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The General Class of String Theories on Orbifolds

arXiv:hep-th/9202083v2 2 Mar 1992KOBE–92–02February 1992The General Class of String Theories on OrbifoldsMakoto SAKAMOTO and Masayoshi TABUSEDepartment of Physics, Kobe UniversityNada, Kobe 657, JapanAbstractWe investigate the following three consistency conditions for constructing stringtheories on orbifolds: i) the invariance of the energy-momentum tensors under twistoperators, ii) the duality of amplitudes and iii) modular invariance of partition func-tions. It is shown that this investigation makes it possible to obtain the general classof consistent orbifold models, which includes a new class of orbifold models.– 1 –

1. IntroductionIn the construction of realistic four-dimensional string models, various approacheshave been proposed [1-8].

Among them, the orbifold compactification [1] is probablythe most efficient method and is believed to provide a phenomenologically realisticstring model. The search for realistic orbifold models has been continued by many au-thors [9-12].

However, only a very small class of orbifold models has been investigatedso far and any satisfactory orbifold models have not yet been found. A more generaland systematic investigation should be required.An orbifold [1] will be obtained by dividing a torus by the action of a discretesymmetry group G of the torus.

A large number of studies have been made on a classof orbifold models in which any group element g of G is represented by [1]g = (U, v),(1 −1)or more generally for asymmetric orbifolds [13]g = (UL, vL; UR, vR),(1 −2)where (UL, UR) are rotation matrices and (vL, vR) are shift vectors. The action of gon a left- and right-moving string coordinate (XL, XR) is given byg(XL, XR)g† = (UL(XL + 2πvL), UR(XR −2πvR)).

(1 −3)However, to the best of our knowledge, there have been few discussions about thequestions whether there might be any other class of consistent orbifold models andwhether the action of g on the string coordinate might in general be given by eq. (1-3).The purpose of this paper is to answer the question what is the most generalclass of consistent bosonic string theories on orbifolds.

We shall show that any groupelement g of G can indeed be specified by eq. (1-2) but that the action of g on thestring coordinate given in eq.

(1-3) is not, in general, correct.In section 2, we describe the basic setup and discuss consistency conditions ofstring theories on orbifolds. In section 3, we investigate the cocycle property of vertexoperators and present an explicit operator representation of cocycle operators, whichare attached to vertex operators to ensure the duality of amplitudes.

In section 4, we– 2 –

discuss the duality of amplitudes in detail. It is shown that the requirement of theduality of amplitudes severely restricts the allowed action of g on the string coordinateand that the transformation (1-3) has to be modified in general.

In this analysis, wesee that the representation of the cocycle operator given in section 3 plays an crucialrole. In section 5, we discuss one loop modular invariance of partition functions andsee that this argument justifies our prescription.

In section 6, we present an exampleof orbifold models, which will give a good illustration of our formalism. Section 7 isdevoted to discussions.

In appendix A, we prove a theorem, which will be used in thetext. In appendix B, we prove that any representation of cocycle operators can reduceto the representation given in section 3 by a suitable unitary transformation (up to aconstant phase).2.

Operator Formalism for Bosonic String Theories on OrbifoldsAn orbifold [1] will be obtained by dividing a torus by the action of a suit-able discrete group G.In the construction of an orbifold model, we start with aD-dimensional toroidally compactified closed bosonic string theory which is specifiedby a (D + D)-dimensional lorentzian even self-dual lattice ΓD,D [14]†, on which theleft- and right-moving momentum (pIL, pIR) (I = 1, · · ·, D) lies. Since an orbifold modelis given by specifying the action of each group element g of G on the left- and right-moving string coordinate (XIL, XIR) (I = 1, · · ·, D), our aim of this paper is to answerthe question what is the most general allowed action of g on the string coordinate.

Todetermine the allowed action of g on the string coordinate, we require the followingthree conditions:(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 [15,16].

(iii) Modular invariance of partition functions; Modular invariance plays an important† The generalization to a lorentzian even self-dual lattice with signature (p, q) willbe straightforward and will not be discussed here.– 3 –

role in the construction of consistent string models [16] and conformally invariantfield theories [17]. Modular invariance may ensure the ultraviolet finiteness andthe anomaly free condition of superstring theories [16,18].

The space-time unitaryalso requires modular invariance [19].Although the first and the third conditions (i) and (iii) have already been con-sidered, little attention has been given to the second condition (ii) so far. As we willsee later, our main results will be obtained from the detailed analysis of the secondcondition (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 −1)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 −2)It follows that the energy-momentum tensors are invariant under the action of g if gacts on (P IL(z), P IR(¯z)) asg(P IL(z), P IR(¯z))g† = (U IJL P JL(z), U IJR P JR(¯z)),(2 −3)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 that orbifolds with UL ̸= UR arecalled asymmetric orbifolds [13].In the untwisted sector, the left- and right-moving string coordinates, XIL(z) andXIR(¯z), are expanded asXIL(z) = xIL −ipILlnz + iXn̸=01nαILnz−n,XIR(¯z) = xIR −ipIRln¯z + iXn̸=01nαIRn¯z−n,(I = 1, · · ·, D),(2 −4)– 4 –

where xIL and pIL (xIR and pIR) are the center of mass coordinate and momentum ofthe left- (right-) mover, respectively. The quantization conditions are given by[xIL, pJL] = iδIJ = [xIR, pJR],[αILm, αJLn] = mδIJδm+n,0 = [αIRm, αJRn],otherwise zeros.

(2 −5)The toroidal compactification means that the momentum (pIL, pIR) lies on a (D + D)-dimensional lorentzian even self-dual lattice ΓD,D [14].In terms of (pIL, αILn) and(pIR, αIRn), eq. (2-3) can be rewritten asg(pIL, αILn)g† = U IJL (pJL, αJLn),g(pIR, αIRn)g† = U IJR (pJR, αJRn).

