On the implementation of the canonical quantum simplicity constraint

In this paper, we discuss several approaches to solve the quadratic and linear simplicity constraints in the context of the canonical formulations of higher dimensional general relativity and supergravity developed in our companion papers. Since the canonical quadratic simplicity constraint operators have been shown to be anomalous in any dimension D ⩾ 3 in Class. Quantum Grav. 30 045003, non-standard methods have to be employed to avoid inconsistencies in the quantum theory. We show that one can choose a subset of quadratic simplicity constraint operators which are non-anomalous among themselves and allow for a natural unitary map of the spin networks in the kernel of these simplicity constraint operators to the SU(2)-based Ashtekar-Lewandowski Hilbert space in D = 3. The linear constraint operators on the other hand are non-anomalous by themselves; however, their solution space is shown to differ in D = 3 from the expected Ashtekar-Lewandowski Hilbert space. We comment on possible strategies to make a connection to the quadratic theory. Also, we comment on the relation of our proposals to the existing work in the spin foam literature and how these works could be used in the canonical theory. We emphasize that many ideas developed in this paper are certainly incomplete and should be considered as suggestions for possible starting points for more satisfactory treatments in the future.