Type IIA orientifolds and orbifolds on non-factorizable tori

We investigate Type II orientifolds on non-factorizable torus with and without its oribifolding. We explicitly calculate the Ramond-Ramond tadpole from string one-loop amplitudes, and confirm that the consistent number of orientifold planes is directly derived from the Lefschetz fixed point theorem. We furthermore classify orientifolds on non-factorizable Z_N x Z_M orbifolds, and construct new supersymmetric Type IIA orientifold models on them.

Most of Type IIA models compactified on six-dimensional spaces have been constructed by orbifold tori given by Z N [7,8], Z 2 × Z 2 [9,10,17] and Z 4 × Z 2 [12,13], whose point group is defined by the Coxeter elements. In the case of Z N orbifold, some models compactified on non-factorizable tori are constructed by Coxeter elements [22], while, in the case of Z N × Z M Coxeter orbifolds [23], the compact spaces are factorized to T 2 × T 2 × T 2 . Recently, however, non-factorizable Z 2 × Z 2 orbifolds were constructed in heterotic string [24][25][26], and in Type IIA string [21]. One of the authors in this paper recently classified Z N × Z M orbifold models on non-factorizable tori [27,28]. Non-factorizable orbifolds possess different geometries from factorizable ones because the number of fixed tori, and the Euler numbers, in six-dimensional spaces can be less than those of the factorizable ones. Such non-factorizable orbifolds can be applied to Type IIA string models, which give rise to rather richer structure by inclusion of D-branes.
For the consistency of theory the tadpole cancellation is required (see [3,[30][31][32][33][34], and for review, [35][36][37]). We explicitly calculate string one-loop amplitudes on the Klein bottle, the annulus and the Möbius strip on non-factorizable tori and orbifolds, and confirm that the consistent number of orientifold planes (O-planes) is directly derived from the Lefschetz fixed point theorem via the cancellations of Ramond-Ramond (RR) tadpole. We give a systematic way to construct various models on non-factorizable orbifolds. Interestingly, we further find new feature of non-factorizable Z 2 × Z 2 orbifolds, in which the numbers of O-planes depend on three-cycles. This paper is organized as follows: In section 2 we describe the tadpole cancellation condition on generic non-factorizable tori. In this analysis the Lefschetz fixed point theorem makes the cancellation condition simplified and provides an intuitive picture. We apply this formula to orientifold models which have been already well-investigated. In section 3 we explicitly construct Type IIA orientifolds on Z 4 × Z 2 and Z 2 × Z 2 orbifolds on the D 6 Lie root lattice. Because the contributions of untwisted sector in orbifolds are given by the same forms of those in tori, the formula, which is derived from the Lefschetz fixed point theorem, provides a necessary condition on non-factorizable orbifolds. We describe general features of orientifold constructions on non-factorizable orbifolds. Section 4 is devoted to the conclusion. In appendix A we explain details of the classification of orientifolds and orbifolds on the Lie root lattices. In appendix B we summarize a set of useful conventions to describe non-factorizable tori in terms of the lattice space and its dual. In appendix C we briefly review the string one-loop amplitudes which are given by the Klein bottle, the annulus and the Möbius strip as the worldsheet topologies.

Orientifold on non-factorizable torus
In this section we will evaluate RR-tadpole cancellation conditions of torus compactification in Type IIA string theory in the presence of D6-branes and orientifold planes (O6-planes). We will show a method to analyze the orientifold models on non-factorizable tori, which can be applied to any kind of torus compactifications. We introduce a set of general formula for the tadpole amplitudes in RR-sector on non-factorizable tori, which are defined by the Lie root lattices. Utilizing the Lefschetz fixed point theorem, we can check the tadpole cancellation condition not only on the usual factorizable tori but also on the non-factorizable ones in a quite simple way. We will further apply this method to orbifold models in section 3.

