scholarly journals Lévy differential operators and Gauge invariant equations for Dirac and Higgs fields

2019 ◽  
Vol 22 (01) ◽  
pp. 1950001 ◽  
Author(s):  
Boris O. Volkov
Keyword(s):  
Differential Operators ◽  
Equations Of Motion ◽  
Parallel Transport ◽  
System Of Differential Equations ◽  
Gauge Invariant ◽  
Yang Mills ◽  
Infinite Dimensional ◽  
Dirac Equations ◽  
Lévy Laplacian ◽  
Higgs Fields

We study the Lévy infinite-dimensional differential operators (differential operators defined by the analogy with the Lévy Laplacian) and their relationship to the Yang–Mills equations. We consider the parallel transport on the space of curves as an infinite-dimensional analogue of chiral fields and show that it is a solution to the system of differential equations if and only if the associated connection is a solution to the Yang–Mills equations. This system is an analogue of the equations of motion of chiral fields and contains the Lévy divergence. The systems of infinite-dimensional equations containing Lévy differential operators, that are equivalent to the Yang–Mills–Higgs equations and the Yang–Mills–Dirac equations (the equations of quantum chromodynamics), are obtained. The equivalence of two ways to define Lévy differential operators is shown.

scholarly journals Stochastic Lévy differential operators and Yang–Mills equations

2017 ◽  
Vol 20 (02) ◽  
pp. 1750008 ◽  
Author(s):  
Boris O. Volkov
Keyword(s):  
Differential Operators ◽  
Parallel Transport ◽  
Directional Derivatives ◽  
Chiral Field ◽  
Action Functional ◽  
Yang Mills ◽  
Infinite Dimensional ◽  
Lévy Laplacian ◽  
Cesaro Averaging ◽  
The Relationship

The relationship between the Yang–Mills equations and the stochastic analogue of Lévy differential operators is studied. The value of the stochastic Lévy–Laplacian is found by means of Cèsaro averaging of directional derivatives on the stochastic parallel transport. It is shown that the Yang–Mills equations and the Lévy–Laplace equation for such Laplacian are not equivalent in contrast to the deterministic case. An equation equivalent to the Yang–Mills equations is obtained. The equation contains the Lévy divergence. It is proved that the Yang–Mills action functional can be represented as an infinite-dimensional analogue of the Direchlet functional of a chiral field. This analogue is also derived using Cèsaro averaging.

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 OFF-SHELL FORMULATION OF N=2 SUPER YANG-MILLS THEORIES COUPLED TO MATTER WITHOUT AUXILIARY FIELDS

1995 ◽  
Vol 10 (27) ◽  
pp. 3937-3950 ◽  
Author(s):  
NICOLA MAGGIORE
Keyword(s):  
Equations Of Motion ◽  
Dimensional Algebra ◽  
Nilpotent Operator ◽  
Gauge Transformations ◽  
Yang Mills ◽  
Infinite Dimensional ◽  
Field Dependent ◽  
External Sources ◽  
Shell Formulation

N=2 supersymmetric Yang-Mills theories coupled to matter are considered in the Wess-Zumino gauge. The supersymmetries are realized nonlinearly and the anticommutator between two susy charges gives, in addition to translations, gauge transformations and equations of motion. The difficulties hidden in such an algebraic structure are well known: almost always auxiliary fields can be introduced in order to put the formalism off-shell, but still the field-dependent gauge transformations give rise to an infinite-dimensional algebra quite hard to deal with. However, it is possible to avoid all these problems by collecting into a unique nilpotent operator all the symmetries defining the theory, namely ordinary BRS, supersymmetries and translations. According to this method the role of the auxiliary fields is covered by the external sources coupled, as usual, to the nonlinear variations of the quantum fields. The analysis is then formally reduced to that of ordinary Yang-Mills theory.

scholarly journals Lévy Laplacians and instantons on manifolds

2020 ◽  
Vol 23 (02) ◽  
pp. 2050008
Author(s):  
Boris O. Volkov
Keyword(s):  
Riemannian Manifold ◽  
Parallel Transport ◽  
Yang Mills ◽  
Infinite Dimensional ◽  
Laplace Equations

The equivalence of the anti-selfduality Yang–Mills equations on the four-dimensional orientable Riemannian manifold and the Laplace equations for some infinite-dimensional Laplacians is proved. A class of modified Lévy Laplacians parameterized by the choice of a curve in the group [Formula: see text] is introduced. It is shown that a connection is an instanton (a solution of the anti-selfduality Yang–Mills equations) if and only if the parallel transport generalized by this connection is a solution of the Laplace equations for some three modified Lévy Laplacians from this class.

