Workshop Programme
for period 4  8 September 2006
Noncommutative Geometry and Physics: Fundamental Structure of Space and Time
4  8 September 2006
Timetable
Monday 4 September  
08:3010:00  Registration  
10:0011:00  Rivasseau, V (Universite d'Orsay)  
The quest for non commutative field theory  Sem 1  
We shall review the ongoing effort to generalize ordinary quantum field theory to non commutative spaces. Scale analysis and renormalization theory lie at the heart of quantum field theory, Even in the simplest Moyal geometry, these concepts require unexpected reworking. Nevertheless we shall argue that they remain essential guidelines both for physics beyond the standard model scale and for condensed matter applications such as a more fundamental theory of the quantum Hall effect. 

11:0011:30  Coffee  
11:3012:30  Schwarz, A (California)  
Space and time from translation symmetry  Sem 1  
It will be shown that the notions of space and time in axiomatic quantum field theory arise from translation symmetry. The possibility of using this construction in string theory will be discussed. The derivation of the notion of space and time can be applied also to the case when the tranlation symmetry is discrete. It seems that in the case of discrete symmetry group it is natural to assume that all basic physical quantities are integer or, at least, rational, and real numbers should be used only for mathematical convenience. If this is the case then one can conjecture that sometimes should be useful to work with padic numbers instead of real numbers. I'll explain how this idea worked in my recent papers written in collaboration with Kontsevich, Vologodsky and Shapiro. 

12:3013:30  Lunch at Wolfson Court  
14:0015:00  Shatashvili, S (Trinity College Dublin)  
Higgs bundles, gauge theories and quantum groups  Sem 1  
15:0015:30  Tea and Poster Session I  
15:3017:30  Connes, A (IHES)  
Noncommutative geometry and the standard model with neutrino mixing  Sem 1  
We show that allowing the metric dimension of a space to be independent of its KOdimension and turning the finite noncommutative geometry F whose product with classical 4dimensional spacetime gives the standard model coupled with gravityinto a space of KOdimension 6 by changing the grading on the antiparticle sector into its opposite, allows to solve three problems of the previous noncommutative geometry interpretation of the standard model of particle physics: The finite geometry F is no longer put in ``by hand" but a conceptual understanding of its structure and a classification of its metrics is given. The fermion doubling problem in the fermionic part of the action is resolved. The action now automatically generates the full standard model coupled with gravity with neutrino mixing and seesaw mechanism for neutrino masses. We shall also discuss three predictions of the model including a relation between boson and fermion masses at unification scale. 

17:3018:30  Welcome Wine Reception  
18:4519:30  Dinner at Wolfson Court (Residents only) 
Tuesday 5 September  
09:0010:00  Kreimer, D (IHES)  
Locality and selfsimilarity  Sem 1  
The talk points out how the quest for local physics and selfsimiilarity combine in DysonSchwinger equations. It points out representation theoretic aspects starting from the Hopf and Liealgebraic approach to quantum fields. It considers SlavnovTaylor identities for couplings as an example how quantum gauge symmetry organizes itself independently of a classical Lagrangian and discusses the principles behind. 

10:0011:00  van Suijlekom, W (MPI, Bonn)  
The Hopf algebra of Feynman graphs in QED  Sem 1  
We discuss the Hopf algebraic description of renormalization theory of quantum electrodynamics. The Ward identities are implemented as linear relations on the Hopf algebra of Feynman graphs of QED. Compatibility of these relations with the Hopf algebra structure is the mathematical formulation of the physical fact that Ward identities are compatible with renormalization. As a result, the counterterms and the renormalized Feynman amplitudes automatically satisfy the Ward identities, which leads in particular to the wellknown identity $Z_1=Z_2$. Related Links


11:0011:30  Coffee  
11:3012:30  Wulkenhaar, R (Westfalischen WilhelmsUniversitat)  
Renormalisation of quantum field theories on noncommutative geometries  Sem 1  
I am going to review the recent progress on the renormalisation programme for quantum field theories on the Moyal plane. There are now three different proofs that the noncommutative \phi^4_4model, extended by a harmonic oscillator potential to reflect the UV/IRentanglement, is renormalisable to all orders in perturbation theory. The behaviour of the \betafunction indicates that a rigorous construction of this model might be possible. Related Links


12:3013:30  Lunch at Wolfson Court  
14:0015:00  VignesTourneret, F (Universite Paris 11)  
Renormalization of the orientable noncommutative GrossNeveu model  Sem 1  
We prove that the noncommutative GrossNeveu model on the twodimensional Moyal plane is renormalizable to all orders. Despite a remaining UV/IR mixing, renormalizability can be achieved. However, in the massive case, this forces us to introduce an additional counterterm of the form "psibar i gamma^{0}gamma^{1} psi". The massless case is renormalizable without such an addition. Related Links 

15:0015:30  Tea and Poster Session I  
15:3016:30  Gayral, V (University of Copenhagen)  
Quantum field theory on projective modules  Sem 1  
I will explain how to construct noncommutative euclidean perturbative vectorial quantum field theories. I will stress on the renormalisation process and treat as a guiding example the case of Heisenberg modules over noncommutative tori. 

