One-loop tests of the supersymmetric higher spin AdS4/CFT3 correspondence

Abstract

We compute one loop free energy for D = 4 Vasiliev higher spin gravities based on Konstein-Vasiliev algebras hu(m; n|4), ho(m; n|4) or husp(m; n|4) and subject to higher spin preserving boundary conditions, which are conjectured to be dual to the U (N ), O(N ) or U Sp(N ) singlet sectors, respectively, of free CFTs on the boundary of AdS 4 . Ordinary supersymmetric higher spin theories appear as special cases of Konstein-Vasiliev theories, when the corresponding higher spin algebra contains OSp(N |4) as subalgebra. In AdS 4 with S 3 boundary, we use a modified spectral zeta function method, which avoids the ambiguity arising from summing over infinite number of spins. We find that the contribution of the infinite tower of bulk fermions vanishes. As a result, the free energy is the sum of those which arise in type A and type B models with internal symmetries, the known mismatch between the bulk and boundary free energies for type B model persists, and ordinary supersymmetric higher spin theories exhibit the mismatch as well. The only models that have a match are type A models with internal symmetries, corresponding to n = 0. The matching requires identification of the inverse Newton's constant G −1 N with N plus a proper integer as was found previously for special cases. In AdS 4 with S 1 × S 2 boundary, the bulk one loop free energies match those of the dual free CFTs for arbitrary m and n. We also show that a supersymmetric double-trace deformation of free CFT based on OSp(1|4) does not contribute to the O(N 0 ) free energy, as expected from the bulk.arXiv:1608.07298v1 [hep-th]

ABJ quadrality

“…First, utilizing the results derived in [33][34][35], one can check that the bulk free energy at one-loop is free of UV divergence [33][34][35]. The coefficient of the log M term also agrees with the expectation from [36], which states that each bulk spin-1 gauge fields obeying the mixed boundary condition contribute to the one loop free energy of the Vasiliev theory by −(1/2) log M .…”

“…Hence we regard our result as prediction to the on-shell action of the N = 5 Vasiliev theory in AdS 4 . As mentioned above, the second term agrees with the one-loop free energy of the Vasiliev theory on AdS 4 , which is free of logarithmic divergences [33][34][35] and equal to (−1/2) times the number of bulk spin-1 gauge fields obeying the mixed boundary condition [36]. In section 6, we summarize and discuss possible extensions of this work.…”

Character integral representation of zeta function in AdSd+1. Part I. Derivation of the general formula

“…Moreover, the CIRZ allowed to tackle the one-loop free energy computation of the stringy AdS theory dual to free matrix model CFTs in the N → ∞ limit [14,[18][19][20]. The CIRZ also proved useful in other vector model dualities: the "colored" higher-spin gravity (where the four-dimensional CIRZ was generalized to fermionic fields) [21] and the type-J higherspin theories (whose conjectured dual are free vector model based on a massless spin-j field) [22]. Some key elements of the CIRZ were also used in [23].…”

“…(4.39) By setting ǫ = +1 or −1, we obtain the formulae derived in [14] and [21] for bosonic and fermionic spectrum, respectively. To compute the secondary contribution, we need the first derivative of µ ǫ,n (z, β):…”

Twisted Flato-Fronsdal theorem for higher-spin algebras

“…Although one of the appealing features of higher-spin holography is the fact that these dualities do not require supersymetry, the four dimensional higher-spin gravity admits a supersymmetric extension with arbitrary N : see [54] for a review (as well as the recent paper [55] where several one-loop tests of these extensions were performed, together with [26] for the 6-dimensional case). Supersymmetric higher-spin algebras 7 were studied in four dimensions in [43,46,57], an analysis later extended to any dimension in [8], where it was also shown that the spectrum of these supersymmetric higher-spin theories is given by the tensor product of the direct sum of the Rac and Di singletons (possibly decorated with Chan-Paton factor, that we will not consider here).…”