Solutions of the KZB equations in genus \geq 1

56 Pages·0.48 MB·English

by  B. Enriquez

SOLUTIONS OF THE KZB EQUATIONS IN GENUS GREATER
THAN ONE
0
0
0
B. ENRIQUEZ AND G. FELDER
2

n
a
J
Introduction. The Knizhnik-Zamolodchikov-Bernard connection ([23, 5, 28])
3
can be viewed as a flat connection over the complement det∗ of the zero-section
g,ν
] in the total space of the determinant line bundle over the moduli space M of
A g,ν·12
genus g curves with ν marked points and tangent vectors ([4]). Projectivization
Q
of this connection is the pull-back of a flat projective connection on M .
. g,ν·12
h The aim of this paper is two-fold. We first construct a flat version of the KZB
t
a connection over the moduli space M , when ν ≥ 1 (we will set ν = n+1). We
g,ν·12
m
thengiveintegralformulasforflatsectionsofthisconnection, usingthefunctional
[
parametrization of conformal blocks and the KZB connection introduced in [10].

2
v Construction of a flat connection. More precisely, we consider the moduli space
8
M of systems m = (X,P ,t,P ,v ) consisting of a curve X of genus g, a
9 g,1∞,n·12 0 i i
1 point P , a jet of coordinate t at P , and n points P with tangent vectors v (i =
0 0 i i
2
1,... ,n). Weassociatetothedataofasemisimple Liealgebrag¯, arepresentation
1
9 V of the Kac-Moody loop algebra g of g¯ and irreducible representations Vi of g¯,
9 the sheaf B of coinvariants over M . We denote by W∞ the subspace
/ V,(Vi) g,1∞,n·12
h of M formed of the systems m such that P is not a Weierstrass point of
g,1∞,n·12 0
at X. The main result of section 1 is the construction of a flat connection ∇V,(Vi) on
m this sheaf, which is smooth over the complement of W∞ (Thm. 1.1). ∇V,(Vi) has
v: a logarithmic singularity around W∞, which we compute in sect. 1.3. Moreover,
i the projectivization of ∇V,(Vi) is smooth and is isomorphic to the connection of
X
[28].
r
a Theproofofthesefactsiscontainedinsect.1. Ourmaintoolisanormalization
of the Sugawara tensor (formula (3)), which makes use of direct sum decomposi-
tions of spaces of the formal functions and formal one-forms at a non-Weierstrass
point of X (Lemma 1.1).
We can therefore think that the curvature of ∇V,(Vi) is a scalar delta-function
concentrated on the Weierstrass locus W∞.
Functional parametrization of ∇V,(Vi). Our next aimisto giveexplicit expressions
for the flat sections of ∇V,(Vi), in the special case where g¯ = sl . For this, we use
2
the functional parametrization of this connection that we introduced in our work
[10]. In sect. 3, we recall the main results of [10] for the sl case.
2
Date: January 2000.
1
2 B. ENRIQUEZ AND G. FELDER
Our work [10] was inspired by [13], where conformal blocks are parametrized
by correlation functions of generating currents of the Lie subalgebra n¯ ⊗C((t))
+
of g, where n¯ is the positive nilpotent subalgebra of g¯. In [10], we consider
+
twisted analogues of these correlation functions; this means that the nilpotent
currents are multiplied by exponentials of elements of h¯⊗C((t)), where h¯ is the
Cartan subalgebra of ¯g. This enabled us in [10] to construct a “functional” sheaf
Fℓ,(Λi) with flat connection ∇Fℓ,(Λi) over a covering Mg(a,1)∞,n·12, and a morphism
of sheaves with connections
corr : (BV,(Vi),∇V,(Vi)) → (Fℓ,(Λi),∇Fℓ,(Λi)).
We specialize V to an induced representation V and the V to a lowest weight
−ℓk i
Verma module V (see 3.1).
