Abstract
We construct classical theories for scalar fields in arbitrary Carroll spacetimes that are invariant under Carrollian diffeomorphisms and Weyl transformations. When the local symmetries are gauge fixed these theories become Carrollian conformal field theories. We show that generically there are at least two types of such theories: one in which only time derivatives of the fields appear and the other in which both space and time derivatives appear. A classification of such scalar field theories in three (and higher) dimensions up to two derivative order is provided. We show that only a special case of our theories arises in the ultra-relativistic limit of a covariant parent theory.
Highlights
Of theories at the fixed points of RG flows, AdS/CFT etc
Just as the conformal algebra can be represented by conformal Killing vectors (CKV) of flat spacetimes, such as R × Sd with line element ds2 = −c2 dt2 + dΩ2d or R1,d, the Galilean and Carrollian conformal algebras can be thought of as appropriately defined conformal algebras of flat Galilei and Carroll spacetimes
The conformal algebra of Minkowski spacetime Md+1 is finite dimensional for d ≥ 2 and obtaining the corresponding Galilean or Carrollian conformal algebras via Inonu-Wigner contraction will necessarily result in finite dimensional algebras for d ≥ 2
Summary
We (re) construct the well-known conformally coupled scalar field equation of motion and its action in detail. Classical (free) scalar field theory in d = 2 + 1 dimensions in a general background with metric gμν This action (2.1) is invariant and the equation of motion (2.2) is covariant under the Weyl transformations: gμν (x). For a linear combination of R Φ and Φ to be covariant all the inhomogeneous terms in the Weyl transformation of that combination should cancel out. We use the same procedure to construct analogous equations of motion and actions for scalar fields on 3-dimensional Carroll manifolds
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