LÉVY–LAPLACIAN AND THE GAUGE FIELDS

2012 ◽  
Vol 15 (04) ◽  
pp. 1250027 ◽  
Author(s):  
BORIS O. VOLKOV
Keyword(s):  
Smooth Manifold ◽  
Parallel Transport ◽  
Second Order ◽  
Gauge Fields ◽  
Directional Derivatives ◽  
Connection Form ◽  
Cesàro Mean ◽  
Yang Mills ◽  
Lévy Laplacian ◽  
Cesaro Mean

The following statement is proved for the Lévy–Laplacian defined as the Cesàro mean of second-order directional derivatives: a connection form on a base Riemannian C3-smooth manifold satisfies the Yang–Mills equations if and only if the parallel transport associated with the connection is Lévy harmonic. This statement is an improvement of the well-known result of L. Accardi, P. Gibilisco and I. V. Volovich2 (see also Ref. 1).

scholarly journals AdS superprojectors

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
E. I. Buchbinder ◽  
D. Hutchings ◽  
S. M. Kuzenko ◽  
M. Ponds
Keyword(s):  
Shell Model ◽  
Differential Operators ◽  
Second Order ◽  
Projection Operators ◽  
De Sitter ◽  
Gauge Invariant ◽  
Gauge Transformations ◽  
Four Dimensions ◽  
Higher Spin

Abstract Within the framework of $$ \mathcal{N} $$ N = 1 anti-de Sitter (AdS) supersymmetry in four dimensions, we derive superspin projection operators (or superprojectors). For a tensor superfield $$ {\mathfrak{V}}_{\alpha (m)\overset{\cdot }{\alpha }(n)}:= {\mathfrak{V}}_{\left(\alpha 1\dots \alpha m\right)\left({\overset{\cdot }{\alpha}}_1\dots {\overset{\cdot }{\alpha}}_n\right)} $$ V α m α ⋅ n ≔ V α 1 … αm α ⋅ 1 … α ⋅ n on AdS superspace, with m and n non-negative integers, the corresponding superprojector turns $$ {\mathfrak{V}}_{\alpha (m)\overset{\cdot }{\alpha }(n)} $$ V α m α ⋅ n into a multiplet with the properties of a conserved conformal supercurrent. It is demonstrated that the poles of such superprojectors correspond to (partially) massless multiplets, and the associated gauge transformations are derived. We give a systematic discussion of how to realise the unitary and the partially massless representations of the $$ \mathcal{N} $$ N = 1 AdS4 superalgebra $$ \mathfrak{osp} $$ osp (1|4) in terms of on-shell superfields. As an example, we present an off-shell model for the massive gravitino multiplet in AdS4. We also prove that the gauge-invariant actions for superconformal higher-spin multiplets factorise into products of minimal second-order differential operators.

scholarly journals Study of the gauge invariant, nonlocal mass operatorTr∫d4xFμν(D2)−1Fμνin Yang-Mills theories

2005 ◽  
Vol 72 (10) ◽  
Author(s):  
M. A. L. Capri ◽  
D. Dudal ◽  
J. A. Gracey ◽  
V. E. R. Lemes ◽  
R. F. Sobreiro ◽  
...  
Keyword(s):  
Gauge Invariant ◽  
Yang Mills

scholarly journals A MANIFESTLY GAUGE INVARIANT AND UNIVERSAL CALCULUS FORSU(N) YANG–MILLS

2006 ◽  
Vol 21 (23n24) ◽  
pp. 4627-4761 ◽  
Author(s):  
OLIVER J. ROSTEN
Keyword(s):  
Perturbation Theory ◽  
Renormalization Group ◽  
Explicit Dependence ◽  
Gauge Invariant ◽  
Yang Mills ◽  
Group A ◽  
Β Function ◽  
Exact Renormalization Group

Within the framework of the Exact Renormalization Group, a manifestly gauge invariant calculus is constructed for SU (N) Yang–Mills. The methodology is comprehensively illustrated with a proof, to all orders in perturbation theory, that the β function has no explicit dependence on either the seed action or details of the covariantization of the cutoff. The cancellation of these nonuniversal contributions is done in an entirely diagrammatic fashion.

scholarly journals The martingale problem for pseudo-differential operators on infinite-dimensional spaces

1999 ◽  
Vol 153 ◽  
pp. 101-118 ◽  
Author(s):  
V. Bogachev ◽  
P. Lescot ◽  
M. Röckner
Keyword(s):  
Differential Operators ◽  
Martingale Problem ◽  
Existence Of A Solution ◽  
Infinite Dimensional ◽  
Pseudo Differential Operators

AbstractA martingale problem for pseudo-differential operators on infinite dimensional spaces is formulated and the existence of a solution is proved. Applications to infinite dimensional “stable-like” processes are presented.

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