---

HERMITIAN vs. ANTI-HERMITIAN 1-MATRIX

arXiv:hep-th/9109046v1 24 Sep 1991IASSNS-HEP-91/59PUPT-1280September, 1991HERMITIAN vs. ANTI-HERMITIAN 1-MATRIXMODELS AND THEIR HIERARCHIESTimothy Hollowood∗, Luis Miramontes† & Andrea PasquinucciJoseph Henry Laboratories, Department of Physics,Princeton University, Princeton, N.J. 08544Chiara NappiInstitute for Advanced Study,Olden Lane, Princeton, N.J. 08540.ABSTRACTBuilding on a recent work of ˇC. Crnkovi´c, M. Douglas and G. Moore, a study of multi-critical multi-cut one-matrix models and their associated sl(2, C) integrable hierarchies,is further pursued.

The double scaling limits of hermitian matrix models with differentscaling ans¨atze, lead, to the KdV hierarchy, to the modified KdV hierarchy and part of thenon-linear Schr¨odinger hierarchy. Instead, the anti-hermitian matrix model, in the two-arcsector, results in the Zakharov-Shabat hierarchy, which contains both KdV and mKdV asreductions.

For all the hierarchies, it is found that the Virasoro constraints act on theassociated tau-functions. Whereas it is known that the ZS and KdV models lead to theVirasoro constraints of an sl(2, C) vacuum, we find that the mKdV model leads to theVirasoro constraints of a highest weight state with arbitrary conformal dimension.∗Address after Oct. 1, 1991: Dept.

of Theoretical Physics, Oxford, U.K.† Address after Oct. 1, 1991: CERN, Geneva, Switzerland.

IntroductionMost of the interesting properties of the matrix model formulation of two dimensionalgravity were originally extracted for the special case of lagrangians with even potentials [1].In the hermitian 1-matrix model it was later shown that the introduction of odd terms inthe potential gives rise to a doubling of the critical degrees of freedom and a doubling of thecritical equations. In the 1-arc sector one gets two decoupled Painlev´e I equations, for thefirst critical point; the underlying integrable structure being two decoupled Korteweg-deVries (KdV) hierarchies [2,3].

The situation, however, turns out to be different in the 2-arcsector of the theory, which, for an even potential, has a Painlev´e II equation as the lowestmulti-critical point; the underlying integrable structure being the modified Korteweg-deVries (mKdV) hierarchy [4,5].Explicit calculations show that the introduction of oddterms do not lead to decoupled equations and to the doubling of the mKdV system [6].This paper originated from our attempt to explore the integrable structures associatedwith the 2-arc sector of the hermitian 1-matrix model, with a generic potential. Indeed, oneof the most interesting features of matrix models is the fact that the known 2d quantumgravity models (both pure and coupled to minimal conformal matter) are described byan integrable hierarchy, supplemented with an additional condition known as the ‘stringequation’ [7].

The common belief is that the (anti-) hermitian n-matrix model shouldcorrespond to a hierarchy associated to the Lie algebra sl(n + 1, C), in the sense that,choosing different scaling ans¨atze for the double scaling limit of the matrix model, onegets a field theory, where the free energy (or a function related to it), satisfies the stringequation of such a hierarchy. Obviously, the case n = 1 is the simplest and also virtuallythe only one where examples can be worked out explicitly.

More general one-matrix modelshave also been considered in ref. [8], which deals with the case for complex matrices.We start by deriving in an explicit way the higher multi-critical points of the her-mitian 1-matrix model in the 2-arc sector with generic polynomials.

We find that theresulting string equations are associated with only the ‘even’ subset of flows of the non-linear Schr¨odinger (NLS) hierarchy. This, indeed, seems to be the hierarchy behind the2-arc sector of the hermitian 1-matrix model, as one can check by computing explicitly theLax operator.Interestingly, we notice that if the odd terms in the potential were purely imaginary,i.e.if the starting matrix model were anti-hermitian instead of hermitian, the multi-critical points in the 2-arc sector reproduce all the string equations associated to the ZShierarchy.

Indeed the Lax operator one gets in this case is that of the Zakharov-Shabat(ZS) hierarchy1. The reason why we do not find the odd string equations of the NLS in ourderivation of multi-critical points of the hermitian matrix model is that those equationsare complex.

Instead all the flows of the ZS hierarchy are real as we will see in section 2.5.While the NLS hierarchy is known to contain, as reduction, the mKdV hierarchy, the1 The connexion of the ZS hierarchy with the (anti-) hermitian matrix model in the 2-arc sectorwas first noticed by the authors of ref. [9].1

ZS hierarchy contains both the mKdV and KdV hierarchies, [10], (which can equivalentlybe obtained directly from the matrix model by restricting the ansatz made for the doublescaling limit). In addition, we find that the string equations of the NLS hierarchy reduceto those of the mKdV hierarchy, and the string equations of ZS to those of the KdV andmKdV hierarchies.

From this one deduces that solutions of the KdV and mKdV theoriescorrespond also to solutions of the NLS and ZS theories, in a particular subspace spannedby the even flows of the two hierarchies. The results are conveniently illustrated in diagram1, which shows how the critical points of the various hierarchies relate both in the 1-arcand 2-arc sectors.KdVրeven p.1-arc ցgeneral p.KdV(double)NLS(even)−→red.mKdVրgeneral p.րherm.

ցeven p.2-arcmKdVցanti-h. րeven p.ցgeneral p.KdVրZSred.ցmKdVDiagram 1We then go on to discuss the tau-function formalism of the ZS hierarchy.In thiscase we find that the partition function of the theory is equal to the tau-function of thehierarchy, rather than to its square, as happens instead in the 1-arc KdV model. As itwas first shown in ref.

[9], the partition function of the matrix model leading to the ZShierarchy, satisfies the Virasoro constraints of an untwisted boson with an, a priori, arbi-trary value for the zero-mode or ‘momentum’. This implies that the Virasoro constraintsact on the tau-function, to mirror the situation for the 1-arc KdV model [11].

However,in the ZS case, the tau-function carries an additional quantum number due to the zero-mode of the untwisted field. This additional quantum number seems to play the rˆole of2

a non-perturbative parameter which labels different sectors of the theory and argumentsconnected with the tau-function formalism suggest that it takes discrete values.An interesting side issue concerns the existence of Virasoro constraints for the modelsdescribed by the mKdV hierarchy; these include both the 2-arc (anti-) hermitian model,with even potential, and the unitary matrix models [12,13]. We find that there are, indeed,Virasoro constraints for the mKdV model; however, the situation is more complicated thanin the KdV case.

In the KdV case, the Virasoro constraints act on the tau-function: that isthe square root of the partition function. For the mKdV model there are two tau-functions;the partition function being the product.

We find that the string equation and the mKdVhierarchy imply a set of Virasoro constraints for each tau-function separately, however, incontrast to the KdV case, only the Ln for n ≥0 appear, and the eigenvalue of L0 is an,a priori, undetermined integration constant. In other words, the tau-functions satisfy theVirasoro constraints of a highest weight state of the conformal algebra.Under the reduction from ZS to KdV, the Virasoro constraints are transformed intothose of a twisted boson acting on the square root of the partition function (as expected[11]).

More interestingly, the reduction to KdV of the t2 = constant, t0 = x, ZS scalingtheory gives rise to the topological point of the KdV hierarchy! In other words, from theanti-hermitian 1-matrix model with a potential of the fourth order it is possible, after thedouble scaling limit, to get Topological Gravity [14].For the reduction to mKdV we get a series of constraints acting on the mKdV partitionfunction which are not Virasoro-like; this is to be expected because they act on the productof the mKdV tau-functions and not on each of them separately.

We also show that the‘conformal dimension’ of the mKdV partition function is fixed under the reduction. Finallyin both reductions the value of the above-mentioned discrete parameter is fixed.Another important issue which arises from this analysis, is the possible existence ofnew continuum theories described by the ‘odd’ critical points of the ZS model.

The firstsuch model was discussed in [9], and it seems to display a topological nature, similar tothe topological point in the 1-arc KdV model [14]. Nevertheless, the theory is sufficientlydifferent from the ‘normal’ topological theory, to make it unclear as to what continuumtheory it describes.

However, like its 1-arc cousin, this topological theory cannot be ob-tained from the matrix model. The possible higher ‘odd’ multi-critical points have yet tobe explored.

We will not address the mathematical technicalities required to prove thatsolutions of these new scaling points exist, however, since the mathematical apparatusexists [15] we hope these questions will be tackled elsewhere.Finally we notice that models with anti-hermitian matrices seem to arise in relationwith topological gravity (the Kontsevich model [16]) and with the Penner model [17].Indeed, whereas hermitian matrix models naturally emerge from the study of quantumgravity as a theory of random surfaces [1], anti-hermitian matrices seem to arise when onetries to make connexions between matrix models and moduli spaces of Riemann surfaces.1. (Anti-) Hermitian 1-Matrix Models and Orthogonal Poly-3

nomialsIn this section we will consider the double scaling limit of a general (anti-) hermitian 1-matrix model. Most of what follows has been already extensively discussed in the literature,so we will recall only the most important results, add some new ones and develop a fewexplicit examples to fix our notation.1.1Double Scaling Limit and String Equations of Hermitian MatrixModelsLet M be a hermitian N × N matrix and considerZN =ZdMe−βtrV (M)(1.1)where V (λ) = g1λ + g22 λ2 + g33 λ3 + .

