Dijkgraaf–Witten

Dijkgraaf–Witten
Name	Dijkgraaf–Witten
Field	Mathematical physics; Topology; Quantum field theory
Introduced	1990
Authors	Robbert Dijkgraaf; Edward Witten
Main idea	Topological quantum field theory defined from finite group gauge theory via group cohomology

Dijkgraaf–Witten

Dijkgraaf–Witten is a family of topological quantum field theories introduced by Robbert Dijkgraaf and Edward Witten in 1990. The theories associate finite-dimensional invariants to closed manifolds and bordisms by using finite groups and cohomology classes, linking constructions from Robbert Dijkgraaf, Edward Witten, Atiyah–Segal axioms, Witten–Reshetikhin–Turaev invariants, and Chern–Simons theory. They provide tractable examples that connect Michael Atiyah's axiomatic perspective, computations in Low-dimensional topology, and advances in Condensed matter physics and Quantum computing.

Introduction

Dijkgraaf–Witten theories arise from the intersection of ideas in Robbert Dijkgraaf's work on topological models, Edward Witten's insights into quantum field theory, and classical results in Group cohomology. Conceived as finite-group analogues of Chern–Simons theory and informed by the Atiyah–Segal axioms for topological quantum field theory, they produce state-sum invariants similar in spirit to the Turaev–Viro invariants and the Reshetikhin–Turaev construction. Early applications involved connections to invariants studied by John Milnor, William Browder, and computational approaches influenced by Vaughan Jones's work on quantum invariants.

Definition and Construction

The Dijkgraaf–Witten construction starts with a finite group G and a cohomology class [ω] ∈ H^n(BG; U(1)) where BG denotes the classifying space of G. For an oriented closed n-manifold M, one sums over isomorphism classes of principal G-bundles (equivalently homotopy classes of maps M → BG) and weights each bundle by the evaluation of ω on the corresponding fundamental class. This procedure mirrors the path-integral heuristic of Feynman path integral formulations and uses techniques developed in the study of Eilenberg–MacLane spaces and Postnikov towers. The state-sum realization connects to categorical formulations stemming from Jean Bénabou's work on bicategories and to algebraic structures like Hopf algebras and Fusion categories.

Topological Quantum Field Theory Interpretation

As a fully local topological quantum field theory in the sense of Michael Atiyah and Graeme Segal, Dijkgraaf–Witten assigns vector spaces to (n−1)-manifolds and linear maps to n-dimensional bordisms. The vector spaces are spanned by equivalence classes of principal G-bundles on the boundary, and the linear maps are constructed by summing over extensions of those bundles to the bordism with ω-weighted amplitudes. This TQFT perspective situates Dijkgraaf–Witten alongside Atiyah–Singer Index Theorem-inspired TQFTs and categorical quantum field theories studied by Jacob Lurie and Kevin Costello, and it relates to modular functors appearing in the work of Gregory Moore and Nathan Seiberg.

Examples and Computations

In three dimensions with G finite and ω ∈ H^3(BG; U(1)), the partition function on a closed 3-manifold recovers invariants analogous to the Chern–Simons invariant for finite gauge groups; explicit computations use triangulations and state sums comparable to the Turaev–Viro state sum or the Crane–Yetter model. For G = Z/n one obtains abelian examples computable via classical techniques from Algebraic topology and Homological algebra, connecting to invariants studied by John H. Conway in low-dimensional combinatorics. Nonabelian computations exploit character theory of Representation theory of finite groups and relations to the Drinfeld double construction and Dijkgraaf–Pasquier–Roche algebraic formulations.

Relation to Group Cohomology and Homotopy Theory

The role of ω ∈ H^n(BG; U(1)) makes Dijkgraaf–Witten a bridge between quantum invariants and classical group cohomology studied by Samuel Eilenberg and Saunders Mac Lane. The classification of twisting classes parallels obstruction theory in the work of J. H. C. Whitehead and relates to Postnikov invariants. Homotopical refinements use the unstable homotopy category and spectral sequence techniques developed by J. Peter May and Jean-Pierre Serre, while equivariant enhancements connect to the machinery of G-equivariant cohomology and fixed-point theories advanced by Atiyah and Bott.

Applications in Condensed Matter and Quantum Computing

Dijkgraaf–Witten models provide lattice realizations of topological phases of matter and symmetry-protected topological phases analyzed in the condensed matter literature by Xiao-Gang Wen, Alexei Kitaev, and Michael Levin. The finite-group TQFTs correspond to exactly solvable lattice Hamiltonians like the Kitaev quantum double models and inform anyon classification relevant to Topological quantum computing proposals studied by Sankar Das Sarma and Chetan Nayak. Connections extend to defect and boundary theories examined in the work of Anton Kapustin and Natalie Read, with implications for braiding statistics, fault-tolerant gate sets, and computational universality criteria influenced by Peter Shor and Alexei Kitaev.

Extensions and Generalizations

Generalizations include higher-group and higher-form symmetries replacing finite G with categorical groups and employing cohomology classes in higher stacks, linking to the higher-TQFT program advocated by Jacob Lurie and formalized via cobordism hypotheses. Continuous-group analogues relate to traditional Chern–Simons theory with compact Lie groups like SU(2) and U(1), while twisted and equivariant versions involve additional data studied by Daniel Freed and Greg Moore. Algebraic extensions connect to modular tensor categories and quantum group constructions developed by Vladimir Drinfeld, Michihisa Wakui, and Nikita Reshetikhin, and homotopical generalizations tie into ongoing research by Dan Freed, Chris Schommer-Pries, and Kevin Walker.

Category:Topological quantum field theories Category:Mathematical physics