Relations for quadratic Hodge integrals via stable maps
Georgios Politopoulos
TL;DR
The paper applies virtual localization on the moduli space of stable maps to $\mathbb{P}^1$ to derive relations among double Hodge integrals. It defines a generating series $P_a(α,t)$ of double Hodge integrals and proves these series are monic polynomials of total degree $|a|$ in the variables $t$ and $α$, with explicit initial values, and discusses a conjecture that the degree is $|a|$ in each variable. The approach uses the localization formula to express GW-invariants in terms of products of Hodge and psi-class integrals and leverages the String and Dilaton equations to obtain structural relations. A key result is that the top coefficient of $P_a$ is 1, and the framework connects Faber–Pandharipande’s localization techniques with the GW theory of $\mathbb{P}^1$ to produce polynomial relations among double Hodge integrals. This provides a concrete, computable mechanism for understanding tautological relations via generating functions.
Abstract
Following Faber-Pandharipande, we use the virtual localization formula for the moduli space of stable maps to $ \mathbb{P}^1 $ to compute relations between Hodge integrals. We prove that certain generating series of these integrals are polynomials.