. .

and λ denotes a (real) eigenvalue of M.As it is well known, one can introduce orthonormal polynomials Pn(λ) such thatZdλe−βV (λ)Pn(λ)Pm(λ) = δn,m(1.2)withλPn(λ) =pRn+1 Pn+1(λ) + SnPn(λ) +pRn Pn−1(λ),(1.3)thenZN = N!N−1Yi=0hi = N! hN0N−1Yi=1RN−ii(1.4)where Rn = hn/hn−1.We will consider, for the moment, the case with real potentials.Notice that thisimplies that both Rn and Sn are real.

Moreover, if V (λ) = V (−λ) then Sn = 0.The most important equation is the so-called ‘string equation’, which can be writtenas:nβ = g2Rn + g3(RnSn + RnSn−1) + · · ·0 = g1 + g2Sn + g3(S2n + Rn + Rn+1) + · · · . (1.5)In the double scaling limit one assumes that β/N →1, n/β = 1 −x/N α andRn = 1 + (−1)n f(x)N γ1 + r(x)N γ2 + j(x)N γ3 + z(x)N γ4 + · · ·Sn = b + (−1)n g(x)N δ1 + s(x)N δ2 + p(x)N δ3 + v(x)N δ4 + · · ·(1.6)where b is an arbitrary constant.4

For an even potential, V (λ) = V (−λ), and V (λ) = g22 λ2 + · · · + g2k2k λ2k, one sets inthe 1-arc sectorf(x) = 0 = g(x) ,α =2k2k + 1 ,γi = δi =i2k + 1(1.7)and in the 2-arc sectorα = 2k −22k −1 ,γi = δi =i2k −1 ,(1.8)and one needs to introduce scaling functions up to the order 2k −2.Let us introduce a such that2a = N12k+1in the 1-arc sectora = N12k−1in the 2-arc sector,(1.9)then eqs. (1.5) become1 −xaǫ = F0(gi) + f(x)aF1(gi) + 1a2 F2(f, g, r, s, f ′, g′; gi) + · · ·0 = G0(gi) + g(x)a G1(gi) + 1a2 G2(f, g, r, s, f ′, g′; gi) + · · ·(1.10)where ǫ = 2k in the 1-arc sector, ǫ = 2k −2 in the 2-arc sector and′ means ∂∂x.

Thefunctions Fi and Gi can be explicitly obtained from eqs. (1.5) and depend on the couplingconstants and on the scaling functions.It is now easy to solve order by order in 1/a the string equation (1.10), for examplethe zeroth order in 1/a (F0 = 0 = G0) fixes the value of g1 and g23:g2 = 1 −2bg3 −(3 + 3b2)g4 −(12b + 4b3)g5 −(10 + 30b2 + 5b4)g6 + · · ·g1 = −bg2 −(2 + b2)g3 −(6b + b3)g4 −(6 + 12b2 + b4)g5+−(30b + 20b3 + b5)g6 + · · ·.

(1.11)Notice that the gi’s depend on the free parameter b.At order 1/a one finds two possible solutions: either f(x) = 0 and g(x) = 0 or onefixes the value of g4. Choosing the solution f(x) = 0 = g(x) one gets the 1-arc sectorstring equations; fixing g4 instead, one gets the 2-arc string equations.

In the case of aneven potential of order 4 (k = 2) these are the well-known Painlev´e I and Painlev´e II stringequations.Moreover, from eq. (1.4), in the double scaling limit, we find the ‘specific heat’ (upto non-universal terms)F ′′ =12f 2(x) −r(x)(1.12)2 This is not the ‘a’ defined by Na2+1/m = 1 usually introduced in the literature.3 Most of the computations of this section have been done with the help of MathematicaTM.5

where Z = exp(−F) and ⟨PP⟩= ∂2x log Z = −F ′′.Let us consider now a real potential V (λ) with both even and odd couplings.Inthe 1-arc sector (f(x) = 0 = g(x)) one gets the well known ‘doubling’ phenomenon. Forexample, fromV (λ) = ( 13b3 −2b)λ + 12(2 −b2)λ2 + 13bλ3 −112λ4(1.13)one has the string equationsx =13χ′′ + χ2 ,x =13χ ′′ + χ 2(1.14)where χ(x) = r(x)−s(x) and χ(x) = r(x)+s(x).

Notice that these equations do not dependon the parameter b and that the g2i+1 are proportional to b. In the same way, in the 1-arcsector one obtains all the string equations of the KdV hierarchy, which are described in thefollowing chapter.

The ‘specific heat’ for these models is given by F ′′ = −r(x) = −12(χ+χ).The 2-arc sector with a general real potential is more interesting, indeed many differentscaling solutions and string equations appear. For example, withV (λ) = (2b −b3)λ + 12(3b2 −2)λ2 −bλ3 + 14λ4(1.15)one getsf ′′ −14f(g2 + f 2) + 12fx = 0g′′ −14g(g2 + f 2) + 12gx = 0(1.16)and fromV (λ) = (b + b3 −12b5)λ + 12(−1 −3b2 + 52b4)λ2 + 13(3b −5b3)λ3+14(−1 + 5b2)λ4 −12bλ5 + 112λ6(1.17)one gets0 = xf −38f 3(g2 + f 2) −38fg2(f 2 + g2) + 52f(f ′)2 + 3gf ′g′+−12f(g′)2 + 52f 2f ′′ + 32g2f ′′ + fgg′′ −f (4)0 = xg −38g3(g2 + f 2) −38gf 2(f 2 + g2) + 52g(g′)2 + 3ff ′g′+−12g(f ′)2 + 52g2g′′ + 32f 2g′′ + fgf ′′ −g(4) .

(1.18)Again these equations do not depend on the parameter b; the g2i+1 are proportional tob and putting g = 0 these equations will turn out to be the first two string equations ofthe mKdV hierarchy [5]. When g is not put to zero, these two string equations will beseen, in the next chapter, to be the second and fourth string equation of the Non-Linear-Schr¨odinger (NLS) hierarchy.

It is not possible to get the first and third string equation ofthe NLS hierarchy from the hermitian 1-matrix model. It seems that the NLS hierarchydoes not have a full ‘matrix model’ realization, in the sense that we do not get the multi-critical points corresponding to the ‘odd’ flows; a fact that will be explained in section§2.5.For all these models the ‘specific heat’ turns out to beF ′′ = 14f 2(x) + g2(x)(1.19)since r(x) = 14(f 2(x) −g2(x)).6

1.2Anti-Hermitian Matrix Models and Their String EquationsLet us first consider, in more detail, the NLS string equations (1.16) and (1.18).Sending g →ig one obtains string equations that are nothing but the string equationsassociated to the Zakharov-Shabat (ZS) hierarchy, as will be apparent from the next chap-ter. But this substitution is incompatible with our ansatz, eq.

(1.6), for the double scalinglimit. Indeed, sending g →ig implies that Sn becomes complex.

But if the potentialis real, V ∗(λ) = V (λ), it is easy to show that Sn must be real. On the other hand, ifV ∗(λ) = V (−λ) then Sn is pure imaginary.

(This can be easily shown using the orthogo-nality of the polynomials Pn(λ) (Pn(λ) = √hnPn(λ)), the fact that Pn(λ) = λn +O(λn−1)and the reality of the partition function (see (1.4)).) Therefore, we are led to consider thepotential V (λ) = ig1λ + 12g2λ2 + 13ig3λ3 + 14g4λ4 + · · · with λ, gk ∈R and some of theg2i+1 different from zero.

The double scaling limit is now done in exactly the same wayas in the previous section except for the ansatz for Sn (eq. (1.6)) which is instead pureimaginary, i.e.Sn = ib + (−1)n ig(x)a+ is(x)a2+ ip(x)a3+ iv(x)a4+ · · ·(1.20)with b, g(x), s(x), .

. .

all real.Notice that we can get rid of the ‘i’ in the potential introducing anti-hermitian ma-trices fM = iM with pure imaginary eigenvalues eλ = iλ. Again the Rn are real and in thesame way it is easy to prove that the Sn are pure imaginary.For notational simplicity in the rest of this section we will continue to use hermitianmatrices with complex potentials instead of using directly anti-hermitian matrices.In the 1-arc sector one gets again the double KdV string equation, now however inthe variables χ = r −is and χ = r + is = χ∗.In the 2-arc sector we found the family of scaling solutions corresponding to the ZShierarchy.

For example, fromV (λ) = i(2b + b3)λ + 12(−2 −3b2)λ2 −ibλ3 + 14λ4(1.21)with ǫ = 2 one gets0 =12xf + f ′′ + 12f(g2 −f 2)0 =12xg + g′′ + 12g(g2 −f 2) . (1.22)FromV (λ) = i−1 + b + b2 −b3 + 12b4 −12b5λ+12−1 −2b + 3b2 −2b3 + 52b4λ2+13i−1 + 3b −3b2 + 5b3λ3 + 14−1 + 2b −5b2λ4+15i 12 −52bλ5 + 112λ6(1.23)and ǫ = 3 one has0 = xf + g′′′ + 32g′ g2 −f 20 = xg + f ′′′ + 32f ′ g2 −f 2.

