Topological Quantum Field Theory — Mathematics Knowledge System

Overview

A topological quantum field theory (TQFT) is a functor from a category of cobordisms to a category of vector spaces, assigning topological invariants to manifolds. TQFTs have no local degrees of freedom -- all observables are topological invariants. Examples include Chern-Simons theory (giving knot invariants like the Jones polynomial), Donaldson theory (4-manifold invariants), and Witten's topological sigma models. Atiyah's axioms for TQFT (1988) formalised the concept. TQFTs are the bridge between physics (quantum gravity, condensed matter) and pure mathematics (low-dimensional topology, category theory).

Intuition

In an ordinary QFT, the partition function Z depends on the metric (geometry) of spacetime. In a TQFT, Z depends only on the topology of spacetime -- it is a topological invariant. For knots in 3D space: Chern-Simons theory on S^3 computes the partition function for knots, and this equals the Jones polynomial (which is purely topological). The mathematical structure is a functor Z: Cob_n -> Vect, where Cob_n is the category of n-dimensional cobordisms (manifolds with boundary) and Vect is the category of vector spaces.

Formal Definition

Definition

An n-dimensional TQFT is a symmetric monoidal functor Z: Cob_n -> Vect_k. Here Cob_n has: objects = closed (n-1)-manifolds (compact, no boundary), morphisms = compact n-manifolds with boundary (cobordisms). The functor assigns: Z(M) = a vector space to each (n-1)-manifold M, Z(W) = a linear map Z(M_in) -> Z(M_out) to each cobordism W: M_in -> M_out, with Z(M_1 square M_2) = Z(M_1) tensor Z(M_2) and Z(M x [0,1]) = identity. The partition function of a closed n-manifold is Z(M) in k (a number).

Z: \mathbf{Cob}_n \to \mathbf{Vect}_k \text{ (symmetric monoidal functor)}

Atiyah's definition of TQFT

Z(M_1 \sqcup M_2) = Z(M_1) \otimes Z(M_2)

Monoidal structure (disjoint union -> tensor product)

\langle J_K, \cdot \rangle_{\mathrm{CS}} = J_K(q) \text{ (Jones polynomial of knot K)}

Chern-Simons gives Jones polynomial

S_{\mathrm{CS}} = \frac{k}{4\pi}\int_M \mathrm{Tr}\left(A\wedge dA + \frac{2}{3}A\wedge A\wedge A\right)

Chern-Simons action

Notation

Notation	Meaning
$\mathbf{Cob}_n$	Category of n-dimensional cobordisms
$Z(M)$	TQFT assigns a vector space to manifold M
$Z(W)$	TQFT assigns a linear map to cobordism W
$S_{\mathrm{CS}}$	Chern-Simons action

Theorems

Theorem 1: Classification of 2D TQFTs

There is a bijection between 2-dimensional TQFTs (Z: Cob_2 -> Vect_k) and commutative Frobenius algebras (A, mu, eta, Delta, epsilon) over k. The bijection: Z(S^1) = A, multiplication mu = Z(pair of pants), comultiplication Delta = Z(upside-down pair of pants), unit eta = Z(cap), counit epsilon = Z(cup). This classification (Dijkgraaf, Abrams) is a complete algebraic invariant of 2D TQFTs.

Theorem 2: Chern-Simons Gives Jones Polynomial

For G = SU(2) Chern-Simons theory on S^3 at level k, the expectation value of a Wilson loop operator W_R(K) in representation R along a knot K equals the Jones polynomial J_K(q) at q = exp(2*pi*i/(k+2)). More precisely, <W_{fund}(K)>_{CS} = J_K(e^{2pi*i/(k+2)}), where J_K is the Jones polynomial normalized so J_{unknot}=1. This is Witten's 1988 discovery.

