**G. Borot**

2d quantum gravity and topological recursion

** **In the early 90s, 3 approaches to were proposed to compute the partition function of 2d quantum gravity: generating series of discrete surfaces (= maps) in the continuum limit; generating series of intersection numbers on M_{g,n} = the moduli space of curves ; certain tau functions of (reductions of) the KP hierarchy. The equivalence between (2) and (3) in the simplest case is the famous conjecture of Witten proved by Kontsevich. In the last 10 years, a theory of loop equations has been developed after the initial work of Eynard and Orantin, and its solutions are described in terms of solutions by the so-called “topological recursion”: it reduces problems of 2d enumerative geometry in all topologies to the enumeration of disks and cylinder topology. Loop equations can be established in (1), (2) and (3), and this allows a proof of the equivalence of these 3 approaches. It was also realized recently, using topological recursion, that “any” generating series of maps can be expressed in terms of integrals on M_{g,n}. I will survey several aspects of these results, partly based on joint work with Bergère, Eynard, and Shadrin.

**S. Carrozza**

**Group field theory, tensor models and the renormalization group**

Group Field Theory (GFT) is a (non-local) quantum field theory formalism aiming at completing the definition of the dynamics of loop quantum gravity. The renormalization group is of central importance in this approach, first to (perturbatively) define the theory, and second to explore its phase structure. In the long run, it should in particular help us understand the effective low energy limit of loop quantum gravity, and determine whether the latter is consistent with general relativity or not. Alternatively, GFTs can be understood as enriched tensor models, which witnessed important developments in the recent years. I will explain how this relationship allowed to define a consistent renormalization group for so-called tensorial GFTs, and will review the main results derived so far. Particular attention will be given to notable differences with respect to ordinary scalar field theories, among which the existence of simple asymptotically free model.

**A. Connes**

**Geometry and the Quantum**

I will present the joint work with A. Chamseddine and S. Mukhanov. Motivated by the construction of spectral manifolds in noncommutative geometry, we introduce a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of scalar fields. This commutation relation appears in two versions, one sided and two sided. It implies the quantization of the volume. In the one-sided case it implies that the manifold decomposes into a disconnected sum of spheres which will represent quanta of geometry. The two sided version in dimension 4 predicts the two algebras $M_2(\mathbb H)$ and $M_4(\mathbb C)$ which are the algebraic constituents of the Standard Model of particle physics. This taken together with the non-commutative algebra of functions allows one to reconstruct, using the spectral action, the Lagrangian of gravity coupled with the Standard Model. We show that any connected Riemannian Spin 4-manifold with quantized volume >4 (in suitable units) appears as an irreducible representation of the two-sided commutation relations in dimension 4 and that these representations give a seductive model of the « particle picture » for a theory of quantum gravity in which both the Einstein geometric standpoint and the Standard Model emerge from Quantum Mechanics.

**J. Cooperman**

**The search for a continuum limit of causal dynamical triangulations**

The causal dynamical triangulations approach aims to construct a quantum theory of gravity as the continuum limit of a lattice-regularized model of dynamical geometry. Already a decade ago the approach began to yield evidence for the existence of valid classical and semiclassical limits on large scales. Only recently have attempts been initiated to investigate the possibility of a continuum limit on small scales. I report on the present status of such renormalization group analyses. Needless to say—or else you would have heard!—we are not there yet.

**Francois David**

**Liouville quantum gravity on the Riemann sphere**

I will present a rigorous probabilistic construction of Liouville Field Theory on the Riemann sphere with positive cosmological constant, as considered by Polyakov in his 1981 seminal work “Quantum geometry of bosonic strings”. Some of the fundamental properties of the theory like conformal covariance, the Weyl anomaly (Polyakov-Ray-Singer) formula, KPZ scaling laws, Seiberg bounds are recovered. This is based on joint works with A. Kupiainen, R. Rhodes and V. Vargas.

**T. Dennen**

**Supergravity as a Consistent Quantum Theory?**

Supergravity theories were once hoped to provide ultraviolet finite theories of quantum gravity without requiring profoundly new physical frameworks. These hopes faded in the 1980s, but renewed efforts in recent years have uncovered some surprising ultraviolet behaviour. In this talk, I will give an overview of the explicit scattering amplitude calculations in perturbative supergravity over the last few years. In particular cases, there are indications of ultraviolet cancellations not accounted for by known symmetry arguments. I will highlight these cases and give some reasons to be optimistic and some reasons to be pessimistic about supergravity as a contender for a quantum theory of gravity.

**A. Eichhorn**

**Unimodular asymptotic safety**

I will discuss a quantum field theory for gravity based on unimodular metrics. I will show new evidence for an asymptotically safe fixed point of the Renormalization Group flow in this setting, and will also discuss the compatibility of this quantum gravity model with the matter degrees of freedom of the Standard Model.

**S. Gielen**

**Spacetime as a condensate in group field theory**

One major challenge of discrete approaches to quantum gravity is to show the emergence of an (approximate) continuum geometric phase, described by geometric variables such as a metric, whose dynamics is governed by something like Einstein’s equations (perhaps with higher order corrections, additional matter fields, etc). I will present one approach towards this issue, in the group field theory (GFT) formalism, in which a continuum spacetime emerges from the effective hydrodynamic description of a ‘condensate’ of a large number of elementary discrete degrees of freedom. The condensate describes a macroscopic, spatially homogeneous universe, making this an interesting setting for cosmology. Using techniques borrowed from the physics of real Bose-Einstein condensates, one can extract an effective quantum cosmology model directly from the fundamental dynamics of quantum gravity given by GFT. I will comment on the implications in the context of GFT as an approach to quantum gravity and on potential links to other approaches.

