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Chiral Supersymmetric Gepner Model Orientifolds

We explicitly construct A-type orientifolds of supersymmetric Gepner models. In order to reduce the tadpole cancellation conditions to a treatable number we explicitly work out the generic form of the one-loop Klein bottle, annulus and Moebius strip amplitudes for simple current extensions of Gepner models. Equipped with these formulas, we discuss two examples in detail to provide evidence that in this setting certain features of the MSSM like unitary gauge groups with large enough rank, chirality and family replication can be achieved.


Introduction
Various ways for constructing phenomenologically semi-realistic four-dimensional string vacua have been followed in the past. After focusing solely on the heterotic string, the realization that D-branes play a very crucial role in string theory opened up the possibility to look out for new string vacua where these D-branes are actually present. The right framework to study such models are orientifold models, where the presence of the D-branes is implied by tadpole cancellation conditions. These orientifold models were the subject of intense study during the last years and it became clear that there are essentially two approaches to get chiral semi-realistic models. One can either get chiral fermions from Dbranes on orbifold singularities [1][2][3][4] or from intersecting D-branes [5][6][7][8][9][10]. Both approaches have been followed extensively and many semi-realistic models have been proposed so far, but it is certainly fair so say that none of them gives indeed rise to physics of the Standard Model, but at least some of its rough data could be realized such as the right gauge group and chiral fermions in three generations.
So far these constructions were mostly limited to toroidal orbifold backgrounds, which, of course, cover only a very small subset of all possible closed string backgrounds. It is therefore desirable to also study orientifold models on more general non -flat Calabi-Yau spaces [11]. One way to approach this problem is to start with a class of exactly solvable superconformal field theories which are known to describe certain points deep inside the Kähler moduli space of Calabi-Yau manifolds. These models are commonly known as Gepner models [12,13]. The construction of boundary [14][15][16][17][18][19][20] and crosscap states [21][22][23][24][25][26][27][28][29][30][31][32] in these models has been under investigation during the last years. However, there have been only quite a few attempts so far to really construct fully fledged orientifolds of Gepner models [33][34][35][36][37]. Whereas in the beginning the construction of these orientifolds was carried out in a case by case study, only very recently [37] it was possible to derive (at least for all levels being odd) quite generic formulas for all one-loop partition functions including the highly non-trivial Möbius strip amplitude containing important sign factors. Moreover, amazingly simple expressions for the tadpole cancellation conditions were derived. We regard these general equations as a useful starting point for an explicit and general study of such models. One has to distinguish two different kinds of orientifolds which are called Atype and B-type. It turned out that for B-type models the number of tadpole cancellation conditions is much smaller than for the A-type models and therefore much easier to solve. This latter point was the main reason why in [37] the primary focus was on B-type models. Even though the tadpole cancellation conditions could easily be solved, the prize one had to pay was that due to general arguments these B-type models were always nonchiral. Therefore from the phenomenological point of view they are not very interesting.
Additionally, from the technical point of view it would have been desirable to confirm that the signs in the Möbius strip amplitude indeed lead to anomaly cancellation of the effective low energy theory. But, of course, without chiral fermions such a check could not be made.
In this paper we continue the study of orientifolds of Gepner models focusing on the A-type models, for which one expects that chiral models should be possible. As we mentioned already, the main obstacle is the huge number of tadpole cancellation conditions (of the order 10 2 ), which made it hard to determine any other solution than the generic one with gauge group SO(4) or SP (4). In order to reduce this number we consider simple current extensions [38] of Gepner models and generalize the explicit results for the one-loop amplitudes of [37] to this case. Note that generic expressions for boundary and crosscap states for simple current extensions were presented in [25], which could also serve as an alternative starting point for such a construction. For the set of simple currents of Gepner models with all levels being odd we derive the generic form of the Klein bottle, annulus and Möbius strip amplitudes. Equipped with these formulas, we study two models in great detail and show that indeed certain aspects of the Standard Model, including chirality, can be realized in this setting. The aim of this paper is not to provide a systematic search for the MSSM in this class of models, but merely to provide the relevant formalism and to give evidence that such a search might be worthwhile carrying out. This paper is organized as follows. In section 2, we review some facts about Gepner models. In section 3, after some comments about the simple current extension of modular invariants for general CFTs, we apply these methods to the diagonal and charge conjugate partition function of the Gepner model. Section 4 contains the derivation of the corresponding orientifolds in terms of the computation of the simple current extended Klein bottle amplitude of A-type. In principle due to the Greene-Plesser construction [39] of the mirror, A-type models would be sufficient, but nevertheless we provide explicit formulas for the B-type Klein bottle amplitude in Appendix A. In section 5, we review some of the important aspects of RS-boundary states including the loop and tree channel annulus amplitudes. The derivation of the Möbius strip amplitudes is the subject of section 6, where for simplicity we restrict ourselves to the explicit computation of the NS sector amplitudes. In section 7, we present the general tadpole cancellation conditions, followed by a general analysis of the massless open string sector including the gauge sector. The techniques developed are employed to discuss a couple of examples in section 8. Finally, section 9 contains our conclusions.
Note: While this work was in its very final stage we became aware of the paper Brunner et. al. [40], which has some overlap with our work.

