Non-SUSY Heterotic String Vacua of Gepner Models with Vanishing Cosmological Constant
aa r X i v : . [ h e p - t h ] F e b Non-SUSY Heterotic String Vacua of Gepner Modelswith Vanishing Cosmological Constant
Koji Aoyama ∗ , Yuji Sugawara † , Department of Physical Sciences, College of Science and Engineering,Ritsumeikan University, Shiga 525-8577, Japan
Abstract
We study a natural generalization of that given in [23] to heterotic string. Namely,starting from the generic Gepner models for Calabi-Yau 3-folds, we construct the non-SUSY heterotic string vacua with the vanishing cosmological constant at the one loop.We especially focus on the asymmetric orbifolding based on some discrete subgroup of thechiral U (1)-action which acts on both of the Gepner model and the SO (32) or E × E -sector. We present a classification of the relevant orbifold models leading to the stringvacua with the properties mentioned above. In some cases, the desired vacua can beconstructed in the manner quite similar to those given in [23] for the type II string, inwhich the orbifold groups contain two generators with the discrete torsions. On the otherhand, we also have simpler models that are just realized as the asymmetric orbifolds ofcyclic groups with only one generator. ∗ [email protected] † [email protected] Introduction and Summary
It has been an interesting subject in superstring theory to explore the non-supersymmetricvacua with the vanishing cosmological constant (at the level of one-loop, at least), probablymotivated by the cosmological constant problem. The consistent type II string vacua with suchnon-trivial property has been first constructed in [1–3] based on some non-abelian orbifolds ofhigher dimensional tori, followed by studies e.g. in [4–9]. More recently, several non-SUSYvacua with this property have been constructed as the asymmetric orbifolds [10] by simplercyclic groups in [11, 12].On the other hand, in heterotic string theory, there have been the studies of the stringvacua with cosmological constants exponentially suppressed with respect to some continuousparameter expressing the ‘distance’ from the SUSY vacua (for instance, the radii of tori ofcompactifications as given in [13]) in [4, 14], and more recently, e.g in [15–22] from the variousviewpoints of model building in string phenomenology.Now, the purpose of current study is to construct the non-SUSY heterotic string vacua withthe vanishing cosmological constant at one-loop based on the non-toroidal models. The methodwe adopt is a natural generalization of those given in our previous work [23]. That is, westart from the generic Gepner models [24] for Calabi-Yau 3-folds, and construct the non-SUSYheterotic string vacua by implementing some asymmetric orbifolding. Since we have various U (1)-symmetries in the Gepner model as well as the SO (32) or E × E -sector in the left-mover(which we assume bosonic), it would be quite natural to make the orbifolding associated tosome cyclic subgroup of these U (1)-actions. Indeed, let us denote the generator of such a cyclicsubgroup as ‘ δ L ’. Then, it is possible to construct the non-SUSY string vacua by making theasymmetric orbifolding defined by the operator b δ ≡ δ L ⊗ ( − F R , where F R denotes the space-time fermion number (in other words, ( − F R acts as the sign-flip on the right-moving Ramond sector). It is obvious that the orbifold projection generatedby the b δ -action completely breaks the bose-fermi cancellation in the untwisted Hilbert space.Moreover, any space-time supercharges cannot be constructed even if incorporating the degreesof 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 = I d ¯ z J αR (¯ z )’, as was addressed in [23]. In this paper, we shall regard the space-time supercharges as the operators acting consistently on the wholeHilbert space of string states and made up of the local perturbative degrees of freedom on the world-sheet. Weuse the term ‘non-SUSY string vacuum’ with the meaning of absence of supercharges under this definition. It isbeyond our scope whether non-perturbative supercharges could exist. In addition, it may be possible that onewould gain some ‘supercharges’ after truncating massive degrees of freedom in the approximation by low energyeffective field theories.
