Let $Q\to M$ be a principal $G$-bundle, and $B_0$ a connection on $Q$. We introduce an infinitesimal homogeneity condition for sections in an associated vector bundle $Q\times_GV$ with respect to $B_0$, and, inspired by the well known Ambrose-Singer theorem, we prove the existence of a connection which satisfies a system of parallelism conditions. We explain how this general theorem can be used to prove the known Ambrose-Singer type theorems by an appropriate choice of the initial system of data.We also obtain new applications, which cannot be obtained using the known formalisms, e.g. a classification theorem for locally homogeneous spinors.
Finally we introduce natural local homogeneity and local symmetry conditions for triples $(g,P\stackrel{p}{\to} M,A)$ consisting of a Riemannian metric on $M$, a principal bundle on $M$, and a connection on $P$. Our main results concern locally homogeneous and locally symmetric triples, and they can be viewed as bundle versions of the Ambrose-Singer and Cartan theorem.
We introduce the concept of a graded bundle which is a natural generalization of the concept of a vector bundle and whose standard examples are higher tangent bundles T^nQ playing a fundamental role in higher order Lagrangian formalisms. Graded bundles are graded manifolds in the sense that we can choose an atlas whose local coordinates are homogeneous functions of degrees 0,1,...,n. We prove that graded bundles have a convenient equivalent description as homogeneity structures, i.e. manifolds with a smooth action of the multiplicative monoid of non-negative reals. The main result states that …
A linear connection on a Lie algebroid is called a Cartan connection if it is suitably compatible with the Lie algebroid structure. Here we show that a smooth connected manifold $M$ is locally homogeneous - i.e., admits an atlas of charts modeled on some homogeneous space $G/H$ - if and only if there exists a transitive Lie algebroid over $M$ admitting a flat Cartan connection that is 'geometrically closed'. It is shown how the torsion and monodromy of the connection determine the particular form of $G/H$. Under an additional completeness hypothesis, local homogeneity becomes global homogeneit…
We slightly extend the notion of a natural fibre bundle by requiring diffeomorphisms of the base to lift to automorphisms of the bundle only infinitesimally, i.e. at the level of the Lie algebra of vector fields. Spin structures are natural only in this extended sense. We classify fibre bundles with this property, assuming a finite dimensional structure group. This includes all spin structures, but only some generalised spin structures. This classification links the gauge group G to the topology of space-time.
We review some basic facts on vector fields, in the complex-analytic setting, thus, obtaining a rationality result and an extension of the Birkhoff-Grothendieck theorem, as follows: (1) Let $Z$ be a compact complex manifold endowed with a very ample line bundle $L$. Denote by $\mathfrak{g}_L$ the extended Lie algebra of infinitesimal automorphisms of $L$. If the representation of $\mathfrak{g}_L$ on the space of holomorphic sections of $L$ is irreducible then $Z$ is rational; (2) Let $P$ be a holomorphic principal bundle over the Riemann sphere, with structural group $G$ whose Lie algebra is n…
We investigate infinitesimal properties of sets of ordered $n$-uples of idempotents in a symmetric Banach $*$-algebra. These sets are called flag manifolds and carry several interesting bundles that hold an important role in some areas of operator theory. In this direction, we introduce and study Stiefel bundles on flag manifolds, which are extensions of the well known Stiefel bundles on Grassmannians. The main ingredient of our investigation is the notion of connection on an infinite-dimensional bundle, and we survey some equivalent ocurrences of such a notion in the literature.
