorphic to V

In order to nd an explicit formula for this connection consider for any V

e e

By denition L the corresp onding g invariant functional H

L for any V V

i k

for n we obtain Using the fact that An

i a i

I I L

k h

The Ward identity gives

a i j

X X

I I

e e

k h z z

i j

j i

a i j

X X

I I

i a

k h z z k h

i j

j i

Thus the action of r on the bundle of invariants is given by

z k h i

BORIS FEIGIN EDWARD FRENKEL AND NIKOLAI RESHETIKHIN

Therefore the at sections of this bundle are solutions of the KZ equation

Note that this construction can b e carried out for arbitrary representations of g

instead of V s In particular we can take the contragradient Verma mo dules M

with arbitrary highest weights i N Then the at sections of the bundle

of invariants will b e solutions of the KZ equation with values in M A similar

i i

analysis of the KZ equation and its generalizations can b e found in

Schechtman and Varchenko have found integral solutions of the KZ equa

N N

tion with values in M and M

i i i

We will now derive the rst SchechtmanVarchenko solution using Wakimoto mo d

ules of noncritical level Essentially this has b een done by Awata Tsuchiya and

Yamada in This derivation provides a clear demonstration of how Bethe vectors

app ear in solutions of the KZ equation

The construction of the Wakimoto mo dules from the previous section can b e gen

eralized to an arbitrary level k Intro duce the Heisenb erg Lie algebra h with

generators b n i l n Z and the central element with the commutation

relations

b n b m k h nhH H i

i j i j nm

For h denote by the Fo ck representation of h which is generated by the

op erators b n n from the vacuum vector v which satises

b nv n b v H v

i i i

for all i l and v v

We dene a homomorphism from g to the lo cal completion of U U h by

replacing h z with b z in the formulas for the homomorphism from Sect and

i i

shifting by k h the constant c in the formula for F z Under the homomor

i i

phism the central element K g maps to k Denote the corresp onding

representation of g in M by W

Let C b e the space C with co ordinates x x without the diagonals and C

p p

b e the space C with co ordinates x x t without the diagonals Denote by A

resp ectively and A the algebras of regular functions on C and C

x W A x A and H U A U Put W

C p p C x

k k

A dt h A The Lie algebra z acts on W x H is a Lie subalgebra of

p x

x the space of invariants of this x Denote by J x hence it also acts on W

p k

action This space is a free mo dule over A which is generated by the H invariant

Avalued functional on W whose value on v v v is equal

p k p

i p

i 1

On the mo dule W x we have a natural action of the op erators

L i p

x i

GAUDIN MODEL BETHE ANSATZ AND CRITICAL LEVEL

In the same way as in the pro of of Lemma we can show that these op erators

commute with each other and normalize the action of the Lie algebra H Thus

these op erators act on the space of invariants J x

In order to nd an explicit formula for the action of the op erators on we

have to compute the action of the op erator L on v W

It is known that the action of L on W coincides with the action of the

op erator

X X

b nb n

k h

where b n are the dual generators to b n

b n b m n

s rs nm

Applying this op erator to v we obtain

b b v

k h

By Ward identity

j r i

X X

b b

i i

y L y L y

p p p

k h x x

i j

j i

i j

k h x x

i j

j i

Consider the following system of equations

i j

x k h x x

i i j

j i

The unique up to a constant factor solution of this system is

k h

i j

x x f

i j

Put x f This is an H invariant element of W x which satises

p x

p k

x L

Now put p N m x z i N and x w

i i i i N j j N j

j m Then we have

k h k

i j

z z z w v

i j N m

N m ij

BORIS FEIGIN EDWARD FRENKEL AND NIKOLAI RESHETIKHIN

where

Y Y

_ _

k h k h

i i i i

j j

w w z w

s j i j

sj ij

Denote by W the subspace of W which is generated from vector v by the

op erators a As a mo dule over the constant subalgebra g of g W is isomorphic

to M cf Sect

z w to the subspace The restriction of

N m

W G v G v

i i

i i 1 i i

denes a linear functional

k i i N

z w w M C

m i

This functional vanishes on all homogeneous comp onents except the one of weight

P P

N m

Therefore it corresp onds to a vector

i i

j i j

k i i N

z w w M

m i

P P

N m

dep ending on z z w w of weight

N m i i

j i j

Using the Ward identity in the same way as in the pro of of Prop osition we

obtain

i i k i k i

m m

1 1

w z w v w w z w

N m

m N m m

k h i i

i j 1

z z w w

i j

by Hence by Lemma

k h i i k i i m

m m

i j 1 1

z z jw w i z w w

i j

m m

Denote by C the space C with co ordinates w w without all diagonals

mz m

w w and all hyp erplanes of the form w z The multivalued function denes

j i j i

a onedimensional lo cal system L on the space C It is clear that for xed z the

m N k

W is a W z w on any vector of the tensor pro duct value of

k k

i i j i N m

pro duct of and a rational function on C

Denote by L L the twisted de Rham complex of L It consists of

dierential forms on C with dierential d acting on a form by the formula

d d d log

It is therefore convenient to represent an element of the twisted de Rham complex

