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8 - Identification of the Space of Conformal Blocks with the Space of Generalized Theta Functions

Published online by Cambridge University Press:  19 November 2021

Shrawan Kumar
Shrawan Kumar
Affiliation:
University of North Carolina, Chapel Hill
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Summary

We prove that the ind-scheme ? consisting of regular maps from an affine curve to a simple, simply-connected group G as closed points is an irreducible and reduced ind-projective variety. By taking the Laurent series expansion at any point at infinity, we can view ? as a subgroup of the loop group. We prove that the central extension of the loop group splits over ? if the affine curve has a single point at infinity. We prove that the space of vacua for any s-pointed smooth curve ? is identified with the space of global sections of the moduli stack of quasi-parabolic G-bundles over ? with respect to a certain line bundle. We also determine the Picard group of this moduli stack. We introduce the notion of theta bundle for a family of G-bundles over ? and determine it for the tautological family of G-bundles over ? parameterized by the infinite Grassmannian in terms of the Dynkin index. The main result of this chapter asserts that there is a canonical identification between the space of global sections of a line bundle over the coarse moduli space of parabolic G-bundles over any s-pointed smooth projective curve ? and the space of vacua.

Keywords

determinant bundletheta bundleDynkin indexspace of vacua and its identification with generalized theta functionsmoduli stack of quasi-parabolic G-bundlesmoduli space of parabolic G-bundlesGrothendieck--Riemann--Roch theorem
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Conformal Blocks, Generalized Theta Functions and the Verlinde Formula , pp. 328 - 378
Publisher: Cambridge University Press
Print publication year: 2021

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