Symplectic (Non-)Invariance of the Free Energy in Topological Recursion

TL;DR

This paper derives a new residue formula for the difference between dual free energies in topological recursion, revealing cases where the free energy is not symplectically invariant and connecting to various geometric and physical theories.

Contribution

It provides an explicit residue-based formula for the difference of free energies under x-y duality for all g ≥ 2, including cases with singularities and trivial duals.

Findings

Derived a residue formula for F_g - F_g^ ext{dual} for all g ≥ 2.

Proved a conjecture relating topological recursion free energies to Nekrasov partition functions.

Computed free energies for specific spectral curves like Gaiotto and Hurwitz curves.

Abstract

Let $F_g$ be the free energy derived from Topological Recursion for a given spectral curve on a compact Riemann surface, and let $F_g^\vee$ be its $x$-$y$ dual, that is, the free energy derived from the same spectral curve with the roles of $x$ and $y$ interchanged. $F_g$ is sometimes called a symplectic invariant due to its invariance under certain symplectomorphisms of the formal symplectic form $dx\wedge dy$. However, the free energy is not generally invariant under the swap of $x$ and $y$; thus, the difference $F_g - F_g^\vee$ is nonzero. We derive a new formula for this difference for all $g\geq 2$ in terms of a residue calculation at the singularities of $x$ and $y$, including cases where $x$ and $y$ have logarithmic singularities. For the derivation, we apply recent developments from $x$-$y$ duality within the theory of (Logarithmic) Topological Recursion. The derived formulas are particularly useful for spectral curves with a trivial $x$-$y$ dual side, meaning those with vanishing $F_g^\vee$. In such cases, one obtains an explicit result for $F_{g\geq 2}$ itself. We apply this to several classes of spectral curves and prove, for instance, a recent conjecture by Borot et al. that the free energies $F_g$ computed by Topological Recursion for the "Gaiotto curve" coincide with the perturbative part (in the $\Omega$-background) of the Nekrasov partition function of $\mathcal{N}=2$ pure supersymmetric gauge theory. Similar computations also provide $F_g$ for the CDO curve related to Hurwitz numbers, or the negative $r$-spin curve related to $\Theta$-class intersection numbers on $\overline{\mathcal{M}}_{g,n}$.