We define and study the category Rep(Q, F1) of representations of a quiver in Vect(F1) the category of vector spaces ”over F1”. Rep(Q, F1) is an F1–linear category possessing kernels, co-kernels, and direct sums. Moreover, Rep(Q, F1) satisfies analogues of the Jordan-Hölder and Krull-Schmidt theorems. We are thus able to define the Hall algebra HQ of Rep(Q, F1), which behaves in some ways like ...

This paper proves that every projective toric variety is the fine moduli space for stable representations of an appropriate bound quiver. To accomplish this, we study the quiver Q with relations R corresponding to the finite-dimensional algebra End (⊕ r i=0 Li ) where L := (OX , L1, . . . , Lr) is a list of line bundles on a projective toric variety X . The quiver Q defines a unimodular, projec...

It is shown that rational points over finite fields of moduli spaces of stable quiver representations are counted by polynomials with integer coefficients. These polynomials are constructed recursively using an identity in the Hall algebra of a quiver.

We provide a quiver setting for quasi-Hopf algebras, generalizing the Hopf quiver theory. As applications we obtain some general structure theorems, in particular the quasi-Hopf analogue of the Cartier theorem and the Cartier-Gabriel decomposition theorem.

We argue that there is an equivalence of M-theory on T 3 ×AN−1 with a fourdimensional non-supersymmetric quiver gauge theory on the Higgs branch. The quiver theory in question has gauge group SU(N)N4N6N8 and is considered in a strong coupling and large N4,6,8 limit. We provide fieldand string-theoretical evidence for the equivalence making use of the deconstruction technique. In particular, we ...

Maximal green sequences are particular sequences of mutations of quivers which were introduced by Keller in the context of quantum dilogarithm identities and independently by Cecotti–Córdova–Vafa in the context of supersymmetric gauge theory. The existence of maximal green sequences for exceptional finite mutation type quivers has been shown by Alim–Cecotti–Córdova–Espahbodi–Rastogi–Vafa except...

We explore several variations of the notion of purity for the action of Frobenius on schemes defined over finite fields. In particular, we study how these notions are preserved under certain natural operations like quotients for principal bundles and also geometric quotients for reductive group actions. We then apply these results to study the cohomology of quiver moduli. We prove that a natura...

We introduce certain quiver analogue of the determinantal variety. We study the Kempf-Lascoux-Weyman’s complex associated to a line bundle on the variety. In the case of generalized Kronecker quivers, we give a sufficient condition on when the complex resolves a maximal Cohen-Macaulay module supported on the quiver determinantal variety. This allows us to find the set-theoretical defining equat...

N = 2 quiver Chern-Simons theory has lately attracted attention as the world volume theory of multiple M2 branes on a Calabi-Yau 4-fold. We study the connection between the stringy derivation of M2 brane theories and the forward algorithm which gives the toric Calabi-Yau 4-fold as the moduli space of the quiver theory. Then the existence of the 3+1 dimensional parent, which is the consistent 3+...

We study Seiberg duality of quiver gauge theories associated to the complex cone over the second del Pezzo surface. Homomorphisms in the path algebra of the quivers in each of these cases satisfy relations which follow from a superpotential of the corresponding gauge theory as F-flatness conditions. We verify that Seiberg duality between each pair of these theories can be understood as a derive...