Non-Lorentzian IIB supergravity from a polynomial realization of SL(2, ℝ)

Abstract

We derive the action and symmetries of the bosonic sector of non-Lorentzian IIB supergravity by taking the non-relativistic string limit. We find that the bosonic field content is extended by a Lagrange multiplier that implements a restriction on the Ramond-Ramond fluxes. We show that the SL(2, ℝ) transformation rules of non-Lorentzian IIB supergravity form a novel, nonlinear polynomial realization. Using classical invariant theory of polynomial equations and binary forms, we will develop a general formalism describing the polynomial realization of SL(2, ℝ) and apply it to the special case of non-Lorentzian IIB supergravity. Using the same formalism, we classify all the relevant SL(2, ℝ) invariants. Invoking other bosonic symmetries, such as the local boost and dilatation symmetry, we show how the bosonic part of the non-Lorentzian IIB supergravity action is formed uniquely from these SL(2, ℝ) invariants. This work also points towards the concept of a non-Lorentzian bootstrap, where bosonic symmetries in non-Lorentzian supergravity are used to bootstrap the bosonic dynamics in Lorentzian supergravity, without considering the fermions.

Non-relativistic ten-dimensional minimal supergravity

“…The latter has recently appeared in [6] and been termed M2T geometry there. T-duality of the SNC type IIA would then connect to the IIB case, whose bosonic sector was obtained in [34].…”

“…while those with half-integer powers m " p3{2, ´3{2, ´9{2q change chirality. After using the equation of motion symmetry to remove divergent terms from the variation of ψ `µ acccording to (32) and hence (34), a straightforward but tedious calculation shows that the finite part of the supersymmetry variations of the fermions take the form:…”

“…In this appendix, we discuss the supersymmetry transformation of λ abcd , which was determined in (43) to have the form: is defined in (33) and (34). The supersymmetry transformation δ 1 ǫ ψ ´µ is defined in (33) at order c 3 and at order c 0 is given in (155).…”