Degree zero Gromov--Witten invariants for smooth curves

For a smooth projective curve, we derive a closed formula for the generating series of its Gromov--Witten invariants in genus $g$ and degree zero. It is known that the calculation of these invariants can be reduced to that of the $\lambda_g$ and $\lambda_{g-1}$ integrals on the moduli space of stable algebraic curves. The closed formula for the $\lambda_g$ integrals is given by the $\lambda_g$ conjecture, proved by Faber and Pandharipande. We compute in this paper the $\lambda_{g-1}$ integrals via solving the degree zero limit of the loop equation associated to the complex projective line.Comment: v2: it was corrected typos and improved the presentation. 13 pages, no figure