CONFORMALLY COVARIANT APPROACH TO THE INTEGRABILITY OF SDYM: LINEAR SYSTEM, β-PLANES, INFINITESIMAL BÄCKLUND TRANSFORMATIONS AND INFINITE-DIMENSIONAL ALGEBRAS

1988 ◽  
Vol 03 (05) ◽  
pp. 1263-1299 ◽  
Author(s):  
J. AVAN ◽  
H.J. de VEGA
Keyword(s):  
Linear System ◽  
The Self ◽  
Bäcklund Transformations ◽  
Dimensional Algebra ◽  
Yang Mills ◽  
Infinite Dimensional ◽  
System A ◽  
Backlund Transformations ◽  
Field Dependent ◽  
Arbitrary Analytic Function

The self-dual Yang-Mills theory is investigated with the help of a new conformally covariant linear system, where the spectral parameter is a projective twistor [Formula: see text]. We derive from this linear system conformally covariant families of β-planes, on which the potential Aµ(x) is a pure gauge. They are parametrized by a dual projective twistor [Formula: see text], orthogonal to the spectral twistor Λ. Conformally covariant infinitesimal Bäcklund transformations (B.T.) are constructed for the gauge group [Formula: see text] or [Formula: see text], and for SU (N). They are characterized by (1) a Lie-algebra index 1≤a≤ dim g; (2) the spectral twistor Λ; (3) a second twistor index 1≤α≤4, (independent of Λ); (4) an arbitrary (analytic) function of the two independent solutions of the free linear system (Aµ=0). The algebra of these infinitesimal B.T. is computed. It turns to close up to a field-dependent gauge transformation, which vanishes for equal twistor indices. The reduction of the number of components of Λ to a single projective parameter [Formula: see text] leads to a loop algebra. In general it yields an infinite-dimensional algebra with five indices.

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.

CONFORMALLY COVARIANT DRESSING TRANSFORMATIONS FOR SUPER SELF-DUALITY EQUATIONS IN D=4

1989 ◽  
Vol 04 (10) ◽  
pp. 971-982
Author(s):  
J. AVAN
Keyword(s):  
Linear System ◽  
Gauge Transformations ◽  
Four Dimensions ◽  
Yang Mills ◽  
Self Duality ◽  
Loop Algebras ◽  
Field Dependent

A set of conformally covariant dressing transformations is constructed for the supersym-metric N=3 self-duality equations in four dimensions, using the associated covariant linear system. They form a closed, 5+6-index algebra, up to field-dependent gauge transformations, containing the previously known loop algebras as a particular subset. This construction generalizes the formerly built set of conformally covariant DT for ordinary self-dual Yang-Mills.

INTEGRABLE ASPECTS OF NONCOMMUTATIVE ANTI-SELF-DUAL YANG-MILLS EQUATIONS

2008 ◽  
Vol 23 (14n15) ◽  
pp. 2237-2238 ◽  
Author(s):  
MASASHI HAMANAKA
Keyword(s):  
Integrable Systems ◽  
Soliton Solutions ◽  
Bäcklund Transformations ◽  
Soliton Theory ◽  
Yang Mills ◽  
Backlund Transformations ◽  
Exact Soliton Solutions

We discuss extension of soliton theory and integrable systems to non-commutative (NC) spaces, focusing on integrable aspects of NC Anti-Self-Dual Yang-Mills (ASDYM) equations. We give exact soliton solutions (with both finite- and infinite-action solutions) by means of Bäcklund transformations. In the construction of NC soliton solutions, one kind of NC determinants, quasideterminants, play crucial roles. This is partially based on collaboration with C. R. Gilson and J. J. C. Nimmo (Glasgow).

scholarly journals Asymptotic symmetries at null-infinity for the Rarita-Schwinger field with magnetic term

2021 ◽  
Author(s):  
Bilyana Lyudmilova Tomova
Keyword(s):  
Magnetic Charge ◽  
Flat Space ◽  
Space Time ◽  
Boundary Term ◽  
Null Infinity ◽  
Dimensional Algebra ◽  
Asymptotically Flat ◽  
Yang Mills ◽  
Infinite Dimensional ◽  
Asymptotic Symmetries

Abstract In this paper we study the magnetic charges of the free massless Rarita-Schwinger field in four dimensional asymptotically flat space-time. This is the first step towards extending the study of the dual BMS charges to supergravity. The magnetic charges appear due to the addition of a boundary term in the action. This term is similar to the theta term in Yang-Mills theory. At null-infinity an infinite dimensional algebra is discovered, both for the electric and magnetic charge.

scholarly journals Bäcklund transformations and the Atiyah–Ward ansatz for non-commutative anti-self-dual Yang–Mills equations

2009 ◽  
Vol 465 (2108) ◽  
pp. 2613-2632 ◽  
Author(s):  
Claire R. Gilson ◽  
Masashi Hamanaka ◽  
Jonathan J. C. Nimmo
Keyword(s):  
Exact Solutions ◽  
Gauge Group ◽  
Bäcklund Transformations ◽  
Twistor Theory ◽  
Yang Mills ◽  
Seed Solution ◽  
Backlund Transformations

We present Bäcklund transformations for the non-commutative anti-self-dual Yang–Mills equation where the gauge group is G = G L (2), and use it to generate a series of exact solutions from a simple seed solution. The solutions generated by this approach are represented in terms of quasi-determinants. We also explain the origins of all the ingredients of the Bäcklund transformations within the framework of non-commutative twistor theory. In particular, we show that the generated solutions belong to a non-commutative version of the Atiyah–Ward ansatz.

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