# Topological Quantum Field Theory Seminar

## Past sessions

Newer session pages: Next 8 7 6 5 4 3 2 1 Newest

### 29/06/2005, 15:30 — 16:30 — Room P3.10, Mathematics Building

João Paulo Santos, *Instituto Superior Técnico*

### Instantons, Holomorphic Bundles and the Bar Construction

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.

### 29/06/2005, 14:00 — 15:00 — Room P3.10, Mathematics Building

Nuno Romão, *University of Adelaide*

### Gauged Vortices in a Background

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:

- N S Manton & P M Sutcliffe, "Topological Solitons", Cambridge Univ. Press, 2004.
- N M Romão, "Gauged vortices in a background",

hep-th/0503014

### 09/06/2005, 15:30 — 16:30 — Room P3.10, Mathematics Building

José Natário, *Instituto Superior Técnico*

### Asymptotic Quasinormal Frequencies for d-Dimensional Black Holes

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

- José Natário, Ricardo Schiappa, On the Classification of Asymptotic Quasinormal Frequencies for d-Dimensional Black Holes and Quantum Gravity,

hep-th/0411267

### 09/06/2005, 14:00 — 15:00 — Room P3.10, Mathematics Building

Ricardo Schiappa, *Instituto Superior Técnico*

### BRST Cohomology and Characters of Pure Spinors

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

- Nathan Berkovits, ICTP Lectures on Covariant Quantization of the Superstring,

hep-th/0209059

- Nathan Berkovits, Covariant Multiloop Superstring Amplitudes,

hep-th/0410079

- Nathan Berkovits, Nikita Nikrasov, The Character of Pure Spinors,

hep-th/0503075

### 11/03/2005, 16:30 — 17:30 — Amphitheatre Pa2, Mathematics Building

João Faria Martins, *Instituto Superior Técnico*

### Categorical Groups, Knots and Knotted Surfaces

We define an invariant of knots and an invariant of knotted surfaces from any finite categorical group (crossed module of groups). We illustrate its non-triviality by calculating an explicit example, namely the Spun Trefoil. The talk will be based on:

[1] João Faria Martins, Categorical Groups, Knots and Knotted Surfaces.

### 11/03/2005, 15:00 — 16:00 — Amphitheatre Pa2, Mathematics Building

Marco Mackaay, *Universidade do Algarve*

### Colored stable Bar-Natan link homology

Khovanov defined several link homologies categorifying the colored Jones polynomial and conjectured relations between them. Unfortunately none of them can be computed with the existing computer programs for link homology. Fortunately Khovanov's constructions are universal in the sense that any Frobenius algebra satisfying Bar-Natan's universal axioms can be plugged into them yielding framed link homologies. Paul Turner and I did this for the stable Bar-Natan Frobenius algebra and computed the colored link homology for this choice completely for any link. In my talk I will review Khovanov's constructions briefly and then explain the results Paul and I obtained for the stable Bar-Natan theory.

[1] Marco Mackaay and Paul Turner, Colored stable Bar-Natan link homology

### 09/02/2005, 14:00 — 15:00 — Room P12, Mathematics Building

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 — Room P3.10, Mathematics Building

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 — Room P3.10, Mathematics Building

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 — Room P3.10, Mathematics Building

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 — Room P3.31, Mathematics Building

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 — Room P3.10, Mathematics Building

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 — Room P3.10, Mathematics Building

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 — Room P3.10, Mathematics Building

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 — Room P3.10, Mathematics Building

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 — Room P3.10, Mathematics Building

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 — Room P3.10, Mathematics Building

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 — Room P3.10, Mathematics Building

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 — Room P3.10, Mathematics Building

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 — Room P3.10, Mathematics Building

Marco Mackaay, *Universidade do Algarve*

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

Current organizers: Roger Picken, Marko Stošić.

FCT Projects PTDC/MAT-GEO/3319/2014, *Quantization and Kähler Geometry*, PTDC/MAT-PUR/31089/2017, *Higher Structures and Applications.*