scholarly journals Chirality, a new key for the definition of the connection and curvature of a Lie-Kac superalgebra

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jean Thierry-Mieg
Keyword(s):  
Scalar Field ◽  
Lie Algebra ◽  
Bianchi Identity ◽  
Parallel Transport ◽  
Lie Superalgebra ◽  
Natural Generalization ◽  
Yang Mills ◽  
Complex Functions ◽  
Definition Of ◽  
Mills Field

Abstract A natural generalization of a Lie algebra connection, or Yang-Mills field, to the case of a Lie-Kac superalgebra, for example SU(m/n), just in terms of ordinary complex functions and differentials, is proposed. Using the chirality χ which defines the supertrace of the superalgebra: STr(…) = Tr(χ…), we construct a covariant differential: D = χ(d + A) + Φ, where A is the standard even Lie-subalgebra connection 1-form and Φ a scalar field valued in the odd module. Despite the fact that Φ is a scalar, Φ anticommutes with (χA) because χ anticommutes with the odd generators hidden in Φ. Hence the curvature F = DD is a superalgebra-valued linear map which respects the Bianchi identity and correctly defines a chiral parallel transport compatible with a generic Lie superalgebra structure.

ANTI-SELF-DUAL SOLUTIONS OF THE YANG-MILLS EQUATIONS IN 4n DIMENSIONS

1992 ◽  
Vol 07 (23) ◽  
pp. 2077-2085 ◽  
Author(s):  
A. D. POPOV
Keyword(s):  
Scalar Field ◽  
Euclidean Space ◽  
Lie Algebra ◽  
Linear Equations ◽  
Dual Solutions ◽  
Gauge Fields ◽  
System Of Equations ◽  
Semisimple Lie Algebra ◽  
Yang Mills ◽  
Self Duality

The anti-self-duality equations for gauge fields in d = 4 and a generalization of these equations to dimension d = 4n are considered. For gauge fields with values in an arbitrary semisimple Lie algebra [Formula: see text] we introduce the ansatz which reduces the anti-self-duality equations in the Euclidean space ℝ4n to a system of equations breaking up into the well known Nahm's equations and some linear equations for scalar field φ.

Field models

2021 ◽  
pp. 42-69
Author(s):  
Iosif L. Buchbinder ◽  
Ilya L. Shapiro
Keyword(s):  
Electromagnetic Field ◽  
Scalar Field ◽  
Spinor Field ◽  
Vector Fields ◽  
Sigma Model ◽  
Complex Scalar ◽  
Yang Mills ◽  
Scalar Field Models ◽  
Complex Scalar Field ◽  
Mills Field

This chapter provides constructions of Lagrangians for various field models and discusses the basic properties of these models. Concrete examples of field models are constructed, including real and complex scalar field models, the sigma model, spinor field models and models of massless and massive free vector fields. In addition, the chapter discusses various interactions between fields, including the interactions of scalars and spinors with the electromagnetic field. A detailed discussion of the Yang-Mills field is given as well.

scholarly journals Decay of a Yang-Mills field coupled to a scalar field

1979 ◽  
Vol 67 (1) ◽  
pp. 51-67 ◽  
Author(s):  
Robert T. Glassey ◽  
Walter A. Strauss
Keyword(s):  
Scalar Field ◽  
Yang Mills ◽  
Mills Field

scholarly journals MAPPING A MASSLESS SCALAR FIELD THEORY ON A YANG–MILLS THEORY: CLASSICAL CASE

2009 ◽  
Vol 24 (30) ◽  
pp. 2425-2432 ◽  
Author(s):  
MARCO FRASCA
Keyword(s):  
Field Theory ◽  
Scalar Field ◽  
Massless Scalar ◽  
Massless Scalar Field ◽  
Field Equations ◽  
Scalar Field Theory ◽  
Classical Level ◽  
Gradient Expansion ◽  
Yang Mills ◽  
Mills Field

We analyze a recent proposal to map a massless scalar field theory onto a Yang–Mills theory at classical level. It is seen that this mapping exists at a perturbative level when the expansion is a gradient expansion. In this limit the theories share the spectrum, at the leading order, that is the one of a harmonic oscillator. Gradient expansion is exploited maintaining Lorentz covariance by introducing a fifth coordinate and turning the theory to Euclidean space. These expansions give common solutions to scalar and Yang–Mills field equations that are so proved to exist by construction, confirming that the selected components of the Yang–Mills field are indeed an extremum of the corresponding action functional.

scholarly journals Nonassociative Algebras: A Framework for Differential Geometry

2003 ◽  
Vol 2003 (60) ◽  
pp. 3777-3795 ◽  
Author(s):  
Lucian M. Ionescu
Keyword(s):  
Differential Geometry ◽  
Lie Algebra ◽  
Vector Fields ◽  
Bianchi Identity ◽  
Affine Connection ◽  
Natural Generalization ◽  
Nonassociative Algebra ◽  
Lie Bracket ◽  
Algebra Of Functions ◽  
Starting Point

A nonassociative algebra endowed with a Lie bracket, called atorsion algebra, is viewed as an algebraic analog of a manifold with an affine connection. Its elements are interpreted as vector fields and its multiplication is interpreted as a connection. This provides a framework for differential geometry on a formal manifold with a formal connection. A torsion algebra is a natural generalization of pre-Lie algebras which appear as the “torsionless” case. The starting point is the observation that the associator of a nonassociative algebra is essentially the curvature of the corresponding Hochschild quasicomplex. It is a cocycle, and the corresponding equation is interpreted as Bianchi identity. The curvature-associator-monoidal structure relationships are discussed. Conditions on torsion algebras allowing to construct an algebra of functions, whose algebra of derivations is the initial Lie algebra, are considered. The main example of a torsion algebra is provided by the pre-Lie algebra of Hochschild cochains of aK-module, with Lie bracket induced by Gerstenhaber composition.

Large-order estimates for perturbation theory of a Yang-Mills field coupled to a scalar field

1979 ◽  
Vol 19 (10) ◽  
pp. 2974-2983 ◽  
Author(s):  
L. N. Lipatov ◽  
A. P. Bukhvostov ◽  
E. I. Malkov
Keyword(s):  
Perturbation Theory ◽  
Scalar Field ◽  
Yang Mills ◽  
Large Order ◽  
Mills Field

scholarly journals Increasing Potentials in Classical Non-Abelian and Abelian Gauge Theories

1997 ◽  
Vol 12 (26) ◽  
pp. 4823-4830 ◽  
Author(s):  
D. Singleton ◽  
A. Yoshida
Keyword(s):  
Exact Solution ◽  
Electromagnetic Field ◽  
Scalar Field ◽  
Coulomb Potential ◽  
Gauge Theories ◽  
Field Energy ◽  
Infinite Field ◽  
Abelian Gauge ◽  
Yang Mills ◽  
Mills Field

An exact solution for an SU(2) Yang–Mills field coupled to a scalar field is given, which has potentials with linear, 1/r and 1/r2 parts. This may be of some interest since some phenomenological QCD studies assume a linear plus Coulomb potential. We also show that in the Nielsen–Olesen Abelian model there is an exact solution in the BPS limit, which has a 1/r electromagnetic field and a logarithmically rising scalar field. Both of these solutions must be cutoff from above to avoid infinite field energy.

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