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Communications in Mathematical Physics

, Volume 313, Issue 1, pp 71–129 | Cite as

Localization of Gauge Theory on a Four-Sphere and Supersymmetric Wilson Loops

  • Vasily PestunEmail author
Article

Abstract

We prove conjecture due to Erickson-Semenoff-Zarembo and Drukker-Gross which relates supersymmetric circular Wilson loop operators in the \({\mathcal N=4}\) supersymmetric Yang-Mills theory with a Gaussian matrix model. We also compute the partition function and give a new matrix model formula for the expectation value of a supersymmetric circular Wilson loop operator for the pure \({\mathcal N=2}\) and the \({\mathcal N=2^*}\) supersymmetric Yang-Mills theory on a four-sphere. A four-dimensional \({\mathcal N=2}\) superconformal gauge theory is treated similarly.

Keywords

Gauge Theory Vector Bundle Matrix Model Wilson Loop Elliptic Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Witten E.: Topological quantum field theory. Commun. Math. Phys. 117, 353 (1988)MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. 2.
    Atiyah M.F., Jeffrey L.: Topological Lagrangians and cohomology. J. Geom. Phys. 7(1), 119–136 (1990)MathSciNetADSzbMATHCrossRefGoogle Scholar
  3. 3.
    Donaldson S.K.: An application of gauge theory to four-dimensional topology. J. Diff. Geom. 18(2), 279–315 (1983)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Nekrasov N.A.: Seiberg-Witten prepotential from instanton counting. Adv. Theor. Math. Phys. 7, 831–864 (2004)MathSciNetGoogle Scholar
  5. 5.
    Nekrasov, N., Okounkov, A.: Seiberg-Witten theory and random partitions. http://arXiV.org/abs/hep-th/0306238v2, 2003
  6. 6.
    Seiberg N., Witten E.: Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD. Nucl. Phys. B 431, 484–550 (1994)MathSciNetADSzbMATHCrossRefGoogle Scholar
  7. 7.
    Seiberg N., Witten E.: Electric—magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory. Nucl. Phys. B 426, 19–52 (1994)MathSciNetADSzbMATHCrossRefGoogle Scholar
  8. 8.
    Karlhede A., Rocek M.: Topological quantum field theory and N = 2 conformal supergravity. Phys. Lett. B212, 51 (1988)MathSciNetADSGoogle Scholar
  9. 9.
    Erickson J.K., Semenoff G.W., Zarembo K.: Wilson loops in N = 4 supersymmetric Yang-Mills theory. Nucl. Phys. B 582, 155–175 (2000)MathSciNetADSzbMATHCrossRefGoogle Scholar
  10. 10.
    Drukker N., Gross D.J.: An exact prediction of N = 4 SUSYM theory for string theory. J. Math. Phys. 42, 2896–2914 (2001)MathSciNetADSzbMATHCrossRefGoogle Scholar
  11. 11.
    Maldacena J.M.: The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2, 231–252 (1998)MathSciNetADSzbMATHGoogle Scholar
  12. 12.
    Witten E.: Anti-de Sitter space and holography. Adv. Theor. Math. Phys. 2, 253–291 (1998)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Gubser S.S., Klebanov I.R., Polyakov A.M.: Gauge theory correlators from non-critical string theory. Phys. Lett. B 428, 105–114 (1998)MathSciNetADSCrossRefGoogle Scholar
  14. 14.
    Maldacena J.M.: Wilson loops in large N field theories. Phys. Rev. Lett. 80, 4859–4862 (1998)MathSciNetADSzbMATHCrossRefGoogle Scholar
