Separation of variables for the eight-vertex SOS ... - Research Collection

Diss. ETH No. 13682

Separation of

variables for the

eight-vertex SOS antiperiodic boundary conditions

model with

A dissertation submitted to the

SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH

for the

degree

of

Doctor of Mathematics

presented by ANKE SCHORR born

Dipl. Phys. ETH September 2, 1972 in Saarbrücken, Bundesrepublik Deutschland

accepted

on

the recommendation of

Prof. Dr. Giovanni Dr.

Felder,

Examiner

Benjamin Enriquez, Chargé de recherches, Co-Examiner

1

To my

parents

Seite Leer / Blank leaf

2

Abstract This work deals

( [14])

group

to

the

on

hand with how to

one

investigate

use

the

representation theory of

statistical mechanical model and

a

to solve the statistical mechanical model

by Sklyanin's

on

a

quantum

the other hand with how

method of

separation of variables.

elliptic quantum group ET^{sl2) established by Felder concretely, [47]. in [29, 25] to investigate the SOS eight-vertex model established by Date, Jimbo, Miwa and Okado [10] with antiperiodic boundary conditions which are the reason that Bethe ansatz fails and we have to use Sklyanin's method of separation of variables [47]. More

The SOS

we use

eight-vertex

[3].

model

the

model is

a

It is related to the

face-model version of Baxter's

elliptic quantum

group

weights We(c,b,a,d\z) by a suitably Re(z,\) defining ET>J](sl2)- This relation reads its Boltzmann

Re(z,

X

=

—2r)d)e[c

—

2_] We(c, b,a,d\z) e[b

d] —

original eight-vertex

ETjTt{sl2),

since

we

rediscover

discretized version of the R-matrix

e[b

a]

c]

—

e[a

—

=

d].

a

The

antiperiodic boundary conditions of the of transfer matrices

family

4.21)

K

[

—

)

.

The

the

The

L-operator

product

(finite) partition

matrix to To find

yield ZM

=

of fundamental

of of

are

a

special are

a

representation of

representations of

ET>ri(sl2)

which is

an

ETv(sl2) at n points. The equivalence of solving this eigenvalue problem instead of the original one is due to the fact that the auxiliary representation and the representation of ET>r)(sl2) which defines the SOS eightvertex model are isomorphic (Theorem 4.44). Let

us

now

troduction

Feigin

briefly state the content -, we briefly present the

and Rubtsov

of variables.

(This

[16],

i.e.

we

serves as an

of

of this work: In the second results

the

chapter

-

after the in¬

Gaudin model

by Enriquez,

state the solutions of this model obtained

by separation

insight

on

into the

elliptic

separation of variables method

as

well

as

seen as a limiting case describing eight-vertex SOS model.) chapter 4, we deal with the eight-vertex SOS model. We first define the basic notions of the eight-vertex SOS model more heuristically. Then, we describe the ba¬ sic representation theory of ET^(sl2). In the next section, we describe the eight-vertex a

model that

can

be

of the

In the

SOS model in terms of the representation

(M(C, V2®n),Lsos,e{z-> A)) ces.

We propose the

theory of

and the commutative

ET>v(sl2) involving

family

auxiliary representation (M(C,

of

the definition of

antiperiodic transfer

V2®71), Lûx e(z, A))

and the

matri¬

emerging

3

family the

of

auxiliary

isomorphism

the final

transfer matrices in the fourth section. In the fifth

between

(M{C,V2®n),LSOs,e{ziA)) main results

and

section,

we

describe

(M(^V2®n),Lûx>e(z,A)). of the

In

description section, eigenvalues eigenvectors of the family of transfer matrices with antiperiodic boundary conditions of the eight-vertex SOS model in terms of the common eigenvalues and eigenvectors of the auxiliary transfer matrices (Proposition 4.54 and Theorem 4.55). In chapter 5, we treat the simplest non-trivial example of the SOS eight-vertex model, 3, to clarify the notions defined in the preceding chapter. namely n We deal with another problem in Appendixl, a problem frequently treated in Sklyanin's papers on separation of variables [47, 46]. There he discusses separation of variables of the XXX model [20, 21, 37], which is related to the representation theory of the Yanwe

state

our

on

the

common

and

=

gian [52] 3^(s£2)- Solving this model involves a procedure analogous to the one for the SOS eight-vertex model: a main problem consists in finding an auxiliary representation (C2,LauXtr(z)) which is isomorphic to the representation (C2,Lxxx{z)) which comes along with

the XXX model.

which differs from what

proposed states in

in the SOS

[46, 44].

Here,

Sklyanin eight-vertex

we

did in case.

propose

[47, 46]

a

version of

obtaining the isomorphism analogy to the isomorphism we results agree with what Sklyanin

and is in

Of course, the

4

Zusammenfassung Diese Arbeit befasst sich

([13])

gruppen

damit,

wie

zum

einen

damit,

wie

Darstellungstheorie kann,

man

auf Modelle der statistischen Mechanik anwenden

von

Quanten¬

zum

anderen

aus entsprechendes Sklyanins Separation der Variablen [47] löst. Konkreter benutzen wir die elliptische Quantengruppe ETjTj(sl2), wie sie von Felder [29, 25] konstruiert wurde, um das SOS

Modell

ein

man

der statistischen Mechanik mit

Methode der

Acht-Vertex-Modell in der Gestalt

tiperiodischen Randbedingungen Bethe-Ansatzes sind und ablen

zu

[10]

mit

an¬

betrachten, die die Ursache für das Versagen des anleiten, Sklyanins Methode der Separation der Vari¬

zu

dazu

uns

Date, Jimbo, Miwa und Okade

von

[47].

benutzen

Das SOS Acht-Vertex-Modell ist eine Version

Baxters

ursprünglichem Acht-Vertex[3] elliptischen Quantengruppe in Zusam¬ menhang, was wir an der folgenden Relation erkennen, die die Boltzmann-Gewichte des Modells We(c,b,a,d\z) mit der geeignet diskretisierten R-Matrix der elliptischen Quan¬ Modell

als "face"-Modell.

tengruppe

Re(z, X)

von

Es steht mit der

verbindet:

Re(z,

X

-2r]d)e[c

=

-

2_j We(c, b, a, d\z) e[b

d] —

e[b

(g>

a]

®

-

e[a

c]

=

d].

—

a

Die

antiperiodischen Randbedingungen des Modells werden dadurch fixiert, dass

spezielle

Familie

von

zéC abhängen, und des

L-Operators

nition

4.21),

Die

zum

Tensorprodukt

(endliche)

(M(C, V2m),LSos,e(z, A))

SOS Acht-Vertex-Modell

mit Twistmatrix K

verschobenen

man

eine

Tsos,e{z, Ao) betrachtet, die von einem Parameter getwistete Spuren über den auxiliären Raum der Quantengruppe

Transfermatrizen

=

j

I

.

Der

fundamentalen

von

L-Operator besteht

Darstellungen

von

aus

sind

(s.

Defi¬

dem n-fachen

ET>ri(sl2).

Partitionsfunktion des SOS Acht-Vertex-Modells mit

antiperiodischen

Randbedingungen ist durch die Transfermatrix gegeben als Zm Tï2M(Tsos,e(zi ô))M Um Eigenwerte und Eigenvektoren der beschriebenen Transfermatrix zu finden, benutzen wir Sklyanins Methode der Separation der Variablen [47]. Mit Hilfe dieser Methode =

gelingt

das

mensionalen

Problem, das ursprünglich das Lösen einer nichtlinearen multidiDifferenzengleichung beinhaltet, auf das Lösen eines Systems von n eindi¬

mensionalen

Differenzengleichungen,

inition

es

uns,

4.52),

der Familie

zurückzuführen.

von

(M{C,V2®n),Lûxfi{z, A)) das

Die

von

Äquivalenz,

die

Wir möchten

Taux>e(z, Xq)

Sklyanins

(Definition 4.33)

,

generischen Punkten zus¬ Darstellung [46, 44], genannt den elliptischen Fall ETjr,(sl2)

an n

auxiliärer

die hier auf

erlaubt, statt des ursprünglichen Problems lösen, beruht darauf, dass die Darstellung von SOS Acht-Vertex-Modell und die auxiliäre Darstellung isomorph sind (The¬

System separierter Gleichungen

ET,r){sl2) zum orem 4.44).

separierten Gleichungen (s. Def¬ Gleichungssystem kommt durch das Auswerten

auxiliären Transfermatrizen

tande. Diese sind Transfermatrizen

erweitert wird.

den sogenannten

Dieses

nun

es

uns

zu

kurz eine Inhaltsübersicht

geben:

Das zweite

Kapitel,

nach der Einfüh¬

rung, enthält ein Resume der Resultate

chen Gaudin-Modell

durch

[16],

Separation der

auch der

aufgefasst

von Enriquez, Feigin und Rubtsov zum elliptis¬ geben die Lösungen dieses Modells an, die die Autoren

Variablen erhalten.

Entwicklung werden

d.h. wir

eines

kann.)

(Dies

Modells, das

dient der Einsicht in diese Methode wie

als Grenzfall des SOS Acht-Vertex-Modells

5

In

Kapitel 4 behandeln

wir das SOS Acht-Vertex-Modell.

Zunächst beschreiben wir die

Grundbegriffe des Modells auf heuristische Art. Dann folgt eine kurze Einführung in die Darstellungstheorie von ET:V{sh)i soweit wir sie benötigen. Im nächsten Abschnitt wird das SOS Acht-Vertex-Modell dann darstellungstheoretisch formuliert, was die Definition von

(M(C,V2®n),Lsos,e(z, A))

fermatrizen umfasst.

und der kommutativen Familie

und die kommutative Familie auxiliärer

Abschnitt 4

(M(C,V2®n),LsosAziX)) mulieren wir

Darstellung (M(C, V2®n),Lûa. e(z, A)) Transfermatrizen, die daraus hervorgeht, in

Im fünften Abschnitt konstruieren wir den

vor.

antiperiodischer Trans¬

Wir stellen die auxiliäre

und

{M{C,V2®n),LSOs,e(z'X))-

Isomorphismus

zwischen

Im letzten Abschnitt for¬

Hauptresultate

zur Beschreibung der gemeinsamen Eigenwerte und Eigenvektoren der Familie von Transfermatrizen der SOS Acht-Vertex-Modells mit an¬ tiperiodischen Randbedingungen mit Hilfe der gemeinsamen Eigenwerte und Eigenvek¬ toren der Familie von auxiliären Transfermatrizen mit antiperiodischen Randbedingungen (Proposition 4.54 und Proposition 4.55). In Kapitel 5 behandeln wir das einfachste nicht-triviale Beispiel, n 3, um das im unsere

=

vorhergehenden Kapitel Hergeleitete zu illustrieren. In Appendix 1 streifen wir ein weiteres Problem,

das in

Sklyanins Artikeln [46, 44] Separation der Variablen oft behandelt wird. Dort erklärt er die Separation der Variablen für die XXX-Kette [20, 21, 37], ein Problem, das mit der Darstellungs¬ theorie des Yangian [52] y{sl2) verbunden werden kann. Die Lösung dieses Modells erfordert ein Vorgehen, das in Analogie zu demjenigen beim SOS Acht-Vertex-Modell über die

betrachtet werden kann:

gian

(C2,Laux,r(z))

(C2 ,Lxxx(z)) Art,

die sich

unserem men

ist.

von

der

Vorgehen

Ein

Hauptproblem

ist es, eine auxiliäre

Darstellung des Yan¬ [46, 44] zu finden, die isomorph au derjenigen zur XXX-Kette Hier konstruieren wir den dazugehörigen Isomorphismus auf eine Herleitung Sklyanins in [46, 44] unterscheidet und in Analogie zu

beim SOS Acht-Vertex-Modell steht. Die erhaltenen Resultate stim¬

selbstverständlich mit denen

Sklyanins

in

[46, 44]

überein.

6

Acknowledgements This work was done during

my time

as

a

teaching

and research assistant at the

Depart¬

ment of Mathematics at the ETH Zürich.

It is my

pleasure

to thank my

supervisor Professor Giovanni Felder for his guidance of

my thesis. Discussions with him to my

goal.

I also

were encouraging and almost always brought me nearer appreciated his providing me with the opportunities to travel and thus

broaden my horizon.

I

especially enjoyed

my

staying

at

ESI, Vienna, facilitated by

an

invitation

by

Professor

A. Alekseev whom I would like to thank at this point. I

obliged

to my co-examiner Dr.

B.

Enriquez for

his support during the final stages influencing the final structure of this work. I also thank A. Rast for proofreading the introductory part of the thesis. I am indebted to my colleagues and friends, inside and outside the math department, for giving me the necessary amount of fun, diversion and understanding. I thank my familiy for their encouragement and support. Above all, I thank Christoph for his constant effort to grapple with my idiosyncrasies am

of my thesis

as

well

and for his love.

as

for his comments

7

Contents 1

Introduction 1.1

The SOS

2

Basic notions

1.1.2

Two different

approaches solving models of statistical mechanics and the SOS

eight-vertex-model Quantum groups, the QISM and different

eight-vertex

The method of

Introduction The

separation of variables

Hamiltonian

22

2.5 2.6

Solutions of the

Completeness

of the Bethe

The SOS

eight-vertex

Basic notions of the SOS

4.2

The

4.2.3 The 4.3.1 4.3.2

4.3.3

26 28 32 32

36

eigenvectors

38

case

model

4.1

4.2.1

24

elliptic Gaudin eigenvalue problem

Introduction to the difference

4.2.2

22

The structure of the solutions

2.6.2

40

eight-vertex model setting corresponding to the SOS eight-vertex

40

model

42

Introduction

Representations,

43 functional

representations, operator algebras

Highest weight representations eigenvalue problem corresponding

...

eight-vertex model The SOS model in terms of the representation theory of ET}V(sl2) The representation attached to the SOS model as a highest-weight representation The family of transfer matrices of the SOS model with antiperiodic boundary conditions .

.

4.7 5

The

Antiperiodic

5.1

A

5.2

Computing

5.3

The

SOS Model:

n

=

53

53

57 58 61

63 .

auxiliary representation antiperiodic SOS model in the case

64 79 83

3

85

preliminary step: Computing the auxiliary representation for the

50 51

results: The

Generalizing Sklyanin's auxiliary representation Introducing the auxiliary representation The auxiliary transfer matrix 4.4.2 4.4.3 Establishing the isomorphism between the SOS and the auxiliary representation abstractly The isomorphism establishing separation of variables for the SOS model Solving the eigenvalue problem of the antiperiodic SOS model Limiting cases of the SOS eight-vertex model

4.6

44

49 to the SOS

4.4.1

4.5

18

22

to

2.6.1

16

20

setting corresponding sfaiC) The setting corresponding to the elliptic Gaudin Hamiltonian The elliptic Gaudin eigenvalue problem Separation of variables for the elliptic Gaudin Hamiltonian

4.4

...

1.2.2

2.2

12 14

forms of the Bethe Ansatz

The connection between quantum groups and statistical mechanics

2.1

4.3

.

model

1.2.1

elliptic Gaudin

2.4

4

9 9

The

The

2.3

3

eight-vertex

1.1.1

1.1.3

1.2

9

model

for n

n

—

=

3

3

n

=

2

.

.

86

93 106

8

6

Appendix described 6.1

The

1:

An alternative

by Sklyanin

7

magnetic

chain

110 to the XXX chain

110

110

functional

Representations, representations, operator algebras A special class of twisted representations 6.1.3 The isomorphism to establish separation of variables for the XXX chain

Appendix

as

Introduction

6.1.2

6.2

to the XXX

[47]

setting corresponding

6.1.1

approach

2:

Spaces

of

elliptic polynomials

.

.

.

Ill

116 .

116 125

9

Introduction

1

In the introduction sis.

First,

will

we

These include for

sequel.

briefly present

the transfer

example

round-a-face models, the Bethe ansatz discovered

presented

the

-,

here

responding

the main themes

pursued throughout

will discuss the main notions of statistical mechanics

we

star-triangle-relation of

can

At the end of the

well

as

pages,

section,

and the

in

as

an

its solid-on-solid

(SOS)

the the¬ in the

using

quantum groups

Yang-Baxter-equation.

[3],

where

abbreviated version also in

we

The main

always

some newer

(IRF)

or

-

what is

an

were

points

cite the

talks,

e.g.

will also present the model treated in the thesis: the

we

vertex model and its interaction-round-a-face

In the second

will be

vertex models and interaction-

matrix,

in its appearance before

-

be found in Baxter's book

course

we

equivalent

cor¬

[40].

eight-

notion

-

version. will

out some general facts about quantum groups algebras and the quantum inverse scattering method (cf. e.g. [14, 23, 22, 47]). Since the main object of the thesis, with regard to quantum groups, is the elliptic quantum group ETAsh) ([29], [25]), we will also discuss its struc¬ or

section,

we

briefly point

Drinfeld-Jimbo quantum affine

ture.

After

ing

this,

we

will

briefly

expose the connection between

of the aforementioned models of statistical

discussed there

be

can

quantum groups (to so-called

Quantum

suitably translated

achieve

a

i.e. how

language

of

some

notions that


treatment). This topic relies on the (QISM) developed by the Faddeev school.

Method

The last part will be devoted to two different realizations of the Bethe ansatz,

the

separation of variables, also The SOS

1.1

eight-vertex

We first have to ask what a

of the

especially

QISM.

model

that interact via their

concerning the

edge

values

model of statistical mechanics is.

a

model of statistical mechanics is

many atoms at the sites of

each

integral part

an

Basic notions

1.1.1

here,

we

certain unification in

Scattering

Inverse

mechanics,

into the

quantum groups and the solv¬

a

spins

a

description

we

or

a

attach

{—1,1}- Solving

of

a

finite lattice

IcC,

a

we

understand it

system consisting of infinitely

infinite lattice aL + ibTL C C in

and columns of the lattice.

rows

of the lattice

G

an

As

with

a

two-dimensional

some

boundary

plane

conditions

simplicity, let us suppose that to denoting a spin, which can only take usually implies the computation of (some of) the For

variable a, e.g.

the model

following quantities:

a)

The

partition function (infinite Z

=

or

finite

êxp(-E(s)/A;T)

or

respectively) ZM

=

s

where the or

of

a

sum

êxp(-.E7(s)/fcT) s

is taken

over

finite lattice with M

all

possible

rows

states

s

of the

and N columns with

spins some

on

the infinite lattice

boundary conditions

(cf. figure below). E(s) is the energy of the system depending on configuration (cf. below) and a possible external field, k is Boltzmann's

w.r.t. the columns

the lattice

of the system

constant and T the

temperature.

10

b)

The free energy of the system which is

F

corresponding

-kTlnZoiF

to whether

we

(

lim

=

start with the

-kT^- \n(ZM) infinite lattice

J

,

or a

finite lattice with

boundary conditions.

some

c)

=

given by

Other

physically interesting quantities

such

as

the

specific heat

and the

magneti¬

zation. In order to understand how

usually calculates these quantities, let us C, a part of which is given in the figure is by means of a vertex model.

one

two-dimensional finite lattice L C way of

approaching

this lattice

We state that the lattice consists of horizontal and vertical An intersection of

to each

edges

and

start with

below.

an arrow

a

One

is attached

edge. by a vertex v. possible physical interpretation is given in [3], p. 127 (though the physical interpre¬ tations of course differ by what combinations of arrows are allowed at each vertex), for horizontal and

a

a

vertical line is indicated

A

the six-vertex model

describing the hydrogen bonding of ice: At each vertex there is an surrounded by four hydrogen ions which are placed at the edges. The atom are attached by a hydrogen bonding. Thus, of every four ions surrounding

oxygen atom

and each ion

the

corresponding atom, signified by an arrow pointing towards the by an arrow pointing away from the atom. (In this case, there also exists a 'non-physical' interpetation of the arrangement of arrows cf. [3] p.165 -, namely the following problem: In how many different ways can the lattice be colored by three different colors if the colors of two faces are adjacent to be different?) An assignment of arrows to a (finite) lattice is called a configuration of an

atom two

atom, and

are near

two

are

farther away from the atom, denoted

-

the lattice. If

look at the finite lattice drawn

we

riodical toroidal

boundary

conditions

lattice aZ + ibZ C C Note that w.r.t.

Let

rows as we

us now

will

always

we

we can

could also

do later

above,

that

by imposing pe¬ representing an infinite impose antiperiodical boundary conditions we can see

think of the part

as

on.

turn to the interaction of the

edges

-

with

arrows

-

on

the lattice. We admit

only nearest-neighbour interactions and interactions of any edge with an external field H. How can we describe the interactions? All types of interactions occurring between

nearest-neighbour edges can be specified by looking at a vertex with some values of the surrounding edges attached to it. To every combination of arrows around a vertex a corresponding weight w(a,b,c,d) exists that describes the statistical occurrence of the

configuration in question, where a, b, c, d are the variables attaches to the sur¬ rounding edges of a given vertex. If we classify all allowed combinations of spins at a vertex with their corresponding weights, we obtain the interactions. Depending on the vertex

model in

question, there

eight-vertex model, to the rule that

are

a

different number of allowed vertex interactions.

the allowed combinations of the

an even

number of

arrows

has to be

arrows are

pointing

drawn

For the

below, according

in and out of the vertex.

11

w(-u,u,u,-u) XX

M

fy+-

w(u,u,-u,-u) Let

us

sum

quantities

indicated

simplify calculations,

To

columns drawn in ditions

one

concerning

(of

the

figurations is then

the

((C2)®w).

End

T G

our

above,

rows

in

Its entries

the next

we

introduces the transfer matrix T.

If

we

have A?"

impose periodic or antiperiodic boundary con¬ this lattice, this is a 2N x 2^ matrix, i.e. in general the

are

If

row.

our

yet normalized

not

-

change

a row

-

probabilities with which 2^ possible con¬

into any of the

finite lattice has M rows, the finite

suitably perform the limit

can

we

partition function

periodic,

that

case

Zm

matrix,

—>

by

,

may do

obtaining the partition partition function, it proves useful cyclicity of the trace, we get in the easiest, we

oo,

the

wnere

>

of the transfer matrix. In this case,

Ai

M

since

Z)î=i ^'

=

Tr2jvT

=

we

the

Az îot i

largest eigenvalue of the matrix. (For a the transfer matrix, cf. e.g. [3], pp. 32.)

The second way of model.

approaching the

instead of

=

1,...

,2N

the

are

eigenvalues

also find the free energy

is the

appearance of

so

the above formula of the

perceive by

the transfer

diagonalize

where

First, the partition function, possible lattice configurations.

given by

function. As to

all

w(a,b,c,d)

finite lattice and

ZM If

we sum over

2^ possible) configuration of on

w(-u,-u,-u,-u)

want to calculate:

we

where

usually

one

-m

w(u,u,u,u)

-

w(-u,u,-u,u)

w(u,-u,-u,u)

turn to the

now

which is the

w(u,-u,u,-u)

w(-u,-u,u,u)

*

lattice is

by

neat

explanation

of the SOS

means

or

of this and the

face

[34]

or

IRF

the vertices the

on Here, dynamical variable is put onto a face of the lattice, face F in the first figure. We call this variable height and adjacent heights are to differ by plus or minus one. The weights describing the interaction, commonly denoted as Boltzmann weights, then indicate an interaction between faces in the manner shown in the figure below.

