Higher Kac-Moody algebras and symmetries of holomorphic field theories

TL;DR

This paper generalizes affine Kac-Moody algebras to higher dimensions using factorization algebras, classifies their cocycles, and explores their role as symmetries in holomorphic quantum field theories, including free field realizations and large N limits.

Contribution

It introduces higher Kac-Moody algebras via factorization algebras, classifies their cocycles, and demonstrates their significance as symmetries in holomorphic quantum field theories.

Findings

Classified local cocycles of higher current algebras.

Generalized free field realization of higher Kac-Moody algebras.

Analyzed large N behavior and connections to non-commutative field theories.

Abstract

We introduce a higher dimensional generalization of the affine Kac-Moody algebra using the language of factorization algebras. In particular, on any complex manifold there is a factorization algebra of "currents" associated to any Lie algebra. We classify local cocycles of these current algebras, and compare them to central extensions of higher affine algebras recently proposed by Faonte-Hennion-Kapranov. A central goal of this paper is to witness higher Kac-Moody algebras as symmetries of a class of holomorphic quantum field theories. In particular, we prove a generalization of the free field realization of an affine Kac-Moody algebra and also develop the theory of q-characters for this class of algebras in terms of factorization homology. Finally, we exhibit the "large N" behavior of higher Kac-Moody algebras and their relationship to symmetries of non-commutative field theories.