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Topological Quantum Field Theory Seminar   RSS

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29/06/2005, 15:30 — 16:30 — Room P3.10, Mathematics Building
, 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 C2 trivialized at , with holomorphic bundles on blow ups of CP2 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
, 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:

  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

09/06/2005, 15:30 — 16:30 — Room P3.10, Mathematics Building
, 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
  1. 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
, 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
  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

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
, 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.
  1. E. Witten, Two-dimensional gauge theories revisited, J. Geom. Phys. 9 (1992), no. 4, 303-368.
    hep-th/9204083
  2. 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:
  1. Dror Bar-Natan, "On Khovanov's categorification of the Jones polynomial", Algebraic and Geometric Topology 2 (2002) 337-370; math.QA/0201043.
  2. Mikhail Khovanov, "A functor-valued invariant of tangles", Algebr. Geom. Topol. 2 (2002) 665-741; math.QA/0103190.
  3. 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:
  1. I. M. Gelfand, M. I. Graev, and N. Ya. Vilenkin. Generalized Functions volume 5, "Integral Geometry and Representation Theory". Academic Press,New York, 1966.
  2. A.A. Kirillov. Elements of the Theory of Representations Springer-Verlag, 1976.
  3. 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:

  1. P. Lopes, Quandles at finite temperatures I, J. Knot Theory Ramifications, 12(2):159-186 (2003), math.QA/0105099
  2. F. M. Dionisio and P. Lopes, Quandles at finite temperatures II, J. Knot Theory Ramifications to appear, math.GT/0205053
  3. 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:

  1. P. Lopes, Quandles at finite temperatures I, J. Knot Theory Ramifications, 12(2):159-186 (2003), math.QA/0105099
  2. F. M. Dionisio and P. Lopes, Quandles at finite temperatures II, J. Knot Theory Ramifications to appear, math.GT/0205053
  3. 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 As of A, known as the symmetric part of A, and of a partial triple product (a,b,c)abc mapping A× As ×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

  1. 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.
  2. 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
, 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

  1. M. Varadarajan, Phys. Rev D. 64 , 104003 (2001); gr-qc/0104051
  2. J.M. Velhinho, Commun. Math. Phys. 227, 541 (2002); math-ph/0107002
  3. A. Ashtekar and J. Lewandowski, Class. Quant. Grav. 18, L117 (2001); gr-qc/0107043
  4. T. Thiemann, gr-qc/0206037
  5. H. Sahlmann, gr-qc/0207112
  6. 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 R2 >N this moduli space is C PN , 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 R2 is close to N.

References

  1. J. M. Baptista and N. S. Manton, The dynamics of vortices on S2 near the Bradlow limit, hep-th/0208001.
  2. T. M. Samols, Vortex Scattering, Commun. Math. Phys. 145, 149 (1992).

19/12/2002, 11:00 — 12:00 — Room P3.10, Mathematics Building
, 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
  1. A. B. Cruzeiro and P. Malliavin -"Renormalized differential geometry on path space: structural equation, curvature", J.Funct. Anal. 139 (1996), p. 119 -181.
  2. 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
, Universidade do Algarve

The coadjoint orbits of 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 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 SL(2,C).

References

  1. I.M. Gelfand, R.A. Minlos, Z.Ya. Shapiro, Representations of the rotation and Lorentz groups and their applications, Pergamon Press, 1963.
  2. A. Kirillov, Elements of representation theory, Springer.
  3. 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

  1. G. Folland, Harmonic Analysis in Phase Space, Princeton University Press.
  2. J. Duistermaat, Fourier Integral Operators, Birkhauser.
  3. 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
, Universidade do Algarve

The coadjoint orbits of SL(2,C)

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