Partition Functions of Holographic Minimal Models (1106.1897v1)

Abstract: The partition function of the W_N minimal model CFT is computed in the large N 't Hooft limit and compared to the spectrum of the proposed holographic dual, a 3d higher spin gravity theory coupled to massive scalar fields. At finite N, the CFT contains additional light states that are not visible in the perturbative gravity theory. We carefully define the large N limit, and give evidence that, at N = infinity, the additional states become null and decouple from all correlation functions. The surviving states are shown to match precisely (for all values of the 't Hooft coupling) with the spectrum of the higher spin gravity theory. The agreement between bulk and boundary is partially explained by symmetry considerations involving the conjectured equivalence between the W_N algebra in the large N limit and the higher spin algebra of the Vasiliev theory.

• The paper provides a detailed computation matching partition functions from large-N W_N minimal models to those in 3D higher spin gravity, establishing their holographic equivalence.
• It employs modular S-matrix techniques and quantum dimension calculations to simplify coset theory results and highlight symmetry matching between the CFT and gravity descriptions.
• The study suggests that the W_N algebra converges to a higher spin algebra at large N, offering fresh insights into holographic duality beyond conventional supersymmetric frameworks.

Analyzing the Partition Functions of Holographic Minimal Models

The paper "Partition Functions of Holographic Minimal Models" by Gaberdiel et al. aims to bridge the gap between the partition functions of $W_N$ minimal models in conformal field theory (CFT) and their proposed holographic duals in three-dimensional (3D) higher spin gravity theories. It is a comprehensive paper of the large $N$ 't Hooft limit of these models and their implications for understanding quantum gravity through the lens of the AdS/CFT correspondence.

The primary contribution of this work is the detailed computation and comparison of the partition functions in the large $N$ limit. On the CFT side, the partition function is derived from the $W_N$ minimal model as represented by coset theories. In the large $N$ limit, these partition functions are simplified using modular $S$-matrix techniques and quantum dimension calculations. The authors demonstrate that these functions involve contributions from the symmetry algebra characterized by box-and-anti-box configurations.

On the gravity side, the paper presents an analogous computation in a 3D higher spin gravity theory. This theory, crucially, includes a tower of massless gauge fields of varying spins and massive scalar fields with specific mass configurations dictated by parameters of the underlying algebra, Vasiliev theory. The authors compute the bulk partition function by considering the contributions of higher spin gravitational fluctuations and the scalar fields' excitations.

The paper establishes a strong agreement between the CFT and the bulk gravity descriptions by associating state configurations in the CFT with particle-like excitations in the gravitational theory. The CFT's low-energy spectrum comprised of specific tableaus of finite size becomes degenerate at large $N$ but decouples appropriately when taking the holographic dual into account. This decoupling indicates a match with the pure gravity contributions only after null states in these higher spin algebras are appropriately accounted for and removed.

The claims of equivalence extend beyond spectra; they also suggest symmetry matching. The authors conjecture that the $W_N$ algebra at large $N$ converges to a higher spin algebra that underlies the bulk description, hs$[#1]\lambda$. The partition functions and symmetry requirements, therefore, provide a fertile ground for exploring the algebraic underpinnings of AdS/CFT holography, specifically in non-supersymmetric, higher spin contexts.

Looking toward future implications, exploring finite $N$ deviations could offer insights into genuinely quantum elements of the Vasiliev theory, particularly concerning black holes and other non-perturbative features. Additionally, understanding the interplay of topological string theory language and knot invariants could further bind these complementary perspectives in holography.

In conclusion, this work develops a precise mathematical framework to explore large $N$ holography involving higher spin symmetries. It provides vital algebraic insights into the holographic duality, enhancing our understanding of the correspondence beyond supersymmetric and string-theoretic paradigms, focusing instead on higher spin gravitational theories. This contributes significantly to the expanding toolkit for tackling the complexities of holographic quantum gravity theories.