Integrable Kondo problems

Gaiotto, Davide; Lee, Ji Hoon; Wu, Jingxiang

doi:10.48550/arxiv.2003.06694

Cited by 9 publications

(59 citation statements)

References 45 publications

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“…if infinitesimal gauge transformations parametrised by ε are to preserve these boundary conditions. 7 We now wish to determine the effective space-time theory. As usual, we begin by pulling back the gauge field to the twistor fibres and expressing it in terms of σ as…”

Section: Trigonometric Actionmentioning

confidence: 99%

“…[3][4][5]. The 4d Chern-Simons theory also treats spectral parameter as part of the geometry, with the striking consequence that, at the quantum level, anomalies and RG flows can be interpreted geometrically [6,7].…”

Section: Introductionmentioning

confidence: 99%

“…This is not the most general choice of l± in g0 ⊕ h, see[24] for a more generic choice 7. A minor technicality: we should really demand that the projection of Ā, and of an infinitesimal gauge transformation ε, onto the subalgebra l± is divisible by π α± π β± .…”

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confidence: 99%

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Twistors, the ASD Yang-Mills equations, and 4d Chern-Simons theory

Roland¹,

Skinner²

2020

Preprint

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We show that the approaches to integrable systems via 4d Chern-Simons theory and via symmetry reductions of the anti-self-dual Yang-Mills equations are closely related, at least classically. Following a suggestion of Kevin Costello, we start from holomorphic Chern-Simons theory on twistor space, defined with the help of a meromorphic (3,0) form Ω. If Ω is nowhere vanishing, the theory is equivalent to anti-self-dual Yang-Mills on spacetime. The twistor action yields space-time actions including the Chalmers & Siegel action for ASD Yang-Mills in terms of an adjoint scalar. Under symmetry reduction, these spacetime actions yield actions for 2d integrable systems. On the other hand, performing the symmetry reduction directly on twistor space reduces the holomorphic Chern-Simons action to the 4d Chern-Simons theory with disorder defects studied by Costello & Yamazaki. Finally we show that similar reductions by a single translation leads to a 5d partially holomorphic Chern-Simons theory describing the Bogomolny equations. Contents 1 Introduction 2 From holomorphic to 4d Chern-Simons theory 2.1 Holomorphic Chern-Simons theory on twistor space 2.2 Effective description as a 4d WZW model 2.3 Symmetry reduction to a principal chiral model with WZW term 2.4 Symmetry reduction to 4d Chern-Simons theory 3 Twistor actions 3.1 Overview 3.2 Chalmers-Siegel action 3.3 Trigonometric action 3.4 4d integrable coupled σ-models 4 Real structures 4.1 Lorentzian and ultrahyperbolic signatures 4.2 Real forms of the gauge group 5 Reductions by 1-dimensional groups of translations 5.1 Reduction to the Bogomolny equations 5.2 The 1+2 dimensional chiral model 6 Conclusions A Notation and conventions for spinors B The twistor correspondence in Euclidean signature C The anti-self-dual Yang-Mills equations D Partially holomorphic structures in 4d and 5d Chern-Simons theory E Calculations for trigonometric actions F Novel actions for anti-self-dual Yang-Mills theory G The twistor correspondence in Lorentzian and ultrahyperbolic signature 1 Here ē0 Ā0 is the component along the base while êA ÂA are the components along the fibres of PT ∼ = O(1) ⊕ O(1) → CP 1 .

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Section: Trigonometric Actionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Twistors, the ASD Yang-Mills equations, and 4d Chern-Simons theory

Roland¹,

Skinner²

2020

Preprint

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“…One may include two kinds of defects, order defects and disorder defects (For the terminology, see [4]). Disorder defects lead to various nonlinear sigma models and their integrable deformations [5][6][7][8][9][10][11][12][13][14][15], and it is closely related to the affine Gaudin formalism [16][17][18][19]. On the other hand, order defects give rise to the models with ultralocal Poisson structures such as the Zakharov-Mikhailov theory [20] and the Faddeev-Reshetikhin model [21].…”

Section: Introductionmentioning

confidence: 99%

Non-Abelian Toda field theories from a 4D Chern-Simons theory

Fukushima,

Sakamoto,

Yoshida

2021

Preprint

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We derive non-abelian Toda field theories (NATFTs) from a 4d Chern-Simons (CS) theory with two order defects by employing a certain asymptotic boundary condition.The 4d CS theory is characterized by a meromorphic 1-form ω . We adopt ω with two simple poles and no zeros, and each of the order defects is located at each pole. As a result, an anisotropy parameter β 2 can be identified with the distance between the two defects. As examples, we can derive the (complex) sine-Gordon model and the Liouville theory.

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“…However, the order-defect case has not been elaborated so much at least so far. For other related works on 4D CS theory, see [14][15][16][17].…”

Section: Introductionmentioning

confidence: 99%