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Partition functions and Casimir energies in higher spin AdSd+1/CFTd

Simone Giombi, Igor R. Klebanov, and Arkady A. Tseytlin
Phys. Rev. D 90, 024048 – Published 18 July 2014

Abstract

Recently, the one-loop free energy of higher spin (HS) theories in Euclidean AdSd+1 was calculated and matched with the order N0 term in the free energy of the large N “vectorial” scalar CFT on the Sd boundary. Here we extend this matching to the boundary theory defined on S1×Sd1, where the length of S1 may be interpreted as the inverse temperature. It has been shown that the large N limit of the partition function on S1×S2 in the U(N) singlet sector of the CFT of N free complex scalars matches the one-loop thermal partition function of the Vasiliev theory in AdS4, while in the O(N) singlet sector of the CFT of N real scalars it matches the minimal theory containing even spins only. We extend this matching to all dimensions d. We also calculate partition functions for the singlet sectors of free fermion CFTs in various dimensions and match them with appropriately defined higher spin theories, which for d>3 contain massless gauge fields with mixed symmetry. In the zero-temperature case R×Sd1 we calculate the Casimir energy in the scalar or fermionic CFT and match it with the one-loop correction in the global AdSd+1. For any odd-dimensional CFT the Casimir energy must vanish on general grounds, and we show that the HS duals obey this. In the U(N) symmetric case, we exhibit the vanishing of the regularized one-loop Casimir energy of the dual HS theory in AdSd+1. In the minimal HS theory the vacuum energy vanishes for odd d while for even d it is equal to the Casimir energy of a single conformal scalar in R×Sd1 which is again consistent with AdS/CFT, provided the minimal HS coupling constant is 1/(N1). We demonstrate analogous results for singlet sectors of theories of N Dirac or Majorana fermions. We also discuss extensions to CFTs containing Nf flavors in the fundamental representation of U(N) or O(N).

  • Received 19 April 2014

DOI:https://doi.org/10.1103/PhysRevD.90.024048

© 2014 American Physical Society

Authors & Affiliations

Simone Giombi1, Igor R. Klebanov1,2, and Arkady A. Tseytlin3,*

  • 1Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
  • 2Princeton Center for Theoretical Science, Princeton University, Princeton, New Jersey 08544, USA
  • 3Blackett Laboratory, Imperial College, London SW7 2AZ, United Kingdom

  • *Also at Lebedev Institute, Moscow, Russia.

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Vol. 90, Iss. 2 — 15 July 2014

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