(2 −6)Since (pIL, pIR) lies on the lattice ΓD,D, the action of g on (pIL, pIR) should be anautomorphism of ΓD,D, i.e.,(U IJL pJL, U IJR pJR) ∈ΓD,Dfor all(pIL, pIR) ∈ΓD,D. (2 −7)Since P IL(z) and P IR(¯z) do not include xIL and xIR, the relation (2-3) or (2-6) doesnot completely determine the action of g on (xIL, xIR).

In fact, the general action of gon (xIL, xIR), which is compatible with the quantization conditions (2-5), may be givenby [20]g(xIL, xIR)g† = (U IJL (xJL + π ∂Φ(pL, pR)∂pJL), U IJR (xJR + π ∂Φ(pL, pR)∂pJR)),(2 −8)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 −9)andgU|0 >= |0 >,(2 −10)where |0 > is the vacuum of the untwisted sector.

Then, the twist operator g whichgenerates the transformations (2-6) and (2-8) will be given byg = eiπΦ(pL,pR)gU. (2 −11)– 5 –

At this stage, Φ(pL, pR) is an arbitrary function of pIL and pIR. In section 4, we willsee that the second condition (ii) severely restricts the form of the phase factor in g.It may be worth while making a comment on the path integral formalism [21]here.

The action of a closed bosonic string theory will be given by †S[X] =ZdτZ π0dσ 12π{ηαβ∂αXI∂βXI + εαβBIJ∂αXI∂βXJ},(2 −12)where τ and σ correspond to the “time” and “space” variables of the world sheet andBIJ (I, J = 1, · · ·, D) is an antisymmetric constant background field [14]. The stringcoordinate XI(τ, σ) in the untwisted sector will be expanded asXI(τ, σ) = xI + (pI −BIJwJ)τ + wIσ + (oscillators),(2 −13)where pI and wI are the center of mass momentum and the winding number, whichare related to the left- and the right-moving momenta, pIL and pIR, as follows:pIL = 12pI + 12(1 −B)IJwJ,pIR = 12pI −12(1 + B)IJwJ.

(2 −14)Let us consider a transformationXI →U IJXJ,(2 −15)where U IJ ∈O(D). (This corresponds to a symmetric orbifold, i.e., UL = UR ≡U.

)Clearly the action (2-12) is not invariant under the transformation (2-15) unless[B, U] = 0. (2 −16)The noncommutativity of BIJ and U IJ might cause a trouble in the path integralformalism because the action (2-12) will not be single-valued on the orbifold.

On theother hand, this noncommutativity seems to cause no trouble in the operator formalismbecause the second term in the action (2-12) is a total divergence and hence the explicitBIJ-dependence does not appear in the energy-momentum tensors (2-1) as well as theequation of motion. The BIJ-dependence can, however, appear in the zero modes as† ηαβ = diag(1, −1) and ε01 = −ε10 = 1.– 6 –

in eqs. (2-13) and (2-14).

As we will see in section 4, the noncommutativity of BIJand U IJ (more generally see eq. (4-6) for asymmetric orbifolds) might cause a troubleeven in the operator formalism, that is, the violation of the duality of amplitudes.

Itsresolution will be our main concern in section 4. In this paper, we will restrict ourconsiderations only to the operator formalism.

We will leave the reinterpretation ofour results from the point of view of the path integral formalism for future work.3. A Representation of Cocycle OperatorsIn this section, we shall investigate cocycle properties of vertex operators andgive an explicit operator representation of cocycle operators.

Let us consider a vertexoperator which describes the emission of a state with the momentum (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 isattached to the vertex operator to ensure the correct commutation relations and theduality of amplitudes [16,22]. 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 cor-responds 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 [15]. 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 modes, 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)– 7 –

The factor (−1)kL·k′L−kR·k′R in eq. (3-4) appears in reversing the order of the nonzeromodes of the vertex operators.

This annoying factor is the reason for the necessity ofthe cocycle operator CkL,kR.The second condition (ii) is now replaced by the statement that the dualityrelation (3-4) has to be preserved under the action of g.To examine this condi-tion, we need to know an explicit operator representation of the cocycle operatorCkL,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.) Toobtain an operator representation of the cocycle operator Ck, let us assume [23,24]Ck = eiπkAMAB ˆpB,(3 −6)where the wedge ∧may be attached to operators to distinguish between c-numbersand q-numbers.

Then, the matrix M AB has to satisfyeiπkA(M−MT )ABk′B = (−1)kAηABk′Bfor allkA, k′A ∈ΓD,D,(3 −7)whereηAB =100−1AB. (3 −8)A solution to this equation may be given byM AB =−12BIJ−12(1 −B)IJ12(1 + B)IJ−12BIJAB,(3 −9)which satisfiesM AB = −M BA.

(3 −10)The BIJ is an antisymmetric constant matrix and is defined as follows: Any (D +D)-dimensional lorentzian even self-dual lattice ΓD,D can be parametrized in terms of aD-dimensional Euclidean lattice Λ and an antisymmetric constant matrix BIJ [14] as†pIL = 12pI + 12(1 −B)IJwJ,pIR = 12pI −12(1 + B)IJwJ,(3 −11)† The variables pI, wI and BIJ are exactly the same as in eq. (2-14).– 8 –

where(pIL, pIR) ∈ΓD,D,pI ∈2Λ∗,wI ∈Λ. (3 −12)Here, Λ∗denotes the dual lattice of Λ. Physically, pI and wI correspond to the centerof mass momentum and the winding number, respectively.Although we have obtained a representation of the cocycle operator Ck, its rep-resentation is not unique.

