An abelianization of SU(2) WZW model

49 Pages·2006·1.04 MB·English

AnnalsofMathematics, 164 (2006),1–49
An abelianization of SU(2) WZW model
By Tomoyoshi Yoshida
1. Introduction
The purpose of this paper is to carry out the abelianization program pro-
posed by Atiyah [1] and Hitchin [9] for the geometric quantization of SU(2)
Wess-Zumino-Witten model.
Let C be a Riemann surface of genus g. Let M be the moduli space of
g
semi-stable rank 2 holomorphic vector bundles on C with trivial determinant.
For a positive integer k, let Γ(M ,Lk) be the space of holomorphic sections of
g
the k-th tensor product of the determinant line bundle L on M . An element
g
of Γ(M ,Lk) is called a rank 2 theta function of level k.
g
Themainresultofourabelianizationistogiveanexplicitrepresentationof
a base of Γ(M ,Lk) as well as its transformation formula in terms of classical
g
Riemann theta functions with automorphic form coefficients defined on the
Prym variety P associated with a two-fold branched covering surface C˜ of C.
Γ(M ,Lk) can be identified with the conformal block of level k of the
g
SU(2) WZW model ([5], [15]). The abelianization procedure enables us to de-
duce the various known results about the conformal block in a uniform way.
Firstly, we construct a projectively flat connection on the vector bundle over
the Teichmu¨ller space with fibre Γ(M ,Lk). Secondly, making use of our ex-
g
plicitrepresentationofrank2thetafunctionsweconstructaHermitianproduct
on the vector bundle preserved by the connection. Also our explicit represen-
tation enables us to prove that Γ(M ,Lk) has the predicted dimension from
g
the Quantum Clebsh-Gordan conditions.
A natural connection on the said vector bundle for the SU(N) WZW
modelwasfirstconstructedbyHitchin[11]. Itwillturnoutthattheconnection
constructed in this paper coincides with the Hitchin connection.
Laszlo [16] showed that the Hitchin connection coincides with the con-
nection constructed by Tsuchiya, Ueno and Yamada [21] through the above
identification. On the other hand Kirillov [13], [14] constructed a Hermitian
product on the conformal block compatible with the Tsuchiya-Ueno-Yamada
connection using the representation theory of affine Lie algebras together with
the theory of hermitian modular tensor categories; cf. [22]. Laszlo’s result
2 TOMOYOSHI YOSHIDA
implies that the Hermitian product of Kirillov defines the one on Γ(M ,Lk)
g
compatible with the Hitchin connection. The author cannot figure out a re-
lation between the Hermitian product constructed in this paper and the one
found by Kirillov.
The paper is organized as follows. In Section 2 we study the topological
propertiesofafamilyof2-foldbranchedcoveringsurfacesC˜ ofafixedRiemann
surface C parametrized by the configuration space of 4g−4 mutually distinct
points on C.
In Sections 3 and 4 we study the Prym variety P of C˜ and the classi-
cal Riemann theta functions defined on it. Especially we will be concerned
with their symmetric properties. That is, the fundamental group of the con-
figuration space induces a finite group action on the space of Riemann theta
functions on P. We call it global symmetry. There is a morphism π : P → M
g
and a pulled back section of Γ(M ,Lk) by π can be expressed by Riemann
g
theta functions of level 2k on P. Then it should satisfy an invariance with
respect to this group action.
In Sections 5 we study the branching divisor of π : P → M . The square
g
root (Pfaffian) of the determinant of π is given by a Riemann theta function
Π of level 4 ([9]). Π plays a central role throughout the paper, and we give a
precise formula for it.
In Section 6 we construct a differential operator D on the space of holo-
morphic sections of the line bundles on the family of Prym varieties P such
that a family ψ˜ of holomorphic sections, which is a pull back by π of a section
ψ ∈ Γ(M ,Lk), satisfies the differential equation Dψ˜= 0.
g
In Section 7 we will show that the global symmetry and the differential
equation Dψ˜= 0 characterize the pull back sections.
