arXiv:2505.10277v1 [hep-th] 15 May 2025
Thermal holographic correlators and KMS
condition
Ilija Burić, Ivan Gusev, Andrei Parnachev
School of Mathematics and Hamilton Mathematics Institute, Trinity College, Dublin 2,
Ireland
E-mail: burici@tcd.ie, gusevi@tcd.ie, parnachev@maths.tcd.ie
Abstract: Thermal two-point functions in holographic CFTs receive contributions
from two parts. One part comes from the identity, the stress tensor and multi-stress
tensors and constitutes the stress-tensor sector. The other part consists of contributions from double-trace operators. The sum of these two parts must satisfy the KMS
condition – it has to be periodic in Euclidean time. The stress-tensor sector can be
computed by analyzing the bulk equations of motions near the AdS boundary and is
not periodic by itself. We show that starting from the expression for the stress-tensor
sector one can impose the KMS condition to fix the double-trace part, and hence the
whole correlator. We perform explicit calculations in the asymptotic approximation,
where the stress-tensor sector can be computed exactly. One can either sum over
the thermal images of the stress-tensor sector and subtract the singularities or solve
for the KMS condition directly and perform the Borel resummation of the resulting
double-trace data – the results are the same.
Contents
1 Introduction and Summary
2
2 Asymptotic model of holography
5
2.1
Thermal two-point functions
6
2.2
Review: multi-stress-tensor OPE coefficients
8
2.3
Sum over images
9
2.4
The asymptotic model
12
2.5
Refinements of the model
14
3 Sum rules
16
3.1
Review: KMS sum rules
16
3.2
Homogeneous equations
18
3.3
Particular solution
19
3.4
Reproducing GFF
21
3.5
Application to the asymptotic model
22
3.6
Application to the refined model
26
4 Discussion
28
A Asymptotic Models for Various ∆ϕ and d
30
Asymptotic Model for ∆ϕ = 52
30
A.1
A.2 Models in six dimensions
31
B Solutions to Sum Rules
33
B.1 Holographic solutions
33
B.2 List of identities
34
C Klein-Gordon equation on the planar AdS black hole
–1–
35
1
Introduction and Summary
Gravitational holography [1–3] provides a powerful framework for understanding
strongly interacting conformal field theories (CFTs) at large central charge CT , and
more generally, quantum field theories. A particularly promising direction is thermal
holography, where CFTs at finite temperature T = 1/β are mapped to asymptotically
anti-de Sitter black holes [4]. In this context, finite-temperature CFT correlators are
related to fields propagating in such spacetimes. A natural question is to investigate
the operator product expansion (OPE) of these correlators and their decomposition
into finite-temperature conformal blocks. It is also natural to introduce the complexified time τ = it + τR and explore questions regarding the analytic behavior of the
OPE series in the complex τ -plane.
At leading order in the inverse central charge expansion, finite-temperature holographic two-point functions of identical primary scalar operators ϕ(xµ ) receive contributions from the identity, the stress tensor and its products (the stress-tensor sector).
In two spacetime dimensions, these contributions constitute the entire correlator.
However, in higher dimensions, double-trace operators [ϕϕ]n,ℓ ≃: ϕ□n ∂µ1 . . . ∂µℓ ϕ :
also play a role (see e.g. [5, 6] for the discussion of such operators in the CFT context
and [7–11] for their appearance in the holographic examples involving asymptotically
AdS black holes).
Any finite-temperature correlator satisfies the KMS condition, stating that the
correlator is periodic in real τ with period β, [12, 13]. Additionally, correlators are
expected to be analytic in the strip 0 < Re τ < β. The simplest example of a
correlator meeting these conditions is provided by the generalized free field (GFF)
model, where the vacuum result is summed over thermal images; see e.g. [7]. The
OPE decomposition of this sum is straightforward: besides the identity’s contribution
(the vacuum term), the sum over thermal images fixes the double-trace operator
contributions. Another perspective is that the double-trace contributions organize
themselves precisely to ensure that the correlator satisfies the KMS condition.
The GFF model serves as the simplest toy model of holography, in which the entire
stress-tensor sector is replaced by just the identity contribution. As explained above,
in this simplified scenario, recovering the double-trace portion of the correlator, which
ensures the KMS periodicity, is straightforward. A natural question arises: does
this reasoning extend to actual holography? Specifically, given the stress-tensor
sector, can one similarly reconstruct the double-trace contributions to the correlator
as straightforwardly as in the GFF case?
This question has been implicitly raised in several recent works [9–11]. Technically, controlling the OPE terms due to double-trace operators is significantly more
challenging compared to terms arising from the stress-tensor sector. The latter can be
effectively obtained by analyzing the bulk equations of motion in the near-boundary
–2–
expansion, [9], with results cross-checked against the momentum space OPE expansion [10, 14, 15]. However, the method of [9] does not determine the double-trace
terms explicitly, although their existence can be inferred from poles in the OPE
coefficients of multi-stress-tensor operators, which appear for integer conformal dimensions of the external scalar fields. These poles signify mixing between multi-stress
tensors and double-trace operators. From a momentum-space perspective, doubletrace contributions correspond to contact terms, making them difficult to recover
directly.
In [11], the asymptotic behavior of the stress-tensor sector terms in the OPE
was analyzed1 . The radius of convergence of the OPE of the stress-tensor sector
was shown to be finite, the latter exhibiting singularities in the strip 0 < Re τ <
β. These singularities, referred to as ‘bouncing singularities’ in [11], correspond
to null geodesics bouncing off the singularities of asymptotically AdS black holes
in the gravitational description2 . Since the complete correlator must not exhibit
such singularities, the double-trace terms must display singular behavior precisely
canceling bouncing singularities arising from the stress-tensor sector.
The finite radius of convergence (β/[2 sin(π/d)] in d spacetime dimensions) makes
it clear that a simple summation over images of the stress-tensor sector is not straightforward, as one needs to analytically continue the stress-tensor sector to get its value
at thermal images of τ = 0. Moreover, the analysis in [11] only captures the leading
singularity of this sector, and the full analytic structure is expected to be considerably more intricate. A natural question arises: how accurately does the asymptotic
behavior derived in [11] approximate the entire stress-tensor sector? While this approximation is clearly accurate near the poles (or equivalently, for multi-stress tensors
[Tµν ]n with n ≫ 1), even at the level of the stress tensor itself, where the asymptotic
approximation error is greatest, the deviation between the asymptotic behavior and
the exact one-point function times the OPE coefficient is relatively small.
This prompts us to introduce the asymptotic model of holography, a model where
all multi-stress-tensor contributions, beginning with the single stress tensor, are captured by the asymptotic formula. As we will see below, this model can often be
explicitly summed, analytically continued, and then summed over thermal images.
While the resulting expression is inherently KMS-periodic, it typically exhibits poles
(bouncing singularities) within the vertical strip. Nonetheless, it is possible to add a
term that affects only the double-trace data, preserves KMS-periodicity, and removes
these poles. The resulting expression provides a significantly improved approximation to the true holographic correlator compared to the GFF formula. Does this
procedure fix the double-trace part uniquely? We believe the answer is yes, under
1
See also [16–29] for additional recent papers where the OPE has been discussed in the context
of holographic dual of asymptotically AdS black holes.
2
See also [10, 30–35] for related work.
–3–
the assumption that we cannot introduce additional double-trace terms which would
be non-analytic in the strip3 .
The asymptotic model just described serves as a good starting point for computing
the full holographic correlator, which, we hope, can be achieved by incorporating
subleading terms in the stress-tensor sector.
One may wonder if there is an alternative procedure to summing over images. A
promising route emerges from analyzing the structure imposed by the KMS condition.
In [25] it was pointed out that the KMS condition, which essentially demands vanishing of the odd derivatives at τ = β/2, forms an infinite system of linear equations for
the thermal one-point coefficients of double-trace operators. However, these equations admit only asymptotic (non-convergent) solutions when the input CFT data is
given by the multi-stress-tensor sector. To make sense of these divergent series, we
employ Borel regularization.
Remarkably, this regularization yields physically meaningful results. When applied to the GFF, Borel summation reproduces the exact double-trace data. We then
apply the same approach to the asymptotic model of holography and find that Borel
summation reconstructs the same double-trace data that follows from summing over
images and canceling the poles associated with bouncing singularities. Our main
results are as follows:
• We explicitly construct the holographic correlator in the asymptotic approximation for half-integer values of ∆ϕ , incorporating the large-n behavior of
multi-stress-tensor coefficients. The resulting correlator satisfies the KMS condition, is analytic in the vertical strip, up to the pole at τ = 0, and captures
the asymptotic stress-tensor sector structure expected from holography.
• We review KMS sum rules, [25], an infinite system of linear equations for thermal OPE coefficients arising from the vanishing of odd derivatives of the twopoint function at τ = β/2. These sum rules generically admit only divergent
solutions when the input data comes from the stress-tensor sector.
• We apply Borel summation to regularize the divergent solutions of the sum
rules. In the case of the GFF model, this reproduces the previously known expressions for the double-trace coefficients. For the asymptotic model of holography, the Borel-resummed solution agrees precisely with the result obtained
by explicit singularity cancellation, providing a second, independent derivation
of the double-trace data.
3
This is because such an additional double-trace piece is a function of τ 2 and hence is invariant
under τ → − τ . It is therefore periodic under τ → τ + β. Hence, if it’s analytic in the strip, it is
also analytic in the entire complex plane and therefore is at most a constant. The constant can be
set to zero by requiring that the correlator decays at large times.
–4–
• We explore refinements of the asymptotic model by incorporating a finite number of exact low-lying multi-stress-tensor coefficients. While these corrections
improve the accuracy of the model’s input, they must be implemented carefully
to preserve analyticity in the complex τ -plane. We find that only a small number of such corrections maintain the desired structure; attempts to go further
lead to divergences of the correlator at Im τ → ∞ unless a more systematic
large-n expansion is employed.
The paper is organized as follows. In Section 2, we introduce the asymptotic model
of holography, beginning with a general discussion of thermal two-point functions
and their conformal block decomposition. We then review the multi-stress-tensor
OPE coefficients and proceed to construct the asymptotic model via resummation
and the method of images. The section concludes with a discussion on possible
refinements of the model by incorporating exact low-lying multi-stress-tensor data.
Section 3 focuses on the derivation and solution of the KMS sum rules. We begin by
reviewing the sum rules and deriving the associated system of equations. We then
solve these equations, both in the homogeneous case and for particular solutions using
Borel summation. This method is applied first to the generalized free field model
and subsequently to the asymptotic holographic model, leading to an independent
reconstruction of the double-trace data. In Section 4, we discuss the implications of
our results and outline possible directions for future refinements of the asymptotic
model. Appendices A–C contain supplementary material: Appendix A presents
asymptotic models for various values of dimension of the external field ∆ϕ and the
spacetime dimension d; Appendix B contains details about solutions to the sum rules,
including identities used throughout the main text; Appendix C provides background
on the Klein-Gordon equation in the AdS-Schwarzschild black hole geometry, which
is central to the computation of the multi-stress-tensor and double-trace coefficients.
2
Asymptotic model of holography
In this section, we construct the asymptotic model of holography. The first subsection reviews the general properties of conformal two-point functions at finite temperature. In the second subsection, we recall the asymptotic form of the multi-stresstensor CFT data. The next two subsections are devoted to the computation of the
asymptotic model two-point function. The final subsection discusses refinements of
the model that take into account more detailed information about the stress-tensor
sector.
–5–
2.1
Thermal two-point functions
The main object of study in this work are conformal two-point functions of scalar
fields at finite temperature T = β −1 ,
gβ (τ, ⃗x) = ⟨ϕ(τ, ⃗x)ϕ(0)⟩β .
(2.1)
Equivalently, these can be thought of as correlators on the geometry Sβ1 × Rd−1 . The
coordinates on Sβ1 × Rd−1 are denoted by (τ, ⃗x) and the Euclidean time is periodic,
τ ∼ τ + β. For the most part, we shall consider the two-point correlation function
evaluated at vanishing spatial separation, that is, at ⃗x = 0. This will be denoted by
gβ (τ ). We begin by listing some of the main properties of (2.1), starting with those
present in any CFT, before turning to the ones specific to theories with holographic
duals.
Periodicity and KMS condition In general, we will allow τ to take complex
values within the vertical strip 0 ≤ Re τ ≤ β. The KMS condition can be written as
β
β
gβ
+ τ, ⃗x = gβ
− τ, ⃗x .
(2.2)
2
2
Conformal block decomposition The two-point function can be expanded in
conformal blocks as, [7],
X
p
λϕϕO bO ( d−2
τ
ℓO !
)
∆O −2∆ϕ
2
C
|x|
,
|x|
=
τ 2 + |⃗x|2 .
gβ (τ, ⃗x) =
ℓO
∆O
ℓO ( d−2 )
β
|x|
2
2 ℓO
O
(2.3)
The sum runs over all operators O that appear in the ϕ × ϕ OPE, their dimensions
(α)
and spins being denoted by (∆O , ℓO ). The Cℓ are Gegenbauer polynomials, λϕϕO
the OPE coefficients of the CFT and bO the thermal one-point coefficients. At zero
spatial separation, we may perform the sum over spins ℓO in (2.3) and write
gβ (τ ) =
X a∆
∆O
O
β ∆O
τ ∆O −2∆ϕ ,
a∆ ≡
( d−2 )
∆X
O =∆
ℓO ! CℓO 2 (1)
O∈ϕ×ϕ
2ℓO ( d−2
)
2 ℓO
λϕϕO bO .
