Algebraic structure of the bosonic string with Newtonian background

Abstract

I derive the symmetry group of the bosonic string with a non-relativistic background which takes the form of a covariant Newton-Cartan geometry. This is done by studying a Polyakov-type action obtained by performing a dimensional reduction of a 1-dimension higher relativistic background along a null Killing vector field. The constraints that arise from this construction then act as additional fields of the theory which have to be taken into account when deriving the symmetries of the non-relativistic bosonic string action. I confirm that the resulting symmetries form a closed group with the resulting Bargmann algebra- the central extension of the Galilean algebra, which correctly describes nonrelativistic motion of objects.