Atiyah–Bott

Atiyah–Bott
Name	Atiyah–Bott
Field	Mathematics
Subfield	Algebraic topology; Differential geometry; Gauge theory; Symplectic geometry
Introduced	1960s–1980s
Main results	Fixed-point theorem for elliptic complexes; Morse theory for Yang–Mills; Localization in equivariant cohomology
Notable contributors	Michael Atiyah; Raoul Bott

Atiyah–Bott

The Atiyah–Bott collaboration produced a collection of results and methods linking Michael Atiyah and Raoul Bott in work that reshaped algebraic topology, differential geometry, gauge theory, and symplectic geometry. Their names designate a fixed-point theorem for elliptic complexes, an equivariant localization technique in cohomology, and a Morse-theoretic analysis of the Yang–Mills equations on Riemann surfaces, influencing research in index theory, moduli spaces, mathematical physics, and representation theory.

Introduction

Atiyah and Bott combined tools from Atiyah–Singer index theorem, Thom isomorphism, Lefschetz fixed-point theorem, and equivariant cohomology to obtain results now collectively described as Atiyah–Bott. Their work connects the names Michael Atiyah, Raoul Bott, Isadore Singer, Raoul Bott's earlier work, and institutions such as Cambridge University, Princeton University, Institute for Advanced Study, and Harvard University. The methods exploit actions of compact Lie groups such as U(1), SU(n), and SO(n) on manifolds studied by people including Henri Cartan, Armand Borel, and Bertram Kostant.

History and Development

The historical thread begins with early fixed-point and index results: Lefschetz, Hermann Weyl, and the development of topological K-theory by Atiyah and Friedrich Hirzebruch. Bott's periodicity theorem and work on Morse theory for loop spaces provided precursors connecting to the Atiyah–Singer index theorem by Atiyah and Isadore Singer. In the 1970s and 1980s, collaborations among Atiyah, Bott, Singer, and contemporaries like Michael Freedman, Edward Witten, and Simon Donaldson fostered applications to Yang–Mills theory and moduli of vector bundles on Riemann surfaces. Seminal papers appeared in venues associated with Proceedings of the National Academy of Sciences, Annals of Mathematics, and conferences at International Congress of Mathematicians gatherings where figures such as Jean-Pierre Serre, Alexander Grothendieck, and Stephen Smale influenced the mathematical milieu.

Atiyah–Bott Fixed-Point Theorem

The Atiyah–Bott fixed-point theorem generalizes the Lefschetz fixed-point theorem and the Atiyah–Singer index theorem to elliptic complexes equivariant under compact group actions such as torus actions by T^n and actions of groups like SU(2). The statement relates equivariant indices computed via K-theory by methods from Michael Atiyah and Isadore Singer to contributions from fixed-point sets analyzed with techniques related to Bott periodicity and the Thom class studied by René Thom. The proof uses localization in equivariant cohomology as developed by Bertram Kostant, Henri Cartan, and later formalized in contexts influenced by Edward Witten's supersymmetric localization. The theorem yields formulas expressing global invariants in terms of local data around fixed-point components studied in work by Nigel Hitchin, Constantin Teleman, and George Lusztig.

Applications and Consequences

Atiyah–Bott methods revolutionized computation in moduli space theory, notably in the study of moduli of stable vector bundles on Riemann surfaces and connections with Narasimhan–Seshadri theorem, Donaldson theory, and Seiberg–Witten theory. Their Morse theory for the Yang–Mills functional on surfaces influenced analyses by Simon Donaldson, Karen Uhlenbeck, and Edward Witten linking to quantum field theory and conformal field theory frameworks used by Alexander Polyakov and Graeme Segal. Localization techniques inform the computation of partition functions and intersection pairings appearing in work by Maxim Kontsevich, Mikhail Gromov, and Anton Alekseev. In representation theory, Atiyah–Bott localization underlies calculations for characters and multiplicities related to Weyl character formula contexts studied by Hermann Weyl and Harish-Chandra.

Related Concepts and Generalizations

Generalizations and related concepts include the Berline–Vergne localization formula, the equivariant index approaches by Donovan–Karoubi and Atiyah–Segal, and extensions to infinite-dimensional settings in loop space and string topology influenced by Chas–Sullivan. Connections appear with the Duistermaat–Heckman formula, the theory of moment maps by Kirillov and Kostant, and the use of equivariant cohomology in the work of Atiyah–Segal and Guillemin–Sternberg. Homotopical and categorical viewpoints bring in contributions from Jacob Lurie, Maxim Kontsevich's homological mirror symmetry circle, and developments in K-theory by Daniel Quillen and Bott periodicity contexts studied by Raoul Bott.

Key Examples and Computations

Key computations include equivariant indices on compact complex manifolds such as flag varietys studied by Hirzebruch and Bernhard Riemann-type moduli on Riemann surfaces leading to explicit Betti number calculations for moduli spaces, pursued by Atiyah, Bott, Narasimhan, and Seshadri. Classic examples treat torus actions on projective varieties like complex projective space and Grassmannians where localization yields formulas related to the Schubert calculus of Hermann Schubert and intersections computed by William Fulton. Gauge-theoretic examples compute Morse stratifications of the space of connections on principal bundles for structure groups U(n), SU(n), and SO(n) with consequences for invariants studied in Donaldson theory and enumerative predictions compared with results by Maxim Kontsevich and Edward Witten.

Category:Mathematics