2025/02/25 by Hock, Alexander · 3 citations
#FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2502.18115
Let Fg be the free energy derived from Topological Recursion for a given spectral curve on a compact Riemann surface, and let Fg^\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. Fg 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 Fg - Fg^\vee is nonzero. We derive a new formula for this difference for all g≥ 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 Fg^\vee. In such cases, one obtains an explicit result for Fg≥ 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 Fg computed by Topological Recursion for the "Gaiotto curve" coincide with the perturbative part (in the Ω-background) of the Nekrasov partition function of N=2 pure supersymmetric gauge theory. Similar computations also provide Fg for the CDO curve related to Hurwitz numbers, or the negative r-spin curve related to Θ-class intersection numbers on Mg,n.