Assignment of a Boltzmann weight

to

four faces

w(a,c,c,b)

If

we

want to calculate the

transfer matrix

bl

(cf. [3],

pp.

we

visualized in the

.

z(n-l)

do

figure

.

+/-al

an

z2

can

+/-M

bn

a2

zl

370),

b2 .

al

partition function

zn

so

by using

the row-to-row-

below. Row-to-row transfer matrix for

an

SOS model

12

The row-to-row transfer matrix of

a

face-model is still

to each face that differs from its

height

2N

a

2N matrix, provided

x

neighbouring height by plus

minus

or

one

a

to each

face. Each entry is of the form n

JJw(&j,a,,aî+i,&î+i), j=i

where an+i odic

±a\ and bn+\

=

conditions

boundary

the

(0,1,0,2,... (&i,... ,bn, bn+i ±61) of faces

assignment ment

±b\ according

=

on

,an, an+i

models the

(cf. [3]

method of

a

363-401).

pp.

Since

Two different

1.1.2

In

notion of the

common

suggests, provides

chapter 8 of

common

we

we

chose

periodic

or

antiperi¬

same

in the lower

row

changes

The statements

row.

the

on

for the vertex models.

as

given

a

into the

assign¬ diagonalization

Note that for SOS

transfer matrix also exists

corner

calculating

a

transfer matrix for

will not need it

here,

we

which, as the name quadrants of the lattice

will not pursue it further.

approaches solving models of statistical mechanics

his book

statistical

mechanics, a family eigenvectors

on

and

eigenvalues

(cf. [3],

±a{)

—

in the upper

—

of the transfer matrix remain the

to whether

An element of the above form tells how

rows.

Baxter

of

proposed

method of

a

finding

of transfer matrices of the six-vertex

140), which involves an ansatz using a vector parametrized by a set of parameters (w\,... ,wm). The vector has to obey certain recursive relations. In order to yield an eigenvector, the set of parameters has to obey a set of equations, model

the vector

[8],

ansatz

-

to the cancellation of

corresponding prevent the

133

pp.

vector

emerging

started with.

one

the set of

"unwanted" terms

some

which,

from the recursion relations from

if

being

they

do not

linear

This idea goes back to Bethe and is known

equations the parameters

to

are

is known

obey

cancel,

dependent as

on

the Bethe

the Bethe ansatz

as

equations. In the ment

chapter

as

9 of the

to when

a

same

family

[3],

book

Baxter

diagonalized solving This "program", formulated on p. 184 -

find

by

next

step is

183)

The transfer

finding

for all

with

of

[3],

seems

involves the

be

(cf. [3], pp. 180 200). following: The first step is to

feasible

-

where every

a

matrix,

matrix

T(u)

Q(u)

i.e. all of its

with

entries, is

for all values of u,

an

entire function of

determinant, obeying the so-called

also

non-zero

an

u.

The

entire function of

Baxter

equation (cf. [3],

u

A(u)

=

($(r?

-

u)Q(u

appearing parameter

of the Boltzmann

weights

be understood for every an

of the model

treat¬

can

commuting family T(u) of transfer matrices, where the variable u G C is obtained reparametrization of the original weights, cf. [3], p. 184 or p. 212, for the eight-

commuting

p.

and hence the

different, fairly general

a

model of statistical mechanics

a

a

a

vertex model.

u,

suggested

of transfer matrices of

and

rf

+

2t/)

A(u)

$(77

is obtained

=

77

—

u)Q(u

+

as a

-

2rfj) /Q(u),

consequence of the

reparametrization

Tri in Baxter's notation. The matrix

diagonal entry

entire scalar function and

+

equation

can

Q(u), cf. pp. 182 in [3]. $(«) is (diagonal) matrix of eigenvalues of the

of the matrix

stands for the

transfer matrix. If

we

every

consider Baxter's

appearing

equation

function is

an

as an

equation of matrix elements, and hence functions, u. Thus, if we consider Q(u) with

entire function of

13

(w\,... ,wm),

zeroes

to vanish.

we

obtain

m

conditions

%

=

1,...

quite another

,

to

and

are

precisely

S,

equation

_ ~

Q(wl Q(wt

-

+

2rj') 2rj>)

the Bethe ansatz

equations, though deduced

in

context.

In Baxter's treatment the matrices matrix

the residues of the above

They read

$(r)-wl) (-n + Wl) for all

forcing

but since

Q(u)

and

V(u)

have to commute with still another

will not need this operator here

we

we

will not go into details.

After the presentation of this program, Baxter proposes some conditions that must be satisfied in order to achieve certain of the above-mentioned steps. For the family of local transfer matrices that later

language known which

as are

on

(cf. [3], p.188),

of Baxter

appeared

as

R-matrices of

Yang-Baxter-relation (here formulated illustrated in the figure below

the

Y^

a

vertex

to be commutative it has to

w(m, a, c, m')w'(n,

c,

b,

in

model, Ul

obey weights according

n")w"(n", m", n', ml)

in the

what later became to

[3],

p.

187),

=

c,m" ,n"

YJ

w"(n, m, n", m")w'(m",

a, m,

n')w(n", c, b, n')

c,m" ,n"

For the row-to-row transfer matrices of

weights

have to

obey

the

an

SOS-model to be commutative, the Boltzmann

(generalized) star-triangle-relation

Y2 wia, 6> c> a")w'(a", c, 6', a')w"(c, b, b", b') c

=

J2 w"(a", a, c, a')w'(a, b, b", c)w(c, b", b', a'), c

where

w,w',w"

are

different Boltzmann weights. It is visualized by the

figure

below.

14

Q(u) obey

In order to have the operator

condition 192

pp.

Q(u),

-

cf.

on

the columns of this operator, named

The

The

was

model

given by

[3]

by

215

-

222

-

a

(cf. [3],

operator

and since

we

0 0

its

eight-vertex model hence its i?-matrix.

weights,

(cf. [27]

and reads

/ a8V(z)

R8v(z)

[3],pp.

e.g.

and the SOS

be described

can

Baxter in

consistent treatment of the

a -

vertex"

a

consider it sufficient to state the condition.

we

eight-vertex-model

eight-vertex

matrix

sequel,

equation, Baxter also formulated

"Propagation through

194). But since so far there has not been [46] and [41], but only explicit examples

will not need it in the

1.1.3

Baxter's

in

comparison

[3],

p.

213)

hv(z) \

0

0

d8V(z) c8V(z)

c&v(z) d8V(z)

0

0

0

a8V(z)

\ b8V(z)

to

The R-

0

J

with

eo(z)Ô0(2V) 90(z-2rl)9o(Oy 0i(z)9o(2ri) e1(z-27])e0(0), e0(z)e1(2r]) 01(3-277)00(0)' 0l(*)0i(2»7) 90(z -2t7)0o(0)"

a8V(z)

b8V(z) c8V(z) d8v(z) It is

an

element of

the standard tensor

e[l]

End

product

e[-l], (0, 0, 1, 0)T

[27]

was

variables of Baxter's

Baxter's

weights by

This model

(cf.

can

Definition

Before

we

matrix

as

where

e[-l] 9q(z)

®

—

ipA Ai>a equal to A. by hv& =

=

tl,

I > A + 1

we

stay in it.

0.

is obvious

here,

since v\ is

product of

e

M

Verma modules

n

of highest weight

EILi î

d2

d

.

(7)

-V^+VïT.

-*

dt2

dt

fw

pW

->

*,,

h®

-+

-2U-1- + K..

(8) (9)

u,tl

1,...

on

,n

,in]/(Eî^+1C[ti,.--

C[*i,...

obey

(iî°) For Y^=1 /i«.

Definition 2.5 erator

=

>*n])-

The generators

for

6l3hW, ^2e«, -^2/W.

=

i

=

=

=

the Verma module

®"=1Va,,

we

(10) (11) (12) may

define

(V[0]) Let ®=1Vai be the Verma module defined V[0] cC[tu... ,t„]/(E"=1^+1C[*i,... ,*„]) as

Definition 2.6

the space

eW,/«,/iW

the commutation relations

[e«,/W] [/jW,eü)] [fc« /Ü)] H°

~

the operators

%

acting

a

=

given by

is

=

to

e

also has to

The tensor

2.4 —

or

an

to check is that

only thing

etA+1

Proposition denoted by V

is

V[0]

=

{f(tU... ,tn)EC[tU...

of h^

—2tl-^-

=

+

A%,

above.

following

Then

we

op¬

define

,in]/(Er=l^+1C[il,--- ,*n])|

H°f(t1,...,tn) Due to the action

the

=

0}.

the space is

equivalently

described

by

n

V[0]

=

{/(ti,

.

.

.

,tn)

E

C[h,

.

.

.

,in]/(]T ifl+1C[ii,

•

•

•

,tn}) I

1=1

f(cti,...ctl,... ,ctn) i.e.

it consists

of complex polynomials

t/ie variables t\,... ,tn.

that

are

=cs"=i"2i7(ii,... ,tn)}, homogeneous of degree

n

m

=

X)[=i

A,

in

24

The

2.3

to the


Gaudin Hamiltonian

elliptic

Synopsis: Here,

we

first define the needed

functions which

elliptic

basic for this

are

chapter

and

chapter group ET^(sh) (Definition 2.7). With the help down the write of these functions, we operators ee(z), fe(z),he(z) and its commutation relations (Definition 2.8 and Proposition 2.9) which are the ones of a generalized elliptic r-matrix algebra (cf. Defintition 2.10). We will need these operators in the following also the

on

the

elliptic quantum

section to formulate the Gaudin Definition 2.7

a)

eigenvalue problem.

(Basic notions)

C, Im(r) > 0. If we define the C/T correspondingly defined by ET

Let

t

E

lattice T

tZ, the elliptic

curve

ET is

be the odd Jacobi Theta

func¬

Z +

=

=

.

b)

Let

9(z)

=

9(z,t)

£neZe(n+^2Te2(n+2)(*+D

=

tion.

Its

transformation properties

9(z We also need two other

transforming

+

1)

1,...

to be

functions

+

r)= e~2mz9(z).

defined by

9'(z

+

1)

9'(z)

9'(z

+

9(z

+

l)

9(z)' 9(z

+

p(z)

—

(-jfQ)' p(z

folllows,

-9(z), 9(z

=

of

means

Theta

functions:

-jn^r

like

and its derivative

In what

given by

are

we

which

l)

+

=

always consider the

9'(z) 9(z)

r)

t)

transforms

p(z

T)

+

as

p(z).

=

product

tensor

.

*"'

V

=

®"=1Vaj5 A,

g

N,

i

=

,n.

Definition 2.8 Let the

(z\,... ,zn)

G

parameter and

Cn z

—

E C

n

points (zx,... ,zn)

diag ,i a

=

complex

1,...

,n

on

coordinate.

i=i

G

the

(ET)n

diag be the projections of elliptic curve ET. Let X E C be some —

Let

9(X)9(z

A 9(X-z

+

-

zt

Zl)9'(0)

__

1=1

9(X)B(Z

-

(l)

Zy)

(14)

n

i=i

Remark: We may define

q(x)6U-z )

~

a\{z

~

zt)

Note that

a$z)

has the transformation prop¬

erties:

ax(z

+

$=ax(z), ax(z

+

r)

=

e2mX.

25

Proposition if

z

^

2.9

The operators

above

defined

obey the following commutation relations:

w

[eeW'/eH]

9%nzW-t)°]){~he{z) +he{w))

=

^

d_e(x + z-w)m W 0(A)0(z-u;) r,

/

n

/

0(A-Z

m

„,9'(z

w"-«-)i

If

z

=

w

,

the relations

are

—

+

w)

d

(t)

.

.

(1 '

'

jj^

lijW(O)

+

,

,_„,

.

.

+/(w^T^

-

(18)

given by n

[ee(z),fe(z)\

-tie(z)-p(X)^^\

=

(20)

i=i

[M*),ee(*)]

=

[he(z),fe(z)]

=

2(^ -^)ee(z), 2f'e(z)+2(e-^ ^)fe(z). -2e'e(z)

(21)

+

+

(22)

+

Proof:

proof is straightforward and uses the fact that two theta functions are equal if their residues, zeroes and transformation properties under z—y-z + l,z—y-z + \ coincide. The

Remark: Note that

by defining

He(z) we

=

He(z, X)

=

X-he(z)

-

(23)

~

,

in order to deal may rewrite the above commutation relations

the Bethe ansatz

(cf.

the

Appendix).

[ee(z)'/e(")]

This

^9(X + z-W)9(M

u

w)

u

6{z-w)B{\)

=

U

\

0

°......

...

e\z

given by

r-matrix is

°

n —

elliptic

P-(z-w)

u

r(z

The

9(z-w)B(X)

e

\z

ei(z-w) J

0

0

Remark: Note that can

quantum case, Proposition 4.3,

be defined that satisfies the modified classical

the structure of the

The

2.4

elliptic

elliptic

r-matrix is also described in

Gaudin

classical

a

elliptic r-matrix

Yang-Baxter relation, more

cf.

[28]. There,

detail.

eigenvalue problem

Synopsis: First,

define the Gaudin Hamiltonians

we

Consider

attach

a

n atoms on the elliptic representation of «/2(C),

sponding These

to its

spin. The ith commute

operators

finding

ET

C/T, each at a site z%. To each atom we V\t, with highest weight Aj G N corre¬

=

Verma module

atom is

interacting

(Proposition 2.12).

eigenvectors

common

Out of these

curve

a

in Definition 2.11.

HI

Hamiltonians,

as

we

noted below then

develop

also commutes with the Hamiltonians

with the other

sition 2.17.

Se(z)

Proposition the

next

section,

next

section).

-"e

-

n

d\

+

2^,^=1

2.12

The

1^2

d{zl-zJ)

n

are

called the

proof of

Remark:

this

n

(cf.

its

imposing

ee(z), fe(z), he(z)

operators

+

9(A)

in

a

Propo¬ in the

the first lines of the

e

J

[HI,Hi]

=

0, for all 1,j

[Hl,H°]

=

0.

proposition is straightforward.

,9r,

(27)

elliptic Gaudin

elliptic Gaudin Hamiltonians

Proof: The

thus

9(\+z%-z,)9'(0) f(i)Ji)\ e(\)9&-z,) r'eKJ>)

Hamiltonians thus obtained

Proposition H°.

V[0],

the space

study

(HI)

I +

n

it is sensible to

perform the separation of variables

since this method is formulated for these

Definition 2.11

The

to

of

problem

operator Se(z) (Definition 2.15) which

(Lemma 2.16). Thus,

us

the Hamiltonian

2.12.

in terms of the operators

This reformulation allows

by

Thus it is feasible to treat the

eigenvalue problem. Here, we restrict ourselves onto condition on possible eigenvalues (Corollary 2.14). We reformulate the operator

ones

=

Hamiltonians.

commute with each other and with

l,...

,n,

(28)

(29)

27

elliptic Gaudin

The fact that the

of

commute allows for their simultaneous

Hamiltonian is

diagonalization,

of every Hamiltonian.

eigenvector eigenvector problem of how to find eigenvalues corresponding to possible energy and eigenvectors of the Hamiltonians, hence levels of the atoms on the elliptic curve of the equations possible solutions (p,t, ip) since

We

an

now

one

an

turn to the

-

-

H\\p(X,ti,... ,tn) To find

a

complete

set of

=

plip(X,ti,... ,tn),

eigenvectors

for

certain

a

for i

given

=

1,...

(30)

,7i.

(//i,... ßn)

set

,

will be

our

aim

chapter.

in this

// restricted

Lemma 2.13

V[0],

the space

on

n

Y,Hl^(X,h,...,tn)

(31)

0.

=

i=i

Proof: This is done

by

2.14

Corollary

straightforward

a

While

working

calculation.

on

V[0], n

£>

(32)

0.

=

2=1

Remark: Note that since

V>(A, t\,... tn)

made the restriction of

to be

formation behaviour of the operators

Since the

elliptic

stead of them

E

z

ET be n

se(z) is

on

,

dependence

The

can

=

a

be

ni

coordinate

(

on

A is dictated

by

simultaneously diagonalized,

on

the

elliptic

the trans¬

we

may in¬

curve

ET.

n

\ _

+ £ *>(* Y, Hi7jjhri a^z z%> ,=i

H£

=

investigate the following operator.

Definition 2.15 Let

where

m

with respect to this variable.

Hle

Gaudin Hamiltonians

V[0], possible eigenfunctions X)I=i ~t m the variables (t\,... tn)

working

of degree

homogeneous polynomial in the variables (t\,... ,tn). ,

and

we

are

~

*)cW

+

^

^33)

,=i

given by

m=it+m>=i{kW^yh{i)hij) Remark:

eigenvalue problem now reads as follows: We want to determine eigenvectors ip(X,ti,... ,tn) to a given set (p,c,\i\,... ,/j,n) of The

Se(z)iJ)

=

Qe(z)

=

qe(z)Tp, A

where \xc

corresponds

complete

9'(z-z%) ^

to the value of

+

A,(A,

+

2)

7

P(z-*))+f*c

%>

H£

set of

(34)

with

1^^0(z_z) i=i

a

on

ip(X, t\,... tn). ,

(35)

28

a) [Se(z),Se(w)} b) [Se(z),H°}

[Se{z),&]

c)

=

0,

0,

=

=

identities holde true

following

The

Lemma 2.16

Q

,tn)

d) jT/i/j(A,ti,

solves the


7/;(A,ti,

,tn)

=

ecXf(h,

,

it

can

be written

(36)

,tn)

Proof:

a)

proof

The

b) H°

and

given

is

H\

m

commute

[26] by Lemma

the commutativity of H° and

c)

The calculation

d)

This

is

is

2

H%

12, H° and cW by

a

short calculation

Proving

straightforward

is

straightforward

Since the operators Se (z) and ^ commute, they can be to the eigenvalue c G C simultaneously The eigenfunctions of the

corollary of c)

a

digonahzed correspond

-^

ecA

to

Remark: Due to the first part of Lemma 2 16

the variable In order to solve the

z

the operators and relations

eigenvalue problem,

Proposition

are

indeed

independent

of

ET

E

use

possible eigenvectors ip

we

need the

developed

in

the first part of this

chapter

to

following

2.17

1,

Se(z)

=

,

,

.

^(ee(z)fe(z)

+

fe(z)ee(z))

( d

+

x2

1.

(^

-

^he(z) j

(37)

Proof:

Se(z) z

is a

Expanding Se(z)

Zj

=

constant

This

by

term,

we

see

into

a

Laurent

series

at

z

=

z% for all

that the difference of the left and the

due to the fact that the difference

is

regular elliptic

are

at most double

meromorphic doubly periodic function with

functions

vanishing

yields

a

at least at

i

=

right

poles

at the

1,

,n up to the

points

hand side vanishes

differential operator whose coefficients

point, thus vanishing everywhere

one

Liouville's Theorem

2.5

Separation

of variables for the

elliptic

Gaudin Hamiltonian

Synopsis: Here,

we

first write down how to obtain out of the

used for the tensor 2

product of Verma modules of s/2,

separated variables the 1 e

the "old" variables

variables

we

(Proposition

18)

To show that this transformation of variables

is

indeed

useful,

we

reformulate the operator

29

Se(z)

(Proposition 2.23) and especially this operator evaluated (z\,... zn). By this evaluation, we obtain the of separated equations (Proposition 2.21, Definition 2.22) which all show the same system structure of a second order differential equation solvable by Lamé's method (cf. [51]). in the

at the

separated

points

n

-

variables

sites of the atoms

-

,

That the solution of this system of differential equations is

eigenvalue problem

Se(z)

for

The main idea of this

/e(z)

while restricted

paragraph relies

-

(C,y±,... ,yn)

Proposition

2.18

are

called

(z\,... ,zn)

Let

U

=

the

C

_

"

•

Iff=1fl(*-y,) U:=iO(z~zi)

(38)

•

separated variables.

(ET)n

E

is shown in

solving Proposition 2.24. to

following identity (cf. [16], [44]):

Afff^HMO) *> ^ 9(X)9(z-Zl)

The variables

V[0]

on

the

on

equivalent

diag

—

The

.

mapping

Il%8{zi-yj)

CU^=iO(zï-zJ)9'(0)

(39)

'

n

\

-Y,(zt-yt)

=

(40)

i=i

defines

bisection

a

between

,tn,X)\Y^tt^0}

{(h,...

and

i=i

{(yi,...,yn,C)ESn(ET)xC/0}. Proof: Note that the z —y z

+

t

for A is obtained

identity

by looking

^ yields e~2m^=^~y'+z') 2.19

.

The y3 for

Note the

j

dVj are

obtained

=

K=i0(z a

1,...

,n

=

the

transformation:

Ê;=i(-z,+yj))uu9^)

%j

hand side

'

(42)

~z*)

the map

right

Sn(ET).

in

i=i

by studying

zy)

are

£

1=1

d

1

Thus,

dX

+

A%,-2,)

d2

z{)

U\

dy3

"

1=1 —

a„.

à

^

"9>(y3-zy)

d

_

.,2

4

ff (Z

-

e(z

-

-

zy)9'(z

-

zi)

0(z-zy)9(z-zl)

ZAff (Z

zy)9(z

-

-

Zl)

Q

Ô

lldtydtt

Zl)

i=i

=

ai?+g-(*-«)*•«:-E^fcoää A A,A; fl'fo aQg'faj Zi) 4 %,-*)%,-*!) -

-

^

9y;

^

.

9'(y1—zl)

%j

"

d

zi) dVo

v^

t=ï

.

.

A. 2

=

31

A A,A; 9'(y3 Zy)9'(y3 zj) 1 +2M%j4 0(j/3-*)%,-*) -

+

yields

This

-

,

^

the desired result.

(Separated

Definition 2.22

obtained all show the

same

differential

structure and

(d

equations) the

define

zt)K\2

^9>(t-

Proposition

,.,

M ««(*)«(*)•

=

2.23

wj-l,=ic

Se(y3) for j

^

r*

+

^=in;=1«(Z-Z))x

{SM

UU,^ &)

x

where

M «(*)

\jrhiW^zi)-2)

thus

differential operators separated differential equation Then

,n is calculated in

1,...

=

ö(E3n=12/3-Er=i^)

£3U cW*>(ifc

-

-

**))

(46)

>

Proposition 2.21.

Proof: n

First,

we

Se (z)

notice that

=

Se(z)

—

2_^c-%'p(z

—

z%)

only simple poles

has

the

on

i=i

fundamental domain F

=

{x

+

yr\

x,y G

[0,1)}

at the

points

z

=

zt,i

=

1,...

,n.

Thus,

n

S®(z) JT 0(2

—

Zj)

is

elliptic polynomial

an

in

Gn(e^>=i^) (cf.

