3-Manifolds, Q-Series, and Topological Strings

Citation

Park, Sunghyuk (2022) 3-Manifolds, Q-Series, and Topological Strings. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/m1fr-6038. https://resolver.caltech.edu/CaltechTHESIS:05252022-010117961

Abstract

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 spin^c 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 F_K(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 F_K(x,q) in terms of topological strings, such as the HOMFLY-PT analogue (i.e., a-deformation) of F_K(x,q) and the holomorphic Lagrangian generalizing the A-polynomial.

Thesis Files