This paper proves a key part of the conformal bootstrap for Liouville conformal field theory on compact surfaces that have an edge or boundary. The authors focus on the spectral (frequency‑like) analysis of a geometric operation they call the half‑annulus semigroup and show how that analysis leads to a concrete “bootstrap” formula. In plain terms, they break the space of possible boundary states into basic building blocks, then use that breakdown to rewrite any correlation function as an integral over simple pieces.

Concretely, the researchers identify the generator of the half‑annulus semigroup with what they call the boundary Hamiltonian — the operator that governs scale changes along a boundary segment. Using scattering methods (a standard kind of argument that studies how waves or states behave at large times or distances), they establish a spectral decomposition of this boundary Hamiltonian. The outcome is a direct‑integral decomposition: the boundary state space splits into a continuum of irreducible Virasoro representations. (Virasoro representations are the mathematical objects that encode the theory’s conformal symmetry; “irreducible” means these pieces cannot be broken down further.) The decomposition is controlled by a boundary spectral measure, which tells how much of each irreducible piece appears.

They then combine this spectral picture with Ward identities. Ward identities are equations that encode how correlation functions change under small deformations of the boundary — in other words, they are the consequences of symmetry. Inserting the spectral decomposition at each curve where a surface is cut (a “gluing” curve) turns correlation functions into explicit integrals over spectral parameters attached to those cuts. The integrands are products of known structure constants (numbers that come from basic three‑point or two‑point building blocks on disks and boundaries) and conformal blocks (special functions that capture how descendants of primary fields contribute). Evaluating amplitudes at primary states recovers the structure constants, while descendants produce the conformal blocks. This completes the bootstrap picture for bordered surfaces: correlators are assembled from structure constants and conformal blocks integrated against the spectral measure.