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 ) = (A n−1 , A k−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 ) = (A 2n , T k−1 ) due to Feigin-Stoyanovsky (n = 1) and Stoyanovsky, and (X, X ) = (A 2n−1 , T 1 ) due to Warnaar-Zudulin. We also conduct computational investigations in classical Andrews-Gordon case i.e. when (X, X ) = (A 1 , T k−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.