A Caltech Library Service

3-Manifolds, Q-Series, and Topological Strings


Park, Sunghyuk (2022) 3-Manifolds, Q-Series, and Topological Strings. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/m1fr-6038.


ẑ is a 3d TQFT whose existence was predicted by S. Gukov, D. Pei, P. Putrov, and C. Vafa in 2017. To each 3-manifold equipped with a spinc structure, ẑ is supposed to assign a q-series with integer coefficients that is categorifiable and provides an analytic continuation of the Witten-Reshetikhin-Turaev invariants. In 2019, S. Gukov and C. Manolescu initiated a program to mathematically construct ẑ via Dehn surgery, and as part of that they conjectured that the Melvin-Morton-Rozansky expansion of the colored Jones polynomials can be re-summed into a two-variable series FK(x,q), which is ẑ for the knot complement. Following those developments, in this thesis we develop further and generalize the theory of ẑ. Some of the main results are:
1. Proof of Gukov-Manolescu conjecture for a big class of links, including all homogeneous braid links, which gives a mathematical definition of ẑ for the complements of those links;
2. Generalization of Gukov-Pei-Putrov-Vafa formula for ẑ for negative-definite plumbed 3-manifolds to general Lie algebra;
3. Various conjectures coming out of the interpretation of FK(x,q) in terms of topological strings, such as the HOMFLY-PT analogue (i.e., a-deformation) of FK(x,q) and the holomorphic Lagrangian generalizing the A-polynomial.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:topological field theory; Chern-Simons theory; topological string theory; quantum topology; q-series
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Awards:Scott Russell Johnson Prize for Excellence in Graduate Studies, 2020. Scott Russell Johnson Prize for Excellence in First-Year Graduate Studies, 2018.
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Gukov, Sergei
Thesis Committee:
  • Ni, Yi (chair)
  • Gukov, Sergei
  • Manolescu, Ciprian
  • Ekholm, Tobias
  • Hutchcroft, Thomas
Defense Date:3 May 2022
Funding AgencyGrant Number
Kwanjeong Educational FoundationUNSPECIFIED
Record Number:CaltechTHESIS:05252022-010117961
Persistent URL:
Related URLs:
URLURL TypeDescription adapted for ch. 5 adapted for ch. 3 adapted for ch. 6 adapted for ch. 3 and 4 adapted for ch. 6
Park, Sunghyuk0000-0002-6132-0871
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:14606
Deposited By: Sunghyuk Park
Deposited On:26 May 2022 21:02
Last Modified:02 Jun 2022 23:25

Thesis Files

[img] PDF - Final Version
See Usage Policy.


Repository Staff Only: item control page