On the Globalization of the Poisson Sigma Model in the BV-BFV Formalism

TL;DR

This paper develops a global quantization framework for the Poisson Sigma Model within the BV-BFV formalism, addressing boundary conditions, anomalies, and the structure of the boundary operator to ensure a consistent quantum theory.

Contribution

It introduces a method to construct a global quantization of the Poisson Sigma Model in the BV-BFV formalism, including handling boundary conditions and anomalies.

Findings

Boundary conditions can cause quantum anomalies.

Adding boundary and corner terms restores the quantum master equation.

The quantum GBFV operator is a differential, ensuring well-defined cohomology.

Abstract

We construct a formal global quantization of the Poisson Sigma Model in the BV-BFV formalism using the perturbative quantization of AKSZ theories on manifolds with boundary and analyze the properties of the boundary BFV operator. Moreover, we consider mixed boundary conditions and show that they lead to quantum anomalies, i.e. to a failure of the (modified differential) Quantum Master Equation. We show that it can be restored by adding boundary terms to the action, at the price of introducing corner terms in the boundary operator. We also show that the quantum GBFV operator on the total space of states is a differential, i.e. squares to zero, which is necessary for a well-defined BV cohomology.