Periodic Reporting for period 3 - eQG (Exceptional Quantum Gravity) Reporting period: 2020-12-01 to 2022-05-31 Summary of the context and overall objectives of the project Motivated by the success of symmetry concepts in formulating basic laws of physics, eQG seeks to develop a new symmetry based approach to the problem of reconciling Quantum Mechanics and Einstein’s General Relativity into a consistent theory of Quantum Gravity. The need for such a theory is evident from the so far unresolved problems of black hole singularities and the Big Bang, and also driven by the search for a consistent “UV completion” of the Standard Model of Particle Physics and the unification of the fundamental interactions. eQG tackles these problems from a new perspective: on the one hand, by paying particular attention to recent advances in our understanding of cosmological singularities and the evidence for novel infinite-dimensional duality symmetries near the singularity that has emerged in supergravity and string theory, and to recent progress in formulating ‘exceptional geometries’ transcending Riemannian geometry; on the other hand, by exploiting insights from modern canonical quantisation towards a better understanding of the basic degrees of freedom and the dynamics of quantum space-time. The main focus of eQG will be the ‘maximally extended’ exceptional hyperbolic Kac–Moody symmetry E10, whose uniquely distinguished status makes it a prime candidate symmetry for unifying the known dualities of string and M theory, for a conceptually precise scenario of emergent (quantum) space and time near the singularity, and finally, for replacing supersymmetry as a guiding principle for unification. Consequently, the principal goal of eQG is to explore how this symmetry can define a theory of quantum gravity, how it acts on its fundamental degrees of freedom, what the special features are of the quantised theory, and what physical predictions can be derived from it. 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. Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far During the third reporting period (2020–2022) of the ERC Project "Exceptional Quantum Gravity" four postdocs (Ana Alonso-Serrano, Franz Ciceri, Andre Coimbra and Victor Leukeu) were employed with ERC funding, after the early departures of Emanuel Malek and Hadi Godazgar. Lars Kreutzer finished is PhD in 2021 (Magna cum Laude); a new PhD student Benedikt König now also works on eQG related projects, but is independently funded by Konrad-Adenauer-Stiftung. 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. Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far) Given the fact that major progress has been achieved in all sub-projects listed in the original eQG proposal, future work within the eQG project and beyond falls into one of two categories. The first one concerns issues that remain conceptually and technically challenging, but where further progress can be foreseen until the end of the eQG project. The second one concerns problems whose solution will require entirely new and difficult to predict new insights, and thus is expected to reach beyond 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! WebHeader eQG
