Harnessing S-Duality in $\mathcal{N}=4$ SYM & Supergravity as
$SL(2,\mathbb{Z})$-Averaged Strings

Scott Collier∘, Eric Perlmutter∙

$\circ$ Princeton Center for Theoretical Science, Princeton University,
Princeton, NJ 08544, USA

$\bullet$ Université Paris-Saclay, CNRS, CEA, Institut de Physique Théorique,
91191, Gif-sur-Yvette, France

<EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

We develop a new approach to extracting the physical consequences of S-duality
of four-dimensional $\mathcal{N}=4$ super Yang-Mills (SYM) and its string
theory dual, based on $SL(2,\mathbb{Z})$ spectral theory.

We observe that CFT observables $\mathcal{O}$, invariant under
$SL(2,\mathbb{Z})$ transformations of a complexified gauge coupling $\tau$,
admit a unique spectral decomposition into a basis of square-integrable
functions. This formulation has direct implications for the analytic structure
of $\mathcal{N}=4$ SYM data, both perturbatively and non-perturbatively in all
parameters. These are especially constraining for the structure of instantons:
$k$-instanton sectors are uniquely determined by the zero- and one-instanton
sectors, and Borel summable series around $k$-instantons have convergence
radii with simple $k$-dependence. In large $N$ limits, we derive the existence
and scaling of non-perturbative effects, in both $N$ and the ’t Hooft
coupling, which we exhibit for certain $\mathcal{N}=4$ SYM observables. An
elegant benchmark for these techniques is the integrated stress tensor
multiplet four-point function, conjecturally determined by [1] for all $\tau$
for $SU(N)$ gauge group; we derive and elucidate its form, and explain how the
$SU(2)$ case is the simplest possible observable consistent with
$SL(2,\mathbb{Z})$-invariant perturbation theory.

These results have ramifications for holography. We explain how
$\langle\mathcal{O}\rangle$, the ensemble average of ${\cal O}$ over the
$\mathcal{N}=4$ supersymmetric conformal manifold with respect to the
Zamolodchikov measure, is cleanly isolated by the spectral decomposition. We
prove that the large $N$ limit of $\langle\mathcal{O}\rangle$ equals the large
$N$, large ’t Hooft coupling limit of $\mathcal{O}$. Holographically speaking,
$\langle\mathcal{O}\rangle={\cal O}_{\rm sugra}$, its value in type IIB
supergravity on AdS${}_{5}\times S^{5}$. This result, which extends to all
orders in $1/N$, embeds ensemble averaging into the traditional AdS/CFT
paradigm. The statistics of the $SL(2,\mathbb{Z})$ ensemble exhibit both
perturbative and non-perturbative $1/N$ effects. We discuss further
implications and generalizations to other AdS compactifications of
string/M-theory.

###### Contents

  1. 1 Introduction
  2. 2 $SL(2,\mathbb{Z})$ Spectral Theory
  3. 3 Spectral Decomposition in $\mathcal{N}=4$ SYM
  4. 4 Integrated Correlator I: Warmup
    1. 4.1 Definition
    2. 4.2 Spectral decomposition
  5. 5 Instantons and the Analytic Structure of $\mathcal{N}=4$ SYM
    1. 5.1 Perturbation theory
    2. 5.2 The analytic structure of CFT spectral overlaps
    3. 5.3 $SL(2,\mathbb{Z})$ Borel transform
    4. 5.4 The redundancy of instantons
    5. 5.5 $\mathcal{N}=4$ SYM observables as Poincaré sums
    6. 5.6 Integrated correlator II: Derivation
    7. 5.7 Including cusp forms
    8. 5.8 On the strong coupling expansion
  6. 6 Large $N$: Generalities
  7. 7 ’t Hooft Limit
    1. 7.1 Non-perturbative effects implied by $SL(2,\mathbb{Z})$
    2. 7.2 Comments + String theory interpretation
    3. 7.3 Integrated correlator III: ’t Hooft limit and D-string instantons from resurgence
    4. 7.4 The general result: Restoring instanton-anti-instanton effects
  8. 8 Very Strongly Coupled Limit
    1. 8.1 Example: $\mathcal{G}_{N}(\tau)$
    2. 8.2 Example with non-perturbative effects: ${\cal F}_{N}(\tau)$
  9. 9 Interlude: Non-perturbative Effects in Anomalous Dimensions at Finite $N$
  10. 10 AdS${}_{5}\times S^{5}$ Supergravity is Ensemble-Averaged String Theory
  11. 11 Statistics of the $SL(2,\mathbb{Z})$ Ensemble
    1. 11.1 Higher moments of CFT observables
    2. 11.2 The variance at large $N$
    3. 11.3 Non-perturbative corrections to the variance
  12. 12 Remarks on the AdS/CFT Paradigm
    1. 12.1 Wormholes in moduli space and emergent averaged holographic duality
    2. 12.2 Generalizations
  13. 13 Conclusion
  14. A A few details for constraining spectral overlaps
    1. A.1 Deriving (5.6)
    2. A.2 Proofs for Section 5.2
  15. B Solutions to the inhomogeneous Laplace equation
    1. B.1 General case
    2. B.2 $s_{1}=s_{2}$
    3. B.3 A worked example: $r=3$, $s_{1}=s_{2}={3\over 2}$

##  1 Introduction

This paper pursues two intertwined endeavors.

The first is to understand how to extract the full implications of S-duality
for the observables of superconformal field theories, focusing specifically on
four-dimensional $\mathcal{N}=4$ super Yang-Mills theory.

The second is to reframe the AdS/CFT Correspondence, namely the original
duality between $\mathcal{N}=4$ super Yang-Mills and type IIB string theory on
AdS${}_{5}\times S^{5}$, by applying new lessons from S-duality.

The $\mathcal{N}=4$ super Yang-Mills (SYM) theory is, at risk of stating the
obvious, a beautiful theory from myriad points of view. Maximal supersymmetry
imposes rigid structure and regularity of CFT data, yet provides a route to
computing certain observables that coincide with those in less supersymmetric
theories. For a finite number of colors $N$, perturbative gauge theory
calculations are complemented by modern bootstrap and supersymmetric
localization methods, providing rigorous and sometimes exact results for local
and non-local quantities. In the large $N$ ’t Hooft limit, new symmetries
emerge, leading to integrability solutions at the planar level and beyond. Not
to be forgotten, the holographic correspondence with type IIB string theory on
AdS${}_{5}\times S^{5}$ furnishes our most explicit definition of a theory of
quantum gravity, shedding light on both sides of the duality.

$\mathcal{N}=4$ SYM also enjoys S-duality [2, 3, 4, 5]. For simply-laced gauge
group parameterized by the complexified gauge coupling $\tau$, this is a self-
duality: namely, an invariance of the theory under $(P)SL(2,\mathbb{Z})$
transformations of $\tau$. Holographically, this symmetry appears as the
$SL(2,\mathbb{Z})$ symmetry of type IIB string theory in AdS${}_{5}\times
S^{5}$ — a background that preserves the $SL(2,\mathbb{Z})$ symmetry of flat
space string theory — where $\tau$ is dual to the axio-dilaton.

There have been numerous checks of S-duality against independent calculations
in $\mathcal{N}=4$ SYM or in type IIB string theory. All have succeeded. The
existence of this symmetry, while no less remarkable, seems without question.
What are the full implications of S-duality for $\mathcal{N}=4$ SYM
observables? In the presence of symmetries in any physical theory, one should
incorporate their effects at the outset of calculations. This will be our
approach: to efficiently process S-duality invariance of $\mathcal{N}=4$ SYM
observables, reducing these data to their dynamical content.

Viewing $\mathcal{N}=4$ SYM observables ${\cal O}(\tau)$ (e.g. conformal
dimensions or correlation functions) as non-holomorphic functions invariant
under $SL(2,\mathbb{Z})$ transformations of $\tau$, we have very limited
“real-world” information about the form of these functions. Being exactly
marginal, $\tau$ parameterizes a one-complex dimensional conformal manifold,
$\mathcal{M}$, that preserves the full $\mathcal{N}=4$ supersymmetry. In terms
of real parameters,

$\tau={\theta\over 2\pi}+{4\pi i\over g^{2}}$ (1.1)

where $g$ is the Yang-Mills coupling and $\theta$ is the topological theta
angle. While perturbative results near $\tau=i\infty$ (possibly with instanton
backgrounds [6, 7, 8, 9]) give some information, the modular structure of
${\cal O}(\tau)$ is inherently non-perturbative. S-duality has been input into
the construction of interpolating functions across moduli space [10, 11, 12,
13], but (as noted there) those interpolating functions lacked a principle for
eliminating ambiguity in the chosen function space. There is a shining recent
exception to this general paucity of data which we discuss, then derive, at
length below.

Our idea is to introduce certain methods from the mathematics literature that
we argue are perfectly tailored to the $\mathcal{N}=4$ SYM context. There
exists a robust spectral theory of $SL(2,\mathbb{Z})$, applicable to
quantities that are square-integrable on the $SL(2,\mathbb{Z})$ fundamental
domain, which we call ${\cal F}=\mathbb{H}/SL(2,\mathbb{Z})$. As we will
explain, $\mathcal{N}=4$ SYM observables ${\cal O}(\tau)$ are in this
category. Employing a unique spectral decomposition into an
$SL(2,\mathbb{Z})$-invariant eigenbasis fully incorporates the S-duality
symmetry. Determining ${\cal O}(\tau)$ then boils down to computing its
spectral overlaps; specializing this mathematical toolkit to this physical
context, the analytic structure of the spectral overlaps is in turn very
strongly constrained by the consistency of weak coupling perturbation theory.

With this framework in place, relatively simple calculations lead to a wealth
of information at finite $N$, described in further detail below. This is
especially true as regards the structure of instantons. The ensuing
calculations look rather different than existing approaches to $\mathcal{N}=4$
SYM observables. We gain confidence by applying the general formalism to the
integrated correlator introduced in [14] and studied in detail in [15, 1] — a
unique observable that is conjecturally known exactly as a function of $N$ and
$\tau$ — leading to a derivation of their result and a crisp accounting of
many of its properties.

Applying this approach at large $N$ opens yet other doors. One may develop the
’t Hooft limit of large $N$ and fixed $\lambda\coloneqq g^{2}N$. At
$\lambda\gg 1$, the theory is famously dual to type IIB supergravity on
AdS${}_{5}\times S^{5}$, endowed with a prescribed series of stringy
$\alpha^{\prime}$ corrections. Perhaps surprisingly, the spectral method has
something fundamental to say about holography: in particular, it suggests a
new picture of AdS/CFT that unifies the traditional holographic paradigm for
UV complete theories with recent ideas on ensemble averaging in lower-
dimensional AdS/CFT.

In the current setting, the ensemble in question is what we call the
$SL(2,\mathbb{Z})$ ensemble, the space of $\mathcal{N}=4$ SYM theories living
on the conformal manifold $\mathcal{M}$. The central result, elaborated upon
in the description of Section 10 below, can be simply stated: the large $N$
limit of ensemble-averaged $\mathcal{N}=4$ SYM is the strong coupling limit of
planar $\mathcal{N}=4$ SYM. This holds at the level of individual observables
${\cal O}(\tau)$. In bulk terms, type IIB supergravity on AdS${}_{5}\times
S^{5}$ is both a low-energy limit of type IIB string theory, and the average
of type IIB string theory over moduli space. Unlike ensemble-averaged
dualities in lower dimensions, the duality between $\mathcal{N}=4$ SYM and
AdS${}_{5}\times S^{5}$ string theory applies for every microscopic instance
of $\mathcal{N}=4$ SYM: the ensemble average is an emergent description of the
strongly coupled, planar limit. This equivalence permits a satisfying
embedding of various developments involving wormholes, factorization, and
ensemble statistics into the quintessential holographic correspondence.

$(\lambda\to\infty)$$(N\to\infty)$ Figure 1: A cartoon depicting the two
equivalent field theory duals of type IIB supergravity on AdS${}_{5}\times
S^{5}$, phrased in terms of the conformal manifold $\mathcal{M}$: as the limit
of large ’t Hooft coupling (depicted by the point approaching the cusp of
$\mathcal{M}$) of planar $\mathcal{N}=4$ SYM (left), and as the large $N$
limit of the ensemble average (denoted by the shading) of $\mathcal{N}=4$ SYM
(right). On the left it is understood that the $N\to\infty$ limit is taken
first.

Let us now give a slightly more detailed description of our results.

In Section 2, we begin with a brief, physicist-oriented introduction to the
theory of harmonic analysis of $SL(2,\mathbb{Z})$. The $L^{2}({\cal F})$
eigenspace contains a continuous subspace spanned by non-holomorphic
Eisenstein series $E_{s}(\tau)$ with $s={1\over 2}+i\mathbb{R}$, and a
discrete subspace spanned by an infinite set of Maass cusp forms
$\phi_{n}(\tau)$ labeled by $n\in\mathbb{Z}_{\geq 0}$ (where $\phi_{0}$ is
constant). Both are eigenfunctions of the Laplacian on the upper-half plane.
The former are well-understood, while the latter are wild objects, of
prevailing interest to mathematicians, exhibiting several signals of chaos. We
depict this wildness with a numerical plot; see Figure 2.

In Section 3, we develop the spectral decomposition of observables in
$\mathcal{N}=4$ SYM. We explain how, and why, the results of the previous
section apply to this setting. A non-perturbatively well-defined observable
${\cal O}(\tau)={\cal O}(\gamma\tau)$, with $\gamma\in SL(2,\mathbb{Z})$,
admits the following spectral decomposition:

$\mathcal{O}(\tau)=\overline{\mathcal{O}}+{1\over 4\pi
i}\int_{\operatorname{Re}s={1\over
2}}ds\,(\mathcal{O},E_{s})E_{s}(\tau)+\sum_{n=1}^{\infty}(\mathcal{O},\phi_{n})\phi_{n}(\tau).$
(1.2)

where $(\cdot\,,\cdot)$ denotes the Petersson inner product. The first term,
$\overline{\mathcal{O}}$, is a constant, the modular average of ${\cal O}$,
defined as the integral of ${\cal O}(\tau)$ over ${\cal F}$ with the
$SL(2,\mathbb{R})$-invariant measure,

$\overline{\cal O}\coloneqq\operatorname{vol}({\cal F})^{-1}\int_{{\cal
F}}{dxdy\over y^{2}}{\cal O}(\tau)\,,\quad\tau\coloneqq x+iy$ (1.3)

The basic statement of the spectral decomposition is that determining ${\cal
O}(\tau)$ thus reduces to computing its average $\overline{\mathcal{O}}$ and
the spectral overlaps $(\mathcal{O},E_{s})$ and $(\mathcal{O},\phi_{n})$. The
spectral overlap $(\mathcal{O},E_{s})$ is a meromorphic function of the
spectral parameter $s$, whose analyticity in the complex $s$ plane is subject
to stringent constraints from the consistency of the weak-coupling expansion,
well-definedness of the CFT observable in the ’t Hooft limit, and the
Eisenstein series itself. As an example, the modified overlap
$\\{\mathcal{O},E_{s}\\}\coloneqq(\mathcal{O},E_{s})/\Lambda(s)$, where
$\Lambda(s)$ is the completed Riemann zeta function defined in (2.12), must
satisfy the functional equation
$\\{\mathcal{O},E_{s}\\}=\\{\mathcal{O},E_{1-s}\\}$.

Without further computation one observes two important consequences. The first
is a redundancy in instanton physics: $k$-instanton effects are fully
determined by $k=0,1$ instanton effects, because the latter can be used to
“invert” this expansion.111We develop a Fourier expansion ${\cal
O}(\tau)={\cal O}_{0}(y)+\sum_{k=1}^{\infty}2\cos(2\pi kx)\,{\cal O}_{k}(y)$,
where $k$ is the total instanton number. The second is a suggestive fact about
the spectral decomposition. For any $N$ and any ${\cal O}(\tau)$, we can
define the ensemble average $\langle{\cal O}\rangle$, as the integral of
${\cal O}(\tau)$ over $\mathcal{M}$ with respect to the Zamolodchikov measure
on the conformal manifold. Thanks to maximal supersymmetry, the modular
average, $\overline{\mathcal{O}}$, equals the ensemble average, $\langle{\cal
O}\rangle$. Therefore, $\langle{\cal O}\rangle$ appears in a very clean way in
the spectral decomposition of ${\cal O}(\tau)$. This foreshadows some results
to follow.

In Section 4, we introduce an object that will be a touchstone for our methods
throughout: the integrated correlator $\mathcal{G}_{N}(\tau)$. This is a four-
point function of the ${\cal O}_{\bf 20^{\prime}}$ operator in the $SU(N)$
theory, integrated over space with a measure that preserves supersymmetry.
Accordingly, $\mathcal{G}_{N}(\tau)$ may also be represented as a
supersymmetric localization integral. In a beautiful paper [1], the authors
conjectured and thoroughly tested an expression for $\mathcal{G}_{N}(\tau)$,
valid for all $N$ and $\tau$, given by a two-dimensional lattice sum over one-
dimensional integrals (see (4.5)). Its $N$-dependence is moreover fixed
recursively by a “Laplace difference equation” (see (4.7)). It is a rare
situation, even in $\mathcal{N}=4$ SYM, to have an exact expression for any
observable at finite $N$, whose functional complexity (e.g. the number of
integrals) is independent of $N$. The evidence collected in [1] for their
conjecture, reviewed later, is abundant. Here we will give the first concrete
application of our spectral methods to $\mathcal{N}=4$ SYM, by writing the
spectral decomposition of $\mathcal{G}_{N}(\tau)$. For $SU(2)$, for example,

$\begin{split}\\{\mathcal{G}_{2},E_{s}\\}={\pi\over\sin\pi
s}s(1-s)(2s-1)^{2}\,,\qquad(\mathcal{G}_{2},\phi_{n})=0\end{split}$ (1.4)

The result for all $N$ is extremely simple, with no cusp form overlap. This
demonstrates both how neatly the $\tau$-dependence is couched in the
eigenbasis, and the non-genericity of $\mathcal{G}_{N}(\tau)$ in the space of
possible observables.

In Section 5, we develop the general theory of spectral decomposition of
$\mathcal{N}=4$ SYM observables. We explain how perturbative expansions
$g^{2}\rightarrow 0$, i.e. $y\rightarrow\infty$, may be developed by contour
deformation of (1.2). Because the cusp forms $\phi_{n}(\tau)$ have the special
property that their $k=0$ Fourier modes vanish, they do not appear in
perturbation theory. This simplifies matters. Indeed, a consistent
perturbative expansion — namely, no logarithms or fractional powers of $g^{2}$
— implies that the key player $({\cal O},E_{s})$ must take a simple functional
form (5.7), with rigid analytic structure (see the discussion surrounding
(5.9)). In particular, there is a clear separation between perturbative and
non-perturbative parts of the overlap, i.e. those which contribute power law
terms $\sim y^{-n}$ and those which contribute instanton-anti-instanton terms
$\sim(q\bar{q})^{n}$, where $q\coloneqq e^{2\pi i\tau}$.

