Topological pseudo entropy

TL;DR

This work defines and analyzes topological pseudo entropy as a counterpart to topological entanglement entropy in three-dimensional Chern-Simons theory with Wilson loops. Using replica techniques and the CS/WZW duality, it derives exact results for states on S^2 and T^2, revealing both universal and knot-dependent contributions through modular S-matrix data and quantum dimensions. A key finding is the equivalence between pseudo entropy in certain CS/CFT setups and interface entropy in two-dimensional CFTs, enabling explicit calculations in compact scalars, free fermions, and RCFT interfaces. The paper also extends the framework to left-right pseudo entanglement entropy in BCFTs, connecting edge and boundary data via Ishibashi and Cardy-type constructions, and discusses geometric interpretations that parallel holographic entanglement entropy. Overall, topological pseudo entropy offers a window into quantum phase differences and edge structure in topological phases, with potential for supersymmetric generalizations and broader spacetime manifolds.

Abstract

We introduce a pseudo entropy extension of topological entanglement entropy called topological pseudo entropy. Various examples of the topological pseudo entropies are examined in three-dimensional Chern-Simons gauge theory with Wilson loop insertions. Partition functions with knotted Wilson loops are directly related to topological pseudo (Rényi) entropies. We also show that the pseudo entropy in a certain setup is equivalent to the interface entropy in two-dimensional conformal field theories (CFTs), and leverage the equivalence to calculate the pseudo entropies in particular examples. Furthermore, we define a pseudo entropy extension of the left-right entanglement entropy in two-dimensional boundary CFTs and derive a universal formula for a pair of arbitrary boundary states. As a byproduct, we find that the topological interface entropy for rational CFTs has a contribution identical to the topological entanglement entropy on a torus.

Figures (20)

• Figure 1: A manifold can be decomposed into two by cutting it a half and attaching hemispheres to each of them.
• Figure 2: The modular transformation in Chern-Simons gauge theory and the evaluations of the partition functions with Wilson loops. The horizontal solid tori have a complex structure $\tau$ while the vertical ones have $-1/\tau$. The dot means the gluing along the torus on the boundaries of two solid tori.
• Figure 3: We can calculate $Z\left[\mathbb{S}^3;R_i,R_j\right]$ by applying \ref{['eq:cutting']} and \ref{['eq:pf1']}.
• Figure 4: We can calculate ${\rm Tr}_A\left[({\rm Tr}_B\ket{\psi}\bra{\psi})^2\right]$ by gluing $B$ with the neighboring $\bar{B}$, corresponding to taking the partial trace over $B$, and $A$ with the neighboring $\bar{A}$, corresponding to the product of $\rho_A$. The last $\bar{A}$ is glued to the first $A$, corresponding to the trace over $A$.
• Figure 5: The difference $\Delta S$ of the pseudo entropy from the averaged entanglement entropy as a function of the levels $k$ when $N=5$. The left panel shows $\Delta S$ of the form \ref{['eq:deltaSgen']} by the second prescription (2) of analytic continuation. The blue, orange, green and red curves represent the cases with $|a-b|=1,2,3,4$ respectively. For comparison, the right panel shows $\Delta S$ of the form \ref{['eq:deltaSgen1']} by a naive prescription (1) when $|a-b|$ is odd. For even $|a-b|$, $\Delta S$ takes the same values as the left panel.