Modular envelopes, OSFT and nonsymmetric (non-$\Sigma$) modular operads

TL;DR

This paper introduces non-$\Sigma$ modular operads, modeling surfaces with open string outputs, and demonstrates their application in describing the modular envelope of the associative operad relevant to open string field theory.

Contribution

It develops the theory of non-$\Sigma$ modular operads and proves that the modular envelope of the associative operad is the linearization of the terminal non-$\Sigma$ modular operad.

Findings

The modular envelope of the associative operad is the linearization of the terminal non-$\Sigma$ modular operad.

Non-$\Sigma$ modular operads model surfaces with open string outputs.

Provides a concise proof relevant to open string field theory.

Abstract

Our aim is to introduce and advocate non-$\Sigma$ (non-symmetric) modular operads. While ordinary modular operads were inspired by the structure of the moduli space of stable complex curves, non-$\Sigma$ modular operads model surfaces with open strings outputs. An immediate application of our theory is a short proof that the modular envelope of the associative operad is the linearization of the terminal operad in the category of non-$\Sigma$ modular operads. This gives a succinct description of this object that plays an important role in open string field theory. We also sketch further perspectives of the presented approach.