AN EXAMPLE OF BERGLUND-HÜBSCH MIRROR SYMMETRY FOR A CALABI-YAU COMPLETE INTERSECTION

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1 AN EXAMPLE OF BERGLUND-HÜBSCH MIRROR SYMMETRY FOR A CALABI-YAU COMPLETE INTERSECTION STEFANO FILIPAZZI, FRANCO ROTA Abstract. We study an example of complete intersection Calabi-Yau threefold due to Libgober and Teitelbaum [LT3], and verify mirror symmetry at a cohomological level. Direct computations allow us to propose an analogue to the Berglund-Hübsch mirror symmetry setup for this example [BH3]. We then follow the approach of Krawitz to propose an explicit mirror map [Kra0]. Contents. Introduction. Hybrid Landau-Ginzburg models 3 3. Description of the example 5 4. Krawitz's mirror map revisited 6 5. The mirror map 8 Appendix A. Tables 3 References 8. Introduction This note is an account of a calculation which we carried out during the Summer School Pragmatic 05, and represents a rst step in our project of generalizing the Krawitz-Chiodo-Ruan cohomological isomorphisms to Berglund-Hübsch mirror pairs of Calabi-Yau complete intersections. In 3, Berglund and Hübsch describe a procedure to construct the mirror manifold of a large class of Calabi-Yau hypersurfaces in weighted projective space, using the formalism of Landau-Ginzburg models [BH3]. Later, Krawitz gives an explicit description of the mirror map between the state spaces of two Berglund-Hübsch mirror LG models [Kra0], and Borisov re-proves and generalizes his statements [Bor3]. Since the work by Berglund and Hübsch, the role of Landau-Ginzburg models in mirror symmetry has increased in importance, particularly after Fan, Jarvis and Ruan dened their quantum invariants in [FJR3] (FJRW theory), and extended the denition to an even more general setting in [FJR5]. In a subsequent work, Chiodo and Nagel recognize the state space of Landau-Ginzburg isolated singularities dened in [FJR3] as an instance of relative orbifold Chen-Ruan cohomology [CN5]. This allows to dene the cohomology of hybrid Landau-Ginzburg models, and to identify it with the cohomology of Calabi-Yau complete intersections in weighted projective space. In this sense, [CN5] extends the LG/CY correspondence, which was proven Date: November 7, 07. The two authors were partially supported by NSF FRG Grant DMS-6585.

2 to hold for hypersurfaces in [CR], to the case of complete intersection Calabi-Yau. The situation in the hypersurface case can be summarized by the following diagram, LG LG CY CY where an arrow indicates an isomorphism of the corresponding state spaces. Existence of the vertical arrows has been established in [CR], while the horizontal one is the object of [Kra0] and [Bor3]. In the case of a complete intersection, [CN5] provides the vertical arrows of the diagram LG LG CY CY by proving an orbifold version of the Thom isomorphism. A related result by Libgober [Lib5] shows the invariance of the elliptic genus for complete intersections and other examples of GIT quotients. The object of this work is to exhibit a horizontal arrow in a particular example by Libgober and Teitelbaum [LT3]. In this way, we aim at making progress towards a denition of a mirror symmetry construction for Calabi-Yau complete intersections of the same type as the Berglund-Hübsch construction. We describe a complete intersection Calabi-Yau threefold and its mirror orbifold using an analogue of the Berglund-Hübsch formalism, and propose a generalization of Krawitz's mirror map. The direction of this paper is suggested by the classical result of Batyrev and Borisov on cohomological mirror symmetry for Calabi-Yau complete intersections [BB6], although at the moment it is unclear how to explicitly relate to their work. Starting from a paper by Libgober and Teitelbaum [LT3], we consider the complete intersection of two cubics in P 5 together with its mirror. We recall the setup for hybrid Landau-Ginzburg models in Section. In Section 3, using the Chiodo-Nagel CY/LG-correspondence [CN5], we rephrase the example of [LT3] in terms of hybrid Landau-Ginzburg models. Then, in Section 4, we generalize the construction of Krawitz for the quintic Fermat hypersurface in P 4 with the language proposed for complete intersections. Finally, in Section 5, we verify that the state spaces of the complete intersection and of its mirror are isomorphic, and present an explicit mirror map generalizing Krawitz's formula. Acknowledgements. We started working on this project during the PRAGMATIC 05 Research School in Algebraic Geometry and Commutative Algebra, Moduli of Curves and Line Bundles, held in Catania, Italy, in July 05. We are very grateful to Alo Ragusa, Francesco Russo, and Giuseppe Zappalà, the organizers of the PRAGMATIC school, for the wonderful environment they created at the school. We thank the lecturers of the school, Alessandro Chiodo and Filippo Viviani. In