(1.24)7

Finally, fromV (λ) = ib −b3 −12b5λ + 12−1 + 3b2 + 52b4λ2 + 13i3b + 5b3λ3+14−1 −5b2λ4 −12ibλ5 + 112λ6(1.25)with ǫ = 4 one has0 = xf + 38f 3 g2 −f 2−38fg2 g2 −f 2+ 52f(f ′)2 −3gf ′g′+12f(g′)2 + 52f 2f ′′ −32g2f ′′ −fgg′′ −f (4)0 = xg −38g3 g2 −f 2+ 38gf 2 g2 −f 2−52g(g′)2 + 3ff ′g′+−12g(f ′)2 −52g2g′′ + 32f 2g′′ + fgf ′′ −g(4) . (1.26)These are the first three string equations of the ZS hierarchy (excluding the topological-like point [9] which cannot be obtained from the matrix model).More precisely, theycorrespond to the points tǫ = constant, t0 = x and ti = 0 for i ̸= {0, ǫ} in the ZS hierarchy(see next section).A few comments are in order.

As usual the string equations do not depend on b. Theg2i+1 are proportional to b except for the case of eq. (1.23).

Indeed, it is not possible toget the string equation eq. (1.24) from a real potential (V ∗(λ) = V (λ)).

Moreover, settingg = 0 one obtains the first two string equations of the mKdV hierarchy (eq. (1.24) vanishesidentically) and setting 12(f +g) = −1 in eqs.

(1.22) and (1.26) one gets the first two stringequations (topological point included) of the KdV hierarchy in the variable ψ = 12(f −g).For all these models it turns out thatF ′′ = −14g2(x) −f 2(x)(1.27)since r(x) = 14[f 2(x) + g2(x)].1.3Lax Operators from Matrix ModelsThe basic idea, due to Douglas [7], is that it is possible to make the double scaling limitnot only on the string equations but also directly on λ and ∂∂λ seen as operators acting onPn(λ). This operator is then related to the Lax operator of the integrable hierarchy whichunderlies the behaviour of the continuum theory associated to the particular sequence ofscaling ans¨atze chosen.

Indeed, under a double scaling limit from eq. (1.3) it is easy to seethat λ becomes a differential operator (bλ) of order 2 in x.

The obvious relation∂∂λ, λ= 1(1.28)after the double scaling limit becomes the string equation [7]. Eq.

(1.3) under the doublescaling limit becomesbλΨ = ΛΨ(1.29)8

where Ψ, which can be a vector, is related to the rescaled polynomials and Λ is a differentialoperator of degree 2 in x. Analogously, there exists an equation of the formc∂∂λΨ = MΨ. (1.30)These two equations can be rewritten asLΨdef=bλ −ΛΨ = 0 , c∂∂λ −M!Ψ = 0(1.31)and then the string equation becomes the compatibility condition for these differentialequations [15], i.e.

"L, c∂∂λ −M!#= 0. (1.32)It is easy to get the explicit expression of L from eq.

(1.29). L is then the Lax operatorof the corresponding hierarchy.

Using the techniques of Zakharov-Shabat and Drinfeld-Sokolov [18,19], one can also explicitly compute M [7,4,9].We now explicitly compute L from eq. (1.3).

Consider first the case of a hermitianmatrix with a real potential. Let Π(x, λ) denote Pn(λ) after the double scaling limit4, thuseq.

(1.3) becomesλ Pn(λ) ∼(2 + b)Π(x) + 1a2 [r(x)Π(x) + s(x)Π(x) + Π′′(x)] + · · · ,(1.33)where for notational simplicity we have dropped the dependence on λ in Π. Setting λ −(b + 2) →bλ/a2 one hasL Π(x, λ) = 0 ,(1.34)whereL = ∂2x + (r(x) + s(x)) −bλ .

(1.35)This is the Lax operator of the KdV hierarchy.Notice that setting Π(x, λ) ∼(−1)nPn(λ) one gets L = ∂2x + (r(x) −s(x)) + bλ. Thusthere are two KdV Lax operators associated to the hermitian 1-matrix model in the 1-arcsector, this is nothing but the doubling phenomenon [2].In a similarly way, for the anti-hermitian 1-matrix model in the 1-arc sector one getsthe previous two Lax operators with s(x) →is(x).Consider now the case of the 2-arc sector with a hermitian matrix and a real potential.In order to get a non trivial Lax operator in the double scaling limit, following [4], weneed to introduce two scaling functions depending on parity, Π(λ, x) ∼(−1)mP2m(λ) andΩ(λ, x −1a) ∼(−1)mP2m+1(λ) where x = x2m.

Thus for n even eq. (1.3) becomesλ (−1)n/2Pn(λ) ∼bΠ(x) + 1a [g(x)Π(x) −f(x)Ω(x) −2Ω′(x)] + · · ·(1.36)4 Actually, as explained in refs.

[7,4,15], under the double scaling limit Π(x, λ)∼e−NV (λ)/2Pn(λ) has a smooth behaviour and should be considered for this computation.9

and for n odd one hasλ (−1)(n−1)/2Pn(λ) ∼bΩ(x) + 1a [−g(x)Ω(x) −f(x)Π(x) + 2Π′(x)] + · · · . (1.37)Thus, setting λ −b →bλ/a one hasL Ψ = 0(1.38)whereL =bλ2 −g(x)2100−1+ f(x)20110+ ∂∂x01−10(1.39)and Ψ =ΠΩ.

Now, after having conjugated L and rotated Ψ, we get eq. (1.38) withL = iσ3bλ + σ2g −σ1f + ∂x(1.40)and [σl, σj] = iǫljkσk.

As we expected, this is the Lax operator of the NLS hierarchy.In the case of an anti-hermitian matrix with pure imaginary eigenvalues eλ one cando the same computation with P2m+1 and Sn pure imaginary (see eq. (1.20)).

Lettingeλ −ib →ibλ/a, one finally getsL = ∂x + iσ2g −σ1f −σ3bλ ,(1.41)and this is the Lax operator of the ZS hierarchy.Notice that the mKdV string equations can be obtained both from the NLS (hermitianmatrix, real potential) and ZS (anti-hermitian matrix, complex potential) hierarchy. Thisis obvious since setting Sn = 0 (b = 0 = g = .

. .) hermitian and anti-hermitian matrixmodels coincide.2.Hierarchies and String EquationsIn this section we will discuss the hierarchies of integrable equations which lie behindthe non-perturbative structure of the one matrix model.

The relevant hierarchies are well-known in the mathematical physics literature, and are the Zakharov-Shabat (ZS), for theLie algebra sl(2, C), the Korteweg-de Vries (KdV), modified Korteweg-de Vries (mKdV)and non-linear Schr¨odinger (NLS) hierarchies [10].Interestingly, all these hierarchies are intimately related to the Lie algebra sl(2, C).This is apparent in the matrix Lax formalism [18] and also the ‘Hirota’ or ‘tau-function’formalism [20]. It is also significant that the complex-ZS hierarchy is the ‘master’ hierar-chy for sl(2, C), since the ZS, NLS, KdV and mKdV hierarchies can all be obtained byappropriate reductions.

This will be explained in more detail below. We shall also discusshow the hierarchies are related to the matrix models.10

2.1The sl(2, C) HierarchiesIn section §1.3 we showed how the matrix models are connected with the integrablehierarchies presented in the Lax formalism. In fact, the most economical way of explicitlyintroducing the hierarchies is via their recursion relations.

These can be extracted fromthe Lax formalism, see for example ref. [18].

In what follows we often write x for theflow t0, of the hierarchy under discussion. For the KdV equation the hierarchy can bepresented, in the following way∂u∂tk= ∂xRk+1k ≥0,(2.1)where Rk is a polynomial in u and its x-derivatives (the Rk’s are known as the Gel’fand-Diki polynomials [21], and are not to be confused with the Rk used in the previous section).The integrability of the hierarchy is summed up in the recursion relation∂xRk+1 = 12∂3x + 2u∂x + u′Rk .

(2.2)Specifying R0 = 1 along with the recursion relation completely determines the hierar-chy. To make contact with the notation used in the previous section, for example for thehermitian matrix model in the 1-arc sector with even potential, one should set u = r.For the complex-ZS hierarchy a similar structure of recursion relations is found.

Fortwo independent complex variables ψ and ψ, if∂ψ∂tk= 12(Fk+1 −Gk+1) ,∂ψ∂tk= 12(Fk+1 + Gk+1) ,(2.3)with k ≥−1, thenFk+1 = G′k + (ψ −ψ)HkGk+1 = F ′k + (ψ + ψ)HkH′k = ψ(Gk −Fk) −ψ(Gk + Fk) . (2.4)Specifying F0 = ψ −ψ and G0 = ψ + ψ along with the recursion relations then determinesthe whole hierarchy.

Notice that the flow t−1 is particularly simple∂ψ∂t−1= −ψ,∂ψ∂t−1= ψ ,(2.5)implying the following dependence on t−1: ψ ∼e−t−1 and ψ ∼et−1. To make contact withthe notation of the previous section ψ = 12(f −g) and ψ = 12(f + g), which also agreeswith the conventions of ref.

