Holonomies of gauge fields in twistor space 3: Gravity as a square of theory

In a recent paper, we show that an S-matrix functional for graviton amplitudes can be described by an N = 8 supersymmetric gravitational holonomy operator in twistor space. In this paper, we obtain an alternative expression for the gravitational holonomy operator such that it can be interpreted as a square of an N = 4 holonomy operator for frame fields, by taking a sum of certain shuffles over ordered indices. The new expression leads to amplitudes of not only spin-2 gravitons but also spin-0 massless particles. We discuss that the squared model is favored as a theory of quantum gravity.