Quasi-modular forms and trace functions associated to free boson and lattice vertex operator algebras (Q2725070)

In the fundamental paper [J. Am. Math. Soc. 9, 237-302 (1996; Zbl 0854.17034)], \textit{Y. Zhu} investigated general correlation functions, which generalize the graded characters for vertex operator algebras. Motivated by Zhu's results, the authors of the present paper study 1-point correlation functions, or trace functions, which arise from certain vertex operator algebras. NEWLINENEWLINENEWLINELet \(V\) be a vertex operator algebra with \({\mathbb Z}\)--grading \(V=\bigoplus_{n \in \mathbb Z} V_n\). If \(v \in V_k\) with vertex operator \(Y(v,z) = \sum_{n \in {\mathbb Z} } v(n) z ^{-n-1}\), then \(o(v) = v(k-1)\) is a linear operator on \(V\) which leaves invariant each homogeneous space \(V_n\) so that one can define the expression: NEWLINE\[NEWLINE Z(v,q) = q ^{ - c /24} \sum_{n \in {\mathbb Z} } (\text{tr}_{V_n} o(v)) q ^{n},NEWLINE\]NEWLINE where \(c\) is the central charge of \(V\). The linear extension of \(Z(v,q)\) on \(V\) is called the (1-point) correlation function determined by \(V\). NEWLINENEWLINENEWLINEIn the paper under review the authors study modular-invariance properties of function \(Z(v,q)\) for the lattice and the free-boson vertex operator algebras. They find a remarkable connection with the theory of quasi-modular forms studied by \textit{M. Kaneko} and \textit{D. Zagier} [in: The moduli space of curves (Texel Island, 1994), Progr. Math. 129, 165-172 (1995; Zbl 0892.11015)]. The relation between spherical harmonics and the highest weight vectors for the Virasoro algebra is also discussed.