We use free boson techniques to investigate AD -E-quiver matrix models. Certain higher spin fields in the free boson formulation give rise to higher order loop equations valid at finite N. These fields form a special kind of W-algebra, called Casimir algebra. We compute explicitly the loop equations for A r and D r quiver models and check that at large N they are related to a deformation of the...

This is a survey paper of our recent results [6]. We present a novel method to detect robust and common topological structures of two geometric objects. The idea is to extend the notion of persistent homology [5, 12] to representations on a commutative triple ladder quiver. Our contributions of this paper are given as follows: (i) We prove that the commutative triple ladder quiver is representa...

Following Braverman–Finkelberg–Feigin–Rybnikov (arXiv: 1008.3655), we study the convolution algebra of a handsaw quiver variety, a.k.a. a parabolic Laumon space, and a finite W -algebra of type A. This is a finite analog of the AGT conjecture on 4-dimensional supersymmetric Yang–Mills theory with surface operators. Our new observation is that the C-fixed point set of a handsaw quiver variety is...

We investigate the Poisson geometry of the Marsden-Weinstein reductions of the moment map associated to the cotangent bundle of the space of representations of a quiver. We show that the stratification by representation type equals the stratification by symplectic leaves. The deformed symplectic quotient singularities spectra of centres of symplectic reflection algebras associated to a wreath p...

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...

We give a quiver representation theoretic interpretation of generalized cluster complexes defined by Fomin and Reading. Using d-cluster categories defined by Keller as triangulated orbit categories of (bounded) derived categories of representations of valued quivers, we define a d-compatibility degree (− ‖−) on any pair of “colored” almost positive real Schur roots which generalizes previous de...

Quiver Grassmannians and quiver flags are natural generalisations of usual Grassmannians and flags. They arise in the study of quiver representations and Hall algebras. In general, they are projective varieties which are neither smooth nor irreducible. We use a scheme theoretic approach to calculate their tangent space and to obtain a dimension estimate similar to the one of Reineke in [Rei02]....

In [3], Bautista and Salmeron explained how, given a finite quiver Z without double arrows or oriented cycles, certain sets of subsets of ZZ (called (P)-generating systems) may, under appropriate conditions, generate algebras whose preprojective components can be embedded in ZZ in a nice way, and which are (P)-algebras [2] whenever they are representation-finite. Here, we define numerical funct...

We show that any quiver Grassmannian associated with a rigid representation of is rational variety using torus localization techniques.