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Volume complexity for the non-supersymmetric Janus AdS$_5$ geometry

Stefano Baiguera, Sara Bonansea, Kristian Toccacelo

arXiv:2105.12743v1hep-th

TL;DR

This paper examines the ultraviolet divergences of CV complexity for non-supersymmetric Janus AdS5. It computes full-space and ball-shaped subregion volumes with two regularization prescriptions, finding a power-law divergence for the full boundary and a universal logarithmic coefficient for the subregion.

  • Problem

    The paper investigates the structure and universality of UV divergences in CV complexity for defect geometries, including whether lower-dimensional Janus results extend to Janus AdS5.

    Source: §1 · pp. 3–4, §3 · p. 7

  • Method

    The authors compute the extremal volume for the full boundary and a symmetric ball-shaped subregion around the interface using single- and double-cutoff regularizations.

    Source: §5 · p. 17

  • Results

    The full-boundary complexity has a power-law divergence, while the subregion has an additional logarithmic divergence and a non-vanishing finite term; only the subregion logarithmic coefficient is regulator-independent.

    Source: §5 · pp. 17–18

  • Takeaways & Limitations

    Logarithmic or finite terms encode universal properties of complexity, but when both occur only the logarithmic coefficient is universal.

    Source: §5 · p. 18

    The calculation chooses vanishing boundary time for convenience, although the constant-time extremal slice is stated to solve the equations of motion at all times.

    Source: Computation of the volume · p. 9

Abstract

from arXiv · show

We compute holographic complexity for the non-supersymmetric Janus deformation of AdS$_5$ according to the volume conjecture. The result is characterized by a power-law ultraviolet divergence. When a ball-shaped region located around the interface is considered, a sub-leading logarithmic divergent term and a finite part appear in the corresponding subregion volume complexity. Using two different prescriptions to regularize the divergences, we find that the coefficient of the logarithmic term is universal.

1 Introduction

The introduction frames holographic complexity as a proposed probe of bulk geometry beyond entropy and motivates studying its ultraviolet divergences. It focuses on CV complexity in defect geometries, where prior lower-dimensional results raise the question of which divergences are universal.

1 Introduction

  • The CV conjecture relates boundary complexity to the volume of a maximal codimension-one bulk submanifold, while CA relates it to gravitational action on a Wheeler-DeWitt patch.The introduction presents these as distinct gravity duals for computational complexity.Source: §1 · p. 2
  • UV divergences in complexity can be classified through curvature invariants integrated over a spatial slice, with regulator-independent terms marking universal information.The introduction connects this question to universal logarithmic or finite terms in entanglement entropy.Source: §1 · p. 3
  • Prior defect studies found a new logarithmic divergence for CV in a thin-brane model, while Janus AdS3 has a logarithmic divergence with a universal coefficient.The BCFT comparison distinguishes CV, whose logarithmic coefficient depends on boundary data, from CA, whose boundary-data dependence occurs in the finite term.Source: §1 · pp. 3–4
  • This paper examines UV divergences and their universality for CV complexity in non-supersymmetric Janus AdS5, particularly in comparison with the lower-dimensional case.The introduction notes that higher-dimensional boundary complexity has a power-law divergence depending on spacetime dimensionality.Source: §1 · p. 4
  • The paper computes complexity of formation for the full spacetime and subregion complexity for a ball centered on the defect, then compares two regularization schemes.Its later sections also discuss the universal structure of UV divergences in the CV approach.Source: §1 · p. 4

2 Five-dimensional Janus AdS geometry

The paper describes defect geometries using AdS slices and introduces the non-supersymmetric Janus solution as a deformation of AdS5. Its interface separates regions with distinct asymptotic dilaton values for generic deformation parameter.

2.1 Geometries with defects

  • A codimension-one planar defect preserves SO(d−1, 2) within the original conformal group SO(d, 2), and the bulk geometry is foliated into AdSd slices.In the Janus dual, the interface is associated with N = 4 SYM on two half-spaces whose coupling varies discontinuously across it.Source: §2.1 · pp. 4–5, §2 · p. 4
  • Near the asymptotic AdSd+1 region, the Fefferman-Graham expansion puts the metric in a form with radial coordinate ξ and boundary-normal coordinate η.The expansion can be used to regularize divergent bulk quantities, although its coordinate transformation may not have a closed form for general defect geometries.Source: §2.1 · p. 5

