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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>3</m:mn> </m:mrow> </m:math> {n\geq 3} . Assuming a logarithmic energy bound in the coupling parameter, we study the asymptotic behaviour of critical points in the London limit. After a convenient choice of the gauge, we show compactness of finite-energy critical points in Sobolev norms. Moreover, thanks to a suitable monotonicity formula, we prove that the energy densities of critical points, rescaled by the logarithm of the coupling parameter, converge to the weight measure of a stationary, rectifiable varifold of codimension 2.