Atiyah–Bott localization

Atiyah–Bott localization

Introduction

Atiyah–Bott localization arises in the intersection of Michael Atiyah's work and Raoul Bott's contributions to topology and index theory, and it connects to the legacy of Atiyah–Singer index theorem, Lefschetz fixed-point theorem, Berline–Vergne localization formula, Hirzebruch–Riemann–Roch theorem, and the program of Equivariant cohomology studied by scholars at Institute for Advanced Study, Princeton University, and Harvard University. The result gives a formula expressing integrals in Equivariant cohomology of a compact Lie group-space in terms of contributions from the fixed point set of a torus action and interacts with techniques from Morse theory used by Marston Morse, Raoul Bott, and M. F. Atiyah in studies related to Yang–Mills theory developed in work of Simon Donaldson and Edward Witten.

Statement of the theorem

The localization theorem, as formulated by Atiyah and Bott, states that for a compact manifold acted on by a compact torus T (often denoted by a copy of S^1^n) with isolated fixed points or components, the inclusion of the fixed locus induces an isomorphism after inverting appropriate elements in the equivariant cohomology ring; this statement refines classical results such as the Lefschetz fixed-point theorem and the Atiyah–Singer index theorem and relates to formulas in Algebraic geometry like the Grothendieck–Riemann–Roch theorem, the Hirzebruch–Riemann–Roch theorem, and localization principles used by Alexander Grothendieck and Jean-Pierre Serre. In the common formulation for a compact Hamiltonian T-space a cohomology class integrates to a sum over fixed components where each summand involves the equivariant Euler class of the normal bundle; this mirrors computations in the work of Vladimir Arnold, André Weil, Harish-Chandra, and techniques connected to the representation theory of Cartan and Weyl group considerations appearing in the work of Élie Cartan and Hermann Weyl.

Proofs and techniques

Atiyah and Bott produced proofs using methods from Morse theory and sheaf-theoretic arguments influenced by Alexander Grothendieck's algebraic formalism and the cohomological methods of Jean Leray and Henri Cartan, while alternative approaches by Berline, Nicole Berline, and Michèle Vergne exploited heat-kernel proofs linked to the Atiyah–Singer index theorem and analytic techniques pioneered by Isadore Singer. Other demonstrations use algebraic geometry tools developed by Pierre Deligne, Jean-Pierre Serre, and David Mumford within the framework of equivariant intersection theory advanced by William Fulton and Roy Smith. Localization proofs often deploy equivariant characteristic classes related to constructions by Chern and Pontryagin and resonate with the algebraic topology constructions of John Milnor and James Stasheff, while connections to fixed-point formulas reflect themes in the works of G. D. Birkhoff and Lefschetz.

Applications and examples

Atiyah–Bott localization has been applied across diverse problems: enumerative counts in Gromov–Witten theory pursued by Maxim Kontsevich and Edward Witten, intersection computations on moduli spaces of bundles as developed by Nigel Hitchin and Simon Donaldson, exact evaluations in Equivariant K-theory studied by Graeme Segal and Friedrich Hirzebruch, and partition function calculations in Supersymmetric gauge theory explored by Seiberg–Witten theory and researchers like Nathan Seiberg and Edward Witten. Specific examples include localization on projective spaces studied by Alexander Grothendieck-inspired methods, fixed-point computations for flag varieties linked to Hermann Schubert and Bernhard Riemann-inspired enumerative geometry, and calculations on quiver varieties related to the work of Hiraku Nakajima and Victor Kac. In mathematical physics, the formula underpins exact results in Chern–Simons theory associated with Edward Witten and in mirror symmetry contexts explored by Kontsevich and Strominger–Yau–Zaslow contributors such as Andrew Strominger, Shing-Tung Yau, and Eric Zaslow.

Generalizations and related results

Generalizations of Atiyah–Bott localization include algebraic formulations by Grigory Faltings and William Fulton in equivariant intersection theory, K-theoretic localizations advanced by Maxim Kontsevich and Ginzburg-school developments, and nonabelian localization approaches studied by Witten and later formalized by researchers affiliated with Harvard University and Princeton University. Related results encompass the Berline–Vergne localization formula, the Duistermaat–Heckman theorem developed by Johannes J. Duistermaat and Geert Heckman, virtual localization techniques of Jun Li and G. Tian, and categorical extensions motivated by the homological program of Alexander Beilinson and Joseph Bernstein. Work connecting to representation theory includes adaptations in the spirit of Harish-Chandra and Bernstein–Gelfand–Gelfand resolutions, while algebro-geometric expansions interact with the paradigms of Serre duality and Grothendieck duality.

Category:Mathematical theorems