Abstract
We discuss a particular non-relativistic limit of NS-NS gravity that can be taken at the level of the action and equations of motion, without imposing any geometric constraints by hand. This relies on the fact that terms that diverge in the limit and that come from the Vielbein in the Einstein-Hilbert term and from the kinetic term of the Kalb-Ramond two-form field cancel against each other. This cancelling of divergences is the target space analogue of a similar cancellation that takes place at the level of the string sigma model between the Vielbein in the kinetic term and the Kalb-Ramond field in the Wess-Zumino term. The limit of the equations of motion leads to one equation more than the limit of the action, due to the emergence of a local target space scale invariance in the limit. Some of the equations of motion can be solved by scale invariant geometric constraints. These constraints define a so-called Dilatation invariant String Newton-Cartan geometry.
Highlights
Of the NR string worldsheet action [16, 17].3 The case of a NR open string in a curved background has been discussed recently in [19, 20]
In [5], the elements of SNC geometry were derived from a NR limit of General Relativity that closely reproduces a formulation of SNC geometry that is obtained from gaugings of underlying ‘String Bargmann or String Newton-Cartan’ space-time symmetry algebras [13]
The emerging dilatation invariance that is present in the NR string worldsheet actions and that we have shown to be preserved in the NR limit of NS-NS gravity, hints that Dilatation invariant String Newton-Cartan (DSNC) geometry is a natural target space geometry of NR string theory
Summary
The worldsheet action for the NR bosonic string in a generic background was derived in [4, 5], by taking a NR limit of the relativistic Polyakov string action, coupled to an arbitrary target space background. In [5], the elements of SNC geometry were derived from a NR limit of General Relativity that closely reproduces a formulation of SNC geometry that is obtained from gaugings of underlying ‘String Bargmann or String Newton-Cartan’ space-time symmetry algebras [13] These symmetry algebras contain a non-central extension ZA, whose corresponding gauge field is given by mμA. This ZA symmetry was argued to be a symmetry of the NR string action, in case the target space SNC geometry obeys the zero torsion constraint (1.2) [5]. When fixing mμA = 0, the bμν field, that can not be viewed as a gauge field of String Bargmann or String Newton-Cartan symmetries, transforms non-trivially under Galilean boosts and becomes part of the fields of SNC geometry. We do not wish to impose the zero torsion constraint and we will not necessarily have the ZA symmetry
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