4

INTRODUCTION

Coordinate-independence

In order to make the connection between vertex algebras and algebraic curves

we first have to make vertex operators coordinate-independent. This is in fact

one of the main goals of this book. To explain what this means, consider the

affine Kac-Moody algebra 9, which is a central extension of the Lie algebra &({£)).

Let X be a smooth projective curve over C, and x a point of X. Denote by 0X

the completed local ring at x, and by %x its field of fractions. Then we have

isomorphisms 0X ~ C[[£]] and %x ~ C((£)), but they are not canonical. To specify

such an isomorphism, we need to fix a formal coordinate t at x, for which we usually

do not have a preferred choice. In applications it is often important to deal with

the central extension of the Lie algebra g 0 Xx, rather than of g((t)). We should

think of this extension as the affine Lie algebra gx, "attached to the point x".

Then we need to know how elements of g 0 %x are realized if we choose a different

coordinate £' at x. This is of course obvious: if an element of g 0 Xx appears as

j 0 / ( t ) with respect to the coordinate £, then with respect to another coordinate t'

it will appear as J0/(/(£')) where p is the change of variables from t' to t such that

t = p{tf). Thus, p is an element of the group AutO of continuous automorphisms

of 0 = C[[£]], which acts simply transitively on the set of all formal coordinates

at x. We conclude that we need to know how elements of the affine Lie algebra

transform under the natural action of the group Aut 0.

In the context of vertex algebras, we look at the generating function

(0.0.2) Ja(z) = YJJ(n)z-n-\

nez

where J?n) = Ja 0 tn E g 0 Xx, for some basis {Ja} of g. These are the vertex

operators generating the corresponding vertex algebra. It follows from the above

discussion that the expression

Ja(t)dt

transforms as a one-form on the punctured

disc D* = SpecXs.

But what is the transformation formula for a general vertex operator Y(A, z)

in a general vertex algebra? Can one describe the entire vertex operation Y in

a coordinate-independent way, so that it makes sense in the neighborhood of any

point on a complex curve, not equipped with a preferred coordinate? In Chapter 6

we answer these questions. The key to the intrinsic description of the vertex algebra

structure is the incorporation of the action of the group of changes of coordinates

Aut 0 (and more generally, the action of the entire Virasoro algebra) into the vertex

algebra structure. In other words, we want the Lie algebra Der 0 of Aut 0 to act on

a vertex algebra V via Fourier coefficients of a vertex operator (i.e., natural basis

elements of Der 0 should act as the linear operators A(n) attached to some A eV,

in the notation of formula (0.0.1)). By exponentiating these Fourier coefficients we

obtain what wre call an action of Aut 0 on V by "internal symmetries".

This naturally leads us to the notion of conformal vertex algebra - one that

contains, among the vertex operators, the generating function of the basis elements

of the Virasoro algebra. To a conformal vertex algebra V we assign a vector bundle

V on an arbitrary smooth curve as follows. Consider the Aut 0-bundle Autx over

X whose fiber over x G X consists of all formal coordinates at x. Define the

associated vector bundle V on X as

V = Autx x V.

AutO