# Seminário de Teoria Quântica do Campo Topológica

## Sessões anteriores

Páginas de sessões mais recentes: Seguinte 8 7 6 5 4 3 2 1 Mais recente

### 09/02/2005, 14:00 — 15:00 — Sala P12, Pavilhão de Matemática

Stephen Sawin, *Fairfield University*

### Witten-Style Nonabelian Localization For a Noncompact Manifold

Witten in [1] offered a clever scheme to express certain integrals over a Hamiltonian (i.e., symplectic, with group action and a moment map) manifold as a sum of local contributions from the critical points of the square of the moment map. In particular this allows one to read off the ring structure of the cohomology of the symplectic reduction (when it is nice enough) from integrating equivariant cohomology classes in the original space. His elegant argument ignores most analytic subtleties and thus is purely heuristic, but Jeffrey and Kirwan in [2] were able to reproduce his key results in the compact case, by relating the question to one accessible by older abelian localization techniques. I will argue that the noncompact case is particularly important by relating to some outstanding cases, and that the abelian localization argument is unlikely to extend here. I will prove Witten's results rigorously using his version of nonabelian localization, and suggest ways to extend these results further. - E. Witten, Two-dimensional gauge theories revisited, J. Geom. Phys. 9 (1992), no. 4, 303-368.

hep-th/9204083 - L. C. Jeffrey, F. Kirwan, Localization for nonabelian group action, Topology 34 (1995) no. 2, 291-327.

alg-geom/9307001

### 21/05/2004, 11:00 — 12:00 — Sala P3.10, Pavilhão de Matemática

Marco Mackaay, *Universidade do Algarve*

### Khovanov's categorification of the Jones polynomial

Following Bar-Natan's down-to-earth approach, I will explain
Khovanov's construction which associates to a knot a certain
complex of graded vector spaces. If two knots are ambient isotopic
their complexes are homotopy equivalent (grading is preserved).
Therefore the cohomology groups of the knot complex are
knot-invariants. It turns out that the Jones polynomial of a knot
equals the graded Euler characteristic of the knot cohomology.
Khovanov derived a more general polynomial from his complexes which
is a more powerful knot invariant, as has been shown by explicit
computations. References:
- Dror Bar-Natan, "On Khovanov's categorification of the Jones
polynomial", Algebraic and Geometric Topology 2 (2002) 337-370;
math.QA/0201043.
- Mikhail Khovanov, "A functor-valued invariant of tangles",
Algebr. Geom. Topol. 2 (2002) 665-741;
math.QA/0103190.
- Mikhail Khovanov, "A categorification of the Jones polynomial",
math.QA/9908171.

### 21/05/2004, 10:00 — 11:00 — Sala P3.10, Pavilhão de Matemática

Pedro Vaz, *Universidade do Algarve*

### Induced representations and geometric quantization of coadjoint
orbits

There is a well known correspondence between the orbit method in
geometric quantization and the theory of unitary irreducible
representations of a Lie group. We show that the pre-quantization
of a coadjoint orbit of a connected Lie group G arises as the
infinitesimal version of an induced representation of G. With the
aid of a polarization, this procedure allow us to construct unitary
irreducible representations that are also the infinitesimal version
of an induced representation. As an example, we construct the
corresponding (infinite dimensional) unitary representations of the
Lie group SL(2,C), the universal cover of the Lorentz group.

References:
- I. M. Gelfand, M. I. Graev, and N. Ya. Vilenkin. Generalized
Functions volume 5, "Integral Geometry and Representation Theory".
Academic Press,New York, 1966.
- A.A. Kirillov. Elements of the Theory of Representations
Springer-Verlag, 1976.
- N. Woodhouse. Geometric Quantization. Oxford, 1991.

### 17/12/2003, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática

J. Scott Carter, *Univ. South Alabama*

### Quandle homology theories and cocycle invariants of knots

Cohomology theories have been developed for certain
self-distributive groupoids called quandles. Variations of
invariants of knots and knotted surfaces have been defined using
quandle cocycles and the state-sum form. We review these
developments, and also discuss quandle modules and their relation
to generalizations of Alexander modules, and topological
applications of these invariants.

### 06/11/2003, 17:00 — 18:00 — Sala P3.31, Pavilhão de Matemática

Pedro Lopes, *Instituto Superior Técnico*

### Nós e os Quandles II

Apresentação dos resultados da tese de
doutoramento.

References:

- P. Lopes, Quandles at finite temperatures I, J. Knot Theory
Ramifications, 12(2):159-186 (2003),
math.QA/0105099
- F. M. Dionisio and P. Lopes, Quandles at finite temperatures
II, J. Knot Theory Ramifications to appear,
math.GT/0205053
- J. Bojarczuk and P. Lopes, Quandles at finite temperatures III,
submitted to J. Knot Theory Ramifications

### 30/10/2003, 16:30 — 17:30 — Sala P3.10, Pavilhão de Matemática

Pedro Lopes, *Instituto Superior Técnico*

### Nós e os Quandles I

Apresentação dos resultados da tese de
doutoramento.

