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1 results

  • 8 - SLE and the Quantum Zipper

  • Nathanaël Berestycki, Universität Wien, Austria, Ellen Powell, Durham University
  • Book:
    Gaussian Free Field and Liouville Quantum Gravity
    Published online:
    20 November 2025
    Print publication:
    04 December 2025, pp 299-333

  • 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.