In Section 8 we construct a basis of Γ(M ,Lk). It will be given in terms
g
of classical Riemann theta functions with automorphic form coefficients. The
result includes the fact that the dimension of Γ(M ,Lk) is equal to the num-
g
ber of the ‘admissible’ spin weights attached to a pant decomposition of the
Riemann surface (Quantum Clebsch-Gordan condition).
In Section 9 we construct a projectively flat connection and a hermitian
product compatible with it on the vector bundle over Teichmu¨ller space with
fibre Γ(M ,Lk).
g
InSection10wegivethetransformationformulaofrank2thetafunctions.
It involves a subtle but important aspect related to the Maslov index.
The author’s hearty thanks go to Professor M. F. Atiyah and Professor
N. J. Hitchin for their encouragement and interest in this work. Also we thank
M. Furuta, A. Tsuchiya and T. Oda for valuable conversations with them. We
are grateful to H. Fujita, S. K. Hansen, and D. Moskovich for their careful
reading of the manuscript.
AN ABELIANIZATION OF SU(2) WZW MODEL 3
2. A family of 2-fold branched covering surfaces
2.1. A family of 2-fold branched covering surfaces. Let C be a closed
Riemann surface of genus g (≥ 2). Let C4g−4(C) be the configuration space of
4g−4 unordered mutually distinct points b = {xj}1≤j≤4g−4 in C; that is,
(cid:1) (cid:2)
C4g−4(C) = C4g−4−∆ /S4g−4
where ∆ denotes the big diagonal of C4g−4 and S is the symmetric group of
n
degree n acting on C4g−4 by permutations of factors.
For b = {xj} in C4g−4(C), let cj denotes the class in H1(C−b,Z2) repre-
sented by the boundary circle of a small disc centered at x in C. Let
j
(1) Hˆ1(C −b,Z ) ≡ {α ∈ H1(C −b,Z ) | (cid:5)α,c (cid:6) = 1 }
2 2 j
where (cid:5) , (cid:6) denotes the evaluation of cohomology classes on homology classes.
Hˆ1(C −b,Z ) is in one-one correspondence with the set of topologically
2
distinct 2-fold branched coverings of C with branch locus b = {x }. Here two
j
branched coverings with branch locus b = {x } are topologically distinct if
j
and only if there is no diffeomorphism between them which is equivariant with
respect to the covering involutions and covers the identity map of C.
Definition 2.1. We call an element of α ∈ Hˆ1(C −b,Z ) a covering type
2
of C.
The family H = {Hˆ1(C − b,Z2)}b∈C4g−4(C) forms a fiber bundle over
C4g−4(C) with finite discrete fiber. Choose a base point bo ∈ C4g−4(C) and let
ρ : π1(C4g−4(C),bo) → Aut(Hˆ1(C −bo,Z2))
be the holonomy representation of the fiber bundle H.
We can describe ρ as follows. For an oriented loop l = {bt = {xtj}}0≤t≤1
basedatbo inC4g−4(C),theunionoforiented4g−4arcs{xtj}formsanoriented
closed curve ¯l in C. For a ∈ H (C −b ,Z ) we can define the Z -intersection
1 o 2 2
number ¯l·a ∈ Z . We obtain the following homomorphism ev which we call
2
the evaluation map
(2) ev : π1(C4g−4(C),bo) → H1(C −bo,Z2).
Clearly ¯l·c = 0 for 1 ≤ c ≤ 4g−4 and we have the following lemma:
j j
Lemma 2.1. Let [l] ∈ π1(C4g−4(C),bo) be the homotopy class represented
by a closed loop l based at b . Then ρ([l]) ∈ Aut(Hˆ1(C −b ,Z )) is given by
o o 2
(3) ρ([l])(α) = α+ev([l])
for α ∈ Hˆ1(C −b ,Z ).
o 2
4 TOMOYOSHI YOSHIDA
Definition 2.2. Let q : B → C4g−4(C) be the covering space of C4g−4(C)
associated with the kernel of ρ. The set B can be identified with the set of
pairs C4g−4(C)×Hˆ1(C −b,Z2) with q the projection to the first factor. We
represent a point ˜b of B by a pair
(4) ˜b = (b,α) for b ∈ C4g−4(C) and α ∈ Hˆ1(C −b,Z2).
