Periods and asymptotics of the 3D-index

The 3D-index of Dimofte-Gaiotto-Gukov is a BPS count of states of a 6-dimensional theory wrapped along an ideally triangulated 3-manifold, decorated by electric and magnetic charges. Mathematically, it is a collection of $q$-series with integer coefficients associated to a suitably triangulated 3-manifold. We show that the asymptotic expansion of these series as $q$ approaches 1 (or a root of unity) is expressed not only in terms of elements of a number field, but also periods of a complex curve associated with the problem. Joint work with Campbell Wheeler, posted in https://arxiv.org/abs/2209.02843