Groupoid crossed products.

Let G be a locally compact group and p: Q -T a principal G-bundle. If A is a C*-algebra with primitive ideal space T, the pull-back p*A of A along p is the balanced tensor product Co((Q) ?C(T) A. If ,3: G -Aut A consists of C(T)-module automorphisms, and -y: G -Aut Co (Q) is the natural action, then the automorphism group y 0X 3 of Co(Q) 0 A respects the balancing and induces the diagonal action p*,3 of G on p*A. We discuss some examples of such actions and study the crossed product p* A x p G. We suggest a substitute D for the fixed-point algebra, prove p*A x G is strongly Morita equivalent to D, and investigate the structure of D in various cases. In particular, we ask when D is strongly Morita equivalent to A-sometimes, but by no means always-and investigate the case where A has continuous trace. Let B be a C*-algebra and G a locally compact group acting on B as a strongly continuous automorphism group a. Our goal here is to study the crossed product C*-algebra B x , G for two classes of diagonal actions for which the induced action of G on B is free. The first class includes actions of the form -y 0X3 on B Co (Q) 0 A, where p: Q -* T is a principal G-bundle, -y is the dual action of G on Co (Q), and 3: G -* Aut A is an action of G on another C*-algebra A. We also consider diagonal actions on algebras which are the pull-backs of another algebra A along a principal bundle p: Q -* T: if A is a C*-algebra with primitive ideal space T, then the pull-back p*A is the balanced tensor product Co(Q) ?Cb(T) A. When 3: G -* Aut A consists of C(T)-module automorphisms, the product action ty 0 /3 preserves the balancing, and the diagonal action p*3 is, by definition, the induced action on p* A. In general, if f: X -* Y and q: PrimA -* Y are continuous, then Cb(Y) acts on Co(X) by composition with f, and on A by composition with q and the DaunsHofmann theorem. We can therefore define the pull-back f*A of A along f as the C*-algebraic tensor product Co(X) ?Cb(y) A. The reason for the name is that when A is the algebra of sections of some C*-bundle E over Y, there is a natural isomorphism of f*A onto the algebra of sections of the pull-back f*E. In ?1 we discuss this and other basic properties of pull-backs and give some evidence to show they are likely to be of interest. In particular, we show that if G is abelian and a: G -* Aut A is locally unitary in the sense of [18], then the crossed product Received by the editors October 20, 1983 and, in revised form, February 15, 1984. 1980 Mathematics Subject Classification. Primary 46L40, 46L55.

In this chapter we treat the theory of principal bundles and connections on them for families of supermanifolds. The most important examples of principal bundles are frame bundles of vector bundles. Many extra structures on vector bundles, such as metrics or almost complex structures can actually be formulated in terms of a reduction of the structure group of the frame bundle of the vector bundle. The theory of connections on principal bundles sheds light on properties of covariant derivatives that are compatible with such extra structures.

Invariance conditions for gauge fields under smooth group actions are interpreted in terms of invariant connections on principal bundles. A classification of group actions on bundles as automorphisms projecting to an action on a base manifold with a sufficiently regular orbit structure is given in terms of group homorphisms and a generalization of Wang’s theorem classifying invariant connections is derived. Illustrative examples on compactified Minkowski space are given.

Given a principal bundle over a closed manifold, \(G \rightarrow P \rightarrow M\), let \(P^{Ad} \rightarrow M\) be the associated adjoint bundle. Gruher and Salvatore (Proc Lond Math Soc 96(3), 78106 2008) showed that the Thom spectrum \((P^\mathrm{Ad})^{-TM}\) is a ring spectrum whose corresponding product in homology is a Chas-Sullivan type string topology product. We refer to this spectrum as the “string topology spectrum of P”, \( \mathcal {S}(P)\). In the universal case when P is contractible, \(\mathcal {S}(P) \simeq LM^{-TM}\) where LM is the free loop space of the manifold. This ring spectrum was introduced by the authors in Cohen et al. (Math Annalen 324, 773–798 2002) as a homotopy theoretic realization of the Chas-Sullivan string topology of M. The main purpose of this paper is to introduce an action of the gauge group of the principal bundle, \(\mathcal {G}(P)\) on the string topology spectrum \(\mathcal {S}(P)\), and to study this action in detail. Indeed we study the entire group of units and the induced representation \(\mathcal {G}(P) \rightarrow GL_1(\mathcal {S}(P))\). We show that this group of units is the group of homotopy automorphisms of the fiberwise suspension spectrum of P. More generally we describe the homotopy type of the group of homotopy automorphisms of any E-line bundle for any ring spectrum E. We import some of the basic ideas of gauge theory, such as the action of the gauge group on the space of connections to the setting of E-line bundles over a manifold and do explicit calculations. We end by discussing a functorial perspective, which describes a sense in which the string topology spectrum \(\mathcal {S}(P)\) of a principal bundle is the “linearization” of the gauge group \(\mathcal {G}(P)\).

