Non-SUSY heterotic string vacua of Gepner models with vanishing cosmological constant

to heterotic strings. Namely, starting from the generic Gepner modelsforCalabi–Yauthree-folds,weconstructnon-SUSYheteroticstringvacuawithvanishing cosmologicalconstantattheone-looplevel.Weespeciallyfocusonasymmetricorbifoldingbasedonsomediscretesubgroupofthechiral U ( 1 ) action which acts on both the Gepner model and the SO ( 32 ) or E 8 × E 8 sector. We present a classiﬁcation of the relevant orbifold models leading to the string vacua with the properties mentioned above. In some cases, the desired vacua can be constructed in a manner quite similar to those given in the previous paper for the type II string, in which the orbifold groups contain two generators with discrete torsions. On the other hand, we also have simpler models that are just realized as asymmetric orbifolds of cyclic groups with only one generator.


Introduction and summary
Exploring non-supersymmetric vacua with the vanishing cosmological constant has been a subject of interest in superstring theory (at the level of one loop, at least), probably motivated by the cosmological constant problem. Consistent type II string vacua with such a non-trivial property were first constructed in Refs. [1][2][3] based on some non-Abelian orbifolds of higher-dimensional tori, followed by studies such as Refs. [4][5][6][7][8][9]. More recently, several non-SUSY vacua with this property have been constructed as asymmetric orbifolds [10] by simpler cyclic groups in Refs. [11,12].
The purpose of the current study is to construct non-SUSY heterotic string vacua with the vanishing cosmological constant at the one-loop level based on non-toroidal models. The method we adopt is a natural generalization of those given in our previous work [27]. That is, we start from the generic Gepner models [28,29] for Calabi-Yau three-folds, and construct non-SUSY heterotic string vacua by implementing some asymmetric orbifolding. Since we have various U (1) symmetries in the Gepner model, as well as SO (32) or the E 8 × E 8 sector in the left mover (which we assume bosonic), it would be quite natural to make the orbifolding associated with some cyclic subgroup of these U (1) PTEP 2021, 033B03 K. Aoyama and Y. Sugawara actions. Indeed, let us denote the generator of such a cyclic subgroup as δ L . Then, it is possible to construct non-SUSY string vacua by making the asymmetric orbifolding defined by the operator where F R denotes the spacetime fermion number (in other words, (−1) F R acts as the sign flip on the right-moving Ramond sector). It is obvious that the orbifold projection generated by the δ action completely breaks the Bose-Fermi cancellation in the untwisted Hilbert space. Moreover, any spacetime supercharges 1 cannot be constructed even if incorporating the degrees of freedom in the twisted sectors, so far as we assume the chiral forms of supercharges, namely, the integrals of conserved world-sheet current Q α = dz J α R (z), as addressed in Ref. [27]. At this point it is crucial that the relevant twisted sectors are associated with the left-moving operator δ L , whereas the possible supercharges should originate from the right-moving degrees of freedom.
In the end, it is enough to ask whether or not the total partition function that contains all the twisted sectors vanishes. We will clarify the criterion for this aim, and present a classification of the relevant orbifold models leading to string vacua with the desired properties. In some cases, the desired vacua can be constructed in a manner similar to those given in Ref. [27] for the type II string, in which the orbifold groups contain two generators equipped with some discrete torsions [30][31][32]. On the other hand, we also find simpler models which are just realized as asymmetric orbifolds of cyclic groups with only one generator, in contrast to the type II string cases.

Preliminaries
We begin with a very brief review of heterotic string vacua, including the Gepner models for the CY 3 compactifications, and describe the notation to be used in the main section.