−Λi
The fiber of F over a point of the moduli space is a space of functions
ℓ,(Λi)
f(λ ,... ,λ |z ,... ,z ), defined on a subset of Cg ×(X −σ−1(P ))N, satisfying
1 g 1 N 0
regularity, symmetry and transformation properties (see sect. 3.3). Here σ : X →
X is the universal cover map of X, and N is given by e
1 e
N = [kℓ+ Λ ].
i
2
i
X
The connection ∇Fℓ,(Λi) is expressed by (20), (21) and (22). In the case N = 0,
the formula for ∇Fℓ,(Λi) is an analogue of the Lam´e equation.
Connections on ΩX. The expression of ∇Fℓ,(Λi) involves a connection D(λ) on the
canonical bundle Ω on X, depending on λ ∈ Cg, which is defined by the formula
X
(D(λ)ω)(z) = −lim (G (z,z′)ω(z′)+G (z′,z)ω(z)),
z′→z λ λ
where G (z,z′) is the twisted Green function on X corresponding to λ (see (11)).
λ
By a connection ∇ on Ω , we understand the data of a divisor D on X and
Ω X
a collection of compatible maps Γ(U,Ω ) → Γ(U,Ω⊗2(D)) for each open subset
X X
of X, satisfying the Leibniz rule.
Any connection ∇ on Ω can be expressed in the form
Ω X
∇ ω = α d(ω/α ),
Ω ∇ ∇
where α is some form on the universal cover of X −D.
∇
The aim of sect. 2 is to compute the form α associated with the connection
∇
D(λ). This is done in Prop. 2.2.
Flat sections of F . As we have seen, to describe flat connections of B is
ℓ,(Λi) V,(Vi)
the same as
(i) describing the flat sections of F , and
ℓ,(Λi)
(ii) characterizing the image of B in F .
V,(Vi) ℓ,(Λi)
In this paper, we only say some words about problem (ii) in the case of inte-
grable modules (see Remark 12).
SOLUTIONS OF THE KZB EQUATIONS IN GENUS GREATER THAN ONE 3
We solve (i) completely in the case N = 0 (sections 4 and 5). Out main
results are Thm. 4.1 and Thm. 5.1, where we show that flat sections correspond
bijectively to solutions of a heat equation. This result is analogous to that of [15]
in the elliptic case g = 1; it is also close to the formulas of Verlinde and Verlinde
([9]). Our formulas also involve functions α(m,P ,v ) and β(m,P ,v ) on the
0 0 0 0
moduli space M , which are determined by differential equations (Thm. 4.1,
g,12
Prop. 5.1). They are homogeneous of fixed degree in the variable v . Their exact
0
meaning is unclear to us, but we propose conjectures about them in Remark 7.
In the case N > 0, we construct flat sections of F as integrals over twisted
ℓ,(Λi)
cycles over powers of X of the functions obtained in Thm. 5.1 (Thm. 6.1). Our
main toolis the result that some flat sections ofB can be constructed as inte-
V,(Vi)
gralsofflatsections ofBV,(Vi),V−2,...,V−2 (Prop.6.1). This result generalizes thoseof
Cherednik ([7]) in the case of genus 0 and Varchenko and the second author ([16])
in thecase g = 1. (An abstract version of thisresult can also befound inthework
[6]). We then obtain our expression of flat sections of F in Thm. 6.1. This
ℓ,(Λi)
expression involves differential-evaluation operators Φ (w ,... ,w |z ,... ,z ),
(Λi) 1 n 1 n
which correspond to the insertion of a product p f(w ) p e(z ) in correla-
j=1 j j=1 j
tion functions (Lemma 6.1). It seems that comparing our formulas with explicit
Q Q
formulas in the case k = 1 would yield integral identities, as it was done in [16].
In section 6.2.2, we write an example of our formulas for a simple case (g = 2).
. Let us now say some words on the topological aspects of our work.
