des L-Operators zum SOS Acht-Vertex-Modell (M(C, V2m),LSos,e(z, A)) sind (s. Defi¬ nition 4.21) ..... The second way of approaching the lattice is by means of the SOS or .... matrix Q(u) with non-zero determinant, also an entire function of u, commuting .... Definition 4.6) by a so-called vertex-IRF transformation (cf. [4], [27]).
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), L^ux 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),L^ux>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 ^o))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),L^uxfi{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),L^ua. 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
representation theory of
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
^exp(-E(s)/A;T)
or
respectively) ZM
=
s
where the or
of
a
sum
^exp(-.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 ^i
d2
d
.
(7)
-V^+VïT.
-*
dt2
dt
fw
pW
->
*,,
h®
-+
-2U-1- + K..
(8) (9)
u,tl
1,...
on
,n
,in]/(E^i^+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
setting corresponding
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
elliptic Gaudin eigenvalue problem
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:
^E;=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.
elliptic Gaudin eigenvalue problem
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'
^At —(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)e^U']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
(_irnecE^ec(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)Y^ai, 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
setting corresponding
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(^o) 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 +
=
>
,
V^e(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,L^e(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
difference operators
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,Li^e(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)
Va^O^o)
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*-^A-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
representation theory of
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
highest weight representation
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
highest weight representation
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
-
^i, 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
Y^ai,...,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
\
T^i )• 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
=
auxiliary representation
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-ua^e^,
£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,^i,--- ,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\, 0^auxte\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^ Tî^i £^A-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
î^A)
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~bau^e (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
Caux^e)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)
baUx^e\Z,
Zi,
,
Zn,
A)
d,aux^e[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 ^i
^e ^e
.
(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)^en\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)L^UXie(z, Zi, For the sake of
/(ft, A) split the
=
into
.
.
—
L®(z, .
simplicity
e^i^+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 -^e_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, h^aizi
-^
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-^a^i
"
„„ ~
\
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:-^i)0(A-27?ft
27?)
+
a(z, X)a
.
g
+
_x
l(zi
-
2t?, A
-
2n)c(zx
-
2n, A)
_
and
O2i(*0
=
(^e/(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-zx^A-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-^;(^^A^) ; d^ux 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)
®d^UXfi(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__^__^i (-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
b^ux e(z,zi,... ,zn,X)
=}_^
zz -
b^ux 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
b^ux 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),Det^ux e(z, z2,... zn) and 7re. zn, X) Ifc^wx^(z, zx,... ,zn,X)
(b^uxfi)i(z,zi,... ,zn,X),
check
,,
.
^aT(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(^i
2n, w'
—
2n, X)c(zx,X
used the fifth RLL relation with z'
~
—
^e%^]%^^) 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)®c^UX}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??)c^A)
+
-2/?'A -2r?)c(zi -^A) 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
g^I^^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 ^-^i+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
^o(^,..)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-2^a;p402Sif1)(z,z2,...,zn+i,A). 3 =2
Thus, the identity involving b^os.el«, Since the other relations
can
zi,...
be shown
,
zn+i,
similarly,
A)
and
this
b^UXfi(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)^o5,e(z,zi, ..,zn,Xo)An)en)
T^UXje(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,^e{-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
auxiliary representation
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
n^iöPfc+z.+A^T^). We get n
rfi
1
^Q^iYlO^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), L^ux 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),L^UX 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
auxiliary representation
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),L^eP,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),L^ux^e(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
—
—
—
auxiliary representation
(M(C,^2),L^e(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 Fx^e^A-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
difference operators
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—~^A+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
a^ux(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
auxiliary representation
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 ^iV^,]
+ 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
L^uxr(z,zx)
is
equal
to
Rr(z
—
(V®2).
(An,r(zi,..- ,zn),An,r(zi,- ,zn)) End
EKU^i
(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+)a^UXjT(z, z2,... zn)a%u^r pi, z2,... p Zi^^pi 2??, Z2, Zn) b---
If
o^uxr(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
o^uxr(z, z2,... zn),
~
V
^+^,=2
*«
2;1^1_^+2r?i^
2n)a^ux>r pi, Z2,
.
.
.
,zn)(aaux^r)
z2,
,
Z„)
X
pl,Z2,... ,Zra),
Zn) 2??p_)aû:u:!.)r p, Z2, {a^uxA^^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)
c^ux 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
c^ux>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) ® a^UXtr(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
L^ux 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)®C^UX!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^^e
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
References G.E.
Andrews, R.J.
Rogers-Ramanujan-type
alized
Eight-vertex SOS-model and gener¬ Stat. Phys. 35, pp. 193 266, 1984.
Baxter and P.J. Forrester. identities. J.
