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Cocycle Properties of String Theories on Orbifolds ∗

arXiv:hep-th/9202084v2 2 Mar 1992KOBE–92–03February 1992Cocycle Properties of String Theories on Orbifolds ∗Tsutomu HORIGUCHI, Makoto SAKAMOTO and Masayoshi TABUSEDepartment of Physics, Kobe UniversityNada, Kobe 657, JapanAbstractWe study cocycle properties of vertex operators and present an operator repre-sentation of cocycle operators, which are attached to vertex operators to ensure theduality of amplitudes. It is shown that this analysis makes it possible to obtain thegeneral class of consistent string theories on orbifolds.∗To appear in the proceedings of YITP Workshop on “Recent Developments inString and Field theory” at Kyoto, Japan on September 9-12, 1991.– 1 –

1. IntroductionOrbifold compactification [1] is believed to provide a realistic four dimensionalstring model.

The search for realistic orbifold models has been continued by manyauthors and various models have been proposed [2-5]. Any satisfactory orbifold modelshave not, however, been found yet.

So far only a very small class of orbifold modelshas been investigated.In the construction of realistic four-dimensional string models, various other ap-proaches have been proposed [6-12]. If string compactification can allow a geometricalinterpretation, orbifold compactification is probably the most efficient method.

AllN = 1 space-time supersymmetric conformally invariant vacua are degenerate. Thedegeneracy should be resolved quantum mechanically and then a true string vacuumwill appear.

If no orbifold models were found to be realistic in spite of thorough in-vestigations, this might indicate that the true string vacuum is far from all conformalinvariant classical vacua and that nonperturbative effects drastically change perturba-tive results [13]. If so, any conformal field theoretical approaches would be useless toconstruct a string model to describe our real world and second quantized string fieldtheoretical approaches [14] might be required.

In any case, more general and thoroughinvestigations of orbifold models would be of great importance and should be done.An orbifold will be obtained by dividing a torus by the action of a discrete symme-try group G of the torus. In ref.

[15], we have clarified the general class of consistentorbifold models: Any element g of G has been shown to be specified byg = (U, v),(1 −1)or more generally for asymmetric orbifolds [16]g = (UL, vL; UR, vR),(1 −2)where (UL, UR) are rotation matrices and (vL, vR) are shift vectors. The correct actionof g on a string coordinate has also been found.

In this paper, we will give some of thedetails of ref. [15], in particular, a geometrical interpretation of our results and variousexamples of orbifolds, which may be good illustrations of our formalism.In section 2, we describe the operator formalism for string theories on orbifoldsand discuss consistency conditions to determine the allowed action of g on a string– 2 –

coordinate. In section 3, we investigate cocycle properties of vertex operators andpresent an explicit operator representation of cocycle operators, which are attached tovertex operators to ensure the duality of amplitudes.

We then see that this analysismakes it possible to obtain the allowed action of g on the string coordinate and hencethe general class of consistent orbifold models.In section 4, we discuss one loopmodular invariance of partition functions and see that this argument justifies ourprescription.In section 5, a geometrical interpretation of our results is discussed.In section 6, we present various examples of orbifold models which may give goodillustrations of our formalism. Section 7 is devoted to discussions.

In appendix A,various useful formulas are given and in appendix B, a part of partition functions isexplicitly evaluated.2. Operator Formaism for Bosonic String Theories on OrbifoldsAn orbifold [1] will be obtained by dividing a torus by the action a suitable discretegroup G. Before the construction of an orbifold, we summarize the basics of stringson a torus.

Let us start with the following action [17]∗S =ZdτZ π0dσ 12π{ηαβ∂αXI∂βXI + ǫαβBIJ∂αXI∂βXJ},(2 −1)where XI(τ, σ) (I = 1, . .

., D) is a string coordinate and BIJ is an antisymmetricconstant background field. Since the second term in eq.

(2-1) is a total divergence,it does not affect the equation of motion. The canonical momentum conjugate toXI(τ, σ), however, becomesP I(τ, σ) = 1π (∂τXI(τ, σ) + BIJ∂σXJ(τ, σ)).

(2 −2)Thereby, the mode expansion of XI(τ, σ) is given byXI(τ, σ) = xI + (pI −BIJwJ)τ + wIσ + i2Xn̸=01n(αILne−2in(τ+σ) + αIRne−2in(τ−σ)),(2 −3)where pI is the center of mass momentum and wI is the winding number.∗ηαβ = diag(1, −1) and ǫ01 = −ǫ10 = 1– 3 –

It is well known that the degree of freedom of the winding number must beincluded in the spectrum of interacting closed strings on a torus. In order to constructthe quantum theory, we will need to introduce a canonical “coordinate” QI conjugateto wI [18].

We now assume the following canonical commutation relations:[xI, pJ] = iδIJ,[QI, wJ] = iδIJ. (2 −4)The string coordinate XI(τ, σ) obeys the boundary conditionXI(τ, σ + π) = XI(τ, σ) + πwI.

(2 −5)A D-dimensional torus T D may be defined by T D = RD/πΛ, where Λ is a D-dimensional lattice. Since XI(τ, σ) is assumed to be a string coordinate on the torus,wI has to lie on the lattice Λ, i.e.,wI ∈Λ.

(2 −6)Since the wave function Ψ(xI) must be periodic, i.e., Ψ(xI + πwI) = Ψ(xI) for anywI ∈Λ, the allowed momentum ispI ∈2Λ∗,(2 −7)where Λ∗is the dual lattice of Λ.For later convenience, we introduce the left- and right-moving coordinatesXI(τ, σ) = 12(XIL(τ + σ) + XIR(τ −σ)),(2 −8)whereXIL(τ + σ) = xIL + 2pIL(τ + σ) + iXn̸=01nαILne−2in(τ+σ),XIR(τ −σ) = xIR + 2pIR(τ −σ) + iXn̸=01nαIRne−2in(τ−σ). (2 −9)The relations between xI, pI, QI, wI and xIL, pIL, xIR, pIR are given byxIL = (1 −B)IJxJ + QI,xIR = (1 + B)IJxJ −QI,– 4 –

pIL = 12pI + 12(1 −B)IJwJ,pIR = 12pI −12(1 + B)IJwJ. (2 −10)Then, the commutation relations are given by[xIL, pJL] = iδIJ = [xIR, pJR],[αILm, αJLn] = mδIJδm+n,0 = [αIRm, αJRn],otherwise zeros.

(2 −11)It follows from the definition (2-10) that the left- and right-moving momentum (pIL, pIR)lies on a (D + D)-dimensional lorentzian even self-dual lattice ΓD,D [17],(pIL, pIR) ∈ΓD,D. (2 −12)This observation is important to one loop modular invariance and also our followingdiscussions.Let us introduce the complex variables z and ¯z defined byz = e−2i(τ+σ),¯z = e−2i(τ−σ).

(2 −13)In terms of z and ¯z, the left- and right-moving string coordinates (2-9) can be writtenasXIL(z) = xIL −ipILlnz + iXn̸=01nαILnz−n,XIR(¯z) = xIR −ipIRln¯z + iXn̸=01nαIRn¯z−n. (2 −14)In the following analysis, the complex variable ¯z will be treated as complex conjugationof z in the sense of Wick rotation.An orbifold is defined by specifying the action of each group element g of G onthe left-and right-moving string coordinate (XIL, XIR) (I = 1, .

. ., D).In order todetermine the allowed action of g on the string coordinate, we require the followingthree conditions:– 5 –

(i) The invariance of the energy-momentum tensors under the action of g; This con-dition guarantees the single-valuedness of the energy-momentum tensors on theorbifold. (ii) The duality of amplitudes; This is one of the important properties of string the-ories [19,20].

(iii) Modular invariance of partition functions; Modular invariance plays an importantrole in the construction of consistent string models [20] and conformally invariantfield theories [21]. Modular invariance may ensure the ultraviolet finiteness andthe anomaly free condition of superstring theories [20,22].

