Holomorphic spectrum of twisted Dirac operators on compact Riemann surfaces

Given a Hermitian line bundle LL with a harmonic connection over a compact Riemann surface (S,g)(S,g) of constant curvature, we study the spectral geometry of the corresponding twisted Dirac operator DD. This problem is analyzed in terms of the natural holomorphic structures of the spinor bundles E±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±E± with the powers of the canonical bundle KSKS, we prove that there exists a certain subset Spechol(D) of the spectrum such that the eigensections associated with λ∈Spechol(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 pp of SS, showing that Spechol(D) does not depend on the spin structure and depends on the line bundle LL only through its degree. This technique provides the whole spectrum of DD for genus p=0p=0 and 1, whereas for genus p>1p>1 we obtain a finite number of eigenvalues.