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Singular matrix models and Kazakov-Migdal Penner model Paniak, Lori Dean 1994

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SINGULAR MATRIX MODELS AND THE KAZAKOV-MIGDALPENNER MODELByLori Dean PaniakB. Sc., The University of Alberta, 1992A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTERS OF SCIENCEinTHE FACULTY OF GRADUATE STUDIESPHYSICSWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAOctober 1994© Lori Dean Paniak, 1994In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)______________________________Department of P t CS.The University of British ColumbiaVancouver, CanadaDate Oc6/ fL7t.(DE.6 (2188)AbstractIn the past few years there has been renewed interest in the large-N solution of QCD dueto a model proposed by Kazkov and Migdal. While this model is formally solvable thereexist only two explicit solutions, namely for the cases of the Gaussian and logarithmic(Penner)-type potentials. Here we give the complete solution for the latter and show thatthe model has critical behaviour for space-time dimension D > 1. As well we point outseveral inconsistencies in the standard method of solution and expectations from basicconsiderations.11Table of ContentsAbstract iiTable of Contents iiiList of Figures vAcknowledgement vii1 Introduction 12 Matrix Models Large N solutions 42.1 Loop Equations 72.2 Orthogonal Polynomials 123 Kazakov-Migdal Penner Model 173.1 DMS Method of Loop Equations 223.1.1 One Matrix Penner Model 273.2 Cut Structure 314 Phases of the KM Penner Model 374.1 lDown lUp 374.2 lUp iDown 434.3 Double Down 495 Critical Behaviour 561115.1 Calculation of Xo 575.2 Criticality in the Different Phases 595.2.1 One down - one up 595.2.2 One up - one downS 615.2.3 Double doww 636 Gauge Field Correlators 656.1 Real Examples 666.1.1 lu-id 666.1.2 idlu 676.1.3 Double down 697 Conclusion 72Bibliography 75ivList of Figures4.1 Effective potential idlu . 404.2 KMM potential (D=4) ldlu . . 414.3 Effective background potential idlu 414.4 Eigenvalue distribution ldlu . . 434.5 Effective potential luld 474.6 KMM potential luld (D=4) 474.7 Effective background potential luld (D=4) . . . 474.8 Eigenvalue distribution luld 484.9 Effective background potential luld (D=4),detail 494.10 Effective potential dbldn 534.11 KMM potential dbldn (D=4) . . 544.12 Background potential dbldn (D=4) 544.13 Eigenvalue distribution support dbldn 544.14 Eigenvalue distribution support(detail) dbldn 555.15 Complex contour for the calculation of susceptibility in ldlu phase . . 595.16 Complex contour for the calculation of susceptibility in mid phase . . 615.17 Numerical comparison of real parts of u (solid) to assumed form (dashed)in the vicinity of the critical point ( c = 1.88346) for a particular example(a=10,b=5) 635.18 Complex contour for the calculation of susceptibility in double down phase 646.19 Correlator normalization as a function of second argument in luid example 67v6.20 Correlation function (ldlu)in the region of the eigenvalue support. Secondargument equal -10.2 about average 686.21 Correlator normalization as a function of second argument in idlu example 686.22 f Cxwp(w)dw — A(A) for ) e [xi, x2] in idlu example 69viAcknowledgementDuring the time I spent working on this thesis I had the pleasure of meeting many peoplewhom I consider friends. While the list of those who helped me in some way over thepast two years is long there are a few who deserve special recognition.First I’d like to thank N. Weiss for his academic, financial and personal support. Itis not an understatement to suggest that this project would not have been successfullycompleted without his help and guidance. As well I thank S. Curnoe and K. Kiers formany discussions about this and other projects. The genesis of many of the ideas herefollow from discussions with R. Szabo who also read the text and made many constructivecomments which are gratefully acknowledged. Finally I recognize the faculty and staffof the U.B.C. Department of Physics for providing a positive atmosphere in which it isa pleasure to work.In a more general way I think that my family and friends in Vancouver and Edmontonhave provided me with the energy and confidence necessary to succeed. Here I wouldlike to express my appreciation for all of their support and encouragement. I hope thatin the future I can return the favour.viiChapter 1IntroductionPerturbation theories, have endowed physics with the ability to get a hold of models ofreality that are too challenging for the available mathematical machinery. Since the earlydays of celestial classical mechanics the idea of calculating corrections to a solvable modelhave led to the most important and realistic views of the world.For example consider the basis of the highly successful quantum electro-dynamics(QED). The ability of this theory to calculate physical parameters to seemingly arbitraryprecision in complete agreement with experiment follows directly from the fact that thecharacteristic correction scale is that of o . This effectively means that higher loopdiagrams in a canonical process series are damped out by an extra power of at eachorder. While in general this kind of series is not convergent it can be shown that it givesexcellent asymptotic precision to high orders of perturbation theory.While such a strategy for expansion in a small parameter has historically been highlysuccessful, there are numerous examples of models where no such parameter is available-at least not in an obvious way. The most important instance of such a theory today isthat of gauge fields and in particular quantum chromo-dynamics (QCD), an SU(3) gaugetheory which is currently the only plausible theory of the strong interactions. Due to thestructure of the equations of motion for such a theory the effective coupling constant canbe absorbed in a redefinition of the fields involved. This effectively eliminates any possibleexpansion based on ioop type corrections as in QED. This handicap was overcome byt’Hooft [1] in 1974 who considered the structure of a Yang-Mills theory like QCD. Since1Chapter 1. Introduction 2the fields effectively couple to each other based on their representation one can obtainsimplifications to the structure of the theory by coilsidering an SU(N) theory for large N.These simplifications effectively reduce the gauge theory to a two dimensional topologyproblem by the machinery of planar diagrams.Clearly the limit of large number of colors simplifies things in the world of QCD butwhat does N —* oc have to do with N=3 reality? We have good reason to believe that thelimiting case, while not exactly correct will give important qualitative information aboutthe behaviour of SU(3) gauge field theories. As an example one can calculate the groundstate energy of a quantum mechanical anharmonic oscillator numerically and one canalso, by the methods of matrix models, calculate the ground state energy of an infinitecollection of anharmonic oscillators. The N —÷ oc approximation is never more than 12% off the true value [3].As described in clear fashion 15 years ago by Witten [2], when one applies large Ntechniques to QCD the gauge fields, whose complicated quantum behaviour gives thetheory it’s rich and difficult character, are replaced by their vacuum expectation values.This observation has two important consequences. First this allows cluster- type decompositions of gauge invariant operator products which will allow for reduction to algebraicequations in the model we will be considering. Second, since vacuum expectation values are Poincaré invariant we are restricted to consider only constant master fields forour large N gauge fields. These qualities make the machinery of D=O matrix modelsappealing in the search for the N —+ oc solution of QCD.The application of matrix models to the exploration of large N QCD began longago with [6] and the counting of topological diagrams for different theories. Much morerecently Kazakov and Migdal proposed a theory of how to generate large N QCD throughthe use of hermitean matrices interacting on a lattice through gauge mediating fields. Inthis thesis we will complete the loop in a sense and use the techniques developed inChapter 1. Introduction 3[3] in order to solve the Kazakov-Migdal model for a particular choice of potential forthe hermitean matrix fields. Originally developed by Dyson and Wigner in the 1950’sto examine level spacing in heavy nuclei, matrix model techniques have been applied toproblems in both physics (2D electron gases, resonant modes of bulk materials, motion ofmicroscopic metal spheres, 2D quantum gravity) and mathematics ( zeros of the Riemannzeta function, orthogonal polynomial theory) with great success. A review of the historyof matrix models by one of the founders, Mehta, is presented in [4].The layout of this thesis is as follows: We will begin with a review of basic matrixmodel machinery including loop equation methods which are fundamental here and orthogonal polynomial basics for completeness. Next we introduce the Kazakov-Migdalmodel (KMM) in general and quickly move to the specification of a KMM-Penner typeof problem. The ideas required to deal with singular matrix models like the KMM-Pennerare founded in investigations of the classical Penner one matrix model and as such wereview the important aspects. Proceeding with the experience of a single well logarithmicpotential we can solve the double well case which arises in the KMM-Penner and showthat the resultant one cut solutions completely span the parameter space as will be shownin three examples. With an exact solution available one can calculate the susceptibilityand critical exponents here which is shown in detail in a section preceding the last whichdeals with the details of the calculation of the gauge field correlators in the KMM-Pennerwhich are a happy by-product of the original method of solution.Chapter 2Matrix Models Large N solutionsThe usefulness of matrix models in calculating the characteristics of complex systemsstems from the fact that the action, or probability density written ill terms of matrices isinvariant under the action of a symmetry group which diagonalizes the matrices. Afterdiagonalization one is left with the N eigenvalues of the matrix variable and we expect asN becomes large a saddle-point approximation to the weight function will become moreaccurate. To be more concrete consider the probability distribution function (pdf) orstate weight function for a polynomial matrix problem:dP[b] exp[—trV(cb)]dq (2.1)V(q) = ab , dq fi dqn>ONow, in general, [4] g is an N x N matrix of hermitean, real orthogonal, or unitarysymplectic type. For each of these choices the distribution function defined above isinvariant under the adjoint action of SU(N),SO(N) or SU(N)xSU(N), respectively. Theimportance of such an invariance is that one has the freedom decompose the space ofmatrices into a product space of (radial) diagonal elements and (angular) symmetryelements. It is informative to consider a concrete example. Let q to be an N x N hermiteanmatrix. Clearly= UAU (2.2)For some U E U(N) and A a diagonal NxN matrix of the real eigenvalues of qS.4Chapter 2. Matrix Models Large N solutionsIn general, implementing such a change of variables in the pdf 2.1 gives:dP[çb] ‘ 4exp[—trV(A)][dU]dA (2.3)where the Jacobian associated with such a change of variables can be shown to be relatedto the Vandermonde determinant (q) of with a dependence on the nature of thematrix:J, = (if is real orthogonal (2.4)is hermitean (2.5)is unitary sympiectic (2.6)Where: () fl( — j) (2.7)i<jjis the ith eigenvaiue ofq (2.8)Combining the Jacobian factor with the exponential one arrives at an expressionfor the weight of a particular configuration in terms of N real variables with a trivialdependence on the symmetry (gauge) group:dP[] = constant exp[— V() + n in (j — )][dU]dA (2.9)i=1 ijwhere n is 1/2, 1, 2 in the orthogonal, hermitean and symplectic ces respectively. Itis this separation of integration over the Lie groups that makes it possible to work withthese types of statistical ensembles. We will assume henceforth that is a hermiteanmatrix.Now that we have probabilities for different configurations we would like to evaluateexpectation values of different functions of the field variable . While this is possible byevaluating sums over the N eigenvalues of the system this is usually not practical as theChapter 2. Matrix Models Large N solutions 6physically interesting applications of such a model occur for large N. In order to simplifythe expressions we must deal with, we make the first assumption of this section andconsider the size of our field variable g5 to be large (ie N —* oc). Taking a large numberof variables qSj leads to integrals over dp[q] which are especially well suited to evaluationby the method of stationary phase. As an example consider the following integral:J dP[] fexp[_V@)+ln@i -)}dA (2.10)i1 ijAt large N this integral may be approximated by its value at the point in the Ndimensional space of eigenvalues where the action:NS = —V(b) +ln( — qj) (2.11)i=1 jr/fhas an extremal value. In order to find this stationary point, we differentiate by one ofthe eigenvalues Sj and look for solutions to the equation:0= -V’() + (2.12)We consider that the distribution of the eigenvalues by indices becomes a continuous function on a bounded interval as the indices become a continuous variable. Moreprecisely we have:(i/N) = i (2.13)dx= dq5(x)(2.14)-* fp(x)dx (2.15)i/N —* x (2.16)Taking this smoothing into account, the saddle-point equation reads:0= -V’()+f(-x)dx(2.17)Chapter 2. Matrix Models Large N solutions 7where the principal value of the integral is the continuous version of >j above. Theproblem is thus reduced to solving for the spectral density p(x).This singular integral equation may be solved by the Riemann-Hilbert method. Fortunately for more complex potentials than the quadratic case we have described herethere are easier ways to solve for the unknown function p(x). We will describe such amethod in the next section.2.1 Loop EquationsAs a more practical, but not general method of solving for the continuous distributionof eigenvalues p(ç) in the limit that the size of matrices () on which we are basing ourpdf on become large, we introduce the method of loop equations. This technique hasmany names in the the literature including reparameterization invariance and SchwingerDyson equations. Regardless of the title this method’s success hinges on the invarianceof moments of the pdf under infinitesimal transformations of the field b.To begin with we consider a pdf over matrices like that above but with an arbitrarypolynomial potential:dP[b] = exp[—trV[b]]dçb (2.18)Since we expect the moments of this distribution to be invariant under a change ofvariable (reparameterization invariance), we have a symmetry to exploit. The most useful(infinitesimal) change of variable is the following:‘S (2.19)V() ,‘ V() + (2.20)d d+E(A)2 (2.21)Chapter 2. Matrix Models Large N solutions 8Substituting such a change of variable in a normalization integral, or partition functionin physics terms, and requiring that the first order terms in vanish, we find:fdge (2.22)= Jd(1+ (A)2X1- eNV’@) +. .)ev (2.23)= J + (2.24)____- jeT+ O(e2) (2.25)(2.26)0= f d[N2(A’ )2 — N(A)le (2.27)The same results may also be obtained by the following generalization which will beuseful in a multi-matrix case. Consider the expectation of the first order variation of thefollowing object in the field variable:a rP[_t[lli 228J uiIf we consider the potential to goto positive infinity for large values of the field(this is not very well defined but taking the norm of q to be large should suffice), as wecan always guarantee by suitable choice of coupling constants, then the previous integralvanishes since it is exact over the manifold of . Hence we are left with a non-trivialequation which can be solved at large N in some cases:0= Jd[(A )j-(2.29)Noting a factorization theorem for gauge invariant objects as N —+ oc [2], we candecompose the first term of the integrand:Id( exp[-NTr(V[])] (2.30)[I d(A )jj exP[_NTr(V[])]] + O() (2.31)Chapter 2. Matrix Models Large N solutions 9In more compact notation our exact integral reads:0=(2.32)Where (2.33)(2.34)And the bracket is defined by (2.35)(F[J) f dTrF[ç] exp[-NTr(V[])] (2.36)The end result of the evaluation of this integral is that we have a quadratic equationfor the quantity E), which is intimately related to the distribution of the eigenvalues ofthe field variable in the large N limit as we will now see. Taking the discontinuity ofthe function EA across its cuts in the complex plane (Hilbert transform) we expect toobtain the probability distribution function of a single eigenvalue, A:jpdx (2.37)— — EA_jE (2.38)= A-+icA--z€(2.39)= —2iirp(A) A e cut EA (2.40)= 0 otherwise (2.41)It should be noted that the +i notation signifies to evaluate E, on either side ofa cut contour on which A may lie. This prescription also demonstrates the fact thatthe cut structure of the generating function EA in the complex variable A is identical tothe support of the eigenvalue distribution for the matrix model problem at large N. Forthe situation described above, namely polynomial potentials of a single field variable,commonly called one matrix models, we can identify this structure. In this case thesediscontinuities are generated by square-root functions which arise in the solution of theChapter 2. Matrix Models Large i\T solutions 10quadratic equation for EA and as we will see choice of the cut placement for these square-roots will be critical for the solution of a more general class of one matrix models- thosewith singular potentials.As an example of this method we consider the toy model with Gaussian potential:V() = with > 0 (2.42)While this particular potential is of trivial physical significance, it allows for an exhibition of the basic machinery involved in large N matrix models.By invariance under infinitesimal transformations we have at large N:= K) (2.43)0 = — cvAEA + c (2.44)The solution of this quadratic with proper asymptotics (EA +...) as A —* oo is givenby:EA =——(2.45)In this case it is easy to see that E is an analytic function of A on the whole complexplane except for a square root cut on x1 = — < A < = x2. It is precisely alongthis cut that the distribution of eigenvalues of the matrix is non-zero. Taking thediscontinuity as discussed above we obtain an explicit form for this distribution:p(A) = -/4 — (2.46)With this distribution in hand we are in a position to specify the entire state ofthe physical system represented by the Gaussian potential in large N matrix models.The most important quantity one can evaluate for a given physical system is the freeChapter 2. Matrix Models Large N solutions 11energy. In matrix models one can define the free energy analogously to that in statisticalmechanics:F = (2.47)Where Z, the partition function is defined as above:Z = fdcbexp [-NV(cb)] (2.48)Written in terms of the distribution of eigenvalues, we have [3] the leading behaviourof the free energy at large N:F°= f2 dwp(w)V(w)—f2 L ddwpQ)p() in — (2.49)But by the saddle-point equations of motion:V’t’X’ fX2 n(w1=] di’ ‘ )e support of p (2.50)2 ).—L4)We can write:F°= j d[V(w) — V(x) + ln x1 w] (2.51)Using this last expression and the symmetry properties of the eigenvalue distributionin this particular case we findF° = 2fd[)2+ln.\]p(.\) (2.52)1 j* d[2 + lnA]4 — (2.53)= const. — ln (2.54)This expression will allow for a comparison between this method of solution and themethod of orthogonal polynomials which is outlined in the next section.Chapter 2. Matrix Models Large N solutions 122.2 Orthogonal PolynomialsAfter the separation of the symmetry group elements from the pdf as described above,one has another option over the Riemann-Hilbert and ioop equation solution methodspresented above. This method is referred to as orthogonal polynomials as it utilizesthe properties of polynomials defined with an inner product whose weight function isexactly the pdf in question [5]. This method has many advantages over the previous onesincluding most importantly the ability for straightforward calculation of corrections tothe N —÷ oc assumption. As an example of this method we will calculate the the partitionfunction for the Gaussian case as in the previous section. This type of system, it shouldbe noted, has relation to an N — oc gas of electrons on a line in two dimensions. Thepartition function for such a system has a familiar form:Z fexp[_+ln( - )]fld (2.55)i=1 ijOrZ (2.56)This last form is interesting in that the Vandermonde determinant can be decomposedinto a determinant of polynomials of varying order over the eigenvalues of the matrix,= det[’] (2.57)= det[Pj_b1)] (2.58)Where : (2.59)j—1P(x) = x + > akxlc (2.60)k=OThe general case of these polynomials were investigated long ago [4] and are explicitlyChapter 2. Matrix Models Large N solutions 13given:m00 m01m10 m11 m1P(x) = const. det (2.61)m(_l)o 1fl(i_fl .1 xWith: (2.62)= Jdxexp[_x2]xx2 (2.63)Since x may be written as a sum of Pk(x) with k j, it is easy to see that:f dxexp[—x2]Pj(x)xi = 0 for i > j (2.64)Now it is obvious that for the case at hand the polynomials Pk(x) are simply the Hermitepolynomials Hk(x).Considering the partition function above to be an inner product of products of thesepolynomials, one can eliminate most integrations by this last orthogonality condition andarrive at the remarkable, and exact, result:N-iZ = N! JJ hk (2.65)k=OWith: (2.66)= fdxexp[_x2JP(x) (2.67)In order to evaluate this considerably simplified partition function, we require recursion relations for the orthogonal polynomialsP2(x). It turns out that these objects satisfya common three term recursion relation of the form:xP(x) = Pi(x) + SP(x) + RP_i(x) (2.68)Chapter 2. Matrix Models Large N solutions 14Here the constant coefficients (Sn, R) carry all the information of the system. Inparticular, as in all cases with symmetric potentials (V() = V(—q) ), here we have0. With this information we can now move on and calculate R in terms of thepotential.First we need to evaluate the normalizations of the polynomials h in terms of therecursion coefficients Rn. To this end consider:h1 = fdxexp[_x2]xPn(x)Pn+i(x) (2.69)= fdxexp[_x2][Pn+2( )+R+iP(x)]P(x) (2.70)= hR1 (2.71)= ha_i(2.72)Rewriting the partition function in terms of R:N-iZ = N! fl h (2.73)n=ON—i z. i t L. 1-.‘n ‘in—i ‘n—2 ‘2 ,i= N! fJ ... (2.74)n=O ‘tn—i ‘n—2 ‘n—3 ‘i ‘ON-i= hUN! fl RnRn_iRn_2.. (2.75)n=1N-i= hUN! fJ R (2.76)As can be seen, this rewrite has just redefined the problem in terms of another setof unknown coefficients. We need to define R in terms of the potential. Fortunately, inour case of a Gaussian potential this may be easily and exactly accomplished. Considerthe identity:flhn = fdxexp[_x2]xP(x)P(x) (2.77)= f dxexp[_x2]P(x)[Pn+i(x) + RnPn_i(X)1 (2.78)Chapter 2. Matrix Models Large I\T solutions 15R f dx exp[— x2}P(x)Pni(x) (2.79)Integrating by parts: (2.80)= f dxexp[_x2]xPni(x)Pn(x) (2.81)= cRh (2.82)(2.83)= (2.84)Here we see that the explicit form of the polynomials Pk(z) is not important. Formore complicated potentials this feature of orthogonal polynomials is the key to themethod’s sucess. Going back to the partition function one more time we have an exactexpression in terms of the parameters of the potential, in this case c:Z = hN!fJR (2.85)N—i= hN!fl()! (2.86)N-i= hUN! H (2.87)Now from this expression we can derive, as in classical statistical mechanics, the freeenergy of the system. Here the only complication arises due to positive powers of Nlurking in the calculation. As a from of renormalization we define the free energy asbefore:F = lnZ (2.88)In the limit as N —* cc we can evaluate the leading contribution to the free energy:F° = urn —--lnZ (2.89)N—oo N2= lim j[ln(hN!) + ln] (2.90)Chapter 2. Matrix Models Large N solutions 16= urn [constamt — mln] (2.91)N—co N n=O1 N(N-1)= urn —[coristarit — lnc] (2.92)N-*ooN 21= constamt—1no (2.93)Comparing with the expression for F° derived by ioop equation methods, we findagreement.Chapter 3Kazakov-Migdal Penner ModelWhile the machinery of large N matrix models was developed mainly for use in 2Dgravity and string theoretic settings we will use it for an investigation of QCD. Utilizingthe principle of universality, Kazakov and Migdal presented a model which was designedto produce the only (known) non-trivial gauge theory in four dimensions namely QCD.Considering a lattice action on which adjoint scalar fields occupy the vertices and gaugefields act on the links as unitary operators, it was