How are pseudo-$q$-traces related to (co)ends?

TL;DR

This paper establishes a deep connection between pseudo-$q$-traces, (co)ends, and conformal blocks in the context of $ ext{VOA}$ modules, confirming conjectures by Gainutdinov-Runkel and Arike-Nagatomo.

Contribution

It proves that the end of modules forms an associative algebra and relates symmetric linear functionals to conformal blocks, confirming key conjectures in VOA theory.

Findings

The end $ ext{E}$ admits a natural associative algebra structure.

The category of left $ ext{E}$-modules is equivalent to $ ext{Mod}( ext{V})$.

Pseudo-$q$-traces give an isomorphism to conformal blocks.

Abstract

Let $\mathbb V$ be an $\mathbb N$-graded $C_2$-cofinite vertex operator algebra (VOA), not necessarily rational or self-dual. Using a special case of the sewing-factorization theorem from [GZ25a], we show that the end $\mathbb E=\int_{\mathbb M\in\mathrm{Mod}(\mathbb V)}\mathbb M\otimes_{\mathbb C}\mathbb M'$ in $\mathrm{Mod}(\mathbb{V}^{\otimes2})$ (where $\mathbb{M}'$ is the contragredient module of $\mathbb{M}$) admits a natural structure of associative $\mathbb C$-algebra compatible with its $\mathbb{V}^{\otimes2}$-module structure. Moreover, we show that a suitable category $\mathrm{Coh}_{\mathrm{L}}(\mathbb E)$ of left $\mathbb E$-modules is isomorphic, as a linear category, to $\mathrm{Mod}(\mathbb V)$, and that the space of vacuum torus conformal blocks is isomorphic to the space $\mathrm{SLF}(\mathbb E)$ of symmetric linear functionals on $\mathbb E$. Combining these results with the main theorem of [GZ25b], we prove a conjecture of Gainutdinov-Runkel: For any projective generator $\mathbb G$ in $\mathrm{Mod}(\mathbb V)$, the pseudo-$q$-trace construction yields a linear isomorphism from $\mathrm{SLF}(\mathrm{End}_{\mathbb V}(\mathbb{G})^{\mathrm{opp}})$ to the space of vacuum torus conformal blocks of $\mathbb V$. In particular, if $A$ is a unital finite-dimensional $\mathbb C$-algebra such that the category of finite-dimensional left $A$-modules is equivalent to $\mathrm{Mod}(\mathbb V)$, then $\mathrm{SLF}(A)$ is linearly isomorphic to the space of vacuum torus conformal blocks of $\mathbb V$. This confirms a conjecture of Arike-Nagatomo.