Modular invariance of (logarithmic) intertwining operators (Q6550962)

Previously, Y. Zhu, in his 1990 PhD thesis (published as [\textit{Y. Zhu}, J. Am. Math. Soc. 9, No. 1, 237--302 (1996; Zbl 0854.17034)]), established a seminal result on the modular invariance of characters for rational \(C_2\)-cofinite vertex algebras, where characters are represented as modified graded traces. In 2004, \textit{M. Miyamoto} [Duke Math. J. 122, No. 1, 51--91 (2004; Zbl 1165.17311)] extended this result to \(C_2\)-cofinite vertex algebras without the rationality requirement, replacing graded traces with more general symmetric functions known as pseudotraces. Around the same time, the author [Commun. Contemp. Math. 7, No. 5, 649--706 (2005; Zbl 1124.11022)] generalized Zhu's result to products of intertwining operators within the rational framework. This current paper offers a unifying approach that broadens Miyamoto's and author's results, providing the most comprehensive modular invariance for products of intertwining operators. More precisely, Huang demonstrates that the spaces generated by analytic extensions of pseudo-\(q\)-traces (where \(q = e^{2 \pi i \tau}\)) shifted by \(-\frac{c}{24}\), of products of geometrically-modified (logarithmic) intertwining operators among grading-restricted generalized V-modules, remain invariant under modular transformations.