Modular Functors with Singularities from Vertex Operator Algebras Beyond Rigidity and Finiteness

Abstract

For a vertex operator algebra $V$ and a suitable category of its modules, we propose a construction for spaces of conformal blocks organized into an open-closed modular functor with singularities. This is inspired by the idea of implementing directly from the start the principle of holomorphic factorization. More precisely, using the strategy of modular extension introduced by Costello and developed further in our previous work, we build for each surface $Σ$ with at least one boundary component per path component and specified boundary labels attached to marked intervals or boundary circles a representation $Ω_V(Σ;-)$ of the mapping class group of $Σ$. The construction can be described explicitly on generating Dehn twists. This approach is a priori independent from other constructions based on algebraic geometry or topological techniques involving e.g. surgery, but we include an overview over the available comparisons. In the special case in which the module category of $V$ is a not necessarily semisimple modular category $\mathcal{A}$, the spaces $Ω_V(Σ)$ are equivalent to the string-net spaces for $\mathcal{A}$ and hence to the modular functor for the Drinfeld center $Z(\mathcal{A})\simeq \bar{\mathcal{A}}\boxtimes\mathcal{A}$. However, the construction of $Ω_V$ in this paper has the advantage of being available beyond rationality, rigidity, self-contragredience and finiteness. Moreover, we prove that $Ω_V$ satisfies excision, is finite-dimensional in the $C_2$-cofinite case and produces representations of surface braid groups generalizing the ones of Brochier-Jordan. We prove for the triplet $\mathcal{W}_{2,3}$ with non-exact fusion product that the boundary conditions introduced by Gaberdiel-Runkel-Wood produce correlation functions, provided that one uses the notion of a modular functor with singularities that we develop.