Differential Equations Associated to The SU(2) WZNW Model on Elliptic Curves

23 Pages·0.23 MB·English

by  Takeshi Suzuki

DIFFERENTIAL EQUATIONS ASSOCIATED TO
THE SU(2) WZNW MODEL ON ELLIPTIC CURVES
Takeshi Suzuki
5
9
9 Research Institute for Mathematical Sciences,
1
Kyoto University, Kyoto, 606-01, Japan

n
e-mail : [email protected]
a
J

4 Abstract. We study the SU(2) WZNW model over a family of elliptic curves.
2
Starting from the formulationdeveloped in [TUY], we derive a system of differential

2 equationswhichcontainstheKnizhnik-Zamolodchikov-Bernardequations[Be1][FW].
v OursystemcompletelydeterminestheN-pointfunctionsandisregardedasanatural
9 elliptic analogue of the system obtained in [TK] for the projective line. We also
1
calculate the system for the 1-point functions explicitly. This gives a generalization
2
2 of the results in [EO2] for sl(2,C)-characters.
1
4 b
9
/
h 0. Introduction.
t §
-
p We consider the Wess-Zumino-Novikov-Witten (WZNW) model. A mathemati-
e
cal formulation of this model on general algebraic curves is given in [TUY], where
h
: the correlation functions are defined as flat sections of a certain vector bundle over
v
i the moduli space of curves. On the projective line P1, the correlation functions are
X
realized more explicitly in [TK] as functions which take their values in a certain
r
a finite-dimensional vector space, and characterized by the system of equations con-
taining the Knizhnik-Zamolodchikov (KZ) equations[KZ]. One aim in the present
paperistohaveaparalleldescriptiononellipticcurves. Namely,wecharacterizethe
N-point functions as vector-valued functions by a system of differential equations
containing an elliptic analogue of the KZ equations by Bernard[Be1]. Furthermore
we write down this system explicitly in the 1-pointed case.
To explain more precisely, first let us review the formulation in [TUY] roughly.
Let g be a simple Lie algebra over C and g the corresponding affine Lie algebra.
We fix a positive integer ℓ (called the level) and consider the integrable highest
weight modules of g of level ℓ. Such modules are parameterized by the set of
b
highest weight P and we denote by the left module corresponding to λ
ℓ λ
H ∈
P . By M we denote the moduli space of N-pointed curves of genus g. For
ℓ g,N b
X M and ~λ = (λ ,... ,λ ) (P )N, we associate the space of conformal
g,N 1 N ℓ
∈ ∈
blocks †(X;~λ). The space †(X;~λ) is the finite dimensional subspace of † :=
Vg Vg H~λ
HomC(Hλ1 ⊗···⊗HλN,C) defined by “the gauge conditions”. Consider the vector
bundle ˜†(~λ) = †(X;~λ) over M . On this vector bundle, projectively
Vg ∪X∈Mg,NVg g,N
flatconnectionsaredefined throughtheKodaira-Spencertheory, andflatsectionsof
˜†(~λ) with respect to these connections are called the N-point correlation functions
g
V
Typeset by AMS-TEX
2 TAKESHI SUZUKI
(orN-pointfunctions). Intherestofthispaperwesetg = sl(2,C) = CE CF CH
⊕ ⊕
for simplicity, where E,F and H are the basis of g satisfying
[H,E] = 2E, [H,F] = 2F, [E,F] = H.
−
We identify P with the set 0, 1,... , ℓ by the mapping λ λ(H).
ℓ { 2 2 } 7→ 2
In the case of genus 0, the space of conformal blocks is injectively mapped into
V~λ† := HomC(Vλ1 ⊗ ···⊗VλN,C) by the restriction map, where Vλ ⊂ Hλ denotes
the finite dimensional irreducible highest weight left g-module with highest weight
λ. This injectivity makes it possible to treat this model in a more explicit way as
above, and the N-point functions are described by the vacuum expectation values
of vertex operators.
On the other hand, in the case of genus 1 this injectivity does not hold, and
in order to recover it we twist the space of conformal blocks by introducing a
new parameter following [Be1,2][EO1][FW]. Because of the twisting, any N-point
function in genus 1 can be calculated from its restriction to V (Proposition 3.3.2).
