Virasoro conjecture for Hodge integrals

Let $X$ be a smooth projective variety, let $F_g^{\mathbb{E}}(\mathbf{t},\mathbf{s})$ be the genus-$g$ generating function of Hodge integrals over moduli spaces of stable maps to $X$, and set

$$\mathcal{D}^{\mathbb{E}}(\mathbf{t},\mathbf{s})=\exp\left(\sum_{g\geq0}\hbar^{2g-2}F_g^{\mathbb{E}}(\mathbf{t},\mathbf{s})\right).$$

For $n\geq-1$, let $L_n^{\mathbb{E}}$ be the differential operators defined in the source, with the displayed difference operators and elementary symmetric functions, satisfying $[L_n^{\mathbb{E}},L_m^{\mathbb{E}}]=(n-m)L_{n+m}^{\mathbb{E}}$.

Virasoro conjecture for Hodge integrals. For any smooth projective variety,

$$L_n^{\mathbb{E}}\mathcal{D}^{\mathbb{E}}(\mathbf{t},\mathbf{s})=0,\qquad n\geq-1.$$

This extends the Virasoro-constraint prediction from descendant Gromov–Witten invariants to Hodge integrals for arbitrary smooth projective targets. The source proves genus-zero constraints for all targets and a genus-one $L_1^{\mathbb{E}}$ constraint with one Hodge-character insertion, while the all-genus assertion remains open.