Modular envelopes, OSFT and nonsymmetric (non-\(\Sigma\)) modular operads (Q740108)

The principal objective of this paper is to introduce non-\(\Sigma\) (non-symmetric) modular operads. Assuming certain familiarity with basic notions of operad theory [\textit{M. Markl} et al., Operads in algebra, topology and physics. Providence, RI: American Mathematical Society (AMS) (2002; Zbl 1017.18001)], this paper shows that the modular envelope of the terminal operad in the category of cyclic (resp. non-\(\Sigma\) cyclic) \(\mathrm{Set}\)-operads is the terminal modular (resp. the terminal non-\(\Sigma\) modular) \(\mathrm{Set}\)-operad. The author shows also that the present setup conveys the ideas in in [\textit{J. Chuang} and \textit{A. Lazarev}, Commun. Number Theory Phys. 1, No. 4, 605--649 (2008; Zbl 1166.18003), Theorem 3.7], [\textit{R. M. Kaufmann} et al., Geom. Topol. 7, 511--568 (2003; Zbl 1034.18009)] and [\textit{R. M. Kaufmann} and \textit{R. C. Penner}, Nucl. Phys., B 748, No. 3, 335--379 (2006; Zbl 1214.32004)].