Hodge Diamonds of the Landau–Ginzburg Orbifolds

. Consider the pairs ( f, G ) with f = f ( x 1 , . . . , x N ) being a polynomial defining a quasihomogeneous singularity and G being a subgroup of SL( N, C ), preserving f . In particular, G is not necessary abelian. Assume further that G contains the grading operator j f and f satisfies the Calabi–Yau condition. We prove that the nonvanishing bigraded pieces of the B-model state space of ( f, G ) form a diamond. We identify its topmost, bottommost, leftmost and rightmost entries as one-dimensional and show that this diamond enjoys the essential horizontal and vertical isomorphisms.

Assume also that x 1 = • • • = x N = 0 is the only critical point of f and d 1 , . . ., d N have no common factor.Then the zero set f (x 1 , . . ., x N ) = 0 defines a degree d 0 quasismooth hypersurface X f in P(d 1 , . . ., d N ).Such hypersurfaces became of great interest in the early 90's in the context of mirror symmetry (cf.[7,8]).In particular, if the Calabi-Yau condition d 0 = N k=1 d k holds, then the first Chern class of X f vanishes and, hence, its canonical bundle is trivial meaning that X f is a Calabi-Yau variety.
The polynomials f above define the so-called quasihomogeneous singularities and can be studied from the point of view of singularity theory.The varieties X f at the same time are the objects of Kähler geometry.To relate the singularity theory properties of f to the Kähler geometry properties of X f is an important problem.This problem is in particular interesting in the context of mirror symmetry.

Hodge diamonds of Calabi-Yau manifolds
The state space of a Calabi-Yau manifolds X is the cohomology ring H * (X).This cohomology ring is naturally bigraded building up a Hodge diamond of size D := dim C X.In particular the following properties hold: (1) H * (X) = p,q∈Z H p,q (X), (2) dim H p,q (X) = 0 if p < 0 or q < 0 or p > D or q > D, (3) dim H 0,0 (X) = dim H D,D (X) = 1, (4) dim H D,0 (X) = dim H 0,D (X) = 1, (5) there is a "horizontal" vector space isomorphism H p,q (X) ∼ = H q,p (X), (6) there is a "vertical" vector space isomorphism H p,q (X) ∼ = H D−q,D−p (X) ∨ , where (−) ∨ stands for the dual vector space In physics, this bigrading is coming from considering Calabi-Yau B-model associated to X and A-model bigrading is obtained from it by the so-called rotation of the diamond by 90 • .On the level of state spaces the mirror symmetry map is an isomorphism of the cohomology vector spaces for a dual pair of Calabi-Yau manifolds switching A-and B-model bigradings.
More, generally if considering Calabi-Yau orbifolds in place of quasismooth varieties, one replaces the ordinary cohomology ring by the Chen-Ruan cohomology ring H * orb .This is an essential question if H * orb forms a Hodge diamond too.Some of the Hodge diamond properties above follow directly from the definitions or could be verified in the similar way to our main Theorem 1.1 below, while the others (like the property (4)) were not investigated in literature up to our knowledge and do not look to be straightforward.

Landau-Ginzburg orbifolds
Another facet of mirror symmetry is given by matching the so-called Landau-Ginzburg orbifolds in place of Calabi-Yau manifolds or orbifolds (cf.[17,22,29,30,31]).Mathematically, these are the pairs (f, G) with f being a quasihomogeneous polynomial with the only critical point 0 ∈ C N and G being a group of symmetries of f .Consider the maximal group of linear symmetries of f It is nontrivial because it contains a nontrivial subgroup J generated by j f j f • (x 1 , . . ., x N ) := e 2π √ −1d 1 /d 0 x 1 , . . ., e 2π √ −1d N /d 0 x N .
Also important is the group SL f := GL f ∩ SL(N, C) consisting of GL f elements preserving the volume form of C N .For any G ⊆ GL f , the pair (f, G) is called a Landau-Ginzburg orbifold.One associates to it the state space, which is the G-equivariant generalization of a Jacobian ring of f , together with A-and B-model bigradings, which again differ by 90 • -rotation from one another provided G acts by transformations with determinant 1.Since these are interdependent, within this paper we will focus on just the B-model, as it has clearer geometric interpretation, much alike bigrading on the cohomology of Calabi-Yau manifold.For this reason, we will call this state space endowed with B-model bigrading by B(f, G).
Up till now, Landau-Ginzburg orbifolds were mostly investigated for the groups G acting diagonally on C N and also for f belonging to a very special class of polynomials -the so-called invertible polynomials (cf.[4,5,6,14,19,20,21]). Also some work was done for the symmetry groups G = S ⋉ G d with G d acting diagonally and S ⊆ S N -a subgroup of a symmetric group on N elements [2,3,10,11,12,18,24,27,32].We relax both conditions in this paper.
Once again, the mirror symmetry attempts to match A-and B-state spaces for dual pairs of Landau-Ginzburg orbifolds.It should be mentioned, that the state space enjoys several other structures besides being just bigraded vector space, like multiplication or bilinear form on both A-and B-sides, which should also be compatible with the mirror map.They will not be considered in the present paper.

