𝒲-Algebras, False Theta Functions and Quantum Modular Forms, I

In this paper, we study certain partial and false theta functions in connection to vertex operator algebras and conformal field theory. We prove a variety of results concerning the asymptotics of modified characters of irreducible modules of certain <f></f>-algebras of singlet type, which allows us in particular to determine their (analytic) quantum dimensions. Our results are fully consistent with the previous conjectures on fusion rings for these vertex algebras. More importantly, we prove quantum modularity (à la Zagier) of the numerator part of irreducible characters of singlet algebra modules, thus demonstrating that quantum modular forms naturally appear in many “sufficiently nice” irrational vertex algebras. It is interesting that quantum modularity persists on the whole set of rationals as in the original Zagier's example coming from Kontsevich's “strange series”. In the last part, slightly independent of all this, we also discuss Nahm-type <f>q</f>-hypergometric series in connection to tails of colored Jones polynomials of certain torus knots and characters of modules for the <f>$(1,p)$</f>-singlet algebra.