Narrow Results

There are two groups which act in a natural way on the bundle $TP$ tangent to the total space $P$ of a principal $G$-bundle $P(M,G)$: the group $Au{t}_{0}TP$ of automorphisms of $TP$ covering the identity map of $P$ and the group $TG$ tangent to the structural group $G$. Let $Au{t}_{TG}TP\subset Au{t}_{0}TP$ be the subgroup of those automorphisms which commute with the action of $TG$. In the paper, we investigate $G$-invariant symplectic structures on the cotangent bundle $T^{\ast }P$ which are in a one-to-one correspondence with elements of $Au{t}_{TG}TP$. Since, as it is shown here, the connections on $P(M,G)$ are in a one-to-one correspondence with elements of the normal subgroup $Au{t}_{N}TP$ of $Au{t}_{0}TP$, so the symplectic structures related to them are also investigated. The Marsden–Weinstein reduction procedure for these symplectic structures is discussed.

Let be a Gerstenhaber algebra generated by and . Given a degree -1 operator D on , we find the condition on D that makes a BV-algebra. Subsequently, we apply it to the Gerstenhaber or BV algebra associated to a Lie algebroid and obtain a global proof of the correspondence between BV-generators and flat connections.

Given a holomorphic line bundle L on a compact complex torus A, there are two naturally associated holomorphic Ω_A-torsors over A: one is constructed from the Atiyah exact sequence for L, and the other is constructed using the line bundle , where α is the addition map on A × A, and p₁ is the projection of A × A to the first factor. In [I. Biswas, J. Hurtvbise and A. K. Raina, Rank one connections on abelian varieties, Internat. J. Math.22 (2011) 1529–1543], it was shown that these two torsors are isomorphic. The aim here is to produce a canonical isomorphism between them through an explicit construction.

The Kubo–Ando theory deals with connections for positive bounded operators. On the other hand, in various analysis related to von Neumann algebras, it is impossible to avoid unbounded operators. In this paper, we try to extend a notion of connections to cover various classes of positive unbounded operators (or unbounded objects such as positive forms and weights) appearing naturally in the setting of von Neumann algebras, and we must keep all the expected properties maintained. This generalization is carried out for the following classes: (i) positive $\tau$-measurable operators (affiliated with a semifinite von Neumann algebra equipped with a trace $\tau$), (ii) positive elements in Haagerup’s $L^p$-spaces and (iii) semifinite normal weights on a von Neumann algebra. Investigation on these generalizations requires some analysis (such as certain upper semi-continuity) on decreasing sequences in various classes. Several results in this direction are proved, which may be of independent interest. Ando studied Lebesgue decomposition for positive bounded operators by making use of parallel sums. Here, such decomposition is obtained in the setting of noncommutative (Hilsum) $L^p$-spaces.

Let $X=\overline{X}-D$ be a smooth quasi-projective curve. We previously constructed a Deligne–Hitchin moduli space with Hecke gauge groupoid for connections of rank $2$. We extend this construction to the case of any rank $r$, although still keeping a genericity hypothesis. The formal neighborhood of a preferred section corresponding to a tame harmonic bundle is governed by a mixed twistor structure.

Let $G$ be a semisimple complex algebraic group with a simple Lie algebra $𝔤$, and let ${{ℳ}}_G^0$ denote the moduli stack of topologically trivial stable $G$-bundles on a smooth projective curve $C$. Fix a theta characteristic $\kappa$ on $C$ which is even in case $dim𝔤$ is odd. We show that there is a nonempty Zariski open substack ${{𝒰}}_{\kappa }$ of ${{ℳ}}_G^0$ such that $H^i(C,\mathrm{\text{ad}}(E_G)\otimes \kappa )=0$, $i=1,2$, for all $E_G\in {{𝒰}}_{\kappa }$. It is shown that any such $E_G$ has a canonical connection. It is also shown that the tangent bundle $T{U}_{\kappa }$ has a natural splitting, where $U_{\kappa }$ is the restriction of ${{𝒰}}_{\kappa }$ to the semi-stable locus. We also produce an isomorphism between two naturally occurring ${\mathrm{\Omega }}_{M_G^{rs}}^1$-torsors on the moduli space of regularly stable $M_G^{rs}$.

We study derivations on a smooth manifold, its twisted de Rham cohomology, generalized connections on vector bundles and their characteristic classes.

Based on the decomposition of SU(2) gauge field, we derive a generalization of the decomposition theory for the SU(N) gauge field. We thus obtain the invariant electromagnetic tensors of SU(N) groups and the extended Wu–Yang potentials. The sourceless solutions are also discussed.

