Holomorphic spectrum of twisted Dirac operators on compact Riemann surfaces

Abstract

Given a Hermitian line bundle L with a harmonic connection over a compact Riemann surface ( S , g ) of constant curvature, we study the spectral geometry of the corresponding twisted Dirac operator D . This problem is analyzed in terms of the natural holomorphic structures of the spinor bundles E ± defined by the Cauchy–Riemann operators associated with the spinorial connection. By means of two elliptic chains of line bundles obtained by twisting E ± with the powers of the canonical bundle K S , we prove that there exists a certain subset Spec hol ( D ) of the spectrum such that the eigensections associated with λ ∈ Spec hol ( D ) are determined by the holomorphic sections of a certain line bundle of the elliptic chain. We give explicit expressions for the holomorphic spectrum and the multiplicities of the corresponding eigenvalues according to the genus p of S , showing that Spec hol ( D ) does not depend on the spin structure and depends on the line bundle L only through its degree. This technique provides the whole spectrum of D for genus p = 0 and 1, whereas for genus p > 1 we obtain a finite number of eigenvalues.