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

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${}_{{}_{\mathrm{T}}}$ of (2,0) tensor multiplets. Anomalies cancel if N$_\rm v$=4 and N${}_{{}_{\mathrm{T}}}$=26 respectively. If the CSG part of the action is dropped and N${}_{\mathrm{v}}$=6+ n${}_{\mathrm{v}}$, the first theory is classically equivalent to the 4d $\mathcal{N}$=4 Poincaré supergravity (PSG) coupled to n${}_{\mathrm{v}}$ vector multiplets, while the second one with N${}_{{}_{\mathrm{T}}}$=5+ n${}_{{}_{\mathrm{T}}}$ is classically equivalent to the 6d (2,0) PSG coupled to n${}_{\mathrm{T}}$ tensor multiplets. We argue that these facts imply that divergences in the 4d PSG with n${}_{\mathrm{v}}$ vectors should be proportional to n${}_{\mathrm{v}}$+2 and similarly in the 6d PSG with n${}_{{}_{\mathrm{T}}}$ tensors to n${}_{{}_{\mathrm{T}}}$-21. These predictions appear to be consistent with known results of explicit scattering amplitude computations in these 4d and 6d PSG theories.

20 pages. v2: references added; v3: minor corrections