[9] when tk →tk+1. The NLS hierarchy is recovered fromthe complex-ZS hierarchy by choosing ψ to be the complex conjugate of ψ (up to a terminvolving t−1), a choice which is easily seen to be consistent with the recursion relations11

of the hierarchy. The ZS hierarchy itself, simply corresponds to the complex-ZS hierarchywith ψ and ψ both real.The mKdV hierarchy is best introduced through its relation to the KdV hierarchy viathe Miura Map.

This map takes a solution of the mKdV ν hierarchy into a solution of theKdV u hierarchy asu = −ν′ −ν2 . (2.6)The pull-back of the KdV flow tKdVkunder the Miura Map is then the mKdV flow tmKdVk.We now make this more explicit.

Defining D = −∂x −2ν and D⋆= ∂x −2ν, the recursionrelation (2.2) becomes ∂xRk+1 = D(−12∂x)D⋆Rk [22]. Since ∂xu = D∂xν, we have∂u∂tKdVk= D∂ν∂tmKdVk= D−12∂xD⋆Rk ,(2.7)and so the mKdV flows are∂ν∂tk= −12∂xD⋆Rkk ≥0,(2.8)where the polynomial Rk = Rk[−ν′ −ν2] is expressed in terms of ν and its x-derivativesby substituting u for ν via the Miura Map.

To make contact with the notation of theprevious section ν = f/2.2.2The ZS, NLS, KDV and MKDV Hierarchies as Reductions of thecomplex-ZS HierarchyIn this section we explain how the the various hierarchies that we have introduced canall be obtained as reductions of the complex-ZS hierarchy. The situation is convenientlysummarized by diagram 2 which shows how the various hierarchies are related by reduction.NLS−→mKdVրcomplex-ZSցKdVրZSցmKdVDiagram 2We have already noted how the complex-ZS hierarchy is reduced to the ZS and NLShierarchies by taking two different ‘real slices’.

For the former one takes ψ and ψ to bereal, whilst for the latter one takes the complex conjugate of ψ to be ψ⋆= e−2t−1ψ.12

The KdV hierarchy is obtained from the ZS hierarchy by settingψ = −et−1,(2.9)the KdV variable being given by u = −ψψ. One can readily prove that F2k+1 +G2k+1 = 0,when evaluated at ψ = −et−1, whereas F2k + G2k ̸= 0.

This means that only the evenflows preserve the condition (2.9), and so only the even flows reduce to flows of the KdVhierarchy. One finds∂(−ψψ)∂t2k⋆= 12 H′2k+1⋆,(2.10)where ⋆means ‘evaluate at ψ = −et−1’.

If one now compares the recursion relations of theZS hierarchy (2.4), evaluated at ψ = −et−1, to those of the KdV hierarchy (2.2), then onededuces thatH2k+1[ψ, ψ = −et−1] = 2k+1Rk+1[−ψψ]. (2.11)This means that the relation between the flow variables is tZS2k = 2−ktKdVk.

It is straight-forward to see that the KdV hierarchy cannot be obtained from the NLS hierarchy by asimilar reduction.The mKdV hierarchy is obtained from both the NLS and ZS hierarchies by settingψ = e2t−1ψ,(2.12)the mKdV variable being given by ν2 = ψψ. One finds that G2k+1 = F2k = H2k = 0,when evaluated at (2.12).

So the situation is similar to that for the KdV reduction, inthat only the even flows preserve the reduction. By pursuing a similar analysis of therecursion relations, one discovers that the flow variables are related via tZS2k = 2−ktmKdVksince F2k+1 = −2k∂xD⋆Rk[−ν′ −ν2], when evaluated at (2.12).2.3The String Equations and Their ReductionsThe hierarchies that we have discussed above admit many types of solution.

However,in applications to matrix models and two-dimensional field theories, very particular solu-tions are required. In addition to boundary conditions, these are specified by adjoining tothe hierarchy an extra condition called the string equation [7].

The string equation mustbe consistent with the flows of the hierarchy, in the sense that it must be preserved bythe flows of the hierarchy. It turns out that the string equation admits certain scaling, ormulti-critical solutions.

It is these solutions which are found in the matrix model, afterthe double scaling limit. They are obtained by restricting to the subspace tk = constant,t0 = x and tj = 0 otherwise, for some k. We call the resulting reduced equation the kthstring equation.

The string equations can be found in general following the analysis of§1.3.The string equation associated to the KdV hierarchy was found originally in [7]. Inour conventions it takes the form∞Xk=1(2k + 1)tk∂u∂tk−1= −1 ,(2.13)13

which may be integrated using (2.1) to give∞Xk=0(2k + 1)tkRk = 0 . (2.14)The kth multi-critical point corresponding to tk = constant, t0 = x, and tj = 0 otherwise,is described by the string equation(2k + 1)tkRk[u] = −x .

(2.15)For the ZS hierarchy, the string equation was found in [9]∞Xk=0(k + 1)tk∂ψ∂tk−1= 0,∞Xk=0(k + 1)tk∂ψ∂tk−1= 0 . (2.16)The kth multi-critical point for which tk =constant, t0 = x and tj = 0 otherwise, isdescribed by the string equation(k + 1)tk(Fk −Gk) = 2xψ,(k + 1)tk(Fk + Gk) = 2xψ .

(2.17)The above equations also apply to the NLS hierarchy by taking ψ⋆= e−2t−1ψ.The string equation of the mKdV hierarchy has been obtained in [4], however, onecan obtain it in a simple way given the string equation of the KdV hierarchy by using theMiura map. The idea is to pull back the string equation of the KdV hierarchy via theMiura map u = −ν′ −ν2; the result is then guaranteed to be consistent with the flows ofthe mKdV hierarchy, because of the Hamiltonian property of the Miura map.

We act on(2.14) with the operator 12∂xD⋆and use eq. (2.8) to obtain∞Xk=1(2k + 1)tk∂ν∂tk= −xν′ −ν .

(2.18)This can be rewritten as∞Xk=0(2k + 1)tk∂ν∂tk+ ν = 0 . (2.19)Using eq.

(2.8) again, we can integrate with respect to x (discarding, an integrationconstant) obtaining the mKdV string equation∞Xk=0(2k + 1)tkD⋆Rk = 0. (2.20)where Rk = Rk[−ν′ −ν2].

The kth multi-critical point for which tk = constant, t0 = xand otherwise tj = 0, is described by the string equation(2k + 1)tkD⋆Rk[−ν′ −ν2] = 2xν . (2.21)14

We now show that the string equation of the ZS hierarchy consistently reduces to thestring equations of the KdV and mKdV hierarchies, respectively, for the reductions wediscussed in §2.2. For the reduction to KdV we evaluate (2.16) at ψ = −et−1, t2k+1 = 0and use the relations obtained in §2.2 to get∞Xk=0(k + 12)tKdVk2−k 12∂2x + ψH2k−1 = 0∞Xk=0(k + 12)tKdVk2−kH2k−1 = 0 .

(2.22)These equations are obviously equivalent to the KdV string equations (2.13) since H2k−1 =2kRk at ψ = −et−1, t2k+1 = 0.Analogously, in the case of the mKdV reduction with ψ = e2t−1ψ, t2k+1 = 0, one ofthe equations (2.16) is trivially solved and the other equation becomes∞Xk=0(k + 12)tmKdVk2−kG2k = 0 . (2.23)But G2k = −2kD∗Rk[−ν′ −ν2] at ψ = e2t−1ψ, t2k+1 = 0, which gives the mKdV stringequation (2.20).The significance of the fact that the string equation of the ZS model reduces to thatof the KdV and mKdV models, is that solutions of the KdV and mKdV string equationsare, by pulling back, solutions of the ZS string equation (on the subspace t2k+1 = 0 ∀k).This will lead us to conclude that the model described by the ZS hierarchy includes themodels described by the KdV and mKdV hierarchies.

We further discuss these facts in§3.4 and §3.5.2.4The Tau-Function FormalismThere is an alternative formalism for constructing integrable hierarchies which wasoriginally developed as a direct solution technique of the non-linear equations of the hi-erarchy. The central objects of this approach are the tau-functions, which satisfy a hier-archy of non-linear ‘Hirota bilinear equations’, see [20] for example.

For the hierarchiesthat we are considering, the Hirota hierarchies are intimately related to the Lie algebrasl(2, C) (= A1). In fact, they use the two vertex operator representations of the basicrepresentations of the affine algebra A(1)1 .

This works in the following way. The basicrepresentations of A(1)1are carried by the Fock space of a scalar field, either twisted oruntwisted.

The Hirota equations for the tau-function are equivalent to the condition thatthe tau-function lies in the orbit of the highest weight state of the group associated to theaffine algebra. The untwisted construction underlies the ZS and NLS hierarchy, whilst thetwisted construction underlies the KdV and mKdV hierarchies [20].15

For the joint KdV and mKdV system (related by the Miura Map) there are two tau-functions τ0 and τ1 arising from the two basic representations of the Kac-Moody algebraA(1)1 . The relationships between these and the functions u and ν areu = 2∂2x log τ0 ,ν = ∂xlogτ1τ0.

