TY - JOUR
T1 - From Harmonic Maps to the Nonlinear Supersymmetric Sigma Model of Quantum Field Theory
T2 - at the Interface of Theoretical Physics, Riemannian Geometry, and Nonlinear Analysis
AU - Jost, Jürgen
AU - Keßler, Enno
AU - Tolksdorf, Jürgen
AU - Wu, Ruijun
AU - Zhu, Miaomiao
N1 - Publisher Copyright:
© 2018, The Author(s).
PY - 2019/3/15
Y1 - 2019/3/15
N2 - 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.
AB - 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.
KW - Dirac-harmonic maps
KW - Gravitino
KW - Harmonic maps
KW - Non-linear sigma model
KW - Supersymmetry
UR - http://www.scopus.com/inward/record.url?scp=85061656875&partnerID=8YFLogxK
U2 - 10.1007/s10013-018-0298-7
DO - 10.1007/s10013-018-0298-7
M3 - Article
AN - SCOPUS:85061656875
SN - 2305-221X
VL - 47
SP - 39
EP - 67
JO - Vietnam Journal of Mathematics
JF - Vietnam Journal of Mathematics
IS - 1
ER -