scholarly journals Twistor Actions for Integrable Systems

2021 ◽  
Vol 2021 (9) ◽  
Author(s):  
Robert F. Penna
Keyword(s):  
Integrable Systems ◽  
Dimensional Reduction ◽  
Vector Bundles ◽  
Sigma Model ◽  
Twistor Space ◽  
Equations Of Motion ◽  
Simons Theory ◽  
Holomorphic Vector Bundles ◽  
Chern Simons ◽  
Chern Simons Theory

Abstract Many integrable systems can be reformulated as holomorphic vector bundles on twistor space. This is a powerful organizing principle in the theory of integrable systems. One shortcoming is that it is formulated at the level of the equations of motion. From this perspective, it is mysterious that integrable systems have Lagrangians. In this paper, we study a Chern-Simons action on twistor space and use it to derive the Lagrangians of some integrable sigma models. Our focus is on examples that come from dimensionally reduced gravity and supergravity. The dimensional reduction of general relativity to two spacetime dimensions is an integrable coset sigma model coupled to a dilaton and 2d gravity. The dimensional reduction of supergravity to two spacetime dimensions is an integrable coset sigma model coupled to matter fermions, a dilaton, and 2d supergravity. We derive Lax operators and Lagrangians for these 2d integrable systems using the Chern-Simons theory on twistor space. In the supergravity example, we use an extended setup in which twistor Chern-Simons theory is coupled to a pair of matter fermions.

REDUCTIONS OF CHERN–SIMONS THEORY RELATED TO INTEGRABLE SYSTEMS WHICH HAVE GEOMETRIC APPLICATIONS

2004 ◽  
Vol 18 (09) ◽  
pp. 1261-1275 ◽  
Author(s):  
PAUL BRACKEN
Keyword(s):  
Integrable Systems ◽  
Equations Of Motion ◽  
Moving Frame ◽  
Simons Theory ◽  
Gauge Potential ◽  
Pure Gauge ◽  
Zero Curvature ◽  
Lagrangian Form ◽  
Chern Simons ◽  
Chern Simons Theory

The Chern–Simons functional is introduced in terms of chiral fields and then studied here. The current can be regarded as a non-Abelian pure gauge potential so that the zero-curvature equations are of Lagrangian form for pure non-Abelian Chern–Simons theory. The equations of motion are developed and a formalism which connects the zero curvature equations with a related moving trihedral is introduced. The moving frame equations are written down for the system and a correspondence between these equations and several related elementary integrable systems is described in the same formalism as well.

scholarly journals 4d Chern-Simons theory as a 3d Toda theory, and a 3d-2d correspondence

2021 ◽  
Vol 2021 (9) ◽  
Author(s):  
Meer Ashwinkumar ◽  
Kee-Seng Png ◽  
Meng-Chwan Tan
Keyword(s):  
Moduli Space ◽  
Sigma Model ◽  
Three Dimensional ◽  
Simons Theory ◽  
Yang Mills ◽  
Pole Type ◽  
Type Boundary ◽  
Chern Simons ◽  
Manifold With Corners ◽  
Chern Simons Theory

Abstract We show that the four-dimensional Chern-Simons theory studied by Costello, Witten and Yamazaki, is, with Nahm pole-type boundary conditions, dual to a boundary theory that is a three-dimensional analogue of Toda theory with a novel 3d W-algebra symmetry. By embedding four-dimensional Chern-Simons theory in a partial twist of the five-dimensional maximally supersymmetric Yang-Mills theory on a manifold with corners, we argue that this three-dimensional Toda theory is dual to a two-dimensional topological sigma model with A-branes on the moduli space of solutions to the Bogomolny equations. This furnishes a novel 3d-2d correspondence, which, among other mathematical implications, also reveals that modules of the 3d W-algebra are modules for the quantized algebra of certain holomorphic functions on the Bogomolny moduli space.

scholarly journals Yang-Baxter deformations of the AdS5×S5 supercoset sigma model from 4D Chern-Simons theory

