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

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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.

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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.

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3. Contrast with Conventional Quantum Field Theories

Feature	Conventional QFT	TQFT
Depends on metric?	Yes	No
Local degrees?	Yes (e.g., particles)	Often no
Sensitive to shape?	Yes	Only to topology
Applications	Particle physics	Knot theory, geometry

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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.

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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

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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.

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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

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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

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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

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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

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11. Observables in TQFT

Observables are topological invariants, such as:

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

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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.

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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.

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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.

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15. TQFT and Knot Theory

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

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16. Modular Tensor Categories

Modular tensor categories classify 3D TQFTs:

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

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17. TQFTs in 2D: Frobenius Algebras

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

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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.

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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

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20. Topological Strings

Topological string theory computes:

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

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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

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22. Open-Closed TQFT

TQFTs with both open and closed strings correspond to:

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

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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

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24. Mathematical Impact

TQFTs have enriched:

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

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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.

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Table of Contents

1. Introduction
2. Historical Context and Maldacena’s Proposal
3. Anti-de Sitter (AdS) Space
4. Conformal Field Theory (CFT) Basics
5. Duality Statement: AdS/CFT
6. D3-Branes and AdS\(_5\) × S\(^5\)
7. \( \mathcal{N}=4 \) Super Yang–Mills Theory
8. Dictionary of the Duality
9. Matching Symmetries
10. Radial Direction as Energy Scale
11. Boundary Operators and Bulk Fields
12. Correlation Functions and Partition Functions
13. Holographic Principle
14. Strong/Weak Coupling Duality
15. Black Holes and Thermodynamics
16. Holographic Entanglement Entropy
17. AdS/CMT and Applications in Condensed Matter
18. Holographic QCD
19. Holographic Renormalization
20. Higher-Spin Generalizations
21. Beyond AdS: de Sitter and Flat Holography
22. Extensions and Other Dualities
23. Open Problems and Limitations
24. Mathematical Impact
25. Conclusion

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1. Introduction

The AdS/CFT correspondence, also known as gauge/gravity duality, is a conjectured relationship between a gravitational theory in anti-de Sitter (AdS) space and a conformal field theory (CFT) defined on its boundary. Proposed by Juan Maldacena in 1997, it provides a non-perturbative definition of quantum gravity and a powerful tool for studying strongly coupled quantum field theories.

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2. Historical Context and Maldacena’s Proposal

Maldacena’s insight came from analyzing low-energy limits of D3-branes in Type IIB string theory. The key idea was that the dynamics near the branes could be described by both:

• Supergravity in AdS\(_5\) × S\(^5\)
• \( \mathcal{N}=4 \) Super Yang–Mills theory in 4D

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3. Anti-de Sitter (AdS) Space

AdS space is a maximally symmetric spacetime with constant negative curvature. For AdS\(_{d+1}\), the metric can be written as:

\[
ds^2 = \frac{R^2}{z^2} (dz^2 + \eta_{\mu\nu} dx^\mu dx^\nu)
\]

Here:

• \( z \): radial coordinate (bulk)
• \( x^\mu \): boundary coordinates

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4. Conformal Field Theory (CFT) Basics

CFTs are invariant under conformal transformations. In 4D, \( \mathcal{N}=4 \) Super Yang–Mills is a well-known CFT with:

• SU(N) gauge symmetry
• 6 real scalars, 4 Majorana fermions
• Exact conformal symmetry at the quantum level

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5. Duality Statement: AdS/CFT

The core statement:

\[
\text{Type IIB string theory on } \text{AdS}_5 \times S^5 \quad \equiv \quad \mathcal{N}=4 \text{ SYM in 4D}
\]

This is a holographic duality: gravity in (d+1) dimensions is equivalent to a QFT in d dimensions.

