Quiver Mutations Research Articles

Exploring Seiberg-like N-alities with eight supercharges

We show that a large subclass of 3d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 quiver gauge theories consisting of unitary and special unitary gauge nodes with only fundamental/bifundamental matter have multiple Seiberg-like IR duals. A generic quiver T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{T} $$\\end{document} in this subclass has a non-zero number of balanced special unitary gauge nodes and it is a good theory in the Gaiotto-Witten sense. We refer to this phenomenon as IR N-ality and the set of mutually IR dual theories as the N-al set associated with the quiver T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{T} $$\\end{document}. Starting from T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{T} $$\\end{document}, we construct a sequence of dualities by step-wise implementing a set of quiver mutations which act locally on the gauge nodes. The associated N-al theories can then be read off from this duality sequence. The quiver T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{T} $$\\end{document} generically has an emergent Coulomb branch global symmetry in the IR, such that the rank of the IR symmetry is always greater than the rank visible in the UV. We show that there exists at least one theory in the N-al set for which the rank of the IR symmetry precisely matches the rank of the UV symmetry. In certain special cases that we discuss in this work, the correct emergent symmetry algebra itself may be read off from this preferred theory (or theories) in addition to the correct rank. Finally, we give a recipe for constructing the 3d mirror associated with a given N-al set and show how the emergent Coulomb branch symmetry of T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{T} $$\\end{document} is realized as a UV-manifest Higgs branch symmetry of the 3d mirror. This paper is the second in a series of four papers on 3d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 Seiberg-like dualities, preceded by the work [1].

Presentations of Braid Groups of Type A Arising from (m+2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(m+2)$$\\end{document}-angulations of Regular Polygons

Coloured quiver mutation, introduced by Buan, A.B., Thomas, H (Adv. Math. 222(3), 971–995 2009), gives a combinatorial interpretation of tilting in higher cluster categories. In type A work of Baur, K., Marsh, B. (Trans. Am. Math. Soc. 360(11), 5789-5803 2008) shows that m-coloured quivers and m-coloured quiver mutations have a nice geometrical description, given in terms of (m+2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(m+2)$$\\end{document}-angulations of a regular polygon, and rotations of an m-diagonal. In this paper, using such correspondence, we describe presentations of braid groups of type A arising from coloured quivers of mutation type A.

Complexity of Ice Quiver Mutation Equivalence

Complexity of Ice Quiver Mutation Equivalence

Quiver combinatorics and triangulations of cyclic polytopes

Motivated by higher homological algebra, we associate quivers to triangulations of even-dimensional cyclic polytopes and prove two results showing what information about the triangulation is encoded in the quiver. We first show that the cut quivers of Iyama and Oppermann correspond precisely to 2d-dimensional triangulations without interior (d+1)-simplices. This implies that these triangulations form a connected subgraph of the flip graph. Our second result shows how the quiver of a triangulation can be used to identify mutable internal d-simplices. This points towards what a theory of higher-dimensional quiver mutation might look like and gives a new way of understanding flips of triangulations of even-dimensional cyclic polytopes.

Exchange graphs for mutation-finite non-integer quivers

Skew-symmetric non-integer matrices with real entries can be viewed as quivers with non-integer arrow weights. Such quivers can be mutated following the usual rules of quiver mutation. Felikson and Tumarkin show that mutation-finite non-integer quivers admit geometric realisations by partial reflections. This allows us to define a geometric notion of seeds and thus to define the exchange graphs for mutation classes. In this paper we study exchange graphs of mutation-finite quivers. The concept of finite type generalises naturally to mutation-finite non-integer quivers. We show that for all non-integer quivers of finite type there is a well-defined notion of an exchange graph, and this notion is consistent with the classical notion of exchange graph of integer mutation types coming from cluster algebras. In particular, exchange graphs of finite type quivers are finite. We also show that exchange graphs of rank 3 affine quivers are finite modulo the action of a finite-dimensional lattice (but unlike the integer case, the rank of the lattice is higher than 1 for non-integer quivers).

