Classical integrability

Торриелли, Алессандро

doi:10.1088/1751-8113/49/32/323001

Cited by 45 publications

(77 citation statements)

References 38 publications

(86 reference statements)

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“…We refer the readers to[46][47][48][49][50] and the references therein for a sampler of beginner's guides to integrability 15. Actually, we meant that the moments are in involution among themselves, which means they Poisson commute.…”

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confidence: 99%

Thermal correlation functions of KdV charges in 2D CFT

Maloney

Ross

et al. 2019

J. High Energ. Phys.

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Two dimensional CFTs have an infinite set of commuting conserved charges, known as the quantum KdV charges, built out of the stress tensor. We compute the thermal correlation functions of the these KdV charges on a circle. We show that these correlation functions are given by quasi-modular differential operators acting on the torus partition function. We determine their modular transformation properties, give explicit expressions in a number of cases, and give a general form which determines an arbitrary correlation function up to a finite number of functions of the central charge. We show that these modular differential operators annihilate the characters of the (2m + 1, 2) family of non-unitary minimal models. We also show that the distribution of KdV charges becomes sharply peaked at large level. Introduction and summary of resultsTwo dimensional conformal field theories (CFTs) have an infinite-dimensional symmetry algebra -the Virasoro algebra of local conformal transformations -which powerfully constrains the structure of the theory. Our goal is to explore the consequences of this symmetry structure for the finite-temperature behaviour of the theory. We will study two dimensional CFTs at finite temperature on the circle. Our interest in this paper is obtaining exact results: we will keep the radius of the circle finite, and will not study the high temperature or large c limits, though these will be discussed in a companion paper [1].Our starting point is the observation that, from the Virasoro algebra, one can define an infinite set of mutually commuting conserved charges [2-4]. In the large c limit these charges generate the KdV hierarchy of differential equations, so they are often referred to as the quantum KdV charges. We will follow the notation of [4] where the charges are denoted I 2m−1 with m = 1, 2, . . . , and the subscript labels the spin of the charge. The first KdV charge I 1 is just L 0 = T (z)dz, the zero mode of the holomorphic part of the stress tensor. The KdV charge I 2m−1 is an m th order polynomial in the Virasoro generators L n .Given this set of mutually commuting charges, one can define a Generalized Gibbs Ensemble (GGE) for two-dimensional CFTs where we introduce a chemical potential for each KdV charge: 1 Z[β, µ 3 , µ 5 , . . .] = Tr [e −βE+µ 3 I 3 +µ 5 I 5 +... ] = Tr [e µ 3 I 3 +µ 5 I 5 +... q L 0 −c/24 ], q ≡ e −β , (1.1)In this equation we have introduced chemical potentials for the left-moving (holomorphic) generators L 0 , I 3 , I 5 , . . . . In general, of course, one should also introduce potentials for the right-moving (anti-holomorphic) generatorsL 0 ,Ī 3 ,Ī 5 , . . . . However, all of our results involve the use just of the Virasoro algebra so the left-and right-moving sectors completely factorize.1 See [5-9] for some recent work on GGEs and [10,11] for recent experimental realisations. This ensemble was studied from a holographic perspective in [12]; see also [13].These KdV charges act within each Virasoro module, so can be simultaneously diagonalized level by level with...

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confidence: 99%

Thermal correlation functions of KdV charges in 2D CFT

Maloney

Ross

et al. 2019

J. High Energ. Phys.

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“…By definition, a Lax integrable system can be presented as an isospectral deformation of the following linear problem [2,31] (L − λ) Ψ = 0,…”

Section: Lax Representationmentioning

confidence: 99%

“…Lax equation (4) provides nontrivial conditions on elements of the L matrix, which give us the corresponding integrable system. It is easy to show, that if equation (4) holds, then [2,31] (…”

Section: Lax Representationmentioning

confidence: 99%

“…where, without loss of generality, B can be set equal to zero. In order to exclude the case of a trivial linear equation, we assume that |f y | 2 + |g yy | 2 = 0 in (31). Another separate case, which we need to consider is the case of T = 0 and RT + 4f 2 y P − 2P 2 y + (2f f y − Q)P y = 0.…”

Section: The Functions a And B Are Defined By The Relationsmentioning

confidence: 99%

“…Therefore, we have found the necessary and sufficient conditions for equation (1) to possess a first integral in the form (17). These conditions splits into six separate cases: generic case (23) and special cases (25), (27) and (28), (29), (31), (32) and (34). Notice that the case when equation (1) degenerates into a linear one (that is, when |f y | 2 + |g yy | 2 = 0) is not considered in this work, since in the compatibility conditions obtained above it is assumed that f y = 0, g yy = 0 or |f y | 2 + |g yy | 2 = 0.…”

Section: The Functions a And B Are Defined By The Relationsmentioning

confidence: 99%

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Lax representation and quadratic first integrals for a family of non-autonomous second-order differential equations

Sinelshchikov

Gaiur

Kudryashov

2019

Journal of Mathematical Analysis and Applications

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We consider a family of non-autonomous second-order differential equations, which generalizes the Liénard equation. We explicitly find the necessary and sufficient conditions for members of this family of equations to admit quadratic, with the respect to the first derivative, first integrals. We show that these conditions are equivalent to the conditions for equations in the family under consideration to possess Lax representations. This provides a connection between the existence of a quadratic first integral and a Lax representation for the studied dissipative differential equations, which may be considered as an analogue to the theorem that connects Lax integrability and Arnold-Liouville integrability of Hamiltonian systems. We illustrate our results by several examples of dissipative equations, including generalizations of the Van der Pol and Duffing equations, each of which have both a quadratic first integral and a Lax representation.

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Integrability Ex Machina

Krippendorf¹,

Lüst

Syvaeri

2021

Fortschritte der Physik

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Determining whether a dynamical system is integrable is generally a difficult task which is currently done on a case by case basis requiring large human input. Here we propose and test an automated method to search for the existence of relevant structures, the Lax pair and Lax connection respectively. By formulating this search as an optimization problem, we are able to identify appropriate structures via machine learning techniques. We test our method on standard systems of classical integrability and find that we can single out some integrable deformations of a system. Due to the ambiguity in defining a Lax pair our algorithm identifies novel Lax pairs which can be easily verified analytically.

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Classical integrability

Abstract: We review some essential aspects of classically integrable systems. The detailed outline of the lectures consists of

Cited by 45 publications

References 38 publications

Thermal correlation functions of KdV charges in 2D CFT

Thermal correlation functions of KdV charges in 2D CFT

Lax representation and quadratic first integrals for a family of non-autonomous second-order differential equations

Integrability Ex Machina

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