## Periodic Reporting for period 3 - eQG (Exceptional Quantum Gravity)

Periodo di rendicontazione: 2020-12-01 al 2022-05-31

According to the original proposal, the main objectives of eQG comprise the following sub-projects:

P1. Duality symmetries of quantum gravity and the special role of E10.

P2. Incorporation of fermions and the R symmetry K(E10).

P3. Arithmetic quantum gravity and quantum cosmology.

P4. Emergence of space-time geometry from the BKL approach.

P5. Canonical treatment and quantisation of exceptional geometry.

P6. N = 8 supergravity in relation to E10 and K(E10).

As explained in the original proposal, these sub-topics are to be investigated in the context of String Theory and Supergravity, of the BKL approach, and of Exceptional Geometry and Exceptional Field Theory. These investigations should also take into account more recent advances (since the submission oft he original ERC proposal in 2016) in generalized geometry (such as L-algebras), and in supergravity (such as issues related to consistent truncations in supergravity compactifications, which rely heavily on exceptional geometry), developments related to higher derivative corrections in supergravity and superstring theory, and finally the Hamiltonian formulation of these theories.

The work performed dealt with all sub-projects of eQG (as listed in PartB1 of the original proposal), with significant advances in all areas. In addition a number of other topics were dealt with that are more indirectly related to these sub-projects but nevertheless relevant to the global aims of eQG.

In particular, further progress can be expected on the following topics: Identification of further and larger unfaithful fermionic representation of K(E10) beyond the s=1/2, 3/2, 5/2 and 7/2 representations obtained so far, together with a better understanding of the known representations (e.g. regarding subgroup decompositions, etc.). In fact, the results obtained in Refs. [1,2] solve this problem for the affine subgroup K(E9) and have provided important hints for K(E10). A better understanding of exceptional geometry for the higher rank groups E7 and E8 has meanwhile been achieved, especially with refs. [15,26,27], but issues such as the development of an exceptional analogue of Riemannian geometry building on earlier work by the PI and H. and M. Godazgar remain to be developed. The insights obtained for consistent truncations for various supergravity compactifications and their implications for AdS/CFT, building on the results obtained by E. Malek and collaborators within eQG, have been successfully exploited in Ref. [6]. Building on this work further progress has meanwhile been achieved by other groups, in particular E. Malek and H. Samtleben. The subject of arithmetic quantum cosmology was greatly advanced with [5], where the canonical Hamiltonian of D=11 supergravity was shown to coincide with the E10 Casimir operator at low levels, and the behavior of the `wave function of the universe’ near the singularity was investigated for various finite-dimensional arithmetic subgroups of E10. The next step will involve the incorporation of fermions, following pioneering work by Damour and Spindel.

In addition there remain several `grand challenges’ where progress will crucially depend on new insights that cannot be foreseen or predicted with certainty: Understanding the emergence of space-time from exceptional symmetries beyond the restriction to first order spatial gradients on the BKL side, and height < 30 roots on the E10 side. A similar challenge concerns the understanding of how space-time symmetries (gauge invariance, general covariance, etc.) can emerge from a purely group theoretic setting. Identifying faithful, hence infinite-dimensional fermionic representations of K(E10). Progress here could not only lead to entirely new insights on E10 and other hyperbolic Kac-Moody algebras, but also enable a better understanding of how the full standard model symmetries including space-time dependent gauge symmetries could emerge from this setup.

More generally: a better understanding and perhaps even a more concrete realization of hyperbolic Kac—Moody algebras (this is a problem that has remained unresolved for more than 50 years), as well as the automorphic structures related to E10 and their implications for solutions of the Wheeler-DeWitt equation.

Further exploring the tantalizing hints obtained already towards a link of the abstract group theory with particle physics and observation, or to state it more simply: finding observational evidence for the ansatz towards unification pursued in eQG!