Quiver Bundles Research Articles

Canonical metrics on holomorphic quiver bundles over compact generalized Kähler manifolds

Canonical metrics on holomorphic quiver bundles over compact generalized Kähler manifolds

On neutral Tannakian subcategories of loop quiver representations

We explored the categories of (twisted) representations of a loop quiver. These representation categories have two choices of tensor structures: Kronecker tensor and Simpson tensor. By studying the rigidity properties, we have provided several examples of (semi-) Tannakian categories using the category of (twisted) representations of a loop quiver for both tensors.We have introduced the concept of essentially finite loop quiver bundles based on the work of Nori, Borne, and Vistoli. As an application, we have given some examples of (semi-) Tannakian categories of equivariant bundles and Hitchin pairs. Additionally, we have defined the notion of H-nflat twisted loop quiver bundles and have established Tannakian category structures for certain classes of varieties.

Canonical Metrics on Twisted Quiver Bundles over a Class of Non-Compact Gauduchon Manifold

The aim of this paper is to prove a theorem for holomorphic twisted quiver bundles over a special non-compact Gauduchon manifold, connecting the existence of (σ,τ)-Hermite–Yang–Mills metric in differential geometry and the analytic (σ,τ)-stability in algebraic geometry. The proof of the theorem relies on the flow method and the Uhlenbeck–Yau’s continuity method.

An elementary discussion of the representation and geometric invariant theory of equioriented quivers of type D with an application to quiver bundles

We first review a constructive proof of Gabriel's theorem on the representation finiteness of Dynkin quivers for the case of an equioriented quiver of type D. Then, we proceed to the analysis of polystable representations of such a quiver, give a direct proof of the characterisationof its semistable representations, and explain how we may recover a theorem of Abeasis and Koike from this characterisation. Finally, we illustrate how these results may be applied to the study of moduli spaces of quiver bundles.

Counting Higgs bundles and type quiver bundles

We prove a closed formula counting semistable twisted (or meromorphic) Higgs bundles of fixed rank and degree over a smooth projective curve of genus $g$ defined over a finite field, when the twisting line bundle degree is at least $2g-2$ (this includes the case of usual Higgs bundles). This yields a closed expression for the Donaldson–Thomas invariants of the moduli spaces of twisted Higgs bundles. We similarly deal with twisted quiver sheaves of type $A$ (finite or affine), obtaining in particular a Harder–Narasimhan-type formula counting semistable $U(p,q)$-Higgs bundles over a smooth projective curve defined over a finite field.

Homology of twisted quiver bundles with relations

We study the Ext modules in the category of left modules over a twisted algebra of a finite quiver over a ringed space (X,OX), allowing for the presence of relations. We introduce a spectral sequence which relates the Ext modules in that category with the Ext modules in the category of OX-modules. Contrary to what happens in the absence of relations, this spectral sequence in general does not degenerate at the second page. We also consider local Ext sheaves. Under suitable hypotheses, the Ext modules are represented as hypercohomology groups.

The Hitchin–Kobayashi Correspondence for Quiver Bundles over Generalized Kähler Manifolds

In this paper, we establish the Hitchin--Kobayashi correspondence for the $I_\pm$-holomorphic quiver bundle $\mathcal{E}=(E,\phi)$ over a compact generalized K\"{a}hler manifold $(X, I_+,I_-,g, b)$ such that $g$ is Gauduchon with respect to both $I_+$ and $I_-$, namely $\mathcal{E}$ is $(\alpha,\sigma,\tau)$-polystable if and only if $\mathcal{E}$ admits an $(\alpha,\sigma,\tau)$-Hermitian--Einstein metric.

Generalized quivers, orthogonal and symplectic representations, and Hitchin–Kobayashi correspondences

We review the theory of quiver bundles over a Kähler manifold, and then introduce the concept of generalized quiver bundles for an arbitrary reductive group [Formula: see text]. We first study the case when [Formula: see text] or [Formula: see text], interpreting them as orthogonal (respectively symplectic) bundle representations of the symmetric quivers introduced by Derksen–Weyman. We also study supermixed quivers, which simultaneously involve both orthogonal and symplectic symmetries. In particular, we completely characterize the polystable forms of such representations. Finally, we discuss Hitchin–Kobayashi correspondences for these objects.

Quiver bundles and wall crossing for chains

Holomorphic chains on a Riemann surface arise naturally as fixed points of the natural C*-action on the moduli space of Higgs bundles. In this paper we associate a new quiver bundle to the Hom-complex of two chains, and prove that stability of the chains implies stability of this new quiver bundle. Our approach uses the Hitchin-Kobayashi correspondence for quiver bundles. Moreover, we use our result to give a new proof of a key lemma on chains (due to \'Alvarez-C\'onsul, Garc\'ia-Prada and Schmitt), which has been important in the study of Higgs bundle moduli; this proof relies on stability and thus avoids the direct use of the chain vortex equations.

