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The Solution Space of the Unitary Matrix

arXiv:hep-th/9112066v1 21 Dec 1991SU-4238-497NSF-ITP-91-133The Solution Space of the Unitary MatrixModel String Equation and theSato GrassmannianKonstantinos N. Anagnostopoulos1 and Mark J. Bowick1Physics DepartmentSyracuse UniversitySyracuse, NY 13244-1130, USAAlbert Schwarz2Department of MathematicsUniversity of CaliforniaDavis, CA 95616, USAAbstractThe space of all solutions to the string equation of the symmetric unitary one-matrixmodel is determined. It is shown that the string equation is equivalent to simple conditionson points V1 and V2 in the big cell Gr(0) of the Sato Grassmannian Gr.

This is a conse-quence of a well-defined continuum limit in which the string equation has the simple form[P, Q−] = 1, with P and Q−2 × 2 matrices of differential operators. These conditions onV1 and V2 yield a simple system of first order differential equations whose analysis deter-mines the space of all solutions to the string equation.

This geometric formulation leadsdirectly to the Virasoro constraints Ln (n ≥0), where Ln annihilate the two modified-KdVτ-functions whose product gives the partition function of the Unitary Matrix Model.December 20, 19911E-mail: Konstant@suhep.bitnet; Bowick@suhep.bitnet.2E-mail: Asschwarz@ucdavis.edu

1. IntroductionMatrix models form a rich class of quantum statistical mechanical systems defined bypartition functions of the formRdM e−Nλ trV (M), where M is an N × N matrix and theHamiltonian trV (M) is some well defined function of M. They were originally introducedto study complicated systems, such as heavy nuclei, in which the quantum mechanicalHamiltonian had to be considered random within some universality class [1,4] .Unitary Matrix Models (UMM), in which M is a unitary matrix U, form a particularlyrich class of matrix models.

When V (U) is self adjoint we will call the model symmetric.The simplest case is given by V (U) = U + U † and describes two dimensional quantumchromodynamics [5–7] with gauge group U(N). The partition function of this theory canbe evaluated in the large-N (planar) limit in which N is taken to infinity with λ = g2Nheld fixed, where g is the gauge coupling.

The theory has a third order phase transitionat λc = 2 [6]. Below λc the eigenvalues eiαj of U lie within a finite domain about α = 0 ofthe form [−αc, αc] with αc < π.

The size of this domain increases as λ increases until theeigenvalues range over the entire circle at λ = λc.In the last two years, matrix models have received extensive attention as discretemodels of two dimensional gravity.In this context, the one-matrix Hermitian MatrixModels (HMM), in which M is a Hermitian matrix, are the clearest to interpret since agiven cellular decomposition of a two dimensional surface is dual to a Feynman diagramof a zero dimensional quantum field theory with action trV (M). In the double scalinglimit of these models, the potential can be tuned to a one parameter family of multicriticalpoints labelled by an integer m. This scaling limit is defined by N going to infinity andλ →λc with t = (1 −nN )N2m2m+1 and y = (1 −λλc )N2m2m+1 held fixed.This requiressimultaneously adjusting m couplings in the potential to their critical values.

At thesemulticritical points the entire partition function (including the sum over topologies) isgiven by a single differential equation (the “string equation”) and can serve as a non-perturbative definition of two dimensional gravity coupled to conformal matter [8–11].This multicriticality may also be described by universal cross-over behaviour in the tail ofthe distribution of the eigenvalues [12].UMM have also been solved in the double scaling limit [13–17] and their generalfeatures are very similar to the HMM. At finite N they exhibit integrable flows in theparameters of the potential similar to the HMM [18–21] and in the double scaling limitthey lie in the same universality class as the double-cut HMM [20–23].

The world sheet1

interpretation of the UMM is not, however, very clear [22].In view of this it seemsworthwhile to explore their structure further.It is well known [24] that the string equation of the (p, q) HMM can be described asan operator equation [P, Q] = 1, where P and Q are scalar ordinary differential operatorsof order p and q respectively. They are the well defined scaling limits of the operators ofmultiplication and differentiation by the eigenvalues of the HMM on the orthonormal poly-nomials used to solve the model.

The set of solutions to the string equation [P, Q] = 1 wasanalyzed in [25] by means of the Sato Grassmannian Gr. It was proved that every solutionof the string equation corresponds to a point in the big cell Gr(0) of Gr satisfying certainconditions.

