Miaomiao Zhu

The boundary value problem for Dirac-harmonic maps

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.

Regularity at the free boundary for Dirac-harmonic maps from surfaces

We establish the regularity theory for certain critical elliptic systems with an anti-symmetric structure under inhomogeneous Neumann and Dirichlet boundary constraints. As applications, we prove full regularity and smooth estimates at the free boundary for weakly Dirac-harmonic maps from spin Riemann surfaces. Our methods also lead to the full interior

Regularity of Solutions of the Nonlinear Sigma Model with Gravitino

\epsilon

Symmetries and conservation laws of a nonlinear sigma model with gravitino

ϵ-regularity and smooth estimates for weakly Dirac-harmonic maps in all dimensions.

A local estimate for super-Liouville equations on closed Riemann surfaces

We construct explicit examples of Dirac-harmonic maps (φ, ψ) between Riemannian manifolds (M, g) and (N, g ′ ) which are non-trivial in the sense that φ is not harmonic. When dim M = 2, we also produce examples where φ is harmonic, but not conformal, and ψ is non-trivial.

From Harmonic Maps to the Nonlinear Supersymmetric Sigma Model of Quantum Field Theory: at the Interface of Theoretical Physics, Riemannian Geometry, and Nonlinear Analysis

We propose a geometric setup to study analytic aspects of a variant of the super symmetric two-dimensional nonlinear sigma model. This functional extends the functional of Dirac-harmonic maps by gravitino fields. The system of Euler–Lagrange equations of the two-dimensional nonlinear sigma model with gravitino is calculated explicitly. The gravitino terms pose additional analytic difficulties to show smoothness of its weak solutions which are overcome using Rivière’s regularity theory and Riesz potential theory.

Harmonic maps from degenerating Riemann surfaces

Let