Asymptotics of Nahm sums at roots of unity

Garoufalidis, Stavros; Zagier, Don

doi:10.1007/s11139-020-00266-x

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Asymptotics of Nahm sums at roots of unity

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Garoufalidis, Stavros
Max Planck Institute for Mathematics, Max Planck Society;

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Zagier, Don
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.1007/s11139-020-00266-x
(Publisher version)

https://doi.org/10.48550/arXiv.1812.07690
(Preprint)

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Citation

Garoufalidis, S., & Zagier, D. (2021). Asymptotics of Nahm sums at roots of unity. Ramanujan Journal, 55(1), 219-238. doi:10.1007/s11139-020-00266-x.

Cite as: https://hdl.handle.net/21.11116/0000-0006-CD39-4

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.