Geodesic
Large N limit of the Yang–Mills measure on compact surfaces II: Makeenko–Migdal equations and the planar master field
Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e17

Voir la notice de l'article provenant de la source Cambridge University Press

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]. 
Dahlqvist, Antoine; Lemoine, Thibaut. Large N limit of the Yang–Mills measure on compact surfaces II: Makeenko–Migdal equations and the planar master field. Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e17. doi: 10.1017/fms.2024.152
@article{10_1017_fms_2024_152,
     author = {Dahlqvist, Antoine and Lemoine, Thibaut},
     title = {Large {N} limit of the {Yang{\textendash}Mills} measure on compact surfaces {II:} {Makeenko{\textendash}Migdal} equations and the planar master field},
     journal = {Forum of Mathematics, Sigma},
     pages = {e17},
     year = {2025},
     volume = {13},
     number = {1},
     doi = {10.1017/fms.2024.152},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2024.152/}
}
TY  - JOUR
AU  - Dahlqvist, Antoine
AU  - Lemoine, Thibaut
TI  - Large N limit of the Yang–Mills measure on compact surfaces II: Makeenko–Migdal equations and the planar master field
JO  - Forum of Mathematics, Sigma
PY  - 2025
SP  - e17
VL  - 13
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.1017/fms.2024.152/
DO  - 10.1017/fms.2024.152
ID  - 10_1017_fms_2024_152
ER  - 
%0 Journal Article
%A Dahlqvist, Antoine
%A Lemoine, Thibaut
%T Large N limit of the Yang–Mills measure on compact surfaces II: Makeenko–Migdal equations and the planar master field
%J Forum of Mathematics, Sigma
%D 2025
%P e17
%V 13
%N 1
%U http://geodesic.mathdoc.fr/articles/10.1017/fms.2024.152/
%R 10.1017/fms.2024.152
%F 10_1017_fms_2024_152

[1] Albeverio, S., Høegh-Krohn, R. and Holden, H., ‘Stochastic Lie group-valued measures and their relations to stochastic curve integrals, gauge fields and Markov cosurfaces’, in Stochastic Processes—Mathematics and Physics (Bielefeld, 1984) (Lecture Notes in Math.) vol. 1158 (Springer, Berlin, 1986), 1–24. Google Scholar | DOI

[2] Albeverio, S., Høegh-Krohn, R. and Holden, H., ‘Stochastic multiplicative measures, generalized Markov semigroups, and group-valued stochastic processes and fields’, J. Funct. Anal. 78(1) (1988), 154–184. Google Scholar | DOI

[3] Anshelevich, M. and Sengupta, A. N., ‘Quantum free Yang–Mills on the plane’, J. Geom. Phys. 62(2) (2012), 330–343. Google Scholar | DOI

[4] Atiyah, M. F. and Bott, R., ‘The Yang-Mills equations over Riemann surfaces’, Philos. Trans. Roy. Soc. London Ser. A 308(1505) (1983), 523–615. Google Scholar

[5] Benaych-Georges, F. and Lévy, T., ‘A continuous semigroup of notions of independence between the classical and the free one’, Ann. Probab. 39(3) (2011), 904–938. Google Scholar | DOI

[6] Biane, P., ‘Free Brownian motion, free stochastic calculus and random matrices’, in Free Probability Theory (Waterloo, ON, 1995) (Fields Inst. Commun.) vol. 12 (Amer. Math. Soc., Providence, RI, 1997), 1–19. Google Scholar

[7] Birman, J. S. and Series, C., ‘Dehn’s algorithm revisited, with applications to simple curves on surfaces’, in Combinatorial Group Theory and Topology (Alta, Utah, 1984) (Ann. of Math. Stud.) vol. 111 (Princeton Univ. Press, Princeton, NJ, 1987), 451–478. Google Scholar | DOI

[8] Bismut, J.-M. and Labourie, F., ‘Symplectic geometry and the Verlinde formulas’, in Surveys in Differential Geometry: Differential Geometry Inspired by String Theory (Surv. Differ. Geom.) vol. 5 (Int. Press, Boston, MA, 1999), 97–311. Google Scholar

[9] Cébron, G., Dahlqvist, A. and Gabriel, F., ‘The generalized master fields’, J. Geom. Phys. 119 (2017), 34–53. Google Scholar | DOI