believed that if the mass of the scalarswas taken to infinity as the continuum limit is taken then a non-trivial gauge theory wouldemerge. The action in question is identical to a Wilson model [7] with the exception thatthe kinetic term for the gauge fields (plaquettes) is left out. This simplification makespossible the use of the techniques of matrix models discussed above in finding a mean-field solution for the distribution of the eigenvalues of the adjoint scalar fields representedby hermitean matrices.The method for solving such a problem is based on the fact that in the limit of alarge number of colors the only Feynman diagrams in the theory that contribute are thosewhich can be drawn on a sphere. These are often referred to as the planar diagrams. Asmight be expected, the next order corrections are those which can be rendered on a torusor a surface with three holes etc. In this way one gets a perturbation series based ontopology. The matrix model methods described above allow one to extract and examineinformation at each order of the topological series. It is precisely for this reason whythese methods were developed for 2D gravity.17Chapter 3. Kazakov-Migdal Penner Model 18The most important features of the Kazakov-Migdal model are related to the correlators of the gauge fields. Since this is really a model of only the gauge bosons of QCD- gluodynamics may be a better term - these are the only physically pertinent objects.As in the case of a standard lattice gauge theory we expect that the correlators of thegauge fields, in the form of plaquettes, to contain information about the critical structureand confinement behaviour of the theory. There are many issues involved with respectto these kinds of calculations [8] - [12] and are beyond the present scope.The only physical importance of the scalars coupled to the gauge fields is to induce theusual F2 kinetic term which was neglected in the original action. This is done throughrenormalization of the scalars. At one loop level it can be shown that the counter-termfor the scalars produces the Maxwell term with the correct sign and can be fine-tunedby adjusting the self-interactions of the scalars [11]. Since it is impossible at this timeto solve the Kazakov- Migdal model for a general scalar potential we are left to considerthe available, soluble potentials.The machinery we will use throughout to solve the KMM is essentially that of [18].This line of attack is essentially a minimal extension of the ioop equation methods introduced earlier to solve the hermitean one matrix problem. To see how such a programmay be implemented here let us begin with the partition function and action of theKazakov-Migdal model [16].Z = f lldb(x) fl[dU(z,y)]e (394)(xy)S = N > TrV[(x)] — > Trb(x)U(x, y)q(y)U(y, x) (3.95)x (y)Here the action is a sum over all local sites x on the D dimensional rectangular latticeof the (identical) scalar potentials ( TrV[b(x)]) and a sum over all the neighbouringChapter 3. Kazakov-Migdal Penner Model 19sites ((xy)) of the gauge interaction (Trcb(x)U(x, y)c/(y)U(y, x)). The hermitean matrixfields q transform under the adjoint representation of SU(N) and [dU] denotes the Haarmeasure of the unitary group. As with all lattice gauge theories, U(x, y) = Ut(y, x).The first step in evaluating such an object comes in the challenge of integrating out thegauge fields U. As we saw before, with a one matrix model the gauge fields easily decouplethrough decomposing the hermitean fields into diagonal and rotation components. In thiscase though such a simple decomposition is not possible due to the prescence of the gaugeinteraction term. Fortunately this term has already been evaluated in closed form [13],[14]. In fact the existence of a closed form for the gauge interaction part of the action isone of the reasons why this particular model was developed in the first place.As outlined in the references there are numerous methods which produce the followingidentities:I[b, x] f[dU] exp [TrbUXUt] (3.96)det exp iXj(3 97(c(x)exp [Tr- + Tr-]det x)2](3.98)Where c, Xj are the ith and jth eigenvalues of the hermitean matrices and xrespectively. The last equality [12] gives a decomposition of the gauge action into aGaussian type background potential for the fields and and interparticle interaction1 399i ,Xj— @)x)This decompostion will be important in the discussion of the solutions of the KazakovMigdal Penner model later.Since we have an expression of the gauge interaction in terms of the eigenvalues ofChapter 3. Kazakov-Migdal Penner Model 20the scalar fields we are in a position to write the effective action of the model for theseeigenvalues. Assuming that the potential is of the form:V[] = (3.100)n>OWe obtain:Seff[cl (3.101)—N[V[(i,x)] +ln((x)) + (3.102)x1nI[(x), (y)}J (3.103)(zy)(3.104)= -N[Tr[V[(i,z)] - 2D(i,x)] + (3.105)lnz(c(x)) + > lnJ[q(x),q5(y)]] (3.106)<,y>With 5(i, x) being the ith eigenvalue of the scalar field at site x.From the last line it is clear that the overall behaviour of the effective action isgoverned by the potential V and an inverted Gaussian-type dimension dependent term.The Vandermonde and ln J[q(x), (y)] contributions are purely interparticle in natureand will only serve to determine the local details of the large N distribution of theeigenvalues (i, x) and not the position of the saddle-point.While the identity 3.98 allows for the explicit expression of the effective action forthe eigenvalues, the result is a particularly difficult many-body problem. Conversely,at large N we expect that correlation functions of the gauge fields reduce to a simpler,factorized form akin to what we noticed in deriving the ioop equations for a 1 matrixmodel in the first section. To review, we note that in the KMM scalar(matrix) fieldslive at the sites of a D dimensional rectangular lattice with gauge fields U, U generatingChapter 3. Kazakov-Migdal Penner Model 21site-site interactions by living on the links of the lattice and coupling to the scalars inan obvious way. The partition function of such a system consists of complicated gaugeinduced connections between many lattice sites. At large N though, when one applies thecondition of a homogeneous, isotropic ‘Master Field’ (all the scalar fields are identicalat each site), then only gauge interactions along a single link are important. It is thisobservation that underlies all known solution methods of the KMM.A more precise statement of this fact is found in [18]: Consider the expectation oftwo gauge fields on the lattice:C[x,y]b1bk = (UiU) (3.107)= f[dqj[dU]UUe_s[1 (3.108)Here C [x, y] gives the joint probability distribution between the ith and jth eigenvalues of the scalar fields on lattice sites x and y. The delta functions come about becauseof gauge invariance. [x, y] is also symmetric in i and j and satisfies a sum rule whichwill be an important test of self-consistency later on:(3.109)As N — oo and the scalar fields are integrated out and replaced by their values atthe extrema of the action, one is left with the site independent correlation function. Itis here that the factorization of the gauge fields occurs. Using the fact that U = Uone gets:C = J[dU]IUi2e_U1 (3.110)This expression can be convoluted with a scalar field at extrema to generate thelogarithmic derivative of the Itzykson-Zuber determinant evaluated at the saddle-point.This equality allows one to write a 1 matrix model type equation of the saddle-point inChapter 3. Kazakov-Migdal Penner Model 22one variable(i):A, = ZC2 (3.111)= (3.112)(3:113)-* 2DA = V’(b) -((3.114)3.1 DMS Method of Loop EquationsIn the mean field limit, N —* cc we have seen that the saddle-point of this partitionfunction 3.94 reduces to that of a 1-matrix model 3.114. In this section we introduce themachinery needed to evaluate such a model and then apply it to the case of a logarithmicPenner type potential. The ioop equations outlined in the first section generalize hereto a set of coupled ioop equations in the variables EA and G [18] which are generating functions for the mean-field distribution of eigenvalues and the correlation functionbetween two gauge fields, respectively. These are defined as:EA=(1);G=(U’U) (3.115)x, U)) f[dU]ddTrF(, x, U) exp[—S(, x, U)] (3.116)By the same reparameterization invariance arguments that were used in the discussionof ioop equation methods in the introduction, we can develop equations for the quantitiesEA and G using the factorization of gauge invariant quantities at large N.BA:1 a —S[q,UJ= N2Zfld fl[dU(x y)] [ (3.117)x 3Chapter 3. Kazakov-Migdal Penner Model 23= J H d(x) fl[dU(x, y)][[ -(x) (3.118)— — U(x, y)(y)U(y, x))—As discussed in the discrete case, in the limit N —* oc, the combination,U(x, y)(y)U(y, x) is essentially given by the logarithmic derivative of the Itzykson- Zuber integral at the saddle-point. Analogously we define the mean-field dependentquantity A() [17], [18] along each of the identical 2 D links that connect to site x:2DA((x)) = lim U(x,y)(y)U(y,x) (3.119)N—oo= U(x,y)2(y) (3.120)Using such a decomposition we can write a tractable expression for the generator ofthe large N distribution of eigenvalues for the scalar field:T 7!E2 121G>0= N2Zfld fl[dU(x,y)] (3.122)x ()- ()U(xY) - ()u(Yx)1i)e= ffldb(x) fl[dU(x,y)]e_UJ (3.123)(cry)1 1 1 1 1- (x)N-(x) - ()U(Yx))— iV’(q(x)— zU(xz)cb(z)U(zx)U( 1N A—(x) XYI_() y,xThe sum over z in the last term again runs over all 2 D neighbours of site x. In thiscase though one of these sites is y and hence its contribution can be reduced. With thisChapter 3. Kazakov-Migdal Penner Model 24reduction and utilizing the definition of A for the remaining 2 D - 1 neighbours we findat large N0 (E + )GA —— ([V—(2D_l)A]@)u (3.124)From the basic definitions one can easily derive the useful asymptotic — oc)expansions of these quantities:+1(3.125)G + ... (3.126)In the expansions it is clear that the equation for G is the more fundamental thanthat of EA. With this in mind one can devise a strategy in which to solve the KazakovMigdal model by establishing G - typically by assuming a form for the quantity [V1 —(2D — 1)A](g). This is precisely what was done in [18] for a Gaussian type problem.Here we will consider the analogous system for a logarithmic type problem. Using theKMM-Penner ansatz [19]:[V1 - (2D - 1)A]() ==___- a (3.127)The equation for G, may be written:— (b— ))E, — (ab+ c)G&,GA— (b — A)(E + ) — (aA + c)(3.128)This quantity may be written in a more useful form by applying the asymptoticconditions (3.125,3.126) and solving for the quantity GbGb(a+)E—1(3.129)ab+cChapter 3. Kazakov-Migdal Penner Model 25Hence:(b— ))EA + 1 —@+it)E,G = (3.130)(b—A)(+EA) —(a+c)By symmetry of the bracket under interchange U —> U, the quantity G is invariantunder interchange of it arguments (G = G). Imposing this condition on the lastexpression and enforcing asymptotic behaviour 3.126,one gets a quadratic equation inEA in terms of the undetermined average of the eigenvalue distribution ():0 = (a+)(b—A)E — [(aA+c)(a-i-)+(1+c—bA)(b—)]EA+(a+b)[+a—b+(g)] (3.131)In order for the solution of this equation to self-consistently satisfy the asymptoticcondition 3.125 it is useful to fix the sign of the square-root function at (real) positiveinfinity and consider a fixed sign of the quantity (a + b). Here the choice of sign of (a + b)will be positive. The other option should result in a completely symmetric solution tothat which we will obtain by this assumption. With these conditions taken into account,one finds:E—_________ __________________________1 2A— 2(a + )(b— ) — 2(a + )(b — )(3. 