The space-time unitaryalso requires modular invariance [23].Although the first and the third conditions (i) and (iii) have already been inves-tigated, no close examination has been made on the second condition (ii) so far. Aswe will see later, our main results will be obtained from the detailed analysis of thesecond condition(ii).Let us first consider the condition (i), that is, the energy-momentum tensors haveto be invariant under the action of g. The energy-momentum tensors of the left- andright-movers are given byTL(z) = limw→z12P IL(w)P IL(z) −D(w −z)2 ,TR(¯z) = lim¯w→¯z12P IR( ¯w)P IR(¯z) −D( ¯w −¯z)2 ,(2 −15)where P IL(z) and P IR(¯z) are the momentum operators of the left- and right-moversdefined byP IL(z) = i∂zXIL(z),P IR(¯z) = i∂¯zXIR(¯z),(I = 1, .

. ., D).

(2 −16)It follows that the energy-momentum tensors are invariant under the action of g ifg(P IL(z), P IR(¯z))g† = (U IJL P JL(z), U IJR P JR(¯z)),(2 −17)where UL and UR are suitable elements of the D-dimensional orthogonal group O(D).Note that UL is not necessarily equal to UR and orbifolds with UL ̸= UR are calledasymmetric orbifolds [16].In terms of (pIL, αILn) and (pIR, αIRn), eq. (2-17) can berewritten asg(pIL, αILn)g† = U IJL (pJL, αJLn),g(pIR, αIRn)g† = U IJR (pJR, αJRn).

(2 −18)– 6 –

Since (pIL, pIR) lies on the lattice ΓD,D, the action of g on (pIL, pIR) has to be anautomorphism of ΓD,D, i.e.,(U IJL pJL, U IJR pJR) ∈ΓD,Dfor all(pIL, pIR) ∈ΓD,D. (2 −19)Since the momentum operators P IL(z) and P IR(¯z) do not include xIL and xIR, eq.

(2-17) does not completely determine the action of g on (xIL, xIR). In fact, the generalaction of g on (xIL, xIR), which is compatible with the quantization conditions (2-11),may be given by [24]g(xIL, xIR)g† = (U IJL (xJL + π ∂Φ(pL, pR)∂pJL), U IJR (xJR + π ∂Φ(pL, pR)∂pJR)),(2 −20)where Φ(pL, pR) is an arbitrary function of pIL and pIR.

Let gU be the unitary operatorwhich satisfiesgU(XIL(z), XIR(¯z))g†U = (U IJL XJL(z), U IJR XJR(¯z)),(2 −21)andgU|0 >= |0 >,(2 −22)where |0 > is the vacuum of the untwisted sector. Then, the twist operator g whichgenerates the transformations (2-18) and (2-20) will be given byg = eiπΦ(pL,pR)gU.

(2 −23)At this stage, Φ is an arbitrary function of pIL and pIR . In the next section, we willsee that the second condition (ii) severely restricts the form of the phase factor in g.3.

Cocycle Properties of Vertex OperatorsIn this section, we shall investigate the second condition (ii), i.e., the duality ofamplitudes, in detail. To this end, it will be necessary to examine cocycle properties ofvertex operators and to give an explicit operator representation of cocycle operators,which may be attached to vertex operators.– 7 –

Let us consider a vertex operator which describes the emission of a state with themomentum (kIL, kIR) ∈ΓD,D,V (kL, kR; z) =: eikL·XL(z)+ikR·XR(¯z)CkL,kR :,(3 −1)where : : denotes the normal ordering and CkL,kR is the cocycle operator which maybe necessary to ensure the correct commutation relations and the duality of amplitudes[20,25]. The product of two vertex operatorsV (kL, kR; z)V (k′L, k′R; z′),(3 −2)is well-defined if |z| > |z′|.

The different ordering of the two vertex operators corre-sponds to the different “time”-ordering . To obtain scattering amplitudes, we mustsum over all possible “time”-ordering for the emission of states.

We must then es-tablish that each contribution is independent of the order of the vertex operators toenlarge the regions of integrations over z variables [19]. Thus the product (3-2), withrespect to z and z′, has to be analytically continued to the region |z′| > |z| and to beidentical toV (k′L, k′R; z′)V (kL, kR; z),(3 −3)for |z′| > |z|.

In terms of the zero mode, the above statement can be expressed asV0(kL, kR)V0(k′L, k′R) = (−1)kL·k′L−kR·k′RV0(k′L, k′R)V0(kL, kR),(3 −4)whereV0(kL, kR) = eikL·xL+ikR·xRCkL,kR. (3 −5)This relation will follow from the following formula:: eikL·XL(z)+ikR·XR(¯z) :: eik′L·XL(z′)+ik′R·XR(¯z′) := (z −z′)kL·k′L(¯z −¯z′)kR·k′R× : eikL·XL(z)+ikR·XR(¯z)+ik′L·XL(z′)+ik′R·XR(¯z′) :,(3 −6)for |z| > |z′|.

The factor (−1)kL·k′L−kR·k′R appearing in eq. (3-4) is the reason for thenecessity of the cocycle operator CkL,kR.The second condition (ii) is now replaced by the statement that the duality re-lation (3-4) has to be preserved under the action of g. To examine this condition,– 8 –

we need to know an explicit operator representation of the cocycle operator CkL,kR.For notational simplicity, we may use the following notations: kA ≡(kIL, kIR),xA ≡(xIL, xIR), . .

. etc.

(A, B, . .

. run from 1 to 2D and I, J, .

. .

run from 1 to D.) To obtainan operator representation of the cocycle operator Ck, let us assume [26,27]Ck = eiπkAMAB ˆpB,(3 −7)where the wedge ∧may be attached to operators to distinguish between c-numbersand q-numbers. It follows from (3-4) that the matrix M AB has to satisfyeiπkA(M−MT )ABk′B = (−1)kAηABk′Bfor allkA, k′A ∈ΓD,D,(3 −8)whereηAB =100−1AB.

(3 −9)A solution to this equation may be given byM AB =−12BIJ−12(1 −B)IJ12(1 + B)IJ−12BIJAB. (3 −10)To see this, first note that M T = −M and consider2kAM ABk′B = −(kL −kR)I((1 + B)IJk′JL + (1 −B)IJk′JR ) + kILk′IL −kIRk′IR= kILk′IL −kIRk′IRmod 2,(3 −11)where we have used the fact thatkIL −kIR ∈Λ,(1 + B)IJk′JL + (1 −B)IJk′JR ∈2Λ∗.

(3 −12)Although we have obtained a representation of the cocycle operator Ck, its rep-resentation is not unique. Indeed, there exist infinitely many other representations ofCk.

In ref. [15], it has, however, been proved that by a suitable unitary transformationany representation of Ck can reduce to eq.

(3-7) with (3-10) up to a constant phase.Thus, it will be sufficient to consider only the representation (3-7) with (3-10) for ourpurpose .– 9 –

To explicitly show the dependence of the cocycle operator in the zero mode partof the vertex operator (3-5), we may writeV0(k; M) ≡eik·ˆxeiπk·M ˆp. (3 −13)Under the action of gU, V0(k; M) transforms asgUV0(k; M)g†U = V0(U T k; U T MU),(3 −14)whereU AB =U IJL00U IJRAB.

(3 −15)It is easy to see that the product of V0(k; M) and V0(k′; U T MU) satisfiesV0(k; M)V0(k′; U T MU) = ξ(−1)k·ηk′V0(k′; U T MU)V0(k; M),(3 −16)whereξ = e−iπk·(M−UT MU)k′. (3 −17)This relation implies that the duality relation (3-4) cannot be preserved under theaction of gU unless ξ = 1.

It does not, however, mean the violation of the dualityrelation under the action of g because the freedom of Φ(p) in g has not been used yet.Under the action of g, V0(k; M) transforms asgV0(k; M)g† = ei(UT k)·ˆxeiπ(UT k)·UT MU ˆp+iπΦ(ˆp+UT k)−iπΦ(ˆp). (3 −18)In order for the duality relation to be preserved, we may require thatgV0(k; M)g† ∝V0(U T k; M),(3 −19)where the proportional constant is required to be a c-number because a q-numberphase will destroy the duality relation.

