QM Research Seminar: Topological recursion, higher quantum Airy structures, and knot invariants

Speaker: Greyson Potter (SDU)

Abstract:

We develop a computational approach to analyzing the generalized volume conjecture, which relates topological recursion and Chern-Simons theory for hyperbolic 3-manifolds with torus boundary. The conjecture states that there is an equality between two series in a formal parameter hbar: the non-perturbative wave function from topological recursion and a state-integral for SL(2,C) Chern-Simons theory. Both series are expressible as sums over graphs. We develop algorithms for efficiently computing the graph sum for the non-perturbative wave function via quantum Airy structures and a software package, called qairy, which implements our algorithms. We further develop tools and techniques which are widely applicable to the calculation and analysis of the non-perturbative wave functions associated to genus one spectral curves. Using these tools, we are able to verify the generalized volume conjecture in several cases up to much higher order in hbar than was previously accessible as well as analyze the arithmetic aspects of the conjecture.