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

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 $g\ge 2$ 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 ${\overline{\mathcal{M}}}_{g,n}$.

31 pages, two more examples included

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• GW/DT invariants and 5D BPS indices for strips from topological recursion
• Whittaker vectors at finite energy scale, topological recursion and Hurwitz numbers
• Quantum Curve for strip geometries, Topological Recursion and open GW/DT invariants