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Variational aspects of the generalized Seiberg-Witten functional

Wanjun Ai, Shuhan Jiang, Jürgen Jost

Abstract

In this paper, as a step towards a unified mathematical treatment of the gauge functionals from quantum field theory that have found profound applications in mathematics, we generalize the Seiberg-Witten functional that in particular includes the Kapustin-Witten functional as a special case. We first demonstrate the smoothness of weak solutions to this generalized functional. We then establish the existence of weak solutions under the assumption that the structure group of the bundle is abelian, by verifying the Palais-Smale compactness.

Variational aspects of the generalized Seiberg-Witten functional

Abstract

In this paper, as a step towards a unified mathematical treatment of the gauge functionals from quantum field theory that have found profound applications in mathematics, we generalize the Seiberg-Witten functional that in particular includes the Kapustin-Witten functional as a special case. We first demonstrate the smoothness of weak solutions to this generalized functional. We then establish the existence of weak solutions under the assumption that the structure group of the bundle is abelian, by verifying the Palais-Smale compactness.
Paper Structure (8 sections, 15 theorems, 130 equations)

This paper contains 8 sections, 15 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. The geometric settings
  3. Dirac G-bundles
  4. Spin G-structures
  5. Weitzenböck formulas
  6. Generalized Seiberg--Witten functional
  7. Regularity of weak solutions of the generalized Seiberg--Witten equations
  8. Compactness and existence theorems for the H-functional

Key Result

Theorem 1.1

Suppose $(A,\sigma)\in W^{1,2}(\Omega^1(\mathrm{ad}\, P_{G/\mathbb{Z}_2}))\times W^{1,2}(\Gamma(S))$ is a weak solution of eq:A and eq:sigma, then there exists a gauge transformation $g$, such that $g(A,\sigma)\mathpunct{:}=(g(A),g^{-1}\sigma)$ is smooth.

Theorems & Definitions (49)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Remark 1.4
  • Remark 1.5
  • Proposition 2.1
  • proof
  • Example 2.2
  • Lemma 2.3
  • proof
  • ...and 39 more