Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

TASI Lectures On Topological Field Theories And Differential Cohomology

Gregory W. Moore, Vivek Saxena

TL;DR

The notes present a structured, pedagogical path from basic quantum mechanics to fully extended topological field theory, establishing a functorial, bordism-based framework (F: Bord_{⟨n−1,n⟩} → VECT) that emphasizes locality, gluing, and dualizability. They derive 1D and 2D TFTs from Frobenius algebras, sew and certify amplitudes via Morse theory, and explain open-closed TFTs through boundary categories, all within a higher-categorical context. The Open-Closed/Extended TFT discussion leads to the cobordism hypothesis, dualizability, and the cobordism-based classification of theories, while the Part II material lays out finite homotopy sigma models, classifying spaces, group cohomology, and DW theory to connect gauge theory with differential cohomology. The work further develops background fields, higher structures, defects, and symmetry actions (e.g., quiche picture), tying together the geometry of field theories with algebra, topology, and category theory for a comprehensive framework. It thus provides both foundational formalism and concrete exemplars (finite groups, Chern-Simons, 2D YM) that illuminate how generalized notions of symmetry and differential cohomology interface with topological and geometric field theories.

Abstract

These are lecture notes expanding upon a set of lectures given by G.M. at the TASI 2023 School. Part I is an introduction to topological field theory, including extended topological field theory. Part II is an introduction to generalized Abelian gauge theories and their relation to differential cohomology.

TASI Lectures On Topological Field Theories And Differential Cohomology

TL;DR

The notes present a structured, pedagogical path from basic quantum mechanics to fully extended topological field theory, establishing a functorial, bordism-based framework (F: Bord_{⟨n−1,n⟩} → VECT) that emphasizes locality, gluing, and dualizability. They derive 1D and 2D TFTs from Frobenius algebras, sew and certify amplitudes via Morse theory, and explain open-closed TFTs through boundary categories, all within a higher-categorical context. The Open-Closed/Extended TFT discussion leads to the cobordism hypothesis, dualizability, and the cobordism-based classification of theories, while the Part II material lays out finite homotopy sigma models, classifying spaces, group cohomology, and DW theory to connect gauge theory with differential cohomology. The work further develops background fields, higher structures, defects, and symmetry actions (e.g., quiche picture), tying together the geometry of field theories with algebra, topology, and category theory for a comprehensive framework. It thus provides both foundational formalism and concrete exemplars (finite groups, Chern-Simons, 2D YM) that illuminate how generalized notions of symmetry and differential cohomology interface with topological and geometric field theories.

Abstract

These are lecture notes expanding upon a set of lectures given by G.M. at the TASI 2023 School. Part I is an introduction to topological field theory, including extended topological field theory. Part II is an introduction to generalized Abelian gauge theories and their relation to differential cohomology.

Paper Structure

This paper contains 127 sections, 9 theorems, 697 equations, 37 figures, 1 table.

Table of Contents

  1. General Plan
  2. Preliminary: Quantum Mechanics
  3. Basic Picture In TFT: Heuristic Motivation
  4. Bordisms
  5. Question:
  6. Answer:
  7. Corollaries:
  8. Example 1: Topological Field Theory In One Dimension
  9. Example 2: Oriented TFT In Dimension Two
  10. Defining Data And Frobenius Algebras
  11. Corollary 1:
  12. Corollary 2:
  13. Corollary 3:
  14. Sewing Theorem And Morse Theory
  15. The Sewing Theorem: Morse Theory Proof
  16. ...and 112 more sections

Key Result

Theorem 6.1

If $(V,m,\theta)$ is a commutative Frobenius algebra, then all amplitudes are well-defined and can be computed by cutting up the bordisms in any way. Put differently: To give an $n=2$ dimensional TFT is to give a commutative finite-dimensional Frobenius algebra.

Figures (37)

  • Figure 1: Picture of "time" evolution. Note that $M_n$ need not be a cylinder, i.e., diffeomorphic to $N_{n-1} \times [0,1]$, and there might not be a globally well-defined vector field of time.
  • Figure 2: An $n$-manifold $M_{n}$ with boundary. The different connected components of the boundary carry an arrow indicating whether they are "ingoing" (i.e., "initial" or "past") spatial components or "outgoing" (i.e., "final" or "future") spatial components. Note well that these arrows, sometimes referred to as a co-orientation, do not imply that $M_n$ or its boundaries are oriented, or even orientable.
  • Figure 3: The propagator.
  • Figure 4: Gluing.
  • Figure 5: Same manifold, different bordism!!
  • ...and 32 more figures

Theorems & Definitions (95)

  • Remark 3.1
  • Definition 4.1: Bordism
  • Remark 4.1
  • Remark 4.2
  • Remark 4.3
  • Definition 6.1: Frobenius algebra
  • Theorem 6.1: Sewing Theorem
  • Remark 6.1
  • Remark 6.2
  • Definition 8.1: Category
  • ...and 85 more