Calabi-Yau/Landau-Ginzburg correspondence
The final piece of matching comes from Calabi-Yau/Landau-Ginzburg correspondence, which relates respective A-models and their state spaces coming from Calabi-Yau and Landau-Ginzburg geometries of (X f , G/J) and (f, G) provided J ⊆ G. Mathematically, up till now this was proved for diagonal symmetry groups by Chiodo and Ruan [9,Theorem 14] and in some special cases with N = 5 in [25].
However, if this correspondence holds, the vector space B(f, G) is also expected to form a diamond.Namely, it should satisfy the properties (1)-( 6) above.Formulated in terms of B(f, G) this becomes a purely singularity theoretic question.It is the main topic of our paper.
x N ] be a quasihomogeneous polynomial satisfying Calabi-Yau condition and defining an isolated singularity.Then for any G ⊆ SL f , such that J ⊆ G the state space B(f, G) forms a diamond of size N − 2 in a sense that it satisfies conditions (1)-( 6) above.
Proof .The proof is summed up in Propositions 5.
In what follows we will say that f is quasihomogeneous with respect to the weights d 0 , d 1 , . . ., d N or the reduced weights q 1 := d 1 /d 0 , . . ., q N := d N /d 0 .
We will say that f defines an isolated singularity at 0 ∈ C N if 0 is the only critical point of f .According to K. Saito [28,Satz 1.3] one may consider without changing the singularity only the quasihomogeneous polynomials, such that 0 < q k ≤ 1/2 for all k = 1, . . ., N .Moreover, we may assume that all variable x k with q k = 1/2 enter f only in monomial x 2 k , in particular, there are no monomials of the form x i x j with i ̸ = j.Then the number of its monomials is not less than N , the number of variables.
Using the word 'type', we mean the certain structure of the monomials without specifying the exponents a i .
It's easy to see that for Fermat, chain and loop type polynomials, the reduced weights q 1 , . . ., q N are defined in a unique way.This is also true for any quasihomogeneous singularity that we assume (cf.[28,Korollar 1.7]).We have where one assumes a 0 := a N , a −1 := a N −1 , a −2 := a N −2 and so on.Given f ∈ C[x 1 , . . ., x N ] and g ∈ C[y 1 , . . ., y M ] both defining quasihomogeneous singularities it follows immediately that f + g ∈ C[x 1 , . . ., x N , y 1 , . . ., y M ] defines a quasihomogeneous singularity too.We will denote such a sum by f ⊕ g.
Example 2.3 ([1, Section 13.2]).If N = 3, then all quasihomogeneous isolated singularities are given by the following polynomials x 1 with some positive a 1 , a 2 , a 3 .The numbers a i are arbitrary for f I , f II , f IV , f V , f VII , however the polynomials f III and f VI are only quasihomogeneous if ε ̸ = 0 and some additional combinatorial condition on a 1 , a 2 , a 3 holds.In particular the least common divisor of (a 2 , a 3 ) should be divisible by a 1 − 1 for f III to exist.Allowed values of ε depend quiet heavily on a i .In particular to define an isolated singularity.

Graph of a quasihomogeneous singularity
Let f ∈ C[x 1 , . . ., x N ] define an isolated singularity.Then for every index j ≤ N the polynomial f has either the summand x a j or a summand x a j x k for some exponent a ≥ 2 and index k ≤ N (cf.[28,Korollar 1.6] and [16,Theorem 2.2]).Construct a map κ : {1, . . ., N } → {1, . . ., N }.Set κ(j) := j in the first case above and κ(j) := k in the second.
Following [16,Section 3] associate to f the graph1 Γ f with N vertices labelled with the numbers 1, . . ., N and the oriented arrows j → κ(j) if j ̸ = κ(j).In other words, the vertices correspond to the variables x i and the arrows to the monomials x a j x k .Example 2.4.The graphs of the N = 3 quasihomogeneous singularities are all listed in Figure 1.
Call a tree oriented if its root has only incoming edges adjacent to it and any other vertex has exactly one outgoing edge and several incoming edges adjacent to it.The following proposition is immediate.In what follows we consider the root of the type (1) graph above as a cycle with one vertex.This merges the two types above.It's easy to see that Γ f ⊕g = Γ f ⊔ Γ g , but it is not true that f decomposes into the ⊕-sum if Γ f has more than one component.For example the graph of f = x 3 1 + x 3 2 + x 3 3 + x 1 x 2 x 3 is just the disjoint union of three vertices without any edges.

