Integration over families of Lagrangian submanifolds in BV formalism

TL;DR

This paper presents a geometric approach to gauge fixing in BV formalism by integrating over families of Lagrangian submanifolds, with applications to string theory amplitudes and analysis of gauge symmetry conditions.

Contribution

It introduces a natural integration procedure over Lagrangian submanifolds in BV formalism, linking gauge fixing to geometric structures and analyzing gauge symmetry conditions for consistency.

Findings

Integration over Lagrangian submanifolds formalizes gauge fixing.

Higher genus string amplitudes interpreted as moduli space integrals.

Gauge symmetry conditions ensure the consistency of the integration procedure.

Abstract

Gauge fixing is interpreted in BV formalism as a choice of Lagrangian submanifold in an odd symplectic manifold. A natural construction defines an integration procedure on families of Lagrangian submanifolds. In string perturbation theory, the moduli space integrals of higher genus amplitudes can be interpreted in this way. We discuss the role of gauge symmetries in this construction. We derive the conditions which should be imposed on gauge symmetries for the consistency of our integration procedure. We explain how these conditions behave under the deformations of the worldsheet theory. In particular, we show that integrated vertex operator is actually an inhomogeneous differential form on the space of Lagrangian submanifolds.