(2.4)
By a slight abuse of terminology, we shall refer to coefficients a∆O rather than bO as
thermal one-point coefficients.
Boundedness In the complex τ -plane, the two-point function is bounded along
the lines parallel to the imaginary axis by its value on the real axis, i.e.
gβ (τ ) ≥ |gβ (τ + iκ)| ,
–6–
τ, κ ∈ R .
(2.5)
We briefly review the derivation of (2.5) (for similar arguments, see for instance
[31]). Recall that, on the cylinder, the Hamiltonian of the theory coincides with the
dilation operator D. In Euclidean time τ , the dilation acts as D = ∂τ , and thus
ϕ(τ ) = eτ D ϕ(0)e−τ D .
(2.6)
Let {|En ⟩} be a basis of Hamiltonian eigenstates of the Hilbert space. By inserting
the identity once, we may write the two-point function in terms of the matrix elements
of the field operator as
X
gβ (τ ) =
e−βEn ⟨En |eτ H ϕ(0)e−τ H |Em ⟩⟨Em |ϕ(0)|En ⟩
(2.7)
m,n
=
X
e−βEn +τ (En −Em ) |ϕmn |2 .
m,n
Here, we have put ϕmn = ⟨Em |ϕ(0)|Em ⟩. Now, let τ and κ be real. By a similar
manipulation as above
X
e−βEn +(τ +iκ)(En −Em ) |ϕmn |2 .
(2.8)
gβ (τ + iκ) =
m,n
But, clearly we have
X
X
e−βEn +(τ +iκ)(En −Em ) |ϕmn |2 | ,
e−βEn +τ (En −Em ) |ϕmn |2 ≥ |
m,n
(2.9)
m,n
since all the terms in the first sum are positive and they get multiplied by phases in
the second. The last statement is precisely the boundedness condition (2.5).
In addition to the above properties, it is expected that the two-point function
gβ (τ ) has no singularities in the complex plane, except the one at the origin, τ = 0,
and its KMS images, τ ∈ βZ.
Holography Our discussion so far applies to any conformal field theory. In the following sections, we further assume that the CFT under consideration is holographic
and that ϕ is a single-trace operator. At leading order in 1/CT , the conformal block
decomposition (2.3) receives contributions from two types of operators only - the
double-trace operators [ϕϕ]n,ℓ and composites of the stress tensor. We denote the
n
latter schematically by Tµν
or [Tµν . . . Tρσ ]. The one-point coefficients a∆ corresponding to multi-stress tensors are obtained by analyzing the Klein-Gordon equation on
the planar AdS-Schwarzschild black hole in a near boundary expansion, [9]. Since
these coefficients play an important role in our analysis, we give more details about
them in Section 2.2.
–7–
Example: Generalized free field An example of a thermal two-point function
that can be computed exactly is that of the generalized free field (GFF) ϕ. The GFF
two-point function is given by the sum of thermal images of the identity exchange
conformal block. At zero spatial separation it can be written in terms of the Hurwitz
zeta function ζH ,
∞
X
τ
τ
ζH 2∆ϕ ,
+ ζH 2∆ϕ , 1 −
.
β
β
(2.10)
Here, ∆ϕ denotes the scaling dimension of ϕ. The operators that appear in the ϕ × ϕ
OPE in (2.10) are the identity operator and the double traces [ϕϕ]n,ℓ - see e.g. [7] for
the corresponding coefficients b[ϕϕ]n,ℓ . The one-point coefficients summed over spins,
that appear in (2.4), read
GFF
g∆
(τ ) =
ϕ
1
1
=
2∆
2∆
|τ + nβ| ϕ
β ϕ
n=−∞
aGFF
[ϕϕ]2∆ +2m =
ϕ
2(2m + 1)2∆ϕ −1
ζ(2∆ϕ + 2m) .
Γ(2∆ϕ )
(2.11)
We have listed those properties of thermal two-point functions that will play a role
in the analysis of this work. Additional interesting properties of thermal holographic
correlators include long time behavior, which is controlled by the quasinormal mode
frequencies (see e.g. [36–60]), pole skipping (see e.g. [61–67]) and bulk-cone singularities (see e.g. [68–72]).
2.2
Review: multi-stress-tensor OPE coefficients
Let us separate the holographic two-point function gβ (τ ) into three pieces corresponding to the contributions of the identity, double-trace operators and multi-stress
tensors, in the conformal block decomposition,
gβ (τ ) = |τ |−2∆ϕ + g[ϕϕ] (τ ) + gT (τ ) .
(2.12)
In the leading order in 1/CT , the operators do not receive anomalous dimensions.
The scaling dimensions of double-trace operators take the form ∆ = 2∆ϕ + 2m,
m ∈ N0 , while the dimensions of multi-stress tensors are ∆ = 4n, with n ∈ N. In
this work, unless stated otherwise, we assume that ∆ϕ is not an integer. Therefore,
the double-trace and the stress-tensor contributions in the decomposition (2.4) do not
mix. The conformal block decomposition of the stress-tensor sector will be denoted
by
4n
∞
X
τ
−2∆ϕ
.
(2.13)
gT (τ ) = |τ |
Λn
β
n=1
The coefficients Λn may be computed exactly by solving the wave equation on the
planar AdS-Schwarzschild black hole, see [9, 11]. The first couple of terms in the
–8–
Λn
cΔϕ (-4)n (4 n)2 Δϕ -3
1.4
1.2
Δ=
1.0
Δ=
0.8
Δ=
0.6
Δ=
0.4
5
4
3
2
5
2
7
2
0.2
10
20
30
40
50
Figure 1. Ratio of the exact stress-tensor coefficients Λn to their asymptotics for different
values of ∆ϕ and α0 = 1 in d=4.
expansion read
1
gT (τ ) = 2∆ϕ
|τ |
π 4 ∆ϕ
40
4
τ
β
(2.14)
π 8 ∆ϕ 63∆4ϕ − 413∆3ϕ + 672∆2ϕ − 88∆ϕ + 144
+
201600(∆ϕ − 4)(∆ϕ − 3)(∆ϕ − 2)
!
8
τ
+ ... .
β
The most important property of Λn -s for our analysis is their behavior at large n.
The latter was computed in [11],
(4n)2∆ϕ −3
,
Λn ∼ Λ0n = c∆ϕ 4n
iπn
√1
e
2
c∆ϕ =
16α0 π∆ϕ (∆ϕ − 1)
∆ϕ
. (2.15)
5
sin(π∆ϕ ) Γ(2∆ϕ + 3/2)
Here, α0 = α0 (∆ϕ ) is a numerical coefficient that is very close to one for ∆ϕ which
is not too large. Figure 1 shows the ratio of the exact and asymptotic stress-tensor
coefficients for various values of ∆ϕ .
2.3
Sum over images
As we have mentioned, the aim of this section is to construct a tentative two-point
function gβ (τ ) that satisfies all the properties reviewed in Section 2.1 and whose
multi-stress-tensor content is given by the asymptotic formula (2.15). Since (2.15)
only agrees with actual holographic multi-stress-tensor coefficients at large n, the
resulting function gβ (τ ) should not be thought of as an exact holographic correlator,
but is expected to capture important features of the latter.
–9–
We will proceed in two stages. By resumming the asymptotic coefficients (2.15),
we will obtain a tentative stress-tensor sector4 of the two-point function, denoted
gT (τ ). The sum over multi-stress-tensor blocks has a finite radius of convergence, but
the function gT (τ ) may be analytically continued beyond the region of convergence
of the sum. Next, using the method of images we will generate the function
∞
X
1
images
gβ
(τ ) =
+ gT (|τ + mβ|) .
(2.16)
|τ + mβ|2∆ϕ
m=−∞
By construction, this function is KMS-invariant and decomposes into identity, doubletrace and stress-tensor sectors. Moreover, the stress-tensor sector of (2.16) receives
contributions only from the zeroth image m = 0 and is given by gT (τ ). However, as
we will see, the function gβimages (τ ) has singularities within the vertical strip, at
τ± =
1±i
β.
2
(2.17)
(Consequently, gβimages (τ ) also violates the boundedness condition discussed in Section
2.1). This brings us to the second stage of the construction, which is concerned with
canceling the singularities of gβimages (τ ) at (2.17).
To this end, one can uniquely isolate the singular part of gβimages (τ ) consisting of
the negative-power terms in the Laurent expansion at each of the singularities (2.17).
This allows us to express gβimages (τ ) as the sum of singular parts centered at (2.17)
and the remaining regular piece
images
images
gβimages (τ ) ≡ greg
(τ ) + gsing
(τ ) .
(2.18)
images
Note that greg
(τ ) is regular at τ± , but still contains the KMS poles at τ = 0, β.
Let K be the order of the pole of gβimages (τ ) at τ+ . By KMS-invariance, the pole at
τ− is of the same order and the Laurent coefficients in expansions around τ+ and τ−
coincide up to a sign. Therefore, the singular piece takes the form
images
gsing
(τ ) =
K X
n=1
(−1)n rn
rn
+
(τ − τ+ )n (τ − τ− )n
.
(2.19)
We would like to drop the singular piece from gβimages (τ ), but this would violate the
KMS condition, since (2.16) respects KMS symmetry, while (2.19) does not. Instead,
we shall subtract from (2.18) the singular part summed over thermal images:
gβ (τ ) ≡ gβimages (τ ) −
∞
X
images
gsing
(τ + mβ) + c ,
(2.20)
m=−∞
4
We will sometimes include the contribution of the identity operator in the stress-tensor sector,
and sometimes not – hopefully this will be clear from the formulas and the context.
– 10 –
where c is a constant defined below. This is our definition of the asymptotic model
for the thermal two-point function. Crucially, one shows that the second term in
(2.20) takes the form of a double-trace sector contribution – it is an even, KMSinvariant meromorphic function which is regular at points τ = 0, β. The asymptotic
model two-point function gβ (τ ) satisfies the KMS condition, has the stress-tensor
content (2.15) and is free of singularities on the vertical strip, except for the OPE
pole at τ = 0 and its KMS image at τ = β. In the following, we shall also verify a
weaker form of the boundedness condition (2.5), namely the absence of singularities
at Im τ → ∞. The constant c in (2.20) is chosen such that
lim gβ (τ ) = 0 .
Im τ →∞
(2.21)
This is a natural property to impose on a two-point function, since the one-point
function ⟨ϕ⟩β is expected to vanish in holography, see e.g. [73, 74] for recent discussion
of this topic. However, one may also consider other values of the constant c, which
we briefly discuss at the end of Section 2.4.
One might also ask whether (2.20) is the unique function satisfying the above
properties. Although we will not attempt a rigorous proof, there is substantial evidence supporting this. In later sections, we show that the same function arises from
the KMS sum rules of [25], solved by using Borel resummation.
We proceed to illustrate the above procedure explicitly on the example of the
scalar field of conformal dimension ∆ϕ = 3/2. The rest of this subsection is dedicated to the construction of gβimages (τ ), while the cancellation of singularities and
computation of gβ (τ ) is carried out in the next. Other half-integer dimensions are
discussed at the end of Section 2.4. From now on, we set β = 1.
For ∆ϕ = 3/2, the asymptotic multi-stress-tensor coefficients read
√
3
96 π
0
∆ϕ = : Λn = −
α0 (−4)n .
2
175
Summing the stress-tensor sector according to (2.13), we obtain
√
√
∞
96 πα0 −3 X √ − iπ 4n 384 πα0
τ
4
=
gT (τ ) = −
τ
2e τ
.
175
175 1 + 4τ 4
n=1
(2.22)
(2.23)
√
We have performed the infinite sum within its radius of convergence, |τ | ≤ 1/ 2.
We regard the right-hand side of (2.23) as the definition of gT (τ ) beyond the radius
of convergence of the sum. Next, we add the identity exchange contribution and
perform the sum over thermal images
√
∞
X
1
3
384 πα0 |τ + m|
images
(τ ) =
∆ϕ = : gβ
+
3
2
|τ
+
m|
175 (1 + 4(τ + m)4 )
m=−∞
√
96i πα0
1
1
GFF
= g∆ϕ =3/2 (τ ) −
−
. (2.24)
175
τ − 1+i
τ − 1−i
2
2
– 11 –
The two terms in the second line are image sums of the two terms in the first,
respectively. The first sum is simply the two-point function of the generalized free
field with ∆ϕ = 3/2. The second sum is exactly evaluated in terms of elementary
functions using the identity
∞
X
|τ + m|
1
.
=
4
2)
1
+
4(τ
+
m)
2
(1
−
2τ
+
2τ
m=−∞
(2.25)
By construction, (2.24) is KMS-invariant. Moreover, it can be seen that only the
zeroth image contributes to the multi-stress-tensor content of gβimages . Hence, this
multi-stress-tensor content is given by gT .
