Black Hole Microstate Geometries and the Fuzzball/Firewall Proposal
Asymptotic safety on an infinite-dimensional coupling space (slides)
Until recently, the asymptotic safety scenario (i.e. the conjecture that a fundamental quantum field theory of gravity exists thanks to a non-trivial UV fixed point) has been always tested by truncating the infinite-dimensional space of all possible gravitational effective actions to finite dimensional subspaces (i.e. a finite number of couplings). In this talk I will discuss recent results in the effort of going beyond such truncations and testing the asymptotic safety conjecture on an infinite-dimensional space of couplings. In particular, I will show how within an f(R) approximation it is possible to prove that the number of relevant perturbations of a fixed
point is finite.
Scalings and large N solutions in random tensor models (slides)
After an introduction to the world of random tensor models and random colored graphs, I will present recent works which aim at exploring the richness of tensor models beyond the so-called melonic sector. Starting from Gurau’s initial 1/N expansion, I will build other 1/N expansions which retain more contributions at large N. Some of them satisfy a surprising universality theorem, while others exhibit interesting non-Gaussian behaviors like random matrix models, with a rich combinatorial structure.
Chiral anomaly and modular invariance
I shall describe the sl2 anomaly in N=4 supergravity coupled to n vector multiplets. Supersymmetry implies that the latter is associated to a supersymmetry invariant that completes the Pontryagin density. I will explain how one can build this invariant as a closed super-form. This anomaly determines the modular transformation of the effective action in supergravity. I will explain how modular invariance can be restored in a unique way, reproducing the Wilsonian effective action of type IIA string theory on K3 X T2.
The nested loop approach to the O(n) model on random maps (slides)
There is currently no consensus on what is the fractal (Hausdorff) dimension of a discrete random surface coupled to a critical matter model. In this talk, we describe a preliminary step in addressing this question, by considering the O(n) loop model on random planar maps (i.e. graphs embedded in the sphere). We explain how an elementary combinatorial decomposition, which consists in cutting the maps along the outermost loops, allows to relate the O(n) model to the simpler problem of counting maps with controlled face degrees (which may be solved using the classical Hermitian one-matrix model). This translates into a functional relation for the “resolvent” of the model, which is exactly solvable in several interesting cases. We then look for critical points of the model: our construction shows that at the so-called non-generic critical points, the O(n) model is related to the “stable” map, of known Hausdorff dimension, introduced by Le Gall and Miermont.
Generalized CDT as a scaling limit of planar maps (slides)
Generalized causal dynamical triangulations (generalized CDT) is a model of two-dimensional quantum gravity in which a limited number of spatial topology changes is allowed to occur. After identifying the model as a scaling limit of random quadrangulations, I will show how it can be solved using a bijection between quadrangulations and trees. Another bijection relating quadrangulations to planar maps allows us to interpret generalized CDT as a scaling limit or random planar maps with a restriction on the number of faces. Finally I will show how this interpretation clarifies certain mysterious identities in generalized CDT amplitudes. (This
talk is largely based on arXiv:1302.1763.)
Towards the continuum limit of spin foams and spin nets: renormalization flow in background independent theories (slides)
We aim to extract the behaviour of spin foam models at scales large compared to the average discretization scale. To this end we first introduce a conceptual framework for renormalization in background independent theories. This will also bring up the main technical tool we will be using, tensor network renormalization flow. We then present first numerical results on the renormalization flow of so-called spin net models.
Counting discrete surfaces of any genus, by topological recursion (slides)
Matrix Theories and Emergent Space
J.F. Le Gall
A continuous limit for large random planar graphs (slides)
Consider a triangulation of the sphere chosen uniformly at random among all triangulations with a fixed number of faces (two triangulations are identified if they correspond via a deformation of the sphere). We equip the vertex set of this triangulation with the usual graph distance. When the number of faces tends to infinity, the (suitably rescaled) resulting metric space converges in distribution towards a random compact metric space called the Brownian map. This result, which confirms a conjecture of Schramm in 2006, holds with the same limit for much more general random graphs drawn on the sphere. The Brownian map thus appears as a universal model of a random surface, which is homeomorphic to the sphere but has Hausdorff dimension 4.
