Flat Space Limit of (Higher-Spin) Cardy Formula
Max Riegler
TL;DR
The paper derives the flat-space limit of the modified (inner-horizon) Cardy formula and shows it reproduces the Galilean conformal field theory counting of flat-space cosmology microstates. By contracting the inner BTZ horizon, it obtains a finite GCFT entropy for FSCs in both Einstein gravity and flat-space chiral gravity, and it clarifies why the outer-horizon Cardy formula diverges in the flat limit. The work further extends to higher-spin charges, predicting a Cardy-like expression for FSCs with spin-3 charges via spin-3 BTZ contractions, and confirms consistency with holonomy arguments. Collectively, these results provide a microscopic, holographic understanding of FSC entropy and its higher-spin generalizations within flat-space holography and GCFTs, with potential generalizations to higher spin and broader flat-space limits.
Abstract
In this note I derive the flat space limit of the modified Cardy formula associated with inner horizons and show that it reproduces the correct Galilean conformal field theory counting of flat space cosmology microstates. l also determine the entropy of flat space cosmologies in flat space chiral gravity in this way. In addition, I derive a Cardy-like expression for flat space cosmologies with spin-3 charges and thus give a prediction for the corresponding Galilean conformal field theory counting of flat space cosmology microstates with spin-3 charges.