Modular properties of affine $\SLA{sl}{2}$ torus $n$-point functions

A. Zuevsky

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arXiv · 2609.01496

Modular properties of affine $\SLA{sl}{2}$ torus $n$-point functions

A. Zuevsky

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Abstract

Krauel, Shafiq and Wood realized torus $1$-point functions for the simple affine \voa{}s $L(k,0)$ built from $\alg{sl}(2)$ as vector-valued modular forms attached to a cyclic $R$-module structure and to the modular tensor category $\rep{L(k,0)}$. We extend this to torus $n$-point functions, defined as traces of chains of $n$ intertwining operators winding once around the $τ$-cycle. Every coefficient of their local (Laurent, or Puiseux) expansion about a diagonal stratum is again a \vvmf{}, lying, whenever a mild lowest-weight hypothesis holds, in the very $R$-modules of \cite{KSW}. Granted a torus-primarity hypothesis verified explicitly below for $\alg{sl}(2)$, the operator product expansion then reduces leading short-distance behaviour to the classified $1$-point theory. For $\alg{sl}(2)$ we classify the fusion-chain conformal block spaces, identify torus $n$-point primary vectors, and treat $n=2$ in detail, obtaining an explicit $(k+1)$-dimensional family of vector-valued Jacobi-type forms generalizing the level-$k$ forms $η^{3k/2}$ of \cite{KSW}. We also extend \KSW's{} categorical picture, representations of $\Bthree=\PMod(\surf{1}{1})$ built from a modular tensor category, to representations, on fusion-chain block spaces, of a mapping-class subgroup of the $n$-punctured torus generated by $S$, $T$, and adjacent braidings, with a Verlinde-type dimension formula and categorical $S$-, $T$-operators. The explicit $\alg{sl}(2)$ matrix form of $S$ beyond this abstract construction remains open.

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BibTeX RIS

A. Zuevsky. 2026-09-01. Modular properties of affine $\SLA{sl}{2}$ torus $n$-point functions. https://arxiv.org/abs/2609.01496

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