2.2 Non-supersymmetric Janus AdS5 geometry

  • The non-SUSY Janus solution is a one-parameter deformation of AdS5 with a nontrivial dilaton, described using a deformation parameter γ in the range 3/4 ≤ γ ≤ 1.At γ = 1 it reduces to AdS5 with constant dilaton, whereas γ = 3/4 gives a linear dilaton.Source: §2.2 · pp. 5–6
  • For generic γ, the Janus geometry has two asymptotically AdS5 regions with different constant dilaton values, and its conformal diagram represents the interface as a boundary joint.The solution is defined between the simple poles of its warp factor.Source: §2.2 · p. 6, Figure 1 · p. 6

3 Volume for the Janus AdS5 geometry

This section studies CV complexity in non-supersymmetric Janus AdS5, motivated by lower-dimensional logarithmic divergences and the dimension-dependent behavior found for BCFTs.

3 Volume for the Janus AdS5 geometry

  • The section computes holographic complexity using the CV conjecture for non-SUSY Janus-deformed AdS5.It notes that the extremal volume in Janus AdS3 and a Randall-Sundrum defect model exhibits a logarithmic divergence.Source: §3 · p. 7
  • Because BCFT CV divergences depend on dimensionality, the section expects analogous behavior for a Janus interface through the unfolding relation between BCFTs and codimension-one defects.The cited lower-dimensional Janus AdS3 and Randall-Sundrum cases exhibit logarithmic divergences.Source: §3 · p. 7
  • The stated aim is to examine the structure of the divergences and their universality in the Janus AdS5 setting.Source: §3 · p. 7

3.1 UV regularizations of the extremal volume

The section presents UV regularization procedures for extremal-volume integrals, contrasting a Fefferman-Graham cutoff with a double-cutoff prescription. The FG method requires a smooth interpolation across the region where its coordinates fail.

3.1 UV regularizations of the extremal volume

  • Extremal volumes integrate the induced metric over a codimension-one spatial slice, with the interface-dependent information encoded in the y and z directions.Integrations along the other spatial directions are trivial.Source: §3.1 · p. 7
  • Fefferman-Graham regularization cuts at ξ = δ, but the FG expansion breaks down near the interface where ξ/η ≪ 1.Source: §3.1 · p. 7
  • The FG cutoff surfaces in the left and right patches are joined by an arbitrary curve in the middle region, constrained to be smooth at y = y0.The figure identifies y0 with the boundary of the region where FG coordinates are ill-defined.Source: §3.1 · p. 7, Figure 2 · p. 8
  • The single-cutoff prescription uses the FG expansion to impose a minimal z value, whereas the double-cutoff method sets z = δ and introduces a separate ε cutoff.The ε cutoff is an intermediate mathematical artifact sent to zero; physical quantities do not depend on it.Source: §3.1 · pp. 7–8

3.2 Extremal volume: single cutoff procedure

The single-cutoff calculation evaluates the extremal volume and subtracts empty AdS5 to isolate the Janus interface contribution. The resulting complexity of formation has a δ^-2 divergence, while its finite part vanishes as zIR→∞.

3.2 Extremal volume: single cutoff procedure

  • The single-cutoff prescription intertwines the UV cutoffs in w and z, with the extremal volume evaluated at zero boundary time.The boundary-time choice is convenient because translational invariance makes the constant-time extremal slice a solution.Source: §3.2 · p. 8, Computation of the volume · p. 9

Determination of the geometric data

  • The cutoff condition uses h(0)=γ, the minimum of h, to fix zmin and the equivalent compact-coordinate cutoff w±.The cutoff w± regularizes the poles of h(w) at w=±w0 and has an expansion in even powers of δ/z.Source: Determination of the geometric data · p. 9

Computation of the volume

  • The remaining divergent terms are extracted by subtracting the lowest orders of the large-τ expansion, and A(γ) and B(γ) are numerically evaluable except at the singular γ=3/4 limit.At γ=3/4, w0→∞.Source: Computation of the volume · pp. 10–11

Subtraction of vacuum AdS5

Total result

  • The interface contribution to complexity of formation has a δ^-2 divergence, while its finite part vanishes as zIR→∞.This follows after subtracting the empty AdS5 volume.Source: Total result · p. 12

3.3 Extremal volume: double cutoff procedure

The double-cutoff prescription computes the Janus complexity of formation with separate z- and w-direction regulators. Its divergent coefficient differs from the single-cutoff result, while the finite terms match and vanish in the smooth IR limit.

