Supergravity one-loop corrections on AdS_7 and AdS_3, higher spins and AdS/CFT

TL;DR

The paper analyzes one-loop corrections to vacuum energy and a-anomalies in AdS/CFT setups beyond the well-studied AdS5 case. Using general higher-spin formulas in AdS7 and KK decompositions for AdS7×S^4 and AdS3×S^3×M^4, it shows that one-loop contributions reproduce minus the tensor multiplet (6d) or minus the corresponding free multiplet (2d) data, yielding precise subleading terms: $a_{(2,0)} = 4 N^{3} - \frac{9}{4} N - \frac{7}{4}$ and $E_c$ relations in 6d, and $E_c = -\frac{1}{2}$ with $c = 6(Q_1 Q_5 + 1)$ in 2d. The authors derive explicit expressions for the Casimir energy and a-anomaly for generic higher-spin fields in AdS$_7$, and show that vectorial AdS$_7$/CFT$_6$ tests require specific shifts of the inverse coupling $G^{-1}$ to match various boundary singleton sectors. They also extend the analysis to AdS$_3$ and demonstrate that the subleading central charge piece in the D1-D5 system is recovered from the one-loop 10d supergravity on AdS$_3\times S^3\times M^4$, consolidating a broader framework for quantum checks of AdS/CFT beyond leading order. Collectively, the work provides general higher-spin tools for AdS$_7$ and concrete one-loop verifications of AdS/CFT in both 6d and 2d boundary theories, with implications for vectorial dualities and the role of tensor multiplets.

Abstract

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.