In this paper, we consider diffeological spaces as stacks over the site of smooth manifolds, as well as the “underlying” diffeological space of any stack. More precisely, we consider diffeological spaces as so-called concrete sheaves and show that the Grothendieck construction sending these sheaves to stacks has a left adjoint: the functor sending any stack to its diffeological coarse moduli space. As an application, we restrict our attention to differentiable stacks and examine the geometry behind the coarse moduli space construction in terms of Lie groupoids and their principal bundles. Within this context, we define a “gerbe”, and show when a Lie groupoid is such a gerbe (or when a stack is represented by one). Additionally, we define basic differential forms for stacks and confirm in the differentiable case that these agree (under certain conditions) with basic differential forms on a representative Lie groupoid. These basic differentiable forms in turn match the diffeological forms on the orbit space.

Dirac introduced the theory of the magnetic monopole which bears his name in 1931 in [9], quite a few years before the advent of either the Yang–Mills theory or the theory of fiber bundles. However, by 1975, Wu and Yang had learned enough about fiber bundles to reformulate Dirac’s theory in terms of a connection on a principal bundle. See [52].

In many areas of theoretical physics we encounter fields defined on a spacetime M taking values in some other space F. For instance, presume F as a real finite-dimensional vector space, then ∅ : M → F is a vector field. More generally, we can consider a family of spaces {Fx}xϵM varying over the points on M, that is ∅(x) ϵ Fx for each x ϵ M. A field ∅ is then understood as a section from the spacetime manifold into the bundle of spaces over M. This is exactly the idea encoded in the mathematical theory of fiber bundles. Namely, fiber bundles provide a tool to describe the global structure of physical fields.

Using the invariant geometrical interpretation of gauge and Higgs fields, a simple derivation is given of the dimensional reduction procedure. The underlying assumption with regard to the Riemannian structure, group orbits and invariant connection are clarified and the critical points of the Higgs potential are shown to have a natural geometrical interpretation.

We prove local existence and uniqueness of static spherically symmetric solutions of the Einstein–Yang–Mills (EYM) equations for an arbitrary compact semisimple gauge group in the so-called regular case. By this we mean the equations obtained when the rotation group acts on the principal bundle on which the Yang–Mills connection takes its values in a particularly simple way (the only one ever considered in the literature). The boundary value problem that results for possible asymptotically flat soliton or black hole solutions is very singular and just establishing that local power series solutions exist at the center and asymptotic solutions at infinity amounts to a nontrivial algebraic problem. We discuss the possible field equations obtained for different group actions and solve the algebraic problem on how the local solutions depend on initial data at the center and at infinity.

Let \Lambda be a finite abelian group. A dynamical system with transformation group \Lambda is a triple (A,\Lambda,\alpha) , consisting of a unital locally convex algebra A , the finite abelian group \Lambda and a group homomorphism \alpha:\Lambda\rightarrow\operatorname{Aut}(A) , which induces an action of \Lambda on A . In this paper we present a new, geometrically oriented approach to the noncommutative geometry of principal bundles with finite abelian structure group based on such dynamical systems.

Invariant Connections in a Non-Abelian Principal Bundle

§ i. General principal bundles i. F-structures Let P=(P,~,B,G) be a principal bundle where P and B are topological spaces and @ is a topological group. Let 0: F+G be a continuous homomorphism from a topological group F onto G with kernel K • 0 will be called central, if (i) K is discrete (ii) K is contained in the center of F Thus, in particular, K is abelian. It is easy to check that (ii) follows from (i) if F is connected. Thus every covering projection P: F÷G (F connected) satisfies the conditions above. Throughout this paper we shall assume that p is central. A r-structure on P is a F-principal bundle P=(P,~,B,F) together with a strong bundle map ~: P÷P which is equivariant under the right actions of the structure groups; that is

On a smooth geometric approach to the dynamics of media with microstructures

A Q-manifold is a graded manifold endowed with a vector field of degree 1 squaring to zero. We consider the notion of a Q-bundle, that is, a fiber bundle in the category of Q-manifolds. To each homotopy class of "gauge fields" (sections in the category of graded manifolds) and each cohomology class of a certain subcomplex of forms on the fiber we associate a cohomology class on the base. As any principal bundle yields canonically a Q-bundle, this construction generalizes Chern–Weil classes. Novel examples include cohomology classes that are locally de Rham differential of the integrands of topological sigma models obtained by the AKSZ-formalism in arbitrary dimensions. For Hamiltonian Poisson fibrations one obtains a characteristic 3-class in this manner. We also relate the framework to equivariant cohomology and Lecomte's characteristic classes of exact sequences of Lie algebras.

In this paper we study continuous bundles of C*-algebras which are non-commutative analogues of principal torus bundles. We show that all such bundles, although in general being very far away from being locally trivial bundles, are at least locally ℛKK-trivial. Using earlier results of Echterhoff and Williams, we shall give a complete classification of principal non-commutative torus bundles up to 𝕋n-equivariant Morita equivalence. We then study these bundles as topological fibrations (forgetting the group action) and give necessary and sufficient conditions for any non-commutative principal torus bundle being ℛKK-equivalent to a commutative one. As an application of our methods we shall also give a K-theoretic characterization of those principal 𝕋n-bundles with H-flux, as studied by Mathai and Rosenberg which possess ‘classical’ T-duals.