Quaternionic Line Research Articles

On the Quaternion Transformation and Field Equations in Curved Space-Time

In this paper, we use four-dimensional quaternionic algebra to describe space-time geodesics in curvature form. The transformation relations of a quaternionic variables are established with the help of basis-transformations of quaternion algebra. We deduce the quaternionic covariant derivative that explains how the quaternion components vary with scalar and vector fields. The quaternionic metric tensor and the geodesic equation are also discussed to describe the quaternionic line element in curved space-time. We examine a quaternionic metric tensor equation for the Riemannian Christoffel curvature tensor. We present the quaternionic Einstein’s field-like equation, which indicates that quaternionic matter and geometry are equivalent. Relevance of the work:In recent decades, hypercomplex algebra, viz., quaternion and octonions, has been widely used to explain various branches of physics. In this way, we have investigated quaternionic transformations and field equations in curved space-time. The present novel work will help to explain the characteristics of the curved space-time universe in terms of quaternion algebra. It can also be used to describe quaternionic gravitational waves, the black hole formulation, and so on.

Pullbacks and nontriviality of associated noncommutative vector bundles

Our main theorem is that the pullback of an associated noncommutative vector bundle induced by an equivariant map of quantum principal bundles is a noncommutative vector bundle associated via the same finite-dimensional representation of the structural quantum group. At the level of K0-groups, we realize the induced map by the pullback of explicit matrix idempotents. We also show how to extend our result to the case when the quantum-group representation is infinite dimensional, and then apply it to the Ehresmann–Schauenburg quantum groupoid. Finally, using noncommutative Milnor’s join construction, we define quantum quaternionic projective spaces together with noncommutative tautological quaternionic line bundles and their duals. As a key application of the main theorem, we show that these bundles are stably non-trivial as noncommutative complex vector bundles.

Dirac’s magnetic monopole and the Kontsevich star product

We examine relationships between various quantization schemes for an electrically charged particle in the field of a magnetic monopole. Quantization maps are defined in invariant geometrical terms, appropriate to the case of nontrivial topology, and are constructed for two operator representations. In the first setting, the quantum operators act on the Hilbert space of sections of a nontrivial complex line bundle associated with the Hopf bundle, whereas the second approach uses instead a quaternionic Hilbert module of sections of a trivial quaternionic line bundle. We show that these two quantizations are naturally related by a bundle morphism and, as a consequence, induce the same phase-space star product. We obtain explicit expressions for the integral kernels of star-products corresponding to various operator orderings and calculate their asymptotic expansions up to the third order in the Planck constant . We also show that the differential form of the magnetic Weyl product corresponding to the symmetric ordering agrees completely with the Kontsevich formula for deformation quantization of Poisson structures and can be represented by Kontsevich’s graphs.

Kreck-Stolz invariants for quaternionic line bundles

We generalise the Kreck-Stolz invariants s 2 s_2 and s 3 s_3 by defining a new invariant, the t t -invariant, for quaternionic line bundles E E over closed spin-manifolds M M of dimension 4 k − 1 4k-1 with H 3 ( M ; Q ) = 0 H^3(M; \mathbb Q) = 0 such that c 2 ( E ) ∈ H 4 ( M ) c_2(E)\in H^4(M) is torsion. The t t -invariant classifies closed smooth oriented 2 2 -connected rational homology 7 7 -spheres up to almost-diffeomorphism, that is, diffeomorphism up to a connected sum with an exotic sphere. It also detects exotic homeomorphisms between such manifolds. The t t -invariant also gives information about quaternionic line bundles over a fixed manifold, and we use it to give a new proof of a theorem of Feder and Gitler about the values of the second Chern classes of quaternionic line bundles over H P k \mathbb H P^k . The t t -invariant for S 4 k − 1 S^{4k-1} is closely related to the Adams e e -invariant on the ( 4 k − 5 ) (4k-5) -stem.

