Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T23:07:40.776Z Has data issue: false hasContentIssue false
  • English
  • Français

BÄCKLUND TRANSFORMATIONS FOR NON-COMMUTATIVE ANTI-SELF-DUAL YANG–MILLS EQUATIONS

Published online by Cambridge University Press:  01 February 2009

CLAIRE R. GILSON ,
MASASHI HAMANAKA  and
JONATHAN J. C. NIMMO
CLAIRE R. GILSON
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK e-mail: crg@maths.gla.ac.uk
MASASHI HAMANAKA
Affiliation:
Department of Mathematics, University of Nagoya, Nagoya 464-8602, Japan e-mail: hamanaka@math.nagoya-u.ac.jp
JONATHAN J. C. NIMMO
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK e-mail: j.nimmo@maths.gla.ac.uk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We present Bäcklund transformations for the non-commutative anti-self-dual Yang–Mills equations where the gauge group is G = GL(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 and belong to a non-commutative version of the Atiyah–Ward ansatz. In the commutative limit, our results coincide with those by Corrigan, Fairlie, Yates and Goddard.

Keywords

35Q5846L5570S15
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 51 , Issue A , February 2009 , pp. 83 - 93
Copyright
Copyright © Glasgow Mathematical Journal Trust 2009

References

REFERENCES

1.Atiyah, M. F. and Ward, R. S., Instantons and Algebraic Geometry, Commun. Math. Phys. 55 (1977), 117.CrossRefGoogle Scholar
2.Brain, S. J., The noncommutative Penrose-Ward transform and self-dual Yang-Mills fields, PhD Thesis (University of Oxford, 2005).Google Scholar
3.Brain, S. J. and Majid, S., Quantisation of twistor theory by cocycle twist, Commun. Math. Phys. 284 (2008), 713.CrossRefGoogle Scholar
4.Corrigan, E. and Fairlie, D. B., Scalar field theory and exact solutions to a classical SU(2) gauge theory, Phys. Lett. B 67 (1977), 69; G. 't Hooft, unpublished; Wilczek, F., in Quark Confinement and Field Theory (Wiley, Princeton, 1977) 211 [ISBN/0-471-02721-9].CrossRefGoogle Scholar
5.Corrigan, E., Fairlie, D. B., Yates, R. G. and Goddard, P., Backlund Transformations and the Construction of the Atiyah-Ward Ansatz for Selfdual SU(2) Gauge Fields, Phys. Lett. B 72 (1978), 354; Commun. Math. Phys. 58 (1978), 223.CrossRefGoogle Scholar
6.Dimakis, A. and Müller-Hoissen, F., From nonassociativity to solutions of the KP hierarchy, J. Phys. A 40 (2007), F321.CrossRefGoogle Scholar
7.Etingof, P., Gelfand, I. and Retakh, V., Factorization of differential operators, quasideterminants, and nonabelian Toda field equations, Math. Res. Lett. 4 (1997), 413; Math. Res. Lett. 5 (1998), 1.CrossRefGoogle Scholar
8.Gelfand, I., Gelfand, S., Retakh, V. and Wilson, R. L., Quasideterminants, Adv. Math. 193 (2005), 56.CrossRefGoogle Scholar
9.Gelfand, I. and Retakh, V., Determinants of matrices over noncommutative rings, Funct. Anal. Appl. 25 (1991), 91; Funct. Anal. Appl. 26 (1992), 231.CrossRefGoogle Scholar
10.Gilson, C. R., Hamanaka, M. and Nimmo, J. J. C., Backlund transformations and the Atiyah-Ward Ansatz for noncommutative anti-self-dual Yang-Mills equations, arXiv:0812.1222.Google Scholar
11.Gilson, C. R. and Nimmo, J. J. C., On a direct approach to quasideterminant solutions of a noncommutative KP equation, J. Phys. A 40 (2007), 3839.CrossRefGoogle Scholar
12.Gilson, C. R., Nimmo, J. J. C. and Ohta, Y., Quasideterminant solutions of a non-Abelian Hirota-Miwa equation, J. Phys. A 40 (2007), 12607.CrossRefGoogle Scholar
13.Gilson, C. R., Nimmo, J. J. C. and Sooman, C. M., On a direct approach to quasideterminant solutions of a noncommutative modified KP equation, J. Phys. A 41 (2008), 085202.CrossRefGoogle Scholar
14.Goncharenko, V. M. and Veselov, A. P., Monodromy of the matrix Schrodinger equations and Darboux transformations, J. Phys. A 31 (1998), 5315.CrossRefGoogle Scholar
15.Hamanaka, M., Noncommutative Ward's conjecture and integrable systems, Nucl. Phys. B 741 (2006), 368.CrossRefGoogle Scholar