3 particular, we are indebted with Alessandro Chiodo for rst suggesting this problem, and then advising us through it. We are also grateful to Jérémy Guéré for helpful insight and comments.. Hybrid Landau-Ginzburg models In this section, we recall the notion of hybrid Landau-Ginzburg models. Our setup follows closely the one in [CN5], whose notation we adopt for the most part. The main dierence is that we consider a broader class of groups of symmetries. This is in line with the groups considered in [LT3]. Let W = {W,..., W r } be a set of quasi-homogeneous polynomials of degrees d,..., d r dening a complete intersection X W := {W =... = W r = 0} P(w,..., w n ) in weighted projective space. Assume furthermore that X W is non-degenerate, i.e. the choice of weights w i is unique; X W is smooth outside the origin. The maximal group of diagonal symmetries of X W is the maximal subgroup max (C ) n of diagonal matrices γ preserving X W. More explicitly, γ has diagonal entries of the form (γ,..., γ n ) = α λ := (α λ w,..., α n λ wn ), where λ C, and α = α is a vector whose entries are all non-zero. By denition, for any i r and admissible choice of α, there exists β i,α C such that W i (α x,..., α n x n ) = β i,α W i (x,..., x n ). By a group of diagonal symmetries we mean a subgroup of max. Example.. The group max contains the following special element. Let d denote the greatest common divisor of the d i, and dene J := e πi w d... wn πi e d. The element J is analogous to the generator of the group µ d in the hypersurface case. The group J will play an important role later. Example.. Among the subgroups of max, one can consider the special linear group of diagonal symmetries of a complete intersection, denoted SL(W ). It contains elements of max of determinant. Denition.3. Let γ be a diagonal symmetry acting on C n as a diagonal matrix with entries exp(πiγ i ), with γ i [0, ). Then, we denote γ by (γ,..., γ n ). If we have γ i = c i d for all i, we write (γ,..., γ n ) = (c d,..., c n ). The age of the element γ, denoted a γ, is the sum i γ i. Unless otherwise stated, we will follow the classic literature (see [CR4, Ch. 3]), and assume that any group of diagonal symmetries contains the torus 0 := { λ = (λ w,..., λ wn ) λ C }. Then, 0 is the connected component of the identity in, and we denote by G the quotient / 0. 3

4 Now, consider C n+r with coordinates (x,..., x n, p,..., p r ). We can extend the action of to C n+r by setting () α λ(x, p) := (α λ w x,..., α n λ wn x n, β,αλ d p,..., β r,αλ dr p r ). It is useful to introduce a notation for the xed points of an element γ. We set C n γ := {x Cn γ x = x}, n γ := dimc n γ, where the action of γ is restricted to the rst set of coordinates. Similarly, we dene C r γ := {p Cr γ p = p}, r γ := dimc r γ. The integer n γ (respectively r γ ) counts the dimension of the space spanned by the x (resp. p) variables that are xed by γ. For a polynomial V C[x, p], we let V γ denote V C n γ C r. γ For an element V C[x, p], its chiral algebra Q V is dened as Q V := dx dp Jac(V ), where we are formally tensoring the Jacobi ring Jac(V ) := C[x,..., x n, p,..., p r ] ( V x,..., V x n, V p,..., V by the top form dx dp := dx... dx n dp... dp r. For brevity, in the following we will often omit the wedge symbol and write dxdp := dx dp. We assign bidegree (D k, k) to the elements of Q V which have degree k in the p variables, where D = n r,. By construction, γ (p i W i ) = p i W i for all γ and all i. Then we can dene the -invariant function W : C n+r C p r ) (x, p) p W (x) p r W r (x). Let M := { } W = t 0 denote the ber over any point t0 0. Consider the open set in C n+r dened as U LG := C n (C r {0}). The quotient of U LG by the action of C is the total space of the vector bundle n [U LG /C ] = O P(d) ( w i ) =: O d ( w). j= Denition.4. The datum (O d ( w), W, ) is the hybrid Landau-Ginzburg model with superpotential W : [O d ( w)/g] C. The generalized state space of the hybrid Landau-Ginzburg model (O d ( w), W ) is H p,q (W,..., W r ) := H p+r,q+r CR ([O d ( w)/g], [F/G]), where F is the quotient stack [M/C ] inside O d ( w). For any two non-negative integers r n we need to consider the bigraded ring dt( n) C[t]/(t r n ), where the element dt( n) t k has bidegree (n + k, n + k). We will make use of the following theorem due to Chiodo and Nagel [CN5]. 4

