Connected sum for modular operads and Beilinson–Drinfeld algebras

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.