Equivariant Bundles Research Articles

Equivariant orientation of vector bundles over disconnected base spaces

AbstractIn this paper, we view the equivariant orientation theory of equivariant vector bundles from the lenses of equivariant Picard spectra. This viewpoint allows us to identify, for a finite group , a precise condition under which an ‐orientation of a ‐equivariant vector bundle is encoded by a Thom class. Consequently, we are able to construct a generalization of the first Stiefel–Whitney class of a “homogeneous” ‐equivariant bundle with respect to an ‐ring spectrum . As an application, we show that the 2‐fold direct sum of any homogeneous bundle is ‐orientable, where is the Burnside Mackey functor. We notice that ‐orientability is equivalent to ‐orientability when the order of is odd. When the order of is even, we show that a ‐equivariant analog of the tautological line bundle over is ‐orientable but not ‐orientable.

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.

On equivariant bundles and their moduli spaces

On equivariant bundles and their moduli spaces

Modular geodesics and wedge domains in non-compactly causal symmetric spaces

We continue our investigation of the interplay between causal structures on symmetric spaces and geometric aspects of Algebraic Quantum Field Theory. We adopt the perspective that the geometric implementation of the modular group is given by the flow generated by an Euler element of the Lie algebra (an element defining a 3-grading). Since any Euler element of a semisimple Lie algebra specifies a canonical non-compactly causal symmetric space M=G/H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M = G/H$$\\end{document}, we turn in this paper to the geometry of this flow. Our main results concern the positivity region W of the flow (the corresponding wedge region): If G has trivial center, then W is connected, it coincides with the so-called observer domain, specified by a trajectory of the modular flow which at the same time is a causal geodesic. It can also be characterized in terms of a geometric KMS condition, and it has a natural structure of an equivariant fiber bundle over a Riemannian symmetric space that exhibits it as a real form of the crown domain of G/K. Among the tools that we need for these results are two observations of independent interest: a polar decomposition of the positivity domain and a convexity theorem for G-translates of open H-orbits in the minimal flag manifold specified by the 3-grading.

Relative connections on principal bundles and relative equivariant structures

We investigate relative holomorphic connections on a principal bundle over a family of compact complex manifolds. A sufficient condition is given for the existence of a relative holomorphic connection on a holomorphic principal bundle over a complex analytic family. We also introduce the notion of relative equivariant bundles and establish its relation with relative holomorphic connections on principal bundles.

Non-connected Lie groups, twisted equivariant bundles and coverings

Let Gamma be a finite group acting on a Lie group G. We consider a class of group extensions 1 rightarrow G rightarrow hat{G} rightarrow Gamma rightarrow 1 defined by this action and a 2-cocycle of Gamma with values in the centre of G. We establish and study a correspondence between hat{G}-bundles on a manifold and twisted Gamma -equivariant bundles with structure group G on a suitable Galois Gamma -covering of the manifold. We also describe this correspondence in terms of non-abelian cohomology. Our results apply, in particular, to the case of a compact or reductive complex Lie group hat{G}, since such a group is always isomorphic to an extension as above, where G is the connected component of the identity and Gamma is the group of connected components of hat{G}.

Equivariant higher Dixmier-Douady theory for circle actions on UHF-algebras

We develop an equivariant Dixmier-Douady theory for locally trivial bundles of C⁎-algebras with fibre D⊗K equipped with a fibrewise T-action, where T denotes the circle group and D=End(V)⊗∞ for a T-representation V. In particular, we show that the group of T-equivariant ⁎-automorphisms AutT(D⊗K) is an infinite loop space giving rise to a cohomology theory ED,T⁎(X). Isomorphism classes of equivariant bundles then form a group with respect to the fibrewise tensor product that is isomorphic to ED,T1(X)≅[X,BAutT(D⊗K)]. We compute this group for tori and compare the case D=C to the equivariant Brauer group for trivial actions on the base space.

A construction of equivariant bundles on the space of symmetric forms

We construct stable vector bundles on the space \mathbb{P}(S^d \mathbb{C}^{n+1}) of symmetric forms of degree d in n+1 variables which are equivariant for the action of \text{SL}_{n+1}(\mathbb{C}) and admit an equivariant free resolution of length 2 . For n=1 , we obtain new examples of stable vector bundles of rank d-1 on \mathbb{P}^d , which are moreover equivariant for \operatorname{SL}_2(\mathbb{C}) . The presentation matrix of these bundles attains Westwick's upper bound for the dimension of vector spaces of matrices of constant rank and fixed size.

