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Unitary And Hermitian Matrices In An External Field II:

arXiv:hep-th/9112069v1 23 Dec 1991PUPT-1282December 4, 1991Unitary And Hermitian Matrices In An External Field II:The Kontsevich Model And Continuum Virasoro Constraints.David J. Gross⋆and Michael J. Newman.Joseph Henry Laboratories, Department of Physics,Princeton University, Princeton, N.J. 08544†ABSTRACTWe give a simple derivation of the Virasoro constraints in the Kontsevich model,first derived by Witten. We generalize the method to a model of unitary matrices,for which we find a new set of Virasoro constraints.

Finally we discuss the solutionfor symmetric matrices in an external field.⋆Research supported in part by NSF grant PHY90-21984† E-mail: gross@puhep1.princeton.edu; newman@puhep1.princeton.edu

1. Introduction.There has recently been considerable progress in the study of matrix models,following the remarkable proof by Kontsevich that the intersection numbers (correla-tion functions) of two-dimensional topological gravity are generated by a new type ofmatrix model [1].

A short time later, Witten showed [2] that the partition functionof this model obeys the Virasoro constraints of the one-matrix model [3, 4], thuscompleting the chain of arguments in a proof of the old conjecture that topologicalgravity and matrix models are equivalent [5].However, Witten’s lengthy proof depends on a cumbersome diagrammatic expan-sion. We present in section 2 a much simpler derivation, which arose out of recentwork into matrix integrals involving an external field [6].‡ This approach allows us toconsider other models; and in section 3 we tackle the case of a unitary matrix in anexternal field [8, 6], deriving a set of Virasoro constraints which describe the contin-uum limit of the multicritical unitary matrix models [9, 10].

These results are new,but after this work was completed, we learned§ that another group [11] has found thesame constraints from the complementary viewpoint of the mKdV flows [10, 12].In section 4 we present the results of some work on symmetric matrices in anexternal field. As far as we know this model has never before been investigated, so westray somewhat from the main subject of this paper and give some general details.Unfortunately this solution is incomplete, and in particular we fail to produce forsymmetric matrices any results generalizing the Virasoro constraints of sections 2and 3.In the final section we discuss our results, and speculate what these might teachus about unitary matrix integrals.‡ See also ref.

7.§ We thank E. Witten for informing us of this.2

2. Virasoro constraints from Schwinger-Dyson equations.Consider the integral over N × N hermitian matricesˆZ =ZD ˆM exp TrX ˆM −13g ˆM3,(2.1)which is a function of the N eigenvalues xa of the hermitian matrix X.¶ In refs.

6, 13,it is shown how to evaluate ˆZ using the Schwinger-Dyson equations of motion,ZD ˆM ∂∂ˆMexp TrX ˆM −13g ˆM3=XT −g ∂2∂X2ˆZ = 0 ,(2.2)which can be recast as a set of N differential equations in terms of the eigenvaluesxa,∂2 ˆZ∂x2a+Xb̸=a1xa −xb ∂ˆZ∂xa−∂ˆZ∂xb!= 1gxa ˆZ . (2.3)Full details of the derivation are given in refs.

8, 6. In ref.

13 these equations wereused to find ˆZ in the spherical approximation, and recently we showed how to extendthis solution to all orders in N [6].∗In this paper we start from these same equations,but the analysis will be very different.Closely related to ˆZ is the integralZ =ZDM exp Tr−12AM2 −13gM3,(2.4)for a positive definite hermitian matrix A. In his ground-breaking paper [1], Kontse-vich showed how the expansion of Z in terms of Feynman diagrams can be interpreted¶ Note that for the purposes of this paper, we define the integral without an N in the exponent.∗The reader is advised that an early preprint version of ref.

6 contained some mistakes in theintroduction.3

as a cell decomposition of the moduli space of Riemann surfaces [14, 15, 16]. By ashift of integration variables, it is easy to see that ˆZ and Z are related byˆZ = exp TrA312g2 × Z ,(2.5)provided we choose X = A24g .

Our goal now is to use (2.3) and (2.5) to show that Zobeys the Virasoro constraints of the one-matrix model [3, 4].Following Kontsevich, we factor Z in the formZ =Ya,b(µa + µb)−1/2 Y ,(2.6)where {µa} are the eigenvalues of A. Then, after a change of variables to λa ≡µ2a =4gxa, eqns.

