Exact construction of noncommutative instantons

Abstract

We discuss the Atiyah-Drinfeld-Hitchin-Manin (ADHM) construction of U(N) instantons in noncommutative (NC) space and give some exact instanton solutions for various noncommutative settings. We also present a new formula which is crucial to show an origin of the instanton number for U(1) and to prove the one-to-one correspondence between moduli spaces of the noncommutative instantons and the ADHM data.

Keywords

Instantons / noncommutative geometry

References

[1] Aganagic M, Gopakumar R, Minwalla S, Strominger A. Unstable solitons in noncommutative gauge theory. J High Energy Phys, 2001, 0104: 001

[2] Atiyah M F, Hitchin N J, Drinfeld V G, Manin Yu I. Construction of instantons. Phys Lett A, 1978, 65(3): 185-187

[3] Belavin A A, Polyakov A M, Schwartz A S, Tyupkin Y S. Pseudoparticle solutions of the Yang-Mills equations. Phys Lett B, 1975, 59: 85-87

[4] Chu C-S. Non-commutative geometry from strings. hep-th/0502167

[5] Corrigan E, Goddard P, Osborn H, Templeton S. Zeta function regularization and multi-instanton determinants. Nucl Phys B, 1979, 159: 469-496

[6] Dorey N, Hollowood T J, Khoze V V, Mattis M P. The calculus of many instantons. Phys Rep, 2002, 371: 231-459

[7] Douglas M R, Nekrasov N A. Noncommutative field theory. Rev Modern Phys, 2002, 73: 977-1029

[8] Furuuchi K. Instantons on noncommutative R⁴and projection operators. Progr Theoret Phys, 2000, 103: 1043-1068

[9] Furuuchi K. Equivalence of projections as gauge equivalence on noncommutative space. Comm Math Phys, 2001, 217: 579-593

[10] Furuuchi K. Topological charge of U(1) instantons on noncommutative R⁴. hep-th/0010006

[11] Furuuchi K. Dp-D(p + 4) in noncommutative Yang-Mills. J High Energy Phys, 2001, 0103: 033

[12] Hamanaka M. Atiyah-Drinfeld-Hitchin-Manin and Nahm Constructions of localized solitons in noncommutative gauge theories. Phys Rev D, 2002, 65: 085022

[13] Hamanaka M. Noncommutative solitons and D-branes. Ph D Thesis, University of Tokyo. 2003, hep-th/0303256

[14] Hamanaka M. Noncommutative Ward’s conjecture and integrable systems. Nuclear Phys B, 2006, 741: 368-389

[15] Hamanaka M, Nakatsu T. ADHM construction and group actions for noncommutative instantons

[16] Hamanaka M, Nakatsu T. Noncommutative instantons revisited. J Phys: Conference Ser (to appear)

[17] Harvey J A. Komaba Lectures On Noncommutative Solitons And D-branes. hep-th/0102076

[18] Ishikawa T, Kuroki S I, Sako A. Instanton number calculus on noncommutative R⁴. J High Energy Phys, 2002, 0208: 028

[19] Konechny A, Schwarz A. Introduction to M(atrix) theory and noncommutative geometry. Phys Rep, 2002, 360: 353-421

[20] Konechny A, Schwarz A. Introduction to M(atrix) theory and noncommutative geometry, II. Phys Rep, 2002, 360: 422-465

[21] Lechtenfeld O. Noncommutative instantons and solitons. Fortschr Phys, 2004, 52: 596

[22] Maeda Y, Sako A. Noncommutative deformation of spinor zero mode and ADHM construction. J Math Phys, 2012, 53: 022303

[23] Mason L J, Woodhouse N M. Integrability, Self-Duality, and Twistor Theory. Oxford: Oxford Univ Press, 1996

[24] Moyal J E. Quantum mechanics as a statistical theory. Proc Cambridge Phil Soc, 1949, 45: 99-124

[25] Nakajima H. Resolutions of moduli spaces of ideal instantons on R⁴. In: Topology, Geometry and Field Theory. Singapore World Sci, 1994, 129-136

[26] Nakajima H.Lectures on Hilbert Schemes of Points on Surfaces. Providence: Amer Math Soc, 1999

[27] Nakajima H, Yoshioka K. Instanton counting on blowup. I. Invent Math, 2005, 162: 313-355

[28] Nekrasov N A. Trieste Lectures on Solitons in Noncommutative Gauge Theories. hep-th/0011095

[29] Nekrasov N A. Seiberg-Witten prepotential from instanton counting. Adv Theor Math Phys, 2004, 7: 831-864

[30] Nekrasov N, Schwarz A. Instantons on noncommutative R⁴, and (2,0) superconformal six dimensional theory. Comm Math Phys, 1998, 198: 689-703

[31] Osborn H. Calculation of multi-instanton determinants. Nuclear Phys B, 1979, 159: 497-511

[32] Penrose R. Twistor algebra. J Math Phys, 1967, 8: 345-366

[33] Sako A. Instanton number of noncommutative U(n) gauge theory. J High Energy Phys, 2003, 04: 023

[34] Schaposnik F A. Noncommutative solitons and instantons. Braz J Phys, 2004, 34: 1349-1357

[35] Seiberg N, Witten E. String theory and noncommutative geometry. J High Energy Phys, 1999, 9909: 032

[36] Szabo R J. Quantum field theory on noncommutative spaces. Phys Rep, 2003, 378: 207-299

[37] Tian Y. Topological charge of ADHM instanton on ℝNC2×ℝ2. Modern Phys Lett A, 2004, 19: 1315-1317

[38] Tian Y, Zhu C J, Song X C. Topological charge of noncommutative ADHM instanton. Modern Phys Lett A, 2003, 18: 1691-1703

[39] Ward R S. Integrable and solvable systems, and relations among them. Philos Trans R Soc Lond Ser A, 1985, 315: 451-457

[40] Ward R S, Wells R O. Twistor Geometry and Field Theory. Cambridge: Cambridge Univ Press, 1990

RIGHTS & PERMISSIONS

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg