More on Non-supersymmetric Asymmetric Orbifolds with Vanishing Cosmological Constant

We explore various non-supersymmetric type II string vacua constructed based on asymmetric orbifolds of tori with vanishing cosmological constant at the one loop. The string vacua we present are modifications of the models studied in arXiv:1512.05155[hep-th], of which orbifold group is just generated by a single element. We especially focus on two types of modifications: (i) the orbifold twists include different types of chiral reflections not necessarily removing massless Rarita-Schwinger fields in the 4-dimensional space-time, (ii) the orbifold twists do not include the shift operator. We further discuss the unitarity and stability of constructed non-supersymmetric string vacua, with emphasizing the common features of them.


Introduction and Summary
Much attention has been currently focused on the string theories on non-geometric backgrounds. A simple and interesting class of such backgrounds are constructed due to the asymmetric orbifolds, in which the orbifold twists act asymmetrically on the left and right movers [1]. Although they look beyond our intuitive picture of space-time, they are well-described as done for geometrical ones by the approach of world-sheet conformal field theory (CFT) in the α ′ -exact fashion.
Above all, one of the natural purposes to study the type II string on asymmetric orbifolds would be the construction of non-supersymmetric (SUSY) string vacua with vanishing cosmological constant motivated by phenomenological or theoretical interests.
It seems evident that the SUSY-breaking realized in any geometric or symmetric orbifolds inevitably gives rise to a non-vanishing cosmological constant already at the oneloop. In this sense, the bose-fermi cancellation without SUSY would only be possible in the suitable non-geometric compactification in superstring theory. The attempts of construction of non-SUSY vacua have been initiated by the works [2][3][4] based on some non-abelian orbifolds, followed by closely related studies e.g. in [5][6][7][8]. Moreover, sharing similar motivations, non-SUSY vacua in heterotic string theory have been investigated e.g. in [9][10][11][12][13].
Recently, in our previous paper [14], we have presented a simple new realization of non-SUSY string vacua with the bose-fermi cancellation based on a cyclic orbifold, that is, the relevant orbifold group is generated by a single element. Hence, this construction looks rather simpler than the previous ones given in the papers quoted above. The crucial point in this construction is the fact that 'chiral reflection' (or the T-duality twist) along the T 4 -directions 1 ; 6,7,8,9), (1.1) is not necessarily involutive when acting on the world-sheet fermions, even in the untwisted sector 2 . Indeed, as illustrated in [14], while it is always involutive on the (rightmoving) NS-fermions in the untwisted sector, we still have two possibilities (i) R 2 = 1, (ii) R 2 = −1 for the R-sector. In other words, even though R 2 obviously commutes with all the world-sheet coordinates; it may still act on the Ramond vacua (or spin fields) as a sign flip. The case (ii) means that R 2 = (−1) F R , where F R (F L ) denotes the 'space-time fermion number' from the right(left)-mover. If taking the second one, which we often call the 'Z 4 -chiral reflection', one finds that the type II string vacuum constructed as the Z 4 -orbifold by σ ≡ (−1) F L ⊗ (−1 R ) ⊗4 possesses the next properties; • All the space-time supercharges arising from the untwisted sector are eliminated by the Z 4 -projection 1 4 r∈Z 4 σ r , since any supercharges in the unorbifolded theory do not commute with both of (−1) F L and (−1) F R .
• All the partition sums in the untwisted sector vanish under the insertion of σ r for ∀ r ∈ Z 4 . Namely, we find (q ≡ e 2πiτ ); 1 Through this paper, X µ ≡ (X µ L , X µ R ) (ψ µ ≡ (ψ µ L , ψ µ R )) denotes the world-sheet bosonic (fermionic) fields in the RNS formalism of type II string theory. The directions µ = 0, . . . , 3 are always identified as the 4-dim. Minkowski space-time M 4 , and we mainly focus on the transverse part µ = 2, . . . , 9. In addition, we will often use the notations λ i ≡ (λ i L , λ i R ) (i = 1, . . . , 2N ) to express the free fermions describing the N -dim. torus with the SO(2N )-symmetry enhancement, which will be denoted as T N [SO(2N )] in the text. 2 It is well-known that the chiral reflections often define order N ≥ 4 orbifolds rather than order 2 due to the non-trivial phase factors appearing in the twisted sectors, even though they act as an involution on the untwisted sector. See e.g. [15]. They are surely nice features for the purpose to realize the non-SUSY string vacua with the bose-fermi cancellation. However, as addressed in [14] and will be demonstrated in section 2 for a detail, it turns out that 8 supercharges eventually emerge in the twisted sector. We thus adopted in [14] the (infinite order) orbifold group generated by the in place of σ, following the spirit of Scherk-Schwarz type compactification [16,17]. Here, T 2πR denotes the shift by 2πR along the 'base' direction, originally identified as a real line R base . The inclusion of shift into (1.2) enables us to naturally identify the twisted sectors with the winding sectors of the 'Scherk-Schwarz circle'. More significantly, it plays the role of removing potential supercharges which might arise from the twisted sectors 3 . We also note that this model would be interpreted as a modification of the simple realizations of the 'T-folds' [18][19][20][21][22][23][24], that is, the orbifolds by the chiral reflection (or the T-duality twist) combined with the shift in the base space. These types of non-geometric backgrounds have been studied by the approach of world-sheet CFT e.g.
in [25][26][27][28][29][30][31][32]. Now, in this paper, we would like to explore a variety of non-SUSY string vacua of this type. We shall especially focus on the next two modifications of (1.2): (i) We replace (−1) F L with (−1 L ) ⊗2 , which acts along the various directions of backgorund tori, and plays the role of breaking the left-moving SUSY.
(ii) We do not include the shift operator T 2πR . Instead, we assume that R ≡ (−1 R ) ⊗4 acts as the Z 4 -chiral reflection also for the world-sheet bosons. This is achieved by utilizing the fermionization of bosonic coordinates X µ , and plays the role of preventing the twisted sectors from providing additional supercharges. 3 At first glance, this fact would look obvious, since the inclusion of shift T2πR generically makes all the Ramond states lying in the twisted sectors massive. However, we often find that additional Ramond massless states appear when choosing the Scherk-Schwarz radius R suitably. Nevertheless, one can show that the space-time SUSY is completely broken for an arbitrary value of R. See [14] for the detail.
Stated more concretely, the models that we shall study in this paper are displayed in Tables 1 and 2. In section 2, we briefly review on the 'previous' one studied in [14], which would be helpful to readers. We then investigate the new six models ('models I to VI') in section 3. We exhibit the relevant orbifold actions in Table 1, while the original backgrounds that we orbifold are summarized in Table 2. In all the models the orbifold groups are generated by a single element denoted as g in Table 1. In Table 2, M 4 expresses the four-dimensional Minkowski space-time. The orbifold twists do not act on M 4 × · · · in each row. The shift T 2πR always acts along R base . Throughout this paper, we use the notation 'T N [SO(2N )]' to express the N -dimensional torus at the symmetry enhancement point of SO(2N ). In other words, they can be described in Let us summarize the aspects of models I to VI on which we will elaborate in section 3. The models I and II are defined by including (−1 L ) ⊗2 instead of (−1) F L | ψ . Combining it with (−1 R ) ⊗4 , some directions of tori are eventually orbifolded by the non-chiral reflection: (X µ L , X µ R ) → (−X µ L , −X µ R ), and we simply denote 'T 2 ' and 'S 1 ' for the corresponding directions. It will be shown that these models are indeed the non-SUSY string vacua with the bose-fermi cancellation as expected. We do not have any tachyonic instability in all the untwisted and twisted sectors, while some winding massless modes emerge at particular values of the Scherk-Schwarz radius R. These features are quite similar to the previous one. However, the physical spectra significantly differ from it. Some Rarita-Schwinger fields survive in the 4-dim. massless spectrum in the models I and II, although not interpreted as the gravitini due to the absence of space-time SUSY.
We recall that, in the previous model, the twist by (−1) F L eliminates all the massless spin 3/2 states in the untwisted sector.
The models III-VI are those not including the shift operator. Instead, we shall modify the right-moving chiral reflections so that their squares yield (−1) F R | λ , that is, the sign flip on the Ramond sector of fermions λ i R that describe T N [SO(2N )]. The left-moving space-time SUSY is broken by (−1) F L | ψ in the model III as in the previous one, while (−1 L ) ⊗2 acts on the various directions of tori in the cases of models IV-VI. By the effect of (−1) F R | λ , the twisted sectors gain extra zero point energies despite the absence of shift operator, thereby preventing additional right-moving supercharges from arising. It then turns out that we achieve the desired non-SUSY vacua. They are simpler than the models I and II for the computations of the torus partition functions. Once again, we do not face any tachyonic instabilities, and massless states appear in the twisted sectors as well as the untwisted sector. Note that these models do not include the modulus R as opposed to the cases of models I and II.
The partition functions for all the models in this paper are found manifestly modular invariant and q-expanded in the way compatible with unitarity. Moreover, they are always free from tachyonic instabilities. These would be common features of the toroidal asymmetric orbifolds of these types, as we will discuss in section 4.
2 Notes on the Non-SUSY Asymmetric Orbifold of [14] In this section, we make a brief sketch of the non-SUSY model constructed in [14] to clarify several points that we will discuss for the new models.
Let us introduce the type II string vacuum in the ten-dimensional flat background; where M 4 (X 0,1,2,3 -directions) denotes the 4-dimensional Minkowski space-time, and is just a real line, identified as the 'base space' of the twisted compactification like Scherk-Schwarz [16,17], and, as already mentioned, Then, as was introduced in section 1, we define the asymmetric orbifold generated by the operator acting on the background (2.1). Recall that T 2πR denotes the shift operator along R base ; We then obtain since σ acts as the sign flip of ψ 6 R , . . . ψ 9 R . Thus we readily find which plays a crucial role in the following discussions. See [14] for more detail.
Let us focus on the partition function on the world-sheet torus to investigate the oneloop cosmological constant and the space-time supersymmetry. The relevant partition function is schematically written in the form as where the integer w labels the twisted sectors, while m indicates the g m -insertions into the trace. As already suggested in section 1, they are identified as the spatial and temporal winding numbers on the base space (or the Scherk-Schwarz circle) because of the inclusion of shift T 2πR into (2.2). Z X (w,m) (τ,τ ) denotes the partition functions of the bosonic sectors, while Z ψ L (w,m) (τ ), Z ψ R (w,m) (τ ) are the partition functions of the left-and right-moving fermionic sectors.
Each partition sum Z (w,m) (τ,τ ) is evaluated in the easiest way as follows. We first calculate the trace over the untwisted sector (w = 0) 4 , and those for the general winding sectors (w, m) are uniquely determined by requiring the modular covariance where S : τ → −1/τ , T : τ → τ + 1 are the modular transformations. We then achieve the partition function (2.6) that is manifestly modular invariant. 4 Here we shall adopt the conventional normalization of the trace for the CFT describing R base ; so that we simply obtain Note that the left and right partition sums of fermionic sectors Z ψ L (w,m) (τ ), Z ψ R (w,m) (τ ) are generically asymmetric. The twist operator σ includes (−1) F L | ψ , and we thus find However, one easily finds (2.11) Thus the total partition function vanishes.
Let us turn our attention to the spectrum in the untwisted sector (w = 0). As already mentioned in section 1, all the space-time supercharges are eliminated by the orbifold For all that, one can observe that the same number of bosonic and fermionic states exist at each mass level of the untwisted sector. Especially, the massless spectrum is summarized in Table   3, which includes 32 bosonic and fermionic states. Note that no gravitino appears in the 4-dim. spectrum, which suggests the absence of space-time SUSY. However, this is not the whole story. It might be possible that new supercharges arise from the twisted sectors. We also note that tachyonic states would potentially emerge in the twisted sectors, as in many examples of the SUSY-breaking models of Scherk-Schwarz type. Furthermore, the unitarity of string spectrum is not necessarily self-evident because of the non-trivial phase factors appearing in the twisted sectors necessary for the modular invariance. It is surely significant to examine these issues for our purpose. A direct way to do so is to decompose the partition functions with respect to the spatial winding w and the spin structures as where Z M 4 ×S 1 denotes the bosonic transverse contribution for the M 4 × S 1 -sector that has nothing to do with the orbifolding. The string spectrum in each Hilbert space with winding w can be examined by making the Poisson resummation with respect to the temporal winding m. In this way the following results have been shown in [14]; • The partition function for each winding w and each spin structure is compatible with unitarity.
• The bose-fermi cancellation is observed at each mass level of the string spectrum.
• The space-time SUSY is completely broken.
• No tachyonic states appear in all the sectors.
• Massless states arise in some twisted sectors at the specific radius R (the modulus related to the shift T 2πR ).
Especially, let us focus on how one can conclude that the space-time SUSY is truly broken. It has been explicitly shown in [14] that the partition functions for the winding sectors have the relations summarized in Table 4. For the odd winding sectors, we have the bose-fermi cancellation compatible only with right-moving SUSY, while the even sectors behave as if we only had left-moving supercharges. It is obvious that any supercharges can never be consistent with both of them at the same time.