(2.3) and (2.5) imply that Y satisfies the set of differential equations∂2Y∂λ2a+ ∂Y∂λa µa4g2 −Zaµa+Xb̸=a1µ2b −µ2a ∂Y∂λb−∂Y∂λa+ Y116µ4a+ t204µ2a= 0 ,(2.7)where we have definedZa =Xb1µa + µb;tk =−12k + 1Xb1µ2k+1b,k ≥0 . (2.8)Next we make good use of Kontsevich’s important observation that the perturbativeexpansion of Y depends on the eigenvalues of A only through the invariants tk.

Thislets us change variables from λa to tk, i.e.,∂∂λa= 12Xk1µ2k+3a∂∂tk,(2.9)and after some algebra we obtain14∞Xk,j=01µ2k+2j+6a∂k∂jY + 18g2∞Xk=01µ2k+2a∂kY+ 12∞Xk=0∂kY(2k + 3)tk+1µ2a+ (2k + 1)tkµ4a+ · · ·t0µ2k+4a+116µ4aY + t204µ2aY = 0 ,(2.10)4

where ∂k ≡∂/∂tk. Note that these equations are formally valid independent of thelarge-N limit, except that only for N →∞are the tk’s truly independent, and forfinite N the sums over k must be truncated.∗∗To make contact with Kontsevich’s results, we set g = i/2.

Then we can pick offthe coefficient of 1/µ4+2nato obtain the equationsn = −1 :∞Xk=1(k + 12)tk∂k−1Y + 14t20Y = 12∂0Y ,n = 0:∞Xk=0(k + 12)tk∂kY + 116Y = 12∂1Y ,n ≥1:∞Xk=0(k + 12)tk∂k+nY + 14nXk=1∂k−1∂n−kY = 12∂n+1Y . (2.11)Given that we have used a different normalization for the tk’s, these are precisely theequations first derived for the Kontsevich model in ref.

2,LnY = 12∂n+1Y,n ≥−1 ,(2.12)i.e., the Virasoro constraints of refs. 3, 4, corresponding to perturbations about them = 1 “topological” point of the one-matrix model.

It is remarkable that the entireset of multicritical potentials should be accessible from the simple cubic potentialof (2.4). (On the other hand, it is becoming increasingly apparent that matrix modelsinvolving an external matrix field possess a rich multicritical structure [6, 17].

)Finally, we complete the correspondence between the Kontsevich model and topo-logical gravity by noting that the familiar selection rule for correlation functions ofscaling operators at genus g⟨Ykτnkk ⟩g ≡Yk ∂∂tknk[log Y ]g(2.13)∗∗For related comments, see section 1.1 of ref. 2.5

(where [· · ·]g means the contribution at genus g), namelyXknk(k −1) = 3g −3 ,(2.14)is readily extracted from results in the appendix of ref. 6.

As is well known, eqn. (2.14)and the L0 constraint are all one needs to find the string susceptibility and the scalingdimensions of operators.3.

Unitary matrices.The success of the analysis so far suggests that one should investigate other modelsinvolving an external field. One such case, considered in another context in refs.

8, 6,is that of a unitary matrix,¯Z =ZDU exp TrA†U + U†A,(3.1)where A is now an arbitrary matrix. As mentioned earlier, we know of no simplegeometrical interpretation for such an integral.

One approach is to write the unitarymatrix in terms of a hermitian matrix, for example as U = eiH [18], or U = (1 +iH)/(1 −iH) [19], but neither of these has a particularly attractive expansion interms of surfaces.The partition function ¯Z satisfies the differential equations∂2 ¯Z∂Aab∂A†bc= δac ¯Z . (3.2)After changing variables in these equations to the eigenvalues λa ≡µ2a of A†A, weget the Schwinger-Dyson equations for ¯Z,∂2 ¯Z∂λ2a+Xb̸=a1λa −λb ∂¯Z∂λa−∂¯Z∂λb= 1λa 1 −Xb∂¯Z∂λb!.

(3.3)Guided by experience, we expect that we should factor ¯Z along the lines of (2.6); and6

indeed, after a small amount of trial and error, we find it useful to write¯Z =Ya,b(µa + µb)−1/2 exp2Xbµb¯Y . (3.4)This factor is what one gets by expanding U about the saddle point of the action asthe exponential of a hermitian matrix and dropping terms higher than quadratic, andin a sense (3.4) is the natural analog of the factorization (2.6).It was seen in ref.

6 that ¯Y shares with Y the property of depending only onthe tk’s. This encourages us to try changing variables in (3.3), and we find that ¯Ysatisfies14∞Xk,j=01µ2k+2j+6a∂k∂j ¯Y +∞Xk=01µ2k+4a∂k ¯Y+ 12∞Xk=0∂k ¯Y(2k + 1)tkµ4a+ · · ·t0µ2k+4a+116µ4a¯Y = 0 ,(3.5)which yields the equationsn = 0:∞Xk=0(k + 12)tk∂k ¯Y + 116 ¯Y = −∂0 ¯Y ,n ≥1:∞Xk=0(k + 12)tk∂k+n ¯Y + 14nXk=1∂k−1∂n−k ¯Y = −∂n ¯Y .