[10] Chandra, A., Chevyrev, I., Hairer, M. and Shen, H., ‘Langevin dynamic for the 2D Yang-Mills measure’, Publ. Math. Inst. Hautes Études Sci. 136 (2022), 1–147. Google Scholar | DOI

[11] Chandra, A., Chevyrev, I., Hairer, M. and Shen, H., ‘Stochastic quantisation of yang–mills–higgs in 3d’, Invent. Math. 237(2) (2024), 541–696. Google Scholar | DOI

[12] Chatterjee, S., ‘Rigorous solution of strongly coupled so(n) lattice gauge theory in the large n limit’, Comm. Math. Phys. 366(1) (2019), 203–268. Google Scholar | DOI

[13] Chevyrev, I., ‘Yang-Mills measure on the two-dimensional torus as a random distribution’, Comm. Math. Phys. 372(3) (2019), 1027–1058. Google Scholar | DOI

[14] Chevyrev, I., ‘Stochastic quantization of Yang-Mills’, J. Math. Phys. 63(9) (2022), Paper No. 091101, 19. Google Scholar | DOI

[15] Chevyrev, I. and Shen, H., ‘Invariant measure and universality of the 2d yang-mills langevin dynamic’, 2023. Google Scholar

[16] Collins, B., Guionnet, A. and Maurel-Segala, E., ‘Asymptotics of unitary and orthogonal matrix integrals’, Adv. Math. 222(1) (2009), 172–215. Google Scholar | DOI

[17] Collins, B. and Śniady, P., ‘Integration with respect to the Haar measure on unitary, orthogonal and symplectic group’, Comm. Math. Phys. 264(3) (2006), 773–795. Google Scholar | DOI

[18] Dahlqvist, A., ‘Free energies and fluctuations for the unitary Brownian motion’, Comm. Math. Phys. 348(2) (2016), 395–444. Google Scholar | DOI

[19] Dahlqvist, A., ‘Integration formulas for Brownian motion on classical compact Lie groups’, Ann. Inst. Henri Poincaré Probab. Stat. 53(4) (2017), 1971–1990. Google Scholar | DOI

[20] Dahlqvist, A. and Lemoine, T., ‘Large N limit of Yang–Mills partition function and Wilson loops on compact surfaces’, Probab. Math. Phys. 4(4) (2023), 849–890. Google Scholar | DOI

[21] Dahlqvist, A. and Norris, J. R., ‘Yang–mills measure and the master field on the sphere’, Comm. Math. Phys. 377(2) (2020), 1163–1226. Google Scholar | DOI

[22] Diaconis, P. and Evans, S. N., ‘Linear functionals of eigenvalues of random matrices’, Trans. Amer. Math. Soc. 353(7) (2001), 2615–2633. Google Scholar | DOI

[23] Driver, B. K., ‘YM: continuum expectations, lattice convergence, and lassos’, Comm. Math. Phys. 123(4) (1989), 575–616. Google Scholar | DOI

[24] Driver, B. K., ‘A functional integral approaches to the Makeenko-Migdal equations’, Comm. Math. Phys. 370(1) (2019), 49–116. Google Scholar | DOI

[25] Driver, B. K., Gabriel, F., Hall, B. C. and Kemp, T., ‘The Makeenko-Migdal equation for Yang-Mills theory on compact surfaces’, Comm. Math. Phys. 352(3) (2017), 967–978. Google Scholar | DOI

[26] Driver, B. K., Hall, B. C. and Kemp, T., ‘Three proofs of the Makeenko-Migdal equation for Yang-Mills theory on the plane’, Comm. Math. Phys. 351(2) (2017), 741–774. Google Scholar | DOI

[27] Goldman, W. M., ‘The symplectic nature of fundamental groups of surfaces’, Adv. Math. 54(2) (1984), 200–225. Google Scholar | DOI

[28] Gopakumar, R. and Gross, D. J., ‘Mastering the master field’, Nuclear Phys. B 451(1–2) (1995), 379–415. Google Scholar | DOI

[29] Gross, D. J. and Matytsin, A., ‘Some properties of large-n two-dimensional yang-mills theory’, Nuclear Phys. B 437(3) (1995), 541–584. Google Scholar | DOI

