Abstract
We give three short proofs of the Makeenko–Migdal equation for the Yang–Mills measure on the plane, two using the edge variables and one using the loop or lasso variables. Our proofs are significantly simpler than the earlier pioneering rigorous proofs given by Lévy and by Dahlqvist. In particular, our proofs are “local” in nature, in that they involve only derivatives with respect to variables adjacent to the crossing in question. In an accompanying paper with Gabriel, we show that two of our proofs can be adapted to the case of Yang–Mills theory on any compact surface.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anshelevich M., Sengupta A.N.: Quantum free Yang–Mills on the plane. J. Geom. Phys 62, 330–343 (2012)
Chatterjee, S.: Rigorous solution of strongly coupled SO(N) lattice gauge theory in the large N limit (2015). arXiv:1502.07719 (preprint)
Dahlqvist A.: Free energies and fluctuations for the unitary Brownian motion. Commun. Math. Phys. 348(2), 395–444 (2016)
Dosch H.G., Müller V.F.: Lattice gauge theory in two spacetime dimensions. Fortschritte der Phys. 27, 547–559 (1979)
Driver B.K.: YM2: 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., Gabriel, F., Hall, B.C., Kemp, T.: The Makeenko–Migdal equation for Yang–Mills theory on compact surfaces (2016). arXiv:1602.03905 (preprint)
Gopakumar R.: The master field in generalised QCD2. Nuclear Phys. B 471, 246–260 (1996)
Gopakumar R., Gross D.: Mastering the master field. Nuclear Phys. B 451, 379–415 (1995)
Gross L., King C., Sengupta A.N.: Two-dimensional Yang–Mills theory via stochastic differential equations. Ann. Phys. 194, 65–112 (1989)
Kazakov A.: Wilson loop average for an arbitary contour in two-dimensional U(N) gauge theory. Nuclear Phys. B 179, 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.: Two-dimensional Markovian holonomy fields, vol. 172, no. 329. Astérisque (2010)
Lévy, T.: The master field on the plane (2011). arXiv:1112.2452 (preprint)
Makeenko Y.M., Migdal A.A.: Exact equation for the loop average in multicolor QCD. Phys. Lett. 88, 135–137 (1979)
Menotti P., Onofri E.: The action of SU(N) lattice gauge theory in terms of the heat kernel on the group manifold. Nuclear Phys. B 190, 288–300 (1981)
Migdal A.A.: Recursion equations in gauge field theories. Sov. Phys. JETP 42, 413–418 (1975)
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), Progr. Math., vol. 131, pp. 263–281. Birkhäuser Boston, Boston (1995)
Sengupta, A.N.: Traces in two-dimensional QCD: the large-N limit. Traces in number theory, geometry and quantum fields, Aspects Math., vol. E38, pp. 193–212. Friedr. Vieweg, Wiesbaden (2008)
’t Hooft G.: A planar diagram theory for strong interactions. Nuclear Phys. B 72, 461–473 (1974)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by S. Zelditch
B.C. Hall was supported in part by NSF Grant DMS-1301534. T. Kemp was supported in part by NSF CAREER Award DMS-1254807.
Rights and permissions
About this article
Cite this article
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, 741–774 (2017). https://doi.org/10.1007/s00220-016-2793-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-016-2793-6