Poincare Series, 3D Gravity and CFT Spectroscopy

TL;DR

This work develops a constructive framework to probe how modular invariance constrains the spectrum of 2D CFTs by building modular-invariant partition functions via Poincaré sums over SL(2, Z). The authors show that one can generate spectra with controlled gaps, including the possibility of gaps as large as (c−1)/12, and they analyze the gravitational interpretation in AdS3/CFT2, including the BTZ black hole states and the fate of pure gravity. They demonstrate that the naive pure gravity partition function obtained from a Poincaré sum yields pathologies (continuous spectrum and negative density) that can be cured by subleading, intrinsically quantum corrections, suggesting a richer gravitational path integral than the classical saddles alone. The work also connects to the modular bootstrap, deriving bounds on the lowest non-vacuum primary and showing that full SL(2, Z) invariance cannot tighten these bounds beyond 2ξ; together, these results illuminate the landscape of consistent holographic CFTs and the role of modular symmetry in shaping their spectra.

Abstract

Modular invariance strongly constrains the spectrum of states of two dimensional conformal field theories. By summing over the images of the modular group, we construct candidate CFT partition functions that are modular invariant and have positive spectrum. This allows us to efficiently extract the constraints on the CFT spectrum imposed by modular invariance, giving information on the spectrum that goes beyond the Cardy growth of the asymptotic density of states. Some of the candidate modular invariant partition functions we construct have gaps of size (c-1)/12, proving that gaps of this size and smaller are consistent with modular invariance. We also revisit the partition function of pure Einstein gravity in AdS3 obtained by summing over geometries, which has a spectrum with two unphysical features: it is continuous, and the density of states is not positive definite. We show that both of these can be resolved by adding corrections to the spectrum which are subleading in the semi-classical (large central charge) limit.

Figures (2)

• Figure 1: The censored region $\mathcal{P}$ with ${\xi}=\frac{c-1}{24}$, and the shifted energy and spin $e=\Delta+{\bar{\Delta}}$, $j=\Delta-{\bar{\Delta}}$.
• Figure 2: The space of modular invariant partition functions, plotting the conformal weight $\Delta_1$ of the lowest primary against the central charge $c$ of the theory. The red region is ruled out by conformal bootstrap methods. For the green region we construct explicit examples of partition functions. The status of the white wedge between is still an open question.