Integrable systems and modular forms of level 2

Ablowitz, Mark J.; Chakravarty, Sarbarish; Hahn, Heekyoung

doi:10.1088/0305-4470/39/50/003

Cited by 22 publications

(81 citation statements)

References 21 publications

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“…which, under the substitution h = y/2, coincides with the equation (4.7) from [2]. The potential V satisfies the equation…”

Section: Proofmentioning

confidence: 58%

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Integrable Lagrangians and modular forms

Ferapontov¹,

Odesskii²

2010

Journal of Geometry and Physics

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We investigate non-degenerate Lagrangians of the form f (u x , u y , u t ) dx dy dt such that the corresponding Euler-Lagrange equations (f ux ) x + (f uy ) y + (f ut ) t = 0 are integrable by the method of hydrodynamic reductions. We demonstrate that the integrability conditions, which constitute an involutive over-determined system of fourth order PDEs for the Lagrangian density f , are invariant under a 20-parameter group of Lie-point symmetries whose action on the moduli space of integrable Lagrangians has an open orbit. The density of the 'master-Lagrangian' corresponding to this orbit is shown to be a modular form in three variables defined on a complex hyperbolic ball. We demonstrate how the knowledge of the symmetry group allows one to linearise the integrability conditions. MSC: 35Q58, 37K05, 37K10, 37K25.

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“…which, under the substitution h = y/2, coincides with the equation (4.7) from [2]. The potential V satisfies the equation…”

Section: Proofmentioning

confidence: 58%

“…We point out that modular forms and non-linear ODEs related to them appear in a variety of problems in mathematical physics, see e.g. [1,2,4,5,9,10,14,15,18] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Integrable Lagrangians and modular forms

Ferapontov¹,

Odesskii²

2010

Journal of Geometry and Physics

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“…Their tangent spaces are given by du = u i dx i , dv = v i dx i where u i , v i can be viewed as local coordinates on Gr(3, 5). The corresponding system Σ(X) reduces to a pair of first-order PDEs for u and v, F (u 1 , u 2 , u 3 , v 1 , v 2 , v 3 ) = 0, G(u 1 , u 2 , u 3 , v 1 , v 2 , v 3 ) = 0, (1) u i = ∂u/∂x i , v i = ∂v/∂x i . Equation (1) specifies a fourfold X ⊂ Gr (3,5).…”

Section: Formulation Of the Problemmentioning

confidence: 99%

“…Equation (1) specifies a fourfold X ⊂ Gr (3,5). The class of systems (1) is invariant under the equivalence group SL(5) that acts by linear transformations on the combined set of variables x 1 , x 2 , x 3 , u, v. Since this action preserves the integrability, all our classification results will be formulated modulo SL(5)-equivalence. Necessary details on the equivalence group are provided in Section 3.…”

Section: Formulation Of the Problemmentioning

confidence: 99%

“…Necessary details on the equivalence group are provided in Section 3. There exists a whole variety of important examples that fall into class (1). One of them appears in the context of the dispersionless Kadomtsev-Petviashvili (dKP) equation, u xt − u x u xx − u yy = 0, one of the most well-studied dispersionless integrable PDEs arising in non-linear acoustics [53] and the theory of Einstein-Weyl structures [19].…”

Section: Formulation Of the Problemmentioning

confidence: 99%

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On integrability in Grassmann geometries: integrable systems associated with fourfolds in Gr(3,5)

Doubrov

Ferapontov

Kruglikov

et al. 2018

Proc. London Math. Soc.

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Let Gr(d, n) be the Grassmannian of d-dimensional linear subspaces of an n-dimensional vector space V n . A submanifold X ⊂ Gr(d, n) gives rise to a differential system Σ(X) that governs d-dimensional submanifolds of V n whose Gaussian image is contained in X. Systems of the form Σ(X) appear in numerous applications in continuum mechanics, theory of integrable systems, general relativity and differential geometry. They include such well-known examples as the dispersionless Kadomtsev-Petviashvili equation, the Boyer-Finley equation, Plebańsky's heavenly equations and so on.In this paper we concentrate on the particularly interesting case of this construction where X is a fourfold in Gr(3, 5). Our main goal is to investigate differential-geometric and integrability aspects of the corresponding systems Σ(X). We demonstrate the equivalence of several approaches to dispersionless integrability such as• the method of hydrodynamic reductions;• the method of dispersionless Lax pairs; • integrability on solutions, based on the requirement that the characteristic variety of system Σ(X) defines an Einstein-Weyl geometry on every solution; • integrability on equation, meaning integrability (in twistor-theoretic sense) of the canonical GL(2, R) structure induced on a fourfold X ⊂ Gr(3, 5).All these seemingly different approaches lead to one and the same class of integrable systems Σ(X). We prove that the moduli space of such systems is six-dimensional. We give a complete description of linearisable systems (the corresponding fourfold X is a linear section of Gr(3, 5)) and linearly degenerate systems (the corresponding fourfold X is the image of a quadratic map P 4 Gr(3, 5)). The fourfolds corresponding to 'generic' integrable systems are not algebraic, and can be parametrised by generalised hypergeometric functions.

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An approach to BPS black hole microstate counting in an N = 2 STU model

Cardoso

Nampuri

Polini

2020

J. High Energ. Phys.

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We consider four-dimensional dyonic single-center BPS black holes in the N = 2 STU model of Sen and Vafa. By working in a region of moduli space where the real part of two of the three complex scalars S, T, U are taken to be large, we evaluate the quantum entropy function for these BPS black holes. In this regime, the subleading corrections point to a microstate counting formula partly based on a Siegel modular form of weight two. This is supplemented by another modular object that takes into account the dependence on Y 0 , a complex scalar field belonging to one of the four off-shell vector multiplets of the underlying supergravity theory. We also observe interesting connections to the rational Calogero model and to formal deformation of a Poisson algebra, and suggest a string web picture of our counting proposal.

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Integrable systems and modular forms of level 2

Cited by 22 publications

References 21 publications

Integrable Lagrangians and modular forms

Integrable Lagrangians and modular forms

On integrability in Grassmann geometries: integrable systems associated with fourfolds in Gr(3,5)

An approach to BPS black hole microstate counting in an N = 2 STU model

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