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Balachandran, Aiyalam Parameswaran (Physics Department,Syracuse University, Syracuse, NY) Noncommutative Geometry:Fuzzy Spaces, the Groenewold-Moyal Plane This is a review talk describing how ideas of noncommutative geometry have been emerging in fundamental physics in recent years from diverse sources like quantum gravity, string physics,regularisation techniques of quantum field theories (qft's)etc.It also outlines the basic ideas of fuzzy physics and qft's on the Groenewold-Moyal (GM) plane.Recent developments regarding the implementation of diffeomorphism groups on the GM plane and its consequences for spin and statistics are alluded to. |
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Forgács, Péter (KFKI RMKI, Budapest) PDF Twisted vortices in extended Abelian Higgs models A new class of twisted, current carrying, stationary, straight string solutions having finite energy per unit length is constructed numerically in extended Abelian Higgs models with a global SU(N) symmetry. The new solutions correspond to deformations of the embedded Abrikosov-Nielsen-Olesen (ANO) vortices. For each value of the winding number n=1,2 (determining the magnetic flux through the plane orthogonal to the string) there are n distinct, one parametric families of solutions. The continuously varying parameter is the twist, or equivalently the induced global current flowing through the string. For a fixed value of the twist, the n distinct solutions have different energies, there is a lowest energy ``fundamental'' string and its n-1 ''excitations''. In the stationary case the relative phase of the two scalar fields rotates with constant velocity, giving rise to angular momentum of the string together with a screened electric field in the plane orthogonal to the string. The static twisted vortex solutions have lower energy than the embedded ANO vortices and could be of considerable importance in various physical systems (from condensed matter to cosmic strings). |
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Gawedzki, Krzysztof (ENS, Lyon) PDF WZW models and gerbes Lagrangian formulation of the Wess-Zumino-Witten models is intimately related to bundles gerbes, geometric objects generalizing line bundles to the stringy context. I will discuss how classification of such objects and of the so called gerbe modules enters the exact solution of the WZW models on the classical and quantum level. |
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Harnad, John (Concordia University and Centre de recherches mathematiques, Universite de Montreal) PDF Matrix Models, Integrable Systems and Riemann-Hilbert Methods This talk will largely be introductory, aimed at those who are relatively unfamiliar with matrix models and their relations to integrable systems. It will aim at giving an overview of selected results from random matrix theory, including some of relatively recent date, on two-matrix models. The subject and its interrelations with other fields is rather vast. The talk will aim to provide at least partial answers to the following questions, depending on the time available: 1) What are matrix models, and how are they related to integrable systems? 2) What are their possible applications? 3) What are the key known results on spectral statistics for finite N models? 4) What is "universality", in the large N limit, and what is the role of the Riemann-Hilbert method in proving this? 5) What are the ongoing problems of interest? |
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Horváthy, Péter (University of Tours) PDF Exotic galilean symmetry and non-commutative mechanics The ``exotic'' particle model with non-commuting position coordinates, associated with the two-parameter central extension of the planar Galilei group, can be used to derive the ground states of the Fractional Quantum Hall Effect. The relation to other NC models is discussed. Anomalous coupling is presented. Similar equations arise for a semiclassical Bloch electron, used to explain the anomalous/spin/optical Hall effects. |
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Jorjadze, George (Razmadze Mathematical Institute) PS Open problems of Liouville theory The scheme of canonical quantization of Liouville theory is based on two canonical transformations. The first one relates the asymptotic in-out fields and the second one maps in (or out) field to the Liouville field. The generating functional of these transformations are known and some properties of the corresponding quantum transformations were also discussed in the literature, but a global compact form of these `unitary' operators is still unknown. The first operator is the Liouville S-matrix and the second one is the Moller matrix, which transforms the free-field exponentials to the Liouville vertex operators. The both operators commute with conformal transformations and we investigate their structures using the coherent states of 2d conformal group. |
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Karowski, Michael (FU-Berlin) PDF Integrable quantum field theories: The form factor program The task of the form factor program for integrable quantum field theories in 1+1 dimensions is to construct a model in terms of its Wightman functions explicitly. |
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Klimcik, Ctirad (Institute de mathematiques de Luminy, Marseille) PDF Gauging the deformed WZW model We review the concept of the (anomalous) Poisson-Lie symmetry in a way that emphasises the notion of Poisson-Lie Hamiltonian. Then we prove that the left and the right actions of a group $G$ on its twisted Heisenberg double $(D,\kappa)$ realize the (anomalous) Poisson-Lie symmetries. Under some additional conditions, we construct also a non-anomalous moment map corresponding to a sort of quasi-adjoint action of $G$ on $(D,\kappa)$. The absence of the anomaly of this "quasi-adjoint" moment map permits to perform the gauging of deformed WZW models. |
