1998 seminars

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.
  1. E. Witten, Two-dimensional gauge theories revisited, J. Geom. Phys. 9 (1992), no. 4, 303-368.
  2. L. C. Jeffrey, F. Kirwan, Localization for nonabelian group action, Topology 34 (1995) no. 2, 291-327.

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

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.
  1. Nathan Berkovits, ICTP Lectures on Covariant Quantization of the Superstring,
  2. Nathan Berkovits, Covariant Multiloop Superstring Amplitudes,
  3. Nathan Berkovits, Nikita Nikrasov, The Character of Pure Spinors,

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

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.


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

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 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.

Michael Paluch, Instituto Superior Técnico
Twisted K-Theory

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 ch(E)*Td(X) from K 0 (X) to H * (X) is a natural transformation of covariant functors. Here K 0 (X) denotes the Grothendieck group of algebraic vector bundles on X , H * (X) denotes a suitable cohomology theory, ch is the Chern character, and Td (X) 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.


  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.

Roger Picken, Instituto Superior Técnico
Gerbes and their Parallel Transport

Gerbes are higher-order generalizations of Abelian bundles. They appear in nature, for instance as obstructions to lifting SO (n) -bundles to Spin (n) or Spin c (n) 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.


  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.