Suppose that Φ(p) is expanded asΦ(p) = φ + 2vAηABpB + 12CABpApB+Xn≥31n!CA1...AnpA1 · · · pAn. (3 −20)Inserting eq.

(3-20) into eq. (3-18) and requiring eq.

(3-19), we may conclude thatCA1...An = 0 for n ≥3 andkACABk′B = kA(M −U T MU)ABk′Bmod 2,(3 −21)– 10 –

for kA, k′A ∈ΓD,D. There are no constraints on φ and vA.

This result is nothingbut the result given in ref. [15], where a slightly different approach has been used.

Itseems that there is no solution to eq. (3-21) because CAB is a symmetric matrix whileM AB is an antisymmetric one.

However, we can always find a symmetric matrix CABsatisfying (3-21). To see this, let us introduce a basis eAa (a = 1, .

. ., 2D) of ΓD,D, i.e.,kA = P2Da=1 kaeAa (ka ∈Z) for kA ∈ΓD,D.

Then, eq. (3-21) may be rewritten asCab = eAa (M −U T MU)ABeBbmod 2,(3 −22)where Cab ≡eAa CABeBb .

Since the matrix M AB satisfies eq. (3-8), we find2kA(M −U T MU)ABk′B = 0mod 2,(3 −23)because U ABkB, U ABk′B ∈ΓD,D and U T U = 1.

This implies thatkA(M −U T MU)ABk′B ∈Z,or equivalentlyeAa (M −U T MU)ABeBb ∈Z. (3 −24)This guarantees the existence of a solution to eq.

(3-22).We have observed that the duality relation can be preserved under the action ofg if Φ(p) in g is chosen asΦ(p) = φ + 2vAηABpB + 12CABpApB,(3 −25)where the symmetric matrix CAB is defined through the relation (3-21) or (3-22).We will see in the next section that modular invariance requires φ = 0 and imposessome constraints on vA. The symmetric matrix CAB seems not to be defined uniquelyin eq.

(3-21) or (3-22).Let C′AB be another choice satisfying eq. (3-21).WritingpA = P paeAa with pa ∈Z and defining Cab = eAa CABeBb , we have12(C′AB −CAB)pApB = 12Xa,b(C′ab −Cab)papb= 12Xa=b(C′aa −Caa)(pa)2 +Xa

where we have used the fact that C′ab −Cab ∈2Z and pa ∈Z. Thus, the differencebetween CAB and C′AB can be absorbed into the redefinition of vA and hence thechoice of CAB is essentially unique.

Therefore, we have found that any twist operatorg can always be parametrized by (UL, vL; UR, vR), as announced in the introduction.4. One Loop Modular InvarianceIn this section ,we will investigate one loop modular invariance of partition func-tions.

Let Z(h, g; τ) be the partition function of the h-sector twisted by g which isdefined, in the operator formalism, byZ(h, g; τ) = Tr[gei2πτ(L0−D24 )−i2π¯τ(¯L0−D24 )]h−sector,(4 −1)where L0(¯L0) is the Virasoro zero mode operator of the left- (right-) mover.Thetrace in eq. (4-1) is taken over the Hilbert space of the h-sector.

Then, the one looppartition function will be of the form,Z(τ) = 1NXg,h∈Ggh=hgZ(h, g; τ),(4 −2)where N is the order of G. In the above summation, only the elements h and g whichcommute each other contribute to the partition function. This will be explained asfollows: On the orbifold, each string obeys a boundary condition such that for someelement h ∈G,(XIL(e2πiz), XIR(e−2πi¯z)) = h · (XIL(z), XIR(¯z)),(4 −3)up to a torus shift.

A string obeying the boundary condition (4-3) is said to belongto the h-sector.If h is not the unit element of G, such string is called a twistedstring. The total Hilbert space Htotal consists of the direct sum of every Hilbert spaceHh(h ∈G),Htotal =Mh∈GHh.

(4 −4)The physical Hilbert space is not the total Hilbert space itself but the G-invariantsubspace of Htotal because any physical state must be invariant under the action ofall g ∈G. Thus, the partition function will be given by– 12 –

Z(τ) =Xh∈GZ(τ)h−sector,(4 −5)whereZ(τ)h−sector = Tr(phys)[ei2πτ(L0−D24 )−i2π¯τ(¯L0−D24 )]h−sector. (4 −6)Here, the trace should be taken over the physical Hilbert space of the h-sector, whichwill be given byH(phys)h= PHh,(4 −7)where P is the projection operator defined byP = 1NXg∈Gg.

(4 −8)By use of the projection operator , the trace formula (4-6) may be rewritten asZ(τ)h−sector = Tr[Pei2πτ(L0−D24 )−i2π¯τ(¯L0−D24 )]h−sector,(4 −9)where the trace is taken over the Hilbert space Hh. Let us consider the action of gon the string coordinate (XIL(z), XIR(¯z)) in the h-sector.

It follows from (4-3) thatg(XIL(z), XIR(¯z))g† obeys the boundary condition of the ghg−1-sector. Let |h > beany state in the h-sector.

The above observation may imply that g|h > belongs tothe ghg−1-sector but not the h-sector (unless g commutes with h). Therefore, in thetrace formula (4-9),Tr[gei2πτ(L0−D24 )−i2π¯τ(¯L0−D24 )]h−sector,(4 −10)will vanish identically unless g commutes with h.One loop modular invariance of the partition function is satisfied providedZ(h, g; τ + 1) = Z(h, hg; τ),(4 −11)Z(h, g; −1τ ) = Z(g−1, h; τ).

(4 −12)Let us first evaluate the partition function of the untwisted sector twisted by g,Z(1, g; τ). It follows from the discussions of the previous section that in the untwistedsector the twist operator g would be of the formg = eiπΦ(p)gU,(4 −13)– 13 –

whereΦ(p) = φ + 2vAηABpB + 12pACABpB. (4 −14)Let n be the smallest positive integer such that gn = 1.

It means thatU n = 1,(4 −15)nφ +n−1Xℓ=0{2v · ηU ℓp + 12p · U −ℓCU ℓp} = 0mod 2for all pA ∈ΓD,D. (4 −16)The zero mode part of Z(1, g; τ) can easily be evaluated and the result isZ(1, g; τ)zero mode =X(kR,kR)∈Γd, ¯dgeiπΦ(k)eiπτk2L−iπ¯τk2R,(4 −17)where Γd, ¯dgis the g-invariant sublattice of ΓD,D, i.e.,Γd, ¯dg= {(kL, kR) ∈ΓD,D|(ULkL, URkR) = (kL, kR)}.

(4 −18)Here, d + ¯d denotes singature of the lorentzian lattice Γd, ¯dg. We now show that thefollowing relation holds for a suitable constant vector v′A :12kACABkB = 2v′AηABkBmod 2,(4 −19)for all kA ∈Γd, ¯dg .

To show this , definef(k) = 12kACABkB. (4 −20)Note thatkACABk′B = kA(M −U T MU)ABk′Bmod 2= 0 mod 2for all kA, k′A ∈Γd, ¯dg ,(4 −21)where we have used eqs.

(3-21) and (4-18) . It follows thatf(k + k′) = f(k) + f(k′)mod 2,(4 −22)for all k, k′ ∈Γd, ¯dg .

This relation ensures the existence of a vector v′A satisfying eq . (4-19) .

Using the relation (4-19), we can write (4-17) asZ(1, g; τ)zero mode =X(kL,kR)∈Γd, ¯dgeiπφ+i2π(v+v′)·ηkeiπτk2L−iπ¯τk2R. (4 −23)– 14 –

It will be useful to introduce a projection matrix PU defined byPU = 1nn−1Xℓ=0U ℓ. (4 −24)Noting that PUk = k for all k ∈Γd, ¯dgand using the Poisson resummation formula, wehaveZ(1, g; −1τ )zero mode= eiπφ (−iτ)d2 (i¯τ)¯d2VΓd, ¯dgX(qL,qR)∈Γd, ¯d∗g−(v∗+v′∗)eiπτq2L−iπ¯τq2R,(4 −25)where v∗+v′∗≡PU(v+v′), VΓ is the unit volume of the lattice Γ and Γd, ¯d∗gis the duallattice of Γd, ¯dg .

