Trivariate Hypergeometric Series Formulas for Pure Partition Functions of Multiple $3$-SLE$_\kappa$

Abstract

Pure partition functions of multiple SLE are characterized by null-state partial differential equations, M\"obius covariance, and boundary asymptotics. After quotienting by M\"obius covariance, the case of three curves is the first genuinely multivariable one: the moduli space has three independent variables, naturally represented by the three unoriented cross-ratios of the three pairs of links. We solve this M\"obius-normalized three-variable problem for the two basic link-pattern types of multiple $3$-SLE$_\kappa$, namely the rainbow and neighbor patterns. Writing $\beta=4/\kappa$, we construct explicit trivariate hypergeometric-series normal forms and identify them with the corresponding pure partition functions {for all $\beta>1/2$ in both cases, equivalently for all $\kappa\in(0,8)$}. The proof is analytic. The null-state PDEs and M\"obius covariance yield recursion relations for the trivariate coefficient arrays. In the rainbow case, coefficient estimates give convergence and boundary regularity on the closed cube. In the neighbor case, Pfaff systems continue the local power series to a neighborhood of $[0,1)^3$, while {boundary analysis and corner propagation} give continuity on $[0,1]^3$ {throughout the full range $\beta>1/2$}. The two-dimensional boundary degenerations are classical Appell $F_1$ and Horn $G_2$ functions. The probabilistic identification uses SLE martingale arguments and It\^o calculus, together with positivity and boundary regularity. We also discuss boundary degenerations, including heuristic connections with boundary Green's functions.