Quantum field theories on algebraic curves. I. Additive bosons

Takhtajan, Leon A.

doi:10.1070/im2013v077n02abeh002640

Cited by 7 publications

(4 citation statements)

References 15 publications

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“…The classical integrable structure of the real branch was studied in [55] but the action of the corresponding model has not yet been identified. The above analysis shows that at the Hamiltonian level this model is described by a dihedral affine Gaudin model associated with the divisor (80), where x ± ∈ R, together with the choice of levels (82). When x + = −x − we obtain a one-parameter deformation described by an integrable gauged WZW-type theory introduced by Sfetsos [80], which interpolates between the WZW model and the non-abelian T -dual of the principal chiral model.…”

Section: Of Part 2]mentioning

confidence: 93%

On Integrable Field Theories as Dihedral Affine Gaudin Models

Vicedo

2018

International Mathematics Research Notices

267

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We introduce the notion of a classical dihedral affine Gaudin model, associated with an untwisted affine Kac–Moody algebra $\widetilde{\mathfrak{g}}$ equipped with an action of the dihedral group $D_{2T}$, $T \geq 1$ through (anti-)linear automorphisms. We show that a very broad family of classical integrable field theories can be recast as examples of such classical dihedral affine Gaudin models. Among these are the principal chiral model on an arbitrary real Lie group $G_0$ and the $\mathbb{Z}_T$-graded coset $\sigma $-model on any coset of $G_0$ defined in terms of an order $T$ automorphism of its complexification. Most of the multi-parameter integrable deformations of these $\sigma $-models recently constructed in the literature provide further examples. The common feature shared by all these integrable field theories, which makes it possible to reformulate them as classical dihedral affine Gaudin models, is the fact that they are non-ultralocal. In particular, we also obtain affine Toda field theory in its lesser-known non-ultralocal formulation as another example of this construction. We propose that the interpretation of a given classical non-ultralocal integrable field theory as a classical dihedral affine Gaudin model provides a natural setting within which to address its quantisation. At the same time, it may also furnish a general framework for understanding the massive ordinary differential equations (ODE)/integrals of motion (IM) correspondence since the known examples of integrable field theories for which such a correspondence has been formulated can all be viewed as dihedral affine Gaudin models.

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Section: Of Part 2]mentioning

confidence: 93%

On Integrable Field Theories as Dihedral Affine Gaudin Models

Vicedo

2018

International Mathematics Research Notices

267

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“…In any case, we are led to speculate from (2.4.3) that both CFT and the theory of vertex operator algebras (and indeed Moonshine itself) may extend quite nicely to the p-adics Q p . Some moves in this direction are [562], [520]. To a number theorist, the usual perturbation about a vacuum would correspond to the infinite prime, but would mysteriously ignore the contributions from all the finite primes.…”

Section: Informal Conformal Field Theorymentioning

confidence: 99%

Moonshine beyond the Monster

Gannon

2023

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Moonshine forms a way of explaining the mysterious connection between the monster finite group and modular functions from classical number theory. The theory has evolved to describe the relationship between finite groups, modular forms and vertex operator algebras. Moonshine beyond the Monster describes the general theory of Moonshine and its underlying concepts, emphasising the interconnections between mathematics and mathematical physics. Written in a clear and pedagogical style, this book is ideal for graduate students and researchers working in areas such as conformal field theory, string theory, algebra, number theory, geometry and functional analysis. Containing more than 100 exercises, it is also a suitable textbook for graduate courses on Moonshine and as supplementary reading for courses on conformal field theory and string theory. Originally published in 2006, this title has been reissued as an Open Access publication on Cambridge Core.

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“…in the second tensor factor, as in (2.1). The Lie subalgebra ι λ R λ (g) ⊂ A λ (g) is maximally isotropic with respect to •, • k , for any k ∈ Z, by the strong residue theorem; see for instance [Ta,Corollary 1].…”

Section: General Setupmentioning

confidence: 99%

Classical Yang-Baxter equation, Lagrangian multiforms and ultralocal integrable hierarchies

Caudrelier¹,

Stoppato²

2022

Preprint

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We cast the classical Yang-Baxter equation (CYBE) in a variational context for the first time, by relating it to the theory of Lagrangian multiforms, a framework designed to capture integrability in a variational fashion. This provides a significant connection between Lagrangian multiforms and the CYBE, one of the most fundamental concepts of integrable systems. This is achieved by introducing a generating Lagrangian multiform which depends on a skew-symmetric classical r-matrix with spectral parameters. The multiform Euler-Lagrange equations produce a generating Lax equation which yields a generating zero curvature equation. The CYBE plays a role at three levels: 1) It ensures the commutativity of the flows of the generating Lax equation; 2) It ensures that the generating zero curvature equation holds; 3) It implies the closure relation for the generating Lagrangian multiform. The specification of an integrable hierarchy is achieved by fixing certain data: a finite set S ∈ CP 1 , a Lie algebra g, a g-valued rational function with poles in S and an r-matrix. We show how our framework is able to generate a large class of ultralocal integrable hierarchies by providing several known and new examples pertaining to the rational or trigonometric class. These include the Ablowitz-Kaup-Newell-Segur hierarchy, the sine-Gordon (sG) hierarchy and various hierachies related to Zakharov-Mikhailov type models which contain the Faddeev-Reshetikhin (FR) model and recently introduced deformed sigma/Gross-Neveu models as particular cases. The versatility of our method is illustrated by showing how to couple integrable hierarchies together to create new examples of integrable field theories and their hierarchies. We provide two examples: the coupling of the nonlinear Schrödinger system to the FR model and the coupling of sG with the anisotropic FR model.

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Quantum field theories on algebraic curves. I. Additive bosons

Cited by 7 publications

References 15 publications

On Integrable Field Theories as Dihedral Affine Gaudin Models

On Integrable Field Theories as Dihedral Affine Gaudin Models

Moonshine beyond the Monster

Classical Yang-Baxter equation, Lagrangian multiforms and ultralocal integrable hierarchies

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