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Workshop Programme

for period 4 - 8 September 2006

Noncommutative Geometry and Physics: Fundamental Structure of Space and Time

4 - 8 September 2006


Monday 4 September
08:30-10:00 Registration
10:00-11:00 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.

11:00-11:30 Coffee
11:30-12:30 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 p-adic numbers instead of real numbers. I'll explain how this idea worked in my recent papers written in collaboration with Kontsevich, Vologodsky and Shapiro.

12:30-13:30 Lunch at Wolfson Court
14:00-15:00 Shatashvili, S (Trinity College Dublin)
  Higgs bundles, gauge theories and quantum groups Sem 1
15:00-15:30 Tea and Poster Session I
15:30-17:30 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 KO-dimension and turning the finite noncommutative geometry F-- whose product with classical 4-dimensional space-time gives the standard model coupled with gravity--into a space of KO-dimension 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 see-saw mechanism for neutrino masses. We shall also discuss three predictions of the model including a relation between boson and fermion masses at unification scale.

17:30-18:30 Welcome Wine Reception
18:45-19:30 Dinner at Wolfson Court (Residents only)
Tuesday 5 September
09:00-10:00 Kreimer, D (IHES)
  Locality and self-similarity Sem 1

The talk points out how the quest for local physics and self-simiilarity combine in Dyson--Schwinger equations. It points out representation theoretic aspects starting from the Hopf- and Lie-algebraic approach to quantum fields. It considers Slavnov--Taylor identities for couplings as an example how quantum gauge symmetry organizes itself independently of a classical Lagrangian and discusses the principles behind.

10:00-11:00 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 well-known identity $Z_1=Z_2$.

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11:00-11:30 Coffee
11:30-12:30 Wulkenhaar, R (Westfalischen Wilhelms-Universitat)
  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_4-model, extended by a harmonic oscillator potential to reflect the UV/IR-entanglement, is renormalisable to all orders in perturbation theory. The behaviour of the \beta-function indicates that a rigorous construction of this model might be possible.

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12:30-13:30 Lunch at Wolfson Court
14:00-15:00 Vignes-Tourneret, F (Universite Paris 11)
  Renormalization of the orientable noncommutative Gross-Neveu model Sem 1

We prove that the non-commutative Gross-Neveu model on the two-dimensional 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.

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15:00-15:30 Tea and Poster Session I
15:30-16:30 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.

16:30-17:00 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 4-dimensional 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 n-point 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.

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17:00-18:30 Poster Session I
18:45-19:30 Dinner at Wolfson Court (Residents only)
Wednesday 6 September
09:00-10:00 Penrose, R (Oxford)
  Spin-networks, twistors, cosmology and all that Sem 1
10:00-11:00 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.

11:00-11:30 Coffee
11:30-12:30 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, non-baryonic 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.

12:30-13:30 Lunch at Wolfson Court
14:00-15:00 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 space-time 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 energy-momentum 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 space-time geometry. These results will be briefly discussed.

15:00-15:30 Tea and Poster Session II
15:30-16:30 Barrett, J (University of Nottingham)
  The physics of three-dimensional quantum gravity Sem 1
16:30-17:00 Goldin, G (Rutgers University)
  Local current algebras for deformed quantum mechanics with fundamental length scales Sem 1

We construct some infinite-dimensional Lie algebras of local currents, compatible with the quantum kinematics of a theory based on a deformed Heisenberg-Poincare 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 Heisenberg-Poincare 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 infinite-dimensional 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 single-particle configuration space, having one dimension more than the original physical space.

Thus, for example, the deformed (1+1)-dimensional theory entails self-adjoint representations of an infinite-dimensional Lie algebra of nonrelativistic, local currents on (2+1)-dimensional space-time. Interestingly, the local operators no longer act in a single irreducible representation of the (global, finite-dimensional) 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 two-space, obeying braid statistics, might also provide local currents for a deformed algebra describing N-particle quantum mechanics in one spatial dimension having a fundamental length scale.

17:00-18:30 Poster Session II
19:30-18:00 Conference Dinner in the Dining Hall at St John's College (Pre-dinner drinks from 19:30)
Thursday 7 September
09:00-10:00 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 self-duality equations for gauge connections.

10:00-11:00 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 Poisson-compatible 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 non-associative 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.

11:00-11:30 Coffee
11:30-12:30 Movshev, M (Stony Brook)
  Supersymmetric Yang-Mills theory and noncommutative supergeometry Sem 1

I will discuss a recent progress in classification and construction of maximally supersymmetric gauge theories.

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12:30-13:30 Lunch at Wolfson Court
14:00-15:00 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 well-known spherical entropy bound.

15:00-15:30 Tea and Poster Session II
15:30-16:00 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 non-formal 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 non-Abelian} Lie groups. The above observation motivates the search of more general universal deformation formulae.

16:00-16:30 Barannikov, S (ENS, Paris)
  Noncommutative Batalin-Vilkovisky geometry, Feynman transform of modular operads and Deligne-Mumford moduli spaces Sem 1

I'll describe the higher genus generalisation of A-infinity algebras and the analogue of Batalin-Vilkovisky geometry arising in this context.

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16:30-17:00 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 1-forms 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 Carnot-Caratheodory (or horizontal) distance.

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18:30-19:30 Dinner at Wolfson Court (Residents only)
19:00-20:00 Pre-event buffet at Emmanuel College (Panel members and guests only)
20:00-22:00 Panel discussion at Emmanuel College
Friday 8 September
09:00-10:00 Douglas, M (Rutgers the State University of NJ)
  Why does space-time 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.

10:00-11:00 Berman, DS (University of London)
  Noncommutative geometry in M-theory Sem 1

We review the applications of noncommutative geometry to M-theory. This also includes extending noncommutative geometry to include the novel geometric situations that occur in M-theory.

11:00-11:30 Coffee
11:30-12:30 Ramgoolam, S (University of London, Queen Mary)
  Large N Expansion of q-deformed two-dimensional Yang-Mills theory and Hecke algebras Sem 1

The q-deformation of the chiral Gross-Taylor holomorphic string large N expansion of two dimensional SU(N) Yang-Mills 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 Schur-Weyl duality between unitary groups and symmetric groups is now played by q-deformed Schur-Weyl 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 hep-th/0603056 , the geometrical story in terms of some q-deformation of Hurwitz spaces is an interesting open question.

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12:30-13:30 Lunch at Wolfson Court
14:30-15:00 Lee, K-M (Korea Institute for Advanced Study)
  The fuzzy space-time Sem 1

We reconsider the fuzzyness of 2-dim noncommutative Minkowski space-time 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 space-time

15:00-15:30 Tea
15:30-16:30 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 "non-geometric" 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.

18:45-19:30 Dinner at Wolfson Court (Residents only)

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