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

TL;DR

The paper resolves a long-standing puzzle in AdS$_3$/CFT$_2$ by providing a bulk mechanism for the finite-$N$ chiral primary spectrum of Sym$^N(\mathcal{M}_4)$. It proposes a grand-canonical residue framework where the finite-$N$ spectrum is obtained from a regulated sum over one-loop supersymmetric partition functions of IIB string theory on orbifolded backgrounds $(\mathrm{AdS}_3\times\mathrm{S}^3)/\mathbb{Z}_k\times\mathcal{M}_4$ and their spectral flows, with alternating signs from BPS negative modes enabling large cancellations to yield the correct finite polynomials $Z_N(y,\bar{y})$. A central technical achievement is the identification of Stokes sectors $\mathcal{S}_{y,\bar{y}}$ that determine which infinite subsets of saddles contribute in a given fugacity region, and the explicit matching of residues $\widehat{Z}_{k}^{\mu}$ to supersymmetric one-loop determinants around these orbifold geometries, including both symmetric and asymmetric orbifolds and a Gauss-law projection that enforces half-integer R-charges. The work provides a bulk explanation of the stringy exclusion principle at finite $N$ and connects to the DMVV formalism, tensionless worldsheet descriptions, and potential extensions to higher-dimensional analogies and brane-based interpretations. Overall, it offers a non-perturbative, bulk-computable account of finite-$N$ chiral primaries in AdS$_3$/CFT$_2$ and opens avenues for contour/duality explorations in holography.

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.

Figures (1)

• Figure 1: Proposal for the Stokes sectors of the BPS partition function $Z_N(y,\bar{y})$ of the IIB theory on asymptotically $\mathrm{AdS}_3 \times \mathrm{S}^3 \times \mathcal{M}_4$ backgrounds, as functions of the fugacities $y,\bar{y}$. The dotted lines denote Stokes lines and the solid lines denote anti-Stokes lines.