Compactified superstrings and torsion

String theories on a background manifold with torsion are constructed and investigated in the light-cone gauge with holonomy group H ⊆ SU(3). The appropriate non-linear sigma model is constructed on a hermitian manifold. The relationship between the Neveu-Schwarz-Ramond and the Green-Schwarz versions of the theory is discussed. Under the assumption that the β-function vanishes identically we show that the only viable compactification is on a manifold with zero torsion. This result is obtained from effective field considerations as well as directly from string considerations.     

<Emphasis Type="Italic">SU</Emphasis>(3)-Instantons and <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>,<Emphasis Type="Italic">Spin</Emphasis>(7)-Heterotic String Solitons

Necessary and sufficient conditions to the existence of a hermitian connection with totally skew-symmetric torsion and holonomy contained in SU(3) are given. A formula for the Riemannian scalar curvature is obtained. Non-compact solution to the supergravity-type I equations of motion with non-zero flux and non-constant dilaton is found in dimension 6. Non-conformally flat non-compact solutions to the supergravity-type I equations of motion with non-zero flux and non-constant dilaton are found in dimensions 7 and 8. A Riemannian metric with holonomy contained in G₂ arises from our considerations and Hitchin’s flow equations, which seems to be new. Compact examples of SU(3),G₂ and Spin(7) instanton satisfying the anomaly cancellation conditions are presented.     

An Efficient Numerical Solution Method for Elliptic Problems in Divergence Form

In this paper the problem $-{\rm div}(a(x,y)\nabla u)=f$ with Dirichlet boundary conditions on a square is solved iteratively with high accuracy for $u$ and $\nabla u$ using a new scheme called "hermitian box-scheme". The design of the scheme is based on a "hermitian box", combining the approximation of the gradient by the fourth order hermitian derivative, with a conservative discrete formulation on boxes of length 2$h$. The iterative technique is based on the repeated solution by a fast direct method of a discrete Poisson equation on a uniform rectangular mesh. The problem is suitably scaled before iteration. The numerical results obtained show the efficiency of the numerical scheme. This work is the extension to strongly elliptic problems of the hermitian box-scheme presented by Abbas and Croisille (J. Sci. Comput., 49 (2011), pp. 239--267).     

Representations of Hermitian Kernels¶by Means of Krein Spaces.¶II. Invariant Kernels

In this paper we study hermitian kernels invariant under the action of a semigroup with involution. We characterize those hermitian kernels that realize the given action by bounded operators on a Kre?n space. This is motivated by the GNS representation of *-algebras associated to hermitian functionals, the dilation theory of hermitian maps on C ^*-algebras, as well as others. We explain the key role played by the technique of induced Kre?n spaces and a lifting property associated to them. Received: 27 March 2000/ Accepted: 5 September 2000     

A Berger-type theorem for metric connections with skew-symmetric torsion

We prove a Berger-type theorem which asserts that if the orthogonal subgroup generated by the torsion tensor (pulled back to a point by parallel transport) of a metric connection with skew-symmetric torsion is not transitive on the sphere, then the space must be locally isometric to a Lie group with a bi-invariant metric or its symmetric dual (we assume the space to be locally irreducible). We also prove that a (simple) Lie group with a bi-invariant metric admits only two flat metric connections with skew-symmetric torsion: the two flat canonical connections. In particular, we get a refinement of a well-known theorem of Cartan and Schouten. Finally, we show that the holonomy group of a metric connection with skew-symmetric torsion on these spaces generically coincides with the Riemannian holonomy.     

Vanishing of the cosmological constant in non-Abelian Kaluza-Klein theories

We present a new approach to the unification of gravity and non-Abelian gauge fields in the framework of Kaluza-Klein theory. It consists in introducing a new connection on the (n + 4)-dimensional manifoldP (metrized principal fiber bundle). This connection is metrical, but with nonvanishing torsion. An enormous cosmological term in the Einstein equations vanishes due to this connection. The new connection simultaneously cancels Planck's mass term in the Dirac equation for the five-dimensional case. The usual interpretation of geodesic equations is still valid.     

On the role of a torsion‐like field in a scenario for the Spin Hall Effect

Starting from a field theory action that describes a Dirac fermion, we propose and analyze a model based on a low‐relativistic Pauli equation coupled to a torsion‐like term to study Spin Hall Effect (SHE). We point out a very particular connection between the modified Pauli equation and the (SHE), where what we refer to torsion as field playing an important role in the spin‐orbit (SO) coupling process. In this scenario, we present a proposal of a spin‐type current, considering the tiny contributions of torsion in connection with intrinsic anisotropy of the crystal electric field.     

Some comments on Chern-Simons gauge theory

Following M. F. Atiyah and R. Bott [AB] and E. Witten [W], we consider the space of flat connections on the trivialSU(2) bundle over a surfaceM, modulo the space of gauge transformations. We describe on this quotient space a natural hermitian line-bundle with connection and prove that if the surfaceM is now endowed with a complex structure, this line bundle is isomorphic to the determinant bundle. We show heuristically how path-integral quantisation of the Chern-Simons action yields holomorphic sections of this bundle.I.M.S. and T.R.R. supported by DOE grant DE-FG02-88ER 25066. J.W. supported by NSF Mathematical Sciences post-doctoral research scholarship 8807291     

Line bundles on super Riemann surfaces

We give the elements of a theory of line bundles, their classification, and their connections on super Riemann surfaces. There are several salient departures from the classical case. For example, the dimension of the Picard group is not constant, and there is no natural hermitian form on Pic. Furthermore, the bundles with vanishing Chern number aren't necessarily flat, nor can every such bundle be represented by an antiholomorphic connection on the trivial bundle. Nevertheless the latter representation is still useful in investigating questions of holomorphic factorization. We also define a subclass of all connections, those which are compatible with the superconformal structure. The compatibility conditions turn out to be constraints on the curvature 2-form.     

