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(0,2) Mirror Symmetry

We generalize the previously established (0,2) triality of exactly solvable models, Landau-Ginzburg theories and Calabi-Yau manifolds to a number of different classes of (0,2) compactifications derived from (2,2) vacua. For the resulting models we show that the known (2,2) mirror constructions induce mirror symmetry in the (0,2) context.


Introduction
The discovery of mirror symmetry among Calabi-Yau manifolds seven years ago [1,2] has led to a number of developments which have recently culminated in the circle of ideas related to strong-weak coupling dualities [3][4][5][6] and the unification of string vacua [7][8][9].
The framework of (2,2) compactifications in which mirror symmetry has been formulated is rather restricted however. It has been believed for some time that the class of (2,2) supersymmetric ground states defines only a small part of the moduli space of all consistent string vacua with N = 1 spacetime supersymmetry. A natural question therefore is whether mirror symmetry can be extended to (0,2) supersymmetric theories or whether it is indeed an artefact of the restriction to the space of (2,2) vacua, irrelevant for more generic vacua.
It is this question which we address in the present paper.
Our discussion will focus primarily on exactly solvable models and Landau-Ginzburg theories. To prepare the ground we first extend the results of [10] concerning the (0,2) triality of string compactification by identifying the Landau-Ginzburg and manifold phases of a subclass of exact (0,2) models constructed in [11]. The (0,2) theories considered there are derived by applying the method of simple currents [12] to the Gepner class of (2,2) tensor models. By choosing the simple currents appropriately the (2,2) world sheet supersymmetry can be broken to (0,2) supersymmetry, necessary and sufficient for N = 1 spacetime supersymmetry.
A second longstanding problem in the context of (0,2) compactification of the heterotic string has been the question whether (0,2) σ−models renormalize to nontrivial conformally invariant fixed points and if so, whether these fixed points are exactly solvable. A related question is whether such (0,2) theories are in fact consistent or whether they are destabilized by instanton corrections [13]. This problem was addressed in [14], where criteria for stabibility were formulated in the context of the linear σ−model.
In [10] we solved this problem by establishing (0,2) triality for a simple (0,2) σ−model whose data can be summarized by its stable bundle structure V (1,1,1,1,1;5) → IP (1,1,1,1,2,2) [ 4 4] (1.1) over a codimension two Calabi-Yau manifold. It was shown there that the underlying conformal field theory of this σ−model is described by the exactly solvable theory derived from the Gepner model of five minimal factors at level three, the (2,2) 'quintic' model, in combination with a supersymmetry breaking simple current. This example thus not only gives an independent proof for the existence of conformal fixed points for this type of models but also provides its precise exactly solvable structure. The results of [10] indicate that for this class of (0,2) models the exactly solvable theory directly determines the bundle structure while the structure of the base space is less apparent. In Sections 2-7 we generalize the considerations of [10] to several classes of models. It will become clear that the application of certain types of simple currents amounts to defining maps which result in (0,2) daughter theories for particular types of (2,2) vacua. These maps can be abstracted from the exactly solvable framework and applied to more general Landau-Ginzburg compactifications. Proceeding in this way we construct the first large class of (0,2) Landau-Ginzburg theories.
Armed with (0,2) triality we turn to the discussion of mirror symmetry. In the context of (2,2) Calabi-Yau compactification mirror symmetry usually is formulated as an isomorphism between conformal field theories which, in the large radius limit, leads to In (0,2) compactifications the gauge connection is not embedded in the spin connection, hence the gauge bundle is not identified with the tangent bundle of the manifold.
We can therefore ask whether for every consistent (0,2) vacuum described by a stable As in the (2,2) framework the implication here is that the underlying conformal field theories are isomorphic. If this is the case we will call the two bundles (V −→ M, V −→ M ) a (0,2) mirror pair.
In the exactly solvable context mirror symmetry can be seen trivially in terms of the order-disorder duality in each of the N = 2 minimal factors, leading to the usual reversal of the charges. The implementation of this operation does not act on the string vacuum per se and therefore does not lead, in itself, to any insight. Since our construction of (0,2) models derives from (2,2) theories it is natural to ask whether the known mirror construction in the (2,2) context induce mirror pairs for the resulting (0,2) framework.
This is indeed the case. Our strategy to obtain mirror pairs of (0,2) models can thus be summarized independently of any particular framework via the diagram We review in Section 8 those of the known (2,2) mirror constructions which we generalize in Section 9 to the framework of (0,2) vacua. We will see that (0,2) theories derived from (2,2) mirror pairs induce (0,2) mirror pairs. In the last Section we discuss some open problems and directions for future work. In an appendix we collect the three generation models that result from our constructions.

