Integrated Correlators in $\mathcal{N}=4$ SYM via $SL(2,\mathbb{Z})$ Spectral
Theory

Hynek Paul, Eric Perlmutter, Himanshu Raj

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

<EMAIL_ADDRESS>

###### Abstract

We perform a systematic study of integrated four-point functions of half-BPS
operators in four-dimensional $\mathcal{N}=4$ super Yang-Mills theory with
gauge group $SU(N)$. These observables, defined by a certain spacetime
integral of
$\langle\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{p}\mathcal{O}_{p}\rangle$
where $\mathcal{O}_{p}$ is a superconformal primary of charge $p$, are known
to be computable by supersymmetric localization, yet are non-trivial functions
of the complexified gauge coupling $\tau$. We find explicit and remarkably
simple results for several classes of these observables, exactly as a function
of $N$ and $\tau$. Their physical and formal properties are greatly
illuminated upon employing the $SL(2,\mathbb{Z})$ spectral decomposition: in
this S-duality-invariant eigenbasis, the integrated correlators are fixed
simply by polynomials in the spectral parameter. These polynomials are
determined recursively by linear algebraic equations relating different $N$
and $p$, such that all integrated correlators are ultimately fixed in terms of
the integrated stress tensor multiplets in the $SU(2)$ theory. Our
computations include the full matrix of integrated correlators at low values
of $p$, and a certain infinite class involving operators of arbitrary $p$. The
latter satisfy an open lattice chain equation for all $N$, reminiscent of the
Toda equation obeyed by extremal correlators in $\mathcal{N}=2$ superconformal
theories. We compute ensemble averages of these observables and analyze our
solutions at large $N$, confirming and predicting features of semiclassical
AdS${}_{5}\,\times$ S5 supergravity amplitudes.

###### Contents

  1. 1 Introduction
  2. 2 Setup
    1. 2.1 Half-BPS operators and their four-point functions
    2. 2.2 Integrated correlators
    3. 2.3 The $SL(2,\mathbb{Z})$ spectral decomposition
    4. 2.4 Prelude: Review of $p=2$ result
  3. 3 Results I: Integrated correlators for $p\leq 5$
    1. 3.1 Perturbative expansion
    2. 3.2 Spectral overlaps and recursion relations
    3. 3.3 Including instantons: completion to the full correlator
    4. 3.4 Laplace difference equations
  4. 4 Ensemble averages
  5. 5 Large $N$
    1. 5.1 Genus expansion of spectral overlaps
    2. 5.2 Ensemble averages at large $N$ and supergravity
    3. 5.3 Integrated correlators at strong coupling
  6. 6 Results II: Integrated maximal-trace correlators for all $p$
    1. 6.1 Comments and connections to $\mathcal{N}=2$ extremal correlators
    2. 6.2 Large $N$ expansion
  7. 7 General $p$ ansatz
  8. 8 Future directions
  9. A Colour factors
  10. B Localization details
    1. B.1 Solving operator mixing on $S^{4}$ and integrated two-point functions
    2. B.2 Integrated four-point functions
    3. B.3 Expectation values in the Gaussian (special) unitary ensemble
  11. C Recursion relations for spectral overlaps involving SPO’s
  12. D One-instanton results
  13. E Matching to two-loop perturbation theory
  14. F Consistency check: Ensemble average vs. genus expansion
  15. G More on the ’t Hooft limit
    1. G.1 Genus-zero spectral overlap for general $p$
    2. G.2 Genus-one correlators for various values of $p$
    3. G.3 Proof of ()

##  1 Introduction

It is reasonable to hope that the four-dimensional maximally-supersymmetric
Yang-Mills theory may be exactly solved. In the large $N$ ‘t Hooft limit, the
theory becomes famously integrable, and Yangian symmetries emerge, which has
led to the exact determination of various planar observables. At finite $N$,
these special properties are obscured; however, the theory enjoys a non-
perturbative S-duality symmetry [1, 2, 3, 4], which for simply-laced gauge
groups is an invariance (up to global identifications [5]) under
$SL(2,\mathbb{Z})$ transformations of the complexified gauge coupling,

$\tau={\theta\over 2\pi}+i{4\pi\over g_{\rm YM}^{2}}\,.$ (1.1)

It seems likely that a solution of this theory (henceforth $\mathcal{N}=4$
SYM) will be possible only with judicious treatment of S-duality. Even so,
determining the exact $\tau$-dependence of $\mathcal{N}=4$ SYM observables
should be very hard in general, precisely because the theory is strongly
interacting over most of its parameter space, even after modding out by
$SL(2,\mathbb{Z})$.

There is a special family of observables in $\mathcal{N}=4$ SYM, introduced in
[6], which sit askew of this expectation: they vary as functions of $\tau$,
and yet are determinable by supersymmetric localization. They are integrated
four-point functions of half-BPS operators, whose definition will be recalled
further below. In this paper, following [6, 7, 8], we find strikingly compact
solutions for an infinite class of these integrated correlators in the $SU(N)$
theory, exact in $N$ and $\tau$, and study their general systematics.
Tantalizingly, they are optimally understood in a complete basis of
$SL(2,\mathbb{Z})$-invariant functions [8]: in this basis, the integrated
correlators are essentially polynomials in the relevant spectral parameter.

To set the context for these results, let us recall the status of
(unintegrated) $\mathcal{N}=4$ SYM four-point functions. We henceforth
specialize to the $SU(N)$ theory. Half-BPS superconformal primary operators in
$\mathcal{N}=4$ SYM, call them ${\cal O}_{p}^{(i)}$ with $p\geq 2$, are
spacetime scalars in the $[0\,p\,0]$ representation of the $SU(4)$ R-symmetry
with dimension $\Delta_{p}=p$. The case $p=2$ is the $\mathbf{20^{\prime}}$ of
the stress tensor multiplet. At $p>3$ there is degeneracy, which is captured
by the label $i$. Consider the family of four-point functions $\langle{\cal
O}_{2}{\cal O}_{2}{\cal O}_{p}^{(i)}{\cal O}_{p}^{(j)}\rangle$. At finite $N$,
this is known only in weak-coupling perturbation theory $(g_{\rm YM}^{2}\ll
1)$ through two loops in general [9], and three loops for $p=2$ [10] (though
integrands are known to four loops [11, 12]). In the planar limit, it is known
through three loops at weak coupling [13] (with their integrands being known
to five-loop order [14] and even through ten loops for $p=2$ [15]), while at
strong coupling holographic and bootstrap approaches determine the leading
strong coupling limit [16, 17, 18, 19, 20, 21, 22, 23], as well as some sub-
leading $1/\lambda$ [24, 6, 25, 26, 27, 28, 29] and $1/N$ [30, 31, 32, 33, 34,
35, 36, 37, 38] corrections. Knowing any of the $\langle{\cal O}_{2}{\cal
O}_{2}{\cal O}_{p}^{(i)}{\cal O}_{p}^{(j)}\rangle$ exactly in the planar limit
would be tantamount to knowing the holographic dual of the
AdS${}_{5}\,\times\,$S5 Virasoro-Shapiro amplitude – that is, the four-point
scattering of AdS${}_{5}\,\times\,$S5 supergravitons in classical string
theory – and, upon developing its operator product expansion, would furnish a
substantial part of the solution of planar $\mathcal{N}=4$ SYM. Unfortunately,
due to the inherent difficulty of intermediate coupling, what is known about
this object is essentially perturbative around either weak or strong coupling.

And yet, rather astoundingly, if one integrates these four-point functions
against a fairly vanilla spacetime measure, these correlators can be
determined exactly and, as we will see, simply. In particular, the integrated
correlators of interest here are defined as follows [6]:

$\displaystyle\mathcal{G}_{p}^{(N|i,j)}(\tau):=-\frac{2}{\pi}\int_{0}^{\infty}dr\int_{0}^{\pi}d\theta\,\frac{r^{3}\sin^{2}\theta}{u^{2}}\,\mathcal{H}^{(N|i,j)}_{p}(u,v;\tau)|_{u=1+r^{2}-2r\cos\theta,\,v=r^{2}}\,,$
(1.2)

where $\mathcal{H}^{(N|i,j)}_{p}(u,v;\tau)$ is the “dynamical” part of the
correlator not fixed by superconformal Ward identities (recalled explicitly in
Section 2.1 below). The magic is that $\mathcal{G}_{p}^{(N|i,j)}(\tau)$ are
computable from derivatives of a free energy on $S^{4}$ deformed by sources,
which in turn is determined by supersymmetric localization. The aim is then to
find simpler expressions upon starting from the localization integrals. In
[6], these were computed exactly in the ‘t Hooft limit for all $p$, yielding
an elegant integral formula for the integrated AdS-Virasoro-Shapiro amplitude.
Our goal here is to study these integrated correlators at finite $N$, for all
$p$.

For $p=2$, this was done in [7], who found a conjecturally exact, recursive
solution at finite $N$, an elegant result subjected to many checks. (See [39,
40, 41, 42] for subsequent related work.) That solution was elucidated in [8]
from another point of view, one which will be crucial for this paper: namely,
that of the $SL(2,\mathbb{Z})$ spectral decomposition. To introduce this
briefly, any $SL(2,\mathbb{Z})$-invariant observable of $\mathcal{N}=4$ SYM
admits, by way of finiteness of free $\mathcal{N}=4$ SYM and the spectral
theorem for $SL(2,\mathbb{Z})$, a decomposition into a complete,
$SL(2,\mathbb{Z})$-invariant eigenbasis of the Laplacian on the upper-half
$\tau$-plane. There are three branches of eigenfunctions: a continuous branch
(the non-holomorphic Eisenstein series, $E_{s}^{*}(\tau)$, with ${\rm
Re\,}s={1\over 2}$), a discrete branch (the Maass cusp forms,
$\phi_{n}(\tau)$), and a constant. The overlap with the constant term is equal
to the ensemble average, with respect to the Zamolodchikov metric, over the
space of $\mathcal{N}=4$ theories parameterized by $\tau$.

This is a powerful tool in the $\mathcal{N}=4$ SYM context because the
complete eigenbasis “decouples” the $\tau$-dependence from the core
information, not made redundant by S-duality, that characterizes a given
observable. The entire content of an observable is characterized by its
overlaps with these basis elements: $\tau$-space is traded for “spectral
space.” As in any other physical context where a complete basis is identified,
it gives a systematic way to understand the possible $\tau$-dependence of
$SL(2,\mathbb{Z})$-invariant $\mathcal{N}=4$ SYM observables, and, therefore,
to classify these dependences according to the functional complexity of the
overlaps.111It also makes connections, so far nascent, between $\mathcal{N}=4$
SYM observables, on the one hand, and mathematical questions in arithmetic
chaos and the analytic structure of the fundamental domain, on the other. See
[43, 44, 45] for some mathematical background.

Applied to the $p=2$ integrated correlator, $\mathcal{G}_{2}^{(N)}(\tau)$, [8]
found that the entire integrated correlator is determined by a single
polynomial. First, the cusp form overlap vanishes. Second, the Eisenstein
overlap is given by a universal gauge-theoretic prefactor times an
$N$-dependent polynomial in the spectral parameter $s$; this polynomial may be
determined solely from the perturbative expansion of
$\mathcal{G}_{p}^{(N|i,j)}(\tau)$ to finite order. These polynomials solve a
short recursion relation in $N$, equivalent to the recursion originally found
by [7]. Finally, the ensemble average, which is simply (one half times) the
aforementioned polynomial evaluated at $s=1$, is also just a polynomial in
$N$. If one prefers to restore the $\tau$-dependence, the result is

$\begin{split}\mathcal{G}_{2}^{(N)}(\tau)={N(N-1)\over 4}+\frac{1}{4\pi
i}\int_{\text{Re}\,s=\frac{1}{2}}ds\,\frac{\pi}{\sin(\pi
s)}s(1-s)f_{2}^{(N)}(s)\,E^{*}_{s}(\tau)\,,\end{split}$ (1.3)

where the full information (including the additive constant) is contained in
the polynomial $f_{2}^{(N)}(s)$. A consequence of this structure is that the
entire $\mathcal{G}_{2}^{(N)}(\tau)$ is fixed by the sector of zero total
instanton number. The origin of the recursion remains mysterious. However,
what the $SL(2,\mathbb{Z})$ decomposition makes clear is that the integrated
correlator is about as simple of a non-trivial $\tau$-dependent observable as
one could hope for.

In this paper, we carry forward this analysis to the general class of
integrated correlators $\mathcal{G}_{p}^{(N|i,j)}(\tau)$. The results are
highly uniform in $p$. We will compute the full matrix of correlators for
$p\leq 5$ – recall that for $p\geq 4$ there are multiple trace structures – as
well as a certain infinite family that exists for all $p$. The results will
again be very simple in the $SL(2,\mathbb{Z})$ spectral decomposition, with
all integrated correlators computed herein determined simply by polynomials in
the spectral parameter $s$. These polynomials obey fascinating recursion
relations relating different values of both $N$ and $p$. They are determined
by a finite number of orders in weak coupling perturbation theory. The
structural rigidity of these results raises obvious questions about why,
exactly, they are so, and begs for a fundamental derivation of these recursion
relations. From the AdS${}_{5}\times$ S5 point of view, the fact that these
are known for all $N$ and $\tau$ means that the quantum type IIB string theory
scattering amplitudes, upon suitable integration over AdS5 boundary points,
are known. To what extent that recasting can be made substantive, by telling
us something non-trivial about type IIB string theory, is an intriguing open
question for the future.

In Section 2 we introduce the necessary tools and background to study the
integrated correlators – half-BPS operators, their correlators both integrated
and unintegrated, and the $SL(2,\mathbb{Z})$ spectral decomposition – and
review the $p=2$ case.

In Section 3 we present our first results, namely, the integrated correlators
for $p\leq 5$. At both $p=4$ and $p=5$ there are two half-BPS operators, one
single-trace and one double-trace, so each case furnishes a $2\times 2$
matrix. The technique is straightforward: using localization, one develops the
perturbative expansion in $g_{\rm YM}^{2}$; infers from this the Eisenstein
overlap by matching to the perturbative expansion of the spectral
decomposition; and then justifies that the cusp form overlap vanishes. The
result, as advertised above, is that the integrated correlators are determined
in terms of a single polynomial, $f_{p}^{(N|i,j)}(s)$, of degree
$2N+2\lfloor{p\over 2}\rfloor-4$ and symmetric under $s\rightarrow 1-s$.222In
fact, we find that $f_{p}^{(N|i,j)}(s)\propto(2s-1)^{2}$, which further
reduces the degree of the undetermined polynomial. That is,
$\mathcal{G}_{p}^{(N|i,j)}(\tau)$ takes the same functional form as (1.3) with
different polynomial and constant terms. Having enumerated the polynomials for
many values of $N$, we search for, and indeed find, linear recursion
relations. Let us reproduce, as an example, the recursion for $p=3$ here:

$N(N+1)\Big{[}f_{3}^{(N+1)}(s)-f_{3}^{(N)}(s)\Big{]}=2N(N-1)\,f_{2}^{(N+1)}(s)+2(N+1)(N+2)\,f_{2}^{(N)}(s)\,.$
(1.4)

This is a very tidy recursion, that relates the $p=2$ and $p=3$ polynomials
$f_{2}^{(N)}(s)$ and $f_{3}^{(N)}(s)$, respectively, such that the latter are
ultimately fixed by the former in the $SU(2)$ theory (plus a trivial initial
condition at $N=1$). This phenomenon persists for all cases that we study.

