  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. 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 $m^2$. 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 $m^0$, which only involves the
background metric and the conformal factor $\sigma$, the various contributions
to the effective gravitational action at higher orders in $m^2$ 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 $m^2$, these contributions
can be re-written as a term $\int e^{2\sigma}\, \sigma$ 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.
