Quillen superconnections and connections on supermanifolds

Given a supervector bundle $E=E_0\oplus E_1\to M$, we exhibit a parametrization of Quillen superconnections on $E$ by graded connections on the Cartan–Koszul supermanifold $(M,\Omega \left(M\right))$. The relation between the curvatures of both kind of connections, and their associated Chern classes, is discussed in detail. In particular, we find that Chern classes for graded vector bundles on split supermanifolds can be computed through the associated Quillen superconnections.     

On the Quillen determinant

We explain the bundle structures of the Determinant line bundle and the Quillen determinant line bundle considered on the connected component of the space of Fredholm operators including the identity operator in an intrinsic way. Then we show that these two are isomorphic and that they are non-trivial line bundles and trivial on some subspaces. Also we remark a relation of the Quillen determinant line bundle and the Maslov line bundle.     

Quillen metrics and perturbed equations

Letters in Mathematical Physics - We come up with infinite-dimensional prequantum line bundles and moment map interpretations of three different sets of equations—the generalised...     

Mathai–Quillen Formalism — From the Point of View of Path Integrals

We present here a path integral derivation of Mathai–Quillen formalism which gives a new interpretation of the Gauss–Bonnet–Chern theorem and Hopf–Poincaré theorem.     

Canonical connections of Finsler metrics and Finslerian connections on Riemannian metrics

A concept of canonical connection of a Finsler metric is developed. Connections that are compatible with Finsler metrics are compared with the canonical connection itself. They are also compared with the corresponding Cartan connection. A necessary and sufficient condition on metric Finsler connections is given for the metric to be Riemannian. This study unearths different ways in which Finsler geometry could be used to generalize the theory of general relativity.     

Operator-valued connections,Lie connections,and Gauge field theory

Noether's first theorem tells us that the global symmetry groupG _r of an action integral is a Lie group of point transformations that acts on the Cartesian product of the space-time manifold with the space of states and their derivatives. Gauge theory constructs are thus required for symmetry groups that act indiscriminately on the independent and dependent variables where the group structure can not necessarily be realized as a subgroup of the general linear group. Noting that the Lie algebra of a general symmetry groupG _r can be realized as a Lie algebrag _r of Lie derivatives on an appropriately structured manifold,G _r -covariant derivatives are introduced through study of connection 1-forms that take their values in the Lie algebrag _r of Lie derivatives (operator-valued connections). This leads to a general theory of operator-valued curvature 2-forms and to the important special class of Lie connections. The latter are naturally associated with the minimal replacement and minimal coupling constructs of gauge theory when the symmetry groupG _r is allowed to act locally. Lie connections give rise to the gauge fields that compensate for the local action ofG _r in a natural way. All governing field equations and their integrability conditions are derived for an arbitrary finite dimensional Lie group of symmetries. The case whereG _r contains the ten-parameter Poincaré group on a flat space-timeM ₄ is considered. The Lorentz structure ofM ₄ is shown to give a pseudo-Riemannian structure of signature 2 under the minimal replacement associated with the Lie connection of the local action of the Poincaré group. Field equations for the matter fields and the gauge fields are given for any system of matter fields whose action integral is invariant under the global action of the Poincaré group.     

Invariant connections and vortices

We study the vortex equations on a line bundle over a compact Kähler manifold. These are a generalization of the classical vortex equations over ². We first prove an invariant version of the theorem of Donaldson, Uhlenbeck and Yau relating the existence of a Hermitian-Yang-Mills metric on a holomorphic bundle to the stability of such a bundle. We then show that the vortex equations are a dimensional reduction of the Hermitian-Yang-Mills equation. Using this fact and the theorem above we give a new existence proof for the vortex equations and describe the moduli space of solutions.     

Quantum phases and connections

Quantum phases and possible reductions of adiabatic connections are discussed. It is argued that only holonomy groups have physical meaning in that context.     

Bilagrangian connections and the reduction of Lax equations

We obtain the Lax equations associated with a dynamical system endowed with a bilagrangian connection and a closed two-form parallel along the dynamical field . The case of Lagrangian dynamical systems is analysed and the nonnoether constants of motion found by Hojman and Harleston are recovered as being associated to a reduced Lax equation. Completely integrable dynamical systems are also shown to be a particular case of these systems.     

