Schramm Loewner Evolution and Liouville Quantum Gravity

Abstract

We conformally weld (via "quantum zipping") two boundary arcs of a Liouville quantum gravity random surface to generate a random curve called the Schramm-Loewner evolution (SLE). We develop a theory of quantum fractal measures (consistent with the Knizhnik-Polyakov-Zamolochikov relation) and analyze their evolution under welding via SLE martingales. As an application, we construct the natural quantum length and boundary intersection measures on the SLE curve itself.

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References in corpus (3)

• Liouville Field Theory -- A decade after the revolution
• Duality and KPZ in Liouville Quantum Gravity
• The Covariant Measure of SLE on the Boundary

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• Field-measure correspondence in Liouville quantum gravity almost surely commutes with all conformal maps simultaneously
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• Exponents for Hamiltonian paths on random bicubic maps and KPZ
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• Hamiltonian cycles on bicolored random planar maps
• Regularity of Schramm-Loewner Evolutions, annular crossings, and rough path theory