arXiv:1411.3780v1 [math.AG] 14 Nov 2014
COMPUTATIONAL TECHNIQUES IN FJRW THEORY WITH APPLICATIONS TO LANDAU-GINZBURG MIRROR SYMMETRY AMANDA FRANCIS
Abstract. The Landau-Ginzburg A-model, given by FJRW theory, defines a cohomological field theory, but in most examples is very difficult to compute, especially when the symmetry group is not maximal. We give some methods for finding the A-model structure. In many cases our methods completely determine the previously unknown A-model Frobenius manifold structure. In the case where these Frobenius manifolds are semisimple, this can be shown to determine the structure of the higher genus potential as well. We compute the Frobenius manifold structure for 27 of the previously unknown unimodal and bimodal singularities and corresponding groups, including 13 cases using a non-maximal symmetry group.
Contents 1. Introduction 2. Background 3. Computational Methods 4. Summary of Computations References
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1. Introduction Mirror symmetry is a phenomenon from physics that has inspired a lot of interesting mathematics. In the Landau-Ginzburg setting, we have two constructions, the A- and Bmodels, each of which depends on a choice of a polynomial with a group of symmetries. Both models yield Frobenius manifolds. The A-model arising from FJRW theory produces a full cohomological field theory. From the cohomological field theory we can construct correlators and assemble these into a potential function. This potential function completely determines the Frobenius manifold and determines much of the structure of the cohomological field theory as well. Although some of these correlators have been computed in special cases, in many cases their computation is quite difficult, especially in the case that the group of symmetries is not maximal (mirror symmetry predicts that these should correspond to an orbifolded B-model). We give some computational methods for computing correlators, including a formula for concave genus-zero, four-point correlators, and show how to extend these results to find other correlator values. In many cases our methods give enough information to compute the Amodel Frobenius manifold. We give the FJRW Frobenius manifold structure for 27 pairs of polynomials and groups, 13 of which are constructed using a non-maximal symmetry group. Date: November 17, 2014. 1
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Conjecture 1.1 (The Landau Ginzburg Mirror Symmetry Conjecture). There exist A- and B-model structures, each constructed from a polynomial W and an associated group, G (see discussion following Definition 2.1), such that the A-model for W and G is isomorphic to the B-model for W T and GT , where W T and GT are dual to the original polynomial and group. Although physics predicted its existence, a mathematical construction of the A-model was not known until 2007 when Fan, Jarvis, and Ruan, following the ideas of Witten, proposed a cohomological field theory AW,G to satisfy Conjecture 1.1 [1, 2, 3]. A basis for the vector space of the A-model consists of pairs of a monomial and a group element. When the group element acts nontrivially on each variable of W, we call this a narrow element, otherwise, it is called broad. The structure in the A-model is determined by certain structure constants called genus-g, k-point correlators, which come from the cohomology of the moduli space M g,k of genus-g curves with k marked points. The Frobenius algebra structure is given by the genus-zero, three-point correlators, the Frobenius manifold structure by the genus-zero, k-point correlators for k ≥ 3, and the higher genus structure by the genus-g, k-point correlators for all nonnegative integers g and k such that 2g − 2 + k > 0. These correlators are defined as integrals of certain cohomology classes over M g,k . Finding the values of these correlators is a difficult PDE problem, which has not been solved in general. They are difficult to compute, especially when they contain broad elements, so in many cases we still do not know how to compute even the A-model Frobenius algebra structure. In most cases we do not know how to compute the Frobenius manifold or higher genus structures. In this paper we make progress along these lines. In 2010, Krawitz [4] proved Conjecture 1.1 at the level of Frobenius algebra for almost every invertible polynomial W and G = Gmax W , the maximal symmetry group. It is more difficult to determine the structure in the A-model when G , Gmax W , because of the introduction of broad elements. In 2011 Johnson, Jarvis, Francis and Suggs [5] proved the conjecture at the Frobenius algebra level for any pair (W, G) of invertible polynomial and admissible symmetry group with the following property: Property (∗). Let W be an invertible, nondegenerate, quasihomogeneous polynomial and let G be an admissible symmetry group for W. We say the pair (W, G) has Property (∗) if PM (1) W can be decomposed as W = i=1 Wi where the Wi are themselves invertible polynomials having no variables in common with any other W j . (2) For any element g of G, where some monomial dm ; gc is an element of HW,G , and for each i ∈ {1, . . . , M}, g fixes either all of the variables in Wi or none of them. Even for W and G satisfying Property (∗), where we know the isomorphism class of the the Frobenius algebra, we usually still cannot compute the entire Frobenius manifold (the genus-zero correlators), nor the higher genus potential for the cohomological field theory. In fact, computing the full structure of either model is difficult, and has only been done in a few cases. The easiest examples of singularities are the so-called “simple” or ADE singularities. Fan, Jarvis, and Ruan computed the full A-model structure for these in [1]. The next examples come from representatives of the “Elliptic” singularities P8 , X9 , J10 , and their transpose singularities. Shen and Krawitz [6] calculated the entire A-model for T certain polynomial representatives of P8 , X9T and J10 , with maximal symmetry group. In 2013, Gu´er´e [7] provided an explicit formula for the cohomology classes Λ when W is an invertible chain type polynomial, and G is the maximal symmetry group for W.
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In 2008, Krawitz, Priddis, Acosta, Bergin, and Rathnakumara [8] worked out the Frobenius algebra structure for quasihomogeneous polynomial representatives of various singularities in Arnol’d’s list of unimodal and bimodal singularities [9]. We expand on their results, giving the Frobenius manifold structure for the polynomials on Arnol’d’s list, as well as their transpose polynomials. In many cases there is more than one admissible symmetry group for a polynomial. In this paper we consider all possible symmetry groups for each polynomial. Previously, almost no computations have been done in the case where the symmetry group is not maximal, because of the introduction of broad elements. However, this case is particularly interesting because the corresponding transpose symmetry group, used for B side computations, will not be trivial. Thus mirror symmetry predicts these cases will correspond to an orbifolded B-model. Much is still unknown about these orbifolded B-models. Knowing the corresponding A-model structures, though, give us a very concrete prediction for what these orbifolded B-models should be. Much of the work that has been done so far in this area has used a list of properties (“axioms”) of FJRW theory originally proved in [1] to calculate the values of correlators in certain cases. Primarily, these axioms have been used to compute genus-zero threepoint correlators, but many of them can be used to find information about higher genus and higher point correlators. One such axiom is the concavity axiom. This axiom gives a formula for some of the cohomology classes Λ of the cohomological field theory in terms of the top Chern class of a sum of derived pushforward sheaves. We use a result of Chiodo [10] to compute the Chern characters of the individual sheaves in terms of some cohomology classes in the moduli stack of W-curves Wg,k in genus zero. Then we use various properties of Chern classes to compute the top Chern class of the sum of the sheaves given the Chern character of each individual sheaf. Much is known about the cohomology M g,k , and not a lot about Wg,k , so we push down the cohomology classes in Wg,k to certain tautological classes ψi , κa and ∆I over M 0,k . In this way, we provide a method for expressing Λ as a polynomial in the tautological classes of M g,k . Correlators can then be computed by integrating Λ over M g,k , which is equivalent to calculating certain intersection numbers. Algorithms for computing these numbers are well established, for example in [11, 12]. Code in various platforms (for example [11] in Maple and [13] in Sage) has been written which computes the intersection numbers we need. I wrote code in Sage which performs each of the steps mentioned above to find the top Chern class of the sum of the derived pushforward sheaves, and then uses Johnson’s intersection code [13] to find intersection numbers. This allows us to compute certain correlator values which were previously unknown. In particular, in Lemma 3.5, we restate an explicit formula found in [1] for computing any concave genus-zero four-point correlators, with a proof not previously given. We also describe how to compute higher point correlators. To do this, we use a strengthened version of the Reconstruction Lemma of [1] to find values of non-concave correlators, with an aim to describe the full Frobenius manifold structure of many pairs (W, G) of singularities and groups. In many cases, these new methods allow us to compute Frobenius manifold structures for certain singularities and groups which were previously unknown. In 2014, Li, Li, Saito, and Shen [14] computed the entire A-model structure for the 14 polynomial representatives of exceptional unimodal singularities listed in [9], and their transpose polynomial representatives, with maximal symmetry group. In fact, for these polynomials and groups, they proved Conjecture 1.1. But there are also three non-maximal symmetry groups that appear in this family of polynomials: the minimal symmetry groups
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T for Z13 , Q12 , and U12 . The A-model for the first of these is computed in this paper (see Section 4.2), for the second, the A-model splits into the tensor product of two known A-models (see Section 4.1). We do not currently know a way to compute the A-model structure for U12 with minimal symmetry group (see Section 4.3). There are 51 pairs of invertible polynomials and admissible symmetry groups corresponding to those listed in [9], whose A-model structure is still unknown. Of these, 16 pairs have FJRW theories which split into the tensor products of theories previously computed (see Section 4.1). For 27 of the remaining pairs, we are able to use our computational methods to find the full Frobenius manifold structure of the corresponding A-model (see Section 4.2), including 13 examples with non-maximal symmetry groups. There are 8 pairs whose theories we still cannot compute using any known methods (see Section 4.3).
1.1. Acknowledgements. I would like to thank my advisor Tyler Jarvis for many helpful discussions that made this work possible. Thank also to those who have worked on the SAGE code which I used to do these computations, including Drew Johnson, Scott Mancuso and Rachel Webb. I would also like to thank Rachel Webb for pointing out some errors in earlier versions of this paper. 2. Background We begin by reviewing key facts about the construction of the A-models. This will require the choice of an admissible polynomial and an associated symmetry group. 2.1. Admissible Polynomials and Symmetry Groups. A polynomial W ∈ C[x1 , . . . , xn ], P Q a where W = i ci nj=1 x j i, j is called quasihomogeneous if there exist positive rational numbers q j (called weights) for each variable x j such that each monomial of W has weighted P degree one. That is, for every i where ci , 0, j q j ai, j = 1. A polynomial W ∈ C[x1 , . . . , xn ] is called nondegenerate if it has an isolated singularity at the origin. Any polynomial which is both quasihomogeneous and nondegenerate will be considered admissible for our purposes. We say that an admissible polynomial is invertible if it has the same number of variables and monomials. In this paper, we focus on computations involving only invertible polynomials. P The central charge cˆ of an admissible polynomial W is given by cˆ = j (1 − 2q j ). Next we define the maximal diagonal symmetry group. Definition 2.1. Let W be an admissible polynomial. The maximal diagonal symmetry n group Gmax W is the group of elements of the form g = (g1 , . . . , gn ) ∈ (Q/Z) such that W(e2πig1 x1 , e2πig2 x2 , . . . , e2πign xn ) = W(x1 , x2 , . . . , xn ). Remark 2.2. If q1 , q2 , . . . , qn are the weights of W, then the element J = (q1 , . . . , qn ) is an element of Gmax W . FJRW theory requires not only a quasihomogeneous, nondegenerate polynomial W, but also the choice of a subgroup G of Gmax W which contains the element J. Such a group is min called admissible. We denote hJi = Gmin and Gmax is W , and any subgroup between G admissible. Example 2.3. Consider the polynomial W = W1,0 = x4 + y6 . In this case that there are two admissible symmetry groups: hJi = h(1/4, 1/6)i, and Gmax W = h(1/4, 0), (0, 1/6)i. We shall use W = W1,0 and G = hJi for the rest of the examples in this section.
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2.2. Vector Space Construction. We now briefly review the construction of the A-model state space, as a graded vector space. We use the notation Ig = {i | g · xi = xi } to denote the set of indices of those variables fixed by an element g, and Fix(g) to denote the subspace of Cn which is fixed by g, Fix(g) = {(a1 , . . . , an )| such that ai = 0 whenever g · xi , xi }. The notation Wg will denote the polynomial W restricted to Fix(g). For the polynomial W with symmetry group G, the state space HW,G is defined in terms of Lefschetz thimbles and is equipped with a natural pairing, h•, •i : HW,G × HW,G → C which we will not define here. Definition 2.4. Let Hg,G be the G-invariants of the middle-dimensional relative cohomology G Hg,G = H mid (Fix(g), (W)−1 g (∞)) , where Wg−1 (∞) is a generic smooth fiber of the restriction of W to Fix(g). The state space is given by M Hg,G . HW,G = g∈G
C[x1 . . . , xn ] E . It has a natural Recall the Milnor Ring QW of the polynomial W is D ∂W ∂W ∂x1 , . . . , ∂xn residue pairing. Lemma 2.5 (Wall). Let ω = dx1 ∧ . . . ∧ dxn , then Ωn QW ω. dW ∧ Ωn−1 are isomorphic as GW -spaces, and this isomorphism respects the pairing on both. H0,G = H mid (Cn , (W)−1 (∞))
The isomorphism in Lemma 2.5 certainly will hold for the restricted polynomials Wg as well. This gives us the useful fact M M G Qg ωg , Hg,G (1) HW,G = g∈G
g∈G
where ωg = dxi1 ∧ . . . ∧ dxis for i j ∈ Ig . Notation 1. An element of HW,G is a linear combination of basis elements. We denote these basis elements by dm ; gc, where m is a monomial in C[xi1 , . . . , xir ] and {i1 , . . . , ir } = Ig . We say that dm ; gc is narrow if Ig = ∅ , and broad otherwise. l k The complex degree degC of a basis element α = m ; (g1 , . . . , gn ) is given by n
degC α =
X 1 Ni + (g j − q j ), 2 j=1
where Ng is the number of variables in Fix(gi ). By fixing an order for the basis, we can create a matrix which contains all the pairing information. This pairing matrix η is given by hD Ei η = αi , α j where {αi } is a basis for HW,G .