Remarks on the supersymmetric cases
It would be worthwhile to figure out what happens in the closely related model with the SUSY unbroken, that is, the asymmetric orbifold defined by without including the shift. We also adopt (2.4) for the action of σ on Ramond vacua, and thus the orbifold twist is still a Z 4 -action. The partition function is then written in the form as In this case, the orbifold projection still removes all the supercharges in the untwisted sector, but the right-moving supercharges revive from the a = 2 twisted sector.
To show this fact explicitly, let us again decompose the partition functions as where the overall factor 1 16 ≡ 1 4 × 1 4 is due to the Z 4 -orbifolding as well as the chiral GSO projection. Then we obtain Obviously, we cannot construct any left-moving supercharges since we find On the other hand, there would exist some right supercharges in the a = 2 sector which realizes the equalities a+2 mod 4 (τ,τ ), (2.17) as found in (2.15), (2.16). In fact, one can explicitly confirm that the a = 2 sector includes the right-moving massless Ramond states, even though all of them are projected out by (−1) F R | ψ in the untwisted sector.
To be more precise, if starting with the type IIA (IIB) string theory, one can construct 8 supercharges that possess the opposite chirality as those in the type IIB (IIA) theory from the a = 2 sector, as discussed e.g. in [33,34]. The massless spectrum in the untwisted sector is the same as that displayed in Table   3, while that lying in the a = 2 sector is summarized in Table 5. These states are combined into the super-multiplets in an N = 2 supersymmetric theory in 4-dimension.

