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Schramm-Loewner evolution and Liouville quantum gravity.

Bertrand Duplantier1, Scott Sheffield

  • 1Institut de Physique Théorique, CEA/Saclay, Gif-sur-Yvette, France.

Physical Review Letters
|October 27, 2011
PubMed
Summary
This summary is machine-generated.

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Conformal welding of Liouville quantum gravity surfaces creates Schramm-Loewner evolution random curves. This study develops quantum fractal measures and applies them to define quantum length and intersection measures on these curves.

Area of Science:

  • Quantum Gravity
  • Stochastic Processes
  • Geometric Measure Theory

Background:

  • Liouville quantum gravity describes random surfaces in 2D
  • Conformal welding is a method to join surfaces along boundary arcs
  • Schramm-Loewner evolution (SLE) models random curves in 2D

Purpose of the Study:

  • To demonstrate that conformal welding of Liouville quantum gravity surfaces results in Schramm-Loewner evolution curves
  • To develop a theory of quantum fractal measures consistent with the Knizhnik-Polyakov-Zamolochikov (KPZ) relation
  • To apply these measures to the resulting SLE curves

Main Methods:

  • Conformal welding of boundary arcs of Liouville quantum gravity random surfaces
  • Development of a theory for quantum fractal measures

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  • Analysis of measure evolution under conformal welding maps related to SLE
  • Construction of quantum length and boundary intersection measures on SLE curves

    • Main Results:

    • The interface formed by welding two boundary arcs of a Liouville quantum gravity random surface is identified as a Schramm-Loewner evolution curve
    • A theory of quantum fractal measures, consistent with the KPZ relation, is established
    • Quantum length and boundary intersection measures are successfully constructed on the Schramm-Loewner evolution curve

    Conclusions:

    • Conformal welding provides a method to generate Schramm-Loewner evolution curves from quantum gravity surfaces
    • The developed quantum fractal measures offer new tools for analyzing the geometry of random curves
    • This work connects quantum gravity, stochastic processes, and fractal geometry through the study of Schramm-Loewner evolution.