This fact was used to give a derivation of the Virasoro and W-constraintsobtained in [26,27] along the lines of [28–31] and to describe the moduli space of solutionsto this string equation. The aim of the present paper is to prove similar results for theversion of the string equation arising in the UMM.

It was shown in [32] that the stringequation of the UMM takes the form [P, Q−] = const., where for the kth multicritical pointP and Q−are 2 × 2 matrices of differential operators of order 2k and 1 respectively. Forevery solution of the string equation one can construct, with this result, a pair of pointsof the Gr(0) obeying certain conditions.

These conditions lead directly to the Virasoroconstraints for the corresponding τ-functions and give a description of the moduli space ofsolutions. We stress that the above results depend solely on the existence of a continuumlimit in which the string equation has the form [P, Q−] = const.

and the matrices of differ-ential operators P and Q−have a particular form to be discussed in detail in subsequentsections. Our results do not depend on other details of the underlying matrix model.The paper is organized as follows.

In section 2 we review the double scaling limit ofthe UMM in the operator formalism [32]. Since the square root of the specific heat flowsaccording to the mKdV hierarchy we note that its Miura transforms flow according to KdVand thus give rise to two τ-functions related by the Hirota bilinear equations of the mKdVhierarchy [33–35].

In section 3 we derive a description of the moduli space of the stringequation in terms of a pair of points in Gr(0) related by certain conditions. In section 4 weshow the correspondence between points in Gr(0) and solutions to the mKdV hierarchy.The Virasoro constraints are derived from invariance conditions on the points of Gr(0)along the lines of [28,29] .

This is most conveniently done in the fermionic representationof the τ-functions of the mKdV hierarchy. Finally in section 5 we determine the modulispace of the string equation.

It is found to be isomorphic to the two fold covering of the2

space of 2 × 2 matricesPij(z), where Pij(z) are polynomials in z such that P01(z) andP10(z) are even polynomials having equal degree and leading terms and P00(z) and P11(z)are odd polynomials of lower degree satisfying the conditions P00(z) + P11(z) = 0.2. The Symmetric Unitary Matrix ModelIn this paper we will study the UMM defined by the one matrix integralZUN =ZDU exp{−Nλ Tr V (U + U †)} ,(1)where U is a 2N × 2N or a (2N + 1) × (2N + 1) unitary matrix, DU is the Haar measurefor the unitary group and the potentialV (U) =Xk≥0gk U k ,(2)is a polynomial in U.

As standard we first reduce the above integral to an integral overthe eigenvalues [6,36] zi of U which lie on the unit circle in the complex z plane.ZUN =Z{Yjdzj2πizj} |∆(z)|2exp{−NλXiV (zi + z∗i )} ,(3)where ∆(z) = Qk

The polynomials c±n (z) are orthogonal with respect to theinner product⟨c+n , c+m⟩=Idz2πiz exp{−Nλ V (z + z∗)} c+n (z)∗c+m(z)= eφ+n δn,m ,⟨c−n , c−m⟩= eφ−n δn,m ,⟨c+n , c−m⟩= 0 . (5)3

The expression for the Vandermonde determinant is|∆(z)|2 =detc−i (zj)c+i (zj) 2,(6)where j = 1, . .

., 2N, i = 12, 32, . .

., N −12 for U(2N) and j = 1, . .

., 2N + 1, i = 0, 1, . .

., Nfor U(2N + 1) (where the line c−0 (z) ≡0 is understood to be omitted). Then the partitionfunction of the model is given by the product of the norms of the orthogonal polynomials[19]ZUN =Yneφ+n eφ−n = τ (+)N τ (−)N.(7)In constructing the continuum limit of the UMM we will also need the orthonormal func-tionsπ±n (z) = e−φ±n /2e−N2λ V (z+)c±n (z)(8)such that⟨π+n (z), π+m(z)⟩=Idz2πiz π+n (z)∗π+m(z)= δn,m ,⟨π−n (z), π−m(z)⟩= δn,m ,⟨π+n (z), π−m(z)⟩= 0 .

