The E 10 Wheeler-DeWitt operator at low levels

We consider the Wheeler-DeWitt operator associated with the bosonic part of the Hamiltonian of D = 11 supergravity in a formulation with only the spatial components of the three-form and six-form fields, and compare it with the E 10 Casimir operator at low levels, to show that these two operators precisely match modulo spatial gradients up to and including 𝔤𝔩 10 level ℓ = 2. The uniqueness of the E 10 Casimir operator eliminates all ordering ambiguities in the quantum Hamiltonian, at least up to the level considered. Beyond ℓ ≥ 3 the two operators are expected to start to differ from each other, as they do so for the classical expressions. We then consider truncations of the E 10 Wheeler-DeWitt operator for various finite-dimensional subgroups of E 10 in order to exhibit the automorphic properties of the associated wave functions and to show that physically sensible wave functions generically vanish at the cosmological singularity, thus providing new and more sophisticated examples of DeWitt’s proposed mechanism for singularity resolution in quantum gravity. Our construction provides novel perspectives on several unresolved conceptual issues with the Wheeler-DeWitt equation, such as the question of observables in quantum gravity, or the issue of emergent space and time in a purely algebraic framework. We also highlight remaining open questions of the E 10 framework.