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Rigorous Solution of Strongly Coupled SO(N) Lattice Gauge Theory in the Large N Limit

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Abstract

The main result of this paper is a rigorous computation of Wilson loop expectations in strongly coupled SO(N) lattice gauge theory in the large N limit, in any dimension. The formula appears as an absolutely convergent sum over trajectories in a kind of string theory on the lattice, demonstrating an explicit gauge-string duality. The generality of the proof technique may allow it to be extended other gauge groups.

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References

  1. Balaban T.: Regularity and decay of lattice Green’s functions. Commun. Math. Phys. 89(4), 571–597 (1983)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Balaban T.: Ultraviolet stability of three-dimensional lattice pure gauge field theories. Commun. Math. Phys. 102, 255–275 (1985)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Balaban T.: The variational problem and background fields in renormalization group method for lattice gauge theories. Commun. Math. Phys. 102(2), 277–309 (1985)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Balaban T.: Renormalization group approach to lattice gauge field theories. I: generation of effective actions in a small field approximation and a coupling constant renormalization in 4D. Commun. Math. Phys. 109, 249–301 (1987)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  5. Bhanot G., Heller U.M., Neuberger H.: The quenched Eguchi–Kawai model. Phys. Lett. B 113(1), 47–50 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  6. Brydges D., Fröhlich J., Seiler E.: On the construction of quantized gauge fields. I. General results. Ann. Phys. 121(1), 227–284 (1979)

    Article  ADS  MathSciNet  Google Scholar 

  7. Brydges D.C., Fröhlich J., Seiler E.: Construction of quantised gauge fields. II. Convergence of the lattice approximation. Commun. Math. Phys. 71(2), 159–205 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  8. Brydges D.C., Fröhlich J., Seiler E.: On the construction of quantized gauge fields. III. The two-dimensional abelian Higgs model without cutoffs. Commun. Math. Phys. 79(3), 353–399 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  9. Charalambous N., Gross L.: The Yang–Mills heat semigroup on three-manifolds with boundary. Commun. Math. Phys. 317(3), 727–785 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Chatterjee, S.: Concentration inequalities with exchangeable pairs. Ph.D. thesis, Stanford University (2005)

  11. Chatterjee S.: Stein’s method for concentration inequalities. Probab. Theory Relat. Fields 138(1-2), 305–321 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chatterjee S., Meckes E.: Multivariate normal approximation using exchangeable pairs. ALEA Lat. Am. J. Probab. Math. Stat. 4, 257–283 (2008)

    MathSciNet  MATH  Google Scholar 

  13. Collins B., Guionnet A., Maurel-Segala E.: Asymptotics of unitary and orthogonal matrix integrals. Adv. Math. 222(1), 172–215 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Driver B.K.: YM 2: continuum expectations, lattice convergence, and lassos. Commun. Math. Phys. 123(4), 575–616 (1989)

    Article  ADS  MATH  Google Scholar 

  15. Dunne, G.V., Ünsal, M.: New Methods in QFT and QCD: From large-N orbifold equivalence to bions and resurgence. Preprint arXiv:1601.03414 (2016)

  16. Eguchi T., Kawai H.: Reduction of dynamical degrees of freedom in the large-N gauge theory. Phys. Rev. Lett. 48(16), 1063 (1982)

    Article  ADS  Google Scholar 

  17. Ercolani N.M., McLaughlin K.D.T.-R.: Asymptotics of the partition function for random matrices via Riemann–Hilbert techniques and applications to graphical enumeration. Int. Math. Res. Not. 2003(14), 755–820 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ercolani N.M., McLaughlin K.D.T-R., Pierce V.U.: Random matrices, graphical enumeration and the continuum limit of Toda lattices. Commun. Math. Phys. 278(1), 31–81 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Eynard, B.: Large-N expansion of the 2-matrix model. J. High Energy Phys. 2003(1), 051, (2003)

  20. Eynard, B.: Topological expansion for the 1-Hermitian matrix model correlation functions. J. High Energy Phys. 2005(11), 031 (2004)

  21. Eynard B., Orantin N.: Invariants of algebraic curves and topological expansion. Commun. Number Theory Phys. 1(2), 347–452 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Eynard, B., Orantin, N.: Topological recursion in enumerative geometry and random matrices. J. Phys. A 42(29), 293001 (2009)

