Quantum Groups and Integralities in Chern-Simons Theory

Author: Chun, Sungbong

Year: 2019

Degree: Dissertation (Ph.D.)

Advisor: Gukov, Sergei

Committee Members: Wise, Mark B.; Ooguri, Hirosi; Gukov, Sergei; Ni, Yi

Option: Physics

DOI: 10.7907/2CW7-5J29

Abstract

In this dissertation, we investigate integralities in Chern-Simons theory. The integralities of interest arise from non-local observables (Wilson lines) in Chern-Simons theory and the partition function itself. In the associated supersymmetric gauge theories (via 3d-3d correspondence), they encode certain BPS spectrum, which are often identified with homological invariants of links and three-manifolds. In this dissertation, we observe that all of them are equipped with non-trivial algebraic structures, such as quantum group actions, modularity, and logarithmic vertex algebras. In the first half of this dissertation, we identify quantum group representations with the dynamics of line operators and their lift to surface operators. In the second half, Chern-Simons partition functions on Seifert manifolds are studied in detail, and its hidden'' integralities are identified with quantum modular forms and the characters of logarithmic vertex operator algebra. From the latter, we also observe that quantum group actions control the dynamics'' of characters.

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