Regularities, Resurgence and R-Matrices in Chern Simons Theory

Citation

Gruen, Angus Fred Wilkinson (2023) Regularities, Resurgence and R-Matrices in Chern Simons Theory. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/dr5q-3074. https://resolver.caltech.edu/CaltechTHESIS:06012023-043437790

Abstract

This thesis aims to address two related but distinct problems in Chern Simons theory:

1. In 2019, Gukov and Manolescu observed that for fixed a knot K, the family of coloured Jones polynomials J_k(K; q) display regularity in colour k and conjectured that this could be captured by a 2 variable series F_K(x, q). Over the subsequent few years, Park proved that, for a large family of knots, F_K(x, q) could be computed using the R-matrix for a particular Verma module.

We will show that it is possible to extend the work of Park to compute the 2 variable series F^N_K(x, q) associated to other lie groups, sl_N, which capture a similar regularity in the quantum invariants P^N_k(K; q). Following on from this we will further show that in many cases these series F^N_K(x, q) themselves display a regularity in N, reminiscent of the HOMFLY-PT polynomial, allowing the construction of a 3 variable series F_K(x, a, q) interpolating F^N_K(x, q) for all N.

2. Complex Chern Simons theory is a rare example of Quantum field theory with both interesting non-perturbative behaviour and whose perturbative expansion can be computed to high order. For a nice class of 3-manifolds, namely surgeries on knot complements, we will show how to predict aspects of the non-perturbative behaviour first semi-classically and then, using resurgence, through studying just the perturbative expansion around the trivial flat connection. Finally, we show that contrary to expectation, these families of 3-manifolds display regularity in the surgery coefficient.

Thesis Files