Appendix B)

which

can

i=i

be calculated to

the

by interpolating

known values of it at the

n

points

z

=

y3, j

=

1,...

,n

yield

which

is in turn useful to calculate

expression

S^(z)

and

Se(z). Se(z)

as

defined in

n

Definition 2.15 transforms

doubly periodic

on

V

-

due to

"S^H\

—

0

-

and

so

does its

i=i

expression

in this

Proposition.

points

y3, j

1,...

z

=

=

Both

expressions

also coincide at the residues and the

n

,n.

Remark: We will need this

Let

us now

expression

proceed

in

in the

formulating

tions in the terms of the

new

sequel, cf. Chapter 4. a

proposition

on

the strucuture of

possible eigenfunc¬

(C, y\,... yn).

variables

,

function if)(X,t\,... ,tn), homogeneous of degree m YH=i if *n the solution is a of qe(z)ip partial differential equation Se(z)tft (t\,... ,tn), C if and only if

Proposition

2.24

A

=

the variables

for

z

E

=

V>(A,*!,... ,tn)

=

Cmu(yi,... ,yn)

(47)

and

-Q-

-J2 '2LJ^~Zk">j

u^---yn)

=

1e(y3)u(yi,...yn)

(48)

32

for

j

=

1,...

Hence, u(yu

,n. ...

yn)

,

=

117=1 v(yt).

Proof: Let

fact that

[Se(z),C-J^j]

erators.

The

ther note that

0,

—

we

Cmu(yi,... ,yn). By using the simultaneously diagonalize both op¬

ip(X,ti,... ,tn)

first describe how to obtain

us

see

that

we

can

=

eigenfunction of C-^q to the eigenvalue a by the homogeneity property of ip and the

C is

E

given by Ca.

Fur¬

(ti,... ,tn) Cfn(yi,... yn))

definition of

in

,yn) we get V(XT=i(?/« *i), Cfi(yi,... yn), C?mV'(Er=i(^ -Zy),fi(yx,... ,yn),... ,fn(yx,... ,yn)). That it suffices for a function u(yi,... ,yn) in order to be a solution of Se(z)Cmu 1,... ,n is seen by the fact qe(z)Cmu to be a solution of Se(y3)u qe(y3)u for j that Yl-i(Se(z) qe(z))u(yi,... ,yn) E ®n(x)> cf. tne Appendix, vanishing at n generic for 1,... n, thus vanishing everywhere. That u(yi,... yn) points y3 j ]X=i v(Vi) (yu

terms of

-

•

•

,

...

,

=

,

=

=

—

—

=

=

,

,

can

be

j

1,...

=

seen

by

separated differential equation corresponding differential operators depend only on

the structure of the

,n the

one

yt.

For all

variable y3.


Solutions of the

2.6

at y

=

Synopsis: Here,

iß E V[0] separated equations.

solutions of the system of We

(Se(z)

—

qe(z))ip

=

0

by studying

non-degenerate (Proposition 2.26) and a degenrate case (Proposition 2.27), degenerate means that poles of the separated equations can be zeroes of the solu¬

study

where

of

show how to obtain solutions

we

a

tions of these

Proposition

equations.

2.28 shows how to construct out of the solutions of the

solutions of the

operator

fe(z)

eigenvalue problem

solutions which

we

Se(z)

that

can

separated equations

also be formulated in terms of the

ingredient in introducing separation of variables. this section, we will explain how to understand completeness of the gave in Proposition 2.28.

which

In the last part of

of

was

the main

The structure of the solutions

2.6.1

Remark:

Here,

we

will look at

Proposition First, 2.22

note that the critical

are

In the

possible

solutions to the

elliptic eigenvalue problem

as

obtained

by

2.24.

given by

following

0 and

Aj

exponents of the separated differential equation of Proposition + 1 at every z% for all i

of solutions to the

we

=

1,...

,

n.

will indicate how to write down two different kinds

propositions, separated equations by Bethe

two

ansatz. Then

we

will show how to obtain

out of these solutions solutions to

Se(z)ip(X,t1,... ,tm)

=

qe(z)ip(X,t1,... ,tm).

The last theorem will be Definition 2.25

on

the

completeness of the Bethe solutions.

(Bethe solution)

A solution

ip(X,ti,... ,tn)

to

Se(z)ip(X, ti,... ,tn)= qe(z)xb(X, ti,... tm) ,

33

is called

Bethe solution

a

if

the

of

it is

form in

iP(X,tu... ,tn)

ecXHfe(wy)vI,

=

(49)

«=i

where vj is

e^

YÜ=i

Remark

element

singular

a

The value

of

of the

Verma

module,

EN must be chosen

m

to

as

i.e.

ensure

it vanishes

that

by

the action

ip(X,ti,... ,tn)

E

of

V[0].

:

We start with the ansatz which Hermite used to solve Lamé's differential

(cf. [51]). such

a

with

i.e.

function

v(y)

function

a

be written

can

®m(x)>

E

where

m

—

E"=i ^-i- ^y

the

equation Appendix,

as

m

v(y)

ecyj\9(y-wk).

=

i=i

Proposition all i

=

1,...

v(y)

Let

2.26

,n and k

@m(x) for

E

1,...

—

This

,

m.

x G F*

some

function

is

&e

given such that

solution to the

a

zt ^ Wk for elliptic Schrödinger

equation

^-Ëy^y-^)j if

and

only if

its

parameters n

^2/Al

Qi —

all k

for

Wk

=

1,... m

=

qe(y)

obey

,777,

the Bethe Ansatz

equations

9'

^2 j(wk-w3)

(wk-zl)-2

i=i

v(y)

(50)

=2c.

j=i,j^k

Proof:

Note first that Wk

^ wi for all k / I for k, I 1,... m, since the only solution of the dif¬ with its derivative at a regular point is the trivial solution. equation vanishing write down the first and second derivative of v(y), they read v'(y) cv(y) + =

,

ferential If

we

=

£ö'(y v"(y) c2v(y) + 2cZT=lêj(y k)v(y) + w3)UT=i,k^0(y wk) Eti nv-Vk) UT=i,^k %->C2/+EÎT=i ET=i,^k 9'(y-wk)9(y-Wj) Ui=i,i^ 8{V-

wi)ecy.

and

-

Evaluated at

a zero

Wk,

=

obtain

we

m

v"(wk)

=

~

2cv'(wk)

+ 2

y

^T jiwk- w3)v'(wk).

j=l,jj^k

If

we

instead look at the

X7=i ^î0-(y k

=

1,...

—

Zy)v'(y)

separated differential equation,

+

r(y)v(y),

where

,777,. Evaluated at Wk, this

is

a

notice that it

yields v" (y)

=

function at the Wk for all

regular

expression yields n

v"(wk)

r(y)

we

=

9'

Ât —(TJJfc

-Zy)v'(wk).

1=1

Since k

=

v'(wk) ^

1,...

,777

0 for all k

yields the

—

1,...

,

m

Bethe Ansatz

the

comparison of the

equations.

two identities for

v"(wk)

for

34

That this is indeed the

proof, we perceive by the following two arguments: A solution differential equation in &m(x) obeys the Bethe Ansatz equation by construction.

of the

(w\,...

On the other hand any set of parameters ecy

IlfcLi &(y

checked

~

wk)

6m(x)i

G

by comparing

Proposition

which in turn of

zeroes

v(y)

obeys

c) gives

rise to

the differential

a

function

v(y)

equation. This

=

be

can

and residues.

{1,... ,77}

2.27 Let I Ç

,wn,

be given. Let wk

zx

=

for

all

1

E I

and wk

7^

zt

if

i

%

2 otherwise

Proof:

First,

we

may show that

v(y)

—

Y[ieI9(y

exponents of the differential equation

v(y)

—

were

1,...

,77, where

By

straightforward calculation, v(y) obeys

a

is

n

or,

by setting Ay

=

a

z%)~~

'~1v(y)

E

solution to the above differential

Aj

I and

Az

=

—Az

—

2 if

1

v(y)

E

=

I,

it

=

equation

obeys

qe(y)v(y)-

=

-

1

equation.

Starting with this differential equation, we get by writing v(y) liptic polynomial solution to the differential equation we started =

the characteristic

2

\'d^~^~2~ë^v~Zlkn

v(y) eC2/II=i% ®m(x) which vanishes

as

+ 1 at z% for all

the alterated differential

V

At ifi ^

©m(x)>

shown to be 0 and

wt)U3ei0(y

~

^)'K+l

=

eCy

I\?=i 0(V

at the z% for i G I up to order

Ax

+ 1.

~

ecy

111=1 &(y

—

Wy)

an

el¬

with. This solution reads

Wy)\[3eI9(y

-

zt)^1

E

35

v(y)

Since

ni=i Q(y

ecy

=

~

wi)

indicated

obeys the alterated differential equation

preceding proposition obey. They read

with the to

71

-

m'

may

we

-

by

,

the

wmi,

same

c)

and

$'

=

as

way

equations these parameters

-Zy) -2^— (wk -w3)

^2K-ßiwk

are

2c

3=1

1=1

for k

above,

obtain the Bethe Ansatz

9'

„

(w\,...

contains the unknown parameters

1,... ,777'.

=

Remark: Let

found in

Proposition

Proposition

2.28

which

obtained

are

by

product of solutions

a

we

2.26 and 2.27.

Let vq

defined

be the operator

a)

Se(z)

look at the solution of

us now

1 be the

=

at the

highest

the Verma module and let

of

vector

fe(z)

beginning.

first kind of solutions with wk 7^ obtained by Proposition 2.26 yields The

for all

zz

i

=

1,...

,77

and k

=

1,...

,777

m

ip(X,h,... ,tm) =a(zi,...

,uim,c)ecXY\_fe(wj)v0

,zn,wi,...

3=1 as

b)

a

of Se(z)ip(X,ti,... ,tm)

Bethe solution

qe(z)tp(X, t\,... ,tm).

=

zt for alii E I for of solutions with wk Bethe solution the 2.27 yields Proposition by

The second kind

obtained

=

V>(A,*i,... ,tm) a(zu...

some

fixed

I

Ç

{1,... ,77}

=

,wm,c)eÛ']iife(^)Uka(fik))Ak+lvo.

,zn,Wl,...

Proof: For the

proof,

cU%9-zi)Vl)

>

we

need the

fel)

=

Res

following

facts

z=M*)

given throughout the preceding section: fe(z)

and A

£?=i(y*

=

"

*.) ßy Proposition 2.15,

where

ip(X, t\,... ,tn) could be written in terms of the new variables v(yt) solved the corresponding differential equation Se(yl)v(yl)

Let

show how to obtain with these results the first

function

us

The second

one

is then obtained

identity

as =

=

the

CmW^_1v(yl),

qe(yi)v(yi)-

written in the

proposition.

similarly. I

m

n

Cml[v(yy)

=

i=i

i=i n

(_irnecEêc(Er=1(^)) JT

m

Cmll [e^l[9(yy \

3=1

9(w3 -yj

CHI PI yi,.9(w3

3=1 \

6(W3 z=l

Zl>

-Zy))

=

J

m

a(zi,...

,zn,w1,...

,wm,c)ecXY\_fe(w3)n

=iß(X,t1,... ,tn).

3=1

Remark: Note that

module,

in

only our

Ylkei(f^)Ak+lvo realisation

with 1

=

0 has

a

nontrivial

projection

C[*i,... tn]/(J2Z=i tfl+1C[ti,... tn]). ,

,

on

the Verma

36

Completeness

2.6.2

Let

us

first

give the

of the Bethe

eigenvectors

necessary definitions to understand the theorem and then write down

the theorem.

(H(x))

Definition 2.29

Let x G T*

6e

given.

Then

1-i(x)

is the

following

space

of

functions

H(x) 4>(X

+

{:

=

A

l) =X(1^(A), 0(A

V[0] | 4> meromorphic

E

A,

=x(r)e-^=1^w0(A)}.

r)

+

in

Remark:

Since the operators

HI

for i

i

1,...

=

=

1,...

n.

,

,n which

E(x)

Thus,

{(/Jc, Ml,

=

•

•

,Pn)

•

Then

(X) for all do

%(x)-

G

given. Then

Cn+1 I there exists

G

H14>(X)

Theorem 2.31 Let x G T* be

a(r)

\he(z)

Let x G T* be

with

=

—

it is sensible to look for solutions of

of the form

are

(£(x))

Definition 2.30

^

ee(z),fe(z),

/j^(A), Hce(X) Let

nontrivial

a

=

G

H.(x)

ßcm1...mnÇïuyl=i(yi~zi))

this is the

only possible pole for the of the differential equation Hze4>(X) p,y4>(X) as

=

This is proven

by the following argument: As v(y%,... ,yn) arated equations for i 1,... ,77, it may only have poles that occur i.e. the at tions, points z% for i 1,... ,77. For a generic choice of Er=i Vi. E^fc,j=i z3 Ï zk, thus avoiding a pole at £)"=1(yt z%) 0. =

=

~

-

=

solves the sep¬ in these equa¬

the y\,... ,yn,

37

The second

v(yi,...

y% ->

Cm

hypothesis ,yt,...

,

'arMan+l

=

—6n+i

—

instead of

solution to

by

-«l

a

the

b\

=

common

6n+ieigen¬

pair

>-,£SOs(z))

,a,n

which solves

TSOS,e{z)Yâi, eSOs(z)(J2au

,anao.i,

,an

K,

anaai,

,an

[al>

•

•

•

•

,

.

an+l

an+l

=

=

-«1 > ~0-\

=

>)

formula J2ai 0,1 > indicates a linear combination an+i anaai, ,aJai;attachments hence of heights to faces, each attachment with the of antiperiodic paths, antiperiodic boundary conditions preserved. In the sequel, we restrict the eigenvalues we This makes sense since are looking for to be elliptic polynomials (cf. Appendix 2). The

=

•

•

—

j

....

Remark:

Why

it is sensible to

study

thesis. To further stress its

this

problem

importance,

we

was

will

emphasized

give

a

in the introduction of the

heuristic definition of the

function in terms of the above defined transfer matrix

partition

42

Definition 4.4

(Partition function)

of

transfer

terms

its row-to-row

matrix

ZsosMz) where the trace is taken

The

is in

given by tr

=

{TlhsM)

,

of antiperiodic paths where which

the space

over

partition function of the SOS model

row-to-row

the partition

following: intepretation transfer matrix is an endomorphism of. function Zsos,N describes the sum over all possible attachments of heights to faces, i.e. over all antiperiodic paths, of (normalized) probabilities of the following events: we start with a given attachment of heights to the n faces (the antiperiodic boundary conditions understood) and after M row-to-row transitions we are to return to the same attachment The

of heights

which

faces

to

we

is the

started with.

Remark: This

be visualized

can

by the following picture for the simplest

W3

case ?7

=

2 and M

•

=

a,b

-

attachements of

heights

2.

W4

By the partition function Zsos,2 we would obtain a sum of all products of n i.e. 02 Boltzmann weights Wl W2 W3 W4 depending on allowed o\ 0

=

=

-

x

M

=

4

± 1 with

to faces.

eigenvectors of the family of antiperi¬ odic SOS transfer matrices enables us to compute the above partition function of the model as well as other physical interesting quantities as for example the magnetization. possession

Being

in

4.2

The

of the

common

eigenvalues


and

to the SOS

eight-vertex

model

Synopsis: In this

section,

ET,n(sh)

which

a

of

we

define

notation

(Definition 4.5)

The latter will be needed

gives

us

group

the basic structure of this quantum group. The R-matrix

representations of ET>ri(sl2) (Definition 4.8), as it defines representation has to obey, the RLL-relations. Then, we give some exam¬ as

for the

representations (Proposition 4.10). The examples

irreducible

some

%—module

a

which

important

relations every

ples

representation theoretical notions concerning

sequel. First,

diagonalizable (Definition 4.5). representation. Then we introduce the R-matrix of the elliptic quantum

(Definition 4.6), is very

will need in the

we

and the notion of to define

will present the basic

we

representations,

because these

are

the

ones

are

that

mostly finite can

dimensional

be used to construct

analogues of) the eight-vertex SOS model. The construction of the representation corresponding to the eight-vertex SOS model also heavily relies on the fact that we can build shifted tensor products of repre¬ representations corresponding

sentations of

ET^(sl2)

to

to obtain

(higher

new

dimensional

representations of

ET^(sl2) (Proposition 4.9).

of the notion of

a representation: the func¬ slight generalization tional representation and its operator algebra (Definition 4.12 and Definition 4.14). For a functional representation the diagonalizable 'H-module is replaced by a suitable space of

We then continue with

a

functions. We introduce the notion of the quantum determinant

(Definition 4.15)

which

43

sequel

will be needed in the

(cf. introduction)

of

a

as we can

any of the four entries of the

replace of

given representation

equivalently describe this representation. We proceed by introducing the notion of

ETjV(sl2) by

L-operator quantum determinant to

the

highest weight representation of Er^(sl2) (Definition 4.16). We discuss this notion for the following reason: we want to show that the representation of Er^(sl2) to be attached to the eight-vertex SOS model is isomorphic to the auxiliary representation. We then state a theorem on the shifted tensor product a

highest weight representations (Proposition 4.18). Fi¬ [25] stating that finite-dimensional irreducible highest weight nally, give representations of ET^(sl2) are isomorphic if their highest weights coincide (Proposition of finite-dimensional irreducible Theorem

a

we

4.19). Introduction

4.2.1

Remark: to the

appearing in this chapter correspond the differential elliptic Gaudin model. The functions

Let %

Vy,

i

Vy

is

=

,77 be modules

called

a

eigenspaces Vy[p] of We can for example take V a

on

V[— 1]

and

C}

E

are =

=

one

generator h. Let

%.

over

is the direct

diagonalizable H-module ifVy h which

{ae[l] |

algebra generated by

Ch be the one-dimensional Lie

1,..

=

chapter

(Basic notions)

Definition 4.5

a)

defined in the

ones

labeled

G

a

of finite of h: Vr

dimensional

eigenvalues p, ®^Vy[p]. disjoint subspaces V[l] C} by identifying

by C2 and split

{a:e[— 1] |

sum

the

=

it into two

=

"=(J _1).«[1] (ï).«»«'«[-1i (î)=

=

b)

Let

Vy,i

,n be

1,...

=

product Vi

®

...

®

diagonalizable

Vn. For

X E

We may consider their tensor

'H-modules.

End(Vy)

we

denote

by X^>

E

End(Vi

®

...®Vn)

the operator

iW

=

X

1®...® tth

If X

c)

Let

v

E

E

Let X

h^v

d)

=

®

V3),

Vi®.. .®Vn =

X(h^\

ptv

for

.

...

all i

we

define X^

We may ,

—

End(V\

E

hSn>) 1,...

,n,

a

be

®

...®Vn) analogously.

End(V\®.. -®Vn) by the above notation. ® Vn). If function taking values in End(Vi ® then X(h^,... ,h^)v X(p,\,... ,pn)v

define hSl>

be

place

E

...

=

.

diagonalizable ri-module and I (V®J). Then we can define A^^+l-n)

Let V

End

End(Vy

®...®1.

a

A(n-j+i...n)

=

identity

the E

End

l8_gI '

s v

first

j copies

of V

matrix

(V®n) by 0 Am

on

it.

Let A G

44

(R-matrix)

Definition 4.6

Let V

V[— 1]©V[1]

—

be

a

two-dimensional

complex

vector

space.

Let the

elliptic R-matrix Re

( 9(z

E

+

End(V

2n)

u = n

U

\

0

the z, X G C be

on

0

9(z)

n

u

(51)

n

U

9(z

+

J

2n)

=

=

=

,

,

and

defined by

\

0

e{z-X)9(2r]) 0(A) 6(z)0(\-2ri) 0(A) 0

B(z, r) with the two parameters r, n E C, Im(r) > 0. (0 1 0 0)T e[-l] (10 0 0)T e[l] ® e[-l] identified e[l] ® e[l] ® (0001)T. e[-l] e[-l]

where

We

0

6(z)6(\+2V) 0(A) 9(X+z)9(2V) 0(A) 0

n

Re(z,X)

V) depending

®

®

e[l]

=

(0010)T

=

Proposition 4.7 (QDYBE) Baxter equation

The

elliptic

R-matrix

obeys

the

dynamical quantum Yang-

2nh^)RW (z, X)RW (w,X- 2nh^) RW(w, X)RW(z, X 2nhW)RW(z -w,X),

Ä(12) (z-WjX=

where the notation is

4.2.2

(52)

-

as

defined above.

Representations,

functional

This relation is

defined

on

End(V®3).

representations, operator algebras

Remark: We

now

looking

at

representations of

the

elliptic quantum

group

done in two different ways. The first definition will deal with the second

one

will be

a

ETjV(sl2).

This will be

diagonalizable K-modules,

slight generalization.

(Representation [25]) A representation of the elliptic quantum group ET^(sl2) is a pair (W,Le), where W is a diagonalizable H-module W ®fj,ecW[p] and h^ h^ is linear with a E ® + Le meromorphic End(V Le(z,X) W) map commuting Definition 4.8

=

=

in z, X E C called the

The

L-operator.

L-operator obeys

the relation

Ri12) (z-w,X- 2t7/7)41) (z, A)42) K A 2nh^ ) L® (w, A)41} (z, X 2nh^)R^ (z -w,X). -

=

This relation is called the

-

dynamical RLL-relation.

Remark: The

L-operator

is

usually

written in the form

(53)

45

ae(z, X),be(z, A), ce(z, A), de(z, X)

where the

dynamical ßLL-relation,

which

ae(z, X)ae(w, 9(z B(z

w

—

w

—

+

+

2n)ae(z, X)be(w, 2n)be(z, X)ae(w,

9(z

2n)ce(w, X)ae(z,

End(TV)

explicitly yields

ß(z

a(z

—

ß(z

w, A

ae(w,X)ae(z, X-2rj),

X

=

be(w,X)ae(z,X

+

+ae(w, X)be(z,

X

—

X +

X

2n

2rj

=

be(w, X)ae(z,

+ae(w, X)be(z,

X +

=

be(w,X)be(z,X

2n

=

ß(z

—

—

ß(z

=

9(z-w 9(z

w

a(z

=

a(z

a(z

=

ß(z

—

9(z-w

w

9(z

w

tic

A

new

(Wi

®

a(z

X + 2n

=

ß(z

2n)de(w, X)be(z, 2n)ae(w, X)ce(z,

X

—

2n

=

a(z

A)ce(2;, X)be(w,

A

X)be(w, X)ce(z,

X +

2rj

+

ß(z

2??

