T -duality transformation and universal structure of noncritical string field theory

T-duality transformation and universal structure of noncritical string field theory

Takashi Asatani,* Tsunehide Kuroki,† Yuji Okawa,‡ Fumihiko Sugino,§ and Tamiaki Yoneyai

Institute of Physics, University of Tokyo, Komaba, Meguro-ku, Tokyo 153, Japan~Received 30 July 1996!

We discuss aT-duality transformation for thec51/2 matrix model for the purpose of studying dualitytransformations in a possible toy example of nonperturbative frameworks of string theory. Our approach is tofirst investigate the scaling limit of the Schwinger-Dyson equations and the stochastic Hamiltonian in terms ofthe dual variables and then compare the results with those using the original spin variables. It is shown that thec51/2 model in the scaling limit isT-duality symmetric in the sphere approximation. In the case of thestandard two-matrix model, however, the duality symmetry is violated when the higher-genus effects are takeninto account, due to the nonsymmetrical appearence of globalZ2 vector fields corresponding to nontrivialhomology cycles. Some universal properties of the stochastic Hamiltonians which play an important role indiscussing the scaling limit and have been discussed in a previous work by Sugino and Yoneya are refined inboth the original and dual formulations. We also report a number of new explicit results for various amplitudescontaining macroscopic loop operators.@S0556-2821~97!03908-8#

PACS number~s!: 11.25.Pm, 11.25.Sq

I. INTRODUCTION

Recently, dual transformations in various different formsare playing increasingly important roles in string theory. His-torically, one of the first useful examples of the dual trans-formations was the Kramers-Wannier duality@1# for the two-dimensional Ising model on the square lattice, which allowsus to locate exactly the critical temperature for the order-disorder phase transition. The familiarT-dual transformationin closed string theories compactified on a torus is basicallythe same as the Kramers-Wannier dual transformation ondiscretized world sheets for a compact U~1!-spin model~anXY model with a Villain-type action, see, e.g.,@2# and Ap-pendix A!, where the circle as the target space becomes thespin variable and the role of the compactification radius isreplaced by the inverse temperature. A remarkable feature ofthe stringT duality is that the duality symmetry is exact forany world sheet of arbitrary fixed genus, due to the sym-metrical roles@3# played by the momentum and windingmodes of closed strings. On the other hand, the conjecturedS-duality transformation of string theory is a duality withrespect to the target space itself and is a generalization of theelectric-magnetic duality of gauge field theories, exchangingthe coupling constant and its inverse as required by the Diraccondition. Complete understanding of all the duality trans-formations, especially theS dualities, in string theory wouldobviously require some nonperturbative framework. The re-cent developments@4# of the idea of Dirichlet branes in factsuggest that theT-dual transformations should also playsome important roles in nonperturbative physics.

In view of this situation, it seems worthwhile to formulate

the duality symmetries in a simplest setting for nonperturba-tive string theory. Only known effective formulation ofstring theory which is in principle nonperturbative is the ma-trix model corresponding toc<1 conformal matter coupledwith the world-sheet metric. In the present paper, as a simpleexercise towards nonperturbative understanding of dualitiesin string theory, we study theT-dual transformation of thedouble-scaled two-matrix model which is the Ising model atthe critical point on random surfaces. Naively, we expect thatthe model should be self-dual, since the Ising system sitsprecisely at the critical point. We will in fact argue that theself-duality is valid in the sphere approximation. In highergenera, however, the duality symmetry is violated due to theeffect of nontrivial homology cycles. When the model is for-mulated only in terms of the spin variables as usual, there isno analogue of the symmetry between the momentum andwinding modes which is responsible for the exact dualitysymmetry in the torus-compactified model. Rather, the dualtheory must be described by vector fields residing on links.Only in the special case of sphere, the vector fields can beexpressed in terms of the dual spin fields. Another motiva-tion for studying this model is to further pursue the remark-able universal structure of noncritical string field theory@5#which is relevant in taking the scaling limit and for studyingbackground independence, using both the original and dualvariables and to complement the results of the previous work@6# by two of the present authors.

The plan of the present paper is as follows. In the nextsection, we first explain the Kramers-Wannier duality trans-formation of the two-matrix model. Although the dual trans-formation of the two-matrix model has been discussed in theliterature,1 we here explain some elementary details of theconstruction for the purpose of making our interpretation tobe sufficiently definite. In particular, we emphasize that inthe standard formulation of the two-matrix model, the non-

*Electronic address: [email protected]†Electronic address: [email protected]‡Electronic address:[email protected]§Present address: KEK Theory Group, Tsukuba, Ibaraki 305,

Japan. Electronic address: [email protected] address: [email protected]

1In particular, we mention Ref.@7# in which the disk amplitude inthe dual basis was derived in the sphere approximation.

PHYSICAL REVIEW D 15 APRIL 1997VOLUME 55, NUMBER 8

550556-2821/97/55~8!/5083~29!/$10.00 5083 © 1997 The American Physical Society

trivial homology cycles may generally violate the dualitysymmetry between the spin and dual-spin variables, sincethey allow global vector fields. In Sec. III, following themethod of the previous paper@6#, we derive the stochasticHamiltonian@8# in terms of the dual variables. In Sec. IV, weproceed to discuss the double scaling limit of the dual matrixmodel by examining the structure of the Schwinger-Dysonequations and the stochastic Hamiltonian using the dual-matrix variables. We will explicitly evaluate several lower-point correlators and determine their scaling behaviors. InSec. V, based on the results of the previous section, we ex-hibit commutativity property between the operator mixingand the merging-splitting interaction of the macroscopic loopfields, in terms of both the original and dual matrix variables.The results generalize those of the previous paper@6#. In Sec.VI, we discuss the duality symmetry of the stochastic Hamil-tonians. We show that the Hamiltonians are duality symmet-ric in the sphere approximation, although they are not sym-metric for higher genus due to the existence of global vectorfields on the world sheets with nontrivial homology cycles,in conformity with the discussion in Sec. II. Section VII isdevoted to concluding remarks. In Appendix A, we present avery brief discussion on how the exactT-duality symmetry isunderstood as the Kramers-Wannier duality on arbitrary dis-cretized world sheets. Some of the computational details arediscussed in Appendices B and C.

II. KRAMERS-WANNIER DUAL TRANSFORMATIONIN THE TWO-MATRIX MODEL

The partition function we study in the present paper is theN3N Hermitian two-matrix model given by

Z5E dN2AdN

2B exp@2S~A,B!#, ~2.1!

S~A,B!5NtrS 12 A211

2B22cAB2

g

3A32

g

3B3D .

~2.2!

As is well known, this represents an Ising system on randomsurface, where the Boltzmann factors are given by1/(12c2) and c/(12c2) for the nearest neighbor Isinglinks, connecting the spin sites with the same or oppositespins, respectively.

The inverse temperatureb for the Ising system is then

b521

2lnc. ~2.3!

Note that the vertices of Feynman diagrams are nothing butthe surface elements in the standard random surface interpre-tation and the Ising spins live on the center of the surfaceelements. Hence the Ising links just correspond to the propa-gators of Feynman diagrams. The geometrical Boltzmannfactor for the elementary surface element isg;e2l with lbeing the bare cosmological constant. In what follows, weuse the terminologies, ‘‘sites,’’ ‘‘links,’’ in the sense of lat-tice spin systems, and ‘‘vertices,’’ ‘‘edges,’’ in the sense oftriangulations of the random surface.

Now let us consider the Kramers-Wannier dual transfor-mation for this system. A standard way of performing the

dual transformation is to first reinterpret the Boltzmann fac-tors e6b for the links as those for the dual linkse6b bymaking theZ2 Fourier transformation as

eb5K~eb1e2b !, ~2.4!

e2b5K~eb2e2b !, ~2.5!

where K is an overall normalization constantK51/(e2b2e22b)1/2. This fixes the dual inverse temperatureb as

b521

2lnc ~2.6!

with

c512c

11c. ~2.7!

For a fixed lattice, the Boltzmann factor expbvl would cor-respond to dual linksl with parallel ~‘‘stick’’: v l 511)dual spins or antiparallel~‘‘flip’’: v l 521) dual spins, re-spectively. In the present case, however, we will not intro-duce dual spin variables for reasons explained below. Usu-ally, the dual spin variables are introduced as the solutionsfor the constraint for the dual link variables after integratingover the original spin configurations. The constraints take theform

)l,C

v l 51 ~2.8!

for all elementary~namely smallest! closed loopsC of duallinks encircling the spin sites of the original lattice. The so-lution for these constraints is locally given asv l 5 sl 1sl 2 by

introducing the dual spin variablessl 1,sl 2 (561) at thecenter of all dual surface elements. The fixed square latticewith Ising spins is then duality symmetric under the inter-changeb↔b.

For the present model, there are at least two reasons thatthe system cannot in general be duality symmetric. The firstreason is due to the nontrivial homology cycles of the gen-eral random surfaces. The constraint~2.8! allows nontrivialsolutions which cannot be reduced to the product of the dualspin variables, due to the existence of globalZ2 vector fields,associated with the nontrivial homology cycles. For this rea-son, we will treat the dual transformed model by dealingonly with the dual link variables without introducing the dualspin variables explicitly. Secondly, the random surfaces ofthe two-matrix model~2.2! are discretized by triangles cor-responding to the cubic vertices. Hence the original spin siteshave always three links attached. After the above dual trans-formation, the dual spin sites can now have an arbitrary num-ber of dual links attached. For this reason, the critical pointcannot be fixed to be the self-dual point in contrary to thecase of the fixed square lattice. We can, however, expect thatin the continuum limit this difference might be washed outand the model should exhibit partial duality symmetry in

5084 55ASATANI, KUROKI, OKAWA, SUGINO, AND YONEYA

some situations where we can neglect the effect of nontrivialhomology cycles. We will argue in later sections that this isindeed the case.

Now in terms of the matrix fieldsA,B, the Boltzmannfactors for the original Ising links are related to the barepropagators as

ÂA&05^BB&05Leb, ~2.9!

ÂB&05Le2b ~2.10!

with L5Ac/(12c2), where^•••&0 indicates the expectationvalue with the quadratic action. Substituting the Fouriertransformations~2.4! and ~2.5!, we have

ÂA&05^BB&051

2A12c2~eb1e2b !, ~2.11!

ÂB&051

2A12c2~eb2e2b !. ~2.12!

This clearly shows that the dual transformation is performedsimply by changing to the new matrix fieldsX andY, whichdiagonalize the kinetic term of the original action, as

X51

A2~A1B!, ~2.13!

Y51

A2~A2B!, ~2.14!

whose propagators just give the Boltzmann factors for thedual model,

^XX&051

A12c2eb5

1

12c, ~2.15!

^YY&051

A12c2e2b5

1

11c. ~2.16!

The new matrix fieldsX,Y can be regarded as living on thedual vertices~surface elements or triangles of theoriginallattice! but their propagators are associated with the dualIsing links ~dual to the edges of the dual surface elements!,having one-to-one correspondence with original Ising links.Thus they should not be confused with the dual spin vari-ables themselves which are supposed to live on dual-spinsites, namely, the center of the dual surface elements corre-sponding to the vertices of the original random surfaces.

In terms of the new matrix fields, the partition function is(g5g/A2)

Z5E dN2XdN

2Yexp@2SD~X,Y!#, ~2.17!

SD~X,Y!5NtrS 12c

2X21

11c

2Y22

g

3~X313XY2! D .

~2.18!

In comparison with the original representation, this represen-tation itself does not show any trace of possible dual sym-metry, reflecting the situation explained above. This simplydefines a mapping of an Ising model on random surfaces to aspecial case of the O~n! model as has been discussed byseveral authors@9#.

The homological property we have discussed can now bedescribed by the configurations of closed loops formed bythe Y field. The globalZ2 vector fields correspond to theclosedY loops winding around the nontrivial homologicalcycles of the surfaces. TheZ2 nature comes from the factthat even numbers of closedY loops winding around non-trivial homology cycles are continuously deformed into nullclosedY loops. In the case of sphere, there are no nontrivialhomology cycles and hence we can introduce dualZ2 spinvariables which reside on the domains bounded by closedY loops. The existence of such closedY loops windingaround nontrivial cycles forbids us to define dual spin do-mains globally. Equivalently, the Hamiltonian description ofthe time development of the random surfaces necessarily re-quires us to introduce the observables containing odd num-bers of theY fields.

Let us next discuss the basis of string field representationsused in the present paper. In the previous work, we used thecomponent expansion of string fields with respect to thenumber of spin domains. In the dual formulation, the basiswhich can be compared with this is the expansion with re-spect to the number ofY fields. Thus the observables weconsider are of the following type:

1

NtrS 1

j2XD[CX~j!, ~2.19!

1

NtrS 1

j12XYD[C1~j1!, ~2.20!

1

NtrS 1

j12XY

1

j22XYD[C2~j1 ,j2!, ~2.21!

1

NtrS 1

j12XY

1

j22XY

1

j32XYD[C3~j1 ,j2 ,j3!,

~2.22!

. . . , etc.

We note that if we could have neglected the closedY loopswinding around the nontrivial homology cycles,C ’s withodd numbers ofY’s were not necessary. We would then havethe interpretation of these quantities in terms of domainswith respect to dual spins, andC2n would correspond to thecomponent of the field withn pairs of domains. Note that thematrix Y just represents the domain boundaries of the dual-spin interpretation. Thus, in studying the duality symmetry,

55 5085T-DUALITY TRANSFORMATION AND UNIVERSAL . . .

