Source-linked AI summary
Divergences in Holographic Complexity
Alan Reynolds, Simon F. Ross
TL;DR
The UV structure of holographic complexity is not well understood, despite the expectation that its divergences reflect field-theory UV contributions. The paper adds a coordinate-invariance term to the null-boundary action and finds that the leading divergence is volume-proportional, while subleading divergences differ between CA and CV.
Problem
The field-theory interpretation of holographic-complexity divergences is not yet detailed, motivating study of whether their holographic form is reasonable.
Source: §1 · p. 2Method
The paper includes the coordinate-invariance term in the null-boundary action and compares its divergences with those of the maximal-volume CV prescription.
Source: §1 · pp. 2–3Results
Takeaways & Limitations
The leading volume divergence is considered reasonable from a field-theory perspective, while subleading terms distinguish the two prescriptions.
Source: §1 · pp. 2–3The argument covers divergences of the form ϵ^-n log ϵ for n > 0; free asymptotic data may contribute at order log ϵ.
Source: §3.1 · p. 9
Abstract
from arXiv · showhide
We study the UV divergences in the action of the "Wheeler-de Witt patch" in asymptotically AdS spacetimes, which has been conjectured to be dual to the computational complexity of the state of the dual field theory on a spatial slice of the boundary. We show that including a surface term in the action on the null boundaries which ensures invariance under coordinate transformations has the additional virtue of removing a stronger than expected divergence, making the leading divergence proportional to the proper volume of the boundary spatial slice. We compare the divergences in the action to divergences in the volume of a maximal spatial slice in the bulk, finding that the qualitative structure is the same, but subleading divergences have different relative coefficients in the two cases.
1 Introduction
The paper examines UV divergences in holographic complexity and compares the CV volume and CA action prescriptions. It finds that adding a coordinate-invariance term makes their leading divergences volume-proportional, while their subleading divergences differ.
- The CV conjecture relates complexity to maximal bulk volume, whereas the CA conjecture relates it to the Wheeler-de Witt patch action.Source: §1 · p. 2
- The divergences are expected to reflect UV contributions to complexity, although their field-theory interpretation remains incomplete.Source: §1 · p. 2
- Adding the coordinate-invariance term makes the CA leading divergence proportional to boundary-slice volume, matching the CV divergence structure, while subleading contributions differ.Source: §1 · pp. 2–3
- Subleading divergences in both prescriptions depend on local invariants of boundary geometry, but generally differ; global AdS illustrates this difference.Source: §1 · p. 3
2 Review of CV and CA
The section reviews the CV and CA proposals and the action prescription for regions with null boundaries. It explains the coordinate dependence addressed by an added term and notes remaining freedom in boundary contributions.
- The CV conjecture identifies complexity with the volume of a maximal bulk slice ending on the boundary spatial slice.Source: §2 · p. 3
- A lattice product reference state makes volume scaling plausible because preparing short-range entanglement in a Hadamard state requires operations growing with field-theory volume.Source: §2 · p. 3
- The CA proposal identifies complexity with the action of the Wheeler-de Witt patch, which is easier to locate than a maximal-volume slice and requires no explicit bulk length scale.Source: §2 · p. 3
- The null-boundary action prescription follows from requiring vanishing on-shell variation with fixed boundary metric, but its original form depends on the null coordinate.Source: §2 · p. 4
- The action assigns direction and sign conventions to its timelike, spacelike, and null boundaries; non-null junction terms do not occur for Wheeler-de Witt patches.Source: §2 · p. 4
- Adding the coordinate-independent term removes that dependence, while scalar functions of intrinsic boundary geometry remain an allowed action ambiguity.Source: §2 · pp. 4–5
3 UV divergences
Adding the null-boundary term removes coordinate dependence and the anomalously strong logarithmic divergence, leaving a volume-proportional leading divergence. Subleading divergences remain geometrically organized but differ between the action and maximal-volume prescriptions.
3 UV divergences
- An alternative cutoff that replaces the corner with a small timelike boundary produces a different set of subleading-divergence coefficients.This is an alternative to the original Wheeler–de Witt patch cutoff at z = ε.Source: §3 · p. 6
- The null-boundary term cancels coordinate dependence and the stronger-than-volume logarithmic divergence, leaving a leading divergence proportional to boundary spatial volume.The uncorrected action depends on affine-parameter normalizations and contains an ε^{-(d−2)} ln ε divergence.Source: §3 · pp. 5, 7
3.1 Subleading contributions
- For d = 1, the null surfaces have no expansion, so the added term cannot cancel the logarithmic divergence; the CV calculation is also logarithmically divergent.The paper notes this case may arise in applications to near-horizon geometries of near-extremal black holes.Source: §3.1 · p. 7
- Adding the term cancels divergences of the form ε^-n log ε for n > 0, leaving power-law divergences unless free asymptotic data enter.At orders reaching free data in the Fefferman–Graham expansion, logarithmic contributions may still occur in either prescription.Source: §3.1 · pp. 8–9
- The first subleading divergences have similar local-geometric structure in CA and CV, but their coefficients differ.The action’s logarithmic term cancels at this order, while its power-law terms combine into the result compared with the CV expression.Source: §3.1 · pp. 9, 11
3.2 Global AdS
- In global AdS, CA and CV share the leading UV divergence but have different θ_cut dependence, making the prescriptions inequivalent.The subleading CA terms depend on how the Wheeler–de Witt patch is cut off, limiting their clear physical interpretation.Source: §3.2 · p. 12