6

INTRODUCTION

the space of

anon-abelian

theta-functions" on the moduli space of G-bundles on

X. It is this type of result that we have in mind when we talk about the interplay

between vertex algebras and algebraic curves.

This brings us to the connection between vertex algebras and moduli spaces.

Sheaves of coinvariants

An interesting question is to understand how the spaces of coinvariants and

conformal blocks change as we vary the positions of the points and the complex

structure on our curve. It turns out that the spaces of coinvariants can be organized

into a sheaf on the moduli space 9Jlg,n of smooth n-pointed projective curves of

genus g. Moreover, this sheaf carries the structure of a (twisted) D-module on

yjlgHn- In other words, we can differentiate sections of this sheaf.

The key property of the moduli space $Jlg,n used in defining the D-module

structure is its "Virasoro uniformization". Consider for simplicity the case n = 1.

It turns out that the Lie algebra DerC((t)) acts transitively on the larger moduli

space $Jlgti of triples (X, x, z), where X is a smooth projective curve of genus #, x is

a point of X, and z is a formal coordinate at x. So 9JTp,i looks like a homogeneous

space for the Lie algebra DerX. More precisely, we can say that Wlg,i carries

a transitive action of the Harish-Chandra pair (DerDC, Aut 0). By applying the

formalism of Harish-Chandra localization to a conformal vertex algebra, we define

a (twisted) D-module structure on the sheaf of coinvariants on dJlg i (see Chapter

17).

This construction may be generalized to other moduli spaces, such as the moduli

space

9JIG(X)

of G-bundles on a smooth projective curve X, or the Picard variety

(the moduli space of line bundles on X). In particular, applying this construction to

affine Kac-Moody vertex algebra of critical level, we obtain D-modules on 9JIG(X)

parameterized by LG-opers on X, where LG is the Langlands dual group to G.

These D-modules are the Hecke eigensheaves whose existence is predicted by the

geometric Langlands conjectures (see § 18.4).

The above localization construction may be further generalized. Suppose we

are given a vertex algebra V and a Harish-Chandra pair (0, S+) of what we call

internal symmetries of V. This means that the action of the Lie algebra 0 is

induced by the Fourier coefficients of vertex operators on V. Suppose also that

9+ (resp., 0) consists of symmetries of certain geometric data £ on the standard

(resp., punctured) disc. Then to any such datum T on a smooth projective curve X

we can attach a twisted space of coinvariants H7(X, x, V). Moreover, these spaces

with varying 7 may be combined into a (twisted) D-module A(V) on the moduli

space 971$ of the data £ on X.

For instance, we can take as $ the complex structure on X, or a G-bundle on X,

or a line bundle on X. Then by this construction, any vertex algebra equipped with

the action of the corresponding Harish-Chandra pair by internal symmetries gives

rise to a D-module on the corresponding moduli space. The above general picture

suggests that vertex algebras play the role of local algebraic objects governing

deformations of curves and various geometric data on them. It raises the possibility

that more exotic vertex algebras, such as the W-algebras introduced in Chapter

15, may also correspond to some still unknown moduli spaces.

In special cases the space of conformal blocks may be identified with the ring

of functions on the formal neighborhood of a point in the corresponding moduli