  23. Fröhlich J., Spencer T.: Massless phases and symmetry restoration in abelian gauge theories and spin systems. Commun. Math. Phys. 83(3), 411–454 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  24. Gonzalez-Arroyo A., Okawa M.: Twisted-Eguchi–Kawai model: a reduced model for large-N lattice gauge theory. Phys. Rev. D 27(10), 2397–2411 (1983)

    Article  ADS  Google Scholar 

  25. Göpfert M., Mack G.: Proof of confinement of static quarks in 3-dimensional U(1) lattice gauge theory for all values of the coupling constant. Commun. Math. Phys. 82(4), 545–606 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  26. Greensite J., Lautrup B.: First-order phase transition in four-dimensional SO(3) lattice gauge theory. Phys. Rev. Lett. 47(1), 9–11 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  27. Gross D.J., Witten E.: Possible third-order phase transition in the large-N lattice gauge theory. Phys. Rev. D 21(2), 446–453 (1980)

    Article  ADS  Google Scholar 

  28. Gross L.: Convergence of U(1)3 lattice gauge theory to its continuum limit. Commun. Math. Phys. 92(2), 137–162 (1983)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Gross L., King C., Sengupta A.: Two dimensional Yang–Mills theory via stochastic differential equations. Ann. Phys. 194(1), 65–112 (1989)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Guionnet A.: First order asymptotics of matrix integrals; a rigorous approach towards the understanding of matrix models. Commun. Math. Phys. 244(3), 527–569 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Guionnet, A.: Random matrices and enumeration of maps. In: International Congress of Mathematicians, vol. III, pp. 623–636. European Mathematical Society, Zürich (2006)

  32. Guionnet A., Jones V.F.R., Shlyakhtenko D., Zinn-Justin P.: Loop models, random matrices and planar algebras. Commun. Math. Phys. 316(1), 45–97 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Guionnet A., Maïda M.: Character expansion method for the first order asymptotics of a matrix integral. Probab. Theory Relat. Fields 132(4), 539–578 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  34. Guionnet, A., Novak, J.: Asymptotics of unitary multimatrix models: the Schwinger–Dyson lattice and topological recursion. Preprint arXiv:1401.2703 (2014)

  35. Guionnet A., Zeitouni O.: Large deviations asymptotics for spherical integrals. J. Funct. Anal. 188(2), 461–515 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  36. Guth A.H.: Existence proof of a nonconfining phase in four-dimensional U(1) lattice gauge theory. Phys. Rev. D 21(8), 2291–2307 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  37. Jaffe A., Witten E.: Quantum Yang–Mills Theory. The Millennium Prize Problems, pp. 129–152. Clay Mathematical Institute, Cambridge, MA (2006)

    MATH  Google Scholar 

  38. Jafarov, J.: Wilson loop expectations in SU(N) lattice gauge theory. Preprint arXiv:1610.03821 (2016)

  39. Kovtun P., Yaffe M., Yaffe L.G.: Volume independence in large N c QCD-like gauge theories. J. High Energy Phys. 2007(6), 019 (2007)

    Article  Google Scholar 

  40. Lévy, T.: Yang–Mills measure on compact surfaces. Mem. Am. Math. Soc. 166(790) (2003)

  41. Lévy T.: Discrete and continuous Yang–Mills measure for non-trivial bundles over compact surfaces. Probab. Theory Relat. Fields 136(2), 171–202 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  42. Lévy, T.: The master field on the plane. Preprint arXiv:1112.2452 (2011)

  43. Lucini B., Panero M.: SU(N) gauge theories at large N. Phys. Rep. 526(2), 93–163 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  44. Lüscher M.: Construction of a self-adjoint, strictly positive transfer matrix for Euclidean lattice gauge theories. Commun. Math. Phys. 54, 283–292 (1977)

    Article  ADS  Google Scholar 

  45. Lüscher, M.: Properties and uses of the Wilson flow in lattice QCD. J. High Energy Phys. 2010(8), 071 (2010)

  46. Magnen J., Rivasseau V., Sénéor R.: Construction of YM 4 with an infrared cutoff. Commun. Math. Phys. 155, 325–383 (1993)