Seiberg-Witten geometry, modular rational elliptic surfaces and BPS quivers

We study the Coulomb branches of four-dimensional supersymmetric quantum field theories with mathcal{N} = 2 supersymmetry, including the KK theories obtained from the circle compactification of the 5d mathcal{N} = 1 En Seiberg theories, with particular focus on the relation between their Seiberg-Witten geometries and rational elliptic surfaces. More attention is given to the modular surfaces, which we completely classify using the classification of subgroups of the modular group PSL(2, ℤ), deriving closed-form expressions for the modular functions for all congruence and some of the non-congruence subgroups. Moreover, in such cases, we give a prescription for determining the BPS quivers from the fundamental domains of the monodromy groups and study how changes of these domains can be interpreted as quiver mutations. This prescription can be also generalized to theories whose Coulomb branches contain ‘undeformable’ singularities, leading to known quivers of such theories.

Quiver mutations, Seiberg duality, and machine learning

We initiate the study of applications of machine learning to Seiberg duality, focusing on the case of quiver gauge theories, a problem also of interest in mathematics in the context of cluster algebras. Within the general theme of Seiberg duality, we define and explore a variety of interesting questions, broadly divided into the binary determination of whether a pair of theories picked from a series of duality classes are dual to each other, as well as the multi-class determination of the duality class to which a given theory belongs. We study how the performance of machine learning depends on several variables, including number of classes and mutation type (finite or infinite). In addition, we evaluate the relative advantages of Naive Bayes classifiers versus Convolutional Neural Networks. Finally, we also investigate how the results are affected by the inclusion of additional data, such as ranks of gauge/flavor groups and certain variables motivated by the existence of underlying Diophantine equations. In all questions considered, high accuracy and confidence can be achieved.

Linearity of stability conditions

For modules over an artin algebra, a linear stability condition is given by a “central charge” and a nonlinear stability condition is given by the wall-crossing sequence of a “green path.” Finite Harder-Narasimhan stratifications of the module category, maximal forward hom-orthogonal sequences and maximal green sequences, defined using Fomin-Zelevinsky quiver mutation are shown to be equivalent to finite nonlinear stability conditions when the algebra is hereditary. This is the first of a series of three papers whose purpose is to determine all maximal green sequences of maximal length for quivers of affine type A and determine which are linear.

Jacobian matrices of Y-seed mutations

For any quiver mutation sequence, we define a pair of matrices that describe a fixed point equation of a cluster transformation determined from the mutation sequence. We give an explicit relationship between this pair of matrices and the Jacobian matrix of the cluster transformation. Furthermore, we show that this relationship reduces to a relationship between the pair of matrices and the C-matrix of the cluster transformation in a certain limit of cluster variables. As an application, we prove that quivers associated with once-punctured surfaces do not have maximal green or reddening sequences.

Quiver mutations and Boolean reflection monoids

In 2010, Everitt and Fountain introduced the concept of reflection monoids. The Boolean reflection monoids form a family of reflection monoids (symmetric inverse semigroups are Boolean reflection monoids of type A). In this paper, we give a family of presentations of Boolean reflection monoids and show how these presentations are compatible with quiver mutations of orientations of Dynkin diagrams with frozen vertices. Our results recover the presentations of Boolean reflection monoids given by Everitt and Fountain and the presentations of symmetric inverse semigroups given by Popova respectively. Surprisingly, inner by diagram automorphisms of irreducible Weyl groups and Boolean reflection monoids can be constructed by sequences of mutations preserving the same underlying diagrams. Besides, we show that semigroup algebras of Boolean reflection monoids are cellular algebras.

