Partial Regularity for a Nonlinear Sigma Model with Gravitino in Higher Dimensions

TL;DR

This paper investigates the regularity of solutions in a nonlinear sigma model with gravitino fields in higher dimensions, establishing conditions under which solutions are smooth or partially regular.

Contribution

It introduces new regularity results for weak solutions of the model, including partial regularity under stationarity and critical gravitino conditions in various dimensions.

Findings

Weak solutions are smooth under small Morrey norm conditions.

Partial regularity holds for stationary solutions with critical gravitino.

Partial regularity extends to dimensions less than 6 for general gravitino fields.

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.