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.