Topological Quantum Field Theory (TQFT)

Table of Contents

1. Introduction
2. What is a TQFT?
3. Contrast with Conventional Quantum Field Theories
4. Topological Invariance
5. Mathematical Definition of TQFT
6. Cobordism and Categories
7. Atiyah–Segal Axioms
8. Examples of TQFTs
9. BF Theory
10. Chern–Simons Theory
11. Donaldson–Witten Theory
12. Observables in TQFT
13. Wilson Loops and Link Invariants
14. Quantum Invariants of 3-Manifolds
15. Path Integral in TQFT
16. TQFT and Knot Theory
17. Modular Tensor Categories
18. TQFTs in 2D: Frobenius Algebras
19. Relation to Conformal Field Theory
20. TQFTs in String Theory and M-Theory
21. Topological Strings
22. TQFTs and Quantum Computing
23. Open-Closed TQFT
24. Extended TQFT and Higher Categories
25. Conclusion

1. Introduction

Topological Quantum Field Theory (TQFT) is a type of quantum field theory in which physical observables depend only on the topology of the underlying manifold, not on its geometric details. These theories are powerful tools in both theoretical physics and mathematics, particularly in topology, geometry, and knot theory.

2. What is a TQFT?

A TQFT is a quantum field theory where correlation functions and amplitudes are topological invariants — they do not change under smooth deformations of the spacetime manifold. TQFTs capture global topological features and often lack local dynamics or propagating degrees of freedom.

3. Contrast with Conventional Quantum Field Theories

4. Topological Invariance

A defining feature of TQFTs is diffeomorphism invariance. Observables remain unchanged under smooth coordinate transformations — i.e., they are independent of the metric or curvature.

5. Mathematical Definition of TQFT

Formally, a TQFT is a symmetric monoidal functor:

• \( \text{Cob}_n \): category of n-dimensional cobordisms
• \( \text{Vect}_\mathbb{C} \): category of complex vector spaces
• To each (n−1)-manifold, assigns a vector space
• To each n-cobordism, assigns a linear map

6. Cobordism and Categories

Two manifolds \( M_0, M_1 \) are cobordant if there exists a manifold \( W \) such that:

\[
\partial W = M_1 – M_0
\]

TQFTs assign data to manifolds and transitions between them in a consistent, functorial way.

7. Atiyah–Segal Axioms

These axioms formalize the structure of TQFTs:

1. Functoriality: Composition of cobordisms corresponds to composition of linear maps
2. Monoidality: Disjoint union corresponds to tensor product
3. Invariance: Results are independent of smooth deformations

8. Examples of TQFTs

BF Theory:

\[
S = \int_M B \wedge F
\]

• \( B \): 2-form
• \( F \): curvature of a connection
• Metric-independent, defined on any d-dimensional manifold

9. Chern–Simons Theory

Defined on a 3-manifold \( M \) with gauge group \( G \):

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

• Observables: Wilson loops
• Applications: knot invariants, quantum Hall effect, WZW models

10. Donaldson–Witten Theory

A TQFT derived from supersymmetric Yang–Mills theory:

• Captures Donaldson invariants of 4-manifolds
• Uses topological twist of \( \mathcal{N}=2 \) SUSY

11. Observables in TQFT

Observables are topological invariants, such as:

• Link invariants
• Intersection numbers
• Characteristic classes (e.g., Chern classes)

12. Wilson Loops and Link Invariants

In Chern–Simons theory, the Wilson loop operator:

\[
W_R(C) = \text{Tr}_R \, \mathcal{P} \exp \left( \oint_C A \right)
\]

yields link invariants such as the Jones polynomial when computed on knots.

13. Quantum Invariants of 3-Manifolds

Chern–Simons theory produces invariants like:

• Witten–Reshetikhin–Turaev invariants
• Turaev–Viro invariants

These generalize classical topological invariants to quantum contexts.

14. Path Integral in TQFT

The path integral becomes a topological invariant:

\[
Z(M) = \int \mathcal{D}\phi \, e^{iS[\phi]}
\]

This integral is often finite-dimensional due to gauge-fixing or localization.

15. TQFT and Knot Theory

TQFTs provide a natural language for knot invariants and knot polynomials, connecting physics with low-dimensional topology.

16. Modular Tensor Categories

Modular tensor categories classify 3D TQFTs:

• Provide fusion and braiding data
• Essential for constructing TQFTs from algebraic data

17. TQFTs in 2D: Frobenius Algebras

2D TQFTs are classified by commutative Frobenius algebras. The multiplication and trace encode the TQFT’s rules.

18. Relation to Conformal Field Theory

Boundary CFTs often induce a bulk TQFT. Chern–Simons theory on a 3-manifold with boundary induces a Wess–Zumino–Witten (WZW) model.

19. TQFTs in String Theory and M-Theory

• Topological strings: A-model and B-model
• Capture enumerative invariants of Calabi–Yau manifolds
• Relate to Gromov–Witten theory and mirror symmetry

20. Topological Strings

Topological string theory computes:

• Gromov–Witten invariants
• Black hole entropy
• F-terms in supergravity

21. TQFTs and Quantum Computing

Topological quantum computing:

• Uses anyons and braiding as computational gates
• Based on 2D TQFTs
• Robust to local errors due to topological protection

22. Open-Closed TQFT

TQFTs with both open and closed strings correspond to:

• D-brane categories (open sector)
• Closed strings as bulk invariants

23. Extended TQFT and Higher Categories

Extended TQFTs assign data not just to manifolds, but to:

• Points, lines, surfaces
• Capture local-to-global structure
• Modeled using higher category theory

24. Mathematical Impact

TQFTs have enriched:

• Low-dimensional topology
• Category theory
• Quantum algebra
• Knot theory

25. Conclusion

Topological quantum field theories offer a bridge between quantum physics and pure mathematics. By focusing on topological aspects, TQFTs bypass complexities of metric dependence and offer powerful tools for understanding quantum invariants, knot theory, and quantum computation. Their influence spans theoretical physics, geometry, and even the design of future quantum technologies.