by a multivalued dierential form The action of d on coincides with the

action of d on

If f is a holomorphic function on C consider the mform f dw dw

mz m

L Since it is holomorphic it denes an element of the mth cohomology group

H C L of the de Rham complex

GAUDIN MODEL BETHE ANSATZ AND CRITICAL LEVEL

There is a natural pairing b etween this group and the mth homology group H C L

m mz

of C with co ecients in the dual lo cal system This pairing is given by integration

over cycles

H C L H C L

mz m mz

ab out dierent choices of cycles and convergence of the corresp onding integrals cf

This pairing has the following prop erty for any dierential form on C

d H C L

m mz

We can now repro duce SchechtmanVarchenko integral formula

Theorem Let be an mdimensional cycle on C with coecients in the dual

local system L The M valued function

k h i i

i j 1

z z w jw i dw dw

i j m

where jw w i is the Bethe vector is a solution of the KZ equation

z w

i j

Proof To avoid confusion we will denote by A or A the op erator which acts

m N

W and W as A on the factor W or W of the tensor pro duct

k k k k

i i j i i i

j j

as the identity on all other factors

From formula we obtain

k i k

z w L z w

N m N m

By restricting this equation we have

G z w G v v

i i

N m 1 i i

i k

z w L G v G v

i i

N m i 1 i

where M

Using the Ward identity we can rewrite the right hand side of equation as

d N

X X

az z k

j i

G I G v z w I v

i i

1 i i

a N m

k h z z

i j

m d

X X X

k z

z w I G v

N m a 1 i

k h z w

i s

n s a

I n G v G v

i i

s m

i i

s m

BORIS FEIGIN EDWARD FRENKEL AND NIKOLAI RESHETIKHIN

To compute the last term of this formula we have to compute I n G v

for n W

Lemma For any a d and i l

I n G v n

and there exists such Y W that

a k

a a

I G v k h Y I G v k h L Y

i a i a

i i

This Lemma will b e proved in the App endix It gives for the sth summand of the

second term of formula

s z k

G Y v G I v z w

i i

i 1 i

a a N m

where

Y L Y

a a

z w z w

i s i s

z w coincides with the action of According to the action of L on

N m

by w Therefore we can replace Y

w z w

s i s

Then formula can b e rewritten as

z k

Y G I G v v z w

a i i

a 1 i i N m

w z w

s i s

For any z and the term in brackets V is a pro duct of the multivalued function

and a rational function on C Consider V dw w as an element of L

mz m

Then we have

V dw dw dV dw dw dw

m s m

Therefore if is an mdimensional cycle on C with co ecients in L we have

by formula

V dw dw

Thus we see that the second term of formula will disapp ear after integration

over

GAUDIN MODEL BETHE ANSATZ AND CRITICAL LEVEL

But the rst term coincides with the action of the op erator k h So we

obtain

k i

z w L G v G v dw dw

i i m

N m 1 i i

z w G v G v dw dw

i i m

1 i i

N m

k h

Hence by

z w G v G v dw dw k h

i i m

N m 1 i i

z w G v G v dw dw

i i i m

N m 1 i i

M for an arbitrary

The last formula can b e rewritten as

Z Z

i k i i k i

m m

1 1

dw dw w z w dw dw w z w k h

m i m

m m

Therefore we obtain

Z Z

i k i i k i

m m

1 1

k h w z w dw dw w z w dw dw

m i m

m m

Theorem now follows from formula

Theorem shows that Bethe vectors enter solutions of the KZ equation This has

b een observed in Wakimoto mo dules provide a to ol for constructing

eigenvectors in Gaudins mo del and for solving the KZ equation thus giving a natural

