[Paper Review] Unravelling Holographic Entanglement Entropy in Higher Spin Theories

Alejandra Castro, Eva Llabrés|arXiv (Cornell University)|Oct 10, 2014

Black Holes and Theoretical Physics32 references4 citations

TL;DR

This paper establishes the equivalence of two distinct proposals for computing holographic entanglement entropy in AdS₃ higher spin theories based on SL(N,ℝ) Chern-Simons gauge theory. It generalizes the massive particle Wilson line approach to arbitrary N and unitary representations, proving equivalence with the composite line operator proposal via two systematic methods—fundamental representation evaluation and small interval expansion—yielding exact agreement with CFT₂ short interval corrections for any N.

ABSTRACT

There are two proposals that compute holographic entanglement entropy in AdS$_3$ higher spin theories based on $SL(N,\mathbb{R})$ Chern-Simons theory. We show explicitly that these two proposals are equivalent. We also designed two methods that solve systematically the equations for arbitrary $N$. For finite charge backgrounds in AdS$_3$, we find exact agreement between our expressions and the short interval correction of the entanglement entropy for an excited state in a CFT$_2$.

Motivation & Objective

• To resolve the apparent discrepancy between two competing proposals for holographic entanglement entropy in SL(N,ℝ) higher spin gravity.
• To generalize the massive particle Wilson line approach to arbitrary N and a broad class of unitary representations carrying higher spin charges.
• To demonstrate that the composite Wilson line construction of [2] is equivalent to the massive particle approach of [1] through explicit computation.
• To develop two systematic methods for evaluating entanglement entropy in finite charge AdS₃ backgrounds for arbitrary N.
• To verify that the resulting expressions match the universal short interval correction of entanglement entropy in 2D CFTs for any N.

Proposed method

• Generalizes the Wilson line construction from SL(3,ℝ) to SL(N,ℝ) Chern-Simons theory, incorporating massive particles and higher spin representations via the fundamental representation of the algebra.
• Derives the saddle point value of the Wilson line using path integral techniques and on-shell action evaluation, ensuring consistency with gauge invariance and geometric constraints.
• Introduces Method I: exact evaluation of the Wilson line using the fundamental representation, valid for all parameter ranges but computationally intensive.
• Introduces Method II: small interval expansion of the composite Wilson line, capturing first-order corrections to relative entropy in CFT₂.
• Uses Weyl group reflections and dominant weight analysis to extract the leading asymptotic behavior of the trace over representations, identifying the dominant contribution in the large charge limit.
• Applies Dynkin label analysis and representation theory to relate the highest weight of the representation to the background charge, enabling extraction of entanglement entropy from the trace.

Experimental results

Research questions

• RQ1 Are the two existing proposals for holographic entanglement entropy in SL(N,ℝ) higher spin gravity equivalent for general N?
• RQ2 Can the massive particle Wilson line approach be generalized beyond SL(3,ℝ) to include higher spin representations in SL(N,ℝ) Chern-Simons theory?
• RQ3 Does the composite Wilson line operator proposed in [2] reproduce the same entanglement entropy as the massive particle approach when evaluated systematically?
• RQ4 How can one systematically compute entanglement entropy in finite charge AdS₃ backgrounds for arbitrary N?
• RQ5 Do the resulting expressions for entanglement entropy match the universal short interval correction predicted by 2D CFT for any N?

Key findings

• The two proposals for holographic entanglement entropy—based on massive particle Wilson lines and composite gauge-invariant lines—are rigorously proven to be equivalent for all N in SL(N,ℝ) higher spin gravity.
• The generalization of the massive particle Wilson line to arbitrary N and unitary representations yields a consistent and computable expression for entanglement entropy.
• Method I provides an exact evaluation of the Wilson line using the fundamental representation, valid across all parameter regimes.
• Method II, based on small interval expansion, successfully captures the first-order correction to relative entropy and matches the universal CFT₂ result for any N.
• The entanglement entropy formula derived via the Weyl group and dominant weight analysis agrees with the proposal in [2] and reproduces the correct short interval behavior in 2D CFT.
• The final expression for entanglement entropy, S_EE = (k / σ₁/₂) log[lim_{ρ₀→∞} Tr_R(M)], is valid for any embedding and includes the principal embedding as a special case.