C2n should be substituted in place of the observables con-sidered in the previous paper, namely,

Fn~z1 ,s1 , . . . ,zn ,sn![1

NtrS )

i

n1

z i2A

1

s i2BD ,~2.23!

after an appropriate symmetrization with respect to two ma-tricesA andB, taking into account the fact that theC2n doesnot discriminate global dual spin directions. A detailed dis-cussion of the duality symmetry of the stochastic Hamil-tonian will be presented later in Sec. VI.

From the above discussion, it is clear at least conceptuallythat the dual transformation is nothing but a change of thebasis of the string fields between Eqs.~2.23! and ~2.19!–~2.22!, etc.2 However, it turns out that performing this trans-formation directly for the string fields in thecontinuumlimitis technically very difficult, because the process of taking thecontinuum limit involves an intricate operator mixing anddoes not seem to be commutative with the above change ofthe basis. Thus the question of theT-duality symmetry forthe Ising model on the random surface is quite nontrivial.The approach in the present paper will, therefore, be some-what indirect such that we perform an independent study ofthe scaling limit of the Schwinger-Dyson equations and thestochastic Hamiltonian using the dual basis and compare theresults with those obtained using the original basis. Onepractical virtue of this approach would be that the ranges ofthe components where we can compute correlation functionsexplicitly are different from each other depending on differ-ent bases, and hence our results are useful for showing theuniversal nature of the structure of, say, the stochasticHamiltonians, as has been emphasized in the previous work@6#. Presenting such a piece of evidence for the universalstructure of the string field theories is another purpose of thepresent paper.

Finally, we fix our notations in what follows. As intro-duced above, the basis operators areCn . We denote theexpectation values ofCn in the sphere approximation byV(n)(•••)5^Cn(•••)&0 with the same arguments and suf-fices. We have used the notationW(2n) for ^Fn& in the caseof the original spin formulation@6#.

III. STOCHASTIC HAMILTONIANIN THE DUAL FORMALISM

Let us first derive the stochastic Hamiltonian using thedual basis of the observablesC ’s introduced in the previoussection. The generating functional of the dual two-matrixmodel is defined by

Z@K#51

ZE dN2XdN

2Ye2SDeKC, ~3.1!

Z5E dN2XdN

2Ye2SD, ~3.2!

KC5E dj

2p iKX~j!CX~j!

1 (n51

` E )i51

ndj i2p i

Kn~j1 , . . . ,jn!Cn~j1 , . . . ,jn!,

~3.3!

where and in what follows the integral contour, unless speci-fied otherwise, runs parallel to the imaginary axis towardsthe positive imaginary infinity on the right-hand side of allthe pole singularities of the observablesCX ,Cn .

As discussed in the previous paper@6#, the stochasticHamiltonian is derived from the identity

052E dN2XdN

2Y(

a51

N2 F ]

]Xae2S

]

]Xa

1]

]Yae2S

]

]YaGeKC. ~3.4!

The notationsXa ,Ya denote the components of the Hermi-tian matrix in the expansion in terms of the basis$ta%@ tr(tatb)5dab#. The derivative operator inside the inte-gral may be regarded as the Laplacian operator for thepresent model. Expressed as a functional differential equa-tion in terms of the sources, this takes the form

05HZ@K#, ~3.5!

H52KSK d

dK D2KS d

dK~

d

dK D21

N2 KFKS `d

dK D G2KT.

~3.6!

Here the functional derivatived/dKn acting on the sourceterms is defined as

dKm~j18 , . . . ,jm8 !

dKn~j1 , . . . ,jn!5dm,n

1

n~2p i !n

3(c

d~j12jc~1!8 !•••d~jn2jc~n!8 !.

~3.7!

Reflecting the cyclic symmetry ofCn with respect to itsvariables$j i%, the summation in the right-hand side is overcyclic permutationsc( i ) of the indicesi51,2, . . . ,n of j i8 .The origin and definition of each term of the Hamiltonian~3.6! are explained below.

~1! The first term, ‘‘kinetic term’’K@K(d/dK)#, comesfrom the product of the first derivatives of the source termand the action (n51,2, . . . ):

SK d

dK DX

~j!5]jFj~12c2gj!d

dKX~j!G2g R dj8

2p i]j

d

dK2~j,j8!, ~3.8!

2We here mention a paper@10# which has discussed theT dualityin the framework of covariant string field theories of criticalbosonic string theory.


SK d

dK Dn

~j1 ,•••,jn!5(j51

n

@22c1~12c2gj j !j j]j j#

d

dKn~j1 , . . . ,jn!

1(j51

n

g R dj

2p i

d

dKn12~j1 , . . . ,j j21 ,j j ,j,j j ,j j11 , . . . ,jn!

2(j51

n

g R dj

2p i

d

dKn~j1 , . . . ,j j21 ,j,j j11 , . . . ,jn!, ~3.9!

where the integralr is over a closed curve encircling all the pole singularities of the observablesC ’s. The structure of thekinetic term is slightly different from that with the original variablesA,B. If for the moment we allow ourselves to use thedual-spin language for convenience, this contains a part which measures the number of dual-spin flip links, corresponding totheY matrices, in addition to a part measuring the length of each domain. The other parts describe the infinitesimal motion ofstring loops, changing their lengths and/or dual spin directions.

~2! The second term,K(d /dK~d /dK), of ~3.6! comes from the second derivative of the source termKC and representsprocesses where a loop splits into two. The symbol (d /dK~d /dK) I stands for the spin configurations of the resultant twoloops from a single loop with spin configurationI :

S d

dK~

d

dK DX

~j!52]j

d2

dKX~j!2,

S d

dK~

d

dK D1

~j1!522d

dKX~j1!]j1

d

dK1~j1!,

S d

dK~

d

dK D2

~j1 ,j2!522(j51

2d

dKX~j j !]j j

d

dK2~j1 ,j2!12SDz~j1 ,j2!

d

dK1~z! D2

12d2

dKX~j1!dKX~j2!,

S d

dK~

d

dK Dn

~j1 , . . . ,jn!522(j51

nd

dKX~j j !]j j

d

dKn~j1 , . . . ,jn!

12(k, l

Dz~jk ,j l !d

dKl2k~z,jk11 , . . . ,j l21!Dz~jk ,j l !

d

dKn2 l1k~j1 , . . . ,jk21 ,z,j l11 , . . . ,jn!

22(j51

nd

dKX~j j !Dz~j j21 ,j j11!

d

dKn22~j1 , . . . ,j j22 ,z,j j12 , . . . ,jn!

12 (1, l2k,n21

Dz~jk11 ,j l !d

dKl2k21~z,jk12 , . . . ,j l21!

3Dz~jk ,j l11!d

dKn2 l1k21~j1 , . . . ,jk21 ,z,j l12 , . . . ,jn!~n>3!. ~3.10!

Here we used the definition of the combinatorial derivative:

Dz~j1 ,j2! f ~z!5f ~j1!2 f ~j2!

j12j2.

Note also that the last term of Eq.~3.10! is absent forn53.~3! The third term (1/N2)K@K(`d /dK)# comes from the square of the first derivative of the source term, and represents

processes in which two loops merge into a single loop. The symbol (`d /dK) I ,J expresses the spin configuration of theresultant single loop into which two loops with spin configurationsI ,J merge. The variables on the left-hand side of asemicolon represent the variables conjugate to the lengths of domains of a loop with configurationI , while those on theright-hand side represent the variables with configurationJ:


S `d

dK DX,X

~j;j8!52]j]j8Dz~j,j8!d

dKX~z!,

S `d

dK Dn,X

~j1 , . . . ,jn ;j8!5S `d

dK DX,n

~j8;j1 , . . . ,jn!~n51,2,3, . . .!

52(j51

n

]j8]j jDz~j8,j j !

d

dKn~j1 , . . . ,j j21 ,z,j j11 , . . . ,jn!,

S `d

dK D1,1

~j1 ;j18!5Dz~j1 ,j18!Dz~j1 ,j18!d

dK2~z,z!2Dz~j1 ,j18!

d

dKX~z!,

S `d

dK Dn,1

~j1 , . . . ,jn ;j18!5S `d

dK D1,n

~j18 ;j1 , . . . ,jn!~n52,3,4, . . .!

5(j51

n

Dz~j18 ,j j !Dz~j18 ,j j !d

dKn11~j1 , . . . ,j j21 ,z,z,j j11 , . . . ,jn!

1(j51

n

Dz~j j ,j18!Dz~j j11 ,z!d

dKn21~j1 , . . . ,j j21 ,z,j j12 , . . . ,jn!,

S `d

dK Dn,m

~j1 , . . . ,jn ;j18 , . . . ,jm!~n,m52,3,4, . . .!

5(j51

n

(k51

m

Dz~j j ,jk8!Dz~j j ,jk8!d

dKn1m~j1 , . . . ,j j21 ,z,jk118 , . . . ,jm8 ,j18 , . . . ,j8k21 ,z,j j11 , . . . ,jn!

1(j51

n

(k51

m

Dz~j j ,jk118 !Dz~jk8 ,j j11!d

dKn1m22~j1 , . . . ,j j21 ,z,jk128 , . . . ,jm8 ,j18 , . . . ,jk218 ,z,j j12 , . . . ,jn!.

~3.11!

~4! The last term, a tadpole term, arises again from theproduct of the first derivative of the source termKXCX andthe action, and describes the process of the annihilation of ashortestloop into nothing:

KT5E dj

2p iKX~j!g. ~3.12!

Similarly to the Hamiltonian of the original formalism, thepresent dual Hamiltonian satisfies a locality property. Forinstance, only the loopsCX with no domain boundary can beannihilated into nothing. Also, only a single pair of domainsor a single pair of the boundaries of the domains can partici-pate in the splitting or merging processes, and other domainsremain unchanged. One of the differences between theHamiltonians in the original and dual formulations arisesfrom the derivatives with respect toY. The Ya derivativesproduce processes which have no counterpart in the Hamil-tonian in terms of the original variables. For instance, the lastterms of the kinetic~3.9! and merging terms~for I ,J>1) inthe above expressions contain the contributions from theYa derivatives. For the splitting interaction, the last~forI52,3) or the last two terms~for I>4) are the contributionsfrom theYa derivatives. From the viewpoint of continuum

theory, these are very singular; they would be measure-zerocontributions. In later sections, we will show that the termsarising from theYa derivatives in fact vanish in the con-tinuum limit.

IV. SCHWINGER-DYSON EQUATIONSAND THE SCALING LIMIT

OF THE STOCHASTIC HAMILTONIAN

Now we consider the scaling limit of the Hamiltonian~3.6!. To do this, we closely follow@6#. Namely, we will firstidentify and subtract the nonuniversal parts of the disk am-plitudes and establish the operator mixing, and then will re-write the Hamiltonian~3.6! in terms of the universal parts ofthe operators.

As we discuss in Appendix B, the original operatorsC I

and their universal partsC I are related by a linear transfor-mation of the form

C I5(JMIJCJ1c I , I5X,1,2, . . . , ~4.1!

whereMIJ is a mixing matrix of upper-triangular formMIJ50 for I,J, andc I is the nonuniversalc-number func-


tion. The upper-triangular form ofM comes from a propertyof the operator mixing that the operators corresponding tosimpler configurations can mix as nonuniversal parts in tak-ing the continuum limit, as has been discussed in the previ-ous work@6#. Note thatc I vanishes by itself for an arbitraryodd I , because there exists theZ2 symmetry underY→2Y. The first few components of Eq.~4.1! are

CX~j!5CX~j!1cX~j!,

C1~j1!5C1~j1!,

C2~j1 ,j2!52

A5c@CX~j1!1CX~j2!#

1C2~j1 ,j2!1c2~j1 ,j2!,

C3~j1 ,j2 ,j3!52

A5c@2Dj~j1 ,j2!C1~j!

2Dj~j2 ,j3!C1~j!2Dj~j3 ,j1!C1~j!#

1C3~j1 ,j2 ,j3!,

C4~j1 ,j2 ,j3 ,j4!

524

5c@Dj~j1 ,j3!CX~j!1Dj~j2 ,j4!CX~j!#

22

A5c$Dj~j1 ,j3!@C2~j,j2!1C2~j,j4!#

1Dj~j2 ,j4!@C2~j1 ,j!1C2~j3 ,j!#%

1C4~j1 ,j2 ,j3 ,j4!1c4~j1 ,j2 ,j3 ,j4!, . . . ,

~4.2!

where

cX~j!52c~c11!

3g21

3@~5c21!j1gj2#,

c2~j1 ,j2!51

5~114s!22Ac

5~j11j2!,

c4~j1 ,j2 ,j3 ,j4!51, . . . , ~4.3!

and c is at the critical value:c5(2112A7)/27, ands511A7. All these results are derived explicitly in Appen-dix B.3

Denoting the connectedk-point correlator for theK50background as

GI1 , . . . ,I k~k! 5^C I1

. . .C I k&, I 1 , . . . ,I k5X,1,2, . . . ,

~4.4!

the generating functional takes the form

Z@K#5expFKG~1!11

2!K~KG~2!!

