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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2502.18115 |
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| _version_ | 1866908556160139264 |
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| author | Hock, Alexander |
| author_facet | Hock, Alexander |
| contents | 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 $Ω$-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 $Θ$-class intersection numbers on $\overline{\mathcal{M}}_{g,n}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_18115 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Symplectic (Non-)Invariance of the Free Energy in Topological Recursion Hock, Alexander Mathematical Physics High Energy Physics - Theory 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 $Ω$-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 $Θ$-class intersection numbers on $\overline{\mathcal{M}}_{g,n}$. |
| title | Symplectic (Non-)Invariance of the Free Energy in Topological Recursion |
| topic | Mathematical Physics High Energy Physics - Theory |
| url | https://arxiv.org/abs/2502.18115 |