Theorem 3: Cobordism Hypothesis (Lurie)

A fully extended framed TQFT Z: Cob_n^{fr} -> C (for C an (inf,n)-category) is determined completely by its value on the positively framed point Z(pt+). More precisely, there is an equivalence of (inf,n)-categories Fun^tensor(Cob_n^{fr}, C) = C^{fd} where C^{fd} is the (inf,n)-category of fully dualizable objects of C. This classifies all fully extended TQFTs.

Worked Examples

1
A commutative Frobenius algebra: A = k, mu: k tensor k -> k (multiplication), eta: k -> k (unit), Delta: k -> k tensor k (comultiplication), epsilon: k -> k (counit).
2
Take mu(a tensor b) = ab (multiplication), eta(1) = 1 (unit), Delta(1) = 1 tensor 1 (comultiplication), epsilon(a) = a (counit).
$\mu(a\otimes b) = ab,\quad \Delta(a) = a\otimes 1 = 1\otimes a$
3
The corresponding TQFT: Z(S^1) = k, Z(pair of pants) = mu, Z(cap) = eta.
4
Z assigns to each closed surface Sigma: Z(Sigma) = Z(Sigma)|_{S^1=k} = k. The value Z(Sigma_g) = 1 for every oriented closed surface of genus g (since k is 1-dim and everything composes to the identity).
$Z(\Sigma_g) = 1 \in k \text{ for all genera } g$

✓ Answer

The trivial Frobenius algebra A=k gives the trivial TQFT where Z(S^1)=k and Z(any closed surface) = 1.

Practice Problems

Hardfree response

Explain why TQFTs have no local degrees of freedom and why this makes them easier to define rigorously than standard QFTs.

Common Mistakes

Common Mistake

Thinking TQFTs are just gauge theories with topological terms.

Not every topological term (like the theta-term in Yang-Mills) gives a TQFT. A TQFT must have no metric dependence at all -- the partition function and correlators are pure topological invariants. Chern-Simons theory is a TQFT because its action is metric-free. Yang-Mills theory with a theta term is NOT a TQFT because the kinetic term F^2 requires a metric.

Quiz

Atiyah's definition of a TQFT is:

A gauge theory with topological actionA symmetric monoidal functor from cobordisms to vector spacesA representation of the mapping class groupA commutative Frobenius algebra

Historical Background

Witten's 1988 papers showed that Chern-Simons gauge theory (a 3D TQFT) produces knot invariants, explaining the Jones polynomial (1984) via physics. Atiyah immediately formalised the concept with his axioms. Donaldson's 1983 work on 4-manifold invariants was retrospectively understood as a 4D TQFT. The classification of 2D TQFTs (= commutative Frobenius algebras) was proved by Dijkgraaf. Extended TQFTs (Lurie's cobordism hypothesis, 2009) classify fully extended TQFTs via the homotopy theory of (infinity,n)-categories.

1983
Donaldson polynomial invariants of 4-manifolds (4D TQFT perspective)
Simon Donaldson
1984
Jones discovers the Jones polynomial for knots
Vaughan Jones
1988
Witten shows Chern-Simons theory gives the Jones polynomial
Edward Witten
1988
Atiyah axiomatises TQFT as a functor from cobordisms
Michael Atiyah
2009
Lurie's cobordism hypothesis classifies fully extended TQFTs
Jacob Lurie

Summary

A TQFT is a functor Z: Cob_n -> Vect assigning vector spaces to (n-1)-manifolds and maps to cobordisms.
No metric dependence: all observables are topological invariants (no UV divergences).
2D TQFTs = commutative Frobenius algebras (Dijkgraaf-Abrams classification).
3D Chern-Simons TQFT gives the Jones polynomial for knots (Witten 1988).

References

BookKock, J. Frobenius Algebras and 2D Topological Quantum Field Theories. Cambridge, 2004.
BookTuraev, V. Quantum Invariants of Knots and 3-Manifolds. de Gruyter, 1994.
WebsiteWikipedia -- Topological quantum field theory