Work performed on the specific eQG projects can be divided into two categories.The first concerns problems that are conceptually and technically challenging, but which could be fully dealt with by the end of the eQG project.The second concerns problems where important progress has been achieved, but whose ultimate scope extends far beyond eQG.The following contains a brief description of selected highlights that have been achieved during the duration of the eQG project. A major step forward was the extension of generalized holonomies to the affine E9 symmetry by the PI and A. Kleinschmidt, highlighting the central role of the involutory subalgebra K(E9) [47]. These results go far beyond what had been understood up that point about generalized holonomies in supergravity, which were all limited to finite-dimensional Lie groups. The more challenging construction for E10 and K(E10) remains an open issue, as a problem that belongs to the second category, even though the results obtained provide important hints. In [30] F.Ciceri and collaborators have been able to construct the bosonic part of gauged maximal supergravity in two dimensions, succeeding in the derivation of a pseudo-Lagrangian. Completing the construction of a supersymmetric gauged theory, with a derivation of the conjectured form of the embedding tensor from local supersymmetry, remains an open problem. Nevertheless, thanks to B. König’s proof that the basic representation of E9 admits infinitely many subspaces that are invariant under K(E9) [53], there are now excellent prospects for completing this task. For K(E9) an infinite set of unfaithful representations of ever growing dimension was obtained [48]. This construction was based on a novel transformation of the filtered basis of K(E9) to a parabolic one, an insight that itself relied on our detailed of knowledge of the integrable structure of D=2 maximal supergravity. The generalization of this construction to K(E10), and the identification of larger unfaithful fermionic representation of K(E10) beyond the known s=1/2, 3/2, 5/2 and 7/2 representations remains a challenge, together with a better understanding of the known representations. Progress in this direction is a necessary prerequisite in understanding in more detail the emergence of space-time dependence of the fermions in the effective emergent field theory. The subject of arithmetic quantum cosmology was greatly advanced in work of the PI and A. Kleinschmidt [46], where the bosonic canonical Hamiltonian of D=11 supergravity was shown to coincide with the E10 Casimir operator at low levels. In particular, building on our previous studies of mini-superspace quantum gravity we studied the behavior of the `wave function of the universe’ near the singularity for various and more general finite-dimensional truncations of the E10 Hamiltonian. We were able to provide concrete examples of DeWitt’s proposed mechanism of singularity resolution in General Relativity, which relies on the vanishing of the `wave function of the universe’ at the singularity. Our results suggest that the DeWitt mechanism is a generic feature of these models, and may thus serve as a general paradigm for the quantum resolution of space-time singularities. The `grand challenge’ at the core of eQG is to understand the emergence of space-time from a purely algebraic construct. A similar challenge concerns the understanding of how space-time geometry (gauge invariance, general covariance, etc.) can emerge from a purely algebraic setting. In our work on rewriting the Wheeler DeWitt Hamiltonian in terms of the E10 Casimir we propose a more concrete mechanism for the emergence of space-time dependence, which requires the proper incorporation of the Newton and Planck constants via the replacement of canonical brackets by quantum commutators [45,46]. Independently, [19] studied quantum corrections in Bianchi II universes, with direct implications for the BKL conjecture.The formulation of exceptional geometry for the exceptional groups E7 and E8 has been completed. The insights obtained for consistent truncations for various supergravity compactifications and their implications for AdS/CFT have been successfully exploited [29,31,32,34]. A. Kleinschmidt and collaborators have been able to formulate E9 and E11 exceptional field theories [28], in a sequel to the work on finite-dimensional exceptional Lie algebras cited above. These results provide hints how the E10 framework may have to be enlarged by the addition of new representations. Perhaps the most exciting progress was achieved in a special sub-project by the PI and K. Meissner, which establishes a direct connections between the fermionic sector of the Standard Model of particle physics and a K(E10) motivated re-interpretation of N=8 supergravity [1]. More specifically it was shown, in work following Gell-Mann’s proposal of identifying the 48 fermions remaining after removal of eight Goldstinos from the 56 spin-1/2 fermions of maximal supergravity with the three generations of quarks and leptons, and how to implement the Standard Model symmetry SU(3) x SU(2) x U(1) on the 48 supergravity fermions, as well as the (B-L) symmetry. With eight supermassive gravitinos (of spin-3/2) this scheme furnishes a novel Dark Matter candidate of an entirely different type from those known and studied so far [11]. This proposal can be experimentally tested in upcoming underground experiments such as JUNO and DUNE [54]. In an intriguing application of this proposal we showed that supermassive gravitinos can also explain the origin of primordial (galactic) black holes in the early universe, via the formation and subsequent gravitational collapse of `gravitino lumps’ in the early universe [16,17].
To conclude: all objectives of eQG have been met, in several instances far beyond the originally stated aims.
