3-Manifolds, Q-Series, and Topological Strings

Author: Park, Sunghyuk

Year: 2022

Degree: Dissertation (Ph.D.)

Advisor: Gukov, Sergei

Committee Members: Ni, Yi; Gukov, Sergei; Manolescu, Ciprian; Ekholm, Tobias; Hutchcroft, Thomas

Option: Mathematics

DOI: 10.7907/m1fr-6038

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 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.

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