~λ
Itisnatural toaskhowtherestrictionsoftheN-pointfunctions arecharacterizedas
†
V -valued functions. It turns out that the restricted N-point functions satisfy the
~λ
equations (E1)–(E3) in Proposition 3.3.3. These equations are essentially derived
by Bernard[Be1] for traces of vertex operators
TrHµ(ϕ1(z1)···ϕN(zN)qL0−c2v4ξH2 ) ∈ V~λ†,
where z ,... ,z ,q,ξ are the variables in C∗ with q < 1, ϕ : V ˆ
1 N | | j λj ⊗Hµj → Hµj−1
(j = 1,... ,N) are the vertex operators for some µ (i = 0,... ,N) with µ = µ =
i 0 N
µ, L is defined in (1.2.1) and c = 3ℓ/(ℓ 2) (for the details, see 3.4). It is
0 v
− §§
proved that the space of restricted N-point functions is spanned by traces of vertex
operators (Theorem 3.4.3) and hence Bernard’s approach is equivalent to ours.
However, the system (E1)–(E3) is not complete since it has infinite-dimensional
solution space.
We will show that the integrability condition
E t−1 ℓ−2λ+1 v¯(λ) = 0
⊗ | i
for the highest weight vecto(cid:0)r v¯(λ) (cid:1) implies the differential equations (E4),
λ
| i ∈ H
which determine the N-point functions completely combining with (E1)–(E3).
For 1-point functions, the equation (E4) can be written down explicitly, and the
system (E1)–(E4) reduces to the two equations (F1)(F2) in Theorem 4.2.4. In the
simplest case, the 1-point functions are given by the characters
1 ℓ
TrH qL0−c2v4ξH2 µ = 0, ,... , ,
µ 2 2
(cid:18) (cid:19)
and our system coincide with the one obtained in [EO2].
Recently Felder and Wieczerkowski give a conjecture on the characterization of
the restricted N-point functions in genus 1 by using the modular properties and
certain additive conditions[FW]. They confirm their conjecture in some cases by
explicit calculations. We partly recover this result by solving the equation (F2)
(Proposition 4.2.5). The equation (F1) can be also integrated when the dimension
of the solution space is small, and we can calculate the 1-point functions explicitly.
THE SU(2) WZNW MODEL ON ELLIPTIC CURVES 3
1. Representation theory for sl(2,C).
§
For the details of the contents in this section, we refer the reader to [Kac].
b
1.1 Integrable highest weight modules.
By C[[x]] and C((x)), we mean the ring of formal power series in x and the field
of formal Laurent series in x, respectively. We put g = sl(2,C). Let h = CH be a
Cartan subalgebra of g and ( , ) : g g C the Cartan-Killing form normalized by
× →
the condition (H,H) = 2. We identify the set P of dominant integral weights with
+
1Z . For λ P , we denote by V the irreducible highest weight left g-module
2 ≥0 ∈ + λ
with highest weight λ and by v(λ) its highest weight vector.
| i
The affine Lie algebra g associated with g is defined by
g = g C((x)) Cc,
b
⊗ ⊕
where c is an central element of g and the Lie algebra structure is given by
b
[X f(x),Y g(x)] = [X,Y] f(x)g(x)+c (X,Y) Res(g(x) df(x)),
b
⊗ ⊗ ⊗ · x=0 ·
for X,Y g, f(x),g(x) C((ξ)). We use the following notations:
∈ ∈
X = X xn, X = X ,
n 0
⊗
g = g C[[x]]x, g = g C[x−1]x−1,
+ −
⊗ ⊗
p = g g Cc.
± ±
⊕ ⊕
b b
Fix a positive integer ℓ (calledbthe lbevel) and put Pℓ = { 0, 21,... , 2ℓ } ⊂ P+. For
λ P , we define the action of p on V by c = ℓ id and a = 0 for all a g ,
ℓ + λ +
∈ × ∈
and put
Mbλ = U(g)⊗p+ Vλ. b
Then is a highest weight left g-module and it has the maximal proper submod-
λ b
ule M, which is generated by the singulabr vector Eℓ−2λ+1 v(λ) :
Jλ −1 | i
b
= U(p )Eℓ−2λ+1 v(λ) .
