In this talk, I will discuss the self-adjointness of the two-dimensional Dirac operator coupled with a singular combination of electrostatic and Lorentz scalar $\delta$-interaction, supported on a closed Lipschitz curve. The main new ingredients are an explicit use of the Cauchy transform on non-smooth curves and a direct link with the Fredholmness of a singular boundary integral operator. This results in a proof of self-adjointness for a new range of coupling constants, which includes and extends all previous results for this class of problems. The study is particularly precise for the case of curvilinear polygons, as the angles can be taken into account in an explicit way. In particular, if the curve is a curvilinear polygon with obtuse angles, then there is a unique self-adjoint realization with domain contained in $H^{1/2}$ for the full range of non-critical coefficients in the transmission condition. The results are based on a joint work with Badreddine Benhellal and Konstantin Pankrashkin.