Integral structure for simple singularities

We compute the image of the Milnor lattice of an ADE singularity under the period map. We also prove that the Milnor lattice can be identified with an appropriate relative $K$-group defined through the Berglund--H\"ubsch dual of the corresponding singularity.


Introduction
Let f ∈ C[x 1 , x 2 , x 3 ] be a homogeneous polynomial representing the germ of a simple singularity of type A, D, or E. Let f T ∈ C[x 1 , x 2 , x 3 ] be the corresponding Berglund-Hübsch dual of f (see Section 1.2). The main motivation for the problem solved in this paper is to find an explicit identification between the Fan-Jarvis-Ruan-Witten (FJRW) invariants of f T and the Kac-Wakimoto hierarchies.
Such an explicit identification is needed if one is interested in constructing a matrix model for the FJRW invariants of f T , similar to the Kontsevich's matrix model in [10].
Let us recall that the connection between FJRW invariants and Kac-Wakimoto hierarchies can be described in two steps. First, Fan-Jarvis-Ruan proved in [7] that the generating function of FJRW invariants of f T coincides with the total descendant potential of the simple singularity f . Second,  and Frenkel-Givental-Milanov [6] proved that the total descendant potential of a simple singularity is a tau-function of the principal Kac-Wakimoto hierarchy of the same type A, D, or E as the singularity f . However, while the state space of FJRW theory is identified explicitly with the Milnor ring of the singularity (see [7]), the identification of the Milnor ring and the Cartan subalgebra of the corresponding simple Lie algebra is given by a period map and it is not explicit.
In order to obtain an explicit identification, we need to determine the image of the root lattice in the Milnor ring of the singularity. This is exactly the problem that we want to solve in this paper.
1.1. Simple singularities. Let us give a precise statement of the problem that we want to solve. Let f (x 1 , x 2 , x 3 ) = g(x 1 , x 2 ) + x 2 3 , where g is one of the polynomials listed in the following table: The polynomial f represents the germ of a simple singularity at x = 0. Let where α ∈ H 2 (V 1 ; C), φ i (x) (1 ≤ i ≤ N ) is a set of polynomials representing a basis of H f , α λ ∈ H 2 (V λ ; C) is obtained from α via a parallel transport along some reference path, and ω df is the socalled Gelfand-Lerey form (see [1]). Note that I (−1) : C \ {0} → H f is a multivalued analytic function.
Let us assign degree c i ∈ Q >0 to x i (1 ≤ i ≤ 3), such that, the polynomial f has degree 1. Then the Milnor ring becomes a graded ring. The highest possible degree of a homogeneous element in H f 7. It turns out that our answer can be stated quite elegantly via relative K-theory. The idea to look for such a description comes from the work of Iritani [9], Chiodo-Iritani-Ruan [4], and Chiodo-Nagel [5].
The polynomials f corresponding to a simple singularity are invertible polynomials in the sense of [3] (see also [11]). Each polynomial is uniquely determined by a 3 × 3 matrix A = (a ij ) 1≤i,j≤3 with non-negative integer coefficients, such that, 3 .
Let G T be the group of diagonal symmetries of f T , that is, 1 t a 2i 2 t a 3i 3 = 1 ∀i}. 2 Let a ij (1 ≤ i, j ≤ 3) be the entries of the inverse matrix A −1 . The group G T is generated by the following elements Our main interest is in the topological relative K-theoretic orbifold group In general, there is no satisfactory definition of K-theory for non-compact spaces. However, in our case the pair (C 3 , V T 1 ) is G T -equivariantly homotopic to a pair of finite CW complexes, so we may think of (C 3 , V T 1 ) is a G T -equivariant pair of finite CW -complexes. We refer to [15] for some background on equivariant topological K-theory.
