Dirac-harmonic maps are a mathematical version (with commuting variables only) of the solutions of the field equations of the non-linear supersymmetric sigma model of quantum field theory. We explain this structure, including the appropriate boundary conditions, in a geometric framework. The main results of our paper are concerned with the analytic regularity theory of such Dirac-harmonic maps. We study Dirac-harmonic maps from a Riemannian surface to an arbitrary compact Riemannian manifold. We show that a weakly Dirac-harmonic map is smooth in the interior of the domain. We also prove regularity results for Dirac-harmonic maps at the boundary when they solve an appropriate boundary value problem which is the mathematical interpretation of the D-branes of superstring theory.
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Jürgen Jost, Qun Chen, Guofang Wang, Miaomiao Zhu, The boundary value problem for Dirac-harmonic maps. J. Eur. Math. Soc. 15 (2013), no. 3, pp. 997–1031DOI 10.4171/JEMS/384