11

3!K@K~KG~3!!#1••• G . ~4.5!

Separation of the universal partsGI1 , . . . ,I k(k) amounts to a lin-

ear transformation

GI~1!5(

JMIJGJ

~1!1c I ,

GI1 , . . . ,I k~k! 5 (

J1 ,•••,JkMI1J1

. . .MI kJkGJ1 , . . . ,Jk

~k! ~k>2!.

Thus, on introducing the transformed sourceK by

KI5(JKJ~M21!JI , ~4.6!

the generating functionalZ@K# in the continuum theory isobtained by a rescalingZ@K#5eKcZ@K#, giving

Z@K#5expF KG~1!11

2!K~KG~2!!

11

3!K@K~KG~3!!#1••• G . ~4.7!

The Hamiltonian acting onZ@K# now becomes

05HZ@K#, ~4.8!

H52~KM21!FKSM d

dK1c D G2~KM21!T

2~KM21!F SM d

dK1c D ~SM d

dK1c D G

21

N2 ~KM21!X~KM21! F`SM d

dK1c D GC.

~4.9!

Next, we try to eliminate the mixing matrix from Eq.~4.9!.We claim the validity of the identities

3As we comment in Eq.~B9! in Appendix B,c4 has possibly alinear term ofy in addition. This term affects higher components ofspin flip amplitudes through the Schwinger-Dyson equations. How-ever, since it changes only the nonuniversal parts of these ampli-tudes, it can be absorbed into the redefinition of the nonuniversalparts and does not change the scaling limit of the Hamiltonian.


2~KM21!FKSM d

dK1c D G2~KM21!T

2~KM21!F SM d

dK1c D ~SM d

dK1c D G

21

N2 ~KM21!@~KM21!~`c!#

52KS F d

dKD 2~KM21!F SM d

dKD ~SM d

dKD G ,~4.10!

~KM21!F SM d

dKD ~SM d

dKD G5KS d

dK~

d

dKD ,~4.11!

~KM21!F ~KM21!S `Md

dKD G5KF KS `

d

dKD G ,

~4.12!

whereF is a part of the kinetic operatorK, representingonly the spin flip processes. Note that the term(KM21)[( KM21)(`c)] does not completely vanish in thedual theory in contrast to the corresponding identity in theoriginal spin formulation@6#. These identities enable us torewrite the Hamiltonian as

H52KS F d

dKD 2KS d

dK~

d

dKD 2

1

N2 KF KS `d

dKD G .~4.13!

Justification of the identities~4.10!–~4.12!. We now try toestablish the above identities.

~1! Kinetic term. In order to make the expression for thekinetic operatorF explicit, we first consider the spin flipprocess in the continuum theory. For the universal parts ofthe disk and cylinder amplitudes, the string fields with amicroscopic domain containing a single flipped spin are ob-tained from the string fields without any microscopic domainby the following rule~for derivation see Appendix C!:

2]jV2~j!52 Rˆ dh

2p i]jV

~2!~j,h!, ~4.14!

V2~3!~j1 ,j2 ,j3 ; !5 Rˆ dh

2p iV~4!~j1 ,j2 ,j3 ,h!, . . . ,

~4.15!

VYcyl~1!~j !5 Rˆ dh

2p iV1u1

cyl~juh!, ~4.16!

V2uYcyl~j1 ;j2!5 Rˆ dh

2p iVY

cyl~3!~j1 ,h,j2!, . . . ,

~4.17!

where the domain corresponding to the variableh has beenshrunk into the microscopic domain by integration. The in-

tegral symbolr(dh/2p i ) is used in the sense of the integralwith respect to the variabley of the continuum theory@h5j* (11ay)#

Rˆ dh

2p i5j* aEC

dy

2p i, ~4.18!

where the contourC encircles around the negative real axisand the singularities of the left half plane, andj*5 1

5

sc21/2. Here and in what followsa is the lattice spacing.Note a slight difference from the original Ising theory in thepoint that there is no finite renormalization factor denoted bys21 in @6#.

As already emphasized in@6#, these relations are naturalsince the spin flip process occurs locally with respect to do-mains; in Eqs.~4.14!, ~4.15!, ~4.16!, and ~4.17! only thehdomain is concerned and the other domains do not change atall. Because of the locality, we can reasonably expect that therelations such as~4.14!–~4.17! should hold for any ampli-tudes with an arbitrary number of handles with generic spinconfigurations, although a completely general proof for thiscannot be given at present. By assuming this, we can rewritethe spin flip process in {K@M(d/dK)1c#} X(j) as

2]j R dh

2p i F SM d

dKD2

~j,h!1c2~j,h!G52 Rˆ dh

2p i]j

d

dK2~j,h!

2]jXS 1g

@~12c2gj!j22cX~j!# D d

dKX~j!C

21

g]j$gj1@~12c2gj!j2cX~j!#cX~j!%,

~4.19!

where the first term is the universal part, and theothers are the nonuniversal parts. Similarly, we canderive the expressions for the spin-flip processes in{K@M(d/dK)1c#} 2(j1 ,j2). However, it must be notedthat the cylinder amplitudeVY

cyl(1)(j) has a nonuniversalc-number term as shown in Eq.~B13! in Appendix B. In thiscase, a special care must be exercised in rewriting{K@M(d /dK)1c#} 1(j1). Namely, from Eqs.~4.7! and~B13!, we can obtain the relations

d

dK1~j!R dh

2p i F SM d

dKD1

~h!1c1~h!G Z@K#UK50

5 K C1~j! R dh

2p iC1~h!L


51

N2

1

A5c1K C1~j! Rˆ dh

2p iC1~h!L

51

N2

1

A5c1

d

dK1~j!Rˆ dh

2p i

d

dK1~h!Z@K#U

K50

,

~4.20!

and therefore the expression requires a slight modification as

R dh

2p i F SM d

dKD1

~h!1c1~h!G5E dh

2p iK1~h!

1

N2

1

A5c1 Rˆ dh

2p i

d

dK1~h!.

~4.21!

The first term in Eq. ~4.21! leads to the nonuniversalc-number term 1/A5c in the cylinder amplitudeVY

cyl(1)(j).

The appearance of such a term may be interpreted as arisingfrom the difference between the two methods of regulariza-tions for theh integral, namely, the lattice regularization~matrix model! and theb-function regularization. For highercomponents, we expect that there is no such modificationcoming from the terms containingKn (n>2), becauseVY

cyl(3) andVYcyl(5) have no nonuniversalc-number terms and

the higher cylinder amplitudes scale with negative powers ofa, according to the results of the cylinder amplitudes~B16!and ~B18! in Appendix B.

Concerning the third term on the left-hand side~LHS! of~4.10!, (`c) I ,J vanishes, as we will show below, except forthe I5J51 component

~`c! I ,J~j1 , . . . ,j I ;j18 , . . . ,jJ8!

5H 13 @ g~j11j18!15c21# I5J51

0 otherwise.~4.22!

Using these results, we arrive at Eq.~4.10! with

KS F d

dKD 5E dj

2p iKX~j!g~2]j! Rˆ dh

2p i

d

dK2~j,h!1E dj1

2p iK1~j1!g Rˆ dh

2p i

d

dK3~j1 ,h,j1!1•••

1E dj12p i

dj22p i

K2~j1 ,j2!g Rˆ dh

2p i S d

dK4~j1 ,h,j1 ,j2!1

d

dK4~j1 ,j2 ,h,j2!D 1•••, ~4.23!

where the ellipsis stands for the subleading terms and theterms containing the higher componentsKn (n>3).

It is noted that the tadpole term is canceled with a contri-bution of the same form from the kinetic term, as in theoriginal Ising theory@6#, and thatc-dependent terms are ab-sent because their contributions from kinetic and splittingterms cancel each other. In particular the leading contribu-tion of (`c)1,1 is precisely canceled with the nonuniversalc-number term of Eq.~4.21!. Although we do not elaboratefurther on determining the explicit continuum limits forhigher components, it is natural, because of the local natureof the spin-flip processes, to suppose that the above expres-sion already indicates the generic structure of the kineticterm, namely the flipping of a single spin on a loop withgeneral spin configurations, and the absence of the tadpoleterms and the terms without spin flipping.

~2! Interaction term. Next, we consider the splitting andmerging processes, Eqs.~4.11! and ~4.12!. In the next sec-tion, we will argue the validity of the following importantrelations, showing the commutativity of the splitting andmerging processes with the operator mixing:

F SM d

dKD ~SM d

dKD G

I

5FMS d

dK~

d

dKD G

I

,

~4.24!

F`SM d

dKD G

I ,J

5(K,LMIKMJLS `

d

dKDK,L

, ~4.25!

for completely general case, provided the mixing obeys somerules which can be explicitly confirmed for several lowernontrivial components.4 This commutativity property ensuresthat the general structure of the splitting and the merging inthe continuum limit are essentially the same as in the latticetheory.

~3! c-number term. Finally, let us turn to Eq.~4.22!.We can check its validity for the components (I ,J)5(X,X), (X,2), (X,2k11), (1,1), (1,3), (1,2k), (2,2),(2,2k11) ~k is integer! by direct calculation. For highercomponents, we give a general proof of Eq.~4.22! in thesame way as@6#. First we show thatc2k must be polynomialfor everyk. For this purpose, we start with the Schwinger-Dyson equations and derive the dual version of Staudacher’srecursion equations@12#. They relateV(k) to the amplitudesV( l ) ( l,k) as

4The proof of the next section is applicable both for the originaland dual theories, generalizing the results for the original theorygiven in the previous work@6#.


V~2k!~j1 ,..., j2k!51

11c2g~j11j2!

3{ V~j1!@2Dj~j2 ,j2k!#V~2k22!~j1 ,j3 ,..., j2k21!1V~j2!@2Dj~j1 ,j3!#V

~2k22!~j,j4 ,•••,j2k!

1 (l52

k21

@Dj~j2 ,j2l !V~2l22!~j,j3 , . . . ,j2l21!#@Dj~j1 ,j2l11!V

~2k22l !~j,j2l12 , . . . ,j2k!#

2gV2~2k21!~j2 , . . . ,j2k ; !2gV2

~2k21!~j3 , . . . ,j2k ,j1 ; !}, ~4.26!

gV2~2k21!~j2 , . . . ,j2k ; !5~21!k11gV2~jk11!1 (

p50

k22

~21!k1p11Dj~jk2p ,jk1p12!

3$@~12c!j2gj22V~jk2p!2V~jk1p12!#V~2p12!~j,jk2p11 , . . . ,jk1p11!%

1 (p50

k22

(l51

p

~21!k1p11@Dj~jk2p ,jk2p12l !V~2l !~j,jk2p11 , . . . ,jk2p12l21!#

3@Dj~jk2p12l ,jk1p12!V~2p22l12!~j,jk2p12l11 , . . . ,jk1p11!#, ~4.27!

andgV2(2k21)(j3 , . . . ,j2k ,j1 ;) has a similar form. Suppose

thatc2k’s are polynomials up to somek. Then using aboveequations we see that the part of the numerator forV(2k)

consisting only ofc is a polynomial, because in general thecombinatorial derivative of a polynomial is also a polyno-mial andc-number functionc in V2(jk11) is a polynomial.The denominator, on the other hand, behaves in the scalinglimit as

11c2g~j11j2!52acs~y11y2!1O~a2!.

Thus, by using the scaled variablesj i5j* (11ayi), c2k canbe written as

c2k5Polynomial of~y1 ,y2 , . . . ,y2k!

y11y2.

However, from the cyclic symmetry with respect toj i ’s inV(2k)(j1 ,j2 , . . . ,j2k), the denominatory11y2 must be can-celed with the numerator, and thusc2k should have the form

c2k5 Polynomial of~y1 ,y2 , . . . ,y2k!,

where the polynomial has the same symmetry asV(2k). Thus,by induction, thec2k must be a polynomial for generalk.

Now from the scaling behavior of the universal partV(2k),

V~2k!~j1 , . . . ,j2k!5^C2k~j1 , . . . ,j2k!&05a7/32~2/3!kv ~2k!

3~y1 ,y2 , . . . ,y2k!

derived in Appendix B, we expect that the relevant part ofc2k takes the form

c2k5H const, k53,

0, k>4.~4.28!

Since every component of c contains the derivative or thecombinatorial derivative, Eq.~4.28! leads to Eq.~4.22!.

We are now ready to take the continuum limit of thestochastic Hamiltonian~4.13!. The scaling property of thestring fields presented in Appendix B enables us to introducethe variables in the continuum theory as

g5g* S 12a2s2

10TD , j5j* ~11ay!,

1

N5a7/3g st,

d

dKX~j!5a4/3j

*21 d

dKX~y!, KX~j!5a27/3KX~y!,

d

dKn~j1 , . . . ,jn!5a~72n!/3j

*2n d

dKn~y1 , . . . ,yn!,

Kn~j1 , . . . ,jn!5a2~712n!/3Kn~y1 , . . . ,yn!,

~n51,2,3, . . .!. ~4.29!

