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.