Motivated by Iritani's Γ-integral structure in quantum cohomology (see [9]), we will now construct a linear map which is a certain Γ-class modification of the orbifold Chern character map. For a G T -equivariant space X and g ∈ G T , let us denote by Fix g (X) := {x ∈ X | gx = x} the set of fixed points. The elements in the relative K-group will be identified with isomorphism classes [E → F] of two-term complexes E d / / F of G T -equivariant vector bundles, such that, the differential d is a morphism of G Tequivariant vector bundles and d| V T

1
: is an isomorphism. Note that for g ∈ G T , the restriction of a vector bundle E| Fix g (C 3 ) decomposes as a direct sum of eigen-subbundles E ζ and that the restriction to Fix g (C 3 ) of every two term complex E d / / F decomposes as a direct sum of two We have the following well known decomposition (e.g. see [2], Theorem 2): The standard Chern character map gives an isomorphism Finally, if G is a finite group acting on a smooth manifold M, such that the quotient groupoid [M/G] is an effective orbifold, then H * (M/G; C) [H * (M; C)] G . Indeed, for a finite group G the operation taking G-invariants is an exact functor from the category of G-vector spaces to the category of vector spaces. Therefore where A * M is the sheaf of smooth differential forms on M with complex coefficients, the first isomorphism is Satake's de Rham theorem for orbifolds (see [14]), and the last one is the de Rham's theorem for the manifold M. Using the long exact sequence of a pair, we get also that H i (M/G, N /G; C) On the other hand, by definition, Therefore, the composition ch := ch • Tr defines a ring homomorphism which is the orbifold version of the Chern character map. Clearly ch is an isomorphism over C.

Remark 2.
Orbifold cohomology H * orb has two natural gradings -standard topological degree grading coming from the topological space underlying the orbit space and Chern-Ruan grading. In this paper we work with the topological grading and the topological cup product.
Let us recall also the definition of the Γ-class.
is an orbifold vector bundle and Tr(E) = g ζ ζE ζ , then each eigenvalue ζ = e 2πiα , where 0 ≤ α < 1 is a rational number and we define and ι * is an involution in orbifold cohomology that exchanges the direct summands corresponding to g and g −1 . Note that the definition of ι * makes sense because Fix g = Fix g −1 . 4

Theorem 3. There exists a linear isomorphism
such that, the map is an isomorphism of lattices.
Unfortunately we do not have a conceptual definition of the map mir. Our definition is on a case by case basis. We expect that H * has a natural identification with the state space of FJRW-theory under which mir is identified with the mirror map of Fan-Jarvis-Ruan (see [7]). Let us point out also that in all cases the following two properties are satisfied: 3 is a homogeneous monomial representing a vector in H f , then its image under mir is in the twisted sector corresponding to g = ρ (2) The map mir is defined over Q, that is, mir provides an isomorphism 2. Period map image of the Milnor lattice 2.1. Suspension isomorphism in vanishing homology. We will reduce the problem of computing periods of the hypersurface V λ to computing periods of the Riemann surfaces . The fibers of this map are given by Suppose now that A ∈ H 1 (M λ ; Z) is any cycle. The following two maps have images that fit together and give a two-dimensional cycle α ∈ V λ , that is, α = ΣA is the suspension of the cycle A. It is known that the above suspension operation Σ : is an isomorphism (see [1]). 5 Note that we may choose the basis of H f to be such that φ i = φ i (x 1 , x 2 ) does not depend on x 3 .

2.2.
Simple singularities and root systems. Let us first recall several well known facts about simple singularities, which will be needed in our computation (see [1]). The analytic continuation of I is the so-called classical monodromy operator. Recalling the definition of Ψ (see formula (1)), we get the following relation: It is known that (α|β) = −α • β, where • is the intersection pairing (see [8,13]). In particular, the form ( | ) takes integer values on the Milnor lattice.
Finally, let us also recall that we have the following remarkable facts (see [1]): (1) The set of vanishing cycles of the singularity f coincides with the set of all α ∈ H 2 (V 1 ; Z) such that (α|α) = 2.
(2) The tripple (Milnor lattice, set of vanishing cycles, pairing ( | )) form a root system of the same type as the type of the singularity f , that is, the set of vanishing cycles corresponds to the roots, the Milnor lattice corresponds to the root lattice, and ( | ) corresponds to the invariant bilinear form.
