Abstract
We explore the notion of isometries in non-Riemannian geometries. Such geometries include and generalise the backgrounds of non-relativistic string theory, and they can be naturally described using the formalism of double field theory. Adopting this approach, we first solve the corresponding Killing equations for constant flat non-Riemannian backgrounds and show that they admit an infinite-dimensional algebra of isometries which includes a particular type of supertranslations. These symmetries correspond to known worldsheet Noether symmetries of the Gomis-Ooguri non-relativistic string, which we now interpret as isometries of its non-Riemannian doubled background. We further consider the extension to supersymmetric double field theory and show that the corresponding Killing spinors can depend arbitrarily on the non-Riemannian directions, leading to “supersupersymmetries” that square to supertranslations.
Highlights
In the context of string theory, an important example of a non-relativistic limit is given by the Gomis-Ooguri string [9, 10], which can be obtained from a relativistic string in flat space using a limit that distinguishes two target space directions, with a compensating divergent B-field added to cancel the rest mass divergence of the string
We review the realisation of these non-Riemannian geometries as the target spacetime background for a string, and show how the infinite-dimensional isometry algebra leads to an infinite set of Noether symmetries on the worldsheet
The generalised metric of double field theory (DFT) provides a unified description of Riemannian and non-Riemannian geometries, and in this paper we showed how this description can be applied to the notion of Killing symmetries
Summary
After reviewing the appropriate notion of Lie derivatives in double field theory (DFT) as well as the Riemannian and non-Riemannian parametrisations of the DFT metric HAB and dilaton d, we solve the Killing equations for a generic flat nonRiemannian background. This gives rise to an infinite-dimensional set of isometries, which are algebraically similar to the supertranslations that arise in the BMS algebra of asymptotic symmetries of flat spacetime. We continue to introduce and solve the corresponding Killing spinor equations on the same flat non-Riemannian background, which leads to a supersymmetric analog of the supertranslations, or ‘supersupersymmetries’
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