Two dimensional QCD is a one dimensional Kazakov-Migdal model

Abstract

We calculate the partition functions of QCD in two dimensions on a cylinder and on a torus in the gauge ${\partial }_0{A}_0=0$ by integrating explicitly over the non zero modes of the Fourier expansion in the periodic time variable. The result is a one dimensional Kazakov-Migdal matrix model with eigenvalues on a circle rather than on a line. We prove that our result coincides with the standard expansion in representations of the gauge group. This involves a non trivial modular transformation from an expansion in exponentials of $g^2$ to one in exponentials of $1/g^2$. Finally we argue that the states of the $U(N)$ or $SU(N)$ partition function can be interpreted as a gas of N free fermions, and the grand canonical partition function of such ensemble is given explicitly as an infinite product.

DFTT 15/93, 17 pages, Latex (Besides minor changes and comments added we note that for U(N) odd and even N have to be treated separately)

References in corpus (12)

• Two Dimensional QCD is a String Theory
• Two Dimensional QCD as a String Theory
• Twists and Wilson Loops in the String Theory of Two Dimensional QCD
• Induced QCD at Large N
• Equivalence of Two Dimensional QCD and the $c=1$ Matrix Model
• Exact Solution of Induced Lattice Gauge Theory at Large $N$
• Area Law and Continuum Limit in "Induced QCD"
• Some Remarks About Induced QCD
• Exact solution of d=1 Kazakov-Migdal induced gauge theory
• The Kazakov-Migdal Model as a High Temperature Lattice Gauge Theory
• Infinite Tension Strings at D>1
• D-dimensional Induced Gauge Theory as a Solvable Matrix Model

Cited by in corpus (21)

• Random matrix theory and symmetric spaces
• Two-dimensional QCD, instanton contributions and the perturbative Wu-Mandelstam-Leibbrandt prescription
• Updating QCD${}_2$
• Toward an analytic determination of the deconfinement temperature in SU(2) L.G.T.
• Loop Correlators and Theta States in 2D Yang-Mills Theory
• The $W_{1+\infty }$ effective theory of the Calogero- Sutherland model and Luttinger systems.
• Matrix string states in pure 2d Yang Mills theories
• Perturbative and non-perturbative aspects of the two-dimensional string/Yang-Mills correspondence
• String Theoretical Interpretation for Finite N Yang-Mills Theory in Two-Dimensions
• Deconfinement Transition in Large N Lattice Gauge Theory
• Two-Dimensional Yang-Mills Theory and Moduli Spaces of Holomorphic Differentials
• Topology on the lattice; 2d Yang-Mills theories with a theta term
• A String Theory for Two Dimensional Yang-Mills Theory I
• Exact beta function from the holographic loop equation of large-N QCD_4
• Gross-Witten-Wadia Phase Transition in Induced QCD on the Graph
• 2D Yang-Mills Theory as a Matrix String Theory
• A Matrix Model for Random Surfaces with Dynamical Holes
• Analytic results in 2+1-dimensional Finite Temperature LGT
• The Path Integral for 1+1-dimensional QCD
• A new large N phase transition in YM2
• One-dimensional dynamics of QCD2 string