3 )X = (a+))(a)+c)+(b—A)(1 -i--c—bA)Taking this solution of the loop equations and comparing its discontinuities in thecomplex plane with those of 3.121, an expression for [V’ — 2DA](q) can be obtained.Taking the difference with the ansatz 3.127, the quantity A() appears:A()= b-c- 1(3.133)From the knowledge of A we can write down the potential of the Kazakov-Migdalaction which corresponds to the assumed form 3.127:VKMM() = —(ab-i-c) ln(b— ) — (2D — 1)(ab+c+ 1) ln(a+q) — ((2D — 1)b+a)g (3.134)Chapter 3. Kazakov-Migdal Penner Model 26In addition to the proper potential of the model, we can also give the effective backgroundpotential seen by the eigenvalues by taking into account the action of the Itzykson-Zuberdeterminent:Veffective() = VKMM(çb) — 2D9 (3.135)The result of this operation is to define the effective 1-matrix model potential [V’ —2DAJ() which can be analyzed using the formalism discussed in the introduction. Moreprecisely we have reduced the multi-matrix problem of 3.98 to that of a single matrix,and now, a single eigenvalue problem. After a change of variables, the potential energyof a single eigenvalue in the mean-field is given by:J[V’ — 2DA](). V = —ln(b— ) +ln(a + ) — (a + b) (3.136)= ab + c (3.137)(3.138)The 1 matrix potential corresponding to this expression contains many interestingaspects. For example it generalizes the Penner potential, which contains a single singularity to one with two singularities and, as will be seen later, it admits a multi-phasesolution space with non-trivial critical behaviour.The first important feature of the potential 3.136 is that it is singular at the points—a, b. These singularities generate apparent singularities in the generating function EAin contradiction to its basic definition. E), written as an integral is clearly finite for A noton the support of the eigenvalues. Assuming this condition it is clear that the quantitiesE_a, Eb remain finite as the mean field solution of the eigenvalue distribution cannotexist at infinite potential. In order to assure that the apparent singularities in EA remainapparent we face some restrictions as to the possible combinations of the signs of c, 3Chapter 3. Kazakov-Migdal Penner Model 27and the square root function in the vicinity of the poles of the potential. At this point itis useful to demonstrate the analogous problem for the case of a potential with a singlesingularity, namely the Penner case.3.1.1 One Matrix Penner ModelIn order to gain an understanding of the loop equation method of solution for non-polynomial potentials it is instructive to examine the case of the generalized Pennermodel. Originally considered for the calculation of Euler characters for moduli spaces,this model gained the interest of physicists when it was found that at criticality it described a closed string at self-dual radius. The classical Penner model case has beenworked out by this method before [15] and the following analysis is in the same spirit.The Penner model is given by the potential over hermitean matrices :V() = [ln(1— ) + ç] (3.139)The derivation of the loop equations for this case follows as in the previous section.The outstanding question of the validity of the discarding of boundary terms in theSchwinger-Dyson formulation of the equation for EA can be addressed by analytic continuation of the coupling constant. Assuming that the boundary terms vanish, or at leastcan be neglected, one obtains the following quadratic equation in E:0 = (A — 1)E — oAE, + aE1 (3.140)Where E1 is a constant which can be determined to be equal to 1 by fixing the squareroot to be positive at positive real infinity and by use of the asymptotic behaviour ofChapter 3. Kazakov-Migdal Penner Model 28urn + + (3.141)A-+ooA AThe solution to this quadratic equation involves a choice of sign which can be unequivocally determined by comparison to the asymptotic requirements above. The resultantexpression for E is______sign(c)/[cA]2 — 4(A — 1)c= 2(A — 1) — 2(A — 1)(3.142)As A approaches the singularity in the potential (A —* 1) the definition above requiresthat the function EA be a finite constant. This condition places a restriction on thecontour which comprises the cut of the square-root function in E>, and, in turn, thesupport of the eigenvalue distribution for the Hermitean field . As A —* 1:_sign()[cA]22(A — 1) — 2(A — 1)+ finite terms (3.143)In order to cancel the infinite parts in the limit, we place the restriction that:sign(c) = sign(c)sign(.J) (3.144)or (3.145)sign(.J) = + (3.146)What is required at this point is a determination of the sign of the square-root functionin the vicinity of the pole. Since the branch points of the square-root in the expressionof EA form the beginning and end points of the cut contour at which sign changes takeplace, their position is critical in satisfying the condition above. Factoring the quadraticunder the square-root in EA we find the branch points A± satisfy:— 2 = ±2”1 — (3.147)Chapter 3. Kazakov-Migdal Penner Model 29Note that the branch points become complex for o> 1 making physical interpretationin this regime difficult.By use of the arithmetic/geometric inequality it can be shown that for < 0 thebranch points lie to the left of the singularity of the potential at A = 1. The sign of thesquare-root function therefore is the same at A —+ +oo, namely positive as required bythe boundedness condition for E1. Clearly one is then free to take the cut contour of thesquare-root to be a continuous deformation of the trivial cut along the real line betweenA and ) .Hence this case, that of ci <0, is completely analogous to that of polynomialmatrix models where one expects the support of the eigenvalue distribution/ cut contourof the square-root function in EA to lie on the real axis.For > 0 the situation is obviously more complicated. In this case, in order to havethe correct behaviour of E at the pole we are forced to take the cut contour to enclosethe pole[15} . This particular contour may be deformed to a closed ioop about the polein union with a contour equivalent to that in the a < 0 case above. This claim may beverified by observing for > 0 that the quadratic equation above returns values for thebranch point A that lie to the right of the singularity in the potential. If one were toconsider the naive square-root cut along the real axis between ) then the boundednesscondition for E would not be met in this case. The correct contour to choose is one thatallows the sign of the square-root function to be that at positive infinity: the contourbegins at one branch point, loops to the left of the singularity at A = 1, and returns tothe second branch point. This type of contour is not only necessary to maintain finiteE1 but also to obtain the correct expression for the free energy of this model [15J.To check the consistency of these choices with the construction up to this point it isnecessary for the distribution of eigenvalues to have proper normalization. Fortunatelyin this case the evaluation of these two disjoint contributions to the normalization of p(A)is straightforward. Along the real contourChapter 3. Kazakov-Migdal Penner Model 30J p(A)d.\ (3.148)sign(a) 4a(;\ — 1) —= 2(A — 1)dA (3.149)jsign(a)[4(1-c) sign(a)/4(1——(3.150)= sgn)du— 4(1—cr) 2K(u + 2 — a)/4(1-a) +/4(1 — a) —_____du (3.151)= L4(l_) 2(u + 2 — a)= —[2— a — (2 — a)2 — 4(1 — a)j (3.152)2”r= 1-[a+IaI] (3.153)Which integrates to 1 for a <0 in accordance with the results above.For a > 0, the contribution from the closed contour about the pole at \ = 1 evaluatesby the method of residues to:p(w)dw (3.154)+1= I dw+E (3.155)J+i —2ir1 a +/[aw]2_4(1)a= dw[ + (3.156)—2ii 2(w—1) 2(w—1)_____+[]2_4Q (3.157)+2(w—1) 2(w—1)= 1(3.158)—2iir +1 (w — 1)—2iir= urn [a&,]2 — 4Q — 1)a (3.159)—2iir w—1= lal (3.160)Where an extra negative sign comes from the clock-wise orientation of the square-rootcut.Chapter 3. Kazakov-Migdal Penner Model 31The sum of these two contributions gives a unit normalization of p(A) which is consistent with the construction of E), and the ioop equations.3.2 Cut StructureIn order to determine the different types of possible one cut solutions in this problem itis instructive to concentrate on the cut characteristics of the logarithms in the potentialand the constraint on the coefficients of these terms. While the general solution for theeigenvalue distribution and the generating functions E), and G by the loop equation(andother common methods) is independent of the prescence of the cuts associated withln(a + )) and ln(b — )) in the effective 1 matrix potential, the necessity of real freeenergy makes this detail important. As seen before, the free energy of a matrix model tospherical(leading in N) order can be expressed as [3]:F° = fd[V(w) — V(xi) +lnjxi —wj] (3.161)As can be seen in this expression, the real-valuedness of the free energy hinges onthe analytic structure of the logarithms in the effective potential. Since the saddle-pointequations depend only on the derivative of the potential, one can effectively add complexconstants to the potential in order to flip the signs of the arguments of the logarithms.This allows the arbitrary choice of whether to orient the cuts of the logarithms in thepositive or negative real direction. This choice can be exercised for each of the log termsresulting in 4 choices for cut configurations. In order to guarantee that the expressionabove for the free energy is real for real eigenvalue support we require that this supportdoes not lie on the same interval as a cut of the logarithms. With this in mind we canimmediately eliminate one of the 4 possibilities namely that which has the cut from -arunning in the positive direction and the cut from b running in the negative direction.Chapter 3. Kazakov-Migdal Penner Model 32This configuration, for general coefficients of the logarithms, would result in complex freeenergy for any real eigenvalue support.As for the three remaining choices, we are left with: (—cc, —a),(—a, b), (b, +oc). Sincethis analysis is carried out in the large N limit through saddle-point approximation, theeigenvalue support must lie in a minima of the effective potential. Since for a + b > 0the potential decreases linearly to the positive side of b, this region cannot support aneigenvalue distribution. Hence the only possible regions which can support extendedeigenvalue distributions are (—cc, —a) and (—a, b). This information together with thefact that an eigenvalue support cannot cross a log cut and that the generating functionEA must be finite at the poles -a,b results in a nearly complete qualitative picture of thesolutions of the KMM Penner model. It would appear that the only difficulty with thispoint of view lies with the possibility of a case where the support of the eigenvalues lies ontwo disjoint intervals. From the consideration of the cut structure of the logarithms in thepotential 3.136 these two square root cuts would have to lie in the same region: (—cc, —a),(—a, b), or (b, cc). By the structure of EA it is clear that such a configuration wouldhave a region of non-positive eigenvalue distribution thereby violating a probabilisticinterpretation.So, we have some motivation to believe that the structure of the support of theeigenvalue distribution is that of a single, connected real interval. As will be seen laterthis condition is also consistent with a single, connected contour in the complex plane.For now though it is sufficient to restrict ourselves to the cases where the discontinuitiesof the generating function E,, occur for real A. Assuming this observation to be true forall values of the parameters a,b,c of the potential 3.136, we will assume a consistent formof the generating function which will allow us to completely solve the problem of thedistribution of the eigenvalues for general a,b,c. This is commonly known as the 1 cutansatz. The implications of this assumption on the generating function of interest 3.132Chapter 3. Kazakov-Migdal Penner Model 33are to reduce the quartic under the square root to the product of a pair of quadratics,one of which has degenerate zeros. By matching asymptotic conditions of this form withthose of the more general case, we can obtain enough conditions to solve for the branchpoints and () in terms of the original parameters of the system.The 1 cut ansatz in this case gives the generating function E), the following form:Ex - (A) + B)/)- n,\ + V3162—2(a + A)(b—A) 2(a + A)(b — A)Where X is defined as in the general solution to the loop equations 3.132. Here u andv are related to the end (branch) points (x1,x2) of the single cut such that u = x1 + x2,v =x12. This choice of parameters simplifies calculations later. A and B are parametersrelated to the location of the degenerate zerosWith two expressions available for EA , namely 3.132 and the one immediately above,one can enforce the asymptotic conditions of the general solution on this specific one.Order by order one obtains polynomial equations which, in principle, can be solved.As one can see this expression contains 4 unknown quantities which we would like