String Theory and String Newton-Cartan Geometry

TL;DR

The paper addresses how to formulate nonrelativistic string theory by taking a controlled $c \to \infty$ limit of General Relativity with a fluxless two-form and of relativistic string theory in curved backgrounds, yielding string Newton-Cartan geometry and a NR string action coupled to a Kalb-Ramond and a dilaton background. It develops the limit systematically, identifies the independent NR fields $(\tau_\\mu{}^A, E_\\mu{}^{A'}, m_\\mu{}^A)$ and a residual boost connection $W_{AB}{}^{A'}$, and derives the NR beta-functions that reproduce the known equations of motion for string Newton-Cartan gravity. The work also derives nonrelativistic T-duality rules as a limit of the Buscher rules, linking NR backgrounds with longitudinal spatial isometries to relativistic backgrounds with lightlike isometries and establishing a principled, first-principles NR duality framework. Altogether, the results validate a consistent NR string theory in a curved string Newton-Cartan background and illuminate the geometric and symmetry structures underlying nonrelativistic holography and DLCQ-related dualities.

Abstract

Nonrelativistic string theory is described by a sigma model with a relativistic worldsheet and a nonrelativistic target spacetime geometry, that is called string Newton-Cartan geometry. In this paper we obtain string Newton-Cartan geometry as a limit of the Riemannian geometry of General Relativity with a fluxless two-form field. We then apply the same limit to relativistic string theory in curved background fields and show that it leads to nonrelativistic string theory in a string Newton-Cartan geometry coupled to a Kalb-Ramond and dilaton field background. Finally, we use our limiting procedure to study the spacetime equations of motion and the T-duality transformations of nonrelativistic string theory. Our results reproduce the recent studies of beta-functions and T-duality of nonrelativistic string theory obtained from the microscopic worldsheet definition of nonrelativistic string theory.