Partition function of free conformal fields in 3-plet representation

Simplest examples of AdS/CFT duality correspond to free CFTs in d dimensions with fields in vector or adjoint representation of an internal symmetry group dual in the large N limit to a theory of massless or massless plus massive higher spins in AdS(d+1). One may also study generalizations when conformal fields belong to higher dimensional representations, i.e. carry more than two internal symmetry indices. Here we consider the case of the 3-fundamental ("3-plet") representation. One motivation is a conjectured connection to multiple M5-brane theory: heuristic arguments suggest that it may be related to an (interacting) CFT of 6d (2,0) tensor multiplets in 3-plet representation of large N symmetry group that has an AdS 7 dual. We compute the singlet partition function Z on S-1 x Sd-1 for a free field in 3-plet representation of U(N) and analyse its novel large N behaviour. The large N limit of the low temperature expansion of Z which is convergent in the vector and adjoint cases here is only asymptotic, reflecting the much faster growth of the number of singlet operators with dimension, indicating a phase transition at very low temperature. Indeed, while the critical temperatures in the vector (T-c similar to N-gamma; gamma > 0) and adjoint (T-c similar to 1) cases are finite, we find that in the 3-plet case T-c similar to (log N)(-1), i.e. it approaches zero at large N. We discuss some details of large N solution for the eigenvalue distribution. Similar conclusions apply to higher p-plet representations of U( N) or O ( N) and also to the free p- tensor theories invariant under [U(N)](p) or [O (N)] p with p >= 3.