It follows from eq. (4-25) that we can easily extract information aboutthe zero mode of the g−1-sector because Z(1, g; τ) should be related to Z(g−1, 1; τ)thorough the modular transformation, i.e.,Z(g−1, 1; τ) = Z(1, g; −1τ ).

(4 −26)It turns out that the degeneracy of the ground state in the g−1-sector may be givenby [16]pdet′(1 −U)VΓd, ¯dg,(4 −27)where the determinant should be taken over the nonzero eigenvalues of 1 −U and thefactorpdet′(1 −U) will come from the oscillators. The eigenvalues of the momentum(qL, qR) in the g−1-sector may be given by(qL, qR) ∈Γd, ¯d∗g−v∗−v′∗.

(4 −28)It should be noted that the momentum eigenvalues in the g−1-sector are notgiven by Γd, ¯d∗g−v∗, which might naively be expected [16]. The origin of the extracontribution −v′∗is the third term in eq.

(4-14), which has been introduced to ensurethe duality relation of vertex operators. As we will see later, this extra contribution tothe momentum eigenvalues becomes important to ensure the left-right level matchingcondition.– 15 –

Information about the zero mode given above is sufficient to obtain Z(g−1, 1; τ)because the oscillator part of Z(g−1, 1; τ) can unambiguously be calculated . In ap-pendix B , we will prove that the relation (4-26) puts a constraint on φ in eq.

(4-14),i.e.,φ = 0. (4 −29)This is desirable because otherwise the vacuum in the untwisted sector would notbe invariant under the action of g and hence would be removed from the physicalHilbert space.

In the point of view of the conformal field theory the vacuum in theuntwisted sector will correspond to the identity operator, which should be included inthe operator algebra.A necessary condition for modular invariance is the left-right level matching con-dition [16,28]Z(g−1, h; τ + n) = Z(g−1, h; τ). (4 −30)It follows from eq.

(4-1) that the level matching condition is satisfied only if2n(L0 −¯L0) = 0mod 2,(4 −31)where L0 (¯L0) is the Virasoro zero mode operator of the left- (right-) mover in theg−1-sector. Since any contribution to L0 and ¯L0 from the oscillators is a fraction ofn, the level matching condition can be written as2n(εg−1 −¯εg−1 + 12q2L −12q2R) = 0mod 2,for all (qL, qR) ∈Γd, ¯d∗g−v∗−v′∗,(4 −32)where (εg−1, ¯εg−1) is the conformal dimension (or the zero point energy) of the groundstate in the g−1-sector and is explicitly given by [1]εg−1 = 14DXa=1ρa(1 −ρa),¯εg−1 = 14DXa=1¯ρa(1 −¯ρa).

(4 −33)Here, exp(i2πρa) and exp(i2π¯ρa) (a = 1, · · ·, D) are the eigenvalues of UL and UR with0 ≤ρa, ¯ρa < 1, respectively. The condition (4-32) can further be shown to reduce to2n(εg−1 −¯εg−1 + 12(v∗L + v′∗L)2 −12(v∗R + v′∗R)2) = 0mod 2.

(4 −34)– 16 –

To see this, we first note that Γd, ¯dg∗can be expressed as [16]Γd, ¯dg∗= PUΓD,D= {qA = PUkA, kA ∈ΓD,D}. (4 −35)This follows from the property that ΓD,D is self-dual.

From eq. (4-35), any momentumq ∈Γd, ¯dg∗−v∗−v′∗can be parametrized asqA = PU(k −v −v′)Afor some kA ∈ΓD,D.

(4 −36)Then, we haven(q2L −q2R) = nq · ηq= nk · ηPUk −2n(v + v′) · ηPUk + n(v∗+ v′∗) · η(v∗+ v′∗),(4 −37)where we have used the relationsPUη = ηPU,P2U = PU,PTU = PU. (4 −38)Since ΓD,D is an even integral lattice and U is an orthogonal matrix satisfying U n = 1,the first term in the right handed side of eq.

(4-37) is easily shown to reduce tonk · ηPUk =k · ηUn2 kmod 2 if n = even,0mod 2 if n = odd. (4 −39)Using the relation (4-19) and nothing that nPUk ∈Γd, ¯dg , we can rewrite the secondterm in the right hand side of eq.

(4-37) as−2n(v + v′) · ηPUk = −2nv · ηPUk −12k · (n−1Xℓ=0U −ℓ)C(n−1Xm=0U m)kmod 2. (4 −40)Replacing p by p + p′ in eq.

(4-16) with eq. (4-29) and then using (4-16) again, we havep ·n−1Xℓ=0U −ℓCU ℓp′ = 0mod 2,(4 −41)for all p, p′ ∈ΓD,D.

For n odd, it is not difficult to show that−2n(v + v′) · ηPUk = 0mod 2. (4 −42)– 17 –

To derive eq. (4-42), we will use eqs.

(4-16),(4-29),(4-40) and (4-41). For n even, wewill find−2n(v + v′)ηPUk = −k · (n2 −1Xℓ=0U −ℓCU ℓ)Un2 kmod 2.

(4 −43)Remembering eqs. (3-8) and (3-21), we can finally find that for n even−2n(v + v′)ηPUk = −k · ηUn2 kmod 2.

(4 −44)Combining the results (4-39), (4-42) and (4-44), we havenk · ηPUk −2n(v + v′) · ηPUk = 0mod 2. (4 −45)This completes the proof of (4-34).We have shown that the left-right level matching condition (4-30) reduces to thecondition (4-34), which may put a constraint on the shift vector v = (vL, vR).

Itshould be noticed that the level matching condition (4-34) is not always satisfied forasymmetric orbifold models but trivially satisfied for symmetric ones because εg−1 =¯εg−1 and (v∗L + v′∗L)2 = (v∗R + v′∗R)2 for symmetric orbifold models. For the case ofCAB = 0 in eq.

(4-14), it has been proved, in refs. [16,28], that the level matchingcondition is a necessary and also sufficient condition for one loop modular invariance.Even in the case of the general twist (4-13) with the phase (4-14), the sufficiency canprobably be shown by arguments similar to refs.

[16,28]. It should be emphasized thatthe third term in eq.

(4-14) plays an important role in the level matching conditionbecause the relation (4-45) might not hold in general if we put v′ to be zero, thatis, CAB to be zero by hand. In section 6, we will see examples of orbifold modelsthat the introduction of the third term in eq.

(4-14) makes partition functions modularinvariant.Before closing this section, we shall make a comment on modular invariance ofcorrelation functions. Our analysis implies that one loop modular invariance of par-tition functions does not in general ensure one loop modular invariance of correlationfunctions because a wrong choice of a twist operator g could destroy the duality rela-tion of vertex operators even though the partition function is modular invariant.

Suchan example will be found in section 6.– 18 –

5. A Geometrical InterpretationWe have found that the string coordinate XA = (XIL, XIR) in the untwisted sectortransforms under the action of g asgXAg† = U AB(XB + 2πηBCvC + πCBCpC).

(5 −1)It seems that the third term in the right hand side of eq. (5-1) has no clear geometricalmeaning.

Although the momentum and vertex operators definitely transform underthe action of g, why does not the string coordinate (XIL, XIR) transform definitely ?The reason may be that in the point of view of the conformal field theory the stringcoordinate is not a primary field and it is not a well-defined variable on a torus. Thus,there is probably no reason why the string coordinate itself should definitely transformunder the action of g. On the other hand, since the momentum and vertex operatorsare primary fields and are well-defined on a torus, they should definitely transformunder the action of g. In fact, they transform asg(P IL(z), P IR(¯z))g† = (U IJL P JL(z), U IJR P JR(¯z)),gV (kL, kR; z)g† = ei2πv·ηUT k+i π2 k·UCUT kV (U TL kL, U TRkR; z).

(5 −2)As mentioned above, not the string coordinate but the momentum and vertexoperators are relevant operators on tori or orbifolds. Since P IL(z) and P IR(¯z) do not in-clude the “center of mass coordinate” (xIL, xIR), it appears only in the vertex operators.The cocycle operator has been shown to be represented asCk = eiπkAMABpB.