We then proceed to derive a bevy of useful results for general ${\cal
O}(\tau)$, first in the simplifying case that $({\cal O},\phi_{n})=0$, later
re-integrating $\phi_{n}(\tau)$. Many observables in perturbative gauge theory
have Borel summable expansions. We introduce the $SL(2,\mathbb{Z})$ Borel
transform, which is a Borel transform specifically tailored to the resummation
of $SL(2,\mathbb{Z})$-invariant functions. Using this we show that not only
are instantons redundant as described earlier, but the radius of convergence
of the $SL(2,\mathbb{Z})$ Borel transform of the perturbative series around
$k$ instantons admits a universal, merely quadratic dependence on $k$ — see
(5.40). We also construct ${\cal O}(\tau)$ as a (regularized)
$SL(2,\mathbb{Z})$ Poincaré sum of its zero mode. This representation is
equivalent to the spectral decomposition. A non-vanishing
$(\mathcal{O},\phi_{n})$ introduces arithmetic quantum chaos [16] to the
$(k\geq 1)$-instanton sectors, thus isolating the chaotic parts of the
instanton data. We provide a sharp diagnostic (5.60) for the presence of cusp
forms in terms of the radius of convergence of the Borel transform of the
expansion around $k\geq 1$ instantons, relying on a result of Kim and Sarnak
[17] toward proving the Ramanujan-Petersson conjecture; in the case that the
cusp forms give factorially-divergent contributions to perturbation theory,
(5.40) generalizes to (5.70), in turn providing a two-way diagnostic.

All of this is then applied to $\mathcal{G}_{N}(\tau)$, which we can
immediately derive and explain as a flagship demonstration of these results.
The lattice-integral representation of [1] is nothing but the Poincaré sum
representation described above, and the integral kernel (called $B_{N}$) is
the $SL(2,\mathbb{Z})$ Borel transform of the $y\rightarrow\infty$
perturbative expansion of the zero mode, $\mathcal{G}_{N,0}(y)$. Furthermore,
the result (1.4) for $\mathcal{G}_{2}(\tau)$ is seen to be the simplest
possible pair of spectral overlaps consistent with
$SL(2,\mathbb{Z})$-invariant weak coupling perturbation theory. We mean
“simplest” in the mathematical, and hopefully uncontroversial, sense that
$(2s-1)^{2}$ is the simplest non-constant, entire function that is even in
$s\rightarrow 1-s$ (a condition required by the functional equation for the
overlap $\\{\mathcal{G}_{2},E_{s}\\}$). The overlaps for $SU(N)$ are then
determined by the recursion relation of [1], or by direct analysis at fixed
$N$. Overall, this illuminates $\mathcal{G}_{N}(\tau)$ as a truly special
object in the space of $\mathcal{N}=4$ SYM observables, and (in our view)
illustrates the clarifying value of the spectral decomposition.

In Section 6, we write the general form of the spectral overlaps at large $N$.

In Section 7, we treat the ’t Hooft limit of large $N$ and fixed
$\lambda\coloneqq g^{2}N$. In this limit, the cusp forms $\phi_{n}(\tau)$ are
suppressed non-perturbatively in $N$. Combining the results of Section 5 with
the double scaling leads to the most general form of the $1/\lambda$ expansion
at strong coupling (see (7.63)). We then observe a powerful consequence of
S-duality:222Subject to a technical assumption tested against examples. the
convergence of the weak-coupling expansion $\lambda\ll 1$ of a CFT observable
${\cal O}(\tau)$ directly leads to the existence of non-perturbative
corrections both at strong coupling $\lambda\gg 1$, and at large $N\gg 1$ and
finite $\lambda$. A remarkable imprint of $SL(2,\mathbb{Z})$ invariance is
that in both cases the strength of the non-perturbative corrections is set by
the radius of convergence of the weak coupling expansion (see (7.20) and
(7.21)). While the $\lambda\gg 1$ effects appear at the non-perturbative scale
$\sim e^{-\sqrt{\lambda}}$, representing fundamental string worldsheet
instantons in AdS${}_{5}\times S^{5}$, S-duality implies that the $N\gg 1$,
fixed $\lambda$ effects appear at the non-perturbative scale $\sim
e^{{-\sqrt{\lambda_{{\mathsf{S}}}}}}$, where $\lambda_{{\mathsf{S}}}=(4\pi
N)^{2}/\lambda$ may be thought of as an “S-dual ’t Hooft coupling”; these
effects are present, and non-perturbative in $N$, in the ordinary ’t Hooft
limit. These reflect D-string instanton effects in AdS${}_{5}\times S^{5}$. We
apply both of these predictions to our prototypical example
$\mathcal{G}_{N}(\tau)$. The non-perturbative effects at strong coupling in
the ’t Hooft limit are consistent with results previously derived by [1]. The
computation of the D-string instanton effects to $\mathcal{G}_{N}(\tau)$ from
resurgence of the weak-coupling expansion is more novel.

In general CFT, the question of whether the $1/N$ expansion of CFT observables
is asymptotic, and whether non-perturbative corrections are needed, is open.
The analysis in Section 7 shows that under certain general conditions in
$\mathcal{N}=4$ SYM, the answer to both is affirmative, and holographically
implies the non-Borel summability of string perturbation theory on
AdS${}_{5}\times S^{5}$.

In Section 8, we treat the “very strongly coupled” (VSC) limit of large $N$
and fixed $g$ [18]. This interesting regime has recently been studied in [14,
19, 20, 21]. Unlike in the ’t Hooft limit, at finite coupling
$SL(2,\mathbb{Z})$ invariance remains manifest, so the $1/N$ expansion of
${\cal O}(\tau)$ involves $SL(2,\mathbb{Z})$-invariant functions at every
order. The constraints on the spectral decomposition in the VSC limit are
similar to (and can be thought of as inherited from) the ’t Hooft limit. As an
example, we explore the VSC limit of an integrated correlator different from
$\mathcal{G}_{N}(\tau)$, which we call ${\cal F}_{N}(\tau)$. This one, studied
in [22, 20] at large $N$, is also supersymmetric. But as quickly becomes
evident upon examining its explicit form, ${\cal F}_{N}(\tau)$ is
substantially more intricate than $\mathcal{G}_{N}(\tau)$. Nevertheless, we
are able to leverage previous results to determine its spectral overlap —
including its overlap with the cusp forms, $({\cal F}_{N},\phi_{n})$ — to the
first few orders in $1/N$.

In Section 9, we pause to flesh out a particularly nice implication of the
large $N$ results for finite $N$ physics. Specifically, we prove that if that
strong ’t Hooft coupling expansion of a CFT observable $\mathcal{O}(\tau)$
contains integer powers of $1/\lambda$, then it receives non-perturbative,
instanton-anti-instanton corrections at finite $N$. This gives an easy
diagnostic of finite $N$ non-perturbative physics. As an application, we show
that unprotected conformal dimensions in $\mathcal{N}=4$ SYM receive non-
perturbative, instanton-anti-instanton corrections.

In Section 10, we return to the ’t Hooft limit and examine the $\lambda\gg 1$
limit in the context of holography. The ensemble average $\langle{\cal
O}\rangle$ now makes its star turn. This quantity must be computed at finite
$N$,333The ’t Hooft double-scaling limit focuses on the cusp of the conformal
manifold, obscuring the underlying $SL(2,\mathbb{Z})$ invariance. but admits a
subsequent $1/N$ expansion. From the spectral decomposition, the $\lambda\gg
1$ expansion of ${\cal O}(\tau)$ takes a general form given in (10.6).
Focusing on the leading term of order $N^{2}$ for simplicity, that result is

${\cal O}(\lambda\gg 1)\approx N^{2}\left(\langle\\!\langle{\cal
O}^{(0)}\rangle\\!\rangle+\sum_{m=0}^{\infty}\mathsf{a}_{m}^{(0)}\lambda^{-{3+m\over
2}}\right)$ (1.5)

where $\mathsf{a}_{m}^{(0)}$ are coefficients. The quantity
$\langle\\!\langle{\cal O}^{(0)}\rangle\\!\rangle$ is the leading large $N$
limit of $\langle{\cal O}\rangle$, i.e.
$\lim_{N\rightarrow\infty}N^{-2}\langle{\cal O}\rangle=\langle\\!\langle{\cal
O}^{(0)}\rangle\\!\rangle$. This equation is familiar, but uncanny: it is the
strong coupling expansion of ${\cal O}$, but with the ensemble average as the
leading term! One deduces that at leading order in large $N$,

$\begin{split}{\cal O}(\lambda\rightarrow\infty)=\langle{\cal
O}\rangle\end{split}$ (1.6)

Of course, by the usual AdS/CFT dictionary, ${\cal
O}(\lambda\rightarrow\infty)={\cal O}_{\rm sugra}$, its value in
AdS${}_{5}\times S^{5}$ supergravity. This leads to the holographic
reformulation

${\cal O}_{\rm sugra}=\langle{\cal O}\rangle$ (1.7)

This relation extends to all orders in $1/N$ in a sense prescribed in (10.6)
and (10.10). We thus have the following picture. On the one hand, holographic
duality between $\mathcal{N}=4$ SYM and AdS${}_{5}\times S^{5}$ string theory
works as it always has. On the other, semiclassical AdS${}_{5}\times S^{5}$
supergravity has two equivalent descriptions via type IIB string theory:
first, as a low-energy limit; and second, as an $SL(2,\mathbb{Z})$ average.

In Section 11 and Section 12, we unfurl some consequences of this.

If AdS${}_{5}\times S^{5}$ supergravity (i.e. strongly coupled planar
$\mathcal{N}=4$ SYM) is also a large $N$ limit of an ensemble average, one is
motivated to study the statistics of the $SL(2,\mathbb{Z})$ ensemble — that
is, the distribution of $\mathcal{N}=4$ SYM observables over moduli space
$\mathcal{M}$. We initiate this study in Section 11. For general observables,
the variance in the $SL(2,\mathbb{Z})$ ensemble admits an expression in terms
of the squares of spectral overlaps:

$\begin{split}{\cal V}(\mathcal{O})&\coloneqq\langle{\cal
O}^{2}\rangle-\langle{\cal O}\rangle^{2}\\\ &=\operatorname{vol}({\cal
F})^{-1}\left({1\over 4\pi i}\int_{\operatorname{Re}s={1\over
2}}ds\,|(\mathcal{O},E_{s})|^{2}+\sum_{n=1}^{\infty}(\mathcal{O},\phi_{n})^{2}\right)\end{split}$
(1.8)

All observables that vary over the conformal manifold $\mathcal{M}$,
supersymmetric or not, necessarily have nonzero variance. At large $N$, the
variance is parametrically suppressed compared to the squared average,

${{\cal V}(\mathcal{O})\over\langle\mathcal{O}\rangle^{2}}\sim{1\over
N}\leavevmode\nobreak\ .$ (1.9)

Using general properties of the large $N$ spectral overlaps, we also find
certain non-perturbative contributions, scaling as positive powers of
$e^{-4\sqrt{\pi N}}$.

Our result exhibits a role for spacetime wormholes even within the
conventional holographic paradigm. (See Figure 4.) A trademark feature of
recently formulated low-dimensional holographic dualities [23, 24, 25, 26]
involving ensemble-averaged boundary duals is the important role played by
spacetime wormholes, which leads to the non-factorization of observables with
multiple distinct boundaries, as the averaging induces correlations between
the boundaries. In a UV complete realization of holography, any wormhole
contributions to the semiclassical bulk path integral must be supplemented by
other contributions which restore factorization of multi-boundary observables.
In our context, the effect of the ensemble average is to project out these UV-
sensitive details; nevertheless, we identify, via the spectral decomposition,
the natural appearance of wormhole-type contributions to (unaveraged) multi-
boundary observables. This is a form of resolution of the “factorization
puzzle” [27, 28] in AdS${}_{5}\times S^{5}$. The spectral decomposition at
large $N$ also provides a clear incarnation of the “half-wormholes” of [29] in
$\mathcal{N}=4$ SYM observables.

We then ask how these results extend to the general holographic context. While
we set out a spectrum of possibilites, the most well-motivated is that large
$N$ averages over more general U-duality symmetries of string theory (i.e.
generalized S-duality symmetries of CFTs), or over exactly marginal directions
dual to the string coupling, may also localize onto supergravity. It is of
clear interest to explore these possibilities further.

In Section 13, we conclude by highlighting some pertinent future directions.

Two appendices contain some technical details complementing the main text. We
highlight Appendix B, which gives a self-contained and general treatment of a
class of functions appearing in the integrated correlators and in other string
theory contexts from the spectral point of view.

##  2 $SL(2,\mathbb{Z})$ Spectral Theory

We will begin with a very brief review444See [30] for a readable introduction.
of the spectral theory of the Laplacian on the fundamental domain domain of
$SL(2,\mathbb{Z})$,

$\mathcal{F}=\mathbb{H}/SL(2,\mathbb{Z})=\left\\{\tau=x+iy\in\mathbb{H}\,\bigg{|}-{1\over
2}\leq x\leq{1\over 2},|\tau|\geq 1\right\\}.$ (2.1)

Functions $f(\tau)$ defined on the upper half-plane $\mathbb{H}$ that are
invariant under the $SL(2,\mathbb{Z})$ action

$f(\gamma\tau)=f(\tau),\quad\gamma\tau={a\tau+b\over
c\tau+d},\quad\gamma=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in
SL(2,\mathbb{Z})$ (2.2)

can be thought of as being defined solely on the fundamental domain
$\mathcal{F}$, as any point outside ${\cal F}$ can be mapped to a point inside
it by a suitable $SL(2,\mathbb{Z})$ transformation. This space is equipped
with the hyperbolic metric

$ds^{2}={dx^{2}+dy^{2}\over y^{2}}$ (2.3)

and the corresponding hyperbolic Laplacian, which acts on scalar functions
as555We note that this is defined with a minus sign compared to the Laplacian
that often appears in the literature. We have defined it this way so that its
spectrum of square-integrable functions is non-negative.

$\Delta_{\tau}=-y^{2}(\partial_{x}^{2}+\partial_{y}^{2}).$ (2.4)

The Laplacian is self-adjoint with respect to the Petersson inner product on
the space $L^{2}(\mathcal{F})$ of square-integrable modular-invariant
functions

$(f,g)\coloneqq\int_{\mathcal{F}}{dxdy\over y^{2}}f(\tau)\overline{g(\tau)}.$
(2.5)

Square-integrability means that the norm defined with respect to the Petersson
inner product is finite,

$(f,f)<\infty.$ (2.6)

Any square-integrable $SL(2,\mathbb{Z})$-invariant function can be expanded in
a complete eigenbasis of the Laplacian, which includes three distinct
components:

$L^{2}({\cal F})=L^{2}_{\rm const}({\cal F})\oplus L^{2}_{\rm cont}({\cal
F})\oplus L^{2}_{\rm disc}({\cal F})$ (2.7)

The expansion is given in (2.25), but let us first present the eigenbasis. The
most trivial is the constant

$\Delta_{\tau}\nu_{0}=0,\quad\nu_{0}=\operatorname{vol}(\mathcal{F})^{-{1\over
2}}=\sqrt{3\over\pi}.$ (2.8)

Next, there is a continuous branch spanned by the real-analytic Eisenstein
series

$\Delta_{\tau}E_{s}(\tau)=\mu(s)E_{s}(\tau),\quad\mu(s)\coloneqq
s(1-s),\quad\operatorname{Re}(s)={1\over 2}.$ (2.9)

The Eisenstein series are defined by a Poincaré series, a sum over
$PSL(2,\mathbb{Z})$ orbits

$E_{s}(\tau)=\sum_{\gamma\in\Gamma_{\infty}\backslash PSL(2,\mathbb{Z})}\rm
Im\leavevmode\nobreak\ (\gamma\tau)^{s},$ (2.10)

where $\Gamma_{\infty}$ is the subgroup of $PSL(2,\mathbb{Z})$ of upper
triangle matrices that fix $y$. This sum converges for
$\operatorname{Re}(s)>1$. However, the Eisenstein series admits a meromorphic
continuation to the entire complex $s$ plane via its Fourier decomposition

$E_{s}(\tau)=y^{s}+\varphi(s)y^{1-s}+\sum_{k=1}^{\infty}4\cos(2\pi
kx){\sigma_{2s-1}(k)\over k^{s-{1\over 2}}\Lambda(s)}\sqrt{y}K_{s-{1\over
2}}(2\pi ky),$ (2.11)

where $\sigma_{n}(x)=\sum_{p|x}p^{n}$ is the divisor function,

$\Lambda(s)\coloneqq\pi^{-s}\Gamma(s)\zeta(2s)=\Lambda({1\over 2}-s)$ (2.12)

is the completed Riemann zeta function, and

$\varphi(s)\coloneqq{\Lambda(1-s)\over\Lambda(s)}.$ (2.13)

In the following application of spectral theory to observables in
$\mathcal{N}=4$ SYM, the Fourier mode number $k$ will represent the total
instanton number. It will be important in what follows that the meromorphic
continuation of the Eisenstein series satisfies the functional equation

$E^{*}_{s}(\tau)=E^{*}_{1-s}(\tau)$ (2.14)

where we’ve defined

$E_{s}^{*}(\tau)\coloneqq\Lambda(s)E_{s}(\tau).$ (2.15)

This is manifest from the Fourier decomposition (2.11). Note from (2.11) that
the Eisenstein series behave perturbatively at the cusp $y=\infty$,

$E_{s}^{*}(\tau)\sim\Lambda(s)y^{s}+\Lambda(1-s)y^{1-s}\qquad(y\to\infty)\,.$
(2.16)

A final comment about the Eisenstein series is that it has a simple pole at
$s=1$ with a constant residue,

$\operatorname*{Res}_{s=1}E_{s}^{*}(\tau)={1\over 2}$ (2.17)

In Appendix B we mention a few details about the higher-order $s\rightarrow 1$
behavior.

Finally, there is a discrete branch of the eigenspectrum of the Laplacian
spanned by the Maass cusp forms. The cusp forms $\nu_{n}(\tau)$, infinite in
number, are labelled by a parameter $t_{n}$ that specifies their eigenvalue,
unbounded from above:

$\Delta_{\tau}\nu_{n}(\tau)=\mu_{n}\nu_{n}(\tau),\quad\mu_{n}\coloneqq\left({1\over
4}+t_{n}^{2}\right),\quad{0}<t_{1}<t_{2}<\ldots$ (2.18)

The $t_{n}$ are a set of sporadic positive real numbers. The cusp forms admit
a Fourier decomposition that is similar to that of the Eisenstein series666We
have here decomposed only the “even” cusp forms, invariant under
$x\rightarrow-x$, ignoring the “odd” cusp forms which are also infinite in
number. Henceforth we treat only even functions ${\cal O}(\tau)$, so all cusp
forms are taken to be even.

