EXACT SOLUTION OF D=1 KAZAKOV-MIGDAL

arXiv:hep-th/9207086v2 3 Aug 1992DFTT 38/92July 1992EXACT SOLUTION OF D=1 KAZAKOV-MIGDALINDUCED GAUGE THEORYM. Caselle, A. D’Adda and S. PanzeriDipartimento di Fisica Teorica dell’Universit`a di TorinoIstituto Nazionale di Fisica Nucleare,Sezione di Torinovia P.Giuria 1, I-10125 Turin,ItalyAbstractWe give the exact solution of the Kazakov-Migdal induced gauge modelin the case of a D=1 compactified lattice with a generic number S of sitesand for any value of N. Due to the peculiar features of the model, the parti-tion function that we obtain also describes the vortex-free sector of the D=1compactified bosonic string, and it coincides in the continuum limit with theone obtained by Boulatov and Kazakov in this context.⋄email address: Decnet=(31890::CASELLE,DADDA,PANZERI)Bitnet=CASELLE(DADDA)(PANZERI)@TORINO.INFN.IT1

1. IntroductionRecently, a new interesting approach to gauge theories has been proposed [1]and solved in the large N limit[1, 2] by Kazakov and Migdal.

The hope is tobe able to describe within this approach, the asymptotically free fixed point ofQCD in 4 dimensions. The Kazakov-Migdal model seem to be a promising tool inthis direction, but several question must be understood in order to reach this goal.Let us mention two of them: first, one should make sure that there is no phasetransition at any finite value of N; for that reason it would be important to havesome example where the exact explicit dependence on N is known.

Second, onewould like to understand the nature of the transition point which appears in themodel at a critical value mc of the mass parameter. Indeed, looking at the inducedgauge theory, due to the fact that matter fields are in the adjoint representationof the U(N) group (see below), one finds a super- confining behaviour [3], and thehope is to reach an ordinary confining phase through the above mentioned phasetransition.

Hence it would be interesting to study models which are simple enoughto be exactly solvable, but still having all the desired non trivial properties.In this letter we will give the exact solution for any value of N in the case of a d=1compactified lattice made of S matter fields, for any value of S. This model fulfillsthe above requirements: it has a non trivial phase transition, and the solution canbe obtained by using simple combinatoric properties of the permutation group, thekey trick being the reduction of the permutation group to its cyclic representations.Moreover, as one would expect from the definition of the model (see below), oursolution also describes the vortex-free sector of the d=1 compactified bosonic string[4]. In this context it is rather interesting to notice that the analytic continuationin the mass parameter from the strong to the weak coupling phase provides thecorrect prescription to obtain the physical properties of the upside-down oscillatorsfrom the standard matrix oscillators [5].This letter is organized as follows: after a brief introduction on the Kazakov-Migdal model (sect.2) , we give in sect.3 the exact solution of the model.The solutionis discussed in sect.

4 which includes also some concluding remark.2. The Kazakov-Migdal modelThe starting point of Kazakov-Migdal suggestion is to induce the Yang-Millsinteraction using massive scalar fields in the adjoint representation of U(N).

Theaction they propose is defined on a generic d-dimensional lattice and has the fol-lowing form:S =XxNTr[m2φ2(x) −Xµφ(x)U(x, x + µ)φ(x + µ)U†(x, x + µ)](1)where φ(x) is an Hermitian N × N matrix defined on the sites x of the lattice andU(x, x + µ) is a Unitary N × N matrix, defined on the links (x, x + µ), and plays2

the role, as in the usual lattice discretization of Yang-Mills theories, of the gaugefield. Integrating over the scalar field φ one can induce an effective action for thegauge field,ZDUDΦexp(−S) ∼ZDUexp(−Sind[U])(2)with:Sind[U] = −12XΓ|TrU[Γ]|2l[Γ]m2l[Γ],(3)where l[Γ] is the length of the loop Γ, U[Γ] is the ordered product of link matricesalong Γ and the summation is over all closed loops.In a similar way, integrating over the gauge fields one can induce an effectiveinteraction for the scalar fields, which turns out to be deeply related to the matrixapproach to 2D Quantum Gravity and noncritical strings.Integration can be achieved by using the well known formula first discovered byHarish-Chandra [7], rediscovered in the context of matrix models by Itzykson andZuber [6] and fully exploited by Mehta [8]:I(φ(x), φ(y)) =ZDU expN trφ(x)Uφ(y)U†∝detij exp(Nλi(x)λj(y))∆(λ(x))∆(λ(y))(4)where λi(x) are the eigenvalues of the matrix φ(x)∆(λ) =Yi

