ESF RESEARCH NETWORKING PROGRAMME on
Quantum Geometry and Quantum Gravity
Quantum Gravity and Quantum Geometry
We organize an intensive one-week School on Quantum Gravity and
related topics in Corfu from Sunday 13 th till Sunday 20 th of
It will provide an up-to date introduction to some of the main
research topics of the Quantum Geometry and Quantum Gravity ESF
Network, including loop quantum gravity, supergravity,
renormalization theory, higher gauge theories and quantum spaces.
A series of Workshop-type seminars by international experts will
provide a link to current research.
George Zoupanos (principal applicant), Harald Grosse, Larisa Jonke.
In addition, there will be a number of seminars given by invited speakers.
- Prof. Abhay Ashtekar
Title: Loop Quantum Gravity
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.
- Prof. John Baez
Title: Categorification in Fundamental Physics
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
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.
- Prof. John Barrett
Title: Spin networks and quantum gravity
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.
- Prof. Vincent Rivasseau
Title: Renormalization in Fundamental Physics
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
Table of contents
- Renormalization in ordinary QFT.
- Constructive Field Theory Primer.
- Noncommutative Field Theory.
- Noncommutative Renormalization.
- Towards renormalizing Quantum Gravity.
- Prof. Carlo Rovelli
Title: Covariant loop quantum gravity and its low-energy
Abstract and content (**)
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.