Connected sum for modular operads and Beilinson-Drinfeld algebras

TL;DR

This paper introduces a connected sum operation for modular operads, leading to a Beilinson-Drinfeld algebra structure on associated function spaces, with applications to quantum master equations.

Contribution

It defines a new connected sum for modular operads and constructs a Beilinson-Drinfeld algebra combining this with existing BV structures.

Findings

Connected sum induces a commutative product on functions

Formation of a Beilinson-Drinfeld algebra from modular operads

Application to quantum master equation analysis

Abstract

Modular operads relevant to string theory can be equipped with an additional structure, coming from the connected sum of surfaces. Motivated by this example, we introduce a notion of connected sum for general modular operads. We show that a connected sum induces a commutative product on the space of functions associated to the modular operad. Moreover, we combine this product with Barannikov's non-commutative Batalin-Vilkovisky structure present on this space of functions, obtaining a Beilinson-Drinfeld algebra. Finally, we study the quantum master equation using the exponential defined using this commutative product.