Jλ − −1 | i
The integrable highest weight left g-module with highest weight λ is defined
b λ
H
as the quotient module / . We denote by v¯(λ) the highest weight vector in
λ λ
M J | i
. We introduce the lowest weight right g-module structure on
λ b
H
Hλ† = HomCb(Hλ,C)
in the usual way, and denote its lowest weight vector by v¯(λ) .
h |
1.2. Segal-Sugawara construction and the filtration on .
λ
H
Fix a weight λ P . On , elements L (n Z) of the Virasoro algebra act
ℓ λ n
∈ H ∈
through the Segal-Sugawara construction
1 1
(1.2.1) L = ◦ H H ◦ +◦E F ◦ +◦F E ◦ ,
n 2(ℓ+2) ◦2 m n−m◦ ◦ m n−m◦ ◦ m n−m◦
Z(cid:26) (cid:27)
X
4 TAKESHI SUZUKI
where ◦ ◦ denotes the standard normal ordering, and the operators L (n Z)
◦ ◦ n
∈
satisfy the following commutation relations:
c
[L ,L ] = (n m)L + v(n3 n)δ ,
n m n+m n+m,0
− 12 −
[L ,X ] = mX for X g,
n m n+m
− ∈
where c is the central charge of Virasoro algebra:
v
3ℓ
(1.2.2) c = .
v
ℓ+2
Put
X(z) = X z−n−1 (X g), T(z) = L z−n−2.
n n
∈
n∈Z n∈Z
X X
The module has the decomposition = (d), where
λ λ d≥0 λ
H H ⊕ H
(d) = u ; L u = (∆ +d) u ,
λ λ 0 λ
H { | i ∈ H | i | i }
λ(λ+1)
(1.2.3) ∆ = .
λ
ℓ+2
We define the filtration on by
• λ
{F } H
= (d)
p λ λ
F H H
d≤p
X
and put ˆ = (d).
Hλ d≥0Hλ
1.3. The LQie algebra g .
N
Put Lg = g C((x)). For a positive integer N, we define a Lie algebra g by
N
⊗ b
gN = ⊕Nj=1Lg(i) ⊕Cc, b
where Lg denotes a copy of Lg and c is a center. The commutation relations are
(i) b
given by
[ N X f , N Y g ] =
⊕j=1 j ⊗ j ⊕j=1 j ⊗ j
N
N [X ,Y ] f g + (X ,Y )Res(g df ) c.
⊕j=1 j j ⊗ j j j j ξj=0 j · j ·
j=1
X
For each ~λ = (λ ,... ,λ ) (P )N a left g -module is defined by
1 N ∈ ℓ N H~λ
= .
H~λ Hλ1 ⊗b···⊗HλN
Similarly a right g -module † is defied by
N H~λ
b † = † ˆ ˆ † ∼= HomC( ~,C).
H H ⊗···⊗H H
THE SU(2) WZNW MODEL ON ELLIPTIC CURVES 5
The g -action on is given by
N H~λ
b c = ℓ id
·
N
( N a ) u u = ρ (a ) u u
⊕j=1 j | 1 ⊗···⊗ Ni j j | 1 ⊗···⊗ Ni
j=1
X
for a Lg (j = 1,... ,N), where we used the notations
j (j)
∈
u u = u u ,
1 N 1 N
| ⊗···⊗ i | i⊗···⊗| i
ρ (a) u u = u a u u
j 1 N 1 j N
| ⊗···⊗ i | ⊗···⊗ · ⊗···⊗ i
for u (i = 1,... ,N) and a Lg. The right action on † is defined simi-
| ii ∈ Hλi ∈ H~λ
larly. The module has the filtration induced from those of (j = 1,... ,N):
H~λ Hλj
= (d),
FpH~λ H~λ
d≤p
X
where
(d) = (d ) (d ).
H~λ Hλ1 1 ⊗···⊗HλN N
d1+··X·+dN=d
We put
V = V V = (0),
~λ λ1 ⊗···⊗ λN ∼ H~λ
Vλ† = HomC(V~λ,C).
2 The WZNW model in genus 0.
§
In this section we review the SU(2) WZNW model on the projective line P1.
2.1. The space of conformal blocks.
In this subsection we define the N-point functions on P1 following [TUY] as
sections of a vector bundle on the manifold
R = (z ,... ,z ) (C∗)N ; z = z if i = j .