RR-tadpole and the Lefschetz fixed point theorem
We consider the Type IIA models compactified on a six-torus T 6 . A six-torus could be regarded as a six-dimensional Euclidean space R 6 divided by a lattice Λ, i.e., T 6 = R 6 /Λ. As we will see, the structure of the lattice Λ plays a central role in the analysis of this paper. Here let us consider orientifolds in Type IIA given in the following way: Type IIA on T 6 ΩR , (2.1) where Ω is the worldsheet parity operator, and R is the orientifold involution which indicates the reflection of three directions in T 6 . Usually the action R can be given as In order to construct consistent effective theories in four-dimensional spacetime, we study the tadpole cancellation condition in the presence of orientifolds. The tadpole amplitude is derived from the string one-loop graphs whose topologies are the Klein bottle, the annulus, and the Möbius strip. These amplitudes are represented as K, A and M, respectively. Here let us explicitly describe their amplitudes in terms of a modulus t in the loop channel as follows: where F and s denote the fermion numbers in the worldsheet and in the spacetime, respectively; the overall coefficient c is given by c ≡ V 4 /(8π 2 α ′ ) 2 , where V 4 is from the integration over momenta in non-compact directions. Since the divergence from the RR-tadpole should be evaluated in the tree channel, which is described by the l-modulus, we should rewrite them via the modular transformation, even though the computations of the amplitudes are easier in the loop channel given by t-modulus. The RR-sectors in the tree channel which we should evaluate in order to see the tadpole cancellation in the presence of orientifold planes and D-branes, correspond to the states with the following insertions in the loop channel [29]: In this paper we calculate these amplitudes for the case cases that D-branes are parallel on O-planes. Then the amplitudes can be written in the form as follows: where the string oscillation modes are represented with respect to the ϑ-function and the Dedekind η-function, while the zero modes are given by L K , L A and L M . The γ matrices are orientifold actions on the Chan-Paton factors in the notation of [34]. Due to the spacetime supersymmetry, the total amplitudes from RR-and NSNS-sectors should be cancelled to each other, as seen the factor (1 RR − 1 NSNS ) on each amplitude in (2.5). The mapping between the two different moduli t and l in these channels is also given as To evaluate the RR-tadpole generated by the orientifold, we extract only the contributions from RR-sector in the tree channel. In the IR limit l → ∞ the divergence from the RR-tadpole should be cancelled,K RR +Ã RR +M RR → 0, (2.7) whereK RR ,Ã RR andM RR are RR-tadpole contributions in the tree channel mapped from K, A and M in the loop channel under the modular transformation, respectively. Now let us evaluate the zero mode contributions L K,A,M in (2.5) given by the momentum modes and the winding modes. p and winding modes w can be written in terms of a set of certain basis vectors {p i } and {w i }, respectively: The zero mode contribution to the loop channel amplitudes is where n i , m i ∈ Z, M ij = p i · p j , W ij = w i · w j and δ = 1 for Klein bottle, δ = 2 for annulus and Möbius strip. Using the generalized Poisson resummation formula, we can rewrite When we move to the tree channel by using (2.6), the zero mode contribution L is which goes to We consider a six-torus T 6 on a lattice Λ. Then different two points in T 6 are identified in terms of the lattice shift vector rα i ∈ Λ as where r is a radius of T 6 . For simplicity we set r = 1 in the following in this paper . Translation operator acting on the momentum states | p is given by Then the momentum modes are expressed by dual vector α * i ∈ Λ * , (2.14) In the Klein bottle amplitude, the momentum modes should be invariant under the action of ΩR. Thus the vector α * i consists of the R invariant sublattice in the dual lattice Λ * , and we have [21] In the same way, the winding modes w i are given by the lattice vector α i invariant under the action −R on the lattice space Λ (with the constant α ′ = 1). Then we obtain One of the simplest way to cancel the RR-tadpole of the O6-plane is to add D6-branes parallel to the O6-planes. Since the O6-planes lie on the R fixed locus, the basis vectors which describe three-cycles of the O6-plane are generated from R-invariant sublattice Λ R,inv . Then, in the case of the annulus amplitude, the momentum modes are described by the vector in the dual lattice (Λ R,inv ) * . The winding modes are related to the distances between these D6-branes, and they are the sublattice projected by −R, i.e., Λ −R,⊥ ≡ 1−R 2 Λ. In the Möbius strip amplitude the momentum modes are same as the ones of the annulus amplitude. On the other hand, the winding modes should be in the invariant sublattice under −ΩR, and it is given by Λ −R,inv . Summarizing the above, we obtain the following descriptions: where we used the following relations: For the contributions to Chan-Paton factors, we have γ 1 = 1 1 so that tr(γ 1 ) = N is the number of D6-branes. Furthermore we require γ −1 ΩR γ T ΩR = 1 1 in order to cancel the RR-tadpole. Now we are ready to obtain the RR-tadpole cancellation condition. The sum of RR-tadpole contributions for large l is asymptoticallỹ 20) where N O6 is the number of the O6-planes according to the Lefschetz fixed point theorem: The equation (2.20) indicates that the RR-tadpole is cancelled by D6-branes whose number is four times as many as that of O6-planes. Therefore we find that it is enough to count the number of O6-planes in (2.21) instead of calculating individual amplitudes. For factorizable models, we have Vol(Λ −R,⊥ )/Vol(Λ −R,inv ) = 1. The condition (2.20) is also expressed as where Π and Π O6 denote three-cycles in D6-branes and O6-planes, respectively. This is the case for O6-planes in Type IIA theory. We can generalize this tadpole cancellation condition to an Oq-plane in type IIA/IIB theory in such a way as where the number of Oq-planes is given by In the case of an O9-plane,the orientifold action is given by Ω, i.e., R = 1 1, and the above equation is ill-defined, however we can calculate it in a same way. Then it is appropriate to set Vol(Λ −R,⊥ )/Vol(Λ −R,inv ) = 1 for O9-plane.