+

ß(z

X)de(z, X)ae(w,

X + 2n

+

ß(z

X)be(w, X)ce(z,

X + 2n

+

a(z

2n)be(w, X)de(z,

X + 2n

—

—

w,

+

+

2n)ce(z, X)de(w,

+

2n)de(z, X)ce(w,

and

ë(Xj 4.9

group

(Shifted

—

—

X

-

—

—

—

—

a(z

—

—

—

—

—

—

L$ (z,

X

is

-

w, X —

X)ce(w, X)be(z, —

w, X

w, 2nh —

2rjh)be(z, X)ce(w,

—

—

w, 2nh

w, 2nh

w,

—

—

X)ae(z, X)de(w,

X)ae(w, X)de(z,

w, 2nh

—

w,2nh -

w, 2nh

=

=

-

w,

—X)ce(w, X)de(z,

de(w,X)de(z,X ^

following

2nh(2))L{2f (z, A)).

+

2n)

X + —

X —

277),

2n) —

277),

2rj)

+

2n)

+

2n)

X +

277),

-

2n) —

277), +

2n),

2n),

two

representations of the ellip¬

(W2,L2>e(z,X))

tensor

X

+

.

product [25]) Let and

X +

2n)

2n)

-X)ce(w, X)de(z, A

ß(z -w,X)de(w, X)ce(z, —

X +

2n)

2n),

X)be(z, X)de(w,

X)de(w, X)ce(z,

w,

—

2n),

2n),

X)de(z, X)be(w, X -

a(z

w,

—

X

X)be(z, X)ce(w, X

w,—X)ae(w,X)de(z,X —

X

X)ae(z, X)ce(w, X

2n) —

2n),

X)ce(z, X)ae(w, —

—

X +

2nh)be(z, X)de(w,

=

—

X

—

X

2nh)de(z, X)be(w, X

2n

p{z, A)

given by

w,

—

—

ce(w,X)ce(z,X-2n),

X + 2n

the

w, X

—

=

X + 2n

tensor

w, X

2?7

ETjT1(sl2) (Wi,LijË(z, A))

representation

W2,

2nh)ae(z, X)de(w,

+

a(z, X)

quantum

2nh)ae(z, X)ce(w,

X

+ 2n

X)de(w, X)ae(z, X

de(z, X)de(w,

Proposition

—

a(z

a(z

where

w, X

—

+

+ß(z —

2n),

2nh)ce(z, X)ae(w, X

—

X + 2t?

ce(z,X)ce(w,X —

—A),

2nh)de(z, X)ae(w,

—

+ß(z 9(z

w,

—

ß(z ~w,-X)ce(w,X)be(z,X-2n),

w,

w,2nh

—

2n)a(z

-t-

+

—

w,X)

—

X + 2n

+

w, 2nh

-A),

w,

X)de(w, X)ae(z,

—

+ß(z —

w,X) -

a(z

A

+a(z —

—

2n)ß(z

2n)ß(z —

—

+

w,

—

w, X

-

X +

+ 2n —

2n)a(z

2n

w,

—

w, X

—

2nh)ce(z, X)be(w,

—

the

=

+a(z —

meromorphic in z, X E C and obey following sixteen conditions:

are

X-2n

be(z,X)be(w,X +

w

—

G

product of

be

given.

the two

representations:

46

Remark:

explicit L-operator of

The

«iS,>> A) *£§,«(*'A) 41,e(^ A) A £% A

=

=

=

=

-

-

-

2nh^)afl(z, A) + b^(z, X 2rlh{2])b{2% A) + &$(*, A 2r//i(2))42(s, A) + dg(*, A 2^(2))42fe A) + d$(*, A

-

-

-

-

2^)^% A), 2nh^)dfl(z, A), 2^(2))^(z, A), 2nhW)d%(z, A).

Re(z

=

—

zq,

A)),

C,

where zq G

fundamental representation

called the

75

ET:Tj(sl2).

Let

Va &e

define

an

of

Va,L

—

by f(h)ek

complex

vector space with basis

=

=

—

2k)ek for f(h)

ek,k

Endiyjy). f(A L^,e(z Zq)) is a representation of ET>v(sl2), module VX.,e(ô) of ET •^n meromorphic —

ET>ri(sl2) ETjV(sl2).

representation of

1

in

Definition 4.12

definition,

we

may be further

first need to define

to the notion of

a

suitable space of functions.

a

(T^) xn,

A)

"

Cn+1

—y

C

I / holomorphic

in

xt

for

1

=

1,...

,77 and

f

A}. (Functional Representation [25])

of all complex valued functions meromorphic

of xi).

generalized

in

Let

X EC and

T^

be the

complex vector space holomorphic in p E C (instead

47

functional representation of ETiV(sl2) is a pair (W,Le) where W Ç T^ and L Le(z,X) is a function holomorphic in z,p EC and meromorphic in X E C: Le(z,p,X) It acts as a difference operator on V ® W, commutes with h ® 1 + 1 ® h and obeys the elliptical RLL-relations. The operator h, the weight, acts by multiplication by the continuous variable p E C : A

=

=

hv(X,p)

pv(X,p),

—

Proposition

a) (W on

l{e(z

—

zq)) defines

zq)

—

i-

i

defined

is

m

x

w

as

9(z-z0

x

>

(

M

,

w

/_

/

+

d(-X

w

(3

i

—

\

2n,p

u\

=

+

2),

\

\

\i

v(X, p)

^

6(z

pn-n)6((A-

y -

z0

-

pn +

n)9(X

=

=

—

The

A)?7)

X

-

(p

+

p)n)

^-^x

A)n)

v(X

+

2n,p),

E J-'.

—

module

+

—

{v E J7 \ v v(p), p E {A b) If we restrict !F^ to Tr the basis of an infinite v(X,A 2k) ek, ek defining zq)) space, the functional representation (Tr, L^Re(z —

n)9((p

—r

=

(dA>e(z,X,h)v)(z,X,p)

where

z-z0+pn-

9(X + zz0-

x

>

(cKe(z,x,h)v)(z,x,p)

xtj(A

+

v(X-2n,p),

2n,p-2),

7N

x

r])9(X- (p- A)t?)

—

=

(bA,e(z,X,h)v)(z,X,p)

xv(X

+ prj +

=

>

,

Vê(zo).

follows

(aA,e(z,X,h)v)(z,X,p)

r

functional representation of ET)TI(sl2) depending

a

zq G C. It is called the universal evaluation module

parameters A,

two

eW.

(Examples [25])

4.13

Ti,Lê(z

=

v(X,p)

where

—

2k\k

E

Z}}

dimensional

C

T\ and

complex

set

vector

the is the evaluation Verma

of ET^(sl2) VA>e(zo).

L-operator

LARe(z

—

action is restricted onto

zq)

looks the

same

as

the operator

defined

in

a),

but its

Tr.

Proof:

a)

The

b)

This part is done

proof mainly

consists in

checking

(cf. [25]).

the .RLL-relations

by comparison.

Remark:

Since the entries of the functional

v(p,X)

E W

L-operator ator

Ç

J7^,

we can

written down

L-operator

difference operators

algebra of the functional representation. representation of £JTj^(sZ2) can also representation,

representation

of

we see

ET^(sl2).

as


on

the elements

write them down this way. The set of entries of the functional

as

Since any tional

act

that this way

we

plus the operator h

be conceived

as a

are

suitably

also obtain the operator

called the oper¬ restricted func¬

algebra

of

a

given

48

(Operator Algebra [25])

Definition 4.14

a)

Let

suppose

us

a

âe(z, X)v(p, X)

=

ce(z,X)v(p,X)

=

functional representation (W Ç !F^,L^(z,X)) as given by ae(z, X, h)(T-2vv(p, A)), be(z, X)v(p, X) be(z, X, h)(T+2r>v(p, A)), =

ce(z,X,h)(Tfr)v(p,X)),de(z,X)v(p,X)

where every operator is

oe(z,X,h), are

o

a,b,c,d

=

of End (W), difference operators

are

in the

functions meromorphic

complex

de(z,X,h)(T+2T>v(p,X)),

hv(p,X)

and

element

an

=

pv(p,X),

=

weight holomorphic

in the

h whose

variable X and

where the

coefficients in all h and

zEC. Then its operator

algebra

is

the

algebra generated by

h,de(X,z),be(X,z),ce(X,z),de(X,z) b) If we pi,

have

..

on a

X and is

pn G C

given

holomorphic

4.15

following

complex variables

in the

=

o

=

of

element

(Wi,Liê(z,X)) tations of ETiV(sl2)

and

Det

=

and the

-

determinant

the operator

(a^z

on

-

=

=

2v)de(z)

is

-

is i

—

a

function

not

a

ce(z

Dete(z,X) two

and

with

central element:

=

-

2n)be(z))

G

End

(55)

.

(W).

finite dimensional irreducible

Det\fi(z,X)

where

depending

Iwz on

denotes the

the

represen¬

Derle(z,X)ly/x

—

identity

and

matrix

weights of the correpond-

1,2.

of (W\®W2,

L\ J (z, X—2nh2)L2

Detrle(z,X) Detr2 e(z, X)lwi®w2; ®

=

Dete(z, X)

2e(^, A)Iwi respectively,

Devie(z,X)

W\

functions meromorphic in X 1,... ,n, operators h^l\ i

[25])

algebra

(W2,L2>e(z,X))

Then the quantum determinant

Dete(z,X)

are

with quantum determinant

ing representation for

matrix

operator algebra

pyv(X,px,... ,pn).

(Quantum

Let

Wy and

p±,... ,pn and z, its

,

coefficients

It is denoted the quantum determinant.

on

complex variable

-

pt whose

fl(A) 9{\-2r,h)

Det2,e(z,X)

in the

=

Dete(z, X)

b)

,

öe(z,h^l\... ,/i(n),A) oe(z,h^\... ,/*("),A)T+2w\ where oe(z, h^1',... h^n\ X) are to be differ¬ a,b,c,d

all other variables

h^v(X,pi,... ,pn)

The

,

operators in the

Proposition

...

=

{—1,1}, for in

continuous

the

the operators

generated by

ence

of

entries

holomorphic

where p E

a)

in terms

complex

weights hSn\X), functional L-operator Le(z, hP->, hSn', A), o a,b, c, d, acts as a difference oper¬ several

oe(z, hP-',... (sub-)space (of) J7^ of all functions meromorphic

of whose

each ator

,

End(W).

E

functional representation involving

a

the operators

where

Iwi®w2

(z, A))

is

given by

denotes the

identity

e

W2-

Proof: This is shown tor

algebra

by commuting the quantum

of the

determinant with all

generators of the

corresponding (functional) representation ETjV(sl2).

opera¬

49

Highest weight representations

4.2.3

Remark:

eigenvalue problem in the differential case, we needed the notion highest weight representation of sfa A similar notion can be defined for the elliptic

To deal with the Gaudin

of

a

quantum group

ETjr)(sl2).

ETtri(sl2)

tum group

W contains

a

representation (W,Le(z,X)) of the elliptic quan¬ highest weight representation if it has the following properties:

(Highest weight)

Definition 4.16

is

a

nontrivial element VhmVJm E W such that

ce(z, X)vh.w.

ae(z, X)vh,w. for

some to

G

A

A^hw(z, X)vh,w,,

=

f(h)vh.m.

A,

triple (to, A~~h (z, X),A+h representation (W, Le(z, X)). w

(z, A))

w

End

E

=

A~hw(z, X)vh.w,

(W).

highest weight of

is called the

(56)

f(w)vh,w.,

=

de(z, X)vh.w.

A^hw(z,X),A~thw(z,X)

C and

The

all z,

for

0

=

the

highest weight

Remark:

generalized to functional highest weight representations of ET7](sl2). The corresponding highest weight triple structurally stays the same, whereas Vh.wXfJ; A) G W Ç Ti vh.w. ^ 0. Vh.w.

The

previous

notion

be

can

=

Proposition

a)

The

(A> b)

4.17

(Examples)

Kh.w.^ A)

The

77),

is

9(z

=

-

z0

-

77), A-^>, A)

=

9(z

-

z0

-

highest weight representation

+ An +

with

with

highest weight representation +

a

is

a

An + with

finite dimensional

highest weight (A,

An +

highest weight

A+h

w

(z, X)

=

«)^f^).

highest weight

n),A-h,wiz,X)=9(z-z0-An +

WA,e(zo)

representation

weight representation

A-^(z, X)

a

Ar/

zo +

-

9(z-z0

=

is

V^6(zq)

representation

For A E N the est

e(z

=

(A,AtA.wiz,X) c)

VaÔô)

representation

n)e-^^

irreducible

6(z

—

zq +

high¬ A77 +

r?)^^).

Proof:

The

a)

proposition

We choose Vh.w.

weight triple

c)

following

way

0. cA,e(z)vhM. on by checking f(h), aA:e(z), dA,e(z) Vh.w.-

We choose Vh.w. obtained

b)

is proven the eç,

=

—

V\. Then

E

=

v(X)o~a,h

WA,e(zo).

The

highest weight triple

E

=

0. The

is

highest

on

The calculation is that of the first item tion of

The

Ti Then CA,e(z)vh.w. by checking f (h), aA,e(z), dA,e(z) VhmW_.

V(A,X)

is obtained

=

irreducibility

remembering

is shown in

the restricted range of defini¬

[25].

Remark:

By

the next

proposition,

representations

is

again

we see a

that

a

tensor

product of finite-dimensional highest weight

highest-weight representation.

50

([25])

(zi,... ,zn)

diag. Let Az E N for all i 1,... ,77. product ofn irreducible, finite dimensional,highest weight representations ®HiWAl,e(zy) is again a finite dimensional irreducible highest weight representation with highest weight

Proposition

4.18

Let

G

Cn

=

-

Then the tensor

n

n

n

(£A,AW*'A)

=

1=1

IIAJi»,(*-*'A-2ï* E ÄÖ))' 1=1

3=1+1

n

n

A-^(*,A)=nAew*-Â-2?? e h(3))) i=l

®=1Vh

and

highest weight

vector

A+h

j(z, X),A~h

>%(z, X)

w

Proof

w

E

"=iVa,,

where the

highest weight functions

Proposition 4.17.

m

([25]):

The statement is proven

weight

>t

w

described

are

(")

.7=1+1

analogously

to the statements of

Proposition 4.17,

the

highest

being H=1 r](sl2)

are

[25])

finite

Two

dimensional irreducible

isomorphic if their highest weights coincide.

Proof:

The

proof

is

The

4.3

given

in

[25].

eigenvalue problem corresponding

the

to

SOS

eight-vertex

model

Synopsis:

chapter,

In this

ponding transfer

the

emphasis

is

on

introducing

two notions:

the representation

corre-

eight-vertex SOS model (Definition 4.21) and the family of commuting matrices of the eight-vertex SOS model with antiperiodic boundary conditions to the

(Definition 4.26). First,

define how to obtain the Boltzmann

weights corresponding to the ones of the of the R-matrix (Definition 4.21 a)). Then means by elliptic [10] we describe the representation that comes along with the SOS model (Definition 4.21 b)), consisting of a tensor product of 77 shifted fundamental representations of ETjT)(sl2). After this, we want to define the family of commuting transfer matrices of the SOS model with antiperiodic boundary conditions. To ensure commutativity we have to we

eight-vertex SOS

choose

due to

Ao

=

model

Vzi,=ihi-

possible poles

Note that

we

can

properly define

of the transfer matrix if Ao

=

this notion

0 which

can

only

only

occur

if if

is odd

77

77

is

even.

We furthermore want the transfer matrix to act

So,

we

first have to define the notion of

that the

an

on a space of antiperiodic paths Pn. antiperiodic path (Definition 4.25 a)), show

antiperiodic paths thus defined form

isomorphic

to the space which

an

a

basis of

SOS transfer matrix

a

space of

naturally

antiperiodic paths

acts

on

and describe

isomorphism (Definition 4.25 b) and c)). We then show that a transfer matrix of the SOS model with antiperiodic boundary conditions is indeed well-defined on the space Pn

the

51

(Proposition 4.26). In Definition

4.30,

we

eigenvalue problem of the family of with antiperiodic boundary conditions.

common

matrices of the SOS model

commuting transfer The last

pose the

explicitly

proposition of the section shows that the family of SOS transfer matrices

is

indeed commutative. Note that in what

section, they

the notions

concerns

coincide with what

by representation theory

The SOS model in terms of the

4.3.1

defined in the first sectio of this

heuristically

define

we

in this


chapter.

Er^(sl2)

Remark:

First,

we

weights

want to redefine the basic notions

In the first the

elliptic

(In

model.

describing

the SOS

model,

i.e. its Boltzmann

and transfer matrix.

subsection,

we

second

a

weights of

show how the Boltzmann

R-matrix and how to thus attach

subsection,

we

a

the model emerge out of

ET^(sl2)

representation of

show that the attached

representation

weight representation and compute its highest weight.) Note that we are treating here the simplest case of the SOS model, i.e. out of fundamental representations of ETi7](sl2) only. It is called of order a

tensor

where

h%

As

will

we

A to be If

-R(0n)(z, A)).

YH=2 h^)) ®

2n

we

a

E

ET}T](sl2)

element

a given 1,... for i =

,77

l,...

we

Y^=i

built

if it involves

involve

77

—

weights ht,

h-i-

(Weights) the tensor

®

...

®

higher (finite)

[11],

would obtain the SOS models of

of by setting

K e[a{\

e[at]

We

product

®

...

basis

e[an]

®

dimensional rep¬

stated in

{—1,1} for i of V®n, e\o~\\ ®... ® e[an], where

attach

can

as

=

n

o%

weights h%

e[o\]

®

...

G

®

=

e[ay]

®

...

®

[27].

1,...

ay E

,77 to

{—1,1}

e[an]

,77.

define the basic notions of the SOS model.

we can

Definition 4.21

a)

n

=

extended the models to include tensor products of

=

case

(V®n,R^(z,X

i.e.

representations

n

highest

also have to discretize the value of the parameter

by Proposition 4.31 we function of the weights, Ao see

Definition 4.20

Now,

The fundamental

the

a

{—1,1}.

resentations of

fori

ET^(sl2),

product of fundamental representations of

to the SOS

is

(Boltzmann Weights, L-operator)

The Boltzmann

weights We(c, b,

a,

d\z) of the

SOS model

are

defined by

the

following

formula

Re(z,X

—2nd) e[c

=

—

Y,aWe(c,b,a,d\z)e[b where the terms

e[a

—

b)

®

e[c

—

d]

c

—

are

d,b

—

c,b

—

a,a

to be considered

—

as

d E

d] -

®

a]

e[b ®

—

e[a

{—1,1}

c] -

and

=

d], z

the standard tensor

(58) E

C.

The

product

expressions

basis

ofV®V.

52

explicitly the Boltzmann weights read

Written down

We(d

l,d

+

We(d+l,d,d

+

2,d

l,d\z)

+

l, d\z)

+

=

-

=

9(z

2n)

+

B(z- 2nd)9(2n) 9 (2nd) 9(z)9(2n(d 1)) -

We(d

+

l,d,d-l,d\z)

=

l,d\z)

=

We(d-l,d,d

+

9(2nd)

We(d-l,d,d-l,d\z) We(d

-

d

1,

-

2,

d

9(z)6(2n(d + l)) 9 (2nd) 9(2nd + z)9(2n)

=

-

9 (2nd)

1, d\z)

=

9(z

2n).

+

weights thus defined coincide with the ones obtained by eight-vertex SOS-model as we defined it in Definition J^.l.

The Boltzmann

al.

b)

[19] for

The

the

the SOS model is

L-operator of

Date et

given by n

LSos,e(z,zi,... ,zn,X)

=

R^V (z

-

zi, X

2n^2hy)

-

1=2 n

Lf2) (z ~z2,X-2nYJK)... R^n) (z

zn,

-

A),

(59)

i=3

(zu

where

...

,

zn)

Cn

E

LSOs,e(z, zl,...

diag.

-

,

zn,

A)

G

End

(V®("+1)).

Remark: The Boltzmann

Proposition 4.7 was

shown in

into the

[27].

will be useful.

defined above translate the

weights

dynamical Yang-Baxter-relation of

star-triangle-relation mentioned

in the

(general) introduction, as Lgos,e(z, z\,... ,zn,X)

For what follows the definition of the operator

To understand this

definition,

we

must

define the

following

space of

functions: Definition 4.22

(M(C,V®n))

M(C, Vm) Definition 4.23

f

(

SOsA

'

=

:

C

->•

Vm,

X

->

/(A) | / meromorphic

in

A}.

(60)

(LSos,e(z,zi,... ,zn,X))

1"--

^

_

'^'Aj-

=

The

{/

so-defined operator

(

dsOS,e(z,zlT-- ,zn,X) ^ csosAz^r,... ,zn,X)

LSOS,e(z, Zi, is

a

matrix

.

.

on

.

,

Zn,

X)

(

Ao

b_soS,e(z,Zi,... Zn, X) dSosAz,zx,... ,zn,X) ,

0

T+2v

V with entries in End

(M(C, V®")).

53

The

4.3.2

representation attached

SOS model

to the

as

a

highest-weight

representation Remark:

Here,

is

a


Proposition

4.24 Let

attached to the SOS

X^r=2 h%)Re

2r\

ducible

representation which

show that the

we

b)

4.21

iz

~~

(z±,... ,zn)

G

—

Cn


sense

with

(z

—

zn,

A)),

is

a

—

(z

Re

—

Z\,X

finite-dimensional

—

irre¬

highest weight

a)=n ßtz

zr+2"), ^sos^

-

of Definition 4.16

diag. Then the representation of ET^(sl2)

—

^r=3 hi)... RÏ

2t7

L a+os(z, a)=n *(*

attached to the SOS model in Definition

(V®n,Lsos,e(z,zi,... ,zn,X)

model, namely z2i A

we

in the

-

*)

9{X0~(x))

(6i)

Proof:

This

proposition is

1,...

,77.

The

4.3.3

family

boundary

corollary

a

of

4.18 with

Proposition

A,

G N set to

A,

of transfer matrices of the SOS model with

=

1 for i

=

antiperiodic

conditions

proceed by showing how the antiperiodic boundary conditions appear. Then, we turn to the definition of an antiperiodic path and show that the space of antiperiodic paths is isomorphic to V®n. Then, we define the family of antiperiodic SOS transfer matrices by representation theory and show that it is an endomorphism of the path space. Finally, we pose the common eigenvalue problem. We

the next

By

the definition of the Boltzmann

proposition

compare the transfer matrix of the SOS model with

given

here with the

In order to make two

special

Let

1,

i

given

complex

1,

i

consider n+1 numbers a,\,... ,an+i G

=

1,...

us =

n.

define

We

fixed

all

first need to define

|,

this

e[ai

1,...

possible

•

•

•

=

defines

,

—a\.

If

an+i >=

a\,... ,an,an+i,

tiperiodic paths Pn.

a2]

-

consider n+1 numbers a\,... and an+i

§

subject

to the conditions

e[an

an+i]6x,2r,an+1

\at—at+i\

=

\ay—at+i\

=

the vector

an+1 >=

,

a± G

K, for

we

V®n.

us

For every Let

vector space

(Pn,ICA)

|ai,...

b)

in Definition 4.2.

of the definition of the SOS transfer matrix,

bases of the

Lemma 4.25

a)

one

sense

weights will be confirmed if we antiperiodic boundary conditions as

a

,

we

e[ai we

®

..