5 Theorem.5 ([CN5, Theorem 4.3]). Let W i, r and n be as introduced above. Then, we have H (W,..., W r ) = γ H γ ( a γ + r), where H γ with its double grading is given by ( ) (a) if r Q Wγ H γ = γ < n γ (b) dt( n γ ) C[t]/(t rγ nγ ) if r γ n γ. and carries the Tate twist ( a γ + r). 3. Description of the example In this section, we introduce the complete intersection Calabi-Yau object of this work. This example was rst described by Libgober and Teitelbaum [LT3]. Consider P 5 with coordinates [x : x : x 3 : X : X : X 3 ]. Let X P 5 be the Calabi-Yau threefold dened by the vanishing of the polynomials W = x 3 + x 3 + x 3 3 3X X X 3 W = X 3 + X 3 + X 3 3 3x x x 3. Since we are in the standard projective space P 5, and we are considering two polynomials of degree 3, the group J introduced in Example. is isomorphic to µ 3. Introduce two new variables, p and p, and form the polynomial W := p W + p W. Extend the action of the group J to C 6 C by letting J act with weights (,...,, 3, 3). Similarly, following equation (), we can extend the action of any group of diagonal symmetries to the variables p and p. According to the hypersurface case of Berglund-Hübsch mirror symmetry (see [CR4, Ch. 3]), to study the generalized state spaces of the Hybrid Landau-Ginzburg model induced by W and W, we have to make choices for a mirror set of polynomials and for a pair of groups. As suggested by Libgober and Teitelbaum, we choose the same set of polynomials for the mirror. Indeed, this is analogous to the Fermat hypersurface case. Then, let := 0 J, and T := max SL(W ). This choice is in line with the Berglund-Hübsch prescription (see Rmk. 3.). By Theorem.5, the only elements of a group of diagonal symmetries Σ contributing to the generalized state space of (O d ( w), W, Σ) are the elements σ xing at least one of the coordinates (x, x, x 3, X, X, X 3, p, p ). By direct inspection, one checks that the only elements of that x at least one of the variables are the elements of J. Analogously, when considering T, one is left with those elements γ SL(W ) that x at least one of the variables. These γ generate a subgroup SL(W ) whose elements g λ,µ,α,β,δ,ɛ act on (x, x, x 3, X, X, X 3, p, p ) via multiplication by (λξ α 3 ξ µ, λξ β 3 ξ µ, λξ µ, λξ δ 3 ξ µ, λξ ɛ 3 ξ µ, λξ µ, λ 3 ξ µ 3, λ 3 ξ µ 3 ). Here ξ 3 is a primitive third root of unity, ξ is a primitive ninth root of unity, and λ µ. Furthermore, the condition is satised. α + β = 3µ = δ + ɛ mod 3 5

6 Remark 3.. The group T also contains 0, and the quotient T / 0 coincides with the group G 8 described in [LT3]. We have / 0 = J. As a consequence of Libgober and Teitelbaum's work [LT3] and an application of Chiodo and Nagel's CY/LG-correspondence [CN5, Thm. 5.], one knows that the generalized state spaces H p,q (W, W ) and H 3 p,q (W T, W ) have the same dimension. The object of this work is to further investigate this correspondence, and propose a possible mirror map. The main idea is summarized in the following. Denition 3.. We dene an explicit mirror map H, (W T, W ) H, (W, W ), generalizing Krawitz's map for hypersurfaces. Remark 3.3. The mirror map in Denition 3. is expected to extend to a map H (W T, W ) H (W, W ). At this point, the main diculty is to choose appropriate representatives among the 73 generators of H, (W, W ) in order to recognize a pattern similar to the one occurring among the generators of H, (W, W ), which are listed in Table. 4. Krawitz's mirror map revisited In this section, we discuss how the map mentioned in Denition 3. is related to the work of Krawitz. More precisely, we show how the approach suggested for Calabi-Yau complete intersections is a generalization of the one already known for hypersurfaces. We will focus on a concrete example, the Fermat quintic X W P 4. This is given as the vanishing locus of the polynomial W = x 5 + x 5 + x x x 5 5. First, we consider the group J as group of diagonal symmetries. Thus, we rst consider the pair (W, J). Berglund-Hübsch mirror symmetry guarantees the existence of a mirror model, denoted by (W T, J T ) [Kra0; CR4]. In our case, W = W T, and J T = SL(W T ) = SL(W ) is a well known group, denoted by G 5. Now, we would like to regard the hypersurface X W as a complete intersection of one hypersurface. Therefore, we introduce one auxiliary variable p, and, as explained in equation (), extend the action of the diagonal symmetries acting on X W to p. We can then regard pw as a function on C 5 C, and consider the induced Landau-Ginzburg model. Among the groups xing X W, there is C J. In particular, C acts on x i with weight, and on p with weight 5. We can compute the corresponding state spaces. By Theorem.5, only elements γ xing some variable contribute to the computations. Furthermore, it is easy to check that, in this case, γ C is forced to satisfy γ =. More precisely, γ is either, or a primitive fth-root of unity. In the rst case, all the variables are xed, while, in the latter one, just p remains unchanged. The element γ = contributes to the state space with a summand of the form ( dpdx C[p, x,..., x 5 ] (px 4,..., px 4 5, 5 i= x5 i ) Its elements carry a bigrading (3 k, k), where k is the degree of the variable p in the element. As dpdx is C invariant, and we are looking for C -invariant elements, we ) C. 6

