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

Abstract

Harmonic maps from Riemann surfaces arise from a conformally invariant variational problem. Therefore, on one hand, they are intimately connected with moduli spaces of Riemann surfaces, and on the other hand, because the conformal group is noncompact, constitute a prototype for the formation of singularities, the so-called bubbles, in geometric analysis. In theoretical physics, they arise from the nonlinear σ -model of quantum field theory. That model possesses a supersymmetric extension, coupling a harmonic map like field with a nonlinear spinor field. In the physical model, that spinor field is anticommuting. In this contribution, we analyze both a mathematical version with a commuting spinor field and the original supersymmetric version. Moreover, this model gives rise to a further field, a gravitino, that can be seen as the supersymmetric partner of a Riemann surface metric. Altogether, this leads to a beautiful combination of concepts from quantum field theory, structures from Riemannian geometry and Riemann surface theory, and methods of nonlinear geometric analysis.

Dirac-harmonic maps with potential

“…For the current state of research on the mathematical aspects of the supersymmetric nonlinear sigma model, we refer to the recent survey article [ 27 ].…”

Dirac-harmonic maps with potential

“…These provide criteria under which a Dirac-harmonic map must be trivial, that is the map part maps to a point and the spinor vanishes identically. For the current state of research on the mathematical aspects of the supersymmetric nonlinear sigma model we refer to the recent survey article [23]. In the physics literature there exists a version of the supersymmetric nonlinear sigma model coupled to a scalar potential.…”

A vanishing result for the supersymmetric nonlinear sigma model in higher dimensions

“…Recently, there has been an intensive study of the supersymmetric nonlinear σ-model coupled to a gravitino [16] which is the superpartner of the metric on the domain. For a recent overview on the mathematical analysis of the supersymmetric nonlinear σ-model see the survey [15]. The action functional of the supersymmetric sigma model is special if the domain manifold is two-dimensional since in this case the action is invariant under conformal transformations and its critical points share special properties such as the removal of isolated singularities.…”