Heterotic string vacua of Gepner models
The Gepner model [28,29] describing some CY 3 compactifications is defined as the superconformal system where M k denotes the N = 2 minimal model of level k,ĉ ≡ c 3 = k k+2 . We set where lcm means the least common multiplier. To describe the building blocks of the torus partition function, we start with the simple products of the characters of the N = 2 minimal model [33][34][35][36] PTEP 2021, 033B03 K. Aoyama and Y. Sugawara in the NS sector: 2 Those for other spin structures are defined by acting the half spectral flows z → z + r 2 τ + s 2 (r, s ∈ Z 2 ): where we setĉ = 3. Note that the label I ≡ {( i , m i )} of the building blocks (and the spectral flow orbits introduced below) expresses the quantum numbers for the NS sector even for F (R) I . To construct the Gepner models, we need to make the chiral Z N × Z N orbifolding by g L ≡ e 2π iJ tot 0 and g R ≡ e 2π iJ tot 0 , where J tot (J tot ) expresses the total N = 2 U (1) current in the left (right) mover acting over ⊗ i M k i . Recall that the zero-mode J tot 0 takes the eigenvalues in 1 N Z for the NS sector. The chiral Z N orbifolding (in the left mover) is represented in a way respecting the modular covariance by considering the "spectral flow orbits" [37] defined as follows: We also use the abbreviated notation F Assuming the standard embedding of spin connection, the SO(32) heterotic string vacuum compactified on CY 3 is described by the following modular invariant partition function: To avoid complexities, we shall assume the modular invariant coefficient N I L ,I R to be diagonal throughout this paper: Here, the summations of σ L and σ R are taken over the chiral spin structures. We also set to describe the free fermion contributions including the SO(32) sector. The E 8 × E 8 heterotic string vacuum is likewise described as where χ E 8 0 (τ ) denotes the character of the basic representation of affine E 8 , written explicitly as

Constructions of non-SUSY heterotic string vacua
In this section we present our main analysis. Namely, we discuss how we can construct non-SUSY string vacua with the vanishing cosmological constant at one loop (or the vanishing torus partition function) based on the heterotic string compactified on CY 3 given in Eqs. (11) and (13). We start by specifying the relevant orbifold action.

Orbifold actions
Let us fix a subsystem of the minimal models ⊗ i∈S M k i , S ⊂ {1, 2, . . . , r}, on which the orbifold operators act non-trivially. We set The total central charge of the subsystem S is written in the form We fix a positive integer L dividing N , and set for later convenience. We will shortly define the orbifold action δ that satisfies δ 4K = 1 on the untwisted sector. We also define S 1 ⊂ S by For the SO(32) (E 8 × E 8 ) heterotic string, we have the SO(26) (SO(10) × E 8 ) symmetry after making the standard embedding of the spin connection. We will adopt the relevant orbifold action as a cyclic subgroup of for the SO(32) case, and for the E 8 × E 8 -case. Now, let us specify the relevant orbifold action. For the SO(32) heterotic string, we define where J L are those for the U (1) s factor in Eq. (19). In other words, δ L acts on the left-moving characters of M k i , i ∈ S, as the integral spectral flow z → z + L(ατ + β): (α, β) ∈ Z N /L × Z N /L , which yields the modular covariant actions on the spectral flow orbits F F R denotes the spacetime fermion number of the right mover. In other words, the operator (−1) F R acts as the sign flip of the right-moving R sector.
On the other hand, δ L acts on the Jacobi theta functions associated with the U (1) s factor as follows: Here, the inclusion of the phase factor e 2π i αβ 8 is necessary for the modular covariance as in the minimal sector, Eq. (23). The explicit forms of Eq. (24) are also summarized in Appendix B.
We similarly define the orbifold action δ in the in the same way as Eq. (21).

5/15
PTEP 2021, 033B03 K. Aoyama and Y. Sugawara Since the δ orbifold action is defined so as to respect the modular covariance, it is easy to write down the modular invariant partition functions of our asymmetric orbifolds. For example, for the SO(32) heterotic string and in the cases of Ks ∈ 2Z + 1, the δ orbifold is found to be of order 8K, and the modular invariant parttion function is written as Here, we set which originate from the GSO phases (σ R ) modified by the (−1) F R actions included in Eq. (21). Also, we again made use of the abbreviated notations θ [NS],(α,β) (τ ) ≡ θ 3,(α,β) (τ ) ≡ θ 3,(α,β) (τ , 0), and so on. The modular invariants in other cases are obtained similarly.

