scholarly journals The 3-Lie algebra (2,0) tensor multiplet and equations of motion on loop space

2011 ◽  
Vol 2011 (5) ◽  
Author(s):  
Constantinos Papageorgakis ◽  
Christian Sämann
Keyword(s):  
Lie Algebra ◽  
Loop Space ◽  
Equations Of Motion ◽  
Tensor Multiplet

scholarly journals A COHOMOLOGICAL INTERPRETATION OF THE MIGDAL–MAKEENKO EQUATIONS

2002 ◽  
Vol 17 (08) ◽  
pp. 481-489 ◽  
Author(s):  
A. AGARWAL ◽  
S. G. RAJEEV
Keyword(s):  
Lie Algebra ◽  
Equations Of Motion ◽  
Generating Functional ◽  
Yang Mills ◽  
Infinite Dimensional ◽  
Change Of Variables ◽  
Anomalous Term ◽  
Large N ◽  
Infinite Dimensional Lie Algebra ◽  
Cohomological Interpretation

The equations of motion of quantum Yang–Mills theory (in the planar "large-N" limit), when formulated in loop-space are shown to have an anomalous term, which makes them analogous to the equations of motion of WZW models. The anomaly is the Jacobian of the change of variables from the usual ones, i.e. the connection one-form A, to the holonomy U. An infinite-dimensional Lie algebra related to this change of variables (the Lie algebra of loop substitutions) is developed, and the anomaly is interpreted as an element of the first cohomology of this Lie algebra. The Migdal–Makeenko equations are shown to be the condition for the invariance of the Yang–Mills generating functional Z under the action of the generators of this Lie algebra. Connections of this formalism to the collective field approach of Jevicki and Sakita are also discussed.

scholarly journals NOISE AND DISSIPATION IN QUANTUM THEORY

1990 ◽  
Vol 02 (02) ◽  
pp. 127-176 ◽  
Author(s):  
LUIGI ACCARDI
Keyword(s):  
Quantum Theory ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Langevin Equation ◽  
Equations Of Motion ◽  
Natural Generalization ◽  
Usual Sense ◽  
Heisenberg Equation ◽  
Generalized Langevin Equation ◽  
Solid State Physics

A model independent generalization of quantum mechanics, including the usual as well as the dissipative quantum systems, is proposed. The theory is developed deductively from the basic principles of the standard quantum theory, the only new qualitative assumption being that we allow the wave operator at time t of a quantum system to be non-differentiable (in t) in the usual sense, but only in an appropriately defined (Sec. 5) stochastic sense. The resulting theory is shown to lead to a natural generalization of the usual quantum equations of motion, both in the form of the Schrödinger equation in interaction representation (Sec. 6) and of the Heisenberg equation (Sec. 8). The former equation leads in particular to a quantum fluctuation-dissipation relation of Einstein’s type. The latter equation is a generalized Langevin equation, from which the known form of the generalized master equation can be deduced via the quantum Feynmann-Kac technique (Secs. 9 and 10). For quantum noises with increments commuting with the past the quantum Langevin equation defines a closed system of (usually nonlinear) stochastic differential equations for the observables defining the coefficients of the noises. Such systems are parametrized by certain Lie algebras of observables of the system (Sec. 10). With appropriate choices of these Lie algebras one can deduce generalizations and corrections of several phenomenological equations previously introduced at different times to explain different phenomena. Two examples are considered: the Lie algebra [q, p]=i (Sec. 12), which is shown to lead to the equations of the damped harmonic oscillator; and the Lie algebra of SO(3) (Sec. 13) which is shown to lead to the Bloch equations. In both cases the equations obtained are independent of the model of noise. Moreover, in the former case, it is proved that the only possible noises which preserve the commutation relations of p, q are the quantum Brownian motions, commonly used in laser theory and solid state physics.

scholarly journals A pure connection formulation with real fields for gravity

2020 ◽  
pp. 2050114
Author(s):  
J. E. Rosales-Quintero
Keyword(s):  
Lie Algebra ◽  
Lagrange Multipliers ◽  
General Class ◽  
Equations Of Motion ◽  
Conformally Flat ◽  
Einstein Manifolds ◽  
De Sitter ◽  
Killing Form ◽  
Dual Approach ◽  
Four Dimensions

