Defects in Nematic Shells: a $\Gamma$-convergence discrete-to-continuum approach

In this paper we rigorously investigate the emergence of defects on Nematic Shells with genus different from one. This phenomenon is related to a non trivial interplay between the topology of the shell and the alignment of the director field. To this end, we consider a discrete $XY$ system on the shell $M$, described by a tangent vector field with unit norm sitting at the vertices of a triangulation of the shell. Defects emerge when we let the mesh size of the triangulation go to zero, namely in the discrete-to-continuum limit. In this paper we investigate the discrete-to-continuum limit in terms of $\Gamma$-convergence in two different asymptotic regimes. The first scaling promotes the appearance of a finite number of defects whose charges are in accordance with the topology of shell $M$, via the Poincar\'e-Hopf Theorem. The second scaling produces the so called Renormalized Energy that governs the equilibrium of the configurations with defects.