Plenary Sessions:

Group and Algebraic Methods and Hamiltonian Systems

(Monday, 04/24)  

R. Coquereaux (CPT, Marseille, France)

Quantum Symmetries of Graphs and Higher Coxeter-Dynkin Systems

Abstract: At the end of the eighties, it was shown by Cappelli, Itzykson, Zuber that conformal field theories like affine WZW models of type SU(2) or minimal (Virasoro) models can be classified in terms of ADE Dynkin diagrams. A few years later, new kinds of diagrams have been introduced by Di Francesco and Zuber in order to classify conformal field theory models of type SU(3). We describe quantum symmetries of Dynkin diagrams or of their higher analogues by associating with every such graph a quantum groupoid (a weak Hopf algebra) and two algebras of characters: the first is related with the set of irreducible representations of quantum groups at roots of unity and the other with the so-called Ocneanu graph of quantum symmetries. To every such quantum groupoid correspond several sets of generalized Racah-Wigner coefficients (Ocneanu cells or generalized 6J symbols). Results can also be interpreted in terms of fusion categories. The aim of this seminar is to present a general introduction to the above ideas.

B. Dubrovin (SISSA, Italy)

Universality problems in Hamiltonian PDEs

Abstract: Certain properties of solutions to nonlinear PDEs turn out to be essentially independent on the choice of initial data; they also manifest remarkable stability with respect to small deformations of the PDE. Such properties were first discovered in the study of eigenvalue distributions in large N random matrices, they also appear in the Fermi - Pasta - Ulam numerical experiments and in the theory of weakly dispersive waves. In the talk it will be presented a general point of view on the universality phenomena using the theory of Hamiltonian perturbations of hyperbolic PDEs.

Integrable Systems and Gauge Theories

(Tuesday, 04/25)

F. Khanna (Univ. of Alberta, Canada)

Ward-Takahashi Relations: Longitudinal and Transverse

Abstract: Ward [1] wrote down a relation between the vertex and the electron propagator in Spinor QED in 1950. This was generalised to arbitrary fields by Takahashi [2] in 1957. These relations provided the only non-perturbative results for a Quantum Field Theory. These relations provided consistency conditions in solutions in Quantum Field Theory and Many-body Physics, where perturbative treatments were used. The results in perturbative treatment were required to satisfy the basic symmetry aspects in the problem. Use of Ward-Takahashi relations was crucial in getting results that satisfied the symmetry of the problem. However it was noticed soon that the results provided only a limitation on the longitudinal part of the vertex function. The transverse part was not constrained by these relations. In 1977, Takahashi[3] proposed as way out of this dilemma. Recently we have developed this technique that allows us to write down non-perturbative relations that restrict the transverse part of the vertex. Some recent results will be presented for a Fermion-Boson vertex at one loop order. This development makes the generalised Ward-Takahashi relations much more useful in the study of covariant Gauge theories.

References

[1] J.C.Ward,Phys.Rev.78(18950)182.
[2] Y.Takahashi,Nuovo Cimento 6(1957)370.
[3] Y.Takahashi,Phys.Rev.d15(1977)1589.

Quantum Field Theory

(Tuesday, 04/25, and Wednesday, 04/26)  

C. Nuñez (IABA, Argentina)

Free fields for non-RCFT?

Abstract: The free field realization of rational conformal field theories has played a crucial role to determine the structure of minimal models and of WZW models on compact groups. In this talk we will discuss the possibility of extending the free field representation of RCFT to non RCFT. We will consider in particular the non compact SL(2,R)-WZW model and compare the exact results of correlation functions of primary fields with those computed in the free field approximation.

O. Piguet (UFES, Brazil)

A Vector Supersymmetry Killing the Infrared Singularity of Gauge Theories in Noncommutative space 

ABSTRACT: We show that the "topological BF-type" term introduced by Slavnov in order to cure the infrared divergences of gauge theories in noncommutative space can be characterized as the consequence of a new symmetry. This symmetry is a supersymmetry, generated by vector charges,of the same type as the one encountered in Chern-Simons or BF topological theories (Work done in collaboration with Daniel N. Blaschke (Technische Universität Wien), François Gieres (Université de Lyon I) and Manfred Schweda (Technische Universität Wien)).

F. L. Williams (Univ. of Massachussetts, EUA)

Remarks on the BTZ Instanton with Conical Singularity

Abstract: We consider the topology,trace formula,effective action,and the Mann-Solodukhin quantum correction to the entropy of the BTZ instanton with a conical singularity at its horizon.The trace formula,in particular,provides for an alternate approach to the Patterson resolvent formula.The effective action and black hole entropy correction are expressed in terms of a suitable zeta function deformation.

Principles of Quantum Theory

(Thursday, 04/27)  

J. P. Gazeau (Univ. Paris VII, France)

A Survey of Recent Results for Quantum Field Theory in de Sitter Space

Abstract: We present a survey of rigourous quantization results obtained in recent works on quantum free fields in de Sitter space-time. For the “massive” cases which are associated to principal series representations of the de Sitter group SO0(1, 4), the construction is based on analyticity requirements on the Wightman two-point function. For the “massless” cases (e.g. minimally coupled or conformal), associated to the discrete series (and possibly to the complementary series), the quantization schemes are of the Gupta-Bleuler-Krein type. Some hints will be given on the question of interacting fields in de Sitter.

