Conformal Blocks, Generalized Theta Functions and the Verlinde Formula

This book gives a complete proof of the Verlinde formula and of its connection to generalized theta functions.

Shrawan Kumar (Author)

9781316518168, Cambridge University Press

Hardback, published 25 November 2021

540 pages
23.6 x 15.8 x 3.6 cm, 0.9 kg

'The author succeeds in giving a masterful, clear, motivated and practically up to date presentation of a subject which is connected to the forefront of the mathematical research. It is addressed both to those researchers who just enter the subject and to those already active in theoretical physics, algebraic geometry, representation theory. This admirable book (which is the third book of Shrawan Kumar) is the first one about a very advanced and technical subject and it will be surely a standard reference.' Nicolae Manolache, zbMATH

In 1988, E. Verlinde gave a remarkable conjectural formula for the dimension of conformal blocks over a smooth curve in terms of representations of affine Lie algebras. Verlinde's formula arose from physical considerations, but it attracted further attention from mathematicians when it was realized that the space of conformal blocks admits an interpretation as the space of generalized theta functions. A proof followed through the work of many mathematicians in the 1990s. This book gives an authoritative treatment of all aspects of this theory. It presents a complete proof of the Verlinde formula and full details of the connection with generalized theta functions, including the construction of the relevant moduli spaces and stacks of G-bundles. Featuring numerous exercises of varying difficulty, guides to the wider literature and short appendices on essential concepts, it will be of interest to senior graduate students and researchers in geometry, representation theory and theoretical physics.

Introduction
1. An introduction to affine Lie algebras and the associated groups
2. Space of vacua and its propagation
3. Factorization theorem for space of vacua
4. Fusion ring and explicit Verlinde formula
5. Moduli stack of quasi-parabolic G-bundles and its uniformization
6. Parabolic G-bundles and equivariant G-bundles
7. Moduli space of semistable G-bundles over a smooth curve
8. Identification of the space of conformal blocks with the space of generalized theta functions
9. Picard group of moduli space of G-bundles
A. Dynkin index
B. C-space and C-group functors
C. Algebraic stacks
D. Rank-level duality (A brief survey) Swarnava Mukhopadhyay
Glossary
Bibliography
Index.

Subject Areas: Mathematical physics [PHU], Topology [PBP], Algebraic geometry [PBMW]