As for the cusp forms, in Subsection 3.3 we provide strong analytical support
for their vanishing overlap from first principles, by performing explicit
instanton calculations on the localization side and checking that they match
the spectral result without cusp forms. These calculations use the instanton
partition function in the presence of sources for half-BPS operators of higher
charge [46], generalizing the original Nekrasov partition function. All told,
this means that $\mathcal{G}_{p}^{(N|i,j)}(\tau)$ is determined completely by
the first $N+\lfloor{p\over 2}\rfloor-2$ orders in perturbation theory. We do
not know the fundamental reason for this remarkable fact. The recursion
relations may be immediately uplifted to Laplace difference equations, which
we show in Subsection 3.4.

In Sections 4 and 5, we further analyze the $p\leq 5$ results. Section 4
collects their ensemble averages. Section 5 analyzes the integrated
correlators of single-particle operators333These operators, dual to single-
particle states in AdS${}_{5}\times$ S5, are linear combinations of ${\cal
O}_{p}^{(i)}$ defined to be orthogonal to all multi-trace operators. See
(2.5). at large $N$, developing the $1/N$ expansion of the spectral overlaps
and the ensemble averages. This may be done algorithmically to arbitrary
genus. We observe a match between the large $N$ ensemble averages and the
strongly coupled planar results, verifying the general relation established in
[8]. Using the same correspondence between ensemble averages and bulk
quantities, we also make a prediction for the one-loop AdS${}_{5}\times$ S5
supergravity result for the integrated correlator at generic $p$ – see (5.15).
We explicitly develop the ‘t Hooft and very strongly coupled large $N$ limits
from the genus expansion of the spectral overlaps, noting some interesting
functional representations along the way.

In Section 6, we present our second slate of results. We study the integrated
correlator for which both operators ${\cal O}_{p}^{(i)}$ and ${\cal
O}_{p}^{(j)}$ are taken to be composites of ${\cal O}_{2}$ with $p/2$
constituents, for arbitrary $p\in 2\mathbb{Z}_{+}$:

${\cal O}_{p}^{(\rm max)}:=[{\cal O}_{2}]^{p/2}\,.$ (1.5)

We call this the maximal-trace family of integrated correlators,
$\mathcal{G}_{p}^{(N|{\rm max})}(\tau)$. These are a type of generalization of
the extremal two-point functions studied in the context of $\mathcal{N}=2$
SCFTs in four dimensions (e.g. [47, 48, 49, 50, 51, 52]). We again find an
explicit recursive solution for all $N$ and $p$, with the same features as
described earlier, but with one key difference: the recursion relation
operates at fixed $N$, only shifting $p$. It takes the form of a semi-infinite
lattice chain equation

$\displaystyle\Delta_{\tau}\widehat{\mathcal{G}}_{2n-2}^{(N|{\rm max})}=$
$\displaystyle\leavevmode\nobreak\
2\left((n-1)^{2}+2nc-c\right)\widehat{\mathcal{G}}_{2n-2}^{(N|{\rm
max})}-n\left(n+2c-1\right)\widehat{\mathcal{G}}_{2n}^{(N|{\rm max})}$
$\displaystyle-(n-1)\left(n+2c-2\right)\widehat{\mathcal{G}}_{2n-4}^{(N|{\rm
max})}+2c\leavevmode\nobreak\ \widehat{\mathcal{G}}_{2}^{(N|{\rm max})}$ (1.6)

where $\widehat{\mathcal{G}}_{p}^{(N|{\rm max})}(\tau)$ refers to an
appropriately normalized ${\mathcal{G}}_{p}^{(N|{\rm max})}(\tau)$ defined in
(6.5), $c=\frac{1}{4}(N^{2}-1)$ is the central charge and $p=2n$. This is
evocative of the Toda chain equation obeyed by the extremal correlators. The
ensemble average admits a closed-form expression in terms of harmonic numbers
– see (6.7). The large $p$ limit of this sequence is a large charge limit, the
details of which will appear in future work. We also develop the $1/N$
expansion of these correlators, with more nice polynomials emerging, see e.g.
(6.10) and (6.20).

In Section 7, we make an ansatz (7.1) for the general integrated correlator
$\mathcal{G}_{p}^{(N|i,j)}(\tau)$ with arbitrary trace structures $i,j$.

In Section 8, we conclude with some open problems and future directions.

Several appendices round out the text with computational details, consistency
checks and collected formulas.

##  2 Setup

In this section, we introduce the half-BPS operators and their four-point
functions; define the integrated correlators and how one computes them from
localisation; recall the rudiments of the $SL(2,\mathbb{Z})$ spectral
decomposition, a central tool in our study of the integrated correlators; and
finally, as a segue to the next sections, review the $p=2$ integrated
correlator studied in [7, 8].

###  2.1 Half-BPS operators and their four-point functions

We will study four-point correlation functions of half-BPS superconformal
primary operators in $\mathcal{N}=4$ SYM theory with gauge group $SU(N)$.
These operators have protected scaling dimension $\Delta=p$ and transform in
the $[0,p,0]$ representation of the R-symmetry group $SU(4)$. The simplest
realisation of such operators is in terms of single-trace operators $T_{p}$,
which we define by

$\displaystyle T_{p}(x,y)=\frac{1}{p}\leavevmode\nobreak\ y_{I_{1}}\cdots
y_{I_{p}}\text{Tr}\big{(}\Phi^{I_{1}}(x)\cdots\Phi^{I_{p}}(x)\big{)},$ (2.1)

where $\Phi^{I}(x)$, with $I=1,\ldots 6$, are the six real scalars of the
theory and we introduced auxiliary $SO(6)$ null-vectors $y_{I}$ (obeying
$y^{2}\equiv y_{I}\,y^{I}=0$) to project onto the symmetric traceless
representation. In addition to these, we will also consider multi-trace
operators, which can be obtained from products of the form

$\displaystyle T_{p_{1},\ldots,p_{n}}(x,y)=\frac{p_{1}\cdots
p_{n}}{p}\leavevmode\nobreak\ T_{p_{1}}(x,y)\cdots T_{p_{n}}(x,y),\qquad 2\leq
p_{1}\leq\ldots\leq p_{n},$ (2.2)

with $p_{1}+\cdots+p_{n}=p$. Due to this degeneracy in the space of half-BPS
operators, we introduce the notation ${\cal O}_{p}^{(i)}$ to distinguish the
different single- and multi-trace operators of a given dimension $p$, assuming
an ordering with increasing number of traces.444For example, since the single-
trace operator always comes first, we simply have ${\cal O}_{p}^{(1)}=T_{p}$.
Within the same number of traces we adopt lexicographic ordering, such that
e.g. $T_{2,4}$ comes before $T_{3,3}$, etc. The species of operators appearing
at weight $p$ are in one-to-one correspondence with restricted integer
partitions of $p$ that do not include one. In what follows we will sometimes
study operators with $p\leq 5$, where we encounter at most double-trace
operators:

$\displaystyle\begin{split}{\cal O}_{2}^{(i)}&\in\\{T_{2}\\},\qquad{\cal
O}_{3}^{(i)}\in\\{T_{3}\\},\qquad{\cal
O}_{4}^{(i)}\in\\{T_{4},T_{2,2}\\},\qquad{\cal
O}_{5}^{(i)}\in\\{T_{5},T_{2,3}\\}.\end{split}$ (2.3)

Since the operators ${\cal O}_{p}^{(i)}$ are half-BPS, their two-point
functions are protected (meaning they are independent of the coupling $\tau$)
and hence can be computed within the free theory. They are of the general form

$\displaystyle\langle\mathcal{O}_{p}^{(i)}(x_{1},y_{1})\mathcal{O}_{p}^{(j)}(x_{2},y_{2})\rangle=g_{12}^{p}\,R^{(i,j)}_{p}(N),$
(2.4)

where $g_{ij}=y_{ij}^{2}/x_{ij}^{2}$ is the superpropagator and the colour-
factors $R^{(i,j)}_{p}(N)=R^{(j,i)}_{p}(N)$ encode the non-trivial
$N$-dependence. For the operators given in (2.3), we record the corresponding
colour-factors in Appendix A. Note that in the trace-basis the two-point
functions are not diagonal.

In the context of the AdS/CFT correspondence it is also useful to consider so-
called single-particle operators (SPO’s), which we denote by ${\cal O}_{p}$
without the superscript.555In the following, any quantity without a
superscript $(i)$ is evaluated in the SPO basis. These operators are dual to
single-particle states in AdS${}_{5}\,\times\,$S5 supergravity, namely the
AdS5 supergraviton multiplet ($p=2$) and higher Kaluza-Klein modes ($p\geq 3$)
on S5. They are given by particular linear combinations of the ${\cal
O}_{p}^{(i)}$ and can be elegantly defined for all $N$ by their orthogonality
to all multi-trace operators [53, 34]:

$\langle{\cal O}_{p}T_{p_{1},\ldots,p_{n}}\rangle=0\,,\leavevmode\nobreak\
\leavevmode\nobreak\ n\geq 2.$ (2.5)

At leading order in large $N$, ${\cal O}_{p}$ coincides with the single-trace
operator $T_{p}$ but receives $1/N$ suppressed multi-trace admixtures.666The
coefficient of an $n$-trace admixture is suppressed by $1/N^{n-1}$ with
respect to the single-trace contribution $T_{p}$. The first few SPO’s are
given by

$\displaystyle{\cal O}_{2}=T_{2}\,,\quad{\cal O}_{3}=T_{3}\,,\quad{\cal
O}_{4}=T_{4}-\frac{2N^{2}-3}{N(N^{2}+1)}T_{2,2}\,,\quad{\cal
O}_{5}=T_{5}-\frac{5(N^{2}-2)}{N(N^{2}+5)}T_{2,3}\,.$ (2.6)

While we do not have an explicit formula for the generic colour-factors
$R_{p}^{(i,j)}(N)$ in (2.4), the $N$-dependence of the two-point function in
the SPO basis is simply given by777Due to our choice of normalisation for the
external operators (2.1), the colour-factor $R_{p}$ used here has a factor of
$p^{2}$ difference with respect to [54], i.e.
$R_{p}|_{\text{here}}=R_{p}|_{\text{there}}/p^{2}$.  [54]

$\displaystyle
R_{p}(N)=(p-1)\bigg{[}\frac{1}{(N-p+1)_{p-1}}-\frac{1}{(N+1)_{p-1}}\bigg{]}^{-1},$
(2.7)

For the first few SPO’s given above, this yields

$\displaystyle\begin{split}R_{2}(N)&=\frac{(N^{2}-1)}{2},\quad
R_{3}(N)=\frac{(N^{2}-1)(N^{2}-4)}{3N},\quad
R_{4}(N)=\frac{(N^{2}-1)(N^{2}-4)(N^{2}-9)}{4(N^{2}+1)},\\\
R_{5}(N)&=\frac{(N^{2}-1)(N^{2}-4)(N^{2}-9)(N^{2}-16)}{5N(N^{2}+5)}\,.\end{split}$
(2.8)

Note that $R_{p}$ vanishes for $N=1,2,\ldots,p-1$, meaning that ${\cal O}_{p}$
vanishes as an operator for those values of $N$, thereby relating various
elements ${\cal O}_{p}^{(i)}$ in (2.3) at fixed $p$. For instance,
$2\,T_{4}=T_{2,2}$ for $N=2,3$ and $6\,T_{5}=5\,T_{2,3}$ for $N=3,4$. At large
$N$, $R_{p}$ scales as $R_{p}(N)\sim N^{p}/p$.

Next, let us introduce four-point correlation functions of the half-BPS
operators defined above. In particular, we will be interested in the class of
correlators given by $\langle{\cal O}_{2}{\cal O}_{2}{\cal O}^{(i)}_{p}{\cal
O}^{(j)}_{p}\rangle$. Superconformal symmetry constrains these correlators to
take the form [55, 56]888Our conventions for the conformal and $SU(4)$
R-symmetry cross-ratios read  $\displaystyle u=x\bar{x}$
$\displaystyle=\frac{x_{12}^{2}x_{34}^{2}}{x_{13}^{2}x_{24}^{2}},$
$\displaystyle
v=(1-x)(1-\bar{x})=\frac{x_{14}^{2}x_{23}^{2}}{x_{13}^{2}x_{24}^{2}},$
$\displaystyle\frac{1}{\sigma}=y\bar{y}$
$\displaystyle=\frac{y_{12}^{2}y_{34}^{2}}{y_{13}^{2}y_{24}^{2}},$
$\displaystyle\frac{\mu}{\sigma}=(1-y)(1-\bar{y})=\frac{y_{14}^{2}y_{23}^{2}}{y_{13}^{2}y_{24}^{2}},$
(2.9) We denote the second R-symmetry cross ratio as $\mu$ instead of the
canonical $\tau$, to avoid confusion with the coupling.

$\displaystyle\langle{\cal O}_{2}{\cal O}_{2}{\cal O}^{(i)}_{p}{\cal
O}^{(j)}_{p}\rangle=g_{12}^{2}g_{34}^{p}\,\Big{[}\mathcal{G}_{p,\text{free}}^{(i,j)}(u,v;\sigma,\mu)+\mathcal{I}\,\mathcal{H}^{(i,j)}_{p}(u,v;\tau)\Big{]},$
(2.10)

The factor $\mathcal{H}^{(i,j)}_{p}$ is the only part of the correlator which
depends on $\tau$, and hence contains all of the non-trivial dynamics. It is
multiplied by the factor $\mathcal{I}$ which is fixed by superconformal Ward
identities to take the factorised form

$\displaystyle\mathcal{I}$
$\displaystyle=\frac{(x-y)(x-\bar{y})(\bar{x}-y)(\bar{x}-\bar{y})}{(y\bar{y})^{2}}\,.$
(2.11)

On the other hand, the first term in (2.10) is the correlator in the free
theory which can be computed by Wick contractions. It takes a particularly
nice form in the SPO basis, which in our conventions reads

$\displaystyle\mathcal{G}_{p,\text{free}}(u,v;\sigma,\mu)=R_{2}R_{p}\left(\Big{[}1+\delta_{2,p}\Big{(}u^{2}\sigma^{2}+\frac{u^{2}\mu^{2}}{v^{2}}\Big{)}\Big{]}+\frac{p}{2c}\Big{[}u\sigma+\frac{u\mu}{v}+(p-1)\frac{u^{2}\sigma\mu}{v}\Big{]}\right),$
(2.12)

where $c$ is the central charge $c=(N^{2}-1)/4$ of the $SU(N)$ theory. While
the structure of $\mathcal{G}^{(i,j)}_{p,\text{free}}$ for generic $(i,j)$
takes a similar form in terms of the cross-ratios $(u,v)$ and $(\sigma,\mu)$,
the $N$ dependence of the different $(\sigma,\mu)$ monomials is in general
very non-trivial and the simplicity of the above result is a consequence of
the orthogonality property of the SPO’s ${\cal O}_{p}$ [54].

