Open modular functors from non-finite tensor categories

TL;DR

The paper develops open modular functors in the non-finite setting by using a framed $E_2$-algebra $\mathcal{A}$ in $\mathsf{Pr}$ with enough compact projective objects, showing that $\mathcal{A}$ yields a cyclic framed $E_2$-algebra and hence an open modular functor $\mathcal{A}_{!}$. The construction is realized via factorization homology: for each connected surface $\Sigma$ with marked intervals, one obtains a central object in $\int_{\Sigma}\mathcal{A}$ and a $\mathsf{Map}(\Sigma)$-action on $\mathrm{Hom}_{\int_{\Sigma}\mathcal{A}}(\mathcal{O}_{\Sigma}, X\rhd \mathcal{O}_{\Sigma})$, giving the open block spaces and a holographic interpretation of conformal blocks. The authors prove excision, skein-theoretic interpretations (ansular functors), and an evaluation on modules over the reflection equation algebra, establishing a three-dimensional perspective on open correlators in the non-finite setting. They further construct open correlators from compact projective symmetric Frobenius algebras in $\mathcal{A}$, showing these Frobenius data encode open sector correlators and providing concrete examples such as the $\beta\gamma$ ghost module categories. Altogether, the work extends the toolbox for vertex operator algebras and non-finite tensor categories beyond the finite modular category setting, preserving a rich open- and holomorphic-factorization structure via factorization homology.

Abstract

We show that a compact rigid balanced braided monoidal category with enough compact projective objects gives rise to a system of mapping class group representations compatible with the gluing along marked intervals. A motivation to consider non-finite tensor categories is the theory of vertex operator algebras where such categories arise as categories of modules. The mapping class group representations presented in this article admit a factorization homology description. In other words, they are of three-dimensional origin and hence obey a holographic principle. A compact projective symmetric Frobenius algebra endows the representations with a pointing that is mapping class group invariant and compatible with the gluing along intervals. This shows that, at least to some extent, many of the tools for the construction and study of spaces of conformal blocks and correlators remain available in a non-finite, but rigid setting.

Figures (2)

• Figure 1: The value of $\mathcal{A}_{\text{\normalfont \bfseries !}}$ on an open surface with three colored marked intervals.
• Figure 2: A picture of $\Sigma \times [0,1]$.

Theorems & Definitions (20)

• Lemma 3.1
• proof
• Proposition 3.2
• proof
• Example 3.3
• Theorem 3.4
• proof
• Corollary 3.5
• proof
• Example 3.6