AdS$_3$ Quantum Gravity and Finite $N$ Chiral Primaries

TL;DR

This paper proposes a bulk holographic prescription to compute the finite N spectrum of chiral primaries in AdS3/CFT2, explaining the stringy exclusion principle through an exact sum over orbifold geometries and their spectral flows.

Contribution

It introduces a novel bulk method involving one-loop partition functions of orbifolded IIB theories to reproduce the finite N chiral primary spectrum, elucidating the stringy exclusion principle.

Findings

Exact reproduction of the finite N spectrum via orbifold sum

Verification using worldsheet tensionless limit partition functions

Explanation of the exclusion principle through large cancellations

Abstract

String theory on AdS$_3$ $\times$ S$^3$ $\times$ $\mathcal{M}_4$ provides a well-studied realization of AdS$_3$/CFT$_2$ holography, but its non-perturbative structure at finite $N \sim 1/G_N^{(3)}$ is largely unknown. A long-standing puzzle concerns the stringy exclusion principle: what bulk mechanism can reproduce the boundary expectation that the chiral primary Hilbert space of the symmetric orbifold contains only a finite number of states at finite $N$? In this work, we present a bulk prescription for computing the finite $N$ spectrum of chiral primary states in symmetric orbifolds of $\mathbb{T}^4$ or K3. We show that the integer spectrum at any $N$ is reproduced exactly by summing over one-loop supersymmetric partition functions of the IIB theory on (AdS$_3$ $\times$ S$^3$)/$\mathbb{Z}_k$ $\times$ $\mathcal{M}_4$ orbifolds and their spectral flows. Using the worldsheet in the tensionless limit, we verify that the terms appearing in our proposal coincide with the partition functions of these orbifold geometries and their asymmetric generalizations. These partition functions contribute with alternating signs due to BPS modes with negative conformal dimensions and charges in twisted sectors. The resulting alternating sum collapses via large cancellations to the finite $N$ polynomials observed in symmetric orbifold CFTs, providing a bulk explanation of the stringy exclusion principle. We identify different Stokes sectors where different infinite subsets of these geometries contribute to the path integral, and propose a classification as functions of the chemical potentials.