(3) The classical monodromy corresponds to a Coxeter transformation. 6 2.3. A N -singularity. Let us fix the following basis of H f : The residue pairing takes the form where h = N + 1 is the Coxeter number. The Riemann surface M µ for µ 0 is a non-singular curve in along L k+1 except that this time we make a small loop C −1 k+1 clockwise around x 1,k+1 . In other words The cycles A k,a satisfy the following relations A k,0 = −A k,1 and N k=0 A k,a = 0. Let us asume a ∈ Z 2 \ {0} and k ∈ Z N +1 \ {0}, then we get N loops whose homology classes, as we will see later on, represent a basis of H 1 (M µ ; Z).
Let us compute the periods of the holomorphic forms along the cycles A k,a . The paths L k and C k can be parametrized as follows: The integrals along the lifts of C k contribute to the period integral terms of order O(ǫ 1 2 ). These terms vanish in the limit ǫ → 0. The periods that we want to compute are independent of ǫ for homotopy reasons. Therefore, by passing to the limit ǫ → 0 we get The integrals along L k,a can be computed as follows: . 7 Let α k,a = ΣA k,a be the suspension. Recalling formula (3) and using that Recalling the formulas for the residue pairing we get Let us point out that formula (4) yields the following formulas for the classical monodromy operator The intersection pairing takes the form where k, l ∈ Z N +1 and the Kronecker delta is also on Z N +1 . Note that (α k,1 |α k,1 ) = 2, so α k,1 is a vanishing cycle. The determinant of the intersection pairing in the basis {α k,1 }, that is, the determinant of the matrix (α k,1 |α l,1 ) N k,l=1 is N + 1, which coincides with the determinant of the Cartan matrix of the simple Lie algebra of type A N . Therefore, {α k,1 } is a Z-basis of the root lattice, that is, and hence their images Ψ(α k,1 ) (see formula (7)) give a basis for the image of the Milnor lattice in H f .

D N -singularity.
Let us fix the following basis of H f : The residue pairing takes the form where h = 2N − 2 is the Coxeter number. The Riemann surface M µ for µ 0 is a non-singular curve in C 2 defined by the equation k is a loop starting at a point on the line segment L k sufficiently close to 0, it goes along the line segment L k , just before hitting the branch point x 2,k it makes a small loop C k around it and it returns back to the starting point along the line segment L k , and finally it makes a small loop C 0 around 0. Clearly, the loop Let us compute the periods of the holomorphic forms Let us parametrize A ′ k as follows: The integrals along the lifts of C 0 and C k contribute to the period integral terms of orders respectively O(ǫ i−1/2 ) and O(ǫ 1/2 ). These terms vanish in the limit ǫ → 0. The two lifts of L k , before and after going around the branch point x 2,k , have parametrizations, such that, where t varies from 0 to 1, and where t varies from 1 to 0. Now it is clear that the period integral, after passing to the limit ǫ → 0, takes the form . The above integral can be expressed in terms of Euler's Beta We get the following formulas: Let α k = ΣA k be the suspension. Recalling formula (3) and using (6), we get Therefore, Proposition 5. The image of the Milnor lattice under the map Ψ is the lattice in H f with Z-basis Proof. Using formula (5), it is straightforward to check that {v i } 1≤i≤N is an orthonormal basis of H f with respect to the intersection pairing, that is, The residue pairing takes the form where h = 12 is the Coxeter number. The Riemann surface M µ for µ 0 is a non-singular curve in C 2 defined by the equation x 3 1 +x 4 2 = µ. The projection (x 1 , x 2 ) → x 2 defines a degree 3 branched covering M µ → C, with branching points x 2,k = µ k be a loop in the x 2 -plane C going around the branch points x 2,k and x 2,k+1 in the following way: the loop starts at 0, it goes along the line segment L k , just before hitting the branch point x 2,k it makes a small loop C k counterclockwise around x 2,k , it