Testing the (0,2) mirror map

We test a proposed mirror map at the level of correlators for linear models describing the (0,2) moduli space of superconformal field theories with a (2,2) locus associated to Calabi-Yau hypersurfaces in toric varieties. We verify in non-trivial examples that the correlators are exchanged by the mirror map and we derive a correspondence between the observables of the A/2- and B/2-twisted theories. We also comment on the global structure of the (0,2) moduli space and present a simple non-renormalization argument for a large class of B/2 model subfamilies.

A particularly prominent role in mirror symmetry investigations has been played by the gauged linear sigma model (GLSM) [4]. This is a two-dimensional supersymmetric gauge theory which, for a suitable choice of its parameters, flows in the IR to a non-linear sigma model (NLSM) with target space a complete intersection Calabi-Yau (CICY) in a toric variety.
Some of the parameters of the IR conformal field theory have natural representatives in the UV linear model. These are encoded in two types of superpotentials: a chiral superpotential encodes the complex structure parameters and a twisted-chiral superpotential encodes the Kähler parameters. These generically describe only a subset of the full space of deformations of the CFT, as they do not include non-polynomial complex structure deformations -turning these on would prevent the CICY to be embedded in the toric variety -or non-toric Kähler deformations -these are associated to divisors on the ambient toric variety that become reducible when restricted to the CICY -.
A remarkable property of (2,2) mirror symmetry is that it respects this toric structure on both sides of the mirror, that is, polynomial complex structure deformations are mapped to toric Kähler deformations of the mirror and vice versa. This is encoded in the monomialdivisor mirror map (MDMM) [5].
(2,2) SCFTs admit deformations which preserve only (0,2) supersymmetry. In a large radius phase these are interpreted as deformations of the tangent bundle over the CICY. For a subset of these there exist corresponding holomorphic deformations of the linear model Lagrangian, and we refer to these as monad deformations. These, together with polynomial complex structure and Kähler deformations, form the toric moduli space of the SCFT. 1 It therefore comes naturally to ask whether mirror symmetry respects the toric structure of this enlarged (0,2) moduli space, that is, if monad deformations of the tangent bundle are exchanged by the mirror map. This turns out not to be the case. In fact, a careful analysis of the GLSM parameters modulo field redefinitions shows that in general even the dimensions of the toric moduli space of the theory and its mirror do not match [8].
However, there exist subfamilies of such theories which appear to be exchanged by mirror symmetry. A proposal for an extension of the MDMM to a class of (0,2) deformations of (2,2) linear models for CY hypersurfaces was presented in [9]. When this sublocus coincides with the full parameter space of the theory, the model is referred to as reflexively plain [8]. It has been shown that the map satisfies non-trivial tests. In fact, the map exchanges the dimensions of the (subloci of the) moduli spaces of the mirror theories, as well as it exchanges the principal component of the singular locus of the A/2-twisted theory with the principal component of the singular locus of the B/2-twisted mirror theory.
In this work, we will further test the mirror map, and we will show that it leads to an isomorphism at the level of the correlators. That is, the map exchanges correlation functions in the A/2-twisted theory [10] with correlation functions in the B/2-twisted mirror theory. As a consequence, we derive a correspondence between the natural observables of the twisted theories on both sides of the mirror. Unfortunately, we still lack the technical tools to carry out such analysis for a generic model. While recently there has been progress on the A model [11,12] and A/2 model [13] side, techniques to evaluate B model correlators mostly rely on the existence of limiting points where the theory can be solved. For this reason, we focus on theories which admit a Landau-Ginzburg orbifold (LGO) phase. As it has been argued that the corresponding B/2 model does not receive instanton corrections [10], the relevant B/2-twisted correlators are therefore independent of the phase structure of the theory, and we can make use of the LGO description to solve the model. As a consequence, we will be able to argue that the B/2 model of any theory to which the mirror map applies does not receive contributions form worldsheet instantons, regardless of the existence of a LGO phase.
The rest of this paper is organized as follows. In section 2, we review just enough basic notions of toric geometry and of linear models in order to present the (0,2) mirror map of [9]. In section 3, we solve a reflexively plain example, while in section 4 we tackle a more general model, where the mirror map acts on appropriate subfamilies of the moduli space. We conclude with some comments on the structure of the (0,2) moduli space as well as some open questions.

