Graeme B. Segal

Unitary Representations of some Infinite Dimensional Groups

We construct projective unitary representations of (a) Map(S1;G), the group of smooth maps from the circle into a compact Lie groupG, and (b) the group of diffeomorphisms of the circle. We show that a class of representations of Map(S1;T), whereT is a maximal torus ofG, can be extended to representations of Map(S1;G),

Classifying spaces and spectral sequences

© Publications mathematiques de l’I.H.E.S., 1968, tous droits reserves. L’acces aux archives de la revue « Publications mathematiques de l’I.H.E.S. » (http://www. ihes.fr/IHES/Publications/Publications.html), implique l’accord avec les conditions generales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systematique est constitutive d’une infraction penale. Toute copie ou impression de ce fichier doit contenir la presente mention de copyright.

Equivariant K-theory

The purpose of this thesis is to present a fairly complete account of equivariant K-theory on compact spaces. Equivariant K-theory is a generalisation of K-theory, a rather well-known cohomology theory arising from consideration of the vector-bundles on a space. Equivariant K-theory, or KG-theory, is defined not on a space but on G-spaces, i.e. pairs (X,α), where X is a space and α is an action of a fixed group G on X, and it arises from consideration of G-vector-bundles on X, i.e. vector-bundles on whose total space G acts in a suitable way (of 3.1). In this thesis G will always be a compact group. But KG-theory does not appear in the first three chapters, which are introductory. Chapter 1 consists of preliminary discussions of little relevance to the sequel, but which permit me to make a few propositions in the later chapters shorter or more elegant. It was intended to be amusing, and the reader may prefer to omit it. Chapter 2 is devoted to the representation-theory of compact groups. When X is a point a G-vector-bundle on X is just a representation-module for G, so the representation-ring, or character-ring, R(G) plays a fundamental role in KG-theory. In chapter 2 I investigate its algebraic structure, and in particular when G is a compact Lie group I determine completely its prime ideals. To do this I have to discuss first the space of conjugacy-classes of a compact Lie group, and outline an induced-representation construction for obtaining finite-dimensional modules for G from modules for suitable subgroups not of finite index. Chapter 3 is a rather full collection of technical results concerning G-vector-bundles: they are all essentially well-known, but have not been stated in the equivariant case. Chapter 4 presents basic equivariant K-theory. I show that it can be defined in three ways: by G-vector-bundles, by complexes of G-vector-bundles, and by Fredholm complexes of infinite-dimensional G-vector-bundles. This chapter also treats the continuity of KG with respect to inverse limits of G-spaces, the Thorn homomorphism for a G-vector-bundle and the periodicity-isomorphism, and the question of extending KG to non-compact spaces. In chapter 5 I obtain for KG(X) a filtration and spectral sequence generalising those of [6], but without dissecting the space X. My method is based on a Cech approach: for each open covering of X I construct an auxiliary space homotopy-equivalent to X which has the natural filtration that X lacks. Also in chapter 5 I prove the localisation-theorem (5.3), which, together with the theory of chapter 6, is one of the most important tools in applied KG-theory. KG(X) is a module over the character-ring R(G), so one can localise it at the prime ideals of R(G), which I have determined in 2.5. The simplest and most important case of the localisation-theorem states that, if β is the prime ideal of characters of G vanishing at a conjugacy-class γ, and if Xγ is the part of X where elements in γ have fixed-points, then the natural restriction-map KG(X) r KG(Xγ) induces an isomorphism when localised at β. In chapter 6 I show how to associate to certain maps f : X r Y of (G-spaces a homomorphism f! : KG(X) r KG(Y). It is the analogue of the Gysin homomorphism in ordinary cohomology-theory; but it can also be regarded as a generalisation of the induced-representation construction of 2.4. In the important special case when f is a fibration whose fibre is a rational algebraic variety I prove that f! is left-inverse to the natural map f! : KG(Y) r KG(X); and I apply that to obtain the general Thom isomorphism theorem. Finally in chapter 7 I prove the theorem towards which my thesis was originally directed. Just as a G-module defines a vector-bundle on the classifying-space BG for G (of [1]), so a G~vector-bundle on X defines a vector-bundle on the space XG fibred over BG with fibre X. Thus one gets a homomorphism α : KG(X) r K(XG). I prove that if KG(X) and K(XG) are given suitable topologies then in certain circumstances K(XG)is complete and α induces an isomorphism of the completion of KG(X) with K(XG). This generalises the theorem of Atiyah-Hirsebruch that R(G)^ ≅ K(BG).

The Definition of Conformal Field Theory

I shall propose a definition of 2-dimensional conformal field theory which I believe is equivalent to that used by physicists.

The topology of spaces of rational functions

A high temperature furnace for use above 2000 DEG C is provided that features fast initial heating and low power consumption at the operating temperature. The cathode is initially heated by joule heating followed by electron emission heating at the operating temperature. The cathode is designed for routine large temperature excursions without being subjected to high thermal stresses. A further characteristic of the device is the elimination of any ceramic components from the high temperature zone of the furnace.

The representation ring of a compact Lie group

© Publications mathématiques de l’I.H.É.S., 1968, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

On equivariant Euler characteristics

The Euler characteristic of an orbifold M/G as used in string theory is identified with the Euler characteristic of equivariant K -theory KG(M).

Floer's infinite dimensional Morse theory and homotopy theory

This paper is a progress report on our efforts to understand the homotopy theory underlying Floer homology. Its objectives are as follows: (A) to describe some of our ideas concerning what exactly the Floer homology groups compute; (B) to explain what kind of an object we think the «Floer homotopy type» of an infinite dimensional manifold should be; (C) to work out, in detail, the Floer homotopy type in some examples.

The cohomology of the space of magnetic monopoles

Denote byXq the reduced space ofSU2 monopoles of chargeq in ℝ3. In this paper the cohomology ofXq, the cohomology with compact supports ofXq, and the image of the latter in the former are all calculated as representations of ℤ/qℤ which acts onX2. This provides a non-trivial “lower bound” for theL2 cohomology ofXq which is compatible with some conjectures of Sen. It is also shown that, granted some assumptions about the metric onXq, itsL2 cohomology does not exceed this bound in the situation referred to in the paper as the “coprime case”.

The geometry of the KdV equation

In this talk I shall give a fairly geometrical account of the main facts about the KdV equation on the circle, explaining in particular how it is related to the group Diff(S1), and why it is a completely integrable Hamiltonian system. In §4 I shall describe the theorem of Drinfeld and Sokolov [1] which shows that the KdV system can be regarded as a symplectic quotient of a coadjoint orbit of the loop group of SL2(R). Finally, in §5 I shall explain how the theory generalizes from second-order equations and SL2 to nth-order equations and SLn: the resulting classical system is the one whose quantization leads to the Zamolodchikov algebra Wn. My aim is purely expository: all the material is well known.