Lattice Models and Conformal Field Theory / Franck Gabriel, Clément Hongler, and Francesco Spadaro.

Format:

Book

Author/Creator:

Gabriel, Franck, 1988- author.

Hongler, Clément, 1985- author.

Spadaro, Francesco, 1992- author.

Series:

Courant lecture notes in mathematics ; Volume 32.

Courant Lecture Notes Series ; Volume 32

Language:

English

Subjects (All):

Conformal invariants.

Quantum field theory.

Lattice theory.

Statistical mechanics.

Physical Description:

1 online resource (219 pages)

Edition:

First edition.

Place of Publication:

New York, New York : Courant Institute of Mathematical Sciences, [2024]

Summary:

This book introduces the mathematical ideas connecting Statistical Mechanics and Conformal Field Theory (CFT). Building advanced structures on top of more elementary ones, the authors map out a well-posed road from simple lattice models to CFTs. Structured in two parts, the book begins by exploring several two-dimensional lattice models, their phase transitions, and their conjectural connection with CFT. Through these lattice models and their local fields, the fundamental ideas and results of two-dimensional CFTs emerge, with a special emphasis on the Unitary Minimal Models of CFT. Delving into the delicate ideas that lead to the classification of these CFTs, the authors discuss the assumptions on the lattice models whose scaling limits are described by CFTs. This produces a probabilistic rather than an axiomatic or algebraic definition of CFTs. Suitable for graduate students and researchers in mathematics and physics, Lattice Models and Conformal Field Theory introduces the ideas at the core of Statistical Field Theory. Assuming only undergraduate probability and complex analysis, the authors carefully motivate every argument and assumption made. Concrete examples and exercises allow readers to check their progress throughout.

Contents:

Cover

Half-Title page

Editors page

Title page

Copyright

Contents

Preface

Acknowledgments

Part 1. Introduction

Chapter 1. Lattice Models, Phase Transitions, and Critical Exponents

1.1. Ising Model

1.2. Blume-Capel Model and Tricritical Ising Model

1.3. 3-Potts Model and Tricritical 3-Potts Model

1.4. Other Models Described by CFT

1.5. Universality of Phase Transitions

1.6. Outlook: Statistical Field Theory

Chapter 2. Statistical and Quantum Field Theories

2.1. Renormalization Group

2.2. The Gaussian Free Field

2.3. The Free Boson Theory

2.4. Scaling Limits and the Ising Model

2.5. Summary and Outlook

Chapter 3. Conformal Field Theory

3.1. Conformal Symmetry

3.2. Key Insights of CFT

3.3. Lattice Approach of CFT

Part 2. Conformal Field Theory: Minimal Models on the Plane

Chapter 4. Local Fields and Correlations

4.1. Choice of Lattice and Lattice Equivalence

4.2. Lattice Local Fields

4.3. Local Fields as Limits of Lattice Local Fields

4.4. Translations, Scalings, Rotations, and Reflections of Fields

4.5. Universality and LADE

4.6. Field Identities and OPEs

4.7. Local Fields: Summary and Conjecture

Chapter 5. Stress-Energy Tensor and Conformal Ward Identities

5.1. CFT: Definition

5.2. Stress Tensor and Action: Geometric and Probabilistic Origin

5.3. Diffeomorphism Equivalence at the Lattice Level

5.4. SFTs with ILACI Property are CFTs: The Ward Identities

5.5. Virasoro Structure

5.6. Local Field Transformations

5.7. Conformal Maps on the Sphere

5.8. Summary

Chapter 6. Unitarity and Radial Quantization

6.1. Unitarity: Definition and Motivation

6.2. Cylinder Transfer Matrix and Reflection-Positivity

6.3. Radial Lattice, Inversion, and Inversion-Positivity

6.4. Operator-State Correspondence

6.5. Unitarity.

Chapter 7. Primary Fields and Conformal Families

7.1. Primaries and Descendants

7.2. Correlations of Descendants

7.3. Splitting Into Conformal Families from Unitarity

7.4. Verma Module and Shapovalov Form

7.5. Kac Determinant Formula

7.6. Singular Vectors

7.7. Shapovalov Form and Unitarity

7.8. Positive (Semi)Definiteness: Proofs and Insights

7.9. Summary

Chapter 8. Unitary Minimal Models, Lattice Models, and Loop Models

8.1. Unitary Minimal Models

8.2. Ising Model as a Unitary Minimal Model

8.3. Other Lattice Models

Bibliography

Index

Published Titles in This Series

Back Cover.

Notes:

Includes bibliographical references and index.

Description based on publisher supplied metadata and other sources.

Description based on print version record.

Other Format:

Print version: Gabriel, Franck Lattice Models and Conformal Field Theory

ISBN:

9781470477936