Graph decomposition of a polynomial
Assume we only know the graph Γ f and not the polynomial f itself.The graph structure indicates some monomials that are the summands of f .Call these monomials graph monomials.In particular, f has only graph monomials if it is of Fermat, chain or loop type or a ⊕-sum of them.
Let f be such that Γ f has only one connected component.Then Γ f has one oriented circle, and p oriented trees with the roots on this circle.We have the decomposition with (2) f 0 having as the summands only those graph monomials of f , that build up the oriented circle or the common root, (3) f k having as the summands only those graph monomials of f , that build up the k-th oriented tree, (4) as the summands all the non-graph monomials of f .This decomposition extends easily to the case of Γ f having several components.Note that we could have had p = 0, but it should always hold that f 0 ̸ = 0. Writing the decomposition above we had to order the trees by the index of f i .This ordering is not important in what follows.

Graph exponents matrix
Let f define a quasihomogeneous singularity.We introduce the matrix E f with the entries in Z ≥0 .It follows from Proposition 2.5 that f has exactly N graph monomials.Let every row of E f correspond to a graph monomial.The components of this row will be (α 1 , . . ., α N ) if and only if the corresponding graph monomial is ε The matrix E f is only defined up to a permutation of the rows.We will call it graph exponents matrix.
Let E ij denote the components of E f .Then for some non-zero constants c k we have Remark 2.7.Such a matrix was previously defined in the literature only for the invertible polynomials (see Section 2.5).We consider it here in a wider context.
Example 2.8.The matrices E f of the Example 2.3 are The graph exponents matrices of loop and chain type polynomials read In general, by Proposition 2.5 if Γ f has only one connected component, the matrix E f after some renumbering of the variables and the rows has the block form.The diagonal blocks are several chain type exponent matrices and exactly one loop type exponents matrix as in equation (2.4), such that for every chain type block there is exactly one additional matrix entry 1 in the first row of this block and the column of a loop type block.All the other matrix entries except listed vanish, where A 0 is a loop type polynomial exponents matrix and A 1 , . . ., A p are chain type polynomial exponent matrices, the matrix U ij is the rectangular matrix with 1 at position (i, j) and all other entries 0.
Assuming the decomposition of equation (2.2), the matrix A 0 is exactly the exponent matrix of f 0 and the matrices A 1 , . . ., A p are defined by f 1 , . . ., f p .Proposition 2.9.Let f define a quasihomogeneous singularity.Then Proof .Let f be quasihomogeneous with the reduced weights (q 1 , . . ., q N ).We show first that these weights are defined uniquely by the graph monomials of f .
Let f be decomposed as in equation (2.2).Then f 0 is of Fermat or loop type and the weights of its variables are defined uniquely.Similarly for any f k with k = 1, . . ., p corresponding to the tree with the root on the oriented circle, the weight of the root's variable is defined by the quasihomogeneity of f 0 , going up the tree of deduces uniquely the weight of every variable of f k corresponding to the consequent vertex.
Introduce two Z N vectors: q := (q 1 , . . ., q N ) T and 1 := (1, . . ., 1) T .Then the quasihomogeneity condition on f is equivalent to the Q N vector equality E f • q = 1.It follows now from Cramer rule that det(E f ) ̸ = 0 because this equation has a unique solution.This completes (i).
The canonical weight set is obtained by taking

Invertible polynomials
The set of all quasihomogeneous singularities contains the following important class.The polynomial f defining an isolated quasihomogeneous singularity having no monomial of the form x i x j with i ̸ = j and as many monomials as the variables is called invertible polynomial and is said to define an invertible singularity.
The following statement can be assumed as a well-known (cf.[23]), we add up the proof for completeness.
Proposition 2.10.Let f be an invertible polynomial.Then after some rescaling and renumbering of the variables we have k) being either of Fermat, chain or loop type.
Proof .Assume Γ f to contain a vertex with two incoming arrows.Then f is of the form Setting The graphs of invertible singularities are disjoint unions of isolated vertices (Fermat types), oriented cycles (loop types) and one branch trees (chain types).
In the notation of Section 2.3, the graph decomposition equation (2.2) of Fermat, loop and chain type polynomials is the following.We have always f add = 0, p = 0 and f 0 = f for Fermat and loop types, but p = 1 and f 0 + f 1 = f for chain type with f 0 = x am m .Example 2.11.The quasihomogeneous singularities with N = 2 are all invertible.The quasihomogeneous singularities with N = 3 are not all invertible.In the notation of Example 2.3, we have

