Aspects of Nonrelativistic Strings

Abstract

We review recent developments on nonrelativistic string theory. In flat spacetime, the theory is defined by a two-dimensional relativistic quantum field theory with nonrelativistic global symmetries acting on the worldsheet fields. This theory arises as a self-contained corner of relativistic string theory. It has a string spectrum with a Galilean dispersion relation, and a spacetime S-matrix with nonrelativistic symmetry. This string theory also gives a unitary and ultraviolet complete framework that connects different corners of string theory, including matrix string theory and noncommutative open strings. In recent years, there has been a resurgence of interest in the non-Lorentzian geometries and quantum field theories that arise from nonrelativistic string theory in background fields. In this review, we start with an introduction to the foundations of nonrelativistic string theory in flat spacetime. We then give an overview of recent progress, including the appropriate target-space geometry that nonrelativistic strings couple to. This is known as (torsional) string Newton–Cartan geometry, which is neither Lorentzian nor Riemannian. We also give a review of nonrelativistic open strings and effective field theories living on D-branes. Finally, we discuss applications of nonrelativistic strings to decoupling limits in the context of the AdS/CFT correspondence.

Generalized Newton–Cartan geometries for particles and strings

“…are each other's inverse, we will then (with slight abuse of terminology) refer to τ µ as the 'inverse time-like Vielbein' and to e a µ as the 'inverse spatial Vielbein' 14 .…”

“…13 The time-like Vielbein τ µ is also often called the 'clock form' in the literature. 14 In calling frame fields inverse Vielbeine and coframe fields Vielbeine, we conform to the physics literature. In the mathematics literature, one usually reserves the term Vielbeine for a section of the frame bundle, i.e.…”

“…The framework of NC geometry can be generalized to manifolds, in which one can describe the movement of extended objects, such as strings and branes, in a degenerate limit that is akin to a non-relativistic one. Here we will focus on so-called non-relativistic strings [7,8,45] (see also [14] for a recent review). These are obtained from relativistic strings by sending the speed of light in the directions transversal to the strings to infinity, while leaving the relativistic character of the worldsheet untouched.…”

“…that are constrained by requiring that the quantum effective action remains non-relativistic [12] and/or requiring supersymmetry [13]. For recent reviews on non-relativistic string theory and non-Lorentzian geometries with more references, see [14,15].…”

Godbillon-Vey invariants of Non-Lorentzian spacetimes and Aristotelian hydrodynamics

“…These non-Lorentzian versions of spacetime geometry are of interest both as approximations of underlying relativistic theories and as interesting theories in their own right which have lately been under intensive investigations. For recent reviews we refer to [1] for applications to non-Lorentzian particle dynamics and field theory, and to [2] for applications to non-relativistic string theory.…”