BRST-Invariant Deformations of Geometric Structures in Topological Field Theories

TL;DR

This paper explores BRST-invariant deformations of geometric structures in topological field theories, linking elliptic genera to spectral functions and describing deformations via Hochschild cohomology of a DG-algebra.

Contribution

It introduces a novel framework connecting BRST cohomology, spectral functions, and Hochschild cohomology to classify deformations in topological field theories.

Findings

Elliptic genera expressed through spectral functions of hyperbolic geometry.

Deformations classified by Hochschild cohomology of a specific DG-algebra.

Identification of harmonic structures with Ext groups via HKR isomorphism.

Abstract

We study a Lie algebra of formal vector fields $W_n$ with its application to the perturbative deformed holomorphic symplectic structure in the A-model, and a Calabi-Yau manifold with boundaries in the B-model. A relevant concept in the vertex operator algebra and the BRST cohomology is that of the elliptic genera (the one-loop string partition function). We show that the elliptic genera can be written in terms of spectral functions of the hyperbolic three-geometry (which inherits the cohomology structure of BRST-like operator). We show that equivalence classes of deformations are described by a Hochschild cohomology theory of the DG-algebra ${\mathfrak A} = (A, Q)$, $Q =\bar{\partial}+\partial_{\rm deform}$, which is defined to be the cohomology of $(-1)^n Q +d_{\rm Hoch}$. Here $\bar{\partial}$ is the initial non-deformed BRST operator while $\partial_{\rm deform}$ is the deformed part whose algebra is a Lie algebra of linear vector fields ${\rm gl}_n$. We discuss the identification of the harmonic structure $(HT^\bullet(X); H\Omega_\bullet(X))$ of affine space $X$ and the group ${\rm Ext}_{X}^n({\cal O}_{\triangle}, {\cal O}_{\triangle})$ (the HKR isomorphism), and bulk-boundary deformation pairing.