returns back to the starting point along L k ; then the loop travels in a similar fashion along L k+1 except that this time we make a small loop C −1 k+1 in clockwise direction around x 2,k+1 . Clearly, the loop along the cycles A k,a . As a byproduct of our computation we will get that the homology classes of these 6 loops form a basis of H 1 (M µ ; Z). Let us parametrize A ′ k as follows: The integrals along the lifts of C k contribute to the period integral terms of orders These terms vanish in the limit ǫ → 0. Therefore, under this limit, the periods of the holomorphic where η 3 := e 2πi 3 and the integral Then, Let α k,a = ΣA k,a be the suspension. Recalling formula (3) and using (6) (I Recalling the formulas for the residue pairing in the basis {φ i } we get Recalling formula (1) and using that by definition θ(φ i ) = Let us also point out that by using formula (4), we get the following formulas for the classical monodromy operator: Recalling formula (5), we get that the intersection pairing Let us identify Z 3 \ {0} = {1, 2} and Z 4 \ {0} = {1, 2, 3}. Every 1 ≤ a ′ ≤ 6 can be written uniquely in the form a ′ = 3(a − 1) + k, where 1 ≤ a ≤ 2 and 1 ≤ k ≤ 3. Let us define α a ′ := α k,a . The intersection pairings (α a ′ |α b ′ ) are straightforward to compute using the formula from above. We get that (α a ′ |α b ′ ) 12 coincides with the (a ′ , b ′ )-entry of the following matrix: The above matrix has determinant 3. Therefore, the set {α k,a | 1 ≤ a ≤ 2, 1 ≤ k ≤ 3} is a set of linearly independent vanishing cycles. Since the set of all vanishing cycles is a root system of type E 6 and the determinant of the Cartan matrix of a root system of E 6 is also 3, we get that the {α k,a } is a set of simple roots. In particular, it is a Z-basis of the Milnor lattice.
2.6. E 7 -singularity. Let us fix the following basis of H f : The residue pairing takes the form 3 ) . We will compute the period integrals along A k,a for 0 ≤ k, a ≤ 2. As a biproduct of our computation we will get that the following set of 7 loops {A 0,1 , A 1,1 , A 2,1 , A 0,2 , A 1,2 , A 2,2 , A 0,0 } represents a basis of H 1 (M µ ; Z). 13 Let us compute the periods of the holomorphic forms along the cycle A k,a . Let us parametrize A ′ k as follows: The integrals along the lifts of C 0 and C k contribute to the period integral terms of orders respectively In the limitǫ, ǫ → 0 all integrals along the loops C 0 and C k vanish except for the integral along C 0 when i = 7. The latter however is straightforward to compute. Therefore, after passing to the limit ǫ,ǫ → 0, we get where η 3 := e 2πi 3 and the integral Let α k,a = ΣA k,a be the suspension. Recalling formula (3) and using (6) (I if i = 7, 14 Recalling the formulas for the residue pairing in the basis {φ i } we get Recalling formula (1) and using that we get Recalling formula (4), we get the following formulas for the classical monodromy operator: Using formula (5), we get that the intersection pairing (α k,a |α l,b ) = 1 π (Ψ(α k,a ), cos(πθ)Ψ(α l,b )) cos(( i 3 − 7 9 )π) sin(( i 3 − 1 9 )π) sin( π 3 ) . Let us identify Z 3 = {0, 1, 2}. Every 1 ≤ a ′ ≤ 7 can be written uniquely in the form a ′ = 3(a − 1) + k + 1, Put α a ′ := α k,a , where 0 ≤ a ≤ 2 is the remainder of a modulo 3. Using the above formula, we get that the intersection pairing in the basis {α a ′ } 1≤a ′ ≤7 takes the form The above matrix has determinant 2. Since the determinant of the Cartan matrix of the root system of type E 7 is also 2, the conclusion is the same as in the case of E 6 -singularity, that is, the cycles 15 (α 1 , . . . , α 7 ) = (α 0,1 , α 1,1 , α 2,1 , α 0,2 , α 1,2 , α 2,2 , α 0,0 ) form a Z-basis of the Milnor lattice and hence their images under Ψ, computed by formula (10) The residue pairing takes the form where h = 30 is the Coxeter number. The Riemann surface M µ for µ 0 is a non-singular curve in C 2 defined by equation x 3 1 + x 5 2 = µ. The projection (x 1 , x 2 ) → x 2 defines a degree 3 branched covering M µ → C, with branching points x 2,k = µ 1 5 η k 5 , k ∈ Z 5 , where η 5 = e 2 5 πi . The method for constructing loops in M µ is almost the same as that for E 6 -singularity. Let us omit the similar narration, i.e., we define the loops A k,a in the same way, except that now a ∈ Z 3 \ {0} and k ∈ Z 5 \ {0}. We will see that the homology classes of these 8 loops form a Z-basis of H 1 (M µ ; Z). Let us compute the periods of the holomorphic forms along the cycle A k,a . Let us parametrize A ′ k as follows: The integrals along the lifts of C k contribute to the period integral terms of orders These terms vanish in the limit ǫ → 0. Therefore, under this limit, the periods of the holomorphic where η 3 := e Then, Let α k,a = ΣA k,a be the suspension. Recalling formula (3) and using (6) (I Recalling the formulas for residue pairing, we get By definition, Therefore, recalling formula (1), we get Recalling formula (4), we get that the following formulas for the classical monodromy operator: Recalling formula (5), the intersection pairing Let us identify Z 3 \ {0} = {1, 2} and Z 5 \ {0} = {1, 2, 3, 4}. Every 1 ≤ a ′ ≤ 8 can be written uniquely in the form a ′ = 4(a − 1) + k, where 1 ≤ a ≤ 2 and 1 ≤ k ≤ 4. Put α a ′ := α k,a . Then the intersection matrix (α a ′ |α b ′ ) takes the following form: The above matrix has determinant 1. Since the determinant of the Cartan matrix of the root system of type E 8 is also 1, the conclusion is the same as in the previous cases.

K-theoretic interpretation
The goal of this section is to prove Theorem 3

Fermat cases. Let us compute explicitly the map ch
3 with a 3 = 2. In fact, our computation works for arbitrary a 3 as well, except for one small technical detail, that is, we will prove that the group K −1 G T (V T 1 ) is torsion free. This fact should be true for any positive integer a 3 , but the argument that we give works only if a 3 = 2. The group If g ∈ G T is such that I = {i | g i = 1} is a non-empty set, then it is easy to see that the map x = (x 1 , x 2 , x 3 ) → (x a i i ) i∈I induces isomorphisms Fix g (C 3 )/G T C I and Fix g (V T 1 )/G T H I , where H I ⊂ C I is the hyperplane i∈I y i = 1. Since the pair (C I , H I ) is contractible the groups If g ∈ G T is such that g i 1 for all i, then Fix g (C 3 ) = {0} and Fix g (V T 1 ) = ∅. Note that the number of such g is N = (a 1 − 1)(a 2 − 1)(a 3 − 1), that is, the multiplicity of the singularity corresponding to the polynomial f .
Let us postpone the proof of this lemma until Section 3.4. Note that K −1 G T (V T ) ⊗ C = 0. Therefore, according to the above Lemma 6, we have K −1 G T (V T 1 ) = 0. The long exact sequence of the pair (C 3 , V T 1 ) yields the following exact sequence On the other hand, K 0 G T (C 3 ) coincides with the representation ring of G T , that is, where L i = C 3 ×C is the trivial bundle with G T -action g ·(x, λ) := (gx, g i λ). Note that T C 3 L 1 +L 2 +L 3 in the category of G T -equivariant bundles. We claim that Koszul complex corresponding to the sequence (s 1 , s 2 , s 3 ) has the form where C is the trivial bundle with trivial G T -action. The sequence (s 1 , s 2 , s 3 ) is regular, so the corresponding Koszul complex is a resolution of the structure sheaf of the zero locus {s 1 = s 2 = s 3 = 0}. The zero locus is {0} and since 0 V T 1 the restriction of the Koszul complex to V T 1 is exact, i.e., the Koszul complex represents an element of K 0 G T (C 3 , V T 1 ). This proves that the RHS of (12) is a Z-submodule of the LHS. Note that both the LHS and the RHS of (12) are free Z-modules of rank N . Therefore, the quotient of LHS by RHS is a finite Abelian group. In order to prove that the quotient is 0, it is sufficient to prove that if g ∈ K 0 G T (C 3 ) and mg belongs to the RHS of (12) for some integer m, then g belongs to the RHS of (12) too. The proof is straightforward so we leave it as an exercise.
