AdS description of induced higher-spin gauge theory

Abstract

We study deformations of three-dimensional large N CFTs by double-trace operators constructed from spin s single-trace operators of dimension ∆. These theories possess UV fixed points, and we calculate the change of the 3-sphere free energy δF = F UV − F IR . To describe the UV fixed point using the dual AdS 4 space we modify the boundary conditions on the spin s field in the bulk; this approach produces δF in agreement with the field theory calculations. If the spin s operator is a conserved current, then the fixed point is described by an induced parity invariant conformal spin s gauge theory. The low spin examples are QED 3 (s = 1) and the 3-d induced conformal gravity (s = 2). When the original CFT is that of N conformal complex scalar or fermion fields, the U (N ) singlet sector of the induced 3-d gauge theory is dual to Vasiliev's theory in AdS 4 with alternate boundary conditions on the spin s massless gauge field. We test this correspondence by calculating the leading term in δF for large N . We show that the coefficient of log N in δF is equal to the number of spin s − 1 gauge parameters that act trivially on the spin s gauge field. We discuss generalizations of these results to 3-d gauge theories including Chern-Simons terms and to theories where s is half-integer. We also argue that the Weyl anomaly a-coefficients of conformal spin s theories in even dimensions d, such as that of the Weyl-squared gravity in d = 4, can be efficiently calculated using massless spin s fields in AdS d+1 with alternate boundary conditions. Using this method we derive a simple formula for the Weyl anomaly a-coefficients of the d = 4 Fradkin-Tseytlin conformal higher-spin gauge fields. Similarly, using alternate boundary conditions in AdS 3 we reproduce the well-known central charge c = −26 of the bc ghosts in 2-d gravity, as well as its higher-spin generalizations.arXiv:1306.5242v3 [hep-th]

ABJ theory in the higher spin limit

“…one loop of the HS theory for the vector models at both UV and IR fixed points [17][18][19][20]. Meanwhile, the collective field method was applied to the vector models, elucidating how the HS theory can be directly reconstructed from the VM as well as providing a new perspective on the origin of the duality as a gauge phenomenon [21][22][23][24][25][26].…”

“…Here, the quantities f 2n (k, t) are defined as the expansion coefficients and their explicit expression is given by (3.16) as 18) JHEP08 (2016)174 where B n are the Bernoulli numbers. Note that f 2n (k, t) is defined so that its 1/k expansion (which is equivalent to the 1/M expansion) starts with an O(k 0 ) term.…”

“…with the boundary ChernSimons term at level k, corresponding to the gauging of the U(N ) symmetry [18,54].…”

“…• The spin-0 ghost field for the spin-1 field with the mixed boundary condition has the ∆ − = 0 boundary condition [18].…”

“…Hence the net contribution to the one-loop partition function comes solely from ζ (∆,s) . In particular, the O(ln M ) correction observed in the ABJ theory comes entirely from the ∆ − spin-0 ghosts for the spin-1 U(N ) gauge fields and the consequence of the "induced gauge symmetry" [18].…”