-
A
quasi-Hopf interpretation of quantum 3j6j-symbols and difference equations. Phys. Lett. B 375, pp. 89 97, 1996.
0.
Babelon,
D.
Billey.
E.
Bernard,
and
-
R.J. Baxter.
Exactly solved models
in statistical mechanics. London: Academic
Press,
1982.
R.J. Baxter.
Eight-vertex model in lattice statistical mechanics, and one-dimensional
anisotropic Heisenberg chain 1,11,111. Ann. Phys. (N.Y.) 76,
1
-
24,
25
-
47,
48
-
71,
1973.
A. A. Beilinson. V.G.
Drinfeld).
Affine algebras
Langlands duality (after Study, 1994.
at the critical level and
Lectures at the Institute for Advanced
A. A. Beilinson and V. G. Drinfeld.
Quantization
of Hitchin's fibration and
A. Belavin and V. G. Drinfeld. Solutions of the classical
simple
Lie
algebras.
H. A. Bethe.
Fund. Anal.
Appl. 16,
Zur Theorie der Metalle:
linearen Atomkette. Z.
Phys.,
71: 205
1. Chrerednik. Some finite dimensional
Appl. 19,
bras. Fund. Anal. E.
Lang-
Preprint, 1994.
land's program.
pp. 77
Date, M. Jimbo, A. Kuniba,
II: Proof of the
Studies in Pure
T.
-
-
159, 1982.
p.
I.
Yang-Baxter-equation for
Eigenwerte und Eigenfunk
tionen der
226, 1931.
representations of generalized Sklyanin alge¬
79, 1985.
Miwa, M. Okado. Exactly Solvable SOS-Models
Star-Triangle-Relation and Mathematics 16 (M. Jimbo,
Combinatorial T.
Miwa, A.
Identities, in Advanced Tsuchiya ed.), pp. 17-123.
Academic Press, 1988. E.
Date, M. Jimbo,
Math.
Phys. 12,
T.
Miwa, M. Okado. Fusion of the eight-vertex SOS-model. Lett.
pp. 209
-
215, 1986.
V. G. Drinfeld. Amer. J. Math. 105, pp. 85
-
114, 1983.
V. G. Drinfeld.
Nauk.
Hopf algebras and the quantum Yang-Baxter equation. SSSR, 283, pp. 1060 1064, 1985.
V. G. Drinfeld. Mathematicians V. G. Drinfeld. B.
Enriquez,
B.
Quantum Groups. Berkeley 1986, pp.
In
Proceedings of
798
-
Quasi-Hopf algebras. Leningrad Feigin
and V. Rubtsov.
B.
Math.
Separation
Enriquez and G. Felder. Elliptic quantum
Math.
Phys. 195,
pp. 651
-
689, 1998.
the International
Congress of
820, Academic Press, 1986.
Calogero systems, preprint q-alg/ 9695930, May
[17]
InDokl. Akad.
-
J., 1:6,
pp. 1419
of variables for
-
1457, 1990.
elliptic Gaudin-
1996.
groups and
quasi-Hopf algebras. Comm.
130
B.
and V. Rubtsov.
Enriquez
alization method
B.
Quantum
(sÏ2-case). Preprint
groups in
Drinfeld's new re¬ 1123, q-alg/9601022.
genus and
higher
Polytechnique,
Ecole
No.
Enriquez and V. Rubtsov. Quasi-Hopf algebras associated with sh and complex Preprint Ecole Polytechnique, No. 1145, q-alpg/9608005.
curves.
Phys. CI,
L.D. Faddeev. Sov. Sei. Rev. Math. L.D. Faddeev and L.A. L.D. Faddeev. Les
Takhtajan.Russian
J.-B.
L.D. Faddeev. In Nankai Lectures
Song),
by
X.-C.
B.
Feigin,
pp. 23
Frenkel,
E.
-
70,
Zuber,
on
ET:„(sl2),
R.
-
155, 1980.
Surveys 34,
p.
186, 1979.
in Recent Adv. in Field
Stora),
pp. 563
-
608,
Theory and
1984.
Physics, Integrable Systems, (ed.
Mathematical
1987.
N. Reshetikhin. Comm. Math.
G. Felder and A. Varchenko. On the group
Math.
Houches, Session XXXIX.
mechanics, (ed. by
Statistical
pp. 107
Comm. Math.
G. Felder and A. Varchenko.
Phys. 166,
pp. 27
-
61, 1994.
Representation theory of the elliptic quantum
Phys. 181,
pp. 741
-
761, 1996.
Integral representation
of solutions of the
Knizhnik-Zamalodchikov-Bernard equation, Int. Math. Res. Notices No. -
5,
elliptic pp. 221
1995.
