---
tag: DAP92B80J2
kind: concept
name: Topological quantum field theory
msc: [57R56, 81T45, 57K16]
sources:
  - url: https://en.wikipedia.org/wiki/Topological_quantum_field_theory
same_as:
  wikidata: Q721094
  wikipedia: Topological quantum field theory
---

## Summary

A topological quantum field theory assigns a vector space to each closed manifold of a fixed dimension $n$.
It assigns a linear map to each $(n+1)$-manifold whose boundary joins two such manifolds.
Gluing manifolds must correspond to composing linear maps, and disjoint union must correspond to the tensor product.
A closed manifold of the top dimension then receives a number, which is an invariant of the manifold.
Atiyah stated these axioms, drawing on work of Witten and Segal in physics.
A main example is [[Chern–Simons theory|F28G886PWD]], which produces [[quantum invariants of 3-manifolds|YGPR4VR3GP]] and of knots.

## Explanations

### L1 human

Think of a closed surface as a moment in time and a solid shape between two surfaces as a history from one moment to the next.
A topological quantum field theory gives each moment a space of possible states and each history a rule that turns old states into new ones.
Doing two histories one after the other must give the same rule as composing the two rules.
The rules only see the shape up to smooth deformation, not lengths or angles.
In the simplest case, with points as moments and curves as histories, a theory picks a vector space $V$, and the circle receives the number $\dim V$.