(3.6)We recognize these as Virasoro constraints, namelyLn ¯Y = −∂n ¯Y ,n ≥0 . (3.7)These equations can be put in the standard form Ln ¯Y = 0 by a shift of t0, whichmakes explicit that (3.7) corresponds to an expansion about the point t0 = 2, ti = 0for i ≥1.

(Contrast this with the usual situation in hermitian matrix models, wheret0 ≡x is a free parameter, and one other tm, for some m > 0, is fixed to a non-zero7

value.) This is an exceedingly trivial point, the m = 0 unitary matrix model, whichcorresponds naively to a matrix model with potential V (U) = constant (the m = 1model has V = U + U†); nevertheless it is well defined, since the integration is overa compact group.The Virasoro constraints are a convenient starting point for extracting usefulinformation about the model.

For instance, we can follow the example of ref. 3 andderive from them recursion relations for the correlation functions of scaling operators.For the m = 0 model, these take the form⟨τnYk∈Sτk⟩g = −Xj∈S(j + 12)⟨τj+nYk̸=jτk⟩g−14nXj=1⟨τj−1τn−jYk∈Sτk⟩g−1 + 12XS=X∪Yg=g1+g2⟨τj−1Yk∈Xτk⟩g1⟨τn−jYk∈Yτn⟩g2.

(3.8)(The genus dependence of these correlation functions has been inserted by hand, so asto correspond to the conventional 1/N expansion.) Note that the recursion relationsare precisely sufficient to determine all the correlation functions as pure numbers –i.e., all the operators are redundant.

In a loose sense, therefore, this model is also“topological.”There is also a selection rule,Xkk nk = g −1⇐⇒Xkk tk∂klog ¯Yg = (g −1)log ¯Yg ,(3.9)which, like (2.14), was found in ref. 6.

As a check, note that (3.9) and (2.14) aremerely different linear combinations of L0 and the dilaton equation Pk nk = 2 −2g.⋆(In general, one finds the selection rule for the mth multicritical model by eliminatingtm between the two equations. )⋆We thank R. Dijkgraaf for pointing this out to us.8

An important feature of (3.7) is that because there is no L−1 equation, the con-stant in L0 is not determined simply by the requirement of closure of the Virasoroalgebra. In ref.

11 the constant is considered a free parameter of the solution, butour derivation singles out the value116.There is another difference between our results and theirs, which we do not reallyunderstand. The constraints found here act directly on ¯Y , whereas in ref.

11 thepartition function is a product of two tau functions, each of which is annihilatedseparately by the Ln’s.4. Symmetric matrices and unoriented surfaces.In this final section we discuss an interesting example of a model which can besolved only partially by the methods we have been using: the real-symmetric matrixmodel.

This has been investigated in several papers [20, 21, 22, 23], and has a well-known geometric interpretation as the sum over unoriented surfaces. It appears thatthis model in an external field has not been looked at before, so we permit ourselvesto stray somewhat from the subject of Virasoro constraints and discuss this case ina fair amount of detail.

Much of the analysis runs parallel to that of the hermitianmatrices, and wherever possible we use the same notationThe starting point is the integral˜Z =ZY1≤i≤j≤NdMij exp NTrXM −13gM3,(4.1)for some real symmetric matrix X. In this section we need to count powers of N inthe topological expansion, so we include an N in the exponent.The derivation of the Schwinger-Dyson equations from eqn.

(4.1) is more subtlethan for hermitian matrices because the matrix elements of M (or X) are no longerindependent, and when we differentiate with respect to off-diagonal elements there is9

an extra factor of 2. For example,∂∂MijTr (XM) =( Xij + Xji = 2Xiji ̸= jXiii = j.

(4.2)As a consequence, the equations of motion have a complicated form when written interms of derivatives with respect to the matrix elements of X. For the same reason wehave to be careful when we change variables to eigenvalues, which requires evaluatingsuch quantities as ∂2xa/∂Xpq∂Xrs.

Remarkably, the final answer is both simple andfamiliar:∂2 ˜Z∂x2a+ 12Xb̸=a1xa −xb ∂˜Z∂xa−∂˜Z∂xb!= N2g xa ˜Z . (4.3)However, there is a major difference between this and eqns.

