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Manifolds and cell complexes Definition

Topological quantum field theory

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Definition

Let n≥0n \ge 0 be an integer and let kk be a field. All manifolds below are smooth, compact and oriented. For closed nn-manifolds Σ0\Sigma_0 and Σ1\Sigma_1, a cobordism from Σ0\Sigma_0 to Σ1\Sigma_1 is an (n+1)(n+1)-manifold with boundary MM with an orientation-preserving diffeomorphism ∂M→Σ0‾⊔Σ1\partial M \to \overline{\Sigma_0} \sqcup \Sigma_1. Here Σ‾\overline{\Sigma} is Σ\Sigma with the opposite orientation. An (n+1)(n+1)-dimensional topological quantum field theory ZZ over kk assigns two kinds of data. To each closed nn-manifold Σ\Sigma it assigns a finite-dimensional vector space Z(Σ)Z(\Sigma) over kk. To each cobordism MM from Σ0\Sigma_0 to Σ1\Sigma_1 it assigns a linear map Z(M) ⁣:Z(Σ0)→Z(Σ1)Z(M) \colon Z(\Sigma_0) \to Z(\Sigma_1). These data satisfy the following axioms.

  1. Each orientation-preserving diffeomorphism f ⁣:Σ→Σ′f \colon \Sigma \to \Sigma' gives a linear isomorphism Z(f) ⁣:Z(Σ)→Z(Σ′)Z(f) \colon Z(\Sigma) \to Z(\Sigma'), with Z(f′∘f)=Z(f′)∘Z(f)Z(f' \circ f) = Z(f') \circ Z(f). Diffeomorphic cobordisms, by a diffeomorphism compatible with the boundaries, give equal maps after these isomorphisms.
  2. Let MM be a cobordism from Σ0\Sigma_0 to Σ1\Sigma_1, and M′M' one from Σ1\Sigma_1 to Σ2\Sigma_2. Let M′∪Σ1MM' \cup_{\Sigma_1} M be the cobordism obtained by gluing them along Σ1\Sigma_1. Then Z(M′∪Σ1M)=Z(M′)∘Z(M)Z(M' \cup_{\Sigma_1} M) = Z(M') \circ Z(M).
  3. For each Σ\Sigma, the cylinder Σ×[0,1]\Sigma \times [0, 1] gives the identity map of Z(Σ)Z(\Sigma).
  4. The empty nn-manifold gives Z(∅)=kZ(\emptyset) = k, and disjoint unions give natural isomorphisms Z(Σ⊔Σ′)≅Z(Σ)⊗Z(Σ′)Z(\Sigma \sqcup \Sigma') \cong Z(\Sigma) \otimes Z(\Sigma') under which Z(M⊔M′)=Z(M)⊗Z(M′)Z(M \sqcup M') = Z(M) \otimes Z(M'), where ⊗\otimes is the tensor product over kk.

Remarks

A closed (n+1)(n+1)-manifold MM is a cobordism from ∅\emptyset to ∅\emptyset, so Z(M)Z(M) is a linear map k→kk \to k, that is, a number Z(M)∈kZ(M) \in k. By clause 1 this number is a diffeomorphism invariant of MM. The axioms imply that Z(Σ‾)Z(\overline{\Sigma}) is the dual space of Z(Σ)Z(\Sigma). In the language of categories, a topological quantum field theory is a symmetric monoidal functor from the category of (n+1)(n+1)-dimensional cobordisms to the category of vector spaces over kk. Example: let n=0n = 0 and let VV be a finite-dimensional vector space. Set Z(+)=VZ(+) = V for the positive point and Z(−)=V∗Z(-) = V^*, and extend to finite sets of points by tensor products. An arc from ∅\emptyset to {+,−}\set{+, -} gives the element of V⊗V∗V \otimes V^* that corresponds to id⁡V\id_V, an arc from {+,−}\set{+, -} to ∅\emptyset gives evaluation V⊗V∗→kV \otimes V^* \to k, and straight arcs give identities. These choices satisfy the axioms, and the circle, which is the composite of the two arcs, gets the number Z(S1)=dim⁡VZ(S^1) = \dim V. Non-example: set Z(Σ)=kZ(\Sigma) = k for all Σ\Sigma and let every cobordism act as multiplication by 22. Then the cylinder Σ×[0,1]\Sigma \times [0, 1] gives multiplication by 22, not the identity, so clause 3 fails. In dimension 1+11 + 1, topological quantum field theories correspond to commutative Frobenius algebras over kk.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Euclidean space object
  2. Function concept
  3. Function composition concept
  4. Manifold with boundary concept
  5. Orientability concept
  6. Vector space definition
  7. Fréchet derivative definition
  8. Function of several variables concept
  9. Bilinear form definition
  10. Linear map definition
  11. Limit of a function of several variables concept
  12. Partial derivative concept
  13. Linear form definition
  14. Space of linear maps definition
  15. Tensor product of vector spaces definition
  16. Differentiable function of several variables definition
  17. Partial derivative definition
  18. Dual space definition
  19. Differentiable on an open set definition
  20. Jacobian matrix definition
  21. Chain rule in several variables statement
  22. Higher partial derivatives definition
  23. Iterated derivative definition
  24. Jacobian determinant definition
  25. Derivative of the inverse function statement
  26. C^k function characterised by continuous derivatives statement
  27. Inverse function theorem statement
  28. Diffeomorphism definition
  29. Topological quantum field theory definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Topological_quantum_field_theory
About this page
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Kinddefinition
WritingW0 stub
MSC57R56, 81T45, 57K16
DefinesTopological quantum field theory
UsesManifold with boundary, Orientability, Diffeomorphism, Vector space, Linear map, Tensor product of vector spaces, Dual space
Changed2 November 2026

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