(9)The action of the operators z± = z ± 1z and z∂z on the π±n (z) basis is given by finite termrecursion relations [19,32]z+ π±n (z) =qR±n+1π±n+1(z) −r±n π±n (z) +pR±n π±n−1(z) ,z−π±n (z) =qQ∓n+1π∓n+1(z) −q±nsQ∓nR±nπ∓n (z) −qQ±n π∓n−1(z) ,z∂zπ±n (z) = −N2λkXr=1(v±z )n,n+rπ∓n+r(z) +nsQ∓nR±n−N2λ(v±z )n,nπ∓n (z)+ N2λkXr=1(v±z )n,n−rπ∓n−r(z) ,(10)where R±n = eφ±n −φ±n−1, Q±n = eφ±n −φ∓n−1, r±n = ∂φ±n∂g1 , q±n =(Q±n+1−Q±n )+(R∓n+1−R±n )r±n −r∓n, and(v±z )n,n−r =Idz2πiz π∓n−r(z)∗{z∂zV (z+)} π±n (z) .4

The double scaling limit corresponding to the kth multicritical point is defined by N →∞and λ →λc, with t = (1 −nN )N2k2k+1 , y = (1 −λλc )N2k2k+1 held fixed. It was shown in [32]that the operators z± and z∂z have a smooth continuum limit given byz+ →2 + N −22k+1 Q+ ,z−→−2N −12k+1 Q−,z∂z →N12k+1 Pk ,(11)where Q± are given byQ−=0∂+ v∂−v0,Q+ =(∂+ v)(∂−v)00(∂−v)(∂+ v)= Q2−,(12)and Pk byPk =0PkP†k0.

(13)Here ∂≡∂∂x and x = t + y. The scaling function v2 is proportional to the specific heat−∂2 ln Z of the model.

The operators Pk are differential operators of order 2k. The sameassertions hold if we introduce sources t2k+1(t1 ≡x) and deform the kth multicriticalpotential Vk to Vk(z) −Plt2l+1Vl(z)N2(k−l)2k+1 .

From [z∂z, z−] = z+ it follows that[Pk, Q−] = 1 ,(14)where Q−has the form (12) and Pk has the form (13). We stress here that this equationholds for the system perturbed away from the multicritical points as well as exactly atmulticriticality.Our main aim is to study equation (14) - the string equation for theUMM.For completeness we will present here some information about the solutions of (14)that was obtained in [32] (or follows from the same analysis).

Most of these facts will alsofollow from the results of Sections 3-5; the reader may go directly to these sections.It is proved in [32] that Pk are given at the kth multicritical point byPk = ˜Pk −x ,(15)where˜Pk = a−1k {(∂+ v)[(∂−v)(∂+ v)]k−1/2}+ ,(16)5

and a−1k= 2(2k +1)kPl=1(−1)l l2kB(k+1,k+1)Γ(k−l+1)Γ(k+l+1). Here Ψ+ denotes the differential part ofa pseudodifferential operator Ψ.

One can give the corresponding expression P = −Pl≥1(2l+1)t2l+1 ˜Pl −x for perturbations from the kth multicritical point. These expressions can beused to get an ordinary differential equation for the specific heat v in the formˆDRk[u] = akvx ,(17)where ˆD = ∂+2v, u = v2 −v′, and Rk[u] are the Gel’fand-Dikii potentials defined throughthe recursion relation∂Rk+1[u] =14∂3 −12(∂u + u∂)Rk[u] ,R0[u] = 12 .

(18)In the non-critical model the analogous equation isXl≥1(2l + 1)t2l+1 ˆDRl[u] = −vx . (19)The equation [z∂z, z+] = z−in the continuum limit becomes [Pk, Q+] = 2Q−and isconsistent with the relation Q2−= Q+.Equation (17) is closely related to the mKdV hierarchy.

Indeed, by slightly modifyingthe calculations of [22,23], one can show that v flows according to the mKdV hierarchy∂v∂t2k+1= −∂ˆDRk[u] . (20)By introducing scaling operators⟨σk⟩=∂∂t2k+1ln Z(21)one can show that⟨σkσ0σ0⟩= 2v∂ˆDRk[u] .

(22)Then ⟨σ0σ0⟩= −v2 and ⟨σkσ0σ0⟩=∂∂t2k+1 ⟨σ0σ0⟩imply equation (20).If v flows according to mKdV, then the functions u1 = v2 + v′ and u2 ≡u = v2 −v′will flow according to KdV, being related to v by the Miura transformation. The flows ofu1 and u2 have associated τ-functions τ1 and τ2 such thatu1 = −2∂2 ln τ1 ,u2 = −2∂2 ln τ2 .

