Open-closed modular operads, Cardy condition and string field theory

TL;DR

This paper establishes that the modular operad of Riemann surfaces with open and closed boundaries, relevant to string field theory, is derived from its genus 0 part through modular completion and the Cardy condition, with a new presentation and algebraic characterization.

Contribution

It demonstrates the modular completion process for open-closed operads, provides a finitary presentation, and characterizes their algebras via Frobenius algebra morphisms, connecting to prior results.

Findings

Modular operad of open-closed Riemann surfaces is the completion of genus 0 part.

Provides a finitary presentation of the modular two-colored operad.

Characterizes algebras via Frobenius algebra morphisms.

Abstract

We prove that the modular operad of diffeomorphism classes of Riemann surfaces with both `open' and `closed' boundary components, in the sense of string field theory, is the modular completion of its genus 0 part quotiented by the Cardy condition. We also provide a finitary presentation of a version of this modular two-colored operad and characterize its algebras via morphisms of Frobenius algebras, recovering some previously known results of Kaufmann, Penner and others. As an important auxiliary tool we characterize inclusions of cyclic operads that induce inclusions of their modular completions.