No CrossRef data available.
Article contents
Large N behaviour of the two-dimensional Yang–Mills partition function
Published online by Cambridge University Press: 30 June 2021
Abstract
Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.
We compute the large N limit of the partition function of the Euclidean Yang–Mills measure on orientable compact surfaces with genus 
$g\geqslant 1$ and non-orientable compact surfaces with genus 
$g\geqslant 2$, with structure group the unitary group 
${\mathrm U}(N)$ or special unitary group 
${\mathrm{SU}}(N)$. Our proofs are based on asymptotic representation theory: more specifically, we control the dimension and Casimir number of irreducible representations of 
${\mathrm U}(N)$ and 
${\mathrm{SU}}(N)$ when N tends to infinity. Our main technical tool, involving ‘almost flat’ Young diagram, makes rigorous the arguments used by Gross and Taylor (1993, Nuclear Phys. B 400(1–3) 181–208) in the setting of QCD, and in some cases, we recover formulae given by Douglas (1995, Quantum Field Theory and String Theory (Cargèse, 1993), Vol. 328 of NATO Advanced Science Institutes Series B: Physics, Plenum, New York, pp. 119–135) and Rusakov (1993, Phys. Lett. B 303(1) 95–98).
Keywords
MSC classification
Information
- Type
- Paper
- Information
- Copyright
- © The Author(s), 2021. Published by Cambridge University Press
References
Anshelevich, M. and Sengupta, A. N. (2012) Quantum free Yang–Mills on the plane. J. Geom. Phys. 62(2) 330–343.CrossRefGoogle Scholar
Boutet de Monvel, A. and Shcherbina, M. (1998) On free energy in two-dimensional
${\rm U}(n)$
-gauge field theory on the sphere. Teoret. Mat. Fiz. 115(3) 389–401.Google Scholar
${\rm U}(n)$
-gauge field theory on the sphere. Teoret. Mat. Fiz. 115(3) 389–401.Google ScholarBröcker, T. and tom Dieck, T. (1995)
Representations of compact Lie groups
, Vol. 98 of Graduate Texts in Mathematics. Springer-Verlag, New York.Google Scholar
Dahlqvist, A. and Norris, J. (2017) Yang–Mills measure and the master field on the sphere. arXiv preprint: https://arxiv.org/abs/1703.10578.Google Scholar
Douglas, M. R. and Kazakov, V. A. (1993) Large N phase transition in continuum QCD
${}_2$
. Phys. Lett. B 319 219–230.CrossRefGoogle Scholar
${}_2$
. Phys. Lett. B 319 219–230.CrossRefGoogle ScholarDouglas, M. R. (1995) Conformal field theory techniques in large N Yang-Mills theory. In Quantum Field Theory and String Theory (Cargèse, 1993), Vol. 328 of NATO Advanced Science Institutes Series B: Physics, Plenum, New York, pp. 119–135.CrossRefGoogle Scholar
Driver, B. K., Gabriel, F., Hall, B. C. and Kemp, T. (2017) The Makeenko-Migdal equation for Yang-Mills theory on compact surfaces. Comm. Math. Phys. 352(3) 967–978.CrossRefGoogle Scholar
Forrester, P. J., Majumdar, S. N. and Schehr, G. (2011) Non-intersecting Brownian walkers and Yang–Mills theory on the sphere. Nuclear Phys. B 844(3) 500–526.CrossRefGoogle Scholar
Fulton, W. and Harris, J. (1991) Representation Theory, Vol. 129 of Graduate Texts in Mathematics. Springer-Verlag, New York. A first course, Readings in Mathematics.Google Scholar
Gross, D. J. (1993) Two-dimensional QCD as a string theory. Nuclear Phys. B 400(1–3) 161–180.CrossRefGoogle Scholar