(2.24)One of the equations of the hierarchy isτ ′′0 τ1 −2τ ′0τ ′1 + τ0τ ′′1 = 0 ,(2.25)from which one can extract the Miura Map u = −ν′ −ν2 and the relationν2 = −∂2x log (τ0τ1) . (2.26)The Hirota hierarchy which leads to the complex-ZS hierarchies has an infinite setof tau-functions.

This is because the relevant vertex operator construction, in this case,involves an untwisted scalar field, which has a zero-mode.In order that the operatorproduct expansions of the vertex operators are local, the zero-mode must be quantized,taking values in the weight lattice of the finite Lie algebra; A1 in this case, whose weightlattice is simply isomorphic to Z. We will label elements of the weight lattice with half-integers, so that the sub-lattice generated by the root consists of the integers.

With thislabelling, the integers and half-integers correspond to the two distinct basic representationsof the affine algebra A(1)1 . To make a connexion with the complex-ZS hierarchy one choosesa fixed element of the weight lattice, that is a half-integer n. Thenψ = ˜τn+1˜τn,ψ = ˜τn−1˜τn,(2.27)where we have used a tilde in order to avoid confusion with the mKdV/KdV tau-functions.In addition, the equations of the hierarchy implyψψ = −∂2x log ˜τn .

(2.28)The hierarchies for different choices of n are isomorphic.2.5From Hierarchies to Matrix ModelsGiven a hierarchy and its string equation, to obtain the field theory describing amatrix model after a double-scaling limit, one must first identify the partition function ofthe field theory with some variable in the hierarchy.If x is the cosmological constant then the ‘specific heat’ is F ′′ = −∂2x log Z. Clearly−F ′′ has a well defined scaling dimension, from a hierarchical point of view. So, in prin-ciple, one can construct all the terms of the correct dimension from the hierarchy, andthen see which can be integrated twice with respect to x.

However, even this would not16

determine the normalization of the ‘specific heat’. In the absence of any additional physi-cal requirements, one has to appeal to the matrix model.

It transpires that the partitionfunction of each particular model is related to the tau-function of the hierarchy in a simpleway:ZZS = ˜τn,ZKdV = τ 20 ,ZmKdV = τ0τ1 ,(2.29)where the tau-functions for the hierarchies where introduced in section §2.4.With x interpreted as the cosmological constant, the ‘specific heat’ is nothing but thecorrelation function −⟨PP⟩= −∂2x log Z, where the operator which couples to the cosmo-logical constant is conventionally denoted P and called the puncture operator. Finally, theinsertion of an operator in a correlation function is given by the corresponding flow in thehierarchy, for example⟨OiPP⟩=∂∂ti⟨PP⟩(2.30)where O0 = P and the right-hand side is computed using the equation of the hierarchy.Moreover, with the identifications (2.29)⟨PP⟩ZS = −ψψ = ⟨PP⟩NLS ,⟨PP⟩KdV = u ,⟨PP⟩mKdV = −ν2 .

(2.31)Let us consider first the hierarchies which should correspond to the anti-hermitianmatrix models and their correlation functions constructed using the prescription above.The anti-hermitian 1-matrix model in the 1-arc sector should correspond to the sum oftwo KdV hierarchies in the variables χ = r −is, χ = r +is = χ∗. Although these variablesare complex (r and s are real functions of x), it turns out that the correlation functionsand the string equations are real.

Indeed, the ‘specific heat’ is given by the sum of the‘specific heats’,⟨PP⟩=12(χ + χ) = r(2.32)and the correlation functions are given by⟨OkPP⟩=12∂x (Rk+1(χ) + Rk+1(χ)) . (2.33)Using the recursion relations of the KdV hierarchy it is easy to show that Rk(χ) = Rk(χ)∗and hence the correlation functions are real.

Moreover, the sum and the difference of thestring equations (2.14) for χ and χ give exactly eqs. (1.5) for this model.The anti-hermitian 1-matrix model in the 2-arc sector corresponds to the ZS hierarchy.First of all, notice that for the ZS and NLS models one has⟨OkP⟩=12Hk+1(2.34)implying, for example, ⟨PP⟩= 12H1 = −ψψ.

For the ZS hierarchy both ψ and ψ are real,and thus the whole hierarchy and all the physical quantities are real. The string equations(2.16) correspond to the sum and difference of the two equations (1.5) where the secondequation is multiplied by i.We now turn our attention to the hermitian 1-matrix model.

In the 1-arc sector wehave a double KdV hierarchy where everything is expressed in terms of real functions. In17

the 2-arc sector, instead, we found only some of the string equations of the NLS hierarchy.Indeed, for the NLS hierarchy, one can easily show that F2k, H2k and G2k+1 are purelyimaginary, whereas F2k+1, H2k+1 and G2k are real. Thus, although the ‘specific heat’ isreal, many correlation functions are complex or pure imaginary.

This obviously forbids aninterpretation of the full NLS hierarchy as a field theory obtained after a double scalinglimit of a hermitian 1-matrix model in the 2-arc sector. Taking the sum and the differenceof the string equations (2.16) and using the recursion relations of the hierarchy one gets[9]∞Xk=0(k + 1)tkGk = 0 ,∞Xk=0(k + 1)tkFk = 0 .

(2.35)The string equations (1.5) should correspond to the multi-critical points t0 = x, tn =constant and tk = 0 otherwise. Thus, if n is even the first equation is real and the secondis pure imaginary and they correspond to eq.

(1.5) where the second equation is multipliedby i. These are the string equations we found in the first chapter.

For n odd instead, bothequations are complex and so they could not arise from the direct study of the matrixmodel.3.Virasoro ConstraintsIn [11] it was shown, for the models described by the KdV hierarchy, that the stringequation, along with the hierarchy equations, could be reformulated as an infinite numberof Virasoro-like constraints on the square-root of the partition function of the model. Theseconstraints have a natural interpretation in terms of the Schwinger-Dyson equations for theloops of the matrix model.

For the KdV model the square-root of the partition function isthe tau-function of the hierarchy. The fact that the Virasoro constraints act on the tau-function of the hierarchy seems to be a universal feature of all the models, as will becomeapparent.

First we briefly review the case for the KdV hierarchy. Using (2.29), (2.24) andthe string equation (2.13), and integrating twice we deduce ∞Xk=1(k + 12)tk∂∂tk−1+ 18t20!τ0 = 0 .

(3.1)Following [11], we use the recursion relations of the hierarchy (2.2) and the relation betweenτ0 and u in (2.24), which together imply∂2x∂∂tk+1log τ0 = 12∂3x + 2u∂x + u′∂x∂∂tklog τ0,(3.2)in order to express the tk derivative in terms of the tk+1 derivative. With this relation,one finds that Lkτ0 = 0 implies Lk+1τ0 = 0, where the Lk are written below.

All of theconstants of integration encountered are set to zero, on the grounds that they would other-wise introduce spurious scales into the theory, except for the constant in the L0 constraint18

which is dimensionless and fixed by the requirement that the algebra of constraints closes.The end result is that the tau-function satisfies an infinite number of constraints of theformLnτ0 = 0,n ≥−1 ,(3.3)where the Ln’s are the Virasoro generators of a Z2-twisted scalar fieldL−1 =∞Xm=1(m + 12)tm∂∂tm−1+ 18t20L0 =∞Xm=0(m + 12)tm∂∂tm+ 116Ln =∞Xm=0(m + 12)tm∂∂tm+n+ 12nXm=1∂2∂tm−1∂tn−m. (3.4)3.1Virasoro Constraints for the MKDV HierarchyAlthough the matrix model which leads to the mKdV hierarchy has been discussedin the literature [4], the analogue of the Virasoro constraints do not seem to be havebeen determined before (although ref.

[12] does discuss Virasoro constraints before takingthe double scaling limit). In this section we find the constraints using the mKdV stringequation (2.20) and the recursion relations for the hierarchy.The mKdV string equation (2.20) is obtained from integrating (2.19).

From (2.19)one easily deduces∞Xk=0(2k + 1)tk∂ν2∂tk+ 2ν2 = 0. (3.5)Recall that the partition function of the mKdV model is equal to the product of the tau-functions τ0 and τ1.

We now express ν in (2.19) in terms of the tau-functions τ0 and τ1,using (2.24), and ν2 in (3.5) using (2.26). The resulting two equations can be decoupledto arrive at∂2x" ∞Xk=0(2k + 1)tk∂∂tklog τj#= 0 ,j = 0, 1 .

(3.6)Integrating twice, and eliminating dimensionful integration constants, the two resultingequations may be written simply asL0τj = µjτj ,j = 0, 1 ,(3.7)where L0 is identical to the Virasoro constraint of the KdV model, eq. (3.4), and µ0 andµ1 are two, a priori undetermined, dimensionless integration constants.

They are not,however, independent as we now show. By substituting the expression for ν in terms ofthe ratio τ1/τ0, in the equation for the flows (2.8), we deduce∂∂tklogτ1τ0= −12D⋆Rk.

(3.8)19

This can now be substituted directly in (2.20) to yieldXk≥0(2k + 1)tk 1τ1∂τ1∂tk−1τ0∂τ0∂tk= 0. (3.9)The above, along with eq.

