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

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.