Orientifold models on the Lie root lattices
Here let us first review the Type IIA orientifold on a factorizable torus T 2 × T 2 × T 2 to fix our notation. There are two ways to implement ΩR of (2.2) in each T 2 . The lattice Λ i which defines the boundary condition of i-th T 2 is given by where, for simplicity, we set r = 1 in (2.12); α j is a simple root of the lattice. Without loss of generality we can define α 2i along the x 2i -direction for the orientifold action ΩR in (2.2), which acts crystallographically on the lattice Λ i . Therefore the complex structure U i on the i-th torus T 2 should satisfy RU i = U i modulo the shift given by Λ i . Then there are only two solutions which indicates that there are two distinct lattices for the R action 2 . The one is called A-type lattice [38], whose lattice vector is given by Notice that in this case the complex structure of the torus is given by U = ia. The other is called B-type lattice, which is given by (2.28) This corresponds to the case U = 1 2 + ia. We can see it by the re-definition of the vector Then we have two distinct theories which depend on the choice of A-type or B-type lattices in Figure 1. For example, the number of fixed loci given by the action of R is two (for the A-type) and one (for the B-type), which associate the total O6-plane charges. Instead of using the B-type lattice, we define an equivalent orientifold by an alternative (2.29) In order to distinguish the actions on non-factorizable tori from the ones on factorizable torus, let us attach a label to the action (2.29) as D, and to the one (2.2) in the previous subsection as C [21]. For example we call the models by following R action CCD model, In appendix A, we can see that these actions provide convenient tools for the classifications of orientifold orbifolds on the Lie root lattices.
First let us consider the RR-tadpole cancellation conditions in the factorizable models. Instead of the direct calculations of the zero mode contribution on each T 2 and of the oscillator modes in the Klein bottle, the annulus and the Möbius strip amplitudes, it is enough to count the number of O6-planes from (2.20): The numbers of O6-planes are N O6 = 8 (for AAA), 4 (for AAB), 2 (for ABB) and 1 (for BBB). The types of the actions in the T 2 × T 2 × T 2 are illustrated in Figure 2. Here we obtain the RR-tadpole cancellation conditions 3  These are trivial results which have already been known. We emphasize that for the classification of orientifold models on non-factorizable tori and orbifolds it is convenient to fix the lattices and distinguish the models with respect to the definitions of R.
Next we analyze some typical models on a non-factorizable 4 tori T 6 , which cannot be expressed as the direct product T 2 × T 2 × T 2 . As an example we consider an orientifold model on a non-factorizable torus given by the Lie root lattice D 6 . In this model the lattice D 6 can be given by the simple roots α i = e i − e i+1 , α 6 = e 5 + e 6 , i = 1, . . . , 5, (2.32) 3 Because these are the models on factorizable tori, and the C-and D-actions lead to the A-and B-models, respectively. 4 In this work a compactified space which cannot be represented as the direct products of two-torus T 2 is called non-factorizable. For example, six-tori on where e i 's are basis of Cartesian coordinates whose normalization is given as e i · e j = δ ij . The orientifold action R of the CCC-model is R : e 2i−1 → e 2i−1 , e 2i → −e 2i , i = 1, 2, 3. (2.33) The number of O6-planes is obtained by means of (2.21). In order to evaluate the Lefschetz fixed point theorem, we should fix the sublattice spaces Λ −R,⊥ and Λ −R,inv . Λ −R,⊥ is a lattice space projected out by −R, and given by whose basis vectors are given by On the other hand, the sublattice Λ −R,inv , which is invariant under −R, is given by α inv,1 = e 2 − e 4 , α inv,2 = e 4 − e 6 , α inv,3 = e 4 + e 6 . (2.36) Then we can easily evaluate the number of the O6-planes for the CCC model as In the same way, we consider the CCD model. The lattices Λ −R,⊥ is given by α ⊥,1 = e 2 , α ⊥,2 = e 4 , α ⊥,3 = 1 2 (e 5 − e 6 ), (2.38) and Λ −R,inv is given by (2.39) Then we obtain N O6 = 2. Substituting these numbers into the RR-tadpole cancellation condition (2.20), we easily obtain the number of D-branes. Here we summarize the data of the orientifolds on the non-factorizable D 6 lattice: These results completely agree with the ones in [21]. The gauge group of these models are SO(16), SO(8), SO(4) and SO (8), respectively. For models on non-factorizable tori, the closed string spectra are the same as that of factorizable models.
We evaluated the the number of O6-planes N O6 according to the Lefschetz fixed point theorem, and from (2.21) this give the necessary and sufficient condition for the RR-tadpole condition. This analysis is generic and provides quite a simple rule to calculate the number of O-planes and D-branes in orientifold models on non-factorizable tori in Type II string theory.

Supersymmetric Z N × Z M orientifold models
In this section let us consider Type IIA supersymmetric orientifold models on orbifolds and describe the way to deal with orientifolds on non-factorizable lattices. Since the contributions of the RR-tadpole from untwisted states are calculated in the same way as the ones of the orientifolds on tori, we can easily count the numbers of D-branes via the Lefschetz fixed point theorem (2.21). We also provide detail calculations of the RR-tadpole cancellation condition on Z 4 × Z 2 and Z 2 × Z 2 orbifolds.

