Stringy Schroedinger truncations

TL;DR

This work constructs a four-parameter family of warped black strings in type IIB string theory via generalized spectral flows and infinite boosts, and embeds their three-dimensional sectors in consistent truncations to derive Schrödinger vacua and warped BTZ black holes. It demonstrates BTZ-like thermodynamics that are largely independent of warping, analyzes asymptotic symmetry algebras, and develops a suite of 3d truncations capturing the low-energy dynamics around these backgrounds. The string-theoretic spectra reveal both universal $T$-modes and propagating $X$-modes, including travelling-wave solutions with unusual boundary behavior; preliminary linear stability analyses suggest no obvious instability for key backgrounds. Overall, the paper provides tractable, string-theoretic models for finite-temperature Schrödinger holography and a framework for exploring holographic renormalization and boundary conditions in warped AdS$_3$/Schrödinger contexts.

Abstract

Motivated by the desire to better understand finite-temperature holography for three-dimensional Schroedinger spacetimes, we: i) construct a four-parameter family of warped black string solutions of type IIB supergravity and ii) find the first consistent truncations of type IIB string theory to three dimensions that admit both supersymmetric Schroedinger solutions and warped generalizations of the BTZ black hole. Our analysis reveals a number of interesting features. One is that the thermodynamic properties of all the warped black strings, as well as the asymptotic symmetry group data, are identical to those of BTZ, in an appropriate parametrization. A more striking feature is that the spectrum of linearized perturbations around the various supersymmetric Schroedinger vacua oftentimes contains modes that carry energy flux through the spacetime boundary, which are usually believed to be unstable. A preliminary analysis indicates that, at least in the case of most interest, these modes do not lead to an instability.

Figures (4)

• Figure 1: Plot of the four $s_k$ as a function of $\lambda\kappa$. Each solution can be identified by its value at $\lambda \kappa=0$. Note that $s_2$ and $s_3$ cross over at $\lambda \kappa = \frac{\sqrt{3}}{2}$.
• Figure 2: Left: plot of the four $\beta_i$ as a function of $x= \lambda \kappa$; Right: zoom near $\lambda \kappa=0$ in $\beta_1$.
• Figure 3: The region in the $\alpha - x$ plane where the weight equation has imaginary eigenvalues. Note that the graph is symmetric about the $\alpha = \frac{1}{2}$ line. The upper dotted line represents S-dual dipole theories ($\alpha=1$), where the range of imaginary weights has shrunk to zero size, yielding the crossover behaviour in Figure 1. The lower dotted line represents the NHEK truncation ($\alpha =-1$), where there is a finite range of $\kappa$ for which the weight is imaginary, as in Figure 2.
• Figure 4: The spectrum of allowed $\omega$ for $\kappa>0$ is given by the intersection of the phase function $\gamma_\eta(\omega)$, plotted in blue above, with the constant line $\gamma = \gamma_+$, plotted in red.