(5 −3)Therefore, we observe that the “center of mass coordinate” xA = (xIL, xIR) alwaysappears as the following combination:xA + πM ABpB. (5 −4)This observation strongly suggests that the combination is a more fundamental vari-able than xA itself.

To see this, let us introduce the variable x′A, which is slightlydifferent from the variable (5-4),x′A ≡xA + πM ′ABpB,(5 −5)– 19 –

whereM ′AB = M AB + 12ηAB. (5 −6)Note that x′A is related to the variable (5-4) by the following unitary transformation:U(xA + πM ABpB)U† = x′A,(5 −7)whereU = e−i π4 pAηABpB.

(5 −8)Hence, we will discuss a geometrical meaning of x′A instead of the variable (5-4) inthe following. We first note that although xA does not transform definitely under theaction of g, x′A does:gx′Ag† ∼U AB(x′B + 2πηBCvC),(5 −9)where ∼means that the right hand side is identical to the left hand side up to a torusshift.∗In terms of the left- and right-moving coordinates, eq.

(5-5) is written as(x′IL, x′IR) = (xIL + π2 (1 −B)IJ(pJL −pJR), xIR + π2 (1 + B)IJ(pJL −pJR)). (5 −10)We may further rewrite the variables xIL, pIL, xIR and pIR into xI, pI, QI and wI, whichwill geometrically be more fundamental than xIL, pIL, xIR and pIR.

Then, we havex′I = xI + π2 wI,(5 −11)Q′I = QI,(5 −12)where x′I and Q′I are related to x′IL and x′IR through the same relations as eq. (2-10).The question is now what geometrical meaning x′I has.Before we answer the question, it may be instructive to make a comment on thecenter of mass coordinate of a string on a torus, which has a clear geometrical meaning∗In fact, x′A transforms asgx′Ag† = U AB(x′B + 2πηBCvC) + πU AB(U T MU −M + C)BCpC.The last term of the right hand side is nothing but a torus shift becausekAU AB(U T MU −M + C)BCpC = 0mod 2for any kA, pA ∈ΓD,D,where we have used eq.

(3-21).– 20 –

if the string has no winding number. The “center of mass coordinate” is, however, ill-defined geometrically if the string winds around the torus.

Thus, xI can be interpretedas the center of mass coordinate in the absence of the winding number but it will loseits geometrical meaning in the presence of the winding number. However, it may stillbe a useful notion on the covering space of the torus.

It turns out that on the coveringspace of the torus the “center of mass coordinate” of the string may be locate at [27]x′I = xI + π2 wI. (5 −13)To see this, consider the string coordinate XI(τ, σ) at τ = 0 given in eq.

(2-3) andintegrate it over the σ-variable. Then we haveZ π0dσπ XI(0, σ) = xI + π2 wI.

(5 −14)The above observation may suggest that the reason why cocycle operators appearin vertex operators is related to the fact that there is no good variable of the “center ofmass coordinate” of a string on a torus and also suggest that the variable x′I definedin eq. (5-11) is more fundamental than xI on a torus as well as on an orbifold becausex′A but not xA definitely transforms under the action of g.6.

Example of OrbifoldsIn this section, we shall investigate a symmetric Z2-orbifold, a nonabelian S3-orbifold and an asymmetric Z3-orbifold, in detail, which will give good illustrations ofour formalism.Let us introduce the root lattice ΛR and the weight lattice of SU(3) asΛR = {pI =2Xi=1niαIi , ni ∈Z},ΛW = {pI =2Xi=1miµiI, mi ∈Z},(6 −1)where αi and µi (i = 1, 2) are a simple root and a fundamental weight satisfyingαi · µj = δji . We will take αi and µi to beα1 = ( 1√2,r32),– 21 –

α2 = ( 1√2, −r32),µ1 = ( 1√2,r16),µ2 = ( 1√2, −r16). (6 −2)Let pI and wI be the center of mass momentum and the winding number, respectively.They are assumed to lie on the following lattices:pI ∈2ΛW ,wI ∈ΛR.

(6 −3)The left- and right-moving momentum (pIL, pIR) is defined by eq. (2-10), i.e.,pIL = 12pI + 12(1 −B)IJwJ,pIR = 12pI −12(1 + B)IJwJ.

(6 −4)The antisymmetric constant matrix BIJ is chosen asBIJ = 0−1√31√30!. (6 −5)Then, it turns out that (pIL, pIR) lies on the following 2+2-dimensional lorentzian evenself-dual lattice:Γ2,2 = {(pIL, pIR)|pIL, pIR ∈ΛW , pIL −pIR ∈ΛR}.

(6 −6)6-1. A Symmetric Z2-OrbifoldWe shall first consider a symmetric SU(3)/Z2-orbifold whose Z2-transformationis defined bygU(XIL, XIR)g†U = (U IJL XJL, U IJR XJR),(I = 1, 2),(6 −7)whereU IJL= U IJR =−1001.

(6 −8)– 22 –

This is an automorphism of Γ2,2,, as it should be. According to our prescription, theZ2-twist operator g will be given byg = ei π2 pACABpBgU,(6 −9)where pA = (pIL, pIR) and the symmetric matrix CAB is defined through the relationpA(M −U T MU)ABp′B = pACABp′Bmod 2,(6 −10)for pA, p′A ∈Γ2,2.

Here, we have taken a shift vector to zero for simplicity and M AB,U AB are defined byM AB =−12BIJ−12(1 −B)IJ12(1 + B)IJ−12BIJAB,U AB =U IJL00U IJRAB. (6 −11)For symmetric orbifolds (UL = UR), the defining relation (6-10) of CAB may bereplaced by12(pL−pR)I(B−U TL BUL)IJ(p′L−p′R)J = (pL−pR)ICIJ(p′L−p′R)Jmod 2, (6 −12)where CAB has been assumed to be of the formCAB =−CIJCIJCIJ−CIJAB.

(6 −13)Then, eq. (6-9) can be written asg = e−i π2 (pL−pR)ICIJ(pL−pR)JgU.

(6 −14)Since pIL −pIR ∈ΛR, the equation (6-12) may be rewritten as12αIi (B −U TL BUL)IJαJj = αIi CIJαJjmod 2. (6 −15)The left hand side of eq.

(6-15) is found to be12αIi (B −U TL BUL)IJαJj =01−10ij,(6 −16)– 23 –

and hence CIJ cannot be chosen to be zero. We may choose CIJ asαIi CIJαJj =0110ij,orCIJ =100−13IJ.

(6 −17)This choice turns out to be consistent with g2 = 1.Let us consider the following momentum and vertex operators of the left-mover:P IL(z) = i∂zXIL(z),VL(α; z) =: eiα·XL(z)Cα :,(6 −18)where α is a root vector of SU(3) and Cα denotes a cocycle operator. These operatorsform level one Kaˇc-Moody algebra dsu(3)k=1 [25].

Under the action of g, they transformasgP IL(z)g† = U IJL P JL (z),gVL(±α1; z)g† = VL(∓α2; z),gVL(±α2; z)g† = VL(∓α1; z),gVL(±(α1 + α2); z)g† = −VL(∓(α1 + α2); z). (6 −19)Thus, the Z2-invariant physical generators may be given byJ3(z) =√32 {P 2L(z) −i√6(VL(α1 + α2; z) −VL(−α1 −α2; z))},J±(z) =1√2(VL(±α1; z) + VL(∓α2; z)),J(z) = 12{P 2L(z) + ir32(VL(α1 + α2; z) −VL(−α1 −α2; z))},(6 −20)which are found to form Kaˇc-Moody algebra dsu(2)k=1 ⊕du(1).We now examine one loop modular invariance of the partition function which willbe given byZ(τ) = 121Xℓ,m=0Z(gℓ, gm; τ),(6 −21)– 24 –

whereZ(gℓ, gm; τ) = Tr[gmei2πτ(L0−D24 )−i2π¯τ(¯L0−D24 )]gℓ−sector. (6 −22)The partition functions of the untwisted sector can easily be evaluated and the resultisZ(1, 1; τ) =1|η(τ)|4X(kL,kR)∈Γ2,2eiπτk2L−iπ¯τk2R,(6 −23)Z(1, g; τ) = |ϑ3(0|τ)ϑ4(0|τ)||η(τ)|4X(kL,kR)∈Γ1,1gei2π(v′LkL−v′RkR)eiπτk2L−iπ¯τk2R,(6 −24)wherev′L = v′R =12√6,Γ1,1g= {(kL, kR) = (√6n + λ,√6n′ + λ), λ = 0, ±r23, n, n′ ∈Z}.