We study an [Formula: see text] pure connection formulation in four dimensions for real-valued fields, inspired by the Capovilla, Dell and Jacobson complex self-dual approach. By considering the CMPR BF action, also, taking into account a more general class of the Cartan–Killing form for the Lie algebra [Formula: see text] and by refining the structure of the Lagrange multipliers, we integrate out the metric variables in order to obtain the pure connection action. Once we have obtained this action, we impose certain restrictions on the Lagrange multipliers, in such a way that the equations of motion led us to a family of torsionless conformally flat Einstein manifolds, parametrized by two numbers. Finally, we show that, by a suitable choice of parameters, self-dual spaces (Anti-) de Sitter can be obtained.

scholarly journals THE PICARD GROUP OF THE LOOP SPACE OF THE RIEMANN SPHERE

2010 ◽  
Vol 21 (11) ◽  
pp. 1387-1399
Author(s):  
NING ZHANG
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Loop Space ◽  
Riemann Sphere ◽  
Loop Group ◽  
Picard Group ◽  
Line Bundles ◽  
Projective Embedding ◽  
Infinite Dimensional ◽  
Holomorphic Line Bundles

The loop space Lℙ1 of the Riemann sphere consisting of all Ck or Sobolev Wk, p maps S1 → ℙ1 is an infinite dimensional complex manifold. We compute the Picard group pic(Lℙ1) of holomorphic line bundles on Lℙ1 as an infinite dimensional complex Lie group with Lie algebra the Dolbeault group H0, 1(Lℙ1). The group G of Möbius transformations and its loop group LG act on Lℙ1. We prove that an element of pic(Lℙ1) is LG-fixed if it is G-fixed, thus completely answering the question of Millson and Zombro about the G-equivariant projective embedding of Lℙ1.

CONFORMALLY REDUCED WZNW THEORY, NEW EXTENDED CHIRAL ALGEBRAS AND THEIR ASSOCIATED TODA TYPE INTEGRABLE SYSTEMS

1992 ◽  
Vol 07 (28) ◽  
pp. 7015-7043 ◽  
Author(s):  
BO-YU HOU ◽  
LIU CHAO
Keyword(s):  
Large Class ◽  
Lie Algebra ◽  
Integrable Systems ◽  
Equations Of Motion ◽  
Chiral Algebras ◽  
Block Diagonal

We propose and analyze a large class of conformal reductions Cons [g(H, d)] of WZNW theory based on the integral gradations of the underlying Lie algebra g. The W bases of the associated W algebras W[g(H, d)] are constructed under the generalized Drinfeld-Sokolov gauge which we call O’Raifeartaigh gauge of the constrained Kac-Moody currents, and the equations of motion of the extended Toda type integrable systems corresponding to these W algebras are also derived. As an example, we construct explicitly the W algebra associated with the (pqp) block diagonal decomposition of sl2p+q, namely W[(pqp)2], and discuss some of the properties thereof.

scholarly journals VERTEX OPERATORS FOR AdS3 WITH RAMOND BACKGROUND

2001 ◽  
Vol 16 (05) ◽  
pp. 812-821
Author(s):  
LOUISE DOLAN
Keyword(s):  
Equations Of Motion ◽  
Vertex Operators ◽  
Linearized Equations ◽  
Tensor Multiplet

The computation of exact vertex operators for the Type IIB superstring in an AdS3×S3 background with Ramond-Ramond flux is described. The components of these vertex operators are shown to satisfy the supergravity linearized equations of motion for the six-dimensional (2,0) theory of a supergravity and tensor multiplet expanded around AdS3×S3 spacetime.

scholarly journals Spin-Valued Four Bosons Electrodynamics

2021 ◽  
Vol 19 ◽  
pp. 93-133
Author(s):  
R. Doria ◽  
I. Soares
Keyword(s):  
Lie Algebra ◽  
Electric Charge ◽  
Lorentz Group ◽  
Vector Fields ◽  
Magnetic Moments ◽  
Equations Of Motion ◽  
Spin Coupling ◽  
Abelian Gauge ◽  
Spin Effects ◽  
Massive Photon