(0,2) Triality
In [11] we proposed a Gepner like construction of string models featuring (0, 2) world sheet supersymmetry. This construction was based on the so-called simple current method to obtain heterotic modular invariant partition functions. We obtained a variety of models with different kinds of gauge groups and massless spectra.
(0,2) vacua have been constructed in a number of different contexts but what has been missing was the analog of (2,2) triality: the identification of the underlying exactly solvable models of (0,2) Calabi-Yau manifolds defined by stable bundles with a Landau-Ginzburg phase. In [10] we described such an identification for a (0,2) daughter of the 3 5 A 5 Gepner model based on the simple current 2 J = (0 5 1)(0 0 0) 4 (1)(0). (2.1) The (0,2) Calabi-Yau theory is defined by the bundle SO(8) , where (l i m i s i ) N =2 are the quantum numbers of the i th N = 2 minimal superconformal factor. constructed from the exact sequence over the threefold configuration IP (1,1,1,1,2,2) [4 4]. Here the subscripts denote the number of generations and antigenerations. Not only did the massless spectra agree but also the 10 · 16 · 16 Yukawa couplings and the spacetime R charges. In [15] the moduli space of this model has been investigated in some detail.
In [10] we confined our analysis to stable bundles defined over Calabi-Yau manifolds which are not only quasismooth but smooth. This left open the problem of establishing (0,2) triality in a more systematic fashion. The first step in the present paper is to remedy this situation. In the process we will construct special classes of (0, 2) models with gauge group SO(10), SU (5) and E 3 , which will turn out to provide a good testing ground for a number of aspects of (0, 2) models, such as mirror symmetry. Before describing a particular class of models we analyze a quasismooth model in some detail.

A quasismooth (0, 2) model
Consider the (0,2) daughter based on the diagonal Gepner model 3 · 8 3 A 4 and the simple current as before The gauge group is SO(10) × SU (2) × U (1) 3 with N 16 = 113 generations and N 16 = 5 antigenerations. More details of the massless spectrum of this model are summarized in . (2.5) These bundle data satisfy c 1 (V (1,1,1,2,5;10) ) = 0 as well as the anomaly cancellation condition c 2 (V (1,1,1,2,5;10) ) = c 2 (T ) and by following [16] one finds that the linear σ−model leads to a well behaved Landau-Ginzburg phase for r ≪ 0. For r ≫ 0 one still obtains a compact parameter space but the threefold has both an orbifold and a hypersurface singularity at (0, 0, 0, 0, 0, 1). An interesting question is, whether one can generalize the algorithm for resolving (0, 2) orbifold singularities presented in [17] in such a way as to obtain agreement with the spectrum in the Landau-Ginzburg phase. The spectrum of the latter phase can be obtained by the methods in [18,19]. In Table 2 we collect the left and right U (1) charges of the chiral fields and Fermi fields in the linear σ−model.  Furthermore we need the ground state energies and charges of the twisted sectors of the Landau-Ginzburg orbifold. These are contained in Table 3.
The left handed 16s in the untwisted sector are given by polynomials of degree ten modulo This leaves exactly 113 generations. Right handed 10s are given by degree twenty polynomials modulo  Table 4.  For the choice of the W j and F a above we get n g = 6 and thus the desired N 1 = 561 massless singlets.
We conclude that the above SCFT based on the 3 · 8 3 A 4 Gepner parent describes a certain point in the LG phase of the linear σ−model with the data (2.5). Inspection of the untwisted 16s reveals that the introduction of the simple current in the partition function corresponds to the following modification of the geometric data. First, choose the bundle data of the would-be (0, 2) model to be the Calabi-Yau data of the (2, 2) model corresponding to the parent Gepner model. In the case at hand the 3 · 8 3 A 4 Gepner model lives on the CY manifold IP (1,1,1,2,5) [10], where the K = 3 tensor factor corresponds to the coordinate of weight k = 2 in the ambient space and we have introduced a trivial factor.
The CY base manifold of the (0, 2) model then is obtained by replacing the coordinate corresponding to the K = 3 tensor factor by two fields whose weight is twice that of the original coordinate. Finally, the two hypersurfaces have equal weight. This operation which defines a (0,2) theory in terms of the data of certain (2,2) theories can be generalized to the class of Gepner models and, more generally, to all Landau-Ginzburg theories with an appropriate scaling field.