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2.3. The Moduli Space and basic properties. The FJRW cohomological field theory arises from the construction of certain cohomology classes on a finite cover of the moduli space of stable curves, the moduli space of W-orbicurves. 2.3.1. Moduli Spaces of Curves. The Moduli Space of stable curves of genus g over C with k marked points M g,k can be thought of as the set of equivalence classes of (possibly nodal) Riemann surfaces C with k marked points, p1 , . . . pk , where pi , p j if i , j. We require an additional stability condition, that the automorphism group of any such curve be finite. This means that 2g − 2 + k > 0 for each irreducible component of C, where k includes the nodal points. π − M g,k . We denote the universal curve over M g,k by C → The dual graph of a curve in M g,k is a graph with a node representing each irreducible component, an edge for each nodal point, and a half edge for each mark. Componenents with genus equal to zero are denoted with a filled-in dot while higher-genus components are denoted by a vertex labeled with the genus. Example 2.6. A nodal curve in M 1,3 and its dual graph are shown below. 1
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3
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2
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For each pair of non-negative integers g and k, with 2g − 2 + k > 0, the FJRW coho⊗k mological field theory produces for each k-tuple (α1 , . . . , αk ) ∈ HW,G a cohomology class ∗ ΛW g,k (α1 , α2 , ..., αk ) ∈ H (M g,k ). The definition of this class can be found in [2]. A genus-g, k-point correlator with insertions α1 , . . . , αk ∈ HW,G is defined by the integral Z
hα1 , . . . , αk ig,n =
M g,k
Λg,k (α1 , . . . , αk ) .
Finding the values of these correlators is a difficult PDE problem, which has not been solved in general. In spite of this, we can calculate enough of them to determine the full Frobenius manifold structure in many cases. Fan, Jarvis and Ruan [1] provide some axioms the Λ classes must satisfy, which, in some cases, allow us to determine their values. For example, these axioms give the following selection rules for non-vanishing correlators. Axiom 2.7 (FJR). Here we let αi = dmi ; gi c, with gi = (g1i , . . . , gni ).
(1) hα1 , . . . , αk ig,k = ασ(1) , . . . , ασ(k) g,k . For any permutation σ ∈ S k . P (2) hα1 , . . . , αk i0,k = 0 unless ki=1 degC αi = cˆ + k − 3. P (3) hα1 , . . . , αk i0,k = 0 unless q j (k − 2) − ki=1 gij ∈ Z for each j = 1, . . . n Notice that the last selection rule above gives the following lemma. Pk−1 Lemma 2.8. The correlator hα1 , . . . , αk i0,k = 0 unless gk = (k − 2) · J − i=1 gi , where max J ∈ GW is the element whose entries are the quasihomogeneous weights of W. The following splitting axiom will also be useful for us. Axiom 2.9 (FJR). if W1 ∈ C[x1 , . . . , xr ] and W2 ∈ C[xr+1 , . . . , xn ] are two admissible polynomials with symmetry groups G1 , and G2 , respectively, then HW1 +W2 ,G1 ⊕G2 HW1 ,G1 ⊗ HW2 ,G2
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where the Λ classes are related by ΛW1 +W2 (α1 ⊗ β1 , . . . , αk ⊗ βk ) = ΛW1 (α1 , . . . , αk ) ⊗ ΛW2 (β1 , . . . , βk ). We omit the rest of the axioms (except the Concavity Axiom, which we will discuss later), but refer the reader to [1] for the axioms, and [8] for a detailed explanation of how to use them to find genus-zero, three-point correlator values. 2.3.2. B-model Structure. For the unorbifolded B-model the Frobenius manifold is given by the Saito Frobenius manifold for a particular choice of primitive form (see [15]). For the orbifolded B-model the Frobenius manifold structure is still unknown. 3. Computational Methods Here we give results for computing concave genus-zero correlators and discuss how to use the reconstruction lemma to find values of other correlators. 3.1. Using the Concavity Axiom. We give a formula for Λ as a polynomial in the tautological classes ψi , κa , and ∆I in H ∗ (M g,k ). We then give a formula for computing concave genus-zero four-point correlators. Axiom 3.1 Suppose that all αi are narrow insertions of the form d1 ; gc (See Notation L(FJR). n 1). If π∗ i=1 Li = 0, then the cohomology class ΛW . . , αk ) can be given in terms of g,k (α1 , .L n 1 the top Chern class of the derived pushforward sheaf R π∗ i=1 Li : n −1 1 M |G|g W D (2) Λg,k (α1 , . . . , αk ) = PDst∗ PD (−1) cD R π∗ Li deg(st) i=1
Recall that integration of top dimensional cohomology classes is the same as pushing them forward to a point, or computing intersection numbers. In this section we express (−1)D cD R1 π∗ (L1 ⊕ . . . ⊕ LN ) as a polynomial f in terms of pullbacks of ψ, κ, and ∆I classes, then, which gives |G|g PDst∗ PD−1 (st∗ f (κ1 , . . . , κD , ψ1 , . . . , ψk , {∆I }I∈I )) hα1 , . . . , αk i = p∗ deg(st) = |G|g p∗ ( f (κ1 , . . . , κD , ψ1 , . . . , ψk , {∆I }I∈I )) , and this allows us to use intersection theory to solve for these numbers. Recall the following well-known property of chern classes from K-theory. ∞ X 1 1 = = (−ct (−E ))i (3) ct (E ) = ct (−E ) 1 − (−ct (−E )) i=0 A vector bundle E is concave when R0 π∗ E = 0, or when R• π∗ E = R0 π∗ E − R1 π∗ E = −R1 π∗ E Equation 3 gives ct (R π∗ 1
M i
j ∞ X M • Li ) = Li ) −ct (R π∗ i
j=0
Also, since the total Chern classes are multiplicative, we have, n M X Y • • (4) ck (R π∗ Li ) = ci j (R π∗ L j ). P i
i j =k
j=1
Together Equations 3 and 4 give a formula for finding the ith chern class of R1 π∗ terms of the chern classes of the R• π∗ L j .
L i
Li in
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It is well-known that Chern characters can be expressed in terms of Chern classes, but it is also possible to express Chern classes in terms of Chern characters (for example, in [16]). We have P i−1 • i ct (R• π∗ Lk ) = exp ∞ i=1 (i − 1)!(−1) chi (R π∗ Lk )t j (5) P∞ 1 P∞ i−1 • i i=1 (i − 1)!(−1) chi (R π∗ Lk )t j=0 j! Thus, there exists a polynomial f such that M (6) cD (R1 π∗ Li ) = f (ch1 (R• π∗ Lk ), . . . , chD (R• π∗ Lk )) i
In Section 3.2 we will use a result of Chiodo [10] to express chi (R• π∗ Lk ) in terms of tautological cohomology classes on M g,n and this will allow us to compute the corresponding correlators, but first we need to discuss some cohomology classes on Wg,n and M g,n and their relations. 3.1.1. Orbicurves. An orbicurve C with marked points p1 , . . . , pk is a stable curve C with orbifold structure at each pi and each node. Near each marked point pi there is a local group action given by Z/mi for some positive integer mi . Similarly, near each node p there is again a local group Z/n j whose action on one branch is inverse to the action on the other branch. In a neighborhood of pi , C maps to C via the map, % : C → C, where if z is the local coordinate on C near pi , and x is the local coordinate on C near pi , then %(z) = zr = x. Let KC be the canonical bundle of C. The log-canonical bundle of C is the line bundle KC,log = KC ⊗ O(p1 ) ⊗ . . . ⊗ O(pk ), where O(pi ) is the holomorphic line bundle of degree one whose sections may have a simple pole at pi . The log-canonical bundle of C is defined to be the pullback to C of the log-canonical bundle of C: KC ,log = %∗ KC,log . Given an admissible polynomial W, a W-structure on an orbicurve C is essentially P a choice of n line bundles L1 , ..., Ln so that for each monomial of W = j M j , with a M j = x1a j,1 . . . xn j,n , we have an isomorphism of line bundles ⊗a j,1
ϕ j : L1
⊗a j,n
. . . Ln
˜ C ,log . −→K
Recall that Fan, Jarvis and Ruan [1] defined a stack WW,g,k = {C , p1 , . . . , pk , L1 , . . . , LN , ϕ1 . . . ϕ s } of stable orbicurves with the additional W-structure, and the canonical morphism, st - Mg,k Wg,k from the stack of W-curves to the stack of stable curves, Mg,k . Notice that %∗ will take global sections to global sections, and a straightforward compu⊗s tation (see [1] §2.1 ) shows that if L on C such that L ⊗r KC , then ,log ⊗s (%∗ L )r = KC,log (−(r − mi )p).
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Recall that if L is the line bundle associated to the sheaf L and the action at an orbifold point pi on L is given in local coordinates by (z, v) 7→ (ζz, ζ mi v), then the action on L will ⊗s take a generator s of L near pi and map it to ζ (r−mi ) s. Thus, if L r KC on a smooth ,log mi orbicurve with action of the local group on L defined by ζ for mi > 0 at each marked point pi , then O r r ⊗s (%∗ L) = |L| = ωC,log ⊗ O((−mi )pi ) . i
Example 3.2. Suppose W and G are as in Example 2.3 with notation as in Notation 1. If, for a curve in M 0,4 the four marked points correspond to the A-model elements d1 ; (1/4, 1/2)c, d1 ; (1/4, 1/2)c, d1 ; (1/4, 1/2)c, and d1 ; (3/4, 5/6)c, then |L x |4 = ωC,log ⊗ O((−1)p1 ) ⊗ O((−1)p2 ) ⊗ O((−1)p3 ) ⊗ O((−3)p4 ) |Ly |6 = ωC,log ⊗ O((−3)p1 ) ⊗ O((−3)p2 ) ⊗ O((−3)p3 ) ⊗ O((−5)p4 ) 3.1.2. Some Special Cohomology Classes in M g,k and Wg,k . For i ∈ {1, . . . , k}, ψi ∈ H q (M g,k ) is the first Chern class of the line bundle whose fiber at (C, p1 , . . . , pk ) is the cotangent space to C at pi . In other words, if πk+1 : M g,k+1 = C → M g,k is the universal curve, and it is also the morphism obtained by forgetting the (k + 1)-st marked point, ωπk+1 = ω is the relative dualizing sheaf, and σi is the section of πk+1 which attaches a genus-zero, three-pointed curve to C at the point pi , and then labels the two remaining marked points on the genus-zero curve i and k + 1, M g,k+1 σi
πk+1 ? M g,k
then, L = σ∗ (ω) is the cotangent line bundle and its first Chern class is ψi : ψi = c1 (σ∗ (ω)), Let Di,k+1 be the image of σi in M g,k+1 , then we define k X K = c1 ω Di,k+1 , i=1
and for a ∈ {1, . . . , 3g − 3 + k},
κa = π∗ (K a+1 ).
Each partition I t J = {1, . . . , k} and g1 + g2 = g of marks and genus such that 1 ∈ I, 2g1 − 2 + |I| + 1 > 0 and 2g2 − 2 + |J| + 1 > 0, gives an irreducible boundary divisor, which we label ∆g1 ,I . These boundary divisors are the nodal curves in M g,k . For example, the boundary divisor ∆1,{1,2} in M 1,5 and its dual graph are given below. 2
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1 2
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We will use the following well-known lemma for M 0,k for expressing ψ classes in terms of boundary divisors.
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Lemma 3.3. ψi =
X
∆I
a∈I b,c 2
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Using the formulas in Equation 8 and the polynomial defined in Equation 6 we can now express Λ as a polynomial in ψ, κ and ∆ classes, ΛW g,k (α1 , . . . , αk ) P P B (γK ) (s/r) (mi /r) = (−1)D f B2(2)! κ1 − ni=1 B2 (2)! ψi + K 2(2)!+ ( jK )∗ (ψ− − ψ+ ) , . . . Pn BD+1 (mi /r) D P BD+1 (γ+K ) P BD+1 (s/r) i j (D+1)! κD − i=1 (D+1)! ψi + K (D+1)! ( jK )∗ ( i+ j=D−1 (−ψ+ ) ψ− ) The following lemma allows us to always choose a γ+K in a way that makes sense in Chiodo’s formula. Lemma 3.4. Let B be a degree one boundary graph, with decorations γ11 , . . . , γ1k1 for the first node and γ21 , . . . , γ2k2 for the second, and genera g1 and g2 , respectively. If a smooth curve with decorations γ11 , . . . , γ1k1 , γ21 , . . . , γ2k2 and genus g1 + g2 has integer line bundle degree, then it is possible to assign decorations to the edge of B such that each node will have integral line bundle degree. Proof. If the line bundle degree of the smooth curve is integral then: X X γ1i − γ2j ∈ Zn S 0 = J(2(g1 + g2 ) − 2 + k1 + k2 ) − i
j
n
Similarly let S 1 and S 2 be the equivalent sums in Q , corresponding to the nodes with decorations γ11 , . . . , γ1k1 and γ21 , . . . , γ2k2 , respectively. To find γ0 we take X γ0 ≡ J(2g1 − 2 + (k1 + 1)) − γ1i i
Then, by Lemma 2.8 this will force the S 1 to be an integer vector. Also, X S 0 = S 1 + γ0 + J(g2 − 2 + (k2 + 1)) − γ2j = S 1 + S 2 j
Which implies that S 2 ∈ Z.
We have now described the virtual class Λ in the concave case explicitly in terms of tautological classes on the moduli of stable curves. Now we will talk about how these can actually be computed. As a special case we have the following explicit formula for concave genus-zero four-point correlators. Lemma 3.5. If hd1 ; g1 c , d1 ; g2 c , d1 ; g3 c , d1 ; g4 ci is a genus-zero, four-point correlator which satisfies parts (2) and (3) of Axiom 2.7, and if it is also satisfies the hypotheses of the concavity axiom, (all insertions are narrow, all line bundle degrees are negative, and all line bundle degrees of the nodes of the boundary graphs ∆1,2 , ∆1,3 , and ∆1,4 are all negative), then N 4 3 X X 1 X j B2 (qi ) − B2 ((g j )i ) + B2 (γ+ ) hd1 ; g1 c , d1 ; g2 c , d1 ; g3 c , d1 ; g4 ci = 2 i=1 j=1 k=1 N 4 3 X X 1 X qi (qi − 1) − = θij (θij − 1) + γ+j (γ+j − 1) 2 i=1 j=1 k=1 where for each j g j = (θ1j , . . . , θnj ), γ+1 ≡ J − g1 − g2 , γ+2 ≡ J − g1 − g3 , and γ+3 ≡ J − g1 − g4 , and B2 is the 2nd Bernoulli polynomial.
