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Large N limit of the Yang-Mills measure on compact surfaces II: Makeenko-Migdal equations and planar master field
Probability · Submitted 15 Jan 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We show their convergence for all loops on the torus and give a conjecture supported by partial results for other surfaces.
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The abstract, sentence by sentence
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 one, with a structure group given by a classical compact matrix Lie group. We show their convergence for all loops on the torus and give a conjecture supported by partial results for other surfaces.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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 one, with a structure group given by a classical compact matrix Lie group. | contribution 0.53 | 0.51 | 0.07 | asserted 0.91 |
| s1 | We show their convergence for all loops on the torus and give a conjecture supported by partial results for other surfaces. | result 0.90 | 0.93 | 0.73 | asserted 0.92 |
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The paper
Original abstract
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 one, with a structure group given by a classical compact matrix Lie group. We show their convergence for all loops on the torus and give a conjecture supported by partial results for other surfaces.
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References
How the literature uses it
5 arXiv papers cite this in 8 sentences
Papers citing it · 3 sentences without a confident label
Cites it as background 5 papers
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Rigorous work in the continuum begins with [ref] on the plane and remains an ongoing area of investigation for compact surfaces, the sphere case being studied in [ref] and more recently, progress on general surfaces in [ref] .
Makeenko-Migdal equations for 2D Yang-Mills: from lattice to continuum · Introduction · 0.99
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We remark that the identities [ref]-[ref] are not new, and are sometimes called the “magic formulas”, see e.g. [ref] for more background and literature, or [ref] .
A new derivation of the finite $N$ master loop equation for lattice Yang-Mills · Proof of the main theorem · 0.98
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In the last case, the Yang–Mills measure in 2D is known to converge to a deterministic limit called the master field [ref] which satisfies the Makeenko–Migdal equations; on lattice much more results can be proved, see [ref] and dynamic approach [ref] .
Large $N$ limit and $1/N$ expansion of invariant observables in $O(N)$ linear $\sigma$-model via SPDE · Introduction · 0.98
What it cites · 4 sentences without a confident label
Compares with 1 paper
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* Although both approaches use the convergence of the partition function of the model, we use in [ref] the Markov property of the Yang–Mills holonomy field in order to prove the convergence for simple loops, then we use the Makeenko–Migdal equations to induce the convergence on a larger class of loops; the latter is actually not needed in the zero volume case. * We only consider the limit of Wilson loops, whereas [ref] proves a $1/N$ expansion. * We prove a convergence in probability whereas [ref] gets a convergence in expectation. * We also consider a larger family of matrix groups, whereas he only treats the unitary case. * In the case $g=1$, the Atiyah–Bott–Goldman measure is ill-defined, hence Magee's paper cannot handle it, but we still find a result when $T>0$, which gives a matrix approximation of an interpolation between classical and free convolution of Haar unitaries.
Random Unitary Representations of Surface Groups II: The large $n$ limit · Atiyah–Bott–Goldman measure · 1.00
Cites as background 1 paper
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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 [ref] , a consequence of the result on the torus in non-commutative probability, and lastly, a sketch of the strategy of the main proofs.
Random Unitary Representations of Surface Groups II: The large $n$ limit · Introduction · 0.85
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