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Report

Teaser, summary, work performed and final results

Periodic Reporting for period 1 - eQG (Exceptional Quantum Gravity)

Teaser

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...

Summary

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:

1. Duality symmetries of quantum gravity and the special role of E10.
2. Incorporation of fermions and the R symmetry K(E10).
3. Arithmetic quantum gravity and quantum cosmology.
4. Emergence of space-time geometry from the BKL approach.
5. Canonical treatment and quantisation of exceptional geometry.
6. 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.
During the first period (covered by this report) substantial progress was achieved on the sub-topics 1,2,5 and 6, as explained in more detail in section B. This progress is mostly due to work of the members of the eQG team supported by this ERC Advanced Grant, that is, the PI himself as well as

Postdocs: Andre Coimbra, Amaury Leonard and Emanuel Malek;

PhD students: Lars Kreutzer and Adriano Vigano (who has meanwhile left for private reasons).

One main advance of more general relevance beyond the scope of the original proposal was in work by the PI and K. Meissner (University of Warsaw) where it was shown how symmetry concepts based on the exceptional mathematical structures mentioned above may lead to a possible explanation of the fermion spectrum of the Standard Model, with three generations of quarks and leptons. Unexpectedly, this work has given rise to an entirely new proposal for Dark Matter (with a very unusual Dark Matter candidate that, unlike more standard candidates like axions or WIMPs, i

Work performed

\"During the first reporting period of the ERC Project \"\"Exceptional Quantum Gravity\"\" three postdocs (A. Coimbra, A. Leonard and E. Malek) and two PhD students (L. Kreutzer, A. Vigano) were employed with ERC funding (A. Vigano has meanwhile left for private reasons). The work performed specifically dealt with the following sub-projects of eQG (as listed in PartB1):

-- P1: duality symmetries and the special role of E10 (refs. [1,2,3]);

-- P2: incorporation of fermions and the R symmetry K(E10) (refs. [6,7,8,9]);

-- P5: canonical treatment and quantisation of exceptional geometry (PhD work
in progress);

-- P6: N=8 supergravity in relation to E10 and K(E10) (refs. [8,9,10]).

The sub-projects P3 (arithmetic quantum gravity and quantum cosmology) and P4 (Emergence of space-time geometry from BKL) will be addressed in the following period, in particular with a new hiring (S. Bramberger).

More specifically, the achievements by the individual members of the team were:

Andre Coimbra’s research at AEI has been mostly focused on the problem of higher derivative corrections to the low energy limit of string theory, in particular finding their flux completion and understanding their mathematical properties, following two approaches: using the computer algebra program Cadabra to apply an algorithm based on a generalisation of the Lichnerowicz formula in eleven-dimensions along the lines of [Coimbra,Minasian, arXiv:1705.04330], and so generate the extra flux terms; and attempting to adapt/combine the E7 generalised geometry and heterotic generalised geometry formalisms of [Coimbra et al., JHEP1411(2014)160] to give a more geometrical description of the R^4 corrections with fluxes of eleven-dimensional supergravity restricted to seven dimensions. In the latter case, it was shown that the corrected Bianchi identity of the seven-form flux implies a gauge algebra that can no longer be given solely in terms of a Courant bracket. Instead, theories of this type have a gauge algebra which should be thought of in the wider terms of an L-infinity algebra. For a Bianchi identity relating the seven-form flux to an eight-form R^4, this algebra is over a graded vector space with seven terms and with multibrackets up to level ten. These results, including the explicit form of the L-infinity multibrackets and the proof that they satisfy the required generalised Jacobi identities, will soon be published [11].

Amaury Leonard’s work [1] has been focused on obtaining the Hamiltonian formalism of free fermionic higher spin gauge fields propagating on a flat background space-time of dimension four. The Lagrangian description of these massless higher spin fermions uses symmetric covariant tensor-spinors enjoying a gauge invariance under linearized generalized diffeomorphisms. As for any gauge system, the Hamiltonian analysis leads to constraints, which were computed and solved locally through the introduction of so–called prepotentials, that is, unconstrained Hamiltonian variables. These prepotentials are symmetric spatial tensor-spinors, and they enjoy a gauge invariance including both linearized generalized diffeomorphisms and, remarkably, Weyl rescaling. This conformal invariance led to a study of the properties of the corresponding curvature: the Cotton tensor. These geometrical tools were essential in solving the constraints. In terms of these prepotentials, the equations of motion take a very simple form equating the time derivative of the Cotton tensor of the prepotentials to its (chirality dual) spatial curl. A similar approach to linearized supergravity was applied over a flat background space-time of arbitrary dimension [2], yielding a first order formalism for both the graviton and the gravitino, which treats the graviton and its dual on the same footing. Again, this required constructing a complete set of curvatures invariant under linearized generalized Weyl rescaling, for tensors of mi\"

Final results

Given the progress achieved so far, future work within the eQG project falls into one of two categories. The first one concerns issues that remain conceptually and technically challenging, but where progress can be foreseen with some confidence following up on earlier eQG achievements so far; the second one concerns problems whose solution will require entirely new and difficult to predict new insights (“quantum leaps”).

Progress can be confidently 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 so far by the PI and A. Kleinschmidt have been taken up independently by group of mathematicians (led by R. Koehl), who have already made some remarkable and independent progress.
• A better understanding of exceptional geometry for the higher rank groups E7 and E8, and concerning issues such as the development of an exceptional analogue of Riemannian geometry (building on earlier work by the PI and H. and M. Godazgar).
• Further exploitation of 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.
• New insights for arithmetic quantum cosmology, especially with incorporation of fermions, following pioneering work by Damour and Spindel (with new postdoc S. Bramberger, who will bring in new and independent expertise in quantum cosmology)
• Progress with the Hamiltonian formulation of exceptional field theories, also for the E7 and E8 theories, and possibly an “Ashtekar-type” reformulation of exceptional field theory in the canonical setting.

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!