The (0,2) mirror symmetry set-up
In this section we review the GLSM construction of mirror pairs for Calabi-Yau hypersurfaces in toric varieties, introducing along the way some relevant toric geometry notions. As this material is quite standard, our discussion will not try to be exhaustive, and for more details the reader can for instance refer to [14].

Polytopes and hypersurfaces
We start with a d-dimensional lattice polytope ∆ ∈ M R ≃ R d , and we define the dual polytope where N R ≃ (M R ) ∨ and the pairing product ·, · : M R × N R → R descends from the standard product in R d . A lattice polytope ∆ is said to be reflexive if and only if its dual ∆ • is also a lattice polytope, and it follows that both ∆ and ∆ • have a unique interior point which we assume, without loss of generality, to be the origin. Let ∆ be a reflexive polytope. Then ∆ can be interpreted as the Newton polytope for a hypersurface {P = 0} ∈ V , where V is the toric variety whose fan Σ V ∈ N R is obtained by taking cones over the faces of ∆ • . To make this more explicit, let ρ 1 , . . . , ρ n be the non-zero points in ∆ • , which correspond to integral generators of the one-dimensional cones of ∆ • . 2 For each of these we introduce a coordinate X ρ ∈ C n , and we present the toric variety V as the holomorphic quotient where G = (C * ) r × H, r = n − d and H is a finite Abelian group. Z(F ) ∈ C n is the subvariety associated to the irrelevant ideal (or Cox ideal) in the homogeneous coordinate ring C[X ρ 1 , . . . , X ρn ], and in particular it depends on the specific triangulation Σ V . The (C * ) r action on the coordinates X ρ is determined in terms of a basis for the kernel of the pairing matrix 3 m, ρ between the non-zero elements of ∆ and ∆ • . The pairing matrix has rank d by construction, and the quotient action is given by where the charges Q a ρ span an integral basis for the kernel of m, ρ . The polynomial defining the CY hypersurface is given in terms of the homogeneous coordinates by (2.4) In particular, under the action (2.3) each monomial in P transforms according to and therefore (2.4) transforms with charge ρ Q a ρ and the Calabi- The fact that ∆ is reflexive is equivalent to the fact that ∆ • is reflexive as well. Therefore, applying the procedure we summarized above while exchanging the roles of ∆ and ∆ • , we obtain a Calabi-Yau hypersurface M • = {P • = 0} ⊂ V • . In particular, let m label the non-zero points in ∆ and n • = |m|, and let us introduce a homogeneous coordinate Y m ∈ C n • for each of these. Then we have . The (C * ) r • action on these is given by where Q a m is an integral basis for the kernel of the pairing matrix ρ, m for non-zero ρ ∈ ∆ • and m ∈ ∆, which is simply the transpose of the pairing matrix m, ρ above. Thus, Q a m parametrize an integral basis for the cokernel of m, ρ . Similarly, one can show that transforms under (2.7) with charge m Q a m , and the Calabi- In the context of a heterotic string theory background, a (0,2) NLSM is constructed by specifying a target space M , which we assume to be of the form described above, together with a holomorphic vector bundle E → M . When E = T M the theory possesses (2,2) supersymmetry and the pairs T M → M and T M • → M • form a Batyrev mirror pair [15] and constitute the starting point for our analysis. These theories admit deformations that preserve only (0,2) supersymmetry. In the geometric phase, these correspond to a bundle E obtained as a deformation of the tangent bundle. As mentioned above, for a given UV GLSM realization of the low-energy SCFT, the associated (0,2) linear model moduli space includes, in addition to the toric Kähler and polynomial complex structure deformations, a subset of the bundle deformations. This is the moduli space to which the (0,2) mirror map applies and to which we turn in the next section.

