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By Y. He

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For a illustrative review upon this elegant subject, the reader is referred to [9]. 1 McKay’s Correspondence Perhaps it is a good point here to introduce the famous McKay correspondence, which will be a major part of Liber III. We shall be brief now, promising to expound upon the matter later. Due to the remarkable observation of McKay in [32], there is yet another justification of naming the classification of the discrete finite subgroups Γ of SU(2) as ADE. Take the defining representation R of Γ, and consider its tensor product with all the irreducible representations Ri : R ⊗ Ri = aij Rj .

From σ we can compute its dual cone σ ∨ as σ ∨ := {u ∈ MIR |u · v ≥ 0∀v ∈ σ} . Subsequently we have a finitely generated monoid Sσ := σ ∨ ∩ M = {u ∈ M|u · σ ≥ 0} . We can finally associate maximal ideals of the monoid algebra of the polynomial ring adjoint Sσ to points in an algebraic (variety) scheme. This is the affine toric variety Xσ associated with the cone σ: Xσ := Spec(C[Sσ ]). 35 To go beyond affine toric varieties, we simply paste together, as co¨ordinate patches, various Xσi for a collection of cones σi ; such a collection is called a fan Σ = i σi and we finally arrive at the general toric variety XΣ .

But pray be patient as this discussion would have to wait until Liber II. 2 du Val-Kleinian Singularities Having digressed some elements of graph and quiver theories, let us return to algebraic geometry. We shall see below a beautiful link between the theory of quivers and that of orbifold of C2 . First let us remind the reader of the classification of the quotient singularities of C2 , these date as far back as F. Klein [30]. The affine equations of these so-called ALE (Asymptotically Locally Euclidean) singularities can be written in C[x, y, z] as An : xy + z n = 0 Dn : x2 + y 2 z + z n−1 = 0 E6 : x2 + y 3 + z 4 = 0 E7 : x2 + y 3 + yz 3 = 0 E8 : x2 + y 3 + z 5 = 0.

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Algebraic Singularities, Finite Graphs and D-Brane Theories by Y. He


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