Trivial Vector Bundle Research Articles

Connections on trivial vector bundles over projective schemes

Connections on trivial vector bundles over projective schemes

Covariant star product on semi-conformally flat noncommutative Calabi–Yau manifolds and noncommutative topological index theorem

A differential geometric statement of the noncommutative topological index theorem is worked out for covariant star products on noncommutative vector bundles. To start, a noncommutative manifold is considered as a product space [Formula: see text], wherein [Formula: see text] is a closed manifold, and [Formula: see text] is a flat Calabi–Yau [Formula: see text]-fold. Also, a semi-conformally flat metric is considered for [Formula: see text] which leads to a dynamical noncommutative spacetime from the viewpoint of noncommutative gravity. Based on the Kahler form of [Formula: see text] the noncommutative star product is defined covariantly on vector bundles over [Formula: see text]. This covariant star product leads to the celebrated Groenewold–Moyal product for trivial vector bundles and their flat connections, such as [Formula: see text]. Hereby, the noncommutative characteristic classes are defined properly and the noncommutative Chern–Weil theory is established by considering the covariant star product and the superconnection formalism. Finally, the index of the ⋆-noncommutative version of elliptic operators is studied and the noncommutative topological index theorem is stated accordingly.

Construction of a photon position operator with commuting components from natural axioms

A general form of the photon position operator with commuting components fulfilling some natural axioms is obtained. This operator commutes with the photon helicity operator, is Hermitian with respect to the Bia\l{}ynicki--Birula scalar product, and is defined up to a unitary transformation preserving the transversality condition. It is shown that, using the procedure analogous to the one introduced by T. T. Wu and C. N. Yang for the case of the Dirac magnetic monopole, the photon position operator can be defined by a flat connection in some trivial vector bundle over ${\mathbb{R}}^{3}\ensuremath{\setminus}{(0,0,0)}$. This observation enables us to reformulate the quantum mechanics of a single photon on $({\mathbb{R}}^{3}\ensuremath{\setminus}{(0,0,0)})\ifmmode\times\else\texttimes\fi{}{\mathbb{C}}^{2}$.

Euler characteristics of tautological bundles over Quot schemes of curves

We compute the Euler characteristics of tautological vector bundles and their exterior powers over the Quot schemes of curves. We give closed-form expressions over punctual Quot schemes in all genera. For higher rank quotients of a trivial vector bundle, we obtain answers in genus 0. We also study the Euler characteristics of the symmetric powers of the tautological bundles, for rank 0 quotients.

Supermanifolds corresponding to the trivial vector bundle over torus

All supermanifolds whose retract $T^{m|n}$ is determined by the trivial bundle of rank $n$ over the torus $T^m$ are $\overline 0$-homogeneous if and only if $T^{m|n}$ is homogeneous.

Recovery of a Time-Dependent Hermitian Connection and Potential Appearing in the Dynamic Schrödinger Equation

We consider, on a trivial vector bundle over a Riemannian manifold with boundary, the inverse problem of uniquely recovering time- and space-dependent coefficients of the dynamic, vector-valued Schrödinger equation from the knowledge of the Dirichlet-to-Neumann (D-to-N) map. We show that the D-to-N map uniquely determines both the connection form and the potential appearing in the Schrödinger equation, under the assumption that the manifold is either (a) two-dimensional and simple or (b) of higher dimension with strictly convex boundary and admits a smooth, strictly convex function.

Numerical characterization of some toric fiber bundles

Given a complex projective manifold X and a divisor D with normal crossings, we say that the logarithmic tangent bundle $$T_X(-\hbox {log}\, D)$$ is R-flat if its pull-back to the normalization of any rational curve contained in X is the trivial vector bundle. If moreover $$-(K_X+D)$$ is nef, then the log canonical divisor $$K_X+D$$ is torsion and the maximally rationally chain connected fibration turns out to be a smooth locally trivial fibration with typical fiber F being a toric variety with boundary divisor $$D_{|F}$$ .

On certain Tannakian categories of integrable connections over Kähler manifolds

AbstractGiven a compact Kähler manifold X, it is shown that pairs of the form $(E,\, D)$ , where E is a trivial holomorphic vector bundle on X, and D is an integrable holomorphic connection on E, produce a neutral Tannakian category. The corresponding pro-algebraic affine group scheme is studied. In particular, it is shown that this pro-algebraic affine group scheme for a compact Riemann surface determines uniquely the isomorphism class of the Riemann surface.

Unipotent factorization of vector bundle automorphisms

We provide unipotent factorizations of vector bundle automorphisms of real and complex vector bundles over finite dimensional locally finite CW-complexes. This generalizes work of Thurston–Vaserstein and Vaserstein for trivial vector bundles. We also address two symplectic cases and propose a complex geometric analog of the problem in the setting of holomorphic vector bundles over Stein manifolds.

Irreducible flat [formula omitted]-connections on the trivial holomorphic bundle

We construct an irreducible holomorphic connection with SL(2,R)–monodromy on the trivial holomorphic vector bundle of rank two over a compact Riemann surface. This answers a question of Calsamiglia, Deroin, Heu and Loray in [5].