Let us fix the following basis of K 0 G T (C 3 , V T 1 ): Let e k 1 ,k 2 ,k 3 = 1 ∈ H 0 (Fix g (C 3 )/G, Fix g (V T 1 )/G), where g = (e 2πik 1 /a 1 , e 2πik 2 /a 2 , e 2πik 3 /a 3 ). We get Let us specialize the above formula to the cases of A N , E 6 , and E 8 singularities. In the first case a 1 = N + 1, a 2 = a 3 = 2. The above formula takes the form Comparing with (7), we get that if we define mir(φ i ) = e i,1,1 (1 ≤ i ≤ N ), then the images of Ψ and ch Γ will coincide. The vanishing cycle α k,a corresponds to (−1) a A k,0,0 .
Suppose now that the singularity is of type E 8 , that is, a 1 = 3, a 2 = 5, and a 3 = 2. The formula takes the form where η 3 = e 2πi/3 and η 5 = e 2πi/5 . Comparing with formula (11) we get that if we define

Proof. By definition
where we used that K i µ 2 (I, ∂I) is isomorphic to Z for i even and 0 for i odd, so the last isomorphism is given by the equivariant Küneth formula (see [12]). 20 Suppose now that Y ⊂ X is a G-invariant CW-subcomplex of X. Using the long exact sequence of the tripple Ë 0 ⊂ ΣY ⊂ ΣX and Lemma 7, it is straightforward to prove the following corollary.

Corollary 8. The exterior tensor product by ℓ induces an isomorphism
3.3. The relative K-ring for D N -singularity. Let us return to the settings of D N -singularity. We Let L i = C 3 × C be the G T -equivariant line bundle for which the action of G T on C is given by the Let us introduce the following N complexes of G T -equivariant vector bundles on C 3 : .
is a tensor product of L i−1 2 , L 1 x 1 / / C , and L 3 x 3 / / C and that the complex E • N is a tensor product of C = 1}. Slightly abusing the notation we denote by L 1 and L 2 the restriction of the vector bundles L 1 and L 2 to C 2 . Note that the operation tensor product by the complex L 3 x 3 / / C is precisely the exterior tensor product by the complex ℓ in the suspension isomorphism from Corollary 8. Therefore, it is sufficient to prove that the complexes M). The long exact sequence of the pair (C 2 , M) yields the following exact sequence: where we used that K −1 A (C 2 ) = 0. We have where the RHS is the representation ring of A. Just like in the Fermat cases it is easy to prove that the image of ρ coincides with the ideal (L 1 −1)K 0 A (C 2 ). Note that Im(ρ) Z N −1 and that ρ( It remains only to prove that K −1 A (M) Z and that Im(δ) is generated as a Z-module by the complex E • N . Let us first prove that K −1 A (M) Z. Let π : M → C * be the map (x 1 , x 2 ) → x 2 1 . The map π is a branched covering with only one branch point, that is, 1 ∈ C * . The corresponding ramification points are R = {(−1, 0), (1, 0)}. Note that R is an A-invariant subset. The idea is to use the long exact sequence of the pair (M, M \ R). The action of A on M \ R is free, so we have Therefore an open neighborhood of the zero section R in ν R . Clearly, the pullback of ν R to U is L 2 and the Thom class of ν R is represented as an element of K 0 is the cyclic subgroup of A generated by (1, η 2 ). Therefore, K −1 A (R) = 0 and K 0 A (R) coincides with the representation ring of B. Since the Thom isomorphism is given by tensor product with the Thom class, we get The long exact sequence of the pair (M, M \ R) takes the form We already proved that We will make use of the following explicit interpretation of the K-group K −1 A ( ). By definition, for any finite CW-complex X, we have K −1 A (X) = K 0 A (Σ(X ⊔ pt)). Since the complement of X in Σ(X ⊔ pt) is contractible, we can think of an element of K −1 A (X) as a representation of A on some vector space C r and an A-equivariant isomorphism φ : X × C r → X × C r , that is, an A-equivariant morphism X → GL r (C). In our case the elements of . The latter is generated by two elements that correspond, in the way described above, to the two maps C \ {0, 1} → C * , t → t and t → 1 − t.