7 need polynomial coecients containing ve among x,..., x 5 for every appearance of p. There is just a one-dimensional family in degree (3, 0), there are spaces with dimension 0 in degree (, ) and (, ), and there is one more one-dimensional family in degree (0, 3). If γ is a primitive fth-root of unity, we have r γ = and n γ = 0. Therefore, each one of these elements contributes with a one-dimensional vector space, generated by one element, denoted by g. Taking into account the grading of the above pieces, we recover the well known Hodge diamond 0 0 Now, we want to consider the mirror of (pw, C J). As W is a polynomial of Fermat type, we will let the mirror polynomial be pw as well. As C is the minimial group of diagonal symmetries containing C, we consider the maximal group of symmetries xing the polynomial pw. Let be such group. It can be checked by direct computation that = C G 5. It acts as (x, x, x 3, x 4, x 5, p) (λx, λξ a x, λξ b x 3, λξ c x 4, λξ a b c x 5, λ 5 p), where λ C, ξ is a primitive fth-root of unity, and a, b, c Z. Remark 4.. In [CN5], Chiodo and Nagel consider the maximal group xing W, and then they extend its action to pw. On the other hand, for our purposes, such a group turns out to be too rigid, and does not recover the right cohomologies. As above, the only elements γ contributing to the state spaces are the ones where λ is a fth-root of unity. In particular, p is xed by any relevant element γ. This highlights how the variable p is not necessary for the computations in the hypersurface case. Since λ needs to be a fth-root of unity, we can absorb it in the contribution of ξ. Therefore, any element γ contributing to the computations can be encoded by a 6-tuple (a, a, a 3, a 4, a 5 ; 0), where the action on x i is given by multiplication by ξ a i. We will use this formalism as a slight modication of the one in Denition.3, in order to distinguish the p and the x variables. Now, we are left with direct computations. We will proceed by cases. Assume that p is the only variable xed. In such case, r γ =, and n γ = 0. Therefore, each element contributes with a one-dimensional vector space. There are 04 of these elements, and their ages groups them in four families of dimension, 0, 0 and respectively. These compute the twisted sector of the cohomology ring. If p and some but not all of the x i are xed, there is no contribution. 7

8 The identity element, represented by the string (0, 0, 0, 0, 0; 0), contributes to the untwisted sector. We have to consider the ring ( ) C G 5 C[p, x,..., x 5 ] dpdx (px 4,..., px 4 5, 5 i= x5 i ). This ring has dimension four as vector space, with basis (dpdx, dpdx px... x 5, dpdx (px... x 5 ), dpdx (px... x 5 ) 3 ). Each one of these elements has a dierent bidegree. If we put the above facts together, we recover the Hodge diamond 0 0 With the elements explicitly listed, we can proceed and rewrite Krawitz's mirror map with this new notation. First, we consider the generators of the untwisted sector of the mirror. These are mapped according to (px... x 5 ) a dpdx id (a, a, a, a, a; 0), where a {,, 3, 4}. Symmetrically, the twisted elements of the mirror are mapped to the untwisted sector of the quintic. The map is given by (3) (a, b, c, d, e; 0) dpdx p a+b+c+d+e 5 x a x b x c 3 x d 4 x e 5 id. Remark 4.. The quantity a+b+c+d+e 5 is the age a γ of the group element γ considered. Therefore, under the mirror map introduced by Krawitz [Kra0], the auxiliary variable p appears with exponent a γ. 5. The mirror map In this section, we construct the mirror map in Denition 3.. As we noticed in the example of the Fermat quintic, there is no harm in dropping the assumption that the diagonal symmetries have determinant. Indeed, the new group elements coming into the picture do not contribute to the computation of the state spaces. Recall that we consider the groups = 0 J and T acting on the same set of polynomials (see Section 3). In the following, we rst compute the state space associated to. Then, we consider the one associated to T, and we conclude describing the mirror map. 5.. The state spaces H p,q (W, W ). We follow the notation of Theorem.5. Notice that for γ, if r γ = n γ = 0, then H γ = 0 gives no contribution to the state space. Then, only the three elements of J give non-trivial contributions. We will proceed by cases. The non-trivial elements have ages and 4 respectively. They both contribute with the cohomology of a projective line P(3, 3) = P, and form the twisted sectors of H (W, W ). 8

