# Seminário de Teoria Quântica do Campo Topológica

## Sessões anteriores

Páginas de sessões mais recentes: Seguinte 6 5 4 3 2 1 Mais recente

### Virtual Knot Theory

New developments in Virtual Knot Theory.

### Dimensional Reduction of Gauge Theories over continuous and fuzzy coset spaces

We review the dimensional reduction of N=1 higher dimensional Gauge Theories over Coset Spaces with emphasis on the possibility to obtain four-dimensional GUTs with chiral fermions and softly broken supersymmetry. Next we consider gauge theories defined in higher dimensions, where the extra dimensions form a fuzzy space (a finite matrix manifold). We emphasize some striking features emerging such as (i) the appearance of non-abelian gauge theories in four dimensions starting from an abelian gauge theory in higher dimensions, (ii) the fact that the spontaneous symmetry breaking of the theory takes place entirely in the extra dimensions and (iii) the renormalizability of the theory.

### From the Chern-Simons path integral to the Reshetikhin-Turaev invariant

The study of the heuristic Chern-Simons path integral by E. Witten inspired (at least) two general approaches to quantum topology. Firstly, the perturbative approach based on the CS path integral in the Lorentz gauge and, secondly, the "quantum group approach" by Reshetikhin/Turaev. While for the first approach the relation to the CS path integral is obvious for the second approach it is not. In particular, it is not clear if/how one can derive the relevant R-matrices or quantum 6j-symbols directly from the CS path integral. In my talk, which summarizes the results of a recent preprint, I will sketch a strategy that should lead to a clarification of this issue in the special case where the base manifold is of product form. This strategy is based on the "torus gauge fixing" procedure introduced by Blau/Thompson for the study of the partition function of CS models. I will show that the formulas of Blau/Thompson can be generalized to Wilson lines and that the evaluation of the expectation values of these Wilson lines leads to the same state sum expressions in terms of which Turaev's shadow invariant is defined. Finally, I will sketch how one can obtain a rigorous realization of the path integral expressions appearing in this treatment.

### Instantons and framed bundles on rational surfaces

The talk concerns a correspondence between framed instantons on the one-point compactification of an affine complex surface $X$, and framed holomorphic bundles on a projective completion of $X$. This correspondence is known for $X$ the affine plane (Donaldson) and $X$ the affine plane blown up at a point (King). After reviewing these cases, I will discuss possible generalizations (basically, when the projective completion is a rational surface). I will also spend some words on instanton countings on these surfaces. Physically this corresponds to studying the Nekrasov partition function for topological super Yang-Mills theories on $X$.

### Algebraic Frobenius manidolds and primitive conjugacy classes in Weyl group

We develop the theory of generalized bi-Hamiltonian reduction. Applying this theory to the loop algebra proved to be equivalent to a generalized Drinfeld-Sokolov reduction. This gives a way to construct new examples of algebraic Frobenius manifolds.

### New methods in renormalization theories - I

The first occurence of the ideas of renormalisation in physics is due to Green, around 1850, who used such methods to study the motion of a pendulum in a fluid. The same kind of methods was proposed by J. Oppenheimer around 1930, to take in account the so-called radiative corrections to the spectral lines of atoms. Like the previous attempts in classical electrodynamics, this approach led to unphysical infinite quantities. As it si well-known, the new methods of Bethe, Schwinger, Tomonaga, Feynman and Dyson solved in principle the problem of infinities around 1950. But a conceptual breakthrough occured ten years ago when A. Connes and D. Kreimer introduced Hopf algebraic methods in this game. We propose to explain our own verion of these methods, emphasizing a certain infinite-dimensional group, the so-called dressing group. A striking feature is the deep analogy with groups introduced by Grothendieck under the name of motivic Galois groups. These lectures shall begin with a short historical review, followed by a description of the standard calculations, and then we shall describe in detail the new methods.

### 23/02/2007, 15:45 — 16:45 — Sala P3.10, Pavilhão de MatemáticaRoger Picken, Instituto Superior Técnico

Sétima palestra dum Mini-Encontro do projecto "Topologia Quântica", onde haverá apresentações curtas e informais de membros do projecto sobre assuntos de interesse actual. Todos os interessados bem-vindos.

### 23/02/2007, 14:45 — 15:45 — Sala P3.10, Pavilhão de MatemáticaRui Carpentier, Instituto Superior Técnico

Sexta palestra dum Mini-Encontro do projecto "Topologia Quântica", onde haverá apresentações curtas e informais de membros do projecto sobre assuntos de interesse actual. Todos os interessados bem-vindos.

### 23/02/2007, 14:00 — 15:00 — Sala P3.10, Pavilhão de MatemáticaJoão Martins, Instituto Superior Técnico

Quinta palestra dum Mini-Encontro do projecto "Topologia Quântica", onde haverá apresentações curtas e informais de membros do projecto sobre assuntos de interesse actual. Todos os interessados bem-vindos.

### 22/02/2007, 16:45 — 17:45 — Sala P3.10, Pavilhão de MatemáticaMarco Mackaay, Universidade do Algarve

Quarta palestra dum Mini-Encontro do projecto "Topologia Quântica", onde haverá apresentações curtas e informais de membros do projecto sobre assuntos de interesse actual. Todos os interessados bem-vindos.

