scholarly journals 4-dimensional Chern-Simons theory and integrable field theories

2022 ◽  
Author(s):  
Sylvain Lacroix
Keyword(s):  
Integrable Field Theories ◽  
Sigma Models ◽  
Field Theories ◽  
Simons Theory ◽  
Chiral Model ◽  
Santiago De Compostela ◽  
Chern Simons ◽  
Classical Integrable ◽  
Integrable Field ◽  
Chern Simons Theory

Abstract These lecture notes concern the semi-holomorphic 4d Chern-Simons theory and its applications to classical integrable field theories in 2d and in particular integrable sigma-models. After introducing the main properties of the Chern-Simons theory in 3d, we will define its 4d analogue and explain how it is naturally related to the Lax formalism of integrable 2d theories. Moreover, we will explain how varying the boundary conditions imposed on this 4d theory allows to recover various occurences of integrable sigma-models through this construction, in particular illustrating this on two simple examples: the Principal Chiral Model and its Yang-Baxter deformation. These notes were written for the lectures delivered at the school “Integrability, Dualities and Deformations”, that ran from 23 to 27 August 2021 in Santiago de Compostela and virtually.

scholarly journals 4D Chern–Simons theory and affine Gaudin models

2021 ◽  
Vol 111 (1) ◽  
Author(s):  
Benoît Vicedo
Keyword(s):  
Gauge Theory ◽  
Integrable Field Theories ◽  
Field Theories ◽  
Simons Theory ◽  
Gaudin Models ◽  
Chern Simons ◽  
Classical Integrable ◽  
Integrable Field ◽  
Chern Simons Theory

AbstractWe relate two formalisms recently proposed for describing classical integrable field theories. The first (Costello and Yamazaki in Gauge Theory and Integrability, III, 2019) is based on the action of four-dimensional Chern–Simons theory introduced and studied by Costello, Witten and Yamazaki. The second (Costello and Yamazaki, in Gauge Theory and Integrability, III, 2017) makes use of classical generalised Gaudin models associated with untwisted affine Kac–Moody algebras.

scholarly journals Asymptotic States in Nonlocal Field Theories

1997 ◽  
Vol 12 (07) ◽  
pp. 493-500 ◽  
Author(s):  
D. G. Barci ◽  
L. E. Oxman
Keyword(s):  
Minkowski Space ◽  
Field Theories ◽  
Simons Theory ◽  
Asymptotic State ◽  
Topological Sector ◽  
Nonlocal Field ◽  
Chern Simons ◽  
Chern Simons Theory

Asymptotic states in field theories containing nonlocal kinetic terms are analyzed using the canonical method, naturally defined in Minkowski space. We apply our results to study the asymptotic states of a nonlocal Maxwell–Chern–Simons theory coming from bosonization in (2+1) dimensions. We show that in this case the only asymptotic state of the theory, in the trivial (non-topological) sector, is the vacuum.

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.

scholarly journals INTEGRABLE FIELD THEORIES DERIVED FROM 4-D SELF-DUAL GRAVITY

1996 ◽  
Vol 11 (07) ◽  
pp. 545-552 ◽  
Author(s):  
TATSUYA UENO
Keyword(s):  
Integrable Field Theories ◽  
Sigma Models ◽  
Einstein Equation ◽  
Differential Form ◽  
The Self ◽  
Higgs Bundle ◽  
Field Theories ◽  
Infinite Dimensional ◽  
Dimensional Group ◽  
Integrable Field

We reformulate the self-dual Einstein equation as a trio of differential form equations for simple two-forms. Using them, we can quickly show the equivalence of the theory and 2-D sigma models valued in infinite-dimensional group, which was shown by Park and Husain earlier. We also derive other field theories including the 2-D Higgs bundle equation. This formulation elucidates the relation among these field theories.

scholarly journals A GENERAL SOLUTION OF THE BV-MASTER EQUATION AND BRST FIELD THEORIES

1993 ◽  
Vol 08 (22) ◽  
pp. 2087-2097 ◽  
Author(s):  
ÖMER F. DAYI
Keyword(s):  
Master Equation ◽  
General Procedure ◽  
Field Theories ◽  
Simons Theory ◽  
Proper Solution ◽  
First Order ◽  
Brst Charge ◽  
Chern Simons ◽  
Chern Simons Theory

For a class of first order gauge theories it was shown that the proper solution of the BV-master equation can be obtained straightforwardly. Here we present the general condition which the gauge generators should satisfy to conclude that this construction is relevant. The general procedure is illustrated by its application to the Chern-Simons theory in any odd dimension. Moreover, it is shown that this formalism is also applicable to BRST field theories when one replaces the role of the exterior derivative with the BRST charge of first quantization.

