Families of gauge conditions in BV formalism

TL;DR

This paper develops a geometric framework in BV formalism linking gauge conditions, Lagrangian submanifolds, and closed forms, enabling the extraction of physical and topological invariants through integration.

Contribution

It introduces a method to construct closed forms on spaces of Lagrangian submanifolds and their quotients, connecting gauge conditions with physical and topological invariants.

Findings

Constructed closed forms on Lagrangian submanifold spaces.

Derived string amplitudes from geometric data.

Produced topological invariants in quantum field theories.

Abstract

In BV formalism we can consider a Lagrangian submanifold as a gauge condition. Starting with the BV action functional we construct a closed form on the space of Lagrangian submanifolds. If the action functional is invariant with respect to some group $H$ and $\Lambda$ is an $H$-invariant family of Lagrangian submanifold then under certain conditions we construct a form on $\Lambda$ that descends to a closed form on $\Lambda/H.$ Integrating the latter form over a cycle in $\Lambda/H$ we obtain numbers that can have interesting physical meaning. We show that one can get string amplitudes this way. Applying this construction to topological quantum field theories one obtains topological invariants.