References

[1] R. Bousso, A. Maloney, A. Strominger, Phys. Rev. D 65, 104039 (2002).
[2] R. Hollands, R. Wald, Commun. Math. Phys., 231, (2002).
[3] P.Bartesaghi, J-P. Gazeau, U. Moschella and M. V. Takook, ”Dirac fields and thermal effects in de Sitter universe”, Class. Quant. Grav., 18, 4373 (2001). J-P. Gazeau and M. V. Takook, ”Massive vector fields in de Sitter space” J. Math. Phys., 41, 5920 (2000) et J. Math. Phys., 43, 6379 (2002). T. Garidi, J-P. Gazeau and M. V. Takook, ”Massive spin-2 field in de Sitter space”, J. Math. Phys., 44, 3938 (2003).
[4] T. Garidi, J-P. Gazeau, and M. V. Takook, “Massless vector field in de Sitter space”, in preparation.
[5] J-P. Gazeau, J. Renaud, M. V. Takook, ”Gupta-Bleuler Quantization for minimaly coupled Scalar Fields in de Sitter Space” Class. Quant. Grav, 17, 1415 (2000).
[6] T. Garidi, E. Huguet and J. Renaud, ”de Sitter waves and the zero curvature limit” Phys. Rev. D., 67, 124028 (2003).
[7] J. Bros, H. Epstein and U. Moschella, “The asymptotic symmetry of de Sitter spacetime,” Phys. Rev. D 65, 084012 (2002). [arXiv:hep-th/0107091].

Classical and Quantum Gravity and Cosmology

(Thursday, 04/27)  

E. Elizalde (IEEC, Spain)

On Zeta Regularization and Some of its Uses in Cosmology

Abstract: Zeta regularization has proved to be a powerful and reliable tool for the regularization of the vacuum energy density in ideal situations. With the additional help of the Hadamard calculus, it can been shown to provide finite (and meaningful) answers too in much involved cases, as when imposing physical boundary conditions (BCs) in two-- and higher--dimensional surfaces (being able to mimic in a very convenient way other ad hoc cut-offs, as non-zero depths). These recent developments will be described in the first part of the presentation. Recently, these techniques have been also used in calculations of the contribution of the vacuum energy of the quantum fields pervading the universe to the cosmological constant (cc). Naive calculations of the absolute contributions of all known fields lead to a value which is off by roughly 120 orders of magnitude, as compared with the results obtained from observational fits, what is known as the new cosmological constant problem. This is very difficult to solve and we are not going to address such issue directly.
What we will consider are the additional contributions to the cc that may come from the possibly non-trivial topology of space and from specific boundary conditions imposed on braneworld models (kind of Casimir effects at a cosmological scale). Assuming one will be able to prove (in the future) that the ground value of the cc is zero (as some have always suspected), we will then be left with this incremental value coming from the topology or Bcs. This value can be shown to acquire the correct order of magnitude -- corresponding to the one coming from the observed acceleration in the expansion of our universe -- in a number of reasonable situations involving small and large compactified scales and/or brane BCs.

M. Novello (ICRA/CBPF, Brazil)

The Cosmological Constant and the Mass of the Graviton

Abstract: Recently, a new formulation to deal with the consistency problem of both massive and massless spin-2 fields in arbitrary curved spacetime was presented. Using three-index tensor (Fierz-frame ) to represent the spin-2 field it was shown how to avoid the arbitrariness and inconsistency that exist in the standard ( Einstein-frame ) formulation of a spin-2 field that deals with a second order symmetric tensor. As a consequence of this a relation between the fundamental (bare) cosmological constant and the mass of the graviton appears. As a by-product it is possible to associate the so called cosmological puzzle --- i.e., the enormous value of the ratio ?pl/ ? vac~ 10 120 - to the total number of gravitons in the observable universe.

Strings, Branes and Supersymmetric Theories

(Friday, 04/28)  

N. Berkovits  (IFT/UNESP, Brazil)

On the Calculation of Superstring Multiloop Amplitudes

Abstract: In this talk, I will review the different methods for computing multiloop amplitudes in superstring theory. These methods include the light-cone Green-Schwarz formalism, the light-cone and covariant Ramond-Neveu-Schwarz formalism, and the covariant pure spinor formalism. The pure spinor formalism is the only method which preserves all symmetries of the target space, and is the most convenient method for analyzing the finiteness and duality properties of the scattering amplitudes.

L. Bonora (SISSA, Italy)

Gravitational Droplets and String Field Theory

Abstract: The talk concerns the analogy between a class of 1/2 BPS solutions of N=4 U(N) SYM theory in 4d and a parallel class of solutions of vacuum bosonic string field theory. The correspondence is striking and is all but accidental. It stems from the possibility to represent 1/2 BPS states by means of fermionic systems and on a perfect isomorphism between systems of fermions and solutions of vacuum string field theory. The large N limit shapes these systems into droplets that are seen to describe to 1/2 BPS supergravity solutions. It is argued that this correspondence fits very well into open-closed string duality. The talk will also analyse the problem of finding analogous solutions in superstring field theory.