N 1 N i j
{ ∈ 6 6 }
For a meromorphic function f(t) on P1 and w C, put
∈
X[f(t)] = Res f(t)X(t w)dt,
w
t=w −
d
T[f(t) ] = Res f(t)T(t w)dt.
w
dt t=w −
If f(t) has an Laurent expansion f(t) = a (t w)n then X[f(t)] is an
n≥M n − w
element of g given by
P
X[f(t)] = a X .
w n n
b ≥
X
6 TAKESHI SUZUKI
For z = (z ,... ,z ) R , we set
1 N N
∈
N
g(z) = H0(P1,g P1( zj)).
⊗O ∗
j=1
X
b
Then we have the following injection:
g(z) g ,
N
→
X f(z) X[f] := N X[f] .
⊗ 7→ ⊕j=1 zj
b b
Through this map we regard g(z) as a subspace of g and the residue theorem
N
implies that g(z) is a Lie subalgebra of g . We also use the following notation
N
b b
T[g] = N T[g]
b ⊕bj=1 zj
for g ∈ H0(P1,ΘP1(∗ Nj=1zj)), where ΘP1 denotes the sheaf of vector fields on P1.
Definition 2.1.1. FPor z = (z ,... ,z ) R and ~λ = (λ ,... ,λ ) (P )N we
1 N N 1 N ℓ
∈ ∈
put
(z;~λ) = /g(z) ,
V0 H~λ H~λ
†(z;~λ) = Ψ † ; Ψ g(z) = 0
V0 { h |b∈ H~λ h | }
∼= HomC( 0(z;~λ),C).
V b
† ~
We call (z;λ) the space of conformal blocks (or the space of vacua) in genus 0
V0
~
attached to (z;λ).
For a vector space V and a complex manifold M, we denote by V[M] the set of
multi-valued, holomorphic V-valued functions on M.
Definition 2.1.2. For ~λ (P )N, an element Φ of †[R ] is called an N-point
∈ ℓ h | H~λ N
~
function in genus 0 attached to λ if the following conditions are satisfied:
(A1) For each z R ,
N
∈
† ~
Φ(z) (z;λ)
h | ∈ V0
(A2) For j = 1,... ,N,
∂ Φ(z) = Φ(z) ρ (L ).
zjh | h | j −1
By F (~λ) we denote the set of N-point functions in genus 0 attached to ~λ.
0
Remark. The condition (A1) implies the following:
(A1′) For each z R ,
N
∈
Φ(z) T[g] = 0
h |
for any g ∈ H0(P1,ΘP1(∗ Nj=1zj)).
P
2.2. Restrictions of the N-point functions to V .
~λ
Aremarkablepropertyofthespaceofconformalblocksingenus0isthefollowing:
THE SU(2) WZNW MODEL ON ELLIPTIC CURVES 7
Lemma 2.2.1. The composition map
~
V ֒ (z;λ)
~λ → H~λ → V0
is surjective. In other words, the restriction map
† ~ †
(z;λ) V
V0 → ~λ
is injective.
This lemma implies that, for an N-point function Φ , we can calculate Φ u for
h | h | i
any u , from the data Φ v ; v V . By Fr(~λ) we denote the image of
| i ∈ H~λ { h | i | i ∈ ~λ } 0
F (~λ) in V†[R ] under the restriction map. It is natural to ask how the set Fr(~λ)
0 ~λ N 0
†
is characterized in V [R ], and the answer is given as follows:
~λ N
Proposition 2.2.2. [TK] The space Fr(~λ) coincides with the solution space of the
0
following system of equations:
(B1) For each X g,
∈
N
φ(z) ρ (X) = 0.
j
h |
j=1
X
(B2) [the Knizhnik-Zamolodchikov equations] For each j = 1,... ,N,
Ω
i,j
(ℓ+2)∂ φ(z) = φ(z) ,
z
jh | h |z z
i j
i6=j −
X
where
1
Ω = ρ (H)ρ (H)+ρ (E)ρ (F)+ρ (F)ρ (E).
