Mariño–Vafa formula in the one-leg specialization

Let ${\mathcal P}_+$ denote the set of nonempty partitions, let $\mu^+$, $\mu^-$, $\nu^+$, and $\nu^-$ be partitions, and let $G_{\mu^+,\mu^-}(\tau)$ be the specialization defined by $\tau=y/x$. Let $\chi_\nu(\mu)$, $z_\mu$, $\kappa_\nu$, ${\mathcal W}_{\nu^+,\nu^-}$, and $p^{\pm}_\mu$ have their standard meanings. Mariño–Vafa formula. The identity

$$\exp\left(\sum_{(\mu^+,\mu^- )\in {\mathcal P}_+^2}G_{\mu^+,\mu^-}(\tau)p^+_{\mu^+}p^-_{\mu^-}\right) = \sum_{|\mu^{\pm}|=|\nu^{\pm}|\geq 0}\frac{\chi_{\nu^+}(\mu^+)}{z_{\mu^+}}\frac{\chi_{\nu^-}(\mu^-)}{z_{\mu^-}}e^{\sqrt{-1}(\kappa_{\nu^+}\tau+\frac{\kappa_{\nu^-}}{\tau})\lambda/2}{\mathcal W}_{\nu^+,\nu^-}p^+_{\mu^+}p^-_{\mu^-}$$

holds. This is presented as the main subject of the paper and is the specialized form of the Hodge-integral identity relating the geometric generating functions to representation theory. The supplied source does not provide an independent status assessment beyond presenting the identity as the main result.