Citation
He, Siqi (2018) The KapustinWitten Equations with Singular Boundary Conditions. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/GMA09Z96. https://resolver.caltech.edu/CaltechTHESIS:05092018094640290
Abstract
Witten proposed a fasinating program interpreting the Jones polynomial of knots on a 3manifold by counting solutions to the KapustinWitten equations with singular boundary conditions.
In Chapter 1, we establish a gluing construction for the Nahm pole solutions to the KapustinWitten equations over manifolds with boundaries and cylindrical ends. Given two Nahm pole solutions with some convergence assumptions on the cylindrical ends, we prove that there exists an obstruction class for gluing the two solutions together along the cylindrical end. In addition, we establish a local Kuranishi model for this gluing picture. As an application, we show that over any compact fourmanifold with S^{3} or T^{3} boundary, there exists a Nahm pole solution to the obstruction perturbed KapustinWitten equations. This is also the case for a fourmanifold with hyperbolic boundary under some topological assumptions.
In Chapter 2, we find a system of nonlinear ODEs that gives rotationally invariant solutions to the KapustinWitten equations in 4dimensional Euclidean space. We explicitly solve these ODEs in some special cases and find decaying rational solutions, which provide solutions to the KapustinWitten equations. The imaginary parts of the solutions are singular. By rescaling, we find some limit behavior for these singular solutions. In addition, for any integer k, we can construct a 5k dimensional family of C^{1} solutions to the KapustinWitten equations on Euclidean space, again with singular imaginary parts. Moreover, we get solutions to the KapustinWitten equation with Nahm pole boundary condition over S^{3} × (0, +∞).
In Chapter 3, we develop a KobayashiHitchin type correspondence for the extended Bogomolny equations on Σ× with Nahm pole singularity at Σ × {0} and the Hitchin component of the stable SL(2, ℝ) Higgs bundle; this verifies a conjecture of Gaiotto and Witten. We also develop a partial KobayashiHitchin correspondence for solutions with a knot singularity in this program, corresponding to the nonHitchin components in the moduli space of stable SL(2, ℝ) Higgs bundles. We also prove the existence and uniqueness of solutions with knot singularities on ℂ × ℝ^{+}. This is joint a work with Rafe Mazzeo.
In Chapter 4, for a 3manifold Y, we study the expansions of the Nahm pole solutions to the KapustinWitten equations over Y × (0, +∞). Let y be the coordinate of (0, +∞) and assume the solution convergence to a flat connection at y → ∞, we prove the subleading terms of the Nahm pole solution is C^{1} to the boundary at y → 0 if and only if Y is an Einstein 3manifold. For Y nonEinstein, the subleading terms of the Nahm pole solutions behave as y log y to the boundary. This is a joint work with Victor Mikhaylov.
Item Type:  Thesis (Dissertation (Ph.D.))  

Subject Keywords:  The KapustinWitten Equations, Singular Boundary Conditions  
Degree Grantor:  California Institute of Technology  
Division:  Physics, Mathematics and Astronomy  
Major Option:  Mathematics  
Thesis Availability:  Public (worldwide access)  
Research Advisor(s): 
 
Thesis Committee: 
 
Defense Date:  2 May 2018  
Record Number:  CaltechTHESIS:05092018094640290  
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:05092018094640290  
DOI:  10.7907/GMA09Z96  
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ORCID: 
 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  10867  
Collection:  CaltechTHESIS  
Deposited By:  Siqi He  
Deposited On:  21 May 2018 22:01  
Last Modified:  04 Oct 2019 00:21 
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