Spacetime topology and geometry in JT Gravity from non-crossing permutations

Abstract: We study how discrete parameters which enumerate non-crossing permutations become geometric quantities in JT gravity. Analyzing non-crossing permutations on the disk together with two classes of annular non-crossing permutations, we find that under suitable continuum scaling limits the number of marked boundary points becomes the thermal boundary length, while the number of through-connections becomes a geodesic length. In this limit, the enumerations reproduce the universal low-temperature behavior of the Schwarzian theory on the disk and the orientable and orientation-reversing double-trumpet geometries. We then relate these constructions to higher-boundary non-crossing diagrams, whose continuum enumeration reproduces the JT result obtained by attaching trumpets to the top-degree part of genus-zero Weil-Petersson volumes, suggesting that the same trumpet--enumeration parameter mapping extends beyond the planar case. Together, these results establish a direct connection between non-crossing permutation enumerations and Weil-Petersson volumes with trumpets attached, a construction that plays a central role in the gravitational path integral in JT gravity, and suggest that non-crossing diagrams provide a combinatorial route to computing multi-boundary partition functions.