<em>q</em>-Series, their Modularity, and Nahm's Conjecture

Abstract

This thesis concerns modularity properties of Nahm sums, a family of <em>q</em>-hypergeometric series, and applications. They appear, for example, as characters of VOAs, knot invariants, and generating functions for classes of partitions. <br /> In the context of conformal field theory, Nahm conjectured that the modularity of Nahm sums is related to the vanishing of solutions of algebraic equations in the Bloch group. A precise formulation by Zagier turned out to be wrong according to counterexamples given by Vlasenko-Zwegers. Based on generalised asymptotics on rays on the upper half-plane, we discuss the examples given by Vlasenko-Zwegers and explain how they can be explained in the context of Nahm's observation. Moreover, we refine the correspondence between the modularity of Nahm sums and the Bloch group by studying their vector-valued modularity under the full modular group. <br /> The tail of the coloured Jones polynomial for an alternating knot is a <em>q</em>-series knot invariant that arises as a limit of the coloured Jones polynomials and has a representation as a generalised Nahm sum. For several knots, the tail of the coloured Jones polynomial is known to be a product of (partial) theta functions and thus (almost) modular. The main result of part II of this thesis is a general formula for the tail of the coloured Jones polynomial in terms of (partial) theta functions for a class of knots. <br /> Part III of this thesis deals with an application of <em>q</em>-series to the theory of partitions. There, we prove a conjecture of Andrews concerning the sign pattern of coefficients of a <em>q</em>-series from Ramanujan's “lost” notebook. This part is based on a preprint that is joint work with Amanda Folsom, Joshua Males, and Larry Rolen.