Topological Excitation in Skyrme Theory

Based on the C-mapping topological current theory and the decomposition of gauge potential theory, we investigate knotted vortex lines and monopoles in Skyrme theory and simply discuss the branch processes （splitting, merging, and intersection） during the evolution of the monopoles.     

Topological Invariants of Dynamical Systems and Spaces of Holomorphic Maps: I

Departing from the symbolic dynamics, we study natural group action on spaces of holomorphic maps and complex subvarieties.     

Invariants of chaotic Hamiltonian systems

The invariants of chaotic bounded Hamiltonian systems and their relation to the solutions of the first variational equations of the equations of motion are studied. We show that these invariants are characterized by the fact that they either lose the property of differentiability as functions on phase space or that a certain formal power series defined in terms of the derivatives of the invariants has zero radius of convergence. For a specific example, we show that the former possibility appears to apply.     

Khovanov-Rozansky Homology and Topological Strings

We conjecture a relation between the sl(N) knot homology, recently introduced by Khovanov and Rozansky, and the spectrum of BPS states captured by open topological strings. This conjecture leads to new regularities among the sl(N) knot homology groups and suggests that they can be interpreted directly in topological string theory. We use this approach in various examples to predict the sl(N) knot homology groups for all values of N. We verify that our predictions pass some non-trivial checks Mathematics Subject Classifications (2000): 57M25, 57M27, 18G60, 18E30, 14J32, 14N35, 81T30, 81T45 Dedicated to the memory of F.A. Berezin     

Twisted K Theory Invariants

An invariant for twisted K theory classes on a 3-manifold is introduced. The invariant is then applied to the twisted equivariant classes arising from the supersymmetric Wess–Zumino–Witten model based on the group SU(2). It is shown that the classes defined by different highest weight representations of the loop group LSU(2) are inequivalent. The results are compatible with Freed–Hopkins–Teleman identification of twisted equivariant K theory as the Verlinde algebra.     

State-Dependent Topological Invariants and Anomalous Bulk-Boundary Correspondence in Non-Hermitian Topological Systems with Generalized Inversion Symmetry

Breakdown of bulk-boundary correspondence in non-Hermitian(NH) topological systems with generalized inversion symmetries is a controversial issue. The non-Bloch topological invariants determine the existence of edge states, but fail to describe the number and distribution of defective edge states in non-Hermitian topological systems. The state-dependent topological invariants, instead of a global topological invariant, are developed to accurately characterize the bulk-boundary correspondence of ...     

Topological Invariants for Lens Spaces and Exceptional Quantum Groups

The Reshetikhin–Turaev invariants arising from the quantum groups associated with the exceptional Lie algebras G2, F4 and E8 at odd roots of unity are constructed and explicitly computed for all the lens spaces.     

Positive Knots From Discrete Dynamical Systems Via Symbolic Dynamics

A procedure to construct positive knots motivated by symbolic dynamics is given. It is proved that the corresponding knots have a special type of positive braids, positive permutation braids. It is proved that the constructed knots are invariant under topological conjugacy, up to period five, hence they can be used to classify discrete dynamical systems. An example is given to show that topological conjugacy failed to be an invariant for closed orbits of period more than five.     

Topological Invariants of Eigenvalue Intersections and Decrease of Wannier Functions in Graphene

We investigate the asymptotic decrease of the Wannier functions for the valence and conduction band of graphene, both in the monolayer and the multilayer case. Since the decrease of the Wannier functions is characterised by the structure of the Bloch eigenspaces around the Dirac points, we introduce a geometric invariant of the family of eigenspaces, baptised eigenspace vorticity. We compare it with the pseudospin winding number. For every value $n \in \mathbb {Z}$ of the eigenspace vorticity, we exhibit a canonical model for the local topology of the eigenspaces. With the help of these canonical models, we show that the single band Wannier function $w$ satisfies $|w(x)| \le {\mathrm {const}} \cdot |x|^{-2}$ as $|x| \rightarrow \infty $ , both in monolayer and bilayer graphene.     

Topological Invariants in Terms of Green's Function for the Interacting Kitaev Chain

A one-dimensional closed interacting Kitaev chain and the dimerized version are studied. The topological invariants in terms of Green's function are calculated by the density matrix renormalization group method and the exact diagonalization method. For the interacting Kitaev chain, we point out that the calculation of the topological invariant in the charge density wave phase must consider the dimerized configuration of the ground states. The variation of the topological invariant is attributed to the poles of eigenvalues of the zero-frequency Green functions. For the interacting dimerized Kitaev chain, we show that the topological invariant defined by Green's functions can distinguish more topological nonequivalent phases than the fermion parity.     

First Integrals and Integral Invariants of Relativistic Birkhoffian Systems

For a relativistic Birkhoffian system, the first integrals and the construction of integral invariants arestudied. Firstly, the cyclic integrals and the generalized energy integral of the system are found by using the perfectdifferential method. Secondly, the equations of nonsimultaneous variation of the system are established by using therelation between the simultaneous variation and the nonsimultaneous variation. Thirdly, the relation between the firstintegral and the integral invariant of the system is studied, and it is proved that, using a first integral, we can construct anintegral invariant of the system. Finally, the relation between the relativistic Birkhoffian dynamics and the relativisticHamiltonian dynamics is discussed, and the first integrals and the integral invariants of the relativistic Hamiltoniansystem are obtained. Two examples are given to illustrate the application of the results.     