3.3 Extremal volume: double cutoff procedure

  • Two distinct regulators, for z and w, make the integrations non-nested; the z-direction UV cutoff is set to z=δ.The w cutoff is determined separately.Source: §3.3 · p. 12

Computation of the volume

  • Changing variables to τ=h(w)/γ and integrating over z leads to the double-cutoff extremal volume.The remaining τ integration follows the method used for the single-cutoff calculation.Source: Computation of the volume · p. 12

Subtraction of vacuum AdS5

  • The undeformed AdS5 extremal volume is evaluated for the vacuum subtraction used to obtain the Janus complexity of formation.Subtracting this vacuum volume from the defect-background volume gives the formation complexity.Source: Subtraction of vacuum AdS5 · p. 13, Total result · p. 13

Total result

  • The double-cutoff result has a different divergent coefficient from Eq. (3.34), but matching finite terms that vanish as zIR→∞ leave no universal information in the full Janus geometry beyond δ^-2 scaling.The finite part is inversely proportional to the IR regulator.Source: Total result · p. 13

4 Volume subregion for the Janus AdS5 geometry

The section computes subregion volume complexity for a ball centered on the Janus interface using single- and double-cutoff regularizations. The leading area-proportional divergence agrees with the total-volume result, while the subregion result also contains a logarithmic divergence and a finite term.

4.1 Ball-shaped subregion on the boundary

  • The ball's RT surface is the same as in empty AdS5 and is a global minimum of the area functional in Janus AdS5.The ball-shaped region is centered on the interface.Source: §4.1 · pp. 13–14

4.2 Extremal volume: single cutoff procedure

  • With the single-cutoff procedure, the integration is expanded through order δ to track all possible divergences in the extremal volume.The section combines the resulting terms before subtracting the vacuum solution.Source: §4.2 · p. 15

Subtraction of the vacuum AdS solution

Total result

  • The subregion and total-volume results have matching δ^-2 divergences, while the subregion result has an additional logarithmic divergence and finite term.The R→∞ comparison suppresses logarithmic and finite terms, isolating the area-proportional divergences.Source: Total result · p. 16

4.3 Extremal volume: double cutoff procedure

Total result

  • In the R→∞ limit, the double-cutoff result matches the δ^-2 divergent part of the total-volume result.Source: Total result · p. 17

5 Conclusions

For the non-SUSY Janus AdS5 geometry, whole-boundary complexity of formation has a power-law divergence, while the symmetric ball subregion has an additional logarithmic divergence and a non-vanishing finite term. Across the two regularization prescriptions, only the subregion logarithmic coefficient is scheme-independent.

5 Conclusions

  • The whole-boundary divergence scales as δ^-2, while the subregion has an additional logarithmic divergence and a non-vanishing finite term.The whole-boundary power law is consistent with the BCFT result.Source: §5 · pp. 17–18
  • Only the subregion logarithmic-divergence coefficient is independent of the single- and double-cutoff prescriptions.The total-volume finite term also matches between prescriptions, but vanishes as zIR → ∞.Source: §5 · pp. 17–18
  • When a logarithmic and finite term coexist, only the logarithmic coefficient is universal because rescaling the UV cutoff shifts the finite part.This is consistent with the Janus AdS3 and static Janus BTZ cases, where distinct regularizations preserve the logarithmic coefficient but differ in the finite part.Source: §5 · p. 18
  • Future work includes classifying Janus AdS5 complexity=action divergences, comparing volume and action results, and computing complexity on the field-theory side.The authors also propose extremal-volume calculations for higher-dimensional generalizations of the time-dependent Janus BTZ black hole.Source: §5 · p. 18

A Weierstrass ℘function

The Weierstrass ℘ function is introduced by a series and described through its periodicity, symmetry, and poles. The passage also notes an alternative definition through an equation.

A Weierstrass ℘function

  • The Weierstrass ℘ function is an elliptic function of order 2 defined by a series.Source: A Weierstrass ℘function · p. 19
  • It is doubly periodic with half-periods ω1 and ω2, and is meromorphic, even, and has double poles at lattice points.Source: A Weierstrass ℘function · p. 19
  • The function can alternatively be defined using invariants g2 and g3, but the passage says a differential equation is simpler here.The invariants can be computed as Eisenstein series involving the half-periods.Source: A Weierstrass ℘function · p. 19
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