Minimal Factorization of Chern-Simons Theory -- Gravitational Anyonic Edge Modes

TL;DR

Minimal Factorization of Chern-Simons Theory develops a minimal edge-mode extension for CS theory, showing topological invariance reduces edge DOFs and that a square-root of the CS Poisson structure yields a WZNW piece coupled to a particle on a quantum group. The main construction identifies a surjective gluing map that minimally extends the phase space with edge degrees transforming under a quantum group, and specializes to 3d gravity described by $SL(2,\mathbb{R}) \times SL(2,\mathbb{R})$, relating entanglement entropy to topological entropy. The work derives the nonlinear edge-charge algebra, including the Sklyanin and Semenov-Tian-Shansky brackets, and demonstrates how gravitational anyonic edge modes emerge from a particle-on-quantum-group sector. The framework provides a route to connect topological entanglement entropy in 3d gravity to quantum-group edge degrees and paves the way for extensions to higher rank groups and higher spin theories.

Abstract

One approach to analyzing entanglement in a gauge theory is embedding it into a factorized theory with edge modes on the entangling boundary. For topological quantum field theories (TQFT), this naturally leads to factorizing a TQFT by adding local edge modes associated with the corresponding CFT. In this work, we instead construct a minimal set of edge modes compatible with the topological invariance of Chern-Simons theory. This leads us to propose a minimal factorization map. These minimal edge modes can be interpreted as the degrees of freedom of a particle on a quantum group. Of particular interest is three-dimensional gravity as a Chern-Simons theory with gauge group SL$(2,\mathbb{R}) \times$ SL$(2,\mathbb{R})$. Our minimal factorization proposal uniquely gives rise to quantum group edge modes factorizing the bulk state space of 3d gravity. This agrees with earlier proposals that relate the Bekenstein-Hawking entropy in 3d gravity to topological entanglement entropy.

Figures (9)

• Figure 1: Left: The three colorful lines are Wilson lines anchoring on two physical boundaries (solid black circles). The dashed black circle is the entangling boundary. Right: Outer subsystem with edge modes (red dots) on the entangling boundary.
• Figure 2: We identify all edge states that only differ by moving the Wilson line endpoint on the entangling surface. All Wilson lines are then equivalently represented to anchor on the same point (red dot) on the entangling boundary.
• Figure 3: The annulus on the left is homotopy equivalent to a disc with a puncture (red dot). The puncture is a retract of the entangling boundary.
• Figure 4: Spatial annulus with orientation of the two boundary circles $S^1_L$ (outer) and $S^1_R$ (inner) depicted.
• Figure 5: Visualization of the crossing Wilson line algebra with a contribution from the intersection number $\varepsilon (x^+_{12}, x^-_{12})$, by cutting the Wilson lines into pieces.