Notes on equivariant bundles

We compare two notions of G-fiber bundles and G-principal bundles in the literature, with an aim to clarify early results in equivariant bundle theory that are needed in current work of equivariant algebraic topology. We also give proofs of some equivariant generalizations of well-known non-equivariant results involving the classifying space.

Equivariant Ulrich bundles on exceptional homogeneous varieties

Abstract We prove that the only rational homogeneous varieties with Picard number 1 of the exceptional algebraic groups admitting irreducible equivariant Ulrich vector bundles are the Cayley plane E 6/P 1 and the E 7-adjoint variety E 7/P 1. From this result,we see that a general hyperplane section F 4/P 4 of the Cayley plane also has an equivariant but non-irreducible Ulrich bundle.

Transversals, duality, and irrational rotation

An early result of Noncommutative Geometry was Connes’ observation in the 1980’s that the Dirac-Dolbeault cycle for the 2 2 -torus T 2 \mathbb {T}^2 , which induces a Poincaré self-duality for T 2 \mathbb {T}^2 , can be ‘quantized’ to give a spectral triple and a K-homology class in K K 0 ( A θ ⊗ A θ , C ) \mathrm {KK}_0(A_\theta \otimes A_\theta , \mathbb {C}) providing the co-unit for a Poincaré self-duality for the irrational rotation algebra A θ A_\theta for any θ ∈ R ∖ Q \theta \in \mathbb {R}\setminus \mathbb {Q} . Connes’ proof, however, relied on a K-theory computation and does not supply a representative cycle for the unit of this duality. Since such representatives are vital in applications of duality, we supply such a cycle in unbounded form in this article. Our approach is to construct, for any non-zero integer b b , a finitely generated projective module L b \mathcal {L}_{b} over A θ ⊗ A θ A_\theta \otimes A_\theta by using a reduction-to-a-transversal argument of Muhly, Renault, and Williams, applied to a pair of Kronecker foliations along lines of slope θ \theta and θ + b \theta + b , using the fact that these flows are transverse to each other. We then compute Connes’ dual of [ L b ] [\mathcal {L}_{b}] and prove that we obtain an invertible τ b ∈ K K 0 ( A θ , A θ ) \tau _{b}\in \mathrm {KK}_0(A_\theta , A_\theta ) , represented by an equivariant bundle of Dirac-Schrödinger operators. An application of equivariant Bott Periodicity gives a form of higher index theorem describing functoriality of such ‘ b b -twists’ and this allows us to describe the unit of Connes’ duality in terms of a combination of two constructions in KK-theory. This results in an explicit spectral representative of the unit – a kind of ‘quantized Thom class’ for the diagonal embedding of the noncommutative torus.

Spinor modules for Hamiltonian loop group spaces

Let $LG$ be the loop group of a compact, connected Lie group $G$. We show that the tangent bundle of any proper Hamiltonian $LG$-space $\mathcal{M}$ has a natural completion $\overline{T}\mathcal{M}$ to a strongly symplectic $LG$-equivariant vector bundle. This bundle admits an invariant compatible complex structure within a natural polarization class, defining an $LG$-equivariant spinor bundle $\mathsf{S}_{\overline{T}\mathcal{M}}$, which one may regard as the Spin$_c$-structure of $\mathcal{M}$. We describe two procedures for obtaining a finite-dimensional version of this spinor module. In one approach, we construct from $\mathsf{S}_{\overline{T}\mathcal{M}}$ a twisted Spin$_c$-structure for the quasi-Hamiltonian $G$-space associated to $\mathcal{M}$. In the second approach, we describe an `abelianization procedure', passing to a finite-dimensional $T\subset LG$-invariant submanifold of $\mathcal{M}$, and we show how to construct an equivariant Spin$_c$-structure on that submanifold.

