BV quantization of the Rozansky-Witten model

TL;DR

This paper explores the BV quantization of the Rozansky-Witten 3d sigma-model using configuration space regularization, revealing the structure of quantum observables and confirming the partition function's relation to Rozansky-Witten invariants.

Contribution

It applies Costello's BV formalism and configuration space methods to the Rozansky-Witten model, establishing the structure of quantum observables and linking the partition function to known invariants.

Findings

Cohomology of local quantum observables matches H^*(X, (∧^* T_X)^{⊗ g})

Partition function reproduces Rozansky-Witten invariants

BV quantization can be achieved via configuration space regularization

Abstract

We investigate the perturbative aspects of Rozansky-Witten's 3d $\sigma$-model using Costello's approach to the Batalin-Vilkovisky (BV) formalism. We show that the BV quantization (in Costello's sense) of the model, which produces a perturbative quantum field theory, can be obtained via the configuration space method of regularization due to Kontsevich and Axelrod-Singer. We also study the factorization algebra structure for quantum observables following Costello-Gwilliam. In particular, we show that the cohomology of local quantum observables on a genus $g$ handle body is given by $H^*(X,(\wedge^*T_X)^{\otimes g})$ (where $X$ is the target manifold), and prove that the partition function reproduces the Rozansky-Witten invariants.