Entanglement entropy and the colored Jones polynomial

Vijay Balasubramanian, Matthew DeCross, Jackson Fliss, Arjun Kar, Robert G. Leigh, Onkar Parrikar

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We study the multi-party entanglement structure of states in Chern-Simons theory created by performing the path integral on 3-manifolds with linked torus boundaries, called link complements. For gauge group SU(2), the wavefunctions of these states (in a particular basis) are the colored Jones polynomials of the corresponding links. We first review the case of U(1) Chern-Simons theory where these are stabilizer states, a fact we use to re-derive an explicit formula for the entanglement entropy across a general link bipartition. We then present the following results for SU(2) Chern-Simons theory: (i) The entanglement entropy for a bipartition of a link gives a lower bound on the genus of surfaces in the ambient S3 separating the two sublinks. (ii) All torus links (namely, links which can be drawn on the surface of a torus) have a GHZ-like entanglement structure — i.e., partial traces leave a separable state. By contrast, through explicit computation, we test in many examples that hyperbolic links (namely, links whose complements admit hyperbolic structures) have W-like entanglement — i.e., partial traces leave a non-separable state. (iii) Finally, we consider hyperbolic links in the complexified S L (2 ℂ) Chern-Simons theory, which is closely related to 3d Einstein gravity with a negative cosmological constant. In the limit of small Newton constant, we discuss how the entanglement structure is controlled by the Neumann-Zagier potential on the moduli space of hyperbolic structures on the link complement.

Original languageEnglish (US)
Article number38
JournalJournal of High Energy Physics
Issue number5
StatePublished - May 1 2018


  • Chern-Simons Theories
  • Conformal Field Theory
  • Topological Field Theories
  • Wilson
  • ’t Hooft and Polyakov loops

ASJC Scopus subject areas

  • Nuclear and High Energy Physics


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