16:3017:00  Steinacker, H (Universitat Wien)  
A nontrivial solvable noncommutative $\phi^3$ model in 4 dimensions  Sem 1  
We study the quantization of a noncommutative \phi^3 model on the 4dimensional quantum plane, by mapping it to the Kontsevich model. The model is shown to be renormalizable, using known results for the Kontsevich model, in particular its relation to integrable systems. This allows to obtain the genus expansion of the free energy and of any npoint function, which are finite for each genus after renormalization. A critical coupling is determined, beyond which the model is unstable. This provides a nontrivial interacting NC field theory in 4 dimensions. The case of 2 and 6 dimensions will also be discussed. Related Links


17:0018:30  Poster Session I  
18:4519:30  Dinner at Wolfson Court (Residents only) 
Wednesday 6 September  
09:0010:00  Penrose, R (Oxford)  
Spinnetworks, twistors, cosmology and all that  Sem 1  
10:0011:00  Laemmerzahl, C (ZARM, University of Bremen)  
Experimental search for quantum gravity effects  Sem 1  
The search for QG effects has many aspects: 1. QG effects should violate the Einstein Equivalence Principle (EEP) or fundamentals of Quantum Theory. 2. QG effects are assumed to be very tiny. 3. Need for an appropriate experimental strategy. 4. Which effects are violations of EEP and other fundamental principles? 5. Which effects are QG effects? Accordingly, a first approach for the search of QG effects is to search for violations of the EEP. We will describe in a systematic approach given by the structure of the EEP the huge variety of experiments which are needed to test the EEP and report on the present status of these experiments. When available predictions from quantum gravity inspired scenarios will be presented, too. We also describe experiments in te quantum domain. After the main experimental part we also make short remarks on the strategies of a search for qantum gravity effects, on the magnitude of the expected effects, and on criteria of what can be regarded as a quantum gravity effect. Finally we give some information about which experimental accuracies can be expected in the future. 

11:0011:30  Coffee  
11:3012:30  Taylor, AN (Edinburgh)  
The standard cosmological model  Sem 1  
In the last few years a Standard Model of Cosmology has emerged which describes obervations from the Cosmic Microwave Background, galaxy redshift surveys and distant supernova. This model assumes Einstein gravity, the standard model of particle physics, nonbaryonic dark matter and a mysterious negative pressure dark energy which dominates the energy budget. In this talk I will review the observational basis of this model, discuss its problems, in particular the dark matter and dark energy problems, and point to future tests. 

12:3013:30  Lunch at Wolfson Court  
14:0015:00  Heller, M (Vatican Observatory)  
A noncommutative closed Friedman world model  Sem 1  
In J. Math. Phys. 46, 2005, 122501, we have proposed a model unifying general relativity and quantum mechanics based on a noncommutative algebra A defined on a groupoid having the framer bundle over spacetime as its base space. The generalized Einstein equation is defined in terms of the algebra A and its derivations; matter sources are assumed to vanish. The closed Friedman world model, when computed in this formalism, exhibits two interesting properties. First, additional components of the generalized Einstein equation turn out to be identical with the components of the energymomentum tensor for the usual Friedman model and the corresponding equation of state. One could say that, in this case, matter is produced out of pure (noncommutative) geometry. Second, owing to probabilistic properties of the model, in the noncommutative regime (on the Planck level) singularities are irrelevant. They emerge in the process of transition to the usual spacetime geometry. These results will be briefly discussed. 

