Holomorphic Chern-Simons-Witten Theory: from 2D to 4D Conformal Field Theories

TL;DR

This paper explores holomorphic analogues of Chern-Simons theories on complex 3D manifolds and their connection to integrable 4D conformal field theories, extending known 2D and 3D relationships.

Contribution

It introduces holomorphic Chern-Simons analogues in six real dimensions and describes their relation to 4D conformal field theories, including algebraic structures and quantization.

Findings

Models are integrable and connected to 4D conformal field theories.

Analogues of Virasoro and affine Lie algebras are constructed.

Quantization and links to string theories are discussed.

Abstract

It is well known that rational 2D conformal field theories are connected with Chern-Simons theories defined on 3D real manifolds. We consider holomorphic analogues of Chern-Simons theories defined on 3D complex manifolds (six real dimensions) and describe 4D conformal field theories connected with them. All these models are integrable. We describe analogues of the Virasoro and affine Lie algebras, the local action of which on fields of holomorphic analogues of Chern-Simons theories becomes nonlocal after pushing down to the action on fields of integrable 4D conformal field theories. Quantization of integrable 4D conformal field theories and relations to string theories are briefly discussed.