In fact, there exist infinitely many other representations ofCk. However, as we will see in appendix B, by a suitable unitary transformation anyrepresentation of Ck can be shown to reduce to eq.

(3-6) with (3-9) up to a constantphase. Thus, it will be sufficient to consider only the representation (3-6) with (3-9)for our purpose.

In the next section, we will see that the representation (3-6) plays acrucial role in investigating the duality of amplitudes.4. The Duality of AmplitudesIn the previous section, we have obtained a representation of the cocycle operatorCk.

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. (4 −1)Under the action of gU, V0(k; M) transforms asgUV0(k; M)g†U = V0(U T k; U T MU),(4 −2)whereU AB =U IJL00U IJRAB.

(4 −3)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),(4 −4)– 9 –

whereξ = e−iπk·(M−UT MU)k′. (4 −5)This relation implies that the duality relation (3-4) cannot be preserved under theaction of gU unless ξ = 1 for all kA, k′A ∈ΓD,D.

It may be worth while noting that ifξ ̸= 1 it means[M, U] ̸= 0. (4 −6)For symmetric orbifolds (i.e., UL = UR), eq.

(4-6) means the noncommutativity of BIJand U IJL(or U IJR ). As mentioned in section 2, this noncommutativity may cause atrouble in the path integral formalism and, as just seen above, also in the operatorformalism it causes a trouble, that is, the violation of the duality relation (3-4) underthe action of gU.We have seen that the duality relation (3-4) cannot be preserved under the actionof gU unless ξ = 1 for all kA, k′A ∈ΓD,D.

It does not, however, mean the violationof the duality relation under the action of g because the freedom of Φ(p) in g has notbeen used yet. DefineV ′0(k; M) ≡gV0(Uk; M)g†= eik·ˆxeiπk·UT MU ˆpeiπΦ(ˆp+k)−iπΦ(ˆp).

(4 −7)It is easy to see thatV0(k; M)V ′0(k′; M) = e−iπΘ(−1)k·ηk′V ′0(k′; M)V0(k; M),(4 −8)whereΘ = kA(M −U T MU)ABk′B + Φ(p −k −k′) −Φ(p −k) −Φ(p −k′) + Φ(p). (4 −9)Thus the duality relation (3-4) requires thatΘ = 0mod 2.

(4 −10)To solve the equation (4-10), it may be convenient to change the basis of the momen-tum pA ∈ΓD,D. Let eAa (a = 1, · · ·, 2D) be a basis of ΓD,D, i.e.,ΓD,D = {pA =2DXa=1paeAa , pa ∈Z}.

(4 −11)– 10 –

Suppose that Φ(p) is expanded asΦ(p) = Φ0(p) + ∆Φ(p),(4 −12)whereΦ0(p) = φ + 2vapa + 12Cabpapb,(4 −13)∆Φ(p) =NXn=21n!∆C(n)a1···anpa1 · · ·pan,(pa ∈Z). (4 −14)Here, N (≥2) is an arbitrary positive integer and the symmetric matrix Cab is definedthrough the relation,Cab = −eAa (M −U T MU)ABeBbmod 2.

(4 −15)At first sight, it seems that there is no solution to eq. (4-15) because Cab is a symmetricmatrix but M AB is an antisymmetric one.

However, we can always find a symmetricmatrix Cab satisfying (4-15) because eq. (3-7) with eq.

(3-10) implies thateAa (M −U T MU)ABeBb ∈Z,(4 −16)which guarantees the existence of a solution to eq.(4-15). Inserting eq.

(4-12) intoeq. (4-9) and using the relation (4-15), we find that the condition (4-10) reduces to∆Φ(p −k −k′) −∆Φ(p −k) −∆Φ(p −k′) + ∆Φ(p) = 0mod 2.

(4 −17)Inserting eq. (4-14) into eq.

(4-17) and comparing the Nth order terms of bothsides of eq. (4-17) with respect to pa, ka and k′a, we have1N!2DXa1,···,aN=1∆C(N)a1···aN {(p −k −k′)a1 · · · (p −k −k′)aN −(p −k)a1 · · · (p −k)aN−(p −k′)a1 · · · (p −k′)aN + pa1 · · · paN } = 0mod 2,(4 −18)for all pa, ka, k′a ∈Z.This equation gives various constraints on the coefficient∆C(N)a1···aN .

For example,∆C(N)a···ab ∈(N −1)! 2Z,∆C(N)a···abb ∈(N −2)!

2! 2Z,– 11 –

∆C(N)a···abc ∈(N −2)! 2Z,∆C(N)a···abbb ∈(N −3)!

3! 2Z,· · ·etc.

(4 −19)Then it is not difficult to show the following equality:1N!2DXa1,···,aN=1∆C(N)a1···aN pa1 · · · paN = 1N!2DXa=1∆C(N)a···a(pa)Nmod 2,(4 −20)with ∆C(N)a···a ∈(N −1)! 2Z.

Let (m, n) be the highest common divisor of m and n andϕ(n) be the Euler function, which is equal to the number of d such that (d, n) = 1 ford = 1, 2, · · ·, n −1. In appendix A, we will prove the following theorem: Let n and pbe arbitrary positive integers.

Thenpn = pn−ϕ(n)mod n.(4 −21)Using this theorem and noting ∆C(N)a···a ∈(N −1)! 2Z, we have1N!2DXa=1∆C(N)a···a(pa)N = 1N!2DXa=1∆C(N)a···a(pa)N−ϕ(N)mod 2.

(4 −22)Thus we have found that the Nth order term of Φ(p) (i.e., the left hand side of eq. (4-20)) can reduce to the (N −ϕ(N))th order term (i.e., the right hand side of eq.

