Weyl symmetry is a local rescaling of the metric, which generalize flat space conformal invariance to curved manifolds. It is well known that this symmetry is anomalous at the quantum level, in the sense that its associated Ward identity gets broken, generically, by quantum fluctuations (i.e. the expectation value of the energy momentum tensor acquires a trace). I will propose a novel way of implementing Weyl symmetry, which is achieved by the introduction of a compensating one form field, naturally identified with the torsion tensor, which elevates Weyl symmetry to a gauge theory. This procedure modifies the Weyl symmetry Ward identities, and allows to cure the anomaly. Indeed, in this theory, the anomalous trace signals the breaking of global scale symmetry, and the appearance of the physical goldstone mode of broken dilatations. This will give a mass to the Weyl field, without truly breaking the gauge symmetry.

##### Jean-Luc Lehners, Max Planck Institute for Gravitational Physics

##### No smooth beginning for spacetime

I will discuss a fundamental obstruction to any theory of the beginning of the universe, formulated as a semiclassical path integral. Hartle and Hawking’s no boundary proposal and Vilenkin’s tunneling proposal are examples of such theories. Each may be formulated as the quantum amplitude for obtaining a final 3-geometry by integrating over 4-geometries. The result is obtained using a new mathematical tool - Picard-Lefschetz theory - for defining the semiclassical path integral for gravity. The Lorentzian path integral for quantum cosmology with a positive cosmological constant is mathematically meaningful in this approach, but the Euclidean version is not. Framed in this way, the resulting framework and predictions are unique. Unfortunately, the outcome is that primordial gravitational wave fluctuations are unsuppressed. One can prove a general theorem to this effect, in a wide class of theories.