It is exactly this kind ofobstruction which doesn’t allow, in the context of the matrix approach, a descrip-tion of d > 1 bosonic strings in terms of the eigenvalues only, and which manifestsitself in the case of the d=1 compactified bosonic string as a vortex contribution.This means that in this last example the Kazakov-Migdal model should exactlycorrespond to the singlet, vortex free, solution of the compactified bosonic string.A further important feature of the Kazakov-Migdal model is the presence ofa phase transition, which should occur at a finite, non-zero value mc of the massparameter, between a strong coupling regime (m > mc) and a weak coupling phase(m < mc). This transition was discussed in [1, 2] in the context of the inducedmatrix model (after integration on the gauge fields) and was conjectured in [3]to be related (in the context of the induced gauge theory, after integration on thematter fields) to the change from ordinary to local confinement.

We will show below3

that this same transition in the d=1 case separates the upside-down oscillator phasefrom the standard matrix oscillator description of the d=1 bosonic string.3. Exact solution for a d=1 compactified latticeThe partition function of the Kazakov-Migdal model defined on a 1d lattice with Ssites (labelled by α) compactified on a circle is:Z=SXα=1Zdφ(α)dU(α, α + 1) e−NTr[m2φ(α)2−φ(α)U(α,α+1)φ(α+1)U(α,α+1)†] .

(6)The model (6) is reduced , by integrating over the unitary matrices on each link, toZ=ZYα,idλ(α)ie−m2N Pα,i λ(α)2i1NSN(N−1)/2YαdeteNλ(α)iλ(α+1)j(7)where λ(α)iis the ith eigenvalue of the matrix φ(α) .It is easy to see that this expression can be rewritten as follows:Z =XPα(−1)P1+···+PSZ Yα,idλ(α)ie−N2PNi=1PSα=1[aλ(α)i−b(Pαλ(α+1))i]21NSN(N−1)/2(8)where the Pα’s are S independent permutations of N objects and we have defined2m2 = a2+b2 with ab = 1 . Here a and b are restricted to be real , hence m > 1.

Theregion with m < 1 ,where the integral (7) is not defined, can be reached by analyticcontinuation as described later. Let us introduce the new variables of integrationˆλ(1)i=λ(1)iˆλ(2)i=(P1λ(2))iˆλ(3)i=(P1P2λ(3))i......ˆλ(S)i=(P1P2 .

. .

PS−1λ(S))iand defineP=P1 · · · PSThen we have4

Z=1NSN(N−1)/2 (N! )S−1 XP(−1)PZ Yα,idˆλ(α)i×exp −N2NXi=1{[aˆλ(1)i−bˆλ(2)i ]2 + [aˆλ(2)i−bˆλ(3)i ]2 + · · · + [aˆλ(S)i−b(P ˆλ(1))i]2}(9)It is natural now to change variables in the integral and defineζ(α)i=aˆλ(α)i−bˆλ(α+1)i(10)where ˆλ(S+1)i= (P ˆλ(1))i.

Now we have to calculate the Jacobian. This is a simpletask if we take a cyclic permutation, say of order r, namely the permutation: 1 →2 →3 →· · · →r →1 and if we order the variables in the following way:ζ(1)1 , ζ(2)1 , · · · ζ(s)1 , ζ(1)2 , ζ(2)2 , · · ·ζ(s)2 , · · · · · ·ζ(1)r , ζ(2)r , · · · ζ(s)r ,then the Jacobian is given by:∂ζ∂ˆλ =a−ba−ba−ba−b..a−b−ba= arS −brS ,(11)where we have chosen a > b due to the absolute value in the Jacobian.