12
AMANDA FRANCIS
Proof. cht (−R1 π∗ Li ) =
X Bd+1 (qi ) d≥0
(d + 1)!
κd
i 4 B X d+1 (θ j )
ψdj (d + 1)! j=1 d−1 X Bd+1 ((γ K )i ) X + +r (−ψ+ )k (ψ− )d−1−k td (pK )∗ (d + 1)! K k=0 −
Which means that ch1 (−R1 π∗ Li ) =
4 B (θi ) X X B2 ((γ+ )K ) 2 j B2 (qi ) κ1 − ψj + r (pK )∗ (1) (2)! (2)! (2)! K j=1
Notice that (pK )∗ (1M 0,3 ) = ∆K , and that for M 0,4 , our choices for K ∈ Γcut are just {1, 2}, {1, 3}, and {1, 4}. Numbering these gives: (γ+ )1 = J − g1 − g2 , (γ+ )2 = J − g1 − g3 , (γ+ )3 = J − g1 − g4 ,
for K = {1, 2}, for K = {1, 3}, and for K = {1, 4}.
Since B2 (x) = x2 − x + 61 , ch1 (−R1 π∗ Li )
1 1 qi (qi − 1) + κ1 2 6 4 X 1 θij (θij − 1) + ψ j − 6 j=1 ! X 1 ˜ K K γ+ (γ+ − 1) + ∆K . +r 6 K
=
The psi and kappa classes in W0,4 are all pullbacks of the equivalent psi and kappa classes in M , 0,4 , and the ∆I classes are scalar multiples of the equivalent classes in M , 0,4 , as in Equation 7 So, 4 X X 1 ch1 (−R1 π∗ Li ) = st∗ qi (qi − 1) κ1 − θij (θij − 1) ψ j + γ+K (γ+K − 1) ∆K . 2 K j=1 We use Equation 5 to convert to chern classes, ∞ X 1 i−1 1 i ct (−R π∗ Li ) = exp (−1) (i − 1)!chi (−R π∗ Li )t i=1 ∞ j ∞ X 1 X i−1 1 = (−1) (i − 1)!cht (−R π∗ Li )ti . j! i=1 j=0 Which means that c0 (−R1 π∗ Li ) = 1 and c1 (−R1 π∗ Li ) = ch1 (−R1 π∗ Li ), then, by Equation 4 since D = 1, X Y c1 (−R1 π∗ ⊕i Li ) = c ji (−R1 π∗ Li )
=
0≤ j1 ,... jN j1 +...+ jN =1 N X
N X
i=1
i=1
c1 (−R1 π∗ Li ) =
ch1 (−R1 π∗ Li ).
COMPUTATIONAL TECHNIQUES IN FJRW THEORY
Finally we recall that ct (R1 π∗ Li ) = −
P
j (ct (−R
1
13
π∗ Li )) j , so,
c1 (R1 π∗ Li ) = −c1 (−R1 π∗ Li ) = −
N X
ch1 (−R1 π∗ Li ).
i=1
And, from Equation 2 N X 1 ch1 (−R1 π∗ Li )) PDst∗ PD−1 (−1)1 (− deg(st) i=1 N X 1 ch1 (−R1 π∗ Li )) = PDst∗ PD−1 deg(st) i=1 ! 4 X X 1 1 1 1 i i = qi (qi − 1) + κ1 − θ j (θ j − 1) + ψ j + r γ+K (γ+K − 1) + ∆˜K . 2 6 6 6 K j=1
Λ0,4 (α1 , . . . , α4 ) =
Next, we notice that if p : M 0,4 → (•) is the map sending all of M 0,4 to a point, then the push forward of any of the cohomology classes mentioned above is equal to 1. That is, p∗ κ1 = p∗ ψi = p∗ ∆K = 1. So, hd1 ; g1 c , d1 ; g2 c , d1 ; g3 c , d1 ; g4 ci N X 1 = p∗ PDst∗ PD−1 ch1 (−R1 π∗ Li )) deg(st) i=1 N 4 X X 1 X K K i i (qi (qi − 1)) κ1 − γ+ (γ+ − 1) ∆K θ j (θ j − 1) ψ j + = p∗ 2 i=1 K j=1 N 4 X 1 X i i 1 1 2 2 3 3 qi (qi − 1) + γ+ (γ+ − 1) + γ+ (γ+ − 1) + γ+ (γ+ − 1) − = θ j (θ j − 1) . 2 i=1 j=1 Example 3.6. We will compute the genus-zero four-point correlator hd1 ; (1/4, 1/2)c , d1 ; (1/4, 1/2)c , d1 ; (1/4, 1/2)c , d1 ; (3/4, 5/6)ci0,4 for the A-model HW1,0 ,hJi . It is straightforward to verify that it satisfies Axiom 2.7, and that each of the degree one nodal degenerations of the dual graph are concave. Thus, 1 2
hd1 ; (1/4, 1/2)c , d1 ; (1/4, 1/2)c , d1 ; (1/4, 1/2)c , d1 ; (3/4, 5/6)ci0,4 = 3 3 3 3 3 3 3 3 5 5 5 5 − 16 − 16 − 16 − 16 + 16 + 16 + 16 + 16 + − 36 − 36 − 36 − 36 + 14 + 14 + 14 + 1 1 1 2 0+ 3 = 6
5 36
There are ten concave four-point correlators in HW1,0 ,hJi and 2 which are not concave. We will see in the next section that we can use the values of the concave correlators to find other correlator values. 3.3. Using the Reconstruction Lemma. In this section we show how to use known correlator values to find unknown correlator values. In some cases our new methods for computing concave correlators will allow us to compute all genus-zero correlators in the A-model. The WDVV equations are a powerful tool which can be derived from the Composition axiom. Applying these equations to correlators, we get the following lemma,
14
AMANDA FRANCIS
Lemma 3.7. [17] Reconstruction Lemma. Any genus-zero, k-point correlator of the form hγ1 , ..., γk−3 , α, β, ? φi0,k where 0 < degC (), degC (φ) < cˆ , can be rewritten as X X D E cI,J hγk∈I , α, , δl i δ0l , φ, β, γ j∈J hγ1 , ..., γk−3 , α, β, ? φi = −
ItJ=[k−3]
l
X
X
ItJ=[k−3] J,∅
l
D E cI,J hγk∈I , α, β, δl i δ0l , φ, , γ j∈J .
where the δl are the elements of Q some basis B and δ0l are the corresponding elements of (ξk )! the dual basis B 0 , and cI,J = Q nI (ξin)!KQ n JQ (ξ j )! . Here nX (ξ x ) refers to the number of elements equal to ξ x in the tuple X. The product nX (ξ x )! is taken over all distinct elements ξ x in X. We say that an element α ∈ HW,G is non-primitive if it can be written ? φ = α for some and φ in HW,G with 0 < degC , degC φ < degC α. Otherwise, we say that α is primitive. Corollary 3.8. A genus-zero, four-point correlator containing a non-primitive insertion can be rewritten: X D E X D E hγ, α, , δl i δ0l , ψ, β + hα, , δl i δ0l , ψ, β, γ hγ, α, β, ? ψi0.4 = l
l
−
X
D E hα, β, δl i δ0l , ψ, , γ .
l
In fact, using the reconstruction lemma, it is possible to write any genus-zero k-point correlator in terms of the pairing, genus-zero three-point correlators and correlators of the form hγ1 , . . . , γk0 i for k0 ≤ k where γi is primitive for i ≤ k0 − 2 (see [2]). We say that a correlator is basic if at most two of the insertions are non-primitive. P Recall that the central charge of the polynomial W is cˆ = j (1 − 2q j ). Lemma 3.9 ([2]). If degC (α) < cˆ for all classes α, P is the maximum complex degree of any primitive class, and P < 1, then all the genus-zero correlators are uniquely determined by the pairing, the genus-zero three-point correlators, and the basic genus-zero k-point correlators for each k that satisfies (9)
k ≤2+
1 + cˆ . 1−P
Example 3.10. In the case of HW1,0 ,hJi , the maximal primitive degree is 7/12 and cˆ = 7/6, so the Frobenius manifold structure is determined by the genus-zero three-point correlators and the basic genus-zero k-point correlators for k ≤ 7. All of the three-point correlators can be found using the methods described in [8]. There are 165 distinct genus-zero three point correlators. The selection rules show that 158 of them vanish. All of the seven remaining correlator values can be found using a pairing axiom found in [1]. There are ten concave and two nonconcave four-point correlators which satisfy Axiom 2.7, which might therefore be nonzero. We can use the values of the concave correlators to find values of the others using Lemma 3.7. For example, if we let l k X = d1 ; (1/4, 1/2)c , Y = d1 ; (1/2, 1/3)c , Z = d1 ; (3/4, 1/6)c , and W = xy2 ; (0, 0) ,
COMPUTATIONAL TECHNIQUES IN FJRW THEORY
15
then the nonconcave four point correlators are hX, W, W, X 2 i and hZ, Z, W, Wi. Lemma 3.7 gives X hX, W, W, X 2 i = 2hX, W, X, δihδ0 , W, Xi − hW, W, δihδ0 , X, X, Xi. δ
The selection rules, methods in [8], and computations from Example 3.6 reduce this to hX, W, W, X 2 i = −hW, W, 1ihXY 2 , X, X, Xi= −
1 1 1 · =− . 24 6 144
To find the value of hZ, Z, W, Wi we have to look for ways to reconstruct it using fivepoint correlators. hY, W, W, XY, XZi = hX, Y, W, XZihX, Z, W, XYi + hX, W, W, X 2 ihY, Z, Z, XYi −hW, XY, XZihX, X, Y, Z, Wi − hY, Y, W, XYihX, Z, W, XYi. The correlators hX, Z, W, XYi and hW, XY, XZi are both equal to zero, which can be shown using the methods in [8]. The correlators hY, Z, Z, XYi, and hX, W, W, X 2 i are both concave. Using Lemma 3.5, it is a straightforward calculation to find their values, which are 16 and 1 4 , respectively. So we have (10)
hY, W, W, XY, XZi = hX, W, W, X 2 ihY, Z, Z, XYi = −
1 1 1 · =− . 144 4 576
We can reconstruct this same correlator in a different way: (11)
hW, Y, XY, W, XZi = hX, Y, W, XZihY, Z, W, X 2 i + hX, Y, X 2 , XYihZ, Z, W, Wi −hW, XY, XZihX, X, Y, Z, Wi − hY, Y, W, XYihX, Z, W, XYi = hX, Y, X 2 , XYihZ, Z, W, Wi = 16 hZ, Z, W, Wi.
1 . Together, Equations 10 and 11 tell us that hZ, Z, W, Wi = − 96 To compute the full Frobenius manifold structure of HW1,0 ,hJi , we still need to find the values of all basic five, six, and seven-point correlators. There are fifteen basic five-point correlators, seven of which are not concave. There are five basic six-point correlators, three of which are not concave, and there is one basic seven-point correlator, which is not concave. All of these correlators can be computed using the methods established here. Their values are given in Section 4.2.
Remark 3.11. In certain examples, if we use Lemma 3.7 together with the computed values of concave genus-zero higher point correlators, we can find values of previously unknown three-point correlators. This allows us to find even Frobenius algebra structures that were previously unknown. 4. Summary of Computations We compute the FJRW theories with all possible admissible symmetry groups coming from the quasihomogeneous polynomial representatives corresponding to an isolated singularity in Arnol’d’s [9] list of singularities. Recall that there are potentially several polynomial representatives corresponding to a singularity. We note here that the polynomial representatives of X9 and J10 which appear in this list of singularities [9] were not considered in [6] for any choice of symmetry group. The non-maximal symmetry groups for P8 , also, have not yet been treated.
