Modular invariance as completeness

TL;DR

The paper investigates how modular invariance in 2d unitary CFTs encodes completeness through Haag duality and the absence of superselection sectors. It links Renyi entropies, replica tricks, and torus modular data to define intrinsic indices (Jones/global indices) that quantify Haag duality violations. For rational CFTs, the crossing asymmetry exposes a finite index mu = sum d_r^2; when mu>1, the theory is incomplete and the partition-function data reveal deviations from modular invariance. The authors extend the framework to higher dimensions, where completeness corresponds to HD duality and the index controls universal thermodynamic corrections, with implications for generalized symmetries and quantum gravity.

Abstract

We review the physical meaning of modular invariance for unitary conformal quantum field theories in d=2. For QFT models, while T invariance is necessary for locality, S invariance is not mandatory. S invariance is a form of completeness of the theory that has a precise meaning as Haag duality for arbitrary multi-interval regions. We present a mathematical proof as well as derive this result from a physical standpoint using Renyi entropies and the replica trick. For rational CFT's, the failure of modular invariance or Haag duality can be measured by an index, related to the quantum dimensions of the model. We show how to compute this index from the modular transformation matrices. The index also appears in a limit of the Renyi mutual informations. Cases of infinite index are briefly discussed. Part of the argument can be extended to higher dimensions, where the lack of completeness can also be diagnosed using the CFT data through the thermal partition function and measured by an index.

Figures (3)

• Figure 1: Projector $P_1^\pm$ lines on the two boundaries $R_1^\pm$ of an open cut along the interval $R_1$ coalesce into the protector $P_{\gamma_1}$ on the closed curved $\gamma_1$ surrounding the interval.
• Figure 2: The $n=2$ Renyi entropy for two intervals can be computed as a partition function on the two-copy plane gluing $R^\pm_{1,2}$ cyclically as shown in the figure of the left. Inserting the corresponding projectors over $\gamma_1$ and $\gamma_2$ computes the Renyi entropy for the submodel. The resulting configuration can be mapped to a genus $g=1$ torus as in the right-hand side of the figure. It is a rectangle with opposite sides identified. Note $\gamma_1,\gamma_2$ are homotopical curves.
• Figure 3: The $n=3$ Renyi entropy calculation for two intervals can be mapped to a genus $2$ surface with three handles and a cyclic $Z_3$ symmetry between them. When pursuing the calculation in a submodel, we have three projectors $P_{\gamma_1}$, $P_{\gamma_2}$ and $P_{\gamma_3}$ appearing as circles around each one of the handles. One of the projectors is always redundant. The marked point $a_1$ is one of the endpoints of the interval $R_1$, and is a branching point of the original geometry. The marked three lines starting at $a_1$ are the three copies of the interval $R_1$, that connect $a_1$ with the other endpoint $b_1$ of $R_1$ (not shown).

Theorems & Definitions (4)

• Proposition 1
• Theorem 2
• proof
• Theorem 3