In the limit a→0, the leading contributions in Eq.~4.13!start withO(a1/3), which come from theXa derivatives inEq. ~3.4!. On the other hand, the contributions from theYaderivatives are subleading and do not survive in the con-tinuum limit. After the finite rescaling

K I→~j* g* !21K I ,d

dK I

→j* g*d

dK I

,

g st2→~j* g* !2g st

2 ,

and absorbing the overall factora1/3j*21g* into the renor-

malization of the fictitious time, we have the final result forthe continuum stochastic Hamiltonian:


HD52KS F d

dKD 2KS d

dK~

d

dKD 2g st

2 KF KS `d

dKD G ,~4.30!

where the inner product is defined by

f g[E2 i`

i` dy

2p if X~y!gX~y!

1 (n51

` E2 i`

i`

)i51

ndyi2p i

f n~y1 , . . . ,yn!gn~y1 , . . . ,yn!.

~4.31!

Each term in Eq.~4.30! has the following structure. The firstterm is the kinetic term consisting only of the spin-flip pro-cesses

S F d

dKDX

~y!52]yEC

dx

2p i

d

dK2~y,x!,

S F d

dKD1

~y1!5EC

dx

2p i

d

dK3~y1 ,x,y1!,

S F d

dKD2

~y1 ,y2!5EC

dx

2p i F d

dK4~y1 ,x,y1 ,y2!

1d

dK4~y1 ,y2 ,x,y2!G , . . . ,

S F d

dKDn

~y1 , . . . ,yn!

5(j51

n EC

dx

2p i

d

dKn12~y1 , . . . ,yj ,x,yj , . . . ,yn!, . . . .

~4.32!

The second and third terms represent the splitting andmerging processes, respectively. Due to the commutativity ofthese processes with the operator mixing, they have the samestructure as the contributions from theXa derivatives in thelattice theory. The components of the splitting term are givenas

S d

dK~

d

dKDX

~y!52]yd2

dKX~y!2,

S d

dK~

d

dKD1

~y1!522d

dKX~y1!]y1

d

dK1~y1!,

S d

dK~

d

dKD2

~y1 ,y2!522(j51

2d

dKX~yj !]yj

d

dK2~y1 ,y2!12S Dz~y1 ,y2!

d

dK1~z!D 2, . . . ,

S d

dK~

d

dKDn

~y1 , . . . ,yn!522 (j51

nd

dKX~yj !]yj

d

dKn~y1 , . . . ,yn!12(

k, lDz~yk ,yl !

d

dK l2k~z,yk11 , . . . ,yl21!

3Dw~yk ,yl !d

dKn2 l1k~y1 , . . . ,yk21 ,w,yl11 , . . . ,yn!, . . . . ~4.33!

Also, the components of the merging term are

S `d

dKDX,X

~y;y8!52]y]y8Dz~y,y8!d

dKX~z!,

S `d

dKDn,X

~y1 , . . . ,yn ;y8!5S `d

dKDX,n

~y8;y1 , . . . ,yn!

52]y8 (j51

n

]yjDz~yj ,y8!d

dKn~y1 , . . . ,yj21 ,z,yj11 , . . . ,yn!~n>1!,

S `d

dKD1,1

~y1 ;y18!5Dz~y1 ,y18!Dw~y1 ,y18!d

dK2~z,w!,


S `d

dKDn,1

~y1 , . . . ,yn ;y18!5S `d

dKD1,n

~y18 ;y1 , . . . ,yn!

5(j51

n

Dz~yj ,y18!Dw~yj ,y18!d

dKn11~y1 , . . . ,yj21 ,z,w,yj11 , . . . ,yn!~n>2!,

S `d

dKDn,m

~y1 , . . . ,yn ;y18 , . . . ,ym8 !5(j51

n

(k51

m

Dz~yj ,yk8!Dw~yj ,yk8!

3d

dKn1m~y1 , . . . ,yj21 ,z,yk118 , . . . ,ym8 ,y18 , . . . ,yk218 ,w,yj11 , . . . ,yn!~n,m>2!.

~4.34!

As we promised earlier, the terms of the Hamiltonian arisingfrom theYa derivatives do not contribute in the continuumlimit. Thus it should be noted that the symbols~ andùsed here do not represent completely identical objects with~ and ` in Eq. ~4.13!, because the latter contains termsvanishing in the continuum limit which originate from theYa derivatives in Eq.~3.4!.

In this Hamiltonian, the interactions including the stringfields with odd domains survive after the continuum limit.This is important for discussing the duality symmetry. Re-member that the string fields with even domains can be in-terpreted in terms of dual spins, while those with odd do-mains cannot be. If we allow only the strings with evendomains in the initial state and consider its time evolution bythis Hamiltonianwithin the tree approximation, the stringswith odd domains never contribute, because the vacuum ex-pectation value of a single string field with odd domainsvanishes owing to theZ2 symmetry. In general, however, thestrings with odd domains necessarily contribute in the inter-mediate states, since there exists such a process that a stringwith even domains splits into two strings with odd domainsand they merge to a single string again. It represents theexcitation of oddY loops along the nontrivial cycles of thehandles. In Sec. VI, we will further examine the correspon-dence between the original and dual theories beyond thelevel of the disk and cylinder amplitudes presented in Ap-pendix B.

V. ALGEBRA OF SPLITTING AND MERGINGINTERACTIONS

We saw that the commutativity of the mixing matrix withsplitting and merging processes plays a crucial role in obtain-ing the continuum Hamiltonian in both the original and thedual models. In view of its potential importance for the de-velopment of string field theories, we devote the present sec-tion to its derivation and further elucidation.

The commutativity was previously confirmed explicitly inseveral simple~but nontrivial! configurations and conjec-tured for general configurations. Although we still cannotgive a complete proof, we shall improve the situation byproposing the general form ofMIJ , and then proving the

commutativity for generic case on the basis of this assump-tion.

The general forms of the mixing matrices are summarizedas follows.

~1! M can be expressed in terms of a matrixR as

MIJ5~eaR! IJ , ~5.1!

wherea is a constant whose value isa5A10c for the origi-nal model anda52/A5c for the dual one.

~2! The definition ofR.~a! The original model:

~Rf !A~z!50, ~Rf !B~s!50,

~Rf !1~z,s!5 f A~z!1 f B~s!,

~Rf !n~z1 ,s1 , . . . ,zn ,sn! ~n>2!

52(j51

n

Dz~z j ,z j11! f n21~z1 , . . . ,s j21 ,z,s j11 , . . . ,sn!

2(j51

n

Ds~s j21 ,s j ! f n21~z1 , . . . ,z j21 ,s,z j11 , . . . ,sn!

~5.2!

~z0[zn , zn11[z1 , s0[sn , sn11[s1!.

~b! The dual model:

~Rf !X~j!50, ~Rf !1~j1!50,

~Rf !2~j1 ,j2!5 f X~j1!1 f X~j2!,

~Rf !n~j1 , . . . ,jn! ~n>3!

52(j51

n

Dz~j j21 ,j j11! f n22

3~j1 , . . . ,j j22 ,z,j j12 , . . . ,jn! ~5.3!

~j0[jn , jn11[j1!.


The validity of these rules is explicitly confirmed for theoriginal model up to theI5A,B,1,2 components ofMIJ asgiven in Eq.~99! of @6#, and for the dual model up to theI5X,1,2,3,4,5 components ofMIJ as given in Eqs.~4.2! and~B18!.

The physical meaning of the operatorR is reducing onedomain of spins. Reduction of one domain in generic spinconfigurations causes merging of two domains which havebeen separated by the reduced domain. This explains theappearance of the combinatorial derivatives inR.

Now assuming the above general structure~5.1! forMIJ , the proof of the commutativity Eqs.~4.24! and ~4.25!is reduced to establish the following equations~5.4! and~5.5!, respectively:

@R~ f~g!# I5@~Rf !~g# I1@ f~~Rg!# I , ~5.4!

@`~Rf !# I ,J5(KRIK~` f !K,J1(

LRJL~` f ! I ,L ,

~5.5!

where we introduced two independent arbitrary vectorsf IandgI to emphasize that the above equations should be validirrespectively of the particular structure of (d /d J) I or(d /dK) I . The relations~5.4! and~5.5! say thatR acts like aderivation on the string fields with respect to the rule of theirmultiplications defined by the merging-splitting interactions.Note also the possibility of interpreting the operatorR as aconserved charge on the world sheet.

The definitions of~ and` for lower components in theoriginal model are given in Eqs.~95! and ~96! of @6#. Forattempting the proof of commutativity, we need more precisedefinitions for general configurations.

~ f~g!A~z!52 f A~z!]zgA~z!2gA~z!]z f A~z!,

~ f~g!B~s!52 f B~s!]sgB~s!2gB~s!]s f B~s!,

~ f~g!n~z1 ,s1 , . . . ,zn ,sn!

52 (k51

n

f A~zk!]zkgn~z1 ,s1 , . . . ,zn ,sn!2 (

k51

n

f B~sk!]skgn~z1 ,s1 , . . . ,zn ,sn!

1(k, l

Dz~zk ,z l ! f l2k~z,sk , . . . ,z l21 ,s l21!Dz8~zk ,z l !gn2 l1k~z1 , . . . ,zk21 ,sk21 ,z8,s l , . . . ,sn!

1(k, l

Ds~sk ,s l ! f l2k~zk11 ,sk11 , . . . ,z l ,s!Ds8~sk ,s l !gn2 l1k~z1 , . . . ,zk ,s8,z l11 ,s l11 , . . . ,sn!

2 (k51

n

gA~zk!]zkf n~z1 ,s1 , . . . ,zn ,sn!2 (

k51

n

gB~sk!]skf n~z1 ,s1 , . . . ,zn ,sn!

1(k, l

Dz~zk ,z l !gl2k~z,sk , . . . ,z l21 ,s l21!Dz8~zk ,z l ! f n2 l1k~z1 , . . . ,zk21 ,sk21 ,z8,s l , . . . ,sn!

1(k, l

Ds~sk ,s l !gl2k~zk11 ,sk11 , . . . ,z l ,s!Ds8~sk ,s l ! f n2 l1k~z1 , . . . ,zk ,s8,z l11 ,s l11 , . . . ,sn!, ~5.6!

~` f !A,A~z;z8!52]z]z8Dz~z,z8! f A~z!,

~` f !A,B~z;s!5~` f !B,A~s;z!50,

~` f !B,B~s;s8!52]s]s8Ds~s,s8! f B~s!,

~` f !n,A~z1 ,s1 , . . . ,zn ,sn ;z8!5~` f !A,n~z8;z1 ,s1 , . . . ,zn ,sn!

52]z8(k51

n

]zkDz~zk ,z8! f n~z1 , . . . ,zk21 ,sk21 ,z,sk , . . . ,sn!,

~` f !n,B~z1 ,s1 , . . . ,zn ,sn ;s8!5~` f !B,n~s8;z1 ,s1 , . . . ,zn ,sn!

52]s8(k51

n

]skDs~sk ,s8! f n~z1 , . . . ,zk ,s,zk11 ,sk11 , . . . ,sn!,


~` f !n,m~z1 ,s1 ,..., zn ,sn ;z18 ,s18 ,..., zm8 ,sm8 !

5 (k51

n

(l51

m

Dz~zk ,z l8!Dw~zk ,z l8! f n1m~z1 ,s1 ,..., sk21 ,z,s l8 ,..., sm8 ,z18 ,..., s l218 ,w,sk ,..., sn!

1 (k51

n

(l51

m

Ds~sk ,s l8!Dt~sk ,s l8! f n1m~z1 ,..., zk ,s,z l118 ,..., zm8 ,sm8 z18 ,..., z l8 ,t,zk11 ,..., zn ,sn!.

~5.7!

Note that we have slightly generalized the original definitionof ~ by introducing two different vectorsf I andgI .

For the dual model, we have already presented the defi-nitions of~ and` in ~3.10! and ~3.11! in Sec. III. Again,we make a slight generalization of the definition of~ asabove. The prescription in this generalization is the same asabove. It is sufficient to explain it using a simple example.For instance, theI51 component for the dual model is de-fined in Eq.~3.10! as

S d

dK~

d

dK D1

~j1!522d

dKX~j1!]j1

d

dK1~j1!.

Correspondingly, we generalize the definition as

~ f~g!1~j1!52 f X~j1!]j1g1~j1!2gX~j1!]j1

f 1~j1!.~5.8!

The prescriptions for generic cases can be easily deducedfrom this example.

Let us now proceed to prove Eqs.~5.4! and~5.5! for bothoriginal and dual models on the basis of the rules~5.1!–~5.3!. Most parts of the proof consist of straightforward cal-culations after plugging all these definitions into the formu-las. However, we need several nontrivial identities for somecases. The set of such identities and the cases to use them arethe same for both original and dual models, signaling theiruniversal nature.

For Eq.~5.4!, we use the following two identities valid forarbitrary functionsF(z), G(z) andH(z,w) :

Dz~z1 ,z2!F~z!]zG~z!

5 (k51

2

F~zk!]zkDz~z1 ,z2!G~z!

1Dz~z1 ,z2!F~z!Dz8~z1 ,z2!G~z8!, ~5.9!

Dz~z1 ,z2!Dz8~z,zk!F~z8!Dz9~z,zk!G~z9!

5Dz~z1 ,zk!Dz8~z,z2!F~z8!Dz9~z1 ,zk!G~z9!

1Dz8~z2 ,zk!F~z8!Dz~z2 ,zk!Dz9~z1 ,z!G~z9!.

~5.10!

For Eq.~5.5!,

Dz~z1 ,z2!]zDz~z,z8!F~z!5]z1Dz~z8,z1!Dz~z,z2!F~z!