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Konno, Hitoshi (Hiroshima University) PDF Elliptic Quantum Groups and Elliptic Lattice Models We discuss the elliptic quantum groups introduced by Jimbo, Konno, Odake and Shiraishi ( Transformation Groups 4, 303-327,1999 ). There are two types of them: the vertex type ${\cal A}_{q,p}(\widehat{sl}_n)$ and the face type ${\cal B}_{q,\lambda}(g)$ ($g$ : an affine Lie algebra ). After reviewing some basic facts on the elliptic quantum groups, we show that any finite dimensional representation of the universal $R$ matrix of ${\cal B}_{q,\lambda}(g)$ coincides with a connection matrix for the solutions of the $q$-KZ equation associated with $U_q(g)$. The latter matrix elements are known to give an elliptic solution to the face type Yang-Baxter equation. We hence establish a connection between ${\cal B}_{q,\lambda}(g)$ and the elliptic face (IRF or SOS) models associated with $g$. We also explain an algorithm of calculating correlation functions of the elliptic lattice models by using the representation theory of the elliptic quantum groups. |
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McMullan, David (University of Plymouth) Charges in Gauge Theories PDF Recent progress in the construction of both electric and magnetic charges in gauge theories will be presented. The topological properties of the charges sectors will be highlighted as well as the applications of this work to confinement and infrared dynamics. |
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Mikovic, Alexandar (Lusofona University, Lisbon) Quantum gravity as a broken symmetry phase of the SO(4,1) BF theory We describe how General Relativity with a positive cosmological constant appears from the SO(4,1) BF theory perturbed by a symmetry breaking potential quadratic in the B field. Then we discuss a spin foam quantization of this BF theory. |
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Moshe, Moshe (Technion -- Israel Inst. of Technology) PS Supersymmetry at Finite Temperature and the Energy Momentum Tensor Breakdown of supersymmetry at finite temperature has been debated for a long time. The controversy involved its appearance, the phase structure and its consequences. The restoration of broken internal symmetries at finite temperature supersymmetric theories were also discussed in the past. Here, softly broken supersymmetry at finite temperature is studied. The energy momentum stress tensor is calculated at zero and finite temperatures. The phase structure is unveiled in an $O(N)$ symmetric supersymmetric model , in the limit $N \\to \\infty$ , in three dimensions. It is pointed out that under certain conditions the trace of the energy momentum tensor vanishes even in the massive phase. At zero temperature it reveals spontaneous breaking of scale invariance with no explicit breaking as is the case in the scalar $O(N)$ vector model in three dimensions. At a certain critical coupling one finds a bound state goldston boson and fermion associated with spontaneous breaking of scale invariance (massless dilaton and dilatino) . The effect of finite temperature on this phenomena is elucidated in this study. |
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Nahm, Werner (Dublin Institute for Advanced Studies) Lochlainn O'Raifeartaigh's legacy The core of O'Raifeartaigh's work is the understanding of nature by clear mathematical analysis. As far as experiments are concerned, the observation of magnetic multimonopoles is far off, but when broken supersymmetry will be discovered at LHC, some of his work will be in the centre of attention again. Much of it, however, will find its lasting place in the history of scientific ideas, when quantum field theory will be accepted as an important domain of mathematics. |
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Olshanetsky, Mikhail (ITEP Moscow) PDF Symplectic Hecke correspondence in Hitchin systems The Hitchin systems describe a wide class of completely integrable classical systems, such as Gaudin systems, Calogero-Moser systems, integrable Euler-Arnold tops. The integrable systems arise as a result of Hamiltonian reduction in a topological gauge theory in dimension 2+1. This approach to integrable systems has some advantages. Using it, one can derive the Lax operators in a canonical way, describe separated variables and the action-angle coordinates. From the geometric point of view the phase space of the Hitchin systems is the cotangent bundle to the moduli space of holomorphic vector bundles over a Riemann curve with marked points (the Higgs bundle). Local coordinates in the moduli space play the role of the coordinates of particles, while the spin variables correspond to the time-like Wilson operators attributed to the marked points. Symplectic Hecke correspondence is a singular gauge transform that change a degree of the Higgs bundle. It was demonstrated recently by Kapustin and Witten that this transform is provided by the t'Hooft operator introduced in the gauge theory. It will be demonstrated that the symplectic Hecke transform allows us to establish an equivalence of different integrable systems. As examples, we prove the equivalence of the elliptic Calogero-Moser system and an integrable version of SL(N) Euler-Arnold top, the Painlev{\'e} VI equation and the non-autonomous SL(2) Zhukovsky-Volterra gyrostat. The talk is based on two papers: nlin.SI/0110045 and math.QA/0508058 |
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Palla, Lászlo (Eötvös University, Budapest) PDF Casimir force between planes as a boundary finite size effect The ground state energy of a boundary quantum field theory is derived in planar geometry in D+1 dimensional spacetime. It provides a universal expression for the Casimir energy which exhibits its dependence on the boundary conditions via the reflection amplitudes of the low energy particle excitations. We demonstrate the easy and straightforward applicability of the general expression by analyzing the free scalar field with Robin boundary condition and by rederiving the most important results available in the literature for this geometry. |
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Ragnisco, Orlando (Universita' Roma TRE) PDF Q-deformations and integrable motions on curved manifolds It is shown how a symplectic realization of a non-standard q-deformation of sl(2) yields integrable (many-particle) motions on manifolds with non-zero curvature. A few examples concerning geodesic and harmonic motions are briefly described. |