todetermine. This is similar to the case in polynomial matrix models with the potential2 + g3 [3] except that here our asymptotic conditions begin at constant order and 4equations will require expansion to O(). While this is easily accomplished by expandingthe general solution 3.132, these conditions introduce a new unknown into the system:(). In order to accommodate this extra unknown we are forced to expand to an extraorder, O(). More explicitly we are presented with the following system of polynomialequations in the unknowns u,v,A,B, (q)From expansion of the general solution for EA 3.132 as A —* cc we obtain the nontrivial moments of the distribution in terms of a,b,c and ():Chapter 3. Kazakov-Migdal Penner Model 34(2) = a+(a2—b1)(q5)+(a+b)c (3.163)a+b/3\ — a2 — b2 + 1— (a + b)(çb))(a + (q) + (a + b)(ab + c)\Y/— (a+b)2+ab(b — a) + ()(a2 — ab + b2)Similarly by expanding the assumed form of the generating fullction as A —* oc weobtain at each respective order:O(A°) : 0= (a+b)—A (3.164)1 = (a+b)(a—b+ +(a+b)1—B (3.165)O(): () = ±[_3(a + b)u2 + 4(b2 — a2 — flu + 4(a + b)v — (3.166)16a + 8b — 8c(a + b)]O(): (2) = [—(a + b)u3 + (2(a — b2) — flu2 + 4(a + b)uv (3.167)+16a2—8ab+8b2+4v+4(a— b+a3 +b3—a2b—ab)u+8c(a2 —b2)JO(): 2(q3) = [—2(a3 — b3 + ab(b — a)) + (3.168)(4uv—U3)(2+ 2(a — b2) + (a + b)u) — (a + b) +(b — a)(1+ a2 — b2 + (a + b)u)(4v — u2) + (a + b)u2 +(a+b)(24uv — 5u4 — 16v2)(ab — a2 — b2) +(b+c(a+b)+(1+a2_b2))}Ill general, the solutions to this system may be presented as rational functions of thequantity u. In turn u is defined as the root of a polynomial equation whose coefficientsare solely functions of the parameters of the original ansatz 3.127. While in general aChapter 3. Kazakov-Migdal Penner Model 35system of this type would require the polynomial in u to be of quintic order, in thiscase the quintic factors into a cubic and a quadratic. From comparison with numericalexperiments and through comparison with the qualitative analysis of placement of thecuts of the generating function above, it appears that only the cubic equation generatesphysically significant solutions.The machinery involved in the calculation of such a system involves the conceptof Gröbner bases and the Buchburger reduction algorithm [20]. In short, the set ofpolynomials above, I, forms a zero-dimensional ideal over the field of real coefficientpolynomials in the variables u,v,(). It can be shown that this is equivalent to I beinga set of non-constant univariate polynomials in each of variables. It is the calculationof the Gröbner basis for I which constructively defines the univariate set and allows forthe use of single variable polynomial solutions in order to get the following results. Asthe solution in u factors, there are two parametric expressions for this quantity. Theother variables are rational functions of u hence we need to choose the correct root. Thisprocess will be discussed in the next section.a = Rootof[(a + b)3Z+ 3(2 + a2 —b3)(a + b)2Z (3.169)+2(a+b)[(a+b)( a—b)2 +2c+ 1)+6(a2— b2 + 1)]Z+ (1 + a(a—b)) + 2(1 — a3(a — b)]u = Rootof[(a + b)Z2 + 2(1 + a2 — b2)Z — 4c(a + b) — 4(b + 2ab(a + b))](3.170)— [(a + b)’u3 + (a2 + 4 —b2)u + 4(a — b)u — 8c — l6ab]4((a+b)u+a —b +2)4u — 8b(3.171)4(a+b)((a+b)u+a2—b2+2)Chapter 3. Kazakov-Migdal Penner Model 36= —[(a + b)2u3+ 3(a + b)(1 + a2 —b2)u (3.172)+ (4c(a + b)2 + 2(a — b2) + 2(a + b)2 + 1O(a2 — b2) + 2)u—4(a+b)(b2— a2 —3)c+8a+4ab+4b16a—4b] x18((a+b)u+a—b+2)A = a+b (3.173)B = (a+b)[+a_b+b] (3.174)The quantities u and v above are related to the branch points of the square-rootfunction x1,X2 X3 = X4 Xdouble in the following way:= _J__v (3.175)= + V’i — v (3.176)= X4Xdoub1e_[+ab+bj (3.177)Chapter 4Phases of the KM Penner ModelWhile the general solution for the limits of the eigenvalue support and the average of theeigenvalue distribution (q) in terms of the parameters of the potential is important, itscomplicated structure tends to hide the details of the different phases associated withthe potential 3.136. In the next few sections we will detail these three possible phasesand comment on parameter restrictions and critical behaviour where applicable.The parameter space of the potential 3.136 is neatly divided into three differentphases through the signs of the coefficients of the logarithmic terms. We will label theseaccording to the orientation of the singularities. For example, for c < —1 the singularityof 3.136 at A = b is negative while the singularity at A = —a is positive. This particularphase is labeled as ‘ldown lup’. The other two phases are similarly labelled ‘lup idown’(a > 0) and ‘double down’ (—1 <c <0).4.1 lDown lUpThe first case is that in which the coefficient in 3.136 is negative as is /3. Withthe restriction that /3 = + 1, this requires that c = ab + c < —1. As always wemaintain the assumption that a + b > 0. With these conditions it is easy to see thatthe effective 1 matrix problem that underlies the KMM problem has a single well lyingto the left of the pole at -a. As well, if we consider the options for the branch cutsof the logarithms associated with this potential, we find that an eigenvalue support onthe interval (—oc, —a) will give real free energy to spherical order. It seems reasonable37Chapter 4. Phases of the KM Penner Model 38therefore to expect that the general solution for this region of parameter space is that ofan eigenvalue distribution on the interval (—oo, —a).The first step in determining the detailed eigenvalue distribution is to show that thesolution is necessarily of the 1-cut variety as discussed, and solved above. We begin bynoting that the square-root function is defined to be positive definite at real +oo andhence that the general solution for EA 3.132 is valid. With these signs fixed we considerthe behaviour of EA in the vicinity of the singularities of the potential, namely E_a, Eb.We begin at A —+2(b— A) —+finite terms (4.178)In order to cancel the leading infinite terms as in the Penner case examined above,we require that the sign of the square-root in the vicinity of b be negative. This requiresa single cut in the square-root to exist on the interval (b, oc). Since we have only fourbranch points, there can exist at most two cuts in the square-root. More specificallythere cannot exist three cuts on (b, oc).A similar expansion at A —* —a gives:______— + finite terms (4.179)2(a+A) 2(a+A)In order to cancel the infinite terms here we require that there exist a single cut inthe square-root function on (—oc, —a) in order to change the sign of the asymptoticallypositive square-root.With these requirements we can now show unambiguously that the solution for theeigenvalue distribution in this range of parameter space is of the one cut variety. Considerthe cut on the interval (b, oc). It may be diffeomorphic to a cut that lies entirely on thereal axis or, since the singularity in the potential at b is unbounded below, it may iooparound b. If we consider the former then by taking the discontinuity of E,, we obtainChapter 4. Phases of the KM Penner Model 39a negative definite density of eigenvalues. The only way around this violation of theprobabilistic interpretation is to require that the branch points in this region coalescethereby producing no contribution to the density of eigenvalues. In the second case, aloop about the singularity at b can easily be shown, as was done for the Penner case, toproduce a contribution to the density of eigenvalues equal to the absolute value of thestrength of the singularity. In this case the contribution would be I°1 > 1, hence theunit normalization of the density would be violated. The conclusion is that the requiredcut in the square-root function on the interval (b, cc) is necessarily generated by a pairof degenerate branch points and that the solution for the distribution of the eigenvaluesfor c < —1 is of the one cut form.Just as a matter of detail, a real cut on (—cc, —a) clearly produces a positive definite density of eigenvalues. This cut cannot loop around the singularity at -a since thecontribution to the free energy of such a configuration would be large and positive. As acheck then, it is clear that the cut in this region has an eigenvalue distribution with realsupport.It is possible to quantify the conditions required for the branch points (xi, x2,x3 == Xdouble) and the average value of the field (qf). These conditions will allow us tochoose unambiguously the correct solution for the quantity u from the respective cubicequation 3.169.u(4.180)<> < —a (4.181)xdouble=[+a_b+b]>b (4.182)As an explicit numerical example we study the choice of parameters in the originalpotential 3.127:a = 10 (4.183)Chapter 4. Phases of the KM Penner Model 40Figure 4.1: Effective potential idlub = 5 (4.184)c = —53 (4.185)These values result in the strengths of the singularities of the effective 1 matrixpotential 3.136 being:= ab + c = —3 (4.186)/3 = 1+a=—2 (4.187)The resulting effective potential (Fig.4.1) has the general form of a negative linearpotential with a positive singularity at \ = —a = —10 and a negative singularity at= b = 5. The net result is to produce a local minimum immediately to the left of thepositive singularity. It is here that one expects the large N solution for the support ofthe eigenvalues to lie.Also of interest is the potential of the original KMM action to which 3.127 corresponds.Since this quantity depends on the dimension of the lattice on which one is working it isnecessary to mention that the graph (Fig.4.2) is for D=4. This result is qualitatively thesame for all D > 1/2, where the KMM at D = 1/2 is essentially a one-link or two-matrixmodel.Chapter 4. Phases of the KM Penner Model 41755Figure 4.2: KMM potential (D=4) ldlu2.51Figure 4.3: Effective background potential idluChapter 4. Phases of the KM Penner Model 42Separating the elements of the Itzykson-Zuber intergal into background and interpartide components 3.98, one finds that the effective background potential that an eigenvaluesees is not given by 3.134 but rather by 3.135. This quantity for this particular choice ofparameters is shown in (Fig.4.3).Plugging this choice of parameters into the general solution generated in the previoussection produces a collection of solutions based on the roots of the cubic and quadraticequations for u. Imposing the constraints discussed above one finds unique values for thedetails of the eigenvalue support and the expectation value of the field ():u = —20.52647854 (4.188)v = 105.2821063 (4.189)() = —10.19741307 (4.190)(4.191)These values of u and v give the end points of the eigenvalue distribution (xi, x2) andthe degenerate branch points (x3 = Xdouble) the following values:x1 =— — v = —10.49121737 (4.192)= + — v = —10.03526117 (4.193)= X4 = xdouble = —[ + a — b + a + b1 = 5.196572603 (4.194)Now that all the parameters of the system have been specified we are in a position tograph the eigenvalue distribution. (Fig.4.4)Comparing the location and form of the eigenvalue distribution and the effective onematrix potential one finds good agreement. The distribution is centered about the minimaChapter 4. Phases of the KM Penner Model 43-li.5 -iiFigure 4.4: Eigenvalue distribution idluof the potential as to be expected in a classical solution. In the case of the effective KMMpotential though there appears to be some problems. If one compares figures (Fig.4.3)and (Fig.4.4) the eigenvalues are smeared across the top of a local maximum. It is notat all obvious that the logarithmic interparticle forces can maintain such a configurationatop what is essentially an inverted Gaussian well.4.2 lUp iDownThe next configuration of the effective one matrix potential 3.136 that we will investigateis that which occurs when > 0. Due to the constraint on the strength of the logarithmicsingularities from the equation 3.124, here we have a potential with minima for only asmall range of parameter space as compared to the previous case. Again, as the solutionswe seek are the classical equations of motion for the system, the extrema of the potentialare the regions of interest. To that end we note that the potential has a real minimumon (—a, b) as long as the condition<((a + b)2 — 1)2 (4.195)— 4(a+b)2is met.Chapter 4. Phases of the KM Penner Model 44This constraint effectively marks the upper bound for the coefficient a for given a, bAs the