The (0,2) linear model
We now turn to the physical theory of interest, namely the GLSM. This is a two dimensional supersymmetric gauge theory, whose gauge group we assume to be Abelian in this work. For a suitable choice of values for the UV parameters, the theory flows in the IR to a NLSM describing the geometric structure we introduced above.
Such theory is constructed with n+1 (0,2) bosonic chiral superfields X 0 , X ρ and the same number of (0,2) Fermi multiplets Γ 0 , Γ ρ . These have gauge charges Q a 0 ≡ − ρ Q a ρ , Q a ρ and they are coupled to a collection of r gauge-neutral chiral supermultiplets Σ a . The chirality conditions for the Fermi fields read where E 0 (X), E ρ (X) are polynomials in the superfields X ρ , and D is a gauge covariant (super)derivative. 4 The action for the theory is determined by the kinetic terms for the gauge fields V a , for the Σ a fields and for the various matter fields, as well as by superpotential interactions where Υ a are the twisted-chiral gauge-invariant field strengths and τ a = ir a + θ a /2π are the complexified F.I. parameters. The parameters q a = e 2πiτ a correspond to the algebraic coordinates on the complexified Kähler moduli space of V [17,18], while P (X) is determined by (2.4). The remaining functions J ρ (X) are generic polynomials in the fields X ρ ′ with gauge charges −Q a 0 − Q a ρ . In particular, J ρ (X) are allowed to contain all the monomials that are allowed to appear in ∂ ρ P , that is (2.11) Note that in this expression ρ ′ = 0 while m sums over all points in ∆, including the origin. The condition that the RHS of (2.11) is divisible by X ρ implies that j mρ = 0 whenever m, ρ = −1. The (2,2) locus is described by j mρ = α m ( m, ρ + 1). The map specified in [9] is restricted to a subset of the GLSM moduli space. This is realized by constraining the polynomials E(X) in (2.9). Following the same notation of [9], we restrict our attention to the following form of the chirality conditions for the left-moving Fermi fields where Σ is a vector with components the fields Σ a , while δ and e ρ are vectors of parameters of dimension r. The GLSM action is (0,2) supersymmetric if and only if which using (2.11) can be recasted as which in turn implies the relations If we assume a triangulation Σ V corresponding to a large radius phase of the linear model, the space of classical vacua is the Calabi-Yau hypersurface M = {X ∈ V |P (X) = 0} ⊂ V , where by a slight abuse of notation we denote by X ρ the lowest components of the corresponding bosonic superfields. The left-moving fermions of the theory, which appear as the lowest components in the Fermi supermultiplets Γ ρ , couple to the holomorphic bundle E defined by the cohomology of the short sequence restricted to M . For the class of models we are considering, the holomorphic vector bundle E → M , which we assume nonsingular, describes a subset of deformations of the tangent bundle T M . Of course, the same construction obtained by exchanging the data as described in the previous section yields the mirror GLSM theory, which flows in the IR to a NLSM for the geometry E • → M • .

