Nonlinear Dirac equations, Monotonicity Formulas and Liouville Theorems
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.
Key words and phrases:nonlinear Dirac equations; monotonicity formulas; Liouville Theorems; Dirac-harmonic maps with curvature term
2010 Mathematics Subject Classification:53C27, 35J61
1. Introduction and results
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. A Dirac equation with vanishing right hand side describes a free massless fermion and linear Dirac equations describe free fermions having a mass. However, to model the interaction of fermions one has to take into account nonlinearities.
In mathematical terms spinors are sections in a vector bundle, the spinor bundle, which is defined on a Riemannian spin manifold. The spin condition is of topological nature and ensures the existence of the spinor bundle . The mathematical analysis of linear and nonlinear Dirac equations comes with two kinds of difficulties: First of all, the Dirac operator is of first order, such that tools like the maximum principle are not available. Secondly, in contrast to the Laplacian, the Dirac operator has its spectrum on the whole real line.
Below we give a list of energy functionals that arise in quantum field theory. Their critical points all lead to nonlinear Dirac equations. To this end let be the classical Dirac operator on a Riemannian spin manifold of dimension and an orthonormal basis of . Furthermore, let be the Clifford multiplication of spinors with tangent vectors and the complex volume form. Moreover, we fix a hermitian scalar product on the spinor bundle.
The Soler model [PhysRevD.1.2766] describes fermions that interact by a quartic term in the action functional. In quantum field theory this model is usually studied on four-dimensional Minkowski space:
The Thirring model [Thirring] describes the self-interaction of fermions in two-dimensional Minkowski space:
The Nambu–Jona-Lasinio model [1961PhRv..122..345N] is a model for interacting fermions with chiral symmetry. It also contains a quartic interaction term and is defined on an even-dimensional spacetime:
Note that this model does not have a term proportional to in the energy functional.
The Gross–Neveu model with N flavors [1974PhRvD..10.3235G] is a model for interacting fermions in two-dimensional Minkowski space:
The spinors that we are considering here are twisted spinors, more precisely .
The nonlinear supersymmetric sigma model consists of a map between two Riemannian manifolds and and a vector spinor . Moreover, is the curvature tensor on and denotes the Dirac operator acting on vector spinors. The energy functional under consideration is
The critical points of this functional became known in the mathematical literature as Dirac-harmonic maps with curvature term.
In the models (1)-(4) from above the real parameter can be interpreted as mass, whereas the real constant describes the strength of interaction. All of the models listed above lead to nonlinear Dirac equations of the form
Note that in the physical literature Clifford multiplication is usually expressed as matrix multiplication with and the complex volume element is referred to as . In contrast to the physical literature we will always assume that spinors are commuting, whereas in the physical literature they are mostly assumed to be Grassmann-valued. For simplicity we will mainly focus on the Soler model.
Several existence results for equations of the form (1.1) are available: In [MR2813439] existence results for nonlinear Dirac equations on compact spin manifolds are obtained. For existence results for nonlinear Dirac equation with critical exponent on compact spin manifolds, that is
with , have been obtained in [MR2733578]. For this equation is known as the spinorial Yamabe equation. In particular, this equation is interesting for since it is closely related to conformally immersed constant mean curvature surfaces in . Moreover, existence results for the spinorial Yamabe equation have been obtained on [MR3299382] and on [MR3037015] for . For a spectral and geometric analysis of the spinorial Yamabe equations we refer to [MR2550205]. The regularity of weak solutions of equations of the form (1.1) can be established with the tools from [MR2661574] and [MR2733578], Appendix A.
This article is organized as follows: In Section 2 we study general properties of nonlinear Dirac equations. In particular, we recall the construction for identifying spinor bundles belonging to different metrics and use this to derive the stress-energy tensor for the Soler model. In Section 3 we study nonlinear Dirac equations on closed manifolds, where we mostly focus on the two-dimensional case. In the fourth section we study nonlinear Dirac equations on complete manifolds: We derive several monotonicity formulas and Liouville theorems. In the last section we focus on Dirac-harmonic maps with curvature term from complete manifolds, for which we again derive several monotonicity formulas and Liouville theorems.
2. Nonlinear Dirac equations on Riemannian manifolds
Let be a Riemannian spin manifold of dimension . A Riemannian manifold admits a spin structure if the second Stiefel-Whitney class of its tangent bundle vanishes.
We briefly recall the basic notions from spin geometry, for a detailed introduction to spin geometry we refer to the book [MR1031992].
We fix a spin structure on the manifold and consider the spinor bundle . On the spinor bundle we have the Clifford multiplication of spinors with tangent vectors denoted by . Moreover, we fix a hermitian scalar product on the spinor bundle and denote its real part by . Clifford multiplication is skew-symmetric
for all and . Moreover the Clifford relations
hold for all . The Dirac operator is defined as the composition of first applying the covariant derivative on the spinor bundle followed by Clifford multiplication. More precisely, it is given by
where is an orthonormal basis of . Sometimes we will make use of the Einstein summation convention and just sum over repeated indices. The Dirac operator is of first order, elliptic and self-adjoint with respect to the -norm. Hence, if is compact the Dirac operator has a real and discrete spectrum.
The square of the Dirac operator satisfies the Schroedinger-Lichnerowicz formula
where denotes the scalar curvature of the manifold .
After having recalled the basic definition from spin geometry will focus on the analysis of the following action functional (which is the first one from the introduction)
Its critical points are given by
It turns out that is the right function space for weak solutions of (2.4).
We call a weak solution if it solves (2.4) in a distributional sense.
The analytic structure of the other energy functionals listed in the introduction is the same as the one of (2.3). Due to this reason many of the results that will be obtained for solutions of (2.4) can easily be generalized to critical points of the other models.
The equation (2.4) is also interesting from a geometric point of view since it interpolates between eigenspinors () and a non-linear Dirac equation () that arises in the study of CMC immersion from surfaces into .
In the following we want to vary the energy functional (2.3) with respect to the metric . To this end let us recall the following construction for identifying spinor bundles belonging to different metrics. For the Riemannian case this was established in [MR1158762] and later on generalized to the pseudo-Riemannian case in [MR2121740]. Here, we will follow the presentation from [MR1738150], Chapter 2.
Suppose we have two spinor bundles and corresponding to different metrics and . There exists a unique positive definite tensor field uniquely determined by the requirement , where . Let and be the oriented orthonormal frame bundles of and . Then induces an equivariant isomorphism via the assignment . We fix a spin structure of and think of it as a -bundle. The pull-back of via the isomorphism induces a -bundle . Moreover, we get a Spin-equivariant isomorphism such that the following diagram commutes: