New variables for classical and quantum gravity in all dimensions: III. Quantum theory

We quantize the new connection formulation of (D + 1)-dimensional general relativity developed in our companion papers by loop quantum gravity (LQG) methods. It turns out that all the tools prepared for LQG straightforwardly generalize to the new connection formulation in higher dimensions. The only new challenge is the simplicity constraint. While its 'diagonal' components acting at edges of spin-network functions are easily solved, its 'off-diagonal' components acting at vertices are non-trivial and require a more elaborate treatment.