The basic idea of the conference is to bring together a mixture of young and well-established researchers, both working at the forefront of quantum field theory and quantum gravity, to discuss their contributions to the recent developments within their respective fields. The fields to be covered are

String Theory

Loop Quantum Gravity

Quantum Field Theory on Curved Space-Time

Non-Commutative Field Theory/Geometry as well as the different approaches to quantum field theory and quantum gravity using

Topos Theory.

The number of participants is limited not to exceed 35 - 40 internationally well-recognized scientists. We hope that such a small number of invited scientists will allow a lively discussion of the subject of the conference.

We plan to have about 12 talks of 45 - 60 minutes each (three in the morning and two in the afternoon session), which are thought to build the bases of the ensuing discussions. The latter are regarded to build the `core' of the meeting.

Following previous work of Hollands and Wald a local version of the renormalization group is developped within the framework of perturbative algebraic quantum field theory. The new formulation admits in particular a detailed analysis of the influence of mass terms.

We summarise the current status of Loop Quantum Gravity. We focus on recent advances concerning the quantum dynamics and the classical limit of the theory.

We start by reviewing some small aspects of the recent work of Kapustin and Witten [KW] on S-duality and the Geometric Langlands Program, especially how this leads to the question of an extension of mirror symmetry to noncommutative spaces. Next, we consider the application of one of the steps of [KW] to the gauge theory on the D5-brane worldvolume in Type IIB string theory. We will see that in this case the inclusion of noncommutative deformations is no longer just an option but is necessary in the generic case. We will show that in a special case one can factorize the fields in such a way that one can reduce everything to questions on mirror symmetry on the noncommutative 2-torus. Following the - by now classical - approach of Dijkgraaf to mirror symmetry on elliptic curves, we will see how q-deformations of special functions arise. We will pose the qustion how this is related to field theories on the noncommutative 2-torus.

In quantum field theory with a non-Gaussian measure, the Green functions are described by Feynman diagrams that are built, not only from a single two-legged free propagator g(x,y), but from n-legged free propagators, where n is any natural number. This increases considerably the combinatorial difficulty of the theory.
Moreover, in the application of non-Gaussian QFT to many-body physics, all n-legged free propagators with n different from 2 are solutions of the free Schroedinger equation. Therefore, we are not allowed to use the inverse of g(x,y) to amputate Green functions because this would kill non-Gaussian contributions to the Green function. As a consequence, computational methods based on the Legendre transformation are not available.
For these reasons, the structure of non-Gaussian Green functions were poorly investigated despite their importance for the calculation of strongly-correlated systems. In this talk, the Hopf algebraic methods developed by Fauser, Frabetti, Oeckl and Mestre will be used to solve this problem.

As an embryonic example of the interplay between geometry and quantum physics in string theory (an example that requires neither knowledge nor appreciation of string theory but also does not do justice to the depth and richness of these ideas in the string theory context), we consider the interplay between geometric properties of plane wave space-time metrics on the one hand and corresponding statements about quantum (gauge) theories on the other.

The principle of the fermionic projector provides a new model of space-time together with the mathematical framework for the formulation of physical equations. After a general introduction to the mathematical setting of fermion systems in discrete space-time, we discuss the underlying physical principles. We set up a variational principle which is symmetric under permutations of the space-time points. We describe a mechanism of spontaneous breaking of this permutation symmetry which leads to relations between the space-time points, in particular to the generation of a so-called discrete causal structure. The connection to the space-time continuum is outlined, and obtained results for the effective continuum theory are given.

Following ideas of Dütsch and Rehren, we give a proof of the equivalence of the two prescriptions of the conformal boundary field in AdS/CFT ("duality"). We use rigorous Euclidean path integrals to define functional delta functions of boundary values and restrictions to the conformal boundary. Both, in the interacting and non-interacting case, divergences arise that have not been observed so far. These divergences can, however, be removed without doing damage to reflection positivity and symmetry. Interactions with IR cut-off are taken into account in D=2 dimensional Euclidean AdS. We comment on some intriguing novel features of the IR-problem which comes from the fact that boundary sources in AdS/CFT lead to shifts in the bulk interaction that create an infinite amount of energy.
This talk is based on joint work with Horst Thaler, math-ph/0611006.

We use Witten's volume formula to calculate the cohomological pairings of the moduli space of flat $SU(3)$ connections. The cohomological pairings of moduli space of flat $SU(2)$ connections is known from the work of Thaddeus-Witten-Donaldson, but for higher holonomy groups these pairings are largely unknown. We make some progress on these problems, and show that the pairings can be expressed in terms of multiple zeta functions.

We present how TQFTs of cohomological type can be described in the context of equivariant cohomology. In this setup, we show how appropriate equivariant localization formulas can be applied to the correlation functions of these theories, yielding to a simpler description of the path integrals involved. In particular, for the topological Yang-Mills theory in four dimensions, we derive (at least formally) Witten's conjecture relating Donaldson and Seiberg-Witten invariants. We conclude by presenting several ideas on how to replace the formal aspects of these techniques with rigorous arguments.

In this talk I will discuss a definition of what a deformation quantization of a surjective submersion should be. Using a purely cohomological approach yields existence and classification up to a natural notion of equivalence. The main application is the deformation quantization of principal fiber bundles which I will discuss in some detail. In particular, they can still be used to associate vector bundles, now in the deformed framework.

Effective actions provide a powerful way to understand low-energy, semiclassical or other regimes of quantum field theories. Although canonical quantum gravity theories, such as loop quantum gravity, are not directly accessible to the usual techniques of computing effective actions, equivalent methods to derive an effective Hamiltonian and effective equations have recently been developed. This provides means to study the semiclassical limit of loop quantum gravity, demonstrating the correct approach to classical behavior and including quantum correction terms. Salient features of loop quantum gravity, its effective equations and applications in cosmological regimes will be discussed.

We compare the structure of the Dyson Schwinger equations (DSE) for three different situations: the case of a renormalizable field theory, a generic non-renormalizable theory and pure gravity. We use the Hochschild cohomology of the underlying Hopf algebra to formulate the relevant DSEs. Remarkably, it turns out that gravity is much better behaved than a generic non-renormalizable theory with respect to its Hochschild cohomology and discuss some consequences.

We review what it means for a field theory to be background-independent, and then analyze how this property may be formulated at the quantum level. We discuss how background independence at the classical level may be broken at the quantum level, and we mention some implications for quantum gravity.

In this talk, I will report on recent work with Chris Isham. The basic idea is that a physical system can be represented by structures within a suitable topos associated with the system. The choice of the topos depends on the type of the system (classical, quantum or even some new kind). Within the topos, there is a state object and a quantity value object, and physical quantitites are suitable morphisms between them. Propositions about physical quantities are represented by subobjects of the state object, which form a Heyting algebra. A tentative set of rules for formulating physical theories within a topos that does not fundamentally depend on the continuum (in the form of the real or complex numbers) is presented.