###  2.2 Integrated correlators

Our main focus henceforth will be on a family of observables defined by a
Euclidean integration of the dynamical part of $\langle{\cal O}_{2}{\cal
O}_{2}{\cal O}^{(i)}_{p}{\cal O}^{(j)}_{p}\rangle$:

$\displaystyle\mathcal{G}_{p}^{(N|i,j)}(\tau):=-\frac{2}{\pi}\int_{0}^{\infty}dr\int_{0}^{\pi}d\theta\,\frac{r^{3}\sin^{2}\theta}{u^{2}}\,\mathcal{H}^{(N|i,j)}_{p}(u,v;\tau)|_{u=1+r^{2}-2r\cos\theta,\,v=r^{2}}\,.$
(2.13)

It was first shown in [6] that $\mathcal{G}_{p}^{(N|i,j)}$ is given by
derivatives of $S^{4}$ partition function of $\mathcal{N}=2^{*}$ gauge theory
deformed by couplings $\tau_{A}^{\prime}$ of higher-weight chiral operators
(here and below $\tau_{A}^{\prime}$ collectively denotes all irrelevant
couplings $\tau_{3}^{\prime},\tau_{4}^{\prime},...$ which couple to chiral
primary operators of dimension $\Delta>2$, whereas $\tau$ denotes the exactly
marginal coupling).999 We note that [6] studied the relation (2.14) in the ’t
Hooft limit where contributions from single-trace operators dominate: that is,
for the case $(i,j)=(1,1)$ in our notation. However, as we will discuss in
more detail below, (2.14) is also valid for multi-trace operators, i.e. for
all pairs $(i,j)$, with the appropriate prescription for multi-trace sources.
In our conventions, this relation reads

$\displaystyle\mathcal{G}_{p}^{(N|i,j)}(\tau)=\frac{R^{(i,j)}_{p}}{4}\leavevmode\nobreak\
\frac{v^{i,\mu}_{p}\bar{v}^{j,\nu}_{p}\partial_{\tau_{\mu}^{\prime}}\partial_{\bar{\tau}_{\nu}^{\prime}}\partial_{m}^{2}\log\mathcal{Z}_{N}(\tau,\tau_{A}^{\prime},m)}{v^{i,\mu}_{p}\bar{v}^{j,\nu}_{p}\partial_{\tau_{\mu}^{\prime}}\partial_{\bar{\tau}_{\nu}^{\prime}}\log\mathcal{Z}_{N}(\tau,\tau_{A}^{\prime},m)}\bigg{|}_{\tau^{\prime}_{A}=m=0}\leavevmode\nobreak\
.$ (2.14)

The $S^{4}$ partition function101010Strictly speaking, for non-vanishing
$\tau^{\prime}_{A}$, $\mathcal{Z}_{N}(\tau,\tau_{A}^{\prime},m)$ should be
viewed as a generating functional for correlators of chiral primary operators.
$\mathcal{Z}_{N}(\tau,\tau^{\prime}_{A},m)$ takes the following form [50, 57,
58, 46]

$\displaystyle\mathcal{Z}_{N}(\tau,\tau_{A}^{\prime},m)=\int$ $\displaystyle
d^{N}a\prod_{i<j}(a_{ij})^{2}\delta\big{(}\sum_{i=1}^{N}a_{i}\big{)}\bigg{|}\exp\big{(}i\pi\tau\sum_{i=1}^{N}a_{i}^{2}+\sum_{p=3}^{\infty}i\pi^{p/2}\tau_{p}^{\prime}\sum_{i=1}^{N}a_{i}^{p}\big{)}\bigg{|}^{2}$
$\displaystyle\big{|}Z_{\text{inst}}(\tau,\tau_{A}^{\prime},m,a)\big{|}^{2}\frac{1}{H(m)^{N}}\prod_{i<j}\frac{H^{2}(a_{ij})}{H(a_{ij}+m)H(a_{ij}-m)}\leavevmode\nobreak\
\leavevmode\nobreak\ .$ (2.15)

The derivatives in (2.14) ensure that $\mathcal{G}_{p}^{(N|i,j)}(\tau)$ are
unambiguous functions of the coupling [50, 59, 60].

Let us unpack these expressions. The integration variables $a_{i}$ are over
the Cartan subalgebra of the gauge group $SU(N)$ – i.e. $a_{i}\in\mathbb{R}$
and are constrained by the condition $\sum_{i}a_{i}=0$ – and
$a_{ij}:=a_{i}-a_{j}$. $m$ is the $\mathcal{N}=2$-preserving mass of the
adjoint hypermultiplet. The function $H(z)$ is given by product of two Barnes
G-functions

$\begin{split}H(z)=e^{-(1+\gamma)z^{2}}G(1+iz)G(1-iz)\leavevmode\nobreak\
.\end{split}$ (2.16)

The instanton factor $Z_{\text{inst}}(\tau,\tau_{A}^{\prime},m,a)$ in (2.2) is
the localized contribution to the partition function from non-trivial
instanton configurations of SYM gauge fields. It admits the decomposition

$\displaystyle
Z_{\text{inst}}(\tau,\tau^{\prime}_{A},m,a)=\sum_{k=0}^{\infty}e^{2\pi i\tau
k}Z_{\text{inst}}^{(k)}(\tau^{\prime}_{A},m,a)\leavevmode\nobreak\ ,$ (2.17)

where $Z_{\text{inst}}^{(k)}(\tau^{\prime}_{A},m,a)$ denotes the contribution
from the $k^{\prime}$th instanton sector. The expansion of
$\big{|}Z_{\text{inst}}\big{|}^{2}$ appearing in (2.2), in powers of the
instanton counting factor $q:=e^{2\pi i\tau}$, is

$\begin{split}\big{|}Z_{\text{inst}}\big{|}^{2}=1+q\leavevmode\nobreak\
Z^{(1)}_{\text{inst}}+\bar{q}\leavevmode\nobreak\
\bar{Z}^{(1)}_{\text{inst}}+q\bar{q}\leavevmode\nobreak\
|Z^{(1)}_{\text{inst}}|^{2}+\cdots\end{split}$ (2.18)

The explicit form of $Z_{\text{inst}}^{(k)}(\tau^{\prime}_{A},m,a)$ is known
for generic $k$. For our purposes we will only need the expression for $k=1$
[46]:

$\displaystyle
Z_{\text{inst}}^{(1)}(\tau^{\prime}_{A},m,a)=-m^{2}\sum_{\ell=1}^{N}\exp\left(-\sum_{p=3}^{\infty}i\pi^{p/2}\tau^{\prime}_{p}\left(a_{\ell}^{p}+(a_{\ell}+2i)^{p}-2(a_{\ell}+i)^{p}\right)\right)\prod_{j\neq\ell}\frac{\left(a_{\ell
j}+i\right)^{2}-m^{2}}{\left(a_{\ell j}+i\right)^{2}+1}$ (2.19)

We note that this expression from [46] is for $U(N)$ gauge group.111111Our
notation is related to those of eq. (2.24) in [46] as follows:
$a_{i}^{\text{there}}=-ia_{i}^{\text{here}},\leavevmode\nobreak\
m^{\text{there}}=-(im^{\text{here}}+1)\leavevmode\nobreak\ ,$
$(-i)^{p}2\pi\tau_{p}^{\text{there}}=-\pi^{p/2}p!\tau^{\prime}_{p}\leavevmode\nobreak\
$. However, it is applicable for computing correlators in $\mathcal{N}=4$ SYM
with $SU(N)$ gauge group by simply imposing the constraint $\sum_{i}a_{i}=0$
on the eigenvalues in the $U(N)$ matrix model.121212This is unlike generic
$\mathcal{N}=2$ theories. We are grateful to Silviu Pufu for a discussion.

At the $\mathcal{N}=4$ conformal point, where $m=\tau_{A}^{\prime}=0$,
$Z_{\text{inst}}(\tau,0,0,a)=H(0)=1$, and $\mathcal{Z}_{N}(\tau,0,0)$ is just
a Gaussian matrix model in the (special) unitary ensemble that can be
explicitly evaluated:

$\displaystyle\mathcal{Z}_{N}^{{}^{\mathcal{N}=4}}(\tau)=\int
d^{N}a_{i}\prod_{i<j}\left(a_{ij}\right)^{2}\delta\big{(}\sum
a_{i}\big{)}\bigg{|}e^{i\pi\tau\sum_{i}a_{i}^{2}}\bigg{|}^{2}=\frac{G(N+2)}{(2\pi)^{N(N-1)/2}\left(2{\rm
Im\,}\tau\right)^{(N^{2}-1)/2}\sqrt{N}}\leavevmode\nobreak\
\leavevmode\nobreak\ .$ (2.20)

An important detail in (2.14) is the operator mixing problem on $S^{4}$. As
explained in [50], in curved space, the source for a chiral operator ${\cal
O}^{(i)}_{p}$ can have non-minimal couplings to lower-dimensional chiral
operators ${\cal O}^{(i)}_{p-2n}$ with $n\in\mathbb{Z}_{+}$, due to non-
vanishing background fields such as the spatial curvature. Therefore, to
compute correlators on $\mathbb{R}^{4}$ from the deformed $S^{4}$ partition
function, one must solve this operator mixing problem. This is achieved
through a Gram-Schmidt orthogonalization process which allows us to compute
the complex vectors $v_{p}^{i,\mu}$ from the matrix of $S^{4}$ two-point
functions. These vectors encode the mixing of weight-$p$ chiral operators of
species $i$ with lower-weight chiral operators. Insertion of a chiral operator
${\cal O}_{p}^{(i)}$ in a correlation function on $\mathbb{R}^{4}$ is then
obtained through the combination of derivatives
$v_{p}^{i,\mu}\partial_{\tau_{\mu}^{\prime}}$, where we sum over the index
$\mu$ which indexes operators in the following set:

$\\{{\cal O}_{p}^{(i)}\\}\leavevmode\nobreak\ \cup\leavevmode\nobreak\
\\{{\cal O}_{p-2}^{(j_{1})}\\}\leavevmode\nobreak\ \cup\leavevmode\nobreak\
\\{{\cal O}_{p-4}^{(j_{2})}\\}\leavevmode\nobreak\ \cup\leavevmode\nobreak\
\cdots\leavevmode\nobreak\ $ (2.21)

where $i$ is a fixed index, whereas the $j_{n}$ indices run over all species
of operators of indicated weight, in accord with the discussion in Subsection
2.1. In a slight (but hopefully intuitive) abuse of notation, we write
$\mu=p^{(j)}$ to refer to the indicated operator ${\cal O}_{p}^{(j)}$ of some
fixed $p$ and $j$.

Each operator ${\cal O}_{p}^{(j)}$ is associated with a source in the
partition function $\mathcal{Z}_{N}$; the meaning of
$\partial_{\tau_{p^{(j)}}^{\prime}}$ is to differentiate with respect to that
source. The derivatives are defined as follows: for a given multi-trace
operator ${\cal O}_{p}^{(j)}\equiv T_{p_{1},\ldots,p_{n}}$ such that
$p_{1}+\cdots+p_{n}=p$,

${\partial\over\partial{\tau_{p^{(j)}}^{\prime}}}:=\prod_{i=1}^{n}{\partial\over\partial_{\tau_{p_{i}}^{\prime}}}\,,$
(2.22)

with the understanding that $\tau^{\prime}_{2}\equiv\tau$. Its action on
$\mathcal{Z}_{N}$ defines differentiation with respect to the source for the
composite operators.

As an example, consider the insertion of the double-trace operator ${\cal
O}_{5}^{(2)}\equiv T_{2,3}$ for which $\mu\in\\{5^{(2)},3^{(1)}\\}$. The
expressions of interest will have derivatives of the form
$\partial/\partial{\tau_{p^{(j)}}^{\prime}}$ acting on $\mathcal{Z}_{N}$.
Applying (2.22), for $p^{(j)}=3^{(1)}$, this is a single-trace operator, so
$\partial\mathcal{Z}_{N}/\partial{\tau_{3^{(1)}}^{\prime}}:=\partial_{\tau_{3}^{\prime}}\mathcal{Z}_{N}$.
Likewise, for $p^{(j)}=5^{(2)}$ we have
$\partial\mathcal{Z}_{N}/\partial{\tau_{5^{(2)}}^{\prime}}:=\partial_{\tau_{3}^{\prime}}\partial_{\tau}\mathcal{Z}_{N}$.
Appendix B contains more details on this procedure for cases up to $p=5$.

###  2.3 The $SL(2,\mathbb{Z})$ spectral decomposition

Many observables of $\mathcal{N}=4$ SYM are $SL(2,\mathbb{Z})$-invariant. Such
observables, call them $\mathbb{O}(\tau)$, admit a spectral decomposition into
a complete $SL(2,\mathbb{Z})$-invariant eigenbasis of the Laplacian on the
upper half plane. Let us introduce the main elements of this decomposition.
More complete explanations may be found in Sections 2-3 of [8] in the
$\mathcal{N}=4$ SYM context, and in [61, 62] in great mathematical detail; see
also [63] for a pertinent review. The fundamental domain for
$SL(2,\mathbb{Z})$ may be defined as

$\mathcal{F}=\mathbb{H}/SL(2,\mathbb{Z})=\left\\{\tau=x+iy\in\mathbb{H}\,\bigg{|}-{1\over
2}\leq x\leq{1\over 2},|\tau|\geq 1\right\\}.$ (2.23)

This space is endowed with the hyperbolic metric

$ds^{2}={dx^{2}+dy^{2}\over y^{2}}$ (2.24)

which defines a Laplacian

$\begin{split}\Delta_{\tau}=-y^{2}(\partial_{x}^{2}+\partial_{y}^{2})\,.\end{split}$
(2.25)

We define inner products of square-integrable $SL(2,\mathbb{Z})$-invariant
functions via the Petersson inner product,

$(f,g)\coloneqq\int_{\mathcal{F}}{dxdy\over y^{2}}f(\tau)\overline{g(\tau)}.$
(2.26)

The observable $\mathbb{O}(\tau)$, which is square-integrable [8], obeys the
invariance equation

$\mathbb{O}(\gamma\tau)=\mathbb{O}(\tau),\quad\gamma\tau={a\tau+b\over
c\tau+d},\quad\gamma=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in
SL(2,\mathbb{Z}).$ (2.27)

Restricting to real-valued $\mathbb{O}(\tau)$, we develop a Fourier
decomposition with respect to $x$,

$\mathbb{O}(\tau)=\mathbb{O}_{0}(y)+\sum_{k=1}^{\infty}2\cos(2\pi
kx)\,\mathbb{O}_{k}(y).$ (2.28)

In $\mathcal{N}=4$ SYM, where $x=\theta/2\pi$, the mode number $k$ is 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 Roelcke-Selberg spectral decomposition takes the following form:

$\displaystyle\mathbb{O}(\tau)=\langle\mathbb{O}\rangle+\frac{1}{4\pi
i}\int_{\text{Re}\,s=\frac{1}{2}}ds\,\\{\mathbb{O},E_{s}\\}\,E^{*}_{s}(\tau)+\sum_{n=1}^{\infty}(\mathbb{O},\phi_{n})\,\phi_{n}(\tau),$
(2.29)

This defines a convergent decomposition for any $\tau\in{\cal F}$. There are
three branches here.

