Towards a quadratic Poisson algebra for the subtracted classical monodromy of symmetric space sine-Gordon theories

Delduc, Francois; Hoare, Ben; Magro, Marc

doi:10.1088/1751-8121/ad1d91

Towards a quadratic Poisson algebra for the subtracted classical monodromy of symmetric space sine-Gordon theories

Francois Delduc
(
- ENS, Lyon, Lab. Phys.
)
,
Ben Hoare
(
- Durham U., Dept. of Math.
)
,
Marc Magro
(
- ENS, Lyon, Lab. Phys.
)

Sep 27, 2023

39 pages

Published in:

J.Phys.A 57 (2024) 6, 065401

Published: Jan 29, 2024

e-Print:

2309.15722 [hep-th]

DOI:

10.1088/1751-8121/ad1d91

View in:

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Abstract: (IOP)

Symmetric space sine-Gordon theories are two-dimensional massive integrable field theories, generalising the sine-Gordon and complex sine-Gordon theories. To study their integrability properties on the real line, it is necessary to introduce a subtracted monodromy matrix. Moreover, since the theories are not ultralocal, a regularisation is required to compute the Poisson algebra for the subtracted monodromy. In this article, we regularise and compute this Poisson algebra for certain configurations, and show that it can both satisfy the Jacobi identity and imply the existence of an infinite number of conserved quantities in involution.

Note:

v2: 39 pages, minor comments added, matches published version

classical integrability
Poisson algebra
massive integrable 2D field theories
symmetric space sine-Gordon models
algebra: Poisson
dimension: 2
field theory: integrability
monodromy
regularization
Jacobi identity

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[1]

Magnons, their Solitonic Avatars and the Pohlmeyer Reduction

- JHEP 04 (2009) 060
•
e-Print:
- 0902.2405
•
DOI:
- 10.1088/1126-6708/2009/04/060

[2]

The Semi-Classical Spectrum of Solitons and Giant Magnons

- JHEP 05 (2011) 062
•
e-Print:
- 1103.3148
•
DOI:
- 10.1007/JHEP05(2011)062

[3]

The Structure of Non-Abelian Kinks

- JHEP 10 (2013) 058
•
e-Print:
- 1306.6651
•
DOI:
- 10.1007/JHEP10(2013)058

[4]

Hamiltonian Methods in the Theory of Solitons

L. Faddeev
,
L.L.A. Takhtajan

•
DOI:
- 10.1007/978-3-540-69969-9

[5]

Introduction to Classical Integrable Systems

DOI:
- 10.1017/CBO9780511535024

[6]

Quantum inverse problem method. I

- Theor.Math.Phys. 40 (1979) 2, 688-706,
- Teor.Mat.Fiz. 40 (1979) 194-220
•
DOI:
- 10.1007/BF01018718

[7]

Quantum version of the method of inverse scattering problem

E.K. Sklyanin
(
- Steklov Math. Inst., St. Petersburg
)

- J.Sov.Math. 19 (1982) 1546-1596,
- Zap.Nauchn.Semin. 95 (1980) 55-128
•
DOI:
- 10.1007/BF01091462

[8]

Classical and Quantum Solitons in the Symmetric Space Sine-Gordon Theories

- JHEP 04 (2011) 119
•
e-Print:
- 1012.0716
•
DOI:
- 10.1007/JHEP04(2011)119

[9]

Kac-moody Algebra and Extended {Yang-Baxter} Relations in the O( $N$ ) Nonlinear $\sigma$ Model

Jean Michel Maillet
(
- Paris U., VI-VII
)

- Phys.Lett.B 162 (1985) 137-142
•
DOI:
- 10.1016/0370-2693(85)91075-5

[10]

New Integrable Canonical Structures in Two-dimensional Models

Jean Michel Maillet
(
- Paris U., VI-VII
)

- Nucl.Phys.B 269 (1986) 54-76
•
DOI:
- 10.1016/0550-3213(86)90365-2

[11]

Alleviating the non-ultralocality of coset sigma models through a generalized Faddeev-Reshetikhin procedure

F. Delduc
(
- Lyon, Ecole Normale Superieure
)
,
M. Magro
(
- Lyon, Ecole Normale Superieure
)
,
B. Vicedo
(
- York U., England
)

- JHEP 08 (2012) 019
•
e-Print:
- 1204.0766
•
DOI:
- 10.1007/JHEP08(2012)019

[12]

A lattice Poisson algebra for the Pohlmeyer reduction of the $AdS_5 \times S^5$ superstring

F. Delduc
(
- Lyon, Ecole Normale Superieure
)
,
M. Magro
(
- Lyon, Ecole Normale Superieure
)
,
B. Vicedo
(
- York U., England
)

- Phys.Lett.B 713 (2012) 347-349
•
e-Print:
- 1204.2531
•
DOI:
- 10.1016/j.physletb.2012.06.028

[13]

Quadratic algebras and integrable systems

L. Freidel
(
- Paris U., VI-VII
)
,
J.M. Maillet
(
- Paris U., VI-VII
)

- Phys.Lett.B 262 (1991) 278-284
•
DOI:
- 10.1016/0370-2693(91)91566-E

[14]

On classical and quantum integrable field theories associated to Kac-Moody current algebras

L. Freidel
(
- Paris U., VI-VII
)
,
J.M. Maillet
(
- Paris U., VI-VII
)

- Phys.Lett.B 263 (1991) 403-410
•
DOI:
- 10.1016/0370-2693(91)90479-A

[15]

On integrable field theories as dihedral affine Gaudin models

Benoit Vicedo
(
- Hertfordshire U.
)

- Int.Math.Res.Not. 2020 (2020) 15, 4513-4601
•
e-Print:
- 1701.04856
•
DOI:
- 10.1093/imrn/rny128

[16]

Integrable Hamiltonian Systems and Interactions Through Quadratic Constraints

K. Pohlmeyer
(
- Hamburg U.
)

- Commun.Math.Phys. 46 (1976) 207-221,
- Commun.Math.Phys.46 (1976) 207-221 and Hamburg Desy - DESY 75-28 (75,REC.AUG) 29p
•
DOI:
- 10.1007/BF01609119