Abstract
We define a `non-relativistic conformal method', based on a Schr\"odinger algebra with critical exponent z = 2, as the non-relativistic version of the relativistic conformal method. An important ingredient of this method is the occurrence of a complex compensating scalar field that transforms under both scale and central charge transformations. We apply this non-relativistic method to derive the curved space Newton-Cartan gravity equations of motion with twistless torsion. Moreover, we reproduce z = 2 Ho\v{r}ava-Lifshitz gravity by classifying all possible Schr\"odinger invariant scalar field theories of a complex scalar up to second order in time derivatives.
Highlights
Method [1,2,3,4] where one makes use of compensating multiplets that transform underconformal transformations — for an introduction see [5]
While HL gravity is rather unrelated to NC gravity as a gravitational theory, it has recently been shown [13] that HL gravity can be reformulated in the same geometric formulation as NC gravity, namely using NC geometry — see e.g. [14,15,16] for early works on the geometric structure of NC
The idea is that conformal field theories (CFT’s) of a real scalar field correspond to a class of Poincare invariants
Summary
Before discussing the non-relativistic case, it is instructive to first review the (bosonic) relativistic conformal construction. In the relativistic conformal construction the aim is to construct general Poincare invariants by using the larger conformal symmetry algebra. The idea is that conformal field theories (CFT’s) of a real scalar field correspond to a class of Poincare invariants. The converse is not necessarily true, see below
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