1t this point it is crucial that the relevant twisted sectors are associated to the left-moving operator δ L , whereas the possible supercharges should originate from the right-moving degreesof freedom.In the end, it is enough to ask whether or not the total partition function that contains allthe twisted sectors vanish. We will clarify the ‘criterion’ to this aim, and present a classificationof the relevant orbifold models leading to the string vacua with the desired properties. In somecases, the desired vacua can be constructed in the manner similar to those given in [23] for thetype II string, in which the orbifold groups contain two generators equipped with some discretetorsions [25–27]. On the other hand, we also find out the simpler models which are just realizedas the asymmetric orbifolds of cyclic groups with only one generator, in contrast to the type IIstring cases. We begin with making a very brief review of the heterotic string vacua including the Gepnermodels for the CY -compactifications, and prepare the notations to be used in the main section. The Gepner model [24] describing some CY is defined as the superconformal system;[ M k ⊗ · · · ⊗ M k r ] | Z N -orbifold , r X i =1 k i k i + 2 = 3 , (2.1)where M k denotes the N = 2 minimal model of level k (ˆ c ≡ c kk + 2 ). We set N := L.C.M. { k i + 2 ; i = 1 , . . . , r } . (2.2)(‘L.C.M.’ means the least common multiplier.) To describe the building blocks of torus partitionfunction, we start with the simple products of the characters of N = 2 minimal model [28, 29]in the NS-sector ; F (NS) I ( τ, z ) := r Y i =1 ch (NS) ℓ i ,m i ( τ, z ) , (cid:0) I ≡ { ( ℓ i , m i ) } , ℓ i + m i ∈ Z , ∀ i (cid:1) . (2.3) We summarize the explicit character formulas as well as the conventions of theta functions in appendix A.We set q := e πiτ , y := e πiz through this paper. z z + r τ + s r, s ∈ Z ): F ( f NS) I ( τ, z ) := F (NS) I (cid:18) τ, z + 12 (cid:19) , (2.4) F (R) I ( τ, z ) := q ˆ c y ˆ c F (NS) I (cid:16) τ, z + τ (cid:17) , (2.5) F ( e R) I ( τ, z ) := q ˆ c y ˆ c F (NS) I (cid:18) τ, z + τ + 12 (cid:19) , (2.6)where we set ˆ c = 3. Note that the label I ≡ { ( ℓ i , m i ) } of the building blocks (and the spectralflow orbits introduced below) expresses the quantum numbers for the NS-sector even for F (R) I and F ( e R) I .To construct the Gepner models, we need to make the chiral Z N × Z N orbifolding by g L ≡ e πiJ tot and g R ≡ e πi ˜ J tot , where J tot ( ˜ J tot ) expresses the total N = 2 U (1)-current in theleft (right) mover acting over ⊗ i M k i . Recall that the zero-mode J tot0 takes the eigen-valuesin 1 N Z for the NS sector. The chiral Z N -orbifolding (in the left mover) is represented in away respecting the modular covariance by considering the ‘spectral flow orbits’ [30] defined asfollows: F (NS) I ( τ, z ) := 1 N X a,b ∈ Z N q ˆ c a y ˆ ca F (NS) I ( τ, z + aτ + b ) , (2.7) F ( f NS) I ( τ, z ) := F (NS) I (cid:18) τ, z + 12 (cid:19) ≡ N X a,b ∈ Z N ( − ˆ ca q ˆ c a y ˆ ca F ( f NS) I ( τ, z + aτ + b ) , (2.8) F (R) I ( τ, z ) := q ˆ c y ˆ c F (NS) I (cid:16) τ, z + τ (cid:17) ≡ N X a,b ∈ Z N ( − ˆ cb q ˆ c a y ˆ ca F (R) I ( τ, z + aτ + b ) , (2.9) F ( e R) I ( τ, z ) := q ˆ c y ˆ c F (NS) I (cid:18) τ, z + τ + 12 (cid:19) ≡ N X a,b ∈ Z N ( − ˆ c ( a + b ) q ˆ c a y ˆ ca F ( e R) I ( τ, z + aτ + b ) . (2.10)We also use the abbreviated notation; F ( σ ) I ( τ ) ≡ F ( σ ) I ( τ, F ( σ ) I ( τ, z ) written in terms of the N = 2 minimal characters.Now, let us focus on the heterotic string. We take the convention;left-mover : 26D bosonic , right-mover : 10D super . SO (32) heterotic string vacuumcompactified on CY is described by the following modular invariant partition function; Z SO (32) Het ( τ ) = √ τ | η | ! · N X σ L ,σ R ǫ ( σ R ) (cid:18) θ [ σ L ] η (cid:19) (cid:18) θ [ σ R ] η (cid:19) × X I L ,I R N I L ,I R F ( σ L ) I L ( τ ) F ( σ R ) I R ( τ ) . (2.11)To avoid complexities, we shall assume the modular invariant coefficient N I L ,I R to be diagonalthrough this paper: N I L ,I R ≡ r Y i =1 δ ℓ i,L ,ℓ i,R δ m i,L ,m i,R , ( I L ≡ { ( ℓ i,L , m i,L ) } , I R ≡ { ( ℓ i,R , m i,R ) } ) . (2.12)Here the summations of σ L , σ R are taken over the chiral spin structures. We also set ǫ (NS) = − ǫ ( f NS) = − ǫ (R) = 1 in the standard fashion, and θ [NS] ≡ θ ( τ, θ [ f NS] ≡ θ ( τ, θ [R] ≡ θ ( τ, (cid:16) θ [ e R] ≡ − iθ ( τ, ≡ (cid:17) to describe the free fermion contributions including the SO (32)-sector.The E × E heterotic string vacuum is likewise described as Z E × E Het ( τ ) = √ τ | η | ! · N X σ L ,σ R ǫ ( σ R ) (cid:18) θ [ σ L ] η (cid:19) χ E (cid:18) θ [ σ R ] η (cid:19) × X I L ,I R N I L ,I R F ( σ L ) I L ( τ ) F ( σ R ) I R ( τ ) , (2.13)where χ E ( τ ) denotes the character of basic representation of affine E , written explicitly as χ E ( τ ) ≡ "(cid:18) θ η (cid:19) + (cid:18) θ η (cid:19) + (cid:18) θ η (cid:19) ( τ ) . (2.14) In this section we present our main analysis. Namely, we discuss how we can construct thenon-SUSY string vacua with the vanishing cosmological constant at one-loop (or the vanishingtorus partition function) based on the heterotic string compactified on CY given in (2.11) and(2.13). We start with specifying the relevant orbifold action.4 .1 Orbifold Actions Let us fix a subsystem of the minimal models ⊗ i ∈ S M k i , S ⊂ { , , . . . , r } , on which the orbifoldoperators non-trivially act. We set N ′ := L.C.M. { k i + 2 : i ∈ S } . (3.1)The total central charge of the subsystem S is written in the form;ˆ c S ≡ X i ∈ S k i k i + 2 ! = MN ′ , ∃ M ∈ Z . ( N ′ ∈ Z ) M ′ N ′ , ∃ M ′ ∈ Z , ( N ′ ∈ Z + 1) (3.2)We fix a positive integer L dividing N ′ , and set4 K := L.C.M { N ′ /L, } , (3.3)for the later convenience. We will soon define the orbifold action b δ that satisfies b δ K = 1 on theuntwisted sector. We also define S ⊂ S by S := (cid:26) i ∈ S : N ′ k i + 2 ∈ Z + 1 (cid:27) . (3.4)Note that S = φ since N ′ is the L.C.M. of { k i + 2 } i ∈ S .For the SO (32) ( E × E ) heterotic string, we have the SO (26) ( SO (10) × E ) symmetryafter making the standard embedding of spin connection. We will adopt the relevant orbifoldaction as a cyclic subgroup of U (1) s or U (1) s × U (1) s given as U (1) s × SO (26 − s ) ⊂ SO (26) , (3.5)for the SO (32)-case, and[ U (1) s × SO (10 − s )] × [ U (1) s × SO (16 − s )] ⊂ SO (10) × E , (3.6)for the E × E -case.Now, let us specify the relevant orbifold action. For the cases of SO (32)-heterotic string, wedefine b δ := ( − F R ⊗ δ L , δ L := e πiL P i ∈ S J ( i ) L, e πi P sj =1 K ( j ) L, , (3.7)where J ( i ) L is the left-moving U (1)-current in M k i , i ∈ S , and K ( j ) L are those for the U (1) s -factorin (3.5). In other words, δ L acts on the left-moving characters of M k i , i ∈ S as the integralspectral flow z z + L ( ατ + β ); δ L, ( α,β ) · ch ( σ ) ℓ i ,m i ( τ, z ) := q ki ki +2) L α y kiki +2 Lα e πi ki ki +2) L αβ ch ( σ ) ℓ i ,m i ( τ, z + L ( ατ + β )) , ( α, β ) ∈ Z N ′ /L × Z N ′ /L , (3.8)5hich yields the modular covariant actions on the spectral flow orbits F ( σ L ) I L ( τ ). We summarizethe explicit forms of ‘ δ L, ( α,β ) · F ( σ L ) I L ( τ )’ in Appendix B. F R denotes the space-time fermionnumber of the right-mover. Namely, the operator ( − F R acts as the sign-flip of the right-moving R-sector.On the other hand, δ L acts on the Jacobi’s theta functions associated to the U (1) s -factor asfollows; δ L, ( α,β ) · θ i ( τ, z ) (cid:0) ≡ θ i, ( α,β ) ( τ, z ) (cid:1) := q α y α e πi αβ θ i (cid:18) τ, z + ατ + β (cid:19) . ( i = 3 , ,