Building blocks
Firstly, we discuss the simple example T 2 [SO(4)], identified as the X 6 , X 7 -directions.
The torus partition function of this system is A convenient description is given by introducing the Majorana-Weyl fermions λ i L , λ i R (i = 1, 2, 3, 4).
In the previous section, for simplicity, it has been assumed that the twist operator σ including chiral reflection acts as an involution on the untwisted sector of the bosonic part. However, once adopting the fermionic description of T 2 [SO(4)], we are aware of another possibility in the manner similar to the world-sheet fermions ψ µ L , ψ µ R . Namely, considering the left-mover for instance, the chiral reflection As illustrated in [14] and already mentioned in section 2 for the world-sheet fermions ψ µ , we still need to define the Ramond vacua of this free fermion system to specify completely the action of (−1 L ) ⊗2 . Here, there are essentially two different cases; One can introduce the spin fields as with the bosonization; and the Ramond vacua |ǫ 1 , ǫ 2 L ≡S ǫ 1 ,ǫ 2 , L (0) |0 L are transformed as One may also bosonize λ 1 L , . . . , λ 4 R in a different way; and define the spin fields as follows; (3.9) and the Ramond vacua We thus simply obtain The above arguments are straightforwardly generalized to the cases of 2N ), and we always have two possibilities; Let us describe the relevant blocks which we will utilize later. In the following, the twist parameters a, b ∈ Z in the subscript always labels the spatial and temporal boundary conditions 5 . In other words, the parameter a labels the twisted sectors, while the parameter b corresponds to the insertion of σ b into the trace.

Fermionic Sector
We next consider the fermionic sector. We first recall that the fermionic part of the partition function of the type II string on 10-dim. flat background is just written as We present the relevant chiral blocks from now on. We only focus on the left-mover, and the right-mover is completely parallel. Although the cases (i) and (ii) are already given e.g. in [14], we dare to present them for the convenience to readers.

(3.27)
We note that the left-chiral blocks have to give rise to the phase −e − πi 3 under the Ttransformation to satisfy the modular covariance relation (3.26). h (a,b) (a ∈ 2Z + 1, or b ∈ 2Z + 1) are non-vanishing, which implies the SUSY breaking in the left-moving sector.

(3.30)
Recall that h ( * , * ) (τ ) is given in (3.27), corresponding to (−1) F L | ψ -twisting. We also need the chiral blocks defined by (−1 L ) ⊗2 -twisting. They are determined in the parallel way as above, although the different phase factors have to be included to ensure the modular covariance.

