Dubrovin’s Superpotential as a Global Spectral curve

We rewrite the (extended) Ooguri-Vafa partition function for colored HOMFLY-PT polynomials for torus knots in terms of the free-fermion (semi-infinite wedge) formalism, making it very similar to the generating function for double Hurwitz numbers. This allows us to conjecture the combinatorial meaning of full expansion of the correlation differentials obtained via the topological recursion on the Brini-Eynard-Mariño spectral curve for the colored HOMFLY-PT polynomials of torus knots.This correspondence suggests a structural combinatorial result for the extended Ooguri-Vafa partition function. Namely, its coefficients should have a quasi-polynomial behavior, where non-polynomial factors are given by the Jacobi polynomials. We prove this quasi-polynomiality in a purely combinatorial way. In addition to that, we show that the (0,1)-and (0,2)-functions on the corresponding spectral curve are in agreement with the extension of the colored HOMFLY-PT polynomials data.

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“…Definition 3.4. The topological Ooguri-Vafa partition function for the torus knot T rQ, P s is defined by (15) Zppq :" Zpp˚,pq…”

Section: Rosso-jones Formula As Free-fermion Averagementioning

confidence: 99%

“…All of them are proved to be connected to ξ-function expansions. Typically we have r ě 1 critical points of x, and there is a natural basis of r ξ-functions, called flat basis due to its connection to flat coordinates in the theory of Frobenius manifold [15,16]. We have [28,27]:…”

mentioning

confidence: 99%

Combinatorial structure of colored HOMFLY-PT polynomials for torus knots

Dunin-Barkowski

Popolitov

Shadrin

et al. 2019

Communications in Number Theory and Physics

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“…The construction of [13] begins with Givental's decomposition of the partition function of (the correlators of) a cohomological field theory (with flat identity) and produces a spectral curve satisfying the restriction that y = Y α for each α = The statement of this theorem is the reverse of the result in [13], but it is easily seen to be reversible on spectral curves satisfying the condition on y, [12]. The translation due to local expansions of y is implicit in the statement in [13] and appears inside an operator 1,β (which they denote byR) where the vector 1 1 = {dy(P (α)}.…”

Section: Remark 45mentioning

confidence: 99%

Topological recursion with hard edges

Chekhov

Norbury

2019

Int. J. Math.

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We prove a Givental type decomposition for partition functions that arise out of topological recursion applied to spectral curves. Copies of the Konstevich-Witten KdV tau function arise out of regular spectral curves and copies of the Brezin-Gross-Witten KdV tau function arise out of irregular spectral curves. We present the example of this decomposition for the matrix model with two hard edges and spectral curve (x 2 − 4)y 2 = 1. CONTENTS

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“…The large-N behavior of these monomial matrix models is interesting for the following reason. If one computes the standard spectral curve (forgetting for now that pure phase correlators are not described by it), one gets y ∼ x r , which is the symplectic dual (x ↔ y) [41] to the spectal curve for the r-Gelfand-Dickey hierarchy (see [42] Theorem 7.3). Since pure phases are very natural from the matrix model point of view one can expect that the relevant generalization of the topological recursion is also natural.…”

Section: The Large N Limitmentioning

confidence: 99%

Orbifolds and Exact Solutions of Strongly-Coupled Matrix Models

Córdova¹,

Heidenreich

Popolitov

et al. 2018

Commun. Math. Phys.

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We find an exact solution to strongly-coupled matrix models with a single-trace monomial potential. Our solution yields closed form expressions for the partition function as well as averages of Schur functions. The results are fully factorized into a product of terms linear in the rank of the matrix and the parameters of the model. We extend our formulas to include both logarthmic and finite-difference deformations, thereby generalizing the celebrated Selberg and Kadell integrals. We conjecture a formula for correlators of two Schur functions in these models, and explain how our results follow from a general orbifold-like procedure that can be applied to any one-matrix model with a single-trace potential.

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Dubrovin’s Superpotential as a Global Spectral curve

Cited by 33 publications

References 33 publications

Combinatorial structure of colored HOMFLY-PT polynomials for torus knots

Combinatorial structure of colored HOMFLY-PT polynomials for torus knots

Topological recursion with hard edges

Orbifolds and Exact Solutions of Strongly-Coupled Matrix Models

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