Sasakian quiver gauge theory on the Aloff–Wallach space X1,1

We consider the SU(3)-equivariant dimensional reduction of gauge theories on spaces of the form Md×X1,1 with d-dimensional Riemannian manifold Md and the Aloff–Wallach space X1,1=SU(3)/U(1) endowed with its Sasaki–Einstein structure. The condition of SU(3)-equivariance of vector bundles, which has already occurred in the studies of Spin(7)-instantons on cones over Aloff–Wallach spaces, is interpreted in terms of quiver diagrams, and we construct the corresponding quiver bundles, using (parts of) the weight diagram of SU(3). We consider three examples thereof explicitly and then compare the results with the quiver gauge theory on Q3=SU(3)/(U(1)×U(1)), the leaf space underlying the Sasaki–Einstein manifold X1,1. Moreover, we study instanton solutions on the metric cone C(X1,1) by evaluating the Hermitian Yang–Mills equation. We briefly discuss some features of the moduli space thereof, following the main ideas of a treatment of Hermitian Yang–Mills instantons on cones over generic Sasaki–Einstein manifolds in the literature.

Differential geometry of moduli spaces of quiver bundles

Let P be a parabolic subgroup of a simple affine algebraic group G defined over C and X a compact connected K\"ahler manifold. L. \'Alvarez-C\'onsul and O. Garc\'ia-Prada associated to these a quiver Q and representations of Q into holomorphic vector bundles on X. Our aim here is to investigate the differential geometric properties of the moduli spaces representations of Q into vector bundles on X. In particular, we construct a Hermitian form on these moduli spaces. A fiber integral formula is proved for this Hermitian form; this fiber integral formula implies that the Hermitian form is K\"ahler. We compute the curvature of this K\"ahler form. Under an assumption which says that X is a complex projective manifold, this K\"ahler form is realized as the curvature of a certain determinant line bundle equipped with a Quillen metric.

Birationality of moduli spaces of twisted $$\mathrm {U}(p,q)$$ U ( p , q ) -Higgs bundles

A $\mathrm{U}(p,q)$-Higgs bundle on a Riemann surface (twisted by a line bundle) consists of a pair of holomorphic vector bundles, together with a pair of (twisted) maps between them. Their moduli spaces depend on a real parameter $\alpha$. In this paper we study wall crossing for the moduli spaces of $\alpha$-polystable twisted $\mathrm{U}(p,q)$-Higgs bundles. Our main result is that the moduli spaces are birational for a certain range of the parameter and we deduce irreducibility results using known results on Higgs bundles. Quiver bundles and the Hitchin-Kobayashi correspondence play an essential role.

Sasakian quiver gauge theories and instantons on Calabi–Yau cones

We consider SU(2)-equivariant dimensional reduction of Yang-Mills theory on manifolds of the form $M\times S^3/\Gamma$, where $M$ is a smooth manifold and $S^3/\Gamma$ is a three-dimensional Sasaki-Einstein orbifold. We obtain new quiver gauge theories on $M$ whose quiver bundles are based on the affine ADE Dynkin diagram associated to $\Gamma$. We relate them to those arising through translationally-invariant dimensional reduction over the associated Calabi-Yau cones $C(S^3/\Gamma)$ which are based on McKay quivers and ADHM matrix models, and to those arising through SU(2)-equivariant dimensional reduction over the leaf spaces of the characteristic foliations of $S^3/\Gamma$ which are Kahler orbifolds of $\mathbb{C} P^1$ whose quiver bundles are based on the unextended Dynkin diagram corresponding to $\Gamma$. We use Nahm equations to describe the vacua of SU(2)-equivariant quiver gauge theories on the cones as moduli spaces of spherically symmetric instantons. We relate them to the Nakajima quiver varieties which can be realized as Higgs branches of the worldvolume quiver gauge theories on D$p$-branes probing D$(p+4)$-branes which wrap an ALE space, and to the moduli spaces of spherically symmetric solutions in putative non-abelian generalizations of two-dimensional affine Toda field theories.

Covariant quiver gauge theories

We consider dimensional reduction of gauge theories with arbitrary gauge group in a formalism based on equivariant principal bundles. For the classical gauge groups we clarify the relations between equivariant principal bundles and quiver bundles, and show that the reduced quiver gauge theories are all generically built on the same universal symmetry breaking pattern. The formalism enables the dimensional reduction of Chern-Simons gauge theories in arbitrary odd dimensionalities. The reduced model is a novel Chern-Simons-Higgs theory consisting of a Chern-Simons term valued in the residual gauge group plus a higher order gauge and diffeomorphism invariant coupling of Higgs fields with the gauge fields. We study the moduli spaces of solutions, which in some instances provide geometric representations of certain quiver varieties as moduli spaces of flat invariant connections. As physical applications, we consider dimensional reductions involving non-compact gauge supergroups as a means for systematically inducing novel couplings between gravity and matter. In particular, we reduce Chern-Simons supergravity to a quiver gauge theory of AdS gravity involving a non-minimal coupling to scalar Higgs fermion fields.

