Modular properties of affine SLAsl2 torus n-point functions
Abstract
Krauel, Shafiq and Wood (\KSW{}) realized torus 1-point functions for the simple affine \voa{}s L(k,0) built from algsl(2) as vector-valued modular forms attached to a cyclic R-module structure and to the modular tensor category repL(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}. Giving a torus-primarity hypothesis verified below for algsl(2), the operator product expansion then reduces leading short-distance behaviour to the classified 1-point theory. For algsl(2) we classify the fusion-chain conformal block spaces, identify torus n-point primary vectors, and treat n=2 in detail, obtaining a (k+1)-dimensional family of vector-valued Jacobi-type forms generalizing the level-k forms η3k/2 of \cite{KSW}. We also extend \KSW{} categorical picture, representations of Bthree=PMod(surf11) 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 algsl(2) matrix form of S beyond this abstract construction remains open.
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BibTeX
@misc{zuevsky2026modular,
title = {{Modular properties of affine \(\SLAsl2\) torus \(n\)-point functions}},
author = {A. Zuevsky},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.01496},
eprint = {2609.01496},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01496},
note = {CC0},
}
RIS
TY - UNPB
TI - Modular properties of affine SLAsl2 torus n-point functions
AU - A. Zuevsky
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2609.01496
UR - https://arxiv.org/abs/2609.01496
AB - Krauel, Shafiq and Wood (\KSW{}) realized torus 1-point functions for the simple affine \voa{}s L(k,0) built from algsl(2) as vector-valued modular forms attached to a cyclic R-module structure and to the modular tensor category repL(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}. Giving a torus-primarity hypothesis verified below for algsl(2), the operator product expansion then reduces leading short-distance behaviour to the classified 1-point theory. For algsl(2) we classify the fusion-chain conformal block spaces, identify torus n-point primary vectors, and treat n=2 in detail, obtaining a (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{} categorical picture, representations of Bthree=PMod(surf11) 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 algsl(2) matrix form of S beyond this abstract construction remains open.
ER -