Orbifolds and orientifolds
In the previous section we showed general expressions for orientifolds on non-factorizable tori (2.1). Here let us consider orientifold models on orbifolds given by An orbifold is defined as a quotient of torus over a discrete set of isometries of the torus [39], called the point group P , i.e., Here S is called the space group, and is the semi-direct product of the point group P and the translation group T . Z N orbifolds on the Lie root lattices have been classified in terms of the Coxeter elements or the generalized Coxeter elements. In the case of Z N × Z M orbifolds, the (generalized) Coxeter elements yield only orbifolds on factorizable lattices. Recently, however, Z N × Z M orbifolds on non-factorizable lattices were investigated in heterotic strings [27]. We apply their analyses to Type IIA orientifold models.
Since the point group P of orbifold must act crystallographically on the lattice, we choose these elements from the group generated by the Weyl reflection (A.4) and the outer automorphisms G out . In the case of the Z N × Z M orbifold on a Lie root lattice, the point group elements of the orbifold can be defined by two commutative elements in the group generated from Weyl group and the outer automorphisms, i.e, On the complex coordinates of the torus T 6 , the point group elements of the orbifold act in such a way as are twists of an orbifold. We consider orientifold models with N = 1 supersymmetry as follows: The requirement of SU(3) holonomy can be phrased as invariance of the (3, 0)-form Ω = dz 1 ∧ dz 2 ∧ dz 3 , and leads to The twists of the Z N × Z M orbifolds which are compatible with N = 1 supersymmetric orientifolds are listed in Table 1. As explained in Appendix A there are twelve distinct classes of non-factorizable lattices, see Table 8. The Z 2 × Z 2 , Z 4 × Z 2 and Z 4 × Z 4 orbifolds are allowed on these non-factorizable lattices (see Table 10 in appendix A). The series of generators θ and φ of the Z N ×Z M orbifold as well as the action ΩR consist of the orientifold group: These elements appear in the following string one-loop amplitudes as insertions [23], After extracting the RR-tadpoles, the insertion of ΩRθ k 1 φ k 2 in the Klein bottle amplitude corresponds to the contribution from O-planes fixed by Rθ k 1 φ k 2 . Since in the ΩRθ k 1 φ k 2 insertion the contributions from untwisted sectors are calculated in the same way as the cases of tori in section 2, we obtain the necessary condition (2.20) for the RR-tadpole cancellation by D-branes parallel to the O-planes. From this necessary condition, we obtain all the numbers of O-planes and D-branes on the orbifold. In the next subsection we will demonstrate a few examples of Z 4 × Z 2 orientifold models, and evaluate the RR-tadpole cancellation condition.

Z 4 × Z 2 model
Here we discuss the Z 4 ×Z 2 orientifold model on the Lie root lattice D 6 (2.32) in detail because in this case all possible subtleties show up.
There exists only one distinct Z 4 × Z 2 orbifold on D 6 , whose point group elements θ and φ are given by θ : or, in matrix representation, by θ : By using the above elements we can show all the orientifold actions which preserve N = 1 supersymmetry by means of C and D actions. For example, the reflection R on the DDC model is given by where we used an abbreviation defined by From the Lefschetz fixed point theorem (2.21), the number of O6-plane fixed by R is given as N O6 = 1. If we put four D-branes parallel to this R-fixed O6-plane, the RR-tadpole of this model will be cancelled. Similarly, the element Rθ = (a, −a, a) gives N O6 = 4, whose tadpole is cancelled by sixteen D-branes parallel to this four Rθ-fixed O6-planes. We similarly evaluate the cases for the other elements of the orientifold group. The relations between the orientifold group elements and the numbers of O-planes are summarized in Table 2.
Since the Z 4 action changes the directions of the O-planes by angle of θ 1/2 in the following way: This action generates the exchange between the action C and D each other. Then we can see that CCC and DDC, CCD and DDD, CDD and DCD models are equivalent with each other, respectively. In the case of the CCC model, for example, two different numbers of O6-planes appear since the orientifold group elements in R, Rθ 2 , Rφ and Rθ 2 φ are given by (±a, ±a, ±a), whereas the elements in Rθ, Rθ 3 , Rθφ and Rθ 3 φ are given by (±b, ±b, ±a).
Analyzing such actions, we obtain all the models for Z 4 × Z 2 orientifolds on D 6 lattice, listed in Table 3. Table 3: All the Z 4 × Z 2 orientifold models on the D 6 lattice.
We estimated the RR-tadpole cancellation by counting the O-planes from the equation (2.20), which is the necessary condition in the case of the orbifold model. However it is expected that the RR-tadpoles are cancelled even in the orbifold model. These countings also give correct results for well-investigated non-factorizable models on Z N orbifolds in [22] and Z 2 × Z 2 orbifolds in [21]. We give the explicit results of the RR-tadpole cancellation for a few models in the following.

Klein bottle amplitude
First let us evaluate the Klein bottle amplitude of Z 4 × Z 2 orientifold model on the D 6 lattice (2.32) with the orientifold action which gives the DCC model. The contribution of the oscillator modes are equal in any insertions of the orientifold group because they act as the unit operator in (3.7a). In the θ n 1 φ n 2 -twisted sector, the oscillator contribution is given by K (n 1 ,n 2 ) ≡ K (n 1 ,k 1 )(n 2 ,k 2 ) (see, for the notation, [23]). We also need the multiplicities χ (n 1 ,k 1 )(n 2 ,k 2 ) K of the θ n 1 φ n 2 -twisted fixed sectors, which are invariant under the insertion ΩRθ n 1 φ n 2 , which can be seen in Table 4. Table 4: Multiplicities of the fixed points for the DCC and CCC models.
When an action θ n 1 φ n 2 does not fix certain directions in the compact space, the Kaluza-Klein momentum modes and the winding modes appear as the zero modes in the θ n 1 φ n 2 -fixed sector. Let us evaluate such zero modes in the θ-twisted sector. The θ invariant sublattice Λ θ is expanded in terms of the basis e 5 + e 6 , e 5 − e 6 . We can see that the R invariant dual sublattice (Λ θ ) * R,inv , whose basis is given by {2e 5 }, and the −R invariant sublattice (Λ θ ) −R,inv , with its basis {2e 6 }, yield the momentum modes and the winding modes in this sector, respectively. However there are two subtleties in this evaluation, one of which is caused by the momentum doubling, and the other from the appearance of the half winding states [22].
The former subtlety is caused by the shifts associated to the ΩRθ k 1 φ k 2 insertions. In the θ-twisted sector we have two fixed tori given by where x ∈ R is a coordinate on the fixed tori. Note that the invariance of fixed points or fixed tori under ΩRθ k 1 φ k 2 is defined modulo the translation generated by the lattice Λ. The R insertion acts on the two fixed tori in such a way as R :  In the latter case, the translation of a lattice shift α 4 = e 4 − e 5 is accompanied. Because a momentum mode | p picks up a phase factor e 2πp·l under the translation by l, we generally need phase factors in the amplitudes. In the case of (3.17), the phase factor is p·l = 2e 5 ·e 5 = 2, and does not affect the amplitudes. If the phase factor is given as −1, the momentum modes are effectively doubled by interference between modes with and without shifts: The latter subtlety occurs in the winding modes. There are special points with the following property: where we used a lattice shift given by e 1 + e 6 . The point does not lie on the θ-fixed tori, whereas this shift does generate the winding modes: There are two points 1 2 (e 1 ± e 2 ) which are invariant under the action R, and the multiplicity is equal to that of the θ-fixed tori which are also invariant under R.
Therefore we conclude that the zero modes in the θ-twisted sector with R insertion are given by the following vectors: where n, m ∈ Z. In the notation of (C.4), the zero mode contributions in the Klein bottle amplitude is L 2, 1 2 . In a similar way we can evaluate the other twisted sectors in the orbifold model. Note that for non-factorizable orbifolds the zero mode contributions depend on the insertion ΩRθ k 1 φ k 2 . In the φ-twisted sector we have L 2, 1 2 for the ΩRθ 2k 1 φ k 2 insertions, and L 4,1 for the ΩRθ 2k 1 +1 φ k 2 insertions.
We obtained all the ingredients to write down the Klein bottle amplitude for the DCC model, which are summarized as Its modular transformation to the tree channel is Note that in the IR limit l → ∞, the zero mode contributionsL

