Effective gravitational action for 2D massive Majorana fermions on arbitrary genus Riemann surfaces

We explore the effective gravitational action for two-dimensional massive Euclidean Majorana fermions in a small mass expansion, continuing and completing the study initiated in a previous paper [1]. We perform a detailed analysis of local zeta functions, heat kernels, and Green's functions of the Dirac operator on arbitrary Riemann surfaces. We obtain the full expansion of the effective gravitational action to all orders in m2. For genus one and larger, this requires the understanding of the role of the zero-modes of the (massless) Dirac operator which is worked out.Besides the Liouville action, at order m0, which only involves the background metric and the conformal factor sigma, the various contributions to the effective gravitational action at higher orders in m2 can be expressed in terms of integrals of the renormalized Green's function at coinciding points of the squared (massless) Dirac operator, as well as of higher Green's functions. In particular, at order m2, these contributions can be re-written as a term integral e2 sigma sigma characteristic of the Mabuchi action, much as for 2D massive scalars, as well as several other terms that are multi-local in the conformal factor sigma and involve the Green's functions of the massless Dirac operator and the renormalized Green's function, but for the background metric only, and certain area-like parameters related to the zero-modes.