Reflection Functor in the Representation Theory of Preprojective Algebras for Quivers and Integrable Systems

A brief overview of the representation theory of quivers and the associated (deformed) preprojective algebras, as well as of the theories of moduli spaces of these algebras, quiver varieties and a reflection functor, is given. It is proven that a bijection between moduli spaces (in particular, between quiver varieties), which is induced by a reflection function, is the isomorphism of symplectic affine varieties. The Hamiltonian systems on quiver varieties are defined, and the application of a reflection functor to them is described. The review of [1], concerning the case of a cyclic quiver is given, and a role of the reflection functor in this case is clarified. The “spin” integrable generalizations of Calogero–Moser systems and their application to the KP hierarchy generalizations are described.     

Quivers,Quasi-Quantum Groups and Finite Tensor Categories

We study finite quasi-quantum groups in their quiver setting developed recently by the first author. We obtain a classification of finite-dimensional pointed Majid algebras of finite corepresentation type, or equivalently a classification of elementary quasi-Hopf algebras of finite representation type, over the field of complex numbers. By the Tannaka-Krein duality principle, this provides a classification of the finite tensor categories in which every simple object has Frobenius-Perron dimension 1 and there are finitely many indecomposable objects up to isomorphism. Some interesting information of these finite tensor categories is given by making use of the quiver representation theory.     

Quiver Varieties and Yangians

We prove a conjecture of Nakajima (for type A it was announced by Ginzburg and Vasserot) giving a geometric realization, via quiver varieties, of the Yangian of type ADE (and more in general of the Yangian associated to every symmetric Kac–Moody Lie algebra). As a corollary, we get that the finite-dimensional representation theory of the quantized affine algebra and that of the Yangian coincide.     

Fusion and braiding in W-algebra extended conformal theories

We define the chiral conformal blocks of integer-spin extended (W-algebra) conformal theories by the fusion of elementary ones. The braid group representation matrices which realize the exchange algebra are computed. They are shown to coincide with the Boltzmann weights — in a certain limit of the spectral parameter — of the critical face models of Jimbo et al. In the unitary cases, where the extended conformal theories can be realized as cosets , we relate the braiding matrices of the former to those of the WZW models. In this article we restrict ourselves to the case corresponding to symmetric tensor representations of A_n.     

Z-Algebra Construction of W-Algebra in the Parafermion Model

We propose a new construction — the Z-algebra construction for the W-algebra which includes the Virasoro algebra as a special case. The Z-algebra associated with the general affine Lie algebra is given. And we suggest a calculating program to make the derivation of the Z-algebra become possible. Using this technique we have derived the Virasoro algebra in general G_k parafermion case and the W-algebra in SU(2)_k parafermion case.     

Superalgebras in N = 1 gauge theories

N = 1 supersymmetric gauge theories with global flavor symmetries contain a gauge invariant W-superalgebra which acts on its moduli space of gauge invariants. With adjoint matter, this superalgebra reduces to a graded Lie algebra. When the gauge group is SO(n_c), with vector matter, it is a W-algebra, and the primary invariants form one of its representation. The same superalgebra exists in the dual theory, but its construction in terms of the dual fields suggests that duality may be understood in terms of a charge conjugation within the algebra. We extend the analysis to the gauge group E₆.     

Twisted quiver bundles over almost complex manifolds

In this paper, we study twisted quiver bundle over general almost complex manifolds. A twisted quiver bundle is a set of J-holomorphic vector bundles over an almost complex manifold, labelled by the vertices of a quiver, linked by a set of morphisms twisted by a fixed collection of J-holomorphic vector bundles, labelled by the arrows. We prove a Hitchin–Kobayashi correspondence for twisted quiver bundles over a compact almost Hermitian regularized manifold, relating the existence of solutions to certain gauge equations to an appropriate notion of stability for the corresponding quivers. This result can be seen as a generalization of that in [2], [9].     

Cluster-enriched Yang–Baxter equation from SUSY gauge theories

We propose a new generalization of the Yang–Baxter equation, where the R-matrix depends on cluster y-variables in addition to the spectral parameters. We point out that we can construct solutions to this new equation from the recently found correspondence between Yang–Baxter equations and supersymmetric gauge theories. The \(S^2\) partition function of a certain 2d \({\mathcal {N}}=(2,2)\) quiver gauge theory gives an R-matrix, whereas its FI parameters can be identified with the cluster y-variables.     

Fractional quiver W-algebras

We introduce quiver gauge theory associated with the non-simply laced type fractional quiver and define fractional quiver W-algebras by using construction of Kimura and Pestun (Lett Math Phys, 2018.  https://doi.org/10.1007/s11005-018-1072-1; Lett Math Phys, 2018.  https://doi.org/10.1007/s11005-018-1073-0) with representation of fractional quivers.     

Integrable Lattice Spin Models from Supersymmetric Dualities

Recently, there has been observed an interesting correspondence between supersymmetric quiver gauge theories with four supercharges and integrable lattice models of statistical mechanics such that the two-dimensional spin lattice is the quiver diagram, the partition function of the lattice model is the partition function of the gauge theory and the Yang–Baxter equation expresses the identity of partition functions for dual pairs. This correspondence is a powerful tool which enables us to generate new integrable models. The aim of the present paper is to give a short account on a progress in integrable lattice models which has been made due to the relationship with supersymmetric gauge theories and make clear notes on the special functions used by several authors.     

New example of infinite family of quiver gauge theories

We construct a new infinite family of quiver gauge theories which blow down to the Xp,q$X^{p,q}$ quiver gauge theories found by Hanany, Kazakopoulos and Wecht. This family includes a quiver gauge theory for the third del Pezzo surface. We show, using Z-minimization, that these theories generically have irrational R-charges. The AdS/CFT correspondence implies that the dual geometries are irregular toric Sasaki–Einstein manifolds, although we do not know the explicit metrics.     

On representations of conformal field theories and the construction of orbifolds

We consider representations of meromorphic bosonic chiral conformal field theories and demonstrate that such a representation is completely specified by a state within the theory. The necessary and sufficient conditions upon this state are derived and, because of their form, we show that we may extend the representation to a representation of a suitable larger conformal field theory. In particular, we apply this procedure to the (untwisted) lattice conformal field theories (i.e. corresponding to the propagation of a bosonic string on a torus), and deduce that Dong's proof of the uniqueness of the twisted representation for the reflection-twisted projection of the Leech lattice conformal field theory generalises to an arbitrary even (self-dual) lattice. As a consequence, we see that the reflection-twisted lattice theories of Dolan, Goddard and Montague are truly self-dual, extending the analogies with the theories of lattices and codes which were being pursued. Some comments are also made on the general concept of the definition of an orbifold of a conformal field theory in relation to this point of view.     

Quivers from Matrix Factorizations

We discuss how matrix factorizations offer a practical method of computing the quiver and associated superpotential for a hypersurface singularity. This method also yields explicit geometrical interpretations of D-branes (i.e., quiver representations) on a resolution given in terms of Grassmannians. As an example we analyze some non-toric singularities which are resolved by a single \({\mathbb{P}^1}\) but have “length” greater than one. These examples have a much richer structure than conifolds. A picture is proposed that relates matrix factorizations in Landau–Ginzburg theories to the way that matrix factorizations are used in this paper to perform noncommutative resolutions.     

Instantons, quivers and noncommutative Donaldson-Thomas theory

We construct noncommutative Donaldson-Thomas invariants associated with abelian orbifold singularities by analyzing the instanton contributions to a six-dimensional topological gauge theory. The noncommutative deformation of this gauge theory localizes on noncommutative instantons which can be classified in terms of three-dimensional Young diagrams with a colouring of boxes according to the orbifold group. We construct a moduli space for these gauge field configurations which allows us to compute its virtual numbers via the counting of representations of a quiver with relations. The quiver encodes the instanton dynamics of the noncommutative gauge theory, and is associated to the geometry of the singularity via the generalized McKay correspondence. The index of BPS states which compute the noncommutative Donaldson-Thomas invariants is realized via topological quantum mechanics based on the quiver data. We illustrate these constructions with several explicit examples, involving also higher rank Coulomb branch invariants and geometries with compact divisors, and connect our approach with other ones in the literature.     