1]z2Dz~z8,z2!Dz~z1 ,z!F~z!,

~5.11!

]z]z8Dz~z,z8!F~z!5Dz~z,z8!Dw~z,z8!Dz~z,w!F~z!,~5.12!

Dz~z1 ,z2!Dz~z,z8!Dw~z,z8!H~z,w!

5Dz~z2 ,z8!Dz~z,z1!Dw~z2 ,z8!H~z,w!

1Dz~z1 ,z8!Dz~z1 ,z8!Dw~z,z2!H~z,w!.

~5.13!

The nature of these identities can be seen pictorially in Figs.1–5.5 For the convenience of the reader, we have includedTable I, indicating precisely where these identities should beused. Identities~5.9!, ~5.11! and ~5.12! have already ap-peared in Appendix E of@6# as Eqs.~E6!, ~E10!, and~E11!,respectively. On the other hand, identities~5.10! and ~5.13!are not mentioned in@6# where simpler spin configurationsare dealt with.

Finally, we will explain a property of the combinatorialderivative which is useful in the proof. Obviously, the com-binatorial derivative is symmetric under interchange of itsarguments:

Dz~z1 ,z2!F~z!5F~z1!

z12z21F~z2!

z22z1. ~5.14!

For two or three successive combinatorial derivatives, wehave

Dz~z1 ,z2!Dz8~z,z3!F~z8!

5F~z1!

~z12z2!~z12z3!1

F~z2!

~z22z3!~z22z1!

1F~z3!

~z32z1!~z32z2!, ~5.15!

5The notation for the original model is used in these figures, butthe argument is also valid for the dual one, after changing the vari-ablesz1 ,s,z2 into j1 ,j2 ,j3.


Dz~z1 ,z2!Dz8~z,z3!Dz9~z8,z4!F~z9!

5Dz~z1 ,z2!Dz8~z3 ,z4!Dz9~z,z8!F~z9!

5F~z1!

~z12z2!~z12z3!~z12z4!

1F~z2!

~z22z3!~z22z4!~z22z1!

1F~z3!

~z32z4!~z32z1!~z32z2!

1F~z4!

~z42z1!~z42z2!~z42z3!, ~5.16!

which are symmetric under interchange amongz1 ,z2 ,z3 or

FIG. 1. The pictorial explanation of the identity~5.9!. This isrequired for the following cases. Suppose there are three successivedomains withz1 ,s,z2. The identity shows that the reduction of thedomain with s commutes with the particular splitting processeswhich occur in thez1 domain or in thez2 domain.

FIG. 2. The pictorial explanation of the identity~5.10!. Considerthe three successive domains withz1 ,s,z2 as in Fig. 1. The identityshows that the reduction of the domain withs commutes with asplitting process between the domain withz1 and another domainwith zk and one between the domain withz2 and the domain withzk .

FIG. 3. The pictorial explanation of the identity~5.11!. Thisidentity is required when one of two strings consists of only onedomain withz8. @By ‘‘only one domain,’’ we meanCX ~not C1)for the dual model.# Consider the three successive domains withz1 ,s,z2 in the other string. Merging processes that the string withone domain merges into thez1 domain or thez2 domain commutewith the reduction of thes domain.

FIG. 4. The pictorial explanation of the identity~5.12!. Whenone of two strings consists of two domains withz8 ands8 and weconsider the reduction of thes8 domain, we need this identity. Weconsider a merging process involving thez8 domain and az domainin the other string. The identity shows that it commutes with thereduction of the domain withs8.

FIG. 5. The pictorial explanation of the identity~5.13!. We fo-cus on the three successive domains withz1 ,s,z2 in a string. Weconsider a merging process between thez1 domain and az8 domainin the other string and one between thez2 domain and thez8 do-main. This identity represents the commutativity of those mergingprocesses with the reduction of thes domain.


z1 ,z2 ,z3 ,z4, respectively. Physically, these properties implythat the order of merging of three or four domains is irrel-evant.

The commutativity we have just proven on the basis ofthe rules~5.1!–~5.3! means that the interaction terms in theHamiltonian in terms of the transformed fields (d /d J) I or(d /dK) I take the same form as those in the bare Hamiltonianin terms of (d /dJ) I or (d /dK) I , with a proviso that some ofthe terms present before the scaling limit can vanish after thescaling limit, as occurring in theYa-derivative terms of ourHamiltonian. Since our proof is based on a general structureof the mixing matrix, being the exponential of a derivationoperatorR, we strongly suspect that the commutativity is auniversal property of the matrix models and their string fieldtheories. Furthermore, we might think a transformation suchas Eq. ~4.1! as a special case of more general symmetrytransformations of the string fields: If a linear transformationof string fields obeys the general rules~5.1!–~5.3!, the inter-action terms of the string field theory are invariant under thetransformation, leaving aside the question of generating ki-netic terms in attempting a background independent formu-lation of the string field theory as discussed in@6#.

VI. DISCUSSION OF DUALITY

In Sec. IV, we have derived the string field Hamiltonianin the dual formalism. The result shows that theT-dualitysymmetry would be broken if the topological excitation ofthe Y loops at the quantum level could not be neglected.Here, in order to check this, we shall calculate explicitly thedisk amplitudes with one handle in both the original and dualtheories. We will find that the topologicalY loops indeedcontribute to the amplitude in the continuum theory. Also, afurther correspondence between the original and dual Hamil-tonians will be discussed.

A. Amplitudes for cylinder and disk with one handlein the original theory

We first derive the amplitude for disk with one handle inthe original theory by taking the continuum limit of theSchwinger-Dyson equation. To do this, it is necessary toknow the form of the cylinder amplitudeFA(z)FA(z8)&0.The Schwinger-Dyson equation determining this is

05F3^FA~z!&0212S 22z12gz21

c

gD ^FA~z!&0

2gK 1N trAL0

112gz1~z2gz2!S z2gz22c

gD 1c3

gzG

3^FA~z8!FA~z!&01F2g^FA~z!&0

1gS z2gz22c

gD 211gzG K FA~z8!1

NtrAL

0

1g~22gz!K FA~z8!1

NtrA2L

0

2g2K FA~z8!1

NtrA3L

0

11

N2 ]z8Dz~z,z8!F ^FA~z!&021^FA~z!&0^FA~z!&0

1S 22z12gz21c

gD ^FA~z!&0G , ~6.1!

whereFA(z)5(1/N) tr@1/(z2A)#, and the symbol •••&hrepresents a connected amplitude withh handles. This isderived using a similar method as in the closed cubic equa-tion for the disk amplitudeW(z) @6#. Equation~6.1! can besolved by using the following two facts. One is that the fac-tor in front of ^FA(z8)FA(z)&0 becomes

a8/33cs8/3

40322/3@w~y!22T4/3#1O~a3! ~6.2!

in the continuum limit, where for the continuum limit in theoriginal theory, we use the notation in Appendix C of Ref.@6#. For instance,

g5g* S 12a2s2

20TD , z5

113c

2g*~11ay!, s521A7,

^FA~z!&05a4/3c1/2s4/3

A10324/3w~y!,

TABLE I. Appearance of nontrivial identities.

Process Original model Dual model Necessary identities

Splitting I5n (n>2) I5n (n>3) ~5.9!

I5n (n>3) I5n (n>4) ~5.10!

Merging (I ,J)5(n,A),(n,B) (n>2) (I ,J)5(n,X) (n>3) ~5.11!

(I ,J)5(2,1) ~5.12!

(I ,J)5(n,1) (n>3) ~5.13!

(I ,J)5(1,1) (I ,J)5(2,2) ~5.12!

(I ,J)5(n,1) (n>2) (I ,J)5(n,2) (n>3) ~5.12!, ~5.13!

(I ,J)5(n,m) (n,m>2) (I ,J)5(n,m) (n,m>3) ~5.13!


andw(y) is the leading term ofw(y):

w~y!5~y1Ay22T!4/31~y2Ay22T!4/3.

The other is that the leading term of Eq.~6.2! vanishes aty50,6AT/2.6 Using the scaling of the cylinder^FA(z8)FA(z)&05O@(1/N2)a22# and the above properties,we can easily see

K FA~z!1

NtrAnL

0

5OS 1N2 a22/3D ~6.3!

for n51,2,3. Then, note that Eq.~6.1! can be written as

05a8/33cs8/3

40•22/3@w~y!22T4/3#^FA~z8!FA~z!&0

2a4/3c2s4/3

24/3w~y!K FA~z8!

1

NtrAL

0

1ac2syg2~y8!1c2g3~y8!11

N2 a2/3

c2s2/3

4322/3]y8

3Fw~y!2w~y8!

y2y8@2w~y!1w~y8!#G1OS 1N2aD , ~6.4!

where

g2~y8!52A7K FA~z8!1

NtrAL

0

2A10cFA~z8!1

NtrA2

0 ,

g3~y8!524K FA~z8!1

NtrAL

0

1A70cK FA~z8!1

NtrA2L

0

210

3cK FA~z8!

1

NtrA3L

0

21

N2

2

31a1/3

1

N2

s1/3

3321/3]yw~y!.

From the consistency of Eq.~6.4!, the nontrivial orders ofg2 andg3 must be at most

g2~y!5OS 1N2a21/3D , g3~y!5OS 1N2a

2/3D . ~6.5!

Now, using the second of the above two facts, solving Eq.~6.4! to the leading order is an easy task. We arrive at

K FA~z!1

NtrAL

0

51

N2a22/3224/3s22/3]y

3Fy21w~y!1T1/3

4~y22T/2!y

3@w~y!21T2/3w~y!22T4/3#G1OS 1N2a

21/3D , ~6.6!

g2~y!521

N2a21/3228/3s21/3]y

3F 1

y22T/2@w~y!21T2/3w~y!22T4/3#G

1OS 1N2a0D , ~6.7!

g3~y!521

N2a2/3

s2/3

12•22/3]yFy21@w~y!222T4/3#

1T

4~y22T/2!y@w~y!21T2/3w~y!22T4/3#G

1OS 1N2aD , ~6.8!

^FA~z!FA~z8!&051

N2a22

10c

s2w~y,y8!1OS 1N2a

25/3D ,

w~y,y8!54

9

1

f ~y,y8!g~y,y8!

z1~y!1/3

z1~y!2/31z2~y!2/31T1/3z1~y8!1/3

z1~y8!2/31z2~y8!2/31T1/3F113T1/3

f ~y,y8!13

z1~y!z1~y8!

g~y,y8! G ,~6.9!

where

z6~y!5y6Ay22T, f ~y,y8!5z1~y!1/3z1~y8!1/31z2~y!1/3z2~y8!1/31T1/3,

g~y,y8!5z1~y!2/31z1~y8!2/31z1~y!1/3z1~y8!1/3.

6The Riemann surface of the diskw(y) consists of a three-sheeted plane. The first sheet has one cut in the regiony,2AT on the real axis,the second one has the two cutsy,2AT andy.AT, and the third has the one cuty.AT. In this statement, it should be considered thaty50 andAT/2 are points on the first sheet buty52AT/2 on the second sheet. Here, we derive the cylinder amplitude by assuming thatit is regular at these three points.


Next, let us proceed to the amplitude for disk with one handle. The Schwinger-Dyson equation we consider is

05F3^FA~z!&0212S 22z12gz21

c

gD ^FA~z!&02gK 1N trAL0


gD1c3

gzG ^FA~z!&1

1F3^FA~z!&01c

g22z12gz2G^FA~z!2&02gK FA~z!

1

NtrAL

0

2g^FA~z!&0K 1N trAL1

11

N2

1

2]z2^FA~z!&01FgS z2gz22

c

gD2c~12gz!112cG K 1N trAL1

2g2K 1N trA3L1

. ~6.10!

Note that the factor in front ofFA(z)&1 is the same as the first term in Eq.~6.1!. This equation can then be solved by the sametechnique as before. The result is

K 1N trAL1

51

N2a22

A1024c1/2s2

T211OS 1N2a24/3D , ~6.11!

K 1N trAL1

2c

3 K 1N trA3L1

521

N2a22/3

A10216327/3c1/2s2/3

T21/31OS 1N2a0D , ~6.12!

^FA~z!&151

N2a210/3

40A10c22/3s10/3

w~1!~y!1OS 1N2a23D , ~6.13!

where

w~1!~y!51

216T

1

@z1~y!2/31z2~y!2/31T1/3#5$12y2177T115T1/3@z1~y!4/31z2~y!4/3#150T2/3@z1~y!2/31z2~y!2/3#%.

~6.14!

B. Amplitudes for cylinder and disk with one handle in the dual theory

On the other hand, in the dual theory, the Schwinger-Dyson equation for the cylinder is

05H 23g^CX~j!&0212@2c~11c!2g2j21~125c!gj#^CX~j!&012g2K 1N trXL

0

2g~123c!

