Modular invariance of (logarithmic) intertwining operators

arXiv2305.15152MaRDI QIDQ6510254

Yi-Zhi Huang

Abstract: 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 ($q=e^{2piiau}$) shifted by $-fracc24$ 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 [F1} and [F2] using the method developed by the author in [H2]. The method that we use to prove this conjecture is based on the theory of the associative algebras $A^N(V)$ for $NinmathbbN$, 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.

Mathematics Subject Classification ID

Two-dimensional field theories, conformal field theories, etc. in quantum mechanics (81T40) Vertex operators; vertex operator algebras and related structures (17B69) Automorphic forms in several complex variables (32N10)

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