For any non-semisimple modular category, special symmetric Frobenius algebras now give all consistent open-closed correlators, with a holographic description and a Batalin-Vilkovisky structure on local operators.
The A-infinity operad and the moduli space of curves
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abstract
The modular envelope of a cyclic operad is the smallest modular operad containing it. A modular operad is constructed from moduli spaces of Riemann surfaces with boundary; this modular operad is shown to be the modular envelope of the A-infinity cyclic operad. This gives a new proof of the result of Harer-Mumford-Thurston-Penner-Kontsevich that a cell complex built from ribbon graphs is homotopy equivalent to the moduli space of curves.
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The Construction of Correlators in Finite Rigid Logarithmic Conformal Field Theory
For any non-semisimple modular category, special symmetric Frobenius algebras now give all consistent open-closed correlators, with a holographic description and a Batalin-Vilkovisky structure on local operators.