Representations of the mapping class group. The following three types of rep-
resentations of the mapping class group (MCG) are usually thought to be closely
related:
(a) in [21], Kohno constructed projective representations of the MCG using
braiding and fusing matrices of the genus zero KZ equations, and the method of
Moore and Seiberg [26] for constructing such representations;
(b) in [24], Lyubashenko, relying on work of Reshetikhin-Turaev ([27]), con-
structed projective representations of the MCG using the representation theory
of quantum groups at roots of unity;
(c) the monodromy of the KZB connection also provides projective represen-
tations of the MCG.
The identification between representations of the MCG of (a) and (b) should
follow from the identification of the fusing and braiding matrices, which can be
checked in genera 0 and 1.
Ontheother hand, our integralformulas might serve tounderstand theconnec-
tion (b)-(c). This could be done using the geometric interpretation of quantum
groups of [8]. The identification (b)-(c) would be an analogue of the Kazhdan-
Lusztig equivalence in genus ≥ 1 ([20]).
Extensions of the mapping class group. As we have seen, the KZB connection
constructed in [4] gives representations of π (det∗ ), which is an extension of the
1 g,ν
MCG. In [18], Harer showed that the MCG is perfect and computed its universal
4 B. ENRIQUEZ AND G. FELDER
central extension (the central extensions of the MCG arising in natural projective
representations were later computed by Masbaum and Roberts [25]). It is natural
to believe that π (det∗ ) identifies with this universal central extension.
1 g,ν
On the other hand, if we denote by W(2) the subspace of M formed of the
g,ν·12
modulisuchthatthefirstpointisnotWeierstrass, Thm.1.1yieldsrepresentations
of π (M −W(2)). π (M −W(2)) is also an extension of MCG.
1 g,ν·12 1 g,ν·12
Projectivizations of bothrepresentations yield coinciding representations ofthe
MCG. It is therefore natural to think that
(a) the fundamental groups π (det∗ ) and π (M −W(2)) coincide
1 g,ν 1 g,ν·12
and
(b) the representations of these groups provided by [4] and Thm. 1.1 coincide.
. We would like to thank B. Feigin for discussions about this paper. The first
authorwouldalsolike tothankA.-S.SznitmanforinvitationstotheFIM(ETHZ)
in 1998 and June 1999, as well as A. Alekseev for an invitation to ESI (Vienna)
in August 1999, during which a part of this work was done.
1. Flat version of the KZB connection and Weierstrass points
Notation . For P a point of a curve X, and ω a formal differential at P , we
0 0
denote by hωi its residue res (ω) at the point P . For ξ a vector and ω(2) a
P0 0
quadratic differential at the neighborhood of P , we set hξ,ω(2)i = hξω(2)i. In the
0
same way, if f is a function at the neighborhood of P , we set hf,ωi = hfωi. If
0
f and ω depend on variables z and t, we will write hf,ωi for res [(fω)(z,t)].
z z=P0
If α is a differential of order k at P and v is a tangent vector at P , we denote
0 0 0
by (α,v ) the evaluation of v on α. This expression is homogeneous of degree k
0 0
in v .
0
Finally, if α is a formal differential of order k at P , we define its valuation as
0
the smallest integer k such that tkα is regular at P , where t is a local coordinate
0
at P , and we denote it by val(α).
0
1.1. Weierstrass points. Let X be a compact complex curve; let us denote its
genus by g. A point P of X is not a Weierstrass point if there are holomor-
0
phic differentials on X with valuations 0,1,··· ,g −1 at P ; in other words, the
0
Wronskian of holomorphic differentials does not vanish at P .
0
Let P be an arbitrary point of X, and let t be a formal coordinate on X at
0
P . Let R be the subring of C((t)), formed of the Laurent expansions at P of the
0 0
functions on X regular outside P . Let Ωout be the subspace of C((t))dt formed
0
by the forms on X, regular outside P .
0
Lemma 1.1. The following statements are equivalent:
1) P is not a Weierstrass point of X.
0
2) We have direct sum decompositions
C((t))dt = Ωout ⊕ Ct−1dt⊕tgC[[t]]dt (1)
(cid:0) (cid:1)
SOLUTIONS OF THE KZB EQUATIONS IN GENUS GREATER THAN ONE 5
and
C((t)) = R⊕ ⊕−1 Cti ⊕tC[[t]] . (2)
i=−g
Proof. Let us show the equivalence(cid:0)of 1) and decomp(cid:1)osition (1). Let us study
Ωout ∩ (Ct−1dt + tgC[[t]]dt). Let ω belong to this intersection. Then ω has the
expansionω t−1dt+ ω tidt. Sinceres (ω) = 0,wehaveω = 0. Therefore
−1 i≥g i P0 −1
the valuation at P of ω is ≥ g. It follows that Ωout∩(Ct−1dt+tgC[[t]]dt) consists
0
P
of all regular forms on X whose valuation at P is ≥ 0. This is zero iff P is not
0 0
Weierstrass.
Let us now show that (1) implies (2). We will first show that
R∩ ⊕−1 Cti ⊕tC[[t]] = 0.
i=−g
Assume that this intersection(cid:0)is nonzero, and let(cid:1)φ be a nonzero function in this
intersection. Let j be its valuation at P . j belongs to {−1,... ,−g}. Since P
0 0
is Weierstrass, we have a holomorphic differential ω on X with valuation at P
0
equal to −j−1. Then φω is a holomorphic differential on X with nonzero residue
at P , a contradiction. It follows that R∩ ⊕−1 Cti ⊕tC[[t]] = 0.
0 i=−g
(2)nowfollowsfromthefactsthatΩout andRononeside,and(Ct−1dt⊕tgC[[t]]dt)
(cid:0) (cid:1)
and ⊕−1 Cti ⊕tC[[t]] on the other side, are each other’s annihilators for the
i=−g
residue pairing.
(cid:0) (cid:1)
1.2. “Weierstrass” normalization of the Sugawara tensor. The residue
defines a nondegenerate pairing between C((t)) and C((t))dt. Assume that P is
0
not a Weierstrass point of X. As we have seen, the spaces R and Ωout, and
⊕−1 Cti ⊕tC[[t]] and Ct−1dt⊕tgC[[t]]dt
i=−g
are annihilators o(cid:0)f each other for th(cid:1)is pairin(cid:0)g. (cid:1)
Let (ω ) and (r ) be bases of
i i
⊕−1 Cti ⊕tC[[t]] and Ct−1dt⊕tgC[[t]]dt ,
i=−g
such that the seq(cid:0)uences (val(r )) an(cid:1)d (val((cid:0)ω )) tend to infinit(cid:1)y, and let (ωi) and
i i i i
(ri) be the dual bases of Ωout and R. We have the equality (in C[[z±1,w±1]]dw)
δ(z,w)dz = ωi(z)r (w)+ ω (z)ri(w),
i i
i i
X X
where δ(z,w)dw = ziw−i−1dw.
i∈Z
Let g¯ be a semisimple Lie algebra with nondegenerate invariant bilinear form
h,i , and define g asPthe canonical central extension of ¯g⊗C((t)); it is the direct
¯g
sum ¯g⊗C((t))⊕CK, endowed with the Lie bracket such that K is central and
[x⊗f,x′ ⊗f′] = [x,x′]⊗ff′ +hx,x′i hdff′iK.
¯g
Let V be any g-module on which the Lie subalgebra g¯⊗C[[t]] acts locally finitely.
We will denote by x[f] the element (x⊗f,0) of g for x in g¯, f in C((t)).