Spinor Network for Loop Gravity and U(N) symmetry for Polyhedra (slides)
Causal Dynamical Triangulations and Time
The search for “quantum spacetime” – the putative quantum description of spacetime at Planckian scales – can be approached in many ways, from semiheuristic mathematical modelling to fully-fledged nonperturbative systems of “quantum gravity”. This latter, dynamical viewpoint potentially opens up new perspectives on the role of time, causality, uniqueness and emergence. Most importantly, once quantitative, statistical mechanical tools are brought in to evaluate suitable quantum observables, these issues can be discussed in concrete terms. I will summarize the insights that have been gained about the nature of quantum spacetime and, more generally, quantum gravity by using Causal Dynamical Triangulations (CDT), and describe new results, obtained in a generalized version of CDT, which further illuminate the issue of time in CDT.
Quantum gravity and E10
Cosmological dynamics from Group Field Theory (slides)
We show that effective cosmological equations for continuum homogeneous geometries can be derived directly from fundamental group field theory (GFT) models, in full generality. The relevant quantum states are GFT condensates, and a form of nonlinear quantum cosmology arises as the hydrodynamics of the system. If time allows, we will also report on recent results on GFT renormalization, aimed at establishing these models as well-defined field theories, and at proving the dynamical realization of spacetime condensation.
Asymptotic Safety (slides)
I will review the status of the asymptotic safety programme and discuss challenges and open problems.
Black hole thermodynamics and loop quantum gravity (slides)
I will review the laws of black hole mechanics and then rewrite them from the point of view of local stationary observers close to the black hole horizon. I will argue that this local perspective provides the basis for a new look at the thermodynamic nature of black holes in the framework of loop quantum gravity.
Covariant Loop Gravity: Radiative Corrections (slides)
I briefly review the state of the covariant formulation of (4d Lorentzian) Loop Quantum Gravity. The theory is finite at all orders, but radiative corrections may be large and invalidate the expansion that defines te theory. I present some first calculations of radiative corrections: vertex correction and (edge) self-energy, recently obtained by Aldo Riello. These first news are good: the vertex is finite in the limit of large cosmological constant and the self energy is proportional to the logaritm of the cosmological constant. These results may allow us to compute the scaling of the theory’s constants from the Planck to the macroscopic scale.
Melons, branched polymers and beyond (slides)
Melonic graphs constitute the family of graphs arising at leading order in the 1/N expansion of tensor models. They were shown to lead to a continuum phase, reminiscent of branched polymers. I shall explain how they are in fact precisely branched polymers, that is, they possess Hausdorff dimension 2 and spectral dimension 4/3.
Moreover, melonic graphs represent just the first step in a larger program to define a double scaling limit for tensor models. I shall also present the next step in the program, an investigation of the next-to-leading order, culminating in an analysis of its critical behaviour.
Obvious and hidden tree structures in random triangulations
A standard paradigm of enumerative combinatorics is that discrete structures that have algebraic generating functions should be in some sense tree-like. I will explain what this means and discuss how it has led to revealing hidden tree structures in various families of maps (or graphs on surfaces) that were first studied by Tutte and by Brezin-Itzykson-Parisi-Zuber. I plan to concentrate on two types of triangulations: maximal boundary 3d-triangulations for which the tree-like structure is obvious, and general 2d-triangulations for which the tree-like structure is more difficult to reveal but allows to study the intrinsic geometry of the associated large random surfaces.
Ultraviolet behavior of half-maximal supergravity theories (slides)
I will begin with an introduction where I review the links between ultraviolet (UV) divergences in field theory and the way string theory regularises them. I will then give an overview of the current status of UV divergences hunt in supergravity theories. After what I will present the perturbative structure of half-maximal supergravity theories from the string theoretical and non-renormalisation theorems point of view.
Aspects of the String-Black Hole Correspondence (slides)
After recalling the correspondence between black holes and certain fundamental strings, dubbed “stringholes”, I will illustrate how such objects arise in high-energy gravitational scattering near the threshold of back-hole production, how they exhibit a surprisingly large amount of quantum hair, and how they could have played a role in replacing the Big Bang singularity with a regular Big Bounce transition.