A note on twisted Dirac operators on closed surfaces.

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. [ABSTRACT FROM AUTHOR]