The n-Category Café

Here are the lecture courses:

• Abhay Ashtekar, Loop quantum gravity.
  
  Abstract: This set of lectures will provide an introduction to loop quantum gravity through the simpler setting of loop quantum cosmology. The goal will be to provide a concise summary of the conceptual framework, salient results and open issues. The time limitation will not permit me to give detailed proofs and technical details for which I will provide a guide to literature.
  
  Table of Contents
  
  
  • Background independence and non-perturbative methods.
  • General relativity in terms of connection variables.
  • Loop quantum cosmology: Kinematics.
  • Loop quantum cosmology: Dynamics.
  • Principal results and open problems of loop quantum gravity.
  
  
• John Baez, Categorification in fundamental physics.
  
  Abstract: Categorification is the process of replacing set-based mathematics with analogous mathematics based on categories or n-categories. In physics, categorification enters naturally as we pass from the mechanics of particles to higher-dimensional field theories. For example, higher gauge theory is a generalization of gauge theory that describes the parallel transport not just of particles, but also strings or higher-dimensional branes. To handle strings, we must categorify familiar notions from gauge theory and consider connections on “principal 2-bundles” with a given “structure 2-group”. One of the simplest 2-groups is the shifted version of U(1). U(1) gerbes are really principal 2-bundles with this structure 2-group, and the $B$ field in string theory can be seen as a connection on this sort of 2-bundle. The relation between U(1) bundles and symplectic manifolds, so important in the geometric quantization, extends to a relation between U(1) gerbes and “2-plectic manifolds”, which arise naturally as phase spaces for 2-dimensional field theories, such as the theory of a classical string. More interesting 2-groups include the “string 2-group” associated to a compact simple Lie group $G$. This is built using the central extension of the loop group of $G$. A closely related 3-group plays an important role in Chern–Simons theory, and it appears that $n$-groups for higher $n$ are important in the study of higher-dimensional membranes.
  
  Table of Contents
  
  
  • Connections on abelian gerbes.
  • Lie n-groups and Lie n-algebras.
  • Multisymplectic geometry and classical field theory.
  • Higher gauge theory, strings and branes.
    
    
• John Barrett, Spin networks and quantum gravity.
  
  Abstract: The series of lectures will be devoted to explaining techniques of spin networks and outlining their use in models of quantum space-time and quantum gravity. The lectures will start with the classical SU(2) spin networks, explaining the diagrammatical techniques and the construction of the Ponzano–Regge model of 3d quantum gravity. Then the q-deformation of spin networks and the Turaev–Viro model are constructed, together with an explanation of the completion to a topological quantum field theory. Next, observables are introduced in these models, and some related models of quantum space-time are also mentioned. Finally, there will be an introduction to some four-dimensional models, both the topological ones, and, briefly, an outline of four dimensional gravity models.
  
  
• Vincent Rivasseau, Renormalization in fundamental physics.
  
  Abstract: Renormalization was first invented to cure the short distance singularities in quantum field theory. Simultaneously constructive field theory developed combinatoric tools to also attack the neglected divergence of perturbation theory. It was later understood that the renormalization group is the correct tool to track the change of physical phenomena under change of observation scale. Then it was realized that the correct notion of scale is not always naively related to short or long distance phenomena, but rather to the spectrum of the propagator. This allowed in the recent years to understand how to renormalize noncommutative field theory, and to attack with a fresh look and new hopes the problem of renormalizing quantum gravity.
  
  Table of contents
  
  
  • Renormalization in ordinary QFT.
  • Constructive Field Theory Primer.
  • Noncommutative Field Theory.
  • Noncommutative Renormalization.
  • Towards renormalizing Quantum Gravity.
    
    
• Carlo Rovelli, Covariant loop quantum gravity and its low-energy limit.
  
  Abstract*: I present a new look on Loop Quantum Gravity, aimed at giving a better grasp on its dynamics and its low-energy limit. Following the highly succesfull model of QCD, general relativity is quantized by discretizing it on a finite lattice, quantizing, and then studying the continuous limit of expectation values. The quantization can be completed, and two remarkable theorems follow. The first gives the equivalence with the kinematics of canonical Loop Quantum Gravity. This amounts to an independent re-derivation of all well known Loop Quantum gravity kinematical results. The second the equivalence of the theory with the Feynman expansion of an auxiliary field theory. Observable quantities in the discretized theory can be identifies with general relativity n-point functions in appropriate regimes. The continuous limit turns out to be subtly different than that of QCD, for reasons that can be traced to the general covariance of the theory. I discuss this limit, the scaling properties of the theory, and I pose the problem of a renormalization group analysis of its large distance behavior.
  
  
  * tentative

The summer school is now accepting applications. Maybe I’ll see you there.