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By Y. He

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And conversely all ALE hyper-K¨ahler four-folds are obtained by such a resolution. We remark that in the metric, ζC corresponds to the complex deformation while 5 ( The steps are as follows: (Q ⊗ End(R))Γ = (Q ⊗ Hom( aik Hom(Rk , Rj ))Γ ⊗ Hom(Cni , Cnj ) = ijk 6 i Ri ⊗ Cni , Hom(Cni , Cnj )))Γ = aij Hom(Cni , Cnj ) by Schur’s Lemma. ij Since dim(Xζ ) = dim(Ξ) − 4dim(G) = 2 ij aij ni nj − 4(|Γ| − 1) = 4|Γ| − 4|Γ| + 1 = 4. 52 ζIR = 0 corresponds to the singular limit C2 /Γ. 4 Self-Dual Instantons on the ALE Kronheimer and Nakajima [39] subsequently applied the ADHM construction on the ALE quotient constructed in the previous section.

The one more condition we obtained, namely G = 0, implies that for r > 0 the fields actually live in a hypersurface in CIP4 . Of course such hypersurface, the homogenenous quintic, is a Calabi-Yau manifold. We note therefore, in the limit of r > 0, certain fields whose masses in the original Lagrangian are determined by r, play no rˆole in recovering the Calabi-Yau and are effectively integrated out. We have therefore obtained, in the IR, a conformal nonlinear sigma model on the CY as a hypersurface in a toric variety.

Now we have the i McKay quiver with extra legs. Between each pair of nodes Vq1 and Vq2 we have the 57 (w0 ) W0 j0 i0 (wr ) (v0) V0 B1 Wr Br jr (vr ) Vr i1 ir (v1) V1 j1 B2 (w ) W1 1 (v2) V2 i2 3 (v3) B j2 V3 (w2 ) W2 i3 j3 (w3 ) W3 Figure 4-1: The Kronheimer-Nakajima quiver for C2 /An , extending the McKay quiver to also encapture the information for the construction of the ALE instanton. map Bh with h the edge between these two nodes. We note of course that due to McKay h is undirected and single-valence for SU(2) thus making specifying merely one map between two nodes sufficient.

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


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