16.Hamanaka, M., Notes on exact multi-soliton solutions of noncommutative integrable hierarchies, JHEP 0702 (2007), 094.CrossRefGoogle Scholar
17.Hannabuss, K. C., Non-commutative twistor space, Lett. Math. Phys. 58 (2001), 153.CrossRefGoogle Scholar
18.Horváth, Z., Lechtenfeld, O. and Wolf, M., Noncommutative instantons via dressing and splitting approaches, JHEP 0212 (2002), 060.CrossRefGoogle Scholar
19.Ihl, M. and Uhlmann, S., Noncommutative extended waves and soliton-like configurations in N = 2 string theory, Int. J. Mod. Phys. A 18 (2003), 4889.CrossRefGoogle Scholar
20.Kapustin, A., Kuznetsov, A. and Orlov, D., Noncommutative instantons and twistor transform, Commun. Math. Phys. 221 (2001), 385.CrossRefGoogle Scholar
21.Konechny, A. and Schwarz, A. S., Introduction to M(atrix) theory and noncommutative geometry, Phys. Rept. 360 (2002), 353; Harvey, J. A., Komaba lectures on noncommutative solitons and D-branes, hep-th/0102076; Douglas, M. R. and Nekrasov, N. A., Noncommutative field theory, Rev. Mod. Phys. 73 (2002), 977; Szabo, R. J., Quantum field theory on noncommutative spaces, Phys. Rept. 378 (2003), 207; Hamanaka, M., Noncommutative solitons and D-branes, PhD Thesis, hep-th/0303256. Chu, C. S., Non-commutative geometry from strings, hep-th/0502167; Szabo, R. J., D-Branes in noncommutative field theory, hep-th/0512054.Google Scholar
22.Kupershmidt, B., KP or mKP : noncommutative mathematics of Lagrangian, Hamiltonian, and integrable systems (AMS, 2000) [ISBN/0821814001]; Hamanaka, M., Noncommutative solitons and integrable systems, [hep-th/0504001]; Tamassia, L., Noncommutative supersymmetric/integrable models and string theory, PhD Thesis, hep-th/0506064; Lechtenfeld, O., Noncommutative solitons, hep-th/0605034; Dimakis, A. and Müller-Hoissen, F., From nonassociativity to solutions of the KP hierarchy, nlin.SI/0608017.Google Scholar
23.Lechtenfeld, O. and Popov, A. D., Noncommutative 't Hooft instantons, JHEP 0203 (2002), 040.CrossRefGoogle Scholar
24.Lechtenfeld, O., Popov, A. D. and Spendig, B., Open N = 2 strings in a B-field background and noncommutative self-dual Yang-Mills, Phys. Lett. B 507 (2001), 317; JHEP 0106 (2001), 011.CrossRefGoogle Scholar
25.Li, C. X. and Nimmo, J. J. C., Quasideterminant solutions of a non-Abelian Toda lattice and kink solutions of a matrix sine-Gordon equation, Proc. Roy. Soc. Lond. A 464 (2008), 951.Google Scholar
26.Mason, L. J. and Woodhouse, N. M., Integrability, self-duality, and twistor theory (Oxford University Press, New York, 1996) (London Mathematical Society monographs, new series: 15), [ISBN/0-19-853498-1].CrossRefGoogle Scholar
27.Nekrasov, N. and Schwarz, A., Instantons on noncommutative R**4 and (2,0) superconformal six dimensional theory, Commun. Math. Phys. 198 (1998), 689.CrossRefGoogle Scholar
28.Nimmo, J. J. C., On a non-Aberian Hirota-Miwa equation, J. Phys. A 39 (2006), 5053.CrossRefGoogle Scholar
29.Saleem, U., Hassan, M. and Siddiq, M., Non-local continuity equations and binary Darboux transformation of noncommutative (anti) self-dual Yang-Mills equations, J. Phys. A 40 (2007), 5205.CrossRefGoogle Scholar
30.Samsonov, B. F. and Pecheritsin, A. A., Chains of Darboux transformations for the matrix Schrödinger equation, J. Phys. A 37 (2004), 239.CrossRefGoogle Scholar
31.Sasa, N., Ohta, Y. and Matsukidaira, J., Bilinear Form Approach to the Self-Dual Yang-Mills Equation and Integrable System in (2+1)-Dimension, J. Phys. Soc. Jap. 67 (1998), 83.CrossRefGoogle Scholar
32.Takasaki, K., Anti-self-dual Yang-Mills equations on noncommutative spacetime, J. Geom. Phys. 37 (2001), 291.CrossRefGoogle Scholar
33.Ward, R. S., Integrable and Solvable Systems, and Relations Among Them, Phil. Trans. Roy. Soc. Lond. A 315 (1985), 451.Google Scholar
34.Yang, C. N., Condition of Selfduality For SU(2) Gauge Fields on Euclidean Four-Dimensional Space, Phys. Rev. Lett. 38 (1977), 1377.CrossRefGoogle Scholar