®

-

of V®n.

basis

f

an+i G

subject

to the conditions

consider the vectors

-

a2]

®

obtain

a

®

e[an

basis

+

ai}5\

of V®n,

_27?ai

called the basis

of

an¬

54

c)

e[an]\ay E 1,... ,n} ofV®n, we can attach an antiperiodic path \a\,... ,an > {—1,1} for all i the means Iqa ' V®n —y Pn, Ica^[o~i]®- -®e[cn] isomorphism lai> of On+i by each

To

product basis {e[a\\

the standard tensor

of

element

®

®

...

=

=

i,—l

1

>, where

—a±

a%

\_,°~3

-(—

=

=

•

•

•

j

n

+

7=1

aj) for

/_,

all i

1,...

=

,77 + 1.

3=%

Remark: It is

to consider the vector

helpful

(even)

77

(odd)

8 and

=

n

=

\a\,...

an+i >

,

as

a

path

f

in

is shown below for

as

7.

n=7

-3/2

--

-5/2

-7/2

Here, the axis labelled In

case

of

n

being

and an+i differ

with

indicates

an a

even, a± and an+i differ

by

odd

an

by

--

possible values of an even

integer

the

at,i If

or zero.

1,...

=

+ 1.

77

odd, the

is

n

,

a\

integer.

Proof: First let

attach to it

a)

remark that V®n is of dimension 2n since V is two-dimensional.

us

basis

a

e[al]

®

start with

Let

us

two

possible values of

four

that

now

possible we

fixed

a

\at

—

each combination

=

i

=

Let as

us

again start

—

—an+i,

=

ai+a„

a,,

—

,i

=

2,...

we

=

an+\,

we

set a\

still relate the

since a%

—

=

0

same

al+i =d%

—

=

|

a%+\,i

=

77

1,

=

—

1

conditions,

{—1,1},

we

times,

i

=

can

get

we

03)

—

=

1

see

we

(ai,... ,an+i) subject

where o~% E

To

implement

to the

attach to

1,...

,

n.

This

implies we

get

that a

e^1]

® ,77.

...

e[o~n]

as

=

to the

,

an+i >

än+i

ai+a2n+1

a± =

—

à\. We

basis of V®71.

and

can

do

Especially, if

|äi,... ön+i vector |ai,...

Note that to the vector

1,...

|ai,...

the additional condi¬

the value of a\ to à,\

,77 + 1. This then

vector

02! get by |o2 —

and construct the 2n vectors

readjust

±an+i.

we

77.

,

|ai

another

Due to these

this for all of the 2n vectors of fixed 01, hence a\

1,...

We may

§.

fixed a\ G

have to

we

=

there,

procedure

e[a{[ ®...® e[an],

some

i

condition

From

shown in the first part of the lemma.

tion a,\ a,y

with

this

1,....

construnction works for every a\ E

b)

{—1,1},

By the

ai ± 1.

Iterating

1,

vector

a

=

|.

£

E

different combinations of

possible

al+\\

ai

a2

:

02

values of 03.

have 2n

conditions

e[an],

®

...

where a1

=

,

,

—öi

>

an+i >,

55

c)

isomorphism is a corollary of the construction of the basis Let us check that oi Pn. —an±i: by definition of the a%, antiperiodic paths

The construction of the

of al

=

=

\ E"=l °3

(Antiperiodic

Definition 4.26

SOS model with

and an+l

=

=

conditions is

=

Ao

to

=

n

transfer

The

YH=i ^

+

0

1

1

0

the

,zn,Xo)

•

(62)

csos,e(z,zi,... ,zn,XQ),

and the matrix K is

K"

of

matrix

given by

EMe{-l,l} znAo)

where X E C is

~\ Ej=l a3-

given by

Proposition 4.27 The previous definition of Tsos,e(z,z\,... ,zn) defines the row-tothe figure in the introduction row transfer matrix of the eight-vertex SOS model (cf. where it in «l > corresponds to Definition 4-2, \a\,... an, an+i 1.1.2) as we defined the height configuration of a row with antiperiodic boundary conditions, =

—

,

Tsos,e(z,zi,... ,zn,A0)|ai,...

( IIwe(ai-t-i,al,6l,&l+i|z) ||6i,...

E

Vi=i

&i,— A>+i=-6i

,&n-t-i

,-a\ >=

~h

=

(63)

>

/

Thus, Tsos,e(z,zi,... ,zn,Xo)

End

G

(Pn).

This coincides with

Definition 4-2.

Proof: Let

us

first note that

the definition of the

A0

=

n

J2=i hi

weights ht.

YÜ=iai

=

~

ai+i)"

This agrees with the

=

fixing

(al

~

an+i)??

of A for

an

-277an+i by antiperiodic path: =

à\,-2r]an+i 1. Furthermore, e*[b a] is defined as the dual basis element to e[b a]: e*[b a]e[b a] With the above conventions, the action of the antiperiodic SOS transfer matrix on a path —

—

is

—

—

given by

Tsos,e(z,zi,... ,zn)\ai,...

,an+i

=

-ay >=

n

YJtrV^Rfl\z-zl,X-2nYJh^)... i=2

jj,

R^ (z-zn,

X)T-2vßö-2Van+1,\e[ai

£ e^*[p}K^R^(z

-

î, A

-

e^[p] -

®

e[ai

-

a2]

bn+l]K^R^(z

®

..

®

•

®

e[an

-

2r?f>«)... R^{z

®

e[an

-

an+1] -

zn,

an+1}5-2rian+i+2nß,x

-zuX-2r,J2 h{%))

•

•

•

R{0n)(z

-

zn,

1=2

bn+l

e(0) [an+l

o2]

-

bn+i\

=

X)

i=2

fi

£ e(0)*K+1

-

®

e[ai

-

a2]

®

...

®

e[an

-

an+i}5_2van+1+2V(an+1-bn +i)A

=

X)

=

56

J2 e(°>[an+1

-

bn+1)K^R^(z

-

21, A

27?5>«)

-

bn+i,bn

We(an+i,an,bn,bn+i\z

1=2

zn)e^[an

-

-

bn]

®

e[ai

a2]

-

®

e[bn

®

...

-

bn+i]

=

n

^2 (II WeK+i, ai, &j, 6î+i[2

=

bi,— ,bn+i

-

2î))e(0)*[an+i

-

6n+i]iY(0)5A

_2j?6„+1

2=1

e(°)[ai

-

&i]

®

e[6i

-

62]

®

e[6„

...

-

6n+i]

=

n

E (II we(ai+i,ay, by, by+i\z bi,.--,bn+i

e(0)*[an+i

zl))öx,-2Vbn+1

1=1

6n+i]e(0)[6i

-

-

-

ax]

®

e[bx -b2]®... e[bn

-

bn+1]

=

n

6i,...,6n+i

if

6n_|_i

=

—6n

is

i=l

obeyed.

Since

Corollary 4.28 phism Ica ' V®n

Pn,

—y

Tsos,e(z> zu...

Tsos,e(z,zi,... ,zn,Xo) A0)

,zn,

(Pn), by

End

E

means

of

the isomor¬

define

can

we

=

IcATsos,e(z, zx,...

,

zn,

X0)ICA

G

End

(V®n).

everywhere Remark: Let

us now

completely

proceed

to the common

similar to the

Definition 4.29 Let character x G

|ai,...

T*,

,an, an+\

as

—

esos(z)

be

elliptic polynomial,

an

Appendix, Pn, where every aai

—a\ >E

(esos(z),

we

want to solve. It is of

course

obtained in Definition 4.3.

one

defined

eigenvalue problem

in the

^2

and

an

^2a

,.an

element

of ®n(x)

a0lj...)an We are looking

with

some

a

E C.

cta1,...,an\a,1,...,an,an+i

=

for

a

pair

-a1>)

ai,... ,a„

obeying

Tsos,e(z,zi,... ,zn,X0)

^

aair..)an|ai,...

(64)

,an+i =-ai >

ai,... ,an

=

^2

csos(z)

ai,.

where

Yâi,...,an a,

,an

,

«n+l

=

-ai >G

Pn-

Remark: The

periodic

case

of the SOS model may be treated

by

Felder and Varchenko in

To

ensure

family

that the solutions thus obtained

of transfer matrices

by algebraic Bethe

ansatz

as

shown

[27].

we

need the

are

indeed

following

common

lemmas.

solutions of

a

commuting

57

(W,Le(z,X))

Lemma 4.30 Let

ETiV(sl2),

be

a

representation

x}_(

ae(z,X) Le^X>-{ce(z,X) iy

r

Let X E C be

fixed

to

fLe (z, A0) is

functional representation of

or

with

commutative.

Ao =

=

be(z, Ao) by

T^ 0

Then, the family of transfer

nh.

K is given

be(z,X) \ ( de(z,X) ){

+

K

ce(z, A0)

=

0

1

1

0

0

\

Tî )• matrices

tr{0)K^Le(z, A0)

for

defined by

z

E

(65)

C

=

Proof: This is shown

by using

the

elliptic RLL relations.

We have to check that

[fLe (z, A0), fLe (w, A0)] Proposition matrices

of

(Commutativity

4.31

the

=

0 for all

z,wEC.

of the SOS transfer

antiperiodic SOS model commute,

matrices)

«/Ao

=

4.4

all z,

w

G

transfer

i.e.

[Tsos,e{z,zi,... ,zn,X0),fsos,e{w,zi,... ,zn,X0)] for

The

=

(66)

0

C,

YÜ=iVhf

Generalizing Sklyanin's

results: The

auxiliary representation

Synopsis: In this

section,

we

introduce the so-called

auxiliary representation of

ETyV(sl2).

It is

our

eigenvalue problem of the SOS transfer matrix with antiperiodic boundary conditions. The origin of this construction will be described in the remark below. First, we give the definition of the auxiliary representa¬ tion and show that it is indeed a functional representation of the elliptic quantum group. Then, we define the corresponding family of transfer matrices, denoted the auxiliary transfer matrices with antiperiodic boundary conditions or just the antiperiodic auxil¬ iary transfer matrices. At last, we construct an isomorphism Ipc that allows to write the auxiliary transfer matrices on "non-functional spaces" and show that the family of transfer matrices is commutative. Thus, it makes sense to treat the common eigenvalue problem of the auxiliary transfer matrices, also denoted the auxiliary eigenvalue problem. In the next section, we will show that the auxiliary representation is isomorphic to the representation attached to the SOS model (cf. Theorem 4.21 b)). This was already sug¬ gested by Proposition 4.19. Also, the family of SOS transfer matrices with antiperiodic boundary conditions will be connected by an isomorphism to the family of antiperiodic auxiliary transfer matrices (cf. Corollary 4.51). This will enable us to perform the sep¬ aration of variables. By the construction of the two isomorphisms, the following section will in a sense complete the one which we just began. main tool in order to achieve the solution of the

Remark: In

[46, 44] Sklyanin

achieved the solution of the XXX model

as

described in

[20, 21, 37]

58

with

various, including periodic and antiperiodic, boundary conditions.

the main tool he

1).

The

of

y(sl2)

To achieve this, auxiliary representation of the Yangian 3^(s^2) (cf. Appendix

is the

uses

auxiliary representation he

uses

is shown to be

isomorphic to the representation corresponding transfer matrices can common eigenvalue problem of the XXX

attached to the XXX model and also the the

by

be connected

isomorphism. Thus,

the

transfer matrices, i.e. the solution of the XXX

model,

is connected to

solving the com¬ At this point he can

eigenvalue problem of the family of auxiliary transfer matrices. advantage of the auxiliary representation: its transfer matrix evaluated at n (ausgezeichnet) points yields a system of n difference equations, the separated equa¬ tions, which are one-dimensional problems. They yield Bethe ansatz type equations in the course of their solution. By a suitable interpolation, we can out of their solution find the common eigenvalue of the auxiliary transfer matrices and by the knowledge of the isomorphism also of the original eigenvalue problem of the XXX transfer matrices. Here, we generalize Sklyanin's ideas to the case of UT:7?(sZ2). The succession of the steps will be the following: introduction of the auxiliary representation and the commuting family of auxiliary transfer matrices in this section, construction of the isomorphism re¬ lating the original SOS and the auxiliary transfer matrix in section 4.5, describing the original and the auxiliary common eigenvalue problem as well as the system of separated equations emerging from the auxiliary eigenvalue problem in section 4.6. To be able to define the auxiliary representation, which we describe here as the operator algebra of a functional representation of LVi7?(sZ2), we first have to define the spaces of mon

the main

use

functions

on

which this representation will act.

Introducing

4.4.1

(Fn°

Definition 4.32

for

%

!,...

=

Let

Si

for

i

t^ j

Let D we

can

-

and all

Ayn, i,j

define

a) Tn°

=

=

the

=

1,...

T^ =Tj'{/

d)

J=^

=

E

following

{f(xi,... ,xn)

=

Ay-q

-

(zi,... ,zn)

Let

(ET)n

E

diag

-

and

At

G N

,

+

2n,...

G

Sy for all spaces

{/

G

E

A^} for

—Zy +

Cn

—y

i

1,... of functions.

C

->

=

C

|/

G

,

n}.

i

1,...

=

=0

for

With these

T*, X

| / holomorphic

T\f(xi,... ,xn)

Tn°/{f

Tn I

:

,

,

n, where

Si

n

Si

—

0

n.

{f(xi,... ,xn,X): Cn+1

c)

e) TD

—Zy

{(xi,... ,xn)\xi

=

b) Tn

,T^,T^ ,TmFD)

,?7.

{—Zy

=


the

all

definitions understood,

is restricted to in Xy

for

(xx,... ,xn)

i E

=

A0

=

Yh=i

xi +

•

•

-,

zrti

A, Ai,.

.

.

j

An)

.

.

.

2^^, A, Ai,

,

(,2,

Zl,

g(A-(z

.

.

.

,

2n

,

X

A+e(xl)T+2"T+2^,

=

Ca-uaê^,

£i,.

.

A, Ai,

sO)

+

.

j

#nj

+ z + xt y, 6(+\ 0(A) h,

,A.n)l^

.

.

2V

.

.

.

A

(r0)

jA^Ji^

gÇs

+

a;,)

(71) A, Ai, +

.

,

.

,

A^ji^

A

2s)

J-1-

g(z + gj) 0(^-2;,)

,

X

A-e(zt)T-2"T-2",

x

ße*aux,e(z,î,--- ,zn,A,Ai,... ,An)

.

n^+^)e(A+E-i(;frZi+At??)TA^,

=

Oa-urc^t2!

,

(72)

=

Detaux>e(z, z\,... ,z„,Ai,... ,A„)

=

JJ 9{z

n

2=

define

an

z, +

-

AzV)6(z

-zz-

AlV

-

2jj)

(73)

.

1

operator algebra obeying the elliptic RLL-relations.

Note that the operator Zn, À, Ai, ,Zn, A, A.\, AnJ AnJx^ o,aux>e\Z, Z\, 0âuxte\Z, Z\, and that all is defined implicitly by Proposition 4-15 appearing operators .

End

.

•

.

•

—

.

,

.

.

..

,

.

,

elements

are

of

(F^).

Proof: Let

us

check the second of the RLL-relations

9(z

—

+

w

2n)ae(z, X)be(z, A

—

2t?)

as

a(z

=

+ß(z The

remaining

de(z, X)de(w, Concerning terms

relations

X +

2n)

=

=

be checked

de(w, X)de(z,

the relation

(bn>e(z,X))y

can

we

X +

by

—

w,

example.

w,

It reads:

X)be(w, X)ae(z, X

+

2n)

-X)ae(w, X)be(z, X-2n).

similar means, where the relation

2n)

want to prove,

-

an

can

we

be deduced

first

can

argue

by the preceding relations. -

since

^f^ Tîî £Â-e(xî)T+2î?T+2"

bn>e(z, X)

acting

as

a

consists of difference

60

in the variable x%

operator

J(z

0

ä{z_w

+ Zr])

only

that it suffices to check the

-

x3)9(w + x3 X + 2rj) e{x)9{x_2r]) 9(2n)9(z w 9(z-w)

+

-

X

for all j

1,...

=

,

n.

left invariant

are

9(w

-

+

9(X)

X +

-

_

missing

x3)9(z

77 sums:

+ x3 +

2rj)

ë(Â)

-

X) 9(w

+

x3)9(z + x3-X 0(A)0(A 2t?)

+

0(A)

Note that the

following

+

X

2n)

-

anfi(z, X)

factors of the operators

and

bn>e(z, X)

action of the difference operators.

by the

/2(^, A) + fs(z, A). Let us first check the transformation properties of each of the summands fy(z, A), i 1,2,3, under A —y X + 1 and A —y X + r. If the first transformation is performed, we obtain fz(z, A +1) —f%(z, X) e-T+2m(\+w+x,) ftfa Aj for « 1,2, 3. The second transformation yields ft(z, X + r) We

formally

can

write the

fi(z,X)

as

sum

=

=

=

=

=

for

7

1,2,3.

=

The residues of the

sum

be taken at values A

can

=

0 and A

2n. The first calclation

=

reduces to

9(z

w)9(w

-

+

whereas the second

6(z

w

-

+

x3)9(z + x3+ one

> -A-U-±9(2n)9(z-w)9(w

„,

277)

-

+

x1)9(z

2n)

+ x1 +

yields

2n)9(z

x3)9(w

+

+

x3)

_

9(z

-

w

2n)9(w

+

x3)9(z

+

x3)

+

~

equations

only

The

0

at A

zero

-9(z

=

which is also and

are

zeroes

-

obviously w

=

true.

+ x3 + 2n leads to

W)9(2n)9(z

of the left and

x0+ 2n)

+

By

true statement.

a

"

0(2?7)2

9(2n) Both

°'

=

8(2n)

+

^)9(z

x3+2n)9(W +x3)9(z

+

9(w

W)

x3)

the coincidence of transformation

hand side of the sum, the

right

+

-

behaviour, residues

of both sides is proven.

equality

This proves the correctness of the indicated RLL relation. Note that to show that every operator Det is to

element of

an

Fq

=

function

{/

G

Fp, it T* \f(x\,...

FöauXie

G

For the operators

xn)

,

(x\,... xn) ,

us

dauxfi(z, z\,...

,

zn,

show that also the operators ,

,xn,X)

Fcaux,e(xi, be

Fq

277,...

E

D}

is

a

a,b,c, belonging

o

function

=

onto another

mapped

^0A.

caux,e(z, z\,... ,zn,X,A1,... An) E

0 for all

=

by each operator

for

A, Ai,...

,

An)

and the quantum determinant this is

change

the value of

(x\,...

,

xn)

G

it is fixed.

once

Let

suffices to show that

shown since the action of those operators does not

easily D

oauxfi(z,z\,... ,zn,X,A\,... ,A„)

a

—

,

function

Zy +

Atn}

Fbaux.ixi>

which

are

vanishing

baUx,e(z,

z\,...

define functions

An) and F-bauxe (xx,... ,xn,X), ,zn,X, A\,...

,

Fq. To this end, let f(x\,... ,xn,X) (x\,... xn) G D, i.e. x% E {—z% Ayr), —z%

elements of

at every

—

,

,77. Then consider the function

for every

7

îÂ)

baux>e(z,zi,... ,zn,X,A1,... ,An)f(xi,... ,xn,X)

=

=

1,...

9(X-(z

V^ V{*~ \z 1=1

X

Xy))

+-TXy))

_

0W

^j

yr

9(z

+

x3)

x

Ä^^.)

K,e(x^2vT^2vf(xi, ...,Xy + 2n,xn,X + 27?))

.

—

Ayin +

61

The

only possible

get

to

cases

point

in D is when at least

Fi

(x\,...

,

uated outside

D,

the coefficient

ing

vanishes at

cal

X)

9(xz

=

Ayfj

+

2n,...

,

xn),

+ zl

—

Y\^=i3^z9(xl

A%n)

+

z3

be

can

to the function

applied

when at least

A+e(xz)

...

—z%

=

in this

as

—

xn,

,

X)

—

E

—zl +

Aln,

thus mak¬

F$,

since it

where the criti¬

Ayn which is exactly

vanishes.

äauxfi(z,zi,X,A),baUx,e(z,zi,X,A), Caux,e(z,z\,X,A),daux,e(z,z\,X,A) E End(Fp) as entries of an L-operator are equal the universal evaluation module VAR(zi). 4.34

n

a

case

hence it has to be eval¬

F5auxe(xi,... ,xn,X),

of the x% takes the value xz

one

the value at which the coefficient

For

Aj7?,

A3n) vanishes,

—

,

cases occur

Corollary

—zl +

,

that does not vanish at —z% +

=

F~bauê (xu... xn, X) vanishes. Hence, Flaux^ (xu every (xi,... ,xn) ED.

argument

same

f(x\,...

A)

a;n,

,

do not know about its value. But in this case, xt

we

A~e(xî)

(xi,...

.Fg

of the x% has the value x%

one

involves

where

that

sure

The

xn,

function

a

1, the operators

=

to

Proof:

If for

n

1

=

we

replace

aaux,e\Z, Zl, A)

z

t

,

/

„

A)

u

caux,e\Z,Zl, A)

7

,

o,aux,e\Z, Z\,

^

A)

f(—)v(ß) If

we

keep

n

by

z, the

9(z

_

operators of Proposition 4.12 reduce

-zi+

si)0(A

to

Air?) rr_2rj

-xi+

—

A

P(\\

9(X-z

,x „

uaux,e\Z,Zi,

-

+

hi)9(hx

+ zx-

'

Ait?) ai+2w^+2^

+

J.^

±Xl

—

9(X

hi)9(-hi

+ z-zx-

Ait?)^_2t?^_2??

+

lXl

—

0(z

,

1^

,

/ti)0(A -h1- AX7?) ^+2V

-zi-

—

A

ûl\\

f(~(xi

=

zi))v(p)

+

=

f(p)v(p),

'

where

v(p)

E

Tr.

in mind the definition of

defintion of the universal evalution

h\ given by the last equation and compare with the module, we get the desired result up to a normalizing

factor.

4.4.2

The

auxiliary

transfer matrix

Remark: Let

us

now

define the

auxiliary transfer

compare it to the transfer matrix of the

Definition 4.35

(Auxiliary

matrix and the

Transfer

Matrix)

(Fjj,LauX}f,(z, z\,... be the

isomorphism enabling

antiperiodic SOS model.

,

zn,

Let

X, Ai,...

An))

,

representation of Proposition 4-12.

Let X E C be Then the

fixed to Ao Y^n=i(^ + zî)auxiliary transfer matrix is given by =

J-aux,e\Z,

Z\,

.

.

.

,

Zn, aq, Ai,

tr^K^Laux>e(z,zi,... —

\Oaux,e

~r

Cauxê)yZ,

Zl,

.

.

.

.

.

.

,

AnJ

=

,zn,X0,Ai,... ,An) ,

Zn, Aq, Ai,

.

.

.

,

AnJ.

us

to

62

Remark:

explicit form of

The

the transfer matrix

-Laux,e[Z,

t ^^r0- fÔ'(* ^

i=i

The

°>

odic SOS-model is the

Let

7?

