## Some analytic aspects of Vafa-Witten twisted N̳ = 4 supersymmetric Yang-Millseory theory

##### Author(s)

Mares, Bernard A., Jr. (Bernard Allen)
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##### Other Contributors

Massachusetts Institute of Technology. Dept. of Mathematics.

##### Advisor

Tomasz S.Mrowka.

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Given an oriented Riemannian four-manifold equipped with a principal bundle, we investigate the moduli spaceMVW of solutions to the Vafa-Witten equations. These equations arise from a twist of N = 4 supersymmetric Yang-Mills theory. Physicists believe that this theory has a well-defined partition function, which depends on a single complex parameter. On one hand, the S-duality conjecture predicts that this partition function is a modular form. On the other hand, the Fourier coefficients of the partition function are supposed to be the "Euler characteristics" of various moduli spacesMASD of compactified anti-self-dual instantons. For several algebraic surfaces, these Euler characteristics were verified to be modular forms. Except in certain special cases, it's unclear how to precisely define the partition function. If there is a mathematically sensible definition of the partition function, we expect it to arise as a gauge-theoretic invariant of the moduli spaces MVW. The aim of this thesis is to initiate the analysis necessary to define such invariants. We establish various properties, computations, and estimates for the Vafa-Witten equations. In particular, we give a partial Uhlenbeck compactification of the moduli space.

##### Description

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2010. This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections. In title on title page, double underscored "N" appears as upper case script. Cataloged from student submitted PDF version of thesis. Includes bibliographical references (p. 118-121).

##### Date issued

2010##### Department

Massachusetts Institute of Technology. Dept. of Mathematics.##### Publisher

Massachusetts Institute of Technology

##### Keywords

Mathematics.