In a previous paper, we addressed the method of Abelian decomposition to the case of SU(N) Yang–Mills theory. Here, we extend the decomposition method further to the general case. With the Cartan–Weyl basis we decompose semisimple group connection and discuss the SO(3,1) group in particular. In terms of the vierbein projection, we propose two two-forms as the U(1) gauge fields in the SO(3,1) gauge theory and show that these two-forms are just the different cosmic string tensors. Meanwhile, these two-forms indicate that the cosmic strings appear naturally in the Lorentz spacetime, i.e. the torsion in the Riemann–Cartan spacetime is not necessary for the cosmic strings.

Faddeev formulation of general relativity (GR) is considered where the metric is composed of ten vector fields or a ten-dimensional tetrad. Upon partial use of the field equations, this theory results in the usual GR.

Earlier we have proposed some minisuperspace model for the Faddeev formulation where the tetrad fields are piecewise constant on the polytopes like 4-simplices or, say, cuboids into which ${ℝ}^4$ can be decomposed.

Now we study some representation of this (discrete) theory, an analogue of the Cartan–Weyl connection-type form of the Hilbert–Einstein action in the usual continuum GR.

Faddeev formulation of general relativity (GR) is considered where the metric is composed of ten vector fields or a ten-dimensional tetrad. Upon partial use of the field equations, this theory results in the usual general relativity (GR).

Earlier, we have proposed first-order representation of the minisuperspace model for the Faddeev formulation where the tetrad fields are piecewise constant on the polytopes like four-simplices or, say, cuboids into which ${ℝ}^4$ can be decomposed, an analogue of the Cartan–Weyl connection-type form of the Hilbert–Einstein action in the usual continuum GR.

In the Hamiltonian formalism, the tetrad bilinears are canonically conjugate to the orthogonal connection matrices. We evaluate the spectrum of the elementary areas, functions of the tetrad bilinears. The spectrum is discrete and proportional to the Faddeev analog ${\gamma }_{\mathrm{\text{F}}}$ of the Barbero–Immirzi parameter $\gamma$. The possibility of the tetrad and metric discontinuities in the Faddeev gravity allows to consider any surface as consisting of a set of virtually independent elementary areas and its spectrum being the sum of the elementary spectra. Requiring consistency of the black hole entropy calculations known in the literature we are able to estimate ${\gamma }_{\mathrm{\text{F}}}$.

We consider the Faddeev formulation of general relativity (GR), which can be characterized by a kind of d-dimensional tetrad (typically d = 10) and a non-Riemannian connection. This theory is invariant w.r.t. the global, but not local, rotations in the d-dimensional space. There can be configurations with a smooth or flat metric, but with the tetrad that changes abruptly at small distances, a kind of “antiferromagnetic” structure.

Previously, we discussed a first-order representation for the Faddeev gravity, which uses the orthogonal connection in the d-dimensional space as an independent variable. Using the discrete form of this formulation, we considered the spectrum of (elementary) area. This spectrum turns out to be physically reasonable just on a classical background with large connection like rotations by $\pi$, that is, with such an “antiferromagnetic” structure.

In the discrete first-order Faddeev gravity, we consider such a structure with periodic cells and large connection and strongly changing tetrad field inside the cell. We show that this system in the continuum limit reduces to a generalization of the Faddeev system. The action is a sum of related actions of the Faddeev type and is still reduced to the GR action.

In the first part of this note, we observe that a non-Riemannian piece in the affine connection (a “dark connection”) leads to an algebraically determined, conserved, symmetric 2-tensor in the Einstein field equations that is a natural dark matter candidate. The only other effect it has, is through its coupling to standard model fermions via covariant derivatives. If the local dark matter density is the result of a background classical dark connection, these Yukawa-like mass corrections are minuscule ($\sim 10^{-31}\mathrm{\text{eV}}$ for terrestrial fermions) and none of the tests of general relativity or the equivalence principle are affected. In the second part of the note, we give dynamics to the dark connection and show how it can be re-interpreted in terms of conventional dark matter particles. The simplest way to do this is to treat it as a composite field involving scalars or vectors. The (pseudo-)scalar model naturally has a perturbative shift-symmetry and leads to versions of the Fuzzy Dark Matter (FDM) scenario that has recently become popular (e.g. arXiv:1610.08297) as an alternative to WIMPs. A vector model with a ${{𝒵}}_2$-parity falls into the Planckian Interacting Dark Matter (PIDM) paradigm, introduced in arXiv:1511.03278. It is possible to construct versions of these theories that yield the correct relic density, fit with inflation and are falsifiable in the next round of CMB experiments. Our work is an explicit demonstration that the meaningful distinction is not between gravity modification and dark matter, but between theories with extra fields and those without.

