Speaker
Description
Let $E \to M$ be a Hermitian complex line bundle with structure group ${\rm U}(1)$ over a closed smooth orientable connected Riemannian manifold $M$. Fix a smooth metric connection ${\rm D}_0$ on $E$ and consider, for $\varepsilon > 0$, the non-self dual ${\rm U}(1)$-Yang–Mills–Higgs energies
$\displaystyle G_\varepsilon(u_\varepsilon, A_\varepsilon) := \int_M \frac{1}{2}| {\rm D}_{A_\varepsilon} u_\varepsilon|^2 + \frac{1}{4\varepsilon^2}\left(1-|u_\varepsilon|^2\right)^2 + \frac{1}{2}|F_{A_\varepsilon}|^2 \, {\rm vol}_g, $
where $(u, A) \in W^{1,2}(M,E) \times W^{1,2}(M,{\rm T}^*M)$, ${\rm D}_A := {\rm D}_0 - i A$, and $F_A$ denotes the curvature form of ${\rm D}_A$. The functionals $G_\varepsilon$ 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 $\Gamma$-convergence result, obtained in collaboration with G. Canevari and G. Orlandi (Università di Verona): as $\varepsilon \to 0$, the rescaled functionals $\frac{G_\varepsilon}{|{\log\varepsilon}|}$ $\Gamma$-converge, in the flat topology of Jacobians, to ($\pi$ times) the codimension two area functional.