(4-22)).Therefore, we can put the Nth order term of Φ(p) to be equal to zero because theright hand side of eq. (4-22) can be absorbed into the (N −ϕ(N))th order term of Φ(p)by suitably redefining the coefficient of the (N −ϕ(N))th order term of Φ(p).Next consider the (N −1)th order term in eq.

(4-17) with respect to pa, ka andk′a. Comparing both sides of eq.

(4-17) and using the theorem (4-21), we can show thefollowing equality:1(N −1)!2DXa1,···,aN−1=1∆C(N−1)a1···aN−1pa1 · · · paN−1=1(N −1)!2DXa=1∆C(N−1)a···a(pa)N−1−ϕ(N−1)mod 2,(4 −23)– 12 –

with ∆C(N−1)a···a∈(N −2)! 2Z.

Thus the (N −1)th order term of Φ(p) can be absorbedinto the (N −1 −ϕ(N −1))th order term of Φ(p) by suitably redefining the coefficientof the (N −1−ϕ(N −1))th order term. Repeating the above argument order by order,we conclude that ∆Φ(p) can be put to be equal to zero, i.e.,∆Φ(p) = 0mod 2.

(4 −24)We have observed that the duality relation can be preserved under the action ofg if Φ(p) in g is chosen asΦ(p) = φ + 2vapa + 12paCabpb,or equivalently,Φ(p) = φ + 2vAηABpB + 12pACABpB,(4 −25)where the symmetric matrix Cab is defined through the relation (4-15) andva = vAηABeBa ,Cab = eAa CABeBb . (4 −26)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.(4-15). Let C′ab be another choice satisfying eq.(4-15).

Then, we find12Xa,b(C′ab −Cab)papb = 12Xa=b(C′aa −Caa)(pa)2mod 2= 12Xa(C′aa −Caa)pamod 2,(4 −27)where we have used the fact that C′ab −Cab ∈2Z and pa ∈Z. Thus the differencebetween C′ab and Cab can be absorbed into the redefinition of va and hence the choiceof Cab is essentially unique.

Therefore, it is concluded that any twist operator g canalways be parametrized by (UL, vL; UR, vR) and that the action (1-3) of g on the stringcoordinate XA = (XIL, XIR) in the untwisted sector is not in general correct butgXAg† = U AB(XB + 2πηBCvC + πCBCpC),(4 −28)as announced in the introduction.– 13 –

Does the third term of Φ(p) in eq. (4-25) affect the physical spectrum?Anyphysical state on the orbifold has to be invariant under the action of g. The thirdterm in eq.

(4-25) contributes to g as a momentum-dependent phase and hence plays animportant role in extracting physical states from the Hilbert space. Although we haveintroduced the third term of Φ(p) in eq.

(4-25) to preserve the duality of amplitudes, wewill see in the next section that modular invariance will also require the introductionof the third term of Φ(p) in eq.(4-25).5. 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,(5 −1)where L0(¯L0) is the Virasoro zero mode operator of the left- (right-) mover. The tracein eq.

(5-1) is taken over the Hilbert space of the h-sector. Then, the one loop partitionfunction will be of the form,Z(τ) = 1NXg,h∈Ggh=hgZ(h, g; τ),(5 −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: To calculate Z(h, g; τ) in the operator formalism, we need to introduce thestring coordinate (XIL(z), XIR(¯z)) in the h-sector, which obeys the boundary condition(XIL(e2πiz), XIR(e−2πi¯z)) = h · (XIL(z), XIR(¯z)),(5 −3)up to torus shifts. Let us consider the action of g on the string coordinate in theh-sector.

Then it turns out that g(XIL(z), XIR(¯z))g† obeys the boundary conditionof the ghg−1-sector. Let |h > be any state in the h-sector.

The above observationimplies that the state g|h > belongs to the ghg−1-sector but not the h-sector (unlessg commutes with h).Therefore, in the trace formula (5-1), Z(h, g; τ) will vanishidentically unless g commutes with h.– 14 –

One loop modular invariance of the partition function is satisfied providedZ(h, g; τ + 1) = Z(h, hg; τ),(5 −4)Z(h, g; −1τ ) = Z(g−1, h; τ). (5 −5)Let us first consider the partition function of the untwisted sector twisted by g, i.e.,Z(1, g; τ).

It follows from the discussions of section 4 that in the untwisted sector thetwist operator g would be of the formg = eiπΦ(p)gU,(5 −6)whereΦ(p) = φ + 2vAηABpB + 12pACABpB. (5 −7)The symmetric matrix CAB is defined through the relation (4-15) orkACABk′B = −kA(M −U T MU)ABk′Bmod 2,(5 −8)for kA, k′A ∈ΓD,D.

Let n be the smallest positive integer such that gn = 1. Then, itfollows thatU n = 1,(5 −9)nφ +n−1Xℓ=0{2vA(ηU ℓ)ABpB + 12pA(U −ℓCU ℓ)ABpB} = 0mod 2,(5 −10)for all pA ∈ΓD,D.

The zero mode part of Z(1, g; τ) can easily be evaluated and theresult isZ(1, g; τ)zero mode =X(kR,kR)∈Γd, ¯dgeiπΦ(k)eiπτk2L−iπ¯τk2R,(5 −11)where Γd, ¯dgis the g-invariant sublattice of ΓD,D, i.e.,Γd, ¯dg= {(kIL, kIR) ∈ΓD,D|(U IJL kJL, U IJR kJR) = (kIL, kIR)}. (5 −12)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,(5 −13)– 15 –

for all kA ∈Γd, ¯dg . To show this , definef(k) ≡12kACABkB.