Review of Gepner models
Let us briefly review some aspects of Gepner models needed in the following. In light-cone gauge, the internal sector of a Type II compactification to four dimensions with N=2 supersymmetry is given by tensor products of the rational models of the N=2 super Virasoro algebra with total central charge c = 9 [12,13]. Space-time supersymmetry is achieved by a GSO projection, which can be described by a certain simple current in the superconformal field theory.
The minimal models are parametrized by the level k = 1, 2, . . . and have central charge Since c < 3, one achieves the required value c = 9 by using tensor products of such minimal with l + m + s = 0 mod 2. Actually, the identification between (l, m, s) and (k − l, m + k + 2, s + 2) reveals that the range (2.2) is a double covering of the allowed representations.
The conformal dimension and charge of the highest weight state with label (l, m, s) is given Note that these formulas are only correct modulo one and two respectively. To obtain the precise conformal dimension h and charge from (2.3) one first shifts the labels into the standard range |m − s| ≤ l by using the shift symmetries m → m + 2k + 4,s → s + 4 and the reflection symmetry. The NS-sector consists of those representations with even s, while the ones with odd s make up to the R-sector.
In addition to the internal N=2 sector, one has the contributions with c = 3 from the two uncompactified directions. The two world-sheet fermions ψ 2,3 generate a U (1) 2 model whose four irreducible representations are labeled by s 0 = −1, . . . 2 with highest weight and charge modulo one and two respectively The GSO projection means in the Gepner case that one projects onto states with odd overall U (1) charge Q tot = q s 0 + r j=1 q l j m j ,s j . Moreover, to have a good space-time interpretation one has to ensure that in the tensor product only states from the NS respectively the R sectors couple among themselves.
These projections are described most conveniently in the following notation. First one defines some multi-labels λ = (l 1 , . . . , l r ), µ = (s 0 ; m 1 , . . . m r ; s 1 , . . . , s r ) (2.5) and the respective characters In terms of the vectors β 0 = (1; 1, . . . , 1; 1, . . . , 1), β j = (2; 0, . . . , 0; 0, . . . , 0, 2 j th , 0, . . . , 0) (2.7) and the following product the projections one has to implement are simply Q tot = 2β 0 • µ ∈ 2Z Z + 1 and β j • µ ∈ Z Z for all j = 1, . . . r. Gepner has shown that the following GSO projected partition function is indeed modular invariant and vanishes due to space-time supersymmetry. Here K = lcm(4, 2k j +4) and β means that the sum is restricted to those λ and µ in the range (2.2) satisfying 2β 0 • µ ∈ 2Z Z + 1 and β j • µ ∈ Z Z. The factor 2 r due to the field identifications guarantees the correct normalization of the amplitude. In the partition function (2.9) states with odd charge are arranged in orbits under the action of the β vectors. Therefore, although the partition function is non-diagonal in the original characters, for all levels odd it can be written as a diagonal partition function in terms of the GSO-orbits under the β-vectors (2.7), which in this case have all equal length 2 r K.
Let us also state the modular S-transformation rules for the characters involved in (2.9). For the SU (2) k Kac-Moody algebra the S-matrix is given by where we have used the convention (l, l ′ ) k = π(l+1)(l ′ +1)

