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Instantons on Gravitons

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Abstract

Yang-Mills instantons on ALE gravitational instantons were constructed by Kronheimer and Nakajima in terms of matrices satisfying algebraic equations. These were conveniently organized into a quiver. We construct generic Yang-Mills instantons on ALF gravitational instantons. Our data are formulated in terms of matrix-valued functions of a single variable, that are in turn organized into a bow. We introduce the general notion of a bow, its representation, its associated data and moduli space of solutions. For a judiciously chosen bow the Nahm transform maps any bow solution to an instanton on an ALF space. We demonstrate that this map respects all complex structures on the moduli spaces, so it is likely to be an isometry, and use this fact to study the asymptotics of the moduli spaces of instantons on ALF spaces.

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References

  1. Hitchin N.J.: Polygons and Gravitons. Math. Proc. Cambridge Phil. Soc. 85, 465 (1979)

    Article  MathSciNet  ADS  Google Scholar 

  2. Hawking S.W.: Gravitational Instantons. Phys. Lett. A 60, 81 (1977)

    Article  MathSciNet  ADS  Google Scholar 

  3. Gibbons G.W., Hawking S.W.: Gravitational Multi - Instantons. Phys. Lett. B 78, 430 (1978)

    Article  ADS  Google Scholar 

  4. Cherkis S.A., Kapustin A.: Singular Monopoles and Gravitational Instantons. Commun. Math. Phys. 203, 713 (1999)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. Cherkis S.A., Hitchin N.J.: Gravitational Instantons of Type D k . Commun. Math. Phys. 260, 299 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. Kronheimer P.B., Nakajima H.: Yang-Mills Instantons on ALE Gravitational Instantons. Math. Ann. 288(2), 263–307 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  7. Losev A., Moore G.W., Nekrasov N. et al.: Four-dimensional Avatars of Two-dimensional RCFT. Nucl. Phys. Proc. Suppl. 46, 130–145 (1996)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  8. Nekrasov N., Schwarz A.S.: Instantons on Noncommutative \({\mathbb{R}^4}\) and (2,0) Superconformal Six-dimensional Theory. Commun. Math. Phys. 198, 689–703 (1998)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. Nakajima H.: Instantons on ALE spaces, Quiver Varieties, and Kac-Moody Algebras. Duke Math. J. 76, 2 (1994)

    Article  MathSciNet  Google Scholar 

  10. Nakajima H.: Instantons and Affine Lie Algebras. Nucl. Phys. Proc. Suppl. 46, 154 (1996)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  11. Lusztig G.: On Quiver Varieties. Adv. in Math. 136, 141–182 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Braverman A., Finkelberg M.: Pursuing the Double Affine Grassmannian I: Transversal Slices via Instantons on A k Singularities. Duke Math. J. 152(2), 175–206 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Licata, A.: Framed Rank r Torsion-free Sheaves on \({\mathbb{C}P^2}\) and Representations of the Affine Lie Algebra \({\widehat{gl(r)}}\) . http://arxiv.org/abs/math/0607690v1 [math.RT], 2006

  14. Nakajima H.: Quiver Varieties and Branching. SIGMA 5, 3 (2009)

    MathSciNet  Google Scholar 

  15. Dijkgraaf R., Hollands L., Sulkowski P., Vafa C.: Supersymmetric Gauge Theories, Intersecting Branes and Free Fermions. JHEP 0802, 106 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  16. Tan M.-C.: Five-Branes in M-Theory and a Two-Dimensional Geometric Langlands Duality. Adv. Theor. Math. Phys. 14, 179–224 (2010)

    MathSciNet  MATH  Google Scholar 

  17. Witten, E.: Geometric Langlands from Six Dimensions. http://arxiv.org/abs/0905.2720v1 [hep-th], 2009

  18. Cherkis S.A.: Instantons on the Taub-NUT Space. Adv. Theor. Math. Phys. 14(2), 609–642 (2010)

    MathSciNet  MATH  Google Scholar 

  19. Atiyah M.F., Hitchin N.J., Drinfeld V.G., Manin Yu.I.: Construction of Instantons. Phys. Lett. A 65, 185 (1978)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  20. Nahm, W.: A Simple Formalism for the BPS Monopole. Phys. Lett. B 90, 413 (1980); Nahm, W.: All Self-dual Multimonopoles for Arbitrary Gauge Group. CERN-TH.3172 (1981); KEK entry; Nahm, W.: Selfdual Monopoles and Calorons. BONN-HE-83-16 SPIRES Talk Presented at 12th Colloq. on Group Theoretical Methods in Physics, Trieste, Italy, Sep 5–10, 1983; W. Nahm, Self-dual Monopoles and Calorons. In: Lecture Notes in Physics 201, New York: Springer, 1984, pp. 189–200

