From harmonic maps to the nonlinear supersymmetric sigma model of quantum field theory. At the interface of theoretical physics, Riemannian geometry and nonlinear analysis

TL;DR

This paper explores the deep connections between harmonic maps, quantum field theory, and Riemann surface geometry, analyzing both mathematical and supersymmetric physical models involving spinor fields and gravitinos.

Contribution

It introduces a comprehensive analysis of the nonlinear supersymmetric sigma model, bridging mathematical and physical perspectives, including commuting and anticommuting spinor fields and the role of gravitinos.

Findings

Analysis of mathematical and supersymmetric models

Insights into singularity formation in harmonic maps

Integration of quantum field theory with geometric 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 $\sigma$-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.