Therefore, the group K −1 A (M \ R) is generated by the two elements that correspond to the two maps M \ R → C * defined by (x 1 , x 2 ) → and that what δ(φ) is. The extensions in our case are straightforward to construct. We get that Note however, that x 1 0 so x 2 1 defines an isomorphism, i.e., the first complex is 0 in K 0 A (M, M \R). In particular, the kernel of the connecting homomorphism δ is Z and it is generated by the element in The map π is a branched covering with only one branching point, that is, 1 ∈ C. The corresponding ramification points are R = {(ξ, 0) | ξ a 1 = 1}. The torsion freeness can be deduced easily from the long exact sequence of the pair (M, M \ R). The action of A on M \ R is free, so we have Using the Thom isomorphism for the normal bundle to R in M, we get . On the other hand, note that R is the orbit of A through the point (1, 0) ∈ M, we get R = A/B, where B ⊂ A is the cyclic subgroup generated by (1, η a 2 ), η a 2 = e 2πi/a 2 . Therefore, K −1 A (R) = K −1 A (A/B) = K −1 (B) = 0. Recalling the long exact sequence of the pair (M, M \ R), we get We get that K −1 A (M) can be embedded as a subgroup of K −1 A (M \ R) Z. The latter is torsion free, so K −1 A (M) must be also torsion free.
Remark 10. The above argument can be continued to give a direct proof of the fact that K −1 A (M) = 0. Namely, using the Thom isomorphism, we can prove that the group K 0 A (M, M \ R) is a free Abelian group of rank a 2 and that the complexes [L i−1 2 x 2 / / L i 2 ] (1 ≤ i ≤ a 2 ) represent a Z-basis. Moreover, the image of the connecting morphism δ in (13) can be computed explicitly as well, that is, it coincides with the sum of the above complexes. In particular, we get that δ is an injective map, and hence that is, both the groups and the differentials can be determined. This allows us to give an alternative proof of formula (12). We leave the details to the interested reader.
3.5. The relative K-ring for E 7 -singularity. The argument from the previous section works also for E 7 -singularity. Let us only state the result. The proof is completely analogous.
3.6. Γ-integral structure for D N -singularity. Let us compute ch Γ (E • i ) for 1 ≤ i ≤ N . After a straightforward computation we get that the relative cohomology group H(Fix g (C 3 ), Fix g (V T )) G T is not zero only in the following two cases: 1) g = (g 1 , g 2 , g 3 ) with g i 1 for all i and 2) g = (1, 1, −1). For the first case, there are N − 1 elements, that is, g = (−1, η 2i−1 , −1) (1 ≤ i ≤ N − 1) and the fixed point subsets are Fix g (C 3 ) = {0} and Fix g (V T ) = ∅. Therefore, H(Fix g (C 3 ), Fix g (V T ); C) G T C is non-trivial only in degree 0 and we denote by e i := 1 the unit of the cohomology group. For the second case, A closed form (ω, α) in degree i, that is, d(ω, α) = 0, defines naturally a linear functional on the space of dimension i relative chains γ ⊂ C 2 with ∂γ ⊂ M, that is, Using the de Rham theorem for C 2 and M, it is easy to prove that the above map induces an isomorphism between the i-th cohomology of the complex (14) and H i (C 2 , M; C) G T . Let us denote by e N ∈ H 2 (C 2 , M; C) the cohomology class corresponding to the form (0, − 1 2πi dx 1 /x 1 ). Suppose now that g = (−1, η 2a−1 , −1), 1 ≤ a ≤ N − 1. Let us compute the component of ch Γ (E • i ) for 1 ≤ i ≤ N − 1 in H 0 (Fix g (C 3 ), Fix g (V T ); C) G T . Note that in this case we have an isomorphism K 0 (Fix g (C 3 ), Fix g (V T )) K 0 (Fix g (C 3 )). The image of ι * Tr(E • i ) is where again we abused the notation by denoting by L i the restriction of L i to