For ˜b = (b,α) ∈ B, let C˜ = C˜ be the associated two-fold branched
˜b
covering surface of C with branch point set b of the covering type α. The
genus g˜ of C˜ is 4g−3. We denote the covering projection by p : C˜ → C and
the covering involution by σ : C˜ → C˜.
Definition 2.3. Let C → B be the fiber bundle over B whose fiber at
˜b = (b,α) ∈ B is the 2-fold branched covering surface C˜ of C.
˜b
Note that B and C are connected.
2.2. Pant decompositions of surfaces. Throughout the paper we use the
following notation;
S : the three-holed 2-dimensional sphere
0
T : the one-holed 2-dimensional torus.
0
Definition 2.4. A pant decomposition Υ = {e ,C } of a Riemann surface
l i
C of genus g is defined to be a set of simple closed curves {el}l=1,···,3g−3 and
surfaces {Ci}i=1,···,2g−2 in C such that
(i) {e } is a family of mutually disjoint and mutually freely nonhomotopic
l
simple closed curves in C,
(cid:3)
(ii) C = C where C = S or C = T . If C = S , then ∂C is a union of
i i 0 i 0 i 0 i
three elements of {e }. If C = T , then ∂C is an element of {e }, and
l i 0 i l
C contains an element of {e } in its interior as an essential simple closed
i l
curve.
(cid:3)
(iii) If we cut C along e , then the resulting surface is a disjoint union of
l l
{Ci∗}1≤i≤2g−2, where Ci∗ = S0 for 1 ≤ i ≤ 2g −2 and, if Ci = S0, then
Ci∗ = Ci and, if Ci = T0, then ∂Ci∗ = el(cid:1) ∪e+l ∪e−l , where el(cid:1) = ∂Ci and
e± are the two copies of the essential curve e ⊂ C .
l l i
Definition 2.5. Let Υ = {e ,C } be a pant decomposition of C,
l i
(i) We define C4g−4(C)Υ to be the open subset of C(cid:3)4g−4(C) consisting of
those points b ∈ C4g−4(C) such that Cio = Ci − lel contains exactly
two points {xi,xi} of b.
1 2
AN ABELIANIZATION OF SU(2) WZW MODEL 5
(ii) We define B to be the open subset of B consisting of those points ˜b =
Υ
(b,α) ∈ B such that b ∈ C4g−4(C)Υ and that (cid:5)α,[el](cid:6) = 0 for 1 ≤
l ≤ 3g − 3, where [e ] is the Z homology class represented by e in
l 2 l
H (C −b,Z ).
1 2
Let C → B be the restriction of C → B to B .
Υ Υ Υ
Definition 2.6. For a pant decomposition Υ of C, let
(5) W = π (B ,˜b) ,
Υ 1 Υ
where ˜b = (b,α) is a base point of B .
Υ
Lemma 2.2. There is an exact sequence of groups
(6) 1 → WΥ → π1(C4g−4(C)Υ,b) → Zg2 → 1.
Proof. If we set C = (Co ×Co −{diagonal})/S and b∩Co = b , where
(cid:3) i i i 2 i i
Co = C − e , the group W is the kernel of the composition map
i i l Υ
(cid:4)
(7) π1(Ci,bi) → π1(C4g−4(C)Υ,b) → H1(C −b,Z2)
i
wherethefirstmapisinducedbytheinclusionandthesecondistheevaluation
map ev.
Now we choose and fix a pant decomposition Υ. We fix an orientation
of e for each l = 1,··· ,3g −3. We write e = C ∩−C if e is a common
l l i j l
boundary of C and C and the orientation of e agrees with that of C .
i j l i
We study the group W .
Υ
Let S be a 3-holed sphere as before. Let e be a boundary circle of S .