The first is the constant function, $\phi_{0}={\rm vol}({\cal F})^{-1/2}$, so
$\langle\mathbb{O}\rangle={\rm vol}({\cal F})^{-1}(\mathbb{O},1)$ is the
normalized average of $\mathbb{O}(\tau)$ over ${\cal F}$:

$\begin{split}\langle\mathbb{O}\rangle:={\rm vol}({\cal F})^{-1}\int_{\cal
F}{dxdy\over y^{2}}\,\mathbb{O}(\tau)\end{split}$ (2.30)

The bracket notation indicates that this is equivalent to the normalized
ensemble average of $\mathbb{O}(\tau)$, that is, the average value of
$\mathbb{O}(\tau)$ over the $\mathcal{N}=4$ supersymmetry-preserving conformal
manifold with respect to the Zamolodchikov measure. This identification
between modular and ensemble averages is special to $\mathcal{N}=4$ SYM, in
which the Zamolodchikov metric is exactly hyperbolic due to maximal
supersymmetry [64], unlike other sub-maximal theories whose conformal
manifolds admit an $SL(2,\mathbb{Z})$ action (e.g. $\mathcal{N}=2$ SQCD in
four dimensions [65, 47]).

The second is the continuous branch of non-holomorphic Eisenstein series,

$\displaystyle\Delta_{\tau}E^{*}_{s}(\tau)=s(1-s)E^{*}_{s}(\tau)\,,\quad{\rm
Re\,}(s)=\frac{1}{2}.$ (2.31)

The star denotes that we use the “completed” Eisenstein series, whose Fourier
expansion with respect to $x$ is

$E^{*}_{s}(\tau)=\Lambda(s)y^{s}+\Lambda(1-s)y^{1-s}+\sum_{k=1}^{\infty}4\cos(2\pi
kx)\frac{\sigma_{2s-1}(k)}{k^{s-\frac{1}{2}}}\sqrt{y}K_{s-\frac{1}{2}}(2\pi
ky),$ (2.32)

where $\Lambda(s):=\pi^{-s}\Gamma(s)\zeta(2s)$ is the completed Riemann zeta
function and $\sigma_{2s-1}(k)$ is the divisor function. This normalization is
convenient because it manifests the reflection symmetry

$\begin{split}E^{*}_{s}(\tau)=E^{*}_{1-s}(\tau)\,.\end{split}$ (2.33)

The overlap with $\mathbb{O}(\tau)$ is then a rescaled Petersson inner
product, which may be simplified to

$\begin{split}\\{\mathbb{O},E_{s}\\}&:=\Lambda(s)^{-1}(\mathbb{O},E_{s})\\\
&=\Lambda(s)^{-1}\int_{0}^{\infty}dy\,y^{-s-1}\,\mathbb{O}_{0}(y)\,.\end{split}$
(2.34)

The second line will be important for the physics: the Eisenstein overlap is
fully determined by the zero mode of $\mathbb{O}(\tau)$, i.e. the projection
of $\mathbb{O}(\tau)$ onto the sector of zero total instanton number. This
sector includes the weak-coupling regime $g_{\rm YM}^{2}\ll 1$. The reflection
symmetry of $E^{*}_{s}(\tau)$ implies the same property of
$\\{\mathbb{O},E_{s}\\}$. Note that the ensemble average may be extracted from
the $s\rightarrow 1$ limit of the Eisenstein overlap,

$\langle\mathbb{O}\rangle={1\over 2}\,\lim_{s\rightarrow
1}\\{\mathbb{O},E_{s}\\}\,,$ (2.35)

which follows from $E_{0}(\tau)=1$.

The third is an infinite discrete branch of Maass cusp forms in the so-called
$L^{2}$ norm, $(\phi_{n},\phi_{n})=1$. These functions, infinite in number,
obey the eigenvalue equation

$\Delta_{\tau}\phi_{n}(\tau)=\left({1\over
4}+t_{n}^{2}\right)\phi_{n}(\tau)\,,\quad n\in\mathbb{Z}_{+}\,,\quad
t_{n}\in\mathbb{R}\,,\quad 0<t_{1}<t_{2}<\ldots$ (2.36)

Reality of $\mathbb{O}(\tau)$ restricts the basis to the even cusp forms,
invariant under $x\rightarrow-x$. The cusp forms $\phi_{n}(\tau)$ have no zero
mode, instead decaying exponentially at the cusp $y\rightarrow\infty$, hence
contributing to neither the perturbative nor ‘t Hooft regimes. The cusp forms
are rich mathematical objects exhibiting various signs of arithmetic chaos
[43, 44, 45]; however, quite intriguingly, the integrated correlators
$\mathcal{G}_{p}^{(N)}(\tau)$ will be seen to have vanishing cusp form
overlap, so we will not say anything further about the cusp forms here.

In summary, the entire content of $\mathbb{O}(\tau)$ is stored in the spectral
overlaps $\\{\mathbb{O},E_{s}\\}$ and $(\mathbb{O},\phi_{n})$.

Looking ahead slightly, our computations will begin from the zero mode,
$\mathbb{O}_{0}(y)$:

$\begin{split}\mathbb{O}_{0}(y)=\langle\mathbb{O}\rangle+\frac{1}{2\pi
i}\int_{\operatorname{Re}s=\frac{1}{2}}ds\,\\{\mathbb{O},E_{s}\\}\Lambda(s)y^{s}\end{split}$
(2.37)

As shown in [8], $\mathcal{N}=4$ SYM observables with a consistent
perturbative expansion in $g_{\rm YM}^{2}$ admit a rather constrained analytic
form for their Eisenstein overlaps:

$\begin{split}\\{\mathbb{O},E_{s}\\}=\frac{\pi}{\sin(\pi
s)}s(1-s)f_{\text{p}}(s)+f_{\text{np}}(s)\end{split}$ (2.38)

The function $f_{\text{p}}(s)$ encodes the complete perturbative part of
$\mathbb{O}(\tau)$, its residues giving the weak coupling ($y\gg 1$) data. The
function $f_{\text{np}}(s)$ encodes non-perturbative, instanton-anti-instanton
corrections. Both functions must be invariant under $s\mapsto 1-s$, and are
regular for $s\in{\mathbb{C}}$ away from $s=0$ and its reflection. For the
observables we consider, it will turn out that $f_{\text{np}}(s)=0$, a fact
whose origin lies in the Borel summability of their perturbative expansions;
this further implies [8] that $f_{\text{p}}(s)$ must be regular at $s=0$ and
is thus entire.

###  2.4 Prelude: Review of $p=2$ result

To illustrate the spectral decomposition (2.29) at work – and to set the stage
for our computations ahead – let us consider the example of the $p=2$
integrated correlator, $\mathcal{G}^{(N)}_{2}(\tau)$. The result is

$\displaystyle\mathcal{G}^{(N)}_{2}(\tau)=\frac{N(N-1)}{4}+\frac{1}{4\pi
i}\int_{\text{Re}\,s=\frac{1}{2}}ds\,\frac{\pi}{\sin(\pi
s)}s(1-s)f^{(N)}_{2}(s)\,E^{*}_{s}(\tau).$ (2.39)

That is, the ensemble average, and Eisenstein and cusp form overlaps, are

$\displaystyle\langle\mathcal{G}^{(N)}_{2}\rangle=\frac{N(N-1)}{4},\quad\\{\mathcal{G}^{(N)}_{2},E_{s}\\}=\frac{\pi}{\sin(\pi
s)}s(1-s)f^{(N)}_{2}(s),\quad(\mathcal{G}^{(N)}_{2},\phi_{n})=0.$ (2.40)

For $N\leq 2$,

$f_{2}^{(1)}(s)=0,\quad f_{2}^{(2)}(s)=(2s-1)^{2}.$ (2.41)

For $N>2$, it turns out that the spectral overlaps obey a powerful three-term
recursion relation [7]

$\displaystyle
N(N-1)f^{(N+1)}_{2}(s)=\Big{[}2(N^{2}-1)-s(1-s)\Big{]}f^{(N)}_{2}(s)-N(N+1)f^{(N-1)}_{2}(s),$
(2.42)

This fully determines $\mathcal{G}^{(N)}_{2}(\tau)$ for any $SU(N)$ in terms
of the $SU(2)$ result alone (and its triviality at $N=1$).

To reiterate, the complete content of $\mathcal{G}^{(N)}_{2}(\tau)$ is
contained in (2.40)-(2.42).

This observable also admits a (formally equivalent) lattice-integral
representation, as originally seen in [7], which may be derived by performing
an $SL(2,\mathbb{Z})$ Poincaré sum of the zero mode of (a Borel resummation
of) $\mathcal{G}^{(N)}_{2}(\tau)$. The same will be true of the $p>2$
integrated correlators.131313It is straightforward to prove [8] that any
$SL(2,\mathbb{Z})$-invariant, square-integrable observable $\mathbb{O}(\tau)$
with $(\mathbb{O},\phi_{n})=0$ admits a lattice-integral representation of the
form admitted by $\mathcal{G}^{(N)}_{2}(\tau)$. On the other hand, a generic
$SL(2,\mathbb{Z})$-invariant observable, with nonzero cusp form overlap, need
not. As discussed in the Introduction and hopefully made clear by our
presentation, instead writing $\mathcal{G}^{(N)}_{2}(\tau)$ in a complete,
$SL(2,\mathbb{Z})$-invariant basis illuminates just how simple it is in the
space of possible $\mathcal{N}=4$ observables:

$\bullet$ The cusp form overlap vanishes for all $N$.

$\bullet$ The non-perturbative part of the Eisenstein overlap vanishes for all
$N$.

$\bullet$ The perturbative part of the Eisenstein overlap, $f^{(N)}_{2}(s)$,
is an even polynomial of order $2N-2$.

The passage between bases also helps to clarify the physical meaning of some
of the mathematical aspects of the lattice-integral representation itself,
such as the identification of the lattice-integral kernel as an
$SL(2,\mathbb{Z})$ Borel transform and the observed properties of
$\mathcal{G}^{(N)}_{2}(\tau)$ under integration and inversion.

Let us make a few further comments. First, the $SU(2)$ result is, rather
literally, the simplest result consistent with the analyticity properties of
the overlap: in particular, $f_{p}^{(N)}(s)$ must be an entire, reflection-
symmetric (i.e. even under $s\rightarrow 1-s$), non-constant function of $s$,
the simplest example of which is the degree-two monomial shown above. Second,
since the cusp form overlap vanishes, the above recursion relation may be
uplifted to a ‘Laplace difference equation’ for the integrated correlator [7],
which reads

$\displaystyle
N(N-1)\mathcal{G}^{(N+1)}_{2}(\tau)=\Big{[}2(N^{2}-1)-\Delta_{\tau}\Big{]}\mathcal{G}^{(N)}_{2}(\tau)-N(N+1)\mathcal{G}^{(N-1)}_{2}(\tau),$
(2.43)

where we have used (2.31). This differential relation is equivalent to the
algebraic relation (2.42). Finally, the large $N$ expansion of
$\mathcal{G}^{(N)}_{2}(\tau)$ was analyzed in [7, 8], in various limits and
including non-perturbative corrections in both $1/\lambda$ [7] and $1/N$ [8],
where $\lambda:=g_{\rm YM}^{2}N$ is the ‘t Hooft coupling. We refer the reader
to those papers for details. Here we just highlight one result, to be
generalized to $p>2$ later in this work: at $\lambda\gg 1$,

$\mathcal{G}^{(N)}_{2}(\lambda)\approx{N^{2}\over
4}\left(1-24\zeta(3)\lambda^{-3/2}+O(\lambda^{-5/2})\right)+O(N^{0})$ (2.44)

The leading term, which gives the value of the integrated correlator in tree-
level AdS${}_{5}\,\times\,$S5 supergravity in our normalization, manifestly
equals the ensemble average in (2.40) at large $N$ [8].

With these observations in place, we now proceed to study the more general
class of integrated correlators $\mathcal{G}_{p}^{(N|i,j)}(\tau)$. The
$SL(2,\mathbb{Z})$ spectral decomposition will greatly simplify the physical
and technical analysis.

##  3 Results I: Integrated correlators for $p\leq 5$

We now derive the integrated correlators $\mathcal{G}_{p}^{(N|i,j)}(\tau)$ for
$p\leq 5$ in the $SL(2,\mathbb{Z})$ spectral decomposition. As we will see,
their general structure is uniform in $p$, essentially identical to that of
the $p=2$ case presented in (2.40)-(2.42). The derivation is a three-step
process:

  1. 1)

In Subsection 3.1, we reconstruct the full zero(-instanton) mode of the
integrated correlator from its weak-coupling expansion. The latter is computed
from localization as outlined in Section 2.2 and Appendix B.

  2. 2)

In Subsection 3.2, we leverage the zero mode to deduce the Eisenstein overlap
(2.38) by matching to the spectral decomposition. Just as for $p=2$, these
will have vanishing non-perturbative part, $f_{\rm np}(s)=0$, and the
remaining piece will be fixed by polynomials obeying powerful recursion
relations in $N$.

  3. 3)

Finally, in Subsection 3.3, we upgrade this to the result for the full
integrated correlator by arguing that the cusp form overlap vanishes. We do so
with several explicit computations. This then implies that the aforementioned
recursion relations are relations for the full integrated correlator. We note
that they may be recast as Laplace difference equations, an exercise we carry
out in Subsection 3.4.

###  3.1 Perturbative expansion

The weak-coupling expansion of $\mathcal{G}_{p}^{(N|i,j)}$ (around a zero-
instanton background), may be extracted from the localization expression
(2.14)-(2.2) by expanding the latter at $y\gg 1$ (recall that $y:={\rm
Im\,}\tau$). Details of this computation may be found in Appendix B for all
$p\leq 5$. The final result may be written as

$\displaystyle\mathcal{G}_{p}^{(N|i,j)}(\tau)\Big{|}_{\text{0-inst}}=\frac{R^{(i,j)}_{p}}{4}\frac{C_{p}^{(N|i,j)}}{D_{p}^{(N|i,j)}}\,,$
(3.1)

where $C_{p}^{(N|i,j)}$ and $D_{p}^{(N|i,j)}$ are given in (B.14) and (B.8)
respectively. Let us copy the result from Appendix B for $p=2,3$ below:

$\displaystyle\mathcal{G}_{2}^{(N)}(\tau)\Big{|}_{\text{0-inst}}=$
$\displaystyle\frac{R_{2}}{4}\bigg{(}\frac{12N\zeta(3)}{y\pi}-\frac{75N^{2}\zeta(5)}{y^{2}\pi^{2}}+\frac{735N^{3}\zeta(7)}{2y^{3}\pi^{3}}-\frac{945N^{2}\left(7N^{2}+2\right)\zeta(9)}{4y^{4}\pi^{4}}$
$\displaystyle+\frac{114345N^{3}\left(N^{2}+1\right)\zeta(11)}{16y^{5}\pi^{5}}-\frac{351351N^{2}\left(11N^{4}+25N^{2}+4\right)\zeta(13)}{128y^{6}\pi^{6}}+\ldots\bigg{)},$
(3.2) $\displaystyle\mathcal{G}_{3}^{(N)}(\tau)\Big{|}_{\text{0-inst}}=$
$\displaystyle\frac{R_{3}}{4}\bigg{(}\frac{18N\zeta(3)}{y\pi}-\frac{90N^{2}\zeta(5)}{y^{2}\pi^{2}}+\frac{735N^{3}\zeta(7)}{2y^{3}\pi^{3}}-\frac{2835N^{2}\left(2N^{2}+1\right)\zeta(9)}{4y^{4}\pi^{4}}$
$\displaystyle+\frac{114345N^{3}\left(3N^{2}+5\right)\zeta(11)}{64y^{5}\pi^{5}}-\frac{117117N^{2}\left(11N^{4}+40N^{2}+9\right)\zeta(13)}{64y^{6}\pi^{6}}+\ldots\bigg{)}.$
(3.3)

The results for $p=4,5$ may similarly be assembled from Appendix B.

