### Abstract

Vasiliev's type A higher spin theories in AdS_{4} have been conjectured to be dual to the U(N) or O (N^{0}) 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 AdS_{4}. For the Vasiliev theory including fields of all integer spin and a scalar with Δ = 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 AdS _{3}/CFT_{2} conjectures by calculating the regularized sum over one loop vacuum energies of higher spin fields in AdS_{3}. 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.

Original language | English (US) |
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Article number | 68 |

Journal | Journal of High Energy Physics |

Volume | 2013 |

Issue number | 12 |

DOIs | |

State | Published - Dec 2013 |

### All Science Journal Classification (ASJC) codes

- Nuclear and High Energy Physics

### Keywords

- 1/N expansion
- AdS-CFT correspondence
- Models of quantum gravity