Modular operads and the quantum open-closed homotopy algebra

Abstract

We verify that certain algebras appearing in string field theory are algebras over Feynman transform of modular operads which we describe explicitly. Equivalent description in terms of solutions of generalized BV master equations are explained from the operadic point of view.

Properads and homological differential operators related to surfaces

“…In Theorem 1 of [1], Barannikov observed that an algebra over the Feynman transform of modular operad P is equivalently described as a solution of a certain master equation in an algebra succinctly defined in terms of P, cf. also Theorem 20 in [2]. In this section, we formulate the corresponding theorem for properads in our formalism and then adapt it to our applications.…”

“…we assume P(C, D, χ) to be nontrivial only if both C and D are non-empty and m + n > 2, for G = 0. In this case, we introduce, similarly to [2], positional derivations…”

“…describing closed, open or open-closed strings. The aim of this article is to continue in the work in [2] by Doubek, Jurčo, and Münster but this time for properads.…”

Odd Structures Are Odd

“…‡ They were originally defined as algebras for a certain monad of decorated graphs. For practical calculations it is however convenient to have a biased definition which appeared much later in [1]. For the convenience of the reader we repeat it below.…”

“…Consequently, the bar construction of a modular operad (called in this context the Feynman transform) is not an ordinary modular operad, but an odd modular operad. † It was shown in [4] and in the forthcoming work [1,6,7] that some algebraic structures relevant for string field theory are algebras over the Feynman transform of a modular operad. This explains the interest in explicit understanding odd modular operads and their algebras.…”