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 Calegari et al. (Bloch groups, algebraic K-theory, units and Nahm’s conjecture. arXiv:1712.04887, 2017) to prove 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.
Highlights
Nahm sums are special q-hypergeometric series whose summand involves a quadratic form, a linear form and a constant
Nahm formulated a very surprising conjecture, that has elicited a lot of interest, relating the question of their modularity to the vanishing of a certain invariant in algebraic K -theory
The coincidence of the asymptotics of Nahm sums at q = 1 and the series of [3] was observed several years ago via an explicit map from Neumann–Zagier data to Nahm data, and leads to a deeper connection between quantum invariants of knots defined on the roots of unity and q-series invariants of knots
Summary
Nahm sums are special q-hypergeometric series whose summand involves a quadratic form, a linear form and a constant. The coincidence of the asymptotics of Nahm sums at q = 1 and the series of [3] was observed several years ago via an explicit map from Neumann–Zagier data to Nahm data, and leads to a deeper connection between quantum invariants of knots defined on the roots of unity (such as the Kashaev invariant) and q-series invariants of knots (such as the 3D-index of Dimofte–Gukov–Gaiotto [6]). This connection will be explained in a later publication [10]. 0, each term of the product of the ψ-terms in (10) has an asymptotic expansion in C[x][[ε1/2]] in which the coefficient of xn is O(εn)
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