Topological Quantum Field Theory & Quantum Algebra
Topological Quantum Field Theory (TQFT) bridges pure mathematics and theoretical physics, providing invariants of manifolds through functorial assignments. Quantum algebras, including quantum groups and their representations, form the algebraic backbone of these theories.
Atiyah's TQFT Axioms
TQFT Definition
A TQFT is a functor $Z: \text{Cob}_n \to \text{Vect}_k$ satisfying:
- Functoriality: $Z(M_1 \sqcup M_2) = Z(M_1) \otimes Z(M_2)$
- Identity: $Z(\emptyset) = k$
- Duality: $Z(\overline{M}) = Z(M)^*$
- Trace: $Z(S^1 \times M) = \text{dim}(Z(M))$
Cobordism Hypothesis (Baez-Dolan, Lurie)
The n-groupoid of fully dualizable objects in a symmetric monoidal (∞,n)-category classifies framed extended TQFTs.
Cobordism Categories
Quantum Groups
Quantum SL(2)
The quantum group $U_q(\mathfrak{sl}_2)$ is generated by $E, F, K, K^{-1}$ with relations:
- $KK^{-1} = K^{-1}K = 1$
- $KEK^{-1} = q^2E$, $KFK^{-1} = q^{-2}F$
- $[E,F] = \frac{K - K^{-1}}{q - q^{-1}}$
Quantum Knot Invariants
Trefoil Knot
Jones Polynomial: $V(q) = q + q^3 - q^4$
Hopf Link
HOMFLY: $P(a,z) = a^{-1}z^{-1} + a^{-1}z$
Jones Polynomial via R-Matrix
The Jones polynomial arises from the R-matrix of $U_q(\mathfrak{sl}_2)$ at $q = e^{2\pi i/k}$:
Chern-Simons Theory
The Chern-Simons action for a connection $A$ on a 3-manifold $M$:
$$S_{CS}[A] = \frac{k}{4\pi} \int_M \text{Tr}\left(A \wedge dA + \frac{2}{3} A \wedge A \wedge A\right)$$
Quantization gives knot invariants and 3-manifold invariants!
Braid Group Representations
Modular Tensor Categories
Fusion Rules for SU(2)_k
For level k, the fusion rules are:
Quantum Dimensions
$d_j = \frac{\sin(\frac{(j+1)\pi}{k+2})}{\sin(\frac{\pi}{k+2})}$
Total Dimension
$\mathcal{D} = \sqrt{\sum_j d_j^2}$
Extended TQFT
Extended TQFTs assign:
- Points → n-categories
- 1-manifolds → (n-1)-categories
- ... continuing down to n-manifolds → numbers
Revolutionary Applications
Quantum Computing
Topological quantum computation uses anyons and braiding for fault-tolerant quantum gates.
Condensed Matter Physics
Topological phases of matter described by TQFTs, including fractional quantum Hall states.
Low-dimensional Topology
Invariants of 3 and 4-manifolds via quantum groups and gauge theory.
String Theory
Topological string theory and topological twisting of supersymmetric theories.
Foundational Papers
Fields Medal winning work connecting Chern-Simons theory to knot invariants.
The axiomatization of TQFT that started the field.
Construction of 3-manifold invariants from quantum groups.
Proof of the cobordism hypothesis using higher category theory.
Leading Research Centers
- Institute for Advanced Study - Edward Witten
- Microsoft Station Q - Topological quantum computing
- Oxford - Atiyah's legacy, quantum topology
- UC Santa Barbara - KITP, topological phases
- Perimeter Institute - Quantum gravity and TQFT