On anomaly free 4d $\mathcal{N}$=4 and 6d (2,0) conformal supergravities and UV finiteness of Poincar\'e supergravities

TL;DR

This paper reviews superconformal anomalies in 4d and 6d supergravity theories, showing anomaly cancellation conditions and implications for UV finiteness of related Poincaré supergravities.

Contribution

It establishes anomaly cancellation conditions and predicts divergence behaviors in 4d and 6d Poincaré supergravity theories based on superconformal anomaly analysis.

Findings

Anomalies cancel for specific numbers of multiplets: N_v=4 in 4d and N_T=26 in 6d.

Divergences in 4d PSG with n_v vectors are proportional to n_v+2.

Divergences in 6d PSG with n_T tensors are proportional to n_T-21.

Abstract

We review the structure of superconformal anomalies in 4d $\mathcal N$=4 conformal supergravity (CSG) coupled to a number N$_\rm v$ of $ \mathcal N$=4 vector multiplets and 6d (2,0) CSG coupled to N$_{_{\rm T}}$ of (2,0) tensor multiplets. Anomalies cancel if N$_\rm v$=4 and N$_{_{\rm T}}$=26 respectively. If the CSG part of the action is dropped and N$_{\rm v}$=6+ n$_{\rm v}$, the first theory is classically equivalent to the 4d $\mathcal N$=4 Poincar\'e supergravity (PSG) coupled to n$_{\rm v}$ vector multiplets, while the second one with N$_{_{\rm T}}$=5+ n$_{_{\rm T}}$ is classically equivalent to the 6d (2,0) PSG coupled to n$_{\rm T}$ tensor multiplets. We argue that these facts imply that divergences in the 4d PSG with n$_{\rm v}$ vectors should be proportional to n$_{\rm v}$+2 and similarly in the 6d PSG with n$_{_{\rm T}}$ tensors to n$_{_{\rm T}}$-21. These predictions appear to be consistent with known results of explicit scattering amplitude computations in these 4d and 6d PSG theories.