Speaker
Description
We present a comprehensive theoretical framework for spectral boundary conditions (SBC) in two-dimensional conformal field theory, with particular focus on free boson models with boundaries. Using gamma-matrix formalism, we introduce a boundary helicity operator β = γ₅∂_σ and analyze its eigenvalue spectrum to classify boundary conditions. We investigate three distinct linear homogeneous boundary conditions: (ΠX)|∂M = 0, ∂σ(1-Π)X|∂M = 0, and (ΠX)|∂M + ∂σ(1-Π)X|∂M = 0, where Π represents various projection operators (Π₊, Π₊ + Π₀, Π_o, Π_e). Through Laurent mode expansions and conformal transformations from the infinite strip to the upper half-plane, we derive explicit boundary conditions for the current modes jₙ and j̄ₙ of the free boson theory. Our analysis reveals how different spectral boundary conditions preserve various aspects of conformal symmetry, with implications for the Virasoro algebra generators Lₙ and L̄ₙ. We compute the corresponding boundary partition function for one of the BCs. This work provides a systematic approach to understanding boundary effects in CFTs and offers new insights into the relationship between spectral properties and conformal invariance in the presence of boundaries.
