Random Unitary Representations of Surface Groups I: Asymptotic expansions

Abstract

In this paper we study random representations of fundamental groups of surfaces into special unitary groups. The random model we use is based on a symplectic form on moduli space due to Atiyah, Bott, and Goldman. Let Σ g denote a topological surface of genus g ≥ 2. We establish the existence of a large n asymptotic expansion, to any fixed order, for the expected value of the trace of any fixed element of π 1 (Σ g ) under a random representation of π 1 (Σ g ) into SU(n). Each such expected value involves a contribution from all irreducible representations of SU(n). The main technical contribution of the paper is effective analytic control of the entire contribution from irreducible representations outside finite sets of carefully chosen rational families of representations.

Large N limit of the Yang-Mills measure on compact surfaces II: Makeenko-Migdal equations and planar master field

“…Another measure on connections is due to [4,23] when g ≥ 2. Recently, the limit of Wilson loops under this measure has been investigated by [40,41], we discuss the relation with our result.…”

“…In higher dimension, an analog 3 of this question for a lattice model has also been considered [10]. Very recently and independently from the current work, it was shown in [41,40] that under the Atiyah-Bott-Goldman measure, which can be understood as the weak limit of the Yang-Mills measure when the area of the surface vanishes, the expectation of Wilson loops converges and has a 1 N expansion when the group belongs to the series of special unitary matrices and the surface is closed, orientable and of genus g ≥ 2. For further details and references on the motivations of this problem, we refer to [16,Sec.…”

“…Since for any word with evaluation γ ∈ Γ g , it can be shown that γ = 1 if and only if the loop w is contractible, the above statement can be understood as the T = 0 case of the conjecture 1.3, with a weaker convergence given in expectation instead of in probability. In [40], it is also shown that E µ ABG,g [W ] admits an asymptotic expansion in powers of 1 N . Our main arguments differ from the one of [41]; the one of [16] relies on the Markovian property of the Yang-Mills measure, while the ones of this paper starts from on the Makeenko-Migdal equations which allow to prove that convergence in probability is stable under a large class of deformations, a problem which does not occur in the zero volume case.…”

“…Our main arguments differ from the one of [41]; the one of [16] relies on the Markovian property of the Yang-Mills measure, while the ones of this paper starts from on the Makeenko-Migdal equations which allow to prove that convergence in probability is stable under a large class of deformations, a problem which does not occur in the zero volume case. Though, a common starting point to [16] and [40] is the convergence of the partition function of the model.…”

“…The paper is organised as follows. The first four following sections of the introduction give respectively an informal definition of the Yang-Mills measure and of the main results, a discussion on the relation with the Atiyah-Bott-Goldman measure and the work [40,41], a consequence of the result on the torus in non-commutative probability, and lastly, a sketch of the strategy of the main proofs. Section 2 recalls and adapts some combinatorial notions of discrete homotopy of loops in embedded graph instrumental to the proof.…”