  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.
