Selected Topics in Lattice Gauge Theories
Lattice gauge theories was proposed 50 years ago by Wilson and Polyakov as a non-perturbative framework to study non-abelian Yang-Mills theories, such as Quantum Chromodynamics. A pure lattice gauge theory is characterised by its gauge group (usually a compact matrix Lie group), the dimension of spacetime, and a parameter known as the gauge coupling constant. Lattice gauge theories allow to study non-perturbative aspects of YM theories such as quark confinement and the spectrum of bound states, as well as to perform numerical simulations. In this course I will focus on the analytic methods such as the strong coupling expansion, mean field problem and large-N techniques. Special attention will be payed to the loop-space formulation of the lattice gauge theory which was used recently in a bootstrap program. The loop-space formulation is based on the Migdal-Makeenko equations for the Wilson loop functional, the basic observable in the pure gauge theory. Some toy models as two-dimensional gauge theory on a Riemann surface and the will be considered as well.
讲师
Ivan Kostov
日期
2025年05月06日 至 06月25日
位置
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| 周二,周三 | 10:40 - 12:15 | A3-2-303 | ZOOM 12 | 815 762 8413 | BIMSA |
修课要求
Basic knowledge of Quantum Mechanics, Quantum Field Theory and Statistical Mechanics is needed, as well as reasonable knowledge of lineal algebra, complex analysis and representation theory of compact Lie groups.
课程大纲
1. Historical overview. Functional integral formulation of Euclidean field theory. Global and local invariance. Gauge fields and their geometrical interpretation as connections.
2. Scalar and gauge fields on the lattice. Link variables, plaquettes and gauge transformations. Lattice discretisation of the functional integral. Wilson action. Strong and weak-coupling regimes. Continuum limit.
3. Haar measure over the SU(N) group. Variant lattice actions. Gauge-invariant observables and Elitzur’s theorem. Wilson loops.
4. Strong coupling methods: Character expansion. Criterion for confinement in terms of Wilson loops: Wilson’s area law. Convergence of the strong coupling expansion. Two-dimensional gauge theory.
5. Mean field approximation. Variational analysis. Saddle point method and corrections. Diagrammatic interpretation of corrections. The ’t Hooft large N limit for lattice gauge theories. Gross-Witten phase transition. External field poroblem and the Brezin-Gross integral.
6. Reflection (Osterwalder-Schrader) positivity. Hilbert space, transfer matrix, and Hamiltonian. Static quarks as external charges. The static quark potential and Polyakov loops. A criterion for confinement at zero and finite temperature. The deconfining transition at finite temperature in 3 + 1 or more dimensions.
7. Lattice gauge theory and loop dynamics. Makeenko-Migdal loop equations in the continuum and on the lattice. Factorisation property in the large N limit. Analytic solution of the loop equations in a two dimensional YM theory.
8. Two-dimensional gauge theory on a Riemann surface of arbitrary topology. Analytic solution of the loop equations. Dual formulation in terms of random surfaces.
9. Lattice gauge theory for random surfaces on hypercubical lattice (the Weingarten model). Random surface Ansatz for the Wilson loop in the large N limit.
10. Bootstrap strategy based on the positivity conditions and loop equations in the large N limit and for finite N.
2. Scalar and gauge fields on the lattice. Link variables, plaquettes and gauge transformations. Lattice discretisation of the functional integral. Wilson action. Strong and weak-coupling regimes. Continuum limit.
3. Haar measure over the SU(N) group. Variant lattice actions. Gauge-invariant observables and Elitzur’s theorem. Wilson loops.
4. Strong coupling methods: Character expansion. Criterion for confinement in terms of Wilson loops: Wilson’s area law. Convergence of the strong coupling expansion. Two-dimensional gauge theory.
5. Mean field approximation. Variational analysis. Saddle point method and corrections. Diagrammatic interpretation of corrections. The ’t Hooft large N limit for lattice gauge theories. Gross-Witten phase transition. External field poroblem and the Brezin-Gross integral.
6. Reflection (Osterwalder-Schrader) positivity. Hilbert space, transfer matrix, and Hamiltonian. Static quarks as external charges. The static quark potential and Polyakov loops. A criterion for confinement at zero and finite temperature. The deconfining transition at finite temperature in 3 + 1 or more dimensions.