Annulus amplitude
In order to cancel the RR-tadpole we introduce D-branes parallel to O-planes. We attach a label (i 1 , i 2 ) to a stock of D-branes which is invariant under the orientifold action Rθ i 1 φ i 2 , and define that (0, 0) denotes D-branes invariant under the action R. The three-cycle wrapped by the brane (0, 0) is given by the R invariant lattice Λ R,inv whose basis is given by e 1 + e 2 , e 3 − e 5 , e 3 + e 5 . (3.29) From (3.13) the brane (1, 0) is rotated by half the angle of θ with respect to the brane (0, 0). The three-cycle wrapped by the brane (1, 0) is given by the Rθ invariant lattice Λ Rθ,inv whose basis is given by e 1 − e 5 , e 3 + e 4 , e 1 + e 5 . An open string stretching from brane (i 1 , i 2 ) to brane (i 1 −n 1 , i 2 −n 2 ) is localized at intersection of D-branes. It is convenient to call such a state the θ n 1 φ n 2 -twisted sector.
The three-cycles of brane (0, 0) and brane (1, 0) share a common direction, and the lattice vector in this direction is given by 2e 5 . The momentum modes are obtained from the dual of the vector 2e 5 in such a way as p = n √ 2 e 5 . Then zero mode contribution of the θ-twisted sector is expressed as L 1,1 .
Let us explain one more case of the φ-twisted sector. The winding modes are given as "half winding-like" modes, and are also given by the projected lattice Λ R,⊥ ∩ Λ Rφ,⊥ whose basis is Then the zero modes of open string stretching between the brane (0, 0) and the brane (0, 1) are given by The zero mode contribution in the annulus amplitude is expressed as L 2, 1 2 . The other zero modes are calculated in a similar way.
Since in the θ n 1 φ n 2 -twisted sector the contributions from the oscillator modes do not depend on branes (i 1 , i 2 ), they are given as A (n 1 ,k 1 )(n 2 ,k 2 ) . The insertions of 1 1, θ 2 , φ and θ 2 φ leave D-branes invariant, and perform non-trivial actions on the Chan-Paton factors described as γ (i 1 ,i 2 ) k 1 ,k 2 , which appear in the amplitude as tr γ (3.35) in the θ n 1 φ n 2 -twisted sector. Sectors of k 1 = 0 or k 2 = 0 cannot be cancelled by the other diagrams. Therefore the Z 2 twisted tadpole cancellation condition is required [23,34]: We should also evaluate the multiplicities χ M of the open string states, which are given by the intersection number of D-branes. The intersection numbers of two branes can be obtained by the determinant of vectors v i and v ′ i giving the three-cycles in respective D-branes [22]. These vectors can be expanded in terms of the lattice basis as v i = v ij α j . Then the intersection number is Owing to the above Z 2 twisted tadpole condition, it is sufficient to consider the intersection number χ M for k 1 = k 2 = 0, which are given in Table 5.
The contribution from the zero modes of the untwisted sector is obtained from (2.17b) and (2.17d). For the brane (0, 0), which is paralell to the R-fixed O6-plane ,it is The contributions from the other branes (i 1 , i 2 ) give the same values. These values appear in prefactors of the amplitude after the modular transformation.
Summarizing the above, we obtain the annulus amplitude for the DCC model 2 )A (2,1) + 2A (3,1) . Table 5: Intersection numbers in the annulus amplitudes in the DCC and CCC models.
where A (n 1 ,n 2 ) ≡ A (n 1 ,0)(n 2 ,0) . The modular transformation to the amplitude in the tree channel yieldsÃ We again observe the complete projector in the IR limit.

