Nahm's conjecture in the Cartan case

Abstract

When is a q-hypergeometric series a modular form? The connection between these two objects dates back to 1913 in Ramanujan’s first letter to Hardy. In general, this is a deep question and out of reach at present. In 1994, Nahm conjectured a criterion for this phenomenon that has become a guiding principle in this area of research. The conjecture connects the modularity of a family of Eulerian series to torsion elements in Bloch groups of number fields determined by the Eulerian series. In 2011, Vlasenko and Zwegers exhibited counter-examples to this conjecture. Despite this, a theorem of Lee in 2013 provides strong evidence the conjecture is true in a special case related to models in conformal field theory. These models are parameterised by a pair (X, X'), where X and X' are Dynkin diagrams of ADET type. When gcd ((n − 1)!, k)=1, we prove Nahm’s conjecture holds in the case (X, X') = (An−1, Ak−1). We make use of string functions coming from the representation theory of affine Kac–Moody algebras. This complements previous results pertaining to (X, X') = (A2n, Tk−1) due to Feigin-Stoyanovsky (n = 1) and Stoyanovsky, and (X, X') = (A2n−1, T1) due to Warnaar–Zudulin. We also conduct computational investigations in classical Andrews–Gordon case i.e. when (X, X') = (A1, Tk−1). In particular, Keegan and Nahm ask whether a special family of modular B-vectors are the only ones giving rise to a modular Eulerian series. We confirm this for k = 3 and k = 4.Chapter 2 will introduce the basic objects needed to state Nahm’s conjecture. This will include details on modular forms, dilogarithms and the Bloch group.In Chapter 3 we will provide a statement of Nahm’s conjecture, its asymptotic motivation and counter-examples due to Vlasenko and Zwegers.In Chapter 4 we adopt a less number theoretic point of view and first discuss affine Kac–Moody algebras and some of their representation theory. From this theory we have access to the string functions, their modularity properties and the Virasoro algebra. The end of this chapter contains the original work of the author.