7. Lattice gauge theory and loop dynamics. Makeenko-Migdal loop equations in the continuum and on the lattice. Factorisation property in the large N limit. Analytic solution of the loop equations in a two dimensional YM theory.
8. Two-dimensional gauge theory on a Riemann surface of arbitrary topology. Analytic solution of the loop equations. Dual formulation in terms of random surfaces.
9. Lattice gauge theory for random surfaces on hypercubical lattice (the Weingarten model). Random surface Ansatz for the Wilson loop in the large N limit.
10. Bootstrap strategy based on the positivity conditions and loop equations in the large N limit and for finite N.
参考资料
[1] K. G. Wilson, “Confinement of Quarks,” Phys. Rev. D 10 (1974) 2445–2459.
[2] J.-M. Drouffe and J.-B. Zuber, “Strong coupling and mean field methods in lattice gauge theories,” Physics Reports 102 (1983), no. 1–2, 1 – 119.
[3] E. Brezin and D. J. Gross, “The External Field Problem in the Large N Limit of QCD,” Phys. Lett. B97 (1980) 120.
[4] A. M. Polyakov, “Gauge Fields as Rings of Glue,” Nucl.Phys. B164 (1980) 171–188.
[5] A. A. Migdal, “Loop equations and 1/N expansion,” Physics Reports 102 (1983), no. 4, 199–290.
[6] V. Kazakov and I. Kostov, “Nonlinear strigs in U(∞) gauge theory,” Nucl. Phys. B176 (1980) 199–215.
[7] V. Kazakov and I. Kostov, “Computation of the Wilson loop functional in two-dimensional U(∞) lattice gauge theory ,” Phys. Lett. B105 (1981) 453–456.
[8] I. Kostov, “U(N) gauge theory and lattice strings,” Nucl. Phys. B415 (1994) 29–70, hep-th/9308158.
[9] V. Kazakov and Z. Zheng, “Bootstrap for lattice Yang-Mills theory,” Physical Review D 107 (Mar., 2023) L051501, 2203.11360.
[10] V. Kazakov and Z. Zheng, “Bootstrap for Finite N Lattice Yang-Mills Theory,” 2024.
[11] Y. Guo, Z. Li, G. Yang, and G. Zhu, “Bootstrapping SU(3) Lattice Yang-Mills Theory,” 2025.
[2] J.-M. Drouffe and J.-B. Zuber, “Strong coupling and mean field methods in lattice gauge theories,” Physics Reports 102 (1983), no. 1–2, 1 – 119.
[3] E. Brezin and D. J. Gross, “The External Field Problem in the Large N Limit of QCD,” Phys. Lett. B97 (1980) 120.
[4] A. M. Polyakov, “Gauge Fields as Rings of Glue,” Nucl.Phys. B164 (1980) 171–188.
[5] A. A. Migdal, “Loop equations and 1/N expansion,” Physics Reports 102 (1983), no. 4, 199–290.
[6] V. Kazakov and I. Kostov, “Nonlinear strigs in U(∞) gauge theory,” Nucl. Phys. B176 (1980) 199–215.
[7] V. Kazakov and I. Kostov, “Computation of the Wilson loop functional in two-dimensional U(∞) lattice gauge theory ,” Phys. Lett. B105 (1981) 453–456.
[8] I. Kostov, “U(N) gauge theory and lattice strings,” Nucl. Phys. B415 (1994) 29–70, hep-th/9308158.
[9] V. Kazakov and Z. Zheng, “Bootstrap for lattice Yang-Mills theory,” Physical Review D 107 (Mar., 2023) L051501, 2203.11360.
[10] V. Kazakov and Z. Zheng, “Bootstrap for Finite N Lattice Yang-Mills Theory,” 2024.
[11] Y. Guo, Z. Li, G. Yang, and G. Zhu, “Bootstrapping SU(3) Lattice Yang-Mills Theory,” 2025.
听众
Graduate
视频公开
公开
笔记公开
不公开
语言
英文
讲师介绍
Ivan Kostov obtained his PhD in 1982 from the Moscow State University, with scientific advisers Vladimir Feinberg and Alexander Migdal. Then he worked in the group of Ivan Todorov at the INRNE Sofia, and since 1990 as a CNRS researcher at the IPhT, CEA-Saclay, France. Currently he is emeritus DR CNRS at IPhT and a visiting professor at UFES, Vitoria, Brazil.