As already mentioned, we consider the function (2.24) inside the vertical strip. In
this domain there are four singularities: the OPE pole at τ = 0 and its KMS image
at τ = 1, together with two further simple poles at τ = (1 ± i)/2. The two terms in
(2.24) correspond to what we call regular and singular pieces in (2.18).
2.4
The asymptotic model
Having obtained gβimages (τ ) and identified its regular and singular parts, it remains
to perform the subtraction (2.20). To this end, we use the identity
∞ i X
1
1
1
1
=
−
.
1+i −
1−i
π(2iτ
+1)
π(2iτ
−1)
2π n=−∞ τ + n − 2
1+e
1+e
τ +n− 2
(2.26)
We shall denote the right hand side of (2.26) by h(τ ). This gives us the final result
for the asymptotic model
√
3
96i πα0
1
1
GFF
∆ϕ = : gβ (τ ) = g∆ϕ =3/2 (τ ) −
−
(2.27)
2
175
τ − 1+i
τ − 1−i
2
2
192π 3/2 α0
1
1
+
−
.
175
1 + eπ(2iτ +1) 1 + eπ(2iτ −1)
The expression (2.27) applies in the strip 0 ≤ Re τ ≤ 1. As advocated in our general
discussion, gβ (τ ) satisfies all of the properties listed in Section 2.1. In particular, it
is free of singularities apart from ones at τ = 0, 1 and bounded as Im τ → ∞. Note
that we may try to add to gβ (τ ) a finite linear combination of functions
fs (τ ) = cos(2πsτ ) ,
s ∈ N0 ,
(2.28)
which are KMS-invariant and decompose into double-trace conformal blocks.5 While
this would not change the analytic structure of (2.27), any such linear combination
5
For ∆ϕ = 3/2, multi-stress-tensor blocks are given by odd powers of τ and double-trace blocks
by even powers.
– 12 –
|gβ (τ)|
|gβ (τ)|
60
14
12
50
10
40
8
30
6
20
4
10
2
0
0.0
0.5
1.0
1.5
2.0
Im[τ]
0.2
0.4
0.6
0.8
1.0
Im[τ]
Figure 2. Modulus of gβ (τ ) along the line Re τ = 12 for ∆ϕ = 3/2 and ∆ϕ = 5/2
diverges as Im τ → ∞ and therefore violates the boundedness condition (2.5). This
is with the exception of the s = 0 case, that is, a constant function. The value of
this additive constant is fixed by the requirement (2.21), which is indeed satisfied by
(2.27). One property that is not automatically built in our construction of (2.27) is
the boundedness condition (2.5), which is stronger than the absence of singularities.
However, it can be verified that (2.5) is indeed satisfied. For instance, the plot on
the left in Figure 2 shows the behavior of |gβ (τ )| along the line Re τ = 1/2. One can
draw similar plots for other values of κ in (2.5) and observe that the boundedness
condition (2.5) always holds in the case ∆ϕ = 3/2. It is not necessarily the case for
other values of ∆ϕ .
The one-point coefficients of the asymptotic model can readily be computed. They
are given by
∆ϕ =
3
:
2
GFF
aAM
(2.29)
[ϕϕ]2∆ϕ +2m = a[ϕϕ]2∆ϕ +2m
√
π
192 πα0
⌊m/2⌋ m
m
2m+1 Li−2m (−e )
+
(−1)
2 + (−1) (2π)
,
175
(2m)!
for m ∈ N, and
3
∆ϕ = :
2
√
192
aAM
[ϕϕ]2∆ϕ = 2ζ(3) +
π π α0 1 − 2π tanh
,
175
2
(2.30)
for m = 0. Here, ⌊x⌋ denotes the integer part of x and Lis (x) the polylogarithm.
Other values of ∆ϕ We end this subsection by discussing the above construction
for other values of the conformal dimension ∆ϕ . Whenever the latter takes the form
∆ϕ = p + 1/2, with p ∈ N, the asymptotic model can be constructed explicitly.
Firstly, resumming the asymptotic stress-tensor data (2.15), we obtain the function
gT (τ ), which is a rational function. The sum over thermal images gβimages (τ ) is then
given by a combination of rational and polygamma functions. Considering it on
the vertical strip, this function has KMS poles at τ = 0, 1, as well as two poles at
(1 ± i)/2. The order of the latter depends on the value of ∆ϕ and equals 2p − 1.
Finally, in order to pass to the asymptotic model we perform the sum (2.19) - this is
– 13 –
given by a linear combination of powers {h0 , h1 , h2 , . . . , h2p−1 } of the function h(τ )
given in (2.26). 6
Let us briefly discuss the possibility of adding a constant to the function gβ (τ ).
We have fixed this constant by demanding that the two-point function vanishes as
Im τ → ∞, (2.21). Another property of the two-point function that is sensitive to
the value of this constant is the boundedness condition (2.5), and, with the above
choice, it may be violated depending on the value of ∆ϕ . For example, this is the case
for ∆ϕ = 5/2, as shown in right plot of Figure 2. Indeed, in contrast to the plot on
the left, the maximum of the function is above the value at Im τ = 0. One may also
choose an additive constant such that (2.5) is satisfied. However, it is expected that
the condition (2.5) is sensitive to fine details of the stress-tensor OPE coefficients
(because it is a property of the exact thermal state), and therefore it is not captured
by the asymptotic model.
2.5
Refinements of the model
While it is not a subject that we will pursue further in the present work, in this
subsection we briefly discuss the important question of how one might try to systematically refine our asymptotic model and make it closer to the actual holographic
two-point function.
Until now our analysis only made use of the asymptotic form of multi-stresstensor one-point coefficients (2.15). It is natural to expect that by including more
information about the exact values of these coefficients into the model, one gets closer
to the holographic two-point function. Here we discuss two possible approaches
to imputing further multi-stress-tensor data. The first one, similar to what was
considered in [10], consists of replacing, one by one, the low-lying coefficients Λ0n by
their exact values Λn . After replacing a finite number N of coefficients, one may try
to construct the corresponding asymptotic model along the same lines as above. Let
us denote the stress-tensor sector that we start with by
(N )
gT (τ ) = gT (τ ) + |τ |−2∆ϕ
N
X
δΛn τ 4n ,
δΛn = Λn − Λ0n .
(2.31)
n=1
The first step of the construction consists in taking the sum over thermal images.
However, we face an immediate difficulty in the fact that the sum over images of the
6
An alternative point of view of what we have described is the following. After obtaining the
function gβimages (τ ), we may try to cancel its poles by a linear combination of fs (τ ). Since fs (τ ) have
no singularities, we need an infinite linear combination. In particular, the function (2.26) arises as
the formal sum
∞
X
h(τ ) =
(−1)s eπ|s| cos(2πsτ ) .
s=−∞
For half-integer ∆ϕ , there is a unique finite linear combination of powers of h that can be added to
gβimages (τ ) in order to cancel singularities within the vertical strip.
– 14 –
second term in (2.31) diverges whenever 4N > 2∆ϕ . Following [10], one can make
the sum finite using ζ-function regularization. To give more details, consider the
generically divergent (for 2n > ∆ϕ ) sum over images of a single conformal block
Bn = δΛn
∞
X
|τ + m|4n−2∆ϕ .
(2.32)
m=−∞
To make sense of the sum, one uses the analytic continuation of the formula
∞
X
|τ + m|γ = ζH (−γ, 1 + τ ) ,
(2.33)
m=1
that is valid for γ < −1. The right-hand side of (2.33) is well defined for all γ ̸= −1.
Plugging this into (2.32), we get that each OPE coefficient correction δΛn contributes
GFF
reg
(τ ) , (2.34)
Bn = δΛn ζH (2∆ϕ − 4n, τ ) + ζH (2∆ϕ − 4n, 1 − τ ) ≡ δΛn g∆
ϕ −2n
to the sum over images. Therefore, we obtain
images,(N )
gβ
(τ ) = gβimages (τ ) +
N
X
GFF
δΛn g∆
(τ ) .
ϕ −2n
(2.35)
n=1
The function added to gβimages (τ ) is entire for half-integer values of ∆ϕ that we consider
images,(N )
in this work. Therefore, the singular part of gβ
(τ ) is the same as that of
images
gβ
(τ ). This leads to the refinement of the asymptotic model
(N )
gβ (τ ) = gβ (τ ) +
N
X
GFF
δΛn g∆
(τ ) .
ϕ −2n
(2.36)
n=1
Let us argue that (2.36) is actually not a good way of adjusting the asymptotic model.
Assume that ∆ϕ = p + 1/2, where p is a positive integer. Then, for 2n > ∆ϕ , the
GFF
function g∆
(τ ) is given in terms of the Bernoulli polynomials of degree 4n − 2p
ϕ −2n
GFF
g∆
(τ ) = ζH (2∆ϕ − 4n, τ ) + ζH (2∆ϕ − 4n, 1 − τ )
ϕ −2n
=−
(2.37)
1
B4n+2∆ϕ +1 (τ ) + B4n+2∆ϕ +1 (1 − τ ) ,
4n + 2∆ϕ + 1
and hence it diverges as Im τ → ∞. Therefore, for a large enough N , the corrections
(2.36) violate the analytic structure that we require of the two-point function. Furthermore, as will be discussed in Section 3.6, the double-trace coefficients associated
with the refined asymptotic model do not converge to a limiting value as N → ∞,
see in particular the equations (3.66) and (3.68).
Having ruled out (2.36) as a systematic procedure that can be extended infinitely
in N , let us note that a certain finite number of terms in (2.36) may be added while
– 15 –
preserving the analytic structure of the two-point function. To see this, note that for
GFF
2n < ∆ϕ − 2, the function g∆
(τ ) decays as Im τ → ∞ and is free from poles on
ϕ −2n
the vertical strip. It follows that adding the finite number of corrections (2.36) with
n = 1, 2, . . . , ⌊p/2⌋ − 1 does not alter the analytic structure of gβ (τ ). The resulting
correlator with N = ⌊p/2⌋−1 has the exact one-point coefficients Λ1 , . . . , Λ⌊p/2⌋−1 for
the first ⌊p/2⌋ − 1 composites of the stress-tensor and asymptotic coefficients (2.15)
for the higher composites.
To make progress beyond the above and obtain a refinement scheme that would
converge to the actual holographic two-point function, one may try to modify all
multi-stress-tensor coefficients at once in a large n-expansion such as
]
Λ[M
= Λ0n
n
M
X
αs
s=0
ns
,
n→∞.
(2.38)
It remains to be seen what precise form of the large n expansion should be used
(e.g. one can try expanding in powers of n−a for any a > 0) and if the resulting
correction procedure converges. This task is challenging primarily because the exact
multi-stress coefficients Λn are computed recursively with increasing n, making the
large-n data not easily accessible. We defer the discussion of large-n refinements of
the asymptotic model to the concluding section.
3
Sum rules
In this section, we analyze the sum rules arising from the KMS condition and show
how they can be solved. After reviewing the sum rules, which take the form of an
infinite system of linear equations, in the first subsection, we go on to derive the
homogeneous and a particular solution in the next two subsections. The particular solution requires regularization, which is performed using the Borel summation
method. We proceed to apply the procedure to the GFF sum rules, recovering the
correct OPE coefficients (2.11). This is followed by solving the sum rules that arise
from the asymptotic multi-stress-tensor CFT data (2.15) (asymptotic holographic
sum rules for short) and showing that the solution coincides with the results given
by the asymptotic model of Section 2. The section ends with a discussion of how these
solutions are affected by refinements of the input CFT data (the multi-stress-tensor
one-point coefficients).
3.1
Review: KMS sum rules
The KMS condition ensures that the thermal correlation functions exhibit periodicity
along the thermal circle. Any two-point function at finite temperature in Euclidean
time satisfies (2.2). Let us review the derivation of the explicit sum rules out of this
– 16 –
condition for the simplest case of zero spatial separation between the operators7 . We
start from the conformal block decomposition of the two-point function (2.4). By
substituting this expansion into the KMS equation (2.2) and applying the binomial
theorem in the form
∆−2∆ϕ X
∆−2∆ϕ −k̃
β
Γ(∆ − 2∆ϕ + 1)
β
± δτ
=
(±δτ )k̃ , (3.1)
2
Γ(∆ − 2∆ϕ − k̃ + 1)Γ(k̃ + 1) 2
k̃=0
we obtain:
X a∆ Γ(∆ − 2∆ϕ + 1)
∆
2∆ Γ(∆ − 2∆ϕ − 2k)
= 0,
k ∈ N0 ,
(3.2)
where we used (2.2), which ensures that only odd values of k̃ = 2k +1 yield nontrivial
condition (3.2). The contribution from the identity operator 1 can be separated from
the rest of the sum, leading to the sum rules
X a∆ Γ(∆ − 2∆ϕ + 1)
∆
2∆ Γ(∆ − 2∆ϕ − 2k)
=
Γ(2∆ϕ + 2k + 1)
,
Γ(2∆ϕ )
k ∈ N0 .
(3.3)
An alternative perspective on equations (3.3) is that expanding the two-point function around the KMS-symmetric point, τ = β2 , is equivalent to imposing the conditions:
∂t2k+1 g(τ, 0)|τ = β = 0 ,
k ∈ N0 .
