Nahm's Conjecture: Asymptotic Computations and Counterexamples

By Masha Vlasenko, Sander Zwegers. 2011

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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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d607c75db4d2f2e076d6a0ffa9b4fb298ade1b62