[30] Gross, D. J. and Taylor, W. Iv, ‘Two-dimensional QCD is a string theory’, Nuclear Phys. B 400(1–3) (1993), 181–208. Google Scholar | DOI

[31] Gross, L., ‘The Maxwell equations for Yang-Mills theory’, in Mathematical Quantum Field Theory and Related Topics (Montreal, PQ, 1987) (Conf. Proc.) vol. 9 (Amer. Math. Soc., Providence, RI, 1988), 193–203. Google Scholar

[32] Gross, L., King, C. and Sengupta, A., ‘Two-dimensional Yang-Mills theory via stochastic differential equations’, Ann. Physics 194(1) (1989), 65–112. Google Scholar | DOI

[33] Hall, B. C., ‘The large- limit for two-dimensional Yang–Mills theory’, Comm. Math. Phys. 363(3) (2018), 789–828. Google Scholar | DOI

[34] Hambly, B. and Lyons, T., ‘Uniqueness for the signature of a path of bounded variation and the reduced path group’, Ann. of Math. (2) 171(1) (2010), 109–167. Google Scholar | DOI

[35] Kassel, A. and Lévy, T., ‘Determinantal probability measures on Grassmannians’, Ann. Inst. Henri Poincaré D 9(4) (2022), 659–732. Google Scholar | DOI

[36] Kazakov, V. A., ‘Wilson loop average for an arbitrary contour in two-dimensional U( 𝑁) gauge theory’, Nuclear Phys. B 179(2) (1981), 283–292. Google Scholar | DOI

[37] Kazakov, V. A. and Kostov, I. K., ‘Nonlinear strings in two-dimensional U(∞) gauge theory’, Nuclear Phys. B 176(1) (1980), 199–215. Google Scholar | DOI

[38] Kenyon, R., ‘Lectures on dimers’, in Statistical Mechanics (IAS/Park City Math. Ser.) vol. 16 (Amer. Math. Soc., Providence, RI, 2009), 191–230. Google Scholar

[39] Lévy, T., ‘Yang-Mills measure on compact surfaces’, Mem. Amer. Math. Soc. 166(790) (2003), xiv+122. Google Scholar

[40] Lévy, T., ‘Discrete and continuous Yang-Mills measure for non-trivial bundles over compact surfaces’, Probab. Theory Related Fields 136(2) (2006), 171–202. Google Scholar | DOI

[41] Lévy, T., ‘Schur-Weyl duality and the heat kernel measure on the unitary group’, Adv. Math. 218(2) (2008), 537–575. Google Scholar | DOI

[42] Lévy, T., ‘Two-dimensional Markovian holonomy fields’, Astérisque 329 (2010), 172. Google Scholar

[43] Lévy, T., ‘The master field on the plane’, Astérisque 388 (2017), ix+201. Google Scholar

[44] Lévy, T., Two-Dimensional Quantum Yang–Mills Theory and the Makeenko–Migdal Equations (Springer International Publishing, Cham, 2020), 275–325. Google Scholar

[45] Liechty, K. and Wang, D., ‘Nonintersecting brownian motions on the unit circle’, Ann. Probab. 44(2) (2016), 1134–1211. Google Scholar | DOI

[46] Liu, K., ‘Heat kernels, symplectic geometry, moduli spaces and finite groups’, in Surveys in Differential Geometry: Differential Geometry Inspired by String Theory (Surv. Differ. Geom.) vol. 5 (Int. Press, Boston, MA, 1999), 527–542. Google Scholar

[47] Magee, M., ‘Random unitary representations of surface groups II: The large limit’, Preprint, 2021, . Google Scholar | arXiv | DOI

[48] Magee, M., ‘Random unitary representations of surface groups I: asymptotic expansions’, Comm. Math. Phys. 391(1) (2022), 119–171. Google Scholar PubMed | DOI

[49] Makeenko, Y. and Migdal, A. A., ‘Exact equation for the loop average in multicolor QCD’, Phys. Lett. B 88 (1979), 135–137. Google Scholar | DOI

[50] Mergny, P. and Potters, M., ‘Rank one HCIZ at high temperature: interpolating between classical and free convolutions’, SciPost Phys. 12 (2022), 022. Google Scholar | DOI