9 The contribution of the identity element is encoded in Q W. This ring carries a bigrading (D k, k), where D = n r = 3, and k {0,,, 3} is the degree in the variables p and p. The relations in Jac(V ) are: x 3 + x 3 + x 3 3 3X X X 3, 3x x x 3 X 3 + X 3 + X 3 3, p x i p x j x k, p X i p X j X k, where all the i, j, k are assumed to be distinct. The form dpdxdx := dp dp dx dx dx 3 dx dx dx 3 is invariant under the action of. Hence, the degree (3, 0) component is generated by dpdxdx. The 73 independent generators of H, (W, W ) are listed in Table. They determine the whole untwisted sector, which is completed by H, (W, W ) and H 0,3 (W, W ), whose dimensions are 73 and respectively. We can summarize the above computations with the following Hodge diamond The state spaces H p,q T (W, W ). Now, we consider the mirror side. We will list the elements of T, and study their contributions to the state spaces. As explained in Denition.3, we will write (γ,..., γ 8 ) if γ acts on C 6 C as a diagonal matrix with entries exp(πiγ i ). As showed in Theorem.5, the contribution of an element γ has two dierent behaviors, depending on how r γ and n γ compare to each other. Therefore, we will proceed by cases. If r γ < n γ, we are in the case given by equation (a). These elements are listed in Table. One of these elements is the identity of T. It corresponds to the untwisted sector of H (W T, W ), which is 4-dimensional and generated by powers of dpdxdx p x 3 in the Jacobi ring. The other four elements all give a two dimensional Jacobi ring, and behave in a similar way. For instance, γ = (0, 0, 0, 3, 3, 3; 0, 0) induces a direct summand H γ = span(dpdxdx x 3, dpdxdx x 3 ), which contributes as two-dimensional subspace of H, (W T, W ). A second one of these elements contributes with another two-dimensional subspace of H, (W T, W ). Similarly, the two remaining elements correspond to two two-dimensional subspaces of H, (W T, W ). If r γ n γ, we are in the case given by equation (b). Each element in Table 3 contributes to the state space with r γ n γ elements. If r γ n γ =, the

10 two generators belong one to and the other to H aγ r+nγ,aγ r+nγ T (W, W ), H aγ+ r+nγ,aγ+ r+nγ T (W, W ). There are four such elements. The rst one of them contributes with one generator to H 0,0 (W T, W ), and with one to H, (W T, W ). Similarly, a second element generates the top cohomology, contributing with one generator to H 3,3 (W T, W ), and with one to H, (W T, W ). Then, the two remaining elements behave in the same way: Each one of them has one generator lying in H, (W T, W ), and one in H, (W T, W ). On the other hand, if r γ n γ =, there is only one generator, which belongs to H aγ r+nγ,aγ r+nγ (W T, W ). The last column in Table 3 counts the elements of the same type. For example, from the element (3, 3, 3, 3, 0, 6; 0, 0) we can obtain 5 more by permuting X, X and X 3 via the action of the symmetric group S 3. These new elements are considered of the same type as the starting one, and are not explicitly listed in Table 3. They are taken into account by counting 6 elements, including their representative, of the type of (3, 3, 3, 3, 0, 6; 0, 0). We can summarize the above computations with the following Hodge diamond The mirror map. We present the explicit isomorphism between the vector spaces H, (W T, W ) and H, (W, W ) mentioned in Denition 3.. Its main feature is that it respects the structure emerging from the lists in Tables, and 3, where it becomes evident that these two vector spaces admit a further interesting subdivisions into direct summands. We follow the notation in [Kra0] and Theorem.5: Given an element σ of a group of diagonal symmetries Σ, we denote by ω σ the element of the state space HΣ determined by ω H σ. Now, we explain the mirror map via its action on some elements listed in Table 3, which represent the dierent possible behaviors. For the complete action of the mirror map, we refer to Table 4 in the Appendix. First, we notice that there is a correspondence between direct summands in H, (W T, W ) and H, (W, W ). For the reader's convenience, we will list them relying on Tables, and 3. In H, (W, W ), we isolated six groups of generators, each one consisting of nine elements. These correspond to the six groups of nine elements γ T having r γ =, n γ = 0, and a γ = 3. 0