(23)6

Thenv2 = −∂2 ln (τ1τ2) ,v = ∂ln τ2τ1(24)The Miura transformation u1 = v2 + v′ yields the simplest bilinear Hirota equation of themKdV hierarchy [33–35], namelyD2 τ1 · τ2 = τ ′′1 τ2 −2τ ′1τ ′2 + τ1τ ′′2 = 0(25)where D denotes the Hirota derivative. The structure of this hierarchy will be examinedfurther in section 4.

Note that (24) shows that the partition function Z of the UMM isgiven byZ = τ1 · τ2(26)with the two mKdV τ functions being related by (25) .3. The Sato GrassmannianThe partition function of the UMM was shown in Section 2 to be the product of twomKdV τ-functions τ1 and τ2.

As will be explained in Section 4, any τ function that canbe represented by a formal power series corresponds to a point of the big cell of the SatoGrassmannian Gr(0). It will be shown that the mKdV flows can be described by the flowsof two points V1, V2 ∈Gr(0) that are related by certain conditions preserved by the flows.The string equation will impose further conditions that will pick out a unique pair (V1, V2).It will further impose constraints on the τ-functions, which turn out to be the expectedVirasoro constraints [22,23].

The treatment described here follows closely that for the caseof the HMM [25–31].Consider the space of formal Laurent seriesH = {Xnanzn ,an = 0forn ≫0 }and its decompositionH = H+ ⊕H−,where H+ = { Pn≥0anzn ,an = 0forn ≫0 }. Then the big cell of the Sato Grass-mannian Gr(0) consists of all subspaces V ⊂H comparable to H+, in the sense that thenatural projection π+ : V →H+ is an isomorphism.7

Consider the space Ψ of pseudodifferential operators W = Pi≤kwi(x)∂i where thefunctions wi(x) are taken to be formal power series (i.e. wi(x) = Pk≥0wikxk , wik = 0 , k ≫0).

W is then a pseudodifferential operator of order k. It is called monic if wk(x) = 1and normalized if wk−1(x) = 0. The space Ψ forms an algebra.

The space of monic,zeroth-order pseudodifferential operators forms a group G.There is a natural action of Ψ on H defined byxm∂n : H →Hφ →(−ddz )m(z)n φ .Then it is well known [38] that every point V ∈Gr(0) can be uniquely represented in theform V = SH+ with S ∈G. This will imply that for every operator Q−we can uniquelyassociate a pair of points V1, V2 ∈Gr(0).Indeed, consider S1 and S2 ∈G such thatˆSQ−ˆS−1 = ˜Q−(27)whereˆS =S100S2, ˜Q−=0∂∂0.

(28)ThenS1(∂+ v)S−12= ∂,S2(∂−v)S−11= ∂,(29)which imply thatS1(∂2 −u1)S−11= ∂2u1 = v2 + v′ ,S2(∂2 −u2)S−12= ∂2u2 = v2 −v′ . (30)The existence of S1 ∈G follows from the general fact [39] that for every monic normalizedpseudodifferential operator L of order n there exists an S such that SLS−1 = ∂n.Given S1, one can determine S2 fromS1(∂+ v) = ∂S2.By taking formal adjoints of (29) and (30), it is easy to show that S1 and S2 be madesimultaneously unitary.

Indeed, from (30) we obtain(S−11 )†(∂2 −˜u)S†1 = ∂2 ⇒(S1S†1)−1∂2(S1S†1) = ∂2 ⇒S1S†1 = f(∂2) ,(31)8

where f is arbitrary. Similarly S2S†2 = g(∂2).

But since (27) implies( ˆS ˆS†)−10∂∂0( ˆS ˆS†) =0∂∂0,(32)then( ˆS ˆS†) =f(∂2)00g(∂2)gives∂g = f∂,∂f = g∂,or, f = g. Therefore S1 and S2 can be simultaneously chosen to be unitary, i.e S1S†1 = 1and S2S†2 = 1.Since V ⊂Gr(0) is given uniquely by V = SH+, the operator Q−determines twospaces V1 = S1H+ and V2 = S2H+. Conversely given spaces V1 and V2 determine Q−uniquely.

The operator Q−, however, is a differential operator and V1, V2 cannot be arbi-trary. Indeed, since every differential operator leaves H+ invariant, we obtain(∂+ v) H+ ⊂H+ ⇔S−11 ∂S2 H+ ⊂H+⇔∂V2 ⊂V1⇔z V2 ⊂V1(33)Similarly, z V2 ⊂V1.The string equation will impose further conditions on V1 and V2.