(3.7) implies that µ0 = µ1 ≡µ.Before we discuss the possible meaning of the constant µ, we first present a simpleargument for determining the higher Virasoro constraints.We already know from theconstruction of the KdV constraints, that Lkτ0 = 0 implies Lk+1τ0 = 0. It is also straight-forward to verify that if L0τ0 = µτ0 then L1τ0 = 0, regardless of the value of µ. Thereforewe deduce that τ0 satisfies the infinite set of constraintsLnτ0 = µτ0 δn,0 ,n ≥0 .

(3.10)To find the constraints satisfied by τ1, we notice that τ1 satisfies exactly the same recursionrelation as τ0, that is (3.2), except that u = 2∂2x log τ0 is replaced with ˜u = 2∂2x log τ1.Therefore, the same arguments that were applied to determine the constraints on τ0 willbe applicable to τ1, hence we deduceLnτj = µτj δn,0 ,n ≥0 ,(3.11)for j = 0 and 1. So the mKdV partition function is the product of two factors whichseparately satisfy a set of Virasoro constraints, however, in contrast to the KdV case thereis no L−1 constraint and the L0 constraint includes an undetermined constant.

Notice thatthe requirement that the constraints form a closed algebra under commutation, does notin any way constrain the value of the constant. It is important to realize that τ0 and τ1 arenot independent, in fact they satisfy a whole hierarchy of equations for which (2.25) is butthe first.

So although, at first sight, the mKdV Virasoro constraints look less restrictivethan the KdV constraints, one must bear in mind the additional equations which tie τ0and τ1 together.The appearance of a parameter, which is not determined from the matrix model,seems, at first sight, to be surprising. However, it is not totally unexpected, indeed, suchan occurrence is found at the first critical point of the mKdV model.

At this point, thesquare root of the specific heat, ν, satisfies the Painlev´e II equation [4]ν′′ −2ν3 + xν = 0 . (3.12)This equation is known to admit a one-parameter family of solutions [23].

The actualsolution which describes the matrix model, requires a scaling behaviour ν ∼zξ, as x →∞.Ref. [24] discusses how, for one particular value of the parameter, such a physical solutiondoes exists and is unique.

It would be natural to suggest that the parameter is related toµ, the eigenvalue of the L0 constraint. Indeed, (3.12) admits the trivial solution ν = 0,which corresponds to the situation when µ =116, for which the Virasoro constraints havethe solution τ = 1 (i.e.

τ being the vacuum of the twisted Fock space). Notice that, forthe KdV model, the Virasoro constraints are those of an sl(2, C) vacuum, whereas, for themKdV model, the Virasoro constraints are those of a highest weight state of conformaldimension µ.20

3.2Virasoro Constraints for the ZS HierarchyThe analogous constraints for the ZS hierarchy and string equation were found in [9].Here, we briefly repeat their derivation which leads to Virasoro type constraints for anuntwisted boson.The string equations are (see eq. (2.16))Xk≥0(k + 1)tkFk = 0 ,Xk≥0(k + 1)tkGk = 0 .

(3.13)Consider first the objectsIj =Xk≥0(k + 1)tkH′k+j =Xk≥0(k + 1)tk(gGk+j −fFk+j) . (3.14)They can be reduced, using the hierarchy, to sums involving only Fk, Gk and their deriva-tives, which are related to the string equations.From I0, integrating twice over x and introducing an arbitrary integration constant5α, we getXk≥1(k + 1)tk⟨Ok−1⟩+ αt02= 0(3.15)which will lead to the L−1 constraint.From I1 we getXk≥0(k + 1)tk⟨Ok⟩+ β = 0 ,(3.16)where we have picked up a new dimensionless integration constant, β.

This leads to theL0 constraint.Using a similar procedure, from I2 we getXk≥0(k + 1)tk⟨Ok+1⟩+ α⟨P⟩= 0(3.17)which leads to the L1 constraint.Finally, with a few more steps from I3 we obtainXk≥0(k + 1)tk⟨Ok+2⟩+ ⟨P⟩2 + ⟨PP⟩+ α⟨O1⟩= 0 . (3.18)Whilst the previous equations involve only first order derivatives of the partition function,the equation coming from I3 has second order terms, which fix the function on which theVirasoro constraints act.

In fact, we can write⟨P⟩2 + ⟨PP⟩= (F ′)2 −F ′′ = 1Z ∂2xZ(3.19)5 We will discard all integration constants which would have non-trivial dimension.21

and, therefore, the Virasoro constraints act on the partition function. Since for this modelthe partition function is equal to the tau-function, we find that the Virasoro constraintsact on the tau-function, mirroring the situation for the KdV model.By consistency, the commutator of two constraints should be a new constraint on thepartition function.

Therefore, using [Ln, L1] ≡(n −1)Ln+1 with n ≥2, we get an infiniteset of constraints acting on the partition function.These constraint are the VirasoroconstraintsLnτ ZS = 0 ,n ≥−1(3.20)whereL−1 =Xk≥1(k + 1)tk∂∂tk−1+ αt02L0 =Xk≥0(k + 1)tk∂∂tk+ α24L1 =Xk≥0(k + 1)tk∂∂tk+1+ α ∂∂t0Ln =Xk≥0(k + 1)tk∂∂tk+n+n−2Xk=0∂2∂tk∂tn−k−2+ α∂∂tn−1(3.21)and β = α2/4 has been fixed by the relation [L1, L−1] = 2(L0 + α2/4 −β) = 2L0.3.3Connexion with the Tau-Function FormalismIt was noticed in ref. [9] that the Virasoro constraints (3.21) are those of an untwistedscalar field.

In the convention for whichϕ(z) = q −ip log z + iXn∈Z̸=0anz−nn,(3.22)and [am, an] = nδn+m,0 and [q, p] = i, we have for k ≥0tk =√2k + 1a−k−1,∂∂tk=1√2ak+1 . (3.23)The zero-mode p is related to the integration constant α via p = α/√2.

The conjugatevariable to p does not appear in the Virasoro operators.The fact that the Virasoro constraints are those of an untwisted scalar field is verynatural from the point of view of the tau-function approach, which we explained in section§2.4 based on ref. [20].

Recall that the zero mode of the scalar field of the constructionof ref. [20] must have the quantized values m/√2, for m ∈Z, in order that the vertexoperators have local expansions.

The tau-function can be projected onto eigenspaces ofthe zero-mode, these are precisely the ˜τn which where introduced in §2.4, with√2n beingthe eigenvalue of the zero-mode.22

From eq. (2.29) the partition function of the ZS model is equal to ˜τn, for some fixedhalf-integer n. It is very natural to identify the scalar field of the Virasoro constraintswith the scalar field of the Hirota equations of [20].

We do not yet have a direct proofof this, however, below we present some arguments which support this view. Given thisidentification, one is led to a relation between the parameter n and the integration constantof the Virasoro constraints α:p =α√2 = −√2n,n ∈12Z .

(3.24)The possibility that α is quantized seems to be consistent with the results that we obtainin §3.4 and §3.5, for the KdV and mKdV reductions which require α = 0 and α = −1,respectively6.To substantiate this identification we now show that if ˜τn satisfies theVirasoro constraints eq. (3.21) with α, then τn±1 satisfy the same constraints but withα →α∓2.

We prove this fact following a similar demonstration as the previous paragraph.Let us consider the objectsYj =Xk≥−1(k + 1)tk∂∂tk+j−1 ˜τn±1˜τn. (3.25)and use the string equations, the hierarchy of Hirota equations for the ˜τn’s, and the factthat ˜τn satisfies Virasoro constraints with α, Lm˜τn ≡Lm(α)˜τn = 0, m ≥−1.

ConsideringY0, Y1 and Y2, it is easy to obtainL−1(α ∓2)˜τn±1 = L0(α ∓2)˜τn±1 = L1(α ∓2)˜τn±1 = 0 . (3.26)Again, the L2 constraint is more tricky.

Considering Y3, it is easy to show thatL2(α ∓2)˜τn±1 = (α ∓2)˜τn ∂∂t1∓∂2∂t20 ˜τn±1˜τn∓2˜τn±1∂2xlog ˜τn(3.27)The right hand side of this equation vanishes because of one of the Hirota equations satisfiedby the tau-functions. In particular, (see ref.

[20] pg. 232 (III)1;n,n+1),L2(α ∓2)˜τn±1 = (α ∓2) 1˜τn(D21 ∓D2)˜τn±1 · ˜τn = 0 .

(3.28)The Di’s are operators of the Hirota calculus which are defined in ref. [20] for example.Therefore, we getLm(α)˜τn = Lm(α ∓2)˜τn±1 = 0 ,m ≥−1 .

(3.29)If we now consider p = α/√2 as the zero-mode, ‘momentum’ operator of the scalar field,with p ˜τn = −√2n˜τn, in accordance with (3.24), then its conjugate variable or ‘position’operator is q = −it−1/√2. This is deduced from equation (2.5) and (3.29).6 However, the ‘topological’ point described in ref.