Quivers and lie superalgebras

Indecomposable representations of quivers are in 1–1 correspondence with positive weight vectors of Kac-Moody algebras. The collection of indecomposable representations of the quiver is tame if the quiver corresponds to a Kac-Moody algebra of polynomial growth. What corresponds to positive roots of Lie algebras of polynomial growth different from Kac-Moody algebras? The classification problem for tame representations of quivers associated to Lie superalgebras is a natural step towards the answer to this question. As an aside we announce a classification of simple graded Lie superalgebras of polynomial growth.     

Brane tilings and their applications

We review recent developments in the theory of brane tilings and four‐dimensional 𝒩 = 1 supersymmetric quiver gauge theories. This review consists of two parts. In part I, we describe foundations of brane tilings, emphasizing the physical interpretation of brane tilings as fivebrane systems. In part II, we discuss application of brane tilings to AdS/CFT correspondence and homological mirror symmetry. More topics, such as orientifold of brane tilings, phenomenological model building, similarities with BPS solitons in supersymmetric gauge theories, are also briefly discussed. This paper is a revised version of the author's master's thesis submitted to Department of Physics, Faculty of Science, the University of Tokyo on January 2008, and is based on his several papers and some works in progress [1–7].     

On Cohen–Macaulayness of Algebras Generated by Generalized Power Sums

Generalized power sums are linear combinations of ith powers of coordinates. We consider subalgebras of the polynomial algebra generated by generalized power sums, and study when such algebras are Cohen–Macaulay. It turns out that the Cohen–Macaulay property of such algebras is rare, and tends to be related to quantum integrability and representation theory of Cherednik algebras. Using representation theoretic results and deformation theory, we establish Cohen–Macaulayness of the algebra of q, t-deformed power sums defined by Sergeev and Veselov, and of some generalizations of this algebra, proving a conjecture of Brookner, Corwin, Etingof, and Sam. We also apply representation-theoretic techniques to studying m-quasi-invariants of deformed Calogero–Moser systems. In an appendix to this paper, M. Feigin uses representation theory of Cherednik algebras to compute Hilbert series for such quasi-invariants, and show that in the case of one light particle, the ring of quasi-invariants is Gorenstein.     

BPS Quivers and Spectra of Complete {\mathcal{N} = 2} Quantum Field Theories

We study the BPS spectra of ${\mathcal{N}=2}$ N = 2 complete quantum field theories in four dimensions. For examples that can be described by a pair of M5 branes on a punctured Riemann surface we explain how triangulations of the surface fix a BPS quiver and superpotential for the theory. The BPS spectrum can then be determined by solving the quantum mechanics problem encoded by the quiver. By analyzing the structure of this quantum mechanics we show that all asymptotically free examples, Argyres-Douglas models, and theories defined by punctured spheres and tori have a chamber with finitely many BPS states. In all such cases we determine the spectrum.     

Crystal Melting and Toric Calabi-Yau Manifolds

We construct a statistical model of crystal melting to count BPS bound states of D0 and D2 branes on a single D6 brane wrapping an arbitrary toric Calabi-Yau threefold. The three-dimensional crystalline structure is determined by the quiver diagram and the brane tiling which characterize the low energy effective theory of D branes. The crystal is composed of atoms of different colors, each of which corresponds to a node of the quiver diagram, and the chemical bond is dictated by the arrows of the quiver diagram. BPS states are constructed by removing atoms from the crystal. This generalizes the earlier results on the BPS state counting to an arbitrary non-compact toric Calabi-Yau manifold. We point out that a proper understanding of the relation between the topological string theory and the crystal melting involves the wall crossing in the Donaldson-Thomas theory.