### 22/02/2007, 16:00 — 17:00 — Sala P3.10, Pavilhão de MatemáticaMarko Stosic, ISR

Terceira palestra dum Mini-Encontro do projecto "Topologia Quântica", onde haverá apresentações curtas e informais de membros do projecto sobre assuntos de interesse actual. Todos os interessados bem-vindos.

### 22/02/2007, 14:45 — 15:45 — Sala P3.10, Pavilhão de MatemáticaPaulo Semião, Universidade do Algarve

Segunda palestra dum Mini-Encontro do projecto "Topologia Quântica", onde haverá apresentações curtas e informais de membros do projecto sobre assuntos de interesse actual. Todos os interessados bem-vindos.

### 22/02/2007, 14:00 — 15:00 — Sala P3.10, Pavilhão de MatemáticaPedro Lopes, Instituto Superior Técnico

Primeira palestra dum Mini-Encontro do projecto "Topologia Quântica", onde haverá apresentações curtas e informais de membros do projecto sobre assuntos de interesse actual. Todos os interessados bem-vindos.

### Obstructions to Quantization 2

Let $\left(L,\nabla \right)$ be a prequantum line bundle over a symplectic manifold $X$, and $S$ its symplectization. Kostant showed that the classical Poisson bracket on $S$ is simply prequantization on $X$. C. Duval and I have taken this a step farther to obtain a quantization of $X$ using a generalized star-product on $S$.

References:
1. Kostant, B. [2003], Minimal coadjoint orbits and symplectic induction, arXiv: SG/0312252.

### Obstructions to Quantization 1

Quantization is not a straightforward proposition, as demonstrated by Groenewold's and Van Hove's discovery, sixty years ago, of an "obstruction" to quantization. Their "no-go theorems" assert that it is in principle impossible to consistently quantize every classical polynomial observable on the phase space ${R}^{2n}$ in a physically meaningful way. Similar obstructions have been recently found for ${S}^{2}$ and ${T}^{*}{S}^{1}$, buttressing the common belief that no-go theorems should hold in some generality. Surprisingly, this is not so-it has just been proven that there are no obstructions to quantizing either ${T}^{2}$ or ${T}^{*}{R}_{+}$. In this talk we conjecture-and in some cases prove-generalized Groenewold-Van Hove theorems, and determine the maximal Lie subalgebras of observables which can be consistently quantized. This requires a study of the structure of Poisson algebras of symplectic manifolds and their representations. To these ends we review known results as well as recent theoretical work. Our discussion is independent of any particular method of quantization; we concentrate on the structural aspects of quantization theory which are common to all Hilbert space-based quantization techniques. (This is joint work with J. Grabowski, H. Grundling and A. Hurst.)

References:
1. Gotay, M. J. [2000], Obstructions to Quantization, in: Mechanics: From Theory to Computation. (Essays in Honor of Juan-Carlos Simo), J. Nonlinear Science Editors, 271-316 (Springer, New York).
2. Gotay, M. J. [2002], On Quantizing Non-nilpotent Coadjoint Orbits of Semisimple Lie Groups. Lett. Math. Phys. 62, 47-50.

### Stress-Energy-Momentum Tensors

J. Marsden and I present a new method of constructing a stress-energy-momentum tensor for a classical field theory based on covariance considerations and Noether theory. Our stress-energy-momentum tensor ${T}^{\mu }{}_{\nu }$ is defined using the (multi)momentum map associated to the spacetime diffeomorphism group. The tensor ${T}^{\mu }{}_{\nu }$ is uniquely determined as well as gauge-covariant, and depends only upon the divergence equivalence class of the Lagrangian. It satisfies a generalized version of the classical Belinfante-Rosenfeld formula, and hence naturally incorporates both the canonical stress-energy-momentum tensor and the "correction terms" that are necessary to make the latter well behaved. Furthermore, in the presence of a metric on spacetime, our ${T}^{\mu \nu }$ coincides with the Hilbert tensor and hence is automatically symmetric.

References:
1. Gotay, M. J. and J. E. Marsden [1992], Stress-energy-momentum tensors and the Belinfante-Rosenfeld formula, Contemp. Math. 132, 367-391.
2. Forger, M. and H. Römer [2004], Currents and the energy-momentum tensor in classical field theory: A fresh look at an old problem, Ann. Phys. 309, 306-389.

### Virtual Knot Theory III

Flat virtuals and long flat virtuals. Khovanov homology and virtual knot theory.

### Crossed Modules and Crossed Complexes in Geometric Topology II

This short course aims at describing some applications of crossed modules and crossed complexes to Geometric Topology, and it is based on results by the author. The background is R. Brown and P.J. Higgins beautiful work on crossed modules and crossed complexes. We will give a lot of attention to applications to knotted embedded surfaces in S^4, and we will make explicit use of movie representations of them. Some of the ideas of this work started from Yetter's Invariant of manifolds and subsequent developments. Full summary and references: http://www.math.ist.utl.pt/~rpicken/tqft/kauffman062006/CMGT.pdf

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Organizadores correntes: Roger Picken, Marko Stošić.

Projecto FCT PTDC/MAT-GEO/3319/2014, Quantization and Kähler Geometry.