Holographic entanglement entropy in Chern-Simons gravity with torsion

Holographic entanglement entropy is a key concept linking quantum information theory and gravity. Since the original conjecture of Ryu and Takayanagi, holographic entanglement entropy has been generalized beyond Einstein–Hilbert gravity to include higher-curvature corrections. In most existing generalizations, however, it is implicitly assumed that the bulk spacetime geometry is Riemannian, i.e. torsion-free. Here we propose a prescription for incorporating torsion into holographic entanglement entropy in the boundary theory dual to five-dimensional Chern–Simons gravity. We argue that the entanglement entropy acquires an additional universal divergent term proportional to the logarithm of the UV cutoff, and that this term is generated solely by torsion.

💡 Research Summary

The paper addresses a gap in the holographic entanglement entropy (HEE) literature: virtually all existing generalizations of the Ryu‑Takayanagi (RT) prescription assume a torsion‑free, pseudo‑Riemannian bulk. By working in five‑dimensional Chern‑Simons (CS) gravity, the authors construct a concrete framework that incorporates torsion—encoded in the independent vielbein eᵃ and spin‑connection ω^{ab} of Riemann‑Cartan (RC) geometry—into the holographic computation of entanglement entropy.

The bulk action is the standard five‑dimensional CS form, which contains curvature‑squared and higher‑order vielbein terms. A particular solution with axial torsion is considered: the AdS₅ metric together with a non‑vanishing torsion two‑form T^{i}=C ε^{ijk}dx^{j}∧dx^{k}, where C is a constant measuring the torsion strength. Performing the Fefferman‑Graham expansion of the bulk fields reveals that the boundary inherits a flat metric but a non‑zero RC curvature: the RC Ricci scalar is \bar R_{RC}=−6C², entirely due to torsion.

Two parallel routes are then proposed to obtain the HEE in the presence of torsion. First, Wald’s entropy formula for higher‑derivative gravities is applied, with the usual Riemann curvature replaced by the RC curvature. This yields an entropy functional that contains an extra term proportional to C². Second, the RT functional is generalized to
S_{HEE}=2πk∫{γ{ext}}√h (1+½ R_{(RC)}),
where R_{(RC)} is the RC Ricci scalar induced on the extremal surface γ_{ext}. Both approaches predict a universal logarithmic divergence of the form
S_{log}=α C² ln(ε),
with ε the UV cutoff.

To verify the claim, the authors compute the entanglement entropy for two representative entangling surfaces in the boundary CFT. For a spherical surface Σ of radius R, the Gauss‑Codazzi equation is extended to RC geometry, giving an induced RC scalar R_{(RC)}=2R−4C². Substituting into the generalized RT formula reproduces a logarithmic term S_{log}= (k/π) R²C² ln(ε). For a cylindrical region, the ordinary Riemannian contribution vanishes, yet the torsion‑induced RC curvature still generates a logarithmic term proportional to C².

A subtle point is that the extremal surface in the torsional theory need not coincide with the torsion‑free minimal surface. However, the logarithmic divergence depends only on the near‑boundary behavior r(ρ)=R−ρ⁴R+…, which is the same as in the torsion‑free case. Consequently, the authors can safely use the torsion‑free profile to extract the log term, showing that the universal coefficient is insensitive to the detailed shape of the true extremal surface.

Finally, the paper emphasizes that the torsion‑induced logarithmic term can be interpreted as a new “torsional central charge” in the dual four‑dimensional CFT, complementing the usual a‑type anomaly while the c‑type anomaly remains absent. This result opens a pathway to study quantum information aspects of theories with non‑trivial affine structure and suggests that torsion may play a significant role in holographic dualities beyond the standard Einstein‑Hilbert paradigm.