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Osamu Fukushima ◽  
Jun-ichi Sakamoto ◽  
Kentaroh Yoshida
Keyword(s):  
Boundary Conditions ◽  
Sigma Model ◽  
Simons Theory ◽  
Chiral Model ◽  
Model Case ◽  
Principal Chiral Model ◽  
Chern Simons ◽  
Chern Simons Theory

Abstract We present homogeneous Yang-Baxter deformations of the AdS5×S5 supercoset sigma model as boundary conditions of a 4D Chern-Simons theory. We first generalize the procedure for the 2D principal chiral model developed by Delduc et al. [5] so as to reproduce the 2D symmetric coset sigma model, and specify boundary conditions governing homogeneous Yang-Baxter deformations. Then the conditions are applicable for the AdS5×S5 supercoset sigma model case as well. In addition, homogeneous bi-Yang-Baxter deformation is also discussed.

The solution structure of the O(3) sigma model in a Maxwell-Chern-Simons theory

2017 ◽  
Vol 58 (7) ◽  
pp. 071503 ◽  
Author(s):  
Sze-Guang Yang ◽  
Zhi-You Chen ◽  
Jann-Long Chern
Keyword(s):  
Solution Structure ◽  
Sigma Model ◽  
Simons Theory ◽  
Chern Simons ◽  
Chern Simons Theory

scholarly journals Deformation of surfaces, integrable systems, and Chern–Simons theory

2001 ◽  
Vol 42 (3) ◽  
pp. 1397 ◽  
Author(s):  
L. Martina ◽  
Kur. Myrzakul ◽  
R. Myrzakulov ◽  
G. Soliani
Keyword(s):  
Integrable Systems ◽  
Simons Theory ◽  
Chern Simons ◽  
Chern Simons Theory

scholarly journals Symmetric space λ-model exchange algebra from 4d holomorphic Chern-Simons theory

2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
David M. Schmidtt
Keyword(s):  
Symmetric Space ◽  
Sigma Model ◽  
Hamiltonian Formalism ◽  
Simons Theory ◽  
Symmetry Principle ◽  
R Matrix ◽  
Exchange Algebra ◽  
Coset Spaces ◽  
Chern Simons ◽  
Chern Simons Theory

Abstract We derive, within the Hamiltonian formalism, the classical exchange algebra of a lambda deformed string sigma model in a symmetric space directly from a 4d holomorphic Chern-Simons theory. The explicit forms of the extended Lax connection and R-matrix entering the Maillet bracket of the lambda model are explained from a symmetry principle. This approach, based on a gauge theory, may provide a mechanism for taming the non-ultralocality that afflicts most of the integrable string theories propagating in coset spaces.

scholarly journals Integrable deformed T1,1 sigma models from 4D Chern-Simons theory

2021 ◽  
Vol 2021 (9) ◽  
Author(s):  
Osamu Fukushima ◽  
Jun-ichi Sakamoto ◽  
Kentaroh Yoshida
Keyword(s):  
Boundary Conditions ◽  
Sigma Models ◽  
Sigma Model ◽  
Target Space ◽  
Simons Theory ◽  
Non Linear ◽  
Linear Sigma Models ◽  
Chern Simons ◽  
Parameter Deformation ◽  
Chern Simons Theory

Abstract Recently, a variety of deformed T1,1 manifolds, with which 2D non-linear sigma models (NLSMs) are classically integrable, have been presented by Arutyunov, Bassi and Lacroix (ABL) [46]. We refer to the NLSMs with the integrable deformed T1,1 as the ABL model for brevity. Motivated by this progress, we consider deriving the ABL model from a 4D Chern-Simons (CS) theory with a meromorphic one-form with four double poles and six simple zeros. We specify boundary conditions in the CS theory that give rise to the ABL model and derive the sigma-model background with target-space metric and anti-symmetric two-form. Finally, we present two simple examples 1) an anisotropic T1,1 model and 2) a G/H λ-model. The latter one can be seen as a one-parameter deformation of the Guadagnini-Martellini-Mintchev model.

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