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Asymptotically Schroedinger Space-Times: TsT Transformations and Thermodynamics

Jelle Hartong, Blaise Rollier

TL;DR

The paper systematically builds a TsT-based dictionary between asymptotically AdS$_5$ and asymptotically Schrödinger space-times in five dimensions, identifying a conformal class of Schrödinger boundaries arising from null dipole deformations. It proves that the on-shell action and black-hole thermodynamics are invariant under TsT, and uses this to derive a Hawking–Page–type phase transition in thermal global Schrödinger space-times at finite chemical potential. By analyzing the Maldacena–Martelli–Tachikawa black hole and its thermodynamics, the authors show that the phase structure mirrors that of AdS, with TsT invariance ensuring the corresponding Schrödinger counterpart. The work provides a robust framework for nonrelativistic holography and suggests future directions for well-posed variational problems, boundary stress tensors, and dual field theory interpretations of Schrödinger-phase transitions.

Abstract

We study the complete class of 5-dimensional asymptotically Schroedinger space-times that can be obtained as the TsT transform of a 5-dimensional asymptotically AdS space-time. Based on this we identify a conformal class of Schroedinger boundaries. We use a Fefferman-Graham type expansion to study the on-shell action for this class of asymptotically Schroedinger space-times and we show that its value is TsT invariant. In the second part we focus on black hole space-times and prove that black hole thermodynamics is also TsT invariant. We use this knowledge to argue that thermal global Schroedinger space-time at finite chemical potential undergoes a Hawking-Page type phase transition.

Asymptotically Schroedinger Space-Times: TsT Transformations and Thermodynamics

TL;DR

The paper systematically builds a TsT-based dictionary between asymptotically AdS and asymptotically Schrödinger space-times in five dimensions, identifying a conformal class of Schrödinger boundaries arising from null dipole deformations. It proves that the on-shell action and black-hole thermodynamics are invariant under TsT, and uses this to derive a Hawking–Page–type phase transition in thermal global Schrödinger space-times at finite chemical potential. By analyzing the Maldacena–Martelli–Tachikawa black hole and its thermodynamics, the authors show that the phase structure mirrors that of AdS, with TsT invariance ensuring the corresponding Schrödinger counterpart. The work provides a robust framework for nonrelativistic holography and suggests future directions for well-posed variational problems, boundary stress tensors, and dual field theory interpretations of Schrödinger-phase transitions.

Abstract

We study the complete class of 5-dimensional asymptotically Schroedinger space-times that can be obtained as the TsT transform of a 5-dimensional asymptotically AdS space-time. Based on this we identify a conformal class of Schroedinger boundaries. We use a Fefferman-Graham type expansion to study the on-shell action for this class of asymptotically Schroedinger space-times and we show that its value is TsT invariant. In the second part we focus on black hole space-times and prove that black hole thermodynamics is also TsT invariant. We use this knowledge to argue that thermal global Schroedinger space-time at finite chemical potential undergoes a Hawking-Page type phase transition.

Paper Structure

This paper contains 12 sections, 86 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Fefferman--Graham coordinates and asymptotically Schrödinger space-times
  3. TsT invariance
  4. The on-shell action
  5. Horizon properties
  6. Phase transitions in global Schrödinger space-time
  7. AAdS$_5$ black holes with an asymptotic null Killing vector
  8. Geometric properties of the MMT black hole
  9. Thermodynamics
  10. Phase transitions
  11. Discussion
  12. TsT transformations

Figures (1)

  • Figure 1: Specific heat and grand potential as a function of the black hole parameters $b$ and $r_H$. The green and blue shaded areas are regions where the specific heat is positive and where the grand potential is negative and positive, respectively. The red curve denotes the extremal black holes so that below this curve there are no black hole solutions. The white areas are places where the specific heat is negative and the grand potential is positive.