In this review article we discuss four recent methods for computing Maurer-Cartan structure equations of symmetry groups of differential equations. Examples include solution of the contact equivalence problem for linear hyperbolic equations and finding a contact transformation between the generalized Hunter-Saxton equation and the Euler-Poisson equation.
We introduce a definition of symmetry generating vector fields on manifolds which are equipped with a first-order reductive Cartan geometry. We apply this definition to a number of physically motivated examples and show that our newly introduced notion of symmetry agrees with the usual notions of symmetry of affine, Riemann-Cartan, Riemannian and Weizenböck geometries, which are conventionally used as spacetime models. Further, we discuss the case of Cartan geometries which can be used to model observer space instead of spacetime. We show which vector fields on an observer space can be interpr…
First-order general relativity in $n$ dimensions ($n \geq 3$) has an internal gauge symmetry that is the higher-dimensional generalization of three-dimensional local translations. We report the extension of this symmetry for $n$-dimensional f(R) gravity with torsion in the Cartan formalism. The new symmetry arises from the direct application of the converse of Noether's second theorem to the action principle of f(R) gravity with torsion. We show that infinitesimal diffeomorphisms can be written as a linear combination of the new internal gauge symmetry, local Lorentz transformations, and terms…
We establish several relationships between the non-relativistic conformal symmetries of Newton-Cartan geometry and the Schrödinger equation. In particular we discuss the algebra $\mathfrak{sch}(d)$ of vector fields conformally-preserving a flat Newton-Cartan spacetime, and we prove that its curved generalisation generates the symmetry group of the covariant Schrödinger equation coupled to a Newtonian potential and generalised Coriolis force. We provide intrinsic Newton-Cartan definitions of Killing tensors and conformal Schrödinger-Killing tensors, and we discuss their respective links to cons…
An explicit AdS/CFT correspondence is shown for the Lie group $SO(4,2)$. The Lie symmetry structures allow for the construction of two physical theories through the tools of Cartan geometry. One is a gravitational theory that has anti-de Sitter symmetry. The other is also a gravitational theory but is conformally symmetric and lives on 8-dimensional biconformal space. These "extra" four dimensions have the degrees of freedom used to construct a Yang-Mills theory. The two theories, based on AdS or conformal symmetry, have a natural correspondence in the context of their Lie algebras alone where…
We realize the simple Lie superalgebra G(3) as supersymmetry of various geometric structures, most importantly super-versions of the Hilbert-Cartan equation (SHC) and Cartan's involutive PDE system that exhibit G(2) symmetry. We provide the symmetries explicitly and compute, via the first Spencer cohomology groups, the Tanaka-Weisfeiler prolongation of the negatively graded Lie superalgebras associated with two particular choices of parabolics. We discuss non-holonomic superdistributions with growth vector (2|4,1|2,2|0) deforming the flat model SHC, and prove that the second Spencer cohomology…
There are six different mathematical formulations of the symmetry group in quantum mechanics, among them the set of pure states $\mathbf{P}$ -- i.e., the set of one-dimensional projections on a complex Hilbert space $H$ -- and the orthomodular lattice $\mathbf{L}$ of closed subspaces of $H$. These six groups are isomorphic when the dimension of $H$ is $\geq 3$. Despite of the difficulties caused by $M_2(\mathbb{C})$, rank two algebras are used for quantum mechanics description of the spin state of spin-$\frac12$ particles, there is a counterexample for Uhlhorn's version of Wigner's theorem for…
In this paper we question the status of TEGR, the Teleparallel Equivalent of General Relativity,as a gauge theory of translations. We observe that TEGR (in its usual translation-gauge view) does not seem to realize the generally admitted requirements for a gauge theory for some symmetry group $G$: namely it does not present a mathematical structure underlying the theory which relates to a principal $G$-bundle and the choice of a connection on it (the gauge field). We point out that, while it is usually presented as absent, the gauging of the Lorentz symmetry is actually present in the theory, …
Definitions of non-relativistic conformal transformations are considered both in the Newton-Cartan and in the Kaluza-Klein-type Eisenhart/Bargmann geometrical frameworks. The symmetry groups that come into play are exemplified by the cosmological, and also the Newton-Hooke solutions of Newton's gravitational field equations. It is shown, in particular, that the maximal symmetry group of the standard cosmological model is isomorphic to the 13-dimensional conformal-Newton-Cartan group whose conformal-Bargmann extension is explicitly worked out. Attention is drawn to the appearance of independent…
Let $\check{X}_0$ be a semi-flat Calabi-Yau manifold equipped with a Lagrangian torus fibration $\check{p}:\check{X}_0 \rightarrow B_0$. We investigate the asymptotic behavior of Maurer-Cartan solutions of the Kodaira-Spencer deformation theory on $\check{X}_0$ by expanding them into Fourier series along fibres of $\check{p}$ over a contractible open subset $U\subset B_0$, following a program set forth by Fukaya in 2005. We prove that semi-classical limits (i.e. leading order terms in asymptotic expansions) of the Fourier modes of a specific class of Maurer-Cartan solutions naturally give rise…
In this article we present the algebraic rearrangement, or matrix inversion of the Dirac equation in a curved Riemann-Cartan spacetime with torsion, the presence of non-vanishing torsion is implied by the intrinsic spin-1/2 of the Dirac field. We then demonstrate how the inversion leads to a reformulation of the fully non-linear and self-interactive Einstein-Cartan-Dirac field equations in terms of Dirac bilinears. It has been known for some decades that the Dirac equation for charged fermions interacting with an electromagnetic field can be algebraically inverted, so as to obtain an explicit …
This is a continuation of a programme, initiated in Part I [arXiv:1706.05682], of geometrisation, compatible with the SUSY present, of the Green-Schwarz $(p+2)$-cocycles coupling to the topological charges carried by $p$-branes on reductive homogeneous spaces of SUSY groups described by GS(-type) super-$σ$-models. In the present part, higher-geometric realisations of the various SUSYs - both global and local - of these field theories are studied at length in the form of - respectively - families of gerbe isomorphisms indexed by the global-SUSY group and equivariant structures with respect to S…
In this paper, we show the compatibility of the so-called "dressing field method", which allows a systematic reduction of gauge symmetries, with the inclusion of diffeomorphisms in the BRST algebra of a gauge theory. The robustness of the scheme is illustrated on two examples where Cartan connections play a significant role. The former is General Relativity, while the latter concerns the second-order conformal structure where one ends up with a BRST algebra handling both the Weyl residual symmetry and diffeomorphisms of spacetime. We thereby provide a geometric counterpart to the BRST cohomolo…
Group contractions and its consequences upon representations of different spatial symmetry groups
本文研究了SO（3）->E_2，SO（3，1）->G_3，SO（5-h，h）->P（3，1）（h=1或2）等各种时空群的群压缩方法及其对这些群表示的影响。遵循严格的量子力学程序，我们特别注意在收缩G -> G '中采用的渐近极限程序，不仅注意相应的代数，而且注意由Cartan子代数的本征向量和标记这些表示空间的本征值所跨越的表示空间。在适当的情况下，对收缩过程给出了物理解释。
