Seminário de Teoria Quântica do Campo Topológica  

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.

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.

Let $(L,\nabla )$ 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.

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.

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.

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

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

Continuing discussion of invariants of virtual knots and links. Biquandles and 0-level Alexander polynomial. Quaternionic biquandle. Weyl algebra and the linear non-commutative biquandles.

In this talk we show that the torus knots $T_{p,q}$ for $3\le p\le q$ are homologically thick. Furthermore, we show that we can reduce the number of twists $q$ without changing a certain part of the homology, and consequently we show that there exists a stable homology for torus knots conjectured in [1]. Also, we calculate the Khovanov homology groups of low homological degree for torus knots, and we conjecture that the homological width of the torus knot $T_{p,q}$ is at least $p$. References: [1] N. Dunfield, S. Gukov and J. Rasmussen: The Superpolynomial for link homologies, arXiv:math.GT/0505056. [2] M. Stosic: Homological thickness of torus knots, arXiv:math.GT/0511532

Introduction to combinatorial knot theory; Reidemeister moves, moves on virtuals; interpretation of virtual knot theory in terms of knots and links in thickened surfaces; bracket polynomial for virtuals, involutory quandle for virtuals.

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

Introductory session for the minicourse on Virtual Knot Theory by Louis Kauffman.

Gerbes are higher-order generalizations of Abelian bundles. They appear in nature, for instance as obstructions to lifting $\mathrm{SO}\left(n\right)$-bundles to $\mathrm{Spin}\left(n\right)$ or ${\mathrm{Spin}}_c\left(n\right)$ bundles. It is possible to endow gerbes with connection 1- and 2-forms and curvature 3-forms, and study aspects of the ensuing differential geometry. In particular, gerbes with connection have holonomies and parallel transports along surfaces, as opposed to along loops and paths. Apart from discussing these features, I hope to describe an interesting recent categorification approach to non-Abelian gerbes due to Baez and Schreiber.

References

1. J. Baez and U. Schreiber, Higher gauge theory: 2-connections on 2-bundles, hep-th/0412325.
2. M. Mackaay and R. Picken, Holonomy and parallel transport for Abelian gerbes, Adv. Math. 170, 287-339 (2002), math.DG/0007053.
3. R. Picken, TQFT's and gerbes, Algebr. Geom. Topol. 4 (2004) 243-272, math.DG/0302065.

In 1945 Samuel Eilenberg and Norman E. Steenrod set forth the essential properties of a homology theory in terms of seven axioms; the last stipulating that the reduced homology of point is trivial. A number of years later (1957) Alexander Grothendieck introduced K-theory and expressed the Riemann-Roch theorem for nonsingular projective varieties by saying that the mapping $E\to ch\left(E\right)*\mathrm{Td}\left(X\right)$ from $K^0\left(X\right)$ to $H^{*}\left(X\right)$ is a natural transformation of covariant functors. Here $K^0\left(X\right)$ denotes the Grothendieck group of algebraic vector bundles on $X$, $H^{*}\left(X\right)$ denotes a suitable cohomology theory, $ch$ is the Chern character, and $\mathrm{Td}\left(X\right)$ is the Todd class of the tangent bundle of $X$. Michael Atiyah and Friedrich Hirzebruch developed K-theory in the context of topological spaces and showed that topological K-theory satisfies the first six axioms of Eilenberg and Steenrod. Using Bott periodicity one readily shows that the K-theory of a point is infinite cyclic in even degrees and vanishes in odd degrees.

Recently Edward Witten has argued that K-theory is relevant to the classification of Ramond-Ramond (RR) charges as well as noncommutative Yang-Mills theory or open string field theory. In order to consider D-branes with a topologically non-trivial Neveu-Schwarts 3-form field $H$, one needs to work with a twisted version of topological K-theory. If $H$ represents a torsion class, one may use the twisted K-theory developed by Peter Donavan and Max Karoubi. In this talk I shall describe two constructions of twisted K-theory one set forth by Michael Atiyah and Graeme Segal and the other by Daniel Freed, Michael J. Hopkins and Constantin Teleman. Due to personal limitations I shall give a braneless presentation.

References

1. M Atiyah and F Hirzebruch, Vector bundles and homogenuous spaces, Proc. of Symposia in Pure Maths vol 3, Differential Geometry, Amer. Math. Soc. 1961, 7-38.
2. M Atiyah and G Segal, Twisted K-theory, math.kt/0407054.
3. A Borel and J-P Serre, Le théorème de Riemann-Roch, Bull. Soc. Math. France 86 (1958) 97-136.
4. P Donavan and M Karoubi, Graded Brauer groups and K-theory with local coefficients, Publ. Math. IHES 38 (1970) 5-25.
5. S Eilenberg and N E Steenrod, Axiomatic approach to homology theory, Proc. Nat. Acad. Sci. USA 31 (1945), 117-120.
6. D Freed, M J Hopkins and C Teleman, Twisted K-theory and loop groups representations I, math.AT/0312155.
7. E Witten, Overview of K-theory Applied to Strings, hep-th/0007175.

One version of the Kobayashi-Hitchin correspondence relates moduli spaces of instantons on blow ups of $C^2$ trivialized at $\infty$, with holomorphic bundles on blow ups of ${\mathrm{CP}}^2$ framed on a line. In the rank stable limit these moduli spaces can be described using a bar construction.

I will discuss how a coupling to an external potential can be used to probe the interactions among gauged vortices on a sphere via symplectic localisation. I shall also illustrate how these results can be applied to statistical mechanics on the moduli space of vortices.

References:

1. N S Manton & P M Sutcliffe, "Topological Solitons", Cambridge Univ. Press, 2004.
2. N M Romão, "Gauged vortices in a background",
   hep-th/0503014
   

We explain what asymptotic quasinormal modes are, why there has been considerable recent interest in computing their frequencies, and how to obtain a complete classification of asymptotic quasinormal frequencies for static, spherically symmetric black hole spacetimes in d dimensions.

References

1. José Natário, Ricardo Schiappa, On the Classification of Asymptotic Quasinormal Frequencies for d-Dimensional Black Holes and Quantum Gravity,
   hep-th/0411267
   

In this talk we shall review the pure spinor approach for the super-Poincaré BRST covariant quantization of the superstring. We will focus on the BRST operator, its cohomology and the computation of central charges in the pure spinor conformal field theory, where the ghosts are constrained to be pure spinors. This will mainly review work due to Nathan Berkovits and will be at a broad/informal level.

References

1. Nathan Berkovits, ICTP Lectures on Covariant Quantization of the Superstring,
   hep-th/0209059
   
2. Nathan Berkovits, Covariant Multiloop Superstring Amplitudes,
   hep-th/0410079
   
3. Nathan Berkovits, Nikita Nikrasov, The Character of Pure Spinors,
   hep-th/0503075