2) (3.9)Here the inclusion of phase factor e πi αβ is necessary for the modular covariance as in theminimal sector (3.8). The explicit forms of (3.9) are also summarized in Appendix B.We similarly define the orbifold action b δ in the E × E -case, in which δ L acts on O i ∈ S M k i O U (1) s ⊗ U (1) s , in the same way as (3.7).Since the b δ -orbifold action is defined so as to respect the modular covariance, it is easy towrite down the modular invariant partition functions of our asymmetric orbifolds. For example,for the SO (32) heterotic string and in the cases of Ks ∈ Z + 1, the b δ -orbifold is found to beorder 8 K , and the modular invariant parttion function is written as Z b δ -orb ( τ ) = √ τ | η | ! · N X σ L ,σ R K X α,β ∈ Z K ǫ ( σ R ; α, β ) (cid:18) θ [ σ L ] , ( α,β ) η (cid:19) s (cid:18) θ [ σ L ] η (cid:19) − s (cid:18) θ [ σ R ] η (cid:19) × X I L ,I R N I L ,I R h δ L, ( α,β ) · F ( σ L ) I L ( τ ) i F ( σ R ) I R ( τ ) . (3.10)Here, we set ǫ (NS; α, β ) := − α, β ∈ Z + 1)1 (otherwise) ǫ ( f NS; α, β ) := α ∈ Z + 1 , β ∈ Z ) − ǫ (R; α, β ) := α ∈ Z , β ∈ Z + 1) − ǫ ( σ R ) modified by the ( − F R -actions included in (3.7).Also, we again made use of the abbreviated notations θ [NS] , ( α,β ) ( τ ) ≡ θ , ( α,β ) ( τ ) ≡ θ , ( α,β ) ( τ, .2 Criterion for the Desired Models At this stage let us clarify the ‘criterion’ to search for the heterotic string vacua with the desiredproperties. To this end, we denote the contributions to the torus partition function from theeach twisted sector as ‘ Z ( α,β ) ( τ )’, ( α, β ∈ Z K ). That is, we define Z ( α,β ) ( τ ) ≡ Tr b δ α -twisted hb δ β q L − ˆ c q ˜ L − ˆ c i , (3.12)for the convenience. By our definition of the orbifold action b δ presented above, the buildingblocks Z ( α,β ) ( τ ) covariantly behave under the modular transformations; Z ( α,β ) (cid:18) − τ (cid:19) = Z ( β, − α ) ( τ ) , Z ( α,β ) ( τ + 1) = Z ( α,α + β ) ( τ ) . (3.13)We require the following conditions; • For the ‘even sectors’ ∀ α, β ∈ Z , each building block Z ( α,β ) ( τ ) separately vanishes; Z ( α,β ) ( τ ) ≡ , (3.14) • The partition function for the untwisted sector does not vanish; Z ( τ ) ≡ K X β ∈ Z K Z (0 ,β ) , (3.15) • For all the twisted sectors of b δ α with α ∈ Z + 1, we require X β ′ Z ( α, β ′ ) ( τ ) ≡ . (cid:0) ∀ α ∈ Z + 1 (cid:1) (3.16)Note that, (3.16) just implies X α ∈ Z +1 or β ∈ Z +1 Z ( α,β ) ( τ ) ≡ , (3.17)due to the modular covariance (3.13). Thus, combining it together with the requirement (3.14),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 the different twisted sectors because of the condition (3.15). On the other hand, the possible space-time supercharges should be of the form such as Q α = I d ¯ z J αR (¯ z ), which is consistent withthe conservation on the world-sheet. However, any operators of this form cannot induce theexpected bose-fermi cancellation, because the relevant twisted sectors are associated to the left-moving operator δ L . In this way, we conclude that we do not have any space-time superchargesas 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 aboverequirements are non-supersymmetric ones. 7 .3 Classification of the Models We here study the aspects of orbifolds of heterotic string vacua (2.11) and (2.13) by the cyclicactions of b δ given in (3.7). We classify the models according to the positive integer N ′ /L .First of all, we note Z ( α,β ) ( τ ) ≡ , ∀ α, β ∈ Z , (3.18)for all the cases we will discuss below, since b δ obviously preserves the space-time supercharges.One can also readily confirm that, for the untwisted sector α = 0, Z ( τ ) ≡ K X β ∈ Z K Z (0 ,β ) ( τ ) ! = 0 , (3.19)in all the cases.Now, let us describe the classification: N ′ /L ≡ mod : In this case we have N ′ L = 4 K .We first focus on the SO (32)-case. The crucial point is as follows: For the product of left-moving minimal characters Y i ∈ S ch ( σ ) ℓ i,L ,m i,L − n L ( τ ) ( n L is the spectral flow momentum) as well asthe each ‘free fermion factor’ (cid:18) θ j η (cid:19) , the orbifold action b δ picks up the next phase factor;exp " πi (X i ∈ S d i ( m i,L − n L ) N ′ Lβ + MN ′ L αβ + 18 c j sαβ ) ≡ exp " πi β K (X i ∈ S d i ( m i,L − n L ) + M Lα + 12 Kc j sα ) , (3.20)for the ( α, β )-twisted sector with β ∈ Z . Here we set d i := N ′ k i + 2 for ∀ i ∈ S , and c j = 1( c j = −
