Nonrelativistic pulsating strings

Dibakar Roychowdhury

doi:10.1007/jhep09(2019)002

Abstract

We explore nonrelativistic (NR) pulsating string configurations over torsion Newton-Cartan (TNC) geometry having topology R × S2 and check the corresponding analytic integrability criteria following Kovacic’s algorithm. In the first part we consider pulsating strings propagating over TNC geometry whose world-sheet theory is described by relativistic CFTs. We compute conserved charges associated with the 2D sigma model and show that the classical phase space corresponding to these NR pulsating string configurations is Liouvillian integrable. Finally, we consider nonrelativisitc scaling associated with the world-sheet d.o.f. and show that the corresponding string configuration allows even simpler integrable structure.

Highlights

We start by exploring various pulsating string configurations embedded in torsion Newton Cartan (TNC) geometry with R × S2 topology [11]
We explore nonrelativistic (NR) pulsating string configurations over torsion Newton-Cartan (TNC) geometry having topology R × S2 and check the corresponding analytic integrability criteria following Kovacic’s algorithm
We show that the semi-classical pulsating string configurations defined over TNC geometry with topology R × S2 are Liouvillian intergrable in the sense of Kovacic’s algorithm

Summary

Formulation

Which is consistent with the fact that the string wraps only the compact direction ζ which results in a non-zero momentum (2.7) along the null isometry direction. The above ansatz (3.1) is consistent with the fact that the string has no winding along the null isometry direction (1.3) which results in a vanishing angular momentum along the additional compact direction (ζ) [10]. The equations of motion that readily follow from (3.2) could be formally expressed as, θ − θψ ̈ + sin θθφ − cos θφ 2κ + ψ − cos θφ. A similar analysis reveals that, after multiplying by sin θ, the equation (3.6) corresponding to φ could be expressed as a total derivative in σ0 which could be integrated further to obtain,.

Oscillation number

Example 1

Example 2

A note on the scaling limit

Summary and final remarks