$x-y$ duality in Topological Recursion for exponential variables via Quantum Dilogarithm
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arXiv
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| Natura: | Preprint |
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2023
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| _version_ | 1866929488775872512 |
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| author | Hock, Alexander |
| author_facet | Hock, Alexander |
| contents | For a given spectral curve, the theory of topological recursion generates two different families $ω_{g,n}$ and $ω_{g,n}^\vee$ of multi-differentials, which are for algebraic spectral curves related via the universal $x-y$ duality formula. We propose a formalism to extend the validity of the $x-y$ duality formula of topological recursion from algebraic curves to spectral curves with exponential variables of the form $e^x=F(e^y)$ or $e^x=F(y)e^{a y}$ with $F$ rational and $a$ some complex number, which was in principle already observed in \cite{Dunin-Barkowski:2017zsd,Bychkov:2020yzy}. From topological recursion perspective the family $ω_{g,n}^\vee$ would be trivial for these curves. However, we propose changing the $n=1$ sector of $ω_{g,n}^\vee$ via a version of the Faddeev's quantum dilogarithm which will lead to the correct two families $ω_{g,n}$ and $ω_{g,n}^\vee$ related by the same $x-y$ duality formula as for algebraic curves. As a consequence, the $x-y$ symplectic transformation formula extends further to important examples governed by topological recursion including, for instance, the topological vertex curve which computes Gromov-Witten invariants of $\mathbb{C}^3$, equivalently triple Hodge integrals on the moduli space of complex curves, orbifold Hurwitz numbers, or stationary Gromov-Witten invariants of $\mathbb{P}^1$. The proposed formalism is related to the issue topological recursion encounters for specific choices of framings for the topological vertex curve. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_11761 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | $x-y$ duality in Topological Recursion for exponential variables via Quantum Dilogarithm Hock, Alexander Mathematical Physics High Energy Physics - Theory Algebraic Geometry Exactly Solvable and Integrable Systems 05A15, 14N10, 14H70, 30F30 For a given spectral curve, the theory of topological recursion generates two different families $ω_{g,n}$ and $ω_{g,n}^\vee$ of multi-differentials, which are for algebraic spectral curves related via the universal $x-y$ duality formula. We propose a formalism to extend the validity of the $x-y$ duality formula of topological recursion from algebraic curves to spectral curves with exponential variables of the form $e^x=F(e^y)$ or $e^x=F(y)e^{a y}$ with $F$ rational and $a$ some complex number, which was in principle already observed in \cite{Dunin-Barkowski:2017zsd,Bychkov:2020yzy}. From topological recursion perspective the family $ω_{g,n}^\vee$ would be trivial for these curves. However, we propose changing the $n=1$ sector of $ω_{g,n}^\vee$ via a version of the Faddeev's quantum dilogarithm which will lead to the correct two families $ω_{g,n}$ and $ω_{g,n}^\vee$ related by the same $x-y$ duality formula as for algebraic curves. As a consequence, the $x-y$ symplectic transformation formula extends further to important examples governed by topological recursion including, for instance, the topological vertex curve which computes Gromov-Witten invariants of $\mathbb{C}^3$, equivalently triple Hodge integrals on the moduli space of complex curves, orbifold Hurwitz numbers, or stationary Gromov-Witten invariants of $\mathbb{P}^1$. The proposed formalism is related to the issue topological recursion encounters for specific choices of framings for the topological vertex curve. |
| title | $x-y$ duality in Topological Recursion for exponential variables via Quantum Dilogarithm |
| topic | Mathematical Physics High Energy Physics - Theory Algebraic Geometry Exactly Solvable and Integrable Systems 05A15, 14N10, 14H70, 30F30 |
| url | https://arxiv.org/abs/2311.11761 |