Geometries of Quantum Field Theories
Abstract
This is an English translation of a book originally published in Japanese by Saiensu-sha in 2015. The book is an introduction to the 3d-3d correspondence, the relation between 3d π©=2 supersymmetric gauge theories and the geometry of 3-manifolds, which arises from the compactification of the 6d π©=(2,0) theory. The presentation is bottom-up rather than top-down: instead of starting from the 6d theory, we begin by asking what a quantum field theory is, and let the geometry emerge on its own. Along the way we discuss renormalization and low-energy effective theories, gauging as an operation which glues field theories together, the resulting Sp(2n, β€)-action on 3d theories, 3d π©=2 supersymmetric theories and their dualities, the squashed three-sphere partition function and supersymmetric localization, duality domain walls of 4d π©=2 theories, quantum Teichmuller theory, complex Chern-Simons theory and the quantization of the moduli space of flat SL(2, β)-connections on 3-manifolds, with quantum dilogarithm functions as a recurring thread. The emphasis throughout is on geometric and algebraic structures which are invisible in a single field theory, and appear only in the theory space of quantum field theories. Appendices summarize supersymmetry in various dimensions, classical and quantum dilogarithm functions, and cluster algebras. Exercises with difficulty ratings are included in each chapter.
Related
- Divergences in Nonpolynomial Quantum Field Theories
- Boundary Quantum Knizhnik-Zamolodchikov Equations and Integrability of Quantum Field Theories with Time-Dependent Bulk and Boundary Coupling Strengths
- Time-dependent integrable quantum field theories with dynamical boundaries
- Discrete C*-Frobenius Algebras and Infinite-Index Extensions in Algebraic Quantum Field Theories
- Integrable Chiral Quantum Field Theories with Spacetime-Dependent Interactions
BibTeX
@misc{yamazaki2026geometries,
title = {{Geometries of Quantum Field Theories}},
author = {Masahito Yamazaki},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28210},
eprint = {2609.28210},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28210},
}
RIS
TY - UNPB
TI - Geometries of Quantum Field Theories
AU - Masahito Yamazaki
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28210
UR - https://arxiv.org/abs/2609.28210
AB - This is an English translation of a book originally published in Japanese by Saiensu-sha in 2015. The book is an introduction to the 3d-3d correspondence, the relation between 3d π©=2 supersymmetric gauge theories and the geometry of 3-manifolds, which arises from the compactification of the 6d π©=(2,0) theory. The presentation is bottom-up rather than top-down: instead of starting from the 6d theory, we begin by asking what a quantum field theory is, and let the geometry emerge on its own. Along the way we discuss renormalization and low-energy effective theories, gauging as an operation which glues field theories together, the resulting Sp(2n, β€)-action on 3d theories, 3d π©=2 supersymmetric theories and their dualities, the squashed three-sphere partition function and supersymmetric localization, duality domain walls of 4d π©=2 theories, quantum Teichmuller theory, complex Chern-Simons theory and the quantization of the moduli space of flat SL(2, β)-connections on 3-manifolds, with quantum dilogarithm functions as a recurring thread. The emphasis throughout is on geometric and algebraic structures which are invisible in a single field theory, and appear only in the theory space of quantum field theories. Appendices summarize supersymmetry in various dimensions, classical and quantum dilogarithm functions, and cluster algebras. Exercises with difficulty ratings are included in each chapter.
ER -