Linear hyperelliptic Hodge integrals

We provide a closed form expression for linear Hodge integrals on the hyperelliptic locus. Specifically, we find a succinct combinatorial formula for all intersection numbers on the hyperelliptic locus with one \(\lambda \)-class, and powers of a \(\psi \)-class pulled back along the branch map. This is achieved by using Atiyah–Bott localization on a stack of stable maps into the orbifold \(\left[ {\mathbb {P}}^1/{\mathbb {Z}}_2\right] \).