minima vanishes, the general solutions for the end points of the eigenvaluedistribution become complex leading to a fermionic matrix model [21]. It should benoted that this move off the real line for the branch points does not occur exactly at thepoint in parameter space where the minima vanishes but rather for values of a even lessthan the constraint above allows. This is due to the repulsive interparticle Vandermondecontribution to the free energy of the system which forces the eigenvalue distribution tohave a finite width. The true condition marking this transition from real to imaginarybranch points is when the degenerate zeros of the one cut solution collide with one of thebranch points of the square-root function (xi) as will be seen later.Now that we have identified that there exists, for at least some parameters of theoriginal ansatz 3.127, a minimum on which to base a real large N one cut solution wemust consider the generating function EA 3.121 in the vicinity of the poles. Since the poleat b has strength a> 0 we see on expansion at this point that the finiteness condition ismet if the square-root function has the same sign at A = b as at real infinity (.ie positive).In this case, the finiteness condition is met for /9 = 1 + a> 1 if the sign of the square-rootis that at negative infinity, namely positive again. With these two conditions we find thatthe square-root function has no cuts (or two cuts) on each of the intervals (—oc, —a) and(b, oc). By free energy arguments we may eliminate the two cut possibility and similarlyrule out the possibility that a cut contour can loop around the singularity at b.From the fact that the distribution of eigenvalues has unit normalization we caneliminate the possibility that the cut of the square-root loops around the singularity at-a. Since the contribution to the density from such a loop is equal to 8j = 1 + al > 1,we see that it would overrun the normalization.The only remaining possibilities involve all four branch points (xi, x2,x3,x4) of thesquare-root lying on the interval (—a, b). Taking the discontinuity of E we see that,Chapter 4. Phases of the KM Penner Model 45in general, the density has a two cut structure. The problem with such a configurationis that the density which lies to the left gives negative definite contributions violatingthe probabilistic interpretation of the density p. As before, the only way to resolve thisunphysical situation is to insist that the pair of branch points that lie to the left aredegenerate (x3 = x4 = Xdouble) . In other words, the only consistent solution to the largeN distribution of eigenvalues in this region of parameter space is of one cut structure.From the arguments above we may derive some quantitative restrictions on the branchpoints which will allow us to unambiguously determine the correct root of the generalsolution. These restrictions may be combined into a list of inequalities:—a<xdouble<xi<<x2<b (4.196)From the general solution of the 1 cut ansatz we have‘IL 1X3X4Xdouble_[+a_b+bl (4.197)Hence a 1 cut solution will exist if and only if the end (branch) points of the eigenvaluedistribution (xi, x2) satisfy:3x1+2 —2[a—b+ab(4.198)with equality being a critical point of the model.In fact it is possible to give an explicit expression for the value of c or i as a functionof a and b at this critical point. In terms of the variable u = x1 + x2:c = — ab (4.199)— 4(a b)2[3(a + b)2u+ 6(a + b)(a2 — b2 + 2)u (4.200)+2(a—b+2)2(a+b)4(a—b2+1)]Chapter 4. Phases of the KM Penner Model 46Here, of course, u has a different definition than in the general 1 cut solution 3.169since one of the independent variables has been made dependent. The solution for arbitrary a and b is:= RootOf[(a + b)3Z+ 3(a2 — b2 + 2)(a + b)2Z (4.201)+ 3(a + b)(a2 — b2 + 2)Z + (a2 — b2 + 2) — (a + b)4]As usual, the correct root is chosen by considering the inequalities above.Using these results it can be shown that the list of inequalities above can be satisfiedonly if (a + b)2 > 1. Hence we have the restrictions that for this configuration of theeffective potential 3.136 to exist the poles of the effective potential cannot be “too closetogether” nor can their strengths be “too strong”. These inequalities give quantitativeweight to the comments of [19].As an explicit example of a case in this particular configuration we choose the followingparameters for our potential:a = 10 (4.202)b = 5 (4.203)c = —48 (4.204)= ab+c = 2 (4.205)= +1=3 (4.206)The effective 1 matrix potential given by 3.136 for this choice of parameters is shownin (F’ig4.5).The KMM potential for D=4 is shown in (Fig4.6).Adding in the contribution to the background potential of the Itzykson-Zuber determinant we are left with the effective background potential in which the eigenvalues liveChapter 4. Phases of the KM Penner Model 4715050Figure 4.5: Effective potential luldFigure 4.6: KMM potential luld (D=4)JI—10 —7.5 —5 —2.5Figure 4.7: Effective background potential luld (D=4)Chapter 4. Phases of the KM Penner Model 484.2 4.4Figure 4.8: Eigenvalue distribution luld(Fig 4.7).Considering the conditions on the possible solutions for the branch points and hencethe eigenvalue distribution for this particular configuration given above, we uniquelychoose the following solution:u = 9.459273164 (4.207)v 22.31463645 (4.208)() = 4.797246832 (4.209)From these values we deduce the positions of the branch points of the square-rootand the position of the eigenvalue distribution:= 4.495487614 (4.210)= 4.963785554 (4.211)= = Xdouble = 9.796304 (4.212)From this information we can determine the distribution of the eigenvalues (Fig4.8).At this point we begin to see an inconsistency with the DMS method of solutionfor the Kazakov-Migdal Penner model. From the figure 4.9 which details the effectiveChapter 4. Phases of the KM Penner Model 494.64.85Figure 4.9: Effective background potential luld (D=4),detailbackground potential that the eigenvalues see in the KMM , we find that the minimahere is not populated at all by eigenvalues. To be more precise, the minima occur for= 4.964177085 and —9.885434538 while the support of the distribution runs from x1to x2 as given above. Clearly, the free energy of the configuration could be lowered bysliding the support to the right. Additionally, there are no eigenvalues in the region of theother minima of the potential centered at -9.88. This phenomena appears to be relatedto the apparent inconsistencies of the previous example and will be considered in moredetail later.4.3 Double DownThe last configuration that is possible in the Kazakov-Migdal Penner model is that forwhich —1 < < 0. This choice forces the coefficients of the logarithms —o, = 1 + ain the potential 3.136 to both be positive. The result is that the effective one matrixpotential has no minima and hence no configurations of stationary action on which tobase a stable eigenvalue distribution. This is completely analogous to the case in thePenner problem when the strength of the logarithm term is positive and as will be seen,the solution here is quite similar as well.Chapter 4. Phases of the KM Penner Model 50As usual we look to the requirement that the values of the generating function atthe singularities of the potential be finite for information as to the possible square-rootcut/eigenvalue support configurations possible in this case. Evaluating B_a, Eb it is clearthat the sign of the square-root at b should be negative for /3 = 1 + c > 0 and positive for—1 <c < 0 at -a in order to maintain boundedness. This implies that the square-rootcut crosses the real axis somewhere on (b, cc) and again on (—a, b). At this point, oneknows very little about the details of the square-root cuts save for these crossing pointsand here direct application of the Penner results is invaluable.In the Penner case for positive logarithm coefficient the eigenvalue support followssome arbitrary curve in the complex plane which carries the support around the singularity. Most importantly, this configuration produces a contribution to the normalizationand moments of the eigenvalue distribution from the singularity proportional to thestrength of the singularity. We note that in this case the sum of the absolute values ofthe logarithm coefficients is unity and this leads to the following conjecture.The first aspect of this configuration to note is that if one a priori assumes that eachpole will contribute the absolute value of its coefficient to the normalization and highermoment calculations of the eigenvalue distribution,as in the Penner case, then the KMMconstraint givesf p(x)dz (4.213)+ p(w)d (4.214)+ (4.215)=2iK a (a + )(A — b)X (4.216)[[a(a + b) + (b- ) - (a + b)(a + A)(b - )]2 -4(a+A)(b- A)(a+b)(+a-b+ ())]Chapter 4. Phases of the KM Penner Model 51—1, dw+2i(a+)(—b)X[[(a + b) + (b-- (a + b)(a + )(b - -= ki+I1+ai (4.217)= 1 (4.218)Similarly,() fxp(x)dx = wp()dw + wp(w)dw (4.219)= Iab—I1+oIa (4.220)= —a + aI(a + b) (4.221)The most important part of this condition states that there are no contributions to thenormalization of the eigenvalue distribution from real cuts in the generating function EAIn other words, the solutions of this region consist of oniy loops about the poles as seenabove in the Penner case with the quartic under the square root in E,,\ having two pairs ofdegenerate zeros. As a check of this, the expression for (q) given by this assumption canbe substituted into the explicit expression for EA derived from asymptotic considerationsand the positions of the degenerate branch points ), solved for:b2 — a2 — 1 + /W= 2(a + b)(4.222)Where W is defined by: (4.223)W = (a + b)4 + 2(1 + 2o)(a + b)2 + 1 (4.224)The positions of these zeros also require that the eigenvalue supports encircle bothpoles in order to maintain finite BA at these points. This requirement leads directly tothe contribution of the coefficients of the logarithms to the moments of the eigenvaluedistribution. Hence, it is clear that our initial assumption is self-consistent.Chapter 4. Phases of the KM Penner Model 52While the above self-consistent argument allows one to derive the location of thebranch points and suggest the contours that the cut of the square-root/eigenvalue supportmust follow it is interesting to note that the same result can be found as a solution to theone-cut ansatz as in the previous sections. This fact seems to suggest that only one-cuttype solutions appear in the Kazakov-Migdal Penner model. Of course for more generalpotentials than 3.136, most simply where a and ,@ are independent, this is not the caseand it can be shown that solutions with eigenvalue support on disjoint complex contoursare possible.While the contours associated with the consistent solution of the double up case arenecessarily in the complex plane, the normalization of the eigenvalue distribution is unityand all of its moments are real. In fact the convolution of the eigenvalue distributionwith any meromorphic function can easily be shown to be real. For these cases where thefunction has no extended cuts in the complex plane the general contours may be collapsedonto the real line and about the poles as described above. Only when convoluted with afunction that has an extended cut(s) does one need to define the eigenvalue distribution onthe contours in the general complex plane and necessarily face the possibility of complexresults. This is precisely what occurs in the calculation of the free energy in this case aswe will see in the next chapter.For concreteness we consider the following example:a = 10 (4.225)b = 6 (4.226)c = —60.5 (4.227)a = ab + c = —0.5 (4.228)Chapter 4. Phases of the KM Penner Model 53Figure 4.10: Effective potential dbldn= + 1 = 0.5 (4.229)From the expressions above 4.224 we can determine the expectation value of the fieldand the location of the degenerate branch points:= = —10.031311 (4.230)= = 5.964644 (4.231)() = —2.0 (4.232)It should be noted that these values derived from the self-consistent argument aboveare precisely reproduced by the one cut ansatz solution 3.169. This solution is the oniyone cut type solution with real free energy.The potentials for this set of parameters are given in figures 4.10,4.11,4.12.The resultant eigenvalue support is given in 4.13. While the support, save for thebranch points, is arbitrary we consider a rectangular configuration for numerical calculations. As can be seen in the detailed view 4.14 the support encircles the singularity in thepotential at —a = —10 and is consistent with the previously discussed sign conditions.Chapter 4. Phases of the KM Penner Model 54200100Figure 4.11: KMM potential dbldn (D=4)Figure 4.12: Background potential dbldn (D=4)—2—4—1250I.