(5 −14)Note thatkACABk′B = −kA(M −U T MU)ABk′Bmod 2= 0 mod 2for all kA, k′A ∈Γd, ¯dg ,(5 −15)where we have used eqs. (5-8) and (5-12).

It follows thatf(k + k′) = f(k) + f(k′)mod 2,(5 −16)for all k, k′ ∈Γd, ¯dg . This relation ensures the existence of a vector v′ satisfying eq.(5-13).

Using the relation (5-13), we can write eq. (5-11) asZ(1, g; τ)zero mode =X(kL,kR)∈Γd, ¯dgeiπφ+i2π(v+v′)·ηkeiπτk2L−iπ¯τk2R.

(5 −17)It will be useful to introduce a projection matrix PU defined byPU = 1nn−1Xℓ=0U ℓ. (5 −18)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,(5 −19)where v∗+ v′∗≡PU(v + v′), VΓ denotes the unit volume of the lattice Γ and Γd, ¯d∗gis the dual lattice of Γd, ¯dg .

It follows from eq. (5-19) that we can easily extract infor-mation about the zero modes of the g−1-sector because Z(1, g; τ) should be related toZ(g−1, 1; τ) through the modular transformation, i.e.,Z(g−1, 1; τ) = Z(1, g; −1τ ).

(5 −20)– 16 –

The degeneracy of the ground state in the g−1-sector may be given by [13]pdet′(1 −U)VΓd, ¯dg,(5 −21)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′∗.

(5 −22)It should be noted that the momentum eigenvalues in the g−1-sector are not givenby Γd, ¯d∗g−v∗, which might naively be expected [13]. The origin of the extra con-tribution −v′∗is the third term in eq.

(5-7), which has been introduced to ensure theduality relation of vertex operators. As we will see later, this extra contribution to themomentum eigenvalues becomes important in the left-right level matching condition.Information about the zero modes 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.

Then, itturns out that the relation (5-20) puts a constraint on φ in eq. (5-7), i.e.,φ = 0.

(5 −23)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 [13,25]Z(g−1, h; τ + n) = Z(g−1, h; τ).

(5 −24),where n is the smallest positive integer such that gn = 1. It follows from eq.

(5-1) thatthe level matching condition is satisfied only if2n(L0 −¯L0) = 0mod 2,(5 −25)– 17 –

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′∗,(5 −26)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).

(5 −27)Here, exp(i2πρa) and exp(i2π¯ρa) (a = 1, · · ·, D) are the eigenvalues of UL and URwith 0 ≤ρa, ¯ρa < 1, respectively.The condition (5-26) 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. (5 −28)To see this, we first note that Γd, ¯dg∗can be expressed as [13]Γd, ¯dg∗= PUΓD,D= {qA = PUkA, kA ∈ΓD,D}.

(5 −29)This follows from the property that ΓD,D is self-dual. From eq.

(5-29), any momentumqA ∈Γd, ¯dg∗−v∗−v′∗can be parametrized asqA = PU(k −v −v′)Afor some kA ∈ΓD,D. (5 −30)Then, we haven(q2L −q2R) = nqAηABqB= nkA(ηPU)ABkB −2n(v + v′)A(ηPU)ABkB + n(v∗+ v′∗)AηAB(v∗+ v′∗)B,(5 −31)– 18 –

where we have used the relationsPUη = ηPU,P2U = PU,PTU = PU. (5 −32)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.

(5-31) is easily shown to reduce tonkA(ηPU)ABkB =kA(ηUn2 )ABkBmod 2 if n = even,0mod 2 if n = odd. (5 −33)Using the relation (5-13) and noting that nPUk ∈Γd, ¯dg , we can rewrite the secondterm in the right hand side of eq.

(5-31) as−2n(v+v′)A(ηPU)ABkB = −2nvA(ηPU)ABkB−12kAn−1Xℓ=0n−1Xm=0(U −ℓCU m)ABkB mod 2. (5 −34)Replacing p by p + p′ in eq.

(5-10) with eq. (5-23) and then using eq.

(5-10) again, wehavepAn−1Xℓ=0(U −ℓCU ℓ)ABp′B = 0mod 2,(5 −35)for all p, p′ ∈ΓD,D. For n odd, it is not difficult to show that−2n(v + v′)A(ηPU)ABkB = 0mod 2.

(5 −36)To derive eq. (5-36), we will use eqs.

(5-10), (5-23), (5-34) and (5-35). For n even, wewill find−2n(v + v′)A(ηPU)ABkB = −kAn2 −1Xℓ=0(U −ℓCU ℓ+ n2 )ABkBmod 2.

(5 −37)Remembering the relations (3-7), (3-10) and (5-8), we can finally find that for n even−2n(v + v′)A(ηPU)ABkB = kA(ηUn2 )ABkBmod 2. (5 −38)Combiningtheresults(5-33), (5-36)and(5-38)andusingthefactthatkA(ηUn2 )ABkB ∈Z, we havenkA(ηPU)ABkB −2n(v + v′)A(ηPU)ABkB = 0mod 2.

(5 −39)– 19 –

This completes the proof of (5-28).We have shown that the left-right level matching condition (5-24) reduces to thecondition (5-28), which puts a restriction on the shift vector v = (vL, vR). It should benoticed that the level matching condition (5-28) is not always satisfied for asymmetricorbifold 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 of CAB = 0in eq. (5-7), it has been proved, in refs.

[13,25], that the level matching condition isa necessary and also sufficient condition for one loop modular invariance. Even forthe case of nonzero CAB, the sufficiency can probably be shown by arguments similarto refs.

