Modular Invariance of (Logarithmic) Intertwining Operators

Abstract Let V be a C₂ C 2 -cofinite vertex operator algebra without nonzero elements of negative weights. We prove the conjecture that the spaces spanned by analytic extensions of pseudo- q -traces (q=e^2 i q = e 2 π i τ) shifted by -c24 - c 24 of products of geometrically-modified (logarithmic) intertwining operators among grading-restricted generalized V -modules are invariant under modular transformations. The convergence and analytic extension result needed to formulate this conjecture and some consequences on such shifted pseudo- q -traces were proved by Fiordalisi (Logarithmic intertwining operator and genus-one correlation functions, 2015) and Fiordalisi (Commun Contemp Math 18: 1650026, 2016) using the method developed in Huang (Commun Contemp Math 7: 649–706, 2005). The method that we use to prove this conjecture is based on the theory of the associative algebras A^N (V) A N (V) for N N N ∈ N, their graded modules and their bimodules introduced and studied by the author in Huang (Associative algebras and the representation theory of grading-restricted vertex algebras, 2020) and Huang (Commun Math Phys 396: 1–44, 2022). This modular invariance result gives a construction of C₂ C 2 -cofinite genus-one logarithmic conformal field theories from the corresponding genus-zero logarithmic conformal field theories.