Atiyah's axioms for topological quantum field theory

Atiyah's axioms for topological quantum field theory
Name	Atiyah's axioms for topological quantum field theory
Introduced	1988
Author	Sir Michael Atiyah
Subject	Topological quantum field theory

Atiyah's axioms for topological quantum field theory provide an axiomatic formulation of topological quantum field theories inspired by ideas in Michael Atiyah, Graeme Segal, and developments in Edward Witten's work. The axioms recast physical notions from Richard Feynman's path integral and Paul Dirac's canonical quantization into a mathematical framework drawing on Bourbaki-style abstraction, connecting Alexander Grothendieck's categorical language with constructions from Michael Freedman and Edward Witten's topological insights. This framework has influenced research at the intersection of University of Cambridge (UK), Institute for Advanced Study, and institutions such as Princeton University and University of Oxford.

Introduction

Atiyah presented his axioms as a list of properties for a map assigning algebraic data to geometric data, echoing themes present in work by Graeme Segal and earlier formulations by Michael Atiyah himself. The axioms formalize how closed manifolds and cobordisms produce vector spaces and linear maps, mirroring constructions used by Edward Witten in his analysis of Chern–Simons theory and connections to Jones polynomial invariants discovered by Vaughan Jones. The formulation influenced collaborations at the Mathematical Sciences Research Institute and seminars involving researchers from Harvard University and University of Chicago.

Formal definition

Atiyah's formal definition treats a topological quantum field theory as a symmetric monoidal functor from a category of cobordisms to a category of finite-dimensional vector spaces, inspired by categorical axioms developed in the tradition of Saunders Mac Lane and Alexander Grothendieck. Objects in the domain category are closed (d-1)-manifolds, while morphisms are d-dimensional cobordisms; composition corresponds to gluing, reflecting ideas in René Thom's cobordism theory and Andrey Kolmogorov-style decomposition. The target assigns to each manifold a vector space and to each cobordism a linear map, satisfying multiplicativity under disjoint unions, functoriality under composition, and compatibility with orientation reversal, paralleling structures studied at Princeton University and Massachusetts Institute of Technology.

Examples and constructions

Classic examples include 2-dimensional TQFTs classified by Frobenius algebras, connecting to results from John Milnor and algebraic structures investigated by Pierre Deligne and Jean-Pierre Serre. Three-dimensional examples arise from Chern–Simons theory and quantum invariants studied by Edward Witten, Vladimir Drinfeld, and Mikhail Khovanov; these relate to constructions by Vladimir Turaev and Leonid Reshetikhin. Two-dimensional sigma-model limits and Gromov–Witten theories tie into work by Maxim Kontsevich and Yuri Manin, while lattice models connect to developments at Bell Labs and research groups at University of California, Berkeley. Finite gauge group examples reflect input from John Conway-era combinatorics and classification efforts in Michael Freedman's program.

Relation to quantum field theory and category theory

Atiyah's axioms bridge mathematical physics and pure mathematics, situating topological quantum field theories within the paradigm advanced by Saunders Mac Lane and Alexander Grothendieck's categorical methods and echoing conceptual foundations laid by Paul Dirac and Richard Feynman. The functorial perspective fostered links with Category Theory developments by William Lawvere and the monoidal frameworks used by Max Kelly and Samuel Eilenberg. Connections to perturbative quantum field theory, renormalization techniques by Kenneth Wilson, and nonperturbative constructions by Edward Witten have motivated categorical refinements receptive to insights from Alain Connes's noncommutative geometry.

Applications and consequences

Consequences include classification results for low-dimensional manifolds influenced by work of Michael Freedman and Ciprian Manolescu, and invariants such as the Jones polynomial and its generalizations analyzed by Vladimir Turaev and Edward Witten. The axioms underpin relationships between knot theory studied by Joan Birman and Louis Kauffman and quantum algebra developed by Vladimir Drinfeld and Michio Jimbo. In mathematical contexts, the framework informs modular tensor categories explored by Alexei Kitaev and Bence Csákány, with implications for quantum computing initiatives at institutions like Google's quantum AI lab and academic groups at Yale University.

Variations and generalizations

Generalizations include extended TQFTs assigning higher-categorical data to manifolds of various codimensions, influenced by Jacob Lurie's cobordism hypothesis and higher category theory research from John Baez and James Dolan. Equivariant and twisted variants draw on gauge-theoretic input from Simon Donaldson and Karen Uhlenbeck, while fermionic and spin-refined versions relate to spin cobordism studies by Ralph Cohen and Michael Hopkins. Field-theoretic extensions intersect with string-theoretic perspectives from Juan Maldacena and dualities studied by Edward Witten and Andrew Strominger.

Historical context and development

Atiyah introduced the axioms in lectures and writings influenced by interactions with Graeme Segal and contemporaneous physical breakthroughs by Edward Witten in the late 1980s, following mathematical foundations set by René Thom and Hirzebruch. The proposal catalyzed research across institutions including University of Cambridge (UK), Institute for Advanced Study, Princeton University, and research centers such as the Mathematical Sciences Research Institute, spawning collaborations among mathematicians like Michael Freedman, Maxim Kontsevich, and Jacob Lurie and physicists including Edward Witten and Michael Green. The axiomatic viewpoint remains central in contemporary work linking topology, quantum algebra, and higher category theory, and continues to inform projects at Simons Foundation-funded institutes and leading universities.

Category:Topological quantum field theory