Toric Duality arises as an ambiguity in computing the quiver gauge theory living on a D3-brane which probes a toric singularity. It is reviewed how, in simple cases Toric Duality is Seiberg Duality. The set of all Seiberg Dualities on a single node in the quiver forms a group which is contained in a larger group given by a set of Picard-Lefschetz transformations. This leads to elements in the g...

Given an Artinian algebra A over a field k, there are several combinatorial objects associated to A. They are the diagram DA as defined by Drozd and Kirichenko, the natural quiver ΔA defined by Li (cf. Section 2), and a generalized version of k-species (A/r, r/r2) with r being the Jacobson radical of A. When A is splitting over the field k, the diagram DA and the well-known Ext-quiver ΓA are th...

This is an expository article on the singularities of nilpotent orbit closures in simple Lie algebras over the complex numbers. It is slanted towards aspects that are relevant for representation theory, including Ma ei's theorem relating Slodowy slices to Nakajima quiver varieties in type A. There is one new observation: the results of Juteau and Mautner, combined with Ma ei's theorem, give a g...

We consider the Bethe Ansatz Equations for orbifolds of N =4 SYM w.r.t. an arbitrary discrete group. Techniques used for the Abelian orbifolds can be extended to the generic non-Abelian case with minor modifications. We show how to make a transition between the different notations in the quiver gauge theory.

The fixed point set under a natural torus action on projectivized moduli spaces of simple representations of quivers is described. As an application, the Euler characteristic of these moduli is computed.

1.1. General formalism and basic examples. Let A = ⊕ i>0Ai be a graded algebra equipped with a Poisson bracket {·, ·} that has degree −1 because {Ai, Aj} ⊂ Ai+j−1. By a quantization of A one means a pair (A, ι), where • A is a filtered associative unital algebra A = ∪ i>0 A6i. • ι : grA → A is an isomorphism of graded Poisson algebras. Recall that grA := ⊕ i>0 A6i/A6i−1 is a graded algebra. Tha...

We state a conjectural relationship between the fermionic form [HKOTY] and the Betti numbers of a Grassmannian over a preprojective algebra or, equivalently , of a lagrangian quiver variety.