  21. Witten E.: Sigma Models and the ADHM Construction of Instantons. J. Geom. Phys. 15, 215 (1995)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  22. Douglas M.R.: Gauge Fields and D-branes. J. Geom. Phys. 28, 255 (1998)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. Douglas, M.R. Moore, G.W.: D-branes, Quivers, and ALE Instantons. http://arxiv.org/abs/hep-th/9603167v1, 1996

  24. Johnson C.V., Myers R.C.: Aspects of Type IIB Theory on ALE Spaces. Phys. Rev. D 55, 6382 (1997)

    Article  MathSciNet  ADS  Google Scholar 

  25. Diaconescu D.E.: D-branes, Monopoles and Nahm Equations. Nucl. Phys. B 503, 220 (1997)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  26. Korepin V.E., Shatashvili S.L.: Rational Parametrization of the Three Instanton Solutions of the Yang-Mills Equations. Sov. Phys. Dokl. 28, 1018 (1983)

    ADS  Google Scholar 

  27. Bianchi, M., Fucito, F., Rossi, G, Martellini, M.: On the ADHM Construction on ALE Gravitational Backgrounds. Phys. Lett. B 359, 49 (1995); Bianchi, M., Fucito, F., Rossi, G., Martellini, M.: Explicit Construction of Yang-Mills Instantons on ALE Spaces. Nucl. Phys. B 473, 367 (1996)

    Google Scholar 

  28. Kraan, T.C., van Baal, P.: New Instanton Solutions at Finite Temperature. Nucl. Phys. A 642, 299 (1998); Kraan, T.C., van Baal, P.: Periodic Instantons with Non-trivial Holonomy. Nucl. Phys. B 533, 627 (1998); Kraan, T.C., van Baal, P.: Exact T-duality between Calorons and Taub - NUT Spaces. Phys. Lett. B 428, 268 (1998)

    Google Scholar 

  29. Lee K.M., Lu C.h.: SU(2) Calorons and Magnetic Monopoles. Phys. Rev. D 58, 025011 (1998)

    Article  MathSciNet  ADS  Google Scholar 

  30. Etesi G., Hausel T.: Geometric Construction of New Taub-NUT Instantons. Phys. Lett. B 514, 189 (2001)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  31. Etesi G., Hausel T.: New Yang-Mills instantons on multicentered gravitational instantons. Commun. Math. Phys. 235, 275–288 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  32. Etesi G., Szabo S.: Harmonic Functions and Instanton Moduli Spaces on the Multi-Taub–NUT Space. Commun. Math. phys. 301, 175–214 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  33. Cherkis S.A.: Moduli Spaces of Instantons on the Taub-NUT Space. Commun. Math. Phys. 290, 719 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  34. Minerbe, V.: On some Asymptotically Flat Manifolds with Non-maximal Volume Growth. http://arxiv.org/abs/0709.1084v1 [math.dg], 2007

  35. Taub, A.H.: Empty Space-Times Admitting a Three Parameter Group of Motions. Ann. Math. 53(3), 472–490 (1951); Newman, E., Tamburino, L., Unti, T.: Empty-Space Generalization of the Schwarzschild Metric. J. Math. Phys. 4, 915 (1963); Hawking, S.W.: Gravitational Instantons. Phys. Lett. A 60, 81 (1977)

    Google Scholar 

  36. Cherkis S.A., Kapustin A.: D k Gravitational Instantons and Nahm Equations. Adv. Theor. Math. Phys. 2(6), 1287 (1999)

    MathSciNet  Google Scholar 

  37. Atiyah M.F., Hitchin N.J.: The Geometry and Dynamics of Magnetic Monopoles. M.B. Porter Lectures. Princeton Univ. Press, Princeton, NJ (1988)

