Atiyah–Segal axioms

Atiyah–Segal axioms
Name	Atiyah–Segal axioms
Field	Michael Atiyah, Graeme Segal, Algebraic topology, Mathematical physics
Introduced	1980s
Related	Topological quantum field theory, Category theory, Cobordism hypothesis

Atiyah–Segal axioms

The Atiyah–Segal axioms form an axiomatic framework for Topological quantum field theory that formalizes how geometric and categorical data assign algebraic invariants to manifolds and cobordisms. Originating in the work of Michael Atiyah and Graeme Segal, the axioms connect ideas from Category theory, Algebraic topology, Differential geometry, and Quantum field theory, influencing research across Mathematical physics, Representation theory, and Algebraic geometry.

Introduction

The axiomatic approach emerged from interactions among Michael Atiyah, Graeme Segal, Edward Witten, Freedman, Donaldson, and Witten's work on Chern–Simons theory to provide a categorical formulation linking Cobordism, Homotopy theory, K-theory, Index theorem, and Moduli space constructions. It builds on precedents in the work of Atiyah–Singer index theorem contributors and draws motivation from physical developments such as Conformal field theory, Quantum electrodynamics, Yang–Mills theory, and applications in String theory and M-theory.

Definition and axioms

Roughly, an axiomatic field theory in this setting is a symmetric monoidal functor from a category of manifolds and cobordisms to a category of vector spaces or more general targets. The formulation uses objects and morphisms derived from Cobordism hypothesis ideas, symmetric monoidal structures as in Mac Lane, and functoriality familiar from Category theory. The axioms require assignment of state spaces to closed (d−1)-manifolds and linear maps to d-dimensional cobordisms, compatibility with disjoint union operations as in Monoidal category theory, behavior under gluing corresponding to composition in Category theory, and duality properties related to Poincaré duality and Spanier–Whitehead duality. These conditions echo structures studied by Segal's axioms for conformal field theory, Atiyah's axioms for topological quantum field theory, and categorical refinements by Jacob Lurie.

Examples and constructions

Standard examples include finite-dimensional 2D theories constructed from Frobenius algebra data linked to Dijkgraaf–Witten theory, 3D Chern–Simons theory derived from compact Lie group representations such as SU(2), and 2D theories related to Gromov–Witten invariants and Floer homology inspired by Andreas Floer and Simon Donaldson. Other constructions arise from K-theory and Index theorem methods associated with Atiyah–Singer index theorem, from Vertex operator algebra approaches connected to Borcherds and Richard Borcherds, and from categorical models based on Modular tensor category structures studied by Vladimir Turaev and Alexei Kitaev. Finite gauge theories exemplified by Dijkgraaf–Witten and lattice models influenced by Kenneth Wilson provide combinatorial realizations compatible with the axioms. Higher-categorical examples appear in work by Jacob Lurie and Chris Schommer-Pries on extended theories and fully local constructions.

Relationship to topological quantum field theory

The axioms are often identified with the definition of a topological quantum field theory, connecting to the Cobordism hypothesis and the classification results obtained by Baez, Dolan, and later Jacob Lurie. They formalize locality and gluing, paralleling principles in Quantum field theory as articulated by Gerard 't Hooft and Kenneth Wilson, and provide a bridge between physically motivated path integral constructions like those of Edward Witten and purely algebraic classifications such as those by Freed and Hopkins. The categorical viewpoint links to Monoidal category theory, Higher category theory, and dualizability conditions central to the Cobordism hypothesis.

Extensions and generalizations

Generalizations include equivariant, twisted, and extended variants that incorporate symmetry groups like SO(n), Spin, and gauge groups such as U(1), SU(N), and discrete groups studied by M. F. Atiyah collaborators. Extended topological field theories refine the source category to include manifolds with corners as developed by John Baez, James Dolan, Jacob Lurie, and Chris Schommer-Pries, while twisted versions relate to Twisted K-theory and anomalies explored by Edward Witten and Dan Freed. Non-topological but closely related frameworks include axiomatizations for Conformal field theory by Graeme Segal and algebraic quantum field theory work by Rudolf Haag.

Applications and impact on mathematics and physics

The axioms have catalyzed progress in classification theorems such as generalized cobordism classifications by Jean-Pierre Serre contemporaries and refined invariants in low-dimensional topology like Jones polynomial applications linked to Vaughan Jones and Chern–Simons theory analyses. They influenced developments in Representation theory through connections with Modular tensor category theory, impacted Algebraic geometry via Gromov–Witten and Mirror symmetry collaborations with Maxim Kontsevich, and informed homotopical perspectives in Stable homotopy theory referenced by Douglas Ravenel and Mike Hopkins. In physics, the framework provided rigorous language for topological phases studied by Xiao-Gang Wen and Alexei Kitaev, for anomaly classification in Quantum field theory and for formal aspects of String theory and M-theory. The axioms continue to guide research linking categorical structures to geometric and physical phenomena across institutions such as Institute for Advanced Study, Princeton University, and University of Cambridge.

Category:Mathematical physics