A note on twisted Dirac operators on closed surfaces

Abstract

We derive an inequality that relates nodal set and eigenvalues of a class of twisted Dirac operators on closed surfaces and point out how this inequality naturally arises as an eigenvalue estimate for the Spin c Dirac operator. This allows us to obtain eigenvalue estimates for the twisted Dirac operator appearing in the context of Dirac-harmonic maps and their extensions, from which we also obtain several Liouville type results.

Stationary solutions for the 2D critical Dirac equation with Kerr nonlinearity

“…It is thus desirable to have good estimates about the nodal sets of such spinor solutions, especially in geometric applications, since nodal sets typically contain important geometric informations about the underlying manifold and on the solutions themselves. The analysis of the nodal sets in concrete backgrounds is carried out, for instance, in [2,13,16,17,45] and the only general results known to us was due to Christian Bär [6,7] which deals with smooth right hand side, also considering more general first-order elliptic operators. He proves sharp estimates for the Hausdorff dimension of the nodal set, and provides an upper bound for the Hausdorff densities.…”

Dirac-harmonic maps with potential

“…Motivated from various variants in the physics literature, several extensions of Dirac-harmonic maps have also been studied from a mathematical point of view: Taking into account a two-form contribution in the action functional one is led to magnetic Dirac-harmonic maps [ 4 ], Dirac-harmonic maps to target spaces with torsion are investigated in [ 6 ]. Adding a curvature term to the action functional, which is quartic in the spinors, one obtains Dirac-harmonic maps with curvature term , which have been studied extensively in [ 5 , 7 , 8 , 11 , 14 , 16 , 29 ]. Recently, another extension of Dirac-harmonic maps receives growing interest: Here, one considers an additional field in the action functional, the so-called gravitino [ 26 ].…”

“…Besides the aforementioned existence results, several Liouville-type results have also been established [ 11 , 12 , 14 , 16 ]. These provide criteria under which a Dirac-harmonic map must be trivial, that is the map part maps to a point and the spinor vanishes identically.…”