Book contents
- Frontmatter
- Contents
- List of participants
- Introduction
- Acknowledgements
- On the deformation theory of moduli spaces of vector bundles
- Stable augmented bundles over Riemann surfaces
- On surfaces in ℙ4 and 3-folds in ℙ5
- Exceptional bundles and moduli spaces of stable sheaves on ℙn
- Floer homology and algebraic geometry
- The Horrocks–Mumford bundle
- Faisceaux semi-stables et systemes coherents
- The combinatorics of the Verlinde formulas
- Canonical and almost canonical spin polynomials of an algebraic surface
- On conformal field theory
On conformal field theory
Published online by Cambridge University Press: 12 January 2010
- Frontmatter
- Contents
- List of participants
- Introduction
- Acknowledgements
- On the deformation theory of moduli spaces of vector bundles
- Stable augmented bundles over Riemann surfaces
- On surfaces in ℙ4 and 3-folds in ℙ5
- Exceptional bundles and moduli spaces of stable sheaves on ℙn
- Floer homology and algebraic geometry
- The Horrocks–Mumford bundle
- Faisceaux semi-stables et systemes coherents
- The combinatorics of the Verlinde formulas
- Canonical and almost canonical spin polynomials of an algebraic surface
- On conformal field theory
Summary
Introduction
Belavin-Polyakov-Zamodolochikov ([BPZ]) initiated conformal field theory as a certain limit of the theory of the two-dimensional lattice model. This theory has a deep relationship with string theory and a rich mathematical structure. It is a two-dimensional quantum field theory invariant under conformal transformations; in fact, as we shall see below, it is invariant under a much bigger group of transformations, and this gives a relationship with the moduli space of algebraic curves ([FS], [EO]).
A typical example of conformal field theory is abelian conformal field theory, the theory of free fermions over a compact Riemann surface. For a mathematically rigorous treatment of abelian conformal field theory we refer the reader to [KNTY]. This theory has a deep relationship with various fields of mathematics, such as the moduli theory of algebraic curves, KP hierarchy, theta functions, complex cobordism rings and formal groups ([KNTY], [KSU2], [KSU3]).
For non-abelian conformal field theory the first mathematically rigorous treatment was given by Tsuchiya-Kanie ([TK]), who constructed the theory over ℙ1. Later Tsuchiya-Ueno-Yamada ([TUY]) generalized this to algebraic curves of arbitrary genus.
Let us explain briefly the main ideas of conformal field theory. It can be decomposed into two parts, holomorphic and anti-holomorphic, and in the following we shall only consider the holomorphic theory. This is often called chiral conformal field theory by the physicists.
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- Vector Bundles in Algebraic Geometry , pp. 283 - 345Publisher: Cambridge University PressPrint publication year: 1995
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