How are pseudo-$q$-traces related to (co)ends?
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913977733218304 |
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| author | Gui, Bin Zhang, Hao |
| author_facet | Gui, Bin Zhang, Hao |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_04532 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | How are pseudo-$q$-traces related to (co)ends? Gui, Bin Zhang, Hao Quantum Algebra Mathematical Physics Representation Theory 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. |
| title | How are pseudo-$q$-traces related to (co)ends? |
| topic | Quantum Algebra Mathematical Physics Representation Theory |
| url | https://arxiv.org/abs/2508.04532 |