    Google Scholar 

  38. Dancer A.S.: A Family of Hyperkahler Manifolds. Quart. J. Math. 45(4), 463–478 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  39. Hurtubise J., Murray M.K.: Monopoles and their Spectral Data. Commun. Math. Phys. 133, 487 (1990)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  40. Hitchin N.J., Karlhede A., Lindström U., Roček M.: Hyperkähler Metrics and Supersymmetry. Commun. 108, 535 (1987)

    Article  ADS  MATH  Google Scholar 

  41. Kronheimer, P.B.: A Hyperkähler Structure on the Cotangent Bundle of a Complete Lie Group. MSRI preprint (1988) available at http://arxiv.org/abs/math/090925302 [math.DE], 2009

  42. Donaldson S.K.: Nahm’s Equations and the Classification of Monopoles. Commun. Math. Phys. 96, 387 (1984)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  43. Hitchin, N.: The Dirac Operator. In: Bridson, M.R., Salamon, S. (eds.) Invitations to Geometry and Topology. Oxford: Oxford University Press, 2002

  44. Gibbons G.W., Rychenkova P.: HyperKaehler Quotient Construction of BPS Monopole Moduli Spaces. Commun. Math. Phys. 186, 585 (1997)

    Article  MathSciNet  ADS  Google Scholar 

  45. Kronheimer P.B.: The Construction of ALE Spaces as HyperKähler Quotients. J. Diff. Geom. 29, 665 (1989)

    MathSciNet  MATH  Google Scholar 

  46. Dancer A.S.: Dihedral singularities and gravitational instantons. J. Geom. Phys. 12, 77 (1993)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  47. Witten E.: Branes, Instantons, and Taub-NUT Spaces. JHEP 0906, 067 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  48. Kronheimer, P.B.: Monopoles and Taub-NUT Metrics. M. Sc. Thesis, Oxford, 1985

  49. Donaldson S.K., Kronheimer P.B.: The Geomerty of Four-Manifolds. Oxford Geometry University Press, Oxford (1990)

    Google Scholar 

  50. Braam P.J., van Baal P.: Nahm’s Transformation for Instantons. Commun. Math. Phys. 122, 267 (1989)

    Article  ADS  MATH  Google Scholar 

  51. Maciocia A.: Metrics on the Moduli Spaces of Instantons over Euclidean Four Space. Commun. Math. Phys. 135, 467 (1991)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  52. Nakajima, H.: Monopoles and Nahm’s equations. SPIRES entry. In: Sanda 1990, Proceedings, Einstein metrics and Yang-Mills connections. T. Mabuchi, S. Mukai, eds., Lect. Notes pure and Appl. Math. 145, New York: Marcel Dekker, 1993, pp.193–211

  53. Biquard O., Jardim M.: Asymptotic Behaviour and the Moduli Space of Doubly-periodic Instantons. J. Europ. Math. Soc. 3(4), 335–375 (2001)

    MathSciNet  MATH  Google Scholar 

  54. Bielawski R.: Hyperkähler Structures and Group Actions. J. London Math. Soc. 55, 400–414 (1997)

    Article  MathSciNet  Google Scholar 

  55. Bielawski R.: Asymptotic Metrics for SU(N)-monopoles with Maximal Symmetry Breaking. Commun. Math. Phys. 199, 297–325 (1998)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  56. Bielawski R.: Monopoles and the Gibbons-Manton Metric. Commun. Math. Phys. 194, 297–321 (1998)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  57. Hurtubise J.: The Classification of Monopoles for the Classical Groups. Commun. Math. Phys. 120, 613 (1989)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  58. Bielawski, R., Cherkis, S.A.: In preparation

  59. Hausel T., Hunsicker E., Mazzeo R.: Hodge Cohomology of Gravitational Instantons. Duke Math. J. 122(3), 485–548 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Hamilton Mathematics Institute, Trinity College Dublin, Dublin 2, Ireland

    Sergey A. Cherkis

  2. Center for Theoretical Physics, Department of Physics, University of California, Berkeley, CA, 94720, USA

    Sergey A. Cherkis

  3. Department of Mathematics, Stanford University, Stanford, CA, 94305, USA

    Sergey A. Cherkis

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  1. Sergey A. Cherkis
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Correspondence to Sergey A. Cherkis.

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Communicated by N.A. Nekrasov

Dedicated to the memory of Israel Moiseevich Gelfand

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Cherkis, S.A. Instantons on Gravitons. Commun. Math. Phys. 306, 449–483 (2011). https://doi.org/10.1007/s00220-011-1293-y

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