(3.4)
2
The sum rules (3.3) constitute an infinite set of linear equations. Let us note that
numerical approaches to solving (3.3) based on truncating the system meet with
difficulties - such approaches are best suited for systems where coefficients a∆ are
assumed to be positive. However, one cannot justify this assumption in generic
CFTs. In this work, we focus instead on analytic approaches, showing that despite
the complexity of the sum rules, some solutions can be found exactly. Specifically,
we will be interested in the sum rules in the context of holographic CFTs, where
the ϕ × ϕ OPE, in the leading order, consists solely of the identity, double-trace and
multi-stress-tensor operators
ϕ × ϕ ∼ 1 + [ϕϕ]2∆ϕ +2m + [Tµν ]n .
(3.5)
In this case, the sum rules take the form
∞
Γ(2∆ϕ + 2k + 1) X Λn Γ(dn − 2∆ϕ + 1)
=
−
.
22∆ϕ +2m Γ(2(m − k))
Γ(2∆ϕ )
2dn Γ(dn − 2∆ϕ − 2k)
n=1
m>k
(3.6)
Although the stress-tensor part in (3.6) involves gamma functions that can develop
poles for certain values of n, all such singularities cancel between the numerator
X a[ϕϕ] 2∆ϕ +2m Γ(2m + 1)
7
See [25] for a similar derivation in the more general case of non-zero spatial separation.
– 17 –
and the denominator. As a result, each term in the series remains finite and the
overall sum rules are well-defined. The coefficients of the double-trace operators are
regarded as unknown variables, whereas those corresponding to multi-stress-tensor
operators are taken to be known. First, we turn to the solution to the homogeneous
system of equations, where the right-hand side is equal to zero.
3.2
Homogeneous equations
In this subsection, we derive the solutions to the homogeneous equations
X a[ϕϕ] 2∆ϕ +2m Γ(2m + 1)
m>k
22∆ϕ +2m Γ(2(m − k))
= 0,
k ∈ N0 .
(3.7)
As stated in [75] for general systems of this kind, the general solution to (3.7) can
be expressed in the form
a[ϕϕ]2∆ϕ +2m =
a(−1)m
,
S m (2m)!
m ∈ N,
(3.8)
where a is an arbitrary real number and S is a constant characteristic number. In
the case at hand, S is determined from the non-linear equation
∞
X
(−1)m
=0.
(4S)m (2m − 1)!
m=1
(3.9)
In order to solve (3.9), we can construct a characteristic function
f (x) =
∞
X
(−1)m xm
m=1
(2m − 1)!
√
√ = − x sin x ,
(3.10)
and find its zeros, f (xn ) = 0,
xn = (πn)2 ,
n ∈ N0 .
(3.11)
1
The corresponding Sn = (2πn)
2 lead to the solutions of the homogeneous system of
equations (3.7)
a(−1)m (2πn)2m
a[ϕϕ]2∆ϕ +2m =
,
m ∈ N,
(3.12)
(2m)!
for any positive integer n and a real number a. Note that the homogeneous solution
(3.12) corresponds to the freedom to add the function a cos(2πnτ ) to the two-point
function gβ (τ ). This is in accord with our discussion around (2.28) in the previous
section.
– 18 –
3.3
Particular solution
Having solved the homogeneous problem, we will now develop a method to solve the
sum rule equations for a general right hand side, denoted by Fk . The corresponding
equation is given by
X a[ϕϕ] 2∆ϕ +2m Γ(2m + 1)
22∆ϕ +2m Γ(2(m − k))
m>k
k ∈ N0 .
= Fk ,
(3.13)
The key to solving this linear system of equations is to recognize that the matrix is
an upper-triangular Toeplitz matrix, allowing for an efficient solution via backward
substitution. We will first outline the method by simplifying the equations and
deriving a recursive procedure to obtain the solution. Then we will solve the resulting
recursion relations. We begin by expressing the system in the standard form:
∞
X
Mm am+k = Fk ,
k ∈ N0 ,
(3.14)
m=1
where we denoted
am =
(2m)! a[ϕϕ] 2∆ϕ +2m
and Mm =
22∆ϕ +2m
1
.
(2m − 1)!
(3.15)
To determine a1 , add the second equation of (3.14), multiplied by p2 = −M2 , to the
first equation. Subsequently, the third equation, multiplied by p3 = −M3 − p2 M2 , is
added to the first equation. Repeating this procedure iteratively yields the expression
a1 =
∞
X
pk+1 Fk ,
(3.16)
k=0
where we define p1 = 1 and for k > 2 the pk satisfy recurrence relations
pk = −Mk −
k−1
X
Mk−n+1 pn .
(3.17)
n=2
We can repeat this procedure for any row, which leads to
am =
∞
X
pk+1 Fm+k−1 ,
m∈N.
(3.18)
k=0
Now we proceed to solve the recurrence relation (3.17). Constructing two generating
functions
P (x) =
∞
X
k=2
k
pk x ,
M (x) =
∞
X
Mk xk =
k=2
– 19 –
√
x sinh
√ x − x,
(3.19)
and switching the summation limits in (3.17) with the change of variables, we can
write the identity
1
p2 x2 + P (x) = −M (x) − M2 x2 − P (x)M (x) .
x
(3.20)
Rearranging, this gives
P (x) = −
M (x)
1 + Mx(x)
=x
√
x csch
By expanding P (x) into a series, we find
2 1 − 22k−3 B2k−2
pk =
,
(2k − 2)!
√ x −1 .
(3.21)
k ∈ N,
(3.22)
where Bk denote the Bernoulli numbers. In the following, we will find it convenient to
use an alternative expression for the coefficients pk by writing the Bernoulli numbers
in terms of the Riemann zeta function,
2
k−1 ζ(2k − 2)
pk = 2(−1)
1 − k−1 ,
k∈N.
(3.23)
π 2k−2
4
In particular, for k = 1, one uses the analytically continued value ζ(0) = − 21 . Putting
everything together, we arrive at the final result
∞
22∆ϕ +2m X
ζ(2k)
a[ϕϕ]2∆ϕ +2m =
(−1)k 2k 2 − 41−k Fk+m−1 ,
(2m)! k=0
π
m∈N.
(3.24)
Note that for the sum to converge, Fk must satisfy the following asymptotic condition:
Fk = o(π 2k ) as k → ∞ .
(3.25)
This is particularly significant, as the structure of the KMS sum rules in fact implies
the factorial growth Fk ∼ (2k)!, rendering (3.24) ill-defined. To make sense of the
divergent sum, one can employ the Borel summation8 , which provides a prescription
for handling such asymptotic series. To this end, consider the generating series
associated with the solution (3.18)
am (z) ≡
∞
X
pk+1 Fm+k−1 z k .
(3.26)
k=0
In order to regularize the series, we perform the generalized Borel transform
2
Bm (t z) =
∞
X
pk+1 Fm+k−1 z k
k=0
(2k)!
8
t2k .
(3.27)
More precisely, a series whose partial sums are bounded by (nk)! C k+1 for some constant C > 0
and integer n is said to be summable in the sense of Mittag-Leffler.
– 20 –
Then the Borel summation of am is given by
Z ∞
e−t Bm (t2 ) dt .
am = am (1) ≡
(3.28)
0
We have slightly abused the notation in denoting both the divergent series (3.18)
and its resummation (3.28) by the same letter. Using the rule (3.28) and the explicit
form of the coefficients pk , the particular solution for the general sum rules (3.13)
can be expressed as
Z
∞
22∆ϕ +2m ∞ −t X (−1)k ζ(2k)
1−k
a[ϕϕ]2∆ϕ +2m =
2
−
4
Fk+m−1 t2k dt , m ∈ N .
e
2k
(2m)! 0
(2k)! π
k=0
(3.29)
This is the main general formula of the present section. In the next two subsections,
we show how (3.29) can be explicitly evaluated in GFF and asymptotic holography.
3.4
Reproducing GFF
To illustrate one of the possible applications of the solution (3.29), we consider the
case of the GFF, where the right hand side is given by
FkGFF =
Γ(2∆ϕ + 2k + 1)
.
Γ(2∆ϕ )
(3.30)
We work with the notation introduced in (3.15). According to (3.23) and (3.27), the
GFF 2
(t ) is
generalized Borel transform Bm
GFF 2
Bm
(t ) =
∞
X
k=0
(−1)k
Γ(2k + 2m + 2∆ϕ − 1) 2k
ζ(2k)
t .
2 − 41−k
2k
π
Γ(2∆ϕ )Γ(2k + 1)
(3.31)
To effectively evaluate this sum, we make use of the following identity:
∞
X
ζ(2k)
(−1)k 2k 2 − 41−k t2k = t csch(t) .
π
k=0
(3.32)
This representation allows for a simpler expression for the Borel transform, which
can be re-written as
1
2m+2∆ϕ −2
GFF 2
Bm
(t ) =
∂t
t2m+2∆ϕ −1 csch(t) .
(3.33)
Γ(2∆ϕ )
Therefore, in the case at hand, the coefficients am are evaluated to
Z ∞
1
2m+2∆ϕ −2
GFF
e−t ∂t
t2m+2∆ϕ −1 csch(t) dt
a∆ ϕ , m =
Γ(2∆ϕ ) 0
Z ∞
1
=
e−t t2m+2∆ϕ −1 csch(t)dt
Γ(2∆ϕ ) 0
= 2−2m−2∆ϕ +1 ζ(2m + 2∆ϕ )
– 21 –
Γ(2m + 2∆ϕ )
,
Γ(2∆ϕ )
(3.34)
where, to get to the second line, we used integration by parts and the facts that
t csch(t) − 1 ∼ t2 as t → 0 and csch(t) is bounded for t → ∞. Plugging this into the
formula for the double-trace coefficients (3.15), we find
aGFF
[ϕϕ]2∆ϕ +2m =
2m+2∆ϕ
aGFF
∆ϕ , m 2
Γ(2m + 1)
=
2(2m + 1)2∆ϕ −1
ζ(2∆ϕ + 2m) .
Γ(2∆ϕ )
(3.35)
The result coincides with the thermal one-point coefficients of the GFF theory, (2.11).
3.5
Application to the asymptotic model
Now we apply the solution (3.29) to the holographic sum rules (3.6). Due to linearity,
it suffices to solve equations (3.6) with the right-hand side consisting only of the
second, multi-stress-tensor term, as the contribution from the identity is given by
the GFF solution. Using the notation introduced in (3.15), the multi-stress-tensor
contribution is expressed as
Fk = −
∞
X
Λn Γ(4n − 2∆ϕ + 1)
n=1
24n Γ(4n − 2k − 2∆ϕ )
.
(3.36)
We consider the case where ∆ϕ is a half-integer. In particular, we analyze the two
distinct scenarios separately:
∆ϕ = 2r −
1
2
and ∆ϕ = 2r +
1
,
2
r∈N.
(3.37)
As an example, consider first the simplest case, ∆ϕ = 23 . Explicit expressions for
Fk are not available for exact holography, where one knows only a finite number of
coefficients Λn . However, if the coefficients are replaced by their large n asymptotics,
Fk can be computed. We will denote these numbers by F̂k . To facilitate the analysis,
it is convenient to analyze the coefficients F̂k in two separate cases
∆ϕ =
3
:
2
∞
1 X Λ0n+k+1 Γ(4n + 4k + 2)
F̂2k = − 4(k+1)
,
4n
2
2
Γ(4n
+
1)
n=0
1
∞
X
Λ0n+k+1 Γ(4n + 4k + 2)
F̂2k−1 = − 4(k+1)
2
n=0
24n
Γ(4n + 3)
.
(3.38)
(3.39)
Explicit expressions for F̂2k and F̂2k−1 are given in Appendix B.2, see equation (B.8).
As we have seen, the solution for the double-trace coefficients is given by an integral
over the Borel-transformed sum
∞
X
F̂k+m−1 2k
ζ(2k)
B̂m (t ) =
(−1)k 2k 2 − 41−k
t ,
π
Γ(2k + 1)
k=0
2
We consider two cases, B̂2m (t2 ) and B̂2m−1 (t2 ), in turn.
– 22 –
m = 0, 1, 2, . . . .
(3.40)
Even B̂2m
The sum (3.40) becomes
2
B̂2m (t ) =
∞
X
ζ(4k)
k=0
−
π 4k
2 − 41−2k
∞
X
ζ(4k − 2)
k=1
π 4k−2
F̂2k+2m−1 4k
t
Γ(4k + 1)
2 − 42−2k
F̂2k+2m−2 4k−2
t
.
Γ(4k − 1)
(3.41)
We will analyze each of the two sums separately. Let us denote the first sum by B1 .
It takes the form
∞ X
∞
0
X
ζ(4k)
1−2k Γ(4n + 4k + 4m + 2) Λn+k+m+1 4k
B1 = −
2
−
4
t . (3.42)
4k
4n+4k+4m+4
π
Γ(4k
+
1)Γ(4n
+
3)
2
k=0 n=0
We now use the explicit form of the asymptotic Λ0n = c3/2 (−4)n from (2.22), as this
greatly simplifies the analysis. Then, (3.42) becomes
∞
∞
X
c3/2
(−4)−n X ζ(4k)
2 Γ(4n + 4k + 4m + 2) 4k
−m
(−4)
1 − 2k
t .
B1 =
4 )k
2
Γ(4n
+
3)
(−4π
4
Γ(4k
+
1)
n=0
k=0
(3.43)
It is convenient to define
∞
X
ζ(4k)
2
S1 (t) ≡
1 − 2k t4k .
4 )k
(−4π
4
k=0
(3.44)
A closed-form expression for S1 (t) can be found in Appendix B.2, see (B.9). In terms
of this function, the first line in B̂2m (t2 ) is
∞
X
(−4)−n 4n+4m+1 4n+4m+1
−m−1
B1 = −2c3/2 (−4)
∂
(t
S1 (t)) .
(3.45)
Γ(4n + 3) t
n=0
The second line in (3.41) can be written similarly as
∞
X
(−4)−n 4n+4m−1 4n+4m−1
−m
∂
(t
S2 (t)) .
B2 = −2c3/2 (−4)
Γ(4n + 1) t
n=0
(3.46)
For more details, see Appendix B.2. In particular, the function S2 (t) is given in
(B.10). By applying the Laplace transform, integrating by parts and summing over
n, we obtain the solution
Z ∞
â2m =
e−t B̂2m (t2 )dt
(3.47)
0
Z ∞X
∞
(−4)−n−1 4n+4m+1
(−4)−n 4n+4m−1
e
t
S1 (t) −
t
S2 (t) dt
Γ(4n + 3)
Γ(4n + 1)
0
n=0
!
Z ∞
−c3/2
(4m)!
1 4m+1
−t 4m
π
=
e t K1 (t)dt = c3/2
− π
Li−4m (−e ) .
(−4)m+1 0
(−4)m+1 2
−2c3/2
=
(−4)m
−t
– 23 –
Here, we have denoted
sinh t − sin t
.
(3.48)
cosh t − cos t
To do the final integral, we made use of the identity (B.11) and further simplified
the expressions. The answer for the double-trace coefficients (recalling (3.15)) reads
!
√
4m+1
3
(2π)
192 π
∆ϕ = : â[ϕϕ]2∆ϕ +4m =
(−4)m +
Li−4m (−eπ ) .
(3.49)
2
175
(4m)!
K1 (t) =
Remarkably, we recover precisely the coefficients of the asymptotic model. In more
detail, recall that F̂k in the discussion above correspond to the asymptotic stresstensor contribution to the sum rules. These are to be added to the identity contribution, which yields the GFF OPE coefficients. In accord with this, we find
3
GFF
∆ϕ = : â[ϕϕ]4m+3 = aAM
(3.50)
[ϕϕ]4m+3 − a[ϕϕ]4m+3 ,
2
GFF
where aAM
[ϕϕ]4m+3 and a[ϕϕ]4m+3 are given by (2.29) and (2.11), respectively.
Odd B̂2m+1
The procedure is very similar as for the even coefficients B̂2m , so we will be brief. In
this case, the sum (3.40) becomes
2
B̂2m+1 (t ) =
∞
X
ζ(4k)
k=0
−
π 4k
2 − 41−2k
∞
X
ζ(4k − 2)
k=1
π 4k−2
F̂2k+2m 4k
t
Γ(4k + 1)
2 − 42−2k
F̂2k+2m−1 4k−2
t
.
Γ(4k − 1)
(3.51)
Following the same steps as above, the solution for the remaining double-trace coefficients can be expressed as
Z ∞
Z ∞
1
−t
2
−m−1
â2m+1 =
e B̂2m+1 (t )dt = − (−4)
e−t t4m+2 K2 (t)dt ,
(3.52)
2
0
0
where
sinh t + sin t
K2 (t) =
.
(3.53)
cosh t − cos t
Evaluating the integral by means of (B.12) and plugging back into the relation (3.15),
we obtain
!
√
4m+3
96 π
2(2π)
3
(−4)m+1 +
Li−4m−2 (−eπ ) . (3.54)
∆ϕ = : â[ϕϕ]2∆ϕ +4m+2 = −
2
175
(4m + 2)!
Again, this exactly reproduces the asymptotic model, i.e.
3
GFF
∆ϕ = : â[ϕϕ]4m+5 = aAM
[ϕϕ]4m+5 − a[ϕϕ]4m+5 ,
2
GFF
with aAM
[ϕϕ]4m+5 and a[ϕϕ]4m+5 given in (2.29) and (2.11).
– 24 –
(3.55)
Solutions for other external scaling dimensions
Having shown how the asymptotic holographic sum rules are solved in the case of
conformal dimension ∆ϕ = 3/2, let us briefly discuss other values of ∆ϕ . In Appendix
B.1 we show how to compute the double-trace coefficients a[ϕϕ]2∆ϕ +2m when the stresstensor coefficients are given by (2.15) with ∆ϕ = 5/2. These again reproduce the
asymptotic model of the same ∆ϕ , that is detailed in Appendix A.1.
For general half-integer dimensions, the sum rules may again be solved explicitly.
In the case ∆ϕ = 2r − 12 , r > 1, we obtain the solution9
−m−5r+4
â2m = −c∆ϕ (−4)
4r−4
X
(1)
si,r I4m+i +
i=0
r−1
X
Λ0
i=1
i
4i
2
aGFF
∆ϕ −2i, 2m ,
4r−4
r−1
X
X
c∆ ϕ
Λ0i GFF
(2)
−m−5r+4
si,r I4m+2+i +
â2m+1 = −
(−4)
a
,
2
24i ∆ϕ −2i, 2m+1
i=0
i=1
(3.56)
(3.57)
while for the half-integer dimensions of the form ∆ϕ = 2r + 12 , r > 0, the solution
reads
â2m = −
r
4r−2
X
X
c∆ ϕ
Λ0i GFF
(2)
(−4)−m−5r+2
a
,
si,r+ 1 I4m+i +
4i ∆ϕ −2i, 2m
2
2
2
i=1
i=0
â2m+1 = −c∆ϕ (−4)−m−5r+1
4r−2
X
(1)
si,r+ 1 I4m+2+i +
2
i=0
r
X
Λ0
i
24i
i=1
aGFF
∆ϕ −2i, 2m+1 .
(3.58)
(3.59)
Here, the si,r denote the coefficients of the polynomial defined by the expression
4r−4
(t∂t )
4r−4
e−t X
e−t
= 4r−2
si,r ti .
t4r−2
t
i=0
They may be extracted explicitly via
−t
1 i t 4r−2
4r−4 e
si,r = ∂t e t
.
(t∂t )
i!
t4r−2 t=0
(1)
(2)
(3.60)
(3.61)
On the other hand, In and In are defined in (B.11) and (B.12). Let us note that
(i)
(3.56)–(3.59) typically consist of two types of terms, the first that involves In and
the second involving the coefficients aGFF
∆ϕ −2i, m . In the example ∆ϕ = 3/2 that we
detailed above, only the first kind of terms is present. For generic half-integer ∆ϕ ,
special care needs to be taken when gamma functions in the sum rules (3.6) exhibit
singularities (as we have mentioned, the poles cancel between the numerator and the
denominator in (3.6)). The proper treatment of these terms in the sum rules gives
9
Note that a[ϕϕ]2∆ϕ is not fixed by the KMS condition.
– 25 –
rise to the second type of contributions in (3.56)–(3.59). To see this explicitly in
the simplest nontrivial case, we refer to the example with ∆ϕ = 5/2 discussed in
Appendix B.1.
Formulas (3.49) and (3.54), as well as their generalizations (3.56)-(3.59) are some
of the main results of the present section. Notice that we have found that the particular solution (3.29), without adding to it any homogeneous piece (3.12), reproduces
the asymptotic model of Section 2. The same was true in the GFF example, where
(3.29) gives the exact GFF double-trace coefficients. From these facts, it is natural
to expect that in any theory, the particular solution (3.29) to the KMS sum rules is
the one among infinitely many solutions that gives the physical OPE coefficients.
3.6
Application to the refined model
In this subsection, we briefly discuss the question: ‘What happens to solutions to
sum rules (3.6) if one adjusts a finite number N of stress-tensor OPE coefficients?’
By linearity, we may take the solution to the asymptotic sum rules obtained above
and add to it a solution to the sum rules with Fk given by a finite version of the sum
(3.36), with the replacement Λn → δΛn = Λn − Λ0n for n = 1, . . . , N and Λn → 0
for n > N . To emphasize their nature as a correction, let us relabel the resulting
coefficients as Fk → δFk . Thus, we consider the sum rules with
δFk = −
N
X
δΛn Γ(4n − 2∆ϕ + 1)
24n Γ(4n − 2k − 2∆ϕ )
n=1
≡
N
X
(n)
δFk
.
(3.62)
n=1
Again by linearity, the sum rules can be solved for each n individually. In the case
where the scaling dimension of the field is given by ∆ϕ = p + 12 , with p ∈ N, we
distinguish between two regimes: (i) 2n < ∆ϕ , and (ii) 2n > ∆ϕ . In the first regime,
both the numerator and the denominator in (3.62) are singular, but we may recast
(n)
δFk in the form which is well-defined
2n < ∆ϕ :
(n)
δFk
=
δΛn Γ(2k − 4n + 2∆ϕ + 1)
.
24n
Γ(2∆ϕ − 4n)
(3.63)
Up to an overall constant, the resulting sum rules coincide with those of the GFF
model with conformal dimension ∆ϕ − 2n. This term contributes to the double-trace
coefficients as
2n < ∆ϕ :
−2m−2∆ϕ +1
δΛn ζ(2m + 2∆ϕ − 4n)(2∆ϕ − 4n)2m .
δa(n)
m = 2
(n)
(3.64)
For the second regime 2n > ∆ϕ , we have δFk = 0 for k > 2n − p − 1, owing to
the singularity of the gamma function appearing in the denominator in (3.62). Resummation methods are not required for the remaining finite system, as the solution
– 26 –
given by (3.24) is already convergent. Thus, in this regime the corrections to the
coefficients are given by
2n−m−p
2n > ∆ϕ :
δa(n)
m =
X
(−1)k
k=0
δa(n)
m = 0,
(n)
ζ(2k)
2 − 41−k δFk+m−1 ,
2k
π
m ≤ 2n − p ,
m > 2n − p .
(3.65)
Putting (3.64) and (3.65) together, the solution to the sum rules (3.62) takes the
form
⌊∆ϕ /2⌋
δam =
X
2−2m−2∆ϕ +1 δΛn ζ(2m + 2∆ϕ − 4n)(2∆ϕ − 4n)2m
n=1
+
N
X
2n−m−p
n=⌊∆ϕ /2⌋+1
k=0
X
(−1)k
(n)
ζ(2k)
1−k
2
−
4
δFk+m−1 ,
π 2k
(3.66)
where, in the second line, whenever 2n−m−p < 0, the sum is simply set to zero. If we
express this result in terms of the original double-trace coefficients a[ϕϕ]2∆ϕ +2m , using
the relation (3.15), and subsequently sum over them with the corresponding powers
τ 2m , we recover the two-point function presented in equation (2.36). This result is
identical to the refined asymptotic model.10 Intuitively, in the limit N ≫ 1, where
a large number of stress-tensor coefficients receive corrections, one might expect the
double-trace coefficients to approach their true holographic values. However, this
expectation is not fulfilled. Let us examine this more closely. First, note that for
(n)
fixed n > ⌊∆ϕ /2⌋, the function δFk exhibits factorial growth of the form (2k)!,
which arises from the presence of Γ(4n − 2k − 2∆ϕ ) in its denominator (see (3.62)).
As a result, within the sum over k in (3.66), the dominant contribution arises from
the term with the largest value of k = 2n − m − p, and
(n)
(n)
δFk+m−1
k=2n−m−p
= δF2n−p−1 = −
δΛn
Γ(4n − 2∆ϕ + 1) ,
24n
2n > ∆ϕ .
(3.67)
On the other hand, the difference |δΛn | = |Λn −Λ0n | is expected to grow exponentially
as ∼ 4n . In turn, this implies that when δam is considered as a function of N ≫ 1,
the sum in the first line of (3.66) can be treated as a constant, while the dominant
contribution in the sum in the second line arises from the term with the largest n,
that is,
ζ(2N )
δam ∼ −2δΛN (−1)N
Γ(4N − 2∆ϕ + 1) ∼ (4N )! .
(3.68)
(4π)2N
We conclude that the asymptotic model refined in the above way is not well-defined
in the limit N → ∞ as the coefficients δam do not converge to a finite limiting value.
10
For a detailed discussion of the analytic properties of the resulting two-point function, see
Section 2.5.
– 27 –
Furthermore, adjusting a finite number of stress-tensor coefficients and subsequently
applying Borel regularization to (3.66) will not lead to a meaningful result in the
limit N → ∞, as
Z ∞
e
lim
N →∞
−t
0
N
(n)
X
δam
n=1
(4n)!
t
4n
= lim
N Z ∞
X
N →∞
n=1
0
e
(n)
−t δam 4n
(4n)!
t
=
N
X
δa(n)
m ,
(3.69)
n=1
diverges. Hence, a more promising way to approach the actual holographic correlator
is to consider subleading large n corrections to the asymptotic model, as discussed
at the end of Section 2.
