Promoteur: Nicolas Boulanger

Résumé de la thèse:

For more than a century and as of writing, Einstein’s theory of General Relativity has been recognized
as the most accurate theory of gravity. This is because it passed all the experimental tests from the
historical ones on perihelion precession of Mercury and de
ection of light by sun in 1919, to the latest
one on the direct detection of gravitational waves by the Advanced LIGO Project in 2016. Unfortunately,
General Relativity is not compatible with Quantum Mechanics as it is non-renormalizable, viewed as a
non-abelian quantum eld theory of a massless spin-two eld in Minkowski spacetime. This makes it
mathematically inconsistent and inoperable at short-distance or high-energy, for instance near a black
hole. Moreover, several independent experiments have shown that the main content of the Universe is
predominantly made of dark matter and dark energy, still unknown substances at the time of writing,
which may indicate the necessity to extend or modify General Relativity.
The purpose of this thesis is to scrutinize the algebraic aspects of some extensions of the theory of
General Relativity in order to address the issues presented above. The largest part of the thesis is devoted
to the calculation of interactions in several extensions of General Relativity using the cohomological
reformulation of the Noether deformation procedure. This method is developed in the antield formalism
due to Batalin and Vilkovisky that was originally designed to quantize non-abelian gauge theories. It
has proved to be a very powerful tool also at the classical level as it allows to classify all the possible
interactions that preserve the number of gauge symmetries of a theory.
A rst extension of General Relativity that is studied through this procedure is the range of su-
pergravity theories. These are descriptions of the massless spin-two eld, i.e. the gravitational eld,
coupled to a set of lower-spin elds and enjoying local supersymmetry, a symmetry relating bosons and
fermions. This kind of theories are interesting as they exhibit a better ultraviolet behaviour than the
non-supersymmetric theories of gravitation. Indeed maximal supergravity, for instance, is known to be
nite up to 5 loops in perturbation theory at the quantum level. Although we expect divergences to
show up at some point because of arguments coming from String Theory, it was not expected that they
appear so late in the quantum h expansion and it is believed that some hidden symmetries of gravity
are responsible for the observed cancellation of ultraviolet divergences. In this PhD work, we classify the
possible supergravity theories under some general assumptions in order to unveil some universal features
of this type of non-abelian gauge theories of gravity.
Another possible extension of General Relativity is obtained by allowing the graviton, the particle
associated with the gravitational interaction, to have a non-zero mass. The partially massless graviton,
with its mass given in unit of the positive cosmological constant in a precise way so as to describe a
number of degrees of freedom intermediate between the massless graviton and a massive one, is thoroughly
considered. Both partially massless and massive gravitons are of interest in theoretical cosmology since
they are good candidates to account for dark matter and/or dark energy. In both cases we analyzed the
possible interacting theories for such elds, thereby (i) recovering and clarifying the gauge structure of
dRGT (de Rham, Gabadadze and Tolley) massive gravity theory, and (ii) strengthening previous results
concerning partially massless graviton theories and discovering the rst consistent and non-trivial theory
for such elds in interaction.
Another algebraic aspect that is treated in this thesis concerns the treatment of electric-magnetic
duality in the gravitational context. More precisely, we presented the linearized equations describing both
a massless and a partially massless graviton as twisted-duality relations, and this, in all the maximally
symmetric cosmological spacetimes of dimension strictly larger than three.