Odd Theories
The application of the supersymmetry breaking simple current to the exactly solvable model can be encoded directly on the Landau-Ginzburg theory by a generalization of the above operation. We will abbreviate our construction by calling it a 'move'. For the class of Gepner models the prescription is as follows. Choose a Gepner model which contains at least one factor with odd K = 2ℓ − 1. Let us assume this factor to be the first one. Let d be the lowest common multiple of the numbers {K 1 +2, K 2 +2, K 3 +2, K 4 +2, K 5 +2}. For models with only four factors set K 5 = 0.
Then the analysis of the chiral ring reveals that a model obtained by using the following simple currents in the diagonal Gepner parent model corresponds to a linear σ−model with the following data Similar to the (2, 2) case the class of all Gepner models is not mirror symmetric [20,21].
To get a mirror symmetric plot one had to consider all phase orbifolds including those with discrete torsion [22].
Streamlining the notation, the SCFT analysis has lead us to consider Calabi-Yau hypersurface configurations IP (k 1 ,k 2 ,k 3 ,k 4 ,k 5 ) [d] in weighted projective fourspace with degree d = i k i , such that at least one of the weights k i divides the degree as d/k i = (2ℓ+1).
We will characterize such coordinates and the configurations which contain them, as being 'odd'. For any odd configuration IP (k 1 ,k 2 ,k 3 ,k 4 ,k 5 ) [d], for which we can arrange the odd coordinate to be z 1 , we define a (0,2) daughter theory by considering the following vector bundle configuration where V (k 1 ,k 2 ,k 3 ,k 4 ,k 5 ;d) describes a stable bundle over the Calabi-Yau threefold with weights k i describing the charges of the gauge fermions defined as sections of this bundle. It can be checked that the anomalies of these (0,2) models cancel.

Pairing of generations and antigenerations
The identification of the model (2.5) was fairly straightforward. The following example shows that more involved things may happen. Consider the Gepner parent model The spectrum of the superconformal field theory is (N 16 , N 16 ) = (119, 11). The Landau-Ginzburg computation reveals that in the l = 18 twisted sector something new appears.
The ground state energy is E = 0 with charges (q l , q r ) = (− 1 9 , − 11 18 ). The zero modes are In the q l = 1 sector we obtain the following sequence with right moving charges Now, choosing the BRST operator relevant in this sector as one gets in this sector exactly one massless field with (q l , q r ) = (1, − 1 2 ). Taking also all the other sectors into account one ends up with the spectrum (N 16 , N 16 ) = (118, 10), which is not expected from the SCFT. However, analogously to the model discussed at the beginning, if we choose the BRST operator to be is not any longer in the image of Q + and the overall massless spectrum becomes (N 16 , N 16 ) = (119, 11). Thus, this example reveals for the first time an effect for (0, 2) models which was long believed to exist. The number of generations and antigenerations is not necessarily constant over the moduli space. This is not an exception, for approximately 13% of all SCFT models have higher numbers (up to 3) of generations than those models obtained by minimal choices of the constraints in the LG models. The number of net generations is constant however.
We can obtain a superconformal field theory in the moduli space of this model by first implementing the simple current which generates the orbifolds and then break the (2,2) supersymmetry as in the previous models. This leads to the modification of the partition Indeed, this SCFT has (N 16 , N 16 ) = (42, 2). This model will be reconsidered in the discussion about mirror symmetry.
Another example is provided by the 1 · 5 · 82 2 A 4 parent model describing a point in the moduli space of the threefold configuration IP (1,1,12,28,42)  .
The superconformal theory of this model is given by the above Gepner parent modified by the simple currents There exist many other examples. Thus, we have found a large class of (0, 2) models featuring the desired CFT/LG/CY triality of description. Further work has to be done to gain a better understanding of the Calabi-Yau phase when the resulting models have hypersurface singularities.