After transformationwith the operator ˆS equation (14) becomes[ ˜P(k), ˜Q−] = 1(34)where ˜P(k) = ˆSP(k) ˆS−1. The solution to (34) is˜P(k) =0−x + ˜fk(∂)−x + ˜fk(∂)0(35)which gives P(k) = S−11−x + ˜fk(∂)S2 and P†(k) = S−12−x + ˜fk(∂)S1.

Consistencyrequires therefore that −x + ˜fk(∂) must be self adjoint ˜fk(∂) = fk(∂2).For the kthmulticritical point P(k) is a differential operator of order 2k. Therefore fk(∂2) = ∂2k +.

. ..By using the freedom to redefine Si by a monic, zeroth-order, pseudodifferential operator9

R = 1+ Pi≥1ri∂−i with constant coefficients ri, it is easy to show that all negative powers infk(∂2) may be eliminated. The proof shows that all powers below ∂−1 can be eliminatedby R, and a ∂−1 term is forbidden by self-adjointness.

Thereforefk(∂2) = ∂2k +X1≤i≤kfi(x)∂2(k−i)(36)By Fourier transforming, the action of ˜P on H is represented by˜P(k) =0AkAk0, whereAk = ddz +kXi=0αiz2iand αi =const. (37)Given the constants αi, we can calculate the operator P(k).Since S2(∂−v)(∂+v)S−12= ∂2 implies S2[(∂−v)(∂+ v)]i−12 S−12= ∂2i−1 then using S1(∂+ v)S−12= ∂weobtainS1(∂+ v)[(∂−v)(∂+ v)]i−12 S−12= ∂2i .

(38)Transforming back to H+ we obtainP(k) = S−11 (−x +kXi=0αi∂2i)S2= S−11 (−x + α0)S2 +kXi=1αiS−11 ∂2iS2= S−11 (−x + α0)S2 +kXi=1αi(∂+ v)[(∂−v)(∂+ v)]i−12(39)Comparing with (16) and since S−11 xS2 = x + Pi≥1qi(x)∂−i, we conclude that at the kthmulticritical point, αk = 1 and αi = 0 for i < k. Moreover, by perturbing away from themulticritical points we see thatαi(t) = −(2i + 1)t2i+1 . (40)The requirement that P be a differential operator is equivalent to the conditionsAk V1 ⊂V2 and Ak V2 ⊂V1.

The space of solutions to the string equation is the space ofoperators Q−such that there exists P(k) with [P(k), Q−] = 1. We conclude that this spaceis isomorphic to the set of elements V1, V2 ⊂Gr(0) that satisfy the conditions:z V1 ⊂V2z V2 ⊂V1Ak V1 ⊂V2Ak V2 ⊂V1(41)10

for some Ak =ddz +kPi=0αiz2i.It is now easy to show that the string equation is compatible with the mKdV flows(20). We will show in the next section that the mKdV flows for the scaling function v areequivalent to the condition∂∂t2k+1Vi = z2k+1 Vi(i = 1, 2) .

(42)Then Vi(t) = exp{Pkt2k+1z2k+1}Vi ≡γ(t, z)Vi and (41) implyz γ(z, t)V1 ⊂γ(t, z)V2 ⇒z V1(t) ⊂V2(t)Ak(t) γ(z, t)V1 ⊂γ(t, z)V2 ⇒Ak(t) V1(t) ⊂V2(t) ,(43)whereAk(t) ≡γAkγ−1 = Ak −Xk(2k + 1)t2k+1z2k(44)and analogous equations with V1 and V2 interchanged. This is clearly consistent with (40).From (41) we see that z2, zA and A2 leave V1,2 invariant.

In the next section we showthat this fact implies Virasoro constraints for the τ-functions associated with the mKdVflows of the UMM.4. The mKdV τ-functions and the Virasoro constraintsIn this section we will describe the τ-function formalism for the mKdV system andgive a derivation of the Virasoro constraints on the τ-functions of the UMM.

These willbe derived from the invariance conditions (41) on the spaces V1 and V2 following the linesof [28,29] for the HMM. The idea is to transform the Virasoro generators into fermionicoperators in the fermionic representation of GL(∞) using the boson-fermion equivalence.Then using the correspondence between GL(∞)-orbits of the vacuum and Gr(0), annihi-lation of the τ-function by the Virasoro constraints Ln is shown to be equivalent to theinvariance of V ∈Gr(0) under the action of operators z2nAKdV .

