the space of
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 $Jlgti of triples (X, x, z), where X is a smooth projective curve of genus #, x is
a point of X, and z is a formal coordinate at x. So 9JTp,i looks like a homogeneous
space for the Lie algebra DerX. More precisely, we can say that Wlg,i carries
a transitive action of the Harish-Chandra pair (DerDC, Aut 0). By applying the
formalism of Harish-Chandra localization to a conformal vertex algebra, we define
a (twisted) D-module structure on the sheaf of coinvariants on dJlg i (see Chapter
This construction may be generalized to other moduli spaces, such as the moduli
of G-bundles on a smooth projective curve X, or the Picard variety
(the moduli space of line bundles on X). In particular, applying this construction to
affine Kac-Moody vertex algebra of critical level, we obtain D-modules on 9JIG(X)
parameterized by LG-opers on X, where LG is the Langlands dual group to G.
These D-modules are the Hecke eigensheaves whose existence is predicted by the
geometric Langlands conjectures (see § 18.4).
The above localization construction may be further generalized. Suppose we
are given a vertex algebra V and a Harish-Chandra pair (0, S+) of what we call
internal symmetries of V. This means that the action of the Lie algebra 0 is
induced by the Fourier coefficients of vertex operators on V. Suppose also that
9+ (resp., 0) consists of symmetries of certain geometric data £ on the standard
(resp., punctured) disc. Then to any such datum T on a smooth projective curve X
we can attach a twisted space of coinvariants H7(X, x, V). Moreover, these spaces
with varying 7 may be combined into a (twisted) D-module A(V) on the moduli
space 971$ 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