  15. 15.
    Berenstein D., Corrado R., Fischler W., Maldacena J.M.: The operator product expansion for Wilson loops and surfaces in the large N limit. Phys. Rev. D 59, 105023 (1999)MathSciNetADSCrossRefGoogle Scholar
  16. 16.
    Rey S.-J., Yee J.-T.: Macroscopic strings as heavy quarks in large N gauge theory and anti-de Sitter supergravity. Eur. Phys. J. C 22, 379–394 (2001)MathSciNetADSzbMATHCrossRefGoogle Scholar
  17. 17.
    Drukker N., Gross D.J., Ooguri H.: Wilson loops and minimal surfaces. Phys. Rev. D 60, 125006 (1999)MathSciNetADSCrossRefGoogle Scholar
  18. 18.
    Semenoff G.W., Zarembo K.: More exact predictions of SUSYM for string theory. Nucl. Phys. B 616, 34–46 (2001)MathSciNetADSzbMATHCrossRefGoogle Scholar
  19. 19.
    Zarembo K.: Supersymmetric Wilson loops. Nucl. Phys. B 643, 157–171 (2002)MathSciNetADSzbMATHCrossRefGoogle Scholar
  20. 20.
    Zarembo K.: Open string fluctuations in AdS(5) x S(5) and operators with large R charge. Phys. Rev. D 66, 105021 (2002)MathSciNetADSCrossRefGoogle Scholar
  21. 21.
    Tseytlin A.A., Zarembo K.: Wilson loops in N = 4 SYM theory: rotation in S(5). Phys. Rev. D 66, 125010 (2002)MathSciNetADSCrossRefGoogle Scholar
  22. 22.
    Bianchi M., Green M.B., Kovacs S.: Instanton corrections to circular Wilson loops in N = 4 supersymmetric Yang-Mills. JHEP 04, 040 (2002)MathSciNetADSCrossRefGoogle Scholar
  23. 23.
    Semenoff G.W., Zarembo K.: Wilson loops in SYM theory: From weak to strong coupling. Nucl. Phys. Proc. Suppl. 108, 106–112 (2002)MathSciNetADSzbMATHCrossRefGoogle Scholar
  24. 24.
    Pestun V., Zarembo K.: Comparing strings in AdS(5)xS(5) to planar diagrams: an example. Phys. Rev. D 67, 086007 (2003)MathSciNetADSCrossRefGoogle Scholar
  25. 25.
    Drukker N., Fiol B.: All-genus calculation of Wilson loops using D-branes. JHEP 02, 010 (2005)MathSciNetADSCrossRefGoogle Scholar
  26. 26.
    Semenoff G.W., Young D.: Exact 1/4 BPS loop: Chiral primary correlator. Phys. Lett. B 643, 195–204 (2006)MathSciNetADSCrossRefGoogle Scholar
  27. 27.
    Drukker N., Giombi S., Ricci R., Trancanelli D.: On the D3-brane description of some 1/4 BPS Wilson loops. JHEP 04, 008 (2007)MathSciNetADSCrossRefGoogle Scholar
  28. 28.
    Drukker N., Giombi S., Ricci R., Trancanelli D.: Wilson loops: From four-dimensional SYM to two-dimensional YM. Phys. Rev. D 77, 047901 (2008)MathSciNetADSCrossRefGoogle Scholar
  29. 29.
    Yamaguchi S.: Semi-classical open string corrections and symmetric Wilson loops. JHEP 06, 073 (2007)ADSCrossRefGoogle Scholar
  30. 30.
    Okuyama K., Semenoff G.W.: Wilson loops in N = 4 SYM and fermion droplets. JHEP 06, 057 (2006)MathSciNetADSCrossRefGoogle Scholar
  31. 31.
    Okuyama K.: ’t Hooft expansion of 1/2 BPS Wilson loop. JHEP 09, 007 (2006)MathSciNetADSCrossRefGoogle Scholar
  32. 32.
    Gomis J., Passerini F.: Wilson loops as D3-branes. JHEP 01, 097 (2007)MathSciNetADSCrossRefGoogle Scholar
  33. 33.
    Tai T.-S., Yamaguchi S.: Correlator of fundamental and anti-symmetric Wilson loops in AdS/CFT correspondence. JHEP 02, 035 (2007)MathSciNetADSCrossRefGoogle Scholar