—

us

—

z% +

("/£), I>aux,e\Zi

Z\,

.

,

Aq)

Zn,

.

will need to treat the

we

Ai

with

1,...

=

=

.

.

,

Zn, A, 1,

.

.

=

C

G

a

[fai...an]

Let

,

1JJ

,

Ai

basis

a

An

=

...

eigenvalue problem of the antiperi¬

{(xi,... ,xn) \x%

=

in this

ain,...

Zn,

=

=

An

= ...

of FD°

=

.

.

.

,

X)j.

1.

given by the 2n equivalence classes of func¬

is

,

where

=

—zn +

crn7?)

and vanishes

where outside D it has to

meromorphic function of

b)

The

function Ipc

Aa

V

->

restrictions,

no

^

for

every

a

e[etauXje(z,zi,... ,zn,X)

J\_9(z-Zy-2n)9(z-Zy + 2n).

=

i=i

daUx,e(z, z\,... ,zn,X)

The operator

minant. The values of the

{-n,n},

i

weights

is defined

are

implicitly by

restricted to D

—

means

of the quantum deter¬

{(—zi

+ x\,...

given

in

—

,

xn)\xy

zn +

E

l,... ,n}.

=

Remark:

a)

Note that this —Zy +

b)

expression coincides with the

Xy —y Xy, z

z, for all i

+ 7? —y

1,...

,

one

Proposition

4.33 with

n.

dauxfi (z, z\,... ,zn, A), bauxfi(z,zi,... ,zn,X), caUx,e(z' zi,... ,zn,X), dauxfi(z, z\,... ,zn,X) as an L-operator Note that

one can

i

t

write

\\

This operator acts

to the tensor

I

aaux,e\zizli

\

C-aux,e\Z, Zi, a

as

c) By Proposition 4.41,

matrix

the

on

(An,e(zi,...

Definition 4.42 Let 7re

=

9~(\+2ri)

7re(2;i '

•

.

•

.

,

zn,

a)

baUxê\Z,

Zi,

,

Zn,

A)

d,auxê[Z,

Z\,

•

.

V with entries in

representation

"^

-

2t?)

^n_1

=

,zn,

.

.

.

.

.

,

Zn, A)

\

,

Zn, A)

J

M(C, V®n).

®^=1R^ (z

—

z%, X

—

2t?

is

isomorphic

Y%=t+i hi))-

isomorphism explicitly.

X),An>e(zi,... ,zn,X))

117=2 î

ê ê

.

(FD,Lauxfi(z,z\,... ,zn,X))

product representation (V®n,

The next two theorems will show this

a)

=

îdentity

~

z%

~

matrix

%rj). on

Let h

V®(n~l\

=

£7=2 ^

Then

and

An,e(zi,

/(A, ft) ,

zn,

X)

=

E

66

End

(V®n)

(M(C,V®n))

End

C

is

An,e{zi,

(

put

we

oaux>.

defined

o

(M(C, V®71))

A2e

,zn, A)

Y^=2X^-

=

7re/(A, ft)a"1(zi

2t?)

End

Zn,

\Zn-i+l,

=

where

•

used the notation

4.43

2t?,

n),

•^y,e

Proposition

-

zn,

•

An,e\zlt

we

.

0

2t?, X)c(zi

-

b) An,e(zi,... ,zn,X)E

where

.

ln-1

V a~1(z1 where

by

given

C(01

n)

.

.

.

,

Zn,

A)(I2

®

An,e)

=

(z,zi,... ,zn,X),

An,,

=

Remark:

If

write down the above

we

identity

for each entry of

LauX,e

(z,...

,

A) separately,

we

get

A-}M](z-ziA-2h)ä^2x/\z,Z2, +bW (z-zuX-

,zn,X)

•

2h)ên\z, z2,.-., zn, X))An,e ä^g/Xz, zi,. A^Me\z -ZUX- 2h)b^g/\z, Z2, +bP(z -Zi,X- 2h)d^g/\z, Z2,..., Zn, X))An,e b^g/\z, Zl, AnïetâHz -Zi,X- 2h)a^g/\z,Z2, + (Z -Zi,X- 2h)c^g/\z, Z2,..., Zn, X))An,e ^g/\z, ZU Klei^ (Z-ZI,X- 2h)b^gx^ (Z, Z2, +d£Hz-zi,X-2h)d^gx/\z,Z2,...,zn,X))An,e d^g/\z,zi,. =

,zn,

,zn,X) Zn,

=

.

,

.

zn,

,

put

An,e(zi,... ,zn,X)

X),

,zn,X)

=

we

a),

,zn,X)

•

=

where

A),

,

Zn,

a),

An,e-

=

Proof:

Throughout

the

proof,

we

will write

We have to check that the

LaUX e(z, Z2, ,zn,X) ® (h ~Ân,e)LÛXie(z, Zi, For the sake of

/(ft, A) split the

=

into

.

.

—

L®(z, .

simplicity

eî^+2v)^ a

=

Ae instead of An,e(zi, ,zn,X), since n stays fixed. L-operator of the tensor product Re(z zi,X 2nh) ®

A) (I2

Zn,

,

we

Also'

part depending

zi,...

put if

on

an

®

coincides with the

—

L-operator

Ai.e)-1-

oaux>e(z,Z2,. °Perator

A and

A-independent part by ot(z).

,zn,X)

one

.

,zn,X)

°(ZA),

e.g

independent

=

o(z, X)

for

o

a%UXje(z,zi,... of this

=

a,b,c,d

,zn,X),

parameter, let

us

and

can

be

denote

67

a^(z, zi,... ,zn,X). First note that with the operator In-i, the inverse of An>e reads by a~l (z, X)t£ n(a(z, A)) Let

us

first check

l(z, A),

a

defined

=

4-1

(

=

°

I-1

V -ê_1/(A, h)~1c(zi -2n,X

n'e

+

TV~lf(X, h)~la(zi -2n,X

2n)

2n)

+

We then obtain

^n.e5®!2»2!'--- .Zn,A)Ai, 0

ln-1

-tiTVCA, /i)_1c(zi

-

2)?, A +

2t?)

wë1f(X, hâizi

-^

9(z-zi)e(\-2r,h+2y)

0(z-*! +277)0(2, A) e(2v)e(z-z1-X+2yh) 6(\~2vh)

\

0

,

e(\-2Vh)

clz.AJ

-

zi

^/(A, /i)"1«^!

2t;)

277, A +

-

27,, h)

-

T\

-

'X

2?), A + 2t?)

\

e(z-zie\{%%h+2v) a(z, X)a^ (zi

277, A

-

J T>

2„)

-

ZT(A^T1 ^'"^Ky-âî

"

„„ ~

\

0

(*)

2

277, A)

0

nef(X

277)0(2, A)

+

»21

-

2r,)a(z, A)

ä21(z) (z

J

0

-ir-1f{\, h)-1^! -zi+

,-.

7re/(A,7j)a'-1(zi

277)

2n, A +

-

ln-1

9(z

,

ayz,A)

0

ln-1

a-1^! -2tj,A)c(zi

2??, A + 2rç)

-

277,

A +

277)0(2, \)a~Hz1 -2V,X- 2n)

) TA

2t)

with

,

-

0(27?)0(^-zi-A

x

Ö21(Z)

=

+

27?ft)

C(Z,A)

0(Ä-2^) 0(2:-î)0(A-27?ft

27?)

+

a(z, X)a

.

g

+

_x

l(zi

-

2t?, A

-

2n)c(zx

-

2n, A)

_

and

O2i(*0

=

(ê/(A, ft))"1 (-0(z

9(2n)6(z-zi-X

2t/)c(«i 2nh)

zi +

-

+

^

0(A-2r?ft)

which

A'

=

by

A +

+

2ri)a^ -2^X

+

2^> A)«_1 (*i

the first RLL relation and the ninth

2t?,

can

be

simplified

022(2)

0(A 0(A

~

-

/(A fe) + -

2t?)0(A

-

6{x 277/1) 9(z

277/1 + 2t?)0(A)

-

=

following entry

f(X-2t],h) 9(z-z{)9(X-2nh =

~

2r?'

2??>A

-

relation, where z'

oW

,

n

-

2n)a(z, A)

+

,.

+ 2??)C(^' A)+

A

2??)c^ -2^ =

Z\

2n,w'

—

to

021(2) We still have to compare the

2r?, A

.

~

e{z~Zle(xt~2^

-

_

+

2A

zi)9{X 6{\

-

-

2n)

of the matrix

,

,

,

.

,,

,,

-*(*i -277,A + 277)a(z,A)a-1(zi

277/1 + 277)

277/1)

0.

r

^

aT(z)w

6{X

-

77

£"

x3 +

6{X

+

-

2t?, A

Y.%2 ^jV

2t?)

=

2r?)

+

—

2t?) '"

(z-zxÂ-E^^+E^x ,

0(A)

-

A.A,7 )77',

-ttr(z).

A)) =

z

and

since xi 1. —n and Ai If we rewrite this as a matrix, =

=

we

get

A^1a^{z,Zi,... 9(2-2x)oT(2)

°

coincide. Let us next calculate the

baux,e(z>z2,---

,

zn,X)

+

\„-2v ^

0(2-21)oT(2)a(A-^;(^Â^) ; dûx e(z, zi,...

compare this to the operator

we

=

e(A) 0

If

,ZnX)Ae

conjugation

bx(z,X

zn,

of the operator

A),

notice that both operators

we

b^(z,zi,... ,zn,X)

®dÛXfi(z,Z2,...

2h)

-

,

ai(z,X

=

1

(7re/(A, /i)-1o(2i

0(2-21+277)6(2, A) 6(2V)B(z-z1-\+2vh)

^ 0(2, Aj

e(X-2vh)

1

7re/(A,/i)a_1(zx

277, A + 277)

+

ï\

X

-

277, A)

0

-(7re/(A,/i)-1)c(2i-277,A + 277)

21

I

0 -

1

(2

=

|X

P+2n

e(z-z1)e{\-2r,h+2ri)h(

y.

dl,z>X)

a_1(2x -277,A)c(zx

-

2t?, A + 277)

-

0

,,

0(Ä=2t7M

+ 277)6(2, A) 021(2)

,Z„,X)An,e

0

-(îre/(A,/i)-1)c(2i-277,A + 277)

0(2

2h) ®

,zn,X): -4n,1e^(2,21,--.

-

—

(7re/(A, /i)"1o(2i

-

A +

2t?,

2n)

|X

0

21

7re/(A

277)

+ 277, h +

8C*"*^(ia"y "^ &(*, A)«-1 (21

277)6(2)

27?, A + 2t?)

-

0

/(A

m*)

+/{zïFv) -~tl{XlT+2v) a^

-

^ A+2^fc(z-

A)fl"1^1

~

2^ A+2^)

with

0(2t?)0(z

~

h^z)

=

-

zi

A +

-

2t?ai)

j(

d{z'X)

0(T32^

9(z-zi)9(X-2nh + 2n)1/

+-

g(A

_

^

2

i,

,,

-6(^ AJa"1^!

,

„ -

2r?,

A

v

„

,

+ 2r?)C(*i

„ -

,

2r?,

A

^

+ 4t?),

and

621(2)

=

TT"1 (/(A,

h))"1 (-0(2

0(277)0(2

-

0(A +

This

0(z-2x)0(A-277/1 0(A be

can

,

-

simplified

M«) =*.

277)

277/1) as

a(zx -277,

A +

277)6(2,

tNN

-

1

0(277)0(2

-

2x

0(A

.

+(

0(z -

-

zx

+

2?7)0(A

öT^T^)

+

477),,

-

21

+

277)c(2i

A + 277/1) -a(z!

277/1)

A)o_1(zx

-

277)6(2, A)

277, A +

-

-277,A

277, A + 277)c(zx

-

+

27?)d(z,A)

277, A + 4t?)

J

follows

Ö(A 277/1 + 277) ( 0(2 ^/(A,,))-A__^__î (-1

-i,,/,

,s

+

zx

-

-

,

.

,

„

,

-

2x+2?7)0(A-277/1)

g(A_j;2?j) 277

+ -

,

-

277ft,

277/1

-

-a(zi -2?7,A

2

+

AW277)

0(a

+

2t7)

-

w/

,

+ 277))

277)

0(zx

,

6^'A + 2r>MZl ~2ri'x + 4r>)

(A +

,

-

a": {zx

,,

^^-277^ + 277)6(2^)

-

+

277)d(2,A)

.

(z'

,

X

+

277, A + 27?)c(zx

v)b(-zi -

"

2v'A) )

277, A +

477))

,

69

where

=

we

used the second RLL relation with z'

.-(/(A,

we

we

A +

2t?)ô(zi

-

A)a-1(zi

2t?,

used the tenth RLL relation with z'

(-d(zi where

-

2n, X)a(zi,X

2n)

+

b(zx

+

-

2n, w'

_l9(X-2nh

*'

2n)(flx

+

=

-

{nA'n))

6(X-2nh)

z\

z\

=

9(z

^_,

A +

2t?,

-

—

zx

9(X

0(A-z

we

that

zi)0(-2n)

+

which is

b2\(z)

baux>e(z,

proportional

=

us now

zt,...

to

TXl

.

whereas the

z\

2n, X

—

+

4t?))

,

2t?,

=

X +

An,

-

2n)a~1 (zx, A),

+

v

VTX

\

(baUX,e)i(z,zi,.

where

and the

of

+

2t?),

If

we

compare

=

,zn,X)

—t?,

we

perceive

is the ith

(-z

-

^1 »

2r})e{xl

+ ^t

-

-

2» + 21

bûx e(z,zi,... ,zn,X)

=}_^

zz -

bûx e(z, z\,...

+ zi

"

flm

-

,zn,

A).

2n)b(z,X),

-A0(A-2 + Zj)

\t

r

term of

corresponding

y^0(A-2 + 2t) ^

z3 +

and take into account that xi

0(A)

t~

-

,

corresponding expression

n

9(zx

bûx e(z, z\,... ,zn,X).

bn(z)

t-i^c

,-A-/,/

,

IFc(baux,e)n(z,Zl,... ,Z„)IFC

=2

X +

=

*,, A)

=

bn(z)=9(z~zi

110/^

2t?, A

/(A, h),Detûx e(z, z2,... zn) and 7re. zn, X) Ifc^wx^(z, zx,... ,zn,X)

(bûxfi)i(z,zi,... ,zn,X),

check

,,

.

âT(z)a^(zi) J]

used the definition of

mand of the operator Let

+

^~

this to the term of

IpQ

X'

-

used Proposition 6.5,

=

where

z,

-

2t?,

X)9(2n) 9(X 2nh) 2n) 0(A)

-

,

we

=

X +

=

w

A +

=

xDetaux,e(z> z2,--- zn)a(z, X where

X'

z,

2??)) a~1(zi, A),

+

2n, w'

=

2t?)c(î

2n, w'

—

2n, X)c(zx,X

used the fifth RLL relation with z'

~

—

ê%^]%^^) e(z, zx,...

zn,

,

X)

X)Ae.

c%UXfi(z, zx,...

have to check that

c|(z,zi,... ,zn,A)

,

zn,

X)

=

«4~1c§(z, zx,...

ci(z,X-2nh)®ae(z,Z2,... di(z,X-2nh)®cÛX}e(z,z2,... ,zn,X) =

,zn,

X)Ae,

,zn,X)+

:

Ae C(g)(,Z, Z\,

.

.

.

,Zn,A)Ae,—

1

0

-7T~1(f(X, ft))"1c(zi /

\

-

2r?, A

+

0(2-2l)0(A-27)fc-27j) C\Z,A) 0(A-277o) /

.s

2r?)

tt-1(/(A, ft))"1^

0(277)0(A-277fe+2-2l) ö(A-2t?/i)

0(z-zi

0 1

a-1(zi -2t?,A)c(zi

+

/

-

xs

2r?, A

J

t-2t?x x

0 -

2t?,

A +

2t?)

7re/(A,ft)a-1(zi

-

2t?)

\

a{Z,A)

2t?)c(z,A)

+

2??, A)

where

71

{

7r~l(f(X,h))-1a(zi-2n,X

-K~l(f(X,h))-lc(zi-2n,X + 2n) vre/(A

cn(z)

-

nef(X-2n,h-2n)9(z-zi

c2i(z)

\

2n)c(z,X)a-1(zi-2n,X-2n)

)

ci2(z) A C22(z) J

cii(z) C2l(z)

2t?, A

-

-

*

J

2rl)

2t?)

2n,h)e^%^z-Zl)a(z,X)a^(zi +

+

T_2,

=

A

T-2r, X

with ,

„

9(z-zi)9(X-2nh-2n)

,

H2")%-thrzila{z-Ma~i{zi C2i(z)

=

9(z

zi +

—

.

2,,a -2,)*1

-

2n)c(z, X)a~l(zi —2n,X

—

A.

2,-A)'

-

2n)c(zi

—

2n, A),

and .

0(z

.

^Z)

=

zi)0(A

-

0{2"]%2xhrzi)a^ x)a~^ ,J(2n)9(X~2iTh

n t/\ r\ ci2(z) =7Te/(A-27?,ft)

C2i(z)

z-zx)

0(A-2 h) i

(7re/(A,ft))-1

=

+

( 9(z

zx)9(X

-

(-

-

0(A-27?ft

2nh + +

c22(z)

zi +

-

2??)a(zi

/(A~^'^~2T?) (0(z

=

-

-

2??, A

zi +

+

+ z-zi+47?) 0(27?)0(A-27?ft l C{ZI 9(X-2Vh + An)

Now let

us

check that all of the four operators

(Co«z,e)v(«>*i.--- >^,A), The

simplest calculation C2i(2)

J^0{X-2nh + +

9{Z~

for i,j is the

=

+

+

,

n ~

2r?)

correspond to of c2i

^c{zi -277,A

47?)a(gl

-^^ +

+

+

0(A

2?7)0(A -

-

2t?/i + 477)

277/1 + 2?;)

-2^

-

°{Zl ~2^X

2??)cÂ)

+

-2/?'A -2r?)c(zi -Â) 2t?, A

2t?)c(zi

-

AN

,

,.

-

2t?, A

-

"

2r?'

A

-

a(zx -2t?,A

+

277)c(2,A)

+

2i)c(2,

*1)C(*, +

A +

j a_1(2x

2v)

A +

2t7)c(zx

-

277, A)

-

277, A

-

-277.A))

277)0(21

r

c(*!

2??)

A))

-

,

2t?)

„A

,

„

(*x

2??,

~

2X]))

its counterpart, the operator

2^,^-^^ -277,A-277)c(21

gÎ^^j (-9(2

^^

L(zx-2n,X-2n),

(z), yielding -

A.

x

277)o(2,A)a-1(^ -277,A-277)c(zx -277.A)

g(A_2??/i zx

-

.

A simplification

,

we use

we use

0(z

-

reads JL

0(2-2,+gj+77)

^ n=2 e{Xi_Zi+2j_Xj)

77)) y

f ^-î+xi

+

77)

2-z,+a:1+7j-2EJn=2^

+

_Zx

Vö(x1-2t+2x -zi)

/)

+ gl+

/z1=n

(

-2si)

JL

-«+*+d) (^fT^ö(f9(z-zi)9(X-2nh-2n)

.

=

-

0(z

zi

+

the fifth relation with z'

9(z

-

zi +

-

2t;)0(A

-

^

zx

=

2n)a(zi,X)c(z,

the ninth relation with w'

2t?)0(A

-

T-2"TA-2" Zj + x3 +

,

,,

E^2 x, 0(2

-

+

X

=

—

.

,

-

zx,

=

zi, A

2n)a~l(zi,X z'

E,n=a n) ^

2, + X, +

_

2t?, w'

—

z,

A'

=

„

,

=

-

A

A

A

_

—

4„

2t?

—

An)

—

"

alx

77) _

n

^

+

+

=

r,)

-+-+"0 T^2"

p—fePd—-«•x)«"i-2'» A

=

where

A)

g(2,WA-2^+.-,L)^ A)c(zi] ^ a-i{zu

+

—

zn,

cxx(z) yields

of

Cll(2)

where

,

z-z,+3:t+77-2E;=2^-2Ji) / m

z3+

0(A

"

—

+

wi,0=e

ô(^,..)ii

*+^+"»•

-

J—2

(c^ e)n(z, zx,...

The operator

-2m=

*>

-

ft AïC-%+1)«*

~*'

This coincides with the coefficient of

The

*-+•• An+lAJ,

A(n+l),e where

=

we

used the definition of the entries of

(^n+l,e)(1 "+1)Pl,--.

where

we

Lsos,e(z, z2,...

,Zn+i,A)6505,eP,2l,...

used the definition of the tensor

T^2v

,

of

7= (01 n+1)/ [Z,ZX,. ^SOS,e

zn+i,

A),

A)i4^+"e+1) (zi,

Zn+X,

product

,

.

.

.

,Zn+l,A),

(L^+f e) .

.

.

,

Zn+1,

,

A)

_ —

n+1

401)(z-zi,A-2â;p402Sif1)(z,z2,...,zn+i,A). 3 =2

Thus, the identity involving bôs.el«, Since the other relations

can

zi,...

be shown

,

zn+i,

similarly,

A)

and

this

bÛXfi(z, zx,...

,

zn+x,X)

completes the proof.

is shown.

76

Remark:

The

following corollary is very important since it involves the transfer matrix of the SOS model with antiperiodic boundary conditions and connects it to the auxiliary antiperi¬ odic transfer matrix.

We will need it in the next section.

Corollary

4.45 For all X E C

n)

(An!e)^

{hoS,e(z, ZX,

x^n,e where

which the

.

.

.

,

Zn

[f)aux,e\zizl,

=

pi,... ,zn,X)

An,e

for

following identity

,A)

+

zn

A)

>

CSOS,e(z, ZX, +

caux,e\Z)

.

.

.

zlt

defined,

is

Zn,

,

•

•

•

,

A))

X

X) J

zn,

An,e

=

Proof:

The Corollary is easily explicitly proved that

proven while

looking

at the

proof

of Theorem

4.44, where

pL"1^1 n)(zi,... ,zn,A)W,e(z,zi,... ,zn,X)An\en\zi,... ,zn,X) °aux,e\Z,

zl,

we

=

,Zn,A)

•

and

(^n,l)(1 n)(zi,---

,zn,X)An\en\zi,...

,zn,X)cSos,e(z,zi,...

caux,e\Z, Zl,

,zn,A)

=

,Zn,A).

•

Remark: To show that

Proposition

4.46

Pl"1^1 n)ô5,e(z,zi, ..,zn,Xo)An)en)

TÛXje(z,zi,...

=

is

obeyed for Ao

we 1-6-

need four

t>aux,e\Z,

EILi xi->

=

ZX,

.

.

.

,

f°r the restriction of A to

1-e

lemmas.

more

They

Zn, A), Caux>eyZ, ZX,

M(C, V®n)\x=x0

or

Lemma 4.47 Let

a

E

^

V®n and

C. Let

(baux,e(z,

Zl,

.

