Conference on Calculus of Variations in Lille - 3rd edition - July 4-6 2022

A Γ-convergence result for non-self dual U(1)-Yang–Mills–Higgs energies of Ginzburg–Landau type

Jul 5, 2022, 4:00 PM

30m

M2 building, Cité Scientifique - Meeting room, 1st floor (Laboratoire Paul Painlevé)

Speaker

Federico Luigi Dipasquale (University of Verona )

Description

Let $E\to M$ be a Hermitian complex line bundle with structure group $\mathrm{U}(1)$ over a closed smooth orientable connected Riemannian manifold $M$. Fix a smooth metric connection ${\mathrm{D}}_0$ on $E$ and consider, for $\epsilon >0$, the non-self dual $\mathrm{U}(1)$-Yang–Mills–Higgs energies

${\displaystyle G_{\epsilon }(u_{\epsilon },A_{\epsilon }):={\int }_M\frac{1}{2}|{\mathrm{D}}_{A_{\epsilon }}{u}_{\epsilon }{|}^2+\frac{1}{4{\epsilon }^2}{(1-|u_{\epsilon }{|}^2)}^2+\frac{1}{2}|F_{A_{\epsilon }}{|}^2{\mathrm{v}\mathrm{o}\mathrm{l}}_g,}$

where $(u,A)\in W^{1,2}(M,E)\times W^{1,2}(M,{\mathrm{T}}^{\ast }M)$, ${\mathrm{D}}_A:={\mathrm{D}}_0-iA$, and $F_A$ denotes the curvature form of ${\mathrm{D}}_A$. The functionals $G_{\epsilon }$ arise as natural generalisation of the usual Ginzburg–Landau energy on domains of ${\mathbb{R}}^n$.

The aim of the talk is to illustrate the following $\mathrm{\Gamma }$-convergence result, obtained in collaboration with G. Canevari and G. Orlandi (Università di Verona): as $\epsilon \to 0$, the rescaled functionals $\frac{G_{\epsilon }}{|\log \epsilon |}$ $\mathrm{\Gamma }$-converge, in the flat topology of Jacobians, to ($\pi$ times) the codimension two area functional.