Constraining boundary conditions in non-rational CFTs

TL;DR

This paper analyzes conformal boundary conditions in the 2D compact free boson, focusing on the Friedan-Janik family of boundary states that appear at irrational radii. Using BCFT formalism, Ishibashi decompositions, and RCFT consistency constraints, the authors derive an explicit density of states $\rho(h)$ for open strings between Friedan-Janik boundaries and reveal a characteristic band-gap structure with integrable divergences. They further demonstrate two key pathologies: a violation of the cluster condition in the presence of Friedan-Janik boundaries when nonzero momentum/winding sectors are involved, and a divergent boundary entropy (the $g$-function) in the irrational-radius limit, implying an infinite number of localized degrees of freedom. Collectively, these results suggest Friedan-Janik states are not physically realized boundary conditions in this CFT and point to broader constraints on non-rational BCFTs, while motivating further exploration in related theories such as Narain c=1 models and orbifolds.

Abstract

We revisit the question of conformal boundary conditions in the compact free boson CFT in two dimensions. Besides the well-known Neumann and Dirichlet cases, there is an additional proposed one-parameter family of boundary states when the radius is an irrational multiple of the self-dual radius. These additional states have a continuous open string spectrum and we give an explicit formula for the density of states. We also discuss several pathologies of these states, including a violation of the cluster condition and that they have a divergent g-function.

Figures (4)

• Figure 1: A typical example of the density of states with a clear band structure. The white regions indicate gaps in the spectrum, while in the shaded regions, $\rho(h)$ indicates a continuous spectrum. Though it's difficult to see, there is an initial gap from $h=0$ to $h=0.0025$. The dashed lines indicate divergences at $\frac{1}{4}(n\pm 0.3)^2$.
• Figure 2: Three examples of the density of states when $\theta_2=\pi-\theta_1$. As $\theta_1$ gets smaller, the gaps get larger and the bands narrower, and the density of states approaches a sum of delta functions.
• Figure 3: Three examples of the density of states when $\theta_1=\theta_2$. As $\theta_1$ gets small, the spectrum approaches a continuous distribution with $\rho(h)\propto\sqrt{2/h}$ (although the divergent points persist for any finite value of $\theta_1$, if we subtract off the continuous piece and integrate $\rho$ over what remains, that quantity also vanishes in the limit).
• Figure 4: The cylinder setup used to define the $g$ function