π― Learning Objectives
TQFTs as Functors
Understand the Atiyah axioms: a TQFT is a symmetric monoidal functor Cob_n β Vect.
Cobordism Categories
Master the cobordism category: objects are manifolds, morphisms are cobordisms between them.
The Cobordism Hypothesis
Understand Lurie's theorem: extended TQFTs are classified by fully dualizable objects.
Factorization Algebras
Learn the Costello-Gwilliam approach to non-topological QFT via locality axioms.
π Key Concepts
The Atiyah-Segal Axioms for TQFT
An n-dimensional topological quantum field theory is a symmetric monoidal functor:
Z : Cob_n β Vect
Where:
- Objects of Cob_n: closed (n-1)-dimensional manifolds
- Morphisms: n-dimensional cobordisms (manifolds with boundary)
- Composition: gluing cobordisms along boundaries
- Tensor: disjoint union
Z assigns:
- To each (n-1)-manifold Ξ£: a vector space Z(Ξ£)
- To each cobordism M: Ξ£β β Ξ£β: a linear map Z(M): Z(Ξ£β) β Z(Ξ£β)2D TQFT Classification
2D TQFTs are equivalent to commutative Frobenius algebras.
Circle SΒΉ β¦ Algebra A
Pair of pants β¦ Multiplication ΞΌ: A β A β A
Disk β¦ Unit Ξ·: β β A
Upside-down pants β¦ Comultiplication Ξ: A β A β A
The Frobenius condition and commutativity follow from
the topology of 2D cobordisms!The Cobordism Hypothesis (Lurie)
For extended (fully local) TQFTs valued in an (β,n)-category C:
Extended n-dimensional TQFTs are classified by fully dualizable objects in C.
This is one of the deepest results connecting topology, higher category theory, and physics. The entire theory is determined by what you assign to a point!
Factorization Algebras
For non-topological QFTs, the factorization algebra approach (Costello-Gwilliam) captures locality:
F: Open(M) β Chain complexes
For disjoint Uβ, ..., Uβ β V:
F(Uβ) β ... β F(Uβ) β F(V)
This axiomatizes:
- Operator product expansion
- Locality of observables
- Renormalization structureπ€ AI Learning Prompts
"Explain topological quantum field theories to someone who knows category theory but not much physics. What is a cobordism? Why is TQFT a functor? What physical systems do TQFTs describe?"
"Walk me through the classification of 2D TQFTs by Frobenius algebras. Show me the cobordisms (pair of pants, disk, etc.) and explain why they give the Frobenius algebra structure. What are some examples of 2D TQFTs?"
"Explain the cobordism hypothesis at an intuitive level. What does it mean that an extended TQFT is 'determined by a point'? What is a fully dualizable object? Why is this theorem so important for physics?"
"What is an extended (or fully local) TQFT? How does it differ from the Atiyah-Segal axioms? Why do we need higher categories for extended TQFTs? What do they assign to points, lines, surfaces, etc.?"
"Explain factorization algebras as an approach to QFT. How do they encode locality? How does the operator product expansion appear? What is the relationship between factorization algebras and vertex algebras in 2D?"
"Explain 3D Chern-Simons theory as a TQFT. What does it assign to surfaces? What are the famous connections to knot invariants (Jones polynomial) and the WZW model? Why is Chern-Simons the paradigmatic example of TQFT?"
"How do TQFTs handle defects, boundaries, and domain walls? What categorical structure describes these? How do branes in string theory relate to categorical structures in TQFT?"
"Explain the Batalin-Vilkovisky (BV) formalism and how it connects to factorization algebras. How does BV quantization work? Why is this the 'right' approach to gauge theory quantization from a type-safe perspective?"
βοΈ Exercises
Exercise 1: 2D Cobordisms
List all connected 2D cobordisms between circles. Show they generate all 2D cobordisms under composition and tensor product.
Exercise 2: Frobenius Algebra
Show that the group algebra β[G] of a finite group is a Frobenius algebra. What 2D TQFT does it correspond to?
Exercise 3: Dimension Formula
For a 2D TQFT with Frobenius algebra A, the value on a closed surface of genus g is dim(A)^(1-g). Verify this for the torus (g=1).