I will discuss the 1d Ising model with long-range interaction, decaying as 1/r^{1+s}. At long distances, the critical model corresponds to a family of 1d conformal field theories whose data depend nontrivially on s, in the range 1/2 ≤ s ≤ 1. The model is known to be described by a generalized free field with quartic interaction, which is weakly coupled near s = 1/2 but strongly coupled near the short-range crossover, at s = 1. I will introduce a dual description which becomes weakly coupled at s = 1, where the model becomes an exactly solvable conformal boundary condition for the 2d free scalar. We perform a number of consistency checks of our proposal and calculate the perturbative CFT data around s = 1 analytically using both 1) our proposed field theory and 2) the analytic conformal bootstrap. Our results show complete agreement between the two methods. Based on arXiv: 2412.12243, and arXiv: 2509.05250, in collaboration with D. Benedetti, D. Mazac, and P. van Vliet.