.

\caux,e\Z, Z\,

are

u(xx,... .

.

.

.

.

,

.

,

(78)

a

function of the set

show that every operator used in the .

,

Zn, a),

preserves functions of A while restricted to

Fp°

,zn,X0)

An\ZX,

fixed to

,xn,

.

.

.

,

Zn,

a)

S.T1Q.

Ao- These functions

Aq)

Zn,

Aq )u(xX,

Zn,

Xq )U{XX,

E

.

Xa

Fd°

.

.

,

Xn,

,

xn)

J\.n\ZX, ,Zn,A), either elements of

(X$)c

value

.

of

.

.

Xg.

Then

,Xn,XQj) EFD

..

.

specific

a

are

(xx,...

isomorphism,

Aq ))

E

,

ir£,

.

Proof:

By definition, bauXte(z,zx,

X%)

u(xi, ,xn,X%) as u(x[,... ,x'n,X% + 2n), EILi xi + ^- Hence, A£ EIU ^ + 2t? + a EI=i < + « (A?)'. Hence, (baUx,e(z,zx,... ,zn, X%)u(xi,... ,xn,X%)) E FD°. The proof concerning cauxfi(z, z\,.. ,zn, A") is analogous, switching +2t? to —2t?. where

EILi
C7

7=1,^

£{-1,1}

n^-^

E

7=1

Limiting

4.7

Synopsis: section,

In this

(cf.

cases

we

/

7—1

;

2

'

'

^

transfer

(An>e)Z:.a: i

n

^

2

2

n

'

'

X

^

,

2

1=1

3=1

eight-vertex

of the

) ^°

/

7=i,ê{-i,i}

t

model

operator Se(Z) in separated variables

want to show how to obtain the

2.23) as a limiting case matrices (cf. Definition 4.35).

Définition

^

E

j=l

of the SOS

+

\7=1

^))

+

te{-i,i} Vi=i n

( IT ^~Z%

family

of

commuting auxiliary antiperiodic

Remark: For this

section,

Proposition Corollary

Ce,aux(z, ing

of

on

need

a

slight generalisation

of the


defined in

4.56

z\,...

,

the space

Let the operators

zn,

äe,aux(z, zx,...

A), de>aux(z, zx,... ,zn,X)

F^-

Then

they define

,zn,

be the

the operator

X),bejaux(z, zx,... ,zn,X),

ones

defined by Proposition 4-12

algebra of

a

act¬

functional representation

ETiV(sl2).

This is we

we

4.33.

a

corollary of Proposition 4-33 in the sense that we may imitate the proof where fact that the weights x% for i 1,... ,xn then took values in a

nowhere needed the

discrete set.

=

84

Definition 4.57 In this

case

the

auxiliary transfer

_^9(X-Xy-z)

-

f^

n

x

9(Xy

+

z3+ A3n)T-2^T-2^

A

J] 9(x% + z3

+

\3=t

Proposition

given by

9(z

+

x

3)

x

n

J]

|

matrix is

A3n)T+^T^

-

]

(83)

.

7=1

For X restricted to

4.58

Ao

E"=i(a;J

=

z%)

+

the

transfer

matrices

defined

above commute. The

proof

The

transfer

is

given

in

[39].

matrices

This reduces the

be considered

can now

transfer

acting

on

a

space

F^0

Fn.

—

matrices to

,^

^9(X-xt-z)

_

^j H

laux,e{Z)-l^

x3) 9(X1-Xy) Ô(z

+

w

n

J] 9(xy

7=1 (n

+

A3n)Tx2^

z3+

+

H 9(xy + z3

-

A3n)T+2^

7=1

Remark:

We want to

analyse the elliptic Gaudin limit.

Proposition

Top)

+

4.59

4t72Ti(z)

+

Let

n

h.o.t.,

—y

for Tauxfi(z).

0

We then obtain

an

expansion

Taux,e(z)

=

where

n

A

n

0'

(^-Ef^+^))2-Ec0)^+^) 7=1

7=1 \

5e(z) J]cWpP (n -

-

where in the last expression

Proposition

we

set y3

=

—x3,j

=

JJ0P

zj

7=1

/

1,...

n

-

zt),

(84)

7=1

,n,

in the

expression Se(z) of

2.23.

Proof: The

proof

is

Proposition

straightforward, taking

into account the

expression Se(z) calculated

To calculate any term,

we

have to look at the expressions \

n

J] 0(xk + zy + Ayn)Tx2* + J] 9(xk + zz

(n 7=1 for k

1,... ,n evaluated dependence on ??. =

in

2.23.

-

7=1

at

n

=

0

only,

since the other

Ayn)T+2^ /

appearing

terms involve

no

85

For the term of second

order,

\~\^=x 9(xk

since the term

2|t?2JP- (n[=i @(xk + zt + Ayrj)TXk v),

it suffices to look at n

-\—Ayn)TXk

+ zz

is

symmetric under

n

—>—?? to the term

nîöPfc+z.+A^T^). We get n

rfi

1

^QîYlO^k \7,7

=

1

7

n

9'

d2

d

K

7=1

7=1

7=

for every k

=

Proposition

2.23 with x%

The

5

In this

,n. This

1,...

=

yields y% for

—

i

—

\ /

=

1

n

7=

/

1

the first

Antiperiodic SOS

chapter,

=

E -J1 j(X* + zJ-gfa +Z3)~Y1 -fp(X* + Z*)

4r?2

V

AyriT-2*')^

+ Zy +

l,...

,

7=1

indicated in the

sum n

1

it also

Model:

n

yields

=

proposition and by

the second

one.

3

to look closer at the

steps of solving the eigenvalue prob¬ antiperiodic boundary conditions with 3 spin-^ particles. Hence, we will work with the auxiliary representation of Proposition 4.33, given by (M(C, V®3), Lûx e(z, zi, z2, z3, A)) with Ai A2 A3 1, the tensor product of three want

we

lem for the SOS model with

=

=

fundamental representations model

We

L-operator of the SOS

(M(C,y®3),L^(z,z1,z2,Z3,A) =Ri°1\z-z1, X-2n(h2+h3))R^2)(z-z2,X(z

2nhs)Re the

=

described in the definition of the

as

A))

and the

corresponding isomorphism of Proposition 4.44 connecting auxiliary representation L-operator of the SOS model. proceed in several steps: first, we construct the auxiliary representation for 77 2 —

z%,

with the

=

(M(C, V2),L^lxep, zi,Z2, A)) correct.

Note that the

problem

of the SOS

example

=

2 is of

no use

isomorphism

of

Proposition solving the antiperiodic eigenvalue only be properly treated for an odd

4.44 is

in

can

fundamental representations.

underlying verify that the isomorphism

iary representation

for

öaua;,e(z,zi,z2,z3, X)

is

77

=

example

of the

of

3. We also

Proposition

4.44

correctly reproduces the

auxil¬

compute the basis of F®3 in which the operator

diagonal.

Note that the representation trivial

n

model, since this problem

number of

We then

and show that the

(M(C, V®3),LÛX ep, zi,z2,Z3, A))

antiperiodic

SOS

eigenvalue problem

is the

treated

simplest

non-

by functional Bethe

ansatz.

Finally, itly.

we

compute

sary and sufficient

n?=i

one

eigenvector

We also show that the

0&

-h-

-

of the

eigenvalue

condition

on

antiperiodic SOS

obtained

by

eigenvalues given

2ri)°(^ ~z3+2v)-

this

model for

77

=

eigenvector obeys the

in Theorem 4.54

e(zy)e(z%

3

explic¬ -

—

neces¬

2n)

=

86

A

5.1

77

Computing

preliminary step:

the


Synopsis: give (M(C, V),Re(z zi, A)), i.e. the basic operator to construct the and representation representation connected to the SOS model from.

We first

Then, We

formulate the

proceed by writing

isomorphism of Theorem 4.44

down the

(M(C,_ V®2), Lf2(012)

phism:

the

—

we

for

2

=

=

L-operators which

R{el)(z

zi, A

-

-

in the

case

n

2nh2)R(e2)(z

by

the isomor¬

A)) (Lemma 5.2)

z2,

-

(Lemma 5.1).

2

=

want to compare

we

auxiliary

and

(M(C,^®2),LêP,zi,z2,Z3,A)) (Lemma 5.3). In Proposition 5.4

a

basis of

basis. Let

case n

=

2. In

they

are

indeed related

Proposition 5.5,

we

V®2 (for A ^ 0) is given and the operator this operator

Diagonalizing

us

then show that

we

Theorem 4.44 in the

was

write down the fundamental

one

representation

in matrix form. Remember that it acts

objectives

is

of the

isomorphism diagonal in this isomorphism.

EV^pfe) (M(C, V),Re(z

the space V which is

on

a

of

this

by

d®2(z, zx,Z2, A)

of the main of

by the isomorphism

show that

—

zx,

two-dimensional

A))

com¬

plex vector space with basis e[— l],e[l]. We need this representation to formulate 2. It is given by isomorphism described by Proposition 4.44 in the case n

the

=

'

0(z

ae(X,

zi

a). Let

This

matrix

ce(A,z-zi)

=

de(X,z-zx)

=

Dete(z-zi)

=

Lemma 5.1 In the is given

=

0

'

+2n

0(A)

-2»7

TI

0

-

zi)

9{X-2n) 0(A)

0

^+277 '

A

)

0(z-zi+2t?)

0

9(z-zx+2n)9(z-zx+2n)Ix,

be calculated

can

XX

r;

9{z-zi+X)9(2n)

0

6(z

rp-2n

e(z-zi)-^à

9{X-z+z1)9{2n) 0(A) 0

coincides with

write down the

A2,e(zx,z2,X)

2t?)

0 =

representation

us now

zi +

o

6e(A,z-zi)

where the determinant

-

=

by using

(M(C,

the formula

given

V),Lûxê(z, zx,X))

isomorphism of Proposition

in

Proposition

by Proposition

4.42 in the

case n

=

4.15

4.41.

2, i.e. the

A2,e(zi,z2X).

case n

=

2 the matrix

A2,e(zi,

z2,

A)

End

G

(V®2)

C

End

(M(C, V®2))

by

-4.2,epl,Z2,A)

Its inverse is

/I

0

0

0

9{X+z1-z2)9(2rf) ö(A+277)ö(^x- -Z2) 0

=

0

0

9(z1-z2-2n)9(X) 9{X+2ri)9{z1-Z2) 0

(85)

0 1

J

given by

(A2,e) 1pl,Z2,A)

/I

0

0

°\

0

1

0

0

=

0

9(X+z1-z2)9(2n) 9(z1-z2-2n)9(X) 0

fl(*i-Z2)g(A+2T7) 9(zi-z2-2ri)9{X) 0

0

1J

(86)

87

Proof: This is proven

by filling

form of the inverse no

into the definition 4.42. The correct

appropriate operators

in the

p42,e)-1pi, z2, A)

by multiplying with

is checked

its inverse

involving

residual calculations at all.

Let

us

now

write down the

V®2),Re(z zx, X 2nh2) ® Re(z operators äf2(z,zx,Z2,X), bf2(z, zx,Z2, A),

representation (M(C,

Lf2(z, zi,z2, A)) which consists of the cf2(z, zi,Z2, A), Jf2(z, zi,z2, A) in order to compare Z2,X)

=

it to the

—

—

—


(M(C,^2),Lê(z,zi,z2,A)). Lemma 5.2

The entries

ofLf2(z,zi,z2,A)

given by

are

af2(z,zi,z2,X) ae(z

—

zi, X

—

2??ft2)

äe(z /

-

&i)6p

+

zi

-

,A

2nh2)

-

®

ce(z

\

0

0

0

0

a22

0

0

0

Û32

«33

0

0

0

0

au

an

V

A)

z2,

=

-27?

J

with an

=.

«22

=

0,32

=

9(z-zi+2n)9(z-z2 + 2n), 9(z -zi + 2n)9(z z2)9(X + 2r?) 0(A) -

(0(2r?))20(A

+

z

0(A 0(z

«33

-

z2 +

z2)0(A

-

+ zi

-

2t?)

2t?)0(A)

-

2t?)0(z

z

-

-

zi)0(A)

=

0(A 2t?) 0P-z1)0(z-z2)0(A 0(A) -

A + 1

/

the residues

of

sum

1

of

the

function

4??)

-

-

vanishes.

-

corollary while treating the antiperiodic SOS model.

antiperiodic

The

5.3

will

we

—y

that the

2r?)0(A z2 + 2t?)0(A z3 0(A zi)0(A z2)0(A z2)

zi +

-

being

V

0

0(A

-

0

0

V where

0 ,V8

0

SOS model in the

case n

=

3

Synopsis: Now let the

look at the

antiperiodic SOS model

auxiliary antiperiodic transfer matrix

fixed A

Then, as a

us

Ao

=

sign

in the

case

case 77 77

=

3

=

3.

We first write down

(cf. below).

Note that

we

commutativity. eigenvector of this transfer matrix in Proposition 5.13. (This serves finding eigenvectors of the auxiliary transfer matrix seems feasible.)

to ensure

describe

we

in the

that

an

5.14, we find the corresponding eigenvalue and show that it indeed obeys the properties of Proposition 4.54 which are sufficient and necessary for it to be a common eigenvalue of the SOS antiperiodic transfer matrices as well. The eigenvector of the SOS transfer matrix corresponding to the one of Proposition 5.13 would then be given by In Lemma

Theorem 4.55. Remark:

Here, Xy E

we

first need to

{-n, ??}, i

=

verify that,

1,2,3

,

A

^

We first want to look at the

since

we

had to restrict A to A

=

Xi + X2 + x3 with

0.

eigenvectors and eigenvalues of the antiperiodic SOS transfer

matrix

Taux,e(ziZliz2,Z3,XQ) where

Ao

=

Xi + X2 + x3, and then

=

use

(bauXfi

+

Caux^)(z, Z\, Z2, Z3, A0),

the obtained results to look at the

antiperiodic

SOS transfer matrix

Tsos,e(z, zi,z2, z3, A0) where

A0

=

n(hi

+

ft2

+

ft3).

=

bSos,e(z, zx, Z2, Z3, Ao)

+

cSos,e(z, Zl, Z2, Z3, Ao),

107

Proposition

5.13

An

eigenvector

auxiliary antiperiodic transfer

to the

matrix is

given

by

1 1 ill

VQ,aux

^2 ^2 ^2

=

called the in EC linear with z is a + ® meromorphic map commuting End(V W)

Definition 6.4

(Representation)

A

=

=

L-operator. The

L-operator obeys the relation

412) (z

-

7i;)413) (z)L^ (w)

=

This condition is called the RLL-relation.

423) (w)L^ (z)R^ (z-w).

(100)

112

Remark: The

is

L-operator

written in the form

usually LJz)

_

f

ar(z), br(z),cr(z),dr(z)

where

tions defined

by

br(z)

ar(z) cr(z) E

End(VF)

End

E

dr(z) are

(101)

(V®W),

meromorphic

in

E

z

C and

obey

the condi¬

the i?LL-relation.

The i?LL-relation written in terms of the above operators

yields

the

following

sixteen

expressions:

(z

—

(z

—

(z

—

(z

—

2n)ar(z)br(w

=

2nar(w)br(z)

w

+

2n)br(z)ar(w

=

(z

br(z)br(w

=

br(w)br(z),

+

w

(z (z

—

—

w)ar(z)cr(w)

w)ar(z)dr(w

+

2ncr(z)br(w)

w)br(z)cr(w

+

2ndr(z)ar(w) (z

—

(z

w)ar(w)br(z)

2rj)cr(w)ar(z

—

+

=

(z

=

w)br(z)dr(w)

w

+

2n)dr(w)br(z

w

+

2n)ar(w)cr(z

=

2nar(z)cr(w)

w)cr(z)br(w

+

2nar(z)dr(w)

w)dr(z)ar(w

+

27?6r(z)cr(iü)

=

p

2n)br(w)dr(z

=

2nbr(z)dr(w)

+

(z

cr(z)cr(w

=

cr(w)cr(z),

=

27?crpü)r{z,h)v){fi)

=

(cA,r(z,h)v)(fi)

=

(dA,r(z, h)v)(n)

=

=

E

C,

is

defined

aA,r(z, h)v(n)

ek, ek

(z

=

bA,r(z,h)T^v(pi) CA,r(z,h)T+2r}v(p) dA,r(z, h)v(ß)

zq)),

follows

-

n)v(ß),

z0 + ßn +

=

(A

=

(A

+ -

^)nv(ß-2), ß)nv(ß

(z-z0-ßn

=

—

+

2),

n)v(ß),

+

f(ß)v(ß)

functional Verma module

V^r(zo).

restrict the above

=

as

Fi,L^r(z

=

Fx.

It is called the

b) If we 2k)

End(Fi), A, zq

(äA,r(z, h)v)(ß)

f(h)v(ß) where

£

Yangian is given by (W

the

representation to Fx FxêÂ-2k\ken}) and defining the basis of an infinite dimensional vector space,

set

=

the evaluation Verma module

V^po) by

of

means

the

we

U(A

—

recover

functional representation

(FD,L^(z-z0)). The

L-operator

FXR

c

looks the

same as one

defined

in

a),

but its action is restricted onto

FX.

Proof:

a)

The statement is proven

b)

This is proven

by checking the rational i?LL-relations.

by comparison.

Remark:

a)

For

a

representation of the Yangian,

we can

define its operator

generated by âr(z,h),br(z,h),cr(z,h),dr(z,h),h

b)

We to

can

generalize

the notion of

operators depending

functions !Fn which

depend

on

functional

a

several

on

weights

End

E

algebra as

(W),

representation

or

operator algebra

C, acting on the space of weights. The operators read

the before-mentioned

=

Proposition

a)

The

algebra

ßX,... ,ßn E

ar(z,hx,... ,hn),br(z,hx,... ,hn),cr(z,hx,... ,hn), dr(z,hx,... ,hn),hy with hyf(p,x,... ßn) ujpi,... ßn) for every f E Fn and i l,...

E

=

,

,

the

where W Ç T.

,

End

(Fn)

n.

(Quantum determinant)

6.10

following

element

of the operator algebra

Detr(z)

=

(dr(z

-

2n)dr(z)

-

is

a

cyp

central element:

-

2n)br(z)).

(106)

It is called the quantum determinant.

b) If we have two finite dimensional irreducible representations of the Yangian named (Vx,Lx(z,hx)) and (V2,L2p,/i2)) with quantum determinants Z)ep(z) Deti(z)\yx and Z)ep(z) -Depp)Iy-2, where Depp) and Depp) are scalar functions and 1,2, are the identity matrices on Vy, then the detrminant of the tensor prod¬ Iy,,i uct representation (Pl®!^, Li^p, hx, /i2)) is given by Det(z)i®2 Detip)-Dep(z)=

=

=

=

•^vx®V2!

where

Ivx®v2

is

the

identity

matrix

on

Vx

®

V^.

115

Proof: This

be checked

can

by explicitly commuting all the generators of 3^(sZ2) with part of the proposition, cf. [47], p.69.

the quan¬

tum determinant. For the second

Proposition 6.11 ([46], pp.19 -20) Let pi,... ,zn) 1,... ,n. Let F® be the space of functions defined before. Let

K,AZ)

J\(z

+ ^ +

A*tj)

and

A+T-p)

=

A%

E

N,i

=

[J(z + z,

-

AtV)

.

7=1

7=1

Let the

and

diag,

-

n

ti

=

E Cn


Y^

E

End(FnJ) for

(Y±f)(xu. .,xn)

i

1,...

=

,n, be given

=

(A±r(xy)T±2y)(xi,...,xn)

=

A^r(xy)f(xi,...

by

=

,Xy±2n,... ,xn).

Then the operators n

daux,r(z,Ai,... ,An,Zi,... ,zn)

JJp + a;*),

=

(107)

7=1

+ X,

Z

baux,r(z,Ai,...,An,zx,...,Zn)

(108)

-J2H—~Â+r(Xy)T+2v,

=

X

7=1 3 j=l

Xy

o

>

n

cauXyr(z,Ai,... ,An,zi,... ,zn)

Z

J2H

=

7=1

*l An,r(^)Tx2v,

(109)

3^1

n

Detaux,r(z, Ai,... ,An,zi,... ,zn)

JJp

=

-

zt

-

A^

2t?)

-

x

7=1

(z-z.

x

define

an

(110)

A.t?)

+

operator algebra obeying the RLL-relations of the Yangian [Vpp).

dr(z,Ax,... ,An,zx,... ,zn)

The operator

is

defined implicitly by

the

quantum determi¬

nant.

Remark:

Taken

together

aaux,r\Z, Ai,

.

.

.

as

,

entries of

An, Zi,

define the operator End

.

.

a

2

x

Laux>r(z, Ax,

matrix, the operators

2

,Zn),...

.

aaux,r\Z, AX,

,

..

,

An,

z\,...

,

.

.

.

An, ZX,

,

zn).

It is

=

.

.

.

,Zn)

matrix

on

V with entries in

(F%).

The above defined representation coincides with the Xy

a

—

y y for every

Corollary Then the

6.12

%

=

Let

1,...

n

=

,n and then consider the

l,z'

=

z

—

n

and xx

=

one

given

representation

—zi +

hin.

operators

äi,rp')

=

(z'

biAz')

=

Pit?

ci,rP')

=

di,r(z')

=

-

in

zi+hin

+

n),

At?)T+2P Pit?-At?)T-2P +

(z'-zi -hxn

+

n),

[46]

if

substitute

we

LauXjr(z)

I

)

.

116

in End

of

the

(!FX)

are

functional

operator algebra associated

Verma module

special

A

6.1.3

the

of Proposition

class of twisted

to the

6.9

finite

quotient module

dimensional

a).

representations

Remark: This type of

is needed to describe the

representation

conditions of the XXX chain The

simplest way to following proposition

as

construct the wanted class of

of the

use

non-periodic boundary

by Sklyanin [47].

formulated

and then make

of

case

representations

is to start with the

Hopf algebra property

of

3^pZ2).

6.13 Let

Proposition

A={*") be

an

element

of GL(2,V). Then (V, A^ ®I)

is

im

a

representation o/^pp).

Proof:

straightforward by checking

The way to prove the statement is

412) P) (A ® I) (I ® A) By writing

out the left and

Corollary

6.14 Let

By

the

of

means

hand side

right

(W,Lr(z))

(I ® A) (A

=

be

a

®

explicitly,

I)412) (*)

we see

A^Lj-

•

that

representation ofy(sl2).

Hopf algebra property ofy(sl2) (W,

the rational RLL-relations

they coincide.

Let A E

(z))

is

a

GL(2,V). representation of

y(sh). Remark: In

Sklyanin [47], the matrix A E GL(2,F) was used to define chain, cf. the following section of this chapter.

the

boundary

conditions of

the XXX

The

6.2

to establish

isomorphism

separation

of variables for the XXX

chain

Synopsis: Here,

we

first write down the

auxiliary representation of Definition 6.11 for At

=

1 with

1,... n, since we want to compare this representation of the Yangian with the nfold tensor product of its fundamental representation (Definition 6.15). i

=

,

Since the

auxiliary representation is a functional representation isomorphism from the space of functions on which it acts to

an

nfold tensored fundamental

Then, since the isomorphism formulate

one

tation with

is

inductive step in =

...