Before extracting the spectral overlaps from these, we note that these results
are in agreement with previous results from two-loop perturbation theory: in
particular, following the strategy of [7], in Appendix E we take the two-loop
results for the unintegrated $\langle{\cal O}_{2}{\cal O}_{2}{\cal
O}_{p}^{(i)}{\cal O}_{p}^{(j)}\rangle$ correlators from [9], integrate them
against the measure (2.13), and compare to the localization result shown
above. The results match, as seen in Appendix E. This provides a non-trivial
check of the method used here. It also probes a novel aspect of the $p=4,5$
cases, in which there is a matrix of integrated correlators involving double-
trace composite operators; the agreement described above verifies that the
localization method works even when the external operators are
composites.141414This is also a further verification that the Konishi operator
does not contribute to $\mathcal{G}_{p}^{(N|i,j)}(\tau)$, as was discussed in
[6] in the ’t Hooft limit and [7] for finite $N$.

###  3.2 Spectral overlaps and recursion relations

To deduce the Eisenstein overlap from the perturbative expansion, one must
develop the latter from the spectral decomposition. This is straightforwardly
done by closing the integration contour in (2.37) to the left,
yielding151515As we will point out momentarily, the non-perturbative part of
the overlap vanishes for all $p$. This allows us to employ an amusing abuse of
notation, here and henceforth, in which we refer to the “$p$” subscript on
$f_{p}(s)$ as the label for the $p$’th integrated correlator.

$\displaystyle\mathcal{G}_{p}^{(N|i,j)}(\tau)\Big{|}_{\text{0-inst}}=-\sum_{n=1}^{\infty}(-1)^{n}n(n+1)\Lambda(n+\tfrac{1}{2})f^{(N|i,j)}_{p}(n+1)\,y^{-n}\leavevmode\nobreak\
.$ (3.4)

where we recall the definition of the completed Riemann zeta function
$\Lambda(n+\tfrac{1}{2})$ below (2.32). This is then compared to the weak-
coupling expansion of $\mathcal{G}_{p}^{(N|i,j)}$, from which
$f^{(N|i,j)}_{p}(n+1)$ is extracted. Using the perturbative expansions derived
in the previous subsection and Appendix B, we find that in all cases we
considered,

$\displaystyle{f_{p}^{(N|i,j)}(s)=(2s-1)^{2}\,g_{p}^{(N|i,j)}(s),}$ (3.5)

where $g_{p}^{(N|i,j)}(s)$ are reflection-symmetric polynomials of degree
$2N+2\lfloor{p\over 2}\rfloor-6$, for all labels $(i,j)$.

The perturbative expansions are Borel summable. This typically (but not always
[66, 67]) implies, in the context of resurgence, that non-perturbative
corrections in $g_{\rm YM}^{2}$ are not present. This was confirmed in [7] for
$p=2$, implying that the non-perturbative part of the Eisenstein overlap
(2.38) vanishes [8]. One can confirm this property for the $p>2$ cases
considered here as well:161616One can see this from the localisation point of
view as follows. From the expansion (2.18) and the partition function (2.19)
it can be observed that the instanton-anti-instanton term
$|Z_{1\text{-inst}}|^{2}$ goes like $m^{4}$, and therefore does not contribute
to the integrated correlators (2.14). Persistence of this property to higher
orders leads to the above conclusion.

${f_{\rm np}^{(N|i,j)}(s)=0}$ (3.6)

Therefore, (3.5) gives the full Eisenstein overlap. Summarizing so far, the
Eisenstein overlaps of $\mathcal{G}_{p}^{(N|i,j)}$ for $p=3,4,5$ take the form

$\boxed{\\{\mathcal{G}_{p}^{(N|i,j)},E_{s}\\}=\frac{\pi}{\sin(\pi
s)}s(1-s)(2s-1)^{2}g_{p}^{(N|i,j)}(s)\,,\quad\text{where}\quad\text{deg}\left[g_{p}^{(N|i,j)}(s)\right]=2N+2\left\lfloor\frac{p}{2}\right\rfloor-6\,.}$
(3.7)

This implies that $\\{\mathcal{G}_{p}^{(N|i,j)},E_{s}\\}$ may be completely
fixed by the first $N+\left\lfloor\frac{p}{2}\right\rfloor-2$ orders in weak-
coupling perturbation theory.

Given this simplicity, and inspired by the $p=2$ case (2.42), we are
encouraged to seek recursive relations for $p>2$. To do so, we use the above
observations on the general structure of the spectral overlaps to construct
the most restrictive polynomial ansatz, allowing us to compute the overlaps
$f_{p}^{(N|i,j)}(s)$ for $p\leq 5$ and $N\leq 20$ from the perturbative
expansions to order $y^{-20}$. As we will discuss in detail in the following,
we are then able to find many interesting relations generalising the $p=2$
recursion to relations not only among theories with different values of $N$,
but also among integrated correlators of different values of $p$.

####  3.2.1 $p=3$

In the $p=3$ case, the perturbative expansion vanishes for $N=1,2$ (due to
vanishing of the operator ${\cal O}_{3}$ itself) and hence we have
$f^{(1)}_{3}(s)=f^{(2)}_{3}(s)=0$. For concreteness, we display the
expressions for the next few values of $N$ which read

$\displaystyle\begin{split}f_{3}^{(3)}(s)&=\tfrac{1}{3}(2s-1)^{2}(s^{2}-s+18),\\\
f_{3}^{(4)}(s)&=\tfrac{1}{12}(2s-1)^{2}(s^{4}-2s^{3}+47s^{2}-46s+264),\\\
f_{3}^{(5)}(s)&=\tfrac{1}{120}(2s-1)^{2}(s^{6}-3s^{5}+95s^{4}-185s^{3}+1824s^{2}-1732s+6240).\end{split}$
(3.8)

Considering many more cases up to $N=20$, we find that the above overlaps
satisfy the following striking recursion relation:

$\begin{split}\boxed{N(N+1)\Big{[}f_{3}^{(N+1)}(s)-f_{3}^{(N)}(s)\Big{]}=2N(N-1)\,f_{2}^{(N+1)}(s)+2(N+1)(N+2)\,f_{2}^{(N)}(s)\,.}\end{split}$
(3.9)

This is a remarkable formula: it allows one to recursively compute the $p=3$
overlaps for generic $N$ from knowledge of the $p=2$ overlaps
$f^{(N)}_{2}(s)$, which in turn are fully determined by the $SU(2)$ theory
alone, together with the trivial initial condition $f^{(1)}_{3}(s)=0$! Let us
note that this recursion is consistent starting from the value $N=1$. In
particular, the fact that $f_{3}^{(2)}(s)$ vanishes is compatible with
$f_{2}^{(2)}(s)\neq 0$ in the first term on the RHS thanks to the presence of
an $(N-1)$ factor.

An interesting feature of the above relation is that it has no explicit
dependence on $s$, in contrast with the $p=2$ recursion relation (2.42). This
is concordant with the fact that the spectral overlaps $f_{p}^{(N)}(s)$ are
polynomials of the same degree for $p=2$ and $p=3$.

This is not the only interesting relation the spectral overlaps
$f^{(N)}_{3}(s)$ obey. At the cost of introducing some $s$-dependent terms, we
find a recursion relation involving only the $p=3$ spectral overlaps:

$\displaystyle\begin{split}(N-2)(N-3)(N+1)f_{3}^{(N+1)}(s)&=\leavevmode\nobreak\
\leavevmode\nobreak\ (N-3)\Big{[}(N^{2}+2N-2)-s(1-s)\Big{]}\,f_{3}^{(N)}(s)\\\
&\,\leavevmode\nobreak\ \leavevmode\nobreak\
+(N+2)\Big{[}(N^{2}-4N+1)-s(1-s)\Big{]}\,f_{3}^{(N-1)}(s)\\\
&\,\leavevmode\nobreak\ \leavevmode\nobreak\
-(N-2)(N+2)(N+1)\,f_{3}^{(N-2)}(s)\,.\end{split}$ (3.10)

Note that due to the factors of $(N-2)(N-3)$ on the RHS, the recursion starts
at $N=4$, and as a consequence of being a four-term recursion relation one
needs to supply $f^{(N)}_{3}(s)$ with $N=2,3,4$ as initial data, making it a
somewhat less powerful statement than the previous relation (3.9).

####  3.2.2 $p=4$

Next, $p=4$ is the first case with a degeneracy in the spectrum of half-BPS
operators: namely, the $T_{4}$ and $T_{2,2}$ operators. As such, we have three
cases to consider: $(i,j)=(1,1),\,(1,2),\,(2,2)$. We will also consider the
case of single-particle operators, $(i,j)\rightarrow\text{(SPO, SPO)}$.

Interestingly, while for $N=1$ the perturbative expansions vanish, we find
that for $N=2,3$ the three expansions degenerate and are simply proportional
to each other termwise in their $1/y$ expansions. In terms of spectral
overlaps, this yields

$\displaystyle\begin{split}N=1:\qquad\,f_{4}^{(1|i,j)}(s)&=0\,,\\\ N=2:\qquad
f_{4}^{(2|2,2)}(s)&=2\,f_{4}^{(2|1,2)}(s)=4\,f_{4}^{(2|1,1)}(s)=(2s-1)^{2}(s^{2}-s+8)\,,\\\
N=3:\qquad
f_{4}^{(3|2,2)}(s)&=2\,f_{4}^{(3|1,2)}(s)=4\,f_{4}^{(3|1,1)}(s)=\tfrac{1}{2}(2s-1)^{2}(s^{2}-s+6)(s^{2}-s+18)\,.\end{split}$
(3.11)

This phenomenon is a non-trivial consequence of considering the $SU(N)$ theory
and is explained by the fact that for low values of $N$ the single-trace
operator $T_{4}$ is no longer linearly independent of the double-trace
operator $T_{2,2}$: for $N=2,3$ one has $T_{4}\propto T_{2,2}$ and therefore
also their correlators become proportional to each other. In view of the
different intricate $N$-dependence of the weak-coupling expansions
(B.2)-(B.2), the degenerations for $N=2,3$ seem highly non-trivial (and
further support our claim that the localisation integrals correctly compute
multi-trace insertions).

For $N\geq 4$, the spectral overlaps $f_{4}^{(N|i,j)}(s)$ start to differ for
different $(i,j)$. For example, the case $(i,j)=(2,2)$, corresponding to the
correlator with two double-trace operators, is the one with simplest
$s$-dependence:

$\displaystyle\begin{split}f_{4}^{(4|2,2)}(s)&=\tfrac{1}{12}(2s-1)^{2}(s^{2}-s+4)(s^{2}-s+18)(s^{2}-s+32)\,,\\\
f_{4}^{(5|2,2)}(s)&=\tfrac{1}{144}(2s-1)^{2}(s^{2}-s+50)(s^{6}-3s^{5}+55s^{4}-105s^{3}+664s^{2}-612s+1440)\,.\end{split}$
(3.12)

We have obtained analogous expressions for the $(i,j)=(1,1)$ and $(1,2)$
cases. However, we refrain from presenting more explicit data since, by
considering many cases of different $N$, we discover that the $p=4$ spectral
overlaps likewise obey interesting recursion relations, which we present now.

#####  $\mathbf{(i,j)=(2,2)}$:

Let us start with the simplest case given by the correlator involving two
double-trace operators. In this case we find the very direct relation

$\displaystyle\boxed{f^{(N|2,2)}_{4}(s)=\Big{[}2N^{2}-s(1-s)\Big{]}\,f_{2}^{(N)}(s)\,.}$
(3.13)

The overlaps $f_{4}^{(N|2,2)}(s)$ are fully determined by the $p=2$ overlaps
$f_{2}^{(N)}(s)$ only. This will be elaborated upon in Section 6.

#####  $\mathbf{(i,j)=(1,2)}$:

For the mixed correlator with one single-trace and one double-trace operator,

$\displaystyle\begin{split}f_{4}^{(N+1|1,2)}(s)-f_{4}^{(N|1,2)}(s)&=\frac{1}{2}\bigg{[}\frac{N^{3}+4N^{2}+13N-2}{N+1}-s(1-s)\bigg{]}\,f_{2}^{(N+1)}(s)\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ -\frac{N^{3}+2N^{2}-4N-6}{N}\,f_{2}^{(N)}(s)\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ +\frac{N(N+1)}{2}\,f_{2}^{(N-1)}(s)\,,\end{split}$ (3.14)

This, too, determines $f_{4}^{(N|1,2)}(s)$ from $f^{(N)}_{2}(s)$
alone.171717Here and in what follows, we leave implicit the trivial vanishing
conditions of the spectral overlaps at $N=0,1$.

#####  $\mathbf{(i,j)=(1,1)}$:

Recursion relations also exist for the overlaps of the correlator involving
two single-trace operators albeit in somewhat more complicated form. For this
case, we find

$\displaystyle\begin{split}f_{4}^{(N+1|1,1)}(s)-f_{4}^{(N|1,1)}(s)&=\frac{1}{4}\bigg{[}\frac{3N^{4}+3N^{3}+23N^{2}-23N+26}{(N+1)^{2}}-s(1-s)\bigg{]}\,f_{2}^{(N+1)}(s)\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ +\frac{3(2N^{2}+2N+3)}{N(N+1)}\,f_{3}^{(N)}(s)\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
-\frac{N^{5}+7N^{4}-18N^{2}-12N-36}{2N^{2}(N+1)}\,f_{2}^{(N)}(s)\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ -\frac{N(N+1)}{4}\,f_{2}^{(N-1)}(s)\,,\end{split}$ (3.15)

The new feature relative to the previous relations is the presence of
$f_{3}^{(N)}(s)$ on the RHS. However, since the latter is fixed by
$f_{2}^{(N)}(s)$ via (3.9), so, ultimately, is $f_{4}^{(N|1,1)}(s)$.181818 We
have also found a recursion relation which does not involve the $p=3$
overlaps, but its structure as well as the $N$-dependence of the coefficients
is more complicated and we refrain from giving it here. Notably, the RHS of
that relation has a pole at $N=3$ and hence the recursion can be used only for
$N\geq 4$.

##### (SPO, SPO):

Lastly, we can assemble the spectral overlap $f_{4}^{(N)}(s)$ of the
correlator of two SPO’s ${\cal O}_{4}$, defined as the linear combination in
(2.6). From that definition, one has

$\displaystyle f_{4}^{(N)}(s)=f_{4}^{(N|1,1)}(s)+2\beta
f_{4}^{(N|1,2)}(s)+\beta^{2}f_{4}^{(N|2,2)}(s)\,,$ (3.16)

with mixing coefficient $\beta=-\frac{2N^{2}-3}{N(N^{2}+1)}$. Note that this
particular linear combination is precisely such that $f_{4}^{(N)}(s)=0$ for
$N=1,2,3$ since, by construction, the operator ${\cal O}_{4}$ itself vanishes
for those values of $N$.191919Because the overlaps $f_{4}^{(N|i,j)}(s)$ become
proportional to each other for $N=2,3$ (cf. (3.11)), the mixing coefficient
$\beta$ must take the same value for $N=2$ and $N=3$ in order to be consistent
with $f_{4}^{(N)}(s)=0$ for those $N$. Indeed, one has
$\beta|_{N=2}=\beta|_{N=3}=-\frac{1}{2}$, which is a somewhat non-trivial
property of $\beta$. By combining the recursion relations for
$f_{p}^{(N|i,j)}(s)$ given in equations (3.13)-(3.15) one can derive a similar
recursion for the overlaps of $f_{p}^{(N)}(s)$, whose precise form we record
in Appendix C.

