Meta-conformal transformations are constructed as sets of time-space transformations which are not angle-preserving but contain time- and space translations, time-space dilatations with dynamical exponent ${z}=1$ and whose Lie algebras contain conformal Lie algebras as sub-algebras. They act as dynamical symmetries of the linear transport equation in $d$ spatial dimensions. For $d=1$ spatial dimensions, meta-conformal transformations constitute new representations of the conformal Lie algebras, while for $d\ne 1$ their algebraic structure is different. Infinite-dimensional Lie algebras of meta-conformal transformations are explicitly constructed for $d=1$ and $d=2$ and they are shown to be isomorphic to the direct sum of either two or three centre-less Virasoro algebras, respectively. The form of co-variant two-point correlators is derived. An application to the directed Glauber-Ising chain with spatially long-ranged initial conditions is described.
We present a relationship between the Calogero-Moser particles confined in harmonic oscillator potentials and a representation theory of the infinite dimensional Lie algebra which is a semi-direct sum of Virasoro algebra and its module. More precisely, it is a correspondence of excited states of the model and singular vectors in Verma modules over the algebra. This is found by a free field realization of the time evolution operator of the model. We investigate the Verma modules and some explicit example of singular vectors are given.
The covariant two-point functions, derived from Ward identities in direct space, can be affected by consistency problems and can become unbounded for large time- or space-separations. This difficulty arises for several extensions of dynamical scaling, for example Schrödinger-invariance, conformal Galilei invariance or meta-conformal invariance, but not for standard ortho-conformal invariance. For meta-conformal invariance in 1+1 dimensions, these difficulties can be cured by going over to a dual space and an extension of these dynamical symmetries through the construction of a new generator in…
We discuss possible notions of conformal Lie algebras, paying particular attention to graded conformal Lie algebras with $d$-dimensional space isotropy: namely, those with a $\mathfrak{co}(d)$ subalgebra acting in a prescribed way on the additional generators. We classify those Lie algebras up to isomorphism for all $d\geq 2$ following the same methodology used recently to classify kinematical Lie algebras, as deformations of the `most abelian' graded conformal algebra. We find 17 isomorphism classes of Lie algebras for $d\neq 3$ and 23 classes for $d=3$. Lie algebra contractions define a part…
By Liouville's theorem, in dimensions 3 or more conformal transformations form a finite-dimensional group, an apparent drastic departure from the 2-dimensional case. We propose a derived enhancement of the conformal Lie algebra which is an infinite-dimensional dg-Lie algebra incorporating not only symmetries but also deformations of the conformal structure. Our approach is based on (derived) deformation theory of the ambitwistor space of complex null-geodesics.
We propose leading order $α'$-corrections to the Poisson-Lie T-duality transformation rules of the metric, $B$-field, and dilaton. Based on Double Field Theory, whose corrections to this order are known, we argue that they map conformal field theories to conformal field theories. Remarkably, Born geometry plays a central role in the construction.
The method of nonlinear realizations is a convenient tool for building dynamical realizations of a Lie group, which relies solely upon structure relations of the corresponding Lie algebra. The goal of this work is to discuss advantages and limitations of the method, which is here applied to construct perfect fluid equations with conformal symmetry. Four cases are studied in detail, which include the Schrodinger group, the l-conformal Galilei group, the Lifshitz group, and the relativistic conformal group.
