Relating non-relativistic string theories

TL;DR

The paper shows that non-relativistic string theories based on torsional Newton–Cartan (TNC) geometry and string Newton–Cartan (SNC) geometry are classically equivalent when the SNC longitudinal direction is an isometry. It achieves this by deriving the TNC action from null reduction with NS-NS fields, analyzing large-$c$ expansions to recover SNC, and establishing an explicit background map between the two formalisms under the isometry condition; it also extends the worldsheet limit to include $B$-field and dilaton couplings, yielding a non-relativistic worldsheet with Galilean conformal symmetry. A key finding is that the foliation constraint, previously required in SNC, can be avoided at the classical level through field redefinitions in the presence of a $B$-field, though quantum beta-function considerations may reintroduce constraints. Overall, the work clarifies the relationship between different non-relativistic string constructions and highlights their relevance to non-relativistic holography and SMT limits, while outlining directions for RR backgrounds, open strings, and quantum consistency.

Abstract

Non-relativistic string theories promise to provide simpler theories of quantum gravity as well as tractable limits of the AdS/CFT correspondence. However, several apparently distinct non-relativistic string theories have been constructed. In particular, one approach is to reduce a relativistic string along a null isometry in target space. Another method is to perform an appropriate large speed of light expansion of a relativistic string. Both of the resulting non-relativistic string theories only have a well-defined spectrum if they have nonzero winding along a longitudinal spatial direction. In the presence of a Kalb--Ramond field, we show that these theories are equivalent provided the latter direction is an isometry. Finally, we consider a further limit of non-relativistic string theory that has proven useful in the context of AdS/CFT (related to Spin Matrix Theory). In that case, the worldsheet theory itself becomes non-relativistic and the dilaton coupling vanishes.