1) for j = 3 , β ∈ Z + 1, the similar phase factor isgained, while the (cid:18) θ η (cid:19) s factor is exchanged with (cid:18) θ η (cid:19) s .Fixing the value α ∈ Z + 1, let us evaluate the summation X β ′ Z ( α, β ′ ) ( τ ) . It acts as theprojection imposing X i ∈ S d i ( m i,L − n ) + M Lα + 12 Kc j sα ≡ K ) . (3.21)8he arguments are almost the same for the E × E -case. We only have to replace the term12 Kc j s in (3.20) with 12 K (cid:16) c (1) j s + c (2) k s (cid:17) for the factor (cid:18) θ j η (cid:19) s (cid:18) θ k η (cid:19) s , where c ( r ) j = ± X i ∈ S d i ( m i,L − n ) + M Lα + 12 K (cid:16) c (1) j s + c (2) k s (cid:17) α ≡ K ) . (3.22)Consequently, we obtain the next classification; • Ks ∈ Z [ SO (32) ] Ks , Ks ∈ Z or Ks , Ks ∈ Z + 2 [ E × E ] :In these cases the aspects are almost parallel to that of [23]. The constraint (3.21) or(3.22) implies X i ∈ S m i,L + LM ≡ , (3.23)where S has been defined in (3.4), that is, S ≡ { i ∈ S : d i ∈ Z + 1 } . We then find X β ′ Z ( α, β ′ ) ( τ ) = 0 , since we generically possess many states satisfying the condition (3.23). This means that b δ -orbifolding cannot satisfy (3.16) by itself.However, as was shown in [23] , we can make it possible by further introducing the Z -orbifold action b γ , which commutes with b δ ; b γ := ( − F L ⊗ γ R ( S ∈ Z + 1) ,γ R ( S ∈ Z ) , γ R := Y i ∈ S ( − ℓ i,R . (3.24)on the right-moving minimal characters Y i ∈ S ch ( σ ) ℓ i,R ,m i,R − n R ( τ ), and ( − F L denotes the sign-flip of the left-moving R-sector. We shall also introduce the discrete torsion [25–27] withrespect to the b γ and b δ -actions; ξ ( a, α ; b, β ) := ( − ( LM − aβ − bα ) , ( a, b ∈ Z , α, β ∈ Z K ) , (3.25)where a, b label the spatial and temporal twistings by b γ , while α, β are those associatedto b δ as above. Then, for any fixed α ∈ Z + 1, we readily obtain X β ′ Z ( α, β ′ ) ( τ ) (cid:12)(cid:12) b γ − orbifold ≡ X β ′ X a,b ∈ Z ξ ( a, α ; b, β ′ ) Z ( a,α ; b, β ′ ) ( τ ) = 0 . (3.26) In comparison with [23], the roles of left and right movers have been exchanged here. b γ and b δ . See [23]for more detailed arguments. • Ks ∈ Z + 2 [ SO (32)] Ks ∈ Z , Ks ∈ Z + 2, or Ks ∈ Z + 2, Ks ∈ Z [ E × E ] :In these cases, the constraint (3.21) or (3.22) yields X i ∈ S m i + LM + 1 ≡ , (3.27)in place of (3.23). Therefore, we can make the criterion (3.16) to be satisfied by takingagain the b δ and b γ orbifolds but with the different discrete torsion ξ ′ ( a, α ; b, β ) := ( − LM ( aβ − bα ) , ( a, b ∈ Z , α, β ∈ Z K ) . (3.28) • Ks ∈ Z + 1 [ SO (32)] K ( s + s ) ∈ Z + 1 [ E × E ] :In these cases, any states cannot satisfy the condition (3.21), and thus the criterion (3.16)is trivially achieved by only making the b δ -orbifolding. • Ks , Ks ∈ Z + 1 [ E × E ] :In these remaining cases, both of (3.23) and (3.27) are possible, depending on which thetafunction factors ( θ j ) s ( θ k ) s the operator b δ acts. Thus, (3.16) cannot be satisfied evenif incorporating the b γ -orbifolding. We conclude that the string vacua with the desiredproperties are not constructed in these cases. N ′ /L mod : In this case, N ′ L ∈ Z + 2 or N ′ L ∈ Z + 1, and K ∈ Z + 1 for the both cases.Again, we first consider the SO (32) case. In the case of N ′ L ∈ Z + 2, the b δ picks up thephase factor; exp " πi (X i ∈ S d i ( m i − n ) N ′ Lβ + MN ′ L αβ + 18 sc j αβ ) ≡ exp " πi β K (X i ∈ S d i ( m i − n ) + 2 M Lα + 12