k+2
. For the N=2 minimal model, the modular S-matrix reads (2.11) For a discussion of the normalization see [37].
The loop-and tree-channel Möbius amplitudes are related by the P-matrix P = T 1 2 S T 2 S T 1 2 , which for just the SU (2) k Kac-Moody algebra is given by and for the N=2 unitary models reads [29] m+m ′ ,0 . stem from the roots of the modular T -matrix in the definition of P .
Since in the following we restrict ourselves to the case of all levels being odd, we present in Table 1   Apparently, for all levels odd the number of tensor factors is either five or nine. Therefore the formulas to be presented in the following sections are derived under the assumption of r = 5, 9 and all levels k j odd. However, we would like to point out that we have evidence for some of them to hold also for the case of even levels [41].

Review of general simple current extension
Recall that for a given conformal field theory there exists a very general way to construct modular invariant partition functions via an extension of the chiral symmetry algebra by some element of the set of simple currents [38]. These simple currents are primary fields J i whose OPE with any other primary field φ i only involves one particular other primary field, i.e.
under fusion. It follows in particular, by associativity of the fusion rules, that the OPE of two simple currents yields again a simple current, so that in a rational CFT the set of simple currents forms a finite abelian group S under the fusion product. Denoting the length of the simple current J a as N a , the set {J a , J 2 a , . . . , J N a a } forms an abelian subgroup of S isomorphic to Z Z N a with (J n a ) C ≡ (J n a ) −1 = J N a −n a . Similarly, every simple current groups the primary fields into orbits The crucial observation is that the action of simple currents in a RCFT implies the existence of a conserved quantity for every primary φ i , the monodromy charge Q The monodromy of the identity being 1, it is clear that Q Here the monodromy parameter r a is defined such that Inspired by the idea of orbifolding the CFT with respect to this world-sheet symmetry, one can prove that a simple current J a with even monodromy parameter r a induces the following even more general modular invariant partition function Note that the proof relies on the fact that r a is even, which can always be arranged for odd N a , with r a being defined only mod N a . This yields two very different types of simple current invariants: If the conformal dimension of J a is integer-valued, i.e. r a = 0 mod 1, Z D can be written as a left-right symmetric combination of orbits of primaries of integer monodromy, each occurring with multiplicity N a N i a . In all other cases, Z D forms an automorphism type invariant in the sense that Z D = l χ l (τ ) χ Π(l) (τ ) for some permutation Π of the primary labels. Furthermore, given two modular invariant matrices M a 1 and M a 2 , it is km χ m is another modular invariant partition function with obvious generalizations for several M a i ; the normalization factor N ensures that the vacuum appears precisely once in Z D . As can be checked from above, (M a ) 2 = 1.
The matrices M are also seen to commute if the respective simple currents J a and J b are mutually local, i.e. if their relative monodromy charge Q (a) (J b ) = 0 mod 1, in which case the various δ-function constraints simplify considerably, as will turn out in our application to the Gepner model below.
The above procedure can equally well be used to construct simple current invariants for partition functions other than the diagonal invariant. In particular, one can verify that the extended charge conjugated partition function is obtained from