  34. 34.
    Giombi S., Ricci R., Trancanelli D.: Operator product expansion of higher rank Wilson loops from D-branes and matrix models. JHEP 10, 045 (2006)MathSciNetADSCrossRefGoogle Scholar
  35. 35.
    Chen B., He W.: On 1/2-BPS Wilson-’t Hooft loops. Phys. Rev. D 74, 126008 (2006)MathSciNetADSCrossRefGoogle Scholar
  36. 36.
    Hartnoll S.A.: Two universal results for Wilson loops at strong coupling. Phys. Rev. D 74, 066006 (2006)MathSciNetADSCrossRefGoogle Scholar
  37. 37.
    Drukker N.: 1/4 BPS circular loops, unstable world-sheet instantons and the matrix model. JHEP 09, 004 (2006)MathSciNetADSCrossRefGoogle Scholar
  38. 38.
    Drukker N., Giombi S., Ricci R., Trancanelli D.: More supersymmetric Wilson loops. Phys. Rev. D. 76, 107703 (2007)MathSciNetADSCrossRefGoogle Scholar
  39. 39.
    Chu C.-S., Giataganas D.: 1/4 BPS Wilson loop in beta-deformed theories. JHEP 0710, 108 (2007)MathSciNetADSCrossRefGoogle Scholar
  40. 40.
    Plefka J., Staudacher M.: Two loops to two loops in N = 4 supersymmetric Yang-Mills theory. JHEP 09, 031 (2001)MathSciNetADSCrossRefGoogle Scholar
  41. 41.
    Arutyunov G., Plefka J., Staudacher M.: Limiting geometries of two circular Maldacena-Wilson loop operators. JHEP 12, 014 (2001)MathSciNetADSCrossRefGoogle Scholar
  42. 42.
    Barnes E.W.: The theory of the double gamma function. Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character 196, 265–387 (1901)ADSzbMATHCrossRefGoogle Scholar
  43. 43.
    Losev A., Nekrasov N., Shatashvili S.L.: Issues in topological gauge theory. Nucl. Phys. B 534, 549–611 (1998)MathSciNetADSzbMATHCrossRefGoogle Scholar
  44. 44.
    Moore G.W., Nekrasov N., Shatashvili S.: Integrating over Higgs branches. Commun. Math. Phys. 209, 97–121 (2000)MathSciNetADSzbMATHCrossRefGoogle Scholar
  45. 45.
    Ooguri H., Strominger A., Vafa C.: Black hole attractors and the topological string. Phys. Rev. D 70, 106007 (2004)MathSciNetADSCrossRefGoogle Scholar
  46. 46.
    Beasley, C., Gaiotto, D., Guica, M., Huang, L., Strominger, A., Yin, X.: Why Z BH = |Z top|2. http://arXiv.org/abs/hep-th/0608021v1, 2006
  47. 47.
    Duistermaat J.J., Heckman G.J.: On the variation in the cohomology of the symplectic form of the reduced phase space. Invent. Math. 69(2), 259–268 (1982)MathSciNetADSzbMATHCrossRefGoogle Scholar
  48. 48.
    Atiyah M.F., Bott R.: The moment map and equivariant cohomology. Topology 23(1), 1–28 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Berline N., Vergne M.: Classes caractéristiques équivariantes. Formule de localisation en cohomologie équivariante. C. R. Acad. Sci. Paris Sér. I Math. 295(9), 539–541 (1982)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Witten, E.: Mirror manifolds and topological field theory. http://arxiv.org/abs/hep-th/9112056v1, 1991
  51. 51.
    Atiyah, M.F.: Elliptic operators and compact groups. Lecture Notes in Mathematics, Vol. 401, Berlin: Springer-Verlag, 1974Google Scholar
  52. 52.
    Singer, I.M.: Recent applications of index theory for elliptic operators. In: Partial Differential Equations (Proc. Sympos. Pure Math., Vol. XXIII, Univ. California, Berkeley, Calif., 1971), Providence, R.I.: Amer. Math. Soc., 1973, pp. 11–31Google Scholar
