Supergravity one-loop corrections on AdS₇ and AdS₃, higher spins and AdS/CFT

As was shown earlier, one-loop correction in 10d supergravity on AdS_5 x S^5 corresponds to the contributions to the vacuum energy and boundary 4d conformal anomaly which are minus the values for one n=4 Maxwell supermultiplet, thus reproducing the subleading term in their N^2-1 coefficient in the dual SU(N) SYM theory. We perform similar one-loop computations in 11d supergravity on AdS_7 x S^4 and 10d supergravity on AdS_3 x S^3 x T^4. In the AdS_7 case we find that the corrections to the 6d conformal anomaly a-coefficient and the vacuum energy are again minus the ones for one (2,0) tensor multiplet, suggesting that the total a-anomaly coefficient for the dual (2,0) theory is 4 N^3 - 9/4 N - 7/4 and thus vanishes for N=1. In the AdS_3 case the one-loop correction to the vacuum energy or 2d central charge turns out to be equal to that of one free (4,4) scalar multiplet, i.e. is c=+6. This reproduces the subleading term in the central charge c= 6(Q_1 Q_5 +1) of the dual 2d CFT describing decoupling limit of D5-D1 system. We also present the expressions for the 6d anomaly a-coefficient and vacuum energy for a general-symmetry higher spin field in AdS_7 and consider their application to tests of vectorial AdS/CFT with the boundary conformal 6d theory represented by free scalars, spinors or rank 2 antisymmetric tensors.