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Large-N quantum gauge theories

arXiv:hep-th/9212090v2 3 Jan 1993TAUP-2012-92December 1992hep-th/9212090Large-N quantum gauge theoriesin two dimensionsB.RusakovSchool of Physics and AstronomyRaymond and Beverly Sackler Faculty of Exact SciencesTel-Aviv University, Tel-Aviv 69978, IsraelAbstract.The partition function of a two-dimensional quantum gauge theory in thelarge-N limit is expressed as the functional integral over some scalar field.The large-N saddle point equation is presented and solved. The free energyis calculated as the function of the area and of the Euler characteristic.There is no non-trivial saddle point at genus g > 0.

The existence of a non-trivial saddle point is closely related to the weak coupling behavior of thetheory. Possible applications of the method to higher dimensions are brieflydiscussed.1

It is well known that quantum gauge theories are exactly soluble in twodimensions. It makes two dimensional model a useful instrument for probinga new methods eventually intended to higher dimensions.

In this letter wedevelop a new large-N approach for the gauge theories, and apply it to thetwo-dimensional pure gauge theory. We consider the model in the latticeformulation, given by K.Wilson [1].

A convenient approach to this model atarbitrary N is the group-theoretical expansion proposed by A.A.Migdal [2].This approach has been applied by the author to calculate loop averages1 andthe partition function of the model in two dimensions at arbitrary finite N[5]. The results were expressed through sums over irreducible representationsof the gauge group.The idea we develop below is that in the large-N limit the finite-N sig-natures (parametrized by the N numbers), can be replaced by a scalar field,while the sum over all signatures can be represented by the functional integralover this scalar field.Using the exact results of [5], we realize this program in two dimensions(we consider the case of arbitrary orientable surfaces).

We solve the large-Nsaddle point equation by the method described in ref. [6] and calculate thefree energy as a function of the area and of the topological characteristics ofthe manifold.In conclusion we discuss the difference between our approach and thatof ref.

[7] and the possibility of applying both of them together to higher-dimensional models.Thus, we consider the lattice model [1]S = β0NXftr [ Uf + U†f],(1)where sum goes over all faces (fundamental polygons) f of the two-dimensionallattice and β0 is the lattice coupling constant.Following [2] we expand the contribution of each face over irreduciblerepresentations, r, of the (compact) gauge group :e β0N tr [ Uf+U†f ] =Xrd rλr(β0N)χr(Uf),(2)where χr(U) and d r = χr(I) are characters and dimensions of r’s respectively.The coefficients of the expansion (2)λr(β0N) =ZDU e β0N tr [ U+U†]χr(U)(3)1All loop averages on a plane were calculated by V.A.Kazakov and I.K.Kostov [3]. Seealso paper of N.Bralic [4].2

have the following asymptotic behavior in the continuum limit ǫ →0 (ǫ isthe area of the face and β = ǫβ0) :λr(βN) ∼exp −C2(r)2βN ǫ!,(4)where C2(r) is the eigenvalue of the quadratic Casimir operator.In ref. [5] the following expression for the partition function has been de-rived2:Z =Xrd ηr exp −C2(r)A2βN!

(5)where A is the area of the surface and η = 2 −2g is the Euler characteristic,with g being a number of the handles (genus). It has been argued in [5] thatZ = 1 in the case of the surface with holes.Now we substitute in eq.

(5) an explicit expressions for dimensions,d r =NYi

. , nN} obeying the domi-nance condition n1 ≥.

. .

≥nN. Hence, eq.

(5) readsZ =∞Xn1. .

.∞XnNN−1Yk=1θ(nk −nk+1) e S,(8)S = −A2βNNXk=1n2k + nk(N −2k + 1)+ η2Xi̸=jlog 1 + ni −njj −i!,(9)where the step function, θ(n) = 1 if n ≥0 and θ(n) = 0 if n < 0, realizes thedominance condition for signatures.For large N, we introduce a continuum time, 0 ≤x = kN ≤1, and replacesum over nk’s by the path integral over the scalar field n(x) :Z =ZY1≤x≤0dn(x) e S(10)S = N22Z 10dx(−Aβn2(x) + n(x)(1 −2x)+ η −Z 10dy log 1 + n(x) −n(y)y −x!). (11)2In ref.

[5] this formula was obtained actually for U(N) and SU(N) gauge groups but,as it can be easily proven, it holds for any compact gauge group.3

We omit here the step function since its contribution to the action is of orderN (i.e., 1/N with respect to expression (11)) 3.Now, we calculate (10) using the saddle point method. First, we replacen(x) by the new field φ(x) = n(x) −x + 12.

Then, the saddle point equationis2ξφ(x) = −Z 10dyφ(x) −φ(y);ξ = Aβη . (12)Introducing the densityρ(φ) = dxdφ(13)which should be positive, even and normalized toZ a−adλ ρ(λ) = 1,(14)we rewrite eq.

