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

Abstract

Abstract. Our aim is to introduce and advocate non-Σ (non-symmetric) modular operads. While ordinary modular operads were inspired by the structure of the moduli space of stable complex curves, non-Σ 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-Σ modular operads. This gives a succinct description of this object that plays an important rôle in open string field theory. We also sketch further perspectives of the presented approach.

Decorated Feynman categories

“…The consequences were also further analyzed in [KLP03, KP06,KWZ15]. The example of CycAss is in other guise discussed in detail in [Mar15]. We assert that the same is now possible for the infinity versions, without much ado.…”

“…In this case, we have i * (CycAss) =: ModAss as a possible decoration. Indeed using this, we recover the definition of [Mar15] of non-sigma modular operads, which is the special case of a brane labelled c/o system, with trivial closed part and one color of [KP06][Appendix A.6], see also [KLP03] the appendix of [Kau09] and [Mar15] for details about the correspondence between stable or almost ribbon graphs and surfaces. M pl := M dec M odAss .…”

“…Here we can understand these constructions in a more general framework. First, the diagram considered in [Mar15] is exactly diagram (3.4). Then the fact that the non-Sigma modular envelope of CycAss is terminal is obvious from Theorem 4.1 and Proposition 4.6.…”

Comprehensive factorisation systems

“…in combinatorics, in computer science or in mathematical physics. One of our motivations in writing this text on comprehensive factorisations was a recent article of Markl [27] in which he defines a new class of modular-like operads, based on the combinatorics of "polycylic orderings", with the advantage over modular operads of having less built-in symmetries. With the comprehensive factorisation in hand, we shall see that Markl's definition is a very natural one and to some extent the only possible.…”

“…A Feynman category is a special kind of symmetric monoidal category, and there is a comprehension scheme assigning to a Feynman category F the category of strong symmetric monoidal set-valued functors on F. The resulting comprehensive factorisation of a Feynman functor sheds light on Markl's recent definition of non-Σ-modular operads [27]. Through the 2-equivalence between Feynman categories and small multicategories (also called coloured operads) we obtain a comprehensive factorisation of a multifunctor which directly extends Street and Walters' comprehensive factorisation of a functor.…”