A [formula omitted]-bundle associated with a harmonic map from a Riemann surface into a sphere

A tt⁎-bundle is constructed by a harmonic map from a Riemann surface into an n-dimensional sphere. This tt⁎-bundle is a high-dimensional analogue of a quaternionic line bundle with a Willmore connection. For the construction, a flat connection is decomposed into four parts by a fiberwise complex structure.

Universal vector bundle over the reals

Let X_R be a geometrically irreducible smooth projective curve, defined over R, such that X_R does not have any real points. Let X= X_R\times_R C be the complex curve. We show that there is a universal real algebraic line bundle over X_R x Pic^d(X_R)$ if and only if $\chi(L)$ is odd for L in Pic^d(X_R)$. There is a universal quaternionic algebraic line bundle over X x Pic^d(X) if and only if the degree d is odd. Take integers r and d such that r > 1, and d is coprime to r. Let M_{X_R}(r,d) (respectively, M_X(r,d)$) be the moduli space of stable vector bundles over X_R (respectively, X) of rank r and degree d. We prove that there is a universal real algebraic vector bundle over X_R x M_{X_R}(r,d) if and only if \chi(E) is odd for E in M_{X_R}(r,d). There is a universal quaternionic vector bundle over X x M_X(r,d) if and only if the degree d is odd. The cases where X_R is geometrically reducible or X_R has real points are also investigated.

The spectral curve of a quaternionic holomorphic line bundle over a 2-torus

A conformal immersion of a 2-torus into the 4-sphere is characterized by an auxiliary Riemann surface, its spectral curve. This complex curve encodes the monodromies of a certain Dirac type operator on a quaternionic line bundle associated to the immersion. The paper provides a detailed description of the geometry and asymptotic behavior of the spectral curve. If this curve has finite genus the Dirichlet energy of a map from a 2-torus to the 2-sphere or the Willmore energy of an immersion from a 2-torus into the 4-sphere is given by the residue of a specific meromorphic differential on the curve. Also, the kernel bundle of the Dirac type operator evaluated over points on the 2-torus linearizes in the Jacobian of the spectral curve. Those results are presented in a geometric and self contained manner.

A product formula for valuations on manifolds with applications to the integral geometry of the quaternionic line

The Alesker–Poincaré pairing for smooth valuations on manifolds is expressed in terms of the Rumin differential operator acting on the cosphere-bundle. It is shown that the derivation operator, the signature operator and the Laplace operator acting on smooth valuations are formally self-adjoint with respect to this pairing. As an application, the product structure of the space of \mathrm{SU}(2) - and translation invariant valuations on the quaternionic line is described. The principal kinematic formula on the quaternionic line ℍ is stated and proved.

Homotopy type of gauge groups of quaternionic line bundles over spheres

Let P be a principal S 3 -bundle over a sphere S n , with n ⩾ 4 . Let G P be the gauge group of P. The homotopy type of G P when n = 4 was studied by A. Kono in [A. Kono, A note on the homotopy type of certain gauge groups, Proc. Roy. Soc. Edinburgh Sect. A 117 (1991) 295–297]. In this paper we extend his result and we study the homotopy type of the gauge group of these bundles for all n ⩽ 25 .

The denominators of Lagrangian surfaces in complex Euclidean plane

A quotient of two linearly independent quaternionic holomorphic sections of a quaternionic holomorphic line bundle over a Riemann surface is a conformal branched immersion from a Riemann surface to four-dimensional Euclidean space. On the assumption that a quaternionic holomorphic line bundle is associated with a Lagrangian-branched immersion from a Riemann surface to complex Euclidean plane, we shall classify the denominators of Lagrangian-branched immersion from a Riemann surface to complex Euclidean plane.

