Abstract
In two dimensions a pure lattice gauge theory with the simple Wilson action can be solved analytically. With open boundary conditions and in the axial gauge, the partition function becomes a product of simple group integrals, and the area law behavior is exact for all values of the gauge coupling β. In this chapter we impose periodic boundary conditions in all directions, adequate for finite temperature and finite volume studies. On a torus the solution is a bit less trivial, and the exact solution can be used as a test bed for new Monte Carlo algorithms. First we study simple Abelian gauge models for which the calculation parallels our treatment of one-dimensional spin models.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
A.A. Migdal, Phase transitions in gauge and spin-lattice systems. JETP 69, 1457 (1975)
K.R. Ito, Analytic study of the Migdal-Kadanoff recursion formula. Comm. Math. Phys. 93, 379 (1984)
J.M. Drouffe, J.B. Zuber, Strong coupling and mean field methods in lattice gauge theories. Phys. Rep. 102, 1–119 (1983) 1
H.J. Rothe, Lattice Gauge Theories: An Introduction (World Scientific, Singapore, 2012)
H.G. Dosch, V.V. Müller, Lattice gauge theory in two spacetime dimensions. Fort. der Physik 27, 547 (1979)
G. Bali, Casimir scaling of SU(3) static potentials. Phys. Rev. D62, 114503 (2000)
O. Philipsen, H. Wittig, String breaking in non-Abelian gauge theories with fundamental matter fields. Phys. Rev. Lett. 81, 4059 (1998)
M. Pepe, U.-J. Wiese, From decay to complete breaking: pulling the strings in SU(2) Yang-Mills theory. Phys. Rev. Lett. 102, 191601 (2009)
B. Wellegehausen, A. Wipf, C. Wozar, Casimir scaling and string breaking in G2 gluodynamics. Phys. Rev. D83, 016001 (2011)
D.J. Gross, E. Witten, Possible third order phase transition in the large N lattice gauge theory. Phys. Rev. D21, 446 (1980)
T. Bröcker, T. tom Dieck, Representations of Compact Lie Groups. Graduate Texts in Physics (Springer, Berlin, 2010)
G. Segal, Lie groups, in Lectures on Lie Groups and Lie Algebras (Cambridge University Press, Cambridge, 1995)
C. Chevalley, Theory of Lie Groups. Dover Books on Mathematics (Dover, Mineola, 2018)
M. Hammermesh, Group Theory and Its Application to Physical Problems (Dover, New York, 2003)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Wipf, A. (2021). Two-Dimensional Lattice Gauge Theories and Group Integrals. In: Statistical Approach to Quantum Field Theory. Lecture Notes in Physics, vol 992. Springer, Cham. https://doi.org/10.1007/978-3-030-83263-6_14
Download citation
DOI: https://doi.org/10.1007/978-3-030-83263-6_14
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-83262-9
Online ISBN: 978-3-030-83263-6
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)