Symmetries
Given a quasihomogeneous polynomial f = f (x 1 , . . ., x N ) consider the maximal group of linear symmetries of f defined by Lemma 3.1.Under our assumptions on f any g ∈ GL f necessarily preserves the weights of the variables, i.e., maps each x i to a linear combination of x j with the same weight.
Proof .The action of g preserves the homogeneous components of f .In particular, the variables in the quadratic terms of f map to linear combinations of variables in the quadratic terms of f and hence weight 1/2 subspace is preserved by f .
We may now assume that f has no quadratic terms.In this case, the argument of [26, Theorem 2.1] applies verbatim to the quasihomogeneous situation to prove that GL f is finite.Let The map χ : GL f → C * is a character and GL f is precisely the kernel of χ, in particular, it is a normal subgroup.Moreover, the condition (2.1) provides an inclusion t : C * → GL f , such that χ • t is a degree d 0 > 0 map.The action by conjugation of connected subgroup t(C * ) on the finite subgroup GL f is necessary trivial, which means that GL f commute with t(C * ).This means that GL f preserves the eigenspaces of t(C * ) as desired.■ Remark 3.2.Notably, the observation of this Lemma seems to be unknown to previous authors (for example, it was imposed as condition in [25] and some subsequent works).
Let G d f ⊆ GL f be the maximal group of diagonal symmetries of f .This is the group of all diagonal elements of GL(N, C) belonging to GL f .We have It's obvious that Note, however, that the same does not necessarily hold for GL f ′ ⊕f ′′ .
In what follows we will use the notation Each element g ∈ G d f has a unique expression of the form where r is the order of g.We adopt the additive notation for such an element g.
Example 3.3.For f = x a 1 1 we have GL f = G d f = ⟨g⟩ with g ∈ C * acting by g(x 1 ) = e 1 a 1 • x 1 .Its order is a 1 and in the additive notation we have g = (1/a 1 ), giving us GL f ∼ = Z/a 1 Z.
In the additive notation g 1 = (1/a 1 , 0) and Let (q 1 , . . ., q N ) be the reduced weight set of f .Then we have In particular, it follows that G d f and GL f are not empty whenever f is quasihomogeneous.Denote by J the group generated by j f : Since g ∈ GL f preserves the weights we see that j f commutes with g.In other words J is the central subgroup of GL f .

Fixed loci of the GL f elements
For each g ∈ GL f , denote by Fix(g) the fixed locus of g This is an eigenvalue 1 subspace of C N and therefore a linear subspace of C N .By N g := dim C Fix(g) denote its dimension and by f g := f | Fix(g) the restriction of f to the fixed locus of g.For g ∈ G d f , this linear subspace is furthermore a span of a collection of standard basis vectors.
For each h ∈ G d f , let In the other words, Fix(h) is indexed by I h .In particular, I id = {1, . . ., N }.More generally, for g ∈ GL f , since g preserves the weight subspaces of C N , the weights of the subspace Fix(g) are well-defined and are the subset of {q 1 , . . ., q N }.Fix a subset I g ⊂ {1, . . ., N } such that q k with k ∈ I g are exactly all the weights of Fix(g), so that, in particular, we have f there is no canonical choice for I g , but the choice made at this step will not impact our results.
Denote by I c h the complement of I h in I id and set d h := N − N h , the codimension of Fix(h).
Proposition 3.5.For any diagonalizable g ∈ GL f with N g > 0 there is a choice of coordinates on Fix(g) linear in x i , such that the polynomial f g also defines a quasihomogeneous singularity.
Proof .Let x 1 , . . ., x N be the coordinates of C N dual to the basis diagonalizing g.In this coordinates the polynomial f g is obtained by setting some of x • to zero.The proof follows now by the same argument as in [13, Proposition 5].
for some p depending on g.The polynomial f g is of chain type again: Denote also This group will be important later on because it preserves the volume form of C N .