####  3.2.3 $p=5$

Similarly to $p=4$, the $p=5$ case presents three cases to consider:
$(i,j)=(1,1),\,(1,2),\,(2,2)$, as well as the case of single-particle
operators, $(i,j)\rightarrow\text{(SPO, SPO)}$.

As for $p=4$, we observe that the perturbative expansions degenerate for low
values of $N$: as a consequence of the two half-BPS operators becoming
proportional to each other, i.e. $T_{5}\propto T_{2,3}$ for $N\leq 4$, we have
that the weak-coupling expansions vanish for $N=1,2$ (due to ${\cal O}_{3}$
which ceases to exist for $N=1,2$), and that for $N=3,4$ the expansions
degenerate. As mentioned for $p=4$, the latter property is not at all obvious
from the different $N$-dependent coefficients in the perturbative expansions,
see equations (B.2)-(B.2). For the spectral overlaps we therefore have

$\displaystyle\begin{split}N=1,2:\qquad&f_{5}^{(N|i,j)}(s)=0\,,\\\\[3.0pt]
N=3:\qquad&f_{5}^{(3|2,2)}(s)=\tfrac{6}{5}\,f_{5}^{(3|1,2)}(s)=\tfrac{36}{25}\,f_{5}^{(3|1,1)}(s)\\\
&\leavevmode\nobreak\
=\tfrac{4}{25}(2s-1)^{2}(3s^{4}-6s^{3}+98s^{2}-95s+498)\,,\\\\[3.0pt]
N=4:\qquad&f_{5}^{(4|2,2)}(s)=\tfrac{6}{5}\,f_{5}^{(4|1,2)}(s)=\tfrac{36}{25}\,f_{5}^{(4|1,1)}(s)\\\
&\leavevmode\nobreak\
=\tfrac{3}{50}(2s-1)^{2}(2s^{6}-6s^{5}+149s^{4}-288s^{3}+2297s^{2}-2154s+7704)\,,\end{split}$
(3.17)

and only from $N\geq 5$ onwards their $s$-dependence differs. Comparing again
cases with varying $N$, we are able to find recursion relations for the $p=5$
spectral overlaps, which we list below:

#####  $\mathbf{(i,j)=(2,2)}$:

$\displaystyle\begin{split}f_{5}^{(N+1|2,2)}(s)&=\frac{12(N-1)}{25(N+1)}\bigg{[}(N^{3}+20N^{2}+24N+18)-6s(1-s)\bigg{]}\,f_{2}^{(N+1)}(s)\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
+\frac{18(N^{2}+2N+18)}{25}\,f_{3}^{(N)}(s)-\frac{36(N+2)(3N^{2}+2N-18)}{25N}\,f_{2}^{(N)}(s)\,.\end{split}$
(3.18)

#####  $\mathbf{(i,j)=(1,2)}$:

$\displaystyle\begin{split}f_{5}^{(N+1|1,2)}(s)-f_{5}^{(N|1,2)}(s)&=\frac{12(N-1)}{5(N+1)}\bigg{[}\frac{2N^{3}+3N^{2}+8N-3}{N+1}-s(1-s)\bigg{]}\,f_{2}^{(N+1)}(s)\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
+\frac{24(N+2)}{5N}\,f_{4}^{(N|1,2)}(s)+\frac{18(N-1)(3N-2)}{5N(N+1)}\,f_{3}^{(N)}(s)\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
-\frac{12(N+2)(2N^{4}+N^{3}-12N^{2}+11N-8)}{5N^{2}(N+1)}\,f_{2}^{(N)}(s)\,.\end{split}$
(3.19)

#####  $\mathbf{(i,j)=(1,1)}$:

$\displaystyle f_{5}^{(N+1|1,1)}(s)-f_{5}^{(N|1,1)}(s)$
$\displaystyle=\frac{2(N-1)}{(N+1)}\bigg{[}\frac{3N^{4}+4N^{3}+14N^{2}+4N+15}{(N+1)^{2}}-s(1-s)\bigg{]}\,f_{2}^{(N+1)}(s)\qquad$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ +\frac{8(N^{2}+N+2)}{N(N+1)}\,f_{4}^{(N|1,1)}(s)$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
-\frac{3(2N^{5}-3N^{4}-4N^{3}-23N^{2}-24N-24)}{N^{2}(N+1)^{2}}\,f_{3}^{(N)}(s)$
(3.20) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
-\frac{2(N^{7}+8N^{6}-18N^{4}-5N^{3}-42N^{2}-24N-48)}{N^{3}(N+1)^{2}}\,f_{2}^{(N)}(s)$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ +4N\,f_{2}^{(N-1)}(s)\,,$

where the above recursion relations start from $N=1$ with initial values
$f_{p}^{(N|i,j)}(s)=0$ for $N\leq 1$. As noted earlier in the $p=3$ case, the
factors of $(N-1)$ in front of the $f_{2}^{(N+1)}(s)$ terms on the RHS are
required by consistency of these formulae for $N=1$.

##### (SPO, SPO):

Lastly, the $p=5$ spectral overlaps of the correlator of SPO’s are given by an
analogous equation as given in (3.21),

$\displaystyle f_{5}^{(N)}(s)=f_{5}^{(N|1,1)}(s)+2\beta
f_{5}^{(N|1,2)}(s)+\beta^{2}f_{5}^{(N|2,2)}(s)\,,$ (3.21)

where the mixing coefficient now reads $\beta=-\frac{5(N^{2}-2)}{N(N^{2}+5)}$.
This is the right linear combination which yields $f_{5}^{(N)}(s)=0$ for
$N=1,2,3,4$.202020Consistency with the proportionality relations (3.17)
demands that for $N=3,4$ the $p=5$ mixing coefficient takes the same value.
This is indeed the case and one finds
$\beta|_{N=3}=\beta|_{N=4}=-\frac{5}{6}$. We can then again use the recursion
relations for $f_{5}^{(N|i,j)}(s)$ to assemble a similar recursion for
$f_{5}^{(N)}(s)$ which we give in Appendix C.

###  3.3 Including instantons: completion to the full correlator

Having computed the zero mode of $\mathcal{G}^{(N)}_{p}(\tau)$, and therefore
the Eisenstein spectral overlap, what remains is to determine the cusp form
overlap. We claim that it vanishes:

$\boxed{\left(\mathcal{G}_{p}^{(N|i,j)},\phi_{n}\right)=0\quad\forall\leavevmode\nobreak\
n>0\,.}$ (3.22)

That is, the full integrated correlator is

$\displaystyle\boxed{{\mathcal{G}_{p}^{(N|i,j)}(\tau)=\langle\mathcal{G}_{p}^{(N|i,j)}\rangle+\frac{1}{4\pi
i}\int_{\text{Re}\,s=\frac{1}{2}}ds\,\frac{\pi}{\sin(\pi
s)}s(1-s)(2s-1)^{2}g_{p}^{(N|i,j)}(s)\,E^{*}_{s}(\tau)\,,}}$ (3.23)

where recall that the average is
$\langle\mathcal{G}_{p}^{(N|i,j)}\rangle={1\over 2}g_{p}^{(N|i,j)}(0)$. Note
how strongly constraining this is: because $g_{p}^{(N|i,j)}(s)$ is an even
polynomial of degree given in (3.7), the integrated correlator may be
completely fixed by the first $N+\left\lfloor\frac{p}{2}\right\rfloor-2$
orders in weak-coupling perturbation theory.

We will provide strong support for (3.22) with explicit instanton computations
below. The basic idea behind the check is as follows. First, one can compute
the perturbative expansion of the integrated correlators in nonzero instanton
sectors as predicted by (3.23). This result is then to be compared with a
direct computation from localisation formula (2.14) using the relevant
instanton partition function. For $p=2$, the absence of cusp forms was
extensively checked in this manner in [8, 7]. In this subsection, we carry out
the check for several $p>2$ integrated correlators using the instanton
partition function (2.19).212121Note that it is sufficient to check this in
the one-instanton sector: as argued in [8] using properties of the
$SL(2,\mathbb{Z})$ eigenbasis, all $k>1$ sectors are fully determined by the
$k=0,1$ sectors. Happily, the two results agree.

####  3.3.1 Calculation

The first step of extracting the perturbative expansion around $k$ instantons
from (3.23) is straightforward. (The anti-instanton result, proportional to
$e^{-2\pi ik\bar{\tau}}$, is identical.) One just inserts the $k$’th Fourier
mode of $E_{s}^{*}(\tau)$ from (2.32) and the $y\gg 1$ expansion

$\begin{split}\sqrt{y}K_{s-\frac{1}{2}}(2\pi ky)=\frac{1}{2\sqrt{k}}e^{-2\pi
ky}\sum_{n=0}^{\infty}\frac{a_{n}(s)}{(2\pi ky)^{n}}\leavevmode\nobreak\
,\qquad a_{n}(s)=\frac{(s)_{n}(1-s)_{n}}{(-2)^{n}n!}\leavevmode\nobreak\
.\end{split}$ (3.24)

Borel summability allows us to swap the $s$ integral in (3.23) with the sum
over $n$ [8]. Closing the contour towards the left, and taking $k=1$, we find

$\displaystyle\mathcal{G}_{p}^{(N|i,j)}(\tau)\Big{|}_{\text{1-inst}}={1\over
2}e^{2\pi i\tau}\sum_{n=0}^{\infty}\frac{1}{(2\pi
y)^{n}}\left(\sum_{s=1}^{\infty}(-1)^{s}s(s-1)a_{n}(s)f_{p}^{(N|i,j)}(s)\right)\leavevmode\nobreak\
,$ (3.25)

where we have used the symmetry property $a_{n}(-m)=a_{n}(1+m)$. For fixed
$n$, the sum in parentheses is divergent, since both $f_{p}^{(N|i,j)}(s)$ and
$a_{n}(s)$ are polynomials in $s$. The sum can be regulated in a standard
manner using an exponential or zeta function regulator.

Let us first apply this strategy to the $p=3$ integrated correlator,
$\mathcal{G}_{3}^{(N)}(\tau)$. Plugging its spectral overlap into (3.25) gives
the prediction

$\displaystyle\mathcal{G}_{3}^{(N)}(\tau)\Big{|}_{\text{1-inst}}=e^{2\pi
i\tau}\frac{R_{3}}{4}\sum_{n=0}^{\infty}\frac{\Gamma\left(N-\frac{1}{2}-n\right)}{\sqrt{\pi}\Gamma(N+3)(\pi
y)^{n}}\,h_{3,n}(N)\,,$ (3.26)

where $h_{3,n}(N)$ are polynomials in $N$ of degree $n+2$. These may be
obtained algorithmically for any $n$ via recursion. For first few values of
$n$,

$\displaystyle h_{3,0}(N)$ $\displaystyle=-6N(4N+3)\,,$ $\displaystyle\qquad
h_{3,1}(N)$ $\displaystyle=-\frac{9}{8}N(4N^{2}-33N-12)\,,$ $\displaystyle
h_{3,2}(N)$ $\displaystyle=\frac{45}{256}N^{2}(4N^{2}+47N-237)\,,$
$\displaystyle\qquad h_{3,3}(N)$
$\displaystyle=-\frac{315}{4096}N^{2}(N+1)(N+2)(4N-9)\,.\ $ (3.27)

We now compare (3.26) to the exact localization calculation of the same
quantity via the definition (2.14). To proceed, we use the expression (2.19)
for the one-instanton partition function and set the higher-weight couplings
$\tau_{p>3}=0$, for which we have

$\displaystyle
Z^{(1)}_{\text{inst}}(\tau_{3},m,a)=-m^{2}\sum_{\ell=1}^{N}e^{6\pi^{3/2}\tau_{3}(ia_{\ell}-1)}\prod_{j\neq\ell}\frac{\left(a_{\ell
j}+i\right)^{2}-m^{2}}{\left(a_{\ell j}+i\right)^{2}+1}\leavevmode\nobreak\ .$
(3.28)

Plugging this expression into (2.2), we develop a weak-coupling (large $y$)
expansion for the $p=3$ integrated correlator in the one-instanton sector
(exactly in the same manner as we did for the zero-instanton computations)
using the definition (2.14). A finite generic $N$ computation is technically
involved due to the form of the instanton partition function. But for low
values of $N$, the computation is straightforward. Performing this calculation
for $N=3,4,5$ through the first several orders in $1/y$, we find precise
agreement with (3.26) as obtained from the spectral decomposition.222222The
authors of [68] have, in work to appear, independently studied the $p=3$
integrated correlator at low values of $N$ and performed numerical checks of
the vanishing cusp form overlap at finite $y$, using the convergent expansion
of the deformed instanton partition function. This further complements our
conclusion. We thank Shai Chester for discussions.

The above strategy applies for any integrated correlators
$\mathcal{G}_{p}^{(N|i,j)}(\tau)$. Analogous results for the full matrix of
$p=4,5$ correlators are deferred to Appendix D for which we have also checked
in this way that the cusp form overlaps vanish at low values of $N$.232323It
would be interesting to reproduce and independently confirm these perturbative
results by explicitly integrating the one-instanton correction to the
unintegrated $\langle{\cal O}_{2}{\cal O}_{2}{\cal O}_{p}^{(i)}{\cal
O}_{p}^{(j)}\rangle$ correlators against the measure (2.13). The latter may be
computed in principle using methods based on the ADHM construction of
instantons [69, 70, 71, 72, 73], though very few explicit results for
instanton corrections to the unintegrated correlator exist in the literature
[69, 74, 75].