[13,25] although to this end we need to know the action of g on the stringcoordinate in every twisted sector.It should be emphasized that the third term in eq. (5-7) plays an important rolein the level matching condition because the relation (5-39) might not hold in generalif we put v′ to be zero, that is, Cab to be zero by hand.

In the next section, we will seesuch an example that the introduction of the third term in eq. (5-7) makes partitionfunctions modular invariant.6.

An ExampleIn this section, we shall investigate a symmetric Z2-orbifold model in detail, whichwill give a good illustration of our formalism. Many other examples can be found inref.

[26].Let us introduce the root lattice ΛR and the weight lattice ΛW 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),– 20 –

α2 = ( 1√2, −r32),µ1 = ( 1√2,r16),µ2 = ( 1√2, −r16). (6 −2)The left- and right-moving momentum (pIL, pIR) (I = 1, 2) is defined by eq.

(3-11), i.e.,pIL = 12pI + 12(1 −B)IJwJ,pIR = 12pI −12(1 + B)IJwJ,(6 −3)where pI and wI are the center of mass momentum and the winding number, respec-tively and are assumed to lie on the following lattices:pI ∈2ΛW ,wI ∈ΛR. (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 lorentzianeven self-dual lattice Γ2,2:Γ2,2 = {(pIL, pIR)|pIL, pIR ∈ΛW , pIL −pIR ∈ΛR}. (6 −6)We consider the following Z2-transformation:gU(XIL, XIR)g†U = (U IJL XJL, U IJR XJR),(6 −7)whereU IJL= U IJR =100−1.

(6 −8)This is an automorphism of Γ2,2,, as it should be. According to our prescription, theZ2-twist operator g will be of the formg = ei π2 pACABpBgU,(6 −9)– 21 –

where pA ≡(pIL, pIR) and the symmetric matrix CAB is defined through the relationpACABp′B = −pA(M −U T MU)ABp′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 by(pL−pR)ICIJ(p′L−p′R)J = 12(pL−pR)I(B−U TL BUL)IJ(p′L−p′R)Jmod 2, (6 −12)where CAB has been assumed to be of the formCAB =CIJ−CIJ−CIJCIJAB. (6 −13)Thus, the twist operator (6-9) can be written asg = ei π2 (pL−pR)ICIJ(pL−pR)JgU.

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

(6-15) is found to be12αIi (B −U TL BUL)IJαJj =01−10ij,(6 −16)and hence CIJ cannot be chosen to be zero. We may choose CIJ asαIi CIJαJj =0110ij,orCIJ =100−13IJ.

(6 −17)– 22 –

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 SU(3) Kaˇc-Moody algebra [22].

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) = 2P 1L(z),J±(z) =√2(VL(±α1; z) + VL(±α2; z)).

(6 −20)These generators are found to form level four SU(2) Kaˇc-Moody algebra [27]. Notethat the vertex operators VL(±(α1 + α2); z) are not invariant under the action of gand hence they are removed from the physical generators although the root vectorα1 + α2 is invariant under the action of gU.We now examine one loop modular invariance of the partition function which willbe given byZ(τ) = 121Xℓ,m=0Z(gℓ, gm; τ),(6 −21)whereZ(gℓ, gm; τ) = Tr[gmei2πτ(L0−224 )−i2π¯τ(¯L0−224 )]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)– 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√2,Γ1,1g= {(kL, kR) = (√2n,√2n′), n, n′ ∈Z}. (6 −25)Here, the shift vector (v′L, v′R) has been introduced through the relation (5-13).

Thefunction η(τ) and ϑa(ν|τ) (a = 1, · · ·, 4) are the Dedekind η-function and the Jacobitheta function:η(τ) = q1/12∞Yn=1(1 −q2n),(q = eiπτ),ϑab(ν|τ) =∞Xn=−∞exp{iπ(n + a)2τ + i2π(n + a)(ν + b)},ϑ1(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). (6 −26)It follows from the arguments given in section 5 that the degeneracy of the groundstate in the g-sector ispdet′(1 −U)VΓ1,1g= 1,(6 −27)and that the momentum eigenvalues in the g-sector are given by(qL, qR) ∈Γ1,1g∗−(v′L, v′R),(6 −28)whereΓ1,1g∗= {(qL, qR) = ( 1√2n, 1√2n′), n, n′ ∈Z}.

(6 −29)This information is sufficient 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,(6 −30)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 −31)– 24 –

It is easily verified that Z(gℓ, gm; τ) satisfies the following desired relations:Z(gℓ, gm; τ + 1) = Z(gℓ, gm+ℓ; τ),Z(gℓ, gm; −1τ ) = Z(g−m, gℓ; τ),(6 −32)and hence the partition function (6-21) is modular invariant. It should be empha-sized that the existence of the shift vector (v′L, v′R) ensures modular invariance of thepartition function: The level matching conditionZ(g, 1; τ + 2) = Z(g, 1; τ),(6 −33)is satisfied because for all (qL, qR) ∈Γ1,1g∗−(v′L, v′R),4(12q2L −12q2R) = 0mod 2.

(6 −34)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. (6-34) dose not hold for (qL, qR) ∈Γ1,1g∗.It is interesting to note that in terms of the theta functions the partition functionobtained above can be expressed asZ(1, 1; τ) =1|η(τ)|4X(kL,kR)∈Γ2,2eiπτk2L−iπ¯τk2R,Z(1, g; τ) = |ϑ3(0|τ)ϑ4(0|τ)|2|η(τ)|4,Z(g, 1; τ) = |ϑ3(0|τ)ϑ2(0|τ)|2|η(τ)|4,Z(g, g; τ) = |ϑ4(0|τ)ϑ2(0|τ)|2|η(τ)|4.

