Abstract
As was shown earlier, the one-loop correction in 10d supergravity on AdS5×S5 corresponds to the contributions to the vacuum energy and 4d boundary conformal anomaly which are minus the values for one N=4 Maxwell supermultiplet, thus reproducing the subleading term in the N2−1 coefficient in the dual SU(N) SYM theory. We perform similar one-loop computations in 11d supergravity on AdS7×S4 and 10d supergravity on AdS3×S3×T4. In the AdS7 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 4N3−9/4N−7/4 and thus vanishes for N=1. In the AdS3 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(Q1Q5+1) of the dual 2d CFT describing decoupling limit of D5–D1 system. We also present the expressions for the 6d a-anomaly coefficient and vacuum energy contributions of general-symmetry higher spin field in AdS7 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.
Highlights
5d supergravity modes, and p = 3, 4, ... for the massive KK levels
In addition to the duality examples based on AdS5 × S5 and AdS7 × S4 supergravity backgrounds there is the duality [1, 12] between string theory in AdS3 × S3 × T 4 space supported by RR 3-form flux and 2d CFT corresponding to gauge theory describing lowenergy limit D5-D1 system
In appendix A we present the expressions for the Casimir energy, a-anomaly and partition function for the fields of the free (2, 0) multiplet in 6d
Summary
Given a generic conformal field in 6d we may associate to it a field in AdS7 correspoonding to the same representation of SO(2, 6). In what follows we shall denote by K+ the two quantities a+ and Ec+ corresponding to AdS7 field in a generic massless SO(2, 6) representation and use K = −2 K+ for the associated boundary conformal field values. Starting with the case of a free conformal scalar boundary 6d theory, the corresponding fields of the dual AdS7 theory (“type A” theory) are massless totally symmetric tensors with spin s, for which we find from (2.10),(2.11). Where Kφ = (aφ, Ec φ) are the real scalar values from (A.1) and (A.5), As discussed in appendix C, the l.h.s. of (4.4) corresponds to the representation content of the tensor product of two scalar singletons and the associated sum of characters is equal to the partition function of the singlet sector of the 6d U (N ) invariant theory of N free complex scalars, see (C.5). Since NS-NS sector is common to IIB and IIA theories, the same result should be found in the corresponding IIA theory
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