In [25,30], it was shownthat AKdV was nothing but the operator P of the HMM acting on Gr(0), and the Virasoroconstraints were proved from the string equation. We summarize below these results andderive the Virasoro constraints for the UMM from the conditions (41).First we introduce the fermionic representation of GL(∞) on the Fock space F of freefermions.

The fermionic operators are defined to satisfy the anticommutation relations{ψi, ψ†j} = δij ,{ψi, ψj} = {ψ†i , ψ†j} = 0(i ∈Z) . (45)11

The vacuum |0 > satisfiesψi|0 >= 0fori > 0 ,ψ†i |0 >= 0fori ≤0 ,(46)and the states ( m > 0)|m >= ψ†m . .

. ψ†1|0 > ,| −m >= ψ−m+1 .

. .

ψ0|0 >(47)are the filled states with charge m and −m respectively. The operators ψ†i and ψi havebeen assigned charges 1 and −1 respectively and the vacuum |0 > charge 0.

The normalordering is defined by: ψ†i ψj : = ψ†i ψj−< ψ†i ψj >=ψ†i ψji > 0−ψjψ†ii ≤0(48)Then the fermionic representation of the algebra gl(∞) is defined by 1rF (a)|χ >=Xi,j: ψ†i aijψj : |χ >a ∈gl(∞)|χ >∈F(49)and of the group GL(∞) byRF (g)ψ†i1ψ†i2 . .

.ψi1ψi2 . .

.| −m >=(ψ†g)i1(ψ†g)i2 . .

. (gψ)i1(gψ)i2 .

. .| −m >(50)for m ≫0 such that (ψ†g)−j = ψ†−j for j > m. In (50), g ∈GL(∞) and (ψ†g)i ≡ψ†jgjiand (gψ)i ≡gijψj.

The above representation conserves the charge and therefore preservesthe decompositionF = ⊕m∈ZF(m)where F(m) is the space of states with charge m. The first step in order to establish theboson-fermion correspondence is to define the current operatorsJn =Xr∈Z: ψ†n−rψr :n ∈Z(51)1Note that this representation of gl(∞) and GL(∞) is equivalent to the infinite wedge repre-sentation [34].12

which satisfy the bosonic commutation relations[Jm, Jn] = mδm,−n . (52)Then we define an isomorphism σ : F →B where the bosonic Fock space B = ⊕m∈ZB(m) ∼=C[t1, t2, .

. ., ; u, u−1] of polynomials in t1, t2, .

. .

, ; u, u−1 by the requirementσ|m >= um ,σJnσ−1 =∂∂tn(n ≥0)σJnσ−1 = −nt−n (n < 0) . (53)Then the state |χ >∈F is represented in B byτ χ(t; u, u−1) =Xm∈Zum < m|ePp≥1 tpJp|χ >≡Xm∈Zumτ χm(t)(54)Note that σ = ⊕m∈Zσm, where σm : F(m) →B(m) ∼= umC[t1, t2, .

. .] and τ(t) =⊕m∈Zτm(t).Then one observes that if the state |g >0 belongs to the GL(∞) orbit of the vacuum(i.e.

|g >0= g|0 > for some g ∈GL(∞)), then Pj∈Zψ†j|g >0 ⊗ψj|g >0= 0 leads to thebilinear Hirota equations for the τ-functions of the KP hierarchy (see [33–35] for details).The KP τ-function belongs to the GL(∞) orbit of the vacuum and is given byτ =< 0|ePp≥1 tpJpg|0 > ∈GL(∞) · 1 . (55)Similar considerations apply for the kth modified KP (mKP) hierarchy.

This is definedby the equation Pj∈Zψ†j|g >k ⊗ψj|g >0 = 0 where |g >k belongs to the GL(∞) orbit ofthe state |k > of (47). Kac and Peterson [33] showed that this is equivalent to the mKPτ-function τ(t) = τk(t) ⊕τ0(t) lying on the GL(∞) orbit of |k > ⊕|0 >.One can go further and observe that the Kac-Moody algebra of sln (thought of asˆslnn, C[u, u−1]) when embedded in gl(∞) has irreducible highest weight representationson the space B(n) = ⊕n−1m=1B(m)(n) where B(m)(n) = C[tj|j ̸= 0 mod n] ⊂B(m).

Therefore onecan restrict the mKP(resp. KP) hierarchies and obtain the so called n-reduced mKP(resp.KP) hierarchies.