—7.5 —5 —2.5 2.5 5 --Figure 4.13: Eigenvalue distribution support dbldnChapter 4. Phases of the KM Penner Model 55—9.9 —9.8 —9.7 —9.6 -9.5Figure 4.14: Eigenvalue distribution support(detail) dbldnChapter 5Critical BehaviourWhile the number of possible forms for potentials in a matrix are uncountable, there is asystematic method for characterizing one matrix models in terms of their behaviour at thecritical point(s). This is important in relation to 2D gravity where the critical behaviourof the discrete model determines the (physical) continuum system one is looking at [5].Here critical behaviour is important since we are looking for a phase transition inthe Kazakov-Migdal Penner model which will represent the change from infinite couplingbetween colour charged particles to the more physical characteristics of an area type lawa la Wilson.Since there are discrepancies with the KMM potential corresponding to the effectiveone matrix problem 3.136, we are forced to consider only the critical behaviour of thelatter. It is questionable to assign the critical behaviour of the one matrix problem tothe KM - Penner situation since such quantities depend directly on the details of thebackground and interparticle potentials before any mean-field approximations. Unfortunately, the DMS method of solving the KMM does not appear to give us this informationin a consistent manner. Regardless of the validity of any application to KMM the following calculations are indeed valid for the double-well Penner one matrix problem given by3.136.56Chapter 5. Critical Behaviour 575.1 Calculation of XoThe classification system discussed above is based on the order of the leading singularterm in the free energy as one approaches the critical point(s) of the model in question.More precisely, if a matrix model is in the ‘y equivalence class then the free energy scalesat criticality as:F ( — )2_7 (5.233)Where o is the parameter in the model which is tuned for criticality and c is thevalue of this parameter at the critical point.In this section we will be calculating the susceptibility of the model which by analogywith classical statistical mechanics [4] is defined as:d2x = ——.F(c) (5.234)In order to calculate such a quantity from the matrix model objects at hand we note[22] that since:Z[c] = e_2F = J[dq5] exp [—TrV(q, cr)] (5.235)With the potential V given here by 3.136:V(,c) = —cdn(b—q) + (c + 1)ln(a+) — (a+ b) (5.236)We can write:d2x = j—F(o) (5.237)= f [d]Tr[ln (b- ) - ln (a + )] exp [-TrV(, )]= (ln(b-)-ln(a+))Since we have only solved this model to spherical (N—cc) order we only haveinformation on the leading behaviour of the susceptibility, Xo, in a expansion. UsingChapter 5. Critical Behaviour 58a complex representation of the bracket over the large N configuration of the system wefind:Xo = [ln(b — — ln(a + (5.238)With:dE 1 1 it(a+b)_)2_2dc — 2(b — w) 2(a + w) 4(a + w)(b — w)(a+b)[W+a—b++f} —itw+i’- 4(a+i)(b-w)and as before:u = (5.240)v = x12 (5.241)=- v (5.242)Dots represent derivatives with respect to c.Here E is the generating function for the one matrix problem defined by the potentialV and the complex contour C encloses the support of the eigenvalue distribution ina counter-clock-wise direction. Typically this contour may be taken to be a circle atI wI —* oo, but here the logarithms have cuts on the real line which must be navigated.For each of the different phases discussed above this leads to a different integration as wewill see. While we cannot give an explicit single formula for the susceptibility in all of thephases we may employ some complex variable techniques and rewrite the last expression:Xo = ---[ln(b—w) —ln(a+w)}---E (5243)I dEJ daThe first term can easily be shown to vanish and hence the calculation of the susceptibility has been reduced to a real integral of the quantity .Chapter 5. Critical Behaviour 59Figure 5.15: Complex contour for the calculation of susceptibility in idlu phase5.2 Criticality in the Different PhasesOf the three phases of the model established in the previous section, namely, one up-onedown, one down- one up and double down, each has a unique critical behaviour. In thissection we will classify these both qualitatively and quantitatively.5.2.1 One down - one up:This phase occurs when a < —1. Here the only type of critical behaviour occurs whenthe branch point to the right, x2, collides with the singularity in the potential at -a. Thisoccurs for a value of = —1 where the strength of this singularity vanishes.In order to prove the statements of the last paragraph we need to evaluate the expression for the susceptibility 5.243. To accomplish this we note that in this phase thesupport of eigenvalues generically lies in the range (—oc, —a) and hence the cuts associated with the 1 matrix potential 3.136 are on the half real line (—a, oc). The contour whichcircumnavigates this discontinuity is shown in (FigS.15). Due to cancellations betweenChapter 5. Critical Behaviour 60the logarithms of the potential, this effectively leaves the real integration in 5.243 to runfrom -a to b. Upon evaluation we find1 16(b2—bu+v)(a+au+v)________Xo = _ln[(2b ( — b)u + 2v +2a2+au + vb2 — bu+ v)21(5.244)The denominator can easily be shown to be non-zero here and the numerator vanishesas either of the endpoints of the eigenvalue distribution x1, X2 approaches either of thesingularities in the potential at -a,b. The only case where this is possible in this phaseis when x2 approaches -a. This occurs precisely when a = —1 as can be seen from thegeneral solution.In order to compare with usual methods of computing critical exponents in matrixmodels we calculate the leading divergent term of the susceptibility as a — —1. Keepingonly terms which are potentially divergent:Xo — ln [a2 + au + v] (5.245)—ln[—a—x2] (5.246)Substituting x2 as a function of a which is known from the general 1 cut solution, wefind:Xo — ln [1 + a] (5.247)This result is not surprising as the same behaviour is evident in the classic Pennermodel 3.139 [22]. In that case the result appears to have a connexion with critical bosonicstrings at self-dual radius. Since the critical behaviour here, normally associated witha ‘-y = 0 classification, is identical modulo a trivial additive shift in the critical pointwe expect that this particular phase of the double Penner model 3.136 has a similarinterpretation.Chapter 5. Critical Behaviour 61Figure 5.16: Complex contour for the calculation of susceptibility in luld phase5.2.2 One up - one down:Here for a> 0 and as long as the other conditions for existence are satisfied ( see chapter4), this phase has two types of critical behaviour. The first is the same as that in the onedown - one up case: the right branch point ,x2 runs into a singularity of the potential.It appears that the scaling involved as a —* 0 is the same as in the previous case.Additionally the model has an unknown critical point at the value of a where thedouble zeros of the one cut solution (x3 = x4) collide with the left lying endpoint, x1, ofthe eigenvalue support. A condition on the position of the endpoints was given in theprevious chapter. The condition on a in terms of a and b appears to be non- trivial.Avoiding the logarithm cut of the potential 3.136 as before, one is left with thecomplex integration contour (Fig5.16). Unlike the previous case there are no cancellationsbetween the terms of the real integration in 5.243 and hence the two integration rangesChapter 5. Critical Behaviour 62here are (—oc, —a) and (b, oc). The evaluation of these quantities gives:1 16(b2 — bu +_v)(a2_+an +_v)__________Xo = ——ln[ ] (5.248)2 (—2ab+(a — b)u+2v— 2i/a +au+v/b2—bu+v)2Again the denominator of the argument of the log is non-zero and the numeratorvanishes for endpoint -singularity collisions, the only possible case here being x2 —* b. Itappears that the unknown critical point which corresponds to a coalescing of the branchpoints in the problem is a third order transition since the susceptibility is finite in thisphase for all but the case above.Again examining the critical limit as a —* 0 perturbatively a familiar result appears:Xo — ln [b2 — bu + v] (5.249)—ln[b—x2J (5.250)—ln[a] (5.251)The leading divergent behaviour of the susceptibility in this phase is of the samecritical exponent as the previous case (‘y = 0) and hence has an interpretation in termsof string theory.Returning to the third order critical point, one finds quickly that the complicatedexpression for the general solution leaves little hope for a precise analytical expansion.In spite of this numerical experiments show that the quantity u in the neighbourhood ofthe critical value of a (c) behaves as:(5.252)An example of this is shown in (Fig5.17).With this ansatz for the form of u it is easy to give an expansion of the argument of thelogarithm near the critical point and we find that the first derivative of the susceptibilityhas a square root singularity at the critical point. This behaviour is exactly that of a 1matrix model with = — and corresponds to pure 2D gravity [5].Chapter 5. Critical Behaviour 63—2. 645—2.65—2.655—2.6651.1882 .1884 1.1886 1.1888 1.189Figure 5.17: Numerical comparison of real parts of u (solid) to assumed form (dashed)in the vicinity of the critical point ( c = 1.88346) for a particular example (a=1O,b=5)5.2.3 Double down:In this unique phase where the eigenvalue support is a general contour in the complexplane, the model appears to be critical for all values of the parameters in the potential.This is due to the fact the the endpoints of the square-root cut contour are degenerateleading to = 0 for all cases in this phase.The necessary contour of integration here is essentially that of the first case namelyldlu modulo a reflection (Fig5.18). The real range of integration runs from -a to b withthe result:1 16(b2—bu+v)(a+au+v)_______Xo— 2ln [(+ (a — b)u + 2v + 2a + an + vb2 — bu + v)21(5.253)Notably, since all examples of this phase have the property that 2 =n2/4 — v = 0 itcan easily be shown that the argument of the log is always unity. Hence the susceptibilityin this phase is always vanishing. This strange behaviour throughout the phase deservesfurther investigation.Chapter 5. Critical Behaviour 64Figure 5.18: Complex contour for the calculation of susceptibility in double down phaseChapter 6Gauge Field CorrelatorsUp to this point we have been concentrating on the large N behaviour of the hermiteanmatrix eigenvalue distribution associated with the Kazakov-Migdal Penner model. Whilethis is of great importance in understanding the model, not to mention the application tothe associated 1 matrix and 2 matrix problems, we must remember that the original goalof this exercise was to see if a non-trivial gauge theory can be induced by coupling it tosuch a scalar field. In order to see if we have succeeded in this cause we must consider thebehaviour not of the scalar sector but rather the gauge sector. Fortunately the machineryof lattice gauge theory has been developed thoroughly enough so that we know what tolook for in our theory. Let us consider the gauge field correlator 3.107 in the large-Nlimit. The continuum limit of can be defined as the discontinuity of the generatingfunction G 3.130. Analogously to the definition of the eigenvalue distribution from thegenerating function EA, we evaluate G, across its cuts in the complex plane.A straightforward calculation assuming no singular contributions from the denominator gives the following result for the gauge-gauge correlator:(a+A)(a+i)= (6.254)With the denominator defined by:D, t) = _22 + (b — a). + ii)Aii + ab\2 + i2) (6.255)— )u(a2 + b2 + 2c+ 1) +( —t)(b — ac+bc) — c2 — b2 + a265Chapter 6. Gauge Field Correlators 66-c+(a+b)()Zeros of the denominator occur for= A(A) + E±jE (6.256)Remember that both A and are on the discontinuity of E,, and hence the secondterm in 6.256 has an imaginary part.Clearly the correlator of a pair of gauge fields is non-singular for an eigenvalue distribution with real support. Hence it appears that for the cases luld and idlu theredoes not exist a viable continuum limit in the gauge sector as the correlation lengths arealways finite.6.1 Real ExamplesHaving found the general solution of for the distribution of the eigenvalues of the scalarfields in the previous sections, it is possible to consider explicit examples of the correlationfunction. Here we test the sum rule 3.109 for C which follows from gauge invariance ofthe action. As we will see there are examples of solutions which do not appear to satisfythis sum rule.6.1.1 lu-idThe first case of that is the phase of the model where the effective one matrix potential3.136 has a negative singularity at -a and a positive one at b. As seen before there existsa proper one cut solution as long as some conditions on the parameters of the potentialare met. Using the data 4.202 for the example in chapter 4, we test the equality:f p(w)Cdw = 1 (6.257)Chapter 6. Gauge Field Correlators 671.151.11 . 