Geometric modular action for disjoint intervals and boundary conformal field theory

TL;DR

The paper investigates geometric modular action for unions of disjoint intervals in chiral conformal QFT and translates these ideas to boundary conformal QFT, linking modular flow to a temperature-like parameter along accelerated orbits $β(t,x)$.It shows that product states on $n$-intervals yield purely geometric modular flows with a well-defined interval flow $f_t(z)=rac{1}{n} old{Lambda_I(-2π t)(z^n)}$, while the vacuum of free Fermi theories exhibits a mixing (Casini-Huerta) between intervals that complicates the geometric picture.In completely rational BCFTs, the extended algebras admit a geometric modular action via a unique conditional expectation, while local temperature analyses reveal non-thermal features and negative energy densities in certain constructions, highlighting the nuanced thermality of modular flows.Overall, the work clarifies how geometry, superselection structure (mixing and charge splitting), and modular theory interplay in QFT on disjoint intervals and BCFT, and it outlines open questions about universality and the precise role of mixing versus charge splitting in a general theory.

Abstract

In suitable states, the modular group of local algebras associated with unions of disjoint intervals in chiral conformal quantum field theory acts geometrically. We translate this result into the setting of boundary conformal QFT and interpret it as a relation between temperature and acceleration. We also discuss aspects ("mixing" and "charge splitting") of geometric modular action for unions of disjoint intervals in the vacuum state.

Figures (3)

• Figure 1: Flow $f_t$ in the $3$-intervals $E=\sqrt[3]{S^1_+}=I_1\cup I_2\cup I_3$ and $E'=\sqrt[3]{S^1_-}$.
• Figure 2: Influence of the boundary. Left: modular orbit of an arbitrary point in the symmetric double cone $O=\{(t,x):A\leq t+x\leq B, -\frac{1}{A}\leq t-x\leq -\frac{1}{B}\}$. Right: a zoom on the modular orbit $(u_s, v_s)$ going through the center of the double cone. The plot represents the curve $(\tilde{u}_s,v_s)$ where $\tilde{u}_s = f*(u_s - u^{\text{diag}}_s) +u^{\text{diag}}_s$, with $(u^{\text{diag}}_s, v_s)$ the straight line joining the two tips of the double cone (a special vacuum modular orbit in the absence of the boundary), and $f=100$ a zoom factor.
• Figure 3: The 6 regions mixed by the vacuum modular flow in boundary CFT. $(u,v)$ is a point in $Q\subset O$. The boost is the distinguished orbit in $O$ as in Sect. \ref{['sec:modtemp']}, and defines $u'=-\frac{1}{u}$ and $v'=-\frac{1}{v}$. If $(u,v)$ lies on the boost, then the points $(v,u')$ and $(v',u)$ lie on the boundary. Consequently, if a double cone $Q\subset O$ around $(u,v)$ intersects the distinguished orbit, then four of the 6 associated double cones $Q_\alpha$ merge with each other, while the other two touch the boundary and degenerate to left wedges. (The flow $f_t$ itself, as in Fig. 2, is suppressed.)