4
Discussion
In this work, we have constructed models of holographic thermal two-point functions
that capture the asymptotic behavior of OPE coefficients of composites of the stress
tensor. As an input, we took the large n asymptotics of thermal one-point coefficients
of operators of the schematic form [Tµν ]n , determined in [11], and as an output
obtained one-point coefficients of double trace-operators [ϕϕ]n,ℓ which constitute the
rest of the ϕ × ϕ OPE at large CT . We have explored two methods, both of which
rely on the KMS invariance of the two-point function. The first approach involves
summing over thermal images of the stress-tensor sector and subsequent cancellation
of the bouncing singularities by double-trace terms. The second method involves
solving the KMS sum rules directly. This requires Borel resummation in order to
regularize divergent formal solutions. Both approaches produce the same results.
In Section 2.1, we listed expected properties of thermal two-point functions, all of
which are satisfied by our asymptotic model.11 Let us note that that the model does
not satisfy the property discussed in [25] in the context of thermal CFT Tauberian
theorems. Namely, one-point coefficients of the model do not become sign-definite
at asymptotically large ∆. This is clear from our input data, as already the multistress-tensor coefficients are asymptotically sign-alternating. We conclude that signdefiniteness does not follow from properties listed in Section 2.1.
Our results open several interesting avenues for further investigation. One of the
main open questions is to move from the asymptotic model studied in this work to
real holography. As was stressed several times in this paper, the models that we
discussed do not produce exact holographic one-point coefficients because the multistress-tensor data used for their construction only captures the asymptotic behavior
of the [Tµν ]n data at large n. Therefore, one would like to systematically improve the
models by imputing more information about the stress-tensor sector. The discussion
11
As discussed in Section 2, the asymptotic model satisfies the weak form of boundedness, namely
|gβ (τ )| < ∞ as Im τ → ∞, and not necessarily (2.5).
– 28 –
of Sections 2.5 and 3.6 suggests that the appropriate way of incorporating this data
is via an asymptotic large-n expansion such as
Λn ≃ Λ0n
∞
X
αs
ns
s=0
.
(4.1)
To this end, the first step is to extract from the exact stress-tensor data the asymptotic expansion coefficients αs appearing in (4.1). Let us note that, in principle, there
may be additional subleading terms in (4.1) with slower exponential growth. While
these terms can provide valuable insights into subleading poles of the correlator, they
are particularly challenging to identify due to the lack of data for a more accurate
approximation. Once the refined stress-tensor asymptotics (4.1) is obtained, one can
construct the corresponding asymptotic model following the methodology outlined
in Section 2.5 or 3.6, and extract the corresponding double-trace coefficients. We
leave this for future work.
An alternative approach to determining the holographic one-point coefficients of
double-trace operators involves numerically solving the wave equation on the AdSSchwarzschild black hole background. We discuss this method, following [10], in
Appendix C. A meaningful comparison with the present work requires both the
refinements described in (4.1) and improved numerical precision on the wave equation
side. A preliminary comparison of the double-trace coefficients shows promising
agreement; see Appendix C.
There are various generalizations one may wish to consider. The models in the
main text were restricted to have half-integer scaling dimension ∆ϕ = p + 1/2 of
the external field, in which case all the computations may be carried out explicitly.
For simplicity, we often displayed results for the special case ∆ϕ = 3/2. For other
half-integer scaling dimensions both the asymptotic model and the solution to the
KMS sum rules can be obtained along the same lines. For the case ∆ϕ = 5/2, the
two computations are detailed in appendices A.1 and B.1 (and again produce the
same two-point function). We believe that explicit expressions for the two-point
function and double-trace coefficients for generic p are not difficult to obtain. It is
an interesting challenge to find solutions for generic real values of ∆ϕ , in which case
the stress-tensor sector develops a branch cut. In another direction, changing the
spacetime dimension introduces minimal modifications in our discussion, as long as
the asymptotic stress-tensor data is available. For instance, the asymptotic model in
d = 6 is described in Appendix A.2.
Another extension is to analyze the two-point function at non-vanishing spatial
separation ⃗x, thus probing the double-trace coefficients of operators of different spin
(as opposed to spin-averaged coefficients considered in this work). One may wonder
if the corresponding generalization of the asymptotic model will exhibit quasinormal
modes similar to the ones of holographic systems.
– 29 –
Going beyond holography, large parts of our discussion apply to general CFTs at
finite temperature. On the one hand, the discussion of Section 2.3 suggests a set of
symmetry, analyticity and spectral properties that might uniquely specify a thermal
two-point function, as well as the construction of the latter via a modification of
the method of images. On the other, the approach of Section 3 demonstrates that
KMS sum rules can in some cases be effectively solved by methods of linear algebra.
Moreover, at least in the models we studied, the obtained solution automatically
has good analytic structure. It is an interesting problem to try and identify other
theories that can be approached by these techniques.
In Section 2.1, we listed expected properties of thermal two-point functions, all of
which are satisfied by our asymptotic model.12 Let us note that that the model does
not satisfy the property discussed in [25] in the context of thermal CFT Tauberian
theorems. Namely, one-point coefficients of the model do not become sign-definite
at asymptotically large ∆. This is clear from our input data, as already the multistress-tensor coefficients are asymptotically sign-alternating. We conclude that signdefiniteness does not follow from properties listed in Section 2.1.
A major ingredient of this paper has been Borel summation of the divergent series
to obtain the KMS-invariant form of the correlator. Recently, similar techniques have
been used in the context of boost-invariant flows [76–83]. It would be interesting to
explore potential connections between our approach and those developments.
Acknowledgements
We wish to thank N. Čeplak, C. Esper, A. Grinenko, M. Kulaxizi, H. Liu, V. Niarchos,
C. Papageorgakis, E. Parisini, F. Russo, S. Solodukhin, S. Valach, B. Withers for
discussions, correspondence and comments on the manuscript. This publication has
emanated from research conducted with the financial support of Taighde Éireann –
Research Ireland under Grant number SFI-22/FFP-P/11444.
A
Asymptotic Models for Various ∆ϕ and d
A.1
Asymptotic Model for ∆ϕ = 25
In this appendix, we give details about the asymptotic model in four dimensions with
∆ϕ = 5/2. By summing the asymptotic stress-tensor coefficients (2.15), we obtain
the stress-tensor sector
√
8192 πα0 1 − 4τ 4
gT (τ ) = −
.
(A.1)
693
τ (1 + 4τ 4 )3
12
As discussed in Section 2, the asymptotic model satisfies the weak form of boundedness, namely
|gβ (τ )| < ∞ as Im τ → ∞, and not necessarily (2.5).
– 30 –
Next, we perform the sum over thermal images of (A.1). To write the result, we
make use of the function
√
1+i
128 πα0
1+i
1−i
(1)
τ−
f (τ ) =
64ψ(τ ) − 32ψ τ −
− 32ψ τ −
− 7iψ
693
2
2
2
!
256τ 5 − 584τ 4 + 712τ 3 − 488τ 2 + 196τ − 42
1−i
(1)
−
,
+ 7iψ
τ−
2
(2τ 2 − 2τ + 1)3
where ψ (k) (z) denotes the polygamma function. In terms of the function f (τ ), the
sum of images over (A.1), with the added contribution from the identity, can be
written as
GFF
gβimages (τ ) = g∆
(τ ) + f (τ ) + f (1 − τ ) .
(A.2)
ϕ =5/2
The sum over images has third order poles at (1 ± i)/2. These are canceled by means
of the subtraction (2.20). In the case at hand, the subtraction term can be written
as a linear combination of h, h2 and h3 , where h(τ ) was given in (2.26). Concretely
gβ (τ ) = gβimages (τ )
+
(A.3)
!
512π 5/2 α0
4πh(τ )3 + (6π coth π − 7)h(τ )2 + (2π − 7 coth π)h(τ )
693
.
This is the asymptotic model for ∆ϕ = 5/2. One shows that the only singularities of
(A.3) on the vertical strip are the KMS poles at τ = 0, 1. The double-trace one-point
coefficients of (A.3) can readily be extracted - they are written in (B.7).
A.2
Models in six dimensions
In this appendix we construct the asymptotic model in six dimensions. The main
difference between four and six dimensions is in the behavior of multi-stress-tensor
one-point coefficients. In d = 6, the asymptotic form of the latter reads
Λ0n = c̃∆ϕ n2∆ϕ −4 .
(A.4)
Although the function c̃∆ϕ is not explicitly known, its precise form is not essential
for the purposes of our analysis. We shall fix ∆ϕ = 25 (recall that the unitarity bound
for scalars in d = 6 is ∆ϕ = 2). Other values of ∆ϕ will be briefly discussed at the
end.
Asymptotic coefficients (A.4) give the stress-tensor sector of the two-point function
∞
X
τ
−2∆ϕ
gT (τ ) = τ
Λ0n τ 6n = c̃ 5
.
(A.5)
2
(1 − τ 6 )2
n=1
– 31 –
To proceed, we add the identity exchange contribution and perform the sum over
thermal images
∞
X
1
|τ + m|
images
(τ ) =
gβ
+ c̃ 5
.
(A.6)
5
2
6 )2
|τ
+
m|
(1
−
(τ
+
m)
m=−∞
The sum over the first term gives the GFF two-point function. The sum over the
second can also be performed exactly in terms of polygamma functions. To write the
iπ
final result, let ω = e 3 . Then the image sum (A.6) gives
GFF
gβimages (τ ) = g∆
(τ ) +
ϕ =5/2
c̃ 5
1
1
3
+
+
2
2
18 τ
(τ − 1)
((τ − ω)(τ − ω −1 ))2
2
4
2
2
4
+
+ 4ψ(τ − 1) + 4ψ(−τ − 1)
+
−
−1
2
−2
(τ − ω)(τ − ω ) τ − ω
τ −ω
τ +1
!
.
(A.7)
− 2ψ(τ − ω) − 2ψ(−τ − ω) − 2ψ τ + ω 2 − 2ψ −τ + ω 2
+
The difference compared to the image sum in four dimensions, arising due to the
different form of asymptotics (A.4) is in the pole structure of (A.7). This function
has poles at sixth roots of unity. In addition, the pole at τ = 0 does not correspond
to the holographic spectrum - it contains a non-physical operator of dimension 3. All
these singularities are canceled by the procedure of Section 2.3. The final result for
the asymptotic model reads
1
1 GFF
8π 2
1
images
2
gβ (τ ) = gβ
(τ ) − c̃ 5
g
(τ ) − 2γ H(τ ) −
+
H(τ ) −
.
2
18 ∆=1
2
9
4
(A.8)
GFF
The subtraction term proportional to g∆=1 (τ ) arises as the sum of images of the
subleading pole of gβimages (τ ) at the origin. The other two terms make use of the
function
√
1 + 2eπ( 3+2iτ ) + e4iπτ
√
√
,
H(τ ) =
2 e−π 3 + e2iπτ eπ 3 + e2iπτ
√ √
γ = π coth π 2 + 2π − 3 ,
(A.9)
which is KMS-invariant and has first order poles at ω ±1 . They arise as image sums
of singular pieces of gβimages (τ ) at ω ±1 . The asymptotic model (A.8) satisfies all conditions of Section 2.1 and has the stress-tensor content given in (A.4). In particular,
one can verify the boundedness property (2.5).
We have tested the above construction for other half-integer values of ∆ϕ = p+1/2,
where all the sums can be performed exactly. We find that the same pattern persists:
the image sum analogous to (A.6) has singularities at ω ±1 , as well as subleading poles
at τ = 0 that correspond to unphysical operators in the spectrum. The unphysical
– 32 –
operators can be removed in a unique way by adding a linear combination of GFF
GFF
two-point functions g∆
with ∆ = 1, . . . , p − 1. On the other hand, the singularities at ω ±1 are canceled uniquely by adding a linear combination of functions
{H, H 2 , . . . , H 2(p−1) }.
B
Solutions to Sum Rules
In this appendix, we solve the KMS sum rules for the asymptotic holographic model
using Borel resummation. We explicitly work out the case ∆ϕ = 52 from Section 3.5,
deriving the corresponding thermal one-point coefficients. We also collect a set of
identities and summation formulas used throughout the main text.
B.1
Holographic solutions
The smallest representative of the second set of conformal dimensions given in (3.37)
is ∆ϕ = 25 . The multi-stress-tensor part of the sum rules is given by
∆ϕ =
5
:
2
Fek = −
∞
X
Λ0
n=1
Γ(4n − 4)
n
,
4n
2 Γ(4n − 2k − 5)
(B.1)
where, once again, one can verify that all terms in the series are finite. Since the
system of equations is linear, the n = 1 term — in which both the numerator and
denominator gamma functions exhibit singularities — can be treated separately and
rewritten as follows:
Λ0
(1)
Fek = 41 Γ(2k + 2) .
(B.2)
2
Its contribution to the double-trace coefficients can be determined independently.
Note that this is proportional to the GFF result for ∆ϕ = 21 (see (3.34)). It is once
again useful to decompose the rest of the terms into two distinct parts
∆ϕ =
5
:
2
∞
1 X Λ0n+k+2 Γ(4n + 4k + 4)
Fe2k = − 4k+8
,
4n
2
2
Γ(4n
+
3)
n=0
∞
1 X Λ0n+k+1 Γ(4n + 4k)
Fe2k−1 = − 4k+4
.
