Moduli spaces of d-connections and difference Painlevé equations

Arinkin, Dima; Borodin, Alexei

doi:10.1215/s0012-7094-06-13433-6

Cited by 34 publications

(148 citation statements)

References 26 publications

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“…[26], and subsequently more systematic approach has been used in [3,6,41,76,107,113,137,140,142]. We will explain below how one can use the geometric method for constructing Lax pairs of the discrete Painlevé equations according to the idea in [140,142].…”

Section: Lax Pairsmentioning

confidence: 99%

“…It is actually a polynomial of bidegree (3,2) since the residues at g = z q , κ 1 z vanish by (7.34), which we denote by P 32 ( f, g). This polynomial is characterized by the vanishing condition at the following 12 points:…”

Section: Sufficiency For Compatibilitymentioning

confidence: 99%

“…, v 8 , with κ T κ 2 , and we write f = T ( f ) and g = T −1 (g). We also include the list of points configuration characterizing the equation L 1 y(z) = 0 as a curve of degree (3,2) in ( f, g).…”

Section: Lax Pairsmentioning

confidence: 99%

“…The curve L 1 = 0 is the unique curve of degree (3,2) 6 ) The corresponding continuous flow is P IV with Hamiltonian…”

Section: D-p(ementioning

confidence: 99%

See 3 more Smart Citations

Geometric aspects of Painlevé equations

Kajiwara¹,

Noumi²,

Yamada³

2017

J. Phys. A: Math. Theor.

125

250

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In this paper a comprehensive review is given on the current status of achievements in the geometric aspects of the Painlevé equations, with a particular emphasis on the discrete Painlevé equations. The theory is controlled by the geometry of certain rational surfaces called the spaces of initial values, which are characterized by eight point configuration on P 1 × P 1 and classified according to the degeneration of points. We give a systematic description of the equations and their various properties, such as affine Weyl group symmetries, hypergeometric solutions and Lax pairs under this framework, by using the language of Picard lattice and root systems. We also provide with a collection of basic data; equations, point configurations/root data, Weyl group representations, Lax pairs, and hypergeometric solutions of all possible cases.

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Section: Lax Pairsmentioning

confidence: 99%

Section: Sufficiency For Compatibilitymentioning

confidence: 99%

Section: Lax Pairsmentioning

confidence: 99%

“…The curve L 1 = 0 is the unique curve of degree (3,2) 6 ) The corresponding continuous flow is P IV with Hamiltonian…”

Section: D-p(ementioning

confidence: 99%

See 2 more Smart Citations

Geometric aspects of Painlevé equations

Kajiwara¹,

Noumi²,

Yamada³

2017

J. Phys. A: Math. Theor.

125

250

View full text Add to dashboard Cite

show abstract

“…Пространство модулей κ-связностей было исследовано в [72]. Сами κ-связности позволяют пе-рейти в квазиклассическом пределе к полям Хиггса Φ = lim κ→0 (κ∂ z +A) ⊗ dz ∈ Ω 0 (Σ g , Lie G ⊗ K), где K -канонический класс на Σ g .…”

Section: из (340) следует чтоunclassified

Classification of isomonodromy problems on elliptic curves

Levin¹,

Ольшанецкий²,

Olshanetsky³

et al. 2014

Uspekhi Mat. Nauk

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Mirror symmetry for moduli spaces of Higgs bundles via p-adic integration

2020

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We prove the Topological Mirror Symmetry Conjecture by Hausel-Thaddeus for smooth moduli spaces of Higgs bundles of type SLn and P GLn. More precisely, we establish an equality of stringy Hodge numbers for certain pairs of algebraic orbifolds generically fibred into dual abelian varieties. Our proof utilises p-adic integration relative to the fibres, and interprets canonical gerbes present on these moduli spaces as characters on the Hitchin fibres using Tate duality. Furthermore we prove for d coprime to n, that the number of rank n Higgs bundles of degree d over a fixed curve defined over a finite field, is independent of d. This proves a conjecture by Mozgovoy-Schiffman in the coprime case.Contents arXiv:1707.06417v2 [math.AG] 8 Jun 2018Proof. We begin the proof by fixing notation. The relative Jacobian of the trivial family of curves X × S A / A will be denoted by J. The relative Jacobian of the universal spectral curve Y / A will be denoted by J. Similarly we denote by J 1 and J 1 the relative moduli spaces of degree 1 line bundles.Henceforth we restrict every A-scheme to the open subset A good . To avoid awkward notation we will omit the corresponding superscript.The relative norm map induces a morphism of abelian A-schemes J Nm / / J. Similarly, pullback of line bundles yields π * : J / / J. We claim that these two morphisms are dual to each other with respect to the canonical isomorphism J ∨ J induced by the Poincaré bundle (and similarly for J). To see this we observe that we have a commutative diagram (the horizontal arrows represent the Abel-Jacobi map)to which we can apply the contravariant Pic 0 (?/ A) functor to obtain the commutative diagram of abelian schemes Furthermore, the top row of the first diagram is the fibre of the vertical arrows and hence the top row of the second diagram is the corresponding cofibre. By explicitly computing the fibres in the second diagram

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Moduli spaces of d-connections and difference Painlevé equations

Cited by 34 publications

References 26 publications

Geometric aspects of Painlevé equations

Geometric aspects of Painlevé equations

Classification of isomonodromy problems on elliptic curves

Mirror symmetry for moduli spaces of Higgs bundles via p-adic integration

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