15:0015:30  Tea and Poster Session II  
15:3016:30  Barrett, J (University of Nottingham)  
The physics of threedimensional quantum gravity  Sem 1  
16:3017:00  Goldin, G (Rutgers University)  
Local current algebras for deformed quantum mechanics with fundamental length scales  Sem 1  
We construct some infinitedimensional Lie algebras of local currents, compatible with the quantum kinematics of a theory based on a deformed HeisenbergPoincare algebra with fundamental length scales, where the positional operators (in greater than 1+1 dimensions) no longer commute. Recent work by Chryssomalakos and Okon discusses the full set of possible stable deformations of HeisenbergPoincare algebra, with explanation of the relevant cohomology theory; we base our present work on a specific proposal of Vilela Mendes. We begin by clarifying the relation of the irreducible representations of a deformed subalgebra to those of the limiting Heisenberg algebra. Our construction of generalized kinetic energy and harmonic oscillator Hamiltonians in this framework leads to an answer different from that suggested by Vilela Mendes. Then we consider two approaches to local current algebra. First, we localize currents with respect to the discrete spectrum of the deformed position operator, and (as expected) see that the resulting Lie algebra necessarily includes elements having arbitrarily wide support. Second, we extend the usual nonrelativistic local current algebra of scalar functions and vector fields (and, correspondingly, the infinitedimensional semidirect product groups of scalar functions and diffeomorphisms), whose irreducible representations describe a wide variety of quantum systems. The result is to localize with respect to an abstract singleparticle configuration space, having one dimension more than the original physical space. Thus, for example, the deformed (1+1)dimensional theory entails selfadjoint representations of an infinitedimensional Lie algebra of nonrelativistic, local currents on (2+1)dimensional spacetime. Interestingly, the local operators no longer act in a single irreducible representation of the (global, finitedimensional) deformed Lie algebra, but connect the reducing subspaces in a direct integral of irreducible representations. Such an approach seems to open up some new possibilities. For example, representations previously interpreted as describing N indistinguishable anyonic particles in twospace, obeying braid statistics, might also provide local currents for a deformed algebra describing Nparticle quantum mechanics in one spatial dimension having a fundamental length scale. 

17:0018:30  Poster Session II  
19:3018:00  Conference Dinner in the Dining Hall at St John's College (Predinner drinks from 19:30) 
Thursday 7 September  
09:0010:00  Landi, G (Universita di Trieste)  
Some SUq(2) instantons on noncommutative spaces  Sem 1  
We construct examples of instantonic vector bundles on noncommutative spheres and having quantum SU(2) as `structure group'. Some of these spheres are endowed with an isospectral noncommutative spin geometry. The latter geometry could be used to study selfduality equations for gauge connections. 

10:0011:00  Majid, S (University of London, Queen Mary)  
Anomaly for noncommutative differentials and the origin of time  Sem 1  
We take a closer look at a phenomenon whereby many noncommutative coordinate algebras of interest do not in fact admit associative differential calculi that deform classical differentials while preserving expected symmetries. In physical terms there is an anomaly and we show that it is expressed by the curvature of a certain Poissoncompatible preconnection. We show that the anomaly is present for all standard quantum groups C_q(G) and for all enveloping algebras U(g) of a semisimple Lie algebra g (viewed as a quantisation of g^*). Due to this anomaly, if one wishes to preserve classical dimensions one must have nonassociative exterior algebras and we construct these. Alternatively, one must neutralise the anomaly by adding `extra dimensions' in the cotangent bundle. We show how this works for C_q(SU_2) and U(su_2) where the extra dimension can be viewed as a spontaneously generated time with respect to which the original fields naturally obey Schroedinger's equation. We argue that this is a fairly general phenomenon whereby any sufficiently noncommutative differential geometry induces its own `time evolution'. We also argue that these induced extra dimensions in another context are intimately bound up with the renormalisation group. 

11:0011:30  Coffee  
11:3012:30  Movshev, M (Stony Brook)  
Supersymmetric YangMills theory and noncommutative supergeometry  Sem 1  
I will discuss a recent progress in classification and construction of maximally supersymmetric gauge theories. Related Links 

12:3013:30  Lunch at Wolfson Court  
14:0015:00  Rideout, D (Imperial)  
Entropy bound from causal set quantum gravity  Sem 1  
The various entropy bounds that exist in the literature suggest that spacetime is fundamentally discrete, and hint at an underlying relationship between geometry and ``information''. The foundation of this relationship is yet to be uncovered, but should manifest itself in a theory of quantum gravity. We present a measure for the maximal entropy of spherically symmetric spacelike regions within the causal set approach to quantum gravity. In terms of the proposal, a bound for the entropy contained in this region can be derived from a counting of potential ``degrees of freedom'' associated to the Cauchy horizon of its future domain of dependence. For different spherically symmetric spacelike regions in Minkowski spacetime of arbitrary dimension, we show that this proposal leads, in the continuum approximation, to Susskind's wellknown spherical entropy bound. 

