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Register a new userElements of spin Hurwitz theory: closed algebraic formulas, blobbed topological recursion, and a proof of the Giacchetto-Kramer-Lewanski conjecture
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Added by
Alexander Alexandrov
Publication date
2021
fields
Physics
and research's language is
English
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In this paper, we discuss the properties of the generating functions of spin Hurwitz numbers. In particular, for spin Hurwitz numbers with arbitrary ramification profiles, we construct the weighed sums which are given by Orlovs hypergeometric solutions of the 2-component BKP hierarchy. We derive the closed algebraic formulas for the correlation functions associated with these tau-functions, and under reasonable analytical assumptions we prove the loop equations (the blobbed topological recursion). Finally, we prove a version of topological recursion for the spin Hurwitz numbers with the spin completed cycles (a generalized version of the Giacchetto--Kramer--Lewanski conjecture).
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We introduce a new matrix model representation for the generating function of simple Hurwitz numbers. We calculate the spectral curve of the model and the associated symplectic invariants developed in [Eynard-Orantin]. As an application, we prove the conjecture proposed by Bouchard and Marino, relating Hurwitz numbers to the spectral invariants of the Lambert curve exp(x)=y exp(-y).
The purpose of this article is to analyze the connection between Eynard-Orantin topological recursion and formal WKB solutions of a $hbar$-difference equation: $Psi(x+hbar)=left(e^{hbarfrac{d}{dx}}right) Psi(x)=L(x;hbar)Psi(x)$ with $L(x;hbar)in GL_2( (mathbb{C}(x))[hbar])$. In particular, we extend the notion of determinantal formulas and topological type property proposed for formal WKB solutions of $hbar$-differential systems to this setting. We apply our results to a specific $hbar$-difference system associated to the quantum curve of the Gromov-Witten invariants of $mathbb{P}^1$ for which we are able to prove that the correlation functions are reconstructed from the Eynard-Orantin differentials computed from the topological recursion applied to the spectral curve $y=cosh^{-1}frac{x}{2}$. Finally, identifying the large $x$ expansion of the correlation functions, proves a recent conjecture made by B. Dubrovin and D. Yang regarding a new generating series for Gromov-Witten invariants of $mathbb{P}^1$.
In this article, a novel description of the hypergeometric differential equation found from Gelfand-Kapranov-Zelevinskys system (referred to GKZ equation) for Giventals $J$-function in the Gromov-Witten theory will be proposed. The GKZ equation involves a parameter $hbar$, and we will reconstruct it as the WKB expansion from the classical limit $hbarto 0$ via the topological recursion. In this analysis, the spectral curve (referred to GKZ curve) plays a central role, and it can be defined as the critical point set of the mirror Landau-Ginzburg potential. Our novel description is derived via the duality relations of the string theories, and various physical interpretations suggest that the GKZ equation is identified with the quantum curve for the brane partition function in the cohomological limit. As an application of our novel picture for the GKZ equation, we will discuss the Stokes matrix for the equivariant $mathbb{C}textbf{P}^{1}$ model and the wall-crossing formula for the total Stokes matrix will be examined. And as a byproduct of this analysis we will study Dubrovins conjecture for this equivariant model.
We prove that the topological recursion formalism can be used to quantize any generic classical spectral curve with smooth ramification points and simply ramified away from poles. For this purpose, we build both the associated quantum curve, i.e. the differential operator quantizing the algebraic equation defining the classical spectral curve considered, and a basis of wave functions, that is to say a basis of solutions of the corresponding differential equation. We further build a Lax pair representing the resulting quantum curve and thus present it as a point in an associated space of meromorphic connections on the Riemann sphere, a first step towards isomonodromic deformations. We finally propose two examples: the derivation of a 2-parameter family of formal trans-series solutions to Painleve 2 equation and the quantization of a degree three spectral curve with pole only at infinity.
The BKMP conjecture (2006-2008), proposed a new method to compute closed and open Gromov-Witten invariants for every toric Calabi-Yau 3-folds, through a topological recursion based on mirror symmetry. So far, this conjecture had been verified to low genus for several toric CY3folds, and proved to all genus only for C^3. In this article we prove the general case. Our proof is based on the fact that both sides of the conjecture can be naturally written in terms of combinatorial sums of weighted graphs: on the A-model side this is the localization formula, and on the B-model side the graphs encode the recursive algorithm of the topological recursion. One can slightly reorganize the set of graphs obtained in the B-side, so that it coincides with the one obtained by localization in the A-model.Then it suffices to compare the weights of vertices and edges of graphs on each side, which is done in 2 steps: the weights coincide in the large radius limit, due to the fact that the toric graph is the tropical limit of the mirror curve. Then the derivatives with respect to Kahler radius coincide due to special geometry property implied by the topological recursion.
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