(6 −35)This partition function is exactly identical to that of a symmetric Z′2-orbifold modelwhose Z′2-transformation is defined byZ′2 : (XIL, XIR) →(−XIL, −XIR),(6 −36)instead of the Z2-transformation (6-7). In this orbifold model, level one SU(3) Kaˇc-Moody algebra can be shown to “break” to level four SU(2) Kaˇc-Moody algebra.

Thus,– 25 –

although the two orbifold models are defined by the different Z2-transformations (6-7)and (6-36), they give the same spectrum and interaction [28].7. DiscussionsIn this paper, we have investigated the following three consistency conditions indetail: (i) the invariance of the energy-momentum tensors under the action of thetwist operators, (ii) the duality of amplitudes and (iii) modular invariance of partitionfunctions.

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

(5-7), which is necessary topreserve the duality of amplitudes under the action of g and which plays an importantrole in modular invariance of partition functions. The second point is that any twistoperator g has been proved to be represented by eq.(1-2).

To show this, we have seenthat the first condition (i) is not sufficient and that the second condition (ii) is crucialto restrict the allowed form of Φ(pL, pR) to eq.(5-7). It should be emphasized that it isvery important to show that by a suitable unitary transformation any representation ofcocycle operators can reduce to the representation (3-6) with eq.

(3-9) up to a constantphase because our analysis has heavily relied on the representation (3-6).We have found that the string coordinate XA = (XIL, XIR) in the untwisted sectortransforms under the action of g asgXAg† = U AB(XB + 2πηBCvC + πCBCpC). (7 −1)It seems that the third term of eq.

(7-1) has no clear geometrical meaning. Althoughthe momentum and vertex operators definitely transform under the action of g, whydoes not the string coordinate transform definitely?

The reason is probably that inthe point of view of the conformal field theory the string coordinate is not a primaryfield and it is not a well-defined variable on a torus. Thus, there may be no reasonwhy the string coordinate itself should definitely transform under the action of g. Onthe other hand, since the momentum and vertex operators are primary fields and arewell-defined on a torus, they should definitely transform under the action of g. In fact,– 26 –

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). (7 −2)It may be worth while pointing out that the “center of mass coordinate” xA = (xIL, xIR)always appears as the following combination:x′A ≡xA + πM ABpB,(7 −3)in the vertex operators and that x′A definitely transform under the action of g, i.e.,gx′Ag† = U AB(x′B + 2πηBCvC),(7 −4)up to torus shifts although xA itself does not.

This observation strongly suggests thatthe variable x′A is more fundamental than xA [24,26].We have succeeded to obtain 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 −5)This follows from the relationZ(g, g; τ) = Z(g, 1; τ + 1).

(7 −6)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. (4-5),– 27 –

(untwisted state emission) vertex operators in any twisted sector have already beenconstructed with correct cocycle operators in ref. [29].In the case of ξ ̸= 1, theprescription given in ref.

[29] will be insufficient to obtain desired vertex operatorsbecause the duality relation will not be satisfied. Some attempts [30] have been madebut the general construction of correct vertex operators is still an open problem.As mentioned in section 2, there might appear a trouble in the path integralformalism unless eq.

(2-16) is satisfied. Our success in the operator formalism, however,probably means that our results can be reinterpreted from the path integral point ofview.

Then the geometrical meaning will become clear.– 28 –

Appendix AIn this appendix, we shall prove the following theorem: Let n be a positive integer.Then, for any positive integer p,pn = pn−ϕ(n)mod n,(A −1)where ϕ(n) is the Euler function which is equal to the number of d such that (d, n) = 1for d = 1, 2, · · ·, n −1. Here, (d, n) denotes the highest common divisor of d and n.The Euler function satisfies the following relation:ϕ(mn) = ϕ(m)ϕ(n)if (m, n) = 1.

(A −2)To prove the theorem (A-1), we start with the Euler’s theorem:pϕ(n) = 1mod nfor (p, n) = 1. (A −3)Suppose that n is decomposed asn = (q1)ℓ1(q2)ℓ2 · · · (qr)ℓr,(A −4)where qi (i = 1, · · ·, r) is a prime number and qi ̸= qj if i ̸= j.

Let ℓmax be themaximum number in the set of {ℓi, i = 1, · · ·, r}. In terms of qi, any positive integerp can be decomposed asp = (q1)ℓ′1 · · · (qr)ℓ′rs,(A −5)where ℓ′i ≥0 and s is an integer such that (s, n) = 1.

Then it is not difficult to showthat(qi)ℓ′i(ϕ(n)+ℓmax) = (qi)ℓ′iℓmaxmod n,sϕ(n)+ℓmax = sℓmaxmod n.(A −6)It follows that for any positive integer ppϕ(n)+ℓmax = pℓmaxmod n.(A −7)The Euler function ϕ(n) satisfiesX1≤d≤nd|nϕ(d) = n,(A −8)– 29 –

where d|n means that d is a divisor of n. For any prime number qi,ϕ(qℓii ) = qℓi−1i(qi −1)≥qℓi−1i≥ℓi. (A −9)Then, it follows from eqs.

(A-8) and (A-9) thatn > ℓmax + ϕ(n). (A −10)Multiplying (A-7) by pn−ϕ(n)−ℓmax and noting n −ϕ(n) −ℓmax > 0, we finally havepn = pn−ϕ(n)mod n.(A −11)Appendix BIn this appendix, we shall prove that by a suitable unitary transformation anyrepresentation of the cocycle operator Ck can reduce toCk = eiπkAMAB ˆpB,(B −1)with eq.