Ksc j α ) , (3.29)10nstead of (3.20). In the case of N ′ L ∈ Z + 1, we similarly obtainexp " πi (X i ∈ S d i ( m i − n ) N ′ Lβ + M ′ N ′ L αβ + 18 sc j αβ ) ≡ exp " πi β K (X i ∈ S d i ( m i − n ) + 2 M ′ Lα + 12 Ksc j α ) . (3.30)In the case of E × E , the term 12 Kc j s in (3.29) and (3.30) is again replaced with12 K (cid:16) c (1) j s + c (2) k s (cid:17) .Combining all the things, we obtain the next classifications; • s Z [ SO (32)] s + s ∈ Z + 1, s ∈ Z , s ∈ Z + 2, or s ∈ Z + 2 , s ∈ Z [ E × E ] :In all these cases we simply obtain X β ′ Z ( α, β ′ ) = 0 , (cid:0) ∀ α ∈ Z + 1 (cid:1) , because 12 Ksc j α (or 12 K (cid:16) c (1) j s + c (2) k s (cid:17) α for E × E ) takes values in Z + 12 or 2 Z + 1,and thus the phase factors (3.29), (3.30) never cancel out. Hence, the criterion (3.16) isagain achieved by making only the b δ -orbifolding. • otherwise :In the remaining cases, we have X β ′ Z ( α, β ′ ) = 0 . Moreover, (3.16) cannot be satisfied evenif the b γ -orbifolding is incorporated with any discrete torsion. The desired string vacua arenot constructed in these cases.To summarize, we have obtained the non-SUSY heterotic string vacua with the property Z ( τ ) ≡ b δ (and b γ in some cases) as follows: (1) Ks ∈ Z + 1 [ SO (32)], K ( s + s ) ∈ Z + 1 [ E × E ] :The desired vacua can be constructed only by making the b δ -orbifolding. The order oforbifolding is 8 K although b δ K = 1 if restricting on the untwisted Hilbert space. (2) Ks ∈ Z + 2 and N ′ /L SO (32)], Ks i ∈ Z + 2, Ks j ∈ Z ( i = j ) and N ′ /L E × E ] :The wanted vacua are again constructed only by b δ -action as in the case (1) . However, weobtain an order 4 K orbifold in this case.11 Ks ∈ Z and N ′ /L ≡ SO (32)], Ks , Ks ∈ Z and N ′ /L ≡ E × E ] :The wanted vacua are constructed as the Z K × Z -orbifold defined by b δ and b γ -actions withthe next discrete torsion included ( a, b ∈ Z for b γ -twists, and α, β ∈ Z K for b δ -twists); ξ ( a, α ; b, β ) = ( − LM + Ks − ) ( aβ − bα ) , [ SO (32)]( − LM + K ( s + s ) − ) ( aβ − bα ) . [ E × E ] (3.31) In this paper, as an extension of our previous work [23], we have studied the construction ofnon-SUSY heterotic string vacua with the vanishing cosmological constant at one-loop, basedon the asymmetric orbifolding of the Gepner models. We would like to add a few comments: • In the string vacua we constructed, we could not make up the space-time superchargesthat are conserved on the world-sheet and consistently realizing the bose-fermi cancella-tion expected from the one-loop partition functions. We would like to here emphasizethat Z one-loop ( τ ) ≡ under the free string limit. Therefore, even if they might induce some low-energy effective field theories with un-broken SUSY, the absence of supercharges in the above sense should imply that theycould not be supersymmetric ones when turning on the string interactions described bygeneral world-sheets with higher genera. It would be thus capable for them to generatesmall non-vanishing cosmological constants after incorporating the (perturbative or non-perturbative) stringy quantum corrections, although such analyses still look very hard tocarry out due to the complexities of spectra arising from various twisted sectors. • When being motivated by the cosmological constant problem, it would be more desirablebut much more non-trivial situations that we have the vanishing one-loop cosmologicalconstant without the bose-fermi cancellation at each mass level (in other words, Z ( τ ) ≡ Z d ττ Z ( τ ) = 0). On the other hand, a characteristic feature of the stringvacua given in the present paper (and those given in [23]) is that we have the bose-fermi cancellation among the different twisted sectors of the relevant orbifolding, as wasemphasized several times. We would like to discuss elsewhere the possibility to realize such‘desirable situations’, at least, in some point particle theories with infinite mass spectra(not necessarily, string theories), by implementing this feature [31].12 ppendix A: Summary of Conventions We summarize the notations and conventions adopted in this paper. We set q ≡ e πiτ , y ≡ e πiz .