Möbius strip amplitude
The amplitude of the Möbius strip (3.7c) includes the insertion of ΩR, and string states should be invariant under these orientifold actions. In θ n 1 φ n 2 -twisted sector, the insertion ΩRθ k 1 φ k 2 acts on open strings stretching from brane (i 1 , i 2 ) to brane (i 1 − n 1 , i 2 − n 2 ) as 41) Therefore in the Z 4 × Z 2 orbifold case the following conditions are required: Then the sectors with n 1 = 0, 2 and n 2 = 0 contribute to the amplitude. The intersection number is obtained in the same way as in the case of annulus. In Table 5, we can see that χ M = 1 for untwisted sectors and χ M = 2 for θ 2 -twisted sectors.
The momentum modes are evaluated in a similar way of subsection 3.2.1, however the winding modes are changed due to the insertions. In the untwisted sector with the ΩRθ insertion, from the condition Since only the sectors with n 1 = 0, 2 and n 2 = 0 contribute to the amplitude, we abbreviate a (n 1 ) These assignments correspond to two different classes of the D-brane configurations in this model, and are sufficient to evaluate the tadpole cancellation conditons for Z 4 × Z 2 models. However we will need more independent variables for Z 2 × Z 2 models.
For the contributions from untwisted sector, we can use the results from the (3.26) and (3.39) owing to the relations (2.17a)-(2.17d).
To summarize, we obtain the Möbius strip amplitude in the loop channel as 3,0 + b

(3.48)
The modular transformation to the tree channel yields 3,0 )L 8,2M (1,0) To obtain the complete projector and to cancel the tadpole [23], we set On the other hand, the element Rθ in the orientifold group is given by As seen in Table 2 The prefactors do not correspond to that from the complete projector. The annulus and the Möbius strip amplitudes are also described as The open string massless spectrum is given in Table 6. The multiplicities of twisted states spectra depend on the intersection numbers [23] (see Table 5). We see that the CCD and DCC models are distinct from the CDD model despite the same numbers of O-planes, and actually these four models have different spectra. For the closed string the numbers of massless states are considerably reduced due to their Hodge numbers in [26,27].
(1, ; 1, ) Table 6: Open string massless spectra of Z 4 × Z 2 orbifold on the D 6 lattice. The symbols "V " and "C" denote the vector and chiral multiplets, respectively. and these elements generate O6-planes respectively. From Table 3 the numbers of O6-planes are read two for each elements.
In the CCD orientifold with R = (a, a, b), we have two distinct pairs of the point group elements: The numbers of O-planes generated by the former orbifold actions are also two. In the latter case, the ΩR and ΩRθ (ΩRφ and ΩRθφ) generate two (four) O6-planes, respectively. We can classify the distinct orientifold models on the Lie root lattices, and the other possible elements on the D 6 lattice are listed in Table 7. We should notice that even though the numbers of O6-planes are the same in any three-cycles in Z 2 × Z 2 orientifold models, those of non-factorizable models can be different. Table 7: Z 2 × Z 2 orbifold models on the D 6 Lie root lattice.
Finally we check the RR-tadpole cancellation in the Z 2 ×Z 2 CCC model on the D 6 lattice. The contribution from φ-and θφ-twisted sectors are the same as θ-sector for the CCC model on the D 6 lattice. The RR-tadpole cancellation is satisfied with N = 4 as we can see the following amplitudes in the tree channel. The Klein bottle amplitude is given as The annulus and the Möbius amplitudes are also given as 1) .
We observe that in any amplitudes the prefactors are given by the complete projector (3.28).

Conclusion
In this paper we studied the RR-tadpole cancellation condition in Type II string models compactified on six-tori given by general Lie root lattices. We obtained a simple derivation to count the orientifold planes lying on the lattice by the use of the Lefschetz fixed point theorem.
As expected the RR-tadpole contributions are cancelled by adding an appropriate number of D-branes parallel to the O-planes. The Lefschetz fixed point theorem provides an intuitive picture to non-factorizable models, and we easily showed a way to construct orientifold models on tori and orbifolds.
In D = 4, N = 1 Z N × Z M orientifolds, mainly the factorizable models on T 2 × T 2 × T 2 have been constructed and investigated. We gave the classifications in Type IIA orientifold models with O6-planes, and obtained many new models. As explained in detail, the Lefschetz fixed point theorem provide intuitive and convenient tools in model construction. Since the condition derived in (2.20) is the necessary condition for orbifolds, we performed explicit calculations for Z 4 × Z 2 and Z 2 × Z 2 orbifold models, and confirmed the RR-tadpole calculations. It is expected that even in other non-factorizable orbifold models the RR-tadpole cancellation should be checked in the same calculation. We further found many non-factorizable Z 2 × Z 2 orbifolds in which the numbers of O-planes depend on the three-cycles left invariant under the orbifold projections in Table 7 and in Table 11. These features are not seen in factorizable models, and will provide new possibilities for model constructions. On the other hand, since the metric of non-factorizable tori is changed to B-field via T-duality, our consideration should be related to compactification with such backgrounds. Actually in heterotic orbifolds there are some coincidences between non-factorizable models and factorizable models with generalized discrete torsion [41]. Our results indicate that there would be a possibility to construct various class of D = 4, N = 1 models with different set of chiral spectra from other well-known (non-)factorizable models. roots α i of these Lie algebras can be given as follows: where e i 's are unit vectors whose scalar product is defined as e i · e j = δ ij . The Dynkin diagrams are drawn in Figure 3. From these diagrams one can easily find a set of equivalence relations (isomorphism) among the simple Lie algebra, We further find the equivalence relations from the Lie root lattice point of view: Here we assumed the most symmetric cases, where the lengths of the shortest roots are equal between the lattices given as direct products. Since we are interested in symmetries of the lattices, the assumption is rational. We often use these equivalence relations in the classification of the six-tori.
Taking the direct products of tori generated from these lattices, we obtain six-tori in terms of the Lie root lattices. We conclude that there are only twelve inequivalent non-factorizable six-tori and four factorizable ones 5 in such a way as in Table 8.

