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

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.