16
AMANDA FRANCIS
4.1. FJRW Theories which split into known tensor products. Recall that if a polynomial W can be written W = W1 + W2 , where W1 ∈ C[x1 , . . . , xl ] and W2 ∈ C[xl+1 , . . . , xn ], and if G can be written G = G1 ⊕ G2 , where for each i, Gi is an admissible symmetry group for Wi , then then the A-model associated to the pair (W, G) splits into the tensor product of the A-models for the pairs (W1 , G1 ) and (W2 , G2 ). The ADE singularities (except for D4 ) with all possible symmetry groups were computed in [1]. The D4 case was done in [17]. For the following pairs of polynomial and group, the A-model structure splits into a product of known FJRW theories. We use the notation AW,G to denote the full FJRW theory associated to the pair (W, G). Symmetry groups which are maximal are denoted with the subscript max. W X9 = x4 + y4 J10 = x3 + y6 Q12 = x3 + y5 + yz2 J3,0 = x3 + y9 W1,0 = x4 + y6 E18 = x3 + y10 E20 = x3 + y11 U16 = x3 + xz2 + y5 U16 = x3 + xz2 + y5 T U16 = x3 z + z2 + y5 W18 = x4 + y7 Q16 = x3 + yz2 + y7 Q16 = x3 + yz2 + y7 QT16 = x3 + z2 + y7 z Q18 = x3 + yz2 + y8
G h(1/4, 0), (0, 1/4)imax h(1/3, 0), (0, 1/6)imax h(1/3, 0, 0), (0, 1/5, 2/5)i h(1/3, 0), (0, 1/9)imax h(1/4, 0), (0, 1/6)imax h(1/3, 0), (0, 1/10)imax h(1/3, 0), (0, 1/11)imax h(1/3, 0, 5/6), (0, 1/5, 0)imax h(1/3, 0, 1/3), (0, 1/5, 0)i h(1/6, 0, 1/2), (0, 1/5, 0)imax h(1/4, 0), (0, 1/7)imax h(1/3, 0, 0), (0, 1/7, 13/14)imax h(1/3, 0, 0), (0, 1/7, 3/7)i h(1/3, 0, 0), (0, 1/14, 1/2)imax h(1/3, 0, 0), (0, 1/8, 7/16)imax
dimension 9 10 12 16 15 18 20 20 16 16 18 26 16 16 18
A-models AA3 ,J ⊗ AA3 ,J AA2 ,J ⊗ AA5 ,J AA2 ,J ⊗ AD6 ,J AA2 ,J ⊗ AA8 ,J AA3 ,J ⊗ AA5 ,J AA2 ,J ⊗ AA9 ,J AA2 ,J ⊗ AA10 ,J AD4 ,Gmax ⊗ AA4 ,J AD4 ,J ⊗ AA4 ,J ADT ,J ⊗ AA4 ,J 4 AA3 ,J ⊗ AA6 ,J AA2 ,J ⊗ AD8 ,Gmax AA2 ,J ⊗ AD8 ,J AA2 ,J ⊗ ADT ,J 8 AA2 ,J ⊗ AD9 ,J
4.2. Previously Unknown FJRW Theories we can compute. Recall that in Lemma 3.9 we saw that the Frobenius manifold structure can be found from the genus-zero threepoint correlators, together with the genus-zero k-point correlators for k as in Equation 9. The value of k depends on the central charge cˆ of the polynomial and on the maximum degree P of any primitive element in the state space. The value of k is given for each of the A-models below, together with all necessary correlators. Correlators marked with a ∗ must be found using the Reconstruction Lemma. Unmarked k-point correlators for k > 3 can be found using the concavity axiom. P8 = x3 + y3 + z3 , G = hγ = (1/3, 0, 2/3), γ2 = (0, 1/3, 2/3)i k ≤ 6, X = e0 , Y = xyze0 . P = 1/2, cˆ = 1 Relations: X 2 = Y 2 = 0; dim = 4 4 pt correlators 3 pt correlators None. 6 pt correlators h1, 1, XYi = 1/27 hX, X, Y, Y, XY, XYi∗ = 0 h1, X, Yi = 1/27 5pt correlators None.
COMPUTATIONAL TECHNIQUES IN FJRW THEORY
P8 = x3 + y3 + z3 , G = hJ = (1/3, 1/3, 1/3)i k ≤ 6, X = e0 , Y = xyze0 . P = 1/2, cˆ = 1 Relations: X 2 = Y 2 = 0; dim = 4 4 pt correlators 3 pt correlators None. 6 pt correlators h1, 1, XYi = 1/27 None. h1, X, Yi = 1/27 5pt correlators None. X9 = x4 + y4 , G = hJ = (1/4, 1/4), γ = (0, 1/2)i k ≤ 6, X = xye0 , Y = e J+γ , Z = e2J , W = e3J+γ . P = 1/2, cˆ = 1 Relations: 16X 2 = YW = Z 2 , X 3 = Z 3 = Y 2 = W 2 = 0. dim = 6 3 pt correlators 5 pt correlators 6 pt correlators h1, 1, Z 2 i = 1 hY, Y, Y, Y, Z 2 i = 81 hY, Y, Y, W, W 2 , W 2 i = 0 1 h1, Y, Wi = 1 hY, Y, Z, Z, W 2 , W 2 i = 32 hY, Y, W, W, Z 2 i = 0 1 h1, Z, Zi = 1 hY, Z, Z, W, W 2 i = 16 hY, W, W, W, W 2 , W 2 i = 0 1 1 h1, X, Xi = 161 hZ, Z, Z, Z, W 2 i = 16 hZ, Z, W, W, W 2 , W 2 i = 32 1 1 2 2 2 hW, W, W, W, W i = 8 hX, X, Y, Y, W , W i∗ = − 512 1 1 4 pt correlators hX, X, X, X, W 2 i∗ = 4096 hX, X, Z, Z, W 2 , W 2 i∗ = − 512 1 hX, X, Y, W, W 2 i∗ = 256 hY, Y, Y, Wi = 0 1 hX, X, Z, Z, W 2 i∗ = − 256 hY, Y, Z, Zi = 1/4 hY, W, W, Wi = 0 hZ, Z, W, Wi = 1/4 hX, X, Y, Yi∗ = −1/64 X9 = x4 + y4 , G = hJ = (1/4, 1/4)i k ≤ 6, X = y2 e0 , Y = xye0 , Z = x2 e0 , W = e2J . P = 1/2, cˆ = 1 Relations: 16XZ = 16Y 2 = W 2 , X 2 = Z 2 = Y 3 = W 3 = 0. dim = 6 3 pt correlators 5 pt correlators 6 pt correlators 1 h1, 1, W 2 i = 1 hW, W, W, W, W 2 i = 16 hX, X, Y, W, W 2 , W 2 i∗ = − 8a 1 h1, W, Wi = 1 hX, X, X, X, W 2 i∗ = 32a2 hY, Z, Z, W, W 2 , W 2 i∗ = 65536a 1 2 h1, X, Zi = 16 hX, X, Z, Z, W i∗ = 0 1 1 h1, Y, Yi = 16 hX, Y, Y, Z, W 2 i∗ = − 4096 1 2 hX, Z, W, W, W i∗ = − 256 1 2 4 pt correlators hY, Y, Y, Y, W i∗ = 4096 1 hY, Y, W, W, W 2 i∗ = − 256 hX, X, Y, Wi∗ = a , 0 1 1 hZ, Z, Z, Z, W 2 i∗ = 134217728a hY, Z, Z, Wi∗ = − 8192a 2
17
18
AMANDA FRANCIS
J10 = x3 + y6 , G = hJ = (1/3, 1/6)i k ≤ 6, X = y3 e0 , Y = xye0 , Z = e4J , W = e2J . P = 1/2, cˆ = 1 Relations: 18XY = ZW, X 2 = Y 2 = Z 2 = W 2 = 1, dim = 6 5 pt correlators 3 pt correlators hY, Y, Y, Y, ZWi = 19 h1, 1, ZWi = 1 hY, Z, Z, Z, ZWi = 181 h1, Z, Wi = 1 hX, X, X, Z, ZWi∗ = − a3 h1, X, Yi = 181 1 hX, Y, W, W, ZWi∗ = − 324 1 hY, Y, Y, Z, ZWi∗ = 314,928a 4 pt correlators hZ, Z, Z, Wi = 1/6 6 pt correlators hW, W, W, Wi = 1/3 hX, X, X, Wi∗ = a , 0 hZ, Z, W, W, ZW, ZWi = 541 1 1 hX, Y, Z, Zi∗ = − 108 hX, X, Y, Y, ZW, ZWi∗ = 17,496 1 1 hY, Y, Y, Wi∗ = − 104,976a hX, Y, Z, W, ZW, ZWi∗ = 1944 T Z13 = x3 + xy4 , G = hJ = (1/3, 1/9)i k ≤ 6, X = e4J , Y = e2J , Z = y5 e0 , W = xy2 e0 . P = 5/9, cˆ = 10/9. Relations: X 2 Y = −6Z 2 = 18W 2 , X 3 = Y 2 = Z 3 = W 3 = 0, dim = 8. 3 pt correlators 4 pt correlators h1, 1, X 2 Yi = 1 h1, X, XYi = 1 hX, X, X, X 2 Yi = 1/9 h1, Y, X 2 i = 1 hX, X, Yi = 1 hX, X, X 2 , XYi = 2/9 h1, Z, Zi = −1/6 h1, W, Wi = 1/18 hX, Y, X 2 , X 2 i = 1/9 hY, Y, Y, XYi = 1/3 5 pt correlators hX, Y, Z, XYi∗ = 0 hX, Y, Y, XY, X 2 Yi = 1/27 hX, Z, Z, X 2 i∗ = 1/54 hX, X, Z, XY, X 2 Yi = 1/27 hX, W, W, X 2 i∗ = −1/162 hX, Y, Z, X 2 , X 2 Yi∗ = 0 2 hY, Y, Z, X 2 i∗ = 0 hX, Z, Z, Z, X Yi∗ = 0 2 hX, Z, Z, Zi∗ = a = ±1/9 hX, Z, W, W, X Yi∗ = −2a/9 hX, Z, W, Wi∗ = a/36 hY, Y, Y, X 2 , X 2 Yi∗ = −a/162 6 pt correlators hY, Y, Z, Z, X 2 Yi∗ = 1/162 hY, Y, W, W, X 2 Yi∗ = −1/486 hX, Y, Z, Z, X 2 Y, X 2 Yi∗ = −1/1458 hY, Z, Z, XY, XYi∗ = 1/162 hX, Y, W, W, X 2 Y, X 2 Yi∗ = 1/4374 hY, W, W, XY, XYi∗ = −1/486 hY, Y, Y, Z, X 2 Y, X 2 Yi∗ = 0 hZ, Z, Z, X 2 , XYi∗ = −2a/9 hZ, Z, Z, Z, XY, X 2 Yi∗ = 1/2916 hZ, W, W, X 2 , XYi∗ = −a/162 hZ, Z, W, W, XY, X 2 Yi∗ = −1/26244 hW, W, W, W, XY, X 2 Yi∗ = 1/26244
COMPUTATIONAL TECHNIQUES IN FJRW THEORY
J3,0 = x3 + y9 , G = hJ = (1/3, 1/9)i k ≤ 6, X = e4J , Y = e2J , Z = y5 e0 , W = xy2 e0 . P = 5/9, cˆ = 10/9. Relations: 27ZW = X 2 Y, ab = −1/531441, X 3 = Y 2 = Z 2 = W 2 = 0, dim = 8. 3 pt correlators 5 pt correlators h1, 1, X 2 Yi = 1 h1, X, XYi = 1 hX, Y, Y, XY, X 2 Yi = 1/27 h1, Y, X 2 i = 1 h1, Z, Wi = 1/27 hX, Z, Z, Z, X 2 Yi∗ = −2a/9 hX, X, Yi = 1 hX, W, W, W, X 2 Yi∗ = −2b/9 hY, Y, Y, X 2 , X 2 Yi = 1/27 4 pt correlators hY, Y, Z, W, X 2 Yi∗ = −1/729 hX, X, X, X 2 Yi = 1/9 hY, Z, W, XY, XYi∗ = −1/729 hX, X, X 2 , XYi = 2/9 hZ, Z, Z, X 2 , XYi∗ = −2a/9 2 2 hX, Y, X , X i = 1/9 hW, W, W, X 2 , XYi∗ = −2b/9 hX, Z, W, X 2 i∗ = −1/243 6 pt correlators hY, Y, Y, XYi = 1/3 hX, Y, Z, W, X 2 Y, X 2 Yi∗ = 1/6561 hY, Z, Z, Zi = a hZ, Z, W, W, XY, X 2 Yi∗ = 4/177147 hY, W, W, Wi = b Z1,0 = x3 y + y7 , Gmax = hγ = (1/21, 6/7)i k ≤ 5, X = e5γ , Y = e13γ . P = 1/3, cˆ = 8/7. Relations: X 7 = −3Y 2 , X 13 = Y 3 = 0, dim = 19. 3 pt correlators 4 pt correlators h1, 1, X 5 Y 2 i = 1 h1, X, X 4 Y 2 i = 1 2 3 2 hX, X, X 5 Y, X 5 Y 2 i = 1/7 h1, X , X Y i = 1 h1, X 3 , X 2 Y 2 i = 1 hX, Y, X 6 , X 5 Y 2 i∗ = 1/7 h1, Y, X 5 Yi = 1 h1, X 4 , XY 2 i = 1 4 5 2 hX, Y, Y 2 , X 4 Y 2 i = −1/21 h1, XY, X Yi = 1 h1, X , Y i = 1 2 3 6 6 hX, Y, XY 2 , X 3 Y 2 i = −1/21 h1, X Y, X Yi = 1 h1, X , X i = −3 hX, Y, X 2 Y 2 , X 2 Y 2 i = −1/21 hX, X, X 3 Y 2 i = 1 hX, X 2 , X 2 Y 2 i = 1 hY, Y, Y, X 5 Y 2 i = 2/7 hX, X 3 , XY 2 i = 1 hX, Y, X 4 Yi = 1 4 2 3 hY, Y, XY, X 4 Y 2 i = 5/21 hX, X , Y i = 1 hX, XY, X Yi = 1 2 5 6 hY, Y, X 2 Y, X 3 Y 2 i = 4/21 hX, XY, X Yi = 1 hX, X , X i = −3 hY, Y, X 3 Y, X 2 Y 2 i = 1/7 hX, X 2 Y, X 2 Yi = 1 hX 2 , X 2 , XY 2 i = 1 hY, Y, X 4 Y, XY 2 i = 2/21 hX 2 , X 3 , Y 2 i = 1 hX 2 , Y, X 3 Yi = 1 hY, Y, Y 2 , X 5 Yi = 1/21 hX 2 , X 4 , X 6 i = −3 hX 2 , XY, X 2 Yi = 1 2 5 5 3 3 6 hX , X , X i∗ = −3 hX , X , X i = −3 5 pt correlators hX 3 , Y, X 2 Yi = 1 hX 3 , X 4 , X 5 i∗ = −3 hX 3 , XY, XYi = 1 hY, Y, X 5 i = 1 None hY, X 4 , XYi = 1 hX 4 , X 4 , X 4 i∗ = −3