SCFTs with gauge group SU(5) and E 3
In [10] we have generalized our SO(10) analysis to models with gauge group SU (5) by considering the (0,2) theory derived from the 3 5 A 5 tensor model via the simple currents The (0, 2) LG theory associated to this exactly solvable model is defined by the following data .
Here we introduce a new constraint F 1 of weight k = 5 which resolves the nontransversality of the coordinate of weight k = 5, implying that the quadratic term λ 1 φ 8 in the superpotential gives no contributions to the cohomology. This constraint furthermore guarantees that the central charge of the internal CFT sector is (c l , c r ) = (11,9), as needed for a rank 5 gauge bundle. The generalization of the move to these models is obvious, one simply applies it twice on each of the coordinates separately.

(5.3)
Another example is provided by the 1 2 · 3 · 7 · 43 A 5 parent with the simple currents , in which the move has been applied three times. Since one needs three odd levels in a Gepner model to get such an E 3 model, there do not exist so many of them, in comparison.

(0,2) Landau Ginzburg Theories
In this Section we abstract the move map formulated above from its exactly solvable origin and generalize it to the class to (2,2) LG models. A list of all possible weight combinations which allow the choice of a superpotential with an isolated singularity, defining quasismooth projective varieties, was presented in [23]. Given one (2,2) LG theory it is in general possible to define a number of different (0,2) models with different gauge groups, depending on the number of implemented moves. We discuss these possibilities in turn.

