Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The cobordism hypothesis

Daniel S. Freed

TL;DR

The paper surveys extended topological quantum field theories and the cobordism hypothesis, outlining two paths—algebraic topology via bordism and Morse theory, and quantum field theory via functorial field theories. It develops the framework of (∞,n)-categories to encode n-manifolds with corners and shows that extended TFTs are classified by fully dualizable objects in the target category. A precise framed cobordism hypothesis is stated, with G-structure refinements, and concrete examples (2D Frobenius algebras, DW theory) illustrate the constructions. The discussion connects to invertible theories, stable homotopy theory, and broad applications in topology, algebra, and representation theory, highlighting the potential of extended field theories to unify ideas across mathematics and physics.

Abstract

In this expository paper we introduce extended topological quantum field theories and the cobordism hypothesis.

The cobordism hypothesis

TL;DR

The paper surveys extended topological quantum field theories and the cobordism hypothesis, outlining two paths—algebraic topology via bordism and Morse theory, and quantum field theory via functorial field theories. It develops the framework of (∞,n)-categories to encode n-manifolds with corners and shows that extended TFTs are classified by fully dualizable objects in the target category. A precise framed cobordism hypothesis is stated, with G-structure refinements, and concrete examples (2D Frobenius algebras, DW theory) illustrate the constructions. The discussion connects to invertible theories, stable homotopy theory, and broad applications in topology, algebra, and representation theory, highlighting the potential of extended field theories to unify ideas across mathematics and physics.

Abstract

In this expository paper we introduce extended topological quantum field theories and the cobordism hypothesis.

Paper Structure

This paper contains 7 sections, 12 theorems, 33 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Algebraic topology
  3. Quantum field theory
  4. Topological quantum field theory
  5. $n$-categories and extended topological quantum field theory
  6. The cobordism hypothesis
  7. Implications, extensions, and applications

Key Result

Theorem 1.1

For $n\ge1$, $\mathop{\mathrm{Bord}}\nolimits^{\mathop{\mathrm{fr}}\nolimits}_{n}$ is the free symmetric monoidal $(\infty ,n)$-category with duals generated by $\mathop{\mathrm{pt}}\nolimits_+$.

Figures (17)

  • Figure 1: A bordism $X\colon Y_0\to Y_1$
  • Figure 2: The Pontrjagin-Thom construction
  • Figure 3: Composition of bordisms
  • Figure 4: An elementary bordism
  • Figure 5: A Morse function
  • ...and 12 more figures

Theorems & Definitions (37)

  • Theorem 1.1: Cobordism hypothesis: heuristic algebro-topological version
  • Theorem 1.2: Cobordism hypothesis: weak quantum field theory version
  • Definition 2.1
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6: A2
  • Remark 2.8
  • Lemma 4.5
  • proof
  • ...and 27 more