i,j i j i j i j
2
(B3) For each j = 1,... ,N,
ℓ
j (z z )−ni φ(z) En1v v(λ ) EnNv
i j 1 j N
~n − h | ⊗···⊗ ⊗···⊗ i
j
n1+··X·+nN=ℓj(cid:18) (cid:19)Yi6=j
= 0
for any v V (i = j). Here ℓ = ℓ 2λ +1, ~n = (n ,... ,n ,n ,... ,n )
| ii ∈ λi 6 j − j j 1 j−1 j+1 N
and ℓj is the multinomial coefficient. (cid:3)
~n
j
Rem(cid:0)ark(cid:1). The equation (B3) is a consequence of the integrability condition
(2.2.1) Eℓ−2λj+1 v¯(λ ) = 0 (j = 1,... ,N),
−1 | j i
for the highest weight vector v¯(λ ) .
j λ
| i ∈ H j
2.3. Vertex operators.
We review the description of N-point functions by vertex operators.
8 TAKESHI SUZUKI
Definition 2.3.1. For (ν,λ,µ) (P )3 a multi-valued, holomorphic, operator
ℓ
∈
valued function ϕ(z ) on the manifold C∗ = C 0 is called a vertex operator of
1
\{ }
type (ν,λ,µ), if
ϕ(z ) : V ˆ
1 λ µ ν
⊗H → H
satisfies the following conditions:
(C1) For X g, v V and m Z,
λ
∈ | i ∈ ∈
[X ,ϕ( v ;z )] = zmϕ(X v ;z ).
m | i 1 1 | i 1
(C2) For v V and m Z,
λ
| i ∈ ∈
d
[L ,ϕ( v ;z )] = zm z +(m+1)∆ ϕ( v ;z ).
m | i 1 1 1dz λ | i 1
(cid:26) 1 (cid:27)
Here ϕ( u ;z ) : ˆ is the operator defined by ϕ( u ;z ) v = ϕ(z ) u v
1 ν µ 1 1
| i H → H | i | i | ⊗ i
for u V and v .
λ ν
| i ∈ | i ∈ H
For vertex operators ϕ (z ) (j = 1,... ,N), the composition ϕ (z ) ϕ (z )
j j 1 1 N N
···
makes sense for z > > z and analytically continued to R .
1 N N
| | ··· | |
Proposition 2.3.2. [TK] The space Fr(~λ) is spanned by the following V†-valued
0 ~λ
functions:
v(0) ϕ (z ) ϕ (z ) v(0) ,
1 1 N N
h | ··· | i
where ϕ (j = 1,... ,N) is the vertex operator of type (µ ,λ ,µ ) for some µ
j j−1 j j i
∈
P (i = 0,... ,N) with µ = µ = 0.
ℓ 0 N
Proposition 2.3.3. [TK] Any nonzero vertex operator
ϕ(z ) : V ˆ
1 λ µ ν
⊗H → H
is uniquely extended to the operator
ˆ
ϕˆ(z ) :
1 λ µ ν
M ⊗H → H
by the following condition:
(2.3.1) ϕˆ(X u ;z ) = Res(w z )nϕˆ( u ;z )X(w)dw,
n 1 1 1
| i w=z − | i
for each u , X g and n Z.
λ
| i ∈ M ∈ ∈
Moreover, ϕˆ has the following properties:
(2.3.2) ∂ ϕˆ( u ;z ) = ϕˆ(L u ;z ) for any u ,
z 1 −1 1 λ
| i | i | i ∈ M
(2.3.3) ϕˆ( u ;z ) = 0 for any u = U(p )Eℓ−2λ+1 v(λ) .
| i 1 | i ∈ Jλ − −1 | i
The property (2.3.3) implies that ϕˆ reduces to the operator
b
ˆ
ϕˆ(z ) : .
1 λ µ ν
H ⊗H → H
THE SU(2) WZNW MODEL ON ELLIPTIC CURVES 9
3 The WZNW model in genus 1.
§
In this section we consider the elliptic analogue of the story in the previous
section. Our aim is to embed the set of N-point functions in genus 1 (Definition
†
3.1.3) into the set of V -valued functions, and to characterize its image by a system
~λ
of differential equations. We also show that the N-point functions are given by the
traces of vertex operators.
