Pandharipande–Solomon–Tessler open Virasoro constraint conjecture

Let $F^c$ and $F^o$ be the closed and open generating series, and define

$$Z=\exp(F^c+F^o).$$

For $n\geqslant -1$, let $chat{\mathscr L}_n$ be the $s$-extended Virasoro operator

$$\hat{\mathscr L}_n=\hat L_n+u^ns\frac{\partial^{n+1}}{\partial s^{n+1}}+\frac{3n+3}{4}u^n\frac{\partial^n}{\partial s^n}.$$

Pandharipande–Solomon–Tessler's open Virasoro constraint conjecture. The partition function satisfies

$$\frac{(2n+3)!!}{2^{n+1}}\partial_{n+1}Z=\hat{\mathscr L}_nZ, \qquad n\geqslant -1.$$

This is the explicit Virasoro part of the proposed open extension of the Witten–Kontsevich theory; it remains conjectural because the full open descendent theory was not constructed.