Hence:a2=m2 +√m4 −1b2=m2 −√m4 −1As each permutation P can be decomposed into products of cycles, each integralat the r.h.s. of (9) decomposes into the product of integrals corresponding to thecycles in the decomposition of P.Any permutation is characterized by a set of numbers {r1, · · ·, rj, · · ·, rN} , rjbeing the number of times that the cycle of order j appears in the decompositionof P. Obviously one has the conditionNXj=1j rj = N(12)Among the N!

permutations there are N!QNj=1( 1j )rj1rj! which are characterized bythe same set of numbers {ri} .5

By putting everything together, we find for the partition function the followingexpression :Z=(N! )SNSN(N−1)/2Xr1,···,rNδ(Xj rj −N)(−1)P(j−1)rjNYj=1 Fj,S,Nj!rj 1rj!=(N!

)SNSN(N−1)/2Z 2π0dθ2π exp −iNθ −∞Xr=1(−1)(r)rFr,S,Neiθr(13)where Fr,S,N is the integral corresponding to a cycle of order r:Fr,S,N =ZrYi=1SYα=1dˆλ(α)ie−N2Pri=1PSα=1(aˆλ(α)i−bˆλ(α+1)i)2(14)where as before ˆλ(S+1)i= ˆλ(1)i+1 .But the r.h.s. of eq.

(14) is a gaussian integral in the ζ(α)ivariables, hence:Fr,S,N =1arS −brSZ Ydζ(α)ie−N2Pζ(α)2i= πN rS21arS −brS(15)By inserting (15) into the expression (13) for Z we obtainZ =(N! )SNSN(N−1)/2Z 2π0dθ2πe−iNθ∞YK=0 1 + eiθ πNa2S/2a−2SK!

(16)In this infinite product we recognize the grand-canonical partition function of aset of fermions where A ≡eiθ πNa2S/2 plays the role of the fugacity. Then theonly effect of the integral over θ is to select in the productQ∞K=0(1 + AXSK) thecoefficient of the AN term.

Such coefficient isXK1

. XSKN =1XNSNYK=1XSK1 −XSK(17)So we find the final expression for the partition functionZ =12π(N!

)SNSN(N−1)/2 πN NS2NYK=1(a−2S)N2/21 −(a−2S)K(18)6

4. Discussion of the results and concluding remarksApart from some irrelevant factors the partition function (18) is of the formZ(N)(q) =qN2/2(1 −q)(1 −q2) · · ·(1 −qN)(19)with q = a−2S .The same partition function has been obtained in a completely different fashionby Boulatov and Kazakov in ref [5] as the one describing the singlet (vortex free)part of the partition function for a 1d string.

This is not surprising as the singletis obtained by integrating over the residual angular variables which play thereforethe role of gauge variables. The crucial difference is that the theory considered in[5] has a continuous compactified target space.

As a consequence the argumentq = a−2S in (19)is identified in [5] with q = e−βω where β is the length of the stringand ω is the frequency of the oscillators. A rescaling of time changes both β and ωin such a way to keep this product constant.

The most convenient way to formulatesuch scaling in our case is to definea2 = eϕ(20)so thatm2 = cosh ϕ(21)The partition function (18) is then left invariant by the following rescaling :S →S′ϕ →SS′ϕ(22)which for the mass of the scalar impliesm2 ≡cosh ϕ →m′2 ≡cosh( SS′ϕ)(23)With an infinite rescaling (S′ →∞) one should recover the continuum theory ofref.[5]. In fact it can be easily checked that the action in (6) becomes in suchlimit :Scont = NTrZ β0 dt12(D ˆφ)2 + 12ω2 ˆφ2(24)where t = αS′β , ˆφ =qβS′φ andβ2ω2 = 2S′2(m′2 −1) = 2S′2[cosh( SS′ϕ) −1] = ϕ2 + O( 1S′2)(25)It is apparent from (25) that the continuum limit always correspond to the criti-cal point m2 = 1, although it leads to two completely different phases according7

to whether the point m2 = 1 is reached from above or from below, the formercorresponding to a real and the latter to an imaginary frequency ω .The fact that , except for a rescaling of a, the same partition function is obtainedirrespective of the value of S is quite remarkable and tells us that the whole infor-mation about the partition function is already contained in the simplest case S = 1where the lattice is reduced to just one site and one link. Such drastic reducibilityin the number of degrees of freedom seems to denote that all relevant quantities,such as for instance the vacuum density of the eigenvalues of the scalar field , arespace independent.The calculation leading to eq.

