Abstract
We construct a non-relativistic limit of ten-dimensional mathcal{N} = 1 supergravity from the point of view of the symmetries, the action, and the equations of motion. This limit can only be realized in a supersymmetric way provided we impose by hand a set of geometric constraints, invariant under all the symmetries of the non-relativistic theory, that define a so-called ‘self-dual’ Dilatation-invariant String Newton-Cartan geometry. The non-relativistic action exhibits three emerging symmetries: one local scale symmetry and two local conformal supersymmetries. Due to these emerging symmetries the Poisson equation for the Newton potential and two partner fermionic equations do not follow from a variation of the non-relativistic action but, instead, are obtained by a supersymmetry variation of the other equations of motion that do follow from a variation of the non-relativistic action. We shortly discuss the inclusion of the Yang-Mills sector that would lead to a non-relativistic heterotic supergravity action.
Highlights
Generalized to arbitrary backgrounds is given by a Newton-Cartan-like geometry with codimension two foliation that is characterized by the following ‘zero torsion constraint’ on the longitudinal Vielbein τμA:1
In this paper we extended our previous work on taking the NR limit of ten-dimensional NSNS gravity to the supersymmetric case
The relation between the string sigma model and the target space effective action is less direct than in the bosonic case. This had the effect that we could not independently check the two fermionic Stückelberg symmetries that we found in the target space effective action at the level of the sigma model description of the superstring
Summary
Taking all of the above into account one can show that the terms in [δQ(ε1), δQ(ε2)]S = 0 at order O(c2) vanish if and only if δDS(0) = 0 This shows that the non-relativistic action is dilatation invariant as a consequence of the divergence structure in the supersymmetry rules and the particular form of the commutator between supersymmetry and fermionic shift symmetries. The NR limit S(0) of the tendimensional N = 1 supergravity action is obtained as the leading order term in the c−2expansion of (2.1), after performing the field redefinition (2.6) This NR action S(0) is invariant under two emerging fermionic S- and T -shift symmetries (3.9), an emerging dilatation symmetry (3.10), as well as under the c → ∞ limit of the bosonic transformation rules (2.11), (2.12). Note that we can leave out the parts in (2.16) that involve η− and ρ− (whose explicit expressions are given in (2.17)) from these NR supersymmetry transformation rules as these take the form of S- and T -symmetries
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