These lectures recount an application of stable homotopy theory to a concrete problem in low energy physics: the classification of special phases of matter. While the joint work of the author and Michael Hopkins is a focal point, a general geometric frame of reference on quantum field theory is emphasized. Early lectures describe the geometric axiom systems introduced by Graeme Segal and Michael Atiyah in the late 1980s, as well as subsequent extensions. This material provides an entry point for mathematicians to delve into quantum field theory.
Author(s): Daniel S. Freed
Series: CBMS Regional Conference Series in Mathematics 133
Publisher: AMS
Year: 2019
Language: English
Commentary: decrypted from A035523F2CF6B8E052F8A0FE976A310D source file
Pages: 202
Cover
Title page
Preface
Introduction
Moduli spaces and deformation classes
Axiom System for field theory
Symmetries
Extended Locality
Invertibility and homotopy theory
Extended unitarity
Non-topological invertible theories
Theorems
Free spinor fields
Anomalies
Final remarks
Lecture 1. Bordism and Topological Field Theories
1.1. Classical bordism
1.2. Topological field theories
1.3. Structures on manifolds; further examples
1.4. Varying the codomain: super vector spaces
1.5. Bordism and homotopy theory
Lecture 2. Quantum Mechanics
2.1. An axiomatic view of Hamiltonian mechanics
2.2. Example: particle on a manifold
2.3. Example: a lattice system (toric code)
2.4. Families of quantum systems
2.5. Wick rotation in quantum mechanics
2.6. The Axiom System in quantum mechanics
Lecture 3. Wick-Rotated Quantum Field Theory and Symmetry
3.1. Axiom System for quantum field theory
3.2. Relativistic quantum field theory
3.3. Wick rotation of relativistic quantum field theory
3.4. Symmetry groups in quantum field theory
3.5. Interlude on differential geometry
3.6. Wick-rotated field theory on compact manifolds
Lecture 4. Classification Theorems
4.1. Review of Morse and Cerf theory
4.2. Classification of 2-dimensional topological field theories
4.3. 2-dimensional area-dependent theories
4.4. Classification of 1-dimensional field theories
Lecture 5. Extended Locality
5.1. Higher categories
5.2. Examples of higher categories
5.3. Extended field theories
5.4. Extended operators
5.5. Algebra structures on spheres
5.6. Cobordism hypothesis
5.7. Extended example
Lecture 6. Invertibility and Stable Homotopy Theory
6.1. Categorical preliminaries
6.2. Invertible field theories
6.3. Geometric realization of 1-dimensional bordism
6.4. Non-extended invertible field theories and Reinhart bordism
6.5. Picard groupoids and spectra
6.6. Madsen-Tillmann and Thom spectra
6.7. Duals to the sphere spectrum
6.8. Invertible field theories as maps of spectra
6.9. Deformation classes of invertible field theories
6.10. Continuous invertible topological field theories
Lecture 7. Wick-Rotated Unitarity
7.1. Positive definite Hermitian vector spaces
7.2. Wick-rotated unitarity in quantum mechanics
7.3. Wick-rotated unitarity in Euclidean quantum field theory
7.4. Reflection structures and naive positivity
7.5. Positivity and doubles
7.6. Introduction to extended reflection structures and positivity
Lecture 8. Extended Positivity and Stable Homotopy Theory
8.1. Naive positivity and stability
8.2. Equivariant spectra
8.3. Complex conjugation
8.4. Higher super lines
8.5. Spaces of invertible field theories; extended positivity
8.6. Main theorems
Lecture 9. Non-Topological Invertible Field Theories
9.1. Short-range entangled lattice systems; \tstar field theories
9.2. The long range limit of 3-dimensional Yang-Mills + Chern-Simons
9.3. Examples of non-topological invertible theories
9.4. Differential cohomology and a conjecture
Lecture 10. Computations for Electron Systems
10.1. The 10-fold way for free electron systems
10.2. The long range effective theory of free fermions
10.3. Computations
10.4. Conclusions
Lecture 11. Anomalies in Field Theory
11.1. Pfaffians of Dirac operators
11.2. Anomalies as invertible field theories
11.3. The anomaly theory of a free spinor field
11.4. Anomalies everywhere
Appendix A. Review of Categories
A.1. Categories and groupoids
A.2. Symmetric monoidal categories and duality
A.3. Symmetric monoidal functors
A.4. Picard groupoids
A.5. Involutions
Bibliography
Index
Back Cover
