Duality and the Knizhnik-Polyakov-Zamolodchikov Relation in Liouville Quantum Gravity
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Duplantier-2009-Duality and the Kniz.pdf
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Author(s)
Duplantier, Bertrand
Sheffield, Scott Roger
Date Issued
April 2009
Journal
Physical Review Letters
Publisher
American Physical Society
Citation
Duplantier, Bertrand , and Scott Sheffield. “Duality and the Knizhnik-Polyakov-Zamolodchikov Relation in Liouville Quantum Gravity.” Physical Review Letters 102.15 (2009): 150603. © 2009 The American Physical Society
Version
Final published version
Abstract
We present a (mathematically rigorous) probabilistic and geometrical proof of the Knizhnik-Polyakov-Zamolodchikov relation between scaling exponents in a Euclidean planar domain D and in Liouville quantum gravity. It uses the properly regularized quantum area measure dμ[subscript γ]=ε[superscript γ[superscript 2]/2]e[superscript γh [subscript ε](z)]dz, where dz is the Lebesgue measure on D, γ is a real parameter, 0≤γ<2, and h[subscript ε](z) denotes the mean value on the circle of radius ε centered at z of an instance h of the Gaussian free field on D. The proof extends to the boundary geometry. The singular case γ>2 is shown to be related to the quantum measure dμγ′, γ′<2, by the fundamental duality γγ′=4.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
http://dx.doi.org/10.1103/PhysRevLett.102.150603