Age of a GL f element
For g ∈ GL f let λ 1 , . . ., λ N be the collection of its eigenvalues.Let 0 ≤ α i < 1 be such that then age of g is defined as the number age(g) : The following properties are clear but will be important in what follows.
(1) For any g ∈ GL f we have (2) For a diagonalizable g ∈ GL f we have age(g) = 0 if and only if g = id.
(3) We have g ∈ SL f if and only if age(g) ∈ N.
3.3 Diagonal symmetries and a graph Γ f Proposition 3.8.Let Γ f be a graph of a quasihomogeneous singularity f and g ∈ GL f .Then if g acts nontrivially on x k , then it acts nontrivially on all x i , such that there is an oriented path from i-th to the k-th vertex.
Proof .We first show the statement for the arrows pointing at k. Having an arrow j → k means that f has a monomial x a j j x k as a summand with a nonzero coefficient.We have g • x k ̸ = x k and therefore the summand can only be preserved under the action of g if g • x j ̸ = x j .Having an oriented path i → j 1 → • • • → j n → k we have by using the previous step that g • x jn ̸ = x jn and then g • x ja ̸ = x ja for all a.Hence, for x i too.
■ Let E f be the graph exponents matrix of f .Consider with E ij being the components of E f .The group G gr f is exactly the maximal group of diagonal symmetries of the difference f − f add .In particular, every element of G gr f preserves all graph monomials of f .
We have This gives yet another characterization of the group It follows that every vector g giving a G gr f -element is a linear combination with integer coefficients of the columns of E −1 f .Following the notation of Krawitz [21] define ρ i as the i-th Denote also The columns of E f generate all relations on ρ 1 , . . ., ρ N .
In particular, for (E 1k , . . ., E N k ) T being a k-th column of E f we have in G gr and all other relations among {ρ k } N k=1 follow from those written above.

Diagonal symmetries of an invertible singularity
In [14] for an invertible f , the authors gave the set S f of all N -tuples (s 1 , . . ., s N ) such that every g ∈ G d f \{id} is written uniquely by and s k = 0 if and only if k ∈ I g .Due to equation (3.1) and Proposition 2.10, it is enough to construct such set for Fermat, loop or chain type polynomials.
Proposition 3.9.For f being of Fermat, chain or loop type the set S f consists of all s = (s 1 , . . ., s N ), such that if N is even.

Diagonal symmetries of a quasihomogeneous singularity
For any quasihomogeneous singularity f , consider its graph decomposition as in equation (2.2).Up to the renumbering and rescaling of the variables, we have

K+L
with the similar expression for f 2 , . . ., f p .Any nontrivial g ∈ G d f 0 extends to an element g ∈ G gr f .Moreover it follows that Fix(g) = 0 and also Fix( g) = 0 as long as g ̸ = id.Similarly any element h ∈ G gr f with Fix(h) = 0 acts nontrivially on x 1 , . . ., x K preserving f 0 .Hence it defines h 0 ∈ G d f 0 by the restriction.At the same time any h ∈ (C * ) L acting diagonally on (x K+1 , . . ., x K+L ) preserving f 1 extends to an element of G gr f assuming it to act trivially on f 0 and all other f 2 , . . ., f p .One notes immediately that such elements h are the elements of chain type polynomial symmetry group.Denote the group of all such elements by G • f 1 .We construct the groups G • f 2 , . . ., G • fp in a similar way.For a nontrivial element g ∈ G gr f and its restriction g 0 ∈ G d f 0 , the extension g 0 is not unique.However, having it fixed, we have by Proposition 3.8 that there is a unique set of g k ∈ G • f k for k = 1, . . ., p, s.t.
We have that every g k acts non-trivially only on the variables of f k preserving all the variables of f 0 identically.
We have Associate to every g 0 , g 1 , . . ., g p an element s 0 , s 1 , . . ., s p as in Proposition 3.9.Composing them in one column s, we have g = E −1 f s T .Note that for s 0 ̸ = 0 and s 1 = 0, . . ., s p = 0, all components of g are nonzero.We follow the convention s 1 ̸ = 0, . . ., s p ̸ = 0 if g is such that g 0 ̸ = id.
The following proposition is very important in what follows.
Proposition 3.11.For any g ∈ G d f such that g = E −1 f s, we have age(g) = (1, . . ., 1)E −1 f s T .Proof .We need to show that the components of g belong to [0, 1).This follows immediately from the equality E f g = s, the bounds on s and the special form of the matrix E f (see equation (2.5)).■