Double quiver gauge theory and nearly Kähler flux compactifications

We consider G-equivariant dimensional reduction of Yang-Mills theory with torsion on manifolds of the form MxG/H where M is a smooth manifold, and G/H is a compact six-dimensional homogeneous space provided with a never integrable almost complex structure and a family of SU(3)-structures which includes a nearly Kahler structure. We establish an equivalence between G-equivariant pseudo-holomorphic vector bundles on MxG/H and new quiver bundles on M associated to the double of a quiver Q, determined by the SU(3)-structure, with relations ensuring the absence of oriented cycles in Q. When M=R^2, we describe an equivalence between G-invariant solutions of Spin(7)-instanton equations on MxG/H and solutions of new quiver vortex equations on M. It is shown that generic invariant Spin(7)-instanton configurations correspond to quivers Q that contain non-trivial oriented cycles.

Geometry and energy of non-Abelian vortices

We study pure Yang–Mills theory on Σ × S2, where Σ is a compact Riemann surface, and invariance is assumed under rotations of S2. It is well known that the self-duality equations in this setup reduce to vortex equations on Σ. If the Yang–Mills gauge group is SU(2), the Bogomolny vortex equations of the Abelian Higgs model are obtained. For larger gauge groups, one generally finds vortex equations involving several matrix-valued Higgs fields. Here we focus on Yang–Mills theory with gauge group \documentclass[12pt]{minimal}\begin{document}$\mathrm{SU}(N)/\mathbb {Z}_N$\end{document} SU (N)/ZN and a special reduction which yields only one non-Abelian Higgs field. One of the new features of this reduction is the fact that while the instanton number of the theory in four dimensions is generally fractional with denominator N, we still obtain an integral vortex number in the reduced theory. We clarify the relation between these two topological charges at a bundle geometric level. Another striking feature is the emergence of nontrivial lower and upper bounds for the energy of the reduced theory on Σ. These bounds are proportional to the area of Σ. We give special solutions of the theory on Σ by embedding solutions of the Abelian Higgs model into the non-Abelian theory, and we relate our work to the language of quiver bundles, which has recently proved fruitful in the study of dimensional reduction of Yang–Mills theory.

Some results on the moduli spaces of quiver bundles

This article is concerned with the study of gauge theory, stability and moduli for twisted quiver bundles in algebraic geometry. We review natural vortex equations for twisted quiver bundles and their link with a stability condition. Then we provide a brief overview of their relevance to other geometric problems and explain how quiver bundles can be viewed as sheaves of modules over a sheaf of associative algebras and why this view point is useful, e.g., in their deformation theory. Next we explain the main steps of an algebro-geometric construction of their moduli spaces. Finally, we focus on the special case of holomorphic chains over Riemann surfaces, providing some basic links with quiver representation theory. Combined with the analysis of the homological algebra of quiver sheaves and modules, these links provide a criterion for smoothness of the moduli spaces and tools to study their variation with respect to stability.

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].

Hitchin–Kobayashi Correspondence, Quivers, and Vortices

A twisted quiver bundle is a set of holomorphic vector bundles over a complex manifold, labelled by the vertices of a quiver, linked by a set of morphisms twisted by a fixed collection of holomorphic vector bundles, labelled by the arrows. When the manifold is Kaelher, quiver bundles admit natural gauge-theoretic equations, which unify many known equations for bundles with extra structure. In this paper we prove a Hitchin--Kobayashi correspondence for twisted quiver bundles over a compact Kaehler manifold, relating the existence of solutions to the gauge equations to a stability criterion, and consider its application to a number of situations related to Higgs bundles and dimensional reductions of the Hermitian--Einstein equations.

Dimensional reduction and quiver bundles

The so-called Hitchin-Kobayashi correspondence, proved by Donaldson, Uhlenbeck and Yau, establishes that an indecomposable holomorphic vector bundle over a compact Kahler manifold admits a Hermitian-Einstein metric if and only if the bundle satisfies the Mumford-Takemoto stability condition. In this paper we consider a variant of this correspondence for G-equivariant vector bundles on the product of a compact Kahler manifold X by a flag manifold G=P, where G is a complex semisimple Lie group and P is a parabolic subgroup. The modification that we consider is determined by a filtration of the vector bundle which is naturally defined by the equivariance of the bundle. The study of invariant solutions to the modified Hermitian-Einstein equation over XG=P leads, via dimensional reduction techniques, to gauge-theoretic equations on X. These are equations for hermitian metrics on a set of holomorphic bundles on X linked by morphisms, defin- ing what we call a quiver bundle for a quiver with relations whose structure is entirely determined by the parabolic subgroup P. Similarly, the corresponding stability condition for the invariant filtration over XG=P gives rise to a stability condition for the quiver bundle on X , and hence to a Hitchin-Kobayashi correspondence. In the simplest case, when the flag manifold is the complex projective line, one recovers the theory of vortices, stable triples and stable chains, as studied by Bradlow, the authors, and others.