Simple currents in the Gepner model
To begin with, let us identify the simple currents for the Gepner model. Given the fusion rules we conclude that the simple currents J α of the Gepner model under consideration can be labeled by the vector To compute the orbit length of the generic primary φ l i (m i ,s i ) under J α , we take into account that the m i -and s i -labels are only defined mod (2k i + 4) and mod 4 respectively, so that for all levels odd (i.e. in the absence of short orbits resulting from the field are integer-valued numbers for all i, j from {1, . . . , r} and {0, . . . , r} respectively. It is clear that the orbit length only depends on the simple current in question and is the same for every primary, N α λ,µ ≡ N α . As indicated above, the simultaneous extension of the chiral algebra by several simple currents is possible as long as they are mutually local. For the construction to make sense, we therefore need to impose first compatibility with the simple currents (2.7) generating the GSO-like Gepner projection, i.e.
Starting with the latter and using (3.3) together with (2.3) for the computation of the monodromy charge, this yields the constraints so that s α 0 + s α i = 0 mod 2 for i = 1, . . . r . Without loss of generality, we can restrict ourselves to the case of even s α 0 , s α i and therefore also even m α i , as required by the usual constraint that l α i + m α i + s α i = 0 mod 2. The point is that this is not really a restriction of the generality of the currents to be used since currents from the Neveu-Schwarz sector are related to those in the Ramond sector by spectral flow, of course.
Furthermore, we can even assume for the following reason that actually s α i = 0 mod 4. Since we will eventually be summing over the orbit generated by J i and J α in the partition function independently (cf. (3.20) and (3.22) below), a non-zero value of s α i can be compensated for by the corresponding contribution from J i . In other words, we can encode the information about the value of s α j by a shift of s α 0 , because with the above procedure the two currents (s α 0 ; m α j ; 2) and (s α 0 + 2; m α j ; 0) are equivalent. Mutual locality with the Gepner current J 0 necessitates that For all levels k i odd, this can only be satisfied if s α 0 = 0 mod 4. To summarize, the most general set of simple currents consistent with the Gepner extension is labeled by the vector for even m α i . In view of (3.12), this means that the orbit length is odd for every simple current under consideration.
Furthermore, compatibility of the simple currents with each other imposes the con- Finally, it is sufficient to consider the cases in which none of the simple currents lies in the cyclic group generated by any of the other simple currents or of an arbitrary product of them. As pointed out above, we do not miss any generically new cases by this requirement, since, e.g., M (J α ) M (J n α ) = 1 for all n = 0.

Diagonal and charge conjugate partition functions for Gepner models
We are now in a position to construct the diagonal invariant for the Gepner model extended by appropriate currents J α , α = 1, . . . , I. Since N α is odd, J α and J 2 α generate the same orbit when acting on the primary φ λ µ , which may therefore be parametrized as To deal with the δ-functions appearing in the extended partition function, we observe that the arguments of the δ (1) -functions as written in (3.7) are of the form whereμ = µ + β =α 2 τ β j β accounts for the twists from the simple currents other than τ α . A straightforward calculation shows that due to mutual locality of the currents Putting things together results in the following extension of the diagonal invariant partition function (2.9) . . . where Likewise, one obtains a similar expression for the charge conjugate partition function . . . (3.22)

Orientifolds of extended Gepner models: The A-type Klein bottle
A-type orientifolds are obtained by projecting the bulk theory as defined by the charge conjugate partition function onto Ω-invariant states, i.e. onto those states coupling symmetrically in Z C . From (3.22) it is evident that this projection requires for states appearing i.e.
To incorporate these projections correctly, it is convenient to read off the δ-constraints directly from the very general from of the simple current extended conjugate partition function (3.9) as Under the Klein bottle projection (4.1), a generic δ-constraint is seen to equal Therefore, the monodromy projections are satisfied identically and we are left with the following A-type Klein bottle (4.5) Transforming this into tree channel with the methods of [37] yields with K ′ = lcm(k j + 2). The crosscap state can be extracted from (4.6) up to additional sign factors (4.8) For completeness, we give a similar derivation of the B-type Klein bottle in the appendix.

