Yang–Mills Measure and the Master Field on the Sphere

Abstract

We study the Yang-Mills measure on the sphere with unitary structure group. In the limit where the structure group has high dimension, we show that the traces of loop holonomies converge in probability to a deterministic limit, which is known as the master field on the sphere. The values of the master field on simple loops are expressed in terms of the solution of a variational problem. We show that, given its values on simple loops, the master field is characterized on all loops of finite length by a system of differential equations, known as the Makeenko-Migdal equations. We obtain a number of further properties of the master field. On specializing to families of simple loops, our results identify the high-dimensional limit, in non-commutative distribution, of the Brownian bridge in the group of unitary matrices starting and ending at the identity.

Large N limit of Yang-Mills partition function and Wilson loops on compact surfaces

“…Remark. The linear extension of a master field Φ Σ to C[L(Σ)] comes automatically with a structure of non-commutative probability space [34,11]. In the case of the plane, this non-commutative distribution can be characterised using free probability ( [34] and [5]).…”

“…Theorem 2.8 ([54, 2, 34] and [11]). The conjecture 2.7 holds true when Σ is an open disc of the plane or a sphere of total area T > 0.…”

“…This paper was inspired by the large N limits considered in quantum gauge theories [12,27,42,39,22,21], which started after the landmark work of t'Hooft [52]. Since then, the Yang-Mills measure in two dimensions has been rigorously defined 1 [23,17,48,30], and the latter question has received the attention of several mathematicians 2 [54,49,2,34,19,18,11,25]. It is now known that the limit indeed exists when the f (g)p t (g)dg = f (1).…”

Large N behaviour of the two-dimensional Yang–Mills partition function

“…The asymptotic behaviour of the partition function on the sphere is very different from the higher genus surfaces and needs more analytical tools. Its free energy was computed rigorously by Boutet de Monvel and Shcherbina [2] and later by Lévy and Maïda [18], as well as Dahlqvist and Norris [4]; in particular, Lévy and Maïda proved a phase transition conjectured by Douglas and Kazakov [5]. We do not consider this case because it needs different tools than the ones we use.…”

“…After that, the idea of studying large N limits of matrix models flourished, in particular not only in the case of Quantum Chromodynamics in two dimensions, or QCD 2 [5,10,12], but also in Conformal Field Theory [6] and in Collective Field Theory [11]. Since then, mathematicians tried to derive rigorously some of the formulae used by these physicists, for instance [1,2,4,7,14,16,17,18,21,22,23,24]. We will focus here on the asymptotics of partition functions of the two-dimensional Yang-Mills model over a compact surface, written as sums over irreducible characters of the structure group.…”