Symmetries and the Calabi-Yau condition
Let the reduced weight set q 1 , . . ., q N of f satisfy N k=1 q k = 1.This equality is called the Calabi-Yau condition and we will say that f satisfies the CY condition.We show in this section that it puts significant restrictions on the symmetries of f .
Let the matrix E T f define a polynomial f T .Namely, if for f we have (2.3), then This polynomial does not necessarily define an isolated singularity.However, it is quasihomogeneous again with some weights q T 1 , . . ., q T N by the same argument as in Proposition 2.9.We call f star-shaped if its graph Γ f consists of N − 1 vertices all adjacent to one vertex.Namely, i+1 .Such a polynomial satisfies the CY condition if and only if Example of such a polynomial is given by with the Milnor number 81.We will treat the star-shaped polynomials separately.
Proposition 3.12.Let f not being a star-shaped polynomial, satisfy the CY condition.Then the weights q T 1 , . . ., q T N are all positive.
Proof .This lemma is obvious for invertible polynomial f and we assume only noninvertible cases in the proof.Let E f be written as in equation (2.5) and A 0 be a K×K loop type matrix as in equation (2.4).It is immediate that q T K+1 , . . ., q T N are positive.For i = 1, . . ., K, denote by A i the sum of all q T j with j > K, s.t.j-vertex of Γ f is adjacent to the i-th vertex.Lemma 3.13.We have 0 ≤ A i < 1 for any 1 ≤ i ≤ K.
Proof .A i is non-negative as the sum of the positive weights.However this sum can be empty.
Let A i ≥ 1 for some i.Let the vertices adjacent to the i-th vertex be labelled by K + 1, . . ., K + m contributing to f with the monomials . If the CY condition holds, then If S > 1, this gives q i ≥ 1 which contradicts the quasihomogeneity condition of f .If S = 1, then q K+1 + • • • + q K+m = 1 − q i and the CY condition can only hold if f is a star-shaped CY polynomial.■ Let us show that q T 1 is positive.The proof for q T 2 , . . ., q T N is similar.Let c ij stand for the components of the K × K matrix A −1 0 .Note that up to a sign these are just the products of a i divided by det A = a 1 • • • a K + (−1) K−1 .In particular, we have Then Under the CY condition we have The bracket on the right hand side is positive because q 1 < 1.This gives the estimate because c 11 is positive.We get then the estimate Introduce the positive numbers T r and P r by Some computations give us and These are positive numbers for a i ≥ 2, what gives the proof after applying the lemma above.■ Proposition 3.14.Let f satisfy the CY condition.Then for any diagonalizable g ∈ GL f such that N g = 0, we have The equality is only reached if g = j f .
Proof .Rewrite f in the coordinates x 1 , . . ., x N dual to the basis diagonalizing g.Then each x k is a linear combination of x 1 , . . ., x N .Moreover, one can renumber the new variables such that the weight of x k is the same as the weight of x k , namely q k .The element j f is represented in the old and the new basis by the same diagonal matrix.The given element g acts of x k just by a rescaling.Therefore it is enough to show the proposition for g belonging to the maximal group of diagonal symmetries.
To prove the propositions for g ∈ G d f it is enough to prove the inequality for any g ∈ G gr f with N g = 0 and f , such that the graph Γ f has only one connected component.We have For a given g assume s, such that g = E −1 f s as in Proposition 3.11.None of s k = 0 because N g = 0. We have First assume f is not star-shaped.Then q T k because every s k ≥ 1 and q T i are all positive.Combining with equation (3.3) we get the inequality claimed.Moreover it is obvious that the equality is only reached if s k = 1 for all k.This is equivalent to the fact that g = j f .Now let f be star-shaped.We have q T 1 = 0 and q T k > 0 for k = 2, . . ., N .By the same reasoning as above it is enough to consider for some s 1 = 1, . . ., a 1 − 1.If f defines an isolated singularity, it should have at least one summand x r 2 2 • • • x r N N for some nonnegative r 2 , . . ., r N .The quasihomogeneity and the g-invariance conditions on this summand give These two conditions can only hold when s 1 = 1.■ Remark 3.15.The proposition above holds for any invertible polynomial without the Calabi-Yau condition too.However for noninvertible polynomial without Calabi-Yau condition the proposition does not hold in general.If particular for f = T we have q = 1 10 , 9 20 , 9 20 , 9 20 and age(g) − 4 k=1 q k = − 1 20 .

The total space
Consider the quotient ring It is a finite-dimensional C-vector space whenever f defines an isolated singularity.Call it Jacobian algebra of f and set µ f := dim C Jac(f ) -the Milnor number of f .We will assume an additional convention: for the constant function f = 0 set Jac(f ) := C, µ f := 1.

