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

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 $\mathbb{E}={\int }_{\mathbb{M}\in \mathrm{Mod}(\mathbb{V})}\mathbb{M}{\otimes }_{\mathbb{C}}{\mathbb{M}}^{\prime }$ in $\mathrm{Mod}({\mathbb{V}}^{\otimes 2})$ (where ${\mathbb{M}}^{\prime }$ is the contragredient module of $\mathbb{M}$) admits a natural structure of associative $\mathbb{C}$-algebra compatible with its ${\mathbb{V}}^{\otimes 2}$-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.

arXiv comment64 pages, many figures