One Loop Tests of Higher Spin AdS/CFT (1308.2337v3)

Abstract: Vasiliev's type A higher spin theories in AdS4 have been conjectured to be dual to the U(N) or O(N) singlet sectors in 3-d conformal field theories with N-component scalar fields. We compare the O(N^0) correction to the 3-sphere free energy F in the CFTs with corresponding calculations in the higher spin theories. This requires evaluating a regularized sum over one loop vacuum energies of an infinite set of massless higher spin gauge fields in Euclidean AdS4. For the Vasiliev theory including fields of all integer spin and a scalar with Delta=1 boundary condition, we show that the regularized sum vanishes. This is in perfect agreement with the vanishing of subleading corrections to F in the U(N) singlet sector of the theory of N free complex scalar fields. For the minimal Vasiliev theory including fields of only even spin, the regularized sum remarkably equals the value of F for one free real scalar field. This result may agree with the O(N) singlet sector of the theory of N real scalar fields, provided the coupling constant in the Vasiliev theory is identified as G_N ~ 1/(N-1). Similarly, consideration of the USp(N) singlet sector for N complex scalar fields, which we conjecture to be dual to the husp(2;0|4) Vasiliev theory, requires G_N ~ 1/(N+1). We also test the higher spin AdS3/CFT2 conjectures by calculating the regularized sum over one loop vacuum energies of higher spin fields in AdS3. We match the result with the O(N^0) term in the central charge of the W_N minimal models; this requires a certain truncation of the CFT operator spectrum so that the bulk theory contains two real scalar fields with the same boundary conditions.

• The paper calculates one-loop vacuum energy contributions in CFTs and matches them to higher spin field energies in AdS to test conjectured dualities.
• It shows that one-loop calculations in Type A non-minimal and minimal Vasiliev theories in AdS_4 align with free energy predictions for certain scalar field CFTs.
• The analysis extends to AdS_3/CFT_2, successfully matching one-loop corrections with the predicted central charges of W_N minimal models.

One Loop Tests of Higher Spin AdS/CFT

The paper by Giombi and Klebanov examines the conjectured dualities between higher spin theories in anti-de Sitter space (AdS) and conformal field theories (CFTs) within the framework of the AdS/CFT correspondence. The focus is on Vasiliev's higher spin theories in four-dimensional AdS space (AdS_4) and their proposed dualities with certain three-dimensional CFTs involving scalar fields.

Key Findings

1. Evaluation of One-Loop Corrections: The authors calculate the one-loop vacuum energy contribution to the free energy on a three-sphere (S^3) in the CFTs and compare it to the regularized sum of one-loop vacuum energies of the higher spin fields in Euclidean AdS_4. The main objective is to test the conjectural dualities by matching these results.
2. Scalar Boundary Conditions: Two types of Vasiliev's theories are examined:
   • Non-minimal Type A Theory: Includes integer spin fields and a scalar field with a specific boundary condition (Δ=1). The regularized sum of vacuum energies vanishes, aligning with the theoretical predictions from the CFT side, where corrections in the U(N) singlet sector of N complex scalar fields are expected to cancel.
   • Minimal Type A Theory: Considers only even spin fields. The derived regularized sum corresponds to the free energy of one free real scalar field, supporting a conjecture of duality with the O(N) singlet sector of N real scalar fields.
3. Higher Spin AdS_3/CFT_2 Checks: The paper extends the analysis to the conjectures relating higher spin theories in AdS_3 to two-dimensional minimal models. The authors compute the one-loop correction in AdS_3 and match it with the predicted central charge of the W_N minimal models, considering specific boundary conditions for the scalar fields.

Implications

• Theoretical Validations: The findings bolster confidence in the higher spin AdS/CFT dualities by elucidating one-loop tests that are consistent with the theoretical expectations of these dualities.
• Generalization Potential: The detailed methods and results for AdS_4 and AdS_3 cases further set the groundwork for exploring other such dualities in different dimensional setups or with additional field content.
• Impacts on Gauge Theory: Since some of the dual field theories involve Chern-Simons terms, the paper highlights intriguing connections between higher spin theories and gauge theories, including potential topological implications.

Future Directions

Given the complexities and potential of higher spin theories, several avenues for future research present themselves:

• Complete Actions for Vasiliev Theories: Current analyses rely on gaps in fully developed actions for Vasiliev’s higher spin theories, which are yet to be resolved. Understanding these would allow for deeper insights and fuller computations beyond one-loop corrections.
• Quantum Corrections in Other Dimensions and Models: Applying similar analytical techniques to other dimensions or vortex theories with intriguing implications on quantum gravity and holography.
• Exploration of Parity Breaking Couplings: Investigating the implications and consistency checks of bulk parity breaking interactions in relation to dual CFTs might yield interesting insights concerning duality mappings.

The paper represents a significant analytical effort to test the robustness of higher spin dualities using one-loop determinants and spectral zeta function analysis, advancing our understanding of the intricate waters of non-supersymmetric holography.