22cj~11c22gj!~12c2gj!J ^CX~j8!CX~j!&01g@2g^CX~j!&022c~32c!14cgj#K CX~j8!1

NtrXL

0

14cg2K CX~j8!1

NtrX2L

0

1g21

N2 ]j8K 1N trS 1

j82XY

1

j2XYD L

0

11

N2 ]j8Dz~j,j8!$2g^CX~z!&022g^CX~j!&0^CX~z!&01@2c~11c!2g2z21~125c!gz#^CX~z!&0%, ~6.15!

which is also derived in a similar way as obtaining Eq.~B2!. Note that the term

g21

N2 ]j8K 1N trS 1

j82XY

1

j2XYD L

0

comes from the following Schwinger-Dyson equation

05E dN2XdN

2Y(

a

]

]XaF tr 1

j82XtrS 1

j2XYtaYDe2SDG , ~6.16!

which reflects the fact that the dual theory does not distinguish overall spin directions. When both domains ofj andj8 are ofspin up~spin down!, such a term does not appear in the original theory. As we will see, this survives in the scaling limit andleads to the difference between the cylinder amplitudes.

Since the scaling limit of the factor in front of^CX(j8)CX(j)&0 becomes


a8/33c5/2s8/3

24/3A5@ T4/32v~y!2#1O~a3!,

Eq. ~6.15! can be solved by the same argument as before. We have the following result:

K CX~j!1

NtrXL

0

51

N2a22/3222/3s22/3]yH y21v~y!1

T1/3

4~y22T/2!y@v~y!21T2/3v~y!22T4/3#J 1OS 1N2a

21/3D , ~6.17!

2108cK CX~j!1

NtrXL

0

1324A5cK CX~j!1

NtrX2L

0

14

N2

51

N2a2/3s2/3221/3]yH v~y!222T4/31

T

4~y22T/2!@v~y!21T2/3v~y!22T4/3#J 1OS 1N2aD , ~6.18!

^CX~j8!CX~j!&051

N2a22s225c@2wD~y,y8!1uD~y,y8!#1OS 1N2a

25/3D , ~6.19!

wherewD(y,y8) is the result of the replacementT→T in w(y,y8), anduD(y,y8) represents the difference of the functionalforms between the cylinder amplitudes:

uD~y,y8!524

9

y

v~y!1T2/3y8

v~y8!1T2/31

~y22y82!2

3$z1~y!8/31 z2~y!8/31 z1~y8!8/31 z2~y8!8/326@ z1~y!4/31 z2~y!4/3#@ z1~y8!4/31 z2~y8!4/3#

25T2/3@ z1~y!4/31 z2~y!4/31 z1~y8!4/31 z2~y8!4/3#16T2/3@ z1~y!2/31 z2~y!2/3#@ z1~y8!2/31 z2~y8!2/3#

14~2y22T!@ z1~y8!2/31 z2~y8!2/3#14@ z1~y!2/31 z2~y!2/3#@2y822T#%11

~y1y8!2, ~6.20!

wherez6(y) is obtained by replacingT to T in z6(y). As will be mentioned later, the difference is interpreted as coming fromthe oddY-loop excitations on a nontrivial homology cycle of the cylinder.

Also, for the disk with one handle, we can solve the Schwinger-Dyson equation

05H 23g^CX~j!&0212@2c~11c!2g2j21~125c!gj#^CX~j!&02g

2K 1N trXL0

2g~123c!22cj~11c22gj!~12c2gj!J3^CX~j!&11@23g^CX~j!&012c~11c!2g2j21~125c!gj#^CX~j!2&012g2K CX~j!

1

NtrXL

0

2g2K S 1N tr1

j2XYD 2L

0

2g1

N2

1

2]j2^CX~j!&01@2g2^CX~j!&012cg~231c12gj!#K 1N trXL

1

14cg2K 1N trX2L1

. ~6.21!

Note the appearance of the term2g2^{(1/N)tr@1/(j2X)]Y} 2&0, which does not allow the interpretation interms of dual spins.

The solution is

K 1N trXL1

51

N2a22

A524c1/2s2

T211OS 1N2a24/3D ,

2108A5c3/2K 1N trXL1

120cK 1N trX2L1

521

N2a22/3s22/3

25

36•22/3T21/31OS 1N2a

0D ,~6.22!

^CX~j!&151

N2a210/3s210/353/2322/3c1/2

3@2w~1!D~y!1u~1!~y!#1OS 1N2a23D ,

~6.23!

wherew(1)D(y) is obtained by replacingT to T in w(1)(y),and the difference from the original theory is

u~1!~y!521

54T1/35@ z1~y!2/31 z2~y!2/3#111T1/3

@ z1~y!2/31 z2~y!2/31T1/3#5.

~6.24!


C. Comparison of the amplitudes

We have obtained the amplitudes for cylinder and diskwith one handle both in the original and dual theories. Theresult of the disk with one handle illustrates that theT-duality symmetry does not hold in the higher genus am-plitudes as is expected from the discussion in Sec. II. On theother hand, the difference between the cylinder amplitudesseems to be puzzling. This is resolved as follows. Since thedual theory can not discriminate overall dual spin directions,it is necessary to take account of the cylinder amplitude^FB(z8)FA(z)&0, in comparing correlators with the originaltheory. The Schwinger-Dyson equation needed to obtain thisis

05F3^FA~z!&0212S 22z12gz21

c

gD ^FA~z!&0

2gK 1N trAL0


gD1c3

gzG

3^FB~z8!FA~z!&01F2g^FA~z!&0

1gS z2gz22c

gD211gzG K FB~z8!1

NtrAL

0

1g~22gz!K FB~z8!1

NtrA2L

0

2g2K FB~z8!1

NtrA3L

0

2c2

g

1

N2 ]z8K 1N trS 1

z2A

1

z82BD L0

. ~6.25!

The way to solve this is the same as before. We obtain

K FB~z!1

NtrAL

0

51

N2a22/3224/3s22/3]yH y21w~y!1

T1/3

4~y22T/2!y

3@w~y!21T2/3w~y!22T4/3#J 1OS 1N2a21/3D ,

~6.26!

2A7K FB~z!1

NtrAL

0

2A10cK FB~z!1

NtrA2L

0

521

N2a21/3228/3s21/3]yH 1

y22T/2@w~y!2

1T2/3w~y!22T4/3#J 1OS 1N2a0D , ~6.27!

24K FB~z!1

NtrAL

0

1A70cK FB~z!1

NtrA2L

0

210

3cK FB~z!

1

NtrA3L

0

21

N2

1

32a1/3

1

N2

s1/3

3324/3]yw~y!

521

N2a2/3

s2/3

12•22/3]yH y21@w~y!222T4/3#

1T

4~y22T/2!y@w~y!21T2/3w~y!22T4/3#J

1OS 1N2aD , ~6.28!

and at the leading order the Schwinger-Dyson equation for^FB(z8)FA(z)&01^FA(z8)FA(z)&0 becomes the sameform as that for CX(z8)CX(z)&0. Thus, we have

^FB~z8!FA~z!&01^FA~z8!FA~z!&0

51

N2a22

10c

s2@2w~y,y8!1u~y,y8!#1OS 1N2a

25/3D ,~6.29!

where u(y,y8) is obtained by replacingT to T inuD(y,y8).

It can now be seen that the mixed cylinder amplitude

^@FA~z!1FB~z!#@FA~z8!1FB~z8!#&0

of the original theory has the same functional form as^CX(j)CX(j8)&0 of the dual theory, which means the dual-ity symmetry in the cylinder level. On the cylinder in thedual theory,Y loops along a nontrivial homology cycle canexit. A natural interpretation is that the amplitude with theeven numberY loops corresponds toFAFA&01^FBFB&0in the original theory, and that the amplitude with the oddnumber ones to the cross term^FAFB&01^FBFA&0. This isconsistent with the picture of the dual spins.

We can thus conclude that, to the leading order of theuniversal parts of the disk and cylinder amplitudes, the fol-lowing identification between the original and dual theoriesholds:

T⇔T,

s

A2⇔ s,

1

A2@FA~z!1FB~z!#⇔CX~j!,

1

A2tr~A1B!⇔ trX.


This can be checked for all the disk and cylinder amplitudeswe have obtained.

D. Comparison of the string field Hamiltonians

Until now, after a long analysis of the Schwinger-Dysonequations, we have shown that theT-duality symmetry holdsin some disk and cylinder amplitudes without handles, andthat it is violated in the higher genus amplitude. Here, wemake sure that in general theT-duality is a symmetry in theplanar approximation, by comparing the forms of the stringfield Hamiltonians.

We start with the Hamiltonian of the original theory,where the numerical constantcs21 in the kinetic term isdropped, since it can be absorbed into the finite renormaliza-tion of the fictitious time by rescaling as

JI→1

cs21 JI ,d

d JI→cs21

d

d JI, gst→cs21gst.

We introduce the string fields which have a definite paritywith respect to the reversal (A↔B) of spin directions:

JX~6 !~y!5 JA~y!6 JB~y!,

J2n~6 !~y1 ,y2 , . . . ,y2n!5 1

2 @ Jn~y1 ,y2 , . . . ,y2n!

6 Jn~y2 , . . . ,y2n ,y1!#,

d

d JX~6 !~y!

51

2 S d

d JA~y!6

d

d JB~y!D ,

d

d J2n~6 !~y1 ,y2 , . . . ,y2n!

51

2 F d

d Jn~y1 ,y2 , . . . ,y2n!

6d

d Jn~y2 , . . . ,y2n ,y1!G

~n51,2,3, . . .!. ~6.30!

Recalling that the variableJn(y1 , . . . ,y2n) in the originaltheory has the cyclic symmetry with respect to the permuta-tions of n pairs (y1 ,y2),(y3 ,y4), . . . ,(y2n21 ,y2n), we seethat the new variablesJ(6)’s and the derivatives with respectto them have a definite symmetric or antisymmetric propertyunder the cyclic permutations of the 2n variablesy1 ,y2 , . . . ,y2n . Note that the symmetric property ofJ(1)’sis identical with that of the string fields in the dual theory.

Then the functional derivatives of the new variables be-come

d JX~P!~y!

d JX~Q!~y8!

52p idP,Qd~y2y8!,

d J2n~1 !~y1 , . . . ,y2n!

d J2n~1 !~y18 , . . . ,y2n8 !

51

2n~2p i !2n(

cd~y12yc~1!8 !•••d~y2n2yc~2n!8 !,

d J2n~2 !~y1 , . . . ,y2n!

d J2n~2 !~y18 , . . . ,y2n8 !

51

2n~2p i !2n(

c~21! ucu

3d~y12yc~1!8 !•••d~y2n2yc~2n!8 !,

~6.31!

and the others vanish, whereP andQ take1 or 2, andcrepresents the cyclic permutations of 1, . . . ,2n. The notation(21)ucu denotes its sign, namely, (11) for even cyclic per-mutations or (21) for odd cyclic permutations.

By using these variables, the Hamiltonian is written in theform

H5H~1 !1H~2 !,

H~1 !52 J~1 !S F d

d J~1 !D 2 J~1 !S d

d J~1 !~

d

d J~1 !D21

2g st2 J~1 !F J~1 !S `

d

d J~1 !D G ,H~2 !52 J~2 !S F d

d J~2 !D 2 J~1 !d

d J~2 !~

d

d J~2 !

2 J~2 !S d

d J~1 !~

d

d J~2 !1@~1 !↔~2 !# D

21

2g st2 J~2 !F J~2 !S `

d

d J~1 !D G21

2g st2 $J~1 !J~2 !1@~1 !↔~2 !#%S `

d

d J~2 !D ,~6.32!

where only even numbers of the (2) fields appear becauseof theZ2 symmetry of the original Hamiltonian.

The above expression shows that if the coupling is re-scaled asg st

2→2g st2 , the form ofH(1) becomes completely

identical with that of the dual HamiltonianHD ~4.30! aftersetting the fields with odd domains to zero. There still remainthe portions that do not match between the original and dualtheories. They areH(2) in the original theory and the termscontaining the fields with odd domains in the dual theory.These contributions give no effects in the tree amplitudeswhen the initial state does not contain the (2) fields, in theoriginal theory, because the vacuum expectation value of asingle (2) field vanishes due to theZ2 symmetry. Similarly,in the dual theory, the string fields with odd domains neverappear for the tree amplitudes when the initial state consistsonly of the fields with even domains. Thus we can concludethat at the level of the tree amplitudes theT-duality symme-try holds, and thatJ2n

(1) in the original theory are identifiedwith K2n in the dual theory.


VII. CONCLUDING REMARKS

We have presented a detailed study of the Schwinger-Dyson equations and the stochastic Hamiltonians for thetwo-matrix model formulated in both the dual and originalvariables. We have first performed the duality transformationat the discrete level and then taking the scaling limit sepa-rately in both the original and the dual formulations. Todeepen our understanding on duality and also to extend ourresults to more general cases, it is desirable to find a directtransformation of the string fields under theT-dual transfor-mation in the continuum limit.

Our two main results were the following.~1! The theory in the scaling limit is duality symmetric in

the sphere approximation. Our results for the disc amplitudewith one handle however indicate that the duality symmetryis violated for higher genus because of the nonsymmetricalappearence of the globalZ2 vector fields. To make the theorydual in higher genus, we would have to introduce some newvariables representing global winding modes in the AB for-malism.

~2! We extended the previous discussion on the remark-able commutativity property between the mixing of thestring fields in the process of taking the scaling limit and themerging-splitting interactions of the string fields, by propos-ing a general rule of the string-field mixing.

