nLab Chiral Algebras

This entry is about the book

• Alexander Beilinson and Vladimir Drinfeld

  Chiral Algebras, Amer. Math. Society 2004

  (webpage)

  • Contents, Introduction, Chapter 1, Chapter 2, Chapter 3, Chapter 4, References, Index

on chiral algebras/factorization algebras in the language of D-schemes.

Introduction

1. Axiomatic patterns

2. The Geometry of $\mathcal{D}$-Schemes

2.1 $\mathcal{D}$-modules: recollection and basics

• D-module

2.2 The compound tensor structure

2.3 $\mathcal{D}_X$-Schemes

• D-scheme

2.3.1 $\mathcal{D}_X$-algebras

2.3.2 Jet schemes

• jet bundle

2.3.20 The calculus of variations

• variational calculus

  • Lagrangian

  • action functional

2.4 The space of horizontal sections

2.5 $Lie^\ast$-algebras and algebroids

2.6 Coisson algebras

• Coisson algebras

2.7 The Tate extension

2.8 Tate structures and characteristic classes

2.8.11

• vertical differential form

• variational bicomplex

2.9 The Harish-Chandra setting and the setting of $c$-stacks

3. Local theory: chiral basics

4. Global theory: chiral homology

Related references

Much complementary content by the same authors is also in the developing book on the geometric Langlands program

• Alexander Beilinson, Vladimir Drinfeld, Quantization of the Hitchin system and Hecke eigensheaves, draft, pdf

A goood accessible survey of or introduction into some of the developments in this book can be found around section 8.3

• Frédéric Paugam, Towards the mathematics of quantum field theory (pdf)

and section 3 of

• Frédéric Paugam, Histories and observables in covariant field theory Journal of Geometry and Physics (2010) (arXiv)

with more details in

• Frédéric Paugam, Homotopical Poisson Reduction of gauge theories (pdf)

In the context of factorization algebra of observables:

• Kevin Costello, Owen Gwilliam, Factorization algebras in perturbative quantum field theory

category: reference