$\nu_{n}(\tau)=\sum_{k=1}^{\infty}a_{k}^{(n)}\cos(2\pi
kx)\sqrt{y}K_{it_{n}}(2\pi ky),$ (2.19)

where the Fourier coefficients $a_{k}^{(n)}$ are yet another set of sporadic
real numbers that obey many interesting proven and conjectured statistical
properties (partly summarized in e.g. [31, 16, 32, 33]). From the Fourier
decompositions (2.11) and (2.19) it is manifest that the cusp forms
$\nu_{n}(\tau)$, like the Eisenstein series $E_{s}^{*}(\tau)$, are real-
valued. However, unlike the Eisenstein series, the cusp forms have no zero
mode,

$\int_{-{1\over 2}}^{1\over 2}dx\,\nu_{n}(\tau)=0\,,$ (2.20)

and decay exponentially at the cusp $y=\infty$,

$\nu_{n}(\tau)\sim e^{-2\pi y}\qquad(y\rightarrow\infty)\,.$ (2.21)

There are two conventional normalizations of the cusp forms in the literature.
In the “$L^{2}$ norm”, one rescales the cusp forms so that they have unit norm
with respect to the Petersson inner product

$\phi_{n}(\tau)\coloneqq{\nu_{n}(\tau)\over\sqrt{(\nu_{n},\nu_{n})}},\quad(\phi_{n},\phi_{n})=1\qquad\text{($L^{2}$
norm)}$ (2.22)

In this paper we will mostly make use of the $L^{2}$ norm in order to minimize
notation, but some profound statistical properties of the cusp form Fourier
coefficients are most naturally stated in the “Hecke norm”, which sets the
first Fourier coefficient to unity,

$a_{1}^{(n)}=1\qquad\text{(Hecke norm)}$ (2.23)

We will utilize one such property, on boundedness of Fourier coefficients with
prime $k$ (the Ramanujan-Petersson conjecture), in Subsection 5.7.

(a)  Figure 2: A plot of the zero mode of the square of the Eisenstein series
$E_{11\over 10}(\tau)$ (rescaled by a factor of $1/100$) and of the first
three even Maass cusp forms, defined as
$(\phi_{i}^{2})_{0}(y)\coloneqq\int_{-{1\over 2}}^{1\over
2}dx\,\phi_{i}(\tau)^{2}$, for $0<y\leq 1$. The data for the cusp forms was
obtained using the first 400 Fourier coefficients, known numerically to over
100 digits, available at [34].

To give a feel for these functions, we have shown a plot involving Eisenstein
series and cusp forms in Figure 2. This puts the chaos of the cusp forms in
rather sharp relief: in contrast to the power-law-dominated falloff of the
Eisenstein series, the zero modes of the squares of the cusp forms have an
irregular set of local extrema that varies with $n$.777Since ${\cal F}$ is
two-dimensional, two distinct cusp forms intersect at some nonempty set of
points as $\tau$ varies over ${\cal F}$. However, for three cusp forms, one
would expect no triple intersections, particularly for chaotic functions. This
is confirmed by the plot. The reader is directed to [32] for a lovely
exposition of Maass cusp forms with more pictures.

We would like to emphasize that even the existence of the (infinitely many)
Maass cusp forms is a non-trivial fact, given that $\mathcal{F}$ is
noncompact. Their existence was first established by Selberg, as an
application of the Selberg trace formula (see e.g. [35] for a review). Indeed,
the resulting version of Weyl’s law establishes that the number of square-
integrable eigenfunctions grows asymptotically linearly with the eigenvalue
[36]

$\displaystyle N(\mu)$ $\displaystyle\coloneqq\left(\text{\\# of Maass cusp
forms with eigenvalue ${1\over 4}+t_{n}^{2}\leq\mu$}\right)$ (2.24)
$\displaystyle\sim{\operatorname{vol}({\cal F})\over
4\pi}\mu,\quad\mu\to\infty$

It is the finiteness of $\operatorname{vol}({\cal F})$ which allows the cusp
forms to exist.

Any square-integrable $SL(2,\mathbb{Z})$-invariant function
$\mathcal{O}(\tau)$ then admits a Roelcke-Selberg decomposition into this
complete eigenbasis:

$\mathcal{O}(\tau)=\overline{\mathcal{O}}+{1\over 4\pi
i}\int_{\operatorname{Re}s={1\over
2}}ds\,(\mathcal{O},E_{s})E_{s}(\tau)+\sum_{n=1}^{\infty}(\mathcal{O},\phi_{n})\phi_{n}(\tau).$
(2.25)

The first term $\overline{\cal O}$ is the constant term in the spectral
decomposition. The spectral overlap $(\mathcal{O},E_{s})$ can be written in
terms of a Mellin transform of the zero mode of $\mathcal{O}$, known as the
Rankin-Selberg (RS) transform [37, 38], which inherits many interesting
properties, including a meromorphic continuation in the complex $s$ plane,
from the Eisenstein series. For $\operatorname{Re}s={1\over 2}$, this is given
by

$(\mathcal{O},E_{s})=R_{1-s}[\mathcal{O}]$ (2.26)

where

$\displaystyle R_{s}[\mathcal{O}]$ $\displaystyle\coloneqq\int_{{\cal
F}}{dxdy\over y^{2}}\mathcal{O}(\tau)E_{s}(\tau)$ (2.27)
$\displaystyle=\int_{0}^{\infty}dy\,y^{s-2}\mathcal{O}_{0}(y)$
$\displaystyle=\mathfrak{M}\left[{\mathcal{O}_{0}(y)\over y};s\right]$

Here $\mathcal{O}_{0}(y)=\int_{-{1\over 2}}^{1\over 2}dx\,\mathcal{O}(\tau)$
is the zero mode of $\mathcal{O}$ and
$\mathfrak{M}[f;s]\coloneqq\int_{0}^{\infty}dy\,y^{s-1}f(y)$ is the Mellin
transform. To arrive at this form, we have made use of the fact that the
Eisenstein series is a Poincaré series as in (2.10) and “unfolded” the
integral [37, 38]. The RS transform also satisfies the same functional
equation as the Eisenstein series

$\Lambda(s)R_{s}[\mathcal{O}]=\Lambda(1-s)R_{1-s}[\mathcal{O}].$ (2.28)

From this we see that the modified spectral overlap

$\\{\mathcal{O},E_{s}\\}\coloneqq{(\mathcal{O},E_{s})\over\Lambda(s)}=\\{\mathcal{O},E_{1-s}\\}$
(2.29)

is invariant under the reflection $s\to 1-s$.

The Mellin transform only converges for $\operatorname{Re}s>1$ (and indeed the
form of the Eisenstein series as a Poincaré sum is only valid in this half-
plane); however, the RS transform inherits a meromorphic continuation to the
entire $s$ plane from the Eisenstein series. In particular, its only
singularity in the half-plane to the right of the critical line
$\operatorname{Re}s={1\over 2}$ is a simple pole at $s=1$ with a constant
residue given by

$\operatorname*{Res}_{s=1}R_{s}[\mathcal{O}]=\operatorname{vol}({\cal
F})^{-1}\int_{{\cal F}}{dxdy\over
y^{2}}\mathcal{O}(\tau)=\overline{\mathcal{O}}.$ (2.30)

We see that the residue of the RS transform at $s=1$ encodes the modular
average of the observable $\mathcal{O}(\tau)$ over the fundamental domain of
$SL(2,\mathbb{Z})$, which is also the constant term in its spectral
decomposition.

##  3 Spectral Decomposition in $\mathcal{N}=4$ SYM

Having introduced some $SL(2,\mathbb{Z})$ spectral technology, we transition
to the physics by recalling the necessary salient features of four-dimensional
$\mathcal{N}=4$ SYM. We consider $\mathcal{N}=4$ SYM with arbitrary simply-
laced gauge group $G$, only later specifying $G=SU(N)$ when treating certain
examples. This theory has a one-complex-dimensional $\mathcal{N}=4$
supersymmetry-preserving conformal manifold, $\mathcal{M}$.888The
$\mathcal{N}=1$-preserving conformal manifold is larger, see [39] for a
thorough exposition. $\mathcal{M}$ may be parameterized by $\tau$, the
complexified gauge coupling,

$\tau={\theta\over 2\pi}+{4\pi i\over g^{2}}$ (3.1)

S-duality of $\mathcal{N}=4$ SYM with simply-laced $G$ means invariance of the
theory under $SL(2,\mathbb{Z})$ transformations of $\tau$. This means, in
turn, that (modulo subtleties for line operators [40]) CFT observables ${\cal
O}(\tau)$ are $SL(2,\mathbb{Z})$ invariant,

${\cal O}(\gamma\tau)={\cal O}(\tau)\,,\quad\gamma\in SL(2,\mathbb{Z})\,.$
(3.2)

We would like to emphasize that the term “CFT observables” includes only those
that are non-perturbatively well-defined. The spectrum of the dilatation
operator, OPE coefficients among dilatation eigenstates, and correlation
functions of protected operators are examples of $SL(2,\mathbb{Z})$-invariant
quantities.999On the other hand, “the anomalous dimension of the Konishi
operator” is not non-perturbatively well-defined, as it makes reference to
elementary fields in the $\mathcal{N}=4$ Lagrangian. However, the anomalous
dimension of the lowest-dimension $SU(4)_{R}$-singlet scalar, being agnostic
about the “composition” of the operator, is well-defined and must be
$SL(2,\mathbb{Z})$ invariant. Indeed, we will show in Section 10 that any
quantity which diverges in the large $N$ ’t Hooft limit at $\lambda\gg 1$
cannot be the limit of an $SL(2,\mathbb{Z})$-invariant observable; this
includes all single-trace anomalous dimensions.

Reality of the coupling $g$ implies that $\tau\in{\mathbb{H}}$. S-duality
invariance then implies that we may restrict $\tau$ to the fundamental domain,
whereupon ${\cal O}(\tau)$ becomes a modular-invariant function of
$\tau\in{\cal F}$. A natural measure on $\mathcal{M}$ is determined by the
Zamolodchikov metric,

$d\mu_{\mathcal{M}}\coloneqq g_{\mu\nu}^{Z}(x)dx^{\mu}dx^{\nu}$ (3.3)

where $g_{\mu\nu}^{Z}(x)$ is defined in general as the matrix of two-point
functions of exactly marginal operators. In $\mathcal{N}=4$ SYM, the
Zamolodchikov metric is, owing to maximal supersymmetry, exactly equal to the
Poincaré metric on ${\mathbb{H}}$, so the measure is the
$SL(2,\mathbb{R})$-invariant measure on ${\mathbb{H}}$:

$d\mu_{\mathcal{M}}={dxdy\over y^{2}}$ (3.4)

This is special, and useful. In the notation of Section 2, we have $y\coloneqq
4\pi/g^{2}$, so $g\rightarrow 0$ at fixed $\theta$ approaches the cusp
$y\rightarrow\infty$ at fixed $x$.

Our observation is the following. Consider an $SL(2,\mathbb{Z})$-invariant
observable ${\cal O}(\tau)$. On general grounds, CFT observables ${\cal
O}(\tau)$ are finite for all $\tau$ in the interior of ${\cal F}$, as
singularities can only occur at boundaries of the moduli space. Moreover, the
theory at the cusp $y\rightarrow\infty$ is the free theory, where ${\cal
O}(\tau)$ converges to its (finite) free value. Therefore,

${\cal O}(\tau)\in L^{2}({\cal F})$ (3.5)

This implies that ${\cal O}(\tau)$ admits the Roelcke-Selberg spectral
decomposition into the $SL(2,\mathbb{Z})$-invariant eigenbasis described in
Section 2:

$\boxed{{\cal O}(\tau)=\overline{{\cal O}}+{1\over 4\pi i}\int_{{\rm
Re\,}s={1\over 2}}ds\,\\{{\cal
O},E_{s}\\}\,E^{*}_{s}(\tau)+\sum_{n=1}^{\infty}({\cal
O},\phi_{n})\,\phi_{n}(\tau)}$ (3.6)

This is a simple observation, but it is a far-reaching one. We will spell out
its consequences in detail. However, some of them are visible from the
fundamental structure of the spectral decomposition alone.

Let us develop a Fourier expansion of ${\cal O}(\tau)$, invariant under
$x\rightarrow-x$:

${\cal O}_{k}(\tau)={\cal O}_{0}(y)+\sum_{k=1}^{\infty}2\cos(2\pi kx)\,{\cal
O}_{k}(y)$ (3.7)

$k$ labels the total instanton number. Instanton-anti-instanton pairs
contribute integer powers of $q\bar{q}=e^{-4\pi y}$, where $q\coloneqq e^{2\pi
i\tau}$, with total instanton number zero. The mode ${\cal O}_{k}(y)$ is
obtained from (3.6) by inserting the $k^{\rm th}$ Fourier modes of
$E_{s}(\tau)$ and $\phi_{n}(\tau)$.

Without performing any detailed computation, one sees that the $k=0$ and $k=1$
sectors uniquely determine all $k>1$ sectors. The proof is as follows. The
overlap functions $({\cal O},E_{s})$ and $({\cal O},\phi_{n})$ in the spectral
decomposition may be determined by the $k=0$ and $k=1$ parts of ${\cal
O}(\tau)$, respectively. The former claim is manifest from the definition of
$({\cal O},E_{s})$ as the RS transform (2.27), which involves ${\cal
O}_{0}(y)$ only; note that $\phi_{n}$ has no $k=0$ component, simplifying
matters. Having fixed $({\cal O},E_{s})$ from ${\cal O}_{0}(y)$, one then
fixes $({\cal O},\phi_{n})$ by “inverting” the Bessel function (2.19) using
orthogonality relations [41].101010This same fact was used in [42], where
these mathematical tools were applied to 2d CFT partition functions, to prove
a notion of “spectral determinacy” for 2d CFTs, in which the instanton number
$k$ was instead interpreted as the spin $j$. We note that the inversion of the
cusp forms is not possible in practice, because the chaotic spectrum of
eigenvalues $t_{n}$ is (interestingly!) not fully known. That instantons with
$k>1$ are fully determined by $k\leq 1$ is irrespective of that, as it relies
only on the existence of the spectral decomposition and the structure of the
Fourier expansions of the basis elements.

In this sense, instantons are “redundant” in $\mathcal{N}=4$ SYM. This
surprising simplicity naively suggests that the non-perturbative structure of
instantons — for instance, whether the weak coupling expansions are
convergent, asymptotic, Borel summable, etc, or whether there exist non-
perturbative instanton-anti-instanton corrections in powers of $q\bar{q}$ —
should be the same for all $k$. We will show that, properly understood, this
is correct. Moreover, in favorable cases, the overlap with cusp forms vanishes
— i.e. $({\cal O},\phi_{n})=0$ for all $n$ — in which case ${\cal O}(\tau)$ is
fully determined by the zero-instanton sector alone. We will provide such
examples in $\mathcal{N}=4$ SYM.

Let us now point out a second general consequence of (3.6). Recall that the
first term is the modular average of ${\cal O}(\tau)$ over ${\cal F}$. In a
general CFT with conformal manifold $\mathcal{M}$ coordinatized by moduli
$x^{\mu}$, one may define the ensemble average of ${\cal O}(x)$ as

$\langle{\cal O}\rangle\coloneqq\int d\mu_{\mathcal{M}}\,{\cal O}(x)$ (3.8)

where the integral is supported on the fundamental domain of $\mathcal{M}$. In
$\mathcal{N}=4$ SYM, the Zamolodchikov metric is simply the Poincaré metric,
cf. (3.4), and the fundamental domain is ${\cal F}$. Therefore, the ensemble
average is the modular average:

$\boxed{\langle{\cal O}\rangle=\overline{\cal O}}$ (3.9)

$\overline{\cal O}$ is, we recall from (2.30), determined as a residue of the
RS transform of ${\cal O}(\tau)$ at $s=1$, and is insensitive to the cusp form
contributions to ${\cal O}(\tau)$. So we see that the spectral decomposition
makes manifest the ensemble average of ${\cal O}(\tau)$, and gives a
straightforward way to compute it.

##  4 Integrated Correlator I: Warmup

Before exploring the spectral decomposition of generic observables ${\cal
O}(\tau)$, we introduce $\mathcal{G}_{N}(\tau)$, the integrated correlator
studied by Dorigoni, Green and Wen [15, 1]. In addition to being a rare and
beautiful object, $\mathcal{G}_{N}(\tau)$ will act throughout as an
affirmative benchmark for our methods.

###  4.1 Definition

$\mathcal{G}_{N}(\tau)$ is the four-point function of the ${\cal
O}_{\mathbf{20^{\prime}}}$ superconformal primary operator in the $SU(N)$ SYM
theory, integrated over Euclidean space with the following supersymmetric
measure:

$\mathcal{G}_{N}(\tau)=-{8\over\pi}\int_{0}^{\infty}dr\int_{0}^{\pi}d\theta{r^{3}\sin^{2}\theta\over
u^{2}}{\cal T}_{N}(u,v;\tau)\,,$ (4.1)

where

$u={x_{12}^{2}x_{34}^{2}\over
x_{13}^{2}x_{24}^{2}}=1+r^{2}-2r\cos\theta\,,\qquad
v={x_{14}^{2}x_{23}^{2}\over x_{13}^{2}x_{24}^{2}}=r^{2}$ (4.2)

To be precise, ${\cal T}_{N}(u,v;\tau)$ is the “dynamical” part of the ${\cal
O}_{\mathbf{20^{\prime}}}$ four-point function (after stripping off a power-
law prefactor) that is not determined by $\mathcal{N}=4$ superconformal Ward
identities, in the conventions of [1]. We indicate the $\tau$-dependence
explicitly to emphasize it. That $\mathcal{G}_{N}(\tau)$ is a protected
quantity follows from its alternative formulation as derivatives of a mass-
deformed free energy on $S^{4}$ [14]

$\mathcal{G}_{N}(\tau)=-{1\over 4}\Delta_{\tau}\partial^{2}_{m}\log
Z_{S^{4}}(\tau;m)\big{|}_{m=0}$ (4.3)

where $Z_{S^{4}}(\tau;m)$ may be computed from localization as a matrix
integral over the Cartan of $SU(N)$

$Z_{S^{4}}(\tau;m)\coloneqq\int
d^{N-1}a\left(\prod_{i<j}{a_{ij}^{2}H^{2}(a_{ij})\over
H(a_{ij}-m)H(a_{ij}+m)}\right)|Z_{\rm inst}(\tau)|^{2}e^{-{2\pi
y}\sum_{i}a_{i}^{2}}$ (4.4)

where $a_{ij}\coloneqq a_{i}-a_{j}$ and $H(x)\coloneqq G(1+x)G(1-x)$, with
$G(x)$ the Barnes $G$-function [43, 44].

The authors of [1] conjectured the following form for $\mathcal{G}_{N}(\tau)$
for all $N$ and $\tau$:111111Our conventions differ slightly from [1]. We
normalize our Eisenstein series such that $E_{s}(\tau)\big{|}_{\rm
there}={2\over\Gamma(s)}E^{*}_{s}(\tau)\big{|}_{\rm here}$. Also, in the
formulas to follow, $c_{s}^{(N)}|_{\rm
here}=(-1)^{s+1}{2\over\Gamma(s)}c_{s}^{(N)}|_{\rm there}$. Finally, $y_{\rm
here}={\rm Im\,}(\tau)=y_{\rm there}/\pi$.

