Quantum Gravity

Trisections were introduced by Gay and Kirby in 2013 as a way to study 4-manifolds. They are similar in spirit to a common tool in a lower dimension: Heegaard splittings of 3-manifolds. In both cases, one understands a manifold by examining the ways that standard building blocks can be put together. They both also have the advantage of changing problems about manifolds into problems about diagrams of curves on surfaces. This talk will be a relaxed introduction to these decompositions.

Tensor models exhibit a melonic large $N$ limit: this is a non trivial family of Feynman graphs that can be explicitly summed in many situations. In $d$ dimensions, they give rise to a new family of conformal field theories and provide interesting examples of the renormalization group flow from a free theory in the UV to a melonic large $N$ CFT in the IR. 

We consider here a bosonic tensor model in rank three and $d<4$ dimensions. After giving a short introduction to tensor models, I will present the renormalization group flow of the model. At leading order in $1/N$ but non perturbatively in the coupling constants, we found a real and infrared fixed point.

What is the black hole in quantum mechanics? We examine this problem in a self-consistent manner. First, we analyze time evolution of a 4D spherically symmetric collapsing matter including the back reaction of particle creation that occurs in the time-dependent spacetime. As a result, a compact high-density star with no horizon or singularity is formed and eventually evaporates. This is a quantum black hole. We can construct a self-consistent solution of the semi-classical Einstein equation showing this structure. In fact, we construct the metric, evaluate the expectation values of the energy momentum tensor, and prove the self-consistency under some assumptions. Large pressure appears in the angular direction to support this black hole, which is consistent with 4D Weyl anomaly. When the black hole is formed adiabatically in the heat bath, integrating the entropy density over the interior volume reproduces the area law. 

Tensor Models provide one of the calculationally simplest approaches to defining a partition function for random discrete geometries. The continuum limit of these discrete models then provides a background-independent construction of a partition function of continuum geometry, which are candadates for quantum gravity. The blue-print for this approach is provided by the matrix model approach to two-dimensional quantum gravity. The past ten years have seen a lot of progress using (un)colored tensor models to describe state sums if discrete geometries in more than two dimensions. However, so far one has not yet been able to find a continuum limit of these models that corresponds geometries with more than two continuum dimensions. This problem can be studied systematically using exact renormalization group techniques. In this talk I will report on joint work with Astrid Eichhorn, Antonio Perreira, Joseph Ben Geloun, Daniele Oriti, Johannes Lumma, Alicia Castro and Victor Mu\~noz in this direction. In a separate part of the talk I will explain that the renormalization group is not only a tool to help investigating the continuum limit, but that it in fact also provides a stand-alone approach to quantum gravity. In particular, I will show how scaling relations follow from cylidrical consistency relations.

We derive Schwarzian correlation functions using the BF formulation of Jackiw-Teitelboim gravity, where bilocal operators are interpreted as boundary-anchored Wilson lines in the bulk. Crossing Wilson lines are associated with OTO-correlators and give rise to 6j-symbols. We discuss the semi-classical bulk black hole physics contained within the correlation functions.

Extensions including bulk defects related to the other coadjoint orbits are discussed.

Gravity theories naturally allow for edge states generated by non-trivial boundary-condition preserving diffeomorphisms. I present a specific set of boundary conditions inspired by near horizon physics, show that it leads to soft hair excitations of black hole solutions and discuss implications for black hole entropy.

I will discuss an attempt at approaching quantum features of spacetime from a microscopic point of view. In combination with quantum gravity, one can consider a scenario in which spacetime has structure at the level of quantum interactions, and classical spacetime emerges at larger scales. This type of scenario would require entangling interactions, and a reliance on information exchange would have effects on the classical field theory.

Quantum gravity may be viewed as a bipartite matter-geometry system. Evolution of matter-geometry entanglement entropy is an interesting question for issues such as the emergence of QFT on curved spacetime from quantum gravity: emergence would require initial states to evolve to product states. We study this question in a cosmological model. We give numerical evidence that matter-geometry entanglement entropy increases indefinitely (with respect to a relational time) for apparently arbitrary initial states. This result raises the question of whether matter and geometry remain entangled in a large Universe.

An assessment of the particle standard model and an alternative formulation of
the model are presented. An ultraviolet complete particle model is constructed
for the observed particles of the standard model. The quantum field theory
associates infinite derivative entire functions with propagators and vertices, which
make perturbative quantum loops finite and maintain Poincaré invariance and
unitarity of the model. The electroweak model SU(2) X U(1) group is treated as a
broken symmetry group with non-vanishing experimentally determined boson
and fermion masses. A spontaneous symmetry breaking of the vacuum by a
scalar Higgs field is not invoked to restore boson and fermion masses to the
initially massless SU(2) X U(1) Lagrangian of the standard model. The hierarchy
naturalness problem of the Higgs boson mass is resolved and the model
contains only experimentally observed masses and coupling constants. The
renormalization group features of the scalar Higgs field are investigated and the
model is shown to be asymptotically safe and free from the triviality problem. It is
demonstrated that the finite nonlocal quantum field theory satisfies
microcausality. The model can predict a stable vacuum evolution. Experimental
tests to distinguish the standard model from the alternative model are proposed.
The finite quantum field theory can be extended to quantum gravity.

I will describe the relevant representation theory that allows to think of all components of fermions of a single generation of the Standard Model as components of a single Weyl spinor of an orthogonal group whose complexification is SO(14,C). There are then only two real forms that do not lead to fermion doubling. One of these real forms is the split signature orthogonal group SO(7,7). I will describe some exceptional phenomena that occur for the orthogonal groups in 14 dimensions, and then specifically for this real form. The real form SO(7,7) suggests a link to generalised geometry, which I will describe. I will also describe why spinors in general are really differential forms in disguise. 

Tensor models are generalizations of vector and matrix models. They have been introduced in quantum gravity and are also relevant in the SYK model. I will mostly focus on models with a U(N)^d-invariance where d is the number of indices of the complex tensor, and a special case at d=3 with O(N)^3 invariance. The interactions and observables are then labeled by (d-1)-dimensional triangulations of PL pseudo-manifolds. The main result of this talk is the large N limit of observables corresponding to 2-dimensional planar triangulations at d=3. In particular, models using such observables as interactions have a large N limit exactly solvable as it is Gaussian. If time permits, I will also discuss interesting questions in the field: models which are non-Gaussian at large N, the enumeration of triangulations of PL-manifolds, matrix model representation of some tensor models, etc.

The goal of this series is to collect various different formulations of General Relativity, with emphasis on four spacetime dimensions and formulations that use differential forms. The (non-exhaustive) list of formulations to be covered is per this plan:

Lecture 1): Motivations, followed by the usual Einstein-Hilbert to start with, first order Palatini, second order pure affine connection Eddington-Schroedinger.

Lecture 2) Cartan's geometry of soldering. First order Einstein-Cartan tetrad formulation, second order pure spin connection formulation, MacDowell-Mansouri formulation.

Lecture 3) Non-chiral BF-type formulations. Explicit pure spin connection Lagrangian, field redefinitions, BF plus potential for the 2-form field formulation.

Lecture 4) Chiral formulations in four dimensions. Chiral Einstein-Cartan, Plebanski formulation, pure SU(2) connection formulation, SU(2) BF plus potential for the 2-form field. Self-dual gravity. Concluding remarks.