A Classification of Modular Functors via Factorization Homology

TL;DR

This work provides a complete classification of modular functors valued in a symmetric monoidal $(2,1)$-category via factorization homology. The central theorem identifies modular functors with connected self-dual balanced braided algebras in the ambient category, with genus-zero data determining the full theory; a genus-one reduction yields a practical criterion for connectedness. The authors develop a universal construction of modular functors from a connected cyclic framed $E_2$-algebra, prove a universal property that makes Surf_A classifying for extensions, and establish uniqueness of extensions up to contractible choices. They also supply readily verifiable sufficient conditions, notably cofactorizability, and demonstrate how Lyubashenko’s modular functor emerges in this framework, as well as modular functors beyond modular categories, including vertex operator algebras and Drinfeld centers. The work links skein-theoretic methods with factorization homology and situates modular functors within higher TFTs and Morita theory, with concrete consequences for spaces of conformal blocks and potential extensions to VOAs and vertex-operator-algebraic contexts.

Abstract

Modular functors are traditionally defined as systems of projective representations of mapping class groups of surfaces that are compatible with gluing. They can formally be described as modular algebras over central extensions of the modular surface operad, with the values of the algebra lying in a suitable symmetric monoidal $(2,1)$-category $\mathcal{S}$ of linear categories. In this paper, we prove that modular functors in $\mathcal{S}$ are equivalent to self-dual balanced braided algebras $\mathcal{A}$ in $\mathcal{S}$ (a categorification of the notion of a commutative Frobenius algebra) for which a condition formulated in terms of factorization homology with coefficients in $\mathcal{A}$ is satisfied; we call such $\mathcal{A}$ connected. The equivalence in one direction is afforded by genus zero restriction. Our construction of the inverse equivalence is entirely topological and can be thought of as a far reaching generalization of the construction of modular functors from skein theory. In order to verify the connectedness condition in practice, we prove that it can be reduced to a single condition in genus one. Moreover, we show that cofactorizability of $\mathcal{A}$, a condition known to be satisfied for modular categories, is sufficient. Therefore, we recover in particular Lyubashenko's construction of a modular functor from a (not necessarily semisimple) modular category and show that it is determined by its genus zero part. Additionally, we exhibit modular functors that do not come from modular categories and outline applications to the theory of vertex operator algebras.

Figures (2)

• Figure 1: On the definition of $\nu(\Gammait)$ and $\pi_0(\Gammait)$.
• Figure 2: On the definition of $H^\varphi$.

Theorems & Definitions (82)

• Definition 2.1: $cyclic$
• Theorem 2.2: cyclicmwansular
• Proposition 2.3
• Lemma 2.4
• proof
• proof : Proof of Proposition \ref{['propisofib']}
• Definition 3.1
• Definition 3.2
• Lemma 3.3
• proof