This PHD thesis is concerned with uncertainty relations in quantum probability theory, state estimation in quantum stochastics, and natural bundles in differential geometry. After some comments on the nature and necessity of decoherence in open systems and its absence in closed ones, we prove sharp, state-independent inequalities reflecting the Heisenberg principle, the necessity of decoherence and the impossibility of perfect joint measurement. These bounds are used to judge how far a particular measurement is removed from the optimal one. We do this for a qubit interacting with the quantized…
We describe the infinitesimal moduli space of pairs $(Y, V)$ where $Y$ is a manifold with $G_2$ holonomy, and $V$ is a vector bundle on $Y$ with an instanton connection. These structures arise in connection to the moduli space of heterotic string compactifications on compact and non-compact seven dimensional spaces, e.g. domain walls. Employing the canonical $G_2$ cohomology developed by Reyes-Carrión and Fernández and Ugarte, we show that the moduli space decomposes into the sum of the bundle moduli $H^1_{\check{d}_A}(Y,\mathrm{End}(V))$ plus the moduli of the $G_2$ structure preserving the i…
In our [Higher-order preconnections in synthetic differential geometry of jet bundles, Beiträge zur Algebra und Geometrie, 45 (2004), 677-696] we have established the affine bundle theorem in the synthetic approach to jet bundles in terms of infinitesimal spaces Dⁿ's. In our succeeding [Synthetic differential geometry of higher-order total differentials, to appear in Cahiers de Topologie et Géométrie Différentielle Catégoriques] we have introduced another synthetic approach to jet bundles in terms of infinitesimal spaces D_{n}'s, and have compared it with the former approach both in the …
Let $(M,g)$ be a smooth Riemannian manifold, $K$ a compact Lie group and $p:P\to M$ a principal $K$-bundle over $M$ endowed with a connection $A$. Fixing a bi invariant inner product on Lie algebra $\mathfrak{k}$ of $K$, the connection $A$ and metric $g$ define a Riemannian metric $g_A$ on $P$. Let $\tilde {X}$ be the horizontal lift of vector field $X$ on $M$ and, let $ξ^ν$ be the vertical field associated with section $ν\in A^0(\mathrm{ad}( P))$ of the adjoint bundle. It is proved that the connection $A$ is invariant under the 1-parameter group of local diffeomorphism generated by $\tilde{ X…
VB-groupoids and algebroids are vector bundle objects in the categories of Lie groupoids and Lie algebroids respectively, and they are related via the Lie functor. VB-groupoids and algebroids play a prominent role in Poisson and related geometries. Additionally, they can be seen as models for vector bundles over singular spaces. In this paper we study their infinitesimal automorphisms, i.e. vector fields on them generating a flow by diffeomorphisms preserving both the linear and the groupoid/algebroid structures. For a special class of VB-groupoids/algebroids coming from representations of Lie…
Heterotic string compactifications on integrable $G_2$ structure manifolds $Y$ with instanton bundles $(V,A), (TY,\tildeθ)$ yield supersymmetric three-dimensional vacua that are of interest in physics. In this paper, we define a covariant exterior derivative $\cal D$ and show that it is equivalent to a heterotic $G_2$ system encoding the geometry of the heterotic string compactifications. This operator $\cal D$ acts on a bundle ${\cal Q}=T^*Y\oplus{\rm End}(V)\oplus{\rm End}(TY)$ and satisfies a nilpotency condition $\check{\cal D}^2=0$, for an appropriate projection of $\cal D$. Furthermore, …
We consider $G_2$-structures with torsion coupled with $G_2$-instantons, on a compact $7$-dimensional manifold. The coupling is via an equation for $4$-forms which appears in supergravity and generalized geometry, known as the Bianchi identity. First studied by Friedrich and Ivanov, the resulting system of partial differential equations describes compactifications of the heterotic string to three dimensions, and is often referred to as the $G_2$-Strominger system. We study the moduli space of solutions and prove that the space of infinitesimal deformations, modulo automorphisms, is finite dime…
We define and study multiplicative connections in the tangent bundle of a Lie groupoid. Multiplicative connections are linear connections satisfying an appropriate compatibility with the groupoid structure. Our definition is natural in the sense that a linear connection on a Lie groupoid is multiplicative if and only if its torsion is a multiplicative tensor in the sense of Bursztyn-Drummond [5] and its geodesic spray is a multiplicative vector field. We identify the obstruction to the existence of a multiplicative connection. We also discuss the infinitesimal version of multiplicative connect…
We use foliations and connections on principal Lie groupoid bundles to prove various integrability results for Lie algebroids. In particular, we show, under quite general assumptions, that the semi-direct product associated to an infinitesimal action of one integrable Lie algebroid on another is integrable. This generalizes recent results of Dazord and Nistor.