The first result, which is a consequence of the absence ofmanifest momentum-winding symmetry, might be importantin thinking aboutT-duality transformation in string theoryfor general manifolds without simple target-space isometry.If we want to treat the dual transformed theories in equalfooting with the original theories as required once we intro-duceD-branes, this would suggest that the fundamental vari-ables are world sheet-vector fields or currents instead of thetarget space coordinates.

On the other hand, however, we have to emphasize thatour result inno way shows that theT-duality symmetry isviolated in the matrix model approach in general, but onlysuggests that the usual multimatrix models using the matrixvariables as spin variables in the standard way would violatetheT-duality symmetry for higher genus. For example, if wereformulate the original spin model from the beginning interms of the link variables instead of the site variables, itmay be possible to construct manifestly duality symmetricmodels by explicitly introducing the globalZ2 variables rep-resenting winding modes. Such a possibility is suggestedfrom, e.g., the discussion in Ref.@7#.7 In fact, it is also easyto define a self-dual lattice model using the same method asa self-dual U~1! spin model as a lattice-regularized version ofa circle-compactified string theory. This is discussed in Ap-pendix A. A difficulty is that such a model in general re-quires us to introduce both the site and link variables simul-taneously8 in order to include the global windinglike modesbefore the dual transformation and cannot be described interms of simple matrix models which aresolublewithin thereach of our present technical machinery. We are planning to

present a general formulation of manifestlyT-duality sym-metric matrix models in a subsequent work~see note added!.

The second result concerning the universal structure ofstring field theories might be useful in discussing possiblesymmetry structure and the background independence ofstring field theories in general, quite independently of theproblem of the operator mixing.

From the viewpoint of our original motivation to thepresent work, a crucial question is whether this kind of theformulation of string field theory is really possible for thehigher-dimensional models, especially, supersymmetry. Ifwe can write down a single Hamiltonian which is supposedto govern all-genus behaviors of the theory at once. it wouldbe a promising starting point for studying the nonperturba-tive symmetries includingS-duality structure.

Note added. We have constructed a new class of mani-festlyT-duality symmetric matrix models@14# along the linediscussed above.

ACKNOWLEDGMENTS

The work of T.Y. was partially supported by a U.S.-JapanCollaborative Program from the Japan Society for the Pro-motion of Science. The work of Y.O. was partially supportedby the Japan Society for the Promotion of Science.

APPENDIX A: STRING T DUALITYAS KRAMERS-WANNIER DUALITY

In this appendix, we briefly discuss how theT dualitysymmetry of circle-compactified string theories is reformu-lated as a special case of the Kramers-Wannier duality,namely that of anXY model with a Villain-type action on ageneral lattice. For convenience, we will use the familiarnotation of square lattice, withDm being the directional dif-ference between nearest-neighbor sites along the directionm. Note, however, that the arguments below are valid forarbitrary lattices with nearest-neighbor interactions.

The partition function of the model for a lattice on anarbitrary two-dimensional surface is defined as

Z5 R @dX#S)s,m

(ms,m

D)s

d~Dmms,n2Dnms,m!

3expF2R2

4p(s,m

~DmXs22pms,m!2G , ~A1!

where the constraintDmms,n2Dnms,m50 is imposed in or-der to suppress thelocal vortex excitations which are forbid-den in the usual continuum formulation of circle-compactified string theory. The target space coordinateX ofthe compactification circle is normalized such that the peri-odicity is 2p and the symbolr@dX# denotes the path inte-gral with respect toX over a single period. Correspondingly,the variablems,m takes integer values, and the radius param-eterR appears in front of the world-sheet action. Thus, theR2 just plays the role of the inverse temperature. Note alsothat, after solving the constraint, the integer-valued vectorfieldms,m just represents the winding modes along nontrivialhomology cycles for an arbitrary surface of generic genus,since the pure gauge mode of the formms,m5Dmms is ab-

7We would like to thank the authors of Ref.@7# for an interestingcomment on this question.8For an example of such a model, see@11#.


sorbed into a shift of the target coordinateXs→Xs12pms , which in turn ensures the periodicity of theintegrand of the partition function after the summation overms .

Now performing the Fourier transformation and introduc-ing auxiliary variableX s to exponentiate the constraint, wehave

Z} R @dX#S)s,m

(ms,m

D R @dX#E S )s ,m

dc s ,mD3exp(

sF iX s

emn

2~Dmms,n2Dnms,m!

1 i1

2pemnc s ,m~DnXs22pms,n!2

1

4pR2c s ,m2 G ,

where the integrals forX s are again over a single period2p, while the integral forc s ,m is the whole real axis. Notealso that all the new variables are introduced on the duallattice; namely,X s on dual sites,c s ,m on dual links, and thecorrespondence between the original sites and the dual sites is fixed uniquely in the first and the second terms on theexponential of the integrand. The summation overms,m leadsto the constraint

c s ,m5DmX s22pns ,m , ~A2!

wherens ,m defined on each dual link takes arbitrary integervalues. After substituting this result and performing theXintegral, we obtain the constraint forns ,m

Dmns ,n2Dnns ,m50, ~A3!

whose solutions are again the global integer-valued vectorfields along nontrivial homology cycles. Thus, the partitionfunction now takes the form

Z} R @dX#S )s ,m

(n s ,m

D)s

d~Dmns ,n2Dnns ,m!

3expF21

4pR2(s,m

~DmX s22pns ,m!2G , ~A4!

which is identical to the original form~A1! with the corre-spondenceR↔1/R. Note that the integer-valued vector fieldns ,m , appearing as the winding mode of the dual coordinateX s can be interpreted as the momentum mode for the origi-nal coordinateXs . The symmetrical role of the winding(mm) and the momentum modes (nm) for arbitrary genus isprecisely the structure of the usualT duality symmetry.

Obviously, the above model can be easily generalized forZ(N) target space preserving the self-dual structure. How-ever, it is not easy to construct a simply solvable matrixmodel corresponding to such a lattice model since it requiresus to introduce both link and site variables simultaneously.

APPENDIX B: DISK AND CYLINDER AMPLITUDESIN THE DUAL TWO-MATRIX MODEL

Here, we obtain various disk and cylinder amplitudes inthe dual two-matrix model, by taking the continuum limit ofthe corresponding Schwinger-Dyson equations. Some of ourresults have already been derived in Ref.@7#. However, formaking this paper self-contained, we will briefly mentionthose results.

At first, we introduce the following notations for the diskamplitudes:

VXnYm5 K 1N trXnYmL0

,

V~j!5 K 1N tr1

j2X L0

,

Vj~j!5 K 1N tr1

j2XYj L

0

,

V~m!~j1 , . . . ,jm!5 K 1N tr1

j12XY•••Y

1

jm2XYL

0

,

Vj~m!~j1 ;j2 , . . . ,jm!

5 K 1N tr1

j12XYj

1

j22XY•••Y

1

jm2XYL

0

,

and for the cylinder amplitudes:

VYuYcyl5^ trY trY&0 ,

Vnumcyl ~juh!5 K tr 1

j2XYntr

1

h2XYmL

0

,

Vnu1cyl~j1 ;j2 , . . . ,jmuh!

5 K trS 1

j12XYn

1

j22XY•••Y

1

jm2XYD tr 1

h2XYL

0

,

VnuYjcyl

~j1 ;j2 , . . . ,jm!

5 K trS 1

j12XYn

1

j22XY•••Y

1

jm2XYD trYj L

0

,

V cyl~mu1!~j1 , . . . ,jmuh1!

5 K trS 1

j12XY•••Y

1

jm2XYD tr 1

h12XYL

0

,

VYncyl~m!

~j1 , . . . ,jm!

5 K trS 1

j12XY•••Y

1

jm2XYD trYnL

0

,

where^•••&0 stands for the connected amplitude on the sur-face with no handle.


1. V„j…, V2„j…

As in Ref. @7#, the closed equation forV(j) is derived bycombining the five Schwinger-Dyson equations obtainedfrom the following identities in the planar limit:

05E dN2XdN

2Y(

a

]

]XatrS 1

j2XtaDe2SD,

05E dN2XdN

2Y(

a

]

]YatrS 1

j2XYtaDe2SD,

05E dN2XdN

2Y(

a

]

]YatrS 1

j2XYXtaDe2SD,

05E dN2XdN

2Y(

a

]

]XatrS 1

j2XYtaYDe2SD,

05E dN2XdN

2Y(

a

]

]XatrS 1

j2XY2taDe2SD. ~B1!

The result is

gV~j!31 f 2V~j!21 f 1V~j!1 f 050, ~B2!

where

f 25g2j21~5c21!gj22c~11c!,

f 154cg2j32~6c22c2!gj212c~12c2!j22g2VX

1g~123c!,

f 05~6c22c224cgj!gVX24cg2VX224cg2j2

1~6c22c2!gj22c~12c2!2g2.

Since the original and dual theories are connected by thetransformationX↔(A1B)/A2, Y↔(A2B)/A2, g↔g/A2,the partition functions of both theories are identical. So thecritical points of the couplingsc,g are also identical:c*5 (2112A7)/27, g*5A10c

*3 , andVX andVX2 can be

written by the amplitudes in the original theoryW15^(1/N)trA&0 andW35^(1/N)trA3&0 whose forms inthe continuum limit are already known in Ref.@6#:

VX5A2W1 , VX2512c2

cgW12

g

cW32

1

c. ~B3!

The critical pointj* is determined by a similar way asP* inthe original theory,

j*5 sA5c*

, s511A7.

Introducing the lattice spacinga and the variables in thecontinuum theory as

g5g* S 12a2s2

10TD , j5j* ~11ay!,

the continuum limit of the solution of Eq.~B2! is given by

V~j!5V non~j!1V~j!,

V non~j!52f 2

3g,

V~j!5a4/3c1/2s4/3

22/3A5@~y1Ay22T!4/3

1~y2Ay22T!4/3#1O~a5/3!

[a4/3c1/2s4/3

22/3A5v~y!1O~a5/3!,

~B4!

where V non and V denote the nonuniversal and universalparts, respectively. Here and below,c appearing in the for-mulas in the continuum limit is understood to be fixed at thecritical point c* .

Also, V2(j) can be derived in the same way as in theoriginal theory:

V2~j!5V2non~j!1V2~j!,

V2non~j!5

1

g$@~12c2gj!j2V non~j!#V non~j!

1gVXnon211c1gj

1@~12c2gj!j22V non~j!#V~j!%,

V2~j!521

gV~j!21VX[a8/3

s8/3

24/335c1/2v2~y!1O~a3!,

v2~y!52v~y!21 32 T

4/3, ~B5!

where the nonuniversal and universal parts ofVX , VXnon, and

VX are known from those ofW1 via Eq. ~B3!. They-independent constant inV2(j) is different from that in thecorresponding amplitude@W1(z)# of the original theory.However, in the rule for identifying the nonuniversal anduniversal parts, the termVX , which can be seen as an am-plitude with a simpler spin configuration thanV2(j), is al-lowed to be absorbed in the nonuniversal part. After adopt-ing such a convention for both original and dual theories,there are no real differences. In fact, since the operator cor-responding to this amplitude is always accompanied by thej derivative in the leading contribution of the Hamiltonian,this change does not affect the continuum Hamiltonian.

2. V „2…„j1 ,j2…, V

„4…„j1 ,j2 ,j3 ,j4…

For the higher amplitudeV(2k), we can derive the dualversion of Staudacher’s recursion relation@Eq. ~20! in Ref.@12##. Here, we consider the continuum limit for the casesk51 and 2. The corresponding Schwinger-Dyson equationsare


V~2!~j1 ,j2!5V~j1!V~j2!2g@V2~j1!1V2~j2!#

11c2g~j11j2!, ~B6!

V~4!~j1 ,j2 ,j3 ,j4!51

11c2g~j11j4!

3$@~12c2gj1!j12V~j1!2V~j3!2V~j4!#Dz~j1 ,j3!V~2!~z,j2!

1@~12c2gj2!j22V~j1!2V~j2!2V~j4!#Dz~j2 ,j4!V~2!~z,j3!

1@12c2g~j11j3!#V~2!~j2 ,j3!1@12c2g~j21j4!#V

~2!~j3 ,j4!1gV2~j2!1gV2~j3!%.

~B7!

Puttingj i5j* (11ayi) and expanding with respect toa, the result ofV(2) becomes

V~2!~j1 ,j2!5V~2! non~j1 ,j2!1V~2!~j1 ,j2!,

V~2! non~j1 ,j2!51

5~114s!22Ac

5~j11j2!1

2

A5c@V~j1!1V~j2!#,

V~2!~j1 ,j2!5a5/3s5/3

10321/32 v~y1!

22 v~y2!22 v~y1!v~y2!13T4/3

y11y2

1a22s

75@18T110s~y1

21y22!215sy1y2#1O~a7/3!

[a5/3s5/3

10321/3v ~2!~y1 ,y2!1O~a2!, ~B8!

wherev(y) is defined by the rescaling ofV(j):

V~j!5a4/3c1/2s4/3

22/3A5v~y!,

and

v ~2!~y1 ,y2!52v~y1!

22v~y2!22v~y1!v~y2!13T4/3

y11y2.