Criterion for the desired models
At this stage let us clarify the "criterion" to search for heterotic string vacua with the desired properties.
To this end, we denote the contributions to the torus partition function from each twisted sector as Z (α,β) (τ ) (α, β ∈ Z 4K ). That is, we define for convenience. By our definition of the orbifold action δ presented above, the building blocks Z (α,β) (τ ) behave covariantly under the modular transformations: We require the following conditions: • For the "even sectors," α, β ∈ 2Z, each building block Z (α,β) (τ ) separately vanishes: • The partition function for the untwisted sector does not vanish: • For all the twisted sectors of δ α with α ∈ 2Z + 1, we require Note that Eq. (31) just implies that due to the modular covariance, Eq. (28). Thus, combining it with the requirement in Eq. (29), we can conclude that the total partition function should vanish.
We also note that, in this situation, the Bose-Fermi cancellation can only occur among different twisted sectors because of the condition in Eq. (30). On the other hand, the possible spacetime supercharges should be of a form such as Q α = dz J α R (z), which is consistent with the conservation on the world-sheet. However, any operators of this form cannot induce the expected Bose-Fermi cancellation, because the relevant twisted sectors are associated with the left-moving operator δ L . In this way, we conclude that we do not have any spacetime supercharges as the operators consistently acting on the whole Hilbert space and conserved on the world-sheet. This is the reason why we claim that the heterotic string vacua that satisfy the above requirements are non-supersymmetric ones.

Classification of the models
We study here aspects of the orbifolds of heterotic string vacua in Eqs. (11) and (13) by the cyclic actions of δ given in Eq. (21). We classify the models according to the positive integer N /L.
First of all, we note that for all the cases we will discuss below, since δ 2 obviously preserves the spacetime supercharges. One can also readily confirm that, for the untwisted sector α = 0, in all cases. Now, let us describe the classification.  for the (α, β)-twisted sector with β ∈ 2Z. Here, we set d i := N k i +2 for all i ∈ S, and c j = 1 (c j = −1) for j = 3, 4 (j = 2). We also note that, when β ∈ 2Z + 1, a similar phase factor is gained, while the θ 3 η s factor is exchanged with θ 4 η s . Fixing the value α ∈ 2Z + 1, let us evaluate the summation β Z (α,2β ) (τ ). It acts as the projection imposing The arguments are almost the same for the E 8 Consequently, we obtain the next classification.
• Ks ∈ 4Z [SO(32)]; Ks 1 , Ks 2 ∈ 4Z or Ks 1 , In these cases the aspects are almost parallel to those of Ref. [27]. The constraint in Eqs. (36) or (37) implies where S 1 was defined in Eq. (18), that is, S 1 ≡ {i ∈ S : d i ∈ 2Z + 1}. We then find that β Z (α,2β ) (τ ) = 0, since we generically possess many states satisfying the condition in Eq. (38). This means that δ orbifolding cannot satisfy Eq. (31) by itself. However, as shown in Ref. [27], 3 we can make it possible by further introducing the Z 2 orbifold action γ , which commutes with δ: on the right-moving minimal characters i∈S ch , and (−1) F L denotes the sign flip of the left-moving R sector. We shall also introduce the discrete torsion [30][31][32] with respect to the γ and δ actions: where a, b label the spatial and temporal twistings by γ , while α, β are those associated with δ as above. Then, for any fixed α ∈ 2Z + 1, we readily obtain In these cases, the constraint in Eqs. (36) or (37) in place of Eq. (38). Therefore, we can make the criterion in Eq. (31) be satisfied by taking again the δ and γ orbifolds but with the different discrete torsion In these cases, no state can satisfy the condition in Eq. (36), and thus the criterion in Eq. (31) is trivially achieved by only making the δ orbifolding.
In these remaining cases, both Eq. (38) and Eq. (42) are possible, depending on which theta function factors (θ j ) s 1 (θ k ) s 2 the operator δ acts. Thus, Eq. (31) cannot be satisfied even if incorporating the γ orbifolding. We conclude that string vacua with the desired properties are not constructed in these cases.
• Otherwise: In the remaining cases, we have β Z (α,2β ) = 0. Moreover, Eq. (31) cannot be satisfied even if the γ orbifolding is incorporated with any discrete torsion. The desired string vacua are not constructed in these cases.
To summarize, we have obtained non-SUSY heterotic string vacua with the property Z 1-loop (τ ) ≡ 0 based on orbifolding by δ (and γ in some cases) as follows: The desired vacua can be constructed only by making the δ orbifolding. The order of orbifolding is 8K, although δ 4K = 1 if restricting on the untwisted Hilbert space.
The desired vacua are again constructed only by δ action as in case (1). However, we obtain an order 4K orbifold in this case. The desired vacua are constructed as the Z 4K × Z 2 orbifold defined by δ and γ actions with the next discrete torsion included (a, b ∈ Z 2 for γ twists, and α, β ∈ Z 4K for δ twists): [SO (32)], (46)

