Publications (7)0 Total impact
-
[show abstract]
[hide abstract]
ABSTRACT: We highlight the general notion of a relative quantum field theory, which
occurs in several contexts. One is in gauge theory based on a compact Lie
algebra, rather than a compact Lie group. This is relevant to the maximal
superconformal theory in six dimensions.
12/2012;
-
[show abstract]
[hide abstract]
ABSTRACT: It is a long-standing question to extend the definition of 3-dimensional Chern-Simons theory to one which associates values to 1-manifolds with boundary and to 0-manifolds. We provide a solution in case the gauge group is a torus. We also develop from different points of view an associated 4-dimensional invertible topological field theory which encodes the anomaly of Chern-Simons. Finite gauge groups are also revisited, and we describe a theory of "finite path integrals" as a general construction for a certain class of finite topological field theories. Topological pure gauge theories in lower dimension are presented as a warm-up. Comment: 39 pages, submission to proceedings of "A Celebration of the Mathematical Legacy of Raoul Bott"; v2 has minor changes
05/2009;
-
[show abstract]
[hide abstract]
ABSTRACT: This is the first in a series of papers investigating the relationship between the twisted equivariant K-theory of a compact Lie group G and the "Verlinde ring" of its loop group. In this paper we set up the foundations of twisted equivariant K-groups, and more generally twisted K-theory of groupoids. We establish enough basic properties to make effective computations. Using the Mayer-Vietoris spectral sequence we compute the twisted equivariant K-groups of a compact connected Lie group G with torsion free fundamental group. We relate this computation to the representation theory of the loop group at a level related to the twisting.
12/2007;
-
[show abstract]
[hide abstract]
ABSTRACT: We give an a priori construction of the two-dimensional reduction of three-dimensional quantum Chern-Simons theory. This reduction is a two-dimensional topological quantum field theory and so determines to a Frobenius ring, which here is the twisted equivariant K-theory of a compact Lie group. We construct the theory via correspondence diagrams of moduli spaces, which we "linearize" using complex K-theory. A key point in the construction is to consistently orient these moduli spaces to define pushforwards; the consistent orientation induces twistings of complex K-theory. The Madsen-Tillmann spectra play a crucial role.
12/2007;
-
[show abstract]
[hide abstract]
ABSTRACT: This is the second in a series of papers investigating the relationship
between the twisted equivariant K-theory of a compact Lie group G and the
"Verlinde ring" of its loop group. We introduce the Dirac family of Fredholm
operators associated to a positive energy representation of a loop group. It
determines a map from isomorphism classes of representations to twisted
K-theory, which we prove is an isomorphism if $G$ is connected with
torsion-free fundamental group. We also introduce a Dirac family for finite
dimensional representations of compact Lie groups; it is closely related to
both the Kirillov correspondence and the equivariant Thom isomorphism.
In Part III (math.AT/0312155) we extend the proof of our main theorem to
arbitrary compact Lie groups G and provide supplements in various directions.
In Part I (arXiv:0711.1906) we develop twisted equivariant K-theory and carry
out some of the computations needed here. We refer to the announcements
math.AT/0312155 and math.AT/0206237 for further expository material and
motivation.
11/2005;
-
[show abstract]
[hide abstract]
ABSTRACT: This is the third paper of a series relating the equivariant twisted $K$-theory of a compact Lie group $G$ to the ``Verlinde space'' of isomorphism classes of projective lowest-weight representations of the loop groups. Here, we treat arbitrary compact Lie groups. In addition, we discuss the relation to semi-infinite cohomology, the fusion product of Conformal Field theory, the r\^ole of energy and the topological Peter-Weyl theorem.
01/2004;
-
[show abstract]
[hide abstract]
ABSTRACT: Using a global version of the equivariant Chern character, we describe the complexified twisted equivariant K-theory of a space with a compact Lie group action in terms of fixed-point data. We apply this to the case of a compact Lie group acting on itself by conjugation, and relate the result to the Verlinde algebra and to the Kac numerator at q=1. Verlinde's formula is also discussed in this context.
07/2002;
