## Garćıa-Compeán, “From Principal Chiral Model to Self-dual Gravity

- H.J.F. Plebański M. Przanowski
- Mod. Phys. Lett
- 1996

- Published 1996

Semiclassical string action is rederived from the reduced SU(N) quenching gauge theory via Wigner-Weyl-Moyal formalism. Moyal deformation of SchildEguchi action is obtained. ⋆ In Memory of Dr. Vı́ctor Mart́ınez Olivé. † E-mail: compean@fis.cinvestav.mx ‡ E-mail: pleban@fis.cinvestav.mx § E-mail: nquiroz@fis.cinvestav.mx Many years ago G. ‘t Hooft has shown that drastic simplification occurs in the gauge structure of SU(NC) Quantum Chromodynamics (QCD), as one take the number of colors NC to be infinite 1. Large-N limit, for instance, contains genuine non-perturbative information of field theory at the classical and quantum levels. Chiral symmetry breaking, mass gap and color confinement are important non-perturbative features which persist in this limit. Large-N techniques have been also applied to describe mesons and baryons in a complete picture2 and more recently to matrix models approach to 2D quantum gravity and 2D string theory ¶ , looking for its non-perturbative formulation. The power of these results enable one to study some large-N limits of various physical systems. For instance, some twodimensional integrable systems seem to be greatly affected in this limit, turning out ‘more integrable’. Example of this ‘induced integrability’ occurs in the large-N limit of SU(N) Nahm equations4. Actually the origin of this drastic simplification in the field equations is not well understood5. The results presented in this paper might will sheed some light about this mysterious limit. The usual transition from SU(N) gauge theory to the SU(∞) ones involves the change of the Lie algebra su(N) by the area-preserving diffeomorphisms algebra, sdiff(Σ), on a two-dimensional manifold (Σ, ω). The last one is an infinite dimensional Lie algebra. As we make only local considerations we assume the space Σ to be a two-dimensional simply connected symplectic manifold with local real coordinates {τ, σ}. This space has a natural local symplectic structure given by the local area form ω = dσ ∧ dτ . sdiff(Σ) is precisely the Lie algebra associated with the infinite dimensional Lie group, SDiff(Σ), which is the group of diffeomorphisms on Σ preserving the symplectic structure ω, i.e. for all g ∈ SDiff(Σ), g∗(ω) = ω. Globally the symplectic form is defined by ω : TΣ → T ∗Σ and inverse ω−1 : T ∗Σ → TΣ. Here TΣ and T ∗Σ are the respective tangent and cotangent bundles to Σ. While the hamiltonian vector fields are UHa = ω(dHa) satisfying the sdiff(Σ) algebra ¶ There are many references about this topic, for an excellent review see reference 3 and references therein. 2 [UHa ,UHb] = U{Ha,Hb}P , for all (a 6= b), (1) where {, }P stands for the Poisson bracket. Locally it can be written as {Ha, Hb}P = ω(dHa, dHb) = ω∂iHa∂jHb, (2) where ∂i ≡ ∂ ∂σ , (i = 0, 1), σ0 = τ, σ1 = σ and Hi = Hi(~σ) = Hi(σ, σ1). The generators of sdiff(Σ) are the hamiltonian vector fields UHa associated to the hamiltonian functions Ha UHa = ∂Ha ∂σ0 ∂ ∂σ1 − ∂Ha ∂σ1 ∂ ∂σ0 . (3) On the other hand, in the Ref. 6, the Lie algebra su(N) is defined which appears to be very useful in our further considerations. The elements of this basis are denoted by Lm, Ln,..., etc., m = (m1, m2), n = (n1, n2),..., etc., and m, n, ... ∈ IN ⊂ Z × Z − {(0, 0) mod Nq} where q is any element of Z × Z. The basic vectors Lm, m ∈ IN , are the N × N matrices satisfying the following commutation relations [Lm, Ln] = N π Sin ( π N m× n ) Lm+n modNq, (4) where m× n := m1n2 −m2n1. Now we let N tend to infinity . In this case I∞ ≡ I = Z × Z − {(0, 0)} and the commutation relations (4) read 3 [Lm, Ln] = (m× n) Lm+n. (5) Consider the set {em}m∈I , em = em(~σ) := exp [ i(m1σ +m2σ 1) ] . One quickly finds that {em, en}P = (m× n) em+n. (6) Thus the mapping F : Lm 7→ em, m ∈ I, defines the isomorphism su(∞) ∼= the Poisson algebra on Σ(= T ) ∼= sdiff(T ), (7) where T 2 is the 2-torus. On the other hand, I. Bars using a series of basic results in reduced and quenched large-N gauge theories was able to derive the string theory action in a particular gauge7. To do that he has used the above mentioned area-preserving diffeomorphisms formalism ⋆ . Very similar approach has been considered by C. Zachos in the context of SU(∞) hamiltonian flows9. There it was derived Nambu’s action from the largeN approach to Yang-Mills gauge theory10. In this derivation the Schild-Eguchi action11 for strings arise in a natural way. The reduced SU(N) gauge theory action is7 Sred = − 1 4 (2π Λ d N g2 d(Λ) Tr ( FμνF ) , (8) where d is the space-time dimension, gd(Λ) is the Yang-Mills coupling constant in d dimensions evaluated at certain cut-off Λ and Tr is an invariant bilinear form ⋆ Large-N -limit was first studied by Hoppe in the context of membrane physics. In that case Σ = S, the two-sphere. 4 on the Lie algebra su(N). The general prescription to obtain the reduced scheme from Yang-Mills gauge theory is Fμν = [iDμ, iDν] ⇒ (Fμν)j ≡ [aμ, aν ]j , (9a) where (·)j denotes an N × N matrix, Dμ is the covariant derivative with respect to the Yang-Mills potential Aμ, iDμ must be replaced by aμ = Pμ + Aμ, (9b) where Pμ is the quenched momentum which is a diagonal matrix 12. Now we will use the previous discussion on area preserving diffeomorphisms to apply it to reduced large-N gauge theory (for details see Refs. 7 and 9). Any solution of the SU(N) reduced gauge theory equations coming from the reduced action (8) can be written in the form

@inproceedings{GaraCompen1996ANO,
title={A Note on the String Action Derivation From Reduced SU(N) Gauge Theory Via Weyl-Wigner-Moyal Formalism},
author={Hugo Gar{\'c}ıa-Compe{\'a}n and Norma Quiroz-P{\'e}rez},
year={1996}
}