Convergence of the self-dual U(1)-Yang-Mills-Higgs energies to the (n-2)-area functional

Given a hermitian line bundle L -> M on a closed Riemannian manifold (Mn, g), the self-dual Yang-Mills-Higgs energies are a natural family of functionals E-epsilon(u, del): = integral(M)(vertical bar del u vertical bar(2) + epsilon(2)vertical bar F-del vertical bar(2) + (1-vertical bar u vertical bar(2))(2)/4 epsilon(2)) defined for couples (u, del) consisting of a section u is an element of Gamma(L) and a hermitian connection del with curvature F-del. While the critical points of these functionals have been well-studied in dimension two by the gauge theory community, it was shown in [52] that critical points in higher dimension converge as... 0 (in an appropriate sense) to minimal submanifolds of codimension two, with strong parallels to the correspondence between the Allen-Cahn equations and minimal hypersurfaces. In this paper, we complement this idea by showing the Gamma-convergence of E-epsilon to (2 pi times) the codimension two area: more precisely, given a family of couples (u(c), del(c)) with sup(epsilon) E-c (u(c), del(c)) < infinity, we prove that a suitable gauge invariant Jacobian J(u(epsilon), del(epsilon)) converges to an integral (n - 2)-cycle G, in the homology class dual to the Euler class c(1)(L), with mass 2 pi M(Gamma) <= lim inf(epsilon -> 0) E-epsilon(u(epsilon), del(epsilon)). We also obtain a recovery sequence, for any integral cycle in this homology class. Finally, we apply these techniques to compare min-max values for the (n - 2)-area from the Almgren-Pitts theory with those obtained from the Yang-Mills-Higgs framework, showing that the former values always provide a lower bound for the latter. As an ingredient, we also establish a Huisken-type monotonicity result along the gradient flow of E-epsilon.

Automated summary

This paper studies the self-dual Yang-Mills-Higgs energies on a closed Riemannian manifold and proves their convergence to minimal submanifolds. The author establishes a connection between the energies and the Euler class by introducing a suitable gauge invariant Jacobian, and shows the existence of a recovery sequence under certain conditions. Furthermore, a comparison between the min-max values obtained from the Almgren-Pitts theory and the Yang-Mills-Higgs framework is made, with the former always providing a lower bound for the latter.