Abstract
We study the qualitative behavior of nonlinear Dirac equations arising in quantum field theory on complete Riemannian manifolds. In particular, we derive monotonicity formulas and Liouville theorems for solutions of these equations. Finally, we extend our analysis to Dirac-harmonic maps with curvature term.
Highlights
Introduction and ResultsIn quantum field theory spinors are employed to model fermions
We study the qualitative behavior of nonlinear Dirac equations arising in quantum field theory on complete Riemannian manifolds
The equations that govern the behavior of fermions are both linear and nonlinear Dirac equations
Summary
In quantum field theory spinors are employed to model fermions. The equations that govern the behavior of fermions are both linear and nonlinear Dirac equations. We give a list of action functionals that arise in quantum field theory Their critical points all lead to nonlinear Dirac equations. (5) The nonlinear supersymmetric sigma model in quantum field theory consists of a map φ between two Riemannian manifolds M and N and a spinor along that map. In Theorem 4.13 we show that critical points of the Soler model on a complete noncompact Riemannian manifold with positive Ricci curvature satisfying an additional energy condition must be trivial. 5 we focus on Dirac-harmonic maps with curvature term from complete manifolds The latter consist of a pair of a map between two Riemannian manifolds and a vector spinor defined along that map. We show that Dirac-harmonic maps with curvature term from complete Riemannian manifolds with positive Ricci curvature to target manifolds with negative sectional curvature must be trivial if a certain energy is finite and a certain inequality relating Ricci curvature and energy holds (Theorem 5.18)
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