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From this we have a tautological bundle T whose endomorphisms are furnished by ξ ∈ T ∗ ⊗IR Endskew(R). This is tautological in the sense that ξ ∈ Ξ and the points of the base Xζ are precisely the endomorphisms of the fibre R. 53 On Xζ we define operators A ⊗ IdT , IdV ⊗ ξ and Ψ ⊗ IdT : S ⊗ V ⊗ T → W ⊗ T . Finally we define the operator (which is a (2k + n)|Γ| × 2k|Γ| matrix because S and Q are of complex dimension 2, V , of dimension k and R and T , of dimension |Γ|) D := (A ⊗ Id − Id ⊗ ξ) ⊕ Ψ ⊗ Id mapping S ⊗ V ⊗ R → Q ⊗ V ⊗ T ⊕ W ⊗ R.

4 r U (ni ), and This is in the sense that the group U (|Γ|) is broken down, by Γ-invariance, to i=0 then further reduced to G by the modding out. 51 We now apply the hyper-K¨ahler construction in the previous subsection to the ALE manifold  Ξ := (Q ⊗ End(R))Γ = {ξ =  aij hom(Cni , Cnj ) = α β  } ij  α −β † ∼ = (T ∗ ⊗IR Endskew(R))Γ = {ξ =   where α and β are |Γ| × |Γ| matrices satisfying  β α † Rγ αRγ −1  }    = Qγ  α   for Rγ βRγ −1 β γ ∈ Γ. Of course this is simply the Γ-invariance condition; or in a physical context, the projection of the matter content on orbifolds.

Nite groups. Moreover, the Cartan matrices will correspond to certain graphs constructable from the latter. 43 Chapter 4 Finite Graphs, Quivers, and Resolution of Singularities We have addressed algebraic singularities, symplectic quotients and orbifolds in relation to finite group representations. It is now time to embark on a journey which would ultimately give a unified outlook. To do so we must involve ourselves with yet another field of mathematics, namely the theory of graphs. 1 Some Rudiments on Graphs and Quivers As we shall be dealing extensively with algorithms on finite graphs in our later work on toric singularities, let us first begin with the fundamental concepts in graph theory.

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