The Geometry of Polygons in \open R5and Quaternions

We consider the moduli space \(\mathcal{M}\)r of polygons with fixed side lengths in five-dimensional Euclidean space. We analyze the local structure of its singularities and exhibit a real-analytic equivalence between \(\mathcal{M}\)r and a weighted quotient of n-fold products of the quaternionic projective line \(\mathbb{H}\mathbb{P}\)1 by the diagonal PSL(2, \(\mathbb{H}\))-action. We explore the relation between \(\mathcal{M}\)r and the fixed point set of an anti-symplectic involution on a GIT quotient Grℂ(2, 4)n/SL(4, ℂ). We generalize the Gel'fand—MacPherson correspondence to more general complex Grassmannians and to the quaternionic context, and realize our space \(\mathcal{M}\)r as a quotient of a subspace in the quaternionic Grassmannian Grℍ(2, n) by the action of the group Sp(1)n. We also give analogues of the Gel'fand—Tsetlin coordinates on the space of quaternionic Hermitean marices and briefly describe generalized action—angle coordinates on \(\mathcal{M}\)r.

Quaternionic holomorphic geometry: Pl�cker formula, Dirac eigenvalue estimates and energy estimates of harmonic 2-tori

The paper develops the fundamentals of quaternionic holomorphic curve theory. The holomorphic functions in this theory are conformal maps from a Riemann surface into the 4-sphere, i.e., the quaternionic projective line. Basic results such as the Riemann-Roch Theorem for quaternionic holomorphic vector bundles, the Kodaira embedding and the Pluecker relations for linear systems are proven. Interpretations of these results in terms of the differential geometry of surfaces in 3- and 4-space are hinted at throughout the paper. Applications to estimates of the Willmore functional on constant mean curvature tori, respectively energy estimates of harmonic 2-tori, and to Dirac eigenvalue estimates on Riemannian spin bundles in dimension 2 are given.

Grassmannians, calibrations and 5-brane intersections

We present a geometric construction of a newclass of hyper-Kähler manifolds with torsion. This involvesthe superposition of the four-dimensional hyper-Kählergeometry with torsion associated with the NS-5-brane along quaternionic planes inℍk. We find the moduli space of these geometries and showthat it can be constructed using the bundle space of thecanonical quaternionic line bundle over a quaternionic projective space.We also investigate several special cases which are associatedwith certain classes of quaternionic planes in ℍk. We then show that the eight-dimensional geometries we have found can be constructed using quaternioniccalibrations. We generalize our construction to superpose the same four-dimensionalhyper-Kähler geometry with torsion along complex planes inℂ2k.We find that the resulting geometry is Kähler with torsion.The moduli space of these geometries is also investigated.In addition, the applications of these new geometries to M-theoryand sigma models are presented.In particular, we find new solutions of IIA supergravity with the interpretation ofintersecting NS-5-branes at {Sp}(2)-angles on a string and show that they preserve 3/32, 1/8, 5/32and 3/16of supersymmetry. We also show that two-dimensional sigma modelswith target spaces above the manifolds have (p,q) extendedsupersymmetry.

Octonionic Yang–Mills instanton on quaternionic line bundle of Spin(7) holonomy

The total space of the spinor bundle on the four-dimensional sphere S 4 is a quaternionic line bundle that admits a metric of Spin(7) holonomy. We consider octonionic Yang–Mills instanton on this eight-dimensional gravitational instanton. This is a higher dimensional generalization of (anti-) self-dual instanton on the Eguchi-Hanson space. We propose an ansatz for Spin(7) Yang–Mills field and derive a system of non-linear ordinary differential equations. The solutions are classified according to the asymptotic behavior at infinity. We give a complete solution when the gauge group is reduced to a product of SU(2) subalgebras in Spin(7). The existence of more general Spin(7) valued solutions can be seen by making an asymptotic expansion.

Quaternionic modular groups

Matrices whose entries belong to certain rings of algebraic integers are known to be associated with discrete groups of transformations of inversive n-space or hyperbolic ( n+1)-space H n+1 . In particular, groups operating in the hyperbolic plane or hyperbolic 3-space may be represented by 2×2 matrices whose entries are rational integers or real or imaginary quadratic integers. The theory is extended here to groups operating in H 4 or H 5 and matrices over one of the three basic systems of quaternionic integers. Quaternionic modular groups are shown to be subgroups of the rotation groups of regular honeycombs of H 4 and H 5. For four-dimensional groups the division ring of quaternions is treated as a Clifford algebra. Results in hyperbolic 5-space derive from the homeomorphism of inversive 4-space and the quaternionic projective line.

