Conjecture on Hidden Superconformal Symmetry of N=4 Supergravity

TL;DR

This work contends that the perturbative UV finiteness of pure $N=4$ supergravity at three loops can be understood as a manifestation of a hidden local $N=4$ superconformal symmetry. By embedding the theory in the $SU(2,2|4)$ conformal framework with six conformal compensator vector multiplets, the authors show that no consistent higher-derivative $N=4$ superconformal invariants exist that could generate dangerous $R^4$-type counterterms, and they connect this to the vanishing of the $3$-loop divergence found by Bern et al. The analysis further argues that consistent local anomalies are not available in the $N=4$ case, providing a potential mechanism for UV finiteness beyond standard Poincaré supergravity. The discussion includes a detailed gauge-fixing path to pure $N=4$ Poincaré supergravity, the explicit CSF model arising from conformal gauge choices, and a critical examination of higher-dimensional analogs (notably $D=6$) to test the robustness and falsifiability of the conjecture.

Abstract

We argue that the observed UV finiteness of the 3-loop extended supergravities may be a manifestation of a hidden local superconformal symmetry of supergravity. We focus on the SU(2,2|4) dimensionless superconformal model. In Poincare gauge where the compensators are fixed to phi^2= 6 M_P^2 this model becomes a pure classical N=4 Einstein supergravity. We argue that in N=4 the higher-derivative superconformal invariants like phi^{-4}W^2 \bar W^2 and the consistent local anomaly delta (ln phi W^2) are not available. This conjecture on hidden local N=4 superconformal symmetry of Poincare supergravity may be supported by subsequent loop computations.