The KP hierarchy characterization of Calabi–Yau triple Hodge integrals

Let $Z_{q,p}({\bf T})$ be the generating function of triple Hodge integrals

$$Z_{q,p}({\bf T})=e^{\sum_{g=0}^\infty \sum_{n=0}^\infty \hbar^{2g-2+n} {\mathcal F}_{g,n}},$$

with

$${\mathcal F}_{g,n}=\sum_{a_1,\dots,a_n}\frac{\prod T_{a_i}}{n!}\int_{\overline{\mathcal{M}}_{g,n}} \Lambda_g (-q) \Lambda_g (-p) \Lambda_g (\frac{pq}{p+q})\psi_1^{a_1} \psi_2^{a_2} \cdots \psi_n^{a_n}.$$

The parameters satisfy the Calabi–Yau condition, and $\mathbf T$ denotes the Hodge-integral variables. KP characterization conjecture. The case of triple Hodge integrals satisfying the Calabi–Yau condition is the most general case of generating functions of Hodge integrals satisfying the KP hierarchy in the variables ${\bf T}$ after a linear change of variables. A direct analog is also expected for the $\Theta$-Hodge integrals. This conjecture is stated after the proven KP-integrability theorem for Calabi–Yau triple Hodge integrals; the claimed maximality and the $\Theta$-Hodge analog are not resolved in the supplied text.