Abstract
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.
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Anshelevich M., Sengupta A.N.: Quantum free Yang–Mills on the plane. J. Geom. Phys. 62, 330–343 (2012)
Dahlqvist A.: Free energies and fluctuations for the unitary Brownian motion. Commun. Math. Phys. 348(2), 395–444 (2016)
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)
Driver B.K.: YM\({_{2}}\): continuum expectations, lattice convergence, and lassos. Commun. Math. Phys. 123, 575–616 (1989)
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)
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)
Fine D.S.: Quantum Yang–Mills on the two-sphere. Commun. Math. Phys. 134, 273–292 (1990)
Fine D.S.: Quantum Yang–Mills on a Riemann surface. Commun. Math. Phys. 140, 321–338 (1991)
Gopakumar R.: The master field in generalised QCD\({_{2}}\). Nucl. Phys. B. 471, 246–260 (1996)
Gopakumar R., Gross D.: Mastering the master field. Nucl. Phys. B. 451, 379–415 (1995)
Kazaokv A.: Wilson loop average for an arbitary contour in two-dimensional U(N) gauge theory. Nucl. Phys. B179, 283–292 (1981)
Kazakov V.A., Kostov I.K.: Non-linear strings in two-dimensional \({U(\infty)}\) gauge theory. Nucl. Phys. B. 176, 199–215 (1980)
Lévy, T.: Yang–Mills measure on compact surfaces. Mem. Am. Math. Soc. 166(790), xiv+122 (2003)
Lévy, T.: Two-dimensional Markovian holonomy fields. Astérisque No. 329, 172 pp (2010)
Lévy, T.: The master field on the plane (preprint). arXiv:1112.2452
Makeenko Y.M., Migdal A.A.: Exact equation for the loop average in multicolor QCD. Phys. Lett. 88B, 135–137 (1979)
Sengupta A.N.: Quantum gauge theory on compact surfaces. Ann. Phys. 221, 17–52 (1993)
Sengupta, A.N.: Gauge theory on compact surfaces. Mem. Amer. Math. Soc. 126(600), viii+85 (1997)
Sengupta A.N.: Yang–Mills on surfaces with boundary: quantum theory and symplectic limit. Commun. Math. Phys. 183, 661–705 (1997)
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)
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)
Witten E.: On quantum gauge theories in two dimensions. Commun. Math. Phys. 141, 153–209 (1991)
Witten E.: Two-dimensional gauge theories revisited. J. Geom. Phys. 9, 303–368 (1992)
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Communicated by S. Zelditch
Franck Gabriel: Supported by ERC Grant, “ Behaviour near criticality,” held by M. Hairer.
Brian C. Hall: Supported in part by NSF Grant DMS-1301534.
Todd Kemp Supported in part by NSF CAREER Award DMS-1254807.
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Driver, B.K., Gabriel, F., Hall, B.C. et al. The Makeenko–Migdal Equation for Yang–Mills Theory on Compact Surfaces. Commun. Math. Phys. 352, 967–978 (2017). https://doi.org/10.1007/s00220-017-2857-2
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DOI: https://doi.org/10.1007/s00220-017-2857-2