  53. 53.
    Drukker N., Giombi S., Ricci R., Trancanelli D.: Supersymmetric Wilson loops on S 3. JHEP 0805, 017 (2008)MathSciNetADSCrossRefGoogle Scholar
  54. 54.
    Kapustin, A.: Holomorphic reduction of N = 2 gauge theories, Wilson-’t Hooft operators, and S-duality. http://arXiv.org/abs/hep-th/0612119v2, 2006
  55. 55.
    Kapustin A.: Wilson-’t Hooft operators in four-dimensional gauge theories and S-duality. Phys. Rev. D 74, 025005 (2006)MathSciNetADSCrossRefGoogle Scholar
  56. 56.
    Kapustin, A., Witten, E.: Electric-magnetic duality and the geometric Langlands program. http://arXiv.org/abs/hep-th/0604151v3, 2007
  57. 57.
    Cecotti S., Vafa C.: Topological antitopological fusion. Nucl. Phys. B367, 359–461 (1991)MathSciNetADSCrossRefGoogle Scholar
  58. 58.
    Brink L., Schwarz J.H., Scherk J.: Supersymmetric Yang-Mills theories. Nucl. Phys. B121, 77 (1977)MathSciNetADSCrossRefGoogle Scholar
  59. 59.
    Baum, H.: Conformal killing spinors and special geometric structures in lorentzian geometry—a survey. http://arXiv.org/abs/math/0202008v1 [math.DG], 2002
  60. 60.
    Vafa C., Witten E.: A strong coupling test of S duality. Nucl. Phys. B 431, 3–77 (1994)MathSciNetADSzbMATHCrossRefGoogle Scholar
  61. 61.
    Berkovits N.: A ten-dimensional superYang-Mills action with off-shell supersymmetry. Phys. Lett. B 318, 104–106 (1993)MathSciNetADSCrossRefGoogle Scholar
  62. 62.
    Evans J.M.: Supersymmetry algebras and Lorentz invariance for d = 10 superYang-Mills. Phys. Lett. B 334, 105–112 (1994)MathSciNetADSCrossRefGoogle Scholar
  63. 63.
    Baulieu L., Berkovits N.J., Bossard G., Martin A.: Ten-dimensional superYang-Mills with nine off-shell supersymmetries. Phys. Lett. B 658, 249–254 (2008)MathSciNetADSCrossRefGoogle Scholar
  64. 64.
    Labastida J.M.F., Lozano C.: Mathai-Quillen formulation of twisted N = 4 supersymmetric gauge theories in four dimensions. Nucl. Phys. B 502, 741–790 (1997)MathSciNetADSzbMATHCrossRefGoogle Scholar
  65. 65.
    Witten E.: Two-dimensional gauge theories revisited. J. Geom. Phys. 9, 303–368 (1992)MathSciNetADSzbMATHCrossRefGoogle Scholar
  66. 66.
    Lavaud, P.: Equivariant cohomology and localization formula in supergeometry. http://arXiV.org/abs/math/0402068v1 [math.DG], 2004
  67. 67.
    Lavaud, P.: http://arXiV.org/abs/math/0402067 [math.GR], 2004]
  68. 68.
    Atiyah M.F., Singer I.M.: The index of elliptic operators. I. Ann. of Math. 87(2), 484–530 (1968)MathSciNetzbMATHCrossRefGoogle Scholar
  69. 69.
    Atiyah M.F., Segal G.B.: The index of elliptic operators. II. Ann. of Math. 87(2), 531–545 (1968)MathSciNetzbMATHCrossRefGoogle Scholar
  70. 70.
    Atiyah M.F., Singer I.M.: The index of elliptic operators. III. Ann. of Math. 87(2), 546–604 (1968)MathSciNetzbMATHCrossRefGoogle Scholar
  71. 71.
    Atiyah M.F., Bott R.: A Lefschetz fixed point formula for elliptic differential operators. Bull. Amer. Math. Soc. 72, 245–250 (1966)MathSciNetzbMATHCrossRefGoogle Scholar
  72. 72.
