Partition functions of higher spin black holes and their CFT duals (1108.2567v2)

Abstract: We find black hole solutions of D=3 higher-spin gravity in the hs[\lambda] + hs[\lambda] Chern-Simons formulation. These solutions have a spin-3 chemical potential, and carry nonzero values for an infinite number of charges of the asymptotic W_{\infty}[\lambda] symmetry. Applying a previously developed set of rules for ensuring smooth solutions, we compute the black hole partition function perturbatively in the chemical potential. At \lambda =0, 1 we compare our result against boundary CFT computations involving free bosons and fermions, and find perfect agreement. For generic \lambda\ we expect that our gravity result will match the partition function of the coset CFTs conjectured by Gaberdiel and Gopakumar to be dual to these bulk theories.

Analyzing Partition Functions in Higher Spin Black Holes and Their CFT Duals

The paper by Per Kraus and Eric Perlmutter examines black hole solutions in three-dimensional higher-spin gravity and explores their conformity with boundary Conformal Field Theories (CFTs). The paper focuses on the hs[λ] ⊕ hs[λ] Chern-Simons formulation, emphasizing spin-3 chemical potential contributions and their implications on asymptotic W [λ] symmetry charges.

Key Findings and Methodology

The authors successfully compute perturbative partition functions for these black hole solutions. The methodology involves ensuring smooth transitions with spin-3 chemical potentials by applying previously developed rules. Their calculations reveal the black hole partition function, expressed to various orders in chemical potential. The key focus is on λ = 0 and λ = 1, wherein the results are perfectly consistent with free bosonic and fermionic CFT computations.

For general λ values, the researchers propose that the partition functions derived from gravity should match those from the coset CFTs, theorized to be dual to bulk theories. This proposition is rooted in the Gaberdiel and Gopakumar conjectures for duality between minimal model coset CFTs and bulk higher-spin theories.

Technical Insights

The paper delves deep into higher-spin theory developed initially by Vasiliev, highlighting its potential in simplifying the complexities of the AdS/CFT correspondence. The authors revisit SL(3,R) Chern-Simons theory to deduce entropy in higher spin black holes. Perturbation theory plays a crucial role, with pivotal inputs from gauge invariant smooth horizons expressed in terms of Chern-Simons connection holonomies. They determine that the entropy formula obtained generalizes Cardy's formula by incorporating higher spin charges.

Results and Speculation on Dualities

The computed partition function: $$\ln Z(\tau, \alpha) = \left(\frac{i\pi k}{2\tau}\right)\left(1 - \frac{4\alpha^2}{3\tau^4} + \frac{400(\lambda - 7)\alpha^4}{27(\lambda - 4)\tau^8} - \cdots \right)$$ demonstrates agreement with CFT computations at λ = 0 and λ = 1. This result aligns gravity solutions with CFT predictions in the high temperature limit, suggesting symmetry governs the outcomes, independent of the λ value. This highlights a prospective universal matching across different λ values.

Implications and Future Directions

The paper offers robust evidence for the proposed AdS/CFT duality, specifically for theories conjectured by Gaberdiel and Gopakumar. The researchers suggest future pathways for testing these conjectures by extending CFT computations to arbitrary λ and incorporating additional chemical potentials. Moreover, there appears to be potential in decoding deeper insights into the symmetry-driven nature of this duality, hinting at universality beyond specific values.

The paper concludes with intriguing speculation, noting that for λ = 0, the theory might be equivalent to free fermions constrained by singlet conditions, thus underscoring the need for further research to examine and generalize these findings to broader asymptotic quantum field structures.

In summary, Kraus and Perlmutter provide substantial contributions toward understanding higher-spin black holes and their connections with CFT duals, laying a foundational framework for symmetry-driven inquiries into quantum gravity research.