Gopakumar-Vafa invariants via vanishing cycles

In this paper, we propose an ansatz for defining Gopakumar-Vafa invariants of Calabi-Yau threefolds, using perverse sheaves of vanishing cycles. Our proposal is a modification of a recent approach of Kiem-Li, which is itself based on earlier ideas of Hosono-Saito-Takahashi. We conjecture that these invariants are equivalent to other curve-counting theories such as Gromov-Witten theory and Pandharipande-Thomas theory. Our main theorem is that, for local surfaces, our invariants agree with PT invariants for irreducible one-cycles. We also give a counter-example to the Kiem-Li conjectures, where our invariants match the predicted answer. Finally, we give examples where our invariant matches the expected answer in cases where the cycle is non-reduced, non-planar, or non-primitive.Comment: 63 pages, many improvements of the exposition following referee comments, final version to appear in Inventione

The Toda hierarchy and the KdV hierarchy

The Toda hierarchy of size $N$ is well known to be analogous to the KdV hierarchy at $N$ goes to infinity. This paper shows that given $f$ a periodic function, there is a canonical way of defining the initial data for the Toda lattice equations so that the evolution of this data under the Toda lattice hierarchy looks asymptotically like the evolution of $f$ under the KdV hierarchy. Further, the conserved quantities of $f$ and those of the Toda hierarchy match.Comment: AMSTe

Rotor eddy-current loss in permanent magnet brushless machines

This paper presents an analysis of the rotor eddy-current loss in modular and conventional topologies of permanent magnet brushless machine. The loss is evaluated both analytically and by time-stepped finite-element analysis, and it is shown that it can be significant in both machine topologies. It is also shown that the loss can be reduced significantly by segmenting the magnets

Ultradiscretization of the solution of periodic Toda equation

A periodic box-ball system (pBBS) is obtained by ultradiscretizing the periodic discrete Toda equation (pd Toda eq.). We show the relation between a Young diagram of the pBBS and a spectral curve of the pd Toda eq.. The formula for the fundamental cycle of the pBBS is obtained as a colloraly.Comment: 41 pages; 7 figure

Reductions of the Volterra and Toda chains

The Volterra and Toda chains equations are considered. A class of special reductions for these equations are derived.Comment: LaTeX, 6 page

Chaos in cosmological Hamiltonians

This paper summarises a numerical investigation which aimed to identify and characterise regular and chaotic behaviour in time-dependent Hamiltonians H(r,p,t) = p^2/2 + U(r,t), with U=R(t)V(r) or U=V[R(t)r], where V(r) is a polynomial in x, y, and/or z, and R = const * t^p is a time-dependent scale factor. When p is not too negative, one can distinguish between regular and chaotic behaviour by determining whether an orbit segment exhibits a sensitive dependence on initial conditions. However, chaotic segments in these potentials differ from chaotic segments in time-independent potentials in that a small initial perturbation will usually exhibit a sub- or super-exponential growth in time. Although not periodic, regular segments typically exhibit simpler shapes, topologies, and Fourier spectra than do chaotic segments. This distinction between regular and chaotic behaviour is not absolute since a single orbit segment can seemingly change from regular to chaotic and visa versa. All these observed phenomena can be understood in terms of a simple theoretical model.Comment: 16 pages LaTeX, including 5 figures, no macros require

Dynamics of broken symmetry nodal and anti-nodal excitations in Bi_{2} Sr_{2} CaCu_{2} O_{8+\delta} probed by polarized femtosecond spectroscopy

The dynamics of excitations with different symmetry is investigated in the superconducting (SC) and normal state of the high-temperature superconductor Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\delta}$ (Bi2212) using optical pump-probe (Pp) experiments with different light polarizations at different doping levels. The observation of distinct selection rules for SC excitations, present in A$_{{\rm 1g}}$ and B$_{{\rm 1g}}$ symmetries, and for the PG excitations, present in A$_{{\rm 1g}}$ and B$_{{\rm 2g}}$ symmetries, by the probe and absence of any dependence on the pump beam polarization leads to the unequivocal conclusion of the existence of a spontaneous spatial symmetry breaking in the pseudogap (PG) state

Integrable Discretizations of Chiral Models

A construction of conservation laws for chiral models (generalized sigma-models on a two-dimensional space-time continuum using differential forms is extended in such a way that it also comprises corresponding discrete versions. This is achieved via a deformation of the ordinary differential calculus. In particular, the nonlinear Toda lattice results in this way from the linear (continuum) wave equation. The method is applied to several further examples. We also construct Lax pairs and B\"acklund transformations for the class of models considered in this work.Comment: 14 pages, Late

Remarks on the Collective Quantization of the SU(2) Skyrme Model

We point out the question of ordering momentum operator in the canonical \break quantization of the SU(2) Skyrme Model. Thus, we suggest a new definition for the momentum operator that may solve the infrared problem that appears when we try to minimize the Quantum Hamiltonian.Comment: 8 pages, plain tex, IF/UFRJ/9

Generalized DPW method and an application to isometric immersions of space forms

Let $G$ be a complex Lie group and $\Lambda G$ denote the group of maps from the unit circle ${\mathbb S}^1$ into $G$, of a suitable class. A differentiable map $F$ from a manifold $M$ into $\Lambda G$, is said to be of \emph{connection order $(_a^b)$} if the Fourier expansion in the loop parameter $\lambda$ of the ${\mathbb S}^1$-family of Maurer-Cartan forms for $F$, namely F_\lambda^{-1} \dd F_\lambda, is of the form $\sum_{i=a}^b \alpha_i \lambda^i$. Most integrable systems in geometry are associated to such a map. Roughly speaking, the DPW method used a Birkhoff type splitting to reduce a harmonic map into a symmetric space, which can be represented by a certain order $(_{-1}^1)$ map, into a pair of simpler maps of order $(_{-1}^{-1})$ and $(_1^1)$ respectively. Conversely, one could construct such a harmonic map from any pair of $(_{-1}^{-1})$ and $(_1^1)$ maps. This allowed a Weierstrass type description of harmonic maps into symmetric spaces. We extend this method to show that, for a large class of loop groups, a connection order $(_a^b)$ map, for $a<0<b$, splits uniquely into a pair of $(_a^{-1})$ and $(_1^b)$ maps. As an application, we show that constant non-zero curvature submanifolds with flat normal bundle of a sphere or hyperbolic space split into pairs of flat submanifolds, reducing the problem (at least locally) to the flat case. To extend the DPW method sufficiently to handle this problem requires a more general Iwasawa type splitting of the loop group, which we prove always holds at least locally.Comment: Some typographical correction