There is yet another class of integrated correlators – to be introduced more
thoroughly in Section 6 – which is especially simple. This is what we call the
maximal-trace family of integrated correlators. For $p\in 2\mathbb{Z}_{+}$,
these are defined by taking both operators ${\cal O}_{p}^{(i)}$ and ${\cal
O}_{p}^{(j)}$ to be multi-trace composites of ${\cal O}_{2}$ with $p/2$
constituents,

$\begin{split}{\cal O}_{p}^{(\rm max)}:=T_{2,\ldots,2}\,.\end{split}$ (3.29)

The lowest of these operators besides $T_{2}$ is ${\cal O}_{4}^{(\rm
max)}=T_{2,2}$, whose corresponding integrated correlator
$\mathcal{G}_{4}^{(N|2,2)}(\tau)$ was studied in Section 3.2 above. Its
Eisenstein overlap $f_{4}^{(N|2,2)}(s)$ was computed in (3.13). Inserting this
into (3.25), we find

$\displaystyle\mathcal{G}_{4}^{(N|2,2)}(\tau)\Big{|}_{\text{1-inst}}=e^{2\pi
i\tau}\,\frac{R_{4}^{(2,2)}}{4}\frac{1}{N^{2}+1}\sum_{n=0}^{\infty}\frac{\Gamma\left(N-\frac{3}{2}-n\right)}{\sqrt{\pi}\Gamma(N+2)(\pi
y)^{n}}h_{4,n}^{(2,2)}(N)\,,$ (3.30)

where $h_{4,n}^{(2,2)}(N)$ are polynomials in $N$ of degree $n+4$. For first
few values of $n$ these are

$\displaystyle\begin{split}h_{4,0}^{(2,2)}(N)&=-\frac{3N}{2}\left(8N^{3}-12N^{2}+3N-12\right),\\\
h_{4,1}^{(2,2)}(N)&=-\frac{9N}{32}\left(8N^{4}-52N^{3}+83N^{2}-57N+80\right),\\\
h_{4,2}^{(2,2)}(N)&=\frac{45}{2^{10}}(N-2)N^{2}(N+1)\left(8N^{2}-12N-21\right),\\\
h_{4,3}^{(2,2)}(N)&=-\frac{315}{2^{14}}N^{2}(N+1)\left(8N^{4}-84N^{3}+219N^{2}-9N-414\right).\end{split}$
(3.31)

We now compare (3.30) to the exact localization calculation of the same
quantity using the instanton partition function with vanishing higher-weight
sources ($\tau_{p\geq 3}=0$),

$\displaystyle
Z^{(1)}_{\text{inst}}(m,a)=-m^{2}\sum_{\ell=1}^{N}\prod_{j\neq\ell}\frac{\left(a_{\ell
j}+i\right)^{2}-m^{2}}{\left(a_{\ell j}+i\right)^{2}+1}\leavevmode\nobreak\ .$
(3.32)

It is convenient to first compute
$\langle\partial_{m}^{2}Z^{(1)}_{\text{inst}}(0,a)\rangle$ in a weak-coupling
expansion (the expectation value is in the ensemble (2.20)). This result was
obtained for finite $N$ in [76, 7], with the first few orders being

$\displaystyle\langle\partial_{m}^{2}Z^{(1)}_{\text{inst}}(0,a)\rangle$
$\displaystyle=\bigg{(}-\frac{4\Gamma\left(N+\frac{1}{2}\right)}{\pi^{1/2}\Gamma(N)}-{1\over
y}\frac{3\Gamma\left(N-\frac{1}{2}\right)}{4\pi^{3/2}\Gamma(N-1)}+{1\over
y^{2}}\frac{15N\Gamma\left(N-\frac{3}{2}\right)}{128\pi^{5/2}\Gamma(N-1)}$
$\displaystyle-{1\over
y^{3}}\frac{105(N-3)N\Gamma\left(N-\frac{5}{2}\right)}{2048\pi^{7/2}\Gamma(N-1)}+{1\over
y^{4}}\frac{945N(N(5N-33)+58)\Gamma\left(N-\frac{7}{2}\right)}{131072\pi^{9/2}\Gamma(N-1)}+\ldots\bigg{)}.$
(3.33)

Plugging the above into (2.14) (using (B.7) for solving the operator mixing
problem) we precisely find (3.30). The same calculation may be repeated for
higher-$p$ maximal-trace integrated correlators, studied further in Section 6;
having done so for many $p>4$ again yields perfect agreement.

Altogether, these results provide substantial evidence of vanishing cusp form
overlap for integrated correlators $\mathcal{G}_{p}^{(N|i,j)}(\tau)$. We leave
a deeper physical understanding of this intriguing mathematical property for
the future.

###  3.4 Laplace difference equations

Having given evidence for vanishing cusp form overlap and thereby completion
to the full correlator, the various recursion relations from Section 3.2
should be regarded as non-trivial identities between the full integrated
correlators. Said another way, relations for $f_{p}^{(N|i,j)}(s)$ simply
‘uplift’ to relations for $\mathcal{G}_{p}^{(N)}(\tau)$. Starting from a given
recursion relation for the spectral overlap $f_{p}^{(N|i,j)}(s)$, this is
achieved by applying the replacement rules

$\displaystyle f_{p}^{(N|i,j)}(s)\leavevmode\nobreak\
\mapsto\leavevmode\nobreak\
\mathcal{G}_{p}^{(N|i,j)}(\tau)-\langle\mathcal{G}_{p}^{(N|i,j)}\rangle\,,\qquad
s(1-s)f_{p}^{(N|i,j)}(s)\leavevmode\nobreak\ \mapsto\leavevmode\nobreak\
\Delta_{\tau}\,\mathcal{G}_{p}^{(N|i,j)}(\tau)\,.$ (3.34)

Prime among these relations is the $p=3$ recursion relation (3.9), which
uplifts to the following powerful difference equation obeyed by the integrated
correlator $\mathcal{G}_{3}^{(N)}(\tau)$:

$\displaystyle{N(N+1)\Big{[}\mathcal{G}_{3}^{(N+1)}(\tau)-\mathcal{G}_{3}^{(N)}(\tau)\Big{]}=2N(N-1)\,\mathcal{G}_{2}^{(N+1)}(\tau)+2(N+2)(N+1)\,\mathcal{G}_{2}^{(N)}(\tau)}\,.$
(3.35)

The absence of an inhomogeneous term coming from the ensemble averages
$\langle\mathcal{G}_{p}^{(N|i,j)}\rangle$ follows from consistency of the
recursion relations evaluated at $s=1$.

One may, if desired, recast all of the previously derived recursion formulas
as Laplace difference equations. Let us just give one more example explicitly,
namely, the very simple relation (3.13) between
$\mathcal{G}_{4}^{(N|2,2)}(\tau)$ and $\mathcal{G}_{2}^{(N)}(\tau)$:

$\displaystyle\mathcal{G}_{4}^{(N|2,2)}(\tau)=-(\Delta_{\tau}-2N^{2})\,\mathcal{G}_{2}^{(N)}(\tau)\,.$
(3.36)

We leave other applications of (3.34) as an exercise for the enthusiastic
reader.

##  4 Ensemble averages

We now consider $\langle\mathcal{G}_{p}^{(N|i,j)}\rangle$, the ensemble
averages of the integrated correlators with respect to the Zamolodchikov
measure, cf. (2.30). As can be seen from equation (2.35), these are obtained
by simply evaluating the Eisenstein spectral overlaps $f^{(N|i,j)}_{p}(s)$ at
$s=1$.

For $p=2$, this yields the result already given in [8], which for convenience
we reproduce here:

$\displaystyle\langle\mathcal{G}_{2}^{(N)}\rangle=\frac{N(N-1)}{4}\,.$ (4.1)

We find that for higher $p$ the ensemble averages continue to be given by
rational functions of $N$. Explicitly, we find (in slight abuse of matrix
notation)

$\displaystyle\langle\mathcal{G}_{3}^{(N)}\rangle$
$\displaystyle=\frac{1}{6}(N-1)(N-2)(2N+3)\,,$ (4.2)
$\displaystyle\langle\mathcal{G}_{4}^{(N|i,j)}\rangle$
$\displaystyle=\begin{pmatrix}\frac{(N-1)(3N^{4}-3N^{3}-17N^{2}+12N+36)}{8N}&\frac{(N-1)(4N^{2}-N-6)}{4}\\\\[5.0pt]
\frac{(N-1)(4N^{2}-N-6)}{4}&\frac{(N-1)N^{3}}{2}\end{pmatrix},$ (4.3)
$\displaystyle\langle\mathcal{G}_{5}^{(N|i,j)}\rangle$
$\displaystyle=\begin{pmatrix}\frac{(N-2)(N-1)(4N^{5}+2N^{4}-17N^{3}+10N^{2}+120N+120)}{10N^{2}}&\frac{3(N-2)(N-1)(3N^{3}+4N^{2}-8N-10)}{5N}\\\\[5.0pt]
\frac{3(N-2)(N-1)(3N^{3}+4N^{2}-8N-10)}{5N}&\frac{3(N-2)(N-1)(3N^{3}+5N^{2}+9N+13)}{25}\end{pmatrix}.$
(4.4)

We observe that at leading order in large $N$ the diagonal entries scale as
$N^{p}$, while the off-diagonal ones scale as $N^{p-1}$:

$\displaystyle\langle\mathcal{G}_{4}^{(N|i,j)}\rangle\leavevmode\nobreak\
\longrightarrow\leavevmode\nobreak\
\begin{pmatrix}\frac{3N^{4}}{8}&N^{3}\\\\[3.0pt]
N^{3}&\frac{N^{4}}{2}\end{pmatrix},\qquad\langle\mathcal{G}_{5}^{(N|i,j)}\rangle\leavevmode\nobreak\
\longrightarrow\leavevmode\nobreak\
\begin{pmatrix}\frac{2N^{5}}{5}&\frac{9N^{4}}{5}\\\\[3.0pt]
\frac{9N^{4}}{5}&\frac{9N^{5}}{25}\end{pmatrix}.$ (4.5)

Furthermore, note that the ensemble averages of correlators with two single-
trace operators (i.e. the $(i,j)=(1,1)$ components) at large $N$ exhibit
$p$-dependence given by

$\displaystyle\langle\mathcal{G}_{p}^{(N|1,1)}\rangle\leavevmode\nobreak\
\xrightarrow[]{\leavevmode\nobreak\ N\to\infty\leavevmode\nobreak\
}\leavevmode\nobreak\ \frac{p-1}{2p}\leavevmode\nobreak\ N^{p}+\ldots\,,$
(4.6)

For the correlators with one single-trace and one double-trace operator, the
available data for the ensemble averages at large $N$ is consistent with the
formula

$\displaystyle\langle\mathcal{G}_{p}^{(N|1,j)}\rangle\leavevmode\nobreak\
\xrightarrow[]{\leavevmode\nobreak\ N\to\infty\leavevmode\nobreak\
}\leavevmode\nobreak\ \frac{p_{1}p_{2}(p-2)}{2p}\leavevmode\nobreak\
N^{p-1}+\ldots\,,$ (4.7)

where the label $j$ denotes a specific double-trace operator $T_{p_{1},p_{2}}$
with $p_{1}+p_{2}=p$.

### Ensemble averages for SPO’s

To make contact with correlators describing scattering amplitudes in
AdS${}_{5}\times$S5, let us also consider the ensemble averages of integrated
correlators involving SPO’s. For $p=2,3$ these are equal to the averages for
single-trace operators already given in equations (4.1) and (4.2) above. On
the other hand, for $p=4,5$ we find

$\displaystyle\begin{split}\langle\mathcal{G}_{4}^{(N)}\rangle&={\Gamma(N)\over\Gamma(N-3)}\,\left(\frac{3N^{6}+12N^{5}+15N^{4}+17N^{3}+16N^{2}-5N-6}{8N(N^{2}+1)^{2}}\right),\\\\[3.0pt]
\langle\mathcal{G}_{5}^{(N)}\rangle&={\Gamma(N)\over\Gamma(N-4)}\,\left(\frac{4N^{7}+30N^{6}+95N^{5}+245N^{4}+475N^{3}+325N^{2}-130N-120}{10N^{2}(N^{2}+5)^{2}}\right).\end{split}$
(4.8)

Due to the gamma factors, these have the expected property that they vanish
for $N=1,2,\ldots,p-1$. While at first sight these expressions look rather
messy, a surprising simplification occurs upon pulling out a factor of the SPO
two-point function normalisation $R_{p}$, given in (2.7), and performing a
decomposition into partial fractions:

$\displaystyle\begin{split}\langle\mathcal{G}_{2}^{(N)}\rangle&=\frac{R_{2}}{2}\,\Big{(}1-\frac{1}{(N+1)}\Big{)}\,,\\\
\langle\mathcal{G}_{3}^{(N)}\rangle&=\frac{R_{3}}{2}\,\Big{(}2-\frac{1}{(N+1)}-\frac{2}{(N+2)}\Big{)}\,,\\\
\langle\mathcal{G}_{4}^{(N)}\rangle&=\frac{R_{4}}{2}\,\Big{(}3-\frac{1}{(N+1)}-\frac{2}{(N+2)}-\frac{3}{(N+3)}+\frac{(N+1)}{(N^{2}+1)}-\frac{1}{N}\Big{)}\,,\\\
\langle\mathcal{G}_{5}^{(N)}\rangle&=\frac{R_{5}}{2}\,\Big{(}4-\frac{1}{(N+1)}-\frac{2}{(N+2)}-\frac{3}{(N+3)}-\frac{4}{(N+4)}+\frac{(N+5)}{(N^{2}+5)}-\frac{1}{N}\Big{)}\,.\end{split}$
(4.9)

Recalling that at large $N$ we have $R_{p}\sim N^{p}/p$, one recognises that
the first term in the brackets is again consistent with the leading
contribution (4.6), which is then followed by a string of $1/N$ suppressed
terms. Moreover, a clear pattern is visible in the above expressions, namely

$\displaystyle\langle\mathcal{G}_{p}^{(N)}\rangle=\frac{R_{p}}{2}\,\bigg{(}(p-1)-\sum_{m=1}^{p-1}\frac{m}{(N+m)}+\frac{(N+x_{p})}{(N^{2}+x_{p})}-\frac{1}{N}\bigg{)}\,,$
(4.10)

with the variable $x_{p}$ taking the values $x_{p}=0,0,1,5$ for $p=2,3,4,5$,
respectively. Let us emphasise that the above pattern is based on data up to
$p=5$ only. It would be interesting to consider higher values of $p$ to either
confirm that this nice structure persists or to find a more general structure
of $1/N$ corrections.

We will further analyze the large $N$ structure of
$\langle\mathcal{G}_{p}^{(N|i,j)}\rangle$ in Section 5.2.

##  5 Large $N$

In this section we study the large $N$ limit of the integrated correlators
$\mathcal{G}_{p}^{(N)}(\tau)$. In order to make contact with the dual
AdS${}_{5}\,\times\,$S5 supergravity, we restrict ourselves in this Section to
correlators of single-particle operators (SPOs) ${\cal O}_{p}$. As such, we
drop the $(i,j)$ labels.

All of the $N$-dependence of a given observable $\mathbb{O}(\tau)$ lies purely
in its spectral overlaps, so the large $N$ expansion of the former follows
from that of the latter. The large $N$ expansion of
$\mathcal{G}_{p}^{(N)}(\tau)$, therefore, follows from that of
$f_{p}^{(N)}(s)$, the Eisenstein overlaps. It was shown in [8] that the $1/N$
expansion takes the form242424The canonical development of the genus expansion
of a general observable $\mathbb{O}(\tau)$ starts at order $N^{2}$. While a
given observable may have vanishing contributions at low genus – for example,
a conformal dimension which starts at order $N^{0}$ – it is nevertheless
convenient to refer to its leading contribution as genus zero, instead
adjusting the leading power of $N$ accordingly. Our conventions for
$\mathcal{G}_{p}^{(N)}(\tau)$ are, for various reasons of clarity,
unnormalized, which gives them a leading power $N^{p}$. Upon normalizing by
the two-point functions of ${\cal O}_{2}$ and ${\cal O}_{p}$ using the colour
factors $R_{2}$ and $R_{p}$ given in (2.7), their leading power would be
$1/N^{2}$, which is the correct scaling of a normalized connected four-point
function in a CFT obeying large $N$ factorization.

$\displaystyle
f_{p}^{(N)}(s)=\sum_{g=0}^{\infty}N^{p-2g}\Big{[}N^{s-1}f_{p}^{(g)}(s)+(s\rightarrow
1-s)\Big{]}\,.$ (5.1)

Anticipating the genus expansion of the ’t Hooft limit in (5.16), we call
$f_{p}^{(g)}(s)$ the genus-$g$ spectral overlap.