$\mathcal{G}_{N}(\tau)=\frac{1}{2}\sum_{(m,n)\in\mathbb{Z}^{2}}\int_{0}^{\infty}d\xi\,B_{N}(\xi)\exp\left(-\pi\xi\,\frac{|m+n\tau|^{2}}{y}\right)\,.$
(4.5)

The function $B_{N}(\xi)$ is a rational function of $\xi$,

$B_{N}(\xi)=\frac{{\cal Q}_{N}(\xi)}{(1+\xi)^{2N+1}}\,,$ (4.6)

where ${\cal Q}_{N}(\xi)$ is a polynomial of order $2N+1$ determined by the
following Laplace difference equation for $\mathcal{G}_{N}(\tau)$:

$\left(\Delta_{\tau}-2\right)\mathcal{G}_{N}(\tau)=N^{2}\Big{[}\mathcal{G}_{N+1}(\tau)-2\mathcal{G}_{N}(\tau)+\mathcal{G}_{N-1}(\tau)\Big{]}-N\Big{[}\mathcal{G}_{N+1}(\tau)-\mathcal{G}_{N-1}(\tau)\Big{]}\,.$
(4.7)

Another useful representation, albeit a formal one that does not converge for
all $\tau\in{\mathbb{H}}$, is as an infinite sum over Eisenstein series,

$\mathcal{G}_{N}(\tau)={N(N-1)\over 8}-{1\over
2}\sum_{s=2}^{\infty}(-1)^{s}c_{s}^{(N)}E^{*}_{s}(\tau)$ (4.8)

For $SU(2)$,

$c_{s}^{(2)}=s(1-s)(2s-1)^{2}$ (4.9)

For $SU(N)$, the $c_{s}^{(N)}$ obey the following recursion relation in $N$:

$N(N-1)c_{s}^{(N+1)}-(2(N^{2}-1)-s(1-s))c_{s}^{(N)}+N(N+1)c_{s}^{(N-1)}=0$
(4.10)

The recursion implies that

$c_{s}^{(N)}=P^{(N)}(s)\,c_{s}^{(2)}$ (4.11)

where $P^{(N)}(s)$ is a polynomial of degree $2N-4$ and $P^{(2)}(s)=1$.

The evidence collected in [1] for this remarkable formula is quite
substantial, including finite $N$ matching to localization; matching to direct
spacetime integration of Feynman integrals at low perturbative order; and
large $N$ matching to expectations from type IIB string theory and S-duality.
Still, it remains conjectural, and its interesting structure begs various
questions. Why does $\mathcal{G}_{N}(\tau)$ have a lattice-integral
representation? Do other $\mathcal{N}=4$ observables have one? What is the
physical meaning of the rational functions $B_{N}(\xi)$? Can the Laplace
difference equation be derived?

Since ${\cal T}_{N}(u,v;\tau)$ is a correlator of protected operators, both
${\cal T}_{N}(u,v;\tau)$ and $\mathcal{G}_{N}(\tau)$ are $SL(2,\mathbb{Z})$
invariant. Focusing on $\mathcal{G}_{N}(\tau)$, what is its spectral
decomposition?

###  4.2 Spectral decomposition

Let us start with the $SU(2)$ case. We claim that $\mathcal{G}_{2}(\tau)$ is
given by the following expression:

$\begin{split}\mathcal{G}_{2}(\tau)={1\over 2}+{1\over 4\pi i}\int_{{\rm
Re\,}s={1\over 2}}ds\,\left({\pi\over\sin\pi
s}s(1-s)(2s-1)^{2}\right)E^{*}_{s}(\tau)\end{split}$ (4.12)

In terms of spectral overlaps, we have

$\begin{split}\\{\mathcal{G}_{2},E_{s}\\}={\pi\over\sin\pi
s}\,c_{s}^{(2)}\,,\qquad(\mathcal{G}_{2},\phi_{n})=0\end{split}$ (4.13)

where $c_{s}^{(2)}$ was defined in (4.9). The claim is that
$\mathcal{G}_{2}(\tau)$ is given exactly by this expression, for all
$\tau\in{\cal F}$: there is no need for regularization or resummation, a fact
which is built into the spectral decomposition, convergent by definition. Note
that the spectral representation (4.12) is functionally rather simpler than
(4.5), being a one-dimensional integral without a lattice sum on top of it.

It is easiest to first recover the formal expression (4.8) for $N=2$ from
(4.12). This is done by contour deformation, with the expansion coefficients
$c_{2}^{(N)}$ encoded as residues of $(\mathcal{G}_{2},E_{s})$ at
$s\in\mathbb{Z}_{>1}$. This deformation also generates a constant contribution
from $s=1$, where $E^{*}_{s}(\tau)$ has a simple pole, cf. (2.17): if we
deform to ${\rm Re\,}s>{1\over 2}$, we pick up a constant term $-1/4$. This
combines to give the correct constant in (4.8). Now, recall that (4.8) is not
convergent, but rather required Borel resummation to become well-defined. This
is precisely encoded in the fact that the integrand of (4.13) diverges
factorially at $s\rightarrow\pm\infty$, due to the completed zeta function
$\Lambda(s)$. The spectral representation (4.12) may thus be thought of as the
Borel resummation of the weak coupling expansion.

If (4.12) is correct, then it should equal the (conjectural) expression (4.5)
too, which is also convergent for all $\tau\in{\cal F}$. We will establish
this equality in Section 5.5 as an example of a more general equality between
spectral decompositions and lattice-integral representations of the same
quantity ${\cal O}(\tau)$, and in doing so, prove the validity of the formula
(4.5) given only the perturbative expansions of the $k=0,1$ modes as input.

The result for $SU(N)$, which follows by recursion (4.10), is the direct
extension of the $SU(2)$ case:

$\begin{split}\boxed{{\mathcal{G}_{N}(\tau)={N(N-1)\over 4}+{1\over 4\pi
i}\int_{{\rm Re\,}s={1\over 2}}ds\,{\pi\over\sin\pi
s}\,c_{s}^{(N)}E^{*}_{s}(\tau)}}\end{split}$ (4.14)

i.e.

$\begin{split}\boxed{\\{\mathcal{G}_{N},E_{s}\\}={\pi\over\sin\pi
s}\,c_{s}^{(N)}\,,\qquad(\mathcal{G}_{N},\phi_{n})=0}\end{split}$ (4.15)

One again easily confirms the match to (4.8) by performing a contour
deformation. Note that the constant term can be determined even without using
the explicit $c_{s}^{(N)}$ as follows. The constant term is determined by

$\begin{split}\operatorname*{Res}_{s=1}R_{s}[\mathcal{G}_{N}]&=P^{(N)}(1)\times\operatorname*{Res}_{s=1}R_{s}[\mathcal{G}_{2}]\\\
&={P^{(N)}(1)\over 2}\end{split}$ (4.16)

So our claim is that $P^{(N)}(1)=N(N-1)/2$. This may be derived for all $N$ by
setting $s=1$ in (4.10), whereby the recursion becomes

$N(N-1)P^{(N+1)}(1)-2(N^{2}-1)P^{(N)}(1)+N(N+1)P^{(N-1)}(1)=0$ (4.17)

This is solved by $P^{(N)}(1)=\alpha N+\beta N^{2}$ for any $\alpha,\beta$.
Using initial data for $N=2,3$ derived from recursion fixes
$\alpha=-\beta=-1/2$, concluding the calculation.

Let us stress two striking features of this result. First, the overlap with
cusp forms vanishes for all $N$. Second, the ensemble average is the constant
term in (4.14):

$\boxed{\langle\mathcal{G}_{N}\rangle={N(N-1)\over 4}}$ (4.18)

Note that at large $N$, we have $\langle\mathcal{G}_{N}\rangle\sim N^{2}/4$.
We explicate these points in much more detail in later sections, including the
systematics of the large $N$ expansion of $\mathcal{G}_{N}(\tau)$.

##  5 Instantons and the Analytic Structure of $\mathcal{N}=4$ SYM

We now systematically explore the consequences of spectral decomposition for
$SL(2,\mathbb{Z})$-invariant observables ${\cal O}(\tau)$. For clarity, we
begin this section by writing formulas assuming $({\cal O},\phi_{n})=0$,
generalizing in due course. We remind the reader that $\phi_{n}(\tau)$ has no
zero mode, so all conclusions about ${\cal O}_{0}(y)$ are independent of this
assumption. It will prove convenient to write out the Fourier modes
explicitly: suppressing the cusp forms in ${\cal O}_{k>0}(y)$,

$\begin{split}{\cal O}_{0}(y)&=\langle{\cal O}\rangle+{1\over 2\pi
i}\int_{{\rm Re\,}s={1\over 2}}ds\,\\{{\cal O},E_{s}\\}\Lambda(s)y^{s}\\\
{\cal O}_{k>0}(y)&={1\over 2\pi i}\int_{{\rm Re\,}s={1\over 2}}ds\,\\{{\cal
O},E_{s}\\}\left({\sigma_{2s-1}(k)\over k^{s-{1\over 2}}}\sqrt{y}K_{s-{1\over
2}}(2\pi ky)\right)\end{split}$ (5.1)

where the mode expansion was developed in (3.7).

###  5.1 Perturbation theory

Our starting point is to understand how to recover weak coupling perturbation
theory from this formalism. This is straightforward, and elegant. Weak
coupling perturbation theory is an expansion around $y\rightarrow\infty$. For
$k=0$, the perturbative expansion is therefore read off from contour
deformation of (5.1) in the appropriate direction. In particular, since
$y\rightarrow\infty$, we must deform $y^{s}$ to the left. This seems in
tension with the fact that the integrand may not fall off sufficiently fast at
$s\rightarrow-\infty$ for this procedure to give a finite result. However,
exactly this fact encodes the convergence, or lack thereof, of the
perturbative expansion of ${\cal O}(\tau)$. For example, if ${\cal O}(\tau)$
admits an asymptotic series, then $({\cal O},E_{s})$ will diverge factorially
at $s\rightarrow-\infty$. Thus, the spectral formalism for gauge theory
observables embeds the asymptotic properties of perturbative expansions in the
$|s|\rightarrow\infty$ asymptotics of spectral overlaps. The virtue of the
spectral decomposition lies in the existence of a convergent expression from
which one can, if desired, develop the weak coupling expansion.

For $k>0$, things work differently. We note the asymptotics

$\sqrt{y}K_{s-{1\over 2}}(2\pi ky)={1\over 2\sqrt{k}}e^{-2\pi
ky}\sum_{n=0}^{\infty}{a_{n}(s)\over(2\pi ky)^{n}}$ (5.2)

where

$a_{n}(s)={(s)_{n}(1-s)_{n}\over(-2)^{n}n!}.$ (5.3)

We highlight that the complete asymptotic expansion of $K_{s-{1\over 2}}(2\pi
ky)$ is a single exponential times a power series in $1/y$, for all $s$.
Inserting in (5.1),

${\cal O}_{k>0}(y)={e^{-2\pi ky}\over 4\pi i}\int_{{\rm Re\,}s={1\over
2}}ds\,\\{{\cal
O},E_{s}\\}\left({\sigma_{2s-1}(k)k^{-s}}\sum_{n=0}^{\infty}{a_{n}(s)\over(2\pi
ky)^{n}}\right)$ (5.4)

Suppose we swap the sum and integral in (5.4),

${\cal O}_{k>0}(y)\approx{e^{-2\pi ky}\over 4\pi
i}\sum_{n=0}^{\infty}y^{-n}\left({1\over(-4\pi k)^{n}n!}\int_{{\rm
Re\,}s={1\over 2}}ds\,\\{{\cal
O},E_{s}\\}{\sigma_{2s-1}(k)k^{-s}}(s)_{n}(1-s)_{n}\right)$ (5.5)

If this is allowed, then it is clear that ${\cal O}_{k}(y)$ receives no non-
perturbative, i.e. instanton-anti-instanton, contributions: such contributions
would dress the $e^{-2\pi ky}$ with extra powers of $q\bar{q}=e^{-4\pi y}$,
but (5.5) is a power series in $1/y$. The factor $\sigma_{2it}(k)k^{-{1\over
2}+it}\in\mathbb{R}$ for $k\in{\mathbb{Z}}$ is oscillatory in $t$ but bounded
in $k$. Therefore, the convergence properties of the bracketed integral are
uniform in $k$. By inspection, convergence for fixed $k$ demands that the
overlap decays faster than any polynomial as $t\rightarrow\infty$.

But clearly, this cannot always hold for $\mathcal{N}=4$ SYM observables,
which can have non-perturbative corrections. Indeed, a short calculation (see
Appendix A) shows that convergence of the spectral decomposition only requires

$\big{|}\\{{\cal O},E_{{1\over 2}+it}\\}\big{|}\lesssim O(e^{\pi
t/2})\qquad(t\rightarrow\infty)$ (5.6)

where we have suppressed power law corrections.

Thus, the presence or absence of non-perturbative corrections to ${\cal
O}_{k>0}(y)$ boils down to whether the integrals in (5.5) converge, which in
turn is determined by whether $\\{{\cal O},E_{{1\over 2}+it}\\}$ decays super-
polynomially as $t\rightarrow\infty$. We show below that this asymptotic is
elegantly determined by the Borel summability of the zero mode, ${\cal
O}_{0}(y)$! The connection, of course, is that ${\cal O}_{0}(y)$ and ${\cal
O}_{k>0}(y)$ are fixed (modulo cusp forms) by the same spectral overlap.

###  5.2 The analytic structure of CFT spectral overlaps

Let us now understand how the mathematical structure of the overlaps is
constrained by our physics problem. We are interested in CFT observables
${\cal O}(\tau)$. These must have a good perturbative expansion: specifically,
we allow only non-negative integer powers of $1/y$, and no powers of $\log y$.
This strongly constrains the form of $\\{{\cal O},E_{s}\\}$, because we can
set $k=0$, and extract the perturbative expansion by contour deformation to
the real axis.121212All formulas for $\\{{\cal O},E_{s}\\}$ that follow are
valid even for $({\cal O},\phi_{n})\neq 0$, since $\phi_{n,0}(y)=0$.

The most general such $\\{{\cal O},E_{s}\\}$ can be written as follows:

$\begin{split}\boxed{\\{{\cal O},E_{s}\\}={\pi\over\sin\pi s}s(1-s)f_{\rm
p}(s)+f_{\rm np}(s)}\end{split}$ (5.7)

The labeling of these functions is hopefully clear. By contour deformation,
$f_{\rm p}(s)$ gives the complete perturbative part of ${\cal O}(\tau)$,
proceeding in integer powers of $1/y$.131313The zero mode of $\mathcal{O}$ may
be recovered from an inverse Mellin transform  ${\cal O}_{0}(y)={1\over 2\pi
i}\int_{\sigma-i\infty}^{\sigma+i\infty}ds\,R_{s}[{\cal O}]y^{1-s}.$ (5.8)
$\sigma\in\mathbb{R}$ is taken in the so-called “fundamental strip,” defined
as the largest open strip $s=\sigma+it$ in which the Mellin transform
converges; since $R_{s}[{\cal O}]$ admits a meromorphic continuation to the
entire complex plane, the fundamental strip is likewise extended to include
$\sigma\in\mathbb{R}$. See e.g. the Direct Mapping Theorem of [45]. There is,
in addition, the possibility of a regular part $f_{\rm np}(s)$ which, having
no poles away from $s=1$, can only contribute to the non-perturbative part of
the modes ${\cal O}_{k}(\tau)$, proceeding in powers of $q\bar{q}=e^{-4\pi
y}$. Unlike the perturbative part, in the non-perturbative part of the
spectral integral we cannot deform the contour to infinity and rewrite it as a
sum over residues.

The functions $f_{\rm p}(s)$ and $f_{\rm np}(s)$ must obey the following
properties:

1) Invariant under reflections, $s\rightarrow 1-s$.

2) Real for $s\in\mathbb{R}$.

3) Regular for $s\in\mathbb{R}$ and $s\notin[0,1]$.

These constraints follow immediately from the preceding exposition. In fact,
these functions obey even stronger constraints, which we prove using number
theory in Appendix A:

3a) $f_{\rm p}(s)$ and $f_{\rm np}(s)$ are regular for all $s\in{\mathbb{C}}$
away from $s=1$ (and its reflection).

3b) At $s=1$,

$\lim_{s\rightarrow 1}\,\\{{\cal O},E_{s}\\}=2\,\overline{{\cal O}}$ (5.9)

3c) On the critical line $s={1\over 2}+it$, $\\{{\cal O},E_{s}\\}$ is finite
for finite $t$.

4) If $f_{\rm np}(s)=0$ (resp. $f_{\rm p}(s)=0$) then $f_{\rm p}(s)$ (resp.
$f_{\rm np}(s)$) is entire.

To summarize, $\\{{\cal O},E_{s}\\}$ is a meromorphic function, regular
everywhere on ${\mathbb{C}}\backslash\\{s\in\mathbb{R}\\}$ with poles at
$s=2,3,\ldots$ and their reflections coming only from the $\sin\pi s$ factor.
Determining the (continuous part of the) spectral decomposition of ${\cal
O}(\tau)$ boils down to finding the two real functions $f_{\rm p}(s)$ and
$f_{\rm np}(s)$ with these fairly rigid properties.

Since $\\{{\cal O},E_{s}\\}$ is defined as a Mellin transform of the zero
mode,

$\\{{\cal O},E_{s}\\}={1\over\Lambda(1-s)}\mathfrak{M}\left[{{\cal
O}_{0}(y)\over y};s\right]$ (5.10)

it is simple to read off the values of $f_{\rm p}(s)$ at $s\in\mathbb{Z}_{>1}$
from the perturbative series: inserting (5.7) into (5.1) and deforming the
contour to develop the $1/y$ expansion gives

${\cal O}_{0}(y\rightarrow\infty)\sim\sum_{n\geq
1}(-1)^{n}c_{n}\,y^{-n}\quad\Rightarrow\quad
c_{n}=-n(n+1)\Lambda\Big{(}n+{1\over 2}\Big{)}f_{\rm p}(n+1)$ (5.11)

As $n\rightarrow\infty$,

$f_{\rm p}(n\rightarrow\infty)\sim-{\pi^{n+{1\over 2}}\over
n^{2}}{c_{n\rightarrow\infty}\over\Gamma(n+{1\over 2})}\,.$ (5.12)

This exhibits the encoding of the asymptotics of the perturbative series
${\cal O}_{0}(y)$ in the $s\rightarrow\infty$ behavior of $f_{\rm p}(s)$.