Critical principal connections and gauge-invariance

Let p:P→X be a principal G-bundle over an oriented manifold. As suggested by the classical Yang-Mills field theory, a certain class of variational problems for connections on P is defined. Global equations are derived for the corresponding critical connections and for the Jacobi fields along a critical connection, which are a very adequate tool for the study of the symplectic theory associated with the said variational problems.     

Flat connections and geometric quantization

Using the space of holomorphic symmetric tensors on the moduli space of stable bundles over a Riemann surface we construct a projectively flat connection on a vector bundle over Teichmüller space. The fibre of the vector bundle consists of the global sections of a power of the determinant bundle on the moduli space. Both Dolbeault and ech techniques are used.     

Flat connections and Wigner-Yanase-Dyson metrics

On the manifold of positive definite matrices, we investigate the existence of pairs of flat affine connections, dual with respect to a given monotone metric. The connections are defined either using the α-embeddings and finding the duals with respect to the metric, or by means of contrast functionals. We show that in both cases, the existence of such a pair of connections is possible if and only if the metric is given by the Wigner-Yanase-Dyson skew information.     

Linear connections on the quantum plane

A general definition has been proposed recently of a linear connection and a metric in noncommutative geometry. It is shown that to within normalization there is a unique linear connection on the quantum plane and there is no metric.     

Differential characters and cohomology of the moduli of flat connections

Let \(\pi {:}\, P\rightarrow M\) be a principal bundle and p an invariant polynomial of degree r on the Lie algebra of the structure group. The theory of Chern–Simons differential characters is exploited to define a homology map \(\chi ^{k} {:}\, H_{2r-k-1}(M)\times H_{k}({\mathcal {F}}/{\mathcal {G}})\rightarrow {\mathbb {R}}/{\mathbb {Z}}\), for \(k<r-1\), where \({\mathcal {F}} /{\mathcal {G}}\) is the moduli space of flat connections of \(\pi \) under the action of a subgroup \({\mathcal {G}}\) of the gauge group. The differential characters of first order are related to the Dijkgraaf–Witten action for Chern–Simons theory. The second-order characters are interpreted geometrically as the holonomy of a connection in a line bundle over \({\mathcal {F}}/{\mathcal {G}}\). The relationship with other constructions in the literature is also analyzed.

     

An application of system identification to flutter testing

It is shown that the equations of motion of an aeroelastic system may be derived from measured response data. The structural and aerodynamic terms are separated by analyzing response measurements at two values of kinetic pressure; the derived equations can then be used to calculate dynamic characteristics of the system at any chosen values of kinetic pressure. Two examples of the application of the analysis to the prediction of flutter characteristics are given.     

一种基于功率谱特征参量的水中目标辐射噪声非母板匹配分类识别方法

本文阐述了一种非母板匹配的水中目标辐射噪声分类识别的思想、技术路线及算法的计算机实现关键技术。该算法基于提取水中目标辐射噪声信号功率谱中的最强谱带位置、线谱位置、线谱根数、线谱平均间隔等特征参量，通过特定处理后，直接给出识别结果，不需要附带庞大数据库。经试验验证，该技术突破了传统的母板匹配目标识别的算法思想，能够适用于复杂海况，做到宽容性匹配。识别效率高，识别率达到试验预期要求。     

Existence of connections on superbundles

We show that the existence of a connection on a super vector bundle or on a principal super fibre bundle is equivalent to the vanishing of a cohomological invariant of the superbundle. This invariant is proved to vanish in the case of a De Witt base supermanifold. Finally, some examples are discussed.     

An application of coherence resonances in molecular transition identification

In ?-type three level configurations having long lived lower levels extremely sharp two photon resonances occur. We want to draw attention to the use of these resonances for distinguishing the hyperfine splitting of lower and upper set of levels of molecular transitions. A new feature in the theoretical model is that the saturator and probe beams are coupled both to transitions rendering possible the appearance of interference between the resonances.