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For any subspace U ⊂ V , we define U 0 := {v ∈ V |(u, v) = 0 ∀u ∈ U} with respect to the symplectic (nondegenerate skew bilinear) form ( , ). Moreover we introduce antilinear maps σ : W → W with σ 2 = 1 and σ : V → V with σ 2 = −1 and impose the conditions (1) ∀Z a = 0, UZ := A(Z)W has dimension k and is isotropic (UZ ⊂ UZ0 ); (2) ∀w ∈ W, σA(Z)w = A(σZ)σw. 1) Then the quotient space EZ := UZ0 /UZ of dimension (2k + n − k) − k = n is precisely the rank n SU(n)-bundle E over IP3 which we seek. 9, whereby giving us the required selfdual instanton.

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. We have not named these ADE by coincidence. The resolutions of such singularities were studied extensively by [31] and one sees in fact that the IP1 -blowups intersect precisely in the fashion of the Dynkin diagrams of the simply-laced Lie algebras ADE.

As a holomorphic quotient, a toric variety is simply a generalisation of the complex projective space IPd := (Cd+1 {0})/C∗ with the C∗ -action being the identification x ∼ λx. A toric variety of complex dimension d is then the quotient (Cn \ F )/C∗(n−d) . 34 Qa Here the C∗(n−d) -action is given by xi ∼ λa i xi (i = 1, . . , n; a = 1, . . , n − d) for some integer matrix (of charges) Qai . Moreover, F ∈ Cn \ C∗n is a closed set of points one must remove to make the quotient well-defined (Hausdorff).

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


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