The SO(10) Move
The simplest case generalizes the case in which there is only one supersymmetry breaking simple current. The resulting map can be formulated independent of any exactly solvable considerations in the LG framework as follows: Move: Suppose one has a (2, 2) LG model, denoted as for which one of the weights, e. g. k 1 , satisfies d k 1 = (2ℓ+1) with ℓ = 1, 2, . . .. Then the choice of the data Whether such a model really does yield a bona fide (0, 2) model with gauge group SO (10) is a more subtle question, since depending on the weights and the constraints effects like nontransversality of the (0, 2) LG model or extra massless gauginos in twisted sectors may occur. The latter has been pointed out in [19] where it was considered as the LG analog of the destabilization of the vacuum by world sheet instantons in the CY phase. We will come back to this point below. Note that a move with ℓ = 1 does not change the model, one simply gets the (2, 2) parent. The same can be observed in the SCFT, where the simple current (2.9) in a K = 1 tensor factor does not change the Gepner spectrum at all.
Concerning the transversality of the (0, 2) LG superpotential, one can make the following general statement. If the (2, 2) superpotential can be chosen as then the choice for the (0, 2) constraints clearly is transverse. Here the coordinates y 1 , y 2 are the new ones of weight κ 1 = 2k 1 and then the following choice is transverse. It may happen that Q does not depend on x 2 . In this case one can either choose another polynomial of the right degree or, if this is not possible, set F 2 = 0. One still has (max+1) nontrivial constraints for (max+1) variables and thus the (0, 2) LG phase makes perfect sense. Most, but not all, such rather rare examples are inconsistent due to the appearance of extra gauginos, however 3 .
In order to calculate the massless spectrum in the Landau-Ginzburg phase in a systematic manner an algorithm is needed which can be translated into a computer program.
The contribution from the individual twisted sectors to the cohomology however depends on the precise form of the constraints W j and F a defining the stable bundle There are 26 such examples among 8027 models, 5 of which are consistent of rank r over a complete intersection space of codimension N c . More useful in this context is the elliptic genus whose contribution in the α th twisted sector is given by its virtue being that it is easily computed. Using the notation of [24] the χ y genus of a bundle of rank r can be written as where | * indicates the evaluation of the q 0 y n , n ∈ {0, 1, . . . , r} terms only and Here the charges of the fields are given by q i = k i /m, ν a = 1 − n a /m and ν r+1+j = d j /m.
Implementing this in a C-code yields for the example discussed in detail in Section 2.1 the result shown in Table 5.  Using l = 2α one recognizes this as exactly the massless R-R spectrum obtained in Section 2.1. For models which do not lead to a pairing of generations and antigenerations and for which there are no massless gauginos coming from twisted sectors one can read off the number of generations and antigenerations directly from the χ y genus.
Unfortunately, the χ y genus is only sensible for the right moving U (1) charge modulo two and consequently can not detect such gauge symmetry enhancement. What is needed is a convenient condition for the consistency of a (0, 2) model. One necessary condition is not hard to formulate. To this end, we also have to take explicitly into account the right moving U (1) charge. Generalizing the χ y genus, the generating function for all states in the LG model before calculating the cohomology is with the abbreviation ρ a = 1 − ν a . (6.11) For simplicity let us choose a bundle with rank 4 in the following. Since the BRST operator Q + has (q l , q r ) = (0, 1), one obtains sequences of the form The BRST operator depends explicitly on the constraints However, if a sequence (6.12) in a sector contains gauginos (q l , q r = ± 3 2 ), for q l = 0, ±1, such that a q r > a q r +1 + a q r −1 then these gauginos are inevitable, independently of the detailed form of the constraints. If this inequality is not satisfied then it appears that one has to check the various possible choices in detail.
In Figure 2 we exhibit all (0, 2) models obtained by applying the move on all odd (2,2) LG models in the table of [23] 4 . We have checked all model for 'inevitable' gauginos.
We have further analyzed a number of models with 'noninevitable' gauginos and have found no examples where the gauginos could not be avoided by appropriate choices for the constraints.  N 16 ). However, we are not claiming that these models really are mirror pairs for the matching of (N 16 , N 16 ) alone can also be a coincidence. Even though only 1759 models are not paired, the plot contains 845 unpaired points, which are 4 The list of (0,2) models leading to Figure 2 (and also those with smaller gauge groups, cf. 48%. As we will discuss in the last Section such an asymmetry is to be expected for our class of models for a number of reasons.

SU (5) and E 3
The construction described above can be generalized to smaller gauge groups SU (5) and E 3 = SU (3) × SU (2) by simply applying the above move iteratively. We obtain 1216 models and 499 distinct spectra with gauge group SU (5) which are shown in Figure 3.
For E 3 there are 81 models and 48 distinct spectra. In this case one needs two or three odd coordinates which have to be preserved by the (2, 2) mirror transformation, hence it is clear that we obtain far fewer mirror pairs than for the SO(10) case. Therefore, in the following discussion we will focus on the latter case. We finally list the 23 SO(10) and 4 SU (5) three generation models in Table A.1 of the appendix.

Codim>1 weighted (0,2) CICYs
The class of Landau-Ginzburg vacua constructed in [1,23] contains potentials with more than five scaling variables and many of these theories have an interpretation as weighted complete intersection Calabi-Yau manifolds described by more than one polynomial. The codimension of these more general weighted CICYs is obtained from the LG theory by the total charge Q = i q i , where the q i are the weights of the variables, normalized such that the weight of the potential is unity. This total charge precisely determines the codimension of the Calabi-Yau manifold (if it exists) [20].