Developing the genus expansion of $f_{p}^{(N)}(s)$ is straightforward because
of the recursion relations of Section 3.2. Solving the recursion in a $1/N$
expansion yields $f_{p}^{(g>0)}(s)$ in terms of $f_{p}^{(0)}(s)$. The latter
may be computed in various ways: for example, by studying the leading-order
$N$-dependence order-by-order in the $1/y$ weak-coupling expansion and
reconstructing the $s$-dependence; or, by using the previously known results
for $p=2$ to seed the recursion.

In Subsection 5.1 we present results for $f_{p}^{(g)}(s)$ through $g=2$,
though it is straightforward to compute to arbitrary genus using the recursion
formulas. Taking their $s\rightarrow 1$ limit, it is then trivial to compute
ensemble averages at large $N$ (cf. (2.35)) and make contact with
AdS${}_{5}\,\times\,$S5 supergravity. We do this in Subsection 5.2. Finally,
plugging in the large $N$ expansion (5.1) of the spectral overlaps into the
spectral decomposition (2.29) allows us to make manifest two different large
$N$ limits of $\mathcal{G}_{p}^{(N)}(\tau)$: the ’t Hooft limit in Subsection
5.3.1 and the very strongly coupled (VSC) limit in Subsection 5.3.2.

###  5.1 Genus expansion of spectral overlaps

At genus zero, we find the following result for $p=2$

$\displaystyle
f_{2}^{(0)}(s)=\frac{2^{2s-1}(2s-1)\Gamma\left(s+\frac{1}{2}\right)}{\sqrt{\pi}\Gamma(s+1)\Gamma(s+2)}\,,$
(5.2)

which reproduces the result of [7, 8]. The arbitrary $p$ generalization of
this result is given by

$\displaystyle{f_{p}^{(0)}(s)=\frac{s+1}{2s-1}\left(1+(-1)^{p}\frac{\Gamma(s-1)\Gamma(s+1)}{\Gamma(s-p)\Gamma(s+p)}\right)\leavevmode\nobreak\
f_{2}^{(0)}(s)\,.}$ (5.3)

For a derivation of the equation we refer the reader to Appendix G. The
presence of simple poles at $s=\frac{1}{2}-m$ for $m\in\mathbb{Z}_{+}$ (in
addition to zeros at $s\in{\mathbb{Z}}_{-}$) is consistent with the allowed
general properties of spectral overlaps in the genus expansion.252525We recall
from [8] that whereas the finite $N$ overlap $f_{p}^{(N)}(s)$ must be regular
for all complex $s\neq 0,1$, the large $N$ expansion allows $f_{p}^{(g)}(s)$
to have poles at $s\in\mathbb{Z}/2$ at fixed genus $g$.

Let us make a curious observation. If we write (5.3) in the form

$\displaystyle
f_{p}^{(0)}(s)=\frac{n^{(0)}_{p}(s)}{(s+2)_{p-2}}\leavevmode\nobreak\
f_{2}^{(0)}(s)\,,$ (5.4)

we find that the numerator polynomials $n^{(0)}_{p}(s)$, of degree
$2\lfloor{p-2\over 2}\rfloor$, obey a reflection symmetry for all $p\geq 2$:
that is,

$n_{p}^{(0)}(1-s)=n_{p}^{(0)}(s)\,.$ (5.5)

This symmetry is not required by $SL(2,\mathbb{Z})$-invariance of the spectral
decomposition.

Having computed the leading order result, we then solve the recursion
relations for the spectral overlaps for $p=2,3,4,5$ in a $1/N$ expansion and
find that the higher-genus overlaps are related to the genus-zero $p=2$
overlap in the following simple manner:

$\displaystyle
f_{p}^{(g)}(s)=\frac{n^{(g)}_{p}(s)}{(-2)^{g}\,(4g)!\,\left(\frac{3}{2}-s\right)_{g}(s+2)_{p-2}}\leavevmode\nobreak\
f_{2}^{(0)}(s)\leavevmode\nobreak\ .$ (5.6)

Here we note that the higher-genus overlaps have additional poles from the
$(\frac{3}{2}-s)_{g}$ factor, which includes a finite number of positive
values, as opposed to genus zero (this point will be important later when we
discuss the ’t Hooft limit). The polynomials $n^{(g)}_{p}(s)$ are of degree
$2\lfloor\frac{p-2}{2}+2g\rfloor$; unlike the genus-zero case, the
$n^{(g>0)}_{p}(s)$ are not reflection-symmetric. To give some explicit
examples, the polynomials $n_{p}^{(1)}(s)$ of the genus-one overlaps read

$\displaystyle\begin{split}n_{2}^{(1)}(s)&=(s-6)(s-1)s(s+1)\,,\\\
n_{3}^{(1)}(s)&=4(s+2)(s^{3}-2s^{2}-51s+72)\,,\\\
n_{4}^{(1)}(s)&=(s+3)(s^{5}+2s^{4}-181s^{3}+622s^{2}-3180s+3816)\,,\\\
n_{5}^{(1)}(s)&=12(s+4)(s^{5}+11s^{4}-387s^{3}+1125s^{2}-3270s+3528)\,,\end{split}$
(5.7)

whereas at genus two we find

$\displaystyle\begin{split}n_{2}^{(2)}(s)&=7(s-4)(s-3)(s-2)(s-1)s(s+1)(5s^{2}-47s+30)\,,\\\
n_{3}^{(2)}(s)&=28(s-1)s(s+1)(s+2)(5s^{4}-42s^{3}-473s^{2}+4374s-7560)\,,\\\
n_{4}^{(2)}(s)&=7(s+3)(5s^{9}+8s^{8}-2084s^{7}+8138s^{6}+62635s^{5}\\\
&\qquad\qquad\quad-13978s^{4}-601996s^{3}+2937672s^{2}-9717120s+8985600)\,,\\\
n_{5}^{(2)}(s)&=84(s+4)(5s^{9}+103s^{8}-4022s^{7}-16146s^{6}+485093s^{5}\\\
&\qquad\qquad\quad\leavevmode\nobreak\
+40775s^{4}-7260020s^{3}+24585444s^{2}-47580480s+35942400)\,.\end{split}$
(5.8)

Let us point out the occurrence of certain zeroes in these polynomials, which
have non-trivial consequences for the large $N$ expansion of the ensemble
averages discussed in the next section. In particular, note that for $p=2$ the
spectral overlap $f_{p}^{(g)}(s)$ vanishes at $s=0,1$ for $g\geq 1$, while for
$p=3$ the same is the case for $g\geq 2$. On the other hand, the $p=4,5$
overlaps do not vanish at $s=0,1$, and the reason for this will be explained
next.

###  5.2 Ensemble averages at large $N$ and supergravity

We now make contact with semiclassical AdS${}_{5}\,\times\,$S5 supergravity.

It was shown in [8] that for any $SL(2,\mathbb{Z})$-invariant observable
$\mathbb{O}(\tau)$ that admits a genus expansion in the ‘t Hooft limit, the
ensemble average $\langle\mathbb{O}\rangle$ and the large ‘t Hooft coupling
limit of $\mathbb{O}(\tau)$ are equal at leading order in large $N$:

$\langle\mathbb{O}\rangle=\mathbb{O}(\lambda\rightarrow\infty)\,.$ (5.9)

This correspondence extends to all genera: defining the leading-order term at
genus-$g$ as

$\begin{split}\langle\\!\langle\mathbb{O}^{(g)}\rangle\\!\rangle&\coloneqq\lim_{N\rightarrow\infty}N^{2g-2}\langle\mathbb{O}^{(g)}\rangle\,,\end{split}$
(5.10)

there is an equivalence

$\langle\\!\langle\mathbb{O}^{(g)}\rangle\\!\rangle=\mathbb{O}^{(g)}(\lambda\rightarrow\infty)\leavevmode\nobreak\
\leavevmode\nobreak\ \forall\leavevmode\nobreak\ g\,.$ (5.11)

Translated into bulk terms, the genus-zero relation implies an equivalence
between the large $N$ ensemble average of $\mathbb{O}(\tau)$ and its value in
tree-level AdS${}_{5}\,\times\,$S5 supergravity, while the $g>0$ relation
relates the higher-genera averages to loop-level AdS${}_{5}\,\times\,$S5
supergravity after string theory regularization of divergences.262626We note
that one can perform a mutual consistency check of this dictionary and our
results: namely, matching the large $N$ expansion of the finite $N$ ensemble
averages $\langle\mathcal{G}_{p}^{(N)}\rangle$ to the genus expansion of the
spectral overlaps $f_{p}(s)$. We do so in Appendix F. (The $g>0$ statement of
(5.11) will be elaborated upon below.)

At $g=0$, our results confirm this equivalence. As foreshadowed in (4.6), we
find

$\langle\\!\langle\mathcal{G}_{p}^{(g=0)}\rangle\\!\rangle=\frac{p-1}{2p}\leavevmode\nobreak\
.$ (5.12)

After accounting for differences in normalisations, this agrees with the
$\lambda\rightarrow\infty$ limit at $g=0$ [6].

At $g=1$, the results (4.9) for $p\leq 5$ are consistent with the following
pattern:

$\langle\\!\langle\mathcal{G}_{p}^{(g=1)}\rangle\\!\rangle=-\frac{(p^{2}-1)(p^{2}-4)}{48}\leavevmode\nobreak\
.$ (5.13)

Via (5.11), this makes a prediction for the finite term in the
$\lambda\rightarrow\infty$ limit at genus one. This amounts to a holographic
computation of the 1-loop supergravity computation of
$\mathcal{G}_{p}^{(N)}(\tau)$. To put this in conventional supergravity
language, we note that the bulk loop expansion proceeds not in powers of $1/N$
but in powers of $1/c=4/(N^{2}-1)$. In addition, let us divide by the product
of colour factors $R_{2}R_{p}$ to normalize the correlator,272727In this
convention, the tree-level supergravity result is
$\widetilde{\mathcal{G}}^{(N)}_{p}(\tau)\approx(p-1)/4c$.

$\widetilde{\mathcal{G}}_{p}^{(N)}(\tau):=\mathcal{G}_{p}^{(N)}(\tau)(R_{2}R_{p})^{-1}\,.$
(5.14)

Putting things together leads to

$\widetilde{\mathcal{G}}^{(N)}_{p}(\tau)\Big{|}_{\text{1-loop sugra}}={1\over
c^{2}}\frac{(p-1)p(p+1)(p+2)}{384}\leavevmode\nobreak\ .$ (5.15)

This sets a benchmark for a direct bulk computation of the integrated
correlators in regularized 1-loop AdS${}_{5}\,\times\,$S5
supergravity.282828The results of [34] imply that (5.15), in conjunction with
the results of [33, 35] and the flat space limit, can be used to determine the
unintegrated 1-loop $\langle{\cal O}_{2}{\cal O}_{2}{\cal O}_{p}{\cal
O}_{p}\rangle$ correlator, as anticipated in [77]. We reiterate that this is
the finite result that remains after the (unambiguous) string theory
regularization of 1-loop supergravity divergences. It is notable that the
ensemble average is sensitive to the UV details of this regularization; in
particular, string theory regularizes divergences unambiguously, choosing a
specific renormalization scheme, and the $SL(2,\mathbb{Z})$ average identifies
precisely this scheme.

Finally, let us make the side remark that (4.7) gives a prediction for tree-
level supergravity, this time for the integrated correlator involving one
single-trace operator $T_{p}$ and one double-trace operator $T_{p_{1},p_{2}}$
with $p_{1}+p_{2}=p$.

###  5.3 Integrated correlators at strong coupling

As emphasized throughout this section, the spectral overlaps in the $1/N$
expansion contain complete information about the large $N$ integrated
correlators themselves: simply plug the formulas for the overlaps
$f_{p}^{(g)}(s)$ into the spectral representation (2.29), and expand in the
desired limit.

Having said that, for convenience of future study, we assemble these
ingredients into explicit expressions for $\mathcal{G}_{p}^{(N)}(\tau)$ in two
strongly coupled limits of interest: the ‘t Hooft limit, and the very strongly
coupled (VSC) limit. We will briefly review salient aspects of these limits in
the spectral language; a systematic treatment of these limits for general
$SL(2,\mathbb{Z})$-invariant observables can be found in [8].

####  5.3.1 ’t Hooft limit

In the ’t Hooft limit, where $g_{\text{YM}}\to 0$ and $N\to\infty$ with
$\lambda:=g^{2}_{\text{YM}}N$ fixed, the $1/N$ expansion organises into a
genus expansion, up to non-perturbative corrections in $1/N$. Because
instantons and anti-instantons are non-perturbatively suppressed in $1/N$, the
genus expansion is an expansion of the zero-instanton mode:

$\begin{split}\mathcal{G}_{p}^{(N)}(\tau)=\sum_{\mathfrak{g}=0}^{\infty}N^{p-2\mathfrak{g}}\,\mathcal{G}_{p,0}^{(\mathfrak{g})}(\lambda)+(\text{non-
perturbative})\leavevmode\nobreak\ .\end{split}$ (5.16)

We henceforth drop the “0” subscript with the understanding that we are
working perturbatively in $1/N$. The parameter $\mathfrak{g}$ differs from the
previous definition of genus in (5.1) because of a non-trivial repackaging of
the $1/N$ expansion by the $SL(2,\mathbb{Z})$ spectral decomposition, a fact
we return to momentarily.

We want to perform a spectral decomposition of this quantity. The zero mode of
completed Eisenstein series may be written in terms of a ‘t Hooft coupling as

$\begin{split}E^{*}_{s,0}(y)=\Lambda(s)N^{s}\tilde{\lambda}^{-s}+\Lambda(1-s)N^{1-s}\tilde{\lambda}^{s-1}\leavevmode\nobreak\
,\end{split}$ (5.17)

where

$\tilde{\lambda}\coloneqq{\lambda\over 4\pi}\leavevmode\nobreak\ .$ (5.18)

Upon inserting (5.17) and the large $N$ expansion of the perturbative overlaps
(5.1) into the spectral decomposition (2.29), one finds

$\displaystyle\mathcal{G}_{p}^{(N)}(\lambda)=$
$\displaystyle\leavevmode\nobreak\ \langle\mathcal{G}_{p}^{(N)}\rangle$ (5.19)
$\displaystyle+$ $\displaystyle\frac{1}{2\pi
i}\int_{\text{Re}\,s=\frac{1}{2}}ds\frac{\pi}{\sin(\pi
s)}s(1-s)\sum_{g=0}^{\infty}N^{p-2g}\Big{[}\Lambda(1-s)\tilde{\lambda}^{s-1}+\Lambda(s)N^{2s-1}\tilde{\lambda}^{-s}\Big{]}f_{p}^{(g)}(s)\leavevmode\nobreak\
,$

As the constant term in the spectral decomposition, the ensemble average
$\langle\mathcal{G}_{p}^{(N)}\rangle$ does not depend on $\lambda$. There are
two terms in brackets, each of a different nature. Large $N$ forces us to
close the contour of the second term in brackets, carrying an $N^{2s-1}$, to
the left; this generates terms with positive powers of $\lambda$, regardless
of the value of $\lambda$. These terms were denoted ‘renormalization terms’ in
[8], due to their strong coupling interpretation as bulk UV divergences
regularized by the string scale cutoff.292929At strong coupling, