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General Results for Higher Spin Wilson Lines and Entanglement in Vasiliev Theory

Ashwin Hegde, Per Kraus, Eric Perlmutter

TL;DR

This work develops a practical framework for Wilson lines in 3D higher spin gravity and uses it to compute entanglement entropy in hs[$\\lambda$] Vasiliev theory. By solving the SL(N) Wilson line for arbitrary N and analytically continuing to hs[$\\lambda$] via $N\\rightarrow -\\lambda$, the authors obtain perturbative entanglement entropies and tests against CFT calculations, including in higher-spin black hole backgrounds. They show that the Wilson line computes the semiclassical $W_N$ vacuum block and connect this to scalar correlators in Vasiliev theory, while also providing a route to Virasoro blocks and insights into W$_N$ minimal model holography. The results advance the holographic dictionary in higher spin gravity, offering concrete, testable predictions for EE and conformal blocks in theories with extended $W_N$ symmetry and its hs[$\\lambda$] continuation. The work lays groundwork for nonperturbative explorations and clarifies the relationships between bulk Wilson lines, vacuum blocks, and boundary CFT data.

Abstract

We develop tools for the efficient evaluation of Wilson lines in 3D higher spin gravity, and use these to compute entanglement entropy in the hs$[λ]$ Vasiliev theory that governs the bulk side of the duality proposal of Gaberdiel and Gopakumar. Our main technical advance is the determination of SL(N) Wilson lines for arbitrary $N$, which, in suitable cases, enables us to analytically continue to hs$[λ]$ via $N \rightarrow -λ$. We apply this result to compute various quantities of interest, including entanglement entropy expanded perturbatively in the background higher spin charge, chemical potential, and interval size. This includes a computation of entanglement entropy in the higher spin black hole of the Vasiliev theory. These results are consistent with conformal field theory calculations. We also provide an alternative derivation of the Wilson line, by showing how it arises naturally from earlier work on scalar correlators in higher spin theory. The general picture that emerges is consistent with the statement that the SL(N) Wilson line computes the semiclassical $W_N$ vacuum block, and our results provide an explicit result for this object.

General Results for Higher Spin Wilson Lines and Entanglement in Vasiliev Theory

TL;DR

This work develops a practical framework for Wilson lines in 3D higher spin gravity and uses it to compute entanglement entropy in hs[] Vasiliev theory. By solving the SL(N) Wilson line for arbitrary N and analytically continuing to hs[] via , the authors obtain perturbative entanglement entropies and tests against CFT calculations, including in higher-spin black hole backgrounds. They show that the Wilson line computes the semiclassical vacuum block and connect this to scalar correlators in Vasiliev theory, while also providing a route to Virasoro blocks and insights into W minimal model holography. The results advance the holographic dictionary in higher spin gravity, offering concrete, testable predictions for EE and conformal blocks in theories with extended symmetry and its hs[] continuation. The work lays groundwork for nonperturbative explorations and clarifies the relationships between bulk Wilson lines, vacuum blocks, and boundary CFT data.

Abstract

We develop tools for the efficient evaluation of Wilson lines in 3D higher spin gravity, and use these to compute entanglement entropy in the hs Vasiliev theory that governs the bulk side of the duality proposal of Gaberdiel and Gopakumar. Our main technical advance is the determination of SL(N) Wilson lines for arbitrary , which, in suitable cases, enables us to analytically continue to hs via . We apply this result to compute various quantities of interest, including entanglement entropy expanded perturbatively in the background higher spin charge, chemical potential, and interval size. This includes a computation of entanglement entropy in the higher spin black hole of the Vasiliev theory. These results are consistent with conformal field theory calculations. We also provide an alternative derivation of the Wilson line, by showing how it arises naturally from earlier work on scalar correlators in higher spin theory. The general picture that emerges is consistent with the statement that the SL(N) Wilson line computes the semiclassical vacuum block, and our results provide an explicit result for this object.

Paper Structure

This paper contains 41 sections, 248 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Summary of results
  3. The explicit SL(N) Wilson line (Sections \ref{['wilson line I']}--\ref{['closed']})
  4. "Deriving" SL(N) Wilson lines from SL(N) Vasiliev theory (Section \ref{['vasilievwilsonlines']})
  5. Entanglement in hs[$\lambda$] Vasiliev theory (Sections \ref{['smallcharge']}--\ref{['shortint']})
  6. Virasoro blocks from $W_N$ blocks (Section \ref{['conformalblocks']})
  7. $W_N$ minimal model data (Section \ref{['wnsec']})
  8. Brief Review of 3D Higher Spin Gravity and Wilson Lines
  9. Wilson lines
  10. $W_N$ vacuum blocks and four-point functions
  11. SL(N) Wilson Line I. A Simpler Formulation
  12. Factorization
  13. SL(N) Wilson Line II. Explicit Expression
  14. Computation of ${\cal G}_R$
  15. A check: thermal entropy from the large interval limit
  16. ...and 26 more sections

Figures (1)

  • Figure 1: For external operators with higher spin charge with respect to some $W$-algebra, the $W$-algebra vacuum block (left side) can be branched into an infinite sum of Virasoro blocks (right side). Pictured are the Virasoro blocks corresponding to vacuum and single current exchanges.