**A. Grassi**

**Exact results in ABJM theory and quantum geometry**

I will show that, for some values of the Chern-Simons level, it is possible to write closed form expressions for the grand potential of these theories and for the generating functional of their partition functions. I will then explore some possible consequences for the quantum geometry of M-theory and for de Sitter extensions. As a generalization of these results I will propose an exact quantization condition for ABJM theory which involves standard and refined topological strings on local P1xP1.

**R. Gurau**

**Non perturbative results in matrix and tensor models**

**K. Krasnov**

**Gravitons and spin 2 representations of Lorentz group**

In the case of gauge fields, Yang-Mills fields satisfy second order PDE’s while self-dual gauge fields satisfy first-order PDE’s. In the case of metric gravity the Einstein condition is second-order in the derivatives of the metric, but the condition that the Riemann curvature is self-dual is still second-order. One can describe self-dual metrics with first-order PDE’s by using a heavier machinery of twistor theory, but this does not generalise to arbitrary Einstein metrics, at least not in any useful way. I will describe a formalism for gravity in which the Einstein condition is a second order PDE and the self-dual Einstein condition is a first-order PDE. This formalism uses a different representation of the Lorentz group to describe gravitons. Moreover, once one uses a different field to encode gravitons, one finds that there is not one theory of gravity with second order field equations, but rather an infinite-parametric family.

**S. Liberati**

**Gravity, an unexpected journey: from thermodynamics to emergent gravity, Lorentz breaking and back…**

I will initially review some striking features of gravity and their implications for its possible fundamental nature. I will then discuss the case of Lorentz breaking models of gravity, describing current constraints and their black hole solutions. Finally I will discuss the surprising survival (so far) of black hole thermodynamics in these models.

**H. Nicolai**

**On Exceptional Geometry and Supergravity**

**K. Noui**

**On the Barbero-Immirzi ambiguity in Loop Quantum Gravity**

The Barbero-Immirzi parameter plays a rather intriguing role in Loop Quantum Gravity (LQG). In fact, at the classical level, the parameter plays no role at all. This strongly contrasts with what happens in the quantum theory where it enters into the expression of the spectrum of geometric operators. As a consequence, the effects of this parameter are very important in the physical predictions of LQG (black holes thermodynamics, quantum cosmology, etc.). Therefore there seems to be a strong contradiction between the classical and the quantum role of the Barbero-Immirzi parameter, and this aspect has remained an important open issue for a long time (source of debates and criticisms). Recently, we have found strong evidences of the fact that the parameter should not play any physical role at the quantum level and should be interpreted as a regulator only. In this talk, we are going to give a precise meaning of such a statement, and we will illustrate it with examples from black holes physics and three-dimensional quantum gravity.

**D. Pranzetti**

**Horizon entropy with loop quantum gravity and CFT techniques**

The standard black hole entropy calculation in loop quantum gravity (LQG) strongly relies on the interplay with the Chern-Simons theory describing the horizon degrees of freedom. Despite the success of this approach, the standard coupling between certain structures of LQG in the bulk and Chern-Simons theory on the horizon presents a number of ambiguities which affect the entropy calculation and are at the core of some of the still open issues (like the role of the Barbero-Immirzi parameter). In this talk I introduce a new analysis of the horizon degrees of freedom in terms of purely LQG methods. In particular, I show that the spherically symmetric isolated horizon can be described in terms of an SU(2) connection and a su(2) valued one form, obeying certain constraints. The horizon symplectic structure is precisely the one of 3d gravity in a first order formulation. I quantize the horizon degrees of freedom in the framework of loop quantum gravity, with methods recently developed for 3d gravity with non-vanishing cosmological constant. Bulk excitations ending on the horizon act very similar to particles in 3d gravity and the analysis turns out to be dual to a CFT description. The Bekenstein-Hawking law is recovered in the limit of imaginary Barbero-Immirzi parameter, with the Virasoro energy generator of the dual CFT defining a phase-space ‘complexifier’ à la Thiemann.

**R. Santachiara**

**Liouville theory with central charge c less than or equal one**

We make a proposal for the spectrum and correlation functions of Liouville theory with a central charge less than or equal one. Our claims are supported by numerical checks of crossing symmetry. This completes the definition of Liouville theory for all complex values of the central charge.

**F. Saueressig**

**Asymptotic Safety and Quantum Gravity
**

Asymptotic Safety is a mechanism for making a quantum field theory well defined at all energies without being perturbatively renormalizable. Its key ingredient is a non-trivial fixed point of the renormalization group flow which provides the UV completion of the theory. In this talk, I will give a concise introduction to the Asymptotic Safety program before surveying the status and perspectives for promoting the fixed points seen within finite-dimensional approximations to the realm of fixed functionals. Moreover, the connection between Asymptotic Safety and Horava-Lifshitz gravity will be discussed.

**R. van der Veen**

**Three-dimensional geometry and the tensor model**

Motivated by Gurau’s tensor model for random triangulations we survey several features specific to three dimensions. We focus on combinatorial analogues of Perelman’s Ricci flow and Turaev’s theory of shadows of 4-manifolds.