A Study of U(N) Lattice Gauge Theory in 2-dimensions

Integration over matrices has been extensively discussed in Nuclear Physics. See, e.g., F.J. Dyson, J. Math. Phys. 3

1967

Earlier work this paper cites.

G. ‘t Hooft, Nucl. Phys. B72

1975

Earlier work this paper cites.

K.G. Wilson, Phys. Rev. D10

1975

Earlier work this paper cites.

See, for example: a) W.A. Bardeen, B.W. Lee and R.E. Schrock, Phys. Rev. D14

1976

Cited alongside, same era.

See, e.g., K.G. Wilson and J. Kogut, Phys. Reports 12C

1977

Cited alongside, same era.

E. Witten Havard Preprints HUTP-79/A007 and HUTP/79/A014

Cited in the paper.

‘Singular Integral Equations’, N.I. Muskhelishvili (P. Noordhoof N.V. - Groningen - Holland)

Cited in the paper.

Suggested by L. Kadanoff

Cited in the paper.

After this work was well begun we received a preprint on U ⁡ ( N ) U(N) lattice gauge theories by I. Bars and F. Green (IAS preprint) which also contains formula (40). In our notation they have computed the determinant for small E = 2 ​ β N E=\displaystyle{2\beta\over N} . We remark that in this limit the determinant can be exactly computed using the classical theorem of Szegö on Toeplitz determinant: For large N N , fixed β \beta log \displaystyle\log det ​ I i − j ​ ( 2 ​ β ) ∼ N ​ g 0 + ∑ n = 1 ∞ n ​ g n ​ g − n = β 2 \displaystyle\!\!\!\!\!~{\rm det}~I_{i-j}(2\beta)\sim Ng_{0}+\displaystyle{\sum^{\infty}_{n=1}}ng_{n}g_{-n}=\beta^{2} g n \displaystyle g_{n} = ∫ 0 2 ​ π d ​ θ 2 ​ π ​ e i ​ n ​ θ ​ log ⁡ ( e 2 ​ β ​ cos ⁡ θ ) \displaystyle\!\!\!=\int^{2\pi}_{0}{d\theta\over 2\pi}e^{in\theta}\log(e^{2\beta\cos\theta}) (See McCoy and Wu, ‘The Two Dimensional Ising Model [Hardvard Univ. Press], for an extensive discussion). Other recent papers that discuss U ⁡ ( N ) U(N) lattice theories: T. Eguchi (Chicago preprint EFI 79/22); D. Weingarten (Indiana preprint)

Cited in the paper.

I thank Mark Sweeny for responding with interest to do the computer calculation

Cited in the paper.

While this work was going on and after the solutions of the integral equations were obtained we learnt of a similar work by D. J. Gross and E. Witten (Princeton preprint)

Original

Cited in the paper.

E. Brézin, C. Itzykson, G. Parisi and J.B. Zuber, Commun. Math. Phys. 59

1978

Later among the works it cites.

J.M. Drouffe and C. Itzykson, Physics Reports 38C

1978

Later among the works it cites.