Modular invariance of trace functions on VOAs in many variables (Q2782393)

Let \(V=\bigoplus^\infty_{j=0} V_j\) be a rational vertex operator algebra with \(\dim V_0 = 1\) and assume that \(V\) satisfies the \(C_2\)-condition. Let \(\{W^1,\dots,W^k\}\) be the set of all inequivalent irreducible \(V\)-modules and \(Z_i(v,\tau)\) be the trace function on \(W^i\) for \(v\in V\). It is known by \textit{Y. Zhu} [J. Am. Math. Soc. 9, 237-302 (1996; Zbl 0854.17034)] that \(\sigma\in \text{SL}_2(\mathbb{Z})\) transforms \(Z_i(v,T)\) into a linear combination of \(Z_1(v,\tau),\dots,Z_k(v,\tau)\) with coefficients being independent of \(v\). In a previous paper [Duke Math. J. 101, 221-236 (2000; Zbl 0988.17021)], the author considered a modified trace function \(Z_i(u,v;\tau)\) on \(W^i\) for \(u,v \in V_1\) and proved the modular transformation formula. In the paper under review the author introduces a multi-variable trace function \(Z_i(v;\tau_1,\dots,\tau_m)\) on \(W^i\) for \(v\in\text{span}\{e^1,\dots,e^m\}\), where \(e^1,\dots,e^m\) are mutually orthogonal conformal vectors, and proves the modular transformation formula. As an example, the modular invariance of the multi-variable trace function for the moonshine vertex operator algebra \(V^\sharp\) is discussed.NEWLINENEWLINEFor the entire collection see [Zbl 0980.00029].