Lawrence and Zagier have shown that the Reshetikhin-Turaev invariant of a Brieskorn sphere is the limiting value of a power series with integer coefficients convergent inside the unit disc, as the parameter tends to a certain root of unity. This power series have interesting modularity properties.
Subsequent work by Gukov, Putrov, Pei and Vafa aimed at generalizing this phenomena to other three-manifolds, leading to the invention of a power-series invariants of three-manifolds, which is now known as the so-called Zed-hat invariant.
These invariants are conjectured to be examples of (higher depth) quantum modular forms, and this is known to be true for some families of three-manifolds.
In this talk, we present a joint result with Jørgen E. Andersen, which shows that for a Seifert fibered homology sphere, this invariant can be computed via Borel-Laplace resummation of the Reshetikhin-Turaev invariant, as proposed (and proven for examples with three singular fibers) by Gukov-Marino-Putrov.