As for $f_{\rm np}(s)$, since by definition it only generates non-perturbative
terms at $y\rightarrow\infty$, which are necessarily of the form
$y^{-\alpha}e^{-4\pi ny}$ for constants $\alpha,n\in\mathbb{Z}_{+}$, it is
just the sum of Mellin transforms

$\begin{split}\mathfrak{M}\left[y^{-\alpha}e^{-4\pi ny};-s\right]=(4\pi
n)^{s+\alpha}\,\Gamma(-s-\alpha)\end{split}$ (5.13)

Parameterizing the most general series of non-perturbative corrections as

${\cal O}_{0}(y)\supset\sum_{\alpha=0}^{\infty}c_{\alpha}({\cal
O})y^{-\alpha}\sum_{n=1}^{\infty}\kappa^{(\alpha)}_{n}({\cal O})e^{-4\pi
ny}\,,$ (5.14)

where $\kappa_{n}({\cal O})$ and $c^{(n)}_{\alpha}({\cal O})$ are constants,
translates into the general formula for the non-perturbative overlap $f_{\rm
np}(s)$:

${f_{\rm np}(s)=\Lambda(s)^{-1}\sum_{\alpha=0}^{\infty}c_{\alpha}({\cal
O})\Phi^{(\alpha)}_{\cal O}(-s-\alpha)\Gamma(-s-\alpha)}$ (5.15)

where we have introduced a Dirichlet series $\Phi^{(\alpha)}_{\cal O}(s)$

$\Phi^{(\alpha)}_{\cal
O}(s)\coloneqq\sum_{n=1}^{\infty}{\kappa^{(\alpha)}_{n}({\cal O})\over(4\pi
n)^{s}}$ (5.16)

This is associated to the non-perturbative corrections $(q\bar{q})^{n}$ around
each perturbative correction to ${\cal O}(\tau)$.

An interesting fact is that the coefficients $c_{\alpha}({\cal O})$ and
$\kappa_{n}^{(\alpha)}({\cal O})$ are constrained by the reflection symmetry
of $f_{\rm np}(s)$, which acts as a functional equation, $f_{\rm np}(s)=f_{\rm
np}(1-s)$. In other words, defining the “completed” function

$\Psi^{(\alpha)}_{\cal O}(s)\coloneqq\Lambda(s)^{-1}\Phi^{(\alpha)}_{\cal
O}(-s-\alpha)\Gamma(-s-\alpha)$ (5.17)

we have

${f_{\rm np}(s)=\sum_{\alpha=0}^{\infty}c_{\alpha}({\cal
O})\Psi^{(\alpha)}_{\cal O}(s)=\sum_{\alpha=0}^{\infty}c_{\alpha}({\cal
O})\Psi^{(\alpha)}_{\cal O}(1-s)}$ (5.18)

A consistent set of non-perturbative contributions to ${\cal O}(\tau)$ must
satisfy this fascinating functional equation.141414One may view this as a
“bootstrap” equation for the non-perturbative data $\kappa_{n}({\cal O})$ and
$c^{(n)}_{\alpha}({\cal O})$; we do not investigate this here, but believe
this deserves further study. Similarly, it would be interesting to consider
the deeper meaning of these Dirichlet series for CFT observables.

###  5.3 $SL(2,\mathbb{Z})$ Borel transform

Suppose that ${\cal O}_{0}(y)$ has a factorially divergent, Borel summable
expansion. Here and throughout, following the literature on resurgence, we
take “Borel summable” to mean that the Borel transform $B[\xi]$ is non-
singular in a wedge of the complex $\xi$ plane that includes
$\mathbb{R}_{+}$.151515Generically, non-perturbative additions are not
required for a Borel resummed quantity. However, there are “non-generic” cases
where non-perturbative contributions are present, indeed necessary, even if
the perturbative series is Borel summable. Some representative examples
include “Cheshire cat resurgence” [46, 47, 48] as well as certain effects in
string perturbation theory [49, 50, 20]. We thank Daniele Dorigoni for
correspondence on the status of this question. This is a standard situation in
$\mathcal{N}=4$ SYM perturbation theory at finite $N$. If ${\cal O}_{0}(y)$
has factorial growth, then $f_{\rm p}(n\rightarrow\infty)$ is sub-factorial:

$c_{n}\sim(\pi R)^{-n}n!\quad\Rightarrow\quad f_{\rm p}(n)\sim-n^{-{3\over
2}}R^{-n}\qquad(n\rightarrow\infty)$ (5.19)

The constant $R$ is the radius of convergence of the $n\gg 1$ expansion of
$f_{\rm p}(n)$.

For resumming $\mathcal{N}=4$ SYM perturbation theory, it is convenient to
introduce a new, non-standard Borel transform, which we call the
$SL(2,\mathbb{Z})$ Borel transform. If ${\cal O}_{0}(y)$ has the expansion
(5.11), then we define its $SL(2,\mathbb{Z})$ Borel transform $B[\xi]$ as

$B[\xi]\coloneqq\sum_{n=0}^{\infty}{(-1)^{n}c_{n}\over\Lambda(n+{1\over
2})}\,\xi^{n+1}\qquad\text{($SL(2,\mathbb{Z})$ Borel transform)}$ (5.20)

This may be written in terms of the overlap using (5.11):

$B[\xi]=\sum_{n=0}^{\infty}(-1)^{n+1}n(n+1)f_{\rm p}(n+1)\,\xi^{n+1}$ (5.21)

Its utility follows upon noticing that

$\Lambda(s)=\mathfrak{M}\left[{\theta_{3}(y)-1\over 2};s\right]$ (5.22)

where $\theta_{3}(y)=\sum_{n\in{\mathbb{Z}}}e^{-\pi n^{2}y}$ is the Jacobi
theta function. The Borel resummation of the original series may then be
neatly inverted:

$\begin{split}{\widehat{\cal O}_{0}(y)}\coloneqq y^{{1\over
2}}\int_{0}^{\infty}{d\xi\over\xi^{3\over 2}}\left({\theta_{3}(y\xi)-1\over
2}\right){B[\xi]}\end{split}$ (5.23)

This can be confirmed to return the original sum term-wise upon using (5.22).
The radius $R$ in (5.19) is then the radius of convergence of the
$SL(2,\mathbb{Z})$ Borel transform.

The $SL(2,\mathbb{Z})$ Borel transform is so named because it accounts for the
$\Lambda(n+{1\over 2})$ factor which appears universally in perturbation
theory for $SL(2,\mathbb{Z})$-invariant functions. In other words, the Jacobi
theta kernel resums perturbation theory while manifestly respecting S-duality.
Whereas an ordinary Borel transform divides by $\Gamma(n+1)$, the
$SL(2,\mathbb{Z})$ Borel transform also properly eliminates the Riemann zeta
factor.161616This answers a challenge posed in [11]. We note that other
commonly used modified Borel transforms divide by $\Gamma(n+1)\zeta(n+1)$ (as
in [51, 1]) whereas here, the Riemann zeta argument is twice that of the gamma
argument, by definition of the completed Riemann zeta function.

Just as for an ordinary Borel transform, $\widehat{\cal O}_{0}(y)$ is analytic
for ${\rm Re\,}(y)>0$ in the complex $y$ plane. This domain of analyticity may
be extended to a wedge by considering the “directional” inverse Borel
transform along a ray $e^{i\theta}\xi$: this yields an analytic function for
${\rm Re\,}(e^{i\theta}y)>0$, i.e. in the domain

$D_{\theta}\coloneqq\\{y\in\mathbb{C}\leavevmode\nobreak\
|\leavevmode\nobreak\ {\rm Re\,}(e^{i\theta}y)>0\\}$ (5.24)

It can be shown by analytic continuation (e.g. [51]) that if $B[\xi]$ is
analytic for $\text{arg}(\xi)\in[\theta_{1},\theta_{2}]$, then $\widehat{\cal
O}_{0}(y)$ is analytic in the union of domains $D_{\theta_{1}}\cup
D_{\theta_{2}}$.

Note that the $n\rightarrow 1-n$ transformation properties of the summand
immediately imply the inversion symmetry

${B[\xi]\over\sqrt{\xi}}=\sqrt{\xi}B[\xi^{-1}]$ (5.25)

This inversion implies the non-trivial integral relation

$\int_{0}^{\infty}d\xi{B[\xi]\over\xi}=\int_{0}^{\infty}{d\xi\over\xi}{B[\xi]\over\xi}$
(5.26)

In fact, using (5.22) and (5.23), this integral is just twice the modular —
i.e. ensemble — average of ${\cal O}$:

$\begin{split}\langle{\cal O}\rangle&={1\over
2}\int_{0}^{\infty}{d\xi}{B[\xi]\over\xi}\end{split}$ (5.27)

Finally, note that171717Both (5.27) and (5.28) can be seen to follow from the
fact that the $SL(2,\mathbb{Z})$ Borel transform can be written in terms of an
inverse Mellin transform of the RS transform of $\mathcal{O}$
$B[\xi]=\xi\,\mathfrak{M}^{-1}\left[{R_{s}[\mathcal{O}]\over\Lambda(1-s)};\xi\right]$

$\int_{0}^{\infty}{d\xi\over\xi^{3\over
2}}B[\xi]=-2\operatorname*{Res}_{s={1\over 2}}R_{s}[{\cal O}]=0$ (5.28)

This follows from square-integrability of ${\cal O}(\tau)$, together with the
definition of the RS transform as an inner product involving $E_{s}(\tau)$
[52], which obeys $E_{1/2}(\tau)=0$.

An interesting implication of (5.28) is that $B[\xi]$ is not sign-definite for
$\xi\geq 0$. The physical meaning of zeros of Borel transforms has not been
properly understood in a QFT context (though see [53] for interesting
observations in 3d $\mathcal{N}=2$ SCFT). It would be nice to understand why
$SL(2,\mathbb{Z})$ Borel transforms of $\mathcal{N}=4$ SYM observables must
have zeros.

###  5.4 The redundancy of instantons

Having better understood the structure of overlaps $\\{{\cal O},E_{s}\\}$ in
$\mathcal{N}=4$ SYM, we are in good position to prove relations among non-
perturbative corrections in different $k$-instanton sectors.

We return to the problem posed earlier, below (5.6). There is a very nice
theorem about the decay of Mellin transforms along strips $s=\sigma+it$ as
$t\rightarrow\infty$ (cf. Proposition 5 of [45]):

Let $f(y)$ be analytic in the region

$S_{\theta}=\\{y\in{\mathbb{C}}\leavevmode\nobreak\ |\leavevmode\nobreak\
0<|y|<\infty\,,\leavevmode\nobreak\
|\text{arg}(y)|\leq\theta\\}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{with}\leavevmode\nobreak\ \leavevmode\nobreak\
0<\theta<\pi$ (5.29)

Assume that $f(y)\sim O(y^{-\alpha})$ as $y\rightarrow 0$ and $f(y)\sim
O(y^{-\beta})$ as $y\rightarrow\infty$ in $S_{\theta}$. Then

$\mathfrak{M}[f(y);\sigma+it]=O(e^{-\theta|t|})\leavevmode\nobreak\
\leavevmode\nobreak\ \text{as}\leavevmode\nobreak\ \leavevmode\nobreak\
|t|\rightarrow\infty$ (5.30)

uniformly for $\sigma$ in every closed subinterval of $(\alpha,\beta)$. This
extends to any subinterval of the strip in which $\mathfrak{M}[f(y);s]$ admits
a meromorphic continuation.

The above theorem shows that $\\{{\cal O},E_{s}\\}$ will decay exponentially
as long as ${\cal O}_{0}(y)$ is analytic in a wedge of the complex $y$ plane.
Happily, this follows directly from Borel summability of ${\cal O}_{0}(y)$:
since ${\cal O}_{0}(y)$ is analytic in a finite wedge
$\arg(y)\in[\theta_{1},\theta_{2}]$ around the positive $y$ axis

$\begin{split}\mathfrak{M}\left[{{\cal O}_{0}(y)\over y};s\right]\sim
e^{-\left({\pi\over
2}+\theta\right)|t|}\,,\quad\theta\coloneqq\text{min}\\{|\theta_{1}|,|\theta_{2}|\\}>0\end{split}$
(5.31)

Crucially, $\theta$ is strictly positive. This holds on the critical line
$\sigma=1/2$.181818Indeed, (5.7) implies a meromorphic continuation of
$\mathfrak{M}\left[{{\cal O}_{0}(y)\over y};s\right]$ to all $s\in\mathbb{C}$,
so this theorem holds for any finite $\sigma\in\mathbb{R}$. Together with the
asymptotics

$\begin{split}\Big{|}{\Lambda({1\over 2}-it)^{-1}}\Big{|}\sim e^{\pi t\over
2}\log|t|\qquad(t\rightarrow\infty)\end{split}$ (5.32)

derived in Appendix A, we arrive the final result

$\begin{split}\Big{|}\\{{\cal O},E_{{1\over 2}+it}\\}\Big{|}\lesssim
e^{-\theta t}\,,\quad\theta>0\qquad(t\rightarrow\infty)\,.\end{split}$ (5.33)

Thus, $\\{{\cal O},E_{{1\over 2}+it}\\}$ decays exponentially as
$t\rightarrow\infty$, at a rate determined by the location of the singularity
of the Borel transform $B[\xi]$. This guarantees finiteness of the integrals
in (5.5), for all $k$. Note that if ${\cal O}_{0}(y)$ has singularities only
on the negative real axis, $y\in{\mathbb{R}}_{-}$, then
$\theta\rightarrow\pi$.

We can in fact go further, by characterizing the $n\rightarrow\infty$
asymptotics of the sum (5.5). The best way to do this is by plugging in (5.7)
and deforming to ${\rm Re\,}s>{1\over 2}$, because then we can compare
directly to $B[\xi]$, the Borel transform of ${\cal O}_{0}(y)$, in (5.21).
Dropping an overall $(-2\pi i)$, the coefficient of $y^{-n}$ is

$S_{n,k}\coloneqq{1\over(-4\pi)^{n}n!}\sum_{s=1}^{\infty}(-1)^{s}s(s-1)f_{\rm
p}(s)(s)_{n}(1-s)_{n}\sigma_{2s-1}(k)k^{-s-n}$ (5.34)

The $SL(2,\mathbb{Z})$ Borel transform of ${\cal O}_{k}(y)$ is, in terms of
$S_{n,k}$,

$B_{k}[\xi]\coloneqq\sum_{n=0}^{\infty}{S_{n,k}\over\Lambda\left(n+{1\over
2}\right)}\xi^{n}$ (5.35)

We must thus extract the $n\gg 1$ asymptotics of $S_{n,k}$. This is
straightforward. Set $k=1$ for the moment. We have

$\begin{split}{S_{n,1}\over\Lambda\left(n+{1\over
2}\right)}&={1\over(-4\pi)^{n}n!\Lambda\left(n+{1\over
2}\right)}\sum_{s=1}^{\infty}(-1)^{s}s(s-1)f_{\rm p}(s)(s)_{n}(1-s)_{n}\\\
&\approx{1\over(-4)^{n}}\sum_{p=0}^{\infty}(-1)^{p+1}(p+n+1)(p+n)f_{\rm
p}(p+n+1){(p+1)_{2n}\over(n!)^{2}}\end{split}$ (5.36)

Here we used that $(1-s)_{n}=0$ for integer $s\leq n$ and
$(-x)_{n}=(x-n)_{n}(-1)^{n}$, shifted the sum, and dropped irrelevant factors
subleading at $n\gg 1$. Taking the zero mode to be $SL(2,\mathbb{Z})$ Borel
summable with radius $R>0$ means (cf. (5.19))

$f_{\rm p}(s\gg 1)\sim R^{-s}$ (5.37)

So we insert this scaling into the above equation, perform the sum over $p$,
and afterwards expand at large $n$. Reinstating $k$-dependence is done by
using the asymptotics

$\sigma_{2s-1}(k)k^{-s-n}\big{|}_{s\gg 1}\sim k^{s-n-1}$ (5.38)

This leads to the insertion of a $k^{p}$ in (5.36). The sum may be performed
exactly. Quoting only the $n\gg 1$ result of interest, and suppressing power
law prefactors,

$\begin{split}{S_{n,k}\over\Lambda\left(n+{1\over
2}\right)}&\sim{\left(-{R\over(R+k)^{2}}\right)^{n}}\qquad(n\rightarrow\infty)\end{split}$
(5.39)

Plugging into (5.35) gives the desired result: Borel summability of ${\cal
O}_{0}(y)$ implies Borel summability of ${\cal O}_{k}(y)$ for all $k$! That
is, $B_{k}[\xi]$ is convergent and free of singularities on $\mathbb{R}_{+}$.
Moreover, we have a very simple polynomial formula for $R_{k}$, the radius of
convergence of ${B}_{k}[\xi]$, for all $k$:

$\begin{split}\boxed{{R_{k}\over R}={\left(1+{k\over
R}\right)^{2}}}\end{split}$ (5.40)

This is a powerful demonstration of the ability of $SL(2,\mathbb{Z})$ to
relate different instanton sectors. Other notable properties of $R_{k}$
include its monotonic increase with $k$, and its quadratic growth at $k\gg 1$.

This result transcends the usual philosophy of resurgence methods. In typical
applications of resurgence, one must perform independent resummations in
different instanton sectors. In the presence of extra symmetries, more is
possible. What we are seeing here is that $SL(2,\mathbb{Z})$-invariance is
strong enough to relate all $k$-instanton sectors in a simple way, with the
$k=0$ sector (and, upon including cusp forms, the $k=1$ sector) being the only
independent data. In the language of the so-called “resurgence triangle” [54,
55], the claim is that $SL(2,\mathbb{Z})$-invariance moves horizontally within
the triangle, determining all columns from a finite subset.

This formula may actually be extended to non-Borel summable ${\cal O}_{0}(y)$
as well, assuming the perturbative series still grows only factorially. By
“non-Borel summable,” we mean that $B[\xi]$ has a singularity at
$\xi=R\in\mathbb{R}_{+}$. Relative to the Borel summable case with a
singularity along $\mathbb{R}_{-}$, this modifies the asymptotic (5.39) by
$R\rightarrow-R$; in addition, while the ensuing resummation acquires non-
perturbative corrections to ensure analyticity of the end result
$\widehat{\cal O}_{0}(y)$, none of these corrections affects the perturbative
expansion. Propagating this through, the same logic as above implies that non-
Borel summability of ${\cal O}_{0}(y)$ implies non-Borel summability of ${\cal
O}_{k}(y)$ for all $k$, where $B_{k}[\xi]$ is singular at $\xi=R_{k}$ subject
to the relation (5.39) with $(R,R_{k})\rightarrow-(R,R_{k})$.

Finally, for the sake of clarity, we remind that these conclusions hold modulo
possible contributions from cusp forms. We will reinstate the cusp forms in
Subsection 5.7, giving a diagnostic for their presence and generalizing (5.40)
to (5.70). Moreover, some observables have no cusp form support, in which case
the above results are unconditional.

###  5.5 $\mathcal{N}=4$ SYM observables as Poincaré sums

The previous section derived quantitative formulas for $k$-instanton dynamics
from the zero-instanton dynamics. That this should be possible followed from
the fundamental determinism baked into the $SL(2,\mathbb{Z})$ spectral
decomposition. In this section we show that starting from the zero mode ${\cal
O}_{0}(y)$, one can construct an interesting representation of the full
function ${\cal O}(\tau)$ — as an $SL(2,\mathbb{Z})$-invariant lattice sum —
which is equivalent to the spectral decomposition.

