Nahm's Conjecture: Asymptotic Computations and Counterexamples

TL;DR

This paper investigates the modularity of certain q-series related to Nahm's conjecture, extending previous results to higher dimensions and providing new examples and counterexamples.

Contribution

It generalizes the approach for r>1, discovering new modular forms and counterexamples to Nahm's conjecture beyond the known r=1 case.

Findings

Found new examples of modular q-series for higher dimensions

Identified counterexamples to Nahm's conjecture in multiple cases

Extended the theoretical framework for analyzing the conjecture

Abstract

We consider certain q-series depending on parameters (A,B,C), where A is a positive definite r times r matrix, B is an r-vector and C is a scalar, and ask when these q-series are modular forms. Werner Nahm conjectured a criterion for which A's can occur, in terms of torsion in the Bloch group. The conjecture was proved by Don Zagier and Michael Terhoeven for r=1. We develop their approach for r>1 and find several new examples of modular forms as well as some counterexamples to Nahm's conjecture.