Some comments
In this paper, as an extension of our previous work in Ref. [27], we have studied the construction of non-SUSY heterotic string vacua with the vanishing cosmological constant at the one-loop level, based on the asymmetric orbifolding of the Gepner models.
In the string vacua we constructed, we could not make up the spacetime supercharges that are conserved on the world-sheet and consistently realize the Bose-Fermi cancellation expected from the one-loop partition functions. We would like to emphasize here that Z one-loop (τ ) ≡ 0 just implies Bosefermi cancellation under the free string limit. Therefore, even if they might induce some low-energy effective field theories with unbroken SUSY, the absence of supercharges in the above sense should imply that they could not be supersymmetric ones when turning on the string interactions described by general world-sheets with higher genera. It would thus be possible for them to generate small nonvanishing cosmological constants after incorporating the (perturbative or non-perturbative) stringy quantum corrections, although such analyses still look very hard to carry out due to the complexities of the spectra arising from various twisted sectors.
When being motivated by the cosmological constant problem, it would be more desirable, though much more non-trivial, to have the vanishing one-loop cosmological constant without the Bose-Fermi cancellation at each mass level (in other words, Z(τ ) ≡ 0, but ≡ d 2 τ τ 2 2 Z(τ ) = 0). On the other hand, a characteristic feature of the string vacua given in the present paper (and those given in Ref. [27]) is that we have the Bose-Fermi cancellation among the different twisted sectors of the relevant orbifolding, as was emphasized several times. We would like to discuss elsewhere 10/15 PTEP 2021, 033B03 K. Aoyama and Y. Sugawara the possibility of realizing such "desirable situations," at least in some point particle theories with infinite mass spectra (not necessarily string theories), by implementing this feature (Y. Satoh and Y. Sugawara, work in progress).

Appendix A. Summary of conventions
We summarize the notations and conventions adopted in this paper. We set q ≡ e 2π iτ , y ≡ e 2π iz . Here, we have set q := e 2π iτ , y := e 2π iz (∀τ ∈ H + , z ∈ C), and used the abbreviations,

Appendix B. Explicit forms of the building blocks and their orbifold twistings
We summarize here the explicit expressions for the spectral flow orbits in Eqs. (7)-(10) playing the role of building blocks of relevant modular invariants. We also describe the orbifold actions δ, γ on the spectral flow orbits, as well as the δ twistings on the theta function factor, denoted as θ i,(α,β) (τ , z). We make use of the abbreviated index I ≡ {( i , m i )} ( i + m i ∈ 2Z) again, and set for convenience. F  On the other hand, the γ twisting of F (σ ) We next describe explicitly the Jacobi theta functions twisted by the δ actions given in Eq. (24), that is, They are explicitly written down as follows: α, β ∈ 2Z: