Painlevé equations, topological type property and reconstruction by the topological recursion

Iwaki, Kohei; Marchal, Olivier; Saenz, Axel

doi:10.1016/j.geomphys.2017.10.009

Cited by 23 publications

(66 citation statements)

References 29 publications

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“…In general, the EOC topological recursion has found many other applications outside the scope of matrix model, including enumerative geometry, isomonodromic systems, quantum cohomology (see [Eyn11,IMS16,DBOSS14]). We define it in its original setting of matrix models [Eyn05] and one can find more general definitions in [BHL + 12,DM14].…”

Section: Eoc Topological Recursionmentioning

confidence: 99%

The KPZ universality class and related topics

Saenz¹

2020

Analytic Trends in Mathematical Physics

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These notes are based on a talk given at the 2018 Arizona School of Analysis and Mathematical Physics. We give a comprehensive introduction to the KPZ universality class, a conjectured class of stochastic process with local interactions related to random growth processes in 1 + 1 dimensions. We describe some of the characteristic properties of the KPZ universality class such as scaling exponents and limiting statistics. In particular, we aim to extract the characteristic properties of the KPZ universality class by understanding the KPZ stochastic partial differential equation by a special discrete approximation given by the asymmetric simple exclusion process (ASEP). The connection with the ASEP is very important as the process enjoys a rich integrability structure that leads to many exact formulas.Contents arXiv:1904.03319v1 [math-ph] 5 Apr 2019 with W (t, x) a Gaussian space-time white noise random variable (i.e. a δ-correlated stationary, Gaussian process with mean zero and covariance E [W (t , x )W (t, x)] = δ(t − t)δ(x − x)). The first term ν∂ 2x h(t, x) is a diffusion term which tends to smooth out the height function as time evolves, and the term λ 2 (∂ x h(t, x)) 2 is a lateral growth term, which makes the surface grow normal to its interface. The SPDE defined by eqn. (1.1) is now called the KPZ equation.It turns out that solving the KPZ equation is highly non-trivial, starting with making the KPZ equation (1.1) well-defined. For instance, through a local scaling argument, one might expect that the height function behaves similar to a Brownian motion due to the space-time white noise, and this means that, a priori, the term ∂ x h(t, x) is a distribution without a well-defined square (∂ x h(t, x)) 2 making the KPZ equation ill-defined. Under certain initial conditions, this issue has been resolved in the work of [BG97, ACQ11] by introducing the Cole-Hopf transformation (see eqn. 2.2) and an appropriate discrete approximation of the KPZ equation, and also, in the work of [Hai11] through the theory of rough paths. In [BG97], Bertini and Giacomoni gave a solution to the KPZ equation for initial conditions h(0, x) that (in average) grow at most linearly, such as flat initial conditions or two-sided Brownian motion initial conditions. In particular, Bertini and Giacomoni approach the KPZ equation by first smoothing out the noise, solving the smoothed KPZ equation via a discrete approximation using a discrete Cole-Hopf transformation, and finally taking the appropriate limits to a solution of the KPZ equation. Then, in [ACQ11], Amir, Corwin and Quaste were able to extend the results of [BG97] by treating the KPZ equation with the infinite narrow wedge initial condition (i.e. h(0, x) = log(δ x=0 )), which is a natural initial condition under the random polymer interpretation of the KPZ equation. In [ACQ11], the authors still use the idea of considering the Cole-Hopf transformation and were able to treat the wedge initial conditions via the exact formulas, introduced by Tracy and Widom in [TW08, TW09], for the asy...

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Section: Eoc Topological Recursionmentioning

confidence: 99%

The KPZ universality class and related topics

Saenz¹

2020

Analytic Trends in Mathematical Physics

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“…A geometric definition of a quantum curve that arises as the quantization of a Hitchin spectral curve is developed in [31], based on the work of [26] that solves a conjecture of Gaiotto [42,43]. The WKB analysis of the quantum curve [27,28,30,57,58] is performed by applying the topological recursion of [38]. The Hermite-Weber differential equation is an example of this geometric theory.…”

Section: Quantum Curvesmentioning

confidence: 99%

An invitation to 2D TQFT and quantization of Hitchin spectral curves

Dumitrescu

Mulase

2018

Banach Center Publ.

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This article consists of two parts. In Part 1, we present a formulation of twodimensional topological quantum field theories in terms of a functor from a category of Ribbon graphs to the endofuntor category of a monoidal category. The key point is that the category of ribbon graphs produces all Frobenius objects. Necessary backgrounds from Frobenius algebras, topological quantum field theories, and cohomological field theories are reviewed. A result on Frobenius algebra twisted topological recursion is included at the end of Part 1.In Part 2, we explain a geometric theory of quantum curves. The focus is placed on the process of quantization as a passage from families of Hitchin spectral curves to families of opers. To make the presentation simpler, we unfold the story using SL 2 (C)-opers and rank 2 Higgs bundles defined on a compact Riemann surface C of genus greater than 1. In this case, quantum curves, opers, and projective structures in C all become the same notion. Background materials on projective coordinate systems, Higgs bundles, opers, and non-Abelian Hodge correspondence are explained. 33 8. Projective structures, opers, and Higgs bundles 36 9. Semi-classical limit of SL 2 (C)-opers 47 10. Non-Abelian Hodge correspondence between Hitchin moduli spaces 48 References 50

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“…In particular, in contrast with [7], the operator δ = exp( d dx ) directly comes in its exponentiated form while in [7], the starting point was the operator d dx defined on the corresponding Lie algebra g (in our case g = sl 2 (C)). In fact, a by-product of this article is also to show, on a simple example, that the reconstruction of the determinantal formulas via the topological recursion developed in [7,12,13] in the Lie algebra setting may be adapted to similar problems defined directly on a Lie group.…”

Section: Generalization To Arbitrary 2 × 2 Difference Systemsmentioning

confidence: 99%

“…As explained earlier, the main difference with the setting of [7,12,13] is that the -difference system (2-5) is defined on the Lie group SL 2 (C) with an exponential operator δ = exp( d dx ) rather than on the corresponding Lie algebra sl 2 (C). At the level of representation matrices L(x; ), we would like to find a 2 × 2 matrix D(x; ) ∈ sl 2 (C) such that:…”

Section: The Corresponding Differential Operator D(x; )mentioning

confidence: 99%

“…In this section, we propose to show how we can reconstruct the matrix D(x; ) from the knowledge of the matrix L(x; ) introduced in the general setting of Definition 2.1. In particular, we show for 2 × 2 systems how to obtain the orders D k (x) with k ≥ 1 by induction using (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19). The construction being completely algebraic, the following results may be used also for the shifted version D λ (x) and L λ (x).…”

Section: The Corresponding Differential Operator D(x; )mentioning

confidence: 99%

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WKB solutions of difference equations and reconstruction by the topological recursion

Marchal

2017

Nonlinearity

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The purpose of this article is to analyze the connection between EynardOrantin topological recursion and formal WKB solutions of a -difference equation:. In particular, we extend the notion of determinantal formulas and topological type property proposed for formal WKB solutions of -differential systems to this setting. We apply our results to a specific -difference system associated to the quantum curve of the Gromov-Witten invariants of 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

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Painlevé equations, topological type property and reconstruction by the topological recursion

Cited by 23 publications

References 29 publications

The KPZ universality class and related topics

The KPZ universality class and related topics

An invitation to 2D TQFT and quantization of Hitchin spectral curves

WKB solutions of difference equations and reconstruction by the topological recursion

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