6 B. ENRIQUEZ AND G. FELDER
Define, for ξ in C((t))∂ ,
∂t
1
T [ξ] = xα[ξωi]xα[r ]+xα[ri]xα[ξω ] , (3)
R i i
2κ
α i
XX(cid:0) (cid:1)
with (xα) an orthonormal basis of g¯ and κ = k+h∨, h∨ the dual Coxeter number
of ¯g. We call T [ξ] the normalized Sugawara tensor.
R
Then we have, for any f in C((t)),
[T [ξ],x[f]] = −x[ξ(f)], (4)
R
where ξ(f) denotes the action of the vector field ξ on the function f (the product
of a form ω and a vector ξ, which is a function, is simply denoted ξω; we have
therefore ξ(f) = ξdf).
Define ¯g(R) as image of g¯ ⊗ R in g by the map x ⊗ f 7→ x[f]; this is a Lie
subalgebra of g.
One of the useful features of T [ξ] is:
R
Lemma 1.2. If ξ is a vector field on X, regular outside P , T [ξ] belongs to
0 R
g¯(R)Ug.
Proof. The follows from the fact that ξωi belongs to R.
1.2.1. Another expression of the normalized Sugawara tensor. Here and below,
d denotes the partial exterior differentiation with respect to the variable z′.
z′
Lemma 1.3. Set
ω¯(z,z′) = ωi(z)dr (z′). (5)
i
i
X
Define T (z) as
ω¯
1
lim xα(z)xα(z′)−k(dimg¯)ω¯(z,z′) .
2κ z′→z" #
α
X
Then the matrix elements of T [ξ] and hξ,T (z)i in any highest weight represen-
R ω¯
tation of g are well-defined and coincide.
Proof. Set
x(z) = x[αi]ǫ (z), x(z) = x[ri]ω (z), x(z) = x[r ]ωi(z),
i z→Ωin i z→Ωout i
i i i
X X X
for (αi),(ǫ ) dual bases of C((t))dt and C((t)). We have
i
2κT (z) = xα(z) xα(z)+xα(z)xα(z)
ω¯ z→Ωout z→Ωin
α
X
+lim [xα(z) ,xα(z′)]−k(dimg¯)ω¯(z,z′).
z′→z z→Ωout
α
X
SOLUTIONS OF THE KZB EQUATIONS IN GENUS GREATER THAN ONE 7
As the limit vanishes, we have
1
T (z) = xα(z) xα(z)+xα(z)xα(z) .
ω¯ z→Ωout z→Ωin
2κ
α
X
We have then
1
hT (z),ξi = xα[ri]xα[ǫj]hω α ,ξi+xα[ǫj]xα[r ]hωiα ,ξi
ω¯ i j i j
2κ
α
X
1
= xα[ri]xα[ξω ]+xα[ξωi]xα[r ] = T [ξ].
i i R
2κ
i
X
1.3. Vector fields on moduli spaces. Let g,n be integers ≥ 0. Let M
g,1∞,n·12
be the space of isomorphism classes of systems
m = (X,t,P ,v ,... ,P ,v )
1 1 n n
of a genus g curve X, a marked point P with a coordinate t at P , and other
0 0
markedpointsP ,... ,P distinctfromP togetherwithtangentvectorsv ,... ,v
1 n 0 1 n
at these points. We will denote M simply as M ; it is the space of
g,1∞,0·12 g,1∞
isomorphism classes of pairs
m = (X,t);
∞
M can also be viewed as the space of subrings R of the Laurent series field
g,1∞
C((t)), such that R is isomorphic to a coordinate ring H0(X −{P },O ), with
0 X
X a curve of genus g, and the inclusion R ⊂ C((t)) is the inclusion of R in the
local field of X at P .
0
In the same way, M can be identified with the space of systems m =
g,1∞,n·12
[R,χ¯ ,... ,χ¯ ], where R ⊂ C((t)) is the same subring of C((t)) as above, and χ¯
1 n i
are morphisms from R to C[η]/(η2). (Indeed, a morphism χ¯ from R to C[η]/(η2)
is the same as the data of a pair (χ,∂ ) of a morphism χ from R to C and a map
χ¯
∂ from R to C, such that ∂ (fg) = f∂ (g)+∂ (f)g; χ corresponds to a point
χ¯ χ¯ χ¯ χ¯
of Spec(R) and ∂ to a tangent vector at this point.)