19
20
AMANDA FRANCIS
Z1,0 = x3 y + y7 , G = hJ = (2/7, 1/7)i k ≤ 7, X = e4J , Y = e2J , Z = y4 e0 , W = xy2 e0 . P = 4/7, cˆ = 8/7. Relations: X 3 = −3Y 2 , 21ZW = XY 2 , a , 0, X 5 = Y 3 = Z 2 = W 2 = 0, dim = 9. 5 pt correlators hX, X, X, XY 2 , XY 2 i = 2/49 hX, Y, Y, Y 2 , XY 2 i = 0 hX, Z, Z, Z, XY 2 i∗ = −2a/7 3 pt correlators hX, Z, W, X 2 , XY 2 i∗ = 1/3087 h1, 1, XY 2 i = 1 h1, X, Y 2 i = 1 hX, Z, W, Y 2 , Y 2 i∗ = 1/1029 h1, Y, XYi = 1 h1, Z, Wi = 1/21 2 2 hX, W, W, W, XY 2 i∗ = −2l/7 h1, X , X i = −3 hX, Y, Yi = 1 hY, Y, Y, XY, XY 2 i = 2/49 hX, X, X 2 i = −3 hY, Y, Z, W, XY 2 i∗ = −2/1029 hY, Z, W, XY, Y 2 i∗ = −1/1029 hZ, Z, X, X 2 , Y 2 i∗ = −2a/7 4 pt correlators hZ, Z, Z, XY, XYi∗ = −2a/7 hX, X, Y, XY 2 i = 1/7 hZ, Z, W, W, Y 2 i∗ = −2/64827 hX, X, XY, Y 2 i = 2/7 hW, W, W, X 2 , Y 2 i∗ = −2/(1361367k) hX, Y, XY, XYi = 1/7 2 2 hW, W, W, XY, XYi∗ = −2l/7 hX, Y, X , Y i∗ = 3/7 hX, Z, W, XYi∗ = −1/147 6 pt correlators hY, Y, Y, Y 2 i = 1/7 hX, Y, Z, W, XY 2 , XY 2 i∗ = 5/7203 hY, Y, X 2 , XYi∗ = 0 hY, Z, Z, Z, Y 2 , XY 2 i∗ = 6a/49 hY, Z, Z, Zi∗ = a hY, W, W, W, Y 2 , XY 2 i∗ = 6a/49 hY, Z, W, X 2 i∗ = −1/147 hZ, Z, W, W, XY, XY 2 i∗ = 4/151263 hY, W, W, Wi∗ = −1/(194, 481a) 7 pt correlators None
COMPUTATIONAL TECHNIQUES IN FJRW THEORY
W1,0 = x4 + y6 , G = hJ = (1/4, 1/6)i k ≤ 7, X = e9J , Y = e2J , Z = e7J , W = xy2 e0 . P = 7/12, cˆ = 7/6. Relations: XZ = Y 2 , X 2 Z = 24W 2 , X 3 = Y 3 = Z 2 = W 3 = 0, dim = 9. 3 pt correlators h1, 1, XY 2 i = 1 h1, X, Y 2 i = 1 4 pt correlators h1, Y, XYi = 1 h1, Z, X 2 i = 1 hX, X, X, XY 2 i = 1/6 h1, W, Wi = 1/24 hX, X, Zi = 1 hX, X, X 2 , Y 2 i = 1/6 hX, Y, Yi = 1 hX, X, XY, XYi = 1/3 5 pt correlators hX, Y, X 2 , XYi = 1/6 2 hX, Z, Z, XZi = 0 hX, X, Z, XZ, X Zi = 0 hX, Z, X 2 , X 2 i = 0 hX, Y, Y, Y 2 , XY 2 i = 1/24 hX, W, W, X 2 i∗ = −1/144 hX, Y, Z, XY, XY 2 i = 1/24 2 2 hY, Y, Z, Y 2 i = 1/4 hX, Z, Z, X , X Zi = 0 2 hY, Y, X 2 , X 2 i = 1/6 hX, Z, W, W, X Zi∗ = 1/576 hY, Z, Z, XYi = 1/4 hX, W, W, XZ, XZi∗ = 0 hZ, Z, Z, X 2 i = 0 hY, Y, Y, XY, XY 2 i = 1/24 2 2 hZ, Z, W, Wi∗ = −1/96 hY, Y, Z, X , X Zi = 1/24 hY, Y, W, W, XY 2 i∗ = −1/576 hY, W, W, XY, Y 2 i∗ = −1/576 6 pt correlators hZ, Z, Z, Z, X 2 Zi = 1/8 hX, Z, Z, Z, X 2 Z, X 2 Zi = 0 hZ, Z, Z, XZ, XZi = 1/8 hX, X, W, W, X 2 Z, X 2 Zi∗ = −1/1728 hZ, W, W, X 2 , XZi∗ = 1/576 hY, Y, Z, Z, XY 2 , XY 2 i = 1/24 hZ, W, W, XY, XYi∗ = 0 hZ, Z, W, W, XZ, X 2 Zi∗ = −1/1152 hW, W, W, W, XZi∗ = 1/13824 hW, W, W, W, X 2 , X 2 Zi∗ = −1/27648 7 pt correlators hZ, W, W, W, W, X 2 Z, X 2 Zi∗ = 1/55296 Q2,0 = x3 + xy4 + yz2 , Gmax = hγ = (1/3, 11/12, 1/24)i k ≤ 5, X = e7γ , Y = e11γ . P = 7/24, cˆ = 7/6. Relations: 2X 7 = Y 3 , X 8 = Y 4 = 0 , dim = 17. 4 pt correlators 3 pt correlators hX, X, X 6 , X 7 Yi = 2/3 h1, 1, X 7 Yi = −2 h1, X, X 7 Yi = −2 hX, X, X 7 , X 6 Yi = 1/3 h1, X 2 , X 6 Yi = −2 h1, Y, X 7 i = −2 3 5 6 hX, Y, X 7 , X 7 i = −1/3 h1, X , X Yi = −2 h1, XY, X i = −2 4 3 2 5 hX, Y, Y 2 , Y 3 i∗ = −4/3 h1, X , X Yi = −2 h1, X Y, X i = −2 hY, Y, XY, X 7 Yi = −2/3 h1, Y 3 , Y 3 i = −4 hX, X, X 5 Yi = −2 hY, Y, X 2 Y, X 6 Yi = −2/3 hX, X 2 , X 4 Yi = −2 hX, Y, X 6 i = −2 3 3 5 hY, Y, X 3 Y, X 5 Yi = −2/3 hX, X , X Yi∗ = −2 hX, XY, X i = −2 4 2 2 5 hY, Y, X 4 Y, X 4 Yi∗ = −2/3 hX, X , X Yi = −2 hX , Y, X i = −2 hX 2 , X 2 , X 3 Yi∗ = −2 hX 2 , XY, X 4 i = −2 5 pt correlators hX 2 , X 3 , X 2 Yi∗ = −2 hY, X 3 , X 4 i = −2 2 hY, Y, Y i = −4 hX 3 , X 3 , XYi∗ = −2 None
21
22
AMANDA FRANCIS
Q2,0 = x3 + xy4 + yz2 , G = hJ = (1/3, 1/6, 5/12)i k ≤ 7, X = e10J , Y = e8J , Z = y3 e6J , W = xye6J . P = 7/12, cˆ = 7/6. Relations: X 3 Y = 8Z 2 = −24W 2 , X 4 = Y 2 = Z 3 = W 3 = 0 a = ±1, dim = 10. 3 pt correlators 4 pt correlators h1, 1, X 3 Yi = −2 h1, X, X 2 Yi = −2 hX, X, X 2 , X 3 Yi∗ = 2/3 h1, Y, X 3 i = −2 h1, X 2 , XYi = 1 hX, X, X 3 , X 2 Yi∗ = 4/3 h1, Z, Zi = −1/4 h1, W, Wi = 1/12 hX, Y, Z, X 2 Yi∗ = 0 2 hX, X, XYi∗ = −2 hX, Y, X i = 1 hX, Y, X 3 , X 3 i = 2/3 5 pt correlators hX, Z, Z, X 3 i∗ = −1/12 2 3 hX, Z, XY, XYi∗ = 0 hX, X, Z, X Y, X Yi∗ = 0 hX, W, W, X 3 i∗ = 1/36 hX, Y, Y, X 2 Y, X 3 Yi = 2/9 hY, Y, Y, X 2 Yi = −2/3 hX, Y, Z, X 3 , X 3 Yi∗ = 0 3 hY, Y, Z, X 3 i∗ = 0 hX, Z, Z, Z, X Yi∗ = a/72 3 hY, Y, XY, XYi∗ = −2/3 hX, Z, W, W, X Yi∗ = a/216 hY, X 2 , Z, XYi∗ = 0 hY, Y, Y, X 3 , X 3 Yi = 2/9 hY, Z, Z, Zi∗ = a/48 hY, Y, Z, Z, X 3 Yi∗ = −1/36 hY, Z, W, Wi∗ = a/144 hY, Y, W, W, X 3 Yi∗ = 1/108 hX 2 , X 2 , Z, Zi∗ = 1/12 hY, Z, Z, XY, X 2 Yi∗ = −1/36 hX 2 , X 2 , W, Wi∗ = 1/36 hY, W, W, XY, X 2 Yi∗ = −1/36 6 pt correlators hX 2 , Z, Z, Z, X 2 Yi∗ = a/72 hX 2 , Z, W, W, X 2 Yi∗ = a/216 hX, Y, Z, Z, X 3 Y, X 3 Yi∗ = −1/108 hZ, Z, Z, XY, X 3 i∗ = a/72 hX, Y, W, W, X 3 Y, X 3 Yi∗ = 1/324 hZ, W, W, XY, X 3 i∗ = a/216 hY, Y, Y, Z, X 3 Y, X 3 Yi∗ = 0 7 pt correlators hZ, Z, Z, Z, XY, X 3 Yi∗ = −1/288 2 3 3 hZ, Z, Z, Z, X 2 Y, X 2 Yi∗ = −1/288 hX , Z, Z, Z, Z, X Y, X Yi∗ = −1/216 2 3 3 hZ, Z, W, W, XY, X 3 Yi∗ = 1/2592 hX , Z, Z, W, W, X Y, X Yi∗ = 1/1944 hZ, Z, W, W, X 2 Y, X 2 Yi∗ = 1/2592 hX 2 , W, W, W, W, X 3 Y, X 3 Yi∗ = −1/1944 hW, W, W, W, XY, X 3 Yi∗ = −1/2592 hW, W, W, W, X 2 Y, X 2 Yi∗ = −1/2592
COMPUTATIONAL TECHNIQUES IN FJRW THEORY
QT2,0 = x3 y + y4 z + z2 ,
Gmax = hJ = (7/24, 1/8, 1/2)i
k ≤ 5, X = e11J , Y = e9J , Z = y3 e8J . P = 5/12, cˆ = 7/6. Relations: X 4 = −3Y 2 , aX 3 Y = Z 2 , X 8 = Y 3 = Z 3 = 0, a , 0, dim = 14. 4 pt correlators hX, X, X 2 Y, X 3 Y 2 i = 1/8 hX, X, X 3 Y, X 2 Y 2 i = 1/8 hX, Y, Y 2 , X 2 Y 2 i = −1/12 3 pt correlators hX, Y, X 3 Y, X 3 Yi = 0 3 2 2 2 hX, Y, XY 2 , XY 2 i = −1/12 h1, 1, X Y i = 1 h1, X, X Y i = 1 2 2 3 hX, Y, X 3 , X 3 Y 2 i∗ = 1/8 h1, X , XY i = 1 h1, Y, X Yi = 1 hX, Z, Z, X 3 Y 2 i∗ = −a/8 h1, X 3 , Y 2 i = 1 h1, XY, X 2 Yi = a hX, Z, YZ, X 3 Yi∗ = −a/8 h1, Z, YZi = 1 hX, X, XY 2 i = 1 2 2 2 hX 2 , Z, Z, X 2 Y 2 i∗ = −a/8 hX, X , Y i = 1 hX, Y, X Yi = 1 3 3 hY, Y, Y, X 3 Y 2 i = 7/24 hX, XY, XYi = 1 hX, X , X i∗ = −3 hY, Y, XY, X 2 Y 2 i = 5/24 hX 2 , Y, XYi = 1 hX 2 , X 2 , X 3 i∗ = −3 hY, Y, X 2 Y, XY 2 i = 1/8 hY, Z, Zi∗ = a hY, Y, Y 2 , X 3 Yi = 1/24 5 pt correlators 3 2 3 2 hY, Y, YZ, YZi∗ = a/4 hX, Y, Y, X Y , X Y i∗ = −1/96 hY, Z, Y 2 , YZi∗ = −a/24 hY, Z, Z, XY 2 , X 3 Y 2 i∗ = −a/96 hZ, Z, X 3 , XY 2 i∗ = −a/8 hY, Z, Z, X 2 Y 2 , X 2 Y 2 i∗ = −a/96 hZ, Z, XY, X 3 Yi∗ = −a/8 hZ, Z, X 2 Y, X 2 Yi∗ = −a/8 hZ, Z, Y 2 , Y 2 i∗ = a/24 S 1,0 = x2 y + y2 z + z5 , Gmax = hγ = (1/20, 9/10, 1/5)i k ≤ 4, X = e7γ , Y = eγ . P = 1/4, cˆ = 6/5. Relations: X 5 = Y 3 , X 9 = Y 4 = 0, dim = 17. 3 pt correlators h1, 1, X 3 Y 3 i = −2 h1, X, X 2 Y 3 i = −2 4 pt correlators 3 2 h1, Y, X Y i = −2 h1, X 2 , XY 3 i = −2 hX, X, X 3 Y, X 3 Y 3 i = −2/5 h1, XY, X 2 Y 2 i = −2 h1, X 3 , Y 3 i = −2 hX, X, X 3 Y 2 , X 3 Y 2 i = −2/5 h1, Y 2 , X 3 Yi = −2 h1, X 2 Y, XY 2 i = −2 hX, Y, X 4 , X 3 Y 3 i = −2/5 h1, X 4 , X 4 i = −2 hX, X, XY 3 i = −2 hX, Y, Y 3 , X 2 Y 3 i∗ = −2/5 hX, Y, X 2 Y 2 i = −2 hX, X 2 , Y 3 i = −2 hX, Y, XY 3 , XY 3 i∗ = 1/5 hX, XY, XY 2 i = −2 hX, Y 2 X 2 Yi = −2 hY, Y, Y 2 , X 3 Y 3 i∗ = 8/5 hX, X 3 , X 4 i = −2 hY, Y, X 3 Yi∗ = −2 hY, Y, XY 2 , X 2 Y 3 i∗ = 6/5 hY, X 2 , XY 2 i = −2 hY, XY, X 2 Yi∗ = −2 hY, Y, Y 3 , X 3 Y 2 i∗ = −2/5 hY, X 3 , Y 2 i = −2 hX 2 , X 2 , X 4 i∗ = −2 hY, Y, X 2 Y 2 , XY 3 i∗ = 4/5 hX 2 , XY, Y 2 i = −2 hX 2 , X 3 , X 3 i∗ = −2 hXY, XY, XYi = −2
