Quantum triangles and imaginary geometry flow lines

Morris Ang

doi:10.1214/24-aihp1531

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Quantum triangles and imaginary geometry flow lines

2026-05-01

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Morris Ang *

Xin Sun

Yu Pu

DOI：10.1214/24-aihp1531

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Abstract

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We define a three-parameter family of random surfaces in Liouville quantum gravity (LQG) which can be viewed as the quantum version of triangles. These quantum triangles are natural in two senses. First, by our definition they produce the boundary three-point correlation functions of Liouville conformal field theory on the disk. Second, it turns out that the laws of the triangles bounded by flow lines in imaginary geometry coupled with LQG are given by these quantum triangles. In this paper we demonstrate the second point for boundary flow lines on a quantum disk. Our method has the potential to prove general conformal welding results with quantum triangles glued in an arbitrary way. Quantum triangles play a basic role in understanding the integrability of SLE and LQG via conformal welding. In this paper, we deduce integrability results for chordal SLE with three force points, using the conformal welding of a quantum triangle and a two-pointed quantum disk. Further applications will be explored in subsequent works.

Keywords:

Schramm-Loewner evolution

Liouville quantum gravity