Variational aspects of the generalized Seiberg-Witten functional (Q6574282)

In the framework of developing a unified mathematical treatment of the gauge functionals from quantum field theory, the authors introduce and study the generalized Seiberg-Witten functional \(\mathcal{GSW}(A,\sigma)=\int_M \mathrm{dvol} (|FA-\frac{1}{2}\tau(\sigma)|^2 +|D\!\!\!\!/{}_A\sigma|^2)\), where \(A\) is a connection of a principal \((G/\mathbb{Z}_2)\)-bundle \(P_{G/\mathbb{Z}_2}\) over the \(n\)-dimensional closed manifold \(M\) equipped with a so-called spin\(^G\)-structure \(P\), \(\sigma\) is a section of an associated vector bundle \(S\) of \(P\), \(D\!\!\!\!/{}_A\) ia a generalized Dirac operator on \(S\), and \(\tau\) is a quadratic map from \(\Gamma(S)\) to \(\Omega^2(\mbox{ad}\;P_{G/\mathbb{Z}_2}):=\Gamma(\Lambda^2 T^* M\otimes \mbox{ad}\;P_{G/\mathbb{Z}_2})\). This generalized functional encompasses several classical functionals, such as the Seiberg-Witten and Kapustin-Witten functionals. The main results prove the smoothness of weak solutions of the Euler-Lagrange equations for \(\mathcal{GSW}(A,\sigma)\). Besides, provided that the structural group \(G\) of the bundle is abelian, the existence of weak solutions is demonstrated, by verifying the Palais-Smale compactness.