Grading
The reduced weights q 1 , . . ., q N of f define the Q-grading on C[x 1 , . . ., x N ].Introduce the Qgrading on Jac(f ) by setting Let ϕ 1 , . . ., ϕ µ be the classes of monomials, generating Jac(f ) as a C-vector space.We say that X ∈ Jac(f ) is of degree κ if it is expressed as a C-linear combination of degree κ elements ϕ • .Denote by Jac(f ) κ the linear subspace of Jac(f ) spanned by the degree κ elements.Let the Hessian of f be defined as the following determinant: hess(f ) := det ∂ 2 f ∂x i ∂x j i,j=1,...,N .
Its class is nonzero in Jac(f ).
Proposition 4.2.For any β, such that 0 ≤ β ≤ c the perfect pairing η f induces an equivalence where (−) ∨ stands for the dual vector space.

The total space
For each g ∈ GL f , fix a generator of a one-dimensional vector space Λ(g) := dg C N /Fix(g) .Denote it by ξ g .For g ∈ G d f , it is standard to choose the generator to be the wedge product of x k with k ∈ I c g taken in increasing order.Define B tot (f ) as the C-vector spaces of dimension g∈GL f dim Jac(f g ) Each direct summand Jac(f g )ξ g will be called the g-th sector.We will write just B tot when the polynomial is clear from the context.
Remark 4.3.Note that for g, h ∈ G, such that Fix(g) = Fix(h), we have f g = f h .Then Jac(f g ) = Jac(f h ), but the formal letters ξ g ̸ = ξ h help to distinguish Jac(f g )ξ g and Jac(f h )ξ h , such that Jac(f g )ξ g ⊕ Jac(f h )ξ h is indeed a direct sum of dimension dim Jac(f g ) + dim Jac(f h ).

B-model group action
Note that an element h ∈ GL f induces a map h : Fix(g) → Fix hgh −1 and hence h : Λ(g) → Λ hgh −1 .
Since we have fixed the generators ξ • , the latter map provides a constant ρ h,g ∈ C * such that h(ξ g ) = ρ h,g ξ hgh −1 .We have Note, that if g, h ∈ G d f or, more generally, if g and h commute ρ h,g is independent of the choice of the generators since g = hgh −1 .More precisely, in this case it could be computed as follows.Let λ k , λ ′ k be the eigenvalues of h and g in their common eigenbasis, then We define the action of GL f on B tot by This is indeed a group action, i.e., (h 2 h 1 ) * = h * 2 • h * 1 .Indeed, using equation (4.2) we get Note that, in particular, if g, h ∈ G d f then h acts on ξ g by for any h ∈ GL f .Similarly for any [p]ξ g with a homogeneous p ∈ C[x 1 , . . ., x N ] and g ∈ GL f we have Remark 4.5.In the literature (see, for example, [25]) a different definition could be found where the sum is taken over the representatives of the conjugacy classes of G and the invariants in each sector are taken with respect to the centralizer of the corresponding g.The two definitions are in fact equivalent in the same way as in [3,Proposition 42].
Example 4.6.Let f = x a 1 1 -the Fermat type polynomial.Assume a 1 = rm and consider G to be generated by g = (1/r).We have x l 1 and the G-invariant monomials are x rn 1 with n ∈ Z.This gives ξ id .

Bigrading
The following operators q l , q r : B tot → Q were first introduced in [17] giving the bigrading we use.
For any homogeneous p ∈ C[x 1 , . . ., x N ] define for [p]ξ g its left charge q l and right charge q r to be (q l , q r ) = deg p − This definition endows B tot with the structure of a Q-bigraded vector space.For u, v ∈ G d f it follows immediately that q This bigrading restricts to B(f, G) because q l , q r commute with the action of h * for any h ∈ GL f , h preserves the weights and age(g) = age hgh −1 .