Fock-Goncharov coordinates for rank two Lie groups

Let G be a simply connected, simple, complex Lie group of rank 2. We give explicit Fock-Goncharov coordinates for configurations of triples and quadruples of affine flags in G. We show that the action on triples by orientation preserving permutations corresponds to explicit quiver mutations, and that the same holds for the flip (changing the diagonal in a quadrilateral). This gives explicit coordinates on higher Teichmuller space, and also coordinates for boundary-unipotent representations of 3-manifold groups. As an application, we compute the (generic) boundary-unipotent representations in $${{\,\mathrm{Sp}\,}}(4,{\mathbb {C}})/\langle -I\rangle $$ for the figure-eight knot complement.

Self-Injective Jacobian Algebras from Postnikov Diagrams

We study a finite-dimensional algebra Λ from a Postnikov diagram D in a disk, obtained from the dimer algebra of Baur-King-Marsh by factoring out the ideal generated by the boundary idempotent. Thus, Λ is isomorphic to the stable endomorphism algebra of a cluster tilting module T ∈CM(B) introduced by Jensen-King-Su in order to categorify the cluster algebra structure of mathbb {C}[text {Gr}_{k}(mathbb {C}^{n})]. We show that Λ is self-injective if and only if D has a certain rotational symmetry. In this case, Λ is the Jacobian algebra of a self-injective quiver with potential, which implies that its truncated Jacobian algebras in the sense of Herschend-Iyama are 2-representation finite. We study cuts and mutations of such quivers with potential leading to some new 2-representation finite algebras.

Higher cluster categories and QFT dualities

We present a unified mathematical framework that elegantly describes minimally SUSY gauge theories in even dimension, ranging from $6d$ to $0d$, and their dualities. This approach combines recent developments on graded quiver with potentials, higher Ginzburg algebras and higher cluster categories (also known as $m$-cluster categories). Quiver mutations studied in the context of mathematics precisely correspond to the order $(m+1)$ dualities of the gauge theories. Our work suggests that these equivalences of quiver gauge theories sit inside an infinite family of such generalized dualities, whose physical interpretation is yet to be understood.

Cluster realization of $$\mathcal {U}_q(\mathfrak {g})$$ U q ( g ) and factorizations of the universal R-matrix

For each simple Lie algebra $$\mathfrak {g}$$ , we construct an algebra embedding of the quantum group $$\mathcal {U}_q(\mathfrak {g})$$ into certain quantum torus algebra $$\mathcal {D}_\mathfrak {g}$$ via the positive representations of split real quantum group. The quivers corresponding to $$\mathcal {D}_\mathfrak {g}$$ is obtained from an amalgamation of two basic quivers, each of which is mutation equivalent to one describing the cluster structure of the moduli space of framed G-local system on a disk with 3 marked points on its boundary when G is of classical type. We derive a factorization of the universal R-matrix into quantum dilogarithms of cluster monomials, and show that conjugation by the R-matrix corresponds to a sequence of quiver mutations which produces the half-Dehn twist rotating one puncture about the other in a twice punctured disk.

Cluster integrable systems, q-Painlev\xe9 equations and their quantization

We discuss the relation between the cluster integrable systems and q-difference Painlevé equations. The Newton polygons corresponding to these integrable systems are all 16 convex polygons with a single interior point. The Painlevé dynamics is interpreted as deautonomization of the discrete flows, generated by a sequence of the cluster quiver mutations, supplemented by permutations of quiver vertices.We also define quantum q-Painlevé systems by quantization of the corresponding cluster variety. We present formal solution of these equations for the case of pure gauge theory using q-deformed conformal blocks or 5-dimensional Nekrasov functions. We propose, that quantum cluster structure of the Painlevé system provides generalization of the isomonodromy/CFT correspondence for arbitrary central charge.

