Abstract
The aim of this chapter is to describe a geometrical background common to physical theories that are invariant under the Schrödinger group or under various subgroups of it arising in different contexts, as explained in details in the Preface. All these are natural groups of symmetries for Newtonian geometry. The Schrödinger–Virasoro group subsequently is introduced as yet another group of symmetries of the same geometric structures, albeit in a weaker sense. The reader not particularly skilled in classical differential geometry may skip it without major inconvenients, since it is independent of the sequel, save for a few basic definitions and formulas concerning the Schrödinger, Virasoro and Schrödinger–Virasoro groups and algebras, in particular the definition of the tensor-density Virasoro modules, the abstract definition of the Schrödinger–Virasoro algebra in one space-dimension, together with its vector-field representations; but all necessary formulas have been gathered at the end of the Introduction.
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Notes
- 1.
In fact, the generator of \({H}^{2}(\mathfrak{G}\mathfrak{a}\mathfrak{l}(d); \mathbb{R})\) given above can be integrated to a group cocycle, which in turn gives the Souriau cocycle in \({H}^{1}(\mathrm{Gal}(d); \mathfrak{G}\mathfrak{a}\mathfrak{l}{(d)}^{{_\ast}})\); this cocycle is then naturally interpreted in the mechanical framework as a first integral called momentum, which gives precisely the total mass of the system. For details on the geometrical and physical arguments, see the book by Souriau [121], pp. 148-154. For generalities on the cohomological computations, see [47], Sect. 2.3.
- 2.
On an analytic level, one should distinguish between the heat equation (\(\mathcal{M}\) real) and the Schrödinger equation (\(\mathcal{M}\) imaginary). On an algebraic level though, this distinction is irrelevant. The present convention, used everywhere except in Chaps. 8–10, prevents unwanted \(\sqrt{ -1}\)-factors.
- 3.
In conformal field theory (see Chap. 6), quantized fields \({({X}_{m})}_{m\in \mathbb{Z}}\) such that \([{L}_{n},{X}_{m}] = ((\mu - 1)n - m){X}_{n+m}\) are called primary operators of weight μ. In our language, the components of such a field generates the tensor-density module \({\mathcal{F}}_{\mu -1}\).
- 4.
A Fréchet Lie group is a group endowed with an infinite-dimensional manifold structure, locally modelled on a Fréchet space (i.e. a complete infinite-dimensional vector space equipped with a countable family of semi-norms), such that the multiplication and inverse mappings are differentiable with respect to this differentiable structure. Groups of diffeomorphisms of differentiable manifolds are typical examples of such structures (see [47], Chap. 4). Here the Fréchet-Lie group structure of Diff(S 1) is induced by its C ∞-topology, i.e by the topology of uniform convergence for the functions representing the vector fields in the Lie algebra and all their derivatives.
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Unterberger, J., Roger, C. (2012). Geometric Definitions of \(\mathfrak{s}\mathfrak{v}\) . In: The Schrödinger-Virasoro Algebra. Theoretical and Mathematical Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22717-2_1
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