Non-supersymmetric Asymmetric orbifolds
We are now ready to study the six new models of non-SUSY vacua exhibited in Tables   1 and 2, including the modifications introduced at the beginning of this section.

Model I :
Firstly, we consider the asymmetric orbifold defined by the orbifold twist We can write down the torus partition function in terms of the building blocks introduced in subsection 3.1 as Looking at their expressions, it is easy to confirm that the partition function (3.37) indeed vanishes in the manner similar to the arguments in section 2.
As noticed in section 1, the non-SUSY chiral reflection (−1 L ) ⊗2 plays the similar role of (−1) F L | ψ in the 'previous model' introduced in section 2, and thus we anticipate to achieve a non-SUSY vacuum with the bose-fermi cancellation. We will later show that this is indeed the case.
Before doing so, let us study the massless spectrum lying in the untwisted sector, which we summarize in Table 6. We express the left-moving Ramond vacua in terms of the spin fields for SO (8); |s L ≡ e i 4 a=1 saH a L |0 L , s a ≡ ± 1 2 .  states, while fixing the right-movers. It is, however, impossible as shown from Table   6. For instance, pick up the states ψ µ R,−1/2 |0 , (µ = 6, ..., 9) from the right-mover. Then, one finds that the degrees of freedom of massless bosons amount to 8, whereas the fermionic one is 16.
One can examine the more detailed spectrum of physical states by making the Poisson resummation of the partition function (3.37). To this aim it is convenient to decompose it with respect to the spatial winding w and the spin structures as in (2.12); After dualizing the temporal winding m into the KK momentum n, we obtain the following results; • For w ∈ 4Z: (3.40) • w ∈ 4Z + 2: (3.42) • For w ∈ 4Z + 1: (3.44) • For w ∈ 4Z + 3: In this case, the result is obtained by replacing (−1) an in the first term of (3.43) with (−1) a(n+1) , and by replacing (−1) a(n+1) in the first term of (3.44) with (−1) an .
All of these partition functions are q-expanded so as to be compatible with unitarity, and we have no tachyonic states as confirmed by looking at the conformal weights read from them. Extra massless excitations appear when the X 5 -direction has some specific radii, as summarized in Table 7. Moreover, it is easy to confirm that the above partition functions satisfy the same relation as given in Table 4 with respect to the winding number w. This fact makes it clear that the model I is indeed a non-SUSY vacuum with the bose-fermi cancellation at each mass level.
In Table 7 the 'relevant equation' indicates which partition function includes the terms corresponding to the massless states in question.

Model II :
The model II is defined by the orbifold twist The corresponding partition function is given as where Z S 1 /Z 2 (w,m) (τ,τ ) denotes the building blocks corresponding to the ordinary reflection −1 : (X 5 L , X 5 R ) → (−X 5 L , −X 5 R ) acting on S 1 5 with an arbitrary radius. The bosonic blocks G and the orbifold twist is obtained simply as We also assume that (−1 R ) ⊗4 | X 6 ,...,X 9 acts on the Ramond vacua of the world-sheet fermions ψ µ R in the same way as (2.4). In total, we obtain rather than σ 2 = (−1) F R | ψ .
As we emphasized before, the shift operator T 2πR plays an important role of SUSY breaking, that is, it prevents the twisted sectors from providing new supercharges. However, we here show that other types of non-SUSY vacua are realized as long as (3.54) is satisfied.
The partition function is just written as where F for the odd sectors.
These relations should be compared with those for the supersymmetric case (2.15) and (2.16). Here we never obtain the equalities such as (2.17), rather find the cancellations as depicted in Table 4. Namely, we see that the left-moving NS-R cancellations for the even sectors, whereas the right-moving ones for the odd sectors. This fact clearly shows that the space-time SUSY is completely broken. Recall that, in the supersymmetric case with g = σ, the right-moving SUSY is unbroken, and the supercharges arise from the a = 2 twisted sector. In the current case, however, the same does not happen because the partition functions Z ( * ,R) 2 (τ,τ ) do not contain any massless states. This is the crucial difference caused by the relation (3.54). In this way, we have successfully achieved a desired non-SUSY vacuum without the shift.
The massless spectrum in the untwisted sector is the same as the model introduced in the previous section. In the twisted sectors, on the other hand, there are additional massless states, while no tachyonic states appear.

