Quantum Gravity

I will review the construction of lattice theories which maintain one or more exact supersymmetries for non zero lattice spacing concentrating in particular on the case of N=4 super Yang-Mills. Such lattice theories may be studied using Monte Carlo techniques borrowed from lattice QCD and can be used to explore issues in holography. In three dimensions the same constructions can be used to formulate a topological theory of gravity which we argue is equivalent to Witten's Chern Simons theory.

We introduce an exactly solvable model to test various proposals for the imposition of the spin foam simplicity constraints. This model is a three-dimensional Holst-Plebanski action for the gauge group SO(4), in which the simplicity constraints mimic the situation of the four-dimensional theory. In particular, the canonical analysis reveals the presence of secondary second class constraints conjugated to the primary ones. We perform the spin foam quantization of the theory in the spirit of the BC and EPRL models, and give arguments for modifying the measure over the holonomies in order to account for the presence of the secondary second class constraints.

During the last couple of years Dupuis, Freidel, Livine, Speziale and Tambornino developed a twistorial formulation for loop quantum gravity.
Constructed from Ashtekar--Barbero variables, the formalism is restricted to SU(2) gauge transformations.
In this talk, I perform the generalisation to the full Lorentzian case, that is the group SL(2,C).
The phase space of SL(2,C) (i.e. complex or selfdual) Ashtekar variables on a spinnetwork graph is decomposed in terms of twistorial variables. To every link there are two twistors---one to each boundary point---attached. The formalism provides a clean derivation of the solution space of the reality conditions of loop quantum gravity.
Key features of the EPRL spinfoam model are perfectly recovered.
If there is still time, I'll scatch my current project concerning a twistorial path integral for spinfoam gravity as well. 

The emergence of fractal features in the microscopic structure of space-time is a common theme in many approaches to quantum gravity. In particular the spectral dimension, which measures the return probability of a fictitious diffusion process on space-time, provides a valuable probe which is easily accessible both in the continuum functional renormalization group and discrete Monte Carlo simulations of the gravitational action. In this talk, I will give a detailed exposition of the fractal properties associated with the effective space-times of asymptotically safe Quantum Einstein Gravity (QEG). Comparing these continuum results to three-dimensional Monte Carlo simulations, we demonstrate that the resulting spectral dimensions are in very good agreement. This comparison also provides a natural explanation for the apparent conflicts between the short distance behavior of the spectral dimension reported from Causal Dynamical Triangulations (CDT), Euclidean Dynamical Triangulations (EDT), and Asymptotic Safety.

We relate the discrete classical phase space of loop gravity to the continuous phase space of general relativity. Our construction shows that the flux variables do not label a unique geometry, but rather a class of gauge-equivalent geometries. We resolve the tension between the loop gravity geometrical interpretation in terms of singular geometry, and the spin foam interpretation in terms of piecewise-flat geometry, showing that both geometries belong to the same equivalence class. We also establish a clear relationship between Regge geometries and the piecewise-flat spin foam geometries. All of this is based on arXiv:1110.4833.

Tensor models appear as the higher dimensional extension of the so-called matrix models describing 2D quantum gravity through the sum over triangulations of surfaces. In the light of the recent $1/N$ expansion for these tensor models, we uncover a new class of tensor models for 4D and 3D gravity which are renormalizable at all orders of perturbation theory. An overview of two papers, [arXiv:1111.4997 [hep-th]] and [arXiv:1201.0176 [hep-th]], on the renormalization of these tensor models and their beta function will be given.

Group field theories show up as a higher dimensional generalization of matrix models in background independent approaches to quantum gravity.
Their Feynman expansion generates simplicial complexes of all topologies weighted by spin foam amplitudes.
In this talk, we will present a dual formulation of these theories as non-commutative quantum fields theories, whose variables have a clear interpretation in terms of simplicial geometry. We will show that it gives a geometrically clear ways to define spin foam models for gravity which can be cast as

To study the continuum limit of a microscopic model of gravity we need microscopic observables that have a clear interpretation in terms of continuum geometry. In general the construction of such observables is notoriously difficult. In the model of causal dynamical triangulations (CDT) it is clear what the microscopic observables are, but at present the only known well-behaved observables with a continuum interpretation are spatial volumes. In this talk I will demonstrate what it takes to go beyond these by introducing the moduli as observables for CDT in 2+1 dimensions with spatial topology of the torus. Measurements of these observables using computer simulations provide valuable clues concerning the effective action describing CDT in the continuum. In particular I will present numerical evidence indicating that the effective kinetic term is well described by a modified Wheeler-De Witt metric like the one appearing in Horava-Lifshitz gravity.

Entanglement is a paradigmatic example of quantum correlations, a presumed reason for the superior performance of quantum computation and an obvious divider of states and processes into classical and quantum. In the last decade all these notions were challenged. Entanglement does not capture the totality of non-classical behavior. Quantum discord (in its different versions) is a more general measure of quantum correlations. It can be related to the advantage in some tasks like the extraction of work from a Szilrad heat engine using Maxwell's demons with various resources. The discord turns out to be the difference between the work extracted from a given bipartite system using a a global and a local strategy. Different strategies relate to different definitions of discord, but all definitions agree on zero, so "classical" systems are universal to all heat engines. One way of identifying a task as quantum or classical is by examining the quantum resources required to implement it. This can be done by examining the entanglement required for a LOCC implementation of the task. Creation (or non-creation) of entanglement as a result of its implementation is not enough to identify these resources. An example is a bi-local implementation of an entangling quantum gate (C-NOT) by LOCC with unentagled input and output states. This lack of entanglement does not guarantee the LOCC implementability. A method to determine if entanglement resources are required is to track the change in quantum discord during the process.

We construct the q-deformed spinfoam vertex amplitude using Chern-Simons theory on the boundary 3-sphere of the 4-simplex. The rigorous definition involves the construction of Vassiliev-Kontsevich invariant for trivalent knot graph. Under the semiclassical asymptotics, the q-deformed spinfoam amplitude reproduce Regge gravity with cosmological constant at nondegenerate critical configurations.