3.1 Functions with quasi-periodicity.
First, we prepare some functions for the later use. Put D∗ = q C∗ ; q < 1
{ ∈ | | }
and introduce the following functions on C∗ D∗:
×
(3.1.1) Θ(z,q) = ( 1)n+1q12n2zn
−
n∈XZ+21
= √ 1z12q18 (1 qn)(1 zqn)(1 z−1qn−1),
− − − − −
n≥1
Y
z∂ Θ(z,q)
z
(3.1.2) ζ(z,q) = .
Θ(z,q)
q∂ η(q)
q
(3.1.3) ℘(z,q) = z∂ ζ(z,q)+2 ,
z
− η(q)
where η(q) is the Dedekind eta function
η(q) = q214 (1 qn).
−
n≥1
Y
The function Θ(z,q) satisfies the heat equation
2q∂ Θ(z,q) = (z∂ )2Θ(z,q).
q z
The function ℘(z,q) satisfies ℘(qz,q) = ℘(z,q), andζ(z,q) havethe following quasi-
periodicity:
(3.1.4) ζ(qz,q) = ζ(z,q) 1.
−
For (z,q) C∗ D∗ and ξ C∗, we put
∈ × ∈
Θ(z−1ξ±1,q)Θ′(1,q)
(3.1.5) σ (z,q,ξ) =
± Θ(z,q)Θ(ξ±1,q)
Here Θ′(z,q) = z∂ Θ(z,q). The function σ (z,q,ξ) have the following properties:
z ±
σ (qz,q,ξ) = ξ±1σ (z,q,ξ),
± ±
(3.1.6)
σ (z−1,q,ξ) = σ (z,q,ξ).
± ∓
−
For ζ(z,q) and σ (z,q,ξ), we have the following expansion at z = 1:
±
1 1
(3.1.7) ζ(z,q) = + 2α(q)(z 1)+O(z 1)2,
z 1 2 − − −
−
1 1
(3.1.8) σ (z,q,ξ) = ζ(ξ,q)+
±
z 1 ∓ 2
−
nξ−1qn nξqn
+ (z 1)+O(z 1)2,
− 1 ξ−1qn 1 ξqn − −
≥ (cid:18) − − (cid:19)
X
10 TAKESHI SUZUKI
where α(q) is given by
q∂ η(q) 1
q
(3.1.9) α(q) = + .
− η(q) 24
3.2. Twisting the space of conformal blocks.
In the case of genus 1 (or > 0), if we work with the formulation of [TUY], an
N-point function is not determined by its restriction on V . In order to resolve this
~λ
difficulty we “twist” the space of conformal blocks following [Be1,2][EO1][FW].
For q D∗, we consider the elliptic curve = C∗/ q , where q is the infinite
q
∈ E h i h i
cyclic group of automorphisms generated by z qz. We denote by [z] the image
q
7→
of a point z C∗ on and put
q
∈ E
T = (z,q) = (z ,... ,z ,q) (C∗)N D∗ ; [z ] = [z ] if i = j .
N 1 N i q j q
{ ∈ × 6 6 }
~
Inthefollowingweomitthesubscriptq in[z] . For(z,q) T andλ = (λ ,... ,λ )
q N 1 N
∈ ∈
P we can define the space of conformal blocks attached to the elliptic curve :
ℓ q
E
†([z],q;~λ) = Ψ † ; Ψ g([z],q) = 0 ,
V1 { h | ∈ H~λ h | }
where
N
g([z],q) = H0( ,g ( [z ])),
q E j
E ⊗O q ∗
j=1
X
b
but for our purpose we need to twist it as follows. We introduce a new variable
ξ C∗, and put
∈
N
g([z],q,ξ) = a(t) H0(C∗,g C∗( qnzj)) ; a(qt) = ξH2 (a(t))ξ−H2 .
 ∈ ⊗O ∗ 
 Xj=1nX∈Z 
b
 
This space is regarded as the space of meromorphic sections of the g-bundle which
is twisted by ξH2 along the cycle [w] q ; w R,q w < 1 . For ξ = 1, we
{ ∈ E ∈ ≤ }
have
N
g([z],q,1) = H0( ,g ( [z ])).
q E j
E ⊗O q ∗
j=1
X
b
As in the previous section we have the following injection:
g([z],q,ξ) g
N
→
X f X[f].
⊗ 7→
b b
By this map we regard g([z],q,ξ) as a subspace of g . Furthermore we can easily
N
have the following lemma.
b b