23
24
AMANDA FRANCIS
S 1,0 = x2 y + y2 z + z5 , G = hJ = (3/10, 2/5, 1/5)i k ≤ 7, X = e2J , Y = e7J , Z = e6J , W = z2 e5J . P = 3/5, cˆ = 6/5. Relations: 2XZ = aY 2 , X 3 = 2aYZ, XYZ = 10W 2 , X 5 = Y 4 = Z 2 = 0, a , 0, dim = 10. 3 pt correlators 4 pt correlators h1, 1, XYZi = 1 h1, X, YZi = 1 hX, X, Y, XY 2 i = 1/5 h1, Y, XZi = 1 h1, Z, XYi = 1 hX, X, XY, YZi = 1/5 h1, W, Wi = 1/10 h1, X 2 , X 2 i = 2a hX, X, XZ, XZi∗ = a/5 hX, Y, Zi = 1 hX, X, X 2 i∗ = 2a hX, Y, X 2 , YZi∗ = 2/5 hY, Y, Yi∗ = 2/a hX, Y, XY, XZi = 1/5 5 pt correlators hX, Z, Z, YZi = 1/10 hX, Z, X 2 , XZi∗ = a/5 hX, X, Z, YZ, XYZi = 1/25 hX, Z, XY, XYi = 0 hX, Y, Y, YZ, XYZi∗ = 0 hX, W, W, XYi = −1/50 hX, Y, Z, XZ, XYZi = −1/50 hY, Y, Z, YZi∗ = −1/(5a) hX, Z, Z, XY, XYZi = 0 hY, Y, X 2 , XZi∗ = 0 hX, Z, W, W, XYZi∗ = −3/500 hY, Y, XY, XYi∗ = 2/(5a) hX, W, W, XZ, YZi∗ = −1/500 hY, Z, Z, XZi = −1/5 hY, Y, Y, XZ, XYZi∗ = −2/(25a) hY, Z, X 2 , XYi∗ = 0 hY, Y, Z, XY, XYZi∗ = −2/(25a) hY, W, W, X 2 i∗ = −1/50 hY, Y, W, W, XYZi∗ = 1/(125a) hZ, Z, Z, XYi = −1/10 hY, Z, Z, X 2 , XYZi∗ = 0 hZ, Z, W, Wi∗ = 1/50 hY, Z, Z, YZ, YZi∗ = 1/(25a) hZ, Z, X 2 , X 2 i∗ = 2/5 hY, W, W, XY, YZi∗ = 1/(250a) 6 pt correlators hY, W, W, XZ, XZi∗ = 1/250 hZ, Z, Z, Z, XYZi = 3/25 hX, Z, Z, Z, XYZ, XYZi = 0 hZ, Z, Z, XZ, YZi = 2/25 hX, X, Z, Z, XYZ, XYZi∗ = 1/625 hZ, W, W, X 2 , YZi∗ = −1/125 hY, Y, Z, Z, XYZ, XYZi∗ = 4/(125a) hZ, W, W, XY, XZi∗ = −1/500 hY, Z, W, W, YZ, XYZi∗ = 1/(2500a) hW, W, W, W, XZi∗ = −1/5000 hZ, Z, W, W, XZ, XYZi∗ = −3/2500 7 pt correlators hW, W, W, W, X 2 , XYZi∗ = −9/25000 hW, W, W, W, YZ, YZi∗ = 3/(12500a) hY, W, W, W, W, XYZ, XYZi = −3/(62500a) E19 = x3 + xy7 , Gmax = hJ = (1/3, 2/21)i k ≤ 5, X = e13J , Y = e11J . P = 2/7, cˆ = 8/7. Relations: −7X 6 = Y 3 , X 7 = Y 5 = 0, dim = 15. 3 pt correlators 4 pt correlators h1, 1, X 6 Yi = 1 h1, X, X 5 Yi = 1 hX, X, X 5 , X 6 Yi = 2/21 6 2 4 h1, Y, X i = 1 h1, X , X Yi = 1 hX, X, X 6 , X 5 Yi = 1/21 h1, XY, X 5 i = 1 h1, X 3 , X 3 Yi = 1 hX, Y, X 6 , X 6 i = −1/21 h1, X 2 Y, X 4 i = 1 h1, Y 2 , Y 2 i = −7 hX, Y, Y 2 , X 6 Yi∗ = 1/3 hX, X, X 4 Yi = 1 hX, Y, X 5 i = 1 hY, Y, XY, X 6 Yi = 1/3 hX, X 2 , X 3 Yi = 1 hX, XY, X 4 i = 1 hY, Y, X 2 Y, X 5 Yi = 1/3 hX, X 3 , X 2 Yi = 1 hX 2 , X 2 , Y 2 i = 1 hY, Y, X 3 Y, X 4 Yi = 1/3 hX 2 , Y, X 2 Yi = 1 hX 2 , XY, XYi = 1 hX 3 , Y, XYi = 1 hY, X 2 , X 4 i = 1 5 pt correlators hY, X 3 , X 3 i = 1 hY, Y, Y 2 i = −7 None 2 2 2 2 3 hX , X , X Yi = 1 hX , XY, X i = 1
COMPUTATIONAL TECHNIQUES IN FJRW THEORY T Z17 = x3 + xy8 , Gmax = hγ = (2/3, 1/24)i k ≤ 5, X = e5γ , Y = eγ . P = 7/24, cˆ = 7/6. Relations: −8X 7 = Y 3 , X 8 = XY 2 = Y 4 = 0, dim = 17. 3 pt correlators 4 pt correlators h1, 1, X 7 Yi = 1 h1, X, X 6 Yi = 1 hX, X, X 6 , X 7 Yi = 1/12 h1, X 2 , X 5 Yi = 1 h1, Y, X 7 i = 1 hX, X, X 7 , X 6 Yi = 1/24 h1, X 3 , X 4 Yi = 1 h1, XY, X 6 i = 1 hX, Y, X 7 , X 7 i = −1/24 h1, X 4 , X 3 Yi = 1 h1, X 2 Y, X 5 i = 1 hX, Y, Y 2 , X 7 Yi∗ = 1/3 h1, Y 2 , Y 2 i = −8 hX, X, X 5 Yi = 1 hY, Y, XY, X 7 Yi = 1/3 hX, X 2 , X 4 Yi = 1 hX, Y, X 6 i = 1 hY, Y, X 2 Y, X 6 Yi = 1/3 hX, X 3 , X 3 Yi = 1 hX, XY, X 5 i = 1 hY, Y, X 3 Y, X 5 Yi = 1/3 hX, X 4 , X 2 Yi = 1 hX 2 , X 2 , X 3 Yi = 1 hY, Y, X 4 Y, X 4 Yi = 1/3 hX 2 , Y, X 5 i = 1 hX 2 , X 3 , X 2 Yi = 1 hX 2 , XY, X 4 i = 1 hX 3 , Y, XYi = 1 5 pt correlators hY, Y, Y 2 i = ±1 hY, X 3 , X 4 i = 1 None 3 3 hX , X , XYi = 1 T Z17 = x3 + xy8 , G = hJ = (1/3, 1/12)i k ≤ 7, X = e4J , Y = e2J , Z = y7 e0 , W = xy3 e0 . P = 7/12, cˆ = 7/6. Relations: X 3 Y = −8Z 2 = 24W 2 , X 4 = Y 2 Z 3 = W 3 , a, b , 0 , dim = 10. 5 pt correlators hX, Y, Y, X 2 Y, X 3 Yi = 1/36 3 pt correlators hX, X, Z, X 2 Y, X 3 Yi∗ = 0 h1, 1, X 3 Y i = 1 h1, X, X 2 Yi = 1 hX, Y, Z, X 3 , X 3 Yi∗ = 0 h1, Y, X 3 i = 1 h1, X 2 , XYi = 1 hX, Z, Z, Z, X 3 Yi∗ = −a/6 h1, Z, Zi = −1/8 h1, W, Wi = 1/24 hX, Z, W, W, X 3 Yi∗ = −b/6 hX, X, XYi = 1 hX, Y, X 2 i = 1 hY, Y, Y, X 3 , X 3 Yi = 1/36 4 pt correlators hY, Y, Z, Z, X 3 Yi∗ = 1/288 hX, X, X 2 , X 3 Yi = 1/12 hY, Y, W, W, X 3 Yi∗ = −1/864 hX, X, X 3 , X 2 Yi = 1/6 hY, Z, Z, XY, X 2 Yi∗ = 1/288 hX, Y, X 3 , X 3 i = 1/12 hY, W, W, XY, X 2 Yi∗ = −1/864 hX, Y, Z, X 2 Yi∗ = 0 hX 2 , Z, Z, Z, X 2 Yi∗ = −a/6 hX, Z, Z, X 3 i∗ = 1/96 hX 2 , Z, W, W, X 2 Yi∗ = −b/6 hX, Z, XY, XYi∗ = 0 hZ, Z, Z, XY, X 3 i∗ = −a/6 hX, W, W, X 3 i∗ = −1/288 hZ, W, W, XY, X 3 i∗ = −b/6 hY, Y, Y, X 2 Yi = 1/3 hY, Y, XY, XYi = 1/3 6 pt correlators hY, Y, Z, X 3 i∗ = 0 hX, Y, Z, Z, X 3 Y, X 3 Yi∗ = −1/3456 2 hY, X , Z, XYi∗ = 0 hX, Y, W, W, X 3 Y, X 3 Yi∗ = 1/10368 hY, Z, Z, Zi∗ = a = ±1/192 hY, Y, Y, Z, X 3 Y, X 3 Yi∗ = 0 hY, Z, W, Wi∗ = b = ±1/576 hZ, Z, Z, Z, XY, X 3 Yi∗ = 1/9216 hX 2 , X 2 , Z, Zi∗ = 1/96 hZ, Z, Z, Z, X 2 Y, X 2 Yi∗ = 1/9216 hX 2 , X 2 , W, Wi∗ = −1/288 hZ, Z, W, W, XY, X 3 Yi∗ = −1/82944 hZ, Z, W, W, X 2 Y, X 2 Yi∗ = −1/82944 7 pt correlators hW, W, W, W, XY, X 3 Yi = 1/82944 None hW, W, W, W, X 2 Y, X 2 Yi = 1/82944
25
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AMANDA FRANCIS
Z19 = x3 y + y9 , Gmax = hJ = (8/27, 1/9)i k ≤ 5, X = e11J , Y = e19J . P = 1/3, cˆ = 32/27. Relations: X 9 = −3Y 2 , X 16 = Y 3 = 0, dim = 25. 4 pt correlators 3 pt correlators hX, X, X 7 Y, X 7 Y 2 i = 1/9 7 2 6 2 h1, 1, X Y i = 1 h1, X, X Y i = 1 hX, X 2 , X 6 Y, X 7 Y 2 i = 1/9 h1, X 2 , X 5 Y 2 i = 1 h1, X 3 , X 4 Y 2 i = 1 hX, X 2 , X 7 Y, X 6 Y 2 i = 1/9 h1, X 4 , X 3 Y 2 i = 1 h1, Y, X 7 Y 2 i = 1 hX, X 3 , X 5 Y, X 7 Y 2 i = 1/9 h1, X 5 , X 2 Y 2 i = 1 h1, XY, X 6 Yi = 1 hX, X 3 , X 6 Y, X 6 Y 2 i = 1/9 h1, X 6 , XY 2 i = 1 h1, X 2 Y, X 5 Yi = 1 hX, X 3 , X 7 Y, X 5 Y 2 i = 1/9 h1, X 7 , Y 2 i = 1 h1, X 3 Y, X 4 Yi = 1 hX, X 4 , X 4 Y, X 7 Y 2 i = 1/9 h1, X 8 , X 8 i = −3 hX, X, X 5 Y 2 i = 1 hX, X 4 , X 5 Y, X 6 Y 2 i = 1/9 hX, X 2 , X 4 Y 2 i = 1 hX, X 3 , X 3 Y 2 i = 1 hX, X 4 , X 6 Y, X 5 Y 2 i = 1/9 hX, X 4 , X 2 Y 2 i = 1 hX, Y, X 6 Yi = 1 hX, X 4 , X 7 Y, X 4 Y 2 i = 1/9 hX, X 5 , XY 2 i = 1 hX, XY, X 5 Yi = 1 2 , X 2 , X 5 Y, X 7 Y 2 i = 1/9 hX hX, X 6 , Y 2 i = 1 hX, X 2 Y, X 4 Yi = 1 2 hX , X 2 , X 6 Y, X 6 Y 2 i = 2/9 hX, X 7 , X 8 i = −3 hX, X 3 Y, X 4 Yi = 1 hX 2 , X 2 , X 7 Y, X 5 Y 2 i = 1/9 hX 2 , X 2 , X 3 Y 2 i = 1 hX 2 , X 3 , X 2 Y 2 i = 1 hX 2 , X 3 , X 4 Y, X 7 Y 2 i = 1/9 hX 2 , X 4 , XY 2 i = 1 hX 2 , Y, X 5 Yi = 1 hX 2 , X 3 , X 5 Y, X 6 Y 2 i = 2/9 hX 2 , X 5 , Y 2 i = 1 hX 2 , XY, X 4 Yi = 1 hX 2 , X 3 , X 6 Y, X 5 Y 2 i = 2/9 hX 2 , X 6 , X 8 i = −3 hX 2 , X 2 Y, X 3 Yi = 1 hX 2 , X 3 , X 7 Y, X 4 Y 2 i = 1/9 hX 2 , X 7 , X 7 i∗ = −3 hX 3 , X 3 , XY 2 i = 1 hX 2 , X 4 , X 3 Y, X 7 Y 2 i = 1/9 hX 3 , X 4 , Y 2 i = 1 hX 3 , Y, X 4 Yi = 1 hX 2 , X 4 , X 4 Y, X 6 Y 2 i = 2/9 hX 