[51] Migdal, A. A., ‘Recursion equations in gauge field theories’, Sov. Phys. JETP (1975) 413–418. Google Scholar

[52] Mingo, J. A. and Speicher, R., Free Probability and Random Matrices (Fields Institute Monographs) vol. 35 (Springer, New York; Fields Institute for Research in Mathematical Sciences, Toronto, ON, 2017). Google Scholar | DOI

[53] Mlotkowski, W., ‘Λ-free probability’, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 7(1) (2004), 27–41. Google Scholar | DOI

[54] Moser, J., ‘On the volume elements on a manifold’, Trans. Amer. Math. Soc. 120 (1965), 286–294. Google Scholar | DOI

[55] Park, M., Pfeffer, J., Sheffield, S. and Yu, P., ‘Wilson loop expectations as sums over surfaces on the plane’, 2023. Google Scholar

[56] Sengupta, A., ‘Gauge theory on compact surfaces’, Mem. Amer. Math. Soc. 126(600) (1997), viii+85. Google Scholar

[57] Sengupta, A., ‘Yang-Mills on surfaces with boundary: quantum theory and symplectic limit’, Comm. Math. Phys. 183(3) (1997), 661–705. Google Scholar | DOI

[58] Sengupta, A. N., ‘The volume measure for flat connections as limit of the yang–mills measure’, J. Geom. Phys. 47(4) (2003), 398–426. Google Scholar | DOI

[59] Sengupta, A. N., ‘Traces in two-dimensional QCD: the large-𝑁 limit’, in Traces in Number Theory, Geometry and Quantum Fields (Aspects Math.) E38 (Friedr. Vieweg, Wiesbaden, 2008), 193–212. Google Scholar

[60] Shen, H., ‘A stochastic PDE approach to large problems in quantum field theory: a survey’, J. Math. Phys. 63(8) (2022), Paper No. 081103, 21. Google Scholar | DOI

[61] Shen, H., Smith, S. A. and Zhu, R., ‘A new derivation of the finite 𝑁 master loop equation for lattice Yang-Mills’, Electron. J. Probab. 29 (2024), Paper No. 29, 18. Google Scholar | DOI

[62] Singer, I. M., ‘On the master field in two dimensions’, in Functional Analysis on the Eve of the 21st Century, Vol. 1 (New Brunswick, NJ, 1993) (Progr. Math.) vol. 131 (Birkhäuser Boston, Boston, MA, 1995), 263–281. Google Scholar

[63] Speicher, R. and Wysoczański, J., ‘Mixtures of classical and free independence’, Arch. Math. (Basel) 107(4) (2016), 445–453. Google Scholar | DOI

[64] Stillwell, J., Classical Topology and Combinatorial Group Theory (Graduate Texts in Mathematics) vol. 72 (Springer-Verlag, New York, second edition, 1993). Google Scholar | DOI

[65] Hooft, G., ‘A planar diagram theory for strong interactions’, Nuclear Phys. B 72(3) (1974), 461–473. Google Scholar | DOI

[66] Voiculescu, D., ‘Limit laws for random matrices and free products’, Invent. Math. 104(1) (1991), 201–220. Google Scholar | DOI

[67] Voiculescu, D., ‘The analogues of entropy and of Fisher’s information measure in free probability theory. VI. Liberation and mutual free information’, Adv. Math. 146(2) (1999), 101–166. Google Scholar | DOI

[68] Voiculescu, D., ‘Lectures on free probability theory’, in Lectures on Probability Theory and Statistics (Saint-Flour, 1998) (Lecture Notes in Math.) vol. 1738 (Springer, Berlin, 2000), 279–349. Google Scholar

[69] Wilson, K. G., ‘Confinement of quarks’, Phys. Rev. 10 (1974). Google Scholar

[70] Witten, E., ‘The expansion in atomic and particle physics’, NATO Sci. Ser. B 59 (1980), 403–419. Google Scholar

[71] Witten, E., ‘On quantum gauge theories in two dimensions’, Comm. Math. Phys. 141(1) (1991), 153–209. Google Scholar | DOI

[72] Xu, F., ‘A random matrix model from two-dimensional Yang–Mills theory’, Comm. Math. Phys. 190(2) (1997), 287–307. Google Scholar | DOI

Cité par Sources :