Lorentzian Dynamics and Factorization Beyond Rationality

TL;DR

This work develops a purely two-dimensional framework in which local operators factor through holomorphic and anti-holomorphic defect operators joined by a topological defect line, extending holomorphic factorization beyond rational CFTs. It establishes a direct link between the conformal Regge limit of four-point functions and the action of defect lines, deriving a spectral-radius formula from an opacity bound and exploring implications for Lorentzian dynamics and chaos. The authors analyze both rational and irrational theories, including the c=1 free boson on torus and orbifold branches, revealing Verlinde-line structures in rational cases and non-compact defect lines with continuous spectra at irrational points, and they illustrate the framework with the Ising model and free-boson examples. The results offer a new, broadly applicable perspective on defect data in 2D CFTs, connecting OPE, modular/invariant data, and Lorentzian dynamics, and they point toward a generalized defect-categorical approach that accommodates non-compact lines and non-semisimplicity.

Abstract

We investigate the emergence of topological defect lines in the conformal Regge limit of two-dimensional conformal field theory. We explain how a local operator can be factorized into a holomorphic and an anti-holomorphic defect operator connected through a topological defect line, and discuss implications on Lorentzian dynamics including aspects of chaos. We derive a formula relating the infinite boost limit, which holographically encodes the "opacity" of bulk scattering, to the action of topological defect lines on local operators. Leveraging the unitary bound on the opacity and the positivity of fusion coefficients, we show that the spectral radii of a large class of topological defect lines are given by their loop expectation values. Factorization also gives a formula relating the local and defect operator algebras, and fusion categorical data. We then review factorization in rational conformal field theory from a defect perspective, and examine irrational theories. On the orbifold branch of the $c = 1$ free boson theory, we find a unified description for the topological defect lines through which the twist fields are factorized; at irrational points, the twist fields factorize through "non-compact" topological defect lines which exhibit continuous defect operator spectra. Along the way, we initiate the development of a formalism to characterize non-compact topological defect lines.

Figures (2)

• Figure 1: The conformal Regge limit depicted in the global $\rho$-coordinate. The dotted lines are the light-cones of the operators $\phi$ and $\phi^\dagger$.
• Figure 2: The conformal block decomposition of the four-point function of the holomorphically-defect-factorized local operator \ref{['eqn:bulkfactorizationO']}.

Theorems & Definitions (17)

• Definition 1: Holomorphic-Defect-Factorization
• Definition 2: Factorizing topological defect line
• Definition 3: Holomorphic-defect-factorization prerequisite
• Proposition 1: Analyticity
• Proposition 2: Closedness of factorized operators
• Proposition 3: Closedness of factorizing topological defect lines
• Proposition 4: Uniqueness of factorization
• Proposition 5: Uniqueness of holomorphic defect operator
• Proposition 6: Commutativity
• Definition 4: Weak Cardy condition