0 0
Let x1,x2 be two points in the interior of S0. Let pe = {pe(s)}0≤s≤1 be the
embedded arc in S connecting p (0) = x and p (1) = x as is depicted in
0 e 1 e 2
Figure 1.
e
.
x
1
.
x
2
Figure 1: Arc p
e
6 TOMOYOSHI YOSHIDA
Definition 2.7. Let e ,e ,e be the three boundary circles of S . We de-
1 2 3 0
fine the following closed loops in the symmetric product (S × S − ∆)/S
0 0 2
in which the lower indices should be understood mod.3 (anti-clockwise in
Figure 2),
(cid:5)(cid:1) (cid:2)(cid:6)
(i) t = p (s), p (1−s) ,
el el+1 el−1 0≤s≤1
(ii) k = t t t .
el el−1 el el+1
Here in Figure 2 the left represents the curve t and the right represents
e
1
the curve k . In the figure the curve with one arrow represents the trajectory
e
1
of x and one with double arrow does that of x corresponding to the paths
1 2
t and k respectively.
e e
l l
e1 e1
.
x.1 x1
.
.
x2 x2
e2 e3
e e
2 3
Figure 2: Curves
(cid:3)
For a pant decomposition Υ = {e ,C } of C, cutting out C along e ,
(cid:3) l i l l
∗ ∗
we obtain the disjoint union C as in (iii) in Definition 2.4. Each C can
i i i
be identified with S . Then the loops t and k in S given in Definition 2.7
0 e e 0
define the corresponding loops tCeli∗ and kleCli∗ resplectively in Ci∗ for el ⊂ ∂Ci∗.
Lemma 2.3. Let Υ = {e ,C } be a pant decomposition of C. Then W is
l i Υ
generated by the following elements.
(cid:7) (cid:8)
(i) tCeli∗(tCelj∗)±1 , where el = Ci∩Cj(i (cid:11)= j),
(cid:7) (cid:8)
(ii) tCi∗(tCi∗)±1 , where C = T and e± is as in Definition 2.4 (iii),
e+ e− i 0 l
l l
(cid:7) (cid:8)
(iii) tCeli∗ , where el ⊂ Ci is separating,
(cid:9) (cid:12)
(cid:10) (cid:11)
(iv) tCeli∗ 2 , where el ⊂ Ci,
(cid:7) (cid:8)
(v) keCli∗ , where el ⊂ Ci.
AN ABELIANIZATION OF SU(2) WZW MODEL 7
Proof. Clearly the listed elements are in the kernel of the evaluation map
ev. Let (C ,b ) be as in the proof of Lemma 2.2. The pure Braid group in
i i
the Braid group π (C ,b ) has index two and is generated by those homotopy
1 i i
classes represented by the loops such that x moves once along the small circle
1
centered at x while x is fixed and x (or x resp.) moves once along the loop
2 2 1 2
parallel to one component of the boundary ∂C while x (x resp.) is fixed. It
i 2 1
can be seen without difficulty that those homotopy classes can be represented
(cid:13)
by combin(cid:5)atio(cid:6)ns of tel. Hence the Braid group iπ1(Ci,bi) is generated by
the loops tCi . It is not difficult to see that Ker(ev) is generated by the
el el⊂∂Ci
listed elements.
2.3. Holonomy action of W . We study the holonomy diffeomorphisms of
Υ
the fibre bundle C → B induced by moves of the branch points along simple
Υ Υ
closed curves in B .
Υ
Let S be the 3-holed 2-sphere with ∂S = e ∪e ∪e . Let S˜ be the
0 0 1 2 3 0
2-fold branched covering space of S with branch locus x ∪x and covering
0 1 2
involution σ.
For each e the curve t in S induces a diffeomorphism τ of S˜ depicted
l e 0 e 0
l l
in Figure 3 where the upper and the lower boundary circles are e˜ and σe˜
l l
respectively and e˜ ∪ σe˜ represents the lifts of e . The diffeomorphism is a
l l l
combination of the half Dehn twists along the four curves in the picture in the
directions indicated by the arrows and the flip of the component of S˜ contain-
0
ing the branch points cutting along the two vertical circles which interchange
the points x and x and the two components e˜ and σe˜. The diffeomorphism
1 2 l l
is the identity on the lifts of the other boundary components.
e˜
1
e˜2 e˜3
x.