(3-9) up to a constant phase.We first note that the following factor:eiπ(θ(ˆp+k)−θ(ˆp)),(B −2)can be removed from the cocycle operator by a suitable unitary transformation becausethe cocycle operator Ck appears always in the combination eik·ˆxCk.The cocycle operator Ck will consist of the zero modes. Since the vertex operator(3-1) should represent the emission of a state with the momentum kA, the cocycleoperator Ck will not depend on ˆxA and be represented in terms of ˆpA as well as kA.We may write the cocycle operator Ck into the formCk = eiπkAMAB ˆpB+iπFk(ˆp),(B −3)– 30 –

where M AB is defined by eq.(3-9). We require that the zero mode part of the vertexoperator (3-5) satisfiesV0(k)V0(k′) = ε(k, k′)V0(k + k′)(B −4)= (−1)k·ηk′V0(k′)V0(k),(B −5)where the phase factor ε(k, k′) is assumed to be c-number.

The above two conditionscan be replaced byFk+k′(ˆp) −Fk(ˆp + k′) −Fk′(ˆp) = ˆp−independentmod 2,Fk(ˆp + k′) + Fk′(ˆp) −Fk′(ˆp + k) −Fk(ˆp) = 0mod 2. (B −6)It will be convenient to use the following basis of the momentum: Let eAa (a =1, · · ·, 2D) be a basis of ΓD,D, i.e., any momentum kA ∈ΓD,D can be expressedaskA =2DXa=1kaeAa ,ka ∈Z.

(B −7)In this basis, we assume that Fk(ˆp) can be expanded in powers of ka and ˆpa as follows:Fk(ˆp) =NXn=22DXa1,···,an=1n−1Xj=11j! (n −j)!∆Ma1···ajaj+1···anka1 · · · kaj ˆpaj+1 · · · ˆpan,(ka, ˆpa ∈Z),(B −8)where N is an arbitrary positive integer and the coefficient ∆Ma1···ajaj+1···an is totallysymmetric with respect to lower indices or upper indices.Our aim of this appendix is now to show that by a suitable unitary transformationFk(ˆp) can always reduce toFk(ˆp) = 0mod 2,(B −9)up to ˆp-independent constant terms.

Before we prove eq. (B-9) for arbitrary N, it maybe instructive to examine the case of N = 3, i.e.,Fk(ˆp) =Xa,b∆Mabkaˆpb +Xa,b,c12!

{∆Mabckaˆpbˆpc + ∆Mabckakbˆpc}. (B −10)– 31 –

Inserting eq. (B-10) into eqs.

(B-6) and comparing the third order terms of both sidesof eqs. (B-6) with respect to ka, k′a and ˆpa, we findXa,b,c{∆Mabckak′bˆpc −∆Mabckak′bˆpc} = 0mod 2,Xa,b,c12!

{∆Mabckak′bk′c −∆Mabck′ak′bkc} = 0mod 2,(B −11)for all ka, k′a, ˆpa ∈Z. From these equations, we can show the following equality:Xa,b,c12!

{∆Mabckaˆpbˆpc + ∆Mabckakbˆpc}=Xa=b=c13!∆Maaa{(ˆpa + ka)3 −(ˆpa)3 −(ka)3}+ (Xa=b

(B-10) can be removed by a suitable unitary transformation (up to ˆp-independent terms).Next comparing the second order terms of both sides of eqs. (B-6) with respect toka, k′a and pa, we findXa,b{∆Mabkak′b −∆Mabk′akb} = 0mod 2.

(B −13)Without loss of generality, we can assume that the matrix ∆Mab is antisymmetricbecause the symmetric part of ∆Mab can be removed by a suitable unitary transfor-mation. Since eq.

(B-13) then means that ∆Mab ∈Z, we can introduce a symmetricmatrix Sab through the relationSab = ∆Mabmod 2. (B −14)In terms of Sab, the first term of eq.

(B-10) can be written asXa,b∆Mabkaˆpb =Xa,bSabkaˆpbmod 2=Xa,b12!Sab{(ˆpa + ka)(ˆpb + kb) −ˆpaˆpb −kakb}mod 2. (B −15)– 32 –

Therefore, we have proved that by suitable unitary transformations Fk(ˆp) given ineq. (B-10) can reduce to eq.

(B-9) up to ˆp-independent terms.Let us prove eq. (B-9) for arbitrary N. By inserting eq.

(B-8) into eqs. (B-6) andby comparing the Nth order terms of both sides of eqs.

(B-6) with respect to ka, k′aand ˆpa, it is not difficult to show the following equality:Xa1,···,aNN−1Xj=11j! (N −j)!∆Ma1···ajaj+1···aN ka1 · · · kaj ˆpaj+1 · · · ˆpaN=Xa{12!

(N −2)! (∆Maaa···a −∆Maa···a)(ka)2(ˆpa)N−2+13!

(N −3)! (∆Maaaa···a −∆Maa···a)(ka)3(ˆpa)N−3+ · · · +1(N −2)!2!

(∆Ma···aaa −∆Maa···a)(ka)N−2(ˆpa)2}mod 2, (B −16)with∆M a···a|{z}jN−jz}|{a···a −∆MaN−1z}|{a···a = 0mod 2(j −1)! (N −j)!,(B −17)where in the right hand side of eq.

(B-16) we have omitted ˆp-independent terms aswell as the terms which can be removed by unitary transformations. It follows fromthe theorem (A-1) that all the terms in the right hand side of eq.

(B-16) can reduceto lower order terms with respect to ka and ˆpa and hence they can be absorbed intolower order terms of Fk(ˆp) by suitably redefining the coefficients of the lower orderterms of Fk(ˆp). Therefore, we conclude that the Nth order terms of Fk(ˆp) can be putto be equal to zero.Repeating the above arguments order by order, we finally come to the conclusion(B-9).– 33 –

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