Minimal factorization of Chern-Simons theory – Gravitational anyonic edge modes

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}) × <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mstyle mathvariant="double-struck"> <mml:mi>ℝ</mml:mi> </mml:mstyle> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mo>×</mml:mo> </mml:mrow> </mml:math> SL (2,\mathbb{R}) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mstyle mathvariant="double-struck"> <mml:mi>ℝ</mml:mi> </mml:mstyle> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> . 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.