(18) has been done in the regime m > 1. Insuch regime the quadratic potential is stable , a and b are real and all integralsare well defined.

In the weak coupling regime (m < 1) the quadratic potential isunstable and such instability manifests itself in the divergence of the integrals overthe eigenvalues. It is easy to see for instance that for a and b on the unit circle theintegral in (14) is divergent.The natural way out is to define the partition function for m < 1 as the analyticcontinuation from m > 1.

In terms of the variable q = a−2S it means an analyticcontinuation from the real axis with q < 1 to the unit circle.As discussed above, in the continuum theory we have m →mc = 1 and two differentphases originate corresponding to real (resp. imaginary) frequency oscillators if mcis approached from above (resp.

below). It was argued in ref.

[5] that the physicalproperties of the oscillators with imaginary frequencies - the so called upside-downoscillators - can be obtained from the ones of ordinary oscillators by the replacementω →iω . It was shown this to be consistent with the introduction of an SU(N)invariant cutoffat large λ ’s.In the discrete theory on the other hand it is possible to analytically continue fromthe strong to the weak coupling regime and then perform the continuum limit.In this way one recovers the correct prescription for the upside-down oscillators,namely that their properties are obtained via the substitution ω →iω .Finally we want to remark that it is very simple in this theory to integrate overthe matrix fields φ and obtain the partition function as a function of the gauge fieldsonly.

By taking advantage of the fact that the partition function is independentfrom S we can do the calculation for S = 1. Gauge invariance can be used to choosethe unitary matrix U to be diagonal and the integral over each matrix element ofthe matrix field φ is gaussian.

The result is then :Z(β)=Z 2π0Yk=1N dθk2π |∆(eiθ)|21m2 −1N/2 Yi

and interpret the θi’s as the invariant angles of the product of the unitary matricesover the plaquette. It should be noticed that eq.

(26) is much more suitable thaneq. (3) to study the weak and strong coupling behaviour of the induced gaugetheory since it can be analytically continued from the strong to the weak couplingregime.Eq.

(26) was also derived, in the continuum limit, in ref. [5] (see eq.

(4.34)) byperforming the gaussian integral over ˆφ(t) and it was used as an intermediate stepin deriving eq. (19).Note addedAfter completing this paper we have become aware of a paper by S. Dalleyentitled ”The Weingarten model `a la Polyakov” and published on Mod.Phys.Lett.

A7 (1992) 1651. Induced gauge theories on a lattice are considered there inthe context of a complex matrix model ; in particular the case D = 1 , that hasthe same physical content as the corresponding Kazakov-Migdal model, is studiedin detail.AcknowledgmentsWe thank F. Gliozzi for many enlightening discussions9

References[1] V.A. Kazakov and A.A. Migdal, ”Induced QCD at Large N”, preprint PUPT- 1322, LPTENS -92/15, May 1992[2] A.A. Migdal, ”Exact Solution of Induced Lattice Gauge Theory at large N”,preprint PUPT - 1323,Revised, June 1992[3] I.I.

Kogan, G.W. Semenoffand N. Weiss, ”Induced QCD and Hidden LocalZN Symmetry”, preprint UBCTP 92-022 June 1992[4] D.J.Gross and I.R.Klebanov, Nucl.

Phys. B344 (1990) 475 and B354 (1991)459.

[5] D. Boulatov and V. Kazakov, ”One-dimensional string theory with vorticesas the upside-down matrix oscillator”, preprint LPTENS 91/ 24, KUNS 1094HE(TH) 91/14, August 1991[6] C.Itzykson and J.B.Zuber, J.Math.Phys. 21 (1980) 411.

[7] Harish-Chandra, Amer.J.Math. 79 (1957) 87.

[8] M.L.Mehta, Comm.Math.Phys. 79 (1981) 327.10

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