Modular properties of affine $\text{SLA}sl2$ torus $n$-point functions

Krauel, Shafiq and Wood realized torus $1$-point functions for the simple affine \voa{}s $L(k,0)$ built from $\text{alg}sl(2)$ as vector-valued modular forms attached to a cyclic $R$-module structure and to the modular tensor category $\text{rep}L(k,0)$. We extend this to torus $n$-point functions, defined as traces of chains of $n$ intertwining operators winding once around the $\tau$-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 $\text{alg}sl(2)$, the operator product expansion then reduces leading short-distance behaviour to the classified $1$-point theory. For $\text{alg}sl(2)$ 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 ${\eta }^{3k\mathrm{/}2}$ of \cite{KSW}. We also extend \KSW’s{} categorical picture, representations of $\text{Bthree}=\text{PMod}(\text{surf}11)$ 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 $\text{alg}sl(2)$ matrix form of $S$ beyond this abstract construction remains open.