(2,2) Mirror Symmetry
Our (0,2) mirror construction is based on mirror symmetry in the context of (2,2) theories. In the present Section we therefore briefly review the known mirror constructions before we proceed to generalize them to the (0,2) framework in the following Section.
There have been roughly four different types of constructions of mirror pairs of (2,2) symmetric string vacua. The most precise, if most narrow, framework is provided by considering orbifolds of exactly solvable theories [2]. More general, if less precise, are the Landau-Ginzburg mirror constructions of [28] and [29]. Finally, there exists a toric formulation [30][31][32] in terms of reflexive polyhedra. The focus in the present paper will be on the first three constructions.
Our first task will be to generalize the orbifolding procedure of the exactly solvable models. As in the (2,2) case we will see that this provides the most precise control over the structure of the mirror. In order to extend the orbifolding analysis of the exactly solvable framework to the more general class of Landau-Ginzburg theories we need some way to extract the weights of the fermions defining the stable bundle for the purported (0,2) mirror Landau-Ginzburg theory. There are two tools available which readily allow us to determine the structure of these gauge bundles. The first is the fractional transform [28] which maps orbifolds with respect to certain discrete group actions into weighted configurations. The second is the transposition construction of Berglund and Hübsch [29] which in particularly simple situations provides the weighted mirror as well.

Orbifold Mirrors of Gepner Models
Mirror symmetry in the exactly solvable context has been established for the class of Gepner models [33], described by tensor products ⊗ r i=1 K i of N = 2 superconformal minimal models. The complete set of these models, which has been constructed in [20,21] contains only very few mirror pairs. It was observed in [2], however, that orbifolding individual models with respect to the maximal discrete symmetry group produces the mirror theory.
An alternative way to obtain the exact mirror is by modifying the partition function with appropriate simple currents. The mirror of the quintic model The direct generalization of the simple currents in (8.3) to this case leads to a model with spectrum (99, 3) instead of (145, 1). We have not found a way to implement this Z Z 2 10 orbifold with only two Z Z 10 simple currents which are local with respect to the GSO projection. If one does not insist on locality, then the following simple currents yield the mirror spectrum. In this case one has to use two GSO projections.
The mirror of this model can be obtained by orbifolding with respect to the group action where Φ i (z, θ,θ) = φ i (z) + · · · are chiral N = 2 superfields. Associated to these superpotentials are affine varieties defined as the zero locus of the polynomials obtained by considering constant lowest components φ i = z i of the superfields.
It was shown in [28] that the mirror symmetry among Landau-Ginzburg theories observed in [1] can be understood by mapping orbifold theories to weighted configurations via the iterated application of the isomorphism given by induced by the fractional transformations 5 Here g ab is the greatest common divisor of a and b and h ab is the greatest common divisor The fractional transform is not restricted to the Fermat type Landau-Ginzburg theories of exactly solvable minimal models however. As an illustration consider the configuration IP (1,1,1,1,3) [7] (2,122) ∋ {z 7 1 + z 7 2 + z 7 3 + z 7 4 + z 4 z 2 5 = 0}. (8.19) After orbifolding by Z Z 2 7 and applying the fractional transform we find the mirror configuration IP (31,35,36,42,108)  The relation (8.12) shows that in general we expect some additional modding to be necessary on the fractional transform of the original orbifold in order to find the isomorphic representation. Even though the necessary modding of the FT image becomes part of the projective equivalence of the image in the majority of mirror theories it does happen that the orbifolding has to be performed. An example in case is the mirror orbifold (8.7) whose fractional transform leads to the weighted configuration IP (3,4,9,9,11) [36]. This configuration has the cohomology (h (1,1) , h (2,1) ) = (20,32). We see that for the resulting weighted space the Z Z 2 action on the weighted image is not trivial and must by modded out. Doing so results in the mirror spectrum, as it must. Thus we find for the mirror the relation IP (2,4,9,9,12) [36] (19,43)  ∼ IP (3,4,9,9,11) [36] (20,32) Z Z 2 : [1 0 0 0 1] (43, 19) . (8.21)

