Partial regularity for a nonlinear sigma model with gravitino in higher dimensions

Abstract

We study the regularity problem of the nonlinear sigma model with gravitino fields in higher dimensions. After setting up the geometric model, we derive the Euler-Lagrange equations and consider the regularity of weak solutions defined in suitable Sobolev spaces. We show that any weak solution is actually smooth under some smallness assumption for certain Morrey norms. By assuming some higher integrability of the vector spinor, we can show a partial regularity result for stationary solutions, provided the gravitino is critical, which means that the corresponding supercurrent vanishes. Moreover, in dimension less than 6, partial regularity holds for stationary solutions with respect to general gravitino fields.

Dirac-harmonic maps with potential

“…In this section, we investigate the regularity of weak solution of the system ( 2.1 ), ( 2.2 ). Our analysis is very similar to the regularity analysis of the nonlinear sigma model coupled to a gravitino which was carried out in [ 31 , Section 2] and to the one for Dirac-harmonic maps with curvature term [ 5 , Section 4.1] for and [ 29 ] for the higher-dimensional case. The methods used to control the gravitino of [ 31 ] can be modified such that they also apply to our setup.…”

Dirac-harmonic maps with potential

“…Our analysis is very similar to the regularity analysis of the nonlinear sigma model coupled to a gravitino which was carried out in [27, Section 2] and to the one for Dirac-harmonic maps with curvature term [5, Section 4.1] for dim = 2 and [25] for the higher-dimensional case. The methods used to control the gravitino of [27] can be modified such that they also apply to our setup.…”