Conical Defects in JT Gravity from BF Theory: Quantization, Fusion, and Weighted Moduli Spaces

Abstract: We construct a gauge-invariant BF representative of a conical defect in Jackiw--Teitelboim gravity, with its elliptic direction fixed by the reconstruction of the gravitational variables. In the gravitational sector, the BF observable reproduces the metric defect insertion and its distributional curvature source, and fixes the holonomy around the defect to an elliptic conjugacy class. For a single defect on the disk, the position integral combines with the residual (\mathrm{PSL}(2,\mathbb R)) quotient, leaving a fixed elliptic sector. The exact boundary quantization of this sector gives the elementary kernel and one-defect disk amplitude directly, without using a matrix model. The same local BF operator describes both sharp and blunt defects. Their difference appears only when several defect positions are integrated. BF source composition makes the fractional deficits additive. Requiring a fused source to remain in the gravitational cone sector gives the admissibility condition for defect collisions, while positivity of the hyperbolic area on each component gives the corresponding stability condition. These conditions coincide with those for Hassett weighted curves. Combining these BF results with the known Hassett compactification and conical Weil--Petersson geometry yields the fixed-order genus-zero contact terms and recovers the cluster expansion of deformed JT gravity. Thus sharp and blunt defects differ through the compactified geometry of their relative positions, rather than through the elementary BF operator.