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Topological quantum field theory and the geometric Langlands correspondence

Citation

Setter, Kevin Luke (2013) Topological quantum field theory and the geometric Langlands correspondence. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechTHESIS:09192012-150137728

Abstract

In the pioneering work of A. Kapustin and E. Witten, the geometric Langlands program of number theory was shown to be intimately related to duality of GL-twisted N=4 super Yang-Mills theory compactified on a Riemann surface. In this thesis, we generalize Kapustin-Witten by investigating compactification of the GL-twisted theory to three dimensions on a circle (for various values of the twisting parameter t). By considering boundary conditions in the three-dimensional description, we classify codimension-two surface operators of the GL-twisted theory, generalizing those surface operators studied by S. Gukov and E. Witten. For t=i, we propose a complete description of the 2-category of surface operators in terms of module categories, and, in addition, we determine the monoidal category of line operators which includes Wilson lines as special objects. For t=1 and t=0, we discuss surface and line operators in the abelian case.

We generalize Kapustin-Witten also by analyzing a separate twisted version of N=4, the Vafa-Witten theory. After introducing a new four-dimensional topological gauge theory, the gauged 4d A-model, we locate the Vafa-Witten theory as a special case. Compactification of the Vafa-Witten theory on a circle and on a Riemann surface is discussed. Several novel two- and three-dimensional topological gauge theories are studied throughout the thesis and in the appendices.

In work unrelated to the main thread of the thesis, we conclude by classifying codimension-one topological defects in two-dimensional sigma models with various amounts of supersymmetry.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Langlands, higher categories, topology, field theory
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Physics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Kapustin, Anton N.
Thesis Committee:
  • Kapustin, Anton N. (chair)
  • Porter, Frank C.
  • Gukov, Sergei
  • Wise, Mark B.
Defense Date:24 August 2012
Author Email:kevinsetter (AT) yahoo.com
Record Number:CaltechTHESIS:09192012-150137728
Persistent URL:http://resolver.caltech.edu/CaltechTHESIS:09192012-150137728
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:7208
Collection:CaltechTHESIS
Deposited By: Kevin Setter
Deposited On:30 Oct 2012 17:40
Last Modified:26 Dec 2012 04:45

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