The idea is simply to sum over $SL(2,{\mathbb{Z}})$ images of ${\cal
O}_{0}(y)$. This uniquely determines the function up to overall normalization,
which is fixed by demanding that it produces the correct ${\cal O}_{0}(y)$.

The starting point is (5.23), the Borel resummation of $\mathcal{O}_{0}(y)$,
which we rewrite as

$\mathcal{O}_{0}(y)=\sqrt{y}\int_{0}^{\infty}{d\xi\over\xi^{3\over
2}}B[\xi]\sum_{n=1}^{\infty}e^{-\pi n^{2}\xi y}.$ (5.41)

We will now proceed to perform a regularized Poincaré sum over images. Since
$\mathcal{O}_{0}(y)$ depends only on $y$, we sum over
$\Gamma_{\infty}\backslash PSL(2,\mathbb{Z})$, modding out by the invariant
action of the modular $T$-transformation on the seed. Let us call the
resulting function $\mathcal{O}(\tau)$:191919These Poincaré series are
divergent, but admit a natural zeta function-like regularization. This is most
clearly reflected in (5.45). See [56, 57] for a discussion of similar Poincaré
series in the context of the partition function of pure three-dimensional
quantum gravity as defined by the sum over smooth saddle points in the
gravitational path integral.

$\mathcal{O}(\tau)\coloneqq\mathcal{N}\sum_{\Gamma_{\infty}\backslash
PSL(2,\mathbb{Z})}\operatorname{Im}(\gamma\tau)^{{1\over
2}}\int_{0}^{\infty}{d\xi\over\xi^{3\over 2}}B[\xi]\sum_{n=1}^{\infty}e^{-\pi
n^{2}\xi\operatorname{Im}(\gamma\tau)},$ (5.42)

where $\mathcal{N}$ is a normalization constant that we will fix later. We now
rewrite this via Poisson summation as

$\displaystyle\mathcal{O}(\tau)$ $\displaystyle={\mathcal{N}\over
2}\sum_{\gamma\in\Gamma_{\infty}\backslash
PSL(2,\mathbb{Z})}\left[\int_{0}^{\infty}{d\xi\over\xi^{2}}B[\xi]\sum_{m\in\mathbb{Z}}\exp\left({-{\pi
m^{2}\over\xi\operatorname{Im}(\gamma\tau)}}\right)-\operatorname{Im}(\gamma\tau)^{{1\over
2}}\int_{0}^{\infty}{d\xi\over\xi^{3\over 2}}B[\xi]\right]\,.$ (5.43)

The last term vanishes identically, cf. (5.28). To proceed, we expand the
exponential in (5.43) and note that

$\sum_{\gamma\in\Gamma_{\infty}\backslash
PSL(2,\mathbb{Z})}\sum_{m=1}^{\infty}\left({m^{2}\over\operatorname{Im}(\gamma(\tau))}\right)^{s}={1\over
2}\sum_{(m,n)\neq(0,0)}\left({|m\tau+n|^{2}\over y}\right)^{s}$ (5.44)

to arrive at

$\mathcal{O}(\tau)={\mathcal{N}\over
2}\left[\sum_{\gamma\in\Gamma_{\infty}\backslash
PSL(2,\mathbb{Z})}\int_{0}^{\infty}{d\xi\over\xi^{2}}B[\xi]+\sum_{(m,n)\neq(0,0)}\int_{0}^{\infty}{d\xi\over\xi^{2}}B[\xi]\exp\left({-{\pi|m\tau+n|^{2}\over\xi
y}}\right)\right].$ (5.45)

The last formal equality we need is that
$\sum_{\gamma\in\Gamma_{\infty}\backslash PSL(2,\mathbb{Z})}=E_{0}(\tau)=1$, a
regularization inherited from the meromorphic continuation of the Eisenstein
series. We can then combine the two terms and make use of the inversion
property (5.25) satisfied by the $SL(2,\mathbb{Z})$ Borel transform to arrive
at

$\mathcal{O}(\tau)={\mathcal{N}\over
2}\sum_{(m,n)\in\mathbb{Z}^{2}}\int_{0}^{\infty}{d\xi}{B[\xi]\over\xi}\exp\left({-{\pi\xi|m\tau+n|^{2}\over
y}}\right)\,.$ (5.46)

This is the final result up to the determination of the normalization constant
$\mathcal{N}$. One may borrow the analysis of e.g. [1], done there for the
specific observable $\mathcal{O}=\mathcal{G}_{N}$ but applicable more
generally, to find $\mathcal{N}=1/2$. Thus we have

$\begin{split}\boxed{{\cal O}(\tau)={1\over
4}\sum_{(m,n)\in{\mathbb{Z}}^{2}}\int_{0}^{\infty}d\xi{B[\xi]\over\xi}\exp\left({-\pi\xi{|m+n\tau|^{2}\over
y}}\right).}\end{split}$ (5.47)

To summarize, we have constructed the observable ${\cal O}(\tau)$ with zero
mode ${\cal O}_{0}(y)$ as a Poincaré sum of ${\cal O}_{0}(y)$. This is not
sensitive to any details of $B[\xi]$. The resulting formula is necessarily
equivalent to its spectral decomposition. In other words, one can construct
any modular function ${\cal O}(\tau)\in L^{2}({\cal F})$ uniquely modulo cusp
forms from its zero mode ${\cal O}_{0}(y)$ in (at least) two equivalent ways:
first, as a Poincaré sum over ${\cal O}_{0}(y)$, with the resulting lattice-
integral expression (5.47); and second, as a spectral decomposition (5.1). Of
course, as we’ve been stressing throughout this work, in the absence of cusp
forms the $k>0$ sectors do not contain any new information.

This construction also makes clear that the resulting function is orthogonal
to cusp forms, because any Poincaré sum over a zero mode has vanishing cusp
form overlap: the “unfolding trick” reduces the overlap to a $y$ integral over
the zero mode of the cusp forms, $\phi_{n,0}=0$, cf. (2.20).

We observe that the constant $(m,n)=(0,0)$ term is half of the ensemble
average of ${\cal O}(\tau)$, cf. (5.27).

The formula also applies to non-Borel summable ${\cal O}_{0}(y)$, with
suitable tweaking of $B[\xi]$. Singularities in $B[\xi]$ for
$\xi\in\mathbb{R}_{+}$ lead to non-perturbative terms in ${\cal O}_{0}(y)$,
which arise from taking $B[\xi]\rightarrow\tilde{B}[\xi]$ for some appropriate
$\tilde{B}[\xi]$ whose resummation is analytic. A typical approach is “median
resummation,” which does so by defining the resummation $\widehat{\cal
O}_{0}(y)$ by integrating ${\rm Re\,}B[\xi]$ instead of $B[\xi]$ — see e.g.
[58]. Since the derivation of (5.47) only used the definition (5.23), it
exists even for non-Borel summable cases upon substituting $\tilde{B}[\xi]$
instead.

###  5.6 Integrated correlator II: Derivation

Everything we have established in the previous subsections is beautifully on
display for the integrated correlator $\mathcal{G}_{N}(\tau)$, whose form we
are able to efficiently explain and derive.

In terms of the overlap functions in (5.7), let us write

$\\{\mathcal{G}_{N},E_{s}\\}={\pi\over\sin\pi s}s(1-s)f_{{\rm p},N}(s)+f_{{\rm
np},N}(s)$ (5.48)

For $SU(2)$, from (4.9) and (4.13) we have

$f_{{\rm p},2}(s)=(2s-1)^{2}\,,\quad f_{{\rm np},2}(s)=0$ (5.49)

with the $SU(N)$ results determined by recursion (4.10). This is extremely
simple. In fact, it is the simplest possible non-trivial overlap consistent
with the necessary constraints! Setting $f_{\rm np}(s)=0$ to eliminate non-
perturbative terms, we recall from Section 5.2 that $f_{\rm p}(s)$ must then
be an entire, real, even function of $s-{1\over 2}$. The simplest entire
functions are polynomials. If $f_{\rm p}(s)$ is constant, the corresponding
$SL(2,\mathbb{Z})$ Borel transform is $B[\xi]\propto 2\xi^{2}/(1+\xi)^{3}$,
which is sign definite for $\xi>0$, contradicting (5.28). The next simplest
possibility is the degree-two monomial, which is the case realized by
$\mathcal{G}_{2}(\tau)$.

The lattice-integral formula for $\mathcal{G}_{N}(\tau)$ was given in (4.5).
Comparing to (5.47), we see immediately that

${B[\xi]\over 2\xi}=B_{N}(\xi)$ (5.50)

The kernel of the lattice-integral is simply the $SL(2,\mathbb{Z})$ Borel
transform of the zero mode. (One can check for sanity, e.g. by integrating
numerically, that the $SL(2,\mathbb{Z})$ Borel resummation of
$\mathcal{G}_{N,0}(y)$ is equal to the Borel resummations in [1] that were
performed with respect to more conventional kernels.)

It is satisfying that this derivation elucidates the meaning of $B_{N}(\xi)$,
which had previously been opaque. We also understand why it must be
independent of the instanton index $k$, and why it obeys the inversion
symmetry

$\sqrt{\xi}B_{N}(\xi)={B_{N}(\xi^{-1})\over\sqrt{\xi}}$ (5.51)

This was seen to be crucial for the $SL(2,\mathbb{Z})$ invariance of the
lattice sum in [1]; we now understand inversion symmetry as a consequence of
$SL(2,\mathbb{Z})$ invariance. Moreover, the integral of $B_{N}(\xi)$ is
simply its ensemble average, as implied by (5.27):

$\langle\mathcal{G}_{N}\rangle=\int_{0}^{\infty}d\xi B_{N}(\xi)={N(N-1)\over
4}$ (5.52)

where the second equality uses the result previously extracted from the
spectral decomposition, cf. (4.18). In addition, due to (5.28)

$\int_{0}^{\infty}{d\xi\over\sqrt{\xi}}B_{N}[\xi]=0$ (5.53)

These two equalities were observed in [1].

In Subsection 5.5 we showed that the lattice-integral representation (5.47) is
derived from the zero mode ${\cal O}_{0}(y)$. Applying this to ${\cal
O}=\mathcal{G}_{N}$ thus furnishes a proof of the formula (4.5), taking only
the perturbative expansion of the zero mode $\mathcal{G}_{N,0}(y)$ as input.
Moreover, (5.47) is manifestly equivalent to the spectral decomposition. To
make the proof truly complete, it remains to show that
$(\mathcal{G}_{N},\phi_{n})=0$. This follows from showing that the one-
instanton mode $\mathcal{G}_{N,1}(y)$ is fully reproduced by the lattice-
integral representation (4.5), so in total, one needs both
$\mathcal{G}_{N,0}(y)$ and $\mathcal{G}_{N,1}(y)$ as input.

Finally, let us check the radius of convergence formula (5.40), which should
hold exactly for $\mathcal{G}_{N}(\tau)$. At $k=0$, one quickly deduces that
$R=1$, so (5.40) predicts that

$R_{k}=(1+k)^{2}$ (5.54)

To test this, it is convenient to use equation (3.22) of [1] at $N=2$,

$\begin{split}\mathcal{G}_{2,k}(y)&=e^{-2\pi
ky}\mathop{\sum_{p,q\,\in\,\mathbb{Z}_{+}}}_{pq=k}z^{2}(p+q)[z\left(11p^{2}+2pq+11q^{2}\right)(p-q)^{2}+2z^{2}(p+q)^{2}(p-q)^{4}\\\
&+9p^{2}-12pq+9q^{2}]-\frac{\sqrt{\pi}}{2}z^{3\over
2}e^{z(p+q)^{2}}\left[4z^{3}\left(p^{2}-q^{2}\right)^{4}+24z^{2}\left(p^{2}+q^{2}\right)\left(p^{2}-q^{2}\right)^{2}\right.\\\
&\left.+\,3z\left(9p^{4}-2p^{2}q^{2}+9q^{4}\right)+3\left(p^{2}+q^{2}\right)\right]\text{erfc}\left(\sqrt{z}(p+q)\right)\end{split}$
(5.55)

where $z\coloneqq\pi y$. The leading $y\gg 1$ behavior comes from the
complementary error function alone, which admits an expansion

$\pi(p+q)\sqrt{z}\,e^{z(p+q)^{2}}\text{erfc}\left(\sqrt{z}(p+q)\right)=\sum_{n=0}^{\infty}\left(-{1\over\pi(p+q)^{2}}\right)^{n}\Gamma\Big{(}n+{1\over
2}\Big{)}y^{-n}$ (5.56)

Therefore, applying our previous definitions for the $SL(2,\mathbb{Z})$ Borel
transform,

$R_{k}=\underset{p,q}{\text{max}}\big{[}(p+q)^{2}\big{]}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{s.t.}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
{p,q\in\mathbb{Z}_{+}}\,,\leavevmode\nobreak\ {pq=k}$ (5.57)

The maximum is given by $(p,q)=(k,1)$, thus proving (5.54). The result for
general $N$ follows from recursion.

####  5.6.1 Comment on integrated $\langle 22pp\rangle$ correlator

One may instead consider the supersymmetric integrated $\langle 22pp\rangle$
correlator, where “$p$” denotes the half-BPS superconformal primary scalar
${\cal O}_{p}$ of dimension $\Delta=p$ in the $[0\,p\,0]$ representation of
$SU(4)_{R}$. Let us call this quantity $\mathcal{G}_{22pp}(\tau)$, slightly
abusing earlier notation and leaving the $N$-dependence implicit.
$\mathcal{G}_{22pp}(\tau)$ may also be computed via localization. Adding
sources for integrated ${\cal O}_{p}$ to $Z_{S^{4}}(\tau;m)$,

$Z_{S^{4}}(\tau,\tau_{p};m)\coloneqq\int
d^{N-1}a\left(\prod_{i<j}{a_{ij}^{2}H^{2}(a_{ij})\over
H(a_{ij}-m)H(a_{ij}+m)}\right)|Z_{\rm
inst}(\tau,\tau_{p})|^{2}\exp\left({-2\sum\limits_{p=2}^{\infty}\pi^{p\over
2}{\rm Im\,}(\tau_{p}^{\prime})\sum_{i}(a_{i})^{p}}\right)$ (5.58)

where $\tau^{\prime}_{2}\coloneqq\tau$, one then differentiates with respect
to sources to obtain

$\mathcal{G}_{22pp}(\tau)=\partial_{\tau_{p}^{\prime}}\partial_{\bar{\tau}_{p}^{\prime}}\partial_{m}^{2}\log
Z_{S^{4}}(\tau,\tau_{p};m)\big{|}_{m=\tau_{p}^{\prime}=0}$ (5.59)

This is slightly schematic due to operator mixing issues on $S^{4}$; see [59,
14] for an exact expression.

What is the spectral decomposition of $\mathcal{G}_{22pp}(\tau)$? In
particular, we are interested in whether the $p$-dependence affects the
structural simplicity seen for $p=2$. We address this by examining
$y\rightarrow\infty$ perturbation theory, setting $Z_{\rm
inst}(\tau,\tau_{p})=1$. Compared to the $p=2$ case, the derivatives of $\log
Z_{S^{4}}(\tau,\tau_{p};m)$ simply bring down $p-2$ extra powers of the
eigenvalues $a_{i}$. These powers multiply $p$-dependent coefficients. These
coefficients do not contain factorials, nor do the manipulations leading to
$\mathcal{G}_{22pp}(\tau)$, including the mixing matrix, lead to any.
Therefore, generalizing the computations for $\mathcal{G}_{N}(\tau)$, we learn
that the perturbative expansion of $\mathcal{G}_{22pp}(\tau)$ is Borel
summable. Applying the logic of resurgence for generic observables would imply
that $f_{\rm np}(s)=0$ for $\mathcal{G}_{22pp}(\tau)$ as well.

As for the cusp forms, we have not tried to formulate a rigorous argument as
to their absence; this would involve the structure of $Z_{\rm
inst}(\tau,\tau_{p})$, whose explicit form is unknown. However, the structural
uniformity in $p$ of the localization integral suggests (though certainly does
not prove) the following stronger statement: $\mathcal{G}_{22pp}(\tau)$ is
Borel summable, and obeys $f_{\rm np}(s)=(\mathcal{G}_{22pp},\phi_{n})=0$ for
all $N$ and $p$.

###  5.7 Including cusp forms

We now re-introduce cusp forms $\phi_{n}(\tau)$ to the preceding analysis. The
upshot is that we identify a sharp diagnostic of the presence of cusp forms,
given in (5.70), which follows rather intriguingly from a deep number-
theoretic property of their Fourier expansions.

Let us first explain, for both physics and math reasons, why determining the
cusp form contribution to ${\cal O}_{k>0}(y)$ is inherently much harder than
determining the continuous, Eisenstein contribution.

Maass cusp forms exhibit arithmetic chaos [16, 32]. The fundamental data
characterizing the cusp forms — namely, the spectral parameters $t_{n}$ and
the Fourier coefficients $a_{k}^{(n)}$ — are not known explicitly, but are
known to obey chaotic distributions. For example, the $t_{n}$ exhibit
Poissonian statistics at $n\gg 1$, while the $a_{k}^{(n)}$ for prime $k$ are
conjecturally asymptotically equidistributed with respect to a semi-circle
distribution, at either large $k$ or large $n$.202020The word “arithmetic”
refers to the arithmeticity of $SL(2,\mathbb{Z})$, or to the existence of an
infinite number of mutually commuting generators, known as Hecke operators, of
which each cusp form is an eigenvalue. This symmetry leads to “milder” forms
of chaos, e.g. Poissonian statistics for the $t_{n}$, than is seen in other
contexts. There is good “experimental” evidence for these claims [32, 60, 61].
We will utilize one such property more explicitly below.

At a technical level, this means that it is infeasible to study the sum over
$\phi_{n}(\tau)$ analytically. One could instead imagine approximating the
discrete sum by treating $({\cal O},\phi_{n})$ distributionally, then
performing contour deformations in the complex $s_{n}$ plane away from the
critical line $s_{n}={1\over 2}+it_{n}$. But besides being rather radical (and
not, to our knowledge, something done in the mathematics literature), since
the overlap $({\cal O},\phi_{n})$ is not determined as a Mellin transform,
unlike the Eisenstein sector, its asymptotics as $n\rightarrow\infty$ are
anyway not obviously constrained by analyticity of the Fourier modes of ${\cal
O}(\tau)$.

Nevertheless, we can diagnose the presence of cusp forms.

First, note that the relation (5.40) gives a one-way diagnostic of the
presence of cusp forms. If (5.40) does not hold for some $k$, then $({\cal
O},\phi_{n})\neq 0$ for some $n$:

${R_{k}\over R}\neq{\left(1+{k\over R}\right)^{2}}\quad\Rightarrow\quad({\cal
O},\phi_{n})\neq 0$ (5.60)

What about the converse? It is not obvious that cusp forms must modify $R_{k}$
at all. For example, their contribution could (as far as we can see) be
convergent or even purely non-perturbative, in which case they give a
vanishing contribution to the Borel resummation, and hence to
$R_{k}$.212121This happens if, for example, $({\cal O},\phi_{n})\neq 0$ for a
finite subset of $\phi_{n}(\tau)$. However, let us consider below the
interesting situations, where the cusp forms give a factorially divergent
contribution to perturbation theory, so that their contributions are not
swamped by everything else. Then in fact, (5.60) holds in both directions, as
we now show.