[9] does not seem to require any particularvalue of α.23

The above result (3.29) also implies that the whole Hirota hierarchy admits a ‘master’string equation. It is most suggestively written in terms of the full tau-function ˜τ, forwhich the ˜τn are the projections onto eigenspaces of the zero-mode.

The master stringequation is the m = −1 version of the the following Virasoro constraintsLm˜τ = 0,m ≥−1 ,where the Lm are the Virasoro generators of the bosonic field. So the final set of constraintsare exactly analogous to those for the 1-arc KdV case, the difference being that there onehas a twisted scalar field, whereas here we have an untwisted scalar field.The appearance of the parameter α is rather mysterious, since it was not manifest inthe matrix model.

It seems to label different sectors in the theory which are not connectedby the flows. It is clearly desirable to have a better understanding of its origin and meaning.3.4KDV Reduction of Virasoro ConstraintsWe have already shown how the the ZS hierarchy can be reduced to the KdV hierar-chy, and how the string equations respect the reduction.

On general grounds, one wouldanticipate that this would extend to all the Virasoro constraints, and we now prove this.The reduction involves taking ψ = −et−1 and t2k+1 →0 ∀k. Then u = −ψψ satisfies theKdV hierarchy withtZS2k ≡2−ktKdVk,τ ZS =τ KdV2(3.30)and∂u∂tKdVk= ∂xRk+1 .

(3.31)Notice that the second equation of (3.30) implies that under the reduction, i.e. on thesubspace t2k+1 = 0 ∀k with ψ = −et−1, ZZS →ZKdV, as it should.

Under such reduction,H2k = −H′2k−1:F2k + G2k = (F2k−1 + G2k−1)′ + (g + f)H2k−1 = (g −f)H2k−10 = F2k+1 + G2k+1 = (F2k + G2k)′ + (g + f)H2k⇒((g + f)H2k−1)′ + (g + f)H2k = 0(3.32)but (g + f)′ = 2ψ ′ = 0, and the result follows.We now show that the ZS Virasoro constraints on τ ZS(= ˜τn) reduce to Virasoroconstraints on τ KdV(= τ0) for a precise value of the zero-mode α (or n).Let us firstconsider the equation corresponding to I0, under the reductionXk≥1(2k + 1)t2kH2k + α = −Xk≥1(2k + 1)t2kH′2k−1 + α = 0(3.33)Using the relations with the correlation functions, and integrating twice over t0 = x, wegetXk≥1(2k + 1)t2k⟨O2k−2⟩−αt204= 0(3.34)24

which will produce the reduced L−1 constraint. The reduction of I1 is direct, and leads toXk≥0(2k + 1)t2k⟨O2k⟩+ α24 = 0(3.35)which produces the L0 constraint.

This constraint is also obtained from I2, the L1 con-straint in the ZS hierarchy. The corresponding equation isXk≥0(k + 1)tkH′k+2 + αH′1 + 2H2 = 0⇒Xk≥0(2k + 1)t2kH′′2k+1 + (2 −α)H′1 = 0(3.36)Using the relations with the correlation functions, and integrating three times over t0 = x,we get∂3xXk≥0(2k + 1)t2k⟨O2k⟩+ (1 + α)F= 0(3.37)which, by consistency with eq.

(3.35), requires α = −1. Therefore, the KdV reduction isonly consistent for this value of the, a priori, arbitrary parameter α.Let us now consider the equation corresponding to I3, which, again, will to fix thefunction on which the constraints act:Xk≥0(2k + 1)t2kH′2k+3 + 4⟨PPP⟩⟨P⟩+ 4⟨PP⟩2 + 2H3 + 2(1 −α)⟨PPPP⟩= 0 .

(3.38)In the usual way, we getXk≥0(2k + 1)t2k⟨O2k+2⟩+ ⟨P⟩2 + (1 −α)⟨PP⟩= 0 . (3.39)This equation will lead to the L1 constraint, and, again, the second order terms fix thefunctional on which the Virasoro constraint act.

In this case α = −1, and⟨P⟩2 + (1 −α)⟨PP⟩= ⟨P⟩2 + 2⟨PP⟩=4√ZKdV∂2xpZKdV ≡4τ KdV ∂2xτ KdV . (3.40)Notice that the required value of α is consistent with the quantization proposed in (3.24).Therefore, the reduced Virasoro constraints act on the square root of the partition function,which is the tau-function of the KdV hierarchy, in agreement with [11].In terms of the variables tk ≡tKdVk= 2ktZS2k, the reduced Virasoro constraints are˜Lnτ KdV = 0, with n ≥−1, where the operators ˜Ln are those of eq.

(3.4).25

3.5MKDV ReductionIn the case of the mKdV reduction we have already shown in section §2.2 that G2k+1,F2k and H2k vanish. It is straightforward to verify that under the reduction, i.e.

on thesubspace t2k+1 = 0 ∀k with ψ = e2t−1ψ, ZZS →ZmKdV. Using these results one can applythe reduction directly on the constraints.From I0 (L−1 constraint), we getXk≥1(2k + 1)t2kH2k + α = 0⇒α = 0 .

(3.41)Therefore, the mKdV reduction requires α = 0, which is clearly consistent with the quan-tization of α proposed in (3.24). From I1, we getXk≥0(2k + 1)t2kH′2k+1 + 2H1 = 0 .

(3.42)But,H′2k+1 = 2⟨PPO2k⟩≡−2∂F ′′∂t2k(3.43)and we get the equationXk≥0(2k + 1)t2k∂F ′′∂t2k+ 2F ′′ = 0(3.44)which has the form of an L0 constraint. This equation can also be rewritten asXk≥0(2k + 1)t2k∂ν∂t2k+ ν = 0(3.45)which, after integration, becomes the string equation of the mKdV hierarchy, eq.

(2.20).If we reduce the L0 constraint of the ZS hierarchy itself, we find that the partition ofthe mKdV model satisfiesXk≥0(2k + 1)tmKdVk∂∂tmKdVkZmKdV = 0 . (3.46)Notice that µ, the parameter of the mKdV Virasoro constraints of §3.1, is determined bythe reduction to be116.In a similar way, from I2k we get equations which are identically zero and do not giverise to any constraint.

Instead, from I3 (L2 constraint) we getXk≥0(2k + 1)tmKdVk∂∂tmKdVk+1+ 12∂2∂x2ZmKdV = 0(3.47)26

where tmKdVk= 2ktZS2k. We can write the above constraint in the following way.

Firstly, weexpress the partition function in terms of the tau-functions ZmKdV = τ0τ1. Then we usethe relation (2.25) to write∂2x(τ0τ1) = 2τ ′′0 τ1 + 2τ0τ ′′1 ,(3.48)from which we deduce that (3.47) may be rewritten as(L1τ0) τ1 + τ0 (L1τ1) = 0 ,(3.49)which is clearly a consequence of the L1 constraints for τ0 and τ1 that we found for themKdV model in §3.1.One could carry on this process of reducing the higher Virasoro constraints.Theresulting constraints would act directly on the partition function of the mKdV model; andhence would not be Virasoro constraints.

Nevertheless, we expect that the constraintson the partition function should be expressible in terms of Virasoro constraints acting oneach tau-function separately, as we found for above for L1. In any case, one would alwaysfind an expression which was compatible with the mKdV Virasoro constraints: this is dueto the fact that the mKdV Virasoro constraints follow directly from the mKdV stringequation and the recursion relations of the hierarchy, which are obtained by the reductionfrom the ZS string equation and hierarchy.4.Discussion and Open ProblemsIn this paper we have attempted to analyse all the possible double scaling limits ofthe hermitian and anti-hermitian 1-matrix models.

As it is clear from the fact that eq. (1.3), after the double scaling limit, gives rise to a differential operator of degree two in x,the hermitian and anti-hermitian 1-matrix models are related to the sl(2, C) hierarchies.Hermitian and anti-hermitian matrix models have many common properties: in the 1-arcsector they both give rise to a KdV hierarchy for an even potential and to a doubled KdVhierarchy for a general potential; in the 2-arc sector, with even potential, they both giverise to the mKdV hierarchy.

Instead they differ in the 2-arc sector with a general potentialwhere the hermitian models give rise to only half of the critical points associated to theNLS hierarchy whereas the anti-hermitian model gives all the critical points associated tothe ZS hierarchy (except for the ‘Topological’ one).For the hermitian models the multi-critical points obtained, although described by aNLS hierarchy, actually have solutions which are described by a mKdV hierarchy. We donot know whether these solutions are the only ones.

Furthermore, these critical pointscorrespond to purely even potentials; in this sense the NLS structure is irrelevant, and oneis really dealing with a mKdV structure. Instead the ZS hierarchy admits also a reductionto KdV, in other words in the 2-arc sector of the anti-hermitian matrix model we found anew series of multi-critical points described by the KdV hierarchy, besides the one alreadyknown from the 1-arc sector.

This set of KdV multi-critical points are not in any simpleway connected with those in the 1-arc sector, since, for example, the topological critical27

point describing topological gravity [14], which cannot be obtained from the 1-arc sector, isobtained from the 2-arc sector with anti-hermitian matrices with a fourth order potential.The situation in the 2-arc sector with anti-hermitian matrices and a general potentialseems to be the richest, being described by the ZS hierarchy. The ‘even’ multi-criticalpoints admit solutions described by KdV and mKdV hierarchies, which require two par-ticular values of the parameter α.

