Asymptotics of Nahm sums at roots of unity

TL;DR

This paper derives detailed asymptotic formulas for Nahm sums at roots of unity, linking their behavior to quantum knot invariants and modularity conjectures, with implications for K-theory and quantum topology.

Contribution

It provides the first all-orders asymptotic expansion of Nahm sums at roots of unity, connecting these to quantum invariants and modularity conjectures.

Findings

Derived explicit asymptotics for Nahm sums at roots of unity

Linked Nahm sum asymptotics to Kashaev invariants of knots

Suggested a deep connection between quantum invariants and Nahm sums

Abstract

We give a formula for the radial asymptotics to all orders of the special $q$-hypergeometric series known as Nahm sums at complex roots of unity. This result is used in~\cite{CGZ} to prove one direction of Nahm's conjecture relating the modularity of Nahm sums to the vanishing of a certain invariant in $K$-theory. The power series occurring in our asymptotic formula are identical to the conjectured asymptotics of the Kashaev invariant of a knot once we convert Neumann-Zagier data into Nahm data, suggesting a deep connection between asymptotics of quantum knot invariants and asymptotics of Nahm sums that will be discussed further in a subsequent publication.