Using this and repeating the same procedure fork52, we have

V~4!~j1 ,j2 ,j3 ,j4!5V~4! non~j1 ,j2 ,j3 ,j4!1V~4!~j1 ,j2 ,j3 ,j4!,

V~4! non~j1 ,j2 ,j3 ,j4!5124

5c@Dz~j1 ,j3!V~z!1Dz~j2 ,j4!V~z!#

22

A5c$Dz~j1 ,j3!@V

~2!~z,j2!1V~2!~z,j4!#

1Dz~j2 ,j4!@V~2!~j1 ,z!1V~2!~j3 ,z!#%,

V~4!(j1 ,j2 ,j3 ,j4)5as

20v ~4!(y1 ,y2 ,y3 ,y4)1O(a4/3)

5as

20H 216

3(y11y21y31y4)1

1

2

1

y12y3

1

y22y4

3{ 2v ~2!(y1 ,y2)[v(y1)1v(y2)12v(y3)12v(y4)]


1v ~2!~y1 ,y4!@v~y1!12v~y2!12v~y3!1v~y4!#

1v ~2!~y2 ,y3!@2v~y1!1v~y2!1v~y3!12v~y4!#

2v ~2!~y3 ,y4!@2v~y1!12v~y2!1v~y3!1v~y4!#} J 1O~a4/3!. ~B9!

V(2)(j1 ,j2) has the same form as the corresponding amplitudeW(2)(z,s) in the original theory, which has been pointed outin Ref. @7#. Also, V(4)(j1 ,j2 ,j3 ,j4) is the same asW(4)(z1 ,s1 ,z2 ,s2), except the linear term ofyi . Since this term isanalytic with respect toyi ’s, it is allowed to be absorbed to the nonuniversal part. So, the difference can be regarded asnonuniversal.

3. V2„3…„j1 ,j2 ,j3 ; …

This amplitude can be obtained from the Schwinger-Dyson equation

05E dN2XdN

2Y(

a

]

]XatrS 1

j12XY

1

j22XY

1

j32XtaDe2SD.

A similar calculation as before leads to the result

V2~3!~j1 ,j2 ,j3 ; !5V2

~3! non~j1 ,j2 ,j3 ; !1V2~3!~j1 ,j2 ,j3 ; !,

V2~3! non~j1 ,j2 ,j3 ; !5

1

g@V non~j1!1V non~j3!2~12c2gj3!j3#Dz~j1 ,j3!V

~2!~z,j2!

22Ac

5

1

g@V~j1!1V~j3!#2

2

A5cDz~j1 ,j3!V2~z!2V2~j2!2

1

g@12c2g~j11j3!#,

V~2!~j1 ,j2!,V2~3!~j1 ,j2 ,j3 ; !5a2

s2

20A5c@v~y1!1v~y3!#Dz~y1 ,y3!v

~2!~z,y2!1O~a7/3!

[a2s2

20A5cv2

~3!~y1 ,y2 ,y3 ; !1O~a7/3!. ~B10!

4. VYzYcyl , VY

cyl„1…„j…, V1z1

cyl„jzj8…

Let us next consider the cylinder amplitudes. As presented in@7#, VYuYcyl andVY

cyl(1)(j) can be obtained from the singleSchwinger-Dyson equation

V~j!2~11c22gj!VYcyl~1!~j !22gVYuY

cyl50 ~B11!

by using the fact that the coefficient ofVYcyl(1)(j) vanishes atj5(11c)/2g5j*1O(a2). The results are

VYuYcyl5

1

2gVS 11c

2gD

51

5c2a4/3

s4/3

5325/3cT2/31O~a2!, ~B12!

VYcyl~1!~j !5

1

A5c2a1/3

s1/3

25/3A5cv~y!1T2/3

y1O~a!

[1

A5c1VY

cyl~1!~j !. ~B13!

For V1u1cyl(juj8), considering the Schwinger-Dyson equation obtained from the identity


05E dN2XdN

2Y(

a

]

]YaF trS 1

j2XtaD trS 1

j82XYDe2SDG ,

we have

V1u1cyl~juj8!5

2Dz~j,j8!V~z!22gVYcyl~1!~j8!

11c22gj

5a22/31

25/3s2/31

yy8 S y8v~y!2yv~y8!

y2y82T2/3D1O~a0!. ~B14!

5. V2zYcyl„j1 ;j2…, VY

cyl„3…„j1 ,j2 ,j3…

We start with the two identities

05E dN2XdN

2Y(

a

]

]XatrS 1

j12Xta

1

j22XYD trYe2SD,

05E dN2XdN

2Y(

a

]

]YatrS 1

j12XY

1

j22XY

1

j32XtaD trYe2SD.

The amplitudesV2uYcyl(j1 ;j2) andVY

cyl(3)(j1 ,j2 ,j3) can be obtained by using the Schwinger-Dyson equations derived from theabove identities.

The results are

V2uYcyl~j1 ;j2!5V2uY

cyl non~j1 ;j2!1V2uYcyl~j1 ;j2!,

V2uYcyl non~j1 ;j2!5

1

g@V non~j1!1V non~j2!2~12c2gj1!j1#Dz~j1 ,j2!VY

cyl~1!~z!

2VYuYcyl2

1

g@12c2g~j11j2!#VY

cyl~1!~j2!,

V2uYcyl~j1 ;j2!52a2/3

s2/3

27/335c@v~y1!1v~y2!#Dz~y1 ,y2!

v~z!1T2/3

z1O~a!, ~B15!

VYcyl~3!~j1 ,j2 ,j3!5VY

cyl~3! non~j1 ,j2 ,j3!1VYcyl~3!~j1 ,j2 ,j3!,

VYcyl~3! non~j1 ,j2 ,j3!5

2

A5c@2Dz~j1 ,j2!2Dz~j2 ,j3!2Dz~j3 ,j1!#VY

cyl~1!~z!,

VYcyl~3!~j1 ,j2 ,j3!5a21/3

s21/3

4321/3A5cF T2/3

y1y2y3v~y1! 2

2T4/3

~y11y2!~y21y3!~y31y1!

2v~y1!v~y2!S 1

y1~y12y3!1

1

y2~y22y3!D 1

y11y2

2v~y1!23y1

22y1y22y2y32y3y1y1~y1

22y22!~y1

22y32!

1 cyclic permutations ofy1 ,y2 ,y3G1O~a0!. ~B16!

6. VYcyl„5…

„j1 ,j2 ,j3 ,j4 ,j5…

Here, we show only the result of the nonuniversal part of this amplitude in order to examine the mixing among the operatorswith odd domains. It is necessary to consider the two Schwinger-Dyson equations originating from


05E dN2XdN

2Y(

a

]

]YaF trS 1

j12XY

1

j22XY

1

j32XY

1

j42XY

1

j52XtaD trYe2SDG ,

05E dN2XdN

2Y(

a

]

]XaF trS 1

j12XY

1

j22XY

1

j32XY

1

j42XtaD trYe2SDG .

Combining these, we obtain

VYcyl~5!~j1 , . . . ,j5!5

1

11c2g~j11j5!

3$@~12c2gj5!j52V~j1!2V~j2!2V~j5!#Dz~j2 ,j5!VYcyl~3!~z,j3 ,j4!

1@~12c2gj4!j42V~j1!2V~j4!2V~j5!#Dz~j1 ,j4!VYcyl~3!~z,j2 ,j3!

1@Dz~j3 ,j5!V~2!~z,j4!1Dz~j2 ,j4!V

~2!~z,j3!#Dw~j1 ,j2!VYcyl~1!~w!

1@Dz~j1 ,j3!V~2!~z,j2!1Dz~j2 ,j4!V

~2!~z,j3!#Dw~j4 ,j5!VYcyl~1!~w!

1@Dz~j1 ,j3!V~2!~z,j2!1V~j3!1V~j4!2~12c2gj4!j4#Dw~j3 ,j4!VY

cyl~1!~w!

1@Dz~j3 ,j5!V~2!~z,j4!1V~j2!1V~j3!2~12c2gj3!j3#Dw~j2 ,j3!VY

cyl~1!~w!

1@12c2g~j11j4!#VYcyl~3!~j1 ,j2 ,j3!1@12c2g~j21j5!#VY

cyl~3!~j2 ,j3 ,j4!

2@12c2g~j21j3!#VYcyl~1!~j2!2@12c2g~j31j4!#VY

cyl~1!~j3!

2Dz~j1 ,j5!V~4!~z,j2 ,j3 ,j4!22gVYuY

cyl %. ~B17!

In the continuum limit, the nonuniversal partVYcyl(5) non becomes

VYcyl~5!~j1 , . . . ,j5!5VY

cyl~5! non~j1 , . . . ,j5!1VYcyl~5!~j1 , . . . ,j5!,

VYcyl~5! non~j1 , . . . ,j5!5

4

5c@Dz~j1 ,j3!Dw~z,j5!1Dz~j1 ,j3!Dw~z,j4!

1Dz~j2 ,j4!Dw~z,j5!1Dz~j1 ,j2!Dw~z,j4!

1Dz~j2 ,j3!Dw~z,j5!#VYcyl~1!~w!

22

A5c@Dz~j1 ,j3!VY

cyl~3!~z,j4 ,j5!1Dz~j2 ,j4!VYcyl~3!~j1 ,z,j5!

1Dz~j3 ,j5!VYcyl~3!~j1 ,j2 ,z!1Dz~j1 ,j4!VY

cyl~3!~z,j2 ,j3!

1Dz~j2 ,j5!VYcyl~3!~z,j3 ,j4!#,

VYcyl~5!~j1 , . . . ,j5!5O~a21!. ~B18!


From the results until now, we can read off the scaling prop-erty of the universal parts of the string fields as follows.First, the universal part of trY scales asO(a2/3). Since thepartition functions both in the original and dual theories areidentical, the double scaling limits are also@1/N5O(a7/3) inboth theories#. Using this, it can be seen that the stringfields scale as CX5O(a4/3), Cn5O(a7/32n/3), wheren51,2,3, . . . . This coincides with the interpretation of theboundary conformal field theory@13# as in the originaltheory.

APPENDIX C: CONTINUUM SPIN-FLIP OPERATORIN THE DUAL FORMALISM

Similarly as in the original theory, a microscopic domainconsisting only of a single flipped spin can be obtained by anintegration*C(dy/2p i ) with respect toy of the macroscopicdomain. We will check this in the case of the disk and cyl-inder amplitudes.

First, for the disk amplitudes the following relations canbe verified by direct calculations using the results in Appen-dix B:

]y2ECdy12p i

v ~2!~y1 ,y2!5]y2v2~y2!,

EC

dy12p i

v ~4!~y1 ,y2 ,y3 ,y4!5v2~3!~y2 ,y3 ,y4 ; !, ~C1!

where the rule of the integral is identical with that in theoriginal theory. The contourC wraps around the negative

real axis and the singularities in the left half plane, and theunintegrated variables, for exampley2 ,y3 ,y4 in the above,are understood to be outside the contour, while2y2 ,2y3 ,2y4 are understood to be inside the contour.

By including the overall normalizations, Eq.~C1! can bewritten as

]j2 Rˆ dj1

2p iV~2!~j1 ,j2!5]j2

V2~j2!1O~a2!,

Rˆ dj12p i

V~4!~j1 ,j2 ,j3 ,j4!5V2~3!~j2 ,j3 ,j4 ; !1O~a7/3!,

~C2!

where the integral symbolr(dj/2p i ) is used in the sense of

Rˆ dj

2p i5aj* EC

dy

2p i.

In contrast to the original theory, no additional factor of afinite renormalization appears here.

Similarly, for the cylinder amplitudes, we can show thatthe following formulas hold:

Rˆ dj

2p iV1u1

cyl~juj8!5VYcyl~1!~j8!1O~a2/3!,

Rˆ dj

2p iVY

cyl~3!~j1 ,j,j2!5V2uYcyl~j1 ;j2!1O~a!. ~C3!

@1# H. A. Kramers and G. H. Wannier, Phys. Rev.60, 252~1941!.@2# A. Shapere and F. Wilczek, Nucl. Phys.B320, 669~1989!, and

references therein.@3# K. Kikkawa and M. Yamasaki, Phys. Lett.149B, 357 ~1984!.@4# For a review, see J. Polchinski, S. Chaudhuri, and C. V.

Johnson, Report No. hep-th/9602062~unpublished!.@5# N. Ishibashi and H. Kawai, Phys. Lett. B314, 190 ~1993!.@6# F. Sugino and T. Yoneya, Phys. Rev. D53, 4448~1996!.@7# S. M. Carroll, M. E. Ortiz, and W. Taylor IV, Nucl. Phys.

B468 @FS#, 420 ~1996!; B468 @FS#, 383 ~1996!.

@8# A. Jevicki and J. Rodrigues, Nucl. Phys.B421, 278 ~1994!.@9# I. Kostov, Mod. Phys. Lett. A4, 217~1989!; B. Duplantier and

I. Kostov, Nucl. Phys.B340, 491 ~1990!.@10# T. Kugo and B. Zwiebach, Prog. Theor. Phys.87, 801 ~1992!.@11# V. A. Kazakov, M. Staudacher, and T. Wynter, Commun.

Math. Phys.177, 451 ~1996!.@12# M. Staudacher, Phys. Lett. B305, 332 ~1993!.@13# J. L. Cardy, Nucl. Phys.B275 @FS17#, 200 ~1986!.@14# T. Kuroki, Y. Okawa, F. Sugino, and T. Yoneya, Report No.

hep-th/9611207~unpublished!.


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T -duality transformation and universal structure of noncritical string field theory