233,
G. Felder and A. Varchenko.
ET,n(sh)-
Nuclear
Physics
B
Algebraic Bethe ansatz 480, 485 503, 1996.
for the
elliptic quantum
group
-
G. Felder and C. Wieczerkowski. Knizhnik-Zamolodchikov-Bernard
Conformai Blocks
equation,
on
Comm. Math.
elliptic Curves and the Phys. 176,
pp. 133
-
162,
1996.
G. Felder.
Physics,
Elliptic Quantum Groups.
Paris
In Xlth International
1994 (D- Ialgonitzer, ed.),
pp. 211
-
Congress of Mathematical 218, International Press, 1995.
Felder, A. Schorr. Separation of Variables for quantum integrable systems elliptic curves, math/9905072. G.
E. Frenkel. Affine
tional
Congress of
642. International M. Gaudin.
1087
-
algebras, Langlands duality Mathematical
Physics,
Paris
on
and Bethe Ansatz. In Xlth Interna¬
1994 (D- Ialgonitzer, ed.),
pp. 606
-
Press, 1995.
Diagonalisation
d'une classe d'hamiltoniens de
spin. J. Physique, 37:
1098, October 1976.
Hermite M.
Jimbo,
A.
Kuniba,
Phys. 119,
pp. 543
M.
T.
Jimbo,
-
Miwa,
sentation of classical 1988.
T. Miwa, 565, 1988.
M. Okado. The
An^
face models.
Comm. Math.
M. Okado. Solvable lattice models related to the vector repre¬
simple Lie algebras. Comm. Math. Phys. 116,
pp. 507
-
525,
131
E.G.
Magnet, preprint hep-th/9412190,
P.P. Kulish and E.
and C. Montonen R. P.
S.
W. Miller Jr..
Kalnins, V.B.Kuznetsov,
Gaudin
Sklyanin.
eds.).
and the XXZ
Integrable Quantum Field Theories, (J. Hietarinta
Physics 151,
Lecture notes in
Langlands.Lecture
Notes in Mathematics
pp. 61
pp. 18
170,
-
-
119.
Springer,
Springer,
61.
1982.
1970.
of Quantum Group Theory. Cambridge University Press,
Foundations
Majid.
In
Separation of Variables
December 1994.
1995.
Algebras. Volume
In
Proceedings
(ed. by
of the ICM Berlin
G. Fischer and U.
N. Yu. Reshetikhin in Lett. Math. M.
21,
Rehmann), Phys. 7,
-
Extra
1998.
pp. 205
-
213, 1983. (???) actions.
group
1986.
Sklyanin, Some algebraic structures connected with the Yang-Baxter-equation. Appl. 16, pp. 27 34, 1982. Fund. Anal. Appl. 17, pp. 273 284, 1983.
Fund. Anal. E.
1998, Documenta Mathematica
Semenov-Tian-Shansky.LVe,s.SOT(/" transformations and Poisson
Publ. Res. Inst. Math. Sei. E.
Representation Theory of Quantum Affine
Solvable Lattice Models and
T. Miwa.
Sklyanin.
-
-
J. Sov. Math. 47, pp. 2473
2488, 1989.
-
Sklyanin, Separation of Variables. New Trends. In Quantum field theory, integrable models and beyond. Progr. Theoret. Phys. Suppl. 118, pp. 35 60, 1995.
E.
-
Sklyanin. Functional Bethe Ansatz. In Integrable and Superintegrable Systems, (B.A. Kuperschmidt, ed.), pp. 8 33. World Scientific, 1990.
E.
-
Sklyanin. Quantum Inverse Scattering Method. Selected Topics. In Quantum Groups and Quantum Integrable Systems, (Mo-Lin Ge, ed.), pp. 63 -97. World Sci¬ entific, 1992. E.
E.
Sklyanin
and T. Takebe.
Separation
of Variables in the
elliptic Gaudin
model.
preprint solv-int/9807908, July 1998. A.
Stoyanovsky.
On
Quantization of
the
geometric Langlands correspondence I,
preprint math/9911108. V. Tarasov and A. Varchenko. tions with
Completeness of
regular singular points,
Bethe Vectors and difference equa¬
Internat. Math. Res. Notices
pp. 637
13,
-
669,
1995.
G.N. Watson and E.T. Whittaker. A Course
Cambridge University Press,
of Modern Analysis,
M. Yoshida. Fuchsian 1987.
Cambridge:
1927.
Yang. Some exact results for the many-body problem in repulsive delta-function interaction. Phys. Rev. Letters 19, pp.
C. N.
4th ed.
one
1312
dimension with -
1314, 1967.
Differential Equations, Braunschweig/Wiesbaden:
Vieweg,