1
ó σe˜3 ó σe˜2
σe˜
ó 1
Figure 3: The induced diffeomorphism
8 TOMOYOSHI YOSHIDA
Likewise the curve k induces the Dehn twist κ of S˜ along the simple
e e 0
l l
closed curve which is the inverse image of the arc p (Figure 1) in S˜ .
e 0
l
Let Υ = {e ,C } be a pant decomposition of C.
l i (cid:3) (cid:3)
∗ ∗
Cutting out C along e to the disjoint union C , where C is iden-
l l i i i
tified with S , let C˜∗ be the 2-fold branched cover of C∗ branched at xi ∪xi.
0 i i 1 2
Then the above diffeomorphisms τ and κ of S˜ are converted to C˜∗; that
is, for el ⊂ ∂Ci∗, the holonomy alongelthe curevle tCeli∗0induces the diffeomoirphism
τeC˜li∗ of C˜i∗ which is τel under the identification Ci∗ = S0, and, for el ⊂ ∂Ci∗, the
holonomy along the curve∗keCli induces the Dehn twist κCe˜li∗ of C˜i∗ which is κel
under the identification C = S .
i 0
Definition 2.8. Let Υ = {e ,C } be a pant decomposition of C. Let b ∈
l i
B and let C˜ = C˜ .
Υ b
(i) For e = ∂C ∩∂C (i (cid:11)= j), we define a diffeomorphism of C˜ by
l i j

 τeC˜i∗ on C˜i
l
(8) τ(el) =  τeC˜lj∗ on C˜j
Id on C˜ −C˜ ∪C˜ .
i j
(ii) Let C = T and let e ∈ Υ be the essential simple closed curve in C .
i 0 l i
We define a diffeomorphism τ(e ) of C˜ by
l
(cid:18)
τC˜i∗τC˜i∗ on C˜
(9) τ(e ) = e± e∓ i
l l Ild on C˜ −C˜ .
i
(iii) For el = ∂Ci ∩∂Cj which is separating in C, let C = C− ∪Ci ∪C+ be
the decomposition of C,where C is the connected component of C −e
+ l
containingCj. LetC˜ = C˜−∪C˜i∪C˜+ bethecorrespondingdecomposition
of C˜. We define a diffeomorphism ν(e ) of C˜ by
 l
 Id on C˜−
(10) ν(el) =  τeC˜li∗ on C˜i
σ on C˜ .
+
(iv) For el ⊂ Ci, keCi induces a diffeomorphism κ(el) of C˜ defined by
l (cid:18)
(11) κ(el) = κICe˜dli∗ oonn CC˜˜i−C˜ .
i
(cid:10) (cid:11)
Lemma 2.4. Let WΥo be the subgroup of WΥ generated by { tCeli∗ 2} and
{keCi∗}. Then there is an exact sequence of groups
l
1 → Wo → W → Z3g−3 → 1.
Υ Υ 2
AN ABELIANIZATION OF SU(2) WZW MODEL 9
Proof. For 1 ≤ l ≤ 3g − 3, the inverse image p−1(e ) consists of two
l
connected components e˜ and σe˜. The diffeomorphisms listed in (i) and (ii) in
l l
Definition 2.8 interchanges these two connected components. Hence the action
of the holonomy diffeomorphisms on the homology classes {[e˜ −σe˜]} (with
l l
e˜ suitably oriented) in H (C˜,R) induces the homomorphism W → Z3g−3 in
l 1 Υ 2
the above sequence in the lemma. Then the exactness of the sequence is an
immediate consequence of the construction.
2.4. Marking and the universal cover of B . Let Υ = {e ,C } be a pant
Υ l i
decomposition of C. Let B be the space defined in Definition 2.5.
Υ
Let˜b = (b,α) ∈ B and let p : C˜ = C˜ → C be the corresponding two-fold
Υ ˜b
branched covering surface of C with covering involution σ.
Since ˜b = (b,α) ∈ BΥ, we may write b = {xi1,xi2}1≤i≤2g−2 for xi1,xi2 ∈ Cio
and C˜ = ∪C˜ ,where C˜ is the 2-fold branched covering surface of C branched
i i i
at xi ∪xi for 1 ≤ i ≤ 2g−2.
1 2
.
. .
.
Figure 4: Marking
Definition 2.9. Let Υ = {e ,C } be a pant decomposition of C. Let ˜b =
l i
(b,α) ∈ B .
