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mology of the Lie algebra of regular differential operators on (Cx of degree ^ 1 .... weight representations of the universal central extension of the Lie algebra Q)F ...
Communications in Mathematical

Commun. Math. Phys. 117, 1-36 (1988)

Physics

© Springer-Verlag 1988

Moduli Spaces of Curves and Representation Theory E. Arbarello 1 , C. DeConcini 2 , V. G. Kac 3 , and C. Procesi 1 1 2 3

Dipartimento di Matematica, Universita Degli Studi di Roma "La Sapienza", Rome, Itab Dipartimento di Matematica, Universita di Roma II "Tor Vergata", Rome, Italy Department of Mathematics, M.I.T., Cambridge, MA 02139, USA

Abstract. We establish a canonical isomorphism between the second cohomology of the Lie algebra of regular differential operators on (Cx of degree ^ 1, and the second singular cohomology of the moduli space &g-γ of quintuples (C, p, z, L, [_φ~\\ where C is a smooth genus g Riemann surface, p a point on C, z a local parameter at /?, L a degree g— 1 line bundle on C, and [φ] a class of local trivializations of L at p which differ by a non-zero factor. The construction uses an interplay between various infinite-dimensional manifolds based on the topological space H of germs of holomorphic functions in a neighborhood of 0 in (Cx and related topological spaces. The basic tool is a canonical map from # 0 _ ! to the infinite-dimensional Grassmannian of subspaces off/, which is the orbit of the subspace H_ of holomorphic functions on (Cx vanishing at oo, under the group AutH. As an application, we give a Lie-algebraic proof of the Mumford formula: λn = (6n2 — 6n + ί)λu where λn is the determinant line bundle of the vector bundle on the moduli space of curves of genus g, whose fiber over C is the space of differentials of degree n on C.

Introduction Consider the Lie algebra Q)¥ (F for finite) of regular differential operators of degree less than or equal to 1 on (Cx and its subalgebra dF of vector fields, so that z\ dj = zj+1 — \ is a basis of Q)¥ and {dj}neZ is a basis of dF. The Lie algebra S)F dz)jeZ acts in a natural way on the space Vn of regular differentials of degree n on (Cx with basis vk = z~kdz", keZ. This gives an inclusion

where a^ is the Lie algebra of matrices ( α ^ j e z also consider the restriction of φn tod F :

s u c n t n a t

^ = 0 for \i—j\^>0. We

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E. Arbarello, C. DeConcini, V. Kac, and C. Procesi

One has the following 2-cocycle on a^ [KP, DJKM]: if ί^O, j>0 otherwise,

{ψiE^Ej^-ψiEj.E^ί \ψ(Eip Ers) = 0

whose cohomology class generates //;?ont(a^,(C) = C Another way of expressing this cocycle is the following. Given (fl^eaζ, write /(z, w) = Σaifι ~Xw ~j a n d l e t 1 /- + = Σ at/- *,'*, / + _= Σ a./^w-K i^ 0 J>0

i>0 j^O

Both /+ _ and /_ + are polynomials, and given

we have

Pulling back the cocycle ψ via φn we get a cocycle φ*(ψ) on ^ F which works out to be

φ*(ψ)(zUk)=-δj,-k(n-$)j(j-l). Restricting to dF we get cocycles ρ%(ψ) which satisfy the relation ρ*(ψ).

(0.1)

Recall that the cohomology class oϊρ$(ψ) generates H2(dF, (C) = (C; a less wellknown fact is that # 2 (^ F ,(C)^(C 3 . On the other hand, let π:%?^>S be a family of genus g compact Riemann surfaces and let ω^/s be the relative dualizing sheaf of π. Denote by λn the determinant line bundle of ω^/s on S. Then, as observed by Mumford [Mu], the Grothendieck-Riemann-Roch theorem for the family π gives the following relation between Chern classes: 2

c l μ j =(6n -6n+l)ciμl).

(0.2)

One of the main objectives of the present paper is to explain the coincidence of (0.1) and (0.2). In order to achieve this it is therefore of central importance to us tofinda relationship between extensions of our Lie algebras and line bundles on moduli spaces. Let us briefly introduce the moduli spaces involved in our construction. First of all the moduli space JίQ of smooth curves of genus g, then the moduli space Jί'^ of triples (C, p, v) when C is a genus g Riemann surface, p a point on C, and v a nonzero tangent vector to C at p. We also consider the moduli space 3Fζ of quadruples (C,p,v,L), where L is a degree h line bundle on C and (C,p, v)e Jig.

Moduli Spaces of Curves and Representation Theory

3

Furthermore, we construct an infinite dimensional complex manifold Jίg which is a moduli space of triples (C, p, z), where z is a local parameter at p. Finally, we construct another infinite dimensional complex manifold β"h parametrizing quintuples (C, p, z, L, [_φ~]\ where C, p, z, L are as above, φisa local trivialization of L at p and [φ] is the class of φ modulo non-zero multiplicative constants. Of course, we have natural projections The first projection induces an isomorphism in second cohomology [actually, Harer, Ann. Math. 121, 215-249 (1985), has proven that Jί'g has the same cohomology as Jig for g large], the remaining two are homotopy equivalences. By using the Kodaira-Spencer deformation theory on the infinite dimensional manifolds Jίg and &g-ι we get natural Lie algebra homomorphisms d-Vect(^), where d and 3) are suitable analytic analogues of dF and & in which dF and Q)? are dense. The above homomorphisms have the property that for every xeJίg (respectively ^ _ i ) the evaluation map

is surjective. From this one gets that the tangent bundle T(Jkg) (respectively T(βFg-3)-^$—>0 we can lift canonically the inclusion to an inclusion 9X^§. For this we use the following two facts: i)

^ = [^,^1,

ii)

ρ\@χ is the trivial extension.

Using the inclusion Θx c> @) we can construct an extension of the tangent bundle Ί(β:g_λ) whose fiber at x is @/@x. Thus dualizing and passing to cohomology classes we get μ.

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E. Arbarello, C. De Concini, V. Kac, and C. Procesi

On the other hand, we have a natural homomorphism associating to each line bundle L the class of the extension where ΣL is the sheaf of differential operators of degree less than or equal to one on L. One of our results is that c and μ have the same image. Once this has been shown we get the following diagram: H\(9%) - ^

Ext1 ( ^ , 0 * , )

I

^-#2(d)

I

where the first two vertical arrows are induced by the canonical section JίQ -+&g_γ, associating to a triple (C,p,z) the quintuple (C,p,z, &((g — l)p), \_z~9+λ~\). Thus we obtain an explicit connection between the cohomology of S (respectively d) and line bundles on &Q -1 (respectively Jkg). To see that μ and c have the same image, we first notice that &g-γ and 3) are acted on by automorphisms τ and t defined by τ((C, p, z, L, [(/>])) = (C, p, z, L®ωc((2 - 2g)p), [φz2*~ Hz\),

which are related by the commutative diagram

•I., h 2

Using this diagram and the fact that H (&)) is a cyclic module over the group k {t }kez with cyclic element ψ0 = φo( — ψ), we are reduced to show that μ(ψ0) lies in the image of c. Indeed, we consider the divisor θ on βg_ x consisting of quintuples (C, p, z, L, [φ]) with L effective, and show that μ(ψo) = c(θ).

(0.4)

To prove this equality we use a global version of a construction due to Krichever and analyzed in [SW], giving an analytic map

where Gr(H) is a suitably defined infinite dimensional Grassmannian. We then have that G( — θ) is the pullback, via W, of the determinant line bundle if on Gτ(H). On the other hand, we have the following commutative diagram -> a r

Moduli Spaces of Curves and Representation Theory

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By an argument similar to the one above we associate to the extension of a^ given by the cocycle ψ an extension of «^Gr(H) by ΘGτ{H) which turns out to be the sheaf Σ# of differential operators of degree less than or equal to 1 on jSf. We can then pull back this extension to an extension of ^ V 1? which by functoriality is the sheaf of differential operators of degree less than or equal to one on Θ{ — θ). The above diagram then gives (0.4). Following diagram (0.3) and our analysis of the homomorphisms t and τ we deduce that

In fact, by use of a result of Harer [H], and a generalization of it for J^"_ t which we explain in Sect. 5, we can then conclude: Theorem. There are canonical homomorphisms, which are isomorphisms for g ^ 5,

such that (i) (ii) The diagram

H2(d)—+

H2{Jίg,. In Sect. 5 we compute H2(^L1) = H2{βg.1), using results of [H]. In the Appendix (Sect. 6) we classify the degenerate and the unitary highest weight representations of the universal central extension of the Lie algebra Q)F. Some of the results of this paper were quoted in [AGR].

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E. Arbarello, C. De Concini, V. Kac, and C. Procesi

1. Notation and Preliminaries I) Curves and Their Moduli For any family of smooth curves π:^->S,

(1.1)

parametrized by an analytic space S, we shall denote by ωS whose fiber over 5 6 $ is PicΛ(Cs). Finally, associated to the relative Picard variety Pic9~ι(π) is a relative theta-divisor θπ, or simply θ, which is a line bundle on Pic^" 1 ^), whose restriction to Picflf~1(Cs) is the theta-divisor Θ(ΘS) (for each 5). A family of pointed curves σ

parametrized by S is a family of curves π equipped with a section σ of π. When Λ dealing with pointed curves we can define a canonical section of p:Pic (π)->S:

~ " sZdlcihσ)) canonical isomorphisms ic π -> ic π , (s,L)^(s,L®ΘCs((k-h)σ(s))), and, for h — g — 1, a translation isomorphism τ2g-2.

ic

π ^ ic π, (s,L)ι-(s,Lωc,((2-2g)σ(s))).

(13)

Moduli Spaces of Curves and Representation Theory

Consider the diagram Pic"- '(π

Consider the relative theta-divisor θ = θπ on Pic^~ ι (π) and set θnτ = τ*nθ,

neZ.

We have the following (1.7) Lemma. ξ*θnx^λ~1®ω that (2n — l)2 ί

"

^ ' , where λn = λn(S) and ω = σ*c% /s . \ Note

| degω is equal to the number of Weierstrass points for ωn.

To give the straightforward proof of this isomorphism we recall the basic properties of Poincare bundles. Given the family of pointed curves (1.6), construct first the fiber product

where nι,σί9pί9ξί are the obvious maps and τ1((q,L)) = (q,τ(L)). A Poincare line bundle ifπ σ , or simply i f is a line bundle on ^ x ^ P i c ^ " 1 ^ ) such that (i)

X\c.«{L) = L>

(ii)

σf cSf = p*σ*(ωvίs) = σ*Pι(ω^,s).

It follows from the definition that detπi(^) = ^

1

(1.8)

and that ξίτΓ(i?) = ω& / s ((2n-l)(l -&)*),

(1-9)

where Δ is the image in H+ , Similarly, given a function g(z~ι,w~γ) holomorphic on (HS is an isomorphism}. It is clear that we have a canonical identification

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E. Arbarello, C. DeConcini, V. Kac, and C. Procesi

obtained by taking graphs, and by identifying H_ with Hs and H+ with H$ in the obvious way. One easily checks that this gives Gr(H) the structure of an infinite dimensional complex manifold. We remark that the group A^ acts transitively on Gr(if) making it a homogeneous space. We now turn to the construction of the determinant line bundle j£? on Gr(H). For WeGτ{H) let Kw = KQrp_\w and Cw = Cokeτp_\w. Set max

&w= /\KW®/\(CW)*.

(1.19)

s

(1.20) Lemma. Let We Gr (H); then we have a canonical isomorphism between J£w and _nφH where —(

> i
0 determines αM m ,rc+ mφO via (2.11), α x _j and a2,-2 determine α π ? _ π via (2.12)]. For example: m—n 2

n(n — 4)

ft3

—n

By a straightforward computation one can then see that a general 2-cocycle for is of the form

2

+

s{z)(f1df2-f2df1) l,

(2.13)

where aι,a2,a3eH\9).