We present several theories of four-dimensional gravity in the Plebanski formulation, in which the tetrads and the connections are the independent dynamical variables. We consider the relation between different versions of gravitational theories: Einsteinian, "topological," "mirror" gravities and gravity with torsion. We assume that our world, in which we live, is described by the self-dual left-handed gravity, and propose that if the Mirror World exists in Nature, then the "mirror gravity" is the right-handed antiself-dual gravity. In this connection, we give a brief review of gravi-weak unification models. In accordance with cosmological measurements, we consider the Universe with broken mirror parity. We also discuss the problems of cosmological constant and communication between visible and mirror worlds. Investigating a special version of the Riemann–Cartan space–time, which has torsion as an additional geometric property, we have shown that in the Plebanski formulation the ordinary and dual "topological" sectors of gravity, as well as the gravity with torsion, are equivalent. Equations of motion are obtained. In this context, we have also discussed a "pure connection gravity" — a diffeomorphism-invariant gauge theory of gravity. Loop Quantum Gravity is also briefly reviewed.

A connection-like objects, termed hom-connections are defined in the realm of non-commutative geometry. The definition is based on the use of homomorphisms rather than tensor products. It is shown that hom-connections arise naturally from (strong) connections in non-commutative principal bundles. The induction procedure of hom-connections via a map of differential graded algebras or a differentiable bimodule is described. The curvature for a hom-connection is defined, and it is shown that flat hom-connections give rise to a chain complex.

Let Λ ⊂ ℂ be the ℤ-module generated by 1 and , where τ is a positive real number. Let Z := ℂ/Λ be the corresponding complex torus of dimension one. Our aim here is to give a general construction of holomorphic principal Z-bundles over a complex manifold X.

Let θ₁ and θ₂ be two C^∞ real closed two-forms on X such that the Hodge type (0, 2) component of the form vanishes, and the elements in H²(X, ℂ) represented by θ₁ and θ₂ are contained in the image of H²(X, ℤ). For such a pair we construct a holomorphic principal Z-bundle over X. Conversely, given any holomorphic principal Z-bundle E_Z over X, we construct a pair of closed differential forms on X of the above type.

We present a construction transforming a general connection Γ on a fibered manifold Y → M and a classical connection Λ on its base M into a classical connection on the total space Y by means of a vertical parallelism Φ and an auxiliary linear connection Δ. The relations to the theory of gauge-natural operators are discussed.

In this work, we propose a study of geometric structures (connections, pseudo-Riemannian metrics) adapted to some fundamental problems of Differential Geometry. Then we find geometrical characteristics of some ODE or PDE of Mathematical Physics. While Sec. 1 contains the general setting, Secs. 2–5 contain our results. In Sec. 2, we introduce a Hessian structure having the same connection as the initial metric. In Sec. 3, we initiate a study on iterative 2D Hessian structures. In Sec. 4, we find pairs (metric, connection) generated by special functions. In Sec. 5, we find geometric characteristics of a PDE.

In this work we investigate a material point model (MP-model) and exploit the geometrical meaning of the "entropy form" introduced by Coleman and Owen. We show that a modification of the thermodynamical phase space (studied and exploited in numerous works) is an appropriate setting for the development of the MP-model in different physical situations. This approach allows to formulate the MP-model and the corresponding entropy form in terms similar to those of homogeneous thermodynamics. Closeness condition of the entropy form is reformulated as the requirement that admissible processes curves belong to the (extended) constitutive surfaces foliating the extended thermodynamical phase space of the model over the space X of basic variables. Extended constitutive surfaces Σ_S,κ are described as the Legendre submanifolds Σ_S of the space shifted by the flow of Reeb vector field. This shift is controlled by the entropy production function κ. We determine which contact Hamiltonian dynamical systems ξ_K are tangent to the surfaces Σ_S,κ, introduce conformally Hamiltonian systems μξ_K where conformal factor μ characterizes the increase of entropy along the trajectories. These considerations are then illustrated by applying them to the Coleman–Owen model of thermoelastic point.

We introduce geometric structures (connections, pseudo-Riemannian metrics) adapted to some fundamental problems of Differential Geometry, and find geometrical characteristics associated to equations of Mathematical Physics. Also, we introduce a geometric study of some boundary problems. Throughout this work, as main tool we employed an adequate Riemannian Hessian structure, suggested in [Int. J. Geom. Meth. Mod. Phys.7(7) (2010) 1104–1113].