References:

- P. Lopes, Quandles at finite temperatures I, J. Knot Theory
Ramifications, 12(2):159-186 (2003),
math.QA/0105099
- F. M. Dionisio and P. Lopes, Quandles at finite temperatures
II, J. Knot Theory Ramifications to appear,
math.GT/0205053
- J. Bojarczuk and P. Lopes, Quandles at finite temperatures III,
submitted to J. Knot Theory Ramifications

### 15/10/2003, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática

Igor Kanatchikov, *Institute of Theoretical Physics, Free University of Berlin*

### Precanonical quantization of the Yang-Mills fields and the mass gap
problem

We overview the ideas of the precanonical quantization approach and
apply it to the Yang-Mills fields. We show how the approach reduces
the mass gap problem to a quantum mechanical spectral problem
similar to that for the magnetic Schroedinger operator with a
Clifford-valued magnetic field. Reference: hep-th/0301001.

### 23/01/2003, 13:30 — 14:30 — Sala P3.10, Pavilhão de Matemática

Lina Oliveira, *Instituto Superior Técnico*

### Ideals of nest algebras

Complex Banach spaces are naturally endowed with an algebraic
structure, other than that of a vector space. The
holomorphic structure of the open unit ball in a complex
Banach space A leads to the existence of a closed subspace
${A}_{s}$ of
$A$, known as the symmetric part of
$A$, and of a partial triple product
$(a,b,c)\to abc$ mapping
$A\times {A}_{s}\times A$ to
$A$. The existence of a Jordan triple identity satisfyied by this algebraic
structure
relates any complex Banach space to the Banach Jordan triple systems
important in infinite-dimensional holomorphy.
A nest algebra, which is a primary example of a non-self-adjoint algebra of
operators, is also an
interesting case of a complex Banach space whose symmetric part is a proper subspace.
The ideals
of nest algebras related to its associative
multiplication have been extensively investigated, and
whilst it is clear
that ideals in the associative sense provide
examples of ideals in the partial triple sense, the converse
assertion remains in general an open problem.
It is the aim of this talk to show that, in a large class
of nest algebras, the weak*-closed ideals in the partial triple
sense are also
weak*-closed ideals in the associative algebra sense.
A brief overview of how the partial triple produtct arises from the
holomorphic structure of the open unit ball of the nest algebra will also be given.
#### References

- Jonathan Arazy,
*An application of infinite dimensional
holomorphy to the geometry of Banach space*, Geometrical aspects
of functional analysis, Lecture Notes in Mathematics,
Vol. 1267, Springer-Verlag, Berlin/Heidelberg/New York, 1987.
- Lina Oliveira,
*Weak*-closed Jordan ideals
of nest algebras*, Math. Nach., to appear.

### 23/01/2003, 11:00 — 12:00 — Sala P3.10, Pavilhão de Matemática

José Velhinho, *Univ. Beira Interior and CENTRA*

### Representations of holonomy algebras and shadow states

It has been argued that the Ashtekar-Lewandowski
representation of the
Ashtekar-Isham holonomy algebra is fundamental, in the sense that any
other
representation can be obtained by a suitable limit procedure.
We propose to clarify that statement, providing, in particular, a
canonical way
of mapping GNS states to a family of vectors of the Ashtekar-Lewandowski
Hilbert space. The so-called family of
*shadow states* thus obtained
converges,
as states of the algebra, to the original GNS state.
#### References

- M. Varadarajan, Phys. Rev D. 64 , 104003 (2001); gr-qc/0104051
- J.M. Velhinho, Commun. Math. Phys. 227, 541 (2002);
math-ph/0107002
- A. Ashtekar and J. Lewandowski, Class. Quant. Grav. 18, L117 (2001);
gr-qc/0107043
- T. Thiemann, gr-qc/0206037
- H. Sahlmann, gr-qc/0207112
- A. Ashtekar, J. Lewandowski and H. Sahlmann,
gr-qc/0211012

### 19/12/2002, 13:30 — 14:30 — Sala P3.10, Pavilhão de Matemática

João Baptista, *Cambridge University*

### Vortex dynamics on the sphere

I will first give a brief overview of the Bogomolny equations for vortices, their moduli space of solutions, and the method of geodesic approximation. In the case
of
$N$ vortices on a sphere of radius
${R}^{2}>N$ this moduli space is
$C{P}^{N}$, but the geodesic method cannot be directly applied, because the
solutions of the Bogomolny equations are
not known explicitly. I will then try to show how to circumvent this problem in the limit where
${R}^{2}$ is close to
$N$.

### References

- J. M. Baptista and N. S. Manton,
*The dynamics of vortices on
${S}^{2}$ near the Bradlow limit*, hep-th/0208001.
- T. M. Samols,
*Vortex Scattering*, Commun. Math. Phys. 145, 149 (1992).

