Generalized Newton–Cartan geometries for particles and strings

Abstract

We discuss the generalized Newton-Cartan geometries that can serve as gravitational background fields for particles and strings. In order to enable us to define affine connections that are invariant under all the symmetries of the structure group, we describe torsionful geometries with independent torsion tensors. A characteristic feature of the non-Lorentzian geometries we consider is that some of the torsion tensors are so-called `intrinsic torsion' tensors. Setting some components of these intrinsic torsion tensors to zero leads to constraints on the geometry. For both particles and strings, we discuss various such constraints that can be imposed consistently with the structure group symmetries. In this way, we reproduce several results in the literature.

p -brane Galilean and Carrollian geometries and gravities

“…More details about this for the case of strings, i.e. p = 1, leading to the notion of an absolute worldsheet, can be found in [10]. We can summarize the above geometrical constraints for p = 0 and p = D − 2 by giving a physics language analogue of theorem 29.…”

“…33 In ordinary Newton-Cartan geometry, the Vielbeine τ µ and eµ a of (0-brane) Galilean geometry are supplemented with an extra one-form bµ that transforms non-trivially into eµ a under local Galilean boosts according to δbµ = λ a eµ b δ ab . In the p-brane analogues of Newton-Cartan geometry that we have in mind here, the Vielbeine τµ A and eµ a of p-brane Galilean geometry ought to be supplemented with an extra (p + 1)-form bµ 0 •••µp that transforms to τµ A and eµ a under p-brane Galilean boosts; see [10] for details on the p = 1 case. Note that bµ 0 •••µp is an extra independent field and thus differs from the (p + 1)-form Ω that was considered in (5.32) and that is a wedge product of all τ A µ .…”

“…They form the natural spacetimes in which non-or ultra-relativistic p-dimensional objects propagate. For instance, the target spacetimes of non-relativistic string theory [6,7] exhibit String Newton-Cartan geometry [8][9][10], an example of p-brane non-Lorentzian geometry with p = 1. Here, we will confine ourselves to p-brane Galilean and Carrollian geometry and mostly focus on the p-brane Galilean case; although we will explain how to read off the results for p-brane Carrollian geometry from those of (D − p − 2)-brane Galilean structures (cf the formal duality map of [11,12]).…”

“…We note that the terminology diverges on the definition of (inverse) Vielbeine, as moving frames are also commonly referred to as Vielbeine, and the term inverse Vielbein is in that case reserved for the moving coframes. We opt for the other convention, as it is in line with[10].…”

Anisotropic compactification of nonrelativistic M-theory

“…These two sectors are related through (membrane) Galilean boosts. This geometry is referred to as the membrane Newton-Cartan geometry, which naturally generalizes Newton-Cartan geometry associated with the covariantization of Newtonian gravity to the higher-dimensional foliation structure [9,10] (also see [11][12][13][14] for more general p-brane Newton-Cartan geometries).…”