(2.3), (3.3): namely, thefactor of 12 multiplying the sum. This small change makes an important qualitativedifference to the genus expansion.Before we solve (4.3), it will prove convenient to rescale variables to λa = 4gxa.Then, following the procedure of ref.

6, we define ˜Z = eN ˜F and rewrite (4.3) as anequation for ˜Fa = ∂˜F/∂λa,⋆1N∂˜Fa∂λa+ ˜F 2a + 12NXb̸=a˜Fa −˜Fbλa −λb=164g4λa . (4.4)In the spherical limit the first term is smaller by a factor of 1/N, and we can drop it.The resulting equation has the solution˜F (0)a= νa8g2 + 14νaσ1 −ˆZa,(4.5)where it is useful to introduce the notationνa =pλa + y ,σk = 1NXb1νkb,ˆZa = 1NXb1νa + νb,(4.6)and y is found from the equation y = −4g2σ1; the superscript on ˜Fa refers to the⋆Note that ˜F is of order N, and ˜Fa is of order 1.10

order in 1/N. Integrating eqn.

(4.5), we find that the free energy is1N˜F (0) =112g2σ−3 + 12σ1σ−1 + g26 σ 31 −14N2Xb,cln (νb + νc) . (4.7)We can attempt to solve (4.4) order by order in 1/N to find the corrections to the freeenergy, as in ref.

6. Just write ˜F = ˜F (0) + 1N ˜F (1) +· · ·, insert into (4.4), and linearize.In the solutions of unitary and hermitian matrices [6], the 1/N terms canceled andthe leading correction was O(1/N2).

In the present case, this cancellation does notoccur, because of the factor of 12; therefore the free energy has a contribution of order1/N, as one would expect from topological arguments [24].The resulting linear equation for ˜F (1)ais νa4g2 + 12νaσ1 −ˆZa˜F (1)a+ 12NXb̸=a˜F (1)a−˜F (1)bλa −λb=−132g2νa+116ν3aσ1 −ˆZa−116NXb1ν2a(νa + νb)2 . (4.8)The right-hand side here is more complicated than anything that arose in the unitaryor hermitian cases.

We have not been able to solve this equation, and unfortunatelyit seems probable that the solution to (4.8) cannot be written in closed form.Ifthis is true, then it is impossible even to write down the equations for higher 1/Ncorrections, let alone solve them.On the other hand there is no obstacle to solving (4.8) perturbatively in g (actu-ally, as a double power series expansion in g and 1/N). Rather than using diagram-matic perturbation theory, we can work directly from the differential equation (4.4).But first, note that we are principally interested in the Kontsevich-type integralZ =ZDM exp NTr−12AM2 −13gM3= eNF ,(4.9)where F and ˜F are related by F = ˜F −Tr A3/12g2.Then one can show that11

Fa = ∂F/∂λa satisfies1N∂Fa∂λa+132g2µa+ F 2a + µa4g2Fa + 12NXb̸=aFa −Fbλa −λb+ Za16g2 = 0 ,(4.10)where µa, λa, and Za are defined in section 2. To derive the perturbation expansion,it is convenient to rearrange this into the formFa = −Za4µa−18Nµ2a−4g2µa12NXb̸=aFa −Fbλa −λb+ F 2a + 1N∂Fa∂λa.

(4.11)With some help from Mathematica, we find that the first few terms of the solutionat order 1/N are1N F (1) = −14NXbln µb + g212s1s2 −14N2Xb,c1µbµc(µb + µc)+g4s 21 s4 + 12s11NXb,c1µ2bµ2c(µb + µc) −16N3Xa,b,c1µaµbµc(µa + µb)(µb + µc)(µc + µa)+ O(g6) ,(4.12)where we have used the notation sk = 1NPb1µkb . This can be written in a more conciseform in terms of the “shifted” eigenvalues νa (≡pµ2a + y),1N F (1) = −14NXbln νb −g24N2Xb,c1νbνc(νb + νc) −g46N3Xa,b,c1νaνbνc(νa + νb)(νb + νc)(νc + νa) + O(g6) .

(4.13)To recover (4.12) from this, use the perturbative expansions of νa,νa = µa1 + 12y 1µ2a+ · · ·,(4.14)12

and of y,y = −4g2 1NXb1(µ2a + y)1/2 = −4g2 s1 −12ys3 + · · ·. (4.15)This latter equation can be solved iteratively to give y in terms of the sk’s.The notable feature of (4.13) is that it cannot be expressed in terms of just thetraces s2k+1, in contrast to the unitary and hermitian cases.

