By Y. He

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This above construction , where in fact the natural connection on the bundle Pζ → Xζ is self-dual, is the celebrated hyper-K¨ ahler quotient construction [33]. Now we present a remarkable fact which connects these moment maps to the previous section. 1) for SU(n) groups into a (perhaps more standard) component form, we have the ADHM data M := {A, B; s, t†|A, B ∈ End(V ); s, t† ∈ Hom(V, W )}, 50 with the ADHM equations [A, B] + ts = 0; ([A, A† ] + [B, B † ]) − ss† + tt† = 0. 1 The moment maps for the triholomorphic SU(n) isometries precisely encode the ADHM equation for the SU(n) self-dual instanton construction.

3) and define the toric variety accordingly, the toroidal C∗(n−d) action is prescribed exactly as Qa λa : xi → λa i xi for xi ∈ Cn . In this description therefore, the moment map defining the toric variety is simply the D-term and the charge matrix of the linear sigma model gives the relations among all the generators of the cone. In the case of the toric variety being singular, the desingularisation thereof simply corresponds to the acquisition of non-zero values of the FI-parametre r. In this way we can describe any toric variety as a gauged linear sigma model with charge matrix Qai whose integer kernel has ZZ-span vi , which are the generators of the cone.

Nn ), the vector of Dynkin labels of the Affine Dynkin graph associated with Γ and let w = 0, then for9 ζ := (ζIR , ζC ) ∈ IR3 ⊗ Z \ θ∈IR+ \{n} IR3 ⊗ Dθ , the manifold Xζ := {B ∈ M(v, 0)|µ(B) = ζ}//G′ is the smooth resolution of C2 /Γ with corresponding ALE metric. For our purposes this construction induces a natural bundle which will give us the required instanton. , r indexing the non-Affine nodes where Cnl is the space acted upon by the irreps of Γ (whose dimensions, by the McKay Correspondence, are precisely the Dynkin labels) such that U(Vq ) acts trivially (by Schur’s Lemma) unless q = l.

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


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