(6 −25)Here, η(τ) is the Dedekind η-function and ϑa(ν|τ) (a = 1, · · ·, 4) is the Jacobi thetafunction. Their definition and properties will be found in appendix A.

The shift vector(v′L, v′R) has been introduced through the relation (4-19).It follows from the arguments given in section 4 that the degeneracy of the groundstate in the g-sector ispdet′(1 −U)VΓ1,1g= 1,(6 −26)and that the momentum eigenvalues will be given by(qL, qR) ∈Γ1,1g∗−(v′L, v′R),(6 −27)whereΓ1,1g∗= {(qL, qR) = (r32n + λ,r32n′ + λ), λ = 0, ± 1√6, n, n′ ∈Z}. (6 −28)This information is enough to obtain Z(g, 1; τ) and Z(g, g; τ),Z(g, 1; τ) = |ϑ3(0|τ)ϑ2(0|τ)|2|η(τ)|4X(qL,qR)∈Γ1,1g∗−(v′L,v′R)eiπτq2L−iπ¯τq2R,Z(g, g; τ) = |ϑ4(0|τ)ϑ2(0|τ)|2|η(τ)|4X(qL,qR)∈Γ1,1g∗−(v′L,v′R)eiπ(q2L−q2R)eiπτq2L−iπ¯τq2R.

(6 −29)– 25 –

It is easily verified from the formulas in appendix A that Z(gℓ, gm; τ) satisfies thefollowing desired relations:Z(gℓ, gm; τ + 1) = Z(gℓ, gm+ℓ; τ),Z(gℓ, gm; −1τ ) = Z(g−m, gℓ; τ),(6 −30)and hence the partition function (6-21) is modular invariant. It should be emphasizedthat the existence of the shift vector (v′L, v′R) makes the partition function modularinvariant: The level matching conditionZ(g, 1; τ + 2) = Z(g, 1; τ),(6 −31)is satisfied because for all (qL, qR) ∈Γ1,1g∗−(v′L, v′R),4(12q2L −12q2R) = 0mod 2.

(6 −32)If we put the shift vector (v′L, v′R) or CIJ in g to be zero by hand, the level matchingcondition might, however, be destroyed because eq. (6-32) dose not hold.6-2.

A Nonabelian S3-OrbifoldThe next example is a nonabelian SU(3)/S3-orbifold, where S3 is the symmetricgroup of order three. The symmetric group S3 consists of six elements Ui (i = 0, · · ·, 5),U0 =1001,U1 =−1001≡U,U2 = −12−√32√32−12!≡V,U3 = V 2,U4 = V U,U5 = UV.

(6 −33)The matrices U1, U4 and U5 correspond to the Weyl reflections of SU(3) with respectto the root vectors α1 + α2, α1 and α2, respectively, and the matrices U2 and U3– 26 –

correspond to the rotation by 2π3 and 4π3 , respectively. The action of gU i (i = 0, · · ·, 5)on the string coordinate is defined bygU i(XIL, XIR)gU i† = U IJi (XJL, XJR), (i = 0, · · ·, 5).

(6 −34)Each element of S3 is an automorphism of ΛR and ΛW and hence Γ2,2. The matricesU and V satisfyU 2 = V 3 = 1,V UV = U.

(6 −35)According to our prescription, we may write the twist operators g1 and g2 whichcorrespond to U1 and U2, respectively, asg1 = ei π2 (pL−pR)ICIJ1 (pL−pR)JgU 1,g2 = ei π2 (pL−pR)ICIJ2 (pL−pR)JgU 2,(6 −36)where the symmetric matrices C1 and C2 are defined byαIi CIJ1 αJj = 12αIi (B −U T1 BU1)IJαJjmod 2,αIi CIJ2 αJj = 12αIi (B −U T2 BU2)IJαJjmod 2,(6 −37)and we have put shift vectors to zero. Other twist operators will be defined by g0 = 1,g3 = (g2)2, g4 = g2g1 and g5 = g1g2.

Explicit calculations show that12αIi (B −U T1 BU1)IJαJj =01−10ij,12αIi (B −U T2 BU2)IJαJj =0000ij. (6 −38)In order for gi’s to form the symmetric group S3, we may choose ∗αIi CIJ1 αJj =0110ij,∗If we choose C1 and C2, in general, asαIi CIJ1 αJj =2m11 + 2m31 + 2m32m2,mi ∈Z,αIi CIJ2 αJj =2n12n32n32n2,ni ∈Z,with m1 +m2 ∈2Z and m2 +n2 ∈2Z+1, gi (i = 0, · · ·, 5) forms the symmetric groupS3 and any choice will lead to the same result.– 27 –

αIi CIJ2 αJj =0002ij. (6 −39)Since the symmetric group S3 is nonabelian, the one loop partition function willbe of the form,Z(τ) = 16Xgi,gj ∈S3gigj =gj giZ(gi, gj; τ).

(6 −40)It is not difficult to show that the following combinations of Z(gi, gj; τ)’s are modularinvariant:1) Z(1, 1; τ),2) Z(1, g1; τ) + Z(g1, 1; τ) + Z(g1, g1; τ),3) Z(1, g4; τ) + Z(g4, 1; τ) + Z(g4, g4; τ),4) Z(1, g5; τ) + Z(g5, 1; τ) + Z(g5, g5; τ),5)Xj=2,3Z(1, gj; τ) +Xj=0,2,3(Z(g2, gj; τ) + Z(g3, gj; τ)). (6 −41)Therefore, the partition function (6-40) is also modular invariant.

∗Note that 1)+2) isnothing but the partition function of the Z2-orbifold discussed in the previous example6-1 up to an overall normalization. The combination 1)+3) ( 1)+4) ) is identical to1) +2) and corresponds to the partition function of the Z2-orbifold associated withthe Weyl reflection with respect to α1 (α2).

The combination 1)+5) corresponds tothe partition function of the Z3-orbifold whose Z3-transformation is generated by g2.6-3. An Asymmetric Z3-OrbifoldThe final example is an asymmetric SU(3)/Z3-orbifold whose Z3-transformationis defined bygU(XIL, XIR)g†U = (U IJL XJL, U IJR XJR),(6 −42)whereU IJL= −12−√32√32−12!,∗In ref.

[29], the authors have not succeeded in obtaining a modular invariant par-tition function of the nonabelian S3-orbifold model because of a wrong choice of thetwist operators.– 28 –

U IJR =1001. (6 −43)This is also an automorphism of Γ2,2.

According to our prescription, we may writethe Z3-twist operator g asg = ei2πvAηABpB+i π2 pACABpBgU,(6 −44)where vA = (vIL, vIR), pA = (pIL, pIR) and the symmetric matrix CAB is defined bypACABp′B = pA(M −U T MU)ABp′Bmod 2,(6 −45)for pA, p′A ∈Γ2,2. The matrices M AB and U AB are defined in eqs.(6-11).

To explicitlydetermine the symmetric matrix CAB, let us introduce a basis of Γ2,2,Γ2,2 = {pA =4Xa=1naeAa , na ∈Z},(6 −46)whereeAi = (µiI, µiI),i = 1, 2,eAi+2 = (0, αIi ),i = 1, 2. (6 −47)In terms of ea (a = 1, · · ·, 4), we findeAa (M −U T MU)ABeBb =00−10000−110000100ab.

