Action-angle Coordinates for Integrable Systems on Poisson Manifolds

AbstractThe Heisenberg supermagnet model is an important supersymmetric integrable system in (1+1)-dimensions. We construct two types of the (2+1)-dimensional integrable Heisenberg supermagnet models with the quadratic constraints and investigate the integrability of the systems. In terms of the gage transformation, we derive their gage equivalent counterparts. Furthermore, we also construct new solutions of the supersymmetric integrable systems by means of the Bäcklund transformations.

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Dispersionless integrable systems and the Bogomolny equations on an Einstein–Weyl geometry background

Theoretical and Mathematical Physics ◽

10.1134/s0040577920100037 ◽

2020 ◽

Vol 205 (1) ◽

pp. 1279-1290

Author(s):

L. V. Bogdanov

Keyword(s):

Integrable Systems ◽

Weyl Geometry

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Separation of variables in AdS/CFT: functional approach for the fishnet CFT

Journal of High Energy Physics ◽

10.1007/jhep06(2021)131 ◽

2021 ◽

Vol 2021 (6) ◽

Author(s):

Andrea Cavaglià ◽

Nikolay Gromov ◽

Fedor Levkovich-Maslyuk

Keyword(s):

Integrable Systems ◽

Density Operator ◽

Numerical Evaluation ◽

Lagrangian Density ◽

Separation Of Variables ◽

Functional Approach ◽

Quantum Integrable Systems ◽

Arbitrary State ◽

Nontrivial Coupling ◽

The One

Abstract The major simplification in a number of quantum integrable systems is the existence of special coordinates in which the eigenstates take a factorised form. Despite many years of studies, the basis realising the separation of variables (SoV) remains unknown in $$ \mathcal{N} $$ N = 4 SYM and similar models, even though it is widely believed they are integrable. In this paper we initiate the SoV approach for observables with nontrivial coupling dependence in a close cousin of $$ \mathcal{N} $$ N = 4 SYM — the fishnet 4D CFT. We develop the functional SoV formalism in this theory, which allows us to compute non-perturbatively some nontrivial observables in a form suitable for numerical evaluation. We present some applications of these methods. In particular, we discuss the possible SoV structure of the one-point correlators in presence of a defect, and write down a SoV-type expression for diagonal OPE coefficients involving an arbitrary state and the Lagrangian density operator. We believe that many of the findings of this paper can be applied in the $$ \mathcal{N} $$ N = 4 SYM case, as we speculate in the last part of the article.

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