Quantization of the topological $\sigma$-model and the master equation of the BF formalism

TL;DR

This paper quantizes a topological sigma-model within the Batalin-Vilkovisky formalism, linking the quantum master equation to the elimination of exact states from BRST-invariant states on a fermionic Kähler supermanifold.

Contribution

It introduces a quantization approach for the topological sigma-model and relates the quantum master equation to the structure of the BV phase space.

Findings

Quantum master equation ensures elimination of exact states

BV phase space is a fermionic Kähler supermanifold

Provides a geometric interpretation of the quantization process

Abstract

We quantize the topological $\sigma$-model. The quantum master equation of the Batalin-Vilkovisky formalism $\Delta_\rho \Psi = 0$ appears as a condition which eliminates the exact states from the BRST invariant states $\Psi$ defined by $Q \Psi = 0$. The phase space of the BV formalism is a supermanifold with a specific symplectic structure, called the fermionic K\"ahler manifold.