Then one can show [33] that the τ-function τ(n) = ⊕n−1k=0τk belongs to theˆSLn orbit of the sum of the highest weight vectors ⊕n−1m=01m. We are mainly interested inthe second reduced mKP hierarchies.

Then the simplest bilinear Hirota equations give forui = −2∂2 ln τi , i = 1, 2 and v = ln τ2τ1 equations (23) and (24), and we obtain the mKdVhierarchy.13

Now we want to establish the relation between elements of Gr(0) and fermionic states.Consider V ∈Gr(0) spanned by the vectors {φi} (i = 0, 1, 2, . .

.) where φi = Pk∈Zφi,kzk ∈H.Associate to every φi ∈V a fermionic operator ψ†[φi] byψ†[φi] =Xk∈Zφi,kψ†k(56)and to every V ∈Gr(0) the state |v > belonging to the GL(∞) orbit of the vacuum andsuch thatψ†[φi]|v >= 0∀i ,(57)where V is spanned by the functions {φi}.

Then because bilinear fermionic operatorsˆa =Xi,j: ψ†i aijψj :(58)satisfy[ψi, ˆa] =Xkaikψk ,[ˆa, ψ†i ] =Xkψ†kaki ,(59)we can associate to them operators a acting on H bya h(z) =Xk Xiakihizk(h(z) ∈H) . (60)Then ifˆa1 ↔a1andˆa2 ↔a2then[ˆa1, ˆa2] ↔[a1, a2] .

(61)Moreover, one can prove [28,29] that if |v > corresponds to V ∈Gr(0), thenˆa|v >= const.|v >⇔a V ⊂V . (62)The proof follows immediately from the remark that [ˆa, ψ†(φ)] = ψ†(aφ) (see (59)).

Thus ifˆa|v >= const.|v > and φ ∈V i.e. ψ†(φ)|v >= 0, then ψ†(aφ)|v >= (ˆaψ†(φ)−ψ†(φ)ˆa)|v >=0 and hence aφ ∈V .

In other words a V ⊂V. In a similar way one can establish theimplication in (62) in the reverse direction.

From the above discussion we see that if V1,2are to describe mKdV flows then they should correspond to states |v1 >∈GL(∞) · |0 >and |v2 >∈GL(∞) · |1 >. Then since |vi >t= exp{ Pp≥1tpJp}|vi > or∂∂t2k+1|vi >t= J2k+1|vi >t ,(63)14

equation (60) yields (42).Consider the Virasoro operatorsLn = 122n−1Xp=−∞JpJ2n−p + 116δn,0n ≥0(64)acting on the τ-functions associated with the states |g >iτi(t) =< i −1| exp{Xp≥1tpJp}|g >ii = 1, 2 . (65)Then shift the times t2i+1 →t2i+1 +αi2i+1 for i ≤k, where the αi are defined in (37).

Thenτi(t) →τ ′i(t) =< i −1| exp{Xp≥1(tp + t(0)p )Jp}|g >i ,Ln →L′n = ekPp=0αp2p+1 J2p+1Lne−kPp=0αp2p+1 J2p+1= Ln +kXp=0αpJ2(n+p)+1 . (66)In [28,29] it was shown that the fermion operators L′n correspond via (60) to the operators12z2n+1A = 12z2n+1 ddz +kXp=0αiz2i!.

(67)Then, because of (62), invariance of V1,2 under z2n+1A (see (41) ) implies that the τ-functions τi are annihilated by the Ln’s for n ≥1 andL0τi = µτi . (68)The constant µ is an arbitrary parameter.

Such a parameter does not appear for Ln(n ≥1)by closure of the Virasoro algebra. As pointed out in [23] it is the same for the two τ-functions and it cannot be determined by the closure of the algebra since, contrary tothe HMM, L−1 is absent.

If one includes boundary conditions then there exists a oneparameter family of solutions to the string equation with the correct scaling behaviour atinfinity [40]. It has been suggested in [23] that the parameter of such a particular solutionis related to µ.The Virasoro constraints are then those of a heighest weight state ofconformal dimension µ.

Although L−1 is absent one should bear in mind the additionalconstraints arising from the interrelation of τ1 and τ2 determined by equation (41).15

5. Algebraic Description of the Moduli SpaceIn this section we attempt to give a complete description of the moduli space of thestring equation (14).