The Berglund-Hübsch Construction
For manifolds defined by polynomials which are not of Fermat type there exists an alternative construction which sometimes allows to determine the weighted representation of the mirror configuration. This method [29] is based on the 'transposition of polynomials'. In general the transpose of a manifold does not suffice to construct the mirror of a hypersurface and must be accompanied by an additional orbifolding. It does happen for certain configurations, however, that this additional modding is not necessary in which case transposition itself already generates the mirror weights. In such circumstances we are not restricted to the use of fractional transformations in order to generate the weights of the mirror stable bundle but instead can use the transposition prescription or a combination thereof.
A simple class of models is described by the n-tadpole polynomials [1] p = z a 1 1 z 2 + z a 2 2 z 3 + · · · + z a n−1 n−1 z n + z a n n , (8.22) whose degree matrix can be written as in which the rows indicate the degrees for each of the monomials. The transposed polynomial [29] is defined by the transpose of its degree matrix 6 0 0 0 0 · · · a n−1 0 0 0 0 0 · · · 1 a n Since the dimension of the manifold is to be unchanged under transposition it is apparent that only those polynomials are amenable to transposition for which the number of monomials is equal to the number of variables, excluding the exceptional types discussed in [1].
As mentioned previously, the variety defined by the transposed polynomial is not, in general, the mirror of the original one defined by the polynomial p. Again the modding of an action is the missing ingredient. The orbifolding which produces mirror pairs for pairs of transposed manifolds is determined by splitting off the cyclic group Z Z d whose order is the degree of the polynomial. The symmetry group D p = Z Z a 1 ···a n thus can be decomposed into the quantum phase group [35] Z Z d and the geometric phase group G p as D p = Z Z d × G p 6 The resulting transformation does not have constant Jacobian. and Z Z d t ×G p t . The prescription for determining the additional orbifolding is that by going from a manifold to its mirror the role of the quantum and the geometric phase groups are exchanged [36]. Toric descriptions of this construction can be found in [31,32].

(0,2) Mirror Symmetry
In the class of (2, 2) vacua a mirror symmetric pair is given by two linear σ−models or two toric varieties, respectively, which lead to isomorphic superconformal fixed points, differing only by the relative sign of the U (1) charges. In the Calabi-Yau phase this leads to an interchange of complex and Kähler deformations. In the (0, 2) setting the first definition still makes sense, except that in the (0,2) Calabi-Yau phase we are lead to more general cohomology groups, such as H We now apply the (2,2) constructions reviewed in the previous Section in order to construct (0,2) mirror pairs. We will first consider the orbifolding procedure of the exactly solvable models and then determine the weighted (0,2) Landau-Ginzburg mirrors by applying fractional transformations and/or transposition in order to obtain the needed weights of the mirror gauge bundle.

(0, 2) Orbifold Mirrors
As in the (2,2) case we can orbifold the exactly solvable tensor model with respect to the cyclic groups Z Z K i +2 in each of the minimal factors to generate the exact (2,2) mirror. By applying the supersymmetry breaking simple current (2.9) to this (2,2) mirror we obtain the (0,2) mirror theory. It is clear from this fact that the space of (0,2) theories which we consider is trivially mirror symmetric by virtue of the order-disorder duality of the individual minimal factors [37].
This observation, however, is not particularly useful because this orbifolding is not an operation on the conformal field theory describing the target space. In order to define the modding procedure on the string ground state proper we need to restrict the symmetry actions to leave invariant the holomorphic threeform of the base space. It can be shown that it is always possible to generate the exact mirror of Gepner models with actions of this type and therefore we arrive at the exact (0,2) mirror after implementing the supersymmetry breaking simple current.

(9.3)
Thus we derive the mirror theory by applying the move to the mirror of the (2,2) theory.

