Lifshitz/Schrödinger D-p-branes and dynamical exponents
Harvendra Singh
TL;DR
This work generalizes a double-limit procedure to all black D$p$-brane backgrounds, producing Lifshitz spacetimes with a $p$-dependent dynamical exponent $a_{Lif}=2{p-6\over p-5}$ and, from boosted AdS-Bubble geometries, Schrödinger spacetimes with $a_{Sch}=2/(p-5)$. It reveals a simple mapping between the two nonrelativistic branches, $a_{Lif}=2-a_{Sch}$, realized via exchanging light-cone metric components, and connects their UV/IR structures through a Weyl rescaling that signals running boundary couplings. The paper also analyzes Lifshitz singularities, finite-temperature Lifshitz black holes, and a thermodynamic duality between distinct condensation regimes in DLCQ, suggesting two complementary holographic ensembles. Together, these results extend Lifshitz/Schrödinger realizations to the full D$p$-brane family, provide a tunable exponents framework, and offer insights into nonrelativistic holography and boundary thermodynamics. The inclusion of Kaigorodov-like Schrödinger spaces (e.g., $a=-1$ for D3) and the explicit exponent relations strengthen the potential for modeling quantum critical behavior in strongly coupled systems.
Abstract
We extend our earlier study of special double limits of `boosted' $AdS_5$ black hole solutions to include all black D$p$-branes of type II strings. We find that Lifshitz solutions can be obtained in generality, with varied dynamical exponents, by employing these limits. We then study such double limits for `boosted' D$p$-brane bubble solutions and find that the resulting non-relativistic solutions instead describe Schrödinger like spacetimes, having varied dynamical exponents. We get a simple map between these Lifshitz & Schrödinger solutions and a relationship between two types of dynamical exponents. We also discuss about the singularities of the Lifshitz solutions and an intriguing thermodynamic duality.