Strings with non-relativistic conformal symmetry and limits of the AdS/CFT correspondence

Lorenzo Menculini,Troels Harmark,Ziqi Yan,Jelle Hartong,Niels A. Obers

doi:10.1007/jhep11(2018)190

Abstract

We find a Polyakov-type action for strings moving in a torsional Newton-Cartan geometry. This is obtained by starting with the relativistic Polyakov action and fixing the momentum of the string along a non-compact null isometry. For a flat target space, we show that the world-sheet theory becomes the Gomis-Ooguri action. From a target space perspective these strings are non-relativistic but their world-sheet theories are still relativistic. We show that one can take a scaling limit in which also the world-sheet theory becomes non-relativistic with an infinite-dimensional symmetry algebra given by the Galilean conformal algebra. This scaling limit can be taken in the context of the AdS/CFT correspondence and we show that it is realized by the ‘Spin Matrix Theory’ limits of strings on AdS5 × S5. Spin Matrix theory arises as non-relativistic limits of the AdS/CFT correspondence close to BPS bounds. The duality between non-relativistic strings and Spin Matrix theory provides a holographic duality of its own and points towards a framework for more tractable holographic dualities whereby non-relativistic strings are dual to near BPS limits of the dual field theory.

Highlights

In this paper is motivated by pursuing this latter route, and in particular by applying it to the realm of the AdS/CFT correspondence
The first step towards uncovering this connection was taken in ref. [5], showing that strings moving in a certain type of non-relativistic target spacetime geometry, described by a non-relativistic world-sheet action, are related to the Spin Matrix Theory (SMT) limits of the AdS/CFT correspondence
2As will be clear in section 2.1 we find in this paper that the torsional Newton-Cartan (TNC) geometry is extended with a periodic target space direction

Summary

Polyakov action for strings on TNC geometry

We consider a (d + 2)-dimensional space-time with a null isometry. One can always put the metric in the following null-reduced form ds2 = GMN dxM dxN = 2τ (du − m) + hμν dxμdxν ,. Our goal is to find an action for a closed string on the TNC geometry given by τμ, mμ and hμν For this reason, we focus on a sector of fixed null (light-cone) momentum P = 0.5 It is convenient to find a dual formulation in which the conservation of. We emphasize that the above rewriting of the world-sheet Lagrangian does not correspond to a T-duality since u is a non-compact null-direction and since we work in a sector of fixed momentum P .6. Using the η EOM which tells us that A1 = ∂1Xu and using that Xu is periodic under σ1 → σ1 + 2π, which follows from the fact that the string has no winding along the u direction before the change of variables to η and Aα We consider another form for the Lagrangian (2.6). We stress that v is not part of the TNC target space geometry but should rather be viewed as an additional target space dimension added to the xμ directions of the TNC manifold

Relation to the Gomis-Ooguri non-relativistic string action

Strings with non-relativistic world-sheet theories

Polyakov action for strings with non-relativistic world-sheet theories

GCA symmetry of the non-relativistic sigma-model

Spin Matrix theory

Discussion

A Relation to string Newton-Cartan geometry

Dictionary from string NC geometry

Findings

Scaling limit of string NC geometry sigma model