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We study a variational model which combines features of the Ginzburg-Landaumodel in 2D and of the Mumford-Shah functional. As in the classicalGinzburg-Landau theory, a prescribed number of point vortices appear in thesmall energy regime; the model allows for discontinuities, and the energypenalizes their length. The novel phenomenon here is that the vortices have afractional degree $1/m$ with $m\geq 2$ prescribed. Those vortices must beconnected by line discontinuities to form clusters of total integer degrees.The vortices and line discontinuities are therefore coupled through atopological constraint. As in the Ginzburg-Landau model, the energy isparameterized by a small length scale $\varepsilon>0$. We perform a complete$\Gamma$-convergence analysis of the model as $\varepsilon\downarrow0$ in thesmall energy regime. We then study the structure of minimizers of the limitproblem. In particular, we show that the line discontinuities of a minimizersolve a variant of the Steiner problem. We finally prove that for small$\varepsilon>0$, the minimizers of the original problem have the same structureaway from the limiting vortices.

A Ginzburg-Landau model with topologically induced free discontinuities