Universal $x$-$y$ duality in equivariant topological recursion and quartic Kontsevich model
Topological recursion is a universal procedure that generates an infinite family of multi-differentials with different interpretations in various areas of mathematics. Recently, a new understanding of topological recursion has emerged from $x$-$y$ duality, which relates two such families through explicit formulae. We study $x$-$y$ duality, its parametrised version, and symplectic duality in the setting of equivariant topological recursion. We explicitly analyse the pole structure of the resulting dual families and provide recursive formulae for the different types of polar parts. In contrast to classical topological recursion, the dual family has additional poles at fixed points of the group action, and at anti-diagonals determined by the group action. We show that the partition function $Z_P(E)=\int_{H_N} e^{-N\,\mathrm{Tr}(EΦ^2+P(Φ))}dΦ$ of the Kontsevich matrix model with arbitrary potential $P$, where $dΦ$ is the Lebesgue measure on Hermitian $N\times N$ matrices and $E$ a positive matrix, admits a Fock space realisation related to the character expansion of the IZHC integral. The Fock space framework provides two mutually $X$-$Y$ dual families of multi-differentials relative to an implicitly defined spectral curve $(X,Y)$. The solution $Y(X)$ succeeds for the quartic Kontsevich model with potential $P(Φ)=\fracλ{4}Φ^4$. We identify a network of duality transformations and symplectic shifts which relate the quartic Kontsevich model with the $\mathbb{Z}/2\mathbb{Z}$-equivariant spectral curve. The proposed construction yields a conjectural all-genus solution to the quartic Kontsevich model, which reproduces recursive structures previously established for genus $g\leq 1$.