11 The two groups of six generators of H, (W, W ) correspond to the two groups of six elements γ T having r γ =, n γ =, and a γ =. In T there are two elements with r γ =, n γ = 3 and a γ =. These contribute with two two-dimensional vector spaces. On the side of H, (W, W ), these correspond to the vector spaces p X, 3 p X 3 and p x 3, p x 3. Notice that, by the relations in the Jacobi ring, these spaces contain p X3 3 and p x 3 3 respectively. To conclude, among the elements γ T with r γ = and n γ = 0, three of these contribute with a one-dimensional subspace of H, T (W, W ) each. These will correspond to the three sporadic polynomials listed at the top of Table. Before getting into the details of the mirror map, it is worth asking the following interesting question. Question 5.. Do the aforementioned direct summands of the spaces H, (W T, W ) and H, (W, W ) have an interpretation from the point of view of geometry or physics? Now, we will proceed to describing the mirror map, analyzing each one of the cases listed above. The two rst groups of elements present a fundamental common feature, namely r γ n γ =. This is essentially the only case occurring in the twisted sector for the hypersurface case [Kra0]. As showed in equation (3), once we introduce a p in the hypersurface case, the exponents of the x i contribute to the exponent of this new variable. We will follow this idea, bearing in mind that p is related to the set of variables x i, while p to the set X i. For the elements with r γ n γ =, we then propose the assignment dt (b, b, b 3, c, c, c 3 ; a, a ) (4) dpdxdx p min{ b j 3 } 3 cj min{ 3 p } x b i 3 min{ b j 3 } i X ci 3 min{ c j 3 } i= i id. Indeed, from Table 3, it is clear that exactly one set of variables has an action which is a rotation of at least e πi 3 on each single coordinate. We record this information choosing the corresponding variable p i to appear in the assigned polynomial. Then, in order to choose the exponent appearing for each variable in the polynomial in output, we compare the action withing each group x i and X i. This is where the normalizing exponents min{ b j } and min{ c j } come from. Finally, we point 3 3 out that looking at quantities of the form n is consistent with distinguishing the 3 action of third-roots of unity from the one of primitive ninth-roots. As example of the assignments described in equation (4), we have dt (3, 3, 3, 3, 0, 6; 0, 0) dpdxdx p X X 3 id, and dt (,, 5, 6, 3, 6; 3, 0) dpdxdx p x 3 X X 3 id.

12 Now, we can study how the mirror map acts on the subspaces corresponding to the elements γ T with r γ =, n γ = 3, and a γ =. The rst element is (0, 0, 0, 3, 3, 3; 0, 0), which contributes with the subspace dpdx x 3, dpdx x 3. Following the ideas already discussed, the output of the mirror map should carry the variable p, in order to record the action on the X i variables. Furthermore, it is natural to keep track of the two distinct generators dpdx x 3 and dpdx x 3 via their polynomial part. Therefore, we dene (5) dpdx x 3 i (0, 0, 0, 3, 3, 3; 0, 0) dpdxdx p x 3 i id. Notice that this assignment is valid also for i = 3. Similarly, we dene (6) dpdx Xi 3 (0, 0, 0, 3, 3, 3; 0, 0) dpdxdx p Xi 3 id Notice that the assignments in equations (5) and (6) could be regarded as following an extension of the rule in equation (4), where we allow some degree of complexity in the polynomial part of the input as well. Thus, we are left with assigning a mirror element for the three elements arising in from γ T, with r γ = and n γ = 0. In the case of the elements dt (3, 3, 3, 6, 6, 6; 0, 0), dt (6, 6, 6, 3, 3, 3; 0, 0) we can follow the algorithm in equation (4). In this way, we get dt (3, 3, 3, 6, 6, 6; 0, 0) dpdxdx p X X X 3 id, and dt (6, 6, 6, 3, 3, 3; 0, 0) dpdxdx p x x x 3 id. Now, we have to consider the element tdt (3, 3, 3, 3, 3, 3; 0, 0). This element does not follow directly in the patter of equation (4). On the other hand, it comes from the most symmetric element γ T \ {0}. Analogously, there is a highly symmetric generator of H, (W, W ), namely dpdxdx p X X X 3 = dpdxdx p x x x 3 = dpdxdx p Xi 3 = dpdxdx p x 3 i. Given the common symmetries, we suggest the choice tdt (3, 3, 3, 3, 3, 3; 0, 0) dpdxdx p X X X 3 id. As the above assignments determine a bijective correspondence between a basis of H, (W T, W ) and one of H, (W, W ), we can extend the map by linearity. This completes the denition of the mirror map in Denition 3..