(6 −48)Thus, we may choose the symmetric matrix CAB aseAa CABeBb =0010000110000100ab. (6 −49)Since we want to construct a Z3-orbifold model, we must require that g3 = 1, whichis equivalent to2Xℓ=0{2v · ηU ℓp + 12p · U −ℓCU ℓp} = 0mod 2,(6 −50)– 29 –

for all p ∈Γ2,2. Let ea∗(a = 1, · · ·, 4) be the dual basis of ea (i.e., ea · eb∗= δba).

Interms of ea∗, we may writevA =4Xa=1yaeaA∗. (6 −51)The condition (6-50) puts a constraint on ya (a = 1, · · ·, 4) and is equivalently writtenasy3 = 13(2ℓ−ℓ′),y4 = 13(−ℓ+ 2ℓ′),ℓ, ℓ′ ∈Z,(6 −52)while y1 and y2 are arbitrary.Let us consider the partition function of the untwisted sector twisted by g,Z(1, g; τ) = Tr[gei2πτ(L0−224 )−i2π¯τ(¯L0−224 )]untwist.

(6 −53)The zero mode part of Z(1, g; τ) is given byZ(1, g; τ)zero mode =XkR∈Γ0,2ge−i2πvR·kRe−iπ¯τk2R,(6 −54)where Γ0,2g= ΛR and vIR can be written, in terms of ya, asvIR = (−1√2(y3 + y4), −1√6(y3 −y4)). (6 −55)Note that the term 12pACABpB in eq.

(6-44) does not contribute to Z(1, g; τ) at allbecause12pACABpB = 0mod 2for all pA ∈Γ0,2g . (6 −56)According to the arguments in section 4, we can know information about the zeromode in the g−1-sector: The degeneracy of the ground state ispdet′(1 −U)VΓ0,2g=√3√3 = 1,(6 −57)and the momentum eigenvalues of the g−1-sector is given by(qL, qR) ∈(0, ΛW −vR).

(6 −58)The left-right level matching condition for Z(g−1, 1; τ) isZ(g−1, 1; τ + 3) = Z(g−1, 1; τ),(6 −59)– 30 –

which is equivalent to the condition3(vIR)2 = 23mod 2. (6 −60)It follows from eqs.

(6-52) and (6-55) that the condition (6-60) can be rewritten as23(ℓ2 + ℓ′2 −ℓℓ′) = 23mod 2. (6 −61)Since the left-right level matching condition (6-59) is always a necessary and alsosufficient condition for any Z3-orbifold model, we conclude that the one loop partitionfunction is modular invariant if the shift vector (vIL, vIR) in eq.

(6-44) satisfies (6-60)with (6-52).In this orbifold model, to ensure modular invariance we need a nonzero shiftvector satisfying (6-60) with (6-52). It is consistent with the argument of ref.[30].

Anexplicit example of the shift vector will be given by(vIL, vIR) = (0, −13αI1),(6 −62)which corresponds to y1 = y2 = 0, ℓ= 1 and ℓ′ = 0. As noted before, the followingchoice of the twist operatorg′ = ei2πvAηABpBgU,(6 −63)would also give a modular invariant partition function because of eq.

(6-56) but it doesnot guarantee modular invariance of correlation functions because the twist operatorg′ destroys the duality relation of vertex operators.7. DiscussionsIn this paper, we have investigated the following three consistency conditionsin detail: (i) the invariance of the energy-momentum tensors under the action ofthe twist operators, (ii) the duality of amplitudes and (iii) modular invariance ofpartition functions.

From the analysis of the second condition (ii), we have obtainedvarious important results. The following two points are probably main results of thispaper: The first point is the discovery of the third term in eq.

(3-25), which has to beincluded as a momentum-dependent phase in the twist operator g of the untwisted– 31 –

sector to preserve the duality of amplitudes under the action of g and which playsan important role in modular invariance of partition functions. The second point isthat the first condition (i) is not sufficient to determine the allowed action of g onthe string coordinate and indeed the condition (i) puts no constraint on Φ(pL, pR) ineq.

(2-20) or (2-23). The second condition (ii) has been found to be crucial to restrictthe allowed form of Φ(pL, pR) to eq.

(3-25).We have succeeded in obtaining the general class of bosonic orbifold models. Thegeneralization to superstring theories will be straightforward because fermionic fieldswill definitely transform under the action of twist operators.We have restricted our considerations mainly to the untwisted sector.

However,much information about twisted sectors, in particular, zero modes, can be obtainedthrough modular transformations. Such information is sufficient to obtain the parti-tion function of the g-sector, Z(g, 1; τ) but not Z(g, h; τ) in general because we havenot constructed twist operators in each twisted sector.

The twist operator g in theg-sector can, however, be found to be of the formg = ei2π(L0−¯L0). (7 −1)This follows from the relationZ(g, g; τ) = Z(g, 1; τ + 1).

(7 −2)To obtain an explicit operator representation of any twist operator in every twistedsector, we may need to construct vertex operators in every twisted sector as in theuntwisted sector. In the construction of vertex operators in twisted sectors, the mostsubtle part is a realization of cocycle operators.

In the case of ξ = 1 in eq. (3-17),(untwisted state emission) vertex operators in any twisted sector have already beenconstructed with correct cocycle operators in ref.

[18].In the case of ξ ̸= 1, theprescription given in ref. [18] will be insufficient to obtain desired vertex operatorsbecause the duality relation will not be satisfied.

Some attempts [31] have been madebut the general construction of correct vertex operators is still an open problem.– 32 –

Appendix AIn this appendix, we present various useful formulas which will be used in thetext.We first introduce the theta functionϑab(ν|τ) =∞Xn=−∞exp{iπ(n + a)2τ + i2π(n + a)(ν + b)}. (A −1)The four Jacobi theta functions are given byϑ1(ν|τ) = ϑ 1212 (ν|τ),ϑ2(ν|τ) = ϑ 12 0(ν|τ),ϑ3(ν|τ) = ϑ00(ν|τ),ϑ4(ν|τ) = ϑ0 12 (ν|τ).

(A −2)They satisfyϑ1(ν + 1|τ) = −ϑ1(ν|τ),ϑ2(ν + 1|τ) = −ϑ2(ν|τ),ϑ3(ν + 1|τ) = ϑ3(ν|τ),ϑ4(ν + 1|τ) = ϑ4(ν|τ),(A −3)ϑ1(ν + τ|τ) = −e−iπ(τ+2ν)ϑ1(ν|τ),ϑ2(ν + τ|τ) = e−iπ(τ+2ν)ϑ2(ν|τ),ϑ3(ν + τ|τ) = e−iπ(τ+2ν)ϑ3(ν|τ),ϑ4(ν + τ|τ) = −e−iπ(τ+2ν)ϑ4(ν|τ),(A −4)ϑ1(ν|τ + 1) = ei π4 ϑ1(ν|τ),ϑ2(ν|τ + 1) = ei π4 ϑ2(ν|τ),ϑ3(ν|τ + 1) = ϑ4(ν|τ),ϑ4(ν|τ + 1) = ϑ3(ν|τ),(A −5)ϑ1(ν/τ| −1/τ) = −i(−iτ)1/2eiπν2/τϑ1(ν|τ),ϑ2(ν/τ| −1/τ) = (−iτ)1/2eiπν2/τϑ4(ν|τ),ϑ3(ν/τ| −1/τ) = (−iτ)1/2eiπν2/τϑ3(ν|τ),ϑ4(ν/τ| −1/τ) = (−iτ)1/2eiπν2/τϑ2(ν|τ). (A −6)– 33 –

It is known that the Jacobi theta functions can be expanded asϑ1(ν|τ) = −2q1/4f(q)sinπν∞Yn=1(1 −2q2ncos2πν + q4n),ϑ2(ν|τ) = 2q1/4f(q)cosπν∞Yn=1(1 + 2q2ncos2πν + q4n),ϑ3(ν|τ) = f(q)∞Yn=1(1 + 2q2n−1cos2πν + q4n−2),ϑ4(ν|τ) = f(q)∞Yn=1(1 −2q2n−1cos2πν + q4n−2),(A −7)whereq = eiπτ,f(q) =∞Yn=1(1 −q2n). (A −8)Another important function is the Dedekind η-functionη(τ) = q1/12∞Yn=1(1 −q2n),(A −9)which satisfiesη(τ + 1) = ei π12 η(τ),η(−1τ ) = (−iτ)1/2η(τ).