The B-model connection and mirror symmetry for Grassmannians

We consider the Grassmannian X=Grn−k(Cn) and describe a ‘mirror dual’ Landau-Ginzburg model (Xˇ∘,Wq:Xˇ∘→C), where Xˇ∘ is the complement of a particular anti-canonical divisor in a Langlands dual Grassmannian Xˇ, and we express W succinctly in terms of Plücker coordinates. First of all, we show this Landau-Ginzburg model to be isomorphic to one proposed for homogeneous spaces in a previous work by the second author. Secondly we show it to be a partial compactification of the Landau-Ginzburg model defined in the 1990's by Eguchi, Hori, and Xiong. Finally we construct inside the Gauss-Manin system associated to Wq a free submodule which recovers the trivial vector bundle with small Dubrovin connection defined out of Gromov-Witten invariants of X. We also prove a T-equivariant version of this isomorphism of connections. Our results imply in the case of Grassmannians an integral formula for a solution to the quantum cohomology D-module of a homogeneous space, which was conjectured by the second author. They also imply a series expansion of the top term in Givental's J-function, which was conjectured in a 1998 paper by Batyrev, Ciocan-Fontanine, Kim and van Straten.

Remarks on the linking theory of shape dynamics

This short note is an attempt to bring out the geometric structures in the linking theory of shape dynamics. Symplectic induction is applied to give a natural construction of the extended phase space used in the linking theory as a trivial vector bundle over the original phase space for canonical gravity. The geometry of the gauge fixing for shape dynamics is analyzed with the assistance of the Lichnerowicz–York equation lifted to the extended phase space. An alternative description is provided to show how the same geometry simply derives from symplectic induction.

An integrable connection on the configuration space of a Riemann surface of positive genus

Let X be a Riemann surface of positive genus. Denote by X(n) the configuration space of n distinct points on X. We use the Betti–de Rham comparison isomorphism on H1(X(n)) to define an integrable connection on the trivial vector bundle on X(n) with fiber the universal algebra of the Lie algebra associated with the descending central series of π1 of X(n). The construction is inspired by the Knizhnik–Zamolodchikov system in genus zero and its integrability follows from Riemann period relations.

Branched Holomorphic Cartan Geometries and Calabi–Yau Manifolds

Abstract We introduce the concept of a branched holomorphic Cartan geometry. It generalizes to higher dimension the definition of branched (flat) complex projective structure on a Riemann surface introduced by Mandelbaum [25]. This new framework is much more flexible than that of the usual holomorphic Cartan geometries. We show that all compact complex projective manifolds admit a branched flat holomorphic projective structure. We also give an example of a non-flat branched holomorphic normal projective structure on a compact complex surface. It is known that no compact complex surface admits such a structure with empty branching locus. We prove that non-projective compact simply connected Kähler Calabi–Yau manifolds do not admit any branched holomorphic projective structure. The key ingredient of its proof is the following result of independent interest: if E is a holomorphic vector bundle over a compact simply connected Kähler Calabi–Yau manifold and E admits a holomorphic connection, then E is a trivial holomorphic vector bundle and any holomorphic connection on E is trivial.

Noncommutative potential theory

We propose to view hermitian metrics on trivial holomorphic vector bundles E → Ω as noncommutative analogs of functions defined on the base Ω, and curvature as the notion corresponding to the Laplace operator or ∂∂. We discuss noncommutative generalizations of basic results of ordinary potential theory, mean value properties, maximum principle, Harnack inequality, and the solvability of Dirichlet problems.

Open quotients of trivial vector bundles

Given an arbitrary topological complex vector space A, a quotient vector bundle for A is a quotient of a trivial vector bundle π2:A×X→X by a fiberwise linear continuous open surjection. We show that this notion subsumes that of a Banach bundle over a locally compact Hausdorff space X. Hyperspaces consisting of linear subspaces of A, topologized with natural topologies that include the lower Vietoris topology and the Fell topology, provide classifying spaces for various classes of quotient vector bundles, in a way that generalizes the classification of locally trivial vector bundles by Grassmannians. If A is normed, a finer hyperspace topology is introduced that classifies bundles with continuous norm, including Banach bundles, and such that bundles of constant finite rank must be locally trivial.

Singularity of the varieties of representations of lattices in solvable Lie groups

For a lattice [Formula: see text] of a simply connected solvable Lie group [Formula: see text], we describe the analytic germ in the variety of representations of [Formula: see text] at the trivial representation as an analytic germ which is linearly embedded in the analytic germ associated with the nilpotent Lie algebra determined by [Formula: see text]. By this description, under certain assumption, we study the singularity of the analytic germ in the variety of representations of [Formula: see text] at the trivial representation by using the Kuranishi space construction. By a similar technique, we also study deformations of holomorphic structures of trivial vector bundles over complex parallelizable solvmanifolds.

A symplectic analog of the Quot scheme

We construct a symplectic analog of the Quot scheme that parameterizes the torsion quotients of a trivial vector bundle over a compact Riemann surface. Some of its properties are investigated.

Lévy Laplacians and instantons

We describe dual and antidual solutions of the Yang–Mills equations by means of L´evy Laplacians. To this end, we introduce a class of L´evy Laplacians parameterized by the choice of a curve in the group SO(4). Two approaches are used to define such Laplacians: (i) the Levy Laplacian can be defined as an integral functional defined by a curve in SO(4) and a special form of the second-order derivative, or (ii) the Levy Laplacian can be defined as the Cesaro mean of second-order derivatives along vectors from the orthonormal basis constructed by such a curve. We prove that under some conditions imposed on the curve generating the Levy Laplacian, a connection in the trivial vector bundle with base R4 is an instanton (or an anti-instanton) if and only if the parallel transport generated by the connection is harmonic for such a Levy Laplacian.

Moduli spaces of meromorphic connections and quiver varieties

We describe the moduli spaces of meromorphic connections on trivial holomorphic vector bundles over the Riemann sphere with at most one (unramified) irregular singularity and arbitrary number of simple poles as Nakajima's quiver varieties. This result enables us to solve partially the additive irregular Deligne–Simpson problem.