χ¯
Define ξ 7→ [ξ] as the map from C((t))∂ to Vect(M ), such that for ξ
∂t g,1∞,n·12
in C((t))∂ , the infinitesimal translate of [R,χ¯ ,... ,χ¯ ] by [ξ] is [(1+ǫξ)(R) ⊂
∂t 1 n
C((t)),χ¯ ◦(1−ǫξ),... ,χ¯ ◦(1−ǫξ)].
1 n
Proposition 1.1. ([1, 3, 19, 22]) ξ 7→ [ξ] defines a Lie algebra morphism from
C((t))∂ to Vect(M ). Let ev be the evaluation of a vector field over
∂t g,1∞,n·12 m
M at the point m ∈ M . The kernel of ξ 7→ ev ([ξ]) is the space
g,1∞,n·12 g,1∞,n·12 m
of vector fields ξ, which admit a regular continuation to X −{P } and vanish to
0
second order at the P .
s
1.4. The flat connection (BV,(Vs),∇V,(Vs)).
8 B. ENRIQUEZ AND G. FELDER
1.4.1. The sheaf BV,(Vs). Let Mg,1∞,n be the space of isomorphism classes of sys-
tems m = (X,t,P ) of a genus g curve X, a point P onit, with a local coordinate
s 0
t, and of n distinct points P , distinct from P . M can be viewed as the
s 0 g,1∞,n
space of systems m = [R,χ ] of a subring R of C((t)) as above, and of a collection
s
of n morphisms (χ ) from R to C. Let p be the morphism from C[η]/(η2)
s s=1,...,n
to C, such that p(η) = 0. We have a natural projection π from M to
2→1 g,1∞,n·12
M , such that π ([R,χ¯ ]) = [R,p◦χ¯ ].
g,1∞,n 2→1 s s
Say that a representation (ρV,V) of g is of the class (C) if the following condi-
tions are satisfied:
1) for any v ∈ V, there exists an integer n(v) such that if n ≥ n(v), ρV(g¯ ⊗
tn)(v) = 0; (this implies that the components of the Sugawara tensor act as
well-defined endomorphisms of V);
2)letT bethezero-modeoftheSugawaratensor. Fornanddinteger numbers,
0
let us set V[n,d] = Ker(ρV(T0)−d)∩Ker(ρV(h[1])−n); then V = ⊕n,d∈ZV[n,d],
and for any integer n, there is d(n) such that for d ≥ d(n), V[n,d] = 0.
Modules of the class (C) contain both twisted Weyl modules over g and in-
tegrable representation of g. Moreover, they are such that the positive Fourier
modes of the Sugawara tensor act locally nilpotently.
Fix a representation V of g of the class (C) and representations (π ,V ) of
s s
g¯. We will assume that the Casimir element (x )2 acts on V as a scalar
α α s
∆ id (this is the case if the V are irreducible modules). We have an evaluation
s Vs s P
morphism ev from g¯(R) = g¯⊗R to g¯, defined as id⊗χ . π ◦ev defines then
Ps s s s
a g¯(R)-module structure on V . We denote by V(Ps) the resulting g¯(R)-module.
s s
Define B0 as the sheaf over M , whose sections over an open subset U
V,(Vs) g,1∞,n
are maps ψ : U → V∗, such that ψ(m) is ¯g(R)-invariant for any m in U, where
V = V⊗(⊗ V(Ps)).