Braid theory and Zipf-Mandelbrot relation used in microparticle dynamics

A study is presented of the dynamics of a few body system of microparticles by using rank-ordering statistics in order to gain insight in the magneto-rheological properties of ferrofluids. This dynamical system is made up of micrometer sized plastic spheres dispersed in a ferrofluid driven by external magnetic fields. The world lines of the microspheres are captured and the dynamical modes are described by mathematical braid theory. Rank-ordering statistics on these modes shows a wide power law region consistent with the Zipf-Mandelbrot relation. We have also performed numerical simulations of the experimental system which show results in agreement with the observations.     

First Integrals and Integral Invariants of Relativistic Birkhoffian Systems

For a relativistic Birkhoflan system, the first integrals and the construction of integral invariants are studied. Firstly, the cyclic integrals and the generalized energy integral of the system are found by using the perfect differential method. Secondly, the equations of nonsimultaneous variation of the system are established by using the relation between the simultaneous variation and the nonsimultaneous variation. Thirdly, the relation between the first integral and the integral invariant of the system is studied, and it is proved that, using a t~rst integral, we can construct an integral invarlant of the system. Finally, the relation between the relativistic Birkhoflan dynamics and the relativistic Hamilton;an dynamics is discussed, and the first integrals and the integral invariants of the relativistic Hamiltonian system are obtained. Two examples are given to illustrate the application of the results.     

Equipartition thresholds in chains of anharmonic oscillators

We perform a detailed numerical study of the transition to equipartition in the Fermi-Pasta-Ulam quartic model and in a class of potentials of given symmetry using the normalized spectral entropy as a probe. We show that the typical time scale for the equipartition of energy among Fourier modes grows linearly with system size: this is the time scale associated with the smallest frequency present in the system. We obtain two different scaling behaviors, either with energy or with energy density, depending on the scaling of the initial condition with system size. These different scaling behaviors can be understood by a simple argument, based on the Chirikov overlap criterion. Some aspects of the universality of this result are investigated: symmetric potentials show a similar transition, regulated by the same time scale.     

Primitive Vassiliev Invariants and Factorization in Chern-Simons Perturbation Theory

The general structure of the perturbative expansion of the vacuum expectation value of a Wilson line operator in Chern-Simons gauge field theory is analyzed. The expansion is organized according to the independent group structures that appear at each order. It is shown that the analysis is greatly simplified if the group factors are chosen in a certain way that we call canonical. This enables us to show that the logarithm of a polynomial knot invariant can be written in terms of primitive Vassiliev invariants only. Received: 2 May 1996 / Accepted: 21 March 1997     

使用基于MAPPING哈密顿的对称准经典动力学方法和基于ML-MCTDH的量子动力学方法研究超快电子转移

Symmetrical quasi-classical (SQC) method based on mapping Hamiltonian is an efficient approach that is potentially useful to treat the nonadiabatic dynamics of very large systems. We try to evaluate the performance of this method in the ultrafast electron transfer processes involving a few of electronic states and a large number of vibrational modes. The multilayer multiconfigurational time-dependent Hartree (ML-MCTDH) method was used to get the accurate dynamical results for benchmark. Although the population dynamics in the longtime limit show differences in the ML-MCTDH and SQC calculations, the SQC method gives acceptable results.     

A Topological Glass

We propose and study a model with glassy behavior. The state space of the model is given by all triangulations of a sphere with n nodes, half of which are red and half are blue. Red nodes want to have 5 neighbors while blue ones want 7. Energies of nodes with other numbers of neighbors are supposed to be positive. The dynamics is that of flipping the diagonal of two adjacent triangles, with a temperature dependent probability. We show that this system has an approach to a steady state which is exponentially slow, and show that the stationary state is unordered. We also study the local energy landscape and show that it has the hierarchical structure known from spin glasses. Finally, we show that the evolution can be described as that of a rarefied gas with spontaneous generation of particles and annihilating collisions.     

Topological Geon Black Holes in Einstein-Yang-Mills Theory

We construct topological geon quotients of two families of Einstein-Yang-Mills black holes. For Künzle??s static, spherically symmetric SU(n) black holes with n?>?2, a geon quotient exists but generically requires promoting charge conjugation into a gauge symmetry. For Kleihaus and Kunz??s static, axially symmetric SU(2) black holes a geon quotient exists without gauging charge conjugation, and the parity of the gauge field winding number determines whether the geon gauge bundle is trivial. The geon??s gauge bundle structure is expected to have an imprint in the Hawking-Unruh effect for quantum fields that couple to the background gauge field.     

Topological pions

We study the collider signatures of new pions, composite particles which emerge from a TeV-scale, confining gauge theory with vector-like matter. Similar to the neutral pion in QCD, these new pions mainly decay into a pair of standard model (SM) gauge bosons via triangular anomaly diagrams. One of the new pions, which decays to a gluon plus a photon, has excellent discovery potential at the LHC.