Hodge diamond of LG orbifolds
Assume N ≥ 3 and the reduced weight set of f to satisfy the CY condition N k=1 q k = 1 (see also Section 3.6).
Proposition 5.1.For f satisfying CY condition and G, such that J ⊆ G ⊆ SL f both left and right charges q l and q r of any Y ∈ B(f, G) are integer.
by Example 4.4.Hence for a class in B(f, G) we have e[−q l + age(g)] = 1 and so q l is integer.■ The following two propositions state that the graded pieces of B(f, G) are organized into a diamond when CY condition holds.Proposition 5.2.Let f be a quasihomogeneous polynomial satisfying the CY condition, let G ⊆ SL f be a finite subgroup, and let V a,b stand for the bidegree (a, b)-subspace of B(f, G).We have Proof .Assume X = [p]ξ g for p being a polynomial fixed by g.
(i) If g = id we have q l (X) = q r (X) = deg p ≥ 0. For g ̸ = id we have age(g) ∈ N ≥1 .Rewriting we see that q l (X) ≥ 0 because k∈I c g q k ≤ N k=1 q k = 1.Similarly for q r (X) by the same argument applied to age g −1 .
(ii) If g = id we have that q l (X) = q r (X) = 0 if and only if deg p = 0.By Propositions 4.1 and 4.2, we have [p] = α [1] in Jac(f ) for some constant α ∈ C.
(iii) If g = id, the statement follows from Proposition 4.1 as given the CY condition we have ĉ = N − 2 k q k = N − 2.
For g ̸ = id, apply the same proposition again to estimate deg p in Jac(f g ).Namely, it gives At the same time, we have Combining this with the inequality above we get One gets in the similar way that q r (X) ≤ N − 2.
Consider the direct sum decomposition of B tot as in equation (4.1).We first extend the ϕ f,β isomorphism of Proposition 4.2 to B tot in the following way.The hessian matrix of f g viewed coordinate free is a bilinear form on the tangent bundle of Fix(g).Therefore, its determinant hess(f g ) is canonically an element of Λ Ng Fix(g) ∨ ⊗2 ⊗ C[Fix(g)].Fix a generator where ι is the interior product operator and let the generator of Λ Ng Fix(g) ∨ ⊗2 to be (ξ ∨ g ) ⊗2 .This choice allows us to fix hess(f g ) as a function on Fix(g) and, hence fix a pairing η f g for the g-sector.As in Proposition 4.2, this in turn defines an isomorphism ϕ f g ,β on each sector.Now the vertical morphism Φ is the direct sum of these isomorphisms acting on each sector of B tot It is an isomorphism restricted to B tot,G for any finite G because each of ϕ f g ,β is an isomorphism.Define the horizontal morphism Ψ to act on the g-th sector by Ψ([p]ξ g ) := [p]ξ g −1 .Extend it by linearity to all B tot .This is an isomorphism because f g = f g −1 and Jac(f g ) = Jac f g −1 .
(1) The maps Φ and Ψ are well defined on B(f, G) for any finite G ⊆ SL f .
(2) For f satisfying CY condition and a finite G ⊆ SL f , let V a,b stand for the bidegree (a, b)subspace of B(f, G).Then the maps Ψ and Φ induce the C-vector spaces isomorphisms Proof . 1) The map Ψ commutes with the G-action since Fix(g) = Fix g −1 .Hence Ψ preserves the invariants.
To see that Φ commute with the G-action, recall first that, f h −1 • x = f (x) and, hence, f g h −1 • x = f hgh −1 (x).Furthermore, since det(h) = 1 we have ρ h,g h * ξ ∨ hgh −1 = ξ ∨ g and we can conclude that This implies the statement.
2) We have directly by the definition that q l (Ψ(X)) = q r (X) and q r (Ψ(X)) = q l (X).The first isomorphism follows.
To verify compatibility of Φ with the grading, note first, that c(f g ) = k∈Ig (1 − 2q k ).Thus, by Proposition 4.2 the left charge of [ϕ f g ,deg p (p)]ξ g is given by q l ([ϕ f g ,deg p (p)]ξ g ) = It follows from the propositions above that for f satisfying CY condition and G such that J ⊆ G ⊆ SL f , the numbers h a,b form a diamond, h 0,0 h 0,1 h 1,0 h 1,1 h 2,0 h 0,2 . . . . . . . . .

Example 2 . 1 .
Fermat, chain and loop type polynomials are examples of quasihomogeneous singularities for any natural

Proposition 2 . 5
(cf.[16, Lemma 3.1]).Any graph Γ f is a disjoint union of the graphs of the following two types (1) oriented tree,(2) oriented circle with the oriented trees having the roots on this oriented circle.

For a finite
G ⊆ GL f put B tot,G := g∈G Jac(f g )ξ g ⊂ B totand define the B-model state space B(f, G) by B(f, G) := (B tot,G ) G .Namely, the linear span of the B tot vectors that are invariant with respect to the action of all elements of G.
sector h-th sector g −1 -th sector h −1 -th sector 2, 5.3 and 5.4.The vertical and horizontal isomorphism are given in Section 4.