    Article  ADS  MATH  Google Scholar 

  47. Makeenko Y.M., Migdal A.A.: Exact equation for the loop average in multicolor QCD. Phys. Lett. B 88(1), 135–137 (1979)

    Article  ADS  Google Scholar 

  48. Maldacena J.M.: The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2, 231–252 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  49. Meckes, E.: An infinitesimal version of Stein’s method of exchangeable pairs. Ph.D. thesis, Stanford University (2006)

  50. Osterwalder K., Seiler E.: Gauge field theories on a lattice. Ann. Phys. 110(2), 440–471 (1978)

    Article  ADS  MathSciNet  Google Scholar 

  51. Seiler E.: Upper bound on the color-confining potential. Phys. Rev. D 18(2), 482–483 (1978)

    Article  ADS  Google Scholar 

  52. Seiler E.: Gauge Theories as a Problem of Constructive Quantum Field Theory and Statistical Mechanics. Springer, Berlin (1982)

    Google Scholar 

  53. Sengupta A.: Quantum gauge theory on compact surfaces. Ann. Phys. 221(1), 17–52 (1993)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  54. Sengupta A.: Gauge Theory on Compact Surfaces. American Mathematical Society, Providence, RI (1997)

    Book  MATH  Google Scholar 

  55. Stein, C.: A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In: Proceedings of the Sixth Berkeley Symposium on Mathematics, Statistics and Probability, vol. II, pp. 583–602. University of California Press, Berkeley, CA (1972)

  56. Stein, C.: Approximate computation of expectations. IMS Lecture Notes—Monograph Series, 7. Institute of Mathematical Statistics, Hayward, CA (1986)

  57. Stein, C.: The accuracy of the normal approximation to the distribution of the traces of powers of random orthogonal matrices. Technical Report No. 470, Stanford University Department of Statistics, 1995 (1995)

  58. ’t Hooft G.: A planar diagram theory for strong interactions. Nucl. Phys. B 72(3), 461–473 (1974)

    Article  ADS  MathSciNet  Google Scholar 

  59. Tomboulis E.T., Yaffe L.G.: Finite temperature SU(2) lattice gauge theory. Commun. Math. Phys. 100(3), 313–341 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  60. Yaffe M., Yaffe L.G.: Center-stabilized Yang–Mills theory: confinement and large N volume independence. Phys. Rev. D 78(6), 065035 (2008)

    Article  ADS  Google Scholar 

  61. Wadia, S.R.: A study of U(N) lattice gauge theory in 2-dimensions. Preprint arXiv:1212.2906 (2012)

  62. Wilson K.G.: Confinement of quarks. Phys. Rev. D 10(8), 2445–2459 (1974)

    Article  ADS  Google Scholar 

  63. Witten E.: Gauge theories and integrable lattice models. Nucl. Phys. B 322(3), 629–697 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  64. Witten E.: Gauge theories, vertex models, and quantum groups. Nucl. Phys. B 330(2), 285–346 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  65. Witten E.: Two dimensional gauge theories revisited. J. Geom. Phys. 9(4), 303–368 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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Acknowledgements

I thank Amir Dembo, Persi Diaconis, Bruce Driver, Len Gross, Alice Guionnet, Jafar Jafarov, Todd Kemp, Herbert Neuberger, Steve Shenker, Lior Silberman, Tom Spencer, and Akshay Venkatesh for many helpful discussions and comments. I am grateful to the referee for a number of useful suggestions, and to H.-T. Yau for his enthusiasm about getting this paper published in CMP.

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  1. Department of Statistics, Stanford University, Sequoia Hall, 390 Serra Mall, Stanford, CA, 94305, USA

    Sourav Chatterjee

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Correspondence to Sourav Chatterjee.

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Communicated by H.-T. Yau

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Research partially supported by NSF Grant DMS-1441513.

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Chatterjee, S. Rigorous Solution of Strongly Coupled SO(N) Lattice Gauge Theory in the Large N Limit. Commun. Math. Phys. 366, 203–268 (2019). https://doi.org/10.1007/s00220-019-03353-3

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