The (0,2) mirror map
Having reviewed the class of models which will be the subject of our study, we are ready to state the (0,2) mirror map proposal of [9]. Upon restricting ourselves to the subset of the parameter space identified by (2.12), it is possible to find a parametrization of the linear model which is invariant under the field redefinitions corresponding to the (C * ) d subgroup of Aut V , whose action simply rescales the coordinates of V . We denote these as "toric" field redefinitions. Of course, for a generic model the automorphism group of V is larger, and the description we review below is clearly an overparametrization of the parameter space. However, this redundancy is mirror symmetric [9] and, as in the (2,2) case, the (0,2) mirror map naturally extends to this redundant description. Hence, this fact does not lead to any difficulties in our task of computing the correlators in both theories. In fact, we will see that allowing for some additional redundancy will make the result look simpler.
The generalizations of invariant "complex structure" and "Kähler" coordinates, respectively, are given by (2.17) Notice that κ a coincide with the complex structure coordinates of the (2,2) theory, while on the (2,2) locus, κ a = q a . For the bundle data, a toric field redefinition invariant quantity is given by subject to the condition that b mρ = −1 whenever m, ρ = −1. Assuming that α m = 0, it follows from the (0,2) SUSY constraints (2.15) that where we defined The vectors δ and γ ρ , which therefore span the kernel of b mρ , are determined by (2.19) only up to a GL(r, C) transformation corresponding to the field redefinitions for the r Σ a multiplets.
In fact, the theory is singular both if rank b mρ > d, which corresponds to a vanishing γ ρ and a free Σ a multiplet, as well as if rank b mρ < d, in which case the B/2-twisted theory develops a singularity. Thus we restrict our attention to b mρ having exactly rank d. This means that dim ker b mρ = r and that the γ ρ are completely determined up to the GL(r, C) field redefinitions. The (0,2) mirror map can then be summarized as follows It has already been shown that the map (2.21) passes some significant tests. First, the dimensions of the moduli spaces of the theory and its mirror coincide. Second, the map correctly exchanges the principal components of the singular loci of the A/2-and B/2-twisted theories with those of the mirror. While these tests are certainly non-trivial, it is desirable to show that the local observables of the A/2 and B/2 models get exchanged under the map and more generally to show an equivalence at the level of the correlators. This is what we will present in the rest of the work.

A reflexively plain model
As recalled above, the mirror map is particularly suggestive for reflexively plain models, as the map exchanges the entire GLSM moduli spaces. We therefore begin with an example of such a class of models.
We start with the pair of reflexively plain polytopes ∆ : which, following the discussion in section 2, completely specify the model. In particular, we have that and we find it more manageable to study the A/2 model for M and the B/2 model for M • . We start with the former.

The M model
It turns out that ∆ • contains no other non-zero lattice point, and the associated fan determines the toric variety V = P 4 /Z 5 , where the action of the Z 5 quotient on the homogeneous coordinates of P 4 is given by where ζ = e 2πi 5 . The hypersurface P is presented in terms of a homogeneous polynomial of degree 5 invariant under the above symmetry This has 26 terms, corresponding to the lattice points of ∆. In particular, although the it is not hard to see that a generic hypersurface P will miss these. Thus, (3.3) acts freely and the corresponding CY hypersurface is smooth. The GLSM realizing this geometry is described in terms of the same field content as the usual one-parameter model for the quintic hypersurface in P 4 , with U(1) gauge charges and the E-parameters are simply proportional to their (2,2) form, in which case δ = −5 and e ρ = 1. The J-deformations are given by (without considering the E · J = 0 constraints) and we count 14 × 5 = 70 parameters, which agrees with the number in [8]. In (3.8) we have assumed a specific ordering of the lattice points of ∆, and we will stick to this choice throughout the rest of this section. Finally, according to (2.17), the invariant Kähler coordinate is given by The phase structure of this model is very simple. At r > 0 we recover the NLSM on E → M , where M = {P = 0} and E is specified by (2.16). At r < 0 instead the field X 0 assumes a non-zero vev and we find a Landau-Ginzburg phase where the orbifold is Z 5 × Z 5 .
Next, we need to determine γ ρ , which we recall are specified by the (0,2) supersymmetry constraints to span the kernel of the 25 × 5 matrix b mρ , which, following our discussion above, must have rank d = 4. Thus, for the purpose of determining γ ρ , we can consider any 5 × 5 minor of b mρ . We choose this submatrix to be determined by the elements associated to j ρρ in (3.8), that is (3.10) This matrix has rank at most 4 when the relation holds. The (0,2) supersymmetry constraints then read We can interpret four of these equations as determining γ ρ up to a rescaling, which can then be uniquely specified by the remaining GL(1, C) field redefinition. However, it turns out that the solution will look prettier if we allow for some redundancy. In fact, a solution to (3.12) is given by and we can make use of the GL(1, C) rescaling to set γ 4 = 1. As advertised, these parameters are not independent, but are related by (3.11). This is just a overly simple manifestation of the fact, mentioned above, that the mirror map assumes a more natural form in an overdetermined parameter space. For all practical purposes, we can carry on as if b 11 , . . . , b 55 were independent parameters and impose (3.11) only after actual calculations are performed. Finally, δ assumes the form where we used (3.11).