The A-type Annulus amplitude
As we can see from the various constraints in (4.6), it is the states coupling diagonally in Z D which contribute to the divergent A-type Klein bottle amplitude. For one-loop consistency of the string spectrum we therefore need to introduce an appropriate amount of D-branes, i.e. A-type RS boundary states canceling the divergent terms from the orientifold plane. The A-type boundary states of the pure Gepner model read where we use the normalization as computed in [37] 1 The authors of [18] pointed out that two boundary states of a theory whose symmetry algebra is extended by a certain group of simple currents are equivalent if they lie in the same orbit under a simple current. Applied to the pure Gepner case, this means that the boundary states related by the action of the simple currents J 0 and J i are equivalent, which is consistent also with the detailed form of the various open string amplitudes as calculated in [37]. Together with the constraint L j + M j + S j = 0 mod 2 and the reflection symmetry (L j , M j , S j ) → (k j − L j , M j + k j + 2, S j + 2) this allows us to bring the independent boundary states to the form S 0 ; (L j , M j , 0) r j=1 A with L j = M j = 0 mod 2. Recall from [37] that boundary states with odd S j are excluded in the orientifold, being inconsistent with the Möbius amplitude, since the crosscap state is formally given by 0, (L j , 0, 0) r j=1 for certain numbers L j and all tensor factors need to lie either in the NS-or the R-sector.
In this paper we impose the stronger condition that all boundary states should also be relatively supersymmetric with respect to the orientifold plane, thus giving rise to the additional condition a, J 1 , . . . , J I A = 1 . . .
Therefore, now the independent boundary states are labeled by simple current orbits of the pure Gepner model boundary states. For simplicity, in the following we will label them by one of the representatives appearing in the respective simple current orbit. Inserting the Gepner model boundary states (5.1) into (5.5), the latter ones can also be written as Therefore, in the boundary states only A-type Ishibashi states with integer monodromy charge appear. This is consistent with the analogous statement for the crosscap states, which clearly holds in view of (4.7).
Forming the overlap between two stacks of branes a and ã with Chan-Paton multiplicities N a and Nã respectively results in the tree-channel annulus amplitudẽ In order to finally read off the massless spectrum, we have to transform into loop channel A Ã a a = N a Nã . . . (5.9)

The A-type Möbius amplitude
In [37], a strategy to determine the signs of the crosscap states was developed: Examining the transformation of a whole orbit generated by simple currents (in the case studied there, the Gepner currents) under world-sheet duality, it turned out to be possible to fix . . .
where h λ µ (ν 0 , ν j , τ α ) are the conformal dimensions of the states appearing in the orbit and h λ µ = h λ µ (0, 0, 0) and the extra signs go back to writing the amplitude in terms of the hatted characters. The latter ones are defined aŝ Note that the primary φ λ µ from which the orbit is generated has to appear in the partition function, of course, to be eligible for the Möbius amplitude at all, i.e. it has to satisfy the Gepner constraints and must have integer monodromy with the integer-spin simple currents. The only open question is in as far the simple current contribution to the total conformal dimension of the states modifies the resummation of the P-transformation. As is shown in the appendix, the signs turn out to be unchanged as compared to [37]. In particular, the orbit-into-orbit condition is seen to be satisfied correctly, including the requirement of integer monodromy of the orbits, and we find eventuallỹ where Extracting the signs from (6.3) and forming the overlap between a crosscap and RS boundary state in the NS-sector yields for the A-typẽ ).

Tadpole cancellation conditions
The massless states lead to divergent terms in the Klein bottle, the annulus and the Möbius amplitude. In light-cone gauge, these are known to be those states with conformal dimension h = 1 2 . Recall that in the N = 2 Super-Virasoro minimal model h is bounded from below by half the U(1)-charge, with equality holding exactly for chiral primaries, so we conclude that the A-type divergent terms stem at most from the fields in the (c,c)-ring.
These are exactly the states (2) (0, 0, 0) 5 as well as (0) Besides, the concrete formulas for the various amplitudes put further constraints on the chiral fields to actually contribute.
As in the pure Gepner case, we introduce stacks of N a A-type RS -boundary states 0; j (L a j , M a j , 0) and also their Ω-image 0; j (L a j , −M a j , 0) . One can then check the δ-function constraints in each of the tree-channel A-Type amplitudes separately, assuming w.l.o.g that m ′ j is even for the Ishibashi states by reflection symmetry (l ′ j , l ′ j , 0) ∼ = (k j − l ′ j , l ′ j + k j + 2, 2). The result is that only the above chiral fields satisfying in addition give a non-vanishing contribution. This is the net effect of the simple current extension and reduces the number of tadpole conditions to be satisfied. The actual tadpole conditions as such are unaltered as compared to the pure Gepner case [37] and take the amazingly simple form for all j and k j = 4n j ± 1. The Chan-Paton factor is just N 1 = 4 and for r = 5 leads to a gauge group SO(4) and for r = 9 to SP (4). The interpretation of this solution is that we have just placed appropriate D-branes right on top of the orientifold plane.