The contribution of $\phi_{n}(\tau)$ to the $y^{-m}$ term of ${\cal O}_{k}(y)$
is given by222222The $L^{2}$-normalized cusp forms $\phi_{n}(\tau)$ being used
throughout this paper have Fourier coefficients
$a_{k}^{(n)}\widetilde{a}_{1}^{(n)}$, where
$\widetilde{a}_{1}^{(n)}:=1/\sqrt{(\nu_{n},\nu_{n})}$ is simply a norm factor.
Recall that $a_{1}^{(n)}=1$, cf. (2.23).

${\cal O}_{k}(y)\big{|}_{y^{-m}}\supset{1\over(-4\pi
k)^{m}m!}\sum_{n=1}^{\infty}a_{k}^{(n)}\widetilde{a}_{1}^{(n)}{({\cal
O},\phi_{n})}\left({1\over 2}+it_{n}\right)_{m}\left({1\over
2}-it_{n}\right)_{m}$ (5.61)

The full $y^{-m}$ term is given by the sum of (5.61) and (5.5). Let us use
$R_{1}^{(\phi)}$ to denote the radius of convergence of the $SL(2,\mathbb{Z})$
Borel transform of the cusp contribution at $k=1$:

$\sum_{n=1}^{\infty}\widetilde{a}_{1}^{(n)}{({\cal O},\phi_{n})}\left({1\over
2}+it_{n}\right)_{m}\left({1\over
2}-it_{n}\right)_{m}\,\sim\,(m!)^{2}4^{m}\Big{(}R_{1}^{(\phi)}\Big{)}^{-m}\qquad(m\rightarrow\infty)\,.$
(5.62)

Likewise, use $R_{1}^{(E)}$ to denote the radius of convergence of the
$SL(2,\mathbb{Z})$ Borel transform of the Eisenstein contribution at $k=1$,
determined previously in (5.40). The total radius of convergence of
$B_{k}[\xi]$ at $k=1$ is the minimum of the convergence radii in the cusp and
Eisenstein sectors,

$R_{1}=\text{min}\left(R_{1}^{(E)},R_{1}^{(\phi)}\right)$ (5.63)

We now ask what is the cusp contribution at $k>1$. Associated to it will be a
radius of convergence $R^{(\phi)}_{k}$, such that $R_{k}$, the total radius of
convergence of $B_{k}[\xi]$, is

$R_{k}=\text{min}\left(R_{k}^{(E)},R^{(\phi)}_{k}\right)$ (5.64)

The Fourier coefficients $a_{k}^{(n)}$ are believed to obey the Ramanujan-
Petersson conjecture: for all $n$,

$|a_{k}^{(n)}|\leq 2\qquad(k\text{ prime})\,.$ (5.65)

This implies that the sole $k$-dependence of $R^{(\phi)}_{k}$ comes from the
$k^{-m}$ term in the Fourier expansion of the Bessel function, because it is
not possible for the sum over cusp forms to generate any parametric
$k$-dependence. This is in distinction to the Eisenstein series where
$a_{k}(s)=\sigma_{2s-1}(k)k^{-s}\rightarrow k^{s-1}$ at $k\gg 1$, and hence
the integral over $s$ generated the $k$-dependence seen earlier. In fact,
instead of invoking Ramanujan-Petersson we can use the best proven bound, of
Kim and Sarnak [17]: for all $n$,

$|a_{k}^{(n)}|\leq 2k^{7/64}\qquad(k\text{ prime})\,.$ (5.66)

This scaling is still insufficient to change the asymptotics because it does
not depend on $n$. Thus, (5.61) implies

$R^{(\phi)}_{k}=kR_{1}^{(\phi)}\qquad(k\text{ prime})\,.$ (5.67)

Comparing $R^{(\phi)}_{k}$ with $R^{(E)}_{k}$, we see that at sufficiently
large prime $k=k_{*}$, the former must become smaller — for any values of the
$k=1$ radii — because it grows linearly, rather than quadratically, in $k$.
Hence,

$R_{k>k_{*}}=kR_{1}^{(\phi)}\qquad(k\text{ prime})\,,$ (5.68)

where $k_{*}$ is defined as the (integral part of the) largest root of the
quadratic equation $k_{*}R_{1}^{(\phi)}={R_{k_{*}}^{(E)}}$, with
$R_{k_{*}}^{(E)}$ defined in (5.40).

Actually, we can drop the prime condition by using Hecke relations (e.g.
[62]). The Hecke operators $T_{k}$ act on cusp forms as

$T_{k}\,\phi_{n}=a_{k}^{(n)}\phi_{n}$ (5.69)

and obey $T_{j}T_{k}=\sum_{d|(j,k)}T_{jk/d^{2}}$. This determines the
$a_{k}^{(n)}$ for non-prime $k$ as a polynomial in the $a_{k}^{(n)}$ for prime
$k$, for every $n$. Therefore, all $a_{k}^{(n)}$ obey a polynomial bound in
$k$ that is independent of $n$.

Altogether, when cusp forms contribute to the Borel transform we have the
following result:

$\begin{split}{R_{k}\over R}={\rm min}\left(\left(1+{k\over
R}\right)^{2}\,,\leavevmode\nobreak\ {k\over
R}\,R_{1}^{(\phi)}\right)\end{split}$ (5.70)

where $R_{1}^{(\phi)}$ is infinite if $({\cal O},\phi_{n})=0$ for all $n$.
This also implies that (5.60) holds in both directions, providing a rigorous,
albeit mildly conditional, two-way diagnostic of the presence of cusp forms.
More conceptually, this subsection demonstrates a simple point: instantons in
$\mathcal{N}=4$ SYM exhibit arithmetic chaos, encoded in the cusp forms. If it
happens that $({\cal O},\phi_{n})=0$, one could rightfully call ${\cal
O}(\tau)$ non-chaotic.

###  5.8 On the strong coupling expansion

To conclude this section, let us ask what the spectral decomposition implies
about the strong coupling expansion, $y\rightarrow 0$. To be clear, S-duality
implies that strong coupling can be mapped to weak coupling. The purpose of
this short coda is to derive some precise consequences of this.

To develop the $y\rightarrow 0$ expansion of ${\cal O}_{0}(y)$, we again start
from the first equation of (5.1), but now deforming to the right, ${\rm
Re\,}s>{1\over 2}$. This yields

${\cal O}_{0}(y)=\langle{\cal
O}\rangle+\sum_{s=2}^{\infty}(-1)^{s}s(s-1)f_{\rm
p}(s)\Lambda(s)y^{s}\qquad(y\rightarrow 0)$ (5.71)

There are a few notable features here. First, there is a universal factor
$\varphi(s)$ relating the coefficients of the weak and strong coupling
expansions of ${\cal O}_{0}(y)$, seen by comparing (5.11) and (5.71). In [1],
this was regarded as a striking observation for $\mathcal{G}_{N}(\tau)$. We
are showing that it is a simple consequence of $SL(2,{\mathbb{Z}})$-invariance
for any square-integrable observable. Note that $\varphi(s\gg 1)\sim
1/\sqrt{s}$, so the asymptotics at $y\rightarrow 0$ and $y\rightarrow\infty$
are essentially identical. Second, there is never an $O(y)$ term, which
follows from $\operatorname*{Res}_{s=1}R_{1-s}[{\cal
O}]=\operatorname*{Res}_{s=1}\left(\varphi(s)^{-1}R_{s}[{\cal O}]\right)=0$.
Finally, the ensemble average equals the zero mode at $y=0$:

$\langle{\cal O}\rangle={\cal O}_{0}(y=0)$ (5.72)

In other words, dialing ${\cal O}_{0}(y)$ from weak to strong coupling adds a
factor of the ensemble average. This is a non-trivial statement,
notwithstanding S-duality: we are taking the strong coupling limit of the zero
mode, which is not $SL(2,\mathbb{Z})$ invariant, not of ${\cal O}(\tau)$,
which is $SL(2,\mathbb{Z})$ invariant. We will see a large $N$ version of this
statement in Section 10, with powerful implications.

##  6 Large $N$: Generalities

We will be concerned with two types of large $N$ limits.

The first limit is the ’t Hooft limit, defined as $N\rightarrow\infty$,
$g\rightarrow 0$ with $\lambda\coloneqq g^{2}N$ fixed. In terms of our
variable $y={\rm Im\,}\tau$,

$N\rightarrow\infty,\leavevmode\nobreak\ \leavevmode\nobreak\ \lambda={4\pi
N\over y}\leavevmode\nobreak\ \text{fixed}\qquad\text{('t Hooft)}$ (6.1)

In the ’t Hooft limit, the perturbative $1/N$ expansion organizes into a genus
expansion. Instantons are non-perturbatively suppressed as

${\cal O}_{k}(y)\propto e^{-2\pi ky}\sim e^{-8\pi^{2}k{N\over\lambda}}$ (6.2)

so the genus expansion applies to the zero mode,

${\cal O}_{0}(y)=\sum_{g=0}^{\infty}N^{2-2g}\,{\cal O}_{0}^{(g)}(\lambda)$
(6.3)

There are also other non-perturbative corrections in $N$ which correct the
zero mode itself at each order in the genus expansion, to be derived below.

The second limit is $N\rightarrow\infty$ with $g$ fixed, sometimes called the
“very strongly coupled limit” [18]:

$N\rightarrow\infty,\leavevmode\nobreak\ \leavevmode\nobreak\
y\leavevmode\nobreak\ \text{fixed}\qquad\text{(very strongly coupled)}$ (6.4)

The genus expansion (6.3) does not hold in the very strongly coupled limit,
but taking $y$ fixed and then sending $y\rightarrow\infty$ recovers the
leading $\lambda\gg 1$ behavior in the ’t Hooft limit.

Before delving into details, let us state one of the main overarching points
here. As first explained in the previous section on perturbation theory at
finite $N$, to develop a perturbative expansion in some parameter from the
spectral decomposition, one first expands the spectral overlap $\\{{\cal
O},E_{s}\\}$, and then performs a contour deformation to pick up poles. The
presence or absence of non-perturbative corrections to the resulting expansion
is encoded in the $|s|\rightarrow\infty$ asymptotics of the overlap. In the
large $N$ expansions to follow, this approach will reveal various corrections.
The power of $SL(2,\mathbb{Z})$ is that perturbative and non-perturbative
physics are intertwined into one and the same automorphic object.

We first need to understand how the spectral overlaps are constrained. Let us
present, then explain, the result. At large $N$, the spectral overlap
$\\{{\cal O},E_{s}\\}$ is of the following form:

$\begin{split}\boxed{\\{{\cal
O},E_{s}\\}=\sum_{g=0}^{\infty}N^{2-2g}\left({\pi\over\sin\pi
s}s(1-s)\left(N^{-s}f_{\rm p}^{(g)}(1-s)+(s\rightarrow
1-s)\right)+\sum_{m=0}^{\infty}N^{-3-{m\over 2}}f_{\rm
np}^{(g,m)}(s)\right)}\end{split}$ (6.5)

where $SL(2,\mathbb{Z})$-invariance requires the reflection symmetry $f_{\rm
np}^{(g,m)}(s)=f_{\rm np}^{(g,m)}(1-s)$.

These two sets of terms are the large $N$ expansions of the perturbative and
non-perturbative parts of (5.7), respectively. Let us now explain (6.5) in
slightly more detail. The reader interested only in its consequences may skip
to the next section.

The first, perturbative term in (6.5) can be understood by considering the ’t
Hooft limit, but applies to any large $N$ limit since $\\{{\cal O},E_{s}\\}$
is independent of $y$. The key observation is that any function $f(\lambda)$
is produced by an overlap with $N^{-s}$ scaling: recalling that $\\{{\cal
O},E_{s}\\}$ is $\Lambda(s)^{-1}$ times the RS transform, we have

$\begin{split}R_{1-s}[f(\lambda)]&=\int_{0}^{\infty}dy\,y^{-1-s}f(\lambda)\\\
&=(4\pi N)^{-s}\int_{0}^{\infty}d\lambda\,\lambda^{s-1}f(\lambda)\\\ &=(4\pi
N)^{-s}\,\mathfrak{M}[f(\lambda);s]\end{split}$ (6.6)

where $\lambda$ is the Mellin integration variable in the final line.
Therefore, the genus expansion (6.3) implies that $f_{\rm p}^{(g)}(1-s)$
contains information of the ’t Hooft expansion at genus $g$, for any
$\lambda$. The details of the $\lambda\ll 1$ and $\lambda\gg 1$ expansions —
including possible non-perturbative corrections in $\lambda$ — are encoded in
the polar structure of $f_{\rm p}^{(g)}(1-s)$ for $s\in\mathbb{R}$. The
presence of the reflected solution in (6.5) follows from the
$SL(2,\mathbb{Z})$-invariance of the spectral decomposition, i.e. the
reflection symmetry of $\\{{\cal O},E_{s}\\}$. Note that since we are
expanding at $N\rightarrow\infty$ first, the polar structure of $f_{\rm
p}^{(g)}(s)$ can be non-trivial, unlike $f_{\rm p}(s)$ at finite $N$.

The second, non-perturbative term in (6.5) generates all exponential terms
$\sim e^{-4\pi ny}$, as discussed in Section 5.2. These are non-perturbative
in the ’t Hooft limit; consistent with the previous paragraph, their
$N$-scaling is independent of $s$. It may not be immediately obvious why, or
how, $f_{\rm np}^{(g,m)}(s)$ generates exponentials. After all, to obtain the
contribution of $f_{\rm np}^{(g,m)}(s)$, we simply integrate along the
critical line $s={1\over 2}+it$; why is this non-perturbative at
$y\rightarrow\infty$? The answer can be nicely understood from basic
properties of the Rankin-Selberg method as adapted by Zagier [52]. At a fixed
order in the $1/N$ expansion, ${\cal O}(\tau)$ must be $SL(2,\mathbb{Z})$
invariant, although it need not be square-integrable. However, consistency
with the genus expansion in the ’t Hooft limit means that the leading power of
$y$ at fixed order in $1/N$ is bounded above. As shown in [52], if an
$SL(2,\mathbb{Z})$-invariant function $F(\tau)$ with bounded power-law growth
at $y\rightarrow\infty$ has a zero mode

$F_{0}(y)=\sum_{i=1}^{m}{c_{i}\over
n_{i}!}y^{\alpha_{i}}\log^{n_{i}}y+\text{(exponential)}$ (6.7)

for some finite $m$, with $n_{i}\in\mathbb{Z}_{\geq 0}$ and
$c_{i},\alpha_{i}\in\mathbb{C}$ with bounded ${\rm Re\,}(\alpha_{i})$, then
$F(\tau)$ can be written as

$F(\tau)\coloneqq\sum_{i\,|\,\alpha_{i}\geq{1\over
2}}\left.c_{i}{\partial^{n_{i}}\over\partial
s^{n_{i}}}E_{s}(\tau)\right|_{s=\alpha_{i}}+F_{\rm spec}(\tau)$ (6.8)

The Eisenstein series subtract the powers of $y^{\alpha_{i}\geq{1\over 2}}$ in
a modular-invariant way to restore square-integrability, such that $F_{\rm
spec}(\tau)\in L^{2}({\cal F})$ and hence admits a spectral decomposition.
Then since the zero mode of the Eisenstein series has no exponential terms in
$y$, any exponential terms in $F_{0}(y)$ must come solely from $F_{\rm
spec}(\tau)$.

We pause to briefly note a small subtlety. Taking $F_{0}(y)$ to be free of
logs, as in perturbative CFT applications, we have

$F_{{\rm spec},0}(y)=\sum_{i\,|\,\alpha_{i}<{1\over
2}}c_{i}y^{\alpha_{i}}-\sum_{i\,|\,\alpha_{i}\geq{1\over
2}}c_{i}\varphi(\alpha_{i})y^{1-\alpha_{i}}+\text{(exponential)}$ (6.9)

If the powers of $y$ do not exactly cancel (and they need not), then $F_{{\rm
spec},0}(y)$ has, in addition to the exponentials, spurious powers of $y$ that
are not present in $F_{0}(y)$ (and vice-versa). In order to obtain these
powers from the spectral integral $F_{\rm spec}(\tau)$, one should shift the
spectral contour, picking up residues of spurious poles of $\\{F_{\rm
spec},E_{s}\\}$ along the way.232323If there is a finite number of such
spurious poles, this shift incurs no residue at infinity. All examples we know
are of this type. This is very neatly exhibited by a certain class of
solutions to inhomogeneous Laplace equations appearing in perturbative string
theory and $\mathcal{N}=4$ SYM correlators, as we recall in detail in Appendix
B. The remaining spectral integral has the form of an inverse Mellin
transform.

Finally, we comment on the range of the sum over $m$ in (6.5). Here, unlike
elsewhere in this work, we use some mild input from holography: in particular,
that the first non-perturbative terms in the AdS${}_{5}\times S^{5}$ effective
action appear at 14-derivative order, via $D^{6}R^{4}$ invariants and their
superpartners. This may be seen as a fact about type IIB string theory, or as
following on general grounds from on-shell constraints in theories of maximal
supergravity combined with the flat space limit of AdS${}_{5}\times S^{5}$
[63, 64].

Summarizing and returning to our problem: the function $f_{\rm np}^{(g,m)}(s)$
contributes non-perturbative terms $(q\bar{q})^{n}$, plus a possible set of
spurious poles. Moreover, since $f_{\rm np}^{(g,m)}(s)$ generates exponentials
in $y$,

$\Lambda(s)f_{\rm np}^{(g,m)}(s)\sim\Gamma(|s|)\qquad(s\rightarrow\pm\infty)$
(6.10)

for every $(g,m)$.

One more comment before moving on. Although above we discussed non-
perturbative contributions to the spectral overlap $\\{\mathcal{O},E_{s}\\}$
between a CFT observable and the Eisenstein series, essentially identical
considerations apply to the overlap with the Maass cusp forms,
$(\mathcal{O},\phi_{n})$. In particular, the cusp forms do not enter the
AdS${}_{5}\times S^{5}$ effective action until 14-derivative order, and so
$(\mathcal{O},\phi_{n})$ is suppressed by a factor of $1/N^{3}$ compared to
the leading perturbative effect $f_{p}(s)$. Moreover, like $f_{\rm np}(s)$,
the large $N$ expansion of $(\mathcal{O},\phi_{n})$ must proceed in half-
integer powers of $N$:

$(\mathcal{O},\phi_{n})=\sum_{g=0}^{\infty}N^{2-2g}\sum_{m=0}^{\infty}N^{-3-{m\over
2}}(\mathcal{O},\phi_{n})^{(g,m)}.$ (6.11)

Indeed, in the concrete example of large $N$ expansion of the integrated
correlator ${\cal F}_{N}(\tau)$ in the very strongly coupled limit in Section
8.2, we will see that these two contributions to the spectral overlap ($f_{\rm
np}(s)$ and $(\mathcal{O},\phi_{n})$) are intertwined, as they descend from a
common modular invariant, the solution to the inhomogeneous Laplace equation
described in Appendix B.

