R^4 terms in d = 4 from string theory

Abstract

We analyze the reduction to four dimensions of the R^4 terms which are part of the tendimensional string effective actions, both at tree level and one loop. We show that there are two independent combinations of R^4 present, at one loop, in the type IIA four dimensional effective action, which means they both have their origin in M-theory. The d = 4 heterotic effective action also has such terms. This contradicts the common belief that there is only one R^4 term in four-dimensional supergravity theories, given by the square of the Bel-Robinson tensor. In pure N = 1 supergravity this new R^4 combination cannot be directly supersymmetrized, but we show that, when coupled to a scalar chiral multiplet (violating the U(1) R-symmetry), it emerges in the action after elimination of the auxiliary fields. We then move to the extended (N = 8) supersymmetrization of this term, where no other coupling can be taken. We show that such supersymmetrization cannot be achieved at the linearized level. This is in conflict with the theory one gets after toroidal compactification of type II superstrings being N = 8 supersymmetric. We interpret this result in face of the recent claim that perturbative supergravity cannot be decoupled from string theory in d = 4, and N = 8, d = 4 supergravity is in the swampland.