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We study the two-dimensional Dirac operator with an arbitrary combination ofelectrostatic and Lorentz scalar $\delta$-interactions of constant strengthssupported on a smooth closed curve. For any combination of the couplingconstants a rigorous description of the self-adjoint realizations of theoperators is given and the qualitative spectral properties are described. Theanalysis covers also all so-called critical combinations of coupling constants,for which there is a loss of regularity in the operator domain. In this case,if the mass is non-zero, the resulting operator has an additional point in theessential spectrum, and the position of this point inside the central gap canbe made arbitrary by a suitable choice of the coupling constants. The analysisis based on a combination of the extension theory of symmetric operators with adetailed study of boundary integral operators viewed as periodicpseudodifferential operators.

Two-dimensional Dirac operators with singular interactions supported on closed curves