=

An

=

as

are

in the

ipir,pZ2)

case

-

1 to

a

tensor =

...

then have to define on

which the

Proposition 6.16.

constructed

Proposition 6.19, thus connecting

auxilary representation with Ai

an

In

Ai

acts. This is achieved in

representation -

we

the space

an

inductively, we auxiliary represen¬

product of a fundamental representation An_i 1, where the parameters zx,...

=

=

and ,zn

fixed.

Proposition 6.20,

tion 6.19

an

we

show how to construct out of the

isomorphism

isomorphism given in Proposi¬ product of fundamental

with respect to which the nfold tensor

117

representations of the Yangian and the auxiliary representation of the Yangian with 1 for i Aï 1,... n are isomorphic. =

—

,

In the quantum case,

we

volved in

the XXX chain of order

constructing

which will be defined

want to find

shortly

isomorphism that

an

maps the

[47] Lr(z,zi,... ,zn)

n

-

representation in¬ End

E

(y®(n+1)),

auxihary representation of Proposition 6.11 with for 1 i A, 1,... ,n (Fn°,LauXtr(z,l,... ,l,zi,... ,zn) LauXir(z,zx,... ,zn))- The auxiliary representation is characterized by the property =

to the

-

=

—

that the operator

aûx(z, zx,... zn)

To construct such

an

Remark

,

we

diagonal.

first have to

specify the results

of

Proposition

6.11.

(Auxiliary Representation):

The definitions of be

isomorphism

is

given by

the

Propopsition

6.11

being understood,

let the


following operators 77,

r(z, Zl,

.

.

.

,

Zn)

=

JJP

Zy + T] +

-

Xy),

1=1

bauxAz,ZX,...,Zn)=±i[*ZZ^V 3-hy

7=1

X%

**

Cav,x,r(z, Zl,...

J7

Zl

_

3^y

7=1

-I-

—

_

^

f[(xt

Xl

"•"

X3

n

J ,Zn)=^2Y[ X*

+

Z3

Z,

^ +

Z3 +

-

n)T^\

J=1 n

4-

II ^

^T +

X3

Zy +

-

Z3

~

V)T^,

3=1

n

Vrtaux,r(z,Zl,... ,Zn)

=

JJp-Zj -2n)(z~ Zy + 27?). 7=1

where the operator

dauX:7.(z,zx,... ,zn)

°aux,r\Z, 1, for =

o

=

Det,

a,

b,

c,

d, and

{pi,... ,xn)\xy

Definition 6.15

E

.

.

.

,

1, Zi,

is defined

.

.

.

,

Zn)

=

implicitly,

Oaux^r [Z, ZX,

the values of the operators

{-7?, 7?}

for alii

(L-operator)

=

put

we

.

.

.

(xt,... xn) ,

,

Zn) E D

1,... ,n}.

Let the

L-operator

Lr(z,zx,...,zn)E End(V®(n+V) be given

by

Lr(z,zx,...,

an

ofV®n.

-n)

=

R^l\z

-

zi)...

RW(z -zy)... R(°n\z

-

zn).

(112)

isomorphism between Lr(z, zx,... zn) and LauXtT(z, z\,... zn), let us first isomorphism Ipc that maps a basis of F^ to the standard tensor product basis

To state the state

znfl

,

,

118

Proposition

a)

A basis

([rai...ffn], Ifc)

6.16

of T^

is

given by

{[»Vi...°v,]

=

[JJ ^,17,0-,]

|

o; E

{-1,1} for

all i

1,...

=

,

n},

7=1

where

(—zi

by

[Iir=i îV^,]

+ ain,...

find

,

the

mean

we

onn)

—zn +

meromorphic représentant of

a

equivalence class of functions which

D and

E

else

everywhere

zero

is

D. Note that

on

one we

at

can

this class.

Fn ~^ V®n zs 9iven by Ipcpi...^] epi] ®...® e[an] for all possible combinations of a% E {—1,1} for i ® e[an] 1,... ,n. Here, e[ax] ® is an element of the standard tensor producv basis ofV®n.

b)

The

isomorphism Ifc

'

=

=

...

Proof:

a) [r0-1...o-n]

['Vi.-.o-jJ Thus,

[/]

=

b) By

zero

Er=l,

(L^)rp,Zi,... of the

means

where I2 E

End

Corollary

6.17

(V)

>

is the

Let

In_i

2ri)

=

E

'

pi,... ,an)

at

E D

and

as

=

P2

identity

®

lFc)LaUx,r(z, ZX,

matrix

6.12 and

on

define

we can

.

.

.

,

Zn)(I2

Ip1),

V.

Proposition 6.16

Lûxr(z,zx)

is

equal

to

Rr(z

—

(V®2).

(An,r(zi,..- ,zn),An,r(zi,- ,zn)) End

EKUî

(V®^~1^)

-%i~

An'r The matrix

Zn)

By Corollary

Then the matrix

b)

[/] E J^jf an)[rax...an\-

one

,zn)):

operator in End

Definition 6.18

a)

•

isomorphism Ipc defined above,

Laux,r(Z> ^>

an

•

•

value

D.

construction.

By

as

a

at all other

write every element

we can

Remark

zi)

yield [rai...o-n] has points of

is constructed to

has value

2rÙ-

be the

Let

us

An,r(zx,... ,zn) =

V o.-\zx

An

>

n =

,

Z*

~

n

Z

ZX)^T JJ 7=2

Z3 + X3 + 7?

—

Xy

^7j=2

*

'-

Zy

X3

n

Zll

3=2

written in terms of the operators

Ipc again

(&a«*,r)2i(«, «i,

(&Lr,r)22p, ZX,...

>*n)

=

zn)

=

,Zn)

=

,

(*

and

z,l

inverting

the

compare this to the

~

zl +

2«)&Lz,r p, Z2,

.

.

.

,

isomorphism

Zn),

,

-

conjugated matrix,

we see

c^(z,zx,... ,zn).

.

.

,

Its

1

conjugation yields

\( (2-2l)0(2)

0 _

(7T-)-1o(2i -277) J \ X

V.

1

0

(a- 1c)(21 -277)

(tt-)o-1(2i -2t?)

f 011(2) V

corresponding coefficients

and their

27?ap)(a_1c)pi (27?a(z)(o~'1c)(zi

C2l(jz)

2t?)

+

p

—

are

given

cup)

=

zi)c(z)

=

zi)c(z)a(zi))a_1(zi) (2nop)cpi) + (z- zi)cp)api))a_1pi) (z zi + 27?)a(zi)c(z)a_1(zi). -

2r?)api)

+

(z

-

=

=

—

^

=

x

(tt-52770(2)0(21 -277) 022(2)

simplifications

—

,Zn)Ar

2770(2) (2-21+277)0(2)

0

zn),

,Zn).

that both coincide.

A~1C%,{z,ZX,... -(tt-)-1c(2i-277)

,

X

-

.

It remains to check the operator

where the

—

2n(-K+)aÛXjT(z, z2,... zn)a%u^r pi, z2,... p Zi^^pi 2??, Z2, Zn) b---

If

oûxr(z, z2,... zn)

-X

~\~

-2nr

zx-Zy -*•-*

-

2rl),

]=1

'

or

z3 +

below.

122

Here

we

used the ninth relation first with z'

(7T_)c2ip)(-2??cpi + (z -zx +

2n)a(zx

due to the fifth relation with z'

c22p)

=

(-2nc(zx

—

we

So the

-

+

-

2n)a(z)

2n, w'

—

(z

-

=

zx

—

-

\

2n)a(zi)c(z)a~1 (zi)

zi +

i

\

n

n

\T^

TT M

t

i>c >

=

zi

2n, w'

=

(Caux,r)l2(z, Zl,

—

2t?

Zy%

0

=

-

.

.

.

.

.

,

Zn)

IpCpT?

=

(caux,r)2l(z,Zi,... ,Zn)

=

(coW)22p, zi,... zn)

=

JJ

—

+

Z7

J-

J^l

X-,

+ 7?

^-—p Jjn

A/1

^

=

X

=

(Caux,r)2l(*,3l,-" »^)

=

(caus,r,)22pj ^l,

=

i^nj as

,

completes the proof.

Zl+Z3

X3

J

X

_Xj

~

Z*

~

2r?))JFC'

3=2

n

II

_

P

caux,r \z,

is defined

-Zy +

z3)

^cget

we

,

Zl +

Yl(2n

+ 2V-2wU~1

oûxr(z, z2,... zn),

~

V

^+^,=2

*«

2;1^1_^+2r?i^

2n)aûx>r pi, Z2,

.

.

.

,zn)(aaux^r)

z2,

,

Z„)

X

pl,Z2,... ,Zra),

Zn) 2??p_)aû:u:!.)r p, Z2, {aûxA^^l 2t?, z2,... zn) .

.

X

,

-

,

°> P —-^lJCa^PZ,

appearing

Since the quantum determinants

daUXjr(z, zx,... zn)

ü^1

'

.

x

entries

\

V

gl^1~^

(~

rewrite this in terms of operators

(Caux,r)l2(z,Zl,... ,Zn)

2??

,Oj

Y\(2n-Zy + z3)Tx2npl,

7=2^=2^

(Caux,r)n(z,Zl,--- ,zn)

Zi +

-

~

n

7fc( ^

x

same

2t?)

0,

x

•

—

=

n

71

,

,Zn)Ar

.

3=2

3=2

and

zx.

z.

=

n

the

=

-

C® \Z, Z\,

z

j^7,;=2

Zy +

-

Zl1

are

=

yield

,

X

These

w'

=

-

2n(ir-)a(z)a~l pi (z zi)c(z)

cûx r(z,zx,... zn)

*=2

we

2t?)

-

z,

—

(caux,r)ll (z,zi,... ,zn)

If

2n)

=

matrix looks like

The entries of the matrix

V

2t?)c(zi

-

-

-

—

0

/

2n, then with z'

2n)c(z))a~1(zx 2n) 2n)a~l(zi 2t?) p Zi)c(z),

•Aj.

(z

—

zx)c(z)a(zx

2n)a(zx

zx)c(z)a(zx

zx

=

z.

=

zx +

-

used the fifth relation with z'

conjugated

(z

-

2n)c(z))a~l(zi

zx

=

2n)a(z) (z

where

-

w'

zx,

=

were

in

Z2,

.

AAc%(z, zx,

shown to be

implicitly by

.

.

...

,Zn). ,

zn)Ar-

multiplicative

means

in

Proposition

6.10

of the quantum

determinant,

this

123

Proposition be the

Let

6.20

identity

matrix

An,r(zi,

on

,zn)

the matrix

defined before

and let

I2

End

E

(V)

V.

Then

(I2

A~Azi,... ,zn))Lr(z,zi,... ,zn)(h®An,r(zi,... ,zn))

®

(116)

LauxAz>zi->--- >zA-

=

Remark:

Written down in the components of both

-^7i,rPi5

•

"71,7-Pl>

•

zn)ar[z, z\,...

,

,

Zn)br\Z,

ZX,

An,r\Zli---

,

Zn)cr{Z,

ZX,

--n,r\zl,

,Zn)U,r\Z,Zi,

•

•

•

.

.

.

.

.

.

,

Zn)An%r\Zl,

,Zn)An^\Zl,...

.

,

Zn)An,r\Zl,

•

aaux^r\z,

=

,

Zn)

,

Zn)

,

Zn)

•

get the following four identities:

we

zn)A.np\zx,... ,zn)

,

.

L-operators

=

=

zx,

baux^{Z, ZX,

.

.

.

Caux^r [Z,

.

.

.

ZX,

0,auxr [Z, ZX,

=

.

.

,

zn),

,

Zn),

,Zn), ,Zn).

.

Proof:

Let

proof

us

the four identities

identity involving

the

wrote down in the remark instead of

L-operators. The proof

-4.2)rpi, z2)

definition

By

we

=

A2)?.pi, z2).

A2\(zx,z2)crp,zi,z2)^42)rpi,z2)

reads

shown in

a

similar

Let

by induction. Let

is

us

prove

just

cûx>r(z,zi,z2),

=

us

identity,

one

proving

start with

e.g. the

n

one

=

the 2.

which

since the other identities

are

manner.

A2j, pi, z2)cr(z, zi,z2)A2,r pi, z2)

=

-42)rPi,z;2)crp,zi,z2)^l2)rpi,z2)

=

^l^(zi,z2)(crp-zi) ®ar(z z2) + dr(z A2l(zi,z2)(cr(z zi) ® aÛXtr(z,z2) + dr(z -

-

zi)

-

-

zi)

®

cr(z

z2))A2,r(zi, z2)

=

c^Xirp,z2))*42,rpi,z2)

=

®

-

caux,r\z, z\iz%)i where

we

and the Let

used the definition of the

Lr(z, zi, z2),

the

identity of Rr(z—zx) and

Lûx r(z, zx)

preceding proposition.

us now assume

that

(A~lr){-2-n+l\z2,

...

,

Zn+X)or(z,

Z2,

°aux,r\Z, holds true for

some

fixed

n

for

o

=

a,

b,

.

.

.

,

Zn+X)An2,r'n+1)(z2,

...

,

Zn+l)

,Zn+l)

z2,

c, d.

We claim that under these circumstances it follows that

An+lAZl^-- ,Zn+l)or(z,Zi,... Zn+X)Antr(zX, ,

=

for

o

Let

=

us

a,b,

zli

•

.

.

,

Zn+l)

,zn+l)

•

c, d.

show it for

the other operators

.{2...n+1),

Ah,r

°aux,r\zi

.

cÇuxrp,zi,... ,zn+i), are

,(l...n+l)/ ,zn+i)An+ijr >(zi,... x

pz2,...

since the

strucuturally completely x

,zn+i)

proofs

similar.

of the identities

First note that

-(1...77+1)/ =

'

An+1^T

"'auXtryZ, Zl,

(zx,... .

.

.

,

'Cll.r(*!>••• iZn+l)(cT(z-ZX)®CÛX!r(z,Z2,...

x ,

zn+x).

Zn_)_iJ

=

,Zn+l) +

by

involving definition

-rj

Hence,

124

+dr(z-Zi)®dauXjr(z,Z2,... ,Z„+1))Aî+l,rPl,--- ,zn+i)

A~\lAZ^--- >^»-t-l)(-^,r)(2"'n+1)(2r2,..dr(z

-

Zi)

®

dr(z, Z2,

.

.

.

,

,Zn+i)prp-Zl)(g)6rp,Z2,... Zn+X)

Zn+l))^2r-n+1)(z2,

.

.

.

,

Zn+i)Aî+l,rPl,

•

•

,

used the

preceding proposition,

°aux,r(z> Z2,... zn+i), ,

the definition of

the

assumption

Lr(z,zx,... zn+i) ,

on

Zn+l)

=

,Zn+X),

,

we

+

,

An+lAZl'--- >zn+l)dr(z,Zi,... Zn+l)An+l,r(zi, where

=

the operators denoted

and of

An+XjT(zx,... zn+x). ,

Remark: The last

with

states the

corollary

arbitrary boundary

Corollary

6.21

Let A E

Then, for Laux,r(z, zx,...

isomorphism between the representation of the XXX-chain

conditions and the

auxiliary representation.

GL(2,V) and I2 be the identity matrix on V. ,zn) E End (7(0) ® F%) and Lr(z, zx,... ,zn)

A^Laux^r(z, ZX,... ,Zn)

=

(A® A~j.pi,... ,zn))Lrp,zi,... ,zn)(h

®

E

End

(h® Ipc)

(V^

®

(H7)

An,r(zi,... zn))(h ,

®

LFc)-

125

Appendix

7

of

Spaces

2:

elliptic polynomials

Definition:

a)

öp)

Let

T

> 0. Let

Z + rZ and F*

=

(Cx)2

~

the group of group

homomorphisms

T-+Cx. Let xF*. Then

define the homomorphism

where 1 and

-

b)

we

®k(x)

For x G T* let

obeying r

+

V*

—y

ET by

be the space of entire

following property: f(z

the

:

x ^ 7;—~(mx(r) 2-k%

:

—

r

hrx(l))

oriented basis of F.

are an

r

+

sr)

+

r

holomorphic functions f(z) of level =

e~k(TS

+2sz\(r + sr)f(z)

k

for all

E T. Hence

st

&k(x) f(z

+

r

+

@k(x)

The dimension of

st)

=

ifiz) holomorphic,

e'mk(Ts2+2sz)X(r + sr)f(z)

=

is 0 if k

|

entire

=

for all

r

0 it is

one

+

E

st

if

V}.

(x)

=

0 and 0

otherwise. For the elements of these function spaces,

Proposition

obtain the

we

following

result:

E.l:

The function of

E

z

C

f(a, wx,...

,

z)

wn,,

eaz

=

Yl 6(z + 7=1

belongs

9n(x)

to

X{r On(x) is

with

function in

Every representation

is

unique

st)

+

up to

Proposition

/ the

dlrig,

counted with

zeroes

of g then

Xg(r

+

ST)

do not

Let

of its

vt\u,

+

wn,

E

v

,

z) for some constant C. This wn) if one requires the Wy to

[0,1)}.

properties of theta functions that the number of zeroes

multiplicities,

wt, i

=

©(x)n

of g E

g(z)/f(z,wi,... ,wn,a) depend on z) and regular, 1,...

,

hi F

ls

If wx,... ,wn denote

n-

doubly periodic (since X/p + st) thus constant. Uniqueness follows as is

and a

is

n, and Xg-

E.2:

ET be the elliptic

curve

determined

power. The map P:

symmetric

,

E.l:

uniquely determined by the Corollary

{u

=

It follows from the transformation

JdF

f(a, wx,...

permutation of the (wx,...

be in the fundamental domain F

Proof of

(-ljM»^8^-2«^^).

=

°^ the form C

zeroes

corresponds

mod

V,

is

at most

Sk(E) subject

(@k(x))

injective (i.e.

one

element of

to the condition that

to

by

~^ a

and,

r

for k

>

Sk(E), sending given

Ofe(x))-

X^=i w3

Its —

set of

0, let

an

zeroes

Sk(E)

EjSk its k @k(x) to the set =

element of

[w[,... ,w'k]

E

Sk(E)

there

image consists of classes [wx,... ,Wk] E 4>{x) + ^^, ^ being the image of (1 + r)/2

\nE.

Theorem E.3: n

Let zi,...

,

zn E

C be pairwise distinct modulo V and x

T* such that

\~_, z%

^ 4>(x)

+ kô

7=1

mod F. Then for any

f(zi)

=

fi,i- 1,...

/1,...

,n.

,

fn

E C there exists

a

unique

function

/

E

@n(x)

sucn

that

126

The

interpolation formula

is

given by

r2-Kta{z-z,)9{z

-

+b)

z3

~

V^ t f(v\ I\z)-l^ê

Q(z- z3)

TT

ll M*.-*.\ 4i*0{z>-z>r

0(h\ w

with

-s;0"x(i)-=) 77

b

~4>ix)

=

-nö +

..

^Zy- k—jj—• 7=1

Proof of Theorem E.3: The function

f(z)

of evaluation

points

has the desired transformation properties. The condition that the

ensures

appearing

denominator does not vanish

The function is

vanishing identically.

at

Let

are

unique since the difference of any two such functions is points zx,... ,zn. By Corollary E.2, since ]C"=i z* ^ ^(x)

turn

us now

that

n

doubly

our

interest to

A±(z)

E.4:

Suppose

A±(z)

that

a

theta function

+

kö,

it vanishes

special classes of difference equations involving coefficients

-

2t?)

+

A+(z)Q(z

+

2n)e(z)Q(z),

en(e=F2n"-2,rt£"=iz«s(-l)r+Ä)

E

non-trivial solution of the above difference

a

sum

identically.

en(e^2msn^-2m^=i^s(-l)r+s).

E

Proposition To obtain

the

periodic functions:

A{z)Q(z with

on

with

n even.

equation,

n

2

Q(z)

=

eaz]j9(z + w3)

E

02(x)

and

ep)

E

9n(e2m^=i^s).

3=1

Q(z)

The character of

is fixed up to

parameter by the Bethe Ansatz equations

one

n

n_

2

2

JJ

A+(-Wy)

9(-wl

w3-2n)

+

=

eAr'aA-(-wl) JJ 9(-wt

J-l,3&

for i An

=

1,...

,

n

and wt

^ w3

,

e[Z)

form

2n),

3=1,3&

explicit formula for ep)

(Q(z),e(z))

+ w3 +

_

mod

is

T, for

^ j.

given by

A+(z)Q(z

"

i

+

2t?)

A_(z)Q(z Q(z) +

-

2n)

elliptic polynomial solution. Conversely, if (e(z),Q(z)) is an ellip¬ polynomial solution of the above difference equation, then there exists a solution wn of the Bethe Ansatz equations such that a,wx,... Q(z) is of the above written form an

tic

,

up to

a

constant C and

Proof of

Proposition

ep)

is also of the above written form.

E.4:

A necessary condition of the above difference is that all terms has to be

are

e2m ^"=iz%.

equation having

theta functions with the

same

a

character.

non-trivial solution

Q(z) ep)

So the character of

127

Q(z) be the above written function. The formula for ep) transforms as requested, but may be singular at the zeroes of Q(z). This is precisely prevented by the system of Bethe Let

Ansatz equations, ensuring that all possible residues of

ep)

Thus, ep) is regular everywhere, leading being an elliptic polynomial solution. Hence, (ep), Q(z)) is an elliptic polynomial solution of the difference equation. Suppose now, that we have an elliptic polynomial solution (e(z),Q(z)) of the difference equation. Since we know that ep) E 0a we know that by Proposition E.l, it can be vanish.

to its

written

Wy,i

=

-

up to

1,...

,

a

|,

constant are

tion vanishes at these Wy, i

=

l,...

,

n

to

the

C

-

zeroes

the way of

Q(z),

we so

write it in the the

right

the Bethe Ansatz

equations.

The

points

nad side of the difference equa¬

points, causing also the left hand side

obey

Proposition. to vanish:

this

yields

the

2.

September

1982

-

1991

1972

Geboren in

Saarbrücken, Bundesrepublik

Staatliches

Gymnasium Wendalinum, Bundesrepublik Deutschland

Mai 1991

Abitur

1991

Studium der

-

1994

St.

-

1996

Wendel,

(Mathematik, Physik, Latein) Physik, Tübingen, Bundesrepublik der Physik,

Universität

1994

Deutschland

Studium

Deutschland

ETH Zürich

Oktober 1996

Dipl. Phys.

1997

Assistentin

-

2000

ETH

am

Departement

Mathematik

der ETH Zürich

1997

-

2000

Promotionsarbeit in mathematischer unter

Leitung

von

Physik

Prof. Dr. G. Felder

129

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