A.2 Weyl reflection and graph automorphism
Next we investigate the automorphisms of the above lattices. Both orbifold and orientifold groups should crystallographically act on the lattices. These groups can be classified in terms of the Weyl reflection and the graph automorphism acting on the simple roots of the Lie root lattice. The Weyl group W is generated by the following Weyl reflections r α k which associate the simple root α k : In the case of the D N Lie root lattice, for instance, the Weyl reflection r α k for k = 1, . . . , N − 1 is given as and α m for m = k − 1, k, k + 1 are unchanged. For k = N it is On the other hand, the outer automorphism g of the Cartan diagram is represented as and the other simple roots are left unchanged. In the unit vector basis, it is In terms of e i , we can easily construct any elements generated from r α k and g. For example a product of two Weyl reflections which do not commute with each other makes up Z 3 element as This is the permutation group S 3 . Similarly the Weyl reflections r α k for k = 1, . . . , N − 1 generate a permutation group S N . Adding the other elements r α N and g to S N , the representation of the group is given by permutations with signs Then the order of the Weyl group W and {W, g} are summarized in Table 9: Table 9: The order of the Weyl group and the graph automorphism.
In the case of the A N −1 Lie root lattice, the Weyl reflections generate permutation group S N in terms of e i . Its outer automorphism of the Dynkin diagram is given by the following permutation We can always permute g to g ′ by the elements of W such that g ′ change the sign of all e i 's, This element is expressed as an identity matrix with negative sign −1 1 N , which means {W, g} = {W, −1 1 N }. Therefore the order of {W, g} is twice as many as that of W, as in Table 9.
Then it is straightforward to obtain all Z 2 elements of the D N lattice, and they are given by the following sub-elements and their permutations. Z 4 elements includes the following sub-elements, We can similarly deal with the A N lattices. Note that the roots of the A N and D N can be given by They are symmetric under the permutations of i and j. Now it is apparent that on the D 6 lattice, (3.9b) is the only inequivalent Z 4 × Z 2 elements, and Z 2 × Z 2 elements can be given by a, b and 1 in (3.12). From Table 8, we have all the point group elements which can be expressed by the Weyl reflections and the outer automorphism (except for the E 6 lattice). However there are a few exceptions owing to additional outer automorphisms as follows.
We shortly explain the Coxeter elements and the generalized Coxeter elements 6 . The Coxeter element of the Lie root lattice is defined by product of all the Weyl reflections which associate with simple roots, The other Coxeter elements, which are generated by different ordering of product, are conjugate to one another, and lead to the same class of orbifolds. There are other elements generated by the Weyl reflections. These orbifolds can be classified by the Carter diagrams [40]. The Coxeter elements of D 4 from the Carter diagrams are, for example, D 4 (a1) = r α 1 r α 2 r α 3 r α 2 +α 3 +α 4 , where r α 2 +α 3 +α 4 is a Weyl reflection associated with the sum of simple roots α 2 + α 3 + α 4 . Then the order of D 4 is six, and that of D 4 (a1) is four. However these elements do not include the outer automorphisms 7 . The generalized Coxeter elements are defined by adding outer automorphisms to the Coxeter elements. For example the D N Lie root lattice has a graph automorphism g which exchanges the simple root α N −1 and α N . The generalized Coxeter element is defined by For instance the generalized Coxeter element of D 4 is .19) and the order of this element is eight.
Actually these (generalized) Coxeter elements and elements from the Cater diagrams are included in the above classification by the use of e i . An exception occurs in the D 4 lattice, which has another outer automorphism g ′ , The generalized Coxeter element of this outer automorphism is defined by This action corresponds to a rotation of (e πi/6 , e 5πi/6 ). For this element the classification in the e i basis is inconvenient (since for example it acts as g ′ : e 1 → (e 1 + e 2 + e 3 + e 4 )/2). We comment that among Z N × Z M orbifolds this element generates new orbifold only for Z 3 × Z 3 , e.g. (C [3] ) 4 is rotation of (e 2πi/3 , e 2πi/3 ) and that of r α 3 g ′ is (1, e 2πi/3 ). Then a torus on the D 4 × A 2 lattice allows a Z 3 × Z 3 orbifold.
In the case that two independent radii of a torus on the A 3 × A 3 lattice are equal to each other, there is an additional outer automorphism g 33 , where α i is a simple root of the first (second) A 3 for i = 1, 2, 3 (i = 4, 5, 6). From the observation of its eigenvalues, these elements do not generate another Z N × Z M elements. However the orientifold action R can be generated from g 33 , which will be explained in the next subsection. Such outer automorphisms also arise in factorizable tori including sublattices (A 2 ) n and (A 1 ) m . For example, (A 2 ) 2 has an outer automorphism as where α i is a simple root of the first A 2 and α ′ i is one of the second A 2 . The eigenvalues of this element are (e πi/2 , e πi/2 ), and generate Z 4 elements. In this case the factorizable tori are actually non-factorizable as orbifolds.
We investigate the other cases similarly, and obtain the allowed Z N ×Z M orbifolds in Table  10. In the next subsection we will explain the orientifold actions which are compatible with these non-factorizable orbifolds.