Clearly, it would desirable to understand the rˆole ofthe parameter α, from the point of view of the matrix model, and also to know whetherthe KdV and mKdV solutions exhaust the possible solutions for the ‘even’ multi-criticalpoints. For instance, are there other solutions for different values of α, and do solutionsexist only for the discrete values suggested by the tau-function formalism?

An interest-ing open question regards the nature of the ‘odd’ multi-critical points of the ZS stringequation. It is now well-known that solutions for the KdV and mKdV systems describ-ing multi-critical behaviour exist [1,12,24]; we do not have any arguments to show thatsolutions can be found for the ‘odd’ scaling points of the ZS hierarchy, except for the first(or ‘topological’) point, corresponding to t1 ̸= 0, which was investigated in [9].

This pointcannot be obtained from the matrix model and, as described in ref. [9], could give riseto a new kind of ‘topological’ theory.

Anyway, since the mathematical apparatus existsfor tackling the issue of the existence of solutions to these non-linear differential equations[15], we hope these questions will be addressed and solved elsewhere.One of the results of this paper is the realization of the rather universal nature of theVirasoro constraints and the fact that they act on the tau-functions of the appropriatehierarchy, and not, necessarily, directly on the partition function. For the KdV and ZShierarchies, the Virasoro constraints areLnτ = 0n ≥−1 ,(4.1)where in both cases τ is an element of the basic representation of the Kac-Moody algebraA(1)1 , for the twisted and untwisted constructions, respectively.

In the mKdV case thepartition function is the product of two tau-functions, which arise from the two basicrepresentations of A(1)1 , which both satisfy the Virasoro constraints of a highest weightvector:Lnτj = µτj δn,0,(4.2)for j = 0 and 1.Near the completion of this work our attention was drawn to refs. [25,26] which discusssimilar Virasoro constraints to those above.AcknowledgmentsWe would like to thank ˇC.

Crnkovi´c, M. Douglas and G. Moore for having sent us thepreprint [9] prior to publication. We would like to thank M. Newman for explaining hisrecent work with D. Gross [25].

The research of T.H. is supported by NSF PHY90–21984,that of J.M.

by a Fullbright/MEC fellowship and that of A.P. by an INFN fellowship.

C.N.is partially supported by the Ambrose Monell Foundation.28

References[1] F. David, “Planar diagrams, two-dimensional lattice gravity and surface models”,Nucl. Phys.

B257 [FS14] (1985) 45.V.A. Kazakov, I.K.

Kostov and A.A. Migdal, “Critical properties of randomly trian-gulated planar random surfaces”, Phys. Lett.

157B (1985) 295.I.K. Kostov and M.L.

Mehta, “Random surfaces of arbitrary genus: exact results ford=0 and -2 dimensions”, Phys. Lett.

189B (1987) 118.M. Douglas and S. Shenker, “Strings in less than one dimension”, Nucl.

Phys. B355(1990) 635.D.

Gross and A. Migdal, “A non-perturbative treatment of two-dimensional quantumgravity”, Nucl. Phys.

B340 (1990) 333.E. Brezin and V.A.

Kazakov, “Exactly solvable field theories of closed strings”, Phys.Lett. 236B (1990) 144.

[2] E. Witten, “Two dimensional gravity and intersection theory on moduli space”,preprint IASSNS-HEP-90/45, May, 1990. [3] C. Bachas and P. Petropoulos, “Doubling of equations and universality in matrixmodels of random surfaces”, Phys.

Lett. 247B (1990) 363.

[4] ˇC. Crnkovi´c and G. Moore, “Multi-critical multi-cut matrix models”, Phys.

Lett.257B (1991) 322. [5] M. Douglas, N. Seiberg and S. Shenker, “Flow and instability in quantum gravity”,Phys.

Lett. B244 (1990) 381.P.

Mathieu and D. S´en´echal, “A well-defined multi-critical series in hermitian onematrix models”, preprint LAVAL-PHY-21/91, February 1991. [6] C. Nappi, “Painlev´e II and odd polynomials”, Mod.

Phys. Lett.

A5 (1990) 2773.P. Petropoulos, “Doubling versus non-doubling of equations and phase space structurein the one-hermitian-matrix models”, Phys.

Lett. 261B (1991) 402.

[7] M. Douglas, Strings in less than one dimension and the generalized KdV hierarchies”,Phys. Lett.

238B (1990) 176. [8] S. Dalley, C. Johnson and T. Morris, “Multicritical complex matrix models andnon-perturbative 2d quantum gravity”, preprint SHEP 90/91-16, February 1991.

[9] ˇC. Crnkovi´c, M. Douglas and G. Moore, “Loop equations and the topological phaseof multi-cut matrix models”, preprint YCTP-P25-91, RU-91-36, August 1991.

[10] See for example: A. Das, “Integrable Models”, Lectures Notes in Physics, Vol. 30,World Scientific, Singapore.

[11] R. Dijkgraaf, E. Verlinde and H. Verlinde, “Loop equations and Virasoro constraintsin non perturbative 2d quantum gravity”, Nucl. Phys.

B348 (1991) 435.M. Fukuma, H. Kawai and R. Nakayama, “Continuum Schwinger-Dyson equationsand universal structures in 2-dimensional quantum gravity”, Int.

J. Mod. Phys.

A6(1991) 1385. [12] ˇC.

Crnkovi´c, M. Douglas and G. Moore, “Physical solutions for unitary-matrix mod-els”, Nucl. Phys.

B360 (1991) 507.29

[13] V. Periwal and D. Shevitz, “Unitary-matrix models as exactly solvable string theo-ries”, Phys. Rev.

Lett. 64 (1990) 1326; “Exactly solvable unitary matrix models:multi-critical potentials and correlations”, Nucl.

Phys. B344 (1990) 731.

[14] E. Witten, “On the Structure of the Topological Phase of two dimensional Gravity”,Nucl. Phys.

B340 (1990) 281.R. Dijkgraffand E. Witten, “Mean field theory, topological field theory andmulti-matrix models”, Nucl.

Phys. B342 (1990) 486.J.

Distler, “2d quantum gravity, topological field theory and multi-critical matrixmodels”, Nucl. Phys.

B342 (1990) 523. [15] G. Moore, “Geometry of the string equations”, Comm.

Math. Phys.

133 (1990) 261;“Matrix models of 2d gravity and isomonodromic deformations”, in Common Trendsin Mathematics and Quantum Field Theories, Prog. Theor.

Phys. Suppl.

102 (1990)255. [16] M. Kontsevich, “Intersection theory on the space of curve moduli”, preprint 1990.E.

Witten, “On the Kontsevich model and other models of two dimensional gravity”,preprint IASSNS-HEP-91/24, June, 1991. [17] R.C.

Penner, “Perturbative series and the moduli space of Riemann surfaces”, J. Diff.Geom. 27 (1988) 35.J.

Harer and D. Zagier, “The Euler characteristic of the moduli space of curves”,Invent. Math.

85 (1986) 457.J. Distler and C. Vafa, “A critical matrix model at c=1”, Mod.

Phys. Lett.

A6(1991) 259.C. Itzykson and J.-B.

Zuber, “Matrix integration and the combinatorics of the modulargroup”, Comm. Math.

Phys. 134 (1990) 197.

[18] V.G. Drinfel’d and V.V.

Sokolov, “Lie algebras and equations of Korteweg-de Vriestype”, J. Sov. Math.

30 (1985) 1975. [19] V.E.

Zakharov and A.B. Shabat, “A scheme of integrating non-linear equations ofmathematical physics by the method of the inverse scattering problem”, Funkts.

Anal.Pril. 8 No.

3 (1974) 54. [20] V.G.

Kac and M. Wakimoto, “Exceptional Hierarchies of Soliton Equations”, Pro-ceedings of Symposia in Pure Mathematics 49 (1989) 191. [21] I.M.Gel’fandandL.A.Dikii,“AsymptoticbehaviouroftheresolventofSturm-Louville equations and the algebra of the Korteweg-de Vries equation”, Russ.Math.

Surv. 30 (1975) 77; “Fractional Powers of Operators and Hamiltonian Sys-tems”, Funkts.

Anal. Pril.

10 (1976) 13. [22] B.A.

Kupershmidt and G. Wilson, “Modifying Lax Equations and the Second Hamil-tonian Structure”, Invent. Math.

62 (1981) 403. [23] S.P.

Hastings and J.B. McLeod, “A boundary value problem associated with thesecond Painlev´e transcendent and the Korteweg-de Vries equation”, Arch. Rat.

Mech.and Anal. 73 (1980) 31.

[24] A. Watterstam, “A solution to the string equation of unitary matrix models”, Phys.Lett. 263B (1991) 51.

[25] D.J. Gross and M. Newman, “Unitary and Hermitian matrices in an external field (2):the Kontsevich model and continuum Virasoro constraints”, preprint PUPT-1282,30

September 1991. [26] S. Dalley, C. Johnson and T. Morris, “Non-perturbative two-dimensional QuantumGravity”, preprint SHEP 90/91-28, June 1991.31

---

출처: arXiv:9109.046 • 원문 보기