General relativity histories theory II: Invariance groups

We show in detail how the histories description of general relativity carries representations of both the spacetime diffeomorphisms group and the Dirac algebra of constraints. We show that the introduction of metric-dependent equivariant foliations leads to the crucial result that the canonical constraints are invariant under the action of spacetime diffeomorphisms. Furthermore, there exists a representation of the group of generalised spacetime mappings that are functionals of the four-metric: this is a spacetime analogue of the group originally defined by Bergmann and Komar in the context of the canonical formulation of general relativity. Finally, we discuss the possible directions for the quantization of gravity in histories theory.Comment: 24 pages, submitted to Class. Quant. Gra

Equivariant Kaehler Geometry and Localization in the G/G Model

We analyze in detail the equivariant supersymmetry of the $G/G$ model. In spite of the fact that this supersymmetry does not model the infinitesimal action of the group of gauge transformations, localization can be established by standard arguments. The theory localizes onto reducible connections and a careful evaluation of the fixed point contributions leads to an alternative derivation of the Verlinde formula for the $G_{k}$ WZW model. We show that the supersymmetry of the $G/G$ model can be regarded as an infinite dimensional realization of Bismut's theory of equivariant Bott-Chern currents on K\"ahler manifolds, thus providing a convenient cohomological setting for understanding the Verlinde formula. We also show that the supersymmetry is related to a non-linear generalization (q-deformation) of the ordinary moment map of symplectic geometry in which a representation of the Lie algebra of a group $G$ is replaced by a representation of its group algebra with commutator $[g,h] = gh-hg$. In the large $k$ limit it reduces to the ordinary moment map of two-dimensional gauge theories.Comment: LaTex file, 40 A4 pages, IC/94/108 and ENSLAPP-L-469/9

Learning SO(3) Equivariant Representations with Spherical CNNs

We address the problem of 3D rotation equivariance in convolutional neural networks. 3D rotations have been a challenging nuisance in 3D classification tasks requiring higher capacity and extended data augmentation in order to tackle it. We model 3D data with multi-valued spherical functions and we propose a novel spherical convolutional network that implements exact convolutions on the sphere by realizing them in the spherical harmonic domain. Resulting filters have local symmetry and are localized by enforcing smooth spectra. We apply a novel pooling on the spectral domain and our operations are independent of the underlying spherical resolution throughout the network. We show that networks with much lower capacity and without requiring data augmentation can exhibit performance comparable to the state of the art in standard retrieval and classification benchmarks.Comment: Camera-ready. Accepted to ECCV'18 as oral presentatio

Symplectic Reduction for Semidirect Products and Central Extensions

This paper proves a symplectic reduction by stages theorem in the context of geometric mechanics on symplectic manifolds with symmetry groups that are group extensions. We relate the work to the semidirect product reduction theory developed in the 1980's by Marsden, Ratiu, Weinstein, Guillemin and Sternberg as well as some more recent results and we recall how semidirect product reduction finds use in examples, such as the dynamics of an underwater vehicle. We shall start with the classical cases of commuting reduction (first appearing in Marsden and Weinstein, 1974) and present a new proof and approach to semidirect product theory. We shall then give an idea of how the more general theory of group extensions proceeds (the details of which are given in Marsden, Misiołek, Perlmutter and Ratiu, 1998). The case of central extensions is illustrated in this paper with the example of the Heisenberg group. The theory, however, applies to many other interesting examples such as the Bott-Virasoro group and the KdV equation

Group Approach to Quantization of Yang-Mills Theories: A Cohomological Origin of Mass

New clues for the best understanding of the nature of the symmetry-breaking mechanism are revealed in this paper. A revision of the standard gauge transformation properties of Yang-Mills fields, according to a group approach to quantization scheme, enables the gauge group coordinates to acquire dynamical content outside the null mass shell. The corresponding extra (internal) field degrees of freedom are transferred to the vector potentials to conform massive vector bosons.Comment: 21 pages, LaTeX, no figures; final for

Continuous symmetry reduction and return maps for high-dimensional flows

We present two continuous symmetry reduction methods for reducing high-dimensional dissipative flows to local return maps. In the Hilbert polynomial basis approach, the equivariant dynamics is rewritten in terms of invariant coordinates. In the method of moving frames (or method of slices) the state space is sliced locally in such a way that each group orbit of symmetry-equivalent points is represented by a single point. In either approach, numerical computations can be performed in the original state-space representation, and the solutions are then projected onto the symmetry-reduced state space. The two methods are illustrated by reduction of the complex Lorenz system, a 5-dimensional dissipative flow with rotational symmetry. While the Hilbert polynomial basis approach appears unfeasible for high-dimensional flows, symmetry reduction by the method of moving frames offers hope.Comment: 32 pages, 7 figure