Gravity theory on Poisson manifold with R‐flux

A novel gravity theory based on Poisson Generalized Geometry is investigated. A gravity theory on a Poisson manifold equipped with a Riemannian metric is constructed from a contravariant version of the Levi‐Civita connection, which is based on the Lie algebroid of a Poisson manifold. Then, we show that in Poisson Generalized Geometry the R‐fluxes are consistently coupled with such a gravity. An R‐flux appears as a torsion of the corresponding connection in a similar way as an H‐flux which appears as a torsion of the connection formulated in the standard Generalized Geometry. We give an analogue of the Einstein‐Hilbert action coupled with an R‐flux, and show that it is invariant under both β‐diffeomorphisms and β‐gauge transformations.     

ON THE HERMITIAN OF THE HAMILTONIAN OF RADIAL EQUATION

The Hamiltonian of a radial equation is defined on a half-line,and there is a close relation between its hermitian and the boundary condition of the wave functions at the origin.If the wave functions are nonvanishing and convergent at the origin,the Hamiltonian has a one-parameter family of self-adjoint extensions which are related with the vanishness of the radial probability current at the origin.In this paper the problem on the hermitian of the Hamiltonian of a radial equation is studied systematically.Some methods for determining the parameter for the fermion moving in the magnetic monopole field are discussed.     

Spinning particles in scalar-tensor gravity

We develop a new model of a spinning particle in Brans-Dicke spacetime using a metric-compatible connection with torsion. The particle's spin vector is shown to be Fermi-parallel (by the Levi-Civita connection) along its worldline (an autoparallel of the metric-compatible connection) when neglecting spin-curvature coupling.     

Almost-Kähler Deformation Quantization

We use a natural affine connection with nontrivial torsion on an arbitrary almost-Kähler manifold which respects the almost-Kähler structure in order to construct a Fedosov-type deformation quantization on this manifold.     

Hyper-parahermitian manifolds with torsion

Necessary and sufficient conditions for the existence of a hyper-parahermitian connection with totally skew-symmetric torsion (HPKT-structure) are presented. It is shown that any HPKT-structure is locally generated by a real (potential) function. An invariant first order differential operator is defined on any almost hyper-paracomplex manifold showing that it is two-step nilpotent exactly when the almost hyper-paracomplex structure is integrable. A local HPKT-potential is expressed in terms of this operator. Examples of (locally) invariant HPKT-structures with closed as well as non-closed torsion 3-form on a class of (locally) homogeneous hyper-paracomplex manifolds (some of them compact) are constructed.     

Quantentheorie der Zeitmessung. II

The possibility is discussed that the observable time may be described by a hermitian operator, which is maximal but not hypermaximal. The special example considered regards systems having a continuous energy spectrum with a lower bound. It is shown that in this case physical states can be constructed which are elements of the domain of definition of the time operator and which approximate its eigenfunctions with arbitrary accuracy. Hence time is observable within the limits of the precision of real measuring devices. The situation is thus very similar to that of physical quantities which correspond to hypermaximal operators with continuous spectrum. This suggests that v.Neumann's axiom stating that there is a one-to-one connection between observables and the hypermaximal operators of the Hilbert space of states, is too restrictive.     

基于超长周期光纤光栅的高灵敏度扭曲传感器

利用高频CO₂激光脉冲写入的周期达数毫米的超长周期光纤光栅(ULPFG)，实验研究了这种新型ULPFG的扭曲特性，发现它的某些高阶谐振波长漂移与扭曲率之间具有良好的线性关系和方向相关性，其灵敏度可达0.2244nm/(rad/m)，是高频CO₂激光脉冲写入法写入的普通LPFG扭曲灵敏度的4倍.初步的理论分析表明，新型ULPFG横截面折变的非对称性以及导模与高阶包层模之间发生的耦合使得扭曲具有方向相关性和很高的灵敏度.基于这种ULPFG独特的扭曲特性，设计了一种可 关键词： 光纤传感 光纤光栅 2激光')" href="#">CO₂激光 扭曲测量 双折射     

Spacetime tangent bundle with torsion

It is demonstrated explicitly that the bundle connection of the Finslerspacetime tangent bundle can be made compatible with Cartan's theory of Finsler space by the inclusion of bundle torsion, and without the restriction that the gauge curvature field be vanishing. A component of the contorsion is made to cancel the contribution of the gauge curvature field to the relevant component of the bundle connection. Also, it is shown that the bundle manifold remains almost complex, and that the almost complex structure can be made to have a vanishing covariant derivative if additional conditions on the torsion are satisfied. However, the Finsler-spacetime tangent bundle remains complex only if the gauge curvature field vanishes.     

The cornerstone role of the torsion in Finslerian physical worlds

It is shown that there are no metric-compatible connections with zero torsion onproperly Finslerian, i.e. post-Riemannian, metrics. Since Finslerian connections exist on Riemannian metrics, the torsion rather than the metric becomes the object which determines whether the geometry is properly Finslerian or not. On the other hand, the solder forms and connection are determined by the torsion if the affine curvature is zero, the torsion then containing all the information about the geometric reality of spacetime. Since the metric curvature may still be Riemannian, the question arises of whether its present central role in spacetime physics is but a consequence of requiring that all the geometric content of spacetime be contained in the metric.