We show that the fixed-point subvariety of a Nakajima quiver variety under a diagram automorphism is a disconnected union of quiver varieties for the ‘split-quotient quiver’ introduced by Reiten and Riedtmann. As a special case, quiver varieties of type D arise as the connected components of fixed-point subvarieties of diagram involutions of quiver varieties of type A. In the case where the qui...

This note provides a specific quiver which does not admit a maximal green sequence, and which is mutation-equivalent to a quiver which does admit a maximal green sequence. This provides a counterexample to the conjecture that the existence of maximal green sequences is invariant under quiver mutation. The proof uses the ‘scattering diagrams’ of Gross-Hacking-Keel-Kontsevich to show that a maxim...

Quivers play an important role in the representation theory of algebras, with a key ingredient being the path algebra and the preprojective algebra. Quiver grassmannians are varieties of submodules of a fixed module of the path or preprojective algebra. In the current paper, we study these objects in detail. We show that the quiver grassmannians corresponding to submodules of certain injective ...

these are notes from introductory survey lectures given at the institute for studies in theoretical physics and mathematics (ipm), teheran, in 2008 and 2010. we present the definition and the fundamental properties of fomin-zelevinsky’s cluster algebras. then, we introduce quiver representations and show how they can be used to construct cluster variables, which are the canonical generators of ...

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...

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 ...

In [2][3], the mathematicians Fomin and Zelevinsky described the mathematical object known as a quiver, and connected it with the theory of cluster algebras. In particular, each quiver can be represented by a seed of a cluster algebra, which couples a set of n variables with the adjacency matrix of the quiver. By performing a transformation on the quiver, we change accordingly the values of the...

These are notes from introductory survey lectures given at the Institute for Studies in Theoretical Physics and Mathematics (IPM), Teheran, in 2008 and 2010. We present the definition and the fundamental properties of Fomin-Zelevinsky’s cluster algebras. Then, we introduce quiver representations and show how they can be used to construct cluster variables, which are the canonical generator...