Speaker : Baptiste Cerclé
Title : Liouville Conformal Field Theory in higher dimensions
Abstract :
Providing a rigorous definition to the two-dimensional
Liouville Quantum Gravity as introduced by Polyakov in his 1981 seminal
work has been a challenging problem over the last few years. In this
fundamental article is introduced a « canonical way » of picking at
random a geometry on a surface with fixed topology, using a generalised
path integral approach involving the Liouville functional. The
mathematical interpretation of this formalism is now rather well
understood, thanks to the introduction of a probabilistic framework in a
work initiated by David, Kupiainen, Rhodes and Vargas.
On the other hand, the study of conformal geometry in dimension higher
than two has considerably developed recently, with the introduction of
higher-dimensional analogues of the Laplace operator and the scalar
curvature : the GJMS operators and the $\mathcal{Q}$-curvature. As we
will see , these operators play a role which is similar to the one of
their two-dimensional analogues in the context of LCFT : it is therefore
natural to expect that one can define LCFT in a higher-dimensional
context.
During this talk I will present some recent developments in
higher-dimensional LCFT.
To do so, we will first consider the classical theory, which corresponds
to providing answers to a higher-dimensional uniformisation problem :
does every compact even-dimensional manifold carry a conformal metric
with constant (negative) $\mathcal{Q}$-curvature ?
The variational formulation of this question consists in finding the
critical points of a higher-dimensional analogue of the Liouville action
functional. This approach will naturally lead us to a path integral
formulation of higher-dimensional LCFT which we rigorously define on the
even-dimensional sphere, which is the simplest case of study.
Eventually we willreturn to the classical theory by studying the
semi-classical limit of the model, which consists in taking the quantum
parameters to zero. As expected, the results obtained by doing so
correspond to the classical ones.