Communications in Mathematical Physics

, Volume 352, Issue 3, pp 967–978 | Cite as

The Makeenko–Migdal Equation for Yang–Mills Theory on Compact Surfaces

  • Bruce K. Driver
  • Franck Gabriel
  • Brian C. Hall
  • Todd KempEmail author


We prove the Makeenko–Migdal equation for two-dimensional Euclidean Yang–Mills theory on an arbitrary compact surface, possibly with boundary. In particular, we show that two of the proofs given by the first, third, and fourth authors for the plane case extend essentially without change to compact surfaces.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. AS.
    Anshelevich M., Sengupta A.N.: Quantum free Yang–Mills on the plane. J. Geom. Phys. 62, 330–343 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. Dahl.
    Dahlqvist A.: Free energies and fluctuations for the unitary Brownian motion. Commun. Math. Phys. 348(2), 395–444 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. DK.
    Daul J.-M., Kazakov V.A.: Wilson loop for large N Yang–Mills theory on a two-dimensional sphere. Phys. Lett. B. 335, 371–376 (1994)ADSMathSciNetCrossRefGoogle Scholar
  4. Dr.
    Driver B.K.: YM\({_{2}}\): continuum expectations, lattice convergence, and lassos. Commun. Math. Phys. 123, 575–616 (1989)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. DHK1.
    Driver B.K., Hall B.C., Kemp T.: The large-N limit of the Segal–Bargmann transform on \({\mathbb{U}_{N}}\). J. Funct. Anal. 265, 2585–2644 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. DHK2.
    Driver B.K., Hall B.C., Kemp T.: Three proofs of the Makeenko–Migdal equation for Yang–Mills theory on the plane. Commun. Math. Phys. 351(2), 741–774 (2017)ADSMathSciNetCrossRefGoogle Scholar
  7. Fine1.
    Fine D.S.: Quantum Yang–Mills on the two-sphere. Commun. Math. Phys. 134, 273–292 (1990)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. Fine2.
    Fine D.S.: Quantum Yang–Mills on a Riemann surface. Commun. Math. Phys. 140, 321–338 (1991)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. Gop.
    Gopakumar R.: The master field in generalised QCD\({_{2}}\). Nucl. Phys. B. 471, 246–260 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. GG.
    Gopakumar R., Gross D.: Mastering the master field. Nucl. Phys. B. 451, 379–415 (1995)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. K.
    Kazaokv A.: Wilson loop average for an arbitary contour in two-dimensional U(N) gauge theory. Nucl. Phys. B179, 283–292 (1981)ADSCrossRefGoogle Scholar
  12. KK.
    Kazakov V.A., Kostov I.K.: Non-linear strings in two-dimensional \({U(\infty)}\) gauge theory. Nucl. Phys. B. 176, 199–215 (1980)ADSCrossRefGoogle Scholar
  13. Levy1.
    Lévy, T.: Yang–Mills measure on compact surfaces. Mem. Am. Math. Soc. 166(790), xiv+122 (2003)Google Scholar
  14. Levy2.
    Lévy, T.: Two-dimensional Markovian holonomy fields. Astérisque No. 329, 172 pp (2010)Google Scholar
  15. Levy3.
    Lévy, T.: The master field on the plane (preprint). arXiv:1112.2452
  16. MM.
    Makeenko Y.M., Migdal A.A.: Exact equation for the loop average in multicolor QCD. Phys. Lett. 88B, 135–137 (1979)ADSCrossRefGoogle Scholar
  17. Sen1.
    Sengupta A.N.: Quantum gauge theory on compact surfaces. Ann. Phys. 221, 17–52 (1993)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. Sen2.
    Sengupta, A.N.: Gauge theory on compact surfaces. Mem. Amer. Math. Soc. 126(600), viii+85 (1997)Google Scholar
  19. Sen3.
    Sengupta A.N.: Yang–Mills on surfaces with boundary: quantum theory and symplectic limit. Commun. Math. Phys. 183, 661–705 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. Sen4.
    Sengupta, A.N.: Traces in two-dimensional QCD: the large-N limit. Traces in Number Theory, Geometry and Quantum fields, pp. 193–212. Aspects Math., E38, Friedr. Vieweg, Wiesbaden (2008)Google Scholar
  21. Sing.
    Singer, I.M.: On the master field in two dimensions. In: Functional analysis on the eve of the 21st century, vol. 1 (New Brunswick, NJ, 1993), pp. 263–281, Progr. Math., 131, Birkhäuser Boston, Boston, MA (1995)Google Scholar
  22. Witt1.
    Witten E.: On quantum gauge theories in two dimensions. Commun. Math. Phys. 141, 153–209 (1991)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. Witt2.
    Witten E.: Two-dimensional gauge theories revisited. J. Geom. Phys. 9, 303–368 (1992)ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California San DiegoLa JollaUSA
  2. 2.Mathematics Institute, Zeeman BuildingUniversity of WarwickCoventryUK
  3. 3.Department of MathematicsUniversity of Notre DameNotre DameUSA

Personalised recommendations