2
24n Γ(4n + 1)
n=0
(B.3)
Our goal is to solve the model with the asymptotic multi-stress-tensor coefficients
given in equation (2.15) and compare it to the asymptotic model. The procedure will
be very similar to the ∆ϕ = 32 case described in the main text. The key difference is
that we now need to differentiate sums in order to extract the n2 factor that appears
in the asymptotic coefficients Λ0n . The double-traces take the form
−t Z ∞
Z ∞
c∆ ϕ
e
2
−t e
2
−m−1
4m+4
e
a2m =
e B2m (t )dt = −
(−4)
K2 (t)t
(t∂t )
dt , (B.4)
2
t4
0
0
– 33 –
and
Z ∞
Z ∞
−t e
e
a2m+1 =
K1 (t)t
(t∂t )
dt .
t4
0
0
(B.5)
The integrals can be evaluated by applying the corresponding identities (B.11),
(B.12), yielding
∆ϕ =
e−t Be2m+1 (t2 )dt = −c∆ϕ (−4)−m−2
5
:
2
4m+6
2
c∆ ϕ
(2)
(2)
(2)
(−4)−m−3 16I4m + 7I4m+1 + I4m+2 ,
2
(1)
(1)
(1)
−m−4
e
a2m+1 = −c∆ϕ (−4)
16I4m+2 + 7I4m+3 + I4m+4 .
e
a2m = −
(B.6)
Eventually, incorporating the n = 1 term that was considered separately, the final
result for the double-trace coefficients takes the form:
∆ϕ =
5
:
2
Λ01 −4m
2
ζ(4m + 1)Γ(4m + 1) ,
24
Λ0
a2m+1 = e
a2m+1 + 41 2−4m−1 ζ(4m + 2)Γ(4m + 2) .
2
a2m = e
a2m +
(B.7)
By re-expressing the result in terms of the original double-trace coefficients—using
the definitions from (3.15)—and summing them with the powers τ 2m , we recover a
two-point function that precisely reproduces the asymptotic model for ∆ϕ = 5/2 (see
(A.3)).
B.2
List of identities
Here we collect the integrals and identities used in the derivations above. In Section 3.5, we consider the sum rules for the case ∆ϕ = 23 . The multi-stress-tensor
contributions F̂k in (3.38) and (3.39) can be written explicitly as
∞
1 X (−4)n+k+1 Γ(4n + 4k + 2)
(−1)k
F̂2k
= − 4(k+1)
=
Γ(4k + 2) cos((4k + 2)θ) ,
c3/2
2
24n
Γ(4n + 1)
4 · 52k+1
n=0
∞
F̂2k−1
1 X (−4)n+k+1 Γ(4n + 4k + 2)
(−1)k
= − 4(k+1)
=
Γ(4k) sin(4kθ) ,
4n
2k
c3/2
2
2
Γ(4n
+
3)
4
·
5
n=0
(B.8)
where θ = arctan(−1/3). The coefficient c∆ϕ =3/2 was defined in (2.15). In the
solution of asymptotic holographic sum rules in Section 3.5, we make use of the
– 34 –
following identities
∞
X
ζ(4k)
2
S1 (t) ≡
1 − 2k t4k
4
k
(−4π )
4
k=0
(B.9)
1+i
sin 1+i
t
+
sinh
t
t
2
2
,
=−
2(1 + i)
cos t − cosh t
∞
X
ζ(4k − 2)
2
2
S2 (t) ≡ π
1 − 2k−1 t4k−2
4
k
(−4π )
4
k=1
1+i
t
−
sinh
t
1 + i t sin 1+i
2
2
=
.
8
cos t − cosh t
(B.10)
The one-point coefficients (3.49) and (3.54) are obtained by means of the integrals
Z ∞
sinh t − sin t
(1)
e−t tn
dt
(B.11)
In ≡
cosh t − cos t
0
n
n
1+i
1+i
1−i
1−i
(n)
(n)
= −i −
ψ
+i −
ψ
− n! ,
2
2
2
2
Z ∞
sinh t + sin t
(2)
e−t tn
In ≡
dt
(B.12)
cosh t − cos t
0
n
n
1+i
1−i
1+i
1−i
(n)
(n)
ψ
ψ
=− −
− −
− n! .
2
2
2
2
The same integrals appear in solutions (3.56)-(3.59) to KMS sum rules.
C
Klein-Gordon equation on the planar AdS black hole
In Section 2.2, we reviewed some facts about OPE coefficients of the stress-tensor
sector gT (τ ) of the thermal two-point function. These are obtained by solving the
eigenvalue equation for the Laplacian on the planar AdS-Schwarzschild black hole
in the near boundary expansion. In order to access coefficients of the double-trace
sector, one needs to go beyond the boundary expansion. A numerical approach to
this problem was developed in [10]. In this appendix, we briefly review the method
before presenting and discussing the results it yields in the context of the models
considered in the main text.
The metric of the five-dimensional Euclidean AdS black hole reads
!
2
4
1
dz
z
ds2 = 2
+ 1 − 4 dτ 2 + δab dxa dxb ,
a, b, = 1, 2, 3 .
z4
z
zH
1 − z4
(C.1)
H
We denoted the coordinates by (z, τ, xa ). The constant zH is related to the temperature by β = πzH and the coordinate τ is periodic with the period β. To analyze the
– 35 –
Laplacian eigenvalue problem, we set r2 = xa xa and introduce coordinates (ρ, φ, R)
related to (z, τ, r) by
z = 1 − ρ2 ,
τ=
φ
,
2
r=
R
.
1 − R2
(C.2)
We look for solutions of the equation
∆Φ = ∆ϕ (∆ϕ − 4)Φ ,
(C.3)
which are spherically symmetric, Φ = Φ(ρ, φ, R), are regular in the bulk, and behave
as a delta distribution on the boundary ∂ = {z = 0},
Φ|∂ = G̃AdS |∂ ,
G̃AdS ≡
z ∆ϕ
(z 2 + f (φ) + r2 )∆ϕ
.
(C.4)
The function G̃AdS is obtained from the AdS propagator
GAdS ≡ z ∆ϕ
2
z 2 + φ4
+ r2
∆ϕ ,
(C.5)
by replacing the term φ2 /4 in the denominator by a 2π-periodic function f (φ) with
the same small-φ behavior. The coordinates (C.2) take values in the parallelepiped
ρ, R ∈ [0, 1], φ ∈ [0, 2π], with one pair of opposite faces being identified (corresponding to the periodic coordinate φ). This makes the Klein-Gordon equation suitable
for numerical study, once the boundary conditions on the faces are determined.
In order to discuss the boundary conditions, we start from the fact that, since
the planar black hole is an asymptotically AdS space, every solution to the wave
equation behaves near the boundary as
Φ ∼ z ∆ϕ Φ+ (τ, ⃗x) + z 4−∆ϕ Φ− (τ, ⃗x) .
(C.6)
We put Φ = Ψ + G̃AdS and solve the equation for Ψ instead of Φ. Let us assume that
Ψ splits in the same way as Φ
Ψ ∼ z ∆ϕ Ψ+ (τ, ⃗x) + z 4−∆ϕ Ψ− (τ, ⃗x) .
(C.7)
Depending on whether ∆ϕ is smaller or larger than two, different terms in the expression (C.7) are leading as z → 0. We confine ourselves to the case ∆ϕ = 3/2 and
put
Ψ = z ∆ϕ H .
(C.8)
We look for a solution with Ψ− = 0. Together with regularity in the bulk, this
implies the following set of Dirichlet and Neumann boundary conditions on H
H|R=1 = ∂R H|R=0 = ∂ρ H|ρ=0 = ∂ρ H|ρ=1 = 0 .
– 36 –
(C.9)
The differential equation for H with boundary conditions (C.9) can be solved numerically. One possibility is to discretise the equation on a grid consisting of N 3 points.
We use the uniform distribution of points in φ- and R-directions and the Chebyshev
collocation in the ρ-direction. Finally, the thermal two-point function is related to
the solution by
1
gβ (τ, r) = H(ρ = 1, φ, r) +
,
(C.10)
(f (φ) + r2 )∆ϕ
which follows from the near-boundary expansion
G̃AdS =
z ∆ϕ
(f (φ) + r2 )∆ϕ
+ ... .
(C.11)
Having summarized the method — which was fully developed in [10] — we now
proceed to discuss its results. After extracting the two-point function gβ (τ ) from the
numerical solution via (C.10), we fit its small-τ behavior to the OPE expansion, with
coefficients kept unknown. The fitting is performed on the interval [τmin , τmax ] for
various values of τmin and τmax . Each choice of the interval gives a fit for coefficients
aO . We collect these different fits and take their minimum and maximum, amin
O and
max
aO . For the grid of size N = 140, the results read as follows
aϕ2 ∈ [1.09, 1.10] ,
a[ϕϕ]2∆ϕ +2 ∈ [8.84, 10.97] .
(C.12)
In the fitting procedure, we have imputed exact values of the stress-tensor and doublestress-tensor one-point coefficients
Λ1 =
3π 4
= 3.6528... ,
80
Λ2 =
479π 8
.
268800
(C.13)
In order to be consistent with [10], we have put β = π in the last expressions, rather
than β = 1 used in the main text. The lowest double-trace coefficient in (C.12) is
consistent with the value found in [10]. Furthermore, had we not used the exact
value for Λ1 from (C.13), but instead treated it as an unknown in the fitting, we
would have obtained
Λfit
(C.14)
1 ∈ [3.30, 3.77] ,
consistent with the actual value. These facts lend support to the numerical procedure
employed. Finally, the asymptotic model value of the second double trace coefficient
is aAM
[ϕϕ]2∆ϕ +2 ≈ 7.55. Both the last number and the numerical result (C.12) can only
serve as rough guides for the true value of the holographic coefficient a[ϕϕ]2∆ϕ +2 . In the
case of the former, this comes from the fact that asymptotic rather than exact stresstensor data is used for the computation of aAM
[ϕϕ]2∆ϕ +2 . As for the latter, the numerical
procedure from above needs improvements in order to give a reliable prediction for
the double stress-tensor coefficient. Nevertheless, we find that the comparison of the
two numbers is promising and invites for improved estimates from both methods.
– 37 –
References
[1] J. M. Maldacena, The Large N limit of superconformal field theories and
supergravity, Adv. Theor. Math. Phys. 2 (1998) 231–252, [hep-th/9711200].
[2] S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Gauge theory correlators from
noncritical string theory, Phys. Lett. B 428 (1998) 105–114, [hep-th/9802109].
[3] E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998)
253–291, [hep-th/9802150].
[4] E. Witten, Anti-de Sitter space, thermal phase transition, and confinement in gauge
theories, Adv. Theor. Math. Phys. 2 (1998) 505–532, [hep-th/9803131].
[5] A. L. Fitzpatrick, J. Kaplan, D. Poland, and D. Simmons-Duffin, The Analytic
Bootstrap and AdS Superhorizon Locality, JHEP 12 (2013) 004, [arXiv:1212.3616].
[6] Z. Komargodski and A. Zhiboedov, Convexity and Liberation at Large Spin, JHEP
11 (2013) 140, [arXiv:1212.4103].
[7] L. Iliesiu, M. Koloğlu, R. Mahajan, E. Perlmutter, and D. Simmons-Duffin, The
Conformal Bootstrap at Finite Temperature, JHEP 10 (2018) 070,
[arXiv:1802.10266].
[8] M. Kulaxizi, G. S. Ng, and A. Parnachev, Black Holes, Heavy States, Phase Shift
and Anomalous Dimensions, SciPost Phys. 6 (2019), no. 6 065, [arXiv:1812.03120].
[9] A. L. Fitzpatrick and K.-W. Huang, Universal Lowest-Twist in CFTs from
Holography, JHEP 08 (2019) 138, [arXiv:1903.05306].
[10] E. Parisini, K. Skenderis, and B. Withers, The ambient space formalism, JHEP 05
(2024) 296, [arXiv:2312.03820].
[11] N. Čeplak, H. Liu, A. Parnachev, and S. Valach, Black hole singularity from OPE,
JHEP 10 (2024) 105, [arXiv:2404.17286].
[12] R. Kubo, Statistical mechanical theory of irreversible processes. 1. General theory
and simple applications in magnetic and conduction problems, J. Phys. Soc. Jap. 12
(1957) 570–586.
[13] P. C. Martin and J. S. Schwinger, Theory of many particle systems. 1., Phys. Rev.
115 (1959) 1342–1373.
[14] R. Karlsson, A. Parnachev, V. Prilepina, and S. Valach, Thermal stress tensor
correlators, OPE and holography, JHEP 09 (2022) 234, [arXiv:2206.05544].
[15] A. Manenti, Thermal CFTs in momentum space, JHEP 01 (2020) 009,
[arXiv:1905.01355].
[16] S. El-Showk and K. Papadodimas, Emergent Spacetime and Holographic CFTs,
JHEP 10 (2012) 106, [arXiv:1101.4163].