s s
Define B as the pull-back sheaf π∗ (B0 ) of B0 to M . Local
V,(Vs) 2→1 V,(Vs) V,(Vs) g,1∞,n·12
sections of this sheaf are maps ψ : [R,χ¯ ,... ,χ¯ ] → [V⊗(⊗ V )]∗, such that for
1 n s s
each v ∈ V, m 7→ hψ(m),vi is smooth, and each ψ([R,χ¯ ,... ,χ¯ ]) is g¯(R)-
1 n
invariant. Here V is endowed with the ¯g(R)-module structure given by π ◦
s s
(id ⊗{p◦χ¯ }) (id ⊗{p◦χ¯ } is a map from ¯g(R) = g¯⊗R to ¯g, and π maps ¯g
¯g s ¯g s s
to End(V )).
s
1.4.2. The locus W∞ of Weierstrass points. Define W as the subset of M
g,1
formed of the pairs (X,P ) such that P is a Weierstrass point of X.
0 0
(a) (a)
Let M , resp. M be the moduli space of systems (X,P ,v ,{A }) of a
g,12 g,1 0 0 a
genus g curve with a marked point, a nonzero tangent vector at this point, and
a system of a-cycles (resp. of systems (X,P ,{A })). Let π′ be the natural
0 a 2→1
(a) (a)
projection from M to M .
g,12 g,1
TheWronskianoftheAbeliandifferentialsdefinesafunctionW(X,P ,v ,{A })
0 0 a
on M(a) . This function is homogeneous of degree g(g+1) in v .
g,12 2 0
SOLUTIONS OF THE KZB EQUATIONS IN GENUS GREATER THAN ONE 9
Then (π′ )−1(W) is defined by the equation W(X,P ,v ,{A }) = 0. It fol-
2→1 0 0 a
lows that (π′ )−1(W) has codimension 1 in the smooth part of M(a) , and that
2→1 g,12
W has codimension 1 in the smooth part of M .
g,1
We will define W∞ as the preimage π−1(W) of W by the natural projection
∞
map π : M , such that
∞ g,1∞,n·12
π (X,t,P ,v ) = (X,P ).
∞ i i 0
1.4.3. The connection ∇V,(Vs). For x in Ug, we will denote by x(0) the element
πV(x)⊗(⊗sidVs) of End(V⊗(⊗sVs)) and by x(s) (sometimes x(Ps)) the element
idV ⊗(⊗s′<sidVs′)⊗πVs(x)⊗(⊗s′>sidVs′).
Lemma 1.4. For m 7→ ψ(m) a local section of B , and ξ in C((t))∂/∂t, the
V,(Vs)
value at m of
∂ ψ(m)−ψ(m)◦T [ξ](0)
[ξ] R
belongs to (V∗)¯g(R) and only depends on the value at m of [ξ].
Proof. Let ǫ be a variable such that ǫ2 = 0. Let us set R = (1+ǫξ)(R). Then
ǫ
g¯(R ) = Ad(1−ǫT[ξ])[g¯(R)]. It follmows that ∂ ψ(m)−ψ(m)◦T [ξ](0) belongs
ǫ [ξ] R
to is ¯g(R)-invariant.
Let R be the subspace of R consisting of the functions vanishing at each P .
(Ps) s
Let g¯(R ) be the image of ¯g⊗R by the inclusion of ¯g⊗R in g. Let us show
(Ps) (Ps)
that for ξ a vector field in C((t))∂/∂t, with a regular prolongation to X −{P }
0
and double zeroes at each P , T [ξ] belongs to g¯(R )Ug. Let (ri ) be a basis
s R (Ps) (Ps)
of R and let (ωi ) be a basis of the space of forms Ω on X, regular except
(Ps) (Ps) (Ps)
at P and the P , and with at most simple poles at the P . R and Ω are
0 s s (Ps) (Ps)
the annihilators of each other by the residue pairing. We may therefore choose
(r ) and (ω ) in K and Ω so that (ri ;r ) and (ω ;ωi ) are dual
i;(Ps) i;(Ps) (Ps) i;(Ps) i;(Ps) (Ps)
bases of K and Ω. As the ξωi and the ri belong to R ,
(Ps) (Ps) (Ps)
1
T′[ξ] = xα[ξωi ]xα[r ]+xα[ri ]xα[ξω ]
2κ (Ps) i;(Ps) (Ps) i;(Ps)
α,i
X
belongs to ¯g(R )Ug.