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.

scholarly journals A unifying 2D action for integrable $$\sigma $$-models from 4D Chern–Simons theory

2020 ◽  
Vol 110 (7) ◽  
pp. 1645-1687 ◽  
Author(s):  
Francois Delduc ◽  
Sylvain Lacroix ◽  
Marc Magro ◽  
Benoît Vicedo
Keyword(s):  
Sigma Models ◽  
Simons Theory ◽  
Chern Simons ◽  
Chern Simons Theory

scholarly journals REPRESENTATIONS OF COMPOSITE BRAIDS AND INVARIANTS FOR MUTANT KNOTS AND LINKS IN CHERN-SIMONS FIELD THEORIES

1995 ◽  
Vol 10 (22) ◽  
pp. 1635-1658 ◽  
Author(s):  
P. RAMADEVI ◽  
T.R. GOVINDARAJAN ◽  
R.K. KAUL
Keyword(s):  
Field Theories ◽  
Simons Theory ◽  
Conformal Field Theories ◽  
Abelian Gauge ◽  
Gauge Groups ◽  
Knot Invariants ◽  
Conformal Field ◽  
Chern Simons ◽  
Knots And Links ◽  
Chern Simons Theory

We show that any of the new knot invariants obtained from Chern-Simons theory based on an arbitrary non-Abelian gauge group do not distinguish isotopically inequivalent mutant knots and links. In an attempt to distinguish these knots and links, we study Murakami (symmetrized version) r-strand composite braids. Salient features of the theory of such composite braids are presented. Representations of generators for these braids are obtained by exploiting properties of Hilbert spaces associated with the correlators of Wess-Zumino conformal field theories. The r-composite invariants for the knots are given by the sum of elementary Chern-Simons invariants associated with the irreducible representations in the product of r representations (allowed by the fusion rules of the corresponding Wess-Zumino conformal field theory) placed on r individual strands of the composite braid. On the other hand, composite invariants for links are given by a weighted sum of elementary multicolored Chern-Simons invariants. Some mutant links can be distinguished through the composite invariants, but mutant knots do not share this property. The results, though developed in detail within the framework of SU(2) Chern-Simons theory are valid for any other non-Abelian gauge groups.

scholarly journals On the Zakharov–Mikhailov action: $$4\hbox {d}$$ Chern–Simons origin and covariant Poisson algebra of the Lax connection

2021 ◽  
Vol 111 (3) ◽  
Author(s):  
Vincent Caudrelier ◽  
Matteo Stoppato ◽  
Benoît Vicedo
Keyword(s):  
Equations Of Motion ◽  
Poisson Algebra ◽  
Simons Theory ◽  
Chiral Model ◽  
Hamilton Equation ◽  
First Time ◽  
Chern Simons ◽  
Special Case ◽  
Rational Type ◽  
Chern Simons Theory

AbstractWe derive the $$2\hbox {d}$$ 2 d Zakharov–Mikhailov action from $$4\hbox {d}$$ 4 d Chern–Simons theory. This $$2\hbox {d}$$ 2 d action is known to produce as equations of motion the flatness condition of a large class of Lax connections of Zakharov–Shabat type, which includes an ultralocal variant of the principal chiral model as a special case. At the $$2\hbox {d}$$ 2 d level, we determine for the first time the covariant Poisson bracket r-matrix structure of the Zakharov–Shabat Lax connection, which is of rational type. The flatness condition is then derived as a covariant Hamilton equation. We obtain a remarkable formula for the covariant Hamiltonian in terms of the Lax connection which is the covariant analogue of the well-known formula “$$H={{\,\mathrm{Tr}\,}}L^2$$ H = Tr L 2 ”.

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