[17] E. Katz, S. Sachdev, E. S. Sørensen, and W. Witczak-Krempa, Conformal field
– 38 –
theories at nonzero temperature: Operator product expansions, Monte Carlo, and
holography, Phys. Rev. B 90 (2014), no. 24 245109, [arXiv:1409.3841].
[18] Y.-Z. Li, Z.-F. Mai, and H. Lü, Holographic OPE Coefficients from AdS Black Holes
with Matters, JHEP 09 (2019) 001, [arXiv:1905.09302].
[19] A. L. Fitzpatrick, K.-W. Huang, D. Meltzer, E. Perlmutter, and D. Simmons-Duffin,
Model-dependence of minimal-twist OPEs in d > 2 holographic CFTs, JHEP 11
(2020) 060, [arXiv:2007.07382].
[20] L. F. Alday, M. Kologlu, and A. Zhiboedov, Holographic correlators at finite
temperature, JHEP 06 (2021) 082, [arXiv:2009.10062].
[21] D. Rodriguez-Gomez and J. G. Russo, Correlation functions in finite temperature
CFT and black hole singularities, JHEP 06 (2021) 048, [arXiv:2102.11891].
[22] D. Rodriguez-Gomez and J. G. Russo, Thermal correlation functions in CFT and
factorization, JHEP 11 (2021) 049, [arXiv:2105.13909].
[23] R. Karlsson, A. Parnachev, and P. Tadić, Thermalization in large-N CFTs, JHEP
09 (2021) 205, [arXiv:2102.04953].
[24] M. Dodelson, C. Iossa, R. Karlsson, and A. Zhiboedov, A thermal product formula,
JHEP 01 (2024) 036, [arXiv:2304.12339].
[25] E. Marchetto, A. Miscioscia, and E. Pomoni, Sum rules & Tauberian theorems at
finite temperature, JHEP 09 (2024) 044, [arXiv:2312.13030].
[26] J. R. David and S. Kumar, Thermal one-point functions: CFT’s with fermions, large
d and large spin, JHEP 10 (2023) 143, [arXiv:2307.14847].
[27] N. Benjamin, J. Lee, H. Ooguri, and D. Simmons-Duffin, Universal asymptotics for
high energy CFT data, JHEP 03 (2024) 115, [arXiv:2306.08031].
[28] J. Barrat, E. Marchetto, A. Miscioscia, and E. Pomoni, The thermal bootstrap for
the critical O(N) model, arXiv:2411.00978.
[29] J. Barrat, B. Fiol, E. Marchetto, A. Miscioscia, and E. Pomoni, Conformal line
defects at finite temperature, SciPost Phys. 18 (2025), no. 1 018,
[arXiv:2407.14600].
[30] P. Kraus, H. Ooguri, and S. Shenker, Inside the horizon with AdS / CFT, Phys. Rev.
D 67 (2003) 124022, [hep-th/0212277].
[31] L. Fidkowski, V. Hubeny, M. Kleban, and S. Shenker, The Black hole singularity in
AdS / CFT, JHEP 02 (2004) 014, [hep-th/0306170].
[32] G. Festuccia and H. Liu, Excursions beyond the horizon: Black hole singularities in
Yang-Mills theories. I., JHEP 04 (2006) 044, [hep-th/0506202].
[33] N. Engelhardt, T. Hertog, and G. T. Horowitz, Holographic Signatures of
Cosmological Singularities, Phys. Rev. Lett. 113 (2014) 121602, [arXiv:1404.2309].
[34] G. T. Horowitz, H. Leung, L. Queimada, and Y. Zhao, Boundary signature of
– 39 –
singularity in the presence of a shock wave, SciPost Phys. 16 (2024), no. 2 060,
[arXiv:2310.03076].
[35] M. De Clerck, S. A. Hartnoll, and J. E. Santos, Mixmaster chaos in an AdS black
hole interior, JHEP 07 (2024) 202, [arXiv:2312.11622].
[36] G. T. Horowitz and V. E. Hubeny, Quasinormal modes of AdS black holes and the
approach to thermal equilibrium, Phys. Rev. D 62 (2000) 024027, [hep-th/9909056].
[37] V. Cardoso and J. P. S. Lemos, Quasinormal modes of Schwarzschild anti-de Sitter
black holes: Electromagnetic and gravitational perturbations, Phys. Rev. D 64 (2001)
084017, [gr-qc/0105103].
[38] D. Birmingham, I. Sachs, and S. N. Solodukhin, Conformal field theory
interpretation of black hole quasinormal modes, Phys. Rev. Lett. 88 (2002) 151301,
[hep-th/0112055].
[39] G. Policastro, D. T. Son, and A. O. Starinets, From AdS / CFT correspondence to
hydrodynamics, JHEP 09 (2002) 043, [hep-th/0205052].
[40] A. Nunez and A. O. Starinets, AdS / CFT correspondence, quasinormal modes, and
thermal correlators in N=4 SYM, Phys. Rev. D 67 (2003) 124013,
[hep-th/0302026].
[41] P. K. Kovtun and A. O. Starinets, Quasinormal modes and holography, Phys. Rev. D
72 (2005) 086009, [hep-th/0506184].
[42] M. Brigante, H. Liu, R. C. Myers, S. Shenker, and S. Yaida, Viscosity Bound
Violation in Higher Derivative Gravity, Phys. Rev. D 77 (2008) 126006,
[arXiv:0712.0805].
[43] G. Festuccia and H. Liu, A Bohr-Sommerfeld quantization formula for quasinormal
frequencies of AdS black holes, Adv. Sci. Lett. 2 (2009) 221–235, [arXiv:0811.1033].
[44] A. Buchel and R. C. Myers, Causality of Holographic Hydrodynamics, JHEP 08
(2009) 016, [arXiv:0906.2922].
[45] A. Buchel, J. Escobedo, R. C. Myers, M. F. Paulos, A. Sinha, and M. Smolkin,
Holographic GB gravity in arbitrary dimensions, JHEP 03 (2010) 111,
[arXiv:0911.4257].
[46] J. de Boer, M. Kulaxizi, and A. Parnachev, AdS(7)/CFT(6), Gauss-Bonnet Gravity,
and Viscosity Bound, JHEP 03 (2010) 087, [arXiv:0910.5347].
[47] E. Berti, V. Cardoso, and A. O. Starinets, Quasinormal modes of black holes and
black branes, Class. Quant. Grav. 26 (2009) 163001, [arXiv:0905.2975].
[48] M. Kaminski, K. Landsteiner, J. Mas, J. P. Shock, and J. Tarrio, Holographic
Operator Mixing and Quasinormal Modes on the Brane, JHEP 02 (2010) 021,
[arXiv:0911.3610].
[49] W. Witczak-Krempa and S. Sachdev, The quasi-normal modes of quantum
criticality, Phys. Rev. B 86 (2012) 235115, [arXiv:1210.4166].
– 40 –
[50] R. Emparan, R. Suzuki, and K. Tanabe, Quasinormal modes of (Anti-)de Sitter
black holes in the 1/D expansion, JHEP 04 (2015) 085, [arXiv:1502.02820].
[51] J. F. Fuini, C. F. Uhlemann, and L. G. Yaffe, Damping of hard excitations in
strongly coupled N = 4 plasma, JHEP 12 (2016) 042, [arXiv:1610.03491].
[52] S. Grozdanov, N. Kaplis, and A. O. Starinets, From strong to weak coupling in
holographic models of thermalization, JHEP 07 (2016) 151, [arXiv:1605.02173].
[53] S. Grozdanov and A. O. Starinets, Adding new branches to the “Christmas tree” of
the quasinormal spectrum of black branes, JHEP 04 (2019) 080,
[arXiv:1812.09288].
[54] S. Grozdanov, P. K. Kovtun, A. O. Starinets, and P. Tadić, The complex life of
hydrodynamic modes, JHEP 11 (2019) 097, [arXiv:1904.12862].
[55] A. Jansen and C. Pantelidou, Quasinormal modes in charged fluids at complex
momentum, JHEP 10 (2020) 121, [arXiv:2007.14418].
[56] H.-S. Jeong, K.-Y. Kim, and Y.-W. Sun, Quasi-normal modes of dyonic black holes
and magneto-hydrodynamics, JHEP 07 (2022) 065, [arXiv:2203.02642].
[57] M. Dodelson, Ringdown in the SYK model, arXiv:2408.05790.
[58] S. Grozdanov and M. Vrbica, Duality Constraints on Thermal Spectra of 3D
Conformal Field Theories and 4D Quasinormal Modes, Phys. Rev. Lett. 133 (2024),
no. 21 211601, [arXiv:2406.19790].
[59] M. De Lescluze and M. P. Heller, Quasinormal modes of nonthermal fixed points,
arXiv:2502.01622.
[60] M. Dodelson, Black holes from chaos, arXiv:2501.06170.
[61] S. Grozdanov, K. Schalm, and V. Scopelliti, Black hole scrambling from
hydrodynamics, Phys. Rev. Lett. 120 (2018), no. 23 231601, [arXiv:1710.00921].
[62] M. Blake, H. Lee, and H. Liu, A quantum hydrodynamical description for scrambling
and many-body chaos, JHEP 10 (2018) 127, [arXiv:1801.00010].
[63] F. M. Haehl and M. Rozali, Effective Field Theory for Chaotic CFTs, JHEP 10
(2018) 118, [arXiv:1808.02898].
[64] M. Blake, R. A. Davison, S. Grozdanov, and H. Liu, Many-body chaos and energy
dynamics in holography, JHEP 10 (2018) 035, [arXiv:1809.01169].
[65] S. Grozdanov, On the connection between hydrodynamics and quantum chaos in
holographic theories with stringy corrections, JHEP 01 (2019) 048,
[arXiv:1811.09641].
[66] M. Blake, R. A. Davison, and D. Vegh, Horizon constraints on holographic Green’s
functions, JHEP 01 (2020) 077, [arXiv:1904.12883].
[67] C. Choi, M. Mezei, and G. Sárosi, Pole skipping away from maximal chaos,
arXiv:2010.08558.
– 41 –
[68] V. E. Hubeny, H. Liu, and M. Rangamani, Bulk-cone singularities & signatures of
horizon formation in AdS/CFT, JHEP 01 (2007) 009, [hep-th/0610041].
[69] J. Maldacena, D. Simmons-Duffin, and A. Zhiboedov, Looking for a bulk point,
JHEP 01 (2017) 013, [arXiv:1509.03612].
[70] M. Dodelson and H. Ooguri, Singularities of thermal correlators at strong coupling,
Phys. Rev. D 103 (2021), no. 6 066018, [arXiv:2010.09734].
[71] M. Dodelson, C. Iossa, R. Karlsson, A. Lupsasca, and A. Zhiboedov, Black hole
bulk-cone singularities, JHEP 07 (2024) 046, [arXiv:2310.15236].
[72] H.-Y. Chen, Y. Hikida, and Y. Koga, AdS gravaster and bulk-cone singularities,
arXiv:2502.11403.
[73] N. Chai, S. Chaudhuri, C. Choi, Z. Komargodski, E. Rabinovici, and M. Smolkin,
Thermal Order in Conformal Theories, Phys. Rev. D 102 (2020), no. 6 065014,
[arXiv:2005.03676].
[74] Z. Komargodski and F. K. Popov, Temperature-Resistant Order in 2+1 Dimensions,
arXiv:2412.09459.
[75] F. M. Fedorov, O. F. Ivanova, and N. N. Pavlov, On the specificities of infinite
systems, Mat. Zamet. SVFU 22 (2015), no. 4 62–78. [Mat. Zamet. SVFU].
[76] M. P. Heller, R. A. Janik, and P. Witaszczyk, Hydrodynamic Gradient Expansion in
Gauge Theory Plasmas, Phys. Rev. Lett. 110 (2013), no. 21 211602,
[arXiv:1302.0697].
[77] M. P. Heller and M. Spalinski, Hydrodynamics Beyond the Gradient Expansion:
Resurgence and Resummation, Phys. Rev. Lett. 115 (2015), no. 7 072501,
[arXiv:1503.07514].
[78] G. Basar and G. V. Dunne, Hydrodynamics, resurgence, and transasymptotics, Phys.
Rev. D 92 (2015), no. 12 125011, [arXiv:1509.05046].
[79] I. Aniceto and M. Spaliński, Resurgence in Extended Hydrodynamics, Phys. Rev. D
93 (2016), no. 8 085008, [arXiv:1511.06358].
[80] W. Florkowski, R. Ryblewski, and M. Spaliński, Gradient expansion for anisotropic
hydrodynamics, Phys. Rev. D 94 (2016), no. 11 114025, [arXiv:1608.07558].
[81] M. Spaliński, On the hydrodynamic attractor of Yang–Mills plasma, Phys. Lett. B
776 (2018) 468–472, [arXiv:1708.01921].
[82] J. Casalderrey-Solana, N. I. Gushterov, and B. Meiring, Resurgence and
Hydrodynamic Attractors in Gauss-Bonnet Holography, JHEP 04 (2018) 042,
[arXiv:1712.02772].
[83] M. P. Heller and V. Svensson, How does relativistic kinetic theory remember about
initial conditions?, Phys. Rev. D 98 (2018), no. 5 054016, [arXiv:1802.08225].
– 42 –