Topics in Topological and Holomorphic Quantum Field Theory

Author: Vyas, Ketan D.

Year: 2010

Degree: Dissertation (Ph.D.)

Advisor: Kapustin, Anton N.

Committee Members: Kapustin, Anton N.; Porter, Frank C.; Schwarz, John H.; Wise, Mark B.

Option: Physics

DOI: 10.7907/QPRX-0V43

Abstract

We investigate topological quantum field theories (TQFTs) in two, three, and four dimensions, as well as holomorphic quantum field theories (HQFTs) in four dimensions. After a brief overview of the two-dimensional (gauged) A and B models and the corresponding the category of branes, we construct analogous three-dimensional (gauged) A and B models and discuss the two-category of boundary conditions. Compactification allows us to identify the category of line operators in the three-dimensional A and B models with the category of branes in the corresponding two-dimensional A and B models. Furthermore, we use compactification to identify the two-category of surface operators in the four-dimensional GL theory at t = 1 and t = i with the two-category of boundary conditions in the corresponding three-dimensional A and B model, respectively.

We construct a four-dimensional HQFT related to N = 1 supersymmetric quantum chromodynamics (SQCD) with gauge group SU(2) and two flavors, as well as a four-dimensional HQFT related to the Seiberg dual chiral model. On closed Kahler surfaces with h(2,0) > 0, we show that the correlation functions of holomorphic SQCD formally compute certain Donaldson invariants. For simply-connected elliptic surfaces (and their blow-ups), we show that the corresponding correlation functions in the holomorphic chiral model explicitly compute these Donaldson invariants.

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