Resurgence of 1-dim sums of quantum factorials
by
Stavros Garoufalidis
Georgia Tech University

Consider the expression

I_a, b, ep(q)=_n=0^ (q)_n^a q^b n(n+1)/2 ep^n

for a=natural number, b=integer, and ep= 1. Consider the correpsonding formal power series F_a, b, ep(x)=I_a, b, ep(e^1/x) and its Borel transform G_a, b, ep(p). We will prove that G_a, b, ep(p) is a resurgent function with singularities given by evaluating the Rogers dilogarithm function to solutions of the "gluing equation"

(1-z)^a z^b ep=1.

As a corollary, this implies resurgence of the formal power series associated to the 3_1 and 4_1 knots. The multisum analogue of I_a, b, e(q) is under consideration. This is joint work with Ovidiu Costin, and will appear in a series of 3 papers. In the talk we will sketch the main ideas.

Date received: March 26, 2007