Yang--Mills measure and the master field on the sphere

Dahlqvist, Antoine and Norris, James R (2020) Yang--Mills measure and the master field on the sphere. Communication in Mathematical Physics. ISSN 1432-0916

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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.

Item Type: Article
Keywords: Yang-Mills measure, Brownian motion on Lie groups, Wilson Loops, Brownian bridges, Master field, Free Probability
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Research Centres and Groups: Probability and Statistics Research Group
Subjects: Q Science > QA Mathematics
Q Science > QA Mathematics > QA0273 Probabilities. Mathematical statistics
Depositing User: Antoine Dahlqvist
Date Deposited: 02 Apr 2020 07:42
Last Modified: 19 Jun 2020 15:00
URI: http://sro.sussex.ac.uk/id/eprint/90641

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Project NameSussex Project NumberFunderFunder Ref
New Frontiers in Random Geometry (RaG)UnsetEngineering and Physical Sciences Research CouncilEP/I03372X/1