The boundary value problem for Dirac-harmonic maps

Abstract

Abstract. 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 Diracharmonic 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.

Analyse des problèmes conformément invariants

“…Ces derniers ne nous intéresserons pas dans cette revue car la plus part des phénomènes étudiés sont purement locaux. Mais il s'agit d'une question très intéressante voir [9]. Voici une définition rigoureuse.…”

Coarse regularity of solutions to a nonlinear sigma-model with $$L^p$$ L p gravitino

“…[22][23][24][25] enable us to utilize the antisymmetric structure of the equations for φ to improve the regularity. Using similar methods, regularity results for weak solutions of the simpler models, namely Dirac-harmonic maps and Dirac-harmonic maps with curvature terms, are achieved in [3,9,27,28]. Here in this more general model, the structure of the system is even more complicated because of the divergence terms and the appearance of the gravitinos.…”

“…If in addition, the curvature terms in the Lagrangian also vanish, this reduces to the Dirac-harmonic map functional introduced in [6,7], which is studied to a great extent in e.g. [9,15,25,27,28].…”

“…For the cases of the simpler models, namely Dirac-harmonic maps and Dirac-harmonic maps with curvature terms, see [3,9,18,25,28].…”

Blow-up analysis for approximate Dirac-harmonic maps in dimension 2 with applications to the Dirac-harmonic heat flow

“…It is known that the equation of φ can be written as an elliptic system with an anti-symmetric potential [5,24,30]:…”