Modular invariance of (logarithmic) intertwining operators

Let $V$ be a $C_2$-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 ($q=e^{2\pi i\tau }$) shifted by $-\frac{c}{24}$ 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 in [F1] and [F2] using the method developed in [H2]. The method that we use to prove this conjecture is based on the theory of the associative algebras $A^N(V)$ for $N\in \mathbb{N}$, their graded modules and their bimodules introduced and studied by the author in [H8] and [H9]. This modular invariance result gives a construction of $C_2$-cofinite genus-one logarithmic conformal field theories from the corresponding genus-zero logarithmic conformal field theories.