15:0015:30  Tea and Poster Session II  
15:3016:00  Bieliavsky, P (Brussels)  
Universal deformation formulae for the `ax+b' group, and noncommutative BTZ black holes  Sem 1  
Rieffel's ``Deformation Quantization for actions of $R^d$'' provides a simple machinery for defining a class of noncommutative Riemannian manifolds within the $C^\star$algebraic framework i.e. within a nonformal context. The basic idea coinsists in viewing Weyl symbol composition formula  in the context of Weyl's quantization of $R^d$ as a {\sl Universal deformation formula}, that is, a formula which defines a $C^\star$ deformation of any $C^\star$algebra endowed with an action of the Abelian Lie group $R^d$. Typical examples obtained from this machinery are noncommutative tori and related noncommutative manifolds. Of course, in many geometrical situations where curvature is involved, one disposes of no action of $R^d$, but rather of actions of {\sl nonAbelian} Lie groups. The above observation motivates the search of more general universal deformation formulae. 

16:0016:30  Barannikov, S (ENS, Paris)  
Noncommutative BatalinVilkovisky geometry, Feynman transform of modular operads and DeligneMumford moduli spaces  Sem 1  
I'll describe the higher genus generalisation of Ainfinity algebras and the analogue of BatalinVilkovisky geometry arising in this context. Related Links


16:3017:00  Martinetti, P (Universita Roma ``la Sapienza")  
Distance in noncommutative geometry  Sem 1  
We shall present an overview of the distance formula in Noncommutative Geometry. In particular we will recall how gauge fields  including a scalar component  naturally acquire a metric interpretation. This will be illustrated on the one side by the Higgs field in the standard model, on the other side by connection 1forms on fibre bundles. In particular we will analyse the link between the distance coming from Noncommutative Geometry and another distance classically associated to connections, namely the CarnotCaratheodory (or horizontal) distance. Related Links


18:3019:30  Dinner at Wolfson Court (Residents only)  
19:0020:00  Preevent buffet at Emmanuel College (Panel members and guests only)  
20:0022:00  Panel discussion at Emmanuel College 
Friday 8 September  
09:0010:00  Douglas, M (Rutgers the State University of NJ)  
Why does spacetime exist?  Sem 1  
We discuss various concepts of space and time coming out of string theory and the related mathematics. Then, to get some insight into the question raised by our title, we make some very speculative comparisons to other imaginable "universes" (formal systems) which might allow for "life," but in which space and time are very different, or do not exist at all. 

10:0011:00  Berman, DS (University of London)  
Noncommutative geometry in Mtheory  Sem 1  
We review the applications of noncommutative geometry to Mtheory. This also includes extending noncommutative geometry to include the novel geometric situations that occur in Mtheory. 

11:0011:30  Coffee  
11:3012:30  Ramgoolam, S (University of London, Queen Mary)  
Large N Expansion of qdeformed twodimensional YangMills theory and Hecke algebras  Sem 1  
The qdeformation of the chiral GrossTaylor holomorphic string large N expansion of two dimensional SU(N) YangMills theory is described. Delta functions on symmetric group algebras are replaced by the corresponding objects (canonical trace functions) for Hecke algebras. The role of the SchurWeyl duality between unitary groups and symmetric groups is now played by qdeformed SchurWeyl duality of quantum groups. For q > 1, the string interpretation of Yang Mills theory relies on the description of Hurwitz spaces in terms of symmetric group data. While the algebraic deformation of the symmetric group data to Hecke algebra data is established in hepth/0603056 , the geometrical story in terms of some qdeformation of Hurwitz spaces is an interesting open question. Related Links 

12:3013:30  Lunch at Wolfson Court  
14:3015:00  Lee, KM (Korea Institute for Advanced Study)  
The fuzzy spacetime  Sem 1  
We reconsider the fuzzyness of 2dim noncommutative Minkowski spacetime in terms of the `harmonic' basis. Fermionic representations of the Lorentz group GL(1,R) appears naturally. The universality of the nonabelain GL(1,R) and GL(N,R) gauge theories are discussed also. We also consider the field theory on this spacetime 

15:0015:30  Tea  
15:3016:30  Hull, C (Imperial)  
Generalisations of geometry and thier role in String theory  Sem 1  
String theory naturally leads to a consideration of backgrounds that are not manifolds but are "nongeometric" with transition functions that include the discrete duality symmetries of string theory. These have an interesting mathematical structure that involves some of the conecpts introduced in Hitchin's Generalised Geometry, but which are more general. 

18:4519:30  Dinner at Wolfson Court (Residents only) 