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6. D3-Branes and AdS\(_5\) × S\(^5\)

D3-branes in Type IIB string theory:

• Source RR flux
• Worldvolume theory is \( \mathcal{N}=4 \) SYM
• Near-horizon limit gives AdS\(_5\) × S\(^5\)

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7. \( \mathcal{N}=4 \) Super Yang–Mills Theory

This theory is:

• Conformal for all values of coupling
• Integrable in the planar limit
• Exhibits exact duality properties (S-duality, T-duality)

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8. Dictionary of the Duality

The AdS/CFT dictionary relates:

• Bulk fields ↔ Boundary operators
• Bulk action ↔ Generating functional
• Mass of bulk scalar ↔ Scaling dimension \( \Delta \) of boundary operator:

\[
\Delta(\Delta – d) = m^2 R^2
\]

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9. Matching Symmetries

• AdS\(_5\) isometries ↔ Conformal group SO(4,2)
• S\(^5\) isometries ↔ SU(4) R-symmetry
• SUSY matching: both sides preserve 32 supercharges

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10. Radial Direction as Energy Scale

The AdS radial coordinate \( z \) maps to energy scale in the CFT:

• \( z \to 0 \): UV
• \( z \to \infty \): IR

This connects renormalization group (RG) flow with holography.

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11. Boundary Operators and Bulk Fields

A bulk field \( \phi(z, x) \) near boundary behaves as:

\[
\phi(z, x) \sim z^{d – \Delta} \phi_0(x)
\]

\[
\left\langle e^{\int \phi_0(x) \mathcal{O}(x)} \right\rangle = Z_{\text{string}}[\phi \to \phi_0]
\]

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12. Correlation Functions and Partition Functions

Field theory correlators are computed via bulk string theory:

\[
\left\langle \mathcal{O}1(x_1) \dots \mathcal{O}_n(x_n) \right\rangle = \frac{\delta^n Z{\text{string}}}{\delta \phi_0(x_1) \dots \delta \phi_0(x_n)}
\]

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13. Holographic Principle

AdS/CFT is a realization of the holographic principle — the idea that a higher-dimensional gravity theory can be encoded by degrees of freedom on a lower-dimensional boundary.

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14. Strong/Weak Coupling Duality

AdS/CFT relates:

• Weakly coupled gravity ↔ Strongly coupled gauge theory
• Enables study of QCD-like systems at strong coupling using classical gravity

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15. Black Holes and Thermodynamics

Black holes in AdS correspond to thermal states in the CFT:

• Hawking–Page transition ↔ Confinement/deconfinement transition
• Holographic entropy matches Bekenstein–Hawking formula

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16. Holographic Entanglement Entropy

Ryu–Takayanagi formula:

\[
S_A = \frac{\text{Area}(\gamma_A)}{4 G_N}
\]

Where \( \gamma_A \) is the minimal surface in AdS homologous to boundary region \( A \). Provides geometric interpretation of entanglement.

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17. AdS/CMT and Applications in Condensed Matter

Applied to strongly correlated systems:

• High-\( T_c \) superconductivity
• Quantum criticality
• Non-Fermi liquids

Gravity duals provide insight into transport and thermodynamics.

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18. Holographic QCD

Holographic models approximate QCD:

• Hard wall and soft wall models
• Chiral symmetry breaking
• Glueballs and mesons via bulk modes

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19. Holographic Renormalization

Regularizes AdS divergences:

• Add boundary counterterms
• Extract finite correlators
• Captures RG flow and anomalies

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20. Higher-Spin Generalizations

Extensions involve Vasiliev theory:

• Dual to vector models
• Challenge standard assumptions of AdS/CFT

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21. Beyond AdS: de Sitter and Flat Holography

Attempts to extend holography to:

• de Sitter space (dS/CFT): cosmological settings
• Flat space holography: asymptotically flat spacetimes

Still under development.

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22. Extensions and Other Dualities

AdS/CFT inspired other dualities:

• ABJM duality (AdS\(_4\)/CFT\(_3\))
• AdS\(_3\)/CFT\(_2\)
• AdS\(_7\)/CFT\(_6\)

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23. Open Problems and Limitations

• Proving AdS/CFT rigorously
• Understanding stringy and quantum corrections
• Realistic QCD duals
• Extension to time-dependent and cosmological backgrounds

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24. Mathematical Impact

AdS/CFT has influenced:

• Geometric analysis
• Representation theory
• Category theory and topological invariants

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25. Conclusion

The AdS/CFT correspondence stands as one of the most profound insights in theoretical physics. It provides a non-perturbative definition of quantum gravity, a holographic view of spacetime, and a powerful toolkit to explore strongly coupled systems. Its implications continue to evolve, shaping the landscape of string theory, gauge theories, and quantum gravity research.

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