We now consider the two large $N$ limits. It will prove beneficial to treat
the ’t Hooft limit first, where consistency with the genus expansion will
strongly constrain the allowed poles of the overlap. For brevity, when
referring to the non-perturbative part of (6.5) we sometimes refer to $f_{\rm
np}(s)$ without the $(g,m)$ superscript, bearing in mind its large $N$
expansion.

##  7 ’t Hooft Limit

The strategy to analyze the ’t Hooft limit is clear: we substitute $y=4\pi
N/\lambda$ in the spectral decomposition (5.1) and analyze it at fixed
$\lambda$, taking further perturbative $\lambda\ll 1$ or $\lambda\gg 1$ limits
afterwards. We introduce a useful notation

$\tilde{\lambda}\coloneqq{\lambda\over 4\pi}$ (7.1)

We safely ignore the cusp forms, which are suppressed as $\sim e^{-N}$ in the
’t Hooft limit. Importantly, while we will generalize in due course (see
Section 7.4), we begin by setting

$f_{\rm np}(s)=0$ (7.2)

This turns off the non-perturbative terms $\sim(q\bar{q})^{n}$. We employ this
strategy because it leads to very strong constraints, and will allow us to
cleanly identify signatures of non-perturbative physics in the large $N$
limit.

In the ’t Hooft limit, the zero mode of the completed Eisenstein series is

$E^{*}_{s,0}(y)=\Lambda(s)N^{s}\tilde{\lambda}^{-s}+(s\rightarrow 1-s)$ (7.3)

Plugging (7.3) and the overlap (6.5) into the spectral decomposition (5.1)
gives

${\cal O}_{0}(\lambda)=\langle{\cal O}\rangle+{1\over 2\pi i}\int_{{\rm
Re\,}s={1\over 2}}ds\,{\pi\over\sin\pi
s}s(1-s)\sum_{g=0}^{\infty}N^{2-2g}\left(\Lambda(s)\tilde{\lambda}^{-s}+\Lambda(1-s)N^{1-2s}\tilde{\lambda}^{s-1}\right)f_{\rm
p}^{(g)}(1-s)$ (7.4)

The two terms under the integral are of a very different nature:

$\bullet$ The first term is manifestly consistent with the ’t Hooft limit, and
can be expanded at weak or strong coupling, or evaluated at finite
$\tilde{\lambda}$. At $\tilde{\lambda}\ll 1$, we deform to the left, while at
$\tilde{\lambda}\gg 1$, we deform to the right.

$\bullet$ The second term has an explicit $N^{1-2s}$ factor. Because we are
taking the large $N$ limit, we are forced to deform to the right, for any
$\tilde{\lambda}$.242424Bearing in mind earlier remarks, we must ask whether
the coefficient has a factorial divergence at $s\rightarrow\infty$, thus
generating potential non-perturbative corrections in $N$. We analyze this
question in the next subsection. This generates two constraints on $f_{\rm
p}^{(g)}(1-s)$:

  1. 1)

Consistency with the genus expansion: For ${\rm Re\,}s>{1\over 2}$, the poles
of the integrand must lie at $s={1\over 2}+m$ with $m\in\mathbb{Z}_{+}$. In
particular, the $\sin\pi s$ poles must be cancelled by zeros,

${f_{\rm p}^{(g)}(s)=0\,,\quad s\in\mathbb{Z}_{-}}$ (7.5)

On the other hand, simple poles for half-integer $s$ are allowed,

$\operatorname*{Res}_{s={1\over 2}-m}\left[f_{\rm
p}^{(g)}\left(s\right)\right]\neq 0\,,\quad m\in\mathbb{Z}_{+}$ (7.6)

In addition, $\Lambda(1-s)$ has a pole at $s=1$, which contributes a constant
term. Taking this all into account, we have, for fixed $\tilde{\lambda}$,

$\begin{split}{\cal O}_{0}(\lambda)=&\left(\langle{\cal O}\rangle-{1\over
2}\sum_{g=0}N^{1-2g}f_{\rm
p}^{(g)}(0)\right)-\sum_{g=0}^{\infty}\sum_{m=1}^{\infty}N^{2-2g-2m}\varphi\Big{(}{1\over
2}+m\Big{)}\mathsf{R}_{m}^{(g)}\tilde{\lambda}^{m-{1\over 2}}\\\ &+{1\over
2\pi i}\sum_{g=0}^{\infty}N^{2-2g}\int_{{\rm Re\,}s={1\over
2}}ds\,{\pi\over\sin\pi s}s(1-s)\Lambda(s)\tilde{\lambda}^{-s}f_{\rm
p}^{(g)}(1-s)\end{split}$ (7.7)

where we define a “residue function”

$\begin{split}\mathsf{R}_{m}^{(g)}&\coloneqq\operatorname*{Res}_{s={1\over
2}+m}\Big{[}{\pi\over\sin\pi s}s(1-s)\Lambda(s)f_{\rm
p}^{(g)}(1-s)\Big{]}\end{split}$ (7.8)

We can now deform this as needed to develop perturbative expansions.

  2. 2)

Consistency with the $\tilde{\lambda}\ll 1$ expansion: These residues at
$s={1\over 2}+m$ generate allowed powers of $1/N$, but multiplied by half-
integer powers of $\tilde{\lambda}$, plainly visible above. These terms are
present for any fixed $\tilde{\lambda}$. They must cancel contributions from
the remaining integral when we go to weak coupling, $\tilde{\lambda}\ll 1$,
where only integer powers of $\tilde{\lambda}$ are allowed. Developing the
$\tilde{\lambda}\ll 1$ expansion by deforming (7.7) to the left, the offending
terms are cancelled if and only if the following condition on the residues is
met:

$\varphi\Big{(}{1\over
2}+m\Big{)}\,\mathsf{R}_{m}^{(g)}=\mathsf{R}_{-m}^{(g+m)}$ (7.9)

This intriguing condition relates perturbative data of different genera.

After the dust settles, we have the following weak coupling expansion

$\begin{split}{{\cal O}_{0}(\lambda\ll
1)\approx\sum_{g=0}^{\infty}N^{2-2g}\sum_{m=1}^{\infty}(-1)^{m+1}m(m+1)\Lambda\Big{(}{1\over
2}+m\Big{)}f_{\rm p}^{(g)}(1+m)\,\tilde{\lambda}^{m}}\end{split}$ (7.10)

In terms of the residue function,

$\begin{split}\boxed{{\cal O}_{0}(\lambda\ll
1)\approx\sum_{g=0}^{\infty}N^{2-2g}\sum_{m=1}^{\infty}\mathsf{R}_{-m-{1\over
2}}^{(g)}\,\tilde{\lambda}^{m}}\end{split}$ (7.11)

Note that the absence of $\log\tilde{\lambda}$ terms, or any other non-integer
powers of $\tilde{\lambda}$, requires that $f_{\rm p}^{(g)}(s)$ is regular for
all ${\rm Re\,}s>{1\over 2}$ away from the half-integer poles appearing in the
cancellation condition (7.9). We have thus completely determined the structure
of allowed poles and zeros of $f_{\rm p}^{(g)}(s)$.

The payoff of this work happens at strong coupling, $\tilde{\lambda}\gg 1$.
Deforming the second line of (7.7) to the right now, and applying the previous
constraints on the polar structure of $f_{\rm p}^{(g)}(s)$, yields the strong
coupling expansion

$\begin{split}{\cal O}_{0}(\lambda\gg
1)&\approx\mathsf{C}(N)-\sum_{g=0}^{\infty}\sum_{m=1}^{\infty}N^{2-2g}\left(N^{-2m}\varphi\Big{(}{1\over
2}+m\Big{)}\mathsf{R}_{m}^{(g)}\tilde{\lambda}^{m-{1\over
2}}+\mathsf{R}^{(g)}_{m}\tilde{\lambda}^{-{1\over 2}-m}\right)\end{split}$
(7.12)

The first term is a constant,

$\mathsf{C}(N)\coloneqq\langle{\cal O}\rangle-{1\over
2}\sum_{g=0}^{\infty}N^{1-2g}f_{\rm p}^{(g)}(0)$ (7.13)

The next group of terms are the terms described earlier, but in the strong
coupling expansion they do not cancel with anything. They may at first seem
peculiar — a finite set of positive powers of $\lambda$ at each even power of
$1/N$ — but they are precisely what is needed for the renormalization of the
$1/N$ expansion at strong coupling. As explained in [65], they are, from the
gravity point of view, the string theory regularization of loop-level
divergences in AdS${}_{5}\times S^{5}$ supergravity, where $g+m=L$ with $L$
the loop level. Consistent with this, they are $1/N^{2}$ suppressed compared
to the leading order.252525At $N^{0}$, for example, $(g,m)=(0,1)$ gives a
$\sqrt{\lambda}$ term, which is precisely the regularization of one-loop
supergravity by $R^{4}$ (and its superpartners) in AdS${}_{5}\times S^{5}$. At
$1/N^{2}$, $(g,m)=(0,2)$ is the regularization of two-loop supergravity by
$D^{4}R^{4}$, while $(1,1)$ is the regularization of one-loop supergravity
corrected by the tree-level $R^{4}$ term. This pattern continues.

What we are mainly interested in is the last group of terms, containing the
conventional genus sum over perturbative series in $1/\tilde{\lambda}$. Only
half-integer powers of $\tilde{\lambda}$ appear, a fact which we have derived
solely from consistency conditions on the weak coupling and genus expansions.
This expression contains much physics, which we analyze in stages, beginning
now with the implications for non-perturbative effects.

###  7.1 Non-perturbative effects implied by $SL(2,\mathbb{Z})$

In this subsection we derive a surprising result: as a direct consequence of
S-duality, the convergence of a weak coupling expansion at $\lambda\ll 1$
directly implies the existence of non-perturbative corrections in both
$\lambda\gg 1$ and in $N\gg 1$, with the precise non-perturbative scales set
by the weak coupling radius of convergence.

#### Non-perturbative corrections in $\lambda$

Let us here reproduce the $\lambda\ll 1$ expansion (7.11) and the $\lambda\gg
1$ expansion (7.12) at fixed genus $g$, dropping terms terms in ${\cal
O}_{0}^{(g)}(\lambda\gg 1)$ that do not contribute asymptotically and writing
the residue function in terms of $f_{\rm p}^{(g)}(s)$:

$\begin{split}{\cal O}_{0}^{(g)}(\lambda\ll
1)&=-\sum_{m=1}^{\infty}(-1)^{m}m(m+1)\Lambda\Big{(}{1\over 2}+m\Big{)}f_{\rm
p}^{(g)}(1+m)\,\tilde{\lambda}^{m}\\\ {\cal O}_{0}^{(g)}(\lambda\gg
1)&=-\sum_{m=1}^{\infty}(-1)^{m}\pi\Big{(}m-{1\over 2}\Big{)}\Big{(}m+{1\over
2}\Big{)}\Lambda\Big{(}{1\over 2}+m\Big{)}\operatorname*{Res}_{s={1\over
2}-m}\left[f_{\rm p}^{(g)}(s)\right]\tilde{\lambda}^{-{1\over
2}-m}\end{split}$ (7.14)

This makes clear that the $\lambda\ll 1$ and $\lambda\gg 1$ convergence are
determined by the $|m|\rightarrow\infty$ asymptotics of $f_{\rm p}^{(g)}(m)$
in opposite directions along $\mathbb{R}$.

Suppose that the $\lambda\ll 1$ expansion is convergent, with radius of
convergence $|\lambda|\leq\lambda_{*}$. This is the typical situation in
planar $\mathcal{N}=4$ SYM. Planar combinatorics determine
$\lambda_{*}=\pi^{2}$ for generic ${\cal O}(\tau)$ [66, 67], but for the sake
of generality we will leave $\lambda_{*}$ explicit. At $m\rightarrow+\infty$,

$\Lambda\Big{(}{1\over
2}+m\Big{)}\approx\Gamma(m+1)\frac{\pi^{-m-\frac{1}{2}}}{\sqrt{m}}\qquad(m\rightarrow\infty)$
(7.15)

This implies that262626Here and henceforth in this subsection, we use the
symbol $\sim$ to denote the asymptotic behavior up to multiplicative power law
corrections and beyond.

$f_{\rm
p}^{(g)}(m+1)\sim\Gamma(m+1)^{-1}\left({4\pi^{2}\over\lambda_{*}}\right)^{m}\qquad(m\rightarrow\infty)$
(7.16)

In particular, $f_{\rm p}^{(g)}(x)$ decays factorially at large positive
argument. We now make a technical assumption, borne out by examples, that the
$|x|\rightarrow\infty$ limit of $f_{\rm p}^{(g)}(x)$ commutes with
$x\rightarrow-x$.272727This is weaker than, but included in, the natural
condition that $f_{\rm p}^{(g)}(s)$ admit a uniform asymptotic expansion for
$s\in{\mathbb{C}}$. This in turn implies

$\operatorname*{Res}_{s={1\over 2}-m}\left[f_{\rm
p}^{(g)}\left(s\right)\right]\sim\Gamma\Big{(}\\!-m+{1\over
2}\Big{)}^{-1}\left({4\pi^{2}\over\lambda_{*}}\right)^{-m}\qquad(m\rightarrow\infty)$
(7.17)

We plug these into (7.14). For large positive integer $m$,

$\begin{split}\Gamma(m+1)\Gamma\Big{(}\\!-m+{1\over
2}\Big{)}^{-1}&\sim(-1)^{m}4^{-m}\Gamma(2m)\qquad(m\rightarrow\infty)\end{split}$
(7.18)

Inserting into the strong coupling expansion (7.14), we see that the
$\lambda\gg 1$ expansion diverges double-factorially without alternating sign!
This is a hallmark of Borel non-summability. A summand scaling asymptotically
as

$\left({4\pi^{2}\lambda\over\lambda_{*}}\right)^{-m}\Gamma(2m)$ (7.19)

requires non-perturbative corrections in powers of
$e^{-{2\pi\over\sqrt{\lambda_{*}}}\sqrt{\lambda}}$. Therefore, the full
expression for the zero mode at $\lambda\gg 1$, perturbatively in $1/N$, is

$\boxed{{\cal O}_{0}(\lambda\gg
1)=\sum_{g=0}^{\infty}N^{2-2g}\sum_{n=0}^{\infty}e^{-{2\pi
n\over\sqrt{\lambda_{*}}}\sqrt{\lambda}}\,{\cal O}_{0,n}^{(g)}(\lambda)}$
(7.20)

where ${\cal O}_{0,n}^{(g)}(\lambda)$ dresses the $n$’th exponential
correction by a perturbative series in $1/\lambda$. The purely perturbative
$n=0$ piece is given explicitly in (7.12), while the non-perturbative $n>0$
terms are obtained by applying resurgence to ${\cal O}_{0,0}^{(g)}(\lambda)$.

It is remarkable that the numerical factor in the exponent is determined by
$\lambda_{*}$, the weak coupling radius of convergence.282828In principle
$\lambda_{*}$ could depend on $g$, though we are not aware of any such
examples. If this happens, one should take
$\lambda_{*}\rightarrow\lambda_{*,g}$ in this formula. For the canonical
radius $\lambda_{*}=\pi^{2}$, the non-perturbative corrections are in powers
of $e^{-2\sqrt{\lambda}}$.

#### Non-perturbative corrections in $N$

The exact same argument that led to (7.20) also applies to the second term in
(7.4). The contour deformation develops a formal series in positive integer
powers of $\lambda/N^{2}$, which diverges double-factorially at
$s\rightarrow\infty$ with the same leading-order asymptotics. The previous
analysis carries over, but with the substitution $\tilde{\lambda}\rightarrow
N^{2}/\tilde{\lambda}$. This yields non-perturbative corrections to the ’t
Hooft limit, for fixed $\tilde{\lambda}$, in powers of
$\exp({-{8\pi^{2}\over\sqrt{\lambda_{*}}}{N\over\sqrt{\lambda}}})$.

Thus, for an $SL(2,\mathbb{Z})$-invariant observable ${\cal O}(\tau)$ with
$f_{\rm np}(s)=0$ whose $\lambda\ll 1$ expansion has radius of convergence
$\lambda_{*}$, the ’t Hooft expansion in $1/N$ for fixed $\lambda$ receives
non-perturbative corrections exponentially small in $N$, of the following
form:

$\begin{split}\boxed{{\cal
O}_{0}(\lambda)=\sum_{g=0}^{\infty}N^{2-2g}\left({\cal
O}_{0}^{(g)}(\lambda)+{\cal O}_{0,{\rm
np}}^{(g)}(N,\lambda)\right)}\end{split}$ (7.21)

where

${{\cal O}_{0,{\rm
np}}^{(g)}(N,\lambda)=\sum_{n=0}^{\infty}e^{-{8\pi^{2}n\over\sqrt{\lambda_{*}}}{N\over\sqrt{\lambda}}}\,{\cal
O}_{0,{\rm np}|n}^{(g)}\left({N^{2}\over\lambda}\right)}$ (7.22)

${\cal O}_{0,{\rm np}|n}^{(g)}\left({N^{2}\over\lambda}\right)$ dresses the
$n$’th exponential correction by a perturbative expansion in ${\lambda\over
N^{2}}\ll 1$. The purely perturbative piece ${\cal O}_{0,{\rm
np}|0}^{(g)}\left({N^{2}\over\lambda}\right)$ was given in the first line of
(7.7),

${\cal O}_{0,{\rm np}|0}^{(g)}\left({N^{2}\over\lambda}\right)=-{1\over
N}\sum_{m=1}^{\infty}\varphi\Big{(}{1\over
2}+m\Big{)}\mathsf{R}_{m}^{(g)}\left({4\pi N^{2}\over\lambda}\right)^{{1\over
2}-m}$ (7.23)

We will compute ${\cal O}_{0,{\rm np}}^{(g)}(N,\lambda)$ for ${\cal
O}(\tau)=\mathcal{G}_{N}(\tau)$ by resurgence in the next
subsection.292929This is a slight abuse of notation since ${\cal O}_{0,{\rm
np}|0}^{(g)}\left({N^{2}\over\lambda}\right)$ is itself perturbative in $N$.
This is the series on which we perform resurgence to determine the $n>1$,
genuinely non-perturbative terms. Note that compared to the computation of
${\cal O}_{0,n}^{(g)}(\lambda)$, the two perturbative series to which we apply
resurgence differ by a factor of $\varphi(s)$. This will slightly modify the
details of the non-perturbative corrections, as we will see applied to
$\mathcal{G}_{N}(\tau)$ in Subsection 7.3. Let us stress once again that
$\lambda_{*}$, the radius of convergence at weak ’t Hooft coupling, controls
the strength of these non-perturbative corrections.

#### Nonzero modes