(0, 2) LG Mirrors via Fractional Transformations
To generalize mirror symmetry to the (0,2) Landau-Ginzburg framework we proceed as follows. Given a (0,2) LG model we first apply the move in order to obtain the (2,2) Calabi-Yau manifold. For this CY space we construct the weighted mirror configuration.
We then apply the move in reverse order to get the (0,2) mirror. We thus realize our general strategy described in the introduction in the present context via (9.7) The first obstacle thus is that the weighted representation of the (2,2) mirror of an odd configuration must itself be odd.
Returning yet again to our example from Section 2 based on 3 · 8 3  with the correct mirror spectrum.
A second possible obstacle is that in general it can happen that one hypersurface gives rise to more than one (0,2) model and hence to more than one mirror relation. In such a situation it is necessary to be able to use different representations of the mirror by mapping different representations of the mirror orbifold via fractional transformations. As an example consider the manifold IP (1,2,6,18,27) [54] which gives rise to the two (0,2) models ℓ A third possible obstacle is that a weighted representation of the (2,2) mirror must be found such that the resulting (0,2) theory is not inconsistent because of gauginos. Consider the (0,2) image of the move with ℓ = 3 of the (2, 2) mirror pair in (8.20). This leads to The latter model is not consistent because there inevitably are gauginos in the untwisted sector. In the next Section we will see that the transposition of bundles provides for a (0,2) mirror configuration.
The last example to discuss in this Section is the (0, 2) image of the mirror pairs in (8.28). In contrast to (9.17), in this case the (0, 2) mirror of In general then the problem of constructing (0,2) Landau-Ginzburg mirror pairs splits into two parts. Given an odd configuration we first need to check whether it is possible to find complete intersection representation of the mirror orbifold. Both the fractional transform and the transposition operation in general need an additional orbifolding on the image theory and and it is a priori not clear whether the necessary orbifolding will be trivial by becoming part of the projective equivalence of the purported weighted mirror configuration. Second, even if one succeeds in finding a hypersurface fractional transform of the orbifold, in order for our construction to work, one still has to make certain that this complete intersection is odd in the way defined above.
It can be checked that for many Landau-Ginzburg theories both of these problems can be solved via the fractional transform, transposition or a combination thereof.

Discussion and open problems
By restricting ourselves to (0, 2) models which have a LG description and originate from (2, 2) models admitting a LG phase we should not expect to get a completely mirror symmetric class of theories. There are a number of reasons for the resulting asymmetry.
Firstly, our focus has been on odd configurations, thereby restricting our considerations to a subclass of Landau-Ginzburg theories. More importantly, however, the class of all (2,2) LG models itself is not mirror symmetric. The missing mirror partners have been shown [32] to be more general hypersurfaces in toric varieties which can formally be written as nontransversal hypersurfaces in weighted projective spaces. Consider, for instance, the manifold IP (5,8,12,15,35) [75] (30,27) contained in the list of [23]. This space has no mirror partner in the class of Landau-Ginzburg theories. It can be described in terms of a reflexive polyhedron as a hypersurface in a toric variety and therefore Batyrev's construction [30] shows that the mirror can be described by its dual polyhedron. It is therefore clear that the weights associated to the dual polyhedron cannot admit a transverse polynomial and must lead to a weighted manifold with hypersurface singularities [32]. It can be shown [29] by combining transposition with fractional transformations that the weighted configuration is given by IP (2,4,5,5,9) [25]. All manifolds in this deformation class have a hypersurface singularity at the point (0, 0, 0, 0, 1). Applying the move yields the possible (0, 2) mirror pair V (5,8,12,15,35;75) → IP (5,8,12,30,30,35) [60 60] (18,30) V (2,4,5,5,9;25) → IP (2,4,5,9,10,10) [ 20 20]. (10.1) The coordinate of weight k = 9 can not appear in the constraints in a transversal way hence this model does not have a (0, 2) LG phase and further analysis has to await a more general description of (0,2) models.

(10.2)
Due to the new constraint of weight d = 10 the coordinate of weight k = 5 can now appear in a transversal way leading to a (0, 2) model which has a bona fide LG phase.
This example indicates that even in the weighted framework the (0,2) class is much richer than the more restricted (2,2) vacua.
A further phenomenon specific to (0,2) models is that, as we have discussed, a (2,2) LG model can lead to a (0, 2) model which is destabilized.
Even though in dealing with (0, 2) models one has to be very careful and prepared for surprises, our results show that the status of (0, 2) models is on a par with (2,2) compactifications. As in the latter class we have established mirror symmetry and shown that it can be understood by extending known (2, 2) mirror constructions to the (0, 2) case.
Thus mirror symmetry does in fact generalize to (0,2) theories.