The Weyl map and bundle gerbes

We introduce the notion of a general cup product bundle gerbe and use it to define the Weyl bundle gerbe on T×SU(n)∕T. The Weyl map from T×SU(n)∕T to SU(n) is then used to show that the pullback of the basic bundle gerbe on SU(n) defined by the second two authors is stably isomorphic to the Weyl bundle gerbe as SU(n)-equivariant bundle gerbes. Both bundle gerbes come equipped with connections and curvings and by considering the holonomy of these we show that these bundle gerbes are not D-stably isomorphic.

A classification of equivariant gerbe connections

Let [Formula: see text] be a compact Lie group acting on a smooth manifold [Formula: see text]. In this paper, we consider Meinrenken’s [Formula: see text]-equivariant bundle gerbe connections on [Formula: see text] as objects in a 2-groupoid. We prove this 2-category is equivalent to the 2-groupoid of gerbe connections on the differential quotient stack associated to [Formula: see text], and isomorphism classes of [Formula: see text]-equivariant gerbe connections are classified by degree 3 differential equivariant cohomology. Finally, we consider the existence and uniqueness of conjugation-equivariant gerbe connections on compact semisimple Lie groups.

Equivariant complex bundles, fixed points and equivariant unitary bordism

We study the fixed points of the universal G-equivariant n-dimensional complex vector bundle and obtain a decomposition formula in terms of twisted equivariant universal complex vector bundles of smaller dimension. We use this decomposition to describe the fixed points of the complex equivariant K-theory spectrum and the equivariant unitary bordism groups for adjacent families of subgroups.

Basic equivariant gerbes on non-simply connected compact simple Lie groups

This paper computes the obstruction to the existence of equivariant extensions of basic gerbes over non-simply connected compact simple Lie groups. By modifying a (finite dimensional) construction of Gaw\c{e}dzki-Reis [J. Geom. Phys. 50(1):28-55, 2004], we exhibit basic equivariant bundle gerbes over non-simply connected compact simple Lie groups.

Equivariant prequantization bundles on the space of connections and characteristic classes

We show how characteristic classes determine equivariant prequantization bundles over the space of connections on a principal bundle. These bundles are shown to generalize the Chern-Simons line bundles to arbitrary dimensions. Our result applies to arbitrary bundles, and it is studied the action of both the gauge group and the automorphisms group. The action of the elements in the connected component of the identity of the group generalizes known results in the literature. The action of the elements not connected with the identity is shown to be determined by a characteristic class by using differential characters and equivariant cohomology. We extend our results to the space of Riemannian metrics and the actions of diffeomorphisms. In dimension 2, a ${\Gamma}_{M}$-equivariant prequantization bundle of the Weil-Petersson symplectic form on the Teichm\"uller space is obtained, where ${\Gamma}_{M}$ is the mapping class group of the surface M.

Bundle gerbes for topological insulators

Bundle gerbes are simple examples of higher geometric structures that show their utility in dealing with topological subtleties of physical theories. I review a recent construction of torsion topological invariants for condensed matter systems via equivariant bundle gerbes. The construction covers static and periodically driven systems with time reversal invariance in 2 and 3 space dimensions. It involves refinements of geometry of gerbes that are discussed in the first lecture, the second one being devoted to the applications to topological insulators.

Categorical models for equivariant classifying spaces

Starting categorically, we give simple and precise models of equivariant classifying spaces. We need these models for work in progress in equivariant infinite loop space theory and equivariant algebraic K-theory, but the models are of independent interest in equivariant bundle theory and especially equivariant covering space theory.

Finite group actions on Lagrangian Floer theory

We construct finite group actions on Lagrangian Floer theory when symplectic manifolds have finite group actions and Lagrangian submanifolds have induced group actions. We first define finite group actions on Novikov-Morse theory. We introduce the notion of a {\em spin profile} as an obstruction class of extending the group action on Lagrangian submanifold to the one on its spin structure, which is a group cohomology class in $H^2(G;\Z/2)$. For a class of Lagrangian submanifolds which have the same spin profiles, we define a finite group action on their Fukaya category. In consequence, we obtain the $s$-equivariant Fukaya category as well as the $s$-orbifolded Fukaya category for each group cohomology class $s$. We also develop a version with $G$-equivariant bundles on Lagrangian submanifolds, and explain how character group of $G$ acts on the theory. As an application, we define an orbifolded Fukaya-Seidel category of a $G$-invariant Lefschetz fibration, and also discuss homological mirror symmetry conjectures with group actions.