050.950.90.854 4:2 4.4 4:6 4:8 5 5:2 5.4Figure 6.19: Correlator normalization as a function of second argument in luld exampleHere w and lie on the support of the eigenvalues.Results of numerical integrations for values of i around the support of the eigenvaluedistribution are shown in (Fig.6.19). It appears that in this case the sum rule is satisfied.6.1.2 idluAs in the previous phase, here we are dealing with a single, real, eigenvalue support. Itseems though that this is about all this phase has in common with luld when it comesto the question of the gauge correlators. We will use the same example 4.183 that wasused previously.First we consider the correlator as a function of a single variable where the secondargument takes a value on the support of the density near the average of the field (q).The results are shown in (Fig.6.20). Clearly the absolute value of this function in thisregion is bounded by a value much less than unity and hence the normalization condition6.257 cannot be satisfied.As a more general test of the correlator sum rule for this example, we numericallyintegrate the sum rule 6.257 for a range of second arguments in the region of the eigenvaluesupport (Fig.6.21). Here it is obvious that there is an inconsistency in our method ofChapter 6. Gauge Field Correlators 68_10_i0.403..1 0 —9. 8 —9. 6Figure 6.20: Correlation function (ldlu)in the region of the eigenvalue support. Secondargument equal -10.2 about average0. 0000 . 000—0. 000—0. 000—0.00W0.00040.0002—0.0002—0.0004—0.0006—0.0008Figure 6.21: Correlator normalization as a function of second argument in idlu exampleChapter 6. Gauge Field Correlators 69—:jQ4 —10.3 —10.2 —10.1Figure 6.22: fCwp(w)dw — A() for e [xi,x2] in idlu examplesolution. The sum rule here fails by more than 3 orders of magnitude.Finally we check to see if there is agreement between the integral, f Cwp(w)dw andthe function which it is supposed to define, A(A). The surprising results are shown in(Fig.6.22). It is clear that the function A has not been self-consistently generated in theDMS method. Utilizing the defining identities and the sum rule 6.257, we can test A.J (6.258)= JfCp(A)dp([L)d[t (6.259)= fitp(it)dt (6.260)= () (6.261)Evaluating the first integral for this particular choice of parameters one finds that theresult is indeed the average value of the field (). This leads to the conclusion that thecorrelator in this phase is not given by 6.254.6.1.3 Double downThe final phase of the model is that where both singularities in the effective 1 matrixpotential 3.136 are negative. We will consider the same example as above. Here theChapter 6. Gauge Field Correlators 70support of the eigenvalue distribution falls on a single complex contour which encircles thesingularities of the potential(See example section in chapter 4). In order to evaluate thesum rule 6.257 we will assume that the contour of integration involved in the convolutionmay be deformed to only include the singular points of the potential, namely -a and b.As we will see, only the singularity at b contributes to the integral due to the structureof the correlator 6.254.The calculation of the normalization of C follows as the normalization of the eigenvalue distribution in this phase:J Cp(w)dw = [+]Cp(w)dw (6.262)= 2iKE— G_] (6.263)(a+[I)f dw— 2iK (a+b)(w—b+a+())((b—w)—aw—c)((a+w)—bw+c+1)(6.264)+ b) + (b- ) - (a + b)(a + w)(b - w)]2 - 4(a + w)(b - w)(a + b)(w + a - b + ())(w—b)— a+ 6265— (a+b)2_((a+b)+1+c_b2)— (a+t)6266- (a+)+It seems that the normalization of the gauge correlator in this phase is correct onlyin the limit that the strength of the singularity at -a, namely /, goes to zero. This resultmay be changed however if the eigenvalue support is such that the singularities of the thecorrelator contribute to the normalization in addition to the singular points of the densityfunction. This may be the case if the contours used in the evaluation above enclose theseChapter 6. Gauge Field Correlators 71singularities. The choice of contour here has been particularly arbitrary but there existsa precise prescription [23] which guarantees that the probability measuredq5 = [dU]fld) .—‘ p(A)d) (6.267)is real and positive for all points on the support of the spectral distribution. Unfortunatelythe implications of this method remain to be evaluated.Chapter 7ConclusionThe Kazakov-Migdal model for a Penner-type potential appears not to be a realizationof large N QCD. While the lofty goal of finding the master field of the only known nontrivial gauge theory in four dimensions has failed, along the way we have answered manyquestions and brought up some new ones. Following the formalism of [18] we reduced theproblem to that of a two pole Penner one matrix model which was shown to have onlyone-cut (real) solutions. Further we explicitly solved for the precise nature of these cutsin terms of the parameters of the original potential. With this new detailed informationwe were able to give exact expressions for the susceptibility of the one matrix modeland derive the perturbative critical exponents of the model which were analogous to thesingle Penner and polynomial one matrix models.While the reduced model 3.136 was completely solved and appeared to be quite familiar, the question of the Kazakov-Migdal model corresponding to this solution was moreproblematic. First there appear to be inconsistencies between the one matrix solutionsand the effective KMM potentials. As seen in two examples previously the one matrixsolution gives an eigenvalue distribution which does not lie, for the KMM case, on alocal extrema of the potential. In short there appears to be a problem with the KMMpotential generated by the DMS method.An investigation of the steps involved in determining the ioop equations and the potentials in the DMS approach reveals that reducing the many degree of freedom saddle-pointequation to one over a single variable smooths out the action of the other eigenvalues.72Chapter 7. Conclusion 73For KMM potentials that are local, or simple polynomials in the scalar variables, thiscomplication is not an issue as the diagonalized potentials are simply sums over identicalterms in each of the eigenvalues. More explicitly we expect this formalism to work forKMM potentials of the form:VKMM(q5)= >Z (7.268)kAn example of such a case is the Gaussian potential which was discussed at length in[18]. Here comparisons between the effective one matrix solutions generated by the ioopequations and the exact result derived by Gross [24] are completely consistent. Here, forthe KM-Penner model, there appears to be a difference.This difference comes about when the simple, local ansatz of the DMS approach (eg.3.127) corresponds to KMM potentials of a more general form than 7.268. Here wepropose that the true form of the KMM potential in this general case is only constrainedby invariance under the adjoint action of the gauge group. This idea is not new as onematrix models in both D = 0, 1 with non-local potentials ((Trq2)have been consideredbefore [25]. It was found that these variants have a unique diagrammatic and criticalstructure in these simple cases and it would be very interesting to see how these resultsgeneralize to models like the KM-Penner with the gauge interaction.While it is apparent that the DMS method of solution breaks down at the levelwhere one tries to determine the original KMM potential which produces the results ofthe ansatz, there were additional problems discovered with the gauge-gauge correlatorsand their definitions. The fact that the gauge correlators determined by this methoddo not appear to work in certain phases (especially ldlu) shows that in order for themodel to be self-consistent that there must be a change in structure of the correlator.This requirement when taken with the hypothesis of [18] that the correlator undergoes achange in analytic form at a true gauge field critical point seems to hint at the prescenceChapter 7. Conclusion 74of such a critical point.Whether the required change in structure of the correlator is connected to the moregeneral non-local potentials discussed above is not known.Bibliography[1] t’Hooft, G. , A Planar Diagram Theory for Strong Interactions; Nuci. Phys. B721974, pg.461t’Hooft, G. , A Two Dimensional Model for Mesons; Nuci. Phys. B75 1974, pg.461[2] Witten, E. , The 1/N Expansion in Atomic and Particle Physics; in “Recent Deviopments in Gauge Theories” eds. t’Hooft et al; Plenum Press New York 1980[3] Brezin, E. , Itzykson, C. ,Parisi, G. , Zuber, J.-B. Planar Diagrams; Comm. Math.Phys. 59 1978, pg. 35[4] Mehta, M. L. Random Matrices 2nd ed; Academic Press Inc. 1991[5] For a review see the following and the references therein:Ginsparg, P. Matrix Models of 2D Gravity; hep-th/91122013Kaku, M. Strings, Conformal Fields, and Topology: An Introduction; Springer-Verlag New York Inc. 1991[6] Bessis, D. , Itzykson, C. , Zuber, J.-B. Quantum Field Theory Techniques in Graphical Enumeration; Advances in Applied Math. 1 Vi 2 1980, pg. 109[7] Wilson, K. Confinement of Quarks; Phys Rev. D10 1974, pg. 2445[8] Kogan, I. , Semenoff, G. , Weiss, N. Induced QCD and Hidden Local ZN Symmetry;Phys. Rev. Lett. 69 (1992), pg. 3435[9] Kogan, I. ,Morozov, A. , Semenoff, G. ,Weiss, N. Area Law and Continuum Limitin Induced QCD; UBCTP 92-026[10] Dobroliubov, M. I. , Kogan, I. , Semenoff, G. , Weiss, N. Induced QCD WithoutLocal Confinement; Phys. Lett. B302 (1993), pg. 283[11] Semenoff, G. , Weiss, N. Symmetry and Observables in Induced QCD; Proceedingsof Mathemtical Physics, String Theory and Quantum Gravity, Rhakov, Ukraine,October 1992[12] Dobroliubov, M. I. , Morozov, A. , Semenoff, G.W. , Weiss, N. Evaluation of Observables in the Gaussian N = oc Kazakov-Migdal Model; hep-th/9312145[13] Itzykson, C. , Zuber, J.-B. Planar Approximation II; J. Math. Phys. V21 3 1980,pg. 41075Bibliography 76[14] Mehta, M. L. A Method of Integration over Matrix Variables; Comm. Math. Phys.79 (1981), pg. 327[15] Chauduri, S. , Dykstra, H. , Lykken, J. The Penner Matrix Model and c=1 Strings;Mod. Phys. Lett. A V6 18 1991, pg. 1665[16] Kazakov, V.A. , Migdal, A.A. Induced Gauge Theory at Large N; Nuci. Phys. B3971993, pg. 214[17] Migdal, A.A. Mixed Models of QCD; Mod. Phys. Lett. A8 1993, pg. 359[18] Dobroliubov, M.I. , Makeenko, Yu. , Semenoff, G.W. Correlators of the KazakovMigdal Model; Mod. Phys. Lett. A8 (1993), pg. 2387[19] Makeenko, Yu. Some Remarks about the Two-Matrix Penner Model and the KazakovMigdal Model; Phys. Lett. B314 (1993), pg. 197[20] Becker, T. , Weispfenning, V. , Kredel, H. Gröbner Bases: A Computational Approach to Commutative Algebra; Springer-Verlag New York Inc. 1993[21] Makeenko, Yu. , Zarembo, K. Adjoint Fermion Matrix Models; hep-th/9309012[22] Ambjorn, J. , Kristjansen, C.F. , Makeenko, Yu. Generalized Penner Models to allGenera; hep-th/9403024[23] David, F. Phases of the Large N Matrix Model and Non-perturbative Effects in 2-DGravity; Nucl. Phys. B348 1991, pg.507[24] Gross, D.J. Some remarks about induced QCD; Phys. Lett. B293 (1992), pg. 181[25] Das, S.R. , Dhar, A. , Sengupta, A.M. , Wadia, S.R. New Critical Behaviour in D=OLarge N Matrix Models; Mod. Phys. Lett. AS 1990, pg. 1041Gubser, S.S. , Klebanov, I.R. A Modified c = 1 Matrix Model with New CriticalBehaviour; hep-th/9407014

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