Electromagnetism is based on electric charge and spin. The study here corresponds to understand on spin effects at a vectorial electrodynamics. Its scenario is a non-linear abelian electromagnetism where the electric charge is transmitted through a four bosons quadruplet, constituted by the usual photon, massive photon and charged massive photons. These four bosons intermediate the charge exchange ΔQ = 0, ±1.The spin is introduced at first principles. A spintronics Lagrangian for four vector fields is performed. Considering that spin is a space-time physical entity derived from Lorentz Group, these vector fields are associated to Lorentz Group, as Lie algebra valued. Similarly to non-abelian gauge theories where Aμ≡ Aμ,ata, one introduces the relationship Aμ≡ Aμ,κλΣκλ where (Σκλ)αβ is the Lorentz Group generator. Thus, based on three fundamentals which are light invariance, electric charge conservation law and vector fields Lie algebra valued through Lorentz Group generators, one derives a spin-valued four vectorial electrodynamics. It is given by the fields quadruplet Aμ1 ≡ {Aμ, Uμ, Vμ±}  where Aμ means the usual photon, Uμ a massive photon and Vμ± massive charged photons. Two novelties appear. The first one is that, new terms are developed into usual four bosons electromagnetism. They contribute to Lagrangian, equations of motion, Noether theorem. The second one is that the equations of motion derive a renormalizable spin coupling with the electric and magnetic fields.There is a spin-1 electrodynamics to be investigated. A neutral electromagnetism is mandatory to be analyzed. Something beyond dipole, quadrupole and so on. Understand the role of spin in the electrical and magnetic properties of particles. A spin vectorial expression S-->  is obtained. It adds EM interactions not depending on electric charge and with spin interactions through electric dipole and magnetic moments.

scholarly journals Spin Current in BF Theory

Physics ◽  
2021 ◽  
Vol 3 (2) ◽  
pp. 427-448
Author(s):  
Malik Almatwi
Keyword(s):  
Lie Algebra ◽  
Equations Of Motion ◽  
Spin Current ◽  
Spin Connection ◽  
Bf Theory ◽  
Cylindrically Symmetric ◽  
Covariant Derivation ◽  
A Current ◽  
Symmetric Systems

In this paper, a current that is called spin current and corresponds to the variation of the matter action in BF theory with respect to the spin connection A which takes values in Lie algebra so(3,C), in self-dual formalism is introduced. For keeping the 2-form Bi constraint (covariant derivation) DBi=0 satisfied, it is suggested adding a new term to the BF Lagrangian using a new field ψi, which can be used for calculating the spin current. The equations of motion are derived and the solutions are dicussed. It is shown that the solutions of the equations do not require a specific metric on the 4-manifold M, and one just needs to know the symmetry of the system and the information about the spin current. Finally, the solutions for spherically and cylindrically symmetric systems are found.

scholarly journals Generalized Loop Space and Evolution of the Light-Like Wilson Loops

2015 ◽  
Vol 37 ◽  
pp. 1560028
Author(s):  
Igor O. Cherednikov ◽  
Tom Mertens
Keyword(s):  
Loop Space ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Wilson Loops ◽  
Conformal Invariant ◽  
Gauge Invariant ◽  
Geometrical Properties ◽  
Group Behaviour ◽  
Non Local ◽  
Invariant Quantum

Equations of motion for the light-like QCD Wilson loops are studied in the generalized loop space (GLS) setting. To this end, the classically conformal-invariant non-local variations of the cusped Wilson exponentials lying (partially) on the light-cone are formulated in terms of the Fréchet derivative. The rapidity and renormalization-group behaviour of the gauge-invariant quantum correlation functions (in particular, the three-dimensional parton densities) are demonstrated to be connected to certain geometrical properties of the Wilson loops defined in the GLS.

scholarly journals The non-abelian tensor multiplet in loop space

2006 ◽  
Vol 2006 (01) ◽  
pp. 165-165 ◽  
Author(s):  
Andreas Gustavsson
Keyword(s):  
Loop Space ◽  
Tensor Multiplet
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