This paper studies connections between the preprojective representations of a valued quiver, the (+)-admissible sequences of vertices, and the Weyl group by associating to each preprojective representation a canonical (+)-admissible sequence. A (+)-admissible sequence is the canonical sequence of some preprojective representation if and only if the product of simple reflections associated to th...

The quantum-event / prime ideal in a category/ noncommutative-point alternative to classical-event / commutative prime ideal/ point is suggested. Ideals in additive categories, prime spectra and representation of quivers are considered as mathematical tools appropriate to model quantum mechanics. The space-time framework is to be reconstructed from the spectrum of the path category of a quiver....

Renormalization group flows of quiver gauge theories play a central role in determining the low-energy properties of string vacua. We demonstrate that useful predictions about the RG dynamics of a quiver gauge theory may be extracted from the global structure of its quiver diagram. For quiver theories of a certain type, we develop an efficient and practical method for determining which superpot...

We construct Nakajima’s quiver varieties of type A in terms of conjugacy classes of matrices and (non-Slodowy’s) transverse slices naturally arising from affine Grassmannians. In full generality quiver varieties are embedded into Beilinson-Drinfeld Grassmannians of type A. Our construction provides a compactification of Nakajima’s quiver varieties and a decomposition of an affine Grassmannian i...

We introduce gauge networks as generalizations of spin networks and lattice gauge fields to almost-commutative manifolds. The configuration space of quiver representations (modulo equivalence) in the category of finite spectral triples is studied; gauge networks appear as an orthonormal basis in a corresponding Hilbert space. We give many examples of gauge networks, also beyond the well-known s...

The purpose of this proposal is to study algebraic symplectic varieties, which arise naturally in algebraic geometry (Hilbert schemes), representation theory (quiver varieties, Springer theory), combinatorics and polyhedral geometry (hypertoric varieties), and string theory (moduli spaces of gauge theories and of Higgs bundles). Our primary interest will be a certain category of sheaves on thes...

Let G be an algebraic torus acting on a smooth variety V . We study the relationship between the various GIT quotients of V and the symplectic quotient of the cotangent bundle of V . Let G be a reductive algebraic group acting on a smooth variety V . The cotangent bundle T V admits a canonical algebraic symplectic structure, and the induced action of G on T V is hamiltonian, that is, it admits ...

Crawley-Boevey and Shaw recently introduced a certain multiplicative analogue of the deformed preprojective algebra, which they called the multiplicative preprojective algebra. In this paper we study the moduli space of (semi)stable representations of such an algebra (the multiplicative quiver variety), which in fact has many similarities to the quiver variety. We show that there exists a compl...

We consider equivariant dimensional reduction of Yang-Mills theory on Kähler manifolds of the form M×CP 1×CP . This induces a rank two quiver gauge theory on M which can be formulated as a Yang-Mills theory of graded connections on M . The reduction of the Yang-Mills equations on M×CP 1×CP 1 induces quiver gauge theory equations on M and quiver vortex equations in the BPS sector. When M is the ...