Model IV :
We next consider the background (3.59) and adopt the modification of (3.33); as the relevant orbifold twisting. σ ′ I again acts by (3.35) for the Ramond vacua of world-sheet fermions ψ µ R , ψ µ L . On the other hand, introducing the free fermions λ i L (R) , (i = 1, . . . , 4) describing T 2 8,9 [SO(4)], we define its action on the Ramond vacua of λ i R as that given in (a) of subsection 3.1.1, that is, We thus obtain the crucial relation (σ ′ The partition function is then written as a,b∈Z 4 also from the twisted sectors, because we have In this way, we conclude that the model IV is still a non-SUSY vacuum with the bose-fermi cancellation. Again we find additional massless states in the twisted sectors, while no tachyons appear.

Model V :
The model V is defined similarly to the model IV on the background; with the twist operator g = σ ′ II which is the modification of σ II given in (3.45) as Namely, σ ′ II acts on the world-sheet fermions ψ µ R , ψ µ L in the same way as The partition function is just given as where g (a,b) (τ ), f (a,b) (τ ) are as above, while G for the odd sectors.
The partition functions for even sectors (3.69) coincide with those for the model III (3.57) up to the common factor Z S 1 , implying that the right SUSY is completely broken.
They are also quite similar to the model IV (3.64), as anticipated.
Even though the odd sectors are also similar to (3.65), we here have a crucial difference. Namely, we find Z , just leading to the fact that the left SUSY is obviously broken.
The massless spectra in the twisted sectors are different from the previous two models.

Discussions about the Unitarity and Stability
We have studied various non-SUSY string vacua realized as asymmetric orbifolds in section 3. The right-moving part of the twist operators always include the reflection The torus partition functions for all of these vacua have been computed in a way showing manifestly the modular invariance, and they are properly q-expanded as to be consistent with unitarity. Moreover, by examining the spectra of physical states read off from the partition functions, we have found all of them to be stable, namely, any tachyonic states do not appear in all the untwisted and twisted sectors. These are likely to be common nice features of the non-SUSY string vacua of these types. In this section, we shall try to clarify why this is the case. There are still various extensions or modifications of the non-SUSY vacua studied in this paper, and the arguments given here would be applicable to them rather generally.
We first recall some non-trivial points that are specific in our models of asymmetric orbifolds. First of all, as we emphasized several times, the building blocks given in subsection 3.1 includes various phase factors. They are necessary to assure the modular covariance, and make the orbifold projections in the twisted sectors to differ non-trivially from that for the untwisted sector. As we already mentioned in section 2, this is a main reason why it would not be self-evident whether our models are unitary.
Secondly, needless to say, the absence of tachyonic instability is not obvious for generic non-SUSY vacua. It is a common feature that non-SUSY orbifolds involving our models would include the 'wrong GSO' NS states in the twisted sectors, which are where s L , (s R ) expresses the left-moving (right-moving) spin structure. We are only interested in the twisted sectors a = 0, since the unitarity and stability for the untwisted sector are obvious by construction.
As addressed above, the building blocks we utilized involve various phase factors.
Consequently, it would be useful to reinterpret the b-summation in (4.1) as that for the modular T-transformation τ → τ + 1; (τ,τ ) possess suitable q-expansions with positive integer coefficients 8 . This is indeed the case for all the models given in subsection 3.2, as can be readily confirmed from the explicit forms of the building blocks. We note that, actually, all the terms appearing in Z (s L ,s R ) (a,0) (τ,τ ) are q-expanded in this way.
How about the stability of the vacua? Namely, we would like to understand why no tachyon appears in all the twisted sectors in spite of the complete SUSY violations. We note • The leading term of each Z (s L ,s R ) (a,0) (τ,τ ) always has a non-negative conformal weight, as is obvious from the building blocks presented in subsection 3.1.
Therefore, the minimum conformal weight of the T-invariant terms appearing in Z (s L ,s R ) (a,0) (τ,τ ) has to be equal h = 1 2 + n, (n ∈ Z ≥0 ). This fact is sufficient to conclude that no tachyonic states emerge due to the observation given above.
We next consider the models including the shift operator T 2πR | base . For our purpose it would be useful to partially make the Poisson resummation of Z R,(w,m) (τ,τ ) (2.8) with respect to the temporal winding m ∈ 4Z and to sum up over ∀ w ∈ a + 4Z. Then, we can obtain a schematic decomposition (τ,τ ), leading to the same conclusion. 8 The factor 1 4 is necessary due to the chiral GSO projection.