Infinite-dimensional meta-conformal Lie algebras in one and two spatial dimensions

TL;DR

This paper constructs and analyzes infinite-dimensional meta-conformal Lie algebras in one and two spatial dimensions, revealing their structure as sums of Virasoro algebras and applying them to specific physical models.

Contribution

It introduces new representations of conformal Lie algebras via meta-conformal transformations and explicitly constructs their infinite-dimensional Lie algebras in 1D and 2D.

Findings

Meta-conformal transformations form new representations of conformal Lie algebras in 1D.

Infinite-dimensional Lie algebras are isomorphic to sums of Virasoro algebras in 1D and 2D.

Derived covariant two-point correlators and applied to Glauber-Ising chain.

Abstract

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.