A.3 Orientifolds on non-factorizable orbifolds
In the previous subsection we gave a way to obtain orbifolds on non-factorizable tori. In order to preserve supersymmetry, the orbifold action θ : (z 1 , z 2 , z 3 ) → (e 2πiv 1 z 1 , e 2πiv 2 z 2 , e 2πiv 3 z 3 ) should satisfy the equation v 1 + v 2 + v 3 = 0. Then only a holomorphic (3, 0)-form Ω = dz 1 ∧ dz 2 ∧ dz 3 and a anti-holomorphic (0, 3)-formΩ are left invariant, and the other three forms on a six-tori are generally projected out. The orientifold action R of O6-plane, which preserve N = 1 supersymmetry, should act as where a, b and c are phase factors. Then the every orientifold group element including R generates fixed loci of O6-planes. In other words, the restriction is that the eigenvalues of each orientifold group element R, Rθ, Rφ and Rθφ should be (−1, −1, −1, 1, 1, 1). Note that there are some equivalent actions due to the symmetry of the lattice. These considerations lead to Table 7 for the Z 2 × Z 2 orbifold models on the D 6 lattice.
(A. 27) In this base Z 2 × Z 2 orbifolds are obtained in a similar manner of the D 6 lattice 9 . Note that the action R = ( * , b, * ), where * is b, a or 1, is forbidden due to the lattice structure. The 8 Note that for this orbifold elements the basis is different from (A.24). 9 It may seem that the classification with b,a and 1 elements is missing the action R : α i → −α i with i = 1, 2, 3, however this action is included in orientifold groups, e.g. the Rθφ action of DCD model on Table  11. outer automorphism between two A 3 's generates an exceptional action R : α i ↔ α i+3 , i = 1, 2, 3. (A.28) If we redefine the base of A 3 × A 3 as

B Comments on lattices
In this appendix we briefly summarize conventions of the (sub-)lattice and its dual lattice space for a Z 2 action R in the following way: Λ R,⊥ : lattice projected out by the action R, Λ R,⊥ ≡ 1 + R 2 Λ Λ R,inv : R invariant sublattice Λ * : dual lattice of Λ, for its base α j · α * i = δ ji , α j ∈ Λ, α * i ∈ Λ * These three lattice spaces are closely related to one another. Introducing a lattice Λ −R,⊥ which is projected out by the −R action on it, then we find the following non-trivial equations: Let us analyze in a more concrete way. For example, we consider the four-dimensional D 4 Lie root lattice Λ and its dual lattice Λ * based on Λ : and we give a Z 2 action R on the D 4 lattice as Then, we can obtain the basis vectors in the lattices Λ R,⊥ , Λ R,inv , Λ * R,⊥ and Λ * R,inv in the following forms: (1, 0, 0, 0) (1, 1, 0, 0) (B.4b) Thus we easily see the relation among various lattice spaces:

C String one-loop amplitudes
In this appendix we summarize descriptions of the string one-loop amplitudes whose topologies are given by the Klein bottle, the annulus and the Möbius strip in the loop channel [23,38]. These are applied to discuss the RR-tadpole amplitudes in the main part of this paper. Here we start from the forms 10 in which the zero mode and the oscillator modes are factorized: where the values K (n 1 ,k 1 )(n 2 ,k 2 ) , A (n 1 ,k 1 )(n 2 ,k 2 ) and M (n 1 ,k 1 )(n 2 ,k 2 ) denote oscillator contributions, and L indicates the zero mode contributions in the amplitudes. They belong to the θ n 1 φ n 2twisted sector with θ k 1 φ k 2 -insertion in the amplitudes. The γ (i 1 ,i 2 ) 's are the matrix representations of the orientifold action on the Chan-Paton factors [34], whose superscript (i 1 , i 2 ) labels the different types of D6-branes on which the open string attaches. The location of the brane (i 1 , i 2 ) is defined by rotating brane (0, 0) by the action θ −i 1 /2 φ −i 2 /2 .
Note that in the Klein bottle amplitude χ K denotes the number of the corresponding fixed points which are left invariant under orientifold group actions Rθ k 1 φ k 2 . In the open string amplitudes χ A gives the intersection number of the D-branes involved.
When we consider string propagating in the torus T 6 = R 6 /Λ, the zero modes contributions L from the momentum modes p = i n i p i and the winding modes w = m i w i are given by where t is the modulus in the loop channel and n i , m i ∈ Z are the quanta in the momentum modes and the winding modes [22]. Note that the matrices M ij and W ij are given by the products of p i and of w i in such a way as M ij = p i · p j , W ij = w i · w j ; we set δ = 1 (the Klein bottle), δ = 2 (the annulus and the Möbius strip). Due to this, in two-dimensional torus T 2 ⊂ T 6 , we can rewrite the above equations (C.2) in the following form: where ρ = r 2 /α ′ . It is worth rewriting this to the one in the tree channel. According to the Poisson resummation formula n∈Z e −πn 2 /t = √ t n∈Z e −πn 2 t , (C. 5) we find that the zero mode contribution in the tree channel is given as This formulation is quite useful not only for factorizable torus T 2 × T 2 × T 2 but also for non-factorizable tori in the main text via a suitable arrangement.