Primary invariants of Hurwitz Frobenius
                    manifolds

Dunin-Barkowski, P.; Norbury, P.; Orantin, N.; Popolitov, A.; Shadrin, Sergey

doi:10.1090/pspum/100/09

Cited by 6 publications

(7 citation statements)

References 25 publications

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“…• The differentials (ω h,n ) h,n≥0 may also be considered as functions of the parameters defining the admissible classical spectral curve, namely as functions on the corresponding Hurwitz space (see [21,25] for example for a more detailed explanation). In particular, one may consider locally the space of classical spectral curves obtained by varying the values of periods { i } g i=1 .…”

Section: Properties Of Differentials Produced By the Topological Recu...mentioning

confidence: 99%

“…Using the complete knowledge of the matrix A(λ, ), one may now look at the last line of the compatibility equation (8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22). We get, for the entry (2, 2),…”

Section: Lax Pair Formulationmentioning

confidence: 99%

“…The same computation may be applied to the entry (2, 1) of the compatibility equation (8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22), and it is equivalent to…”

Section: Lax Pair Formulationmentioning

confidence: 99%

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Quantization of classical spectral curves via topological recursion

Eynard¹,

Garcia-Failde²,

Marchal³

et al. 2021

Preprint

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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 Painlevé 2 equation and the quantization of a degree three spectral curve with pole only at infinity.

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Section: Properties Of Differentials Produced By the Topological Recu...mentioning

confidence: 99%

Section: Lax Pair Formulationmentioning

confidence: 99%

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Quantization of classical spectral curves via topological recursion

Eynard¹,

Garcia-Failde²,

Marchal³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…If these agree, then the full higher genus potentials agree, and the higher genus ancestor invariants of

X

are computed by the topological recursion on

S_{X}

by (). Happily, it is a result of Shramchenko that in non‐equivariant GW theory this is always precisely the case [112] (see also [46, Theorem 7]):

R_{GW, X} = R_{normalCEO} false(S_{X} false) .

In other words, the

R

‐calibration

R_{normalCEO} false(S_{X} false)

, which is uniquely specified by the Bergmann kernel of a family of spectral curves

S_{X}

whose prepotential coincides with the genus zero GW potential of a projective variety

X

, coincides with the

R

‐calibration

R_{normalGW, X}

uniquely picked by the de Rham grading in the (non‐equivariant) quantum cohomology of

X

. We get to the following.…”

Section: Application Ii: the Truenormale8̂ Frobenius Manifoldmentioning

confidence: 99%

E8 spectral curves

Brini

2020

Proc. London Math. Soc.

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I provide an explicit construction of spectral curves for the affine E8 relativistic Toda chain. Their closed-form expression is obtained by determining the full set of character relations in the representation ring of E8 for the exterior algebra of the adjoint representation; this is in turn employed to provide an explicit construction of both integrals of motion and the actionangle map for the resulting integrable system. I consider two main areas of applications of these constructions. On the one hand, I consider the resulting family of spectral curves in the context of the correspondences between Toda systems, five-dimensional Seiberg-Witten theory, Gromov-Witten theory of orbifolds of the resolved conifold, and Chern-Simons theory to establish a version of the B-model Gopakumar-Vafa correspondence for the slN Lê-Murakami-Ohtsuki invariant of the Poincaré integral homology sphere to all orders in 1/N. On the other, I consider a degenerate version of the spectral curves and prove a one-dimensional Landau-Ginzburg mirror theorem for the Frobenius manifold structure on the space of orbits of the extended affine Weyl group of type E8 introduced by Dubrovin-Zhang (equivalently, the orbifold quantum cohomology of the type-E8 polynomial CP 1 orbifold). This leads to closed-form expressions for the flat coordinates of the Saito metric, the prepotential, and a higher genus mirror theorem based on the Chekhov-Eynard-Orantin recursion. I will also show how the constructions of the paper lead to a generalisation of a conjecture of Norbury-Scott to ADE P 1-orbifolds, and a mirror of the Dubrovin-Zhang construction for all Weyl groups and choices of marked roots. Contents

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“…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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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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Primary invariants of Hurwitz Frobenius manifolds

Cited by 6 publications

References 25 publications

Quantization of classical spectral curves via topological recursion

Quantization of classical spectral curves via topological recursion

E8 spectral curves

Combinatorial structure of colored HOMFLY-PT polynomials for torus knots

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