Conformal blocks for AdS5 singletons

TL;DR

This work provides a quantum-mechanical derivation of the conformal blocks for the singleton sector in AdS${}_5$/CFT${}_4$ holography for compactifications on $X_5\times Y_5$, retaining second-derivative terms in the BC sector and solving for the boundary $U(1)$ partition function via harmonic data. The construction yields Siegel–Narain theta-function blocks labeled by harmonic lattice classes $\beta$, organizes them under the magnetic translation group, and demonstrates their modular properties under $SL(2,\mathbb{Z})$, with the normalization matching the boundary one-loop determinant. The analysis explains how duality and spin structure of the boundary manifold $M_4$ control anomalies and invariances, and it extends to line operators through a refined Gauss-law lifting and holonomy structure. Overall, the paper makes precise the bulk-boundary correspondence for the free singleton sector and clarifies normalization, duality, and line-operator effects within a Cheeger–Simons/line-bundle framework.

Abstract

We give a simple derivation of the conformal blocks of the singleton sector of compactifications of IIB string theory on spacetimes of the form X5 x Y5 with Y5 compact, while X5 has as conformal boundary an arbitrary 4-manifold M4. We retain the second-derivative terms in the action for the B,C fields and thus the analysis is not purely topological. The unit-normalized conformal blocks agree exactly with the quantum partition function of the U(1) gauge theory on the conformal boundary. We reproduce the action of the magnetic translation group and the SL(2,Z) S-duality group obtained from the purely topological analysis of Witten. An interesting subtlety in the normalization of the IIB Chern-Simons phase is noted.