Open-closed homotopy algebras

Topological types of string worldsheets can be organised into a modular operad QP. For closed strings, algebras over this operad are essentially topological quantum field theories formalized by monoidal functors from cobordisms into vector spaces or by commutative Frobenius algebras. There is an analogous picture for open and open/closed strings. In the closed case, QP can be freely generated from certain cyclic suboperad P via modular envelope: in fact P=Com. In the open case, analogous result holds for P=Ass. In open-closed case, this problem is subtler. For a particular choice of P, we propose two solutions: either we insists on cyclicity of P, and then we show to what extent QP fails to be the modular envelope of P; or we replace the category of cyclic operads by a bigger category of pre-modular operads, where P also lives, and obtain a positive result.

To justify giving this talk on "Seminar on Higher Structures", we also discuss the significance of algebras over the Feynman transform of QP - these are equivalently solutions to Zwiebach's Batalin-Vilkoviski master equation, parallel to the Maurer-Cartan equation for ordinary operads.