Character integral representation of zeta function in AdS_d+1. Part II. Application to partially-massless higher-spin gravities

We compute the one-loop free energies of the type-A_ℓ and type-B_ℓ higher-spin gravities in (d + 1)-dimensional anti-de Sitter (AdS_d+1) spacetime. For large d and ℓ, these theories have a complicated field content, and hence it is difficult to compute their zeta functions using the usual methods. Applying the character integral representation of zeta function developed in the companion paper [arXiv:1805.05646] to these theories, we show how the computation of their zeta function can be shortened considerably. We find that the results previously obtained for the massless theories (ℓ = 1) generalize to their partially-massless counterparts (arbitrary ℓ) in arbitrary dimensions.