13 Appendix A. Tables Here we collect the tables referenced above. Generator # p X X X 3 = p x x x 3 = p X 3 i = p x 3 i p x x x 3 = p X 3 i p X X X 3 = p x 3 i p X 3 p X 3 p x 3 p x 3 p x i x j X k p x i X j X k p X i X j x k = p X l x k p x i X j = p x j x k X j p X i x j p x i X j p x i x j = p x jx k 6 p X i X j = p X i X k 6 Table. A basis of H, (W, W ). All indices i, j, k, l range from to 3, distinct indices have distinct values. On every row there are all the possible ways of writing a generator, and the number of generators of that form. 3

14 r γ n γ element type a γ # r γ n γ element a γ 6 (0,...,0) 0 3 (0, 0, 0, 3, 3, 3; 0, 0) 3 (3, 3, 3, 0, 0, 0; 0, 0) 3 (0, 0, 0, 6, 6, 6; 0, 0) 3 (0, 0, 0, 6, 6, 6; 0, 0) Table. Elements of T for which r γ < n γ 0 (3, 3, 3, 3, 3, 3; 0, 0) 0 (6, 6, 6, 3, 3, 3; 0, 0) 3 0 (3, 3, 3, 6, 6, 6; 0, 0) 3 0 (6, 6, 6, 6, 6, 6; 0, 0) 4 (3, 3, 3, 3, 0, 6; 0, 0) 6 (3, 0, 6, 3, 3, 3; 0, 0) 6 (6, 6, 6, 3, 0, 6; 0, 0) 3 6 (3, 0, 6, 6, 6, 6; 0, 0) (,, 5, 6, 3, 6; 3, 0) 3 0 (6, 3, 6,,, 5; 0, 3) 3 0 (3, 6, 3,, 7, ; 0, 6) 3 0 (, 7,, 3, 6, 3; 6, 0) 3 0 (3, 6, 3, 4, 4, ; 0, 6) 3 0 (4, 4,, 3, 6, 3; 6, 0) 3 0 (6, 6, 3, 8,, 8; 0, 3) 4 0 (8,, 8, 6, 6, 3; 3, 0) 4 0 (6, 6, 3, 5, 5, 8; 0, 3) 4 0 (5, 5, 8, 6, 6, 3; 3, 0) 4 0 (7, 4, 7, 3, 3, 6; 6, 0) 4 0 (3, 3, 6, 7, 4, 7; 0, 6) 4 Table 3. Elements of T for which r γ n γ. Table 4 H, T (W, W ) H, (W, W ) tdt (3, 3, 3, 3, 3, 3; 0, 0) p x 3 id dt (3, 3, 3, 6, 6, 6; 0, 0) p X X X 3 id dt (6, 6, 6, 3, 3, 3; 0, 0) p x x x 3 id dpdxx 3 (0, 0, 0, 3, 3, 3; 0, 0) p x 3 id dpdxx 3 (0, 0, 0, 3, 3, 3; 0, 0) p x 3 id 4

