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 \(g\ge 2\) 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 \(F_{g\ge 2}\) 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 \(\overline{\mathcal {M}}_{g,n}\) .
