On Relative Entropy and Global Index
Feng Xu
TL;DR
This work analyzes the duality between relative entropy in vacuum states and geometric data for conformal nets, introducing a deficit $D_{\frak A}(I)$ and a competing quantity $\hat{D}_{\frak A}(I)$ tied to finite-index inclusions. A central result is that the difference $D_{\frak A}(I)-\hat{D}_{\frak A}(I)$ remains invariant under finite-index inclusions, enabling rigorous criteria for when duality holds: it occurs precisely for holomorphic (i.e., $\mu_{\frak A}=1$) nets and, in two dimensions, for modular-invariant theories. Consequently, for nets chain-related to the free fermion net, duality holds and $D_{\frak A}=0$ iff $\mu_{\frak A}=1$, with extensions to even lattices and to a broad class of 2D CFTs. The paper also derives several limiting behaviors of relative entropy, linking them to global indices and modular properties, and provides a robust framework to study entropy under interval contractions and singular limits in both 1D and 2D conformal field theories.
Abstract
Certain duality of relative entropy can fail for chiral conformal net with nontrivial representations. In this paper we quantify such statement by defining a quantity which measures the failure of such duality, and identify this quantity with relative entropy and global index associated with multi-interval subfactors for a large class of conformal nets. In particular we show that the duality holds for a large class of conformal nets if and only if they are holomorphic. The same argument also applies to CFT in two dimensions. In particular we show that the duality holds for a large class of CFT in two dimensions if and only if they are modular invariant. We also obtain various limiting properties of relative entropies which naturally follow from our formula.