As already mentioned, the space of solutions to (14) is isomorphicto the set of points V1, V2 of Gr(0) that satisfy the conditions (41). Therefore we will startby describing the spaces V1, V2.First choose vectors φ1(z), φ2(z) ∈V1, such thatφ1(z) = 1 + lower order terms ,φ2(z) = z + lower order termsThen the condition z2 V1 ⊂V1 and π+(V1) ∼= H+ shows that we can choose a basis for V1φ1, φ2, z2φ1, z2φ2, .

. .Since z V1 ⊂V2 and π+(V2) ∼= H+ we can choose a basis for V2 to beψ, zφ1, zφ2, z3φ1, z3φ2, .

. .where ψ(z) = 1 + lower order terms.

Using z V2 ⊂V1 we have zψ = αφ1 + βφ2. Chooseφ1, φ2 such that zψ = φ2.

Then we obtain the following basis for V1, V2 (φ ≡φ1):V1 :φ, zψ, z2φ, z3ψ, . .

.V2 :ψ, zφ, z2ψ, z3φ, . .

. (69)Then it is clear that φ, ψ specify the spaces V1, V2.

Using the conditions AV1 ⊂V2 andAV2 ⊂V1 we obtain( ddz + fk(z2))φ = P00(z)φ + P01(z)ψ( ddz + fk(z2))ψ = P10(z)φ + P11(z)ψ . (70)The polynomials P00(z) and P11(z) are odd whereas P01(z), P10(z) are even.

Comparingboth sides of (70) we find that because deg(fk) = 2k, deg(P01(z)) = deg(P10(z)) = 2k anddeg(P11(z)), deg(P00(z)) < 2k and that the coefficients of the leading terms of P01(z) andP10(z) are equal to αk.Equations (70) can be rewritten in the formDχ = B2k(z)χ(71)16

where χ =φψ,D = ddz00ddz,B2k(z) =P00(z) −fk(z2)P01(z)P10(z)P11(z) −fk(z2). (72)The requirement that φ, ψ be solutions of the form 1 + (lower order terms), rather thanexponential, puts further constraints on the matrix B2k(z).

It requires that the eigen-values λ(z) of B must vanish up to O(z−2), i.e λ(z) =Pi≥1λiz−i−1.Indeed thenχ ∼expR z λ(z′)dz′ ∼exp −λ1z∼1 −λ1z−1 + . .

., as desired.But then detB2k(z) isof O(z−4) andf2k(z2) = 12(P00(z) + P11(z)) ±r14(P00(z) + P11(z))2 −∆+ O(z−4)(73)where ∆(z) = P00(z)P11(z) −P01(z)P10(z). Since f(z2) is an even function of z, the oddparity of P00(z) and P11(z) determine that P00(z) + P11(z) = 0.Conversely, given a 2 × 2 matrixPij(z)with P01(z), P10(z) even polynomials ofdegree 2k and P00(z), P11(z) odd polynomials of degree < 2k such that P00(z) + P11(z) =0, we will show that we obtain exactly two solutions to the string equation (34).

Theeigenvalues λ(1,2)(z) ofPij(z)are given byλ(1,2)(z) = ±p−∆(z)(74)and λ(i)(z) =kPj=−∞λ(i)j z2j(i = 0, 1). Then the matrix B2k of (72) withf (i)k (z2) =kXm=−∞α(i)m z2mα(i)m −λ(i)m = 0m ≥0̸= 0at least for 0 ≫m(75)will have determinant at most of O(z−4).

Then the system (70) will have solutions φ(z)and ψ(z) of the form φ(z), ψ(z) = const. + lower order terms.

We can set the constant toone by requiring that the leading terms of the polynomials P01(z) and P10(z) are equal.Since we know from the discussion at the end of section 3 that the m < 0 terms of theoperator A can be gauged away, we see that each eigenvalue λ(i)(z) specifies a uniquesolution to the string equation (34).Hence the space of solutions to the string equation (14) is the two fold covering of thespace of matricesPij(z)with polynomial entries in z such that P01(z) and P10(z) are17

even polynomials having equal degree and leading terms and P00(z) and P11(z) are oddpolynomials satisfying the conditions P00(z) + P11(z) = 0 and degP00(z) < degP01(z).AcknowledgementsThe research of K.A. and M.B.

was supported by the Outstanding Junior InvestigatorGrant DOE DE-FG02-85ER40231, NSF grant PHY 89-04035 and a Syracuse UniversityFellowship. A.S. would like to thank Michael Douglas for useful conversations.

The authorswould like to thank the Institute for Theoretical Physics and its stafffor providing thestimulating environment in which this work was begun.18

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