Abstracts of Lectures

Chern classes of singular varieties, graph hypersurfaces and Feynman integrals
Paolo Aluffi (Florida State University, USA)    

Numerical evidence indicates that the values of individual contributions of graphs to Feynman integrals are multiple zeta values. This fact can be interpreted as a statement concerning certain hypersurfaces of projective space determined by graphs. I will report on work aimed at understanding this phenomenon. I will particularly focus on the role played by characteristic classes in this study, as they give a concrete measure of the singularity of graph hypersurfaces, and lead to an algebro-geometric construction of invariants satisfying Feynman rules. The necessary prerequisites on characteristic classes of singular varieties will be covered.

 Noncommutative spacetimes and quantum physics
A.P. Balachandran (Syracuse University, New York, USA)

In these talks, the motivation for noncommutative spacetimes coming from quantum gravity will be reviewed. The implementation of symmetries coming from theory of Hopf algebras on such spacetimes and the attendant twist of quantum statistics will be discussed in detail. These ideas will be applied to construct covariant quantum fields on such spacetimes and develop quantum field theories including gauge theories. The formalism will then be applied to cosmic microwave background and quantum gravity. It will be explained as to how these models predict apparent non-Pauli transitions. Using experimental data, limits on the scale of noncommutativity will be derived.

Index theory and geometric quantization of non-compact manifolds
Maxim Braverman (Northeastern University, Boston, USA)

The course will present recent attempts to extend the notion of geometric quantization to non-compact manifolds. The plan of the lectures:
1. Introduction to the Atiyah-Singer index theorem
2. Index theorem for transversally elliptic operators
3. From classical to quantum mechanics. Geometric quantization.
4. Symmetries in classical and quantum mechanics. Noether theorem and symplectic reduction. Quantization commutes with reduction theorem.
5. Index theory on non-compact manifolds.
6. Geometric quantization of non-compact manifolds and applications.

Supergeometry and applications to Poisson geometry
Alberto Cattaneo (Universität Zürich, Switzerland)

The langauge of supergeometry allows for a compact description of properties of Poisson manifolds as well as other geometric structures (like courant algebroids and generalized complex structures) together with their reduction theories and possibly quantizations.

Compactifications of string theory and generalized geometry
Mariana Graña (Institut de Physique Théorique CEA-Saclay, France)

In this course we will begin by reviewing the low energy limit of string theory where only the massless excitations of the string are kept. The theory is defined consistently in ten dimensions, and we will discuss compactifications on six-dimensional manifolds giving  rise to four-dimensional effective theories. We will show how these are nicely described by a generalized version of riemannian geometry, namely generalized complex (and generalized exceptional) geometry, the first form of which was introduced by Hitchin. We will see how the four-dimensional effective action and its vacua have a covariant description in generalized geometry, and also how generalized geometry naturally describes “non-geometric compactifications”, where the   internal space does not have a geometrical interpretation.

Spectral Geometry
Bruno Iochum (Centre de Physique Théorique, Marseille, France)

The goal of these lectures is to present the few fundamentals of noncommutative geometry looking around its spectral approach. Strongly motivated by physics, in particular by relativity and quantum mechanics, Chamseddine and Connes have defined an action based on spectral considerations, the so-called spectral action. The idea is to review the necessary tools which are behind this spectral action to be able to compute it first in the case of Riemannian manifolds (Einstein-Hilbert action). Then, all primary objects defined for manifolds will be generalized to reach the level of noncommutative geometry via spectral triples, with the concrete analysis of the noncommutative torus which is a deformation of the ordinary one. The basics of different ingredients will be presented and studied like, Dirac operators, heat equation asymptotics, zeta functions and then, how to get within the framework of operators on Hilbert spaces, the notion of noncommutative residue, Dixmier trace, pseudodifferential operators, etc. These notions are appropriate in noncommutative geometry to tackle the case where the space is swapped with an algebra like for instance the noncommutative torus.

Noncommutative geometry models in particle physics and cosmology
Matilde Marcolli (California Institute of Technology, Pasadena, USA)

Noncommutative metric geometry (spectral triples) and the spectral action functional can be used to construct particle physics models in which the matter lagrangian, coupled to gravity, is computed from the asymptotic expansion of the spectral action. In recent work with Chamseddine and Connes, we developed a version of these NCG models that accommodate, beyond the minimal standard model, additional terms with right handed neutrino with majorana mass terms and neutrino mixing. In more recent work with Pierpaoli, we investigated cosmological predictions on the early universe based on these noncommutative geometry models of gravity coupled to matter. Using the renormalization group analysis for the standard model with right handed neutrinos and Majorana mass terms, we analyzed the behavior of the coefficients of the gravitational and cosmological terms in the lagrangian derived from the asymptotic expansion of the spectral action functional of noncommutative geometry. One can find in this way various cosmological phenomena: Emergent hoyle-narlikar and conformal gravity at the see-saw scales, and a running effective gravitational constant, which affects the propagation of gravitational waves and the evaporation law of primordial black holes and provides linde models of negative gravity in the early universe. The same renormalization group analysis also governs the running of the effective cosmological constant of the model. The model also provides a Higgs based slow-roll inflationary mechanism, for which one can explicitly compute the slow-roll parameters. The particle physics content allows for dark matter models based on sterile neutrinos with Majorana mass terms.

Expansions: A loosely tied traverse from Feynman diagrams to quantum algebra
Dror Bar Natan (University of Toronto, Canada)

Assuming lots of luck, in six classes we will talk about:
1. Perturbed Gaussian integration in Rn and Feynman diagrams.
2. Chern-Simons theory, knots, holonomies and configuration space integrals.
3. Finite type invariants, chord and Jacobi diagrams and "expansions".
4. Drinfel'd associators and knotted trivalent graphs.
5. w-Knotted objects and co-commutative Lie bi-algebras.
6. Dreams on virtual knots and and quantization of Lie bi-algebras.
Each class will be closely connected to the next, yet the first and last will only be very loosely related.
http://wwth.toto.edu/~drorbn/Talks/Colombia-1107/
Integrability and the AdS/CFT correspondence
Matthias Staudacher (Humboldt Universität zu Berlin and Max-Planck-Institut für Gravitationsphysik, Germany)

I will give an introduction to the quantum integrability of spin chains and two-dimensional quantum field theories. I will then apply these methods to the AdS/CFT correspondence. This allows to obtain exact results, both perturbative and non-perturbative, for supersymmetric N=4 Yang-Mills gauge theory, a four-dimensional quantum field theory. Finally it enables us to gain insight into strongly coupled gauge theory, and therefore, via the AdS/CFT correspondence, into superstring theory on the anti-de-Sitter background.

 Introductory lectures on Chern-Simons theories in physics
Jorge Zanelli (Centro de Estudios Científicos, Valdivia, Chile)

In this broad introduction, the role of Chern-Simons (CS) theories in physics will be reviewed. The simplest, 0+1 CS system, describing a point charge coupled to an electromagnetic field (minimal coupling), will be used as the prototype for general CS systems in higher dimensions and for non-abelian connections. The relevance of these systems to describe classical and quantum mechanics, as well as more elaborate gauge field theories, including gravity and supergravity, will also be discussed.