A/2 correlators
In order to solve the A/2-twisted M model we employ the strategy developed in [17] for (2,2) theories and later extended to a class of (0,2) linear models in [10]. Briefly, the idea is that correlators of the A/2-twisted M model can be extracted from the correlators of the A/2-twisted V model by applying the (0,2) version of the quantum restriction formula. These latter correlators, in turn, are much easier to compute as one can rely on the power of toric geometry techniques. Thus, we begin by solving the A/2-twisted V model. The A/2-twisted V model is completely determined by the chirality conditions (3.7), which we express in matrix form as where ρ, ρ ′ = 1, . . . , 5. The V model can be easily solved with Coulomb branch techniques [19]. The effective superpotential for σ -the lowest component in the multiplet Σ -on the Coulomb branch is given by The Coulomb branch vacua are solutions to J = 0, leading to the quantum cohomology relations  which we expressed in terms of the invariant coordinates. Thus, the generic V model correlator is given by It is natural to normalize the correlators such that where the extra factor of 5 −1 follows from the quotient by H = Z 5 [17]. Next, we turn to the A/2-twisted M model, and we make use of the quantum restriction formula, that for this example reads In particular, it follows from (3.17) that and therefore This formula passes two important checks. First, it reduces to the known expression on the (2,2) locus. Second, the correlators are singular where the denominator vanishes, and this reproduces the formula for the principal component of the discriminant locus of the A/2-twisted theory [9].
Finally, we substitute the expressions (3.13) and (3.14) for γ ρ and δ into our correlators and we obtain The natural observables of the A/2 model are represented by It is now simple to compute the full list of correlators where i, j, k = 1, . . . , 5 and we introduced β mρ ≡ b mρ + 1.

The M • model
We now tackle the B/2-twisted theory for the mirror model, the corresponding GLSM being a fairly untreatable 21-parameter model. As sketched above, we can still solve the model using the following two facts. First, it is known that the B/2-twisted model does not receive worldsheet instanton corrections [10]. This implies that we can perform the computation at any point in the Kähler moduli space. Second, one of the phases of the linear model admits a Landau-Ginzburg orbifold description. Thus, we can make use of the explicit LGO theory to completely solve the full B/2 model. The M • LG model is described in terms of the chiral superfields Y 1 , . . . , Y 5 , to which we assign R-charge 1 5 , and we take the orbifold by Z 3 5 , generated by where ζ = e  27) and the unique invariant complex structure coordinate reads κ = α 1 · · · α 5 α 5 0 . (3.28) Lastly, the (0,2) superpotential is determined by A few comments are in order here. First, our notation is such that b mm is to be understood as b m,ρ=m = (b ρ,m=ρ ) ⊤ , ρ = 1, . . . , 5. Second, although in the full GLSM the index m assumes the values m = 1, . . . , 26, only a subset of the corresponding variables Y m actively plays a role in the LG phase, the other coordinates being massive in the limit and we can integrate them out. In particular, our choice of basis for the matrix b in the original model is such that these correspond to m = 1, . . . , 5.

B/2 correlators
The natural observables in the B/2-twisted LGO theory are The formula for B/2-twisted LG correlators [20][21][22] is fairly straightforward to implement and is expressible in terms of a local Grothendieck residue where m 1,2,3 = 0, . . . , 5, and the integral is determined in terms of a cycle Γ = {Y | |J • m | 2 > ǫ m }, ǫ m > 0. Explicitly, such integrals can be evaluated by taking advantage of the transformation law for local residues [23]. As these techniques are quite standard, 5 we simply present the full list of correlators 32) 5 For a review, the reader can refer to [24].
where i, j, k = 1, . . . , 5 and we defined β ρm ≡ b ρm + 1. The formula (3.31) provides unrenormalized correlators, and we do not have an independent way to determine the normalization. Nonetheless, it is natural to normalize the correlators as follows so that the result depends explicitly only on invariant quantities. By comparing these with the A/2 correlators in (3.25) we find the relation Thus, we find complete agreement, up to the relative normalization, between the two sets of correlators under the mirror map if the observables are exchanged according to µ m ↔ H m . Notice that b 44 = −1 corresponds to a singularity of the theory. For example, in the LGO description, at this locus the ideal J • fails to be zero-dimensional. Thus, the relative normalization between the correlators is not allowed to vanish and can be absorbed, for instance, through a non-singular redefinition of the fields.