We introduce a symplectic structure on the space of connections in a G-principal bundle over a four-manifold and the Hamiltonian action on it of the group of gauge transformations which are trivial on the boundary. The symplectic reduction becomes the moduli space of flat connections over the manifold. On the moduli space of flat connections we shall construct a hermitian line bundle with connection whose curvature is given by the symplectic form. This is the Chern-Simons prequantum line bundle. The group of gauge transformations on the boundary of the base manifold acts on the moduli space of…
Let $M$ be a differentiable manifold and $K$ a Lie group. A locally homogeneous triple with structure group $K$ on $M$ is a triple $(g, P\stackrel{p}{\to} M,A)$, where $p:P\to M$ is a principal $K$-bundle on $M$, $g$ is Riemannian metric on $M$, and $A$ is connection on $P$ such that the following locally homogeneity condition is satisfied: for every two points $x$, $x'\in M$ there exists an isometry $φ:U\to U'$ between open neighborhoods $U\ni x$, $U'\ni x'$ with $φ(x)=x'$, and a $φ$-covering bundle isomorphism $Φ:P_U\to P_{U'}$ such that $Φ^*(A_{U'})=A_U$. If $(g,P\stackrel{p}{\to} M,A)$ is …
We describe invariant principal and Cartan connections on homogeneous principal bundles and show how to calculate the curvature and the holonomy; in the case of an invariant Cartan connection we give a formula for the infinitesimal automorphisms. The main result of this paper is that the above calculations are purely algorithmic. As an example of an homogeneous parabolic geometry we treat a conformal structure on the product of two spheres.
We present a new formulation of some basic differential geometric notions on a smooth manifold M, in the setting of nonstandard analysis. In place of classical vector fields, for which one needs to construct the tangent bundle of M, we define a prevector field, which is an internal map from *M to itself, implementing the intuitive notion of vectors as infinitesimal displacements. We introduce regularity conditions for prevector fields, defined by finite differences, thus purely combinatorial conditions involving no analysis. These conditions replace the more elaborate analytic regularity condi…
We study heterotic supergravity at $\mathcal{O}(α')$, first described in detail in 1989 by Bergshoeff and de Roo. In particular, we discuss an ambiguity of a connection choice on the tangent bundle. It is well known that at $\mathcal{O}(α')$ the Hull connection gives a consistent supergravity theory with supersymmetry transformations given in the usual way. We consider deformations of this connection corresponding to field redefinitions, and the necessary corrections to the supersymmetry transformations. We are interested in the moduli space of such field redefinitions which allow for supersym…
We describe the first order moduli space of heterotic string theory compactifications which preserve $N=1$ supersymmetry in four dimensions, that is, the infinitesimal parameter space of the Strominger system. We establish that if we promote a connection on $TX$ to a field, the moduli space corresponds to deformations of a holomorphic structure $\bar D$ on a bundle $\cal Q$. The bundle $\cal Q$ is constructed as an extension by the cotangent bundle $T^*X$ of the bundle $E= {\rm End}(V) \oplus {\rm End}(TX) \oplus TX$ with an extension class $\cal H$ which precisely enforces the anomaly cancela…
Field Theory via Higher Geometry II: Thickened Smooth Sets as Synthetic Foundations
NYUAD Research Institute, New York University Abu Dhabi（阿布扎比纽约大学研究院）；Center for Quantum and Topological Systems, NYUAD Research Institute, New York University Abu Dhabi（阿布扎比纽约大学研究院量子与拓扑系统中心）；The Courant Institute for Mathematical Sciences, NYU（纽约大学库朗数学科学研究所）
