**Contemporary Mathematics**

Volume: 718;
2018;
258 pp;
Softcover

MSC: Primary 18; 53; 55; 81;

**Print ISBN: 978-1-4704-4243-9
Product Code: CONM/718**

List Price: $117.00

AMS Member Price: $93.60

MAA Member Price: $105.30

**Electronic ISBN: 978-1-4704-4941-4
Product Code: CONM/718.E**

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# Topology and Quantum Theory in Interaction

Share this page *Edited by *
*David Ayala; Daniel S. Freed; Ryan E. Grady*

This volume contains the proceedings of the NSF-CBMS Regional
Conference on Topological and Geometric Methods in QFT, held from July
31–August 4, 2017, at Montana State University in Bozeman,
Montana.

In recent decades, there has been a movement to axiomatize quantum
field theory into a mathematical structure. In a different direction,
one can ask to test these axiom systems against physics. Can they be
used to rederive known facts about quantum theories or, better yet, be
the framework in which to solve open problems? Recently, Freed and
Hopkins have provided a solution to a classification problem in
condensed matter theory, which is ultimately based on the field theory
axioms of Graeme Segal.

Papers contained in this volume amplify various aspects of the
Freed–Hopkins program, develop some category theory, which lies behind
the cobordism hypothesis, the major structure theorem for topological
field theories, and relate to Costello's approach to perturbative
quantum field theory. Two papers on the latter use this framework to
recover fundamental results about some physical theories:
two-dimensional sigma-models and the bosonic string. Perhaps it is
surprising that such sparse axiom systems encode enough structure to
prove important results in physics. These successes can be taken as
encouragement that the axiom systems are at least on the right track
toward articulating what a quantum field theory is.

#### Readership

Graduate students and research mathematicians interested in topology, geometry, and mathematical physics.

# Table of Contents

## Topology and Quantum Theory in Interaction

- Cover Cover11
- Title page i2
- Contents iii4
- Preface v6
- Introduction vii8
- Geometry and physics: An overview 114
- 1. Dirac quantization 114
- 2. Missed opportunities 316
- 3. Yang–Mills theory and connections on fiber bundles 417
- 4. Unreasonable effectiveness 518
- 5. Anomalies and index theory 720
- 6. Donaldson invariants 821
- 7. Topological quantum field theory 922
- 8. Seiberg–Witten theory 1023
- 9. Conclusions 1124
- Acknowledgments 1124
- References 1124

- An introduction to spin systems for mathematicians 1528
- Introduction 1528
- 1. General principles of quantum mechanics: a reminder 1730
- 2. A few words about field theories 2437
- 3. From field theories to spin systems 2740
- 4. An example: The Heisenberg spin chain and integrability 3043
- 5. Questions to ask about systems, or, what is a phase of matter? 3548
- 6. Stacking 4053
- 7. Topological order and the toric code 4154
- Acknowledgments 4659
- References 4659

- The Arf-Brown TQFT of pin⁻ surfaces 4962
- A guide for computing stable homotopy groups 89102
- Flagged higher categories 137150
- How to derive Feynman diagrams for finite-dimensional integrals directly from the BV formalism 175188
- Homotopy RG flow and the non-linear 𝜎-model 187200
- The holomorphic bosonic string 213226
- 1. Introduction 213226
- 2. The classical holomorphic bosonic string 217230
- 3. Deformations of the theory and string backgrounds 223236
- 4. Quantizing the holomorphic bosonic string on a disk 228241
- 5. OPE and the string vertex algebra 238251
- 6. The holomorphic string on closed Riemann surfaces 250263
- 7. Looking ahead: Curved targets 254267
- Appendix A. Calculation of anomaly 255268
- Acknowledgments 256269
- References 257270

- Back Cover Back Cover1274