Mariño–Vafa formula for two-part Hodge integrals

Let ${\mathcal P}$ be the set of partitions, let $\mu^+$ and $\mu^-$ be partitions, and let $G_{\mu^+,\mu^-}(x,y)$, $p^+_{\mu^+}$, and $p^-_{\mu^-}$ denote the corresponding generating functions and power-sum variables. For a partition $\nu$, write $\chi_\nu(\mu)$ for the irreducible character of the symmetric group indexed by $\nu$ evaluated on the conjugacy class of type $\mu$, $z_\mu$ for the standard centralizer factor, $\kappa_\nu$ for the partition statistic, and ${\mathcal W}_{\nu^+,\nu^-}$ for the associated two-part representation-theoretic quantity. Mariño–Vafa conjecture. The identity

$$\exp\left(\sum_{(\mu^+,\mu^-)\in {\mathcal P}^2}G_{\mu^+,\mu^-}(x,y)p^+_{\mu^+}p^-_{\mu^-}\right) = \sum_{|\mu^{\pm}|=|\nu^{\pm}|}\frac{\chi_{\nu^+}(\mu^+)}{z_{\mu^+}}\frac{\chi_{\nu^-}(\mu^-)}{z_{\mu^-}}e^{\sqrt{-1}(\kappa_{\nu^+}\frac{y}{x}+\kappa_{\nu^-}\frac{x}{y})\lambda/2}{\mathcal W}_{\nu^+,\nu^-}p^+_{\mu^+}p^-_{\mu^-}$$

holds. The formula relates Hodge-integral generating functions to Wess–Zumino–Witten theory and representation theory. The source states that the formula is completed by a proof using cut-and-join equations and initial values, so the conjectural identity is solved.