2D Integrable systems, 4D Chern–Simons theory and affine Higgs bundles

Levin, A.; Olshanetsky, M.; Zotov, A.

doi:10.1140/epjc/s10052-022-10553-0

Cited by 8 publications

(7 citation statements)

References 33 publications

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“…A natural question at this point is whether there exist a systematic way of constructing (d, •, • , E, p z ) such that these conditions are automatically satisfied. The recent results of [81,84] indicate that such a construction is provided by the formalisms of 4d Chern-Simons theory [85] and affine Gaudin models [86][87][88], which are known to be deeply connected one to another [89,90]. Here, we will use the language of affine Gaudin models (AGM), which is more directly related to the one used in this paper.…”

Section: Integrable σ-Models Affine Gaudin Models and Their Quantisationmentioning

confidence: 99%

On a class of conformal $$ \mathcal{E} $$-models and their chiral Poisson algebras

Lacroix

2023

J. High Energ. Phys.

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In this paper, we study conformal points among the class of $$ \mathcal{E} $$ E -models. The latter are σ-models formulated in terms of a current Poisson algebra, whose Lie-theoretic definition allows for a purely algebraic description of their dynamics and their 1-loop RG-flow. We use these results to formulate a simple algebraic condition on the defining data of such a model which ensures its 1-loop conformal invariance and the decoupling of its observables into two chiral Poisson algebras, describing the classical left- and right-moving fields of the theory. In the case of so-called non-degenerate $$ \mathcal{E} $$ E -models, these chiral sectors form two current algebras and the model takes the form of a WZW theory once realised as a σ-model. The case of degenerate $$ \mathcal{E} $$ E -models, in which a subalgebra of the current algebra is gauged, is more involved: the conformal condition yields a wider class of theories, which includes gauged WZW models but also other examples, seemingly different, which however sometimes turn out to be related to gauged WZW models based on other Lie algebras. For this class, we build non-local chiral fields of parafermionic-type as well as higher-spin local ones, forming classical $$ \mathcal{W} $$ W -algebras. In particular, we find an explicit and efficient algorithm to build these local chiral fields. These results (and their potential generalisations discussed at the end of the paper) open the way for the quantisation of a large class of conformal $$ \mathcal{E} $$ E -models using the standard operator formalism of two-dimensional CFT.

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Section: Integrable σ-Models Affine Gaudin Models and Their Quantisationmentioning

confidence: 99%

On a class of conformal $$ \mathcal{E} $$-models and their chiral Poisson algebras

Lacroix

2023

J. High Energ. Phys.

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“…Integrable 1+1 field theories are actively studied at the classical and quantum levels due to their deep relation to gauge theories, strings and conformal field theories, existence of soliton solutions and many other remarkable properties [14,16,17,24,25,28,37]. We hope to include into consideration a large class of models obtained as 1+1 field generalizations of the widely known finite-dimensional integrable systems of Calogero-Ruijsenaars family.…”

Section: Introductionmentioning

confidence: 99%

Non-ultralocal classical r-matrix structure for 1+1 field analogue of elliptic Calogero–Moser model

Zotov

2024

J. Phys. A: Math. Theor.

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We consider 1+1 ﬁeld generalization of the elliptic Calogero-Moser model. It is shown that the Lax connection satisﬁes the classical non-ultralocal r-matrix structure of Maillet type. Next, we consider 1+1 ﬁeld analogue of the spin Calogero-Moser model and its multipole (or multispin) extension. Finally, we discuss the ﬁeld analogue of the classical IRF-Vertex correspondence, which relates utralocal and non-ultralocal r-matrix structures.

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“…It has been extensively applied to reproduce many existing 2d integrable field theories and also construct a wide variety of new ones [2,[8][9][10][11][12][13][14][15][16][17][18][19][20][21]. See also [1,[22][23][24][25][26][27][28][29][30] for further developments in relation to 4d Chern-Simons theory.…”

Section: Introductionmentioning

confidence: 99%

Integrable degenerate $\mathcal E$-models from 4d Chern-Simons theory

Liniado¹

2023

Preprint

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We present a general construction of integrable degenerate E-models on a 2d manifold Σ using the formalism of Costello and Yamazaki based on 4d Chern-Simons theory on Σ × CP 1 . We begin with a physically motivated review of the mathematical results of [1] where a unifying 2d action was obtained from 4d Chern-Simons theory which depends on a pair of 2d fields h and L on Σ subject to a constraint and with L depending rationally on the complex coordinate on CP 1 . When the meromorphic 1-form ω entering the action of 4d Chern-Simons theory is required to have a double pole at infinity, the constraint between h and L was solved in [2] to obtain integrable non-degenerate E-models. We extend the latter approach to the most general setting of an arbitrary 1-form ω and obtain integrable degenerate E-models. To illustrate the procedure we reproduce two well known examples of integrable degenerate E-models: the pseudo dual of the principal chiral model and the bi-Yang-Baxter σ-model.

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2D Integrable systems, 4D Chern–Simons theory and affine Higgs bundles

Cited by 8 publications

References 33 publications

On a class of conformal $$ \mathcal{E} $$-models and their chiral Poisson algebras

On a class of conformal $$ \mathcal{E} $$-models and their chiral Poisson algebras

Non-ultralocal classical r-matrix structure for 1+1 field analogue of elliptic Calogero–Moser model

Integrable degenerate $\mathcal E$-models from 4d Chern-Simons theory

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