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Roger, Claude (University of Lyon) About Euler equations for infinite dimensional Lie Algebras We consider a Lie algebra generalizing the Virasoro algebra to the case of two space variables, simply taking the loop algebra over Vect(S1), centrally extended with suitable terms. We study its coadjoint representation and calculate the corresponding Euler equations. In particular, we obtain a bi-Hamiltonian system that leads to an integrable non-linear partial differential equation. This equation is an analogue of the Kadomtsev--Petviashvili (of type B) equation. |
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Ruelle, Philippe (University of Louvain) PDF Non-local finite size effects in the dimer model The two-dimensional dimer model exhibits peculiar finite-size effects. In particular the finite-size correction terms for the free energy depend in a crucial way on the parity of the horizontal and vertical lengths of the grid on which it is defined. After reviewing these effects, we show that this unusual dependence can be fully explained in a conformal field theoretic description, by noting that a change of parity of one of the lengths of the grid induces a change of boundary condition. |
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Sorba, Paul (LAPTH Annecy) Bosonisation and vertex representations with an impurity Bosonisation is considered in the case of a point-like defect in 2 dim. and in the framework of the reflexion-transmission algebras (R-T algebras). The main features of the corresponding vertex algebra are established, and first apply to the massless Thirring model with defect. The vertex representation of the Sl(2) Kac-Moody algebra is then constructed, and the interplay mediated by the defect between the left and right sectors studied in some detail. |
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Sreedhar, Vinnakota (Indian Institute of Technology) On the topological origin of quantum entanglement in Ising spin systems It is well-known that purely quantum mechanical features, without any classical analogues, are often associated with the nontrivial topological aspects of the system under consideration. In the same spirit, we trace the roots of quantum entanglement in spin systems to a topological origin, using the three-dimensional Ising model as a prototypical example. |
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Straumann, Norbert (Universty of Zurich, Switzeland) PDF Updates on Dark Energy I shall review some recent observational results and theoretical developments related to the Dark Energy problem. |
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Tsutsui, Izumi (High Energy Accelerator Research Organization, KEK) PDF Singularity in Quantum Mechanics and the Calogero Model Recent progress on the study of quantum singularity is reviewed, starting from the basics on the general treatment of singularity on a line. After mentioning briefly the physical application in quantum well and quantum computing, we discuss the implication of singularity in a solvable model; the N=3 Calogero model. The content of this talk supplements the report presented in the previous LOR symposium held in DIAS in 2002 by the same speaker. |
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Tuite, Michael Patrick (National University of Ireland, Galway) PDF The Virasoro Algebra and Exceptional Lie and Finite Groups We discuss some Casimir-like states in a Vertex Operator Algebra (VOA) or chiral CFT with an invertible invariant form (Zamalodchikov metric). These states are, by construction, necessarily invariant under the automorphism group of the VOA. In the cases where these Casimir states are Virasoro descendents of the vacuum, we discuss some resulting novel relationships between the Kac determinant and Deligne's exceptional series of Lie groups $A_1$,$A_2$,$G_2$,$F_4$,$E_6$,$E_7$,$E_8$. We also discuss novel relationships between the Kac determinant and the Monster group and some other exceptional finite groups. |
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Visinescu, Mihai (National Institute for Physics and Nuclear Engineering, Bucharest) PDF Fermions on curved spaces, symmetries, and quantum anomalies Investigating the Dirac equation on curved backgrounds we point out the role of the Killing-Yano tensors in the construction of the Dirac-type operators. The gravitational and axial anomalies are studied for generalized Euclidean Taub-NUT metrics which admit hidden symmetries analogous to the Runge-Lenz vector of the Kepler-type problem. Using the Atiyah-Patodi-Singer index theorem for manifolds with boundaries, it is shown that these metrics make no contribution to the axial anomaly. |
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Wess, Julius (LM University Munich) Gauge theory and gravity on deformed spaces There is a natural way to introduce a differential calculus on a space with noncommuting coordinates. This is the basis for a deformed differential geometry. Aplications of this formalism are the deformed gaugetheories and gravity. The formalism can be developed to obtain phenomenological predictions. |
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Wipf, Andreas (Friedrich-Schiller-University Jena) PDF Phases of generalized Potts-Models and their Relevance for Gauge Theories Field-theoretic generalizations of Potts-Models describe the dynamics of Polyakov-Loops at finite temperature. These models display a rich phase structure with tricritical points. Recent analytical and numerical are discussed and compared. |
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Yoneyama, Hiroshi (Saga University, Japan) PDF Lattice Field Theory with Sign Problem and the Maximum Entropy Method In the numerical study of the lattice field theory such as finite density QCD and QCD with the theta term, one confronts a notorious problem called sign problem, which is due to negative or complex Boltzmann weights when updating configurations. We consider the problem in terms of the Maximum Entropy Method (MEM), which is based on Baysian probability theory. We apply the MEM to the CP(N-1) model with the theta term and other toy models, where the sign problem appears as flattening phenomenon of the free energy. We carefully study flattening and compare the results with those of the standard method. |