A non-relativistic limit of NS-NS gravity

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.

Automated summary

The discussion focuses on a specific non-relativistic limit of NS-NS gravity that can be achieved at the level of the action and equations of motion without imposing any geometric constraints manually. The cancellation of divergences between terms that arise from the Vielbein in the Einstein-Hilbert term and from the kinetic term of the Kalb-Ramond two-form field is key to this limit. This leads to the emergence of a local target space scale invariance in the limit, resulting in an extra equation in the equations of motion compared to the action limit. The geometric constraints that can be solved by scale-invariant constraints define a Dilatation invariant String Newton-Cartan geometry.