Mirror subfamilies in a non-reflexively plain model
In this section we study the mirror map in a non-reflexively plain model, namely the twoparameter model describing in its large radius phase an octic hypersurface in the toric resolution of the weighted projective space P 4 11222 [17]. It has been shown that the full linear model for this example is not mirror symmetric [8], but there exist subfamilies of the model and its mirror, identified by the restriction to the "diagonal" form of the E-parameters (2.12), which are conjectured to be exchanged by the mirror map [9]. In this section we will show that this is the case at the level of the correlators.

The M model
The model is specified by seven chiral multiplets X 0 , X ρ , ρ = 1, . . . , 6, coupled to the U(1) ⊕2 gauge group via The polynomial defining the octic hypersurface assumes the form The left-moving fermions Γ 0 , Γ ρ couple to the gauge group with the same charges as in (4.1) and have chirality determined by where we have introduced the vectors The J-deformations take the following form and the invariant Kähler coordinates are The matrix b mρ for this example has dimension 104 × 6, and it would be pointless to write it down in its entirety. We instead assume an ordering of the lattice points in ∆ (or equivalently, an ordering of the rows of b mρ ) corresponding to the choice (4.5), and consider the 6× 6 minor For generic values of the parameters this matrix has full rank. However, when the parameters satisfy the relations the rank of (4.7) reduces to at most 4. We will however assume that the remaining parameters are generic enough such that the rank of (4.7) will be exactly 4. Thus, of the 10 parameters in (4.7) only 6 are independent. 6 As in the previous example, the solution takes a much prettier form if we again allow for some redundancy. In particular, it is convenient to eliminate the explicit dependence of b 65 and b 66 through the constraints (4.8), and regarding all the other quantities as "independent" parameters. With this strategy, the kernel of (4.7) assumes a simple form where we used the GL(2, C) remaining field redefinitions to set, in particular, γ 1 2 = 0 and γ 6 to its (2,2) value. Notice that b 64 does not enter our solution, neither explicitly nor implicitly through (4.8). Therefore the correlators will depend on 5 parameters, instead of the naive 6.
We will see when we study the mirror that this is the correct behavior.

A/2 correlators
At this point everything is in place, and we can proceed to solve the model. As in the previous example, we first tackle the A/2-twisted V model and then derive the M model correlators by taking advantage of the quantum restriction formula.

(4.21)
It is not hard to evaluate this expression for any a, t. As an example we present the correlators for t = 1, which read (4.22) In particular, our choice of normalization is such that the correlator σ 3 1 σ 2 = 1 assumes throughout the whole parameter space the same value as on the (2,2) locus.

The M • model
We now turn to the B/2-twisted mirror model. Again, we make use of two properties of the M • theory, namely that the model exhibits a LGO phase, and that the B/2-twisted model is not corrected by worldsheet instantons [10]. Thus, we are able to solve the B/2 model in the LGO phase.
In this phase the model is described in terms of coordinates Y 1,2,3 with R-charge 1 4 , and coordinates Y 4,5 with R-charge 1 8 , supplemented by a Z 8 × Z ⊕3 4 orbifold with action where ζ = e 2πi 8 . The most generic polynomial of R-charge 1 and invariant under (4.35) is given by The invariant complex structure coordinates are according to (2.17) while the J-deformations take the form (4.38) Before we proceed with the computation of the correlators, we point out that already at this level we find an important consistency check with the original model. In fact, the (0,2) superpotential (4.38) depends explicitly on 7 parameters b mm , m = 1, . . . , 5, as well as b 64 and b 65 . This matches precisely the dependence we found in the A/2 model. Once we take into consideration the (mirror) relations (4.8), we find that the above parameters satisfy two relations and the B/2 model depends on 5 bundle coordinates.