(2.15)

f

(2.16) Lemma. For any s and s , ί? = ί*. Proof. It clearly suffices to show that the automorphism

defined by σ s (/δ + g)=/@ be defined as above, then ί?(«i) = αi (2.18)

Moduli Spaces of Curves and Representation Theory

17

Proof. Using the above lemma we can assume s = 0; let ί 0 = ί, then it is clear that ί*(α1) = α 1 . Let us compute: ί*(«2)(/,θ+gl,/2a+g2)= = α 2 + Res {Jxάft -hάf'O = cc2 + 2 α t ,

z= 0

z =0

= a3 — Res

f2"- Rιes(

= oc3 - o c 2 -

Q .E. D.

=0

z =0

2 =

g^ -f2dg'2)

It will be convenient for what follows to introduce a new basis for H2(β) and write t* with respect to this basis. Note that Q) acts naturally on H by

So we get a representation Φo : ^ - + a 0 0 .

(2.19)

In the preceding section we defined a canonical 2-cocycle ψ for a^. Let ψo = φ$( — ψ). It is an easy computation to see that

We now take as a new basis for H2(β) the set {7,φ 0 , f^o}? where y = — ^ocι. It is then immediate to verify the following (2.20) Corollary. With respect to the basis {7,1^05^*^0} the homomorphism t* \U2{β)-^H2{β) is represented by the matrix:

t*=

/I

0

0

0

\0

1

12\ -1

.

(2.21)

2/

In the sequel we shall also use the following basis of H2(&): 2

(2.22)

Finally, notice that the Lie algebra d acts, by Lie bracket, on the ring of pseudodifferential operators

Psd={ and that it preserves the canonical filtration of Psd given by Psdπ= 1= ~

00

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E. Arbarello, C. DeConcini, V. Kac, and C. Procesi

We then get an action of d on the span of differentials of order n: n

n

n

Ω = Hdz = Hδ~ = Psd n /Psd π + , , and hence a representation ρπ:d->aoo. n

It is not hard to verify that ρn = φ0°t straightforward computation gives

(2.23)

°ί, where \Ά-*Q) is the inclusion. A ρ*(ψ) = ρt-n(ψ)-

(2-24)

3. The Basic Varieties and the Krichever Construction We are now going to construct two infinite dimensional varieties Jίg and &g-γ. The first one will parametrize triples (C, p, z), where C is a smooth curve of genus g, p a point on C, and z a local coordinate defined near p and vanishing at p. The second one will parametrize quintuples (C, p, z, L, []), where (C, p, z) are as above, L is a degree g — 1 line bundle on C, and [] is an equivalence class of local trivializations of L near p, differing from each other by a non-zero multiplicative constant. Let us start with a definition. Consider a family

of pointed curves of genus g parametrized by T. We say that $ is a family of pointed curves with local parameters if there exists a neighbourhood % of the section σ(T) and a holomorphic function Z on ^ , vanishing on σ(T), such that for every t in T the function zt = Z\UnSt is a local coordinate around the point σ(ή on the Riemann surface Sv The notion of isomorphism between families of pointed curves with local parameter is the obvious one. Notice also that given a smooth curve C a point p on C and a local parameter z around p, the triple (C, p, z) admits only the trivial automorphism. Therefore, we may define a deformation of the triple (C, p, z), simply as a family of pointed curves with local parameters {β, π, T, σ, Z) together with a point t0 G T and an identification of the "central data" (® to ,σ(ί 0 ),z fo ) = (C,p,z),

(3.1)

an isomorphism between deformations being simply an isomorphism between families (of pointed curves with local parameters) which is compatible with the identification of the central data. Let us now consider a family of pointed curves

Given a point s e S choose a small neighbourhood V of s over which ^ trivializes in the C 0 0 sense. Look at a tubular neighbourhood °U of the section σ(V)C^\v, and think of it as a family discs parametrized by V. As such it must be holomorphically

Moduli Spaces of Curves and Representation Theory

19

trivial. Let {

}

be a trivialization of °U such that Z(σ(v)) = (0, v), veV. For every υeV the pointed curve (