Υ
We define a marking m = {f ,e ,T} of C associated with Υ as follows:
l l
(i) For 1 ≤ l ≤ 3g −3 such that e = C ∩C (1 ≤ i (cid:11)= j ≤ 3g −3), f is
l i j l
an embedded arc in C ∪C connecting xi and xj such that f ∩e =
i j 1 1 l l
{a point}.
(ii) For 1 ≤ l ≤ 3g−3 such that e is an essential curve in a 1-holed torus C ,
l i
f is an essential simple closed curve in C such that f ∩e = {a point}.
l i l l
(iii) For 1 ≤ l (cid:11)= l(cid:5) ≤ 3g −3, fl ∩fl(cid:1) is empty or xi1, where the latter case
occurs exactly when el ∪el(cid:1) ⊂ Ci.
(iv) T isamaximal treewhichisa1-complexwhoseverticesare{xi1}1≤i≤2g−2
and {fl ∩el}1≤l≤3g−3 and whose edges are arcs in {fl ∩Ci} connecting
xi and f ∩e in C for 1 ≤ i ≤ 2g−2.
1 l l i
10 TOMOYOSHI YOSHIDA
The set of pairs (˜b,m) for ˜b ∈ B and a marking m associated with Υ
Υ
serves as the universal covering space B˜ of B .
Υ Υ
2.5. The σ-anti-invariant homology group, the Lagrangian (cid:11)˜and the lat-
tices Λ and Λ. Let Υ = {e ,C } be a pant decomposition of C. For the
0 l i
covering surface p : C˜ → C associated with ˜b = (b,α) ∈ B , let
Υ
(12) H1(C˜,R) = H1(C˜,R)+⊕H1(C˜,R)−
be the decomposition into the invariant (+) and anti-invariant (−) subspaces
of the involution σ∗ on H1(C˜,R) induced by the covering involution σ. Then
H1(C˜,R)+ is isomorphic to H1(C,R) and dimRH1(C˜,R)− = 6g−6.
Definition 2.10. We define a symplectic form ω on H1(C˜,R)−, for a,b ∈
H1(C˜,R)−, by
1
ω(a,b) = (cid:5)a,b(cid:6),
2
where (cid:5)·,·(cid:6) denotes the symplectic form induced by the intersection pairing
on C˜.
Let e˜ be a connected component of p−1(e ) (1 ≤ l ≤ 3g − 3). Then
l l
p−1(e ) = e˜ ∪σe˜. We choose and fix an orientation of e˜.
l l l l
Let(cid:11)˜bethesubspaceinH1(C˜,R)− spannedby{[e˜l−σe˜l]}1≤l≤3g−3. Then
(cid:11)˜is Lagrangian with respect to ω.
Definition 2.11. Let C ∈ Υ.
i
(i) Assume C = S with ∂C = e ∪e ∪e . We set
i 0 i li li li
1 2 3
(cid:19) (cid:20)
1
(13) Ei = −(e˜ −σe˜ )+(e˜ −σe˜ )+(e˜ −σe˜ ) ,
1 2 l1i l1i l2i l2i l3i l3i
(cid:19) (cid:20)
1
Ei = (e˜ −σe˜ )−(e˜ −σe˜ )+(e˜ −σe˜ ) ,
2 2 l1i l1i l2i l2i l3i l3i
(cid:19) (cid:20)
1
Ei = (e˜ −σe˜ )+(e˜ −σe˜ )−(e˜ −σe˜ ) ,
3 2 l1i l1i l2i l2i l3i l3i
(cid:19) (cid:20)
1
Ei = (e˜ −σe˜ )+(e˜ −σe˜ )+(e˜ −σe˜ ) .
0 2 l1i l1i l2i l2i l3i l3i
(ii) Assume C = T . Let e = ∂C and let e ∈ Υ be the essential simple
i 0 li i li
1 2
closed curve in C . We set
i
1
(14) Ei = − [e˜ −σe˜ ]+[e˜ −σe˜ ],
1 2 l1i l1i l2i l2i
1
Ei = [e˜ −σe˜ ],
2 2 l1i l1i
1
Ei = [e˜ −σe˜ ]+[e˜ −σe˜ ].
0 2 l1i l1i l2i l2i