Stably extendible vector bundles over the quaternionic projective spaces

We show that, if a quaternionic k-dimensional vector bundle l' over the quaternionic projective space Hpn is stably extendible and its non-zero top Pontrjagin class is not zero mod 2, then l' is stably equivalent to the Whitney sum of k quaternionic line bundles provided k S; n.

The quaternionic geometry of four-dimensional conformal field theory

We show that four-dimensional conformal field theory is most naturally formulated on Kulkarni 4-folds, i.e. real 4-folds endowed with an integrable quaternionic structure. This leads to a formalism that parallels very closely that of two-dimensional conformal field theory on Riemann surfaces. In this framework, the notion of Fueter analyticity, the quaternionic analog of complex analyticity, plays an essential role. Conformal fields appear as sections of appropriate either harmonic real or Fueter holomorphic quaternionic line bundles. In the free case, the field equations are statements of either harmonicity or Fueter holomorphicity of the relevant conformal fields. We obtain compact quaternionic expressions of such basic objects as the energy-momentum tensor and the gauge currents for some basic models in terms of Kulkarni geometry. We also find a concise expression of the conformal anomaly and a quaternionic four-dimensional analog of the Schwarzian derivative describing the covariance of the quantum energy-momentum tensor. Finally, we analyze the operator product expansions of free fields.

Norms and determinants of quaternionic line bundles

We investigate the relationship between the determinants det (P) and norms N ( P ) of right modules P of rank one over associative composition algebras C. We show: \( {\rm det} (P) \cong N(P) \otimes N(P) \) if C is a quaternion algebra, and \( {\rm det}\,(P) \cong N(P) \otimes {\rm det}\, (C)\) if C is a quadratic etale algebra.

Super multi-instantons in conformal chiral superspace.

We reformulate self-dual supersymmetric theories directly in conformal chiral superspace, where superconformal invariance is manifest. The superspace can be interpreted as the generalization of the usual Atiyah-Drinfel'd-Hitchin-Manin (ADHM) twistors (the quaternionic projective line), the real projective light cone in six dimensions, or harmonic superspace, but can be reduced immediately to four-dimensional chiral superspace. As an exsample, we give the 't Hooft and ADHM multi-instanton constructions for self-dual super Yang-Mills theory. In both cases, all the parameters are represented as a single, irreducible, constant tensor.

HyperK�hler and quaternionic K�hler geometry

 A quaternion-Hermitian manifold, of dimension at least 12, with closed fundamental 4-form is shown to be quaternionic Kahler. A similar result is proved for 8-manifolds. HyperKahler metrics are constructed on the fundamental quaternionic line bundle (with the zero-section removed) of a quaternionic Kahler manifold (indefinite if the scalar curvature is negative). This construction is compatible with the quaternionic Kahler and hyperKahier quotient constructions and allows quaternionic Kahler geometry to be subsumed into the theory of hyperKahler manifolds. It is shown that the hyperKahler metrics that arise admit a certain type of SU (2)- action, possess functions which are Kahler potentials for each of the complex structures simultaneously and determine quaternionic Kahler structures via a variant of the moment map construction. Quaternionic Kahler metrics are also constructed on the fundamental quaternionic line bundle and a twistor space analogy leads to a construction of hyperKahler metrics with circle actions on complex line bundles over Kahler-Einstein (complex) contact manifolds. Nilpotent orbits in a complex semi-simple Lie algebra, with the hyperKahler metrics defined by Kronheimer, are shown to give rise to quaternionic Kahler metrics and various examples of these metrics are identified. It is shown that any quaternionic Kahler manifold with positive scalar curvature and sufficiently large isometry group may be embedded in one of these manifolds. The twistor space structure of the projectivised nilpotent orbits is studied.