### 19/12/2002, 11:00 — 12:00 — Sala P3.10, Pavilhão de Matemática

Ana Bela Cruzeiro, *Instituto Superior Técnico*

### Differential geometry on the path space and applications

When trying to construct a Riemannian geometry on the path space of
a Riemannian manifold several approaches could be thought about.
The local chart approach, considering the path space as an infinite
dimensional manifold and the basic tangent space the Cameron-Martin
Hilbert space, leads to the study of the so-called Wiener-Riemann
manifolds. Several difficulties appear in this study, namely the
difficulty of finding an atlas such that the change of charts is
compatible with the probabilistic structure (preserves the class of
Wiener measures together with the Cameron-Martin type tangent
spaces) and the non-availability of an effective computational
procedure in the local coordinate system. Indeed, in infinite
dimensions, the summation operators of differential geometry become
very often divergent series. But the path space is more than a
space endowed with a probability: time and the corresponding
Itô filtration provide a much richer structure. In
particular, the parallel transport over Brownian paths can be
naturally defined by a limiting procedure from ODEs to SDEs. The
stochastic parallel transport defines a canonical moving frame on
the path space: the point of view we have adopted is the one of
replacing systematically the machinery of local charts by the
method of moving frames (as in Cartan theory). In this way it is
possible to transfer geometrical quantities of the path space to
the classical Wiener space and use Itô calculus to
renormalize the apriori divergent expressions. An effective
computational procedure is then achieved, where Stochastic Analysis
and Geometry interact, not only on a technical level, but in a
deeper way: Stochastic Analysis makes it possible to define
geometrical quantities, Geometry implies new results in Stochastic
Analysis. An application to assymptotics of the vertical
derivatives of the heat kernel associated to the horizontal
Laplacian on the frame bundle is discussed.
##### References

- A. B. Cruzeiro and P. Malliavin -"Renormalized differential
geometry on path space: structural equation, curvature", J.Funct.
Anal. 139 (1996), p. 119 -181.
- A. B. Cruzeiro, P. Malliavin and S. Taniguchi - "Ground state
estimations in gauge theory", Bull Sci.Math. 125, 6-7 (2001), p.
623-640.

### 14/11/2002, 13:30 — 14:30 — Sala P3.10, Pavilhão de Matemática

Marco Mackaay, *Universidade do Algarve*

### The coadjoint orbits of
$\mathrm{SL}(2,C)$ - II

In my talks (17/10 and 14/11) I will sketch the construction of the
principal series of unitary representations of
$\mathrm{SL}(2,C)$ and the coadjoint
orbits they correspond to. I will also try to explain the geometric
quantization of these orbits and show that the infinitesimal
representations thus obtained are indeed the ones corresponding to the
principal series. If there is any time left I might say something about
the Kirillov character formula for
$\mathrm{SL}(2,C)$.
### References

- I.M. Gelfand, R.A. Minlos, Z.Ya. Shapiro,
*Representations of the
rotation and
Lorentz groups and their applications*, Pergamon Press, 1963.
- A. Kirillov,
*Elements of representation theory*, Springer.
- G.W. Mackey,
*Induced representations of locally compact groups I*, Ann. Math.
55(1):101-139, 1952.

### 14/11/2002, 11:00 — 12:00 — Sala P3.10, Pavilhão de Matemática

Jorge Drumond Silva, *Instituto Superior Técnico*

### How the Weyl quantization brings about an improvement in Egorov's Theorem

Using ideas from the Weyl quantization, we show how the classical
theorem by Y. Egorov, on conjugation of pseudo-differential operators by
Fourier integral operators, can have its accuracy increased.

### References

- G. Folland,
*Harmonic Analysis in Phase Space*, Princeton University Press.
- J. Duistermaat,
*Fourier Integral Operators*, Birkhauser.
- L. Hormander,
*The Analysis of Linear Partial Differential Operators*, Vol.
III, IV, Springer-Verlag.

### 17/10/2002, 13:30 — 14:30 — Sala P3.10, Pavilhão de Matemática

Marco Mackaay, *Universidade do Algarve*

### The coadjoint orbits of SL(2,C)

### 17/10/2002, 11:00 — 12:00 — Sala P3.10, Pavilhão de Matemática

Roger Picken, *Instituto Superior Técnico*

### Introduction to the Chas-Sullivan bracket

### 11/04/2002, 14:45 — 15:45 — Sala P3.10, Pavilhão de Matemática

Roger Picken, *Instituto Superior Técnico*

### Twisted vector bundles and twisted K-theory

### 14/03/2002, 16:00 — 17:00 — Sala P3.10, Pavilhão de Matemática

Pedro Castelo Ferreira, *University of Sussex e
Instituto Superior Técnico*

### From Compact Maxwell Chern-Simons to Azbel Hofstadter

### 14/03/2002, 14:15 — 15:15 — Sala P3.10, Pavilhão de Matemática

Carlos Florentino, *Instituto Superior Técnico*

### The Verlinde algebra in conformal field theory, algebraic
geometry and K-theory - II

### 24/01/2002, 16:00 — 17:00 — Sala P3.10, Pavilhão de Matemática

Carlos Florentino, *Instituto Superior Técnico*

### The Verlinde algebra in conformal field theory, algebraic
geometry and K-theory.

Páginas de sessões mais antigas: Anterior 10 11 12 Mais antiga

Organizadores correntes: Roger Picken, Marko Stošić.

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