**CBMS Regional Conference Series in Mathematics**

Volume: 133;
2019;
186 pp;
Softcover

MSC: Primary 55; 81; 82; 57;

**Print ISBN: 978-1-4704-5206-3
Product Code: CBMS/133**

List Price: $55.00

AMS Member Price: $44.00

MAA Member Price: $49.50

**Electronic ISBN: 978-1-4704-5391-6
Product Code: CBMS/133.E**

List Price: $55.00

AMS Member Price: $44.00

MAA Member Price: $49.50

#### Supplemental Materials

# Lectures on Field Theory and Topology

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*Daniel S. Freed*

A co-publication of the AMS and CBMS

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. Classification
theorems in low dimensions are proved to illustrate the framework.
The later lectures turn to more specialized topics in field theory,
including the relationship between invertible field theories and
stable homotopy theory, extended unitarity, anomalies, and
relativistic free fermion systems. The accompanying mathematical
explanations touch upon (higher) category theory, duals to the sphere
spectrum, equivariant spectra, differential cohomology, and Dirac
operators.

The outcome of computations made using the Adams spectral
sequence is presented and compared to results in the condensed matter
literature obtained by very different means. The general perspectives
and specific applications fuse into a compelling story at the
interface of contemporary mathematics and theoretical physics.

#### Readership

Graduate students and researchers interested in the interation between geometry, topology (homotopy theory), and theoretical physics (quantum field theory and condensed matter theory).

#### Table of Contents

# Table of Contents

## Lectures on Field Theory and Topology

- Cover Cover11
- Title page iii4
- Preface xi12
- Introduction 114
- Lecture 1. Bordism and Topological Field Theories 1326
- Lecture 2. Quantum Mechanics 2942
- Lecture 3. Wick-Rotated Quantum Field Theory and Symmetry 4356
- Lecture 4. Classification Theorems 5568
- Lecture 5. Extended Locality 7184
- Lecture 6. Invertibility and Stable Homotopy Theory 8598
- 6.1. Categorical preliminaries 8699
- 6.2. Invertible field theories 8699
- 6.3. Geometric realization of 1-dimensional bordism 88101
- 6.4. Non-extended invertible field theories and Reinhart bordism 90103
- 6.5. Picard groupoids and spectra 92105
- 6.6. Madsen-Tillmann and Thom spectra 93106
- 6.7. Duals to the sphere spectrum 98111
- 6.8. Invertible field theories as maps of spectra 100113
- 6.9. Deformation classes of invertible field theories 101114
- 6.10. Continuous invertible topological field theories 102115

- Lecture 7. Wick-Rotated Unitarity 105118
- Lecture 8. Extended Positivity and Stable Homotopy Theory 119132
- Lecture 9. Non-Topological Invertible Field Theories 131144
- Lecture 10. Computations for Electron Systems 143156
- Lecture 11. Anomalies in Field Theory 153166
- Appendix A. Review of Categories 165178
- Bibliography 175188
- Index 185198
- Back Cover Back Cover1202