3 , X 5 , X 8 i = −3 hX 3 , XY, X 3 Yi = 1 hX 2 , X 4 , X 5 Y, X 5 Y 2 i = 2/9 hX 3 , X 6 , X 7 i∗ = −3 hX 3 , X 2 Y, X 2 Yi = 1 hX 2 , X 4 , X 6 Y, X 4 Y 2 i = 2/9 hX 4 , X 4 , X 8 i = −3 hX 4 , Y, X 3 Yi = 1 hX 2 , X 4 , X 7 Y, X 3 Y 2 i = 1/9 hX 4 , X 5 , X 7 i∗ = −3 hX 4 , XY, X 2 Yi = 1 hX 3 , X 3 , X 3 Y, X 7 Y 2 i = 1/9 hX 4 , X 6 , X 6 i∗ = −3 hY, Y, X 7 i = 1 hX 3 , X 3 , X 4 Y, X 6 Y 2 i = 2/9 hY, X 5 , X 2 Yi = 1 hY, XY, X 6 i = 1 hX 3 , X 3 , X 5 Y, X 5 Y 2 i = 1/3 hX 5 , X 5 , X 6 i∗ = −3 hX 5 , XY, XYi = 1 hX 3 , X 3 , X 6 Y, X 4 Y 2 i = 2/9 hX 3 , X 3 , X 7 Y, X 3 Y 2 i = 1/9 5 pt correlators hX 3 , X 4 , X 2 Y, X 7 Y 2 i = 1/9 hX 4 , X 4 , X 4 , X 6 Y 2 , X 7 Y 2 i = 4/81 hX 3 , X 4 , X 3 Y, X 6 Y 2 i = 2/9 hX 3 , X 4 , X 4 , X 7 Y 2 , X 7 Y 2 i∗ = 2/81 hX 3 , X 4 , X 4 Y, X 5 Y 2 i = 1/3 hX 3 , X 4 , X 5 Y, X 4 Y 2 i = 1/3 hX 3 , X 4 , X 6 Y, X 3 Y 2 i = 2/9 hX 3 , X 4 , X 7 Y, X 2 Y 2 i = 1/9 hX 4 , X 4 , XY, X 7 Y 2 i = 1/9 hX 4 , X 4 , X 2 Y, X 6 Y 2 i = 2/9 hX 4 , X 4 , X 3 Y, X 5 Y 2 i = 1/3 hX 4 , X 4 , X 4 Y, X 4 Y 2 i = 4/9 hX 4 , X 4 , X 5 Y, X 3 Y 2 i = 1/3 hX 4 , X 4 , X 6 Y, X 2 Y 2 i = 2/9 hX 4 , X 4 , X 7 Y, XY 2 i = 1/9
COMPUTATIONAL TECHNIQUES IN FJRW THEORY T Z19 = x3 + xy9 , Gmax = hJ = (1/3, 2/27)i k ≤ 5, X = e16J , Y = e14J . P = 8/27, cˆ = 32/27. Relations: −9X 8 = Y 3 , X 9 = Y 5 = 0, dim = 19. 3 pt correlators h1, 1, X 8 Yi = 1 h1, X, X 7 Yi = 1 2 6 4 pt correlators h1, X , X Yi = 1 h1, Y, X 8 i = 1 hX, X, X 7 , X 8 Yi = 2/27 h1, X 3 , X 5 Yi = 1 h1, XY, X 7 i = 1 hX, X, X 8 , X 7 Yi = 1/27 h1, X 4 , X 4 Yi = 1 h1, X 2 Y, X 6 i = 1 5 3 2 2 hX, Y, X 8 , X 8 i = −1/27 h1, X , X Yi = 1 h1, Y , Y i = −9 hX, Y, Y 2 , X 8 Yi∗ = 1/3 hX, X, X 6 Yi = 1 hX, X 2 , X 5 Yi = 1 hY, Y, XY, X 8 Yi = 1/3 hX, Y, X 7 i = 1 hX, X 2 , X 5 Yi = 1 6 3 4 hY, Y, X 2 Y, X 7 Yi = 1/3 hX, XY, X Yi = 1 hX, X , X Yi = 1 2 5 4 3 hY, Y, X 3 Y, X 6 Yi = 1/3 hX, X Y, X i = 1 hX, X , X Yi = 1 hY, Y, X 4 Y, X 5 Yi = 1/3 hX 2 , X 2 , X 4 Yi = 1 hX 2 , Y, X 6 i = 1 5 pt correlators hX 2 , X 3 , X 3 Yi = 1 hX 2 , XY, X 5 i = 1 hX 2 , X 4 , X 2 Yi = 1 hY, Y, Y 2 i = −9 None hY, X 3 , X 5 i = 1 hY, X 4 , X 4 i = 1 hX 3 , X 3 , X 2 Yi = 1 hX 3 , XY, X 4 i = 1
W17 = x4 + Xy5 , Gmax = hJ = (1/4, 3/20)i k ≤ 4, X = e14J , Y = e9J . P = 1/5, cˆ = 6/5. Relations: X 4 = −5Y 4 , X 7 = Y 5 = 0, dim = 16. 3 pt correlators h1, 1, X 2 Y 4 i = 1 h1, X, XY 4 i = 1 4 pt correlators h1, Y, X 2 Y 3 i = 1 h1, X 2 , Y 4 i = 1 3 h1, XY, XY i = 1 h1, Y 2 , X 2 Y 2 i = 1 hX, X, X 2 Y, X 2 Y 4 i = 1/4 h1, X 2 Y, Y 3 i = 1 h1, XY 2 , XY 2 i = 1 hX, X, X 2 Y 2 , X 2 Y 3 i = 1/4 h1, X 3 , X 3 i = −5 hX, X, Y 4 i = 1 hX, Y, X 3 , X 2 Y 4 i∗ = 1/4 hX, Y, XY 3 i = 1 hX, X 2 , X 3 i = −5 hX, Y, Y 4 , XY 4 i = −1/20 hX, XY, Y 3 i = 1 hX, Y 2 , XY 2 i = 1 hY, Y, Y 3 , X 2 Y 4 i = 3/20 2 2 2 3 hY, Y, X Y i = 1 hY, X , Y i = 1 hY, Y, XY 3 , XY 4 i = 1/10 hY, XY, XY 2 i = 1 hY, Y 2 , X 2 Yi = 1 hY, Y, Y 4 , X 2 Y 3 i = 1/20 2 2 2 2 2 hX , Y , Yi = 1 hX , X , X i∗ = −5 hXY, XY, Y 2 i = 1 T W17 = x4 y + y5 , Gmax = hγ = (1/20, 4/5)i k ≤ 4, X = e3γ , Y = e9γ . P = 1/4, cˆ = 6/5. Relations: X 5 = −4Y 3 , X 8 = Y 4 = 0, dim = 17. 3 pt correlators h1, 1, X 3 Y 3 i = 1 h1, X, X 2 Y 3 i = 1 4 pt correlators 3 2 h1, Y, X Y i = 1 h1, X 2 , XY 3 i = 1 hX, X, X 3 Y, X 3 Y 3 i = 1/5 2 2 3 3 h1, XY, X Y i = 1 h1, X , Y i = 1 hX, X, X 3 Y 2 , X 3 Y 2 i = 1/5 h1, Y 2 , X 3 Yi = 1 h1, X 2 Y, XY 2 i = 1 hX, Y, X 4 , X 3 Y 3 i∗ = 1/5 h1, X 4 , X 4 i = −4 hX, X, XY 3 i = 1 hX, Y, Y 3 , X 2 Y 3 i = 1/20 hX, Y, X 2 Y 2 i = 1 hX, X 2 , Y 3 i = 1 hX, Y, XY 3 , XY 3 i = −1/20 hX, XY, XY 2 i = 1 hX, X 3 , X 4 i = −4 hY, Y, Y 2 , X 3 Y 3 i = 1/5 hX, Y 2 , X 2 Yi = 1 hY, Y, X 3 Yi = 1 hY, Y, XY 2 , X 2 Y 3 i = 3/20 hY, X 2 , XY 2 i = 1 hY, XY, X 2 Yi = 1 hY, Y, Y 3 , X 3 Y 2 i = 1/20 hY, X 3 , Y 2 i = 1 hX 2 , XY, Y 2 i = 1 hY, Y, X 2 Y 2 , XY 3 i = 1/20 hX 2 , X 2 , X 4 i = −4 hX 2 , X 3 , X 3 i∗ = −4 hXY, XY, XYi = 1
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AMANDA FRANCIS T W17 = x4 y + y5 , G = hγ = (1/10, 3/5)i k ≤ 7, X = eγ , Y = e4γ , Z = e7γ , W = xy2 e0 . P = 3/5, cˆ = 6/5. Relations: XZ = Y 2 , X 3 = −4YZ, XYZ = 20W 2 , X 5 = Y 5 = Z 2 = W 3 = 0, dim = 10. 3 pt correlators 4 pt correlators h1, 1, XYZi = 1 h1, X, YZi = 1 hX, X, Y, XYZi = 1/5 h1, Y, XZi = 1 h1, Z, XYi = 1 hX, X, XY, YZi = 1/5 h1, W, Wi = 1/20 h1, X 2 , X 2 i = −4 hX, X, XZ, XZi = 2/5 2 hX, X, X i = −4 hX, Y, Zi = 1 hX, Y, XY, XZi = 1/5 hY, Y, Yi = 1 hX, Y, X 2 , YZi∗ = 2/5 5 pt correlators hX, Z, Z, YZi = 1/5 hX, X, Z, YZ, XYZi = −1/25 hX, Z, XY, XYi = 0 hX, Y, Y, YZ, XYZi = 0 hX, Z, X 2 , XZi∗ = 2/5 hX, Y, Z, XZ, XYZi = 1/50 hX, W, W, XYi∗ = −1/100 hX, Z, Z, XY, XYZi = 0 hY, Y, Z, YZi = 1/10 hX, Z, W, W, XYZi∗ = 3/1000 hY, Y, XY, XYi = 1/5 hX, W, W, XZ, YZi∗ = 1/1000 hY, Y, X 2 , XZi∗ = 0 hY, Y, Y, XZ, XYZi = 1/25 hY, Z, Z, XZi = 1/5 hY, Y, Z, XY, XYZi = 1/25 hY, Z, X 2 , XYi∗ = 0 hY, Y, W, W, XYZi∗ = −1/500 hY, W, W, X 2 i∗ = −1/100 hY, Z, Z, YZ, YZi = 1/50 hZ, Z, Z, XYi = 1/10 hY, Z, Z, X 2 , XYZi∗ = 0 hZ, Z, W, Wi∗ = −1/100 hY, W, W, XY, YZi∗ = −1/1000 hZ, Z, X 2 , X 2 i∗ = 4/5 hY, W, W, XZ, XZi∗ = −1/500 6 pt correlators hZ, Z, Z, Z, XYZi = 3/25 hX, X, W, W, XYZ, XYZi∗ = −1/1250 hZ, Z, Z, XZ, YZi = 2/25 hX, Z, Z, Z, XYZ, XYZi = 0 hZ, W, W, X 2 , XZi∗ = 1/250 hY, Y, Z, Z, XYZ, XYZi = 2/125 hZ, W, W, XY, XZi∗ = 1/1000 hY, Z, W, W, YZ, XYZi∗ = 1/10000 hW, W, W, W, XYZi∗ = 1/20000 hZ, Z, W, W, XZ, XYZi∗ = −3/5000 7 pt correlators hW, W, W, W, X 2 , XYZi∗ = −9/100000 hY, W, W, W, W, XYZ, XYZi∗ = −3/500000 hW, W, W, W, YZ, YZi∗ = 3/100000 hZ, Z, W, W, W, XYZ, XYZi∗ = 0
Q17 = x3 + xy5 + yz2 , Gmax = hJ = (1/3, 2/15, 13/30)i k ≤ 5, X = e10J , Y = e8J . P = 3/10, cˆ = 6/5. Relations: 5X 9 = 2Y 3 , X 10 = Y 4 = 0, dim = 21. 3 pt correlators h1, 1, X 9 Yi = −2 h1, X, X 8 Yi = −2 h1, X 2 , X 7 Yi = −2 h1, Y, X 9 i = −2 4 pt correlators 3 6 h1, X , X Yi = −2 h1, XY, X 8 i = −2 hX, X, X 8 , X 9 Yi = 8/15 h1, X 4 , X 5 Yi = −2 h1, X 5 , X 4 Yi = −2 hX, X, X 9 , X 8 Yi = 4/15 h1, X 2 Y, X 7 i = −2 h1, X 3 Y, X 6 i = −2 hX, Y, Y 2 , X 9 Yi∗ = 2/3 h1, Y 2 , Y 2 i = −5 hX, X, X 7 Yi = −2 hX, Y, X 9 , X 9 i = −4/15 hX, X 2 , X 6 Yi = −2 hX, Y, X 8 i = −2 hY, Y, XY, X 9 Yi = −2/3 hX, X 3 , X 5 Yi = −2 hX, XY, X 7 i = −2 hY, Y, X 2 Y, X 8 Yi = −2/3 hX, X 4 , X 4 Yi∗ = −2 hX, X 5 , X 3 Yi = −2 hY, Y, X 3 Y, X 7 Yi = −2/3 hX, X 2 Y, X 6 i = −2 hX 2 , X 2 , X 5 Yi = −2 hY, Y, X 4 Y, X 6 Yi = −2/3 hX 2 , Y, X 7 i = −2 hX 2 , X 3 , X 4 Yi∗ = −2 hY, Y, X 5 Y, X 5 Yi = −2/3 hX 2 , XY, X 6 i = −2 hX 2 , X 4 , X 3 Yi∗ = −2 5 pt correlators hX 2 , X 2 Y, X 5 i = −2 hY, Y, Y 2 i∗ = −5 None 3 6 4 5 hY, X , X i = −2 hY, X , X i = −2 3 3 3 3 5 hX , X X Yi∗ = −2 hX , XY X i∗ = −2 hX 3 , X 4 X 2 Yi = −2 hXY, X 4 X 4 i∗ = −2
COMPUTATIONAL TECHNIQUES IN FJRW THEORY