    Atiyah M.F., Bott R.: A Lefschetz fixed point formula for elliptic complexes. I. Ann. of Math. 86(2), 374–407 (1967)MathSciNetzbMATHCrossRefGoogle Scholar
  73. 73.
    Atiyah M.F., Bott R.: A Lefschetz fixed point formula for elliptic complexes. II. Applications. Ann. of Math. 88(2), 451–491 (1968)MathSciNetzbMATHCrossRefGoogle Scholar
  74. 74.
    Berline N., Vergne M.: L’indice équivariant des opérateurs transversalement elliptiques. Invent. Math. 124(1–3), 51–101 (1996)MathSciNetADSzbMATHCrossRefGoogle Scholar
  75. 75.
    Berline, N., Vergne, M.: The equivariant Chern character and index of G-invariant operators. Lectures at CIME, Venise 1992. In: D-modules, representation theory, and quantum groups (Venice, 1992), Vol. 1565 of Lecture Notes in Math., Berlin: Springer, 1993, pp. 157–200Google Scholar
  76. 76.
    Witten, E.: Holomorphic Morse inequalities. In: Algebraic and Differential Topology—Global Differential Geometry, Vol. 70 of Teubner-Texte Math., Leipzig: Teubner, 1984, pp. 318–333Google Scholar
  77. 77.
    Flume R., Poghossian R.: An algorithm for the microscopic evaluation of the coefficients of the Seiberg-Witten prepotential. Int. J. Mod. Phys. A 18, 2541 (2003)MathSciNetADSzbMATHCrossRefGoogle Scholar
  78. 78.
    Bruzzo U., Fucito F., Morales J.F., Tanzini A.: Multi-instanton calculus and equivariant cohomology. JHEP 05, 054 (2003)MathSciNetADSCrossRefGoogle Scholar
  79. 79.
    Nakajima, H., Yoshioka, K.: Instanton counting on blowup. I. 4-dimensional pure gauge theory. http://arXiv.org/abs/math/0306198v2 [math.AG], 2005
  80. 80.
    Nakajima, H., Yoshioka, K.: Lectures on instanton counting. http://arXiv.org/abs/math/0311058v1 [math.AG], 2003
  81. 81.
    Okuda, T., Pestun, V.: On the instantons and the hypermultiplet mass of N = 2* super Yang-Mills on S 4. http://arXiv.org/abs/1004.1222v1 [hep-th], 2010
  82. 82.
    Bachas C.P., Bain P., Green M.B.: Curvature terms in D-brane actions and their M-theory origin. JHEP 05, 011 (1999)MathSciNetADSCrossRefGoogle Scholar
  83. 83.
    Losev, A.S., Marshakov, A., Nekrasov, N.A.: Small instantons, little strings and free fermions. http://arXiv.org/abs/hep-th/0302191v3, 2003
  84. 84.
    Flume R., Fucito F., Morales J.F., Poghossian R.: Matone’s relation in the presence of gravitational couplings. JHEP 04, 008 (2004)MathSciNetADSCrossRefGoogle Scholar
  85. 85.
    Argyres P.C., Seiberg N.: S-duality in N = 2 supersymmetric gauge theories. JHEP 0712, 088 (2009)ADSGoogle Scholar
  86. 86.
    Lu H., Pope C.N., Rahmfeld J.: A construction of Killing spinors on S n. J. Math. Phys. 40, 4518–4526 (1999)MathSciNetADSCrossRefGoogle Scholar
  87. 87.
    Goodman M.W., Witten E.: Global symmetries in four-dimensions and higher dimensions. Nucl. Phys. B271, 21 (1986)MathSciNetADSGoogle Scholar
  88. 88.
    Atiyah, M.F.: K-theory. Lecture notes by D. W. Anderson. New York-Amsterdam: W. A. Benjamin, Inc., 1967Google Scholar

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© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Physics DepartmentPrinceton UniversityPrincetonUSA

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