(A −10)We finally give the Poisson resummation formula, which will play a key role in themodular transformation τ →−1τ . Let Γd, ¯d be a d + ¯d-dimensional lorentzian latticeand Γd, ¯d∗be its dual lattice.

Then, the formula is given byX(kL,kR)∈Γd, ¯de−i πτ (kL+vL)2+i π¯τ (kR+vR)2= (−iτ)d/2(i¯τ) ¯d/2VΓd, ¯dX(qL,qR)∈Γd, ¯d∗eiπτq2L−iπ¯τq2R+i2π(vL·qL−vR·qR),(A −11)where (vL, vR) is an arbitrary d + ¯d-dimensional constant vector and VΓ is the unitvolume of the lattice Γ.– 34 –

Appendix BIn this appendix, we shall explicitly evaluate Z(1, g; τ) and Z(g−1, 1; τ) and showthat the constant phase φ in eq. (4-13) has to be zero.For our purpose, it will be sufficient to consider the case of UR = 1, i.e.,gU(XIL, XIR)g†U = (U IJL XJL, XIR).

(B −1)The generalization will be straightforward. Without loss of generality, we can assumethat the orthogonal matrix UL has the following form:U IJL=δab00V ijIJ,(B −2)where V is a d×d orthogonal matrix which has no eigenvalues of 1, i.e., det(1−V ) ̸= 0.Here, I, J, · · · run from 1 to D, a, b, · · · from 1 to D −d and i, j, · · · from 1 to d.We first calculate Z(1, g; τ):Z(1, g; τ) = Tr[gei2πτ(L0−D24 )−i2π¯τ(¯L0−D24 )]untwist,(B −3)where the trace is taken over the Hilbert space of the untwisted sector.

Let exp(i2πρa)be the eigenvalues of V , where 0 < ρa < 1 and a = 1, 2, · · ·, d. Since V is an orthogonalmatrix, the set of eigenvalues {ei2πρa} is identical to the set of {e−i2πρa}. Thus, wemay write the eigenvalues of V as ∗{ei2πρa and e−i2πρa, a = 1, · · ·, d2}.

(B −4)The Virasoro zero mode operators L0 and ¯L0 in the untwisted sector are given byL0 = 12(pIL)2 +∞Xn=1αIL−nαILn,¯L0 = 12(pIR)2 +∞Xn=1αIR−nαIRn. (B −5)In section 3, we have seen that the twist operator g would be of the formg = eiπΦ(p)gU,(B −6)∗Here, we have assumed that the number of the eigenvalue −1 (i.e., ρa = 12) iseven for simplicity.– 35 –

whereΦ(p) = φ + 2vAηABpB + 12pACABpB. (B −7)The action of gU is defined as follows:gU(XIL(z), XIR(¯z))g†U = (U IJL XJL(z), XIR(¯z)),(B −8)gU|0 >= |0 >,(B −9)where |0 > denotes the vacuum of the untwisted sector.The zero mode part ofZ(1, g; τ) can easily be evaluated and the result isZ(1, g; τ)zero mode =X(kL,kR)∈ΓD−d,Dgeiπφ+i2π(vL+v′L)·kL−i2π(vR+v′R)·kReiπτk2L−iπ¯τk2R,(B −10)where (v′L, v′R) is defined in eq.

(4-19) and ΓD−d,Dgis the g-invariant sublattice of ΓD,D,i.e.,ΓD−d,Dg= {(kL, kR) ∈ΓD,D | (ULkL, kR) = (kL, kR)}. (B −11)Since the twist operator g acts on the oscillators asg(αILn, αIRn)g† = (U IJL αJLn, αIRn),(B −12)and the eigenvalues of V are given by (B-4), the remaining oscillator part of Z(1, g; τ)will be given byZ(1, g; τ)osc= |q|−D6∞Yn=1(1 −q2n)−(D−d)(1 −¯q2n)−Dd/2Ya=1∞Yn=1[(1 −ei2πρaq2n)(1 −e−i2πρaq2n)]−1,(B −13)where q = eiπτ.

Thus, Z(1, g; τ) can be written asZ(1, g; τ) =eiπφ|η(τ)|2Dd/2Ya=1[−2sin(πρa)(η(τ))3ϑ1(ρa|τ)]×X(kL,kR)∈ΓD−d,Dgei2π(vL+v′L)·kL−i2π(vR+v′R)·kReiπτk2L−iπ¯τk2R,(B −14)where ϑ1(ν|τ) and η(τ) are the Jacobi theta function and the Dedekind η-functiondefined in appendix A.– 36 –

Let us next consider Z(g−1, 1; τ)Z(g−1, 1; τ) = Tr[ei2πτ(L0−D24 )−i2π¯τ(¯L0−D24 )]g−1−sector,(B −15)where the trace is taken over the Hilbert space of the g−1-sector. As discussed insection 4, the degeneracy of the ground state in the g−1-sector is given bypdet(1 −V )VΓD−d,Dg,(B −16)and the eigenvalues of the momentum (qL, qR) in the g−1-sector are of the form(qL, qR) ∈ΓD−d,Dg∗−(v∗L + v′L∗, v∗R + v′R∗),(B −17)where(v∗L + v′L∗, v∗R + v′R∗) ≡PU · (vL + v′L, vR + v′R).

(B −18)Here, PU is the projection matrix defined in eq.(4-24). This information about thezero mode in the g−1-sector is sufficient to obtain Z(g−1, 1, ; τ).

The zero mode partof Z(g−1, 1; τ) will be given byZ(g−1, 1; τ)zero mode=pdet(1 −V )VΓD−d,Dgei2πτεg−1X(qL,qR)∈ΓD−d,Dg∗−(v∗L+v′L∗,v∗R+v′R∗)eiπτq2L−iπ¯τq2R,(B −19)where εg−1 denotes the zero point energy (of the left mover) of the ground state inthe g−1-sector,εg−1 = 2d/2Xa=114ρa(1 −ρa). (B −20)In the g−1-sector, d of D oscillators of the left mover are twisted with the phases(B-4).

Thus, the remaining oscillator part of Z(g−1, 1; τ) will be given byZ(g−1, 1; τ)osc= |q|−D6∞Yn=1(1 −q2n)−(D−d)(1 −¯q2n)−Dd/2Ya=1∞Yn=1[(1 −q2(n−ρa))(1 −q2(n−1+ρa))]−1. (B −21)– 37 –

Therefore, Z(g−1, 1; τ) can be written asZ(g−1, 1; τ) =pdet(1 −V )VΓD−d,Dgei2πτεg−1|η(τ)|2Dd/2Ya=1[−ie−iπτρa(η(τ))3ϑ1(ρaτ|τ)]×X(qL,qR)∈ΓD−d,Dg∗−(v∗L+v′L∗,v∗R+v′R∗)eiπτq2L−iπ¯τq2R. (B −22)From the expressions (B-14) and (B-22), it is easy to see thatZ(1, g; −1τ )|φ=0 = Z(g−1, 1; τ).

(B −23)This proves that the phase φ of the twist operator g in the untwisted sector has tovanish. To show eq.

(B-23), we may use the formulas in appendix A. We can easilyfindZ(1, g, 1; −1τ ) =eiπφ|η(τ)|2Dd/2Ya=1[−2isin(πρa)(η(τ))3eiπτ(ρa)2ϑ1(ρaτ|τ) ]×1VΓD−d,DgX(qL,qR)∈ΓD−d,Dg∗−(v∗L+v′L∗,v∗R+v′R∗)eiπτq2L−iπ¯τq2R,(B −24)where we have used the relation(vL + v′L) · kL −(vR + v′R) · kR = (v∗L + v′L∗) · kL −(v∗R + v′R∗) · kR,(B −25)for (kL, kR) ∈ΓD−d,Dg.

Using eq. (B-20) and the relationpdet(1 −V ) =d/2Ya=1(2sin(πρa)),(B −26)we finally obtain the relation (B-23).– 38 –

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