Large N limit of the Yang–Mills measure on compact surfaces II: Makeenko–Migdal equations and the planar master field
This paper considers the large N limit of Wilson loops for the two-dimensional Euclidean Yang–Mills measure on all orientable compact surfaces of genus larger or equal to $1$ , with a structure group given by a classical compact matrix Lie group. Our main theorem shows the convergence of all W...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2025-01-01
|
| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S205050942400152X/type/journal_article |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832583751359528960 |
|---|---|
| author | Antoine Dahlqvist Thibaut Lemoine |
| author_facet | Antoine Dahlqvist Thibaut Lemoine |
| author_sort | Antoine Dahlqvist |
| collection | DOAJ |
| description | This paper considers the large N limit of Wilson loops for the two-dimensional Euclidean Yang–Mills measure on all orientable compact surfaces of genus larger or equal to
$1$
, with a structure group given by a classical compact matrix Lie group. Our main theorem shows the convergence of all Wilson loops in probability, given that it holds true on a restricted class of loops, obtained as a modification of geodesic paths. Combined with the result of [20], a corollary is the convergence of all Wilson loops on the torus. Unlike the sphere case, we show that the limiting object is remarkably expressed thanks to the master field on the plane defined in [3, 39], and we conjecture that this phenomenon is also valid for all surfaces of higher genus. We prove that this conjecture holds true whenever it does for the restricted class of loops of the main theorem. Our result on the torus justifies the introduction of an interpolation between free and classical convolution of probability measures, defined with the free unitary Brownian motion but differing from t-freeness of [5] that was defined in terms of the liberation process of Voiculescu [67]. In contrast to [20], our main tool is a fine use of Makeenko–Migdal equations, proving uniqueness of their solution under suitable assumptions, and generalising the arguments of [21, 33]. |
| format | Article |
| id | doaj-art-e00afa6716a04f9e8086a0a48e0e741f |
| institution | Kabale University |
| issn | 2050-5094 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-e00afa6716a04f9e8086a0a48e0e741f2025-01-28T07:00:47ZengCambridge University PressForum of Mathematics, Sigma2050-50942025-01-011310.1017/fms.2024.152Large N limit of the Yang–Mills measure on compact surfaces II: Makeenko–Migdal equations and the planar master fieldAntoine Dahlqvist0https://orcid.org/0000-0002-8095-7843Thibaut Lemoine1https://orcid.org/0009-0009-5445-5063University of Sussex, School of Mathematical and Physical Sciences, Pevensey 3 Building, Brighton, UK;Collège de France, 3, rue d’Ulm, Paris, 75005, France; E-mail:This paper considers the large N limit of Wilson loops for the two-dimensional Euclidean Yang–Mills measure on all orientable compact surfaces of genus larger or equal to $1$ , with a structure group given by a classical compact matrix Lie group. Our main theorem shows the convergence of all Wilson loops in probability, given that it holds true on a restricted class of loops, obtained as a modification of geodesic paths. Combined with the result of [20], a corollary is the convergence of all Wilson loops on the torus. Unlike the sphere case, we show that the limiting object is remarkably expressed thanks to the master field on the plane defined in [3, 39], and we conjecture that this phenomenon is also valid for all surfaces of higher genus. We prove that this conjecture holds true whenever it does for the restricted class of loops of the main theorem. Our result on the torus justifies the introduction of an interpolation between free and classical convolution of probability measures, defined with the free unitary Brownian motion but differing from t-freeness of [5] that was defined in terms of the liberation process of Voiculescu [67]. In contrast to [20], our main tool is a fine use of Makeenko–Migdal equations, proving uniqueness of their solution under suitable assumptions, and generalising the arguments of [21, 33].https://www.cambridge.org/core/product/identifier/S205050942400152X/type/journal_article60B1560B2081T1381T3246L5457M1057R56 |
| spellingShingle | Antoine Dahlqvist Thibaut Lemoine Large N limit of the Yang–Mills measure on compact surfaces II: Makeenko–Migdal equations and the planar master field Forum of Mathematics, Sigma 60B15 60B20 81T13 81T32 46L54 57M10 57R56 |
| title | Large N limit of the Yang–Mills measure on compact surfaces II: Makeenko–Migdal equations and the planar master field |
| title_full | Large N limit of the Yang–Mills measure on compact surfaces II: Makeenko–Migdal equations and the planar master field |
| title_fullStr | Large N limit of the Yang–Mills measure on compact surfaces II: Makeenko–Migdal equations and the planar master field |
| title_full_unstemmed | Large N limit of the Yang–Mills measure on compact surfaces II: Makeenko–Migdal equations and the planar master field |
| title_short | Large N limit of the Yang–Mills measure on compact surfaces II: Makeenko–Migdal equations and the planar master field |
| title_sort | large n limit of the yang mills measure on compact surfaces ii makeenko migdal equations and the planar master field |
| topic | 60B15 60B20 81T13 81T32 46L54 57M10 57R56 |
| url | https://www.cambridge.org/core/product/identifier/S205050942400152X/type/journal_article |
| work_keys_str_mv | AT antoinedahlqvist largenlimitoftheyangmillsmeasureoncompactsurfacesiimakeenkomigdalequationsandtheplanarmasterfield AT thibautlemoine largenlimitoftheyangmillsmeasureoncompactsurfacesiimakeenkomigdalequationsandtheplanarmasterfield |