Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Acknowledgements
- 1 Definition and Properties of the GFF
- 2 Liouville Measure
- 3 Gaussian Multiplicative Chaos
- 4 Statistical Physics on Random Planar Maps
- 5 Introduction to Liouville Conformal Field Theory
- 6 Gaussian Free Field with Neumann Boundary Conditions
- 7 QuantumWedges and Scale-Invariant Random Surfaces
- 8 SLE and the Quantum Zipper
- 9 Liouville Quantum Gravity as a Mating of Trees
- Appendix A Chordal Loewner Chains and Chordal SLE
- Appendix B Reverse Loewner Flow and Reverse SLE
- Appendix C Radial Loewner Chains and Radial SLE
- Appendix D Convergence of Random Variables in the Space of Distributions
- References
- Notation and Symbols
- Index
8 - SLE and the Quantum Zipper
Published online by Cambridge University Press: 20 November 2025
Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Acknowledgements
- 1 Definition and Properties of the GFF
- 2 Liouville Measure
- 3 Gaussian Multiplicative Chaos
- 4 Statistical Physics on Random Planar Maps
- 5 Introduction to Liouville Conformal Field Theory
- 6 Gaussian Free Field with Neumann Boundary Conditions
- 7 QuantumWedges and Scale-Invariant Random Surfaces
- 8 SLE and the Quantum Zipper
- 9 Liouville Quantum Gravity as a Mating of Trees
- Appendix A Chordal Loewner Chains and Chordal SLE
- Appendix B Reverse Loewner Flow and Reverse SLE
- Appendix C Radial Loewner Chains and Radial SLE
- Appendix D Convergence of Random Variables in the Space of Distributions
- References
- Notation and Symbols
- Index
Summary
We describe couplings between Schramm–Loewner Evolution (SLE) curves and variants of the Gaussian free field (GFF). In particular, we give a complete proof of Sheffield’s construction of γ-quantum boundary length along an SLEγ2
curve, as measured by an independent underlying GFF. The main input for this proof is a rigorous construction of the so-called quantum gravity zipper, which is a stationary dynamic on quantum surfaces (defined using a GFF) decorated by SLE. Another consequence of this construction is that drawing an SLE curve on top of an appropriate independent quantum surface splits the surface into two independent and identically distributed (sub)-surfaces, glued according to boundary length. In particular, this shows that SLE curves are solutions of natural random conformal welding problems.
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- Gaussian Free Field and Liouville Quantum Gravity , pp. 299 - 333Publisher: Cambridge University PressPrint publication year: 2025
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