(12) as equation for ρ:2ξλ = −Z a−adµρ(µ)λ −µ;|λ| ≤a . (15)The solution of eq.

(15) is 4ρ(λ) = 2πqξ(1 −ξλ2)(16)witha = 1√ξ . (17)Now, we transform (11) intoS = N22( A12β + 32η −AβZ a−adλ ρ(λ)λ2 −ξ−1 −Z a−adµ ρ(µ) log |λ −µ|).

(18)Then, integrating (15) with respect to λ and defining the free energy asF =2N2 log Z we haveF =A12β + 32η + ηZ a−adλ ρ(λ) log |λ|(19)and, finally,F =A12β + η 1 + 12 log ηβ4A!. (20)3More detailed discussion of this point will be given in [8].4The method of solution of such an equations can be found in ref.

[6].4

We see from (16),(17) that there is no non-trivial large-N saddle pointfor genus g > 0 (ξ and η become negative). The appearance of this phe-nomenon is clear already from formula (5).

A saddle point exists only whenthe topological (entropy) term in the effective action (this term arises frompowers of d r’s) acts in the opposite direction to the area (energy) term.In the case of a torus (g = 1) there is no topological term. The corre-sponding free energy is F =A12β.At higher genera, η < 0, there are only negative powers of d r in (5)and the topological term acts in the same direction as the area term.

Thepartition function (5) in the large-N limit is then dominated by the trivialrepresentation (d r = 1, C2(r) = 0) and corresponding free energy equal tozero.The non-trivial large-N behavior occurs only for a sphere (g = 0). In thiscase, the positive powers of d r’s give a positive contribution to the effectiveaction and compensate (at the saddle point) the negative contribution of thearea term.To summarize, we write the free energy of the large-N quantum gaugetheory on an orientable surface with g handles and with h holes,F =A12β + 2 + log β2A,sphere : g = 0 and h = 0A12β,torus : g = 1 and h = 00,g > 1 or (and) h > 0(21)This formula sums all planar diagrams of the quantum gauge theory intwo dimensions.Note, that the non-trivial large-N saddle point exists only when in theweak coupling limit (β →∞in our notation) the free energy is divergent.Apparently, this is the case in higher dimensions and we can hope that ourmethod will be relevant there.Possible application of our approach to higher dimensions is intimatelyconnected to the problem of the (one-link) integration over unitary matrixwhich is already not so simple as in two dimensions, even at large N. Thisproblem has been studied by D.Gross and E.Witten in [7] and a third orderphase transition was found.

The technical reason for such a phase transi-tion is the unitarity constraint: the eigenvalue density (the analogue of ourquantity (13)),ρ(α) = 2πβ cos(α)s12β −sin2(α)(22)(where α(x) is the scalar field coming from the unitary matrix eigenvalues)depends on the functions bounded by 1, and, consequently, there are two5

different types of behavior for coupling constant β > 12 and for β < 12. Fromthe point of view of our approach this phenomenon seems a lattice artifact,since considering the model, where integration over unitary matrices (overα’s at large N) is performed from very beginning [5], we do not realize aphase transition with respect to β.To conclude, the following proposal for higher dimensions can be made.The large-N limit could be described in terms of both a scalar field n(x)arising from the signatures and a scalar field α(x) arising from eigenvalues ofthe unitary matrix, with a properly defined integration over α’s.

Introduc-ing two scalar fields may seems like an unnecessary complication, but thiscomplication could give more freedom to solve the problem.Acknowledgements.I am grateful to A.A.Migdal for drawing my attention to this problem and fordiscussion. I thank also M.Karliner, N.Marcus, J.Sonnenschein and S.Yankielowiczfor discussion and D.V.Boulatov for valuable comments.

This research hasbeen supported in part by the Basic Research Foundation administered bythe Israel Academy of Sciences and Humanities, by a grant from the UnitedStates–Israel Binational Science Foundation (BSF), Jerusalem, Israel, andalso by the Israel Ministry of Absorption.References[1] K.Wilson, Phys.Rev. D10, 2445 (1974).

[2] A.A.Migdal, ZhETF 69 (1975) 810 (Sov.Phys.JETP 42 413). [3] V.A.Kazakov, I.K.Kostov, Nucl.Phys.

B176 (1980) 199;V.A.Kazakov, Nucl.Phys. B179 (1981) 283.

[4] N.Bralic, Phys.Rev. D22 (1980) 3090.

[5] B.Rusakov, Mod.Phys.Lett. A5 (1990) 693.

[6] E.Brezin, C.Itzykson, G.Parisi, J.B.Zuber, Comm.Math.Phys. 59 (1978) 35.

[7] D.J.Gross, E.Witten, Phys.Rev. D21 (1980) 446.

[8] B.Rusakov, in progress.6

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