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

TL;DR

The paper links UV divergences in half-maximal supergravities to superconformal anomaly cancellation, revealing parallel 4d and 6d stories. It shows that anomaly-free points occur at ${\rm N_{v}}=4$ in 4d and ${\rm N_{T}}=26$ in 6d, leading to quantum equivalences with PSGs that predict divergence scaling as ${\rm n_{v}}+2$ and ${\rm n_{T}}-21$. This framework explains known loop amplitude results and identifies finite theories: a unitary 4d PSG with ghost-like multiplets and a unitary 6d PSG with 21 tensor multiplets, the latter connected to type IIB on K3. The results motivate higher-loop checks and suggest a deep connection between anomaly cancellation and UV finiteness in these supergravity theories.

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 Poincaré 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.