(Ps)
On the other hand, let ω′i and r′ be lifts to Ωout and Rout of dual bases
(Ps) i;(Ps) (Ps)
of Ωout /Ωout and Rout/Rout . T [ξ]−T′[ξ] is equal to
(Ps) (Ps) R
1
[xα[ξω′i ],xα[r′ ]].
2κ (Ps) i;(Ps)
α,i
X
This is proportional to hξω′i dr′ i, which is zero because the ξω′i be-
i (Ps) i;(Ps) (Ps)
long to Ωout. It follows that T [ξ] is equal to T′[ξ], and therefore belongs to
P R
g¯(R )Ug.
(Ps)
For x in g¯(R ) and v in V, ψ(m)(xv) vanishes, therefore ψ(m) ◦ T [ξ](0)
(Ps) R
vanishes if ξ belongs to the kernel of ξ 7→ ev ([ξ]). This proves the Lemma.
m
10 B. ENRIQUEZ AND G. FELDER
It follows from Lemma 1.4 that the formula
∇V,(Vs)ψ(m) = ∂ ψ(m)−ψ(m)◦T [ξ](0)
[ξ] [ξ] R
defines a connection ∇V,(Vs) over the restriction of BV,(Vs) to Mg,1∞ −W∞.
1.4.4. Flatness of ∇V,(Vs). We will show:
Theorem 1.1. ∇V,(Vs) is a flat connection on the restriction of BV,(Vs) to Mg,1∞−
W∞.
Proof. Let us compute the curvature of ∇V,(Vs). We have
[∇V,(Vs),∇V,(Vs)]−∇V,(Vs) ψ = ψ◦(−T [[ξ,η]]−[T [ξ],T [η]]−∂ T [η]+∂ T [ξ])(0).
ξ η [ξ,η] R R R ξ R η R
W(cid:16) e may assume that r an(cid:17)d ω are independent of R. Then
i i
∂ ri = ξ(ri)− hξ(ri),ω irj,
ξ j
j
X
and
∂ ωi = d(ξωi)− hd(ξωi),r iωj.
ξ j
j
X
Then
1
∂ T [η] = xα ηd(ξωi)− hd(ξωi),r iηωj xα[r ]
ξ R j i
2κ
" #
α i j
XX X
+xα ξ(ri)− hξ(ri),ω irj xα[ηω ].
j i
" #
j
X
On the other hand, since ξ(ηω)−η(ξω) = [ξ,η]ω, we have
1
[T [ξ],T [η]]+T [[ξ,η]] = − xα[η(ξωi)]xα[r ]+xα[ηωi]xα[ξ(r )]
R R R i i
2κ
α i
XX
+xα[ξ(ri)]xα[ηω ]+xα[ri]xα[η(ξω )].
i i
It follows that
−2κ(∇V,(Vs) −[∇V,(Vs),∇V,(Vs)])ψ
[ξ,η] ξ η
= ψ ◦πV{− {xα[η(ξωi)]xα[ri]+xα[ηωi]xα[ξ(ri)]+xα[ξ(ri)]xα[ηωi]
α i
XX
+xα[ri]xα[η(ξω )]}
i
+ xα ηd(ξωi)− hd(ξωi),r iηωj xα[r ]+xα ξ(ri)− hξ(ri),ω irj xα[ηω ]
j i j i
" # " #
α i j j
XX X X
− xα ξd(ηωi)− hd(ηωi),r iξωj xα[r ]+xα η(ri)− hη(ri),ω irj xα[ξω ]}}.
j i j i
" # " #
α i j j
XX X X