Denote by Vx the fiber of V at a given point x G X. The coordinate-free approach
of this book is based on the interpretation of the vertex operation Y in a conformal
vertex algebra V as an intrinsic section tyx of the dual bundle V* on the punctured
disc D£ with values in EndVx.
It is easier to describe the corresponding linear map
defined by the formula s t- Res^y^, 5). Namely, if we choose a coordinate z at a;,
then we can trivialize V\DX and identify Vx with V. Then
= Resz=0 Y(A,
for any A G V and n G Z.
Furthermore, we show that the vector bundle V carries a flat connection, with
respect to which the section }$x is horizontal.
Conformal blocks and coinvariants
Having reformulated the vertex operation in a coordinate-independent way, we
can make contact with the global structure of algebraic curves and give a global
geometric meaning to vertex operators. This naturally leads us to the concepts of
conformal blocks and coinvariants.
Let us illustrate this concept on the example of affine Kac-Moody algebras.
Denote by ^out(^) the Lie algebra of g-valued regular functions on X\x. Then
SoutOz) naturally embeds into g 0 Xx, and even into gx. Given a gx-module Mx,
we define its space of coinvariants as the quotient Mx/(gout(#) * 3Vt) and its dual,
the space of conformal blocks, as Homgout(a,)(Mx,C). Now recall that the basis
elements of g can be organized into vertex operators
given by formula (0.0.2),
is naturally a one-form on D£. Using the residue theorem, we can
reformulate the notion of conformal blocks as follows: (p G M* is a conformal block
if and only if for any A G Mx the one-forms ((p,Ja(z) • A)dz, which are a priori
defined only on ££, may be extended to regular one-forms on the entire X\x.
The vertex operators in a general vertex algebra transform in a more compli-
cated way than mere one-forms. Therefore it makes sense to consider all of them
at once, i.e., the entire vertex operation yx, which transforms as a section of V*.
We will say that a linear functional p on V^ is a conformal block if for any A G Vx,
the section ((f^x • A) G r(jD^,V*), which is a priori defined only on £*, can be
extended to a regular section of V* on X\x.
This definition may be generalized to a broader setting when we have several
distinct points on X, with a V-module attached to each of them (see Chapters 9
The spaces of conformal blocks play an important role in the theory of vertex
algebras. Even the vertex operation Y itself may be viewed as a special case of
conformal blocks: in the situation when we have three points on F 1 with two
insertions of V and one insertion of the contragredient module Vy. From this point
of view, various properties of vertex algebras may be interpreted geometrically in
terms of degenerations of the pointed projective line (see § 10.4).
In general, the spaces of conformal blocks may be thought of as invariants of
pointed algebraic curves, attached to a vertex algebra. For instance, in the case
of the affine algebra g, the space of conformal blocks corresponding to a curve X
with a single insertion of an integrable representation of g may be identified with