Complex $\Gamma$-convergence and magnetic Dirichlet Laplacian in bounded thin tubes

The resolvent convergence of self-adjoint operators via the technique of $\Gamma$-convergence of quadratic forms is adapted to incorporate complex Hilbert spaces. As an application, we find effective operators to the Dirichlet Laplacian with magnetic potentials in very thin bounded tubular regions in space built along smooth closed curves; relatively weak regularity is asked for the potentials, and the convergence is in the norm resolvent sense as the cross sections of the tubes go uniformly to zero.