About the Absence of Exotics and the Coulomb Branch Formula

The absence of exotics is a conjectural property of the spectrum of BPS states of four--dimensional $\mathcal{N}=2$ supersymmetric QFT's. In this letter we revisit the precise statement of this conjecture, and develop a general strategy that, if applicable, entails the absence of exotic BPS states. Our method is based on the Coulomb branch formula and on quiver mutations. In particular, we obtain the absence of exotic BPS states for all pure SYM theories with simple simply--laced gauge group $G$, and, as a corollary, of infinitely many other lagrangian $\mathcal{N}=2$ theories.

Connected quantized Weyl algebras and quantum cluster algebras

For an algebraically closed field K, we investigate a class of noncommutative K-algebras called connected quantized Weyl algebras. Such an algebra has a PBW basis for a set of generators {x1,…,xn} such that each pair satisfies a relation of the form xixj=qijxjxi+rij, where qij∈K⁎ and rij∈K, with, in some sense, sufficiently many pairs for which rij≠0. For such an algebra it turns out that there is a single parameter q such that each qij=q±1. Assuming that q≠±1, we classify connected quantized Weyl algebras, showing that there are two types linear and cyclic. When q is not a root of unity we determine the prime spectra for each type. The linear case is the easier, although the result depends on the parity of n, and all prime ideals are completely prime. In the cyclic case, which can only occur if n is odd, there are prime ideals for which the factors have arbitrarily large Goldie rank.We apply connected quantized Weyl algebras to obtain presentations of two classes of quantum cluster algebras. Let n≥3 be an odd integer. We present the quantum cluster algebra of a Dynkin quiver of type An−1 as a factor of a linear connected quantized Weyl algebra by an ideal generated by a central element. We also consider the quiver Pn+1(1) identified by Fordy and Marsh in their analysis of periodic quiver mutation. When n is odd, we show that the quantum cluster algebra of this quiver is generated by a cyclic connected quantized Weyl algebra in n variables and one further generator. We also present it as the factor of an iterated skew polynomial algebra in n+2 variables by an ideal generated by a central element. For both classes, the quantum cluster algebras are simple noetherian.We establish Poisson analogues of the results on prime ideals and quantum cluster algebras. We determine the Poisson prime spectra for the semiclassical limits of the linear and cyclic connected quantized Weyl algebras and show that, when n is odd, the cluster algebras of An−1 and Pn+1(1) are simple Poisson algebras that can each be presented as a Poisson factor of a polynomial algebra, with an appropriate Poisson bracket, by a principal ideal generated by a Poisson central element.

Braid groups and quiver mutation

We describe presentations of braid groups of type ADE and show how these presentations are compatible with mutation of quivers, building on work of Barot and Marsh for Coxeter groups. In types A and D these presentations can be understood geometrically using triangulated surfaces. We then give a categorical interpretation of the presentations, with the new generators acting as spherical twists at simple modules on derived categories of Ginzburg dg-algebras of quivers with potential.

Quiver Mutation Sequences and $q$-Binomial Identities

In this paper, first we introduce a quantity called a partition function for a quiver mutation sequence. The partition function is a generating function whose weight is a $q$-binomial associated with each mutation. Then, we show that the partition function can be expressed as a ratio of products of quantum dilogarithms. This provides a systematic way of constructing various $q$-binomial multisum identities.

BPS spectra of [formula omitted] SO7 and SP4 models

Extending the folding method of ADE Dynkin diagrams of Lie algebras to BPS quivers of 4d N=2 supersymmetric gauge theory with ADE type gauge invariance, we study the BPS spectra for gauge symmetries with non-simply laced Lie algebras. Focussing on the 4d N=2SO7 and SP4 models, we derive the BPS states of the strong chambers of these theories. We find that for both gauge groups Gnsl=SO7 and SP4≃SO(5), the number of BPS states of the strongly coupled chamber is 2dim⁡Gnsl versus 2dim⁡Gsl−2rankGsl for the cousin gauge symmetries Gsl=SO8 and SU4≃SO(6). The relationship between the Gsl and Gnsl types of BPS quiver mutations is derived. Other features are also studied.