interpretation of this phenomenon

Appendix

In this App endix we give the pro of of Lemma and Lemma following

App endix B

First we prove Lemma in the case g sl This is a direct computation using

the explicit formulas for the homomorphism and hence given in Sect

We have Gz az E z From the commutation relations of sl we obtain

E n G H n G Gn F n G H n k

Since

E nv Gnv n H nv F nv n

and H v v we conclude that

E n Gv H n Gv n

BORIS FEIGIN EDWARD FRENKEL AND NIKOLAI RESHETIKHIN

and

F n Gv n

We also have

F Gv k v

and

F Gv GF v H v

aa aa bv k L v

This proves Lemma in the case g sl

Recall that in Sect generalized Wakimoto mo dules over g asso ciated

to an arbitrary parab olic subalgebra of g were dened Denote by p the parab olic

subalgebra of g which is obtained by adjoining the generator F of g to the Borel

subalgebra of g Denote by sl the Lie subalgebra of g generated by E H and F

i i i

i i

and ab elian We have an orthogonal decomp osition of p into its semisimple part sl

part h

Denote by the Heisenb erg Lie algebra with generators a n a n n

Z and the same commutation relations as in g cf Sect Denote by M the Fo ck

representation of which is generated by a vacuum vector v satisfying conditions

The Wakimoto mo dules over g of level k asso ciated to p are realized in the tensor

i i i

pro duct M L Here is the Fo ck representation of the Heisenb erg Lie algebra

and L is a representation of sl of level k such that asso ciated to h

i i

k h k

Remark We would like to take this opp ortunity to correct some errors in Sect

First in the exact sequences on page the mo dule E nd should b e replaced by

Second in formula on page one should take into account the dierence in

normalization of the invariant scalar pro duct h i for a Lie algebra and its subalgebra

Let b e the ratio b etween the scalar pro duct on the Lie algebra g and its Lie

subalgebra g we use the notation from Then formula should b e rewritten

k c k

(i)

p p

Since k k c this gives the following condition on levels of representations of

g and g

k c k c

(i)

i g

In particular in our case p p so and we obtain formula

i i

GAUDIN MODEL BETHE ANSATZ AND CRITICAL LEVEL

y C This is a mo dule over Let us cho ose as L over sl the mo dule W

y k

the Heisenb erg algebra generated by a n a n b n n Z It is clear that the

resulting mo dule over g is isomorphic to the standard Wakimoto mo dule W on

y k

which the op erators Gn act as a n In other words we cho ose such co ordinates

x on the big cell U of the ag manifold of g in which G x In particular

we see that the action of the op erators E n H n and F n with n on the

i i i

W is the same as the action of the op erators E n H n subspace W

y k y k

i i

and F n on the sl mo dule W

y k

Hence the action of the op erators E n H n and F n on G v coincides

i i i i

with that in the case of sl Thus we obtain

E n G v H n G v n

i i i i

i i

n F n G v

i i

and

F G v k h v

i i

k h L v F G v

i i

Now consider the op erators E n H n and F n with j i We rst compute

j j j

the nitedimensional commutation relations of the dierential op erators E H

j j

and F with the vector eld G cf Sect These commutation relations read

j i

cf eg Sect

E G H G H G

j i j i i j i

and

G F G H x

i j i i j

Using these formulas we can nd the commutation relations of E n H n F n

j j j

z there can b e j i with G m using the Wick theorem Since G z a

i i

no double contractions and there is no contribution from the cob oundary term

z Therefore this computation amounts to the computation of all single c a

contractions But those are uniquely dened by the nitedimensional commutation

relations ab ove So we obtain

E n G z H n G z z H G z

j i j i i j i

and

z G z F n G z z H a

i j i i j

These formulas immediately give

E n G v G E nv n

j i i j

i i

BORIS FEIGIN EDWARD FRENKEL AND NIKOLAI RESHETIKHIN

H n G v G H nv H G n v

j i i j i j i

i i i

a r G sv F n G v G F nv H

i j i i j i j

i i i

sr n

The last two formulas clearly hold for n and also for n b ecause H v

H v and F v H a v

i j j i j

i i i

We can compute the action of any other element of g of the form I n n on

G v using its presentation as a commutator of E ms or F ms dep ending

i i i

on whether I b elongs to the upp er or lower nilp otent subalgebra of g resp ectively

if I h then it is a linear combinations of H s and there is nothing to prove

Clearly in the rst case we obtain In the second case we can realize I n as a

successive commutator of E n n and E j i The statement of Lemma

i j

then follows from formulas and the relation L A for any A g

Lemma can b e proved along the same lines We nd

E n G v H n G v

j i j i

and

F n G v n F G v H v

j i j i ij i

for any j l and n Therefore G is a singular vector of imaginary

degree if and only if H

i i i i

Acknowledgements We thank A Beilinson V Ginzburg D Kazhdan and E

Sklyanin for useful discussions

The research of E F was supp orted by a Junior Fellowship from the So ciety of

Fellows of Harvard University and by NSF grant DMS The research of N

R was supp orted by Alfred PSloan fellowship and by NSF grant DMS

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