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. | Vietnam J. Math. (2019) 47:39–67 From Harmonic Maps to the Nonlinear Supersymmetric Sigma Model of Quantum Field Theory: at the Interface of Theoretical Physics, Riemannian Geometry, and Nonlinear Analysis ¨ ¨ Jurgen Jost1 · Enno Keßler1 · Jurgen Tolksdorf1,2 · 1 3 Ruijun Wu · Miaomiao Zhu Received: 1 October 2017 / Accepted: 28 February 2018 / Published online: 28 June 2018 © The Author(s) 2018 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 Dedicated to the memory of Eberhard Zeidler. J¨urgen Jost jjost@ Enno Keßler kessler@ J¨urgen Tolksdorf Ruijun Wu Miaomiao Zhu mizhu@ 1 Max-Planck-Institut f¨ur Mathematik in den Naturwissenschaften, Inselstraße 22, 04103 Leipzig, Germany 2 Institut f¨ur Theoretische Physik, Universit¨at Leipzig, Br¨uderstrasse 16, 04103 Leipzig, Germany 3 School of Mathematical Sciences, Shanghai Jiao Tong University, Dongchuan Road 800, 200240 Shanghai, People’s Republic of China 40 J. Jost