The matter sector
Additional massless matter can arise from all possible intersections of the boundary states. One has to compute how many massless states of the form  Table 2.  The index I oa can be considered as the intersection number between the D-brane |a and the orientifold plane and is determined entirely by the Möbius strip amplitude. In addition one finds some adjoint non-chiral matter in the A aa open string sectors.

Examples
In this section we will exploit the formulas derived in the last sections for the construction of a number of explicit chiral models.

A simple current extension of the (3) 5 model
We start with the simplest Gepner model with levels ( where the L a j denote any of the labels listed in Appendix C. These states carry Chan-Paton indices N 3i with i ∈ {0, . . . , 31}. The second class contains the 32 states with Chan-Paton indices N 3i+1 . Finally, the third class are the states with Chan-Paton indices N 3i+2 .
The next step is to evaluate the six tadpole cancellation conditions in terms of the 96 Chan-Paton factors. After a little algebra one can bring these conditions into a form with explicitly integer valued coefficients.
• Condition 1: • Condition 3: • Condition 4: • Condition 5: Clearly, it is not so easy to classify all possible solutions to these six equations.
Before we display at least a couple of non-trivial solutions, we would like to point out that the intersection numbers between pairs of the 96 boundary states do not always vanish. Therefore, in contrast to the B-type orientifold studied in [37],

A chiral model
The choice N 10 = 4, N 17 = 2, N 25 = 2 and N 49 = 2 with the remaining Chan-Paton indices vanishing satisfies all six tadpole cancellation conditions. All four boundary states are not invariant under the Ω projection, so that we get a gauge group of rank rk(G) = 10. Computing the massless spectrum we find the chiral spectrum displayed in Table 3. deg.

5
(1, Adj, 1, 1)  with the labels L a j taken from the list in Appendix C. Since we are more interested in unitary gauge groups in the following, we only consider these latter boundary states. Again after some algebra, the six independent tadpole cancellation conditions can be brought into a form with only integer coefficients and read: • Condition 1: • Condition 3: • Condition 4: • Condition 5: • Condition 6: Again one can check quite in general that as long as these six tadpole cancellation conditions are satisfied the non-abelian gauge anomalies do all cancel.

A chiral model
Choosing N 35 = 4, N 13 = 2, N 15 = 2 and N 19 = 2 with the remaining Chan-Paton indices vanishing satisfies all six tadpole cancellation conditions. The gauge group is the same as in the first example (8.22) but as shown in Table 5 the chiral massless spectrum is completely different.
Consequently, the Ishibashi expansion of the crosscap states extracted from the Klein bottle is modified only by the analogous projections on integer monodromy: . . .
where again we need to extract the correct sign from the consistent S-transformation of the Möbius-amplitude and the normalization factor is altered by the prefactors of the Lagrange multipliers as we perform a P-transformation on the hatted characters by means of the P-matrices as given in [37].
Collecting the various contributions carefully, we find that we need to evaluate the . . .
where Q U(1) (J α ) denotes the U(1)-charge of J α and The sum over ν 0 yields the constraint The sum over ν j is performed precisely as in [37], leading to (−1) s 2 As for the evaluation of the sum involving the τ α , we note that due to mutual locality and in particular the factors of 2 appearing in front of the τ α , the quadratic part equals 1, and we find immediately the desired constraint Q (α) λ ′ ,µ ′ = 0 mod 1, (B.7) taking into account that all other terms in x α are integral anyways. Collecting the other trivial factors from the P-matrices, we are lead to the result given in the text.