15 dpdxx 3 (3, 3, 3, 0, 0, 0; 0, 0) p X id 3 dpdxx 3 (3, 3, 3, 0, 0, 0; 0, 0) p X id 3 dt (3, 3, 3, 3, 0, 6; 0, 0) p X X3 id dt (3, 3, 3, 0, 3, 6; 0, 0) p X X3 id dt (3, 3, 3, 0, 6, 3; 0, 0) p XX 3 id dt (3, 3, 3, 6, 0, 3; 0, 0) p X X 3 id dt (3, 3, 3, 6, 3, 0; 0, 0) p X X id dt (3, 3, 3, 3, 6, 0; 0, 0) p X X id dt (0, 3, 6, 3, 3, 3; 0, 0) p X X3 id dt (3, 0, 6, 3, 3, 3; 0, 0) p X X3 id dt (6, 0, 3, 3, 3, 3; 0, 0) p XX 3 id dt (0, 6, 3, 3, 3, 3; 0, 0) p X X 3 id dt (3, 6, 0, 3, 3, 3; 0, 0) p X X id dt (6, 3, 0, 3, 3, 3; 0, 0) p XX id dt (,, 5, 6, 3, 6; 3, 0) p x 3 X X 3 id dt (,, 5, 3, 6, 6; 3, 0) p x 3 X X 3 id dt (,, 5, 6, 6, 3; 3, 0) p x 3 X X id dt (, 5,, 6, 3, 6; 3, 0) p x X X 3 id dt (, 5,, 3, 6, 6; 3, 0) p x X X 3 id dt (5,,, 6, 3, 6; 3, 0) p x X X 3 id dt (5,,, 3, 6, 6; 3, 0) p x X X 3 id dt (, 5,, 6, 6, 3; 3, 0) p x X X id dt (5,,, 6, 6, 3; 3, 0) p x X X id dt (6, 3, 6,,, 5; 0, 3) p x x 3 X 3 id dt (3, 6, 6,,, 5; 0, 3) p x x 3 X 3 id dt (6, 3, 6,, 5, ; 0, 3) p x x 3 X id dt (6, 3, 6, 5,, ; 0, 3) p x x 3 X id dt (3, 6, 6,, 5, ; 0, 3) p x x 3 X id dt (3, 6, 6, 5,, ; 0, 3) p x x 3 X id dt (6, 6, 3,,, 5; 0, 3) p x x X 3 id dt (6, 6, 3,, 5, ; 0, 3) p x x X id dt (6, 6, 3, 5,, ; 0, 3) p x x X id 5

16 dt (3, 6, 3,, 7, ; 0, 6) p x Xid dt (3, 6, 3, 7,, ; 0, 6) p x Xid dt (6, 3, 3,, 7, ; 0, 6) p x Xid dt (6, 3, 3, 7,, ; 0, 6) p x Xid dt (3, 3, 6,,, 7; 0, 6) p x 3 X3id dt (3, 6, 3,,, 7; 0, 6) p x X3id dt (6, 3, 3,,, 7; 0, 6) p x X3id dt (3, 3, 6,, 7, ; 0, 6) p x 3 Xid dt (3, 3, 6, 7,, ; 0, 6) p x 3 Xid dt (3, 6, 3, 4, 4, ; 0, 6) p x X X id dt (6, 3, 3, 4, 4, ; 0, 6) p x X X id dt (3, 3, 6, 4,, 4; 0, 6) p x 3 X X 3 id dt (3, 3, 6,, 4, 4; 0, 6) p x 3 X X 3 id dt (3, 6, 3, 4,, 4; 0, 6) p x X X id dt (3, 6, 3,, 4, 4; 0, 6) p x X X 3 id dt (6, 3, 3, 4,, 4; 0, 6) p x X X 3 id dt (6, 3, 3,, 4, 4; 0, 6) p x X X 3 id dt (3, 3, 6, 4, 4, ; 0, 6) p x 3 X X id dt (,, 7, 3, 3, 6; 6, 0) p x 3X 3 id dt (, 7,, 3, 6, 3; 6, 0) p x X id dt (, 7,, 6, 3, 3; 6, 0) p x X id dt (7,,, 3, 6, 3; 6, 0) p x X id dt (7,,, 6, 3, 3; 6, 0) p x X id dt (,, 7, 3, 6, 3; 6, 0) p x 3X id dt (,, 7, 6, 3, 3; 6, 0) p x 3X id dt (, 7,, 3, 3, 6; 6, 0) p x X 3 id dt (7,,, 3, 3, 6; 6, 0) p x X 3 id dt (4, 4,, 3, 6, 3; 6, 0) p x x X id dt (4, 4,, 6, 3, 3; 6, 0) p x x X id dt (4,, 4, 3, 3, 6; 6, 0) p x x 3 X 3 id dt (, 4, 4, 3, 3, 6; 6, 0) p x x X 3 id dt (4, 4,, 3, 3, 6; 6, 0) p x x X 3 id dt (4,, 4, 3, 6, 3; 6, 0) p x x 3 X id 6

17 dt (4,, 4, 6, 3, 3; 6, 0) p x x 3 X id dt (, 4, 4, 3, 6, 3; 6, 0) p x x 3 X id dt (, 4, 4, 6, 3, 3; 6, 0) p x x 3 X id Table 4. The complete description of the mirror map on the generators of H,, grouped according to the subdivision described T above 7

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