B/2 correlators
The natural observables of the B/2-twisted LGO theory are Again, we can straightforwardly employ the formula (3.31) for evaluating the relevant cubic correlators. For example, we present the correlator Again, we find complete agreement up to a non-vanishing 7 relative normalization.

Discussion
In this work we have provided evidence in support of the mirror map for deformations of (2,2) theories proposed in [9]. In particular, we have shown in two key examples that the map exchanges the A/2 model with the B/2 model of the mirror theory at the level of correlators. Moreover, we found that the equivalence in question is a fairly simple one: the map exchanges the natural observables on the two sides of the mirror where γ 0 ≡ δ and M labels monomials in the equation defining the CY hypersurface. In particular, (5.1) holds without requiring, for instance, a parameter dependent redefinition. Notice that the relative normalization between the two sets of correlators can be absorbed through the GL(r, C) field redefinitions of the σ fields, that is, by appropriately rescaling γ ρ and δ. In this final section we employ our results to derive some consequences on the structure of the moduli space for the theories under study. We start by showing that the (0,2) moduli space does not exhibit the sort of splitting that would generalize the structure on the (2,2) locus. A counterexample to such splitting is provided, for instance, by the reflexively plain quintic model discussed in section 3. Although solving the full B/2-twisted theory for the M model is beyond the purpose of this work, we can restrict our attention to the subset of the J-deformations defined by  [10] that for linear models with a Landau-Ginzburg phase these corrections do not occur, but what about more general models? The mirror map provides us with a partial answer. It is apparent from the Coulomb branch computations we adopted in this work 8 that the A/2 model admits dependence on the Kähler parameters κ and the bundle moduli b, while no dependence on κ is possible. Thus, according to the mirror map, the mirror B/2 model will depend only on the complex structure parameters κ and the bundle moduli b, and it cannot admit instanton corrections. Applying the same reasoning to the mirror model, we reach the conclusion that the B/2-twisted theory of the original model is classical as well. Hence, the moduli space of any model to which these techniques apply does in fact exhibit a partial splitting: while the bundle moduli generically play a role in both twisted theories, complex structure and Kähler parameters remain a prerogative of the B/2 and A/2 models, respectively. This simple argument, although quite powerful, is subject to two caveats. First, in nonreflexively plain models, we are forced to work on a subfamily of the full moduli space. On the B/2 model side, this is manifest through the fact that at the LGO point, where we performed such computations, some of the complex structure/bundle moduli are forced to be frozen, as the corresponding operators appear in twisted sectors. In order to study a larger subset of the moduli space it appears necessary to employ, if available, a different description of the same CFT where at least some of such moduli are not frozen. A systematic study along this lines might help unveiling the structure of the non-reflexive subset of the moduli space. For instance, techniques to evaluate B and B/2 model correlators in hybrid models [25,26] have been recently developed [24], and these could be employed to gain insights into this larger set of theories. While we expect a dependence on non-diagonal, but linear, E-parameters, nonlinear E-parameters seem not to affect A/2-twisted V model correlators [27][28][29]. However, the situation is more subtle for A/2-twisted M models, where the supersymmetry constraint relates E and J parameters.
Second, the mirror map, as currently formulated, comprises only hypersurfaces in toric varieties. It is tempting to conjecture that also for more general models a subset of the B/2 moduli space is protected by worldsheet instanton effects, and it would be desirable to test this prediction. Finally, it should be possible, through a deeper understanding of the combinatorics involved, to extend the mirror map to (0,2) deformations of CICY in toric varieties [30,31].

A Correlators for the octic model
In this appendix we collect the full list of correlators in both the A/2 model and B/2 mirror model for the example we solved in section 4.

A.2 B/2 model
We introduce here as well a similar notation as we did in the previous section. We define the parameters β ρm ≡ b ρm + 1, for ρ, m = 0, with β 00 ≡ 1, as well as The principal component of the discriminant locus reads