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Regularities, Resurgence and R-Matrices in Chern Simons Theory


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.


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 Jk(K; q) display regularity in colour k and conjectured that this could be captured by a 2 variable series FK(x, q). Over the subsequent few years, Park proved that, for a large family of knots, FK(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 FNK(x, q) associated to other lie groups, slN, which capture a similar regularity in the quantum invariants PNk(K; q). Following on from this we will further show that in many cases these series FNK(x, q) themselves display a regularity in N, reminiscent of the HOMFLY-PT polynomial, allowing the construction of a 3 variable series FK(x, a, q) interpolating FNK(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.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Quantum Topology; Knot Theory; Chern-Simons theory; Resurgence; Quantum Invariants
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Awards:Scott Russell Johnson Graduate Dissertation Prize in Mathematics, 2023. Apostol Award for Excellence in Teaching in Mathematics, 2022.
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Gukov, Sergei
Thesis Committee:
  • Ni, Yi (chair)
  • Gukov, Sergei
  • Marcolli, Matilde
  • Ekholm, Tobias
Defense Date:16 May 2023
Record Number:CaltechTHESIS:06012023-043437790
Persistent URL:
Related URLs:
URLURL TypeDescription adapted for Chapter 3 adapted for Chapter 4 adapted for Chapter 4
Gruen, Angus Fred Wilkinson0000-0003-0284-009X
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:15256
Deposited By: Angus Gruen
Deposited On:08 Jun 2023 15:29
Last Modified:16 Jun 2023 18:11

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