Quasiclassical Lian-Zuckerman Homotopy Algebras, Courant Algebroids and Gauge Theory

TL;DR

This paper introduces a quasiclassical limit of the Lian-Zuckerman homotopy BV algebra on light modes of VOA cohomology, linking Courant algebroids and Yang-Mills equations through a new algebraic framework.

Contribution

It constructs a functor from VOAs with a formal parameter to A-infinity algebras, revealing a novel connection between Courant algebroids and homotopy algebras in gauge theory.

Findings

Derived a deformation of the BRST differential parametrized by a tensor.

Established a functor from VOAs to A-infinity algebras.

Linked Courant algebroids with homotopy algebra structures in Yang-Mills theory.

Abstract

We define a quasiclassical limit of the Lian-Zuckerman homotopy BV algebra (quasiclassical LZ algebra) on the subcomplex, corresponding to "light modes", i.e. the elements of zero conformal weight, of the semi-infinite (BRST) cohomology complex of the Virasoro algebra associated with vertex operator algebra (VOA) with a formal parameter. We also construct a certain deformation of the BRST differential parametrized by a constant two-component tensor, such that it leads to the deformation of the $A_{\infty}$-subalgebra of the quasiclassical LZ algebra. Altogether this gives a functor the category of VOA with a formal parameter to the category of $A_{\infty}$-algebras. The associated generalized Maurer-Cartan equation gives the analogue of the Yang-Mills equation for a wide class of VOAs. Applying this construction to an example of VOA generated by $\beta$-$\gamma$ systems, we find a remarkable relation between the Courant algebroid and the homotopy algebra of the Yang-Mills theory.