In fact, it would appearthat at each order in g2 we encounter new and more complicated invariants of theeigenvalues. Because the solution in its concise form (4.13) is so simple, one is temptedto guess the general term at order g2n.

But this simplicity is misleading, and even atO(g6) we do not know the form of the solution. The obvious ansatz, namely1N4Xa,b,c,d1νaνbνcνd(νa + νb) · · · (νc + νd) ,(4.16)can be ruled out on dimensional grounds, as we need a term of order 1/ν9, not 1/ν10.We finish by asking whether it is possible to find Virasoro-type constraints forunoriented surfaces.

We have the first ingredient, namely the Schwinger-Dyson equa-tions (4.4). Next we must identify how the integral (4.9) depends on the eigenvaluesof A, and “change variables.” In the previous examples we were lucky: a perturbativeanalysis showed that the integrals were functions solely of the tk’s, and the changeof variables was simple.

In the present case, we don’t even know the complete set ofvariables; but we do know that it is complicated, and the change of variables wouldlikely present a thorny challenge. For these reasons, we think that further progressin this direction is impossible.13

5. Discussion.In this paper we have demonstrated an efficient method of deriving Virasoro con-straints satisfied by the Kontsevich model, which we generalized to the unitary matrixmodel (3.1).

Now we ask whether this has revealed anything about the geometricalinterpretation of unitary matrices. To answer this, we consider first the hermitiancase (2.4), which is well understood.There is an important point about the Kontsevich model which we wish to empha-size.

The great majority of papers on matrix models involve studying them near theircritical points, which is where the perturbation expansion diverges and the Feynmandiagrams have an interpretation as continuum surfaces discretized by large numbersof infinitesimal triangles. This is not the situation in the Kontsevich model, whereone interprets a Feynman diagram of low order in perturbation theory as a surfacewith a finite number of punctures (equal to the number of faces on the diagram).

Thisis a topological picture, in which all surfaces of given genus and number of puncturesare represented by a single “ribbon graph”; and it yields topological information, i.e.,the intersection numbers. In addition, the integral (2.4) also possesses a conventionalcritical point, studied in ref.

6, with an interpretation in terms of continuum surfaces.This situation is reminiscent of the well-known Penner model, whose perturbative“topological” limit encodes the virtual Euler characteristic of surfaces [14, 15, 16],whereas the continuum limit corresponds to a c = 1 string in a compactified targetspace of a particular radius [25].Now we conjecture that the unitary matrix model of section 3 is a third exampleof this type. The integral certainly possesses the two appropriate limits.

From the“perturbative” limit we derive Virasoro constraints. The work of Hollowood et al.

[11]gives an independent confirmation that these are the correct constraints to describethe multicritical unitary matrix models, which we wish to interpret as some ensembleof surfaces coupled to matter. The non-trivial step is to suggest that this structurearises here because the integral (3.1) has a representation, analogous to that of the14

Kontsevich integral, in terms of the “moduli space” of the same (unknown) ensemble.However, we leave open the question of what this ensemble might be.It is very disappointing that we were unable to repeat the analysis for symmetricmatrices, but the moduli space of unoriented surfaces is known to be a lot morecomplicated than that of oriented surfaces.⋆Nevertheless, it has been instructive tostudy this model for the subtleties it reveals in the hermitian and unitary matrixmodels.As a possible generalization of this work, one could try to extend these resultsto the d = −2 external field model [17]. This would not be trivial, however, as thatmodel was solved by a different method.The original motivation for the research that led to this paper was to find ageometrical picture for unitary matrix models.

This is a goal that has long beensought by many others (e.g. [12, 18, 27]).

Unfortunately a satisfactory answer eludesus still, but we offer our insights in the hope that others can continue this work tocompletion.Note added: While this paper was being prepared, we learned that A. Marshakov,A. Mironov, and A. Morozov [28], and M. Kontsevich [29], have independently derivedthese same results for hermitian matrices.

In addition, S. Kharchev, A. Marshakov,A. Mironov, A. Morozov, and A. Zabrodin [30] have recently proposed a generalizedKontsevich model to describe all multi-matrix models.Acknowledgements: We are grateful to the authors of ref.

11 for sharing their resultswith us prior to publication, and in particular to Tim Hollowood and Andrea Pasquin-ucci for valuable discussions. We would also like to thank Ulf Danielsson, RobbertDijkgraaf, Jacques Distler, Miguel Martin-Delgado, Herman Verlinde, and Ed Wittenfor frequent help and explanations along the way.

M. N. is especially indebted to MarkDoyle for many useful conversations, and in particular for his thorough explanationof the paper of Kontsevich.⋆For example, see ref. 26.15

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