Gross, D. J. and Matytsin, A. (1995) Some properties of large-N two-dimensional Yang-Mills theory. Nuclear Phys. B 437(3) 541–584.CrossRefGoogle Scholar
Gross, D. J. and Taylor, IV, W. (1993) Two-dimensional QCD is a string theory. Nuclear Phys. B 400(1–3) 181–208.CrossRefGoogle Scholar
Guionnet, A. and Maïda, M. (2005) Character expansion method for the first order asymptotics of a matrix integral. Probab. Theory Related Fields 132(4) 539–578.CrossRefGoogle Scholar
Hall, B. C. (2018) The large-N limit for two-dimensional Yang–Mills theory. Comm. Math. Phys. 363(3) 789–828.CrossRefGoogle Scholar
Hooft, G. (1974) A planar diagram theory for strong interactions. Nuclear Phys. B 72(3) 461–473.CrossRefGoogle Scholar
Lévy, T. (2003) Yang-Mills measure on compact surfaces. Mem. Am. Math. Soc. 166(790) xiv+122.CrossRefGoogle Scholar
Lévy, T. and Maïda, M. (2015) On the Douglas-Kazakov phase transition. Weighted potential theory under constraint for probabilists. In Modélisation Aléatoire et Statistique—Journées MAS 2014, Vol. 51 of ESAIM Proceedings and Surveys, EDP Sciences, Les Ulis, pp. 89–121.CrossRefGoogle Scholar
Massey, W. S. (1991)
A Basic Course in Algebraic Topology
, Vol. 127 of Graduate Texts in Mathematics. Springer-Verlag, New York.Google Scholar
Rusakov, B. (1993) Large-N quantum gauge theories in two dimensions. Phys. Lett. B 303(1) 95–98.CrossRefGoogle Scholar
Sengupta, A. (1992) The Yang-Mills measure for
$S^2$
. J. Funct. Anal. 108(2) 231–273.CrossRefGoogle Scholar
$S^2$
. J. Funct. Anal. 108(2) 231–273.CrossRefGoogle ScholarSengupta, A. (1997) Yang-Mills on surfaces with boundary: quantum theory and symplectic limit. Comm. Math. Phys. 183(3) 661–705.CrossRefGoogle Scholar
Sengupta, A. N. (2008) Gauge theory in two dimensions: topological, geometric and probabilistic aspects. In Stochastic Analysis in Mathematical Physics, World Scientific Publishing, Hackensack, NJ, pp. 109–129.Google Scholar
Sengupta, A. N. (2008) Traces in two-dimensional QCD: the large-N limit. In Traces in Number Theory, Geometry and Quantum Fields, Vol. E38 of Aspects of Mathematics, Friedrich Vieweg, Wiesbaden, pp. 193–212.Google Scholar
Stanley, R. P. (1999)
Enumerative Combinatorics. Vol. 2
, Vol. 62 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge.Google Scholar
Tate, T. and Zelditch, S. (2003) Counterexample to conjectured SU(N) character asymptotics. arXiv preprint: https://arxiv.org/abs/hep-th/0310149.Google Scholar
Vershik, A. M. and Okounkov, A. Yu. (2004) A new approach to representation theory of symmetric groups. II. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 307(Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 10) 57–98, 281.Google Scholar
Witten, E. (1991) On quantum gauge theories in two dimensions. Comm. Math. Phys. 141(1) 153–209.CrossRefGoogle Scholar
Zagier, D. (1994) Values of zeta functions and their applications. In First European Congress of Mathematics, Vol. II (Paris, 1992), Vol. 120 of Progress in Mathematics, Birkhäuser, Basel, pp. 497–512.CrossRefGoogle Scholar
Zelditch, S. (2004) Macdonald’s identities and the large N limit of
$\rm YM_2$
on the cylinder. Comm. Math. Phys. 245(3) 611–626.CrossRefGoogle Scholar
$\rm YM_2$
on the cylinder. Comm. Math. Phys. 245(3) 611–626.CrossRefGoogle Scholar
You have
Access
- 4
- Cited by