QT17 = x3 y + y5 z + z2 , Gmax = hγ = (1/30, 9/10, 1/2)i k ≤ 5, X = e7γ , Y = e19γ , Z = y4 e10γ . P = 1/3, cˆ = 6/5. Relations: X 5 = −3Y 2 , aX 4 Y = Z 2 , X 10 = Y 3 = Z 3 = 0, a , 0, dim = 17. 4 pt correlators 3 pt correlators hX, X, X 3 Y, X 4 Y 2 i = 1/10 h1, 1, X 4 Y 2 i = 1 h1, X, X 3 Y 2 i = 1 hX, X, X 4 Y, X 3 Y 2 i = 1/10 h1, X 2 , X 2 Y 2 i = 1 h1, Y, X 4 Yi = 1 hX, Y, Y 2 , X 3 Y 2 i = −1/15 3 2 3 h1, X , XY i = 1 h1, XY, X Yi = 1 hX, Y, XY 2 , X 2 Y 2 i = −1/15 4 2 2 2 h1, X , Y i = 1 h1, X Y, X Yi = 1 hX, Y, X 4 Y, X 4 Yi = 0 h1, Z, YZi = a hX, X, X 2 Y 2 i = 1 hX, Y, X 4 , X 4 Y 2 i∗ = 1/10 hX, X 2 , XY 2 i = 1 hX, Y, X 3 Yi = 1 hX, Z, Z, X 4 Y 2 i∗ = −a/10 3 2 2 hX, X , Y i = 1 hX, XY, X Yi = 1 hX, Z, YZ, X 4 Yi∗ = −a/10 4 4 2 2 2 hX, X , X i∗ = −3 hX , X , Y i = 1 hX 2 , Z, Z, X 3 Y 2 i∗ = −a/10 hX 2 , Y, X 2 Yi = 1 hX 2 , X 3 , X 4 i∗ = −3 hY, Y, Y, X 4 Y 2 i = 3/10 hY, Y, X 4 i = 1 hX 2 , XY, XYi = 1 hY, Y, XY, X 3 Y 2 i = 7/30 hY, X 3 , XYi = 1 hY, Z, Zi∗ = a hY, Y, X 2 Y, X 2 Y 2 i = 1/6 3 3 3 hX , X , X i∗ = −3 hY, Y, Y 2 , X 4 Yi = 1/30 hY, Y, X 3 Y, XY 2 i = 1/10 hY, Y, YZ, YZi∗ = a/3 hY, Z, Y 2 , YZi∗ = a/30 hX 3 , Y, Y 2 , X 4 Yi∗ = −a/10 5 pt correlators hZ, Z, XY, X 4 Yi∗ = −a/10 hX, Y, Y, X 4 Y 2 , X 4 Y 2 i = −1/150 hZ, Z, X 4 , XY 2 i∗ = −a/10 hY, Z, Z, XY 2 , X 4 Y 2 i = −a/150 2 2 3 2 hZ, Z, X 2 Y, X 3 Yi∗ = −a/10 hY, Z, Z, X Y , X Y i = −a/150 hZ, Z, Y 2 , Y 2 i∗ = a/30 S 16 = x2 y + xz4 + y2 z, Gmax = hJ = (5/17, 7/17, 3/17)i k ≤ 5, X = e8J , Y = e7J , Z = e6J . P = 7/17, cˆ = 21/17. Relations: −2XZ = Y 2 , X 4 = −2YZ, −4X 3 Y = Z 2 , X 4 = Y 3 = Z 3 = 0, dim = 16. 4 pt correlators 3 pt correlators hX, X, X 2 Y, X 3 YZi = 3/17 hY, Y, YZ, X 3 Zi∗ = 1/17 h1, 1, X 3 YZi = 1 hX, X, X 3 Y, X 2 YZi = 1/17 hY, Y, X 2 Z, XYZi∗ = 3/17 h1, X, X 2 YZi = 1 hX, X, X 3 Z, X 3 Zi∗ = −2/17 hY, Y, X 3 Y, X 3 Yi∗ = −2/17 h1, Y, X 3 Zi = 1 2 2 hX, Y, YZ, X YZi = −2/17 hY, Z, XZ, X 4 Yi = −4/17 h1, X , XYZi = 1 3 3 3 hX, Y, X Y, X Zi = 1/17 hY, Z, YZ, X 4 Yi = 1/17 h1, Z, X Yi = 1 hX, Y, XYZ, XYZi = −2/17 hY, Z, X 2 Z, X 4 Yi = −4/17 h1, XY, X 2 Zi = 1 hX, Y, X 3 , X 3 YZi∗ = 3/17 hY, Z, XY, X 4 Yi∗ = 5/17 h1, X 3 , YZi = 1 2 2 hX, Z, XZ, X YZi = 1/17 hY, Z, X 2 Y, X 4 Yi∗ = 3/17 h1, XZ, X Yi = 1 3 hX, Z, YZ, X Zi = 1/17 hX 2 , Z, Z, X 2 YZi∗ = 6/17 hX, X, XYZi = 1 hX, Z, X 2 Z, XYZi = 1/17 hZ, Z, XY, X 3 Zi∗ = 1/17 hX, Y, X 2 Zi = 1 hX, Z, X 3 Y, X 3 Yi = −2/17 hZ, Z, X 3 , XYZi∗ = 7/17 hX, X 2 , YZi = 1 3 2 hX, Z, Z, X YZi∗ = 5/17 hZ, Z, XZ, X 3 Yi∗ = 3/17 hX, ZX Yi = 1 hY, Y, Z, X 3 YZi∗ = 7/17 hZ, Z, X 2 Y, X 2 Zi∗ = 2/17 hX, XY, XZi = 1 hY, Y, XZ, X 2 YZi∗ = 5/17 hZ, Z, YZ, YZi∗ = −4/17 hX, X 3 , Y 3 i∗ = −2 2 5 pt correlators hY, X , XZi = 1 hY, Z, X 3 i = 1 hX, X, Z, X 3 YZ, X 3 YZi∗ = −4/289 hY, Y, X 2 Yi∗ = −2 hX, Y, Y, X 3 YZ, X 3 YZi∗ = 8/289 hY, XY, XYi∗ = −2 hY, Z, Z, XYZ, X 3 YZi∗ = −4/289 hX 2 , Z, XYi = 1 hY, Z, Z, X 2 YZ, X 2 YZi∗ = −8/289 2 2 3 hX , X , X i∗ = −2 hZ, Z, Z, X 2 Z, X 3 YZi∗ = 20/289 hZ, Z, Zi∗ = −4 hZ, Z, Z, X 3 Z, X 2 YZi∗ = 12/289
29
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AMANDA FRANCIS
S 17 = x2 y + y2 z + z6 , Gmax = hJ = (7/24, 5/12, 1/6)i k ≤ 5, X = e8J , Y = e7J . P = 1/4, cˆ = 5/4. Relations: X 6 = Y 3 , X 11 = Y 4 = 0, dim = 21. 3 pt correlators 4 pt correlators h1, 1, X 10 i = −2 h1, X, X 9 i = −2 hX, X, X 4 Y, X 10 i = −1/3 h1, Y, X 4 Y 2 i = −2 h1, X 2 , X 8 i = −2 3 2 hX, X, X Y , X 3 Y 2 i∗ = 1/(3k) h1, XY, X 3 Y 2 i = −2 h1, X 3 , X 7 i = −2 hX, Y, X 5 , X 10 i∗ = −1/3 h1, X 2 Y, X 2 Y 2 i = −2 h1, X 4 , X 6 i = −2 hX, Y, X 6 , X 9 i = −1/3 h1, Y 2 , X 4 Yi = −2 h1, X 3 Y, XY 2 i = −2 hX, Y, X 7 , X 8 i = −1/3 h1, X 5 , X 5 i = −2 hX, X, X 8 i = −2 hY, Y, Y 2 , X 10 i∗ = 5/3 hX, Y, X 3 Yi = −2 hX, X 2 , X 7 i = −2 hY, Y, XY 2 , X 9 i∗ = −1/3 hX, XY, X 2 Y 2 i = −2 hX, X 3 , X 6 i = −2 hY, Y, X 6 , X 3 Y 2 i∗ = 1/3 hX, X 2 Y, XY 2 i = −2 hX, Y 2 , X 3 Yi = −2 hY, Y, X 2 Y 2 , X 8 i∗ = 1 hX, X 4 , X 5 i = −2 hY, Y, X 4 Yi∗ = −2 hY, Y, Y 2 , X 4 Yi∗ = 1/18 hY, XY, X 3 Yi∗ = −2 hY, X 2 , X 2 Y 2 i = −2 hY, Y, X 7 , X 3 Y 2 i∗ = −1/3 hY, X 3 , XY 2 i = −2 hY, X 2 Y, X 2 Yi∗ = −2 hY, X 4 , Y 2 i = −2 hX 2 , X 2 , X 6 i = −2 hX 2 , XY, XY 2 i = −2 hX 2 , X 3 , X 5 i = −2 5 pt correlators hX 2 , X 2 Y, Y 2 i = −2 hX 2 , X 4 , X 4 i∗ = −2 None 3 2 2 hXY, X , Y i = −2 hXY, XY, X Yi∗ = −2 3 3 4 hX , X , X i∗ = −2
4.3. FJRW Theories which we still cannot compute. Unfortunately, there are some Amodels which we are still unable to compute. In these examples, there are not enough relations between concave correlators (whose values we can compute) and other correlators (whose values we can only find using the Reconstruction Lemma). W P8 = x3 + y3 + z3 U12 = x3 + y3 + z4 T S 1,0 = x2 + xz2 + y5 z Z17 = x3 y + y8 Z18 = x3 y + xy6 T W17 = x4 y + y5 T Q17 = x3 y + y5 z + z2 T S 17 = x2 + y6 z + z2 x T S 17 = x2 + y6 z + z2 x
G h(1/3, 0, 0), (0, 1/3, 1/3)i hJi hJimax hJimax hJimax hJi hJi Gmax hJi
dimension 8 12 14 22 18 8 7 17 7
Remark 4.1. The code which we used to make our computations is available by email request from the author. We made the computations in SAGE [18]. The SAGE computations depend on code written by Drew Johnson [13] for computing intersection numbers of classes on M g,n . References [1] H. Fan, T. J. Jarvis, and Y. Ruan. The Witten equation, mirror symmetry and quantum singularity theory. ArXiv e-prints, December 2007. [2] H. Fan, T. J. Jarvis, and Y. Ruan. Geometry and analysis of spin equations. Communications on Pure and Applied Mathematics, 61(6):745–788, 2008. [3] H. Fan, T. J. Jarvis, and Y. Ruan. The Witten equation and its virtual fundamental cycle. ArXiv e-prints, December 2007. [4] M. Krawitz. FJRW rings and Landau-Ginzburg Mirror Symmetry. ArXiv e-prints, June 2009. [5] A. Francis, T. Jarvis, D. Johnson, and R. Suggs. Landau-Ginzburg Mirror Symmetry for Orbifolded Frobenius Algebras. ArXiv e-prints, November 2011.
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[6] M. Krawitz and Y. Shen. Landau-Ginzburg/Calabi-Yau Correspondence of all Genera for Elliptic Orbifold $\mathbb{p}ˆ1$. ArXiv e-prints, June 2011. [7] J. Gu´er´e. A Landau–Ginzburg mirror theorem without concavity. ArXiv e-prints, July 2013. [8] M. Krawitz, N. Priddis, P. Acosta, N. Bergin, and H. Rathnakumara. FJRW-Rings and Mirror Symmetry. Communications in Mathematical Physics, 296:145–174, May 2010. [9] V. I. Arnold, V. V. Goryunov, O. V. Lyashko, and V. A. Vasilliev. Singularity theory. I. In Dynamical Systems. VI, volume 6 of Encycl. Math. Sci., pages iv+245. Springer-Verlag, New York, 1993. [10] A. Chiodo. Towards an enumerative geometry of the moduli space of twisted curves and r-th roots. ArXiv Mathematics e-prints, July 2006. [11] C. Faber. Algorithms for computing intersection numbers on moduli spaces of curves, with an application to the class of the locus of Jacobians. In eprint arXiv:alg-geom/9706006, page 6006, June 1997. [12] E. Arbarello and M. Cornalba. Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves. In eprint arXiv:alg-geom/9406008, page 6008, June 1994. [13] Drew Johnson. Top Intersections on Mbarg,n, August 2011. https://bitbucket.org/drew j/top-intersectionson-mbar g-n. [14] C. Li, S. Li, K. Saito, and Y. Shen. Mirror symmetry for exceptional unimodular singularities. ArXiv e-prints, May 2014. [15] K Saito. Primitive forms for a universal unfolding of a function with an isolated critical point. J. Fac. Sci. Univ. Tokyo Sect. IA 28, 1982. [16] D. Edidin, T. J. Jarvis, and T. Kimura. Chern Classes and Compatible Power Operations in Inertial K-theory. ArXiv e-prints, September 2012. [17] H. Fan, A. Francis, T. J. Jarvis, E. Merrell, and Y. Ruan. Witten’s D 4 Integrable Hierarchies Conjecture. ArXiv e-prints, August 2010. [18] W. A. Stein et al. Sage Mathematics Software (Version 6.1). The Sage Development Team, 2014. http://www.sagemath.org. Department of Mathematics, Brigham Young University, Provo, UT 84602, USA E-mail address:
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