1. Theta Functions θ ( τ, z ) := i ∞ X n = −∞ ( − n q ( n − / / y n − / ≡ πz ) q / ∞ Y m =1 (1 − q m )(1 − yq m )(1 − y − q m ) , (A.1) θ ( τ, z ) := ∞ X n = −∞ q ( n − / / y n − / ≡ πz ) q / ∞ Y m =1 (1 − q m )(1 + yq m )(1 + y − q m ) , (A.2) θ ( τ, z ) := ∞ X n = −∞ q n / y n ≡ ∞ Y m =1 (1 − q m )(1 + yq m − / )(1 + y − q m − / ) , (A.3) θ ( τ, z ) := ∞ X n = −∞ ( − n q n / y n ≡ ∞ Y m =1 (1 − q m )(1 − yq m − / )(1 − y − q m − / ) . (A.4)Θ m,k ( τ, z ) := ∞ X n = −∞ q k ( n + m k ) y k ( n + m k ) , (A.5) η ( τ ) := q / ∞ Y n =1 (1 − q n ) . (A.6)Here, we have set q := e πiτ , y := e πiz ( ∀ τ ∈ H + , ∀ z ∈ C ), and used abbreviations, θ i ( τ ) ≡ θ i ( τ,
0) ( θ ( τ ) ≡ m,k ( τ ) ≡ Θ m,k ( τ,
2. Character Formulas for N = 2 Minimal Model
The character formulas of the level k N = 2 minimal model (ˆ c = k/ ( k + 2)) [28, 29] aredescribed as the branching functions of the Kazama-Suzuki coset [32] SU (2) k × U (1) U (1) k +2 definedby χ ( k ) ℓ ( τ, w )Θ s, ( τ, w − z ) = X m ∈ Z k +2) ℓ + m + s ∈ Z χ ℓ,sm ( τ, z )Θ m,k +2 ( τ, w − z/ ( k + 2)) ,χ ℓ,sm ( τ, z ) ≡ , for ℓ + m + s ∈ Z + 1 , (A.7)where χ ( k ) ℓ ( τ, z ) is the spin ℓ/ SU (2) k ; χ ( k ) ℓ ( τ, z ) = Θ ℓ +1 ,k +2 ( τ, z ) − Θ − ℓ − ,k +2 ( τ, z )Θ , ( τ, z ) − Θ − , ( τ, z ) ≡ X m ∈ Z k c ( k ) ℓ,m ( τ )Θ m,k ( τ, z ) . (A.8)The branching function χ ℓ,sm ( τ, z ) is explicitly calculated as follows; χ ℓ,sm ( τ, z ) = X r ∈ Z k c ( k ) ℓ,m − s +4 r ( τ )Θ m +( k +2)( − s +4 r ) , k ( k +2) ( τ, z/ ( k + 2)) . (A.9)13hen, the character formulas of unitary representations are written asch (NS) ℓ,m ( τ, z ) = χ ℓ, m ( τ, z ) + χ ℓ, m ( τ, z ) , ch ( f NS) ℓ,m ( τ, z ) = χ ℓ, m ( τ, z ) − χ ℓ, m ( τ, z )ch (R) ℓ,m ( τ, z ) = χ ℓ, m ( τ, z ) + χ ℓ, m ( τ, z )ch ( e R) ℓ,m ( τ, z ) = χ ℓ, m ( τ, z ) − χ ℓ, m ( τ, z ) . (A.10) Appendix B: Explicit Forms of Building Blocks and TheirOrbifold Twistings
In Appendix B, we summarize the explicit expressions of spectral flow orbits (2.7)-(2.10)playing the role of building blocks of relevant modular invariants. We also describe the orbifoldactions 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 ≡ { ( ℓ i , m i ) } ( ℓ i + m i ∈ Z ) again, and set Q ( I ) ≡ Q ( { ( ℓ i , m i ) } ) := r X i =1 m i k i + 2 (cid:18) ∈ N Z (cid:19) , (B.1)for the convenience. F ( σ ) I ( τ, z ) obviously vanishes for Q ( I ) Z by the definitions (2.7)-(2.10),and we obtain the following expressions in the case of Q ( I ) ∈ Z , F (NS) I ( τ, z ) = X n ∈ Z N r Y i =1 ch (NS) ℓ i ,m i − n ( τ, z ) ≡ X n ∈ Z N F (NS) s n ( I ) ( τ, z ) , (B.2) F ( f NS) I ( τ, z ) = ( − Q ( I ) X n ∈ Z N ( − (ˆ c + r ) n r Y i =1 ch ( f NS) ℓ i ,m i − n ( τ, z ) ≡ X n ∈ Z N ( − (ˆ c + r ) n F ( f NS) s n ( I ) ( τ, z ) , (B.3) F (R) I ( τ, z ) = X n ∈ Z N r Y i =1 ch (R) ℓ i ,m i − n − ( τ, z ) ≡ X n ∈ Z N F (R) s n ( I ) ( τ, z ) , (B.4) F ( e R) I ( τ, z ) = ( − Q ( I )+ r X n ∈ Z N ( − (ˆ c + r ) n r Y i =1 ch ( e R) ℓ i ,m i − n − ( τ, z ) ≡ X n ∈ Z N ( − (ˆ c + r ) n F ( e R) s n ( I ) ( τ, z ) , (B.5)where we introduced the notation s n ( I ) := { ( ℓ i , m i − n ) } (for I ≡ { ( ℓ i , m i ) } ) . δ -twisting for F ( σ ) I ( τ ) are expressed explicitly as δ ( α,β ) · F ( σ ) { ( ℓ i ,m i ) } ( τ ) = ζ ˆ c S L ( σ ; α, β ) e πi L MN ′ αβ X n ∈ Z N e πi P i ∈ S L ( mi − n ) ki +2 β F ( σ ) { ( ℓ i ,m ′′ i − n ) } ( τ ) ≡ ζ LM/N ′ ( σ ; α, β ) e πi L MN ′ αβ X n ∈ Z N e πi LN ′ P i ∈ S d i ( m i − n ) β F ( σ ) { ( ℓ i ,m ′′ i − n ) } ( τ ) , (cid:18) d i ≡ N ′ k i + 2 (cid:19) , (B.6)where we introduced the phase factor ζ κ (NS; α, β ) = 1 , ζ κ ( f NS; α, β ) = e iπκα , ζ κ (R; α, β ) = e iπκβ , ζ κ ( e R; α, β ) = e iπκ ( α + β ) , (B.7)and set m ′′ i := m i − Lα i ∈ S,m i otherwise . On the other hand, the γ -twisting of F ( σ ) I ( τ ) is expressed as γ ( a,b ) · F ( σ ) { ( ℓ i ,m i ) } ( τ ) = ( − b P i ∈ S ℓ i F ( σ ) { ( ℓ i ,m i ) } ( τ ) , ( a = 0)( − b P i ∈ S ( ℓ +1) F ( σ ) { ( ℓ ′ i ,m i ) } ( τ ) , ( a = 1) (B.8)where we set ℓ ′ i := k i − ℓ i i ∈ S ,ℓ i otherwise . We next describe explicitly the Jacobi’s theta functions twisted by the δ -actions given in(3.9), that is, θ i, ( α,β ) ( τ ) := q α e πi αβ θ i (cid:18) τ, ατ + β (cid:19) , ( i = 3 , , , . (B.9)They are explicitly written down as follows; (i) α, β ∈ Z : θ , ( α,β ) ( τ ) = ( − αβ θ ( τ ) ,θ , ( α,β ) ( τ ) = ( − αβ + α θ ( τ ) ,θ , ( α,β ) ( τ ) = ( − αβ + β θ ( τ ) ,θ , ( α,β ) ( τ ) = ( − αβ + α + β θ ( τ ) ≡ , (B.10) Here we omit the subscripts ‘ L ’ and ‘ R ’ used in the main text. ii) α ∈ Z , β ∈ Z + 1 : θ , ( α,β ) ( τ ) = e iπ ( αβ + α ) θ ( τ ) ,θ , ( α,β ) ( τ ) = e iπ αβ θ ( τ ) ,θ , ( α,β ) ( τ ) = e iπ ( αβ + α + β +12 ) θ ( τ ) ≡ ,θ , ( α,β ) ( τ ) = e iπ ( αβ + β − ) θ ( τ ) , (B.11) (iii) α ∈ Z + 1 , β ∈ Z : θ , ( α,β ) ( τ ) = e iπ αβ θ ( τ ) ,θ , ( α,β ) ( τ ) = e iπ ( αβ + α ) θ ( τ ) ≡ ,θ , ( α,β ) ( τ ) = e iπ ( αβ + β ) θ ( τ ) ,θ , ( α,β ) ( τ ) = e iπ ( αβ + α + β ) θ ( τ ) , (B.12) (iv) α, β ∈ Z + 1 : θ , ( α,β ) ( τ ) = e iπ ( αβ + α ) θ ( τ ) ≡ ,θ , ( α,β ) ( τ ) = e iπ αβ θ ( τ ) ,θ , ( α,β ) ( τ ) = e iπ ( αβ + α + β +12 ) θ ( τ ) ,θ , ( α,β ) ( τ ) = e iπ ( αβ + β − ) θ ( τ ) . (B.13)16 eferences [1] S. Kachru, J. Kumar and E. 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