The Yang-Mills-Higgs functional on complex line bundles: asymptotics for critical points

Abstract: We consider a gauge-invariant Ginzburg-Landau functional (also known as Abelian Yang-Mills-Higgs model) on Hermitian line bundles over closed Riemannian manifolds of dimension $n \geq 3$. Assuming a logarithmic energy bound in the coupling parameter, we study the asymptotic behaviour of critical points in the non-self dual scaling, as the coupling parameter tends to zero. After a convenient choice of the gauge, we show compactness of finite-energy critical points in Sobolev norms. Moreover, %independently of the gauge andthanks to a suitable monotonicity formula,we prove that the energy densities of critical points, rescaled by the logarithm of the coupling parameter, concentrate towards the weight measure of a stationary, rectifiable varifold of codimension~2.