Fredholm determinant representation of the homogeneous Painlevé II τ-function

Desiraju, Harini

doi:10.1088/1361-6544/abf84a

Cited by 3 publications

(1 citation statement)

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“…When the isomonodromic problem is related to some of the Painlevé equations, this problem has a positive answer see e.g. [14,20,21,26], and also the applications in [7,8].…”

Section: Introductionmentioning

confidence: 99%

Integrable operators, ∂ ― -problems, KP and NLS hierarchy

Bertola,

Grava,

Orsatti

2024

Nonlinearity

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We develop the theory of integrable operators K acting on a domain of the complex plane with smooth boundary in analogy with the theory of integrable operators acting on contours of the complex plane. We show how the resolvent operator is obtained from the solution of a ∂ ― -problem in the complex plane. When such a ∂ ― -problem depends on auxiliary parameters we define its Malgrange one form in analogy with the theory of isomonodromic problems. We show that the Malgrange one form is closed and coincides with the exterior logarithmic differential of the Hilbert–Carleman determinant of the operator K . With suitable choices of the setup we show that the Hilbert–Carleman determinant is a τ-function of the Kadomtsev–Petviashvili (KP) or nonlinear Schrödinger hierarchies.

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“…When the isomonodromic problem is related to some of the Painlevé equations, this problem has a positive answer see e.g. [14,20,21,26], and also the applications in [7,8].…”

Section: Introductionmentioning

confidence: 99%

Integrable operators, ∂ ― -problems, KP and NLS hierarchy

Bertola,

Grava,

Orsatti

2024

Nonlinearity

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Painlevé Kernels and Surface Defects at Strong Coupling

François,

Grassi

2024

Ann. Henri Poincaré

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It is well established that the spectral analysis of canonically quantized four-dimensional Seiberg–Witten curves can be systematically studied via the Nekrasov–Shatashvili functions. In this paper, we explore another aspect of the relation between $${\mathcal {N}}=2$$ N = 2 supersymmetric gauge theories in four dimensions and operator theory. Specifically, we study an example of an integral operator associated with Painlevé equations and whose spectral traces are related to correlation functions of the 2d Ising model. This operator does not correspond to a canonically quantized Seiberg–Witten curve, but its kernel can nevertheless be interpreted as the density matrix of an ideal Fermi gas. Adopting the approach of Tracy and Widom, we provide an explicit expression for its eigenfunctions via an $${{\,\mathrm{O(2)}\,}}$$ O ( 2 ) matrix model. We then show that these eigenfunctions are computed by surface defects in $${{\,\mathrm{SU(2)}\,}}$$ SU ( 2 ) super Yang–Mills in the self-dual phase of the $$\Omega $$ Ω -background. Our result also yields a strong coupling expression for such defects which resums the instanton expansion. Even though we focus on one concrete example, we expect these results to hold for a larger class of operators arising in the context of isomonodromic deformation equations.

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