Abstract
We show that our previous work on Galilei and Carroll gravity, apt for particles, can be generalized to Galilei and Carroll gravity theories adapted to p-branes (p = 0, 1, 2, ⋯). Within this wider brane perspective, we make use of a formal map, given in the literature, between the corresponding p-brane Carroll and Galilei algebras where the index describing the directions longitudinal (transverse) to the Galilei brane is interchanged with the index covering the directions transverse (longitudinal) to the Carroll brane with the understanding that the time coordinate is always among the longitudinal directions. This leads among other things in 3D to a map between Galilei particles and Carroll strings and in 4D to a similar map between Galilei strings and Carroll strings. We show that this formal map extends to the corresponding Lie algebra expansion of the Poincaré algebra and, therefore, to several extensions of the Carroll and Galilei algebras including central extensions. We use this formal map to construct several new examples of Carroll gravity actions. Furthermore, we discuss the symmetry between Carroll and Galilei at the level of the p-brane sigma model action and apply this formal symmetry to give several examples of 3D and 4D particles and strings in a curved Carroll background.
Highlights
Limit of general relativity, an ultra-relativistic version of gravity, called Carroll gravity, has been constructed [11]
In this work we exploited a formal relation between Galilei and Carroll symmetries, first observed in [19], to obtain new results on Carroll gravity theories and the couplings of particles and strings to a curved Carroll background by making use of known results in the Galilei case
A crucial ingredient was to consider these symmetries from a brane perspective instead of particles only
Summary
We show the details of the formal map between the Carroll and Galilei algebras mentioned in the introduction and point out how this formal map extends to the corresponding Lie algebra expansion of the Poincare algebra. In terms of the decomposed generators, the non-zero commutators of the Poincare algebra are given by [JAB, JCD] = 4η[A[C JD]B] , Both the Galilei p-brane algebra and Carroll (D − p − 2)-brane algebras are obtained through a contraction of the Poincare algebra induced by the decomposition g = V0 ⊕ V1 ,. From the above formulae it is clear that there is a formal map between the p-brane Galilei contraction and the (D − p − 2)-brane Carroll contraction given by the exchange of the longitudinal and transverse indices:. Where it is understood that HA ↔ Pa. where it is understood that HA ↔ Pa This defines the following map between the p-brane Galilei and (D − p − 2)-brane Carroll algebras [19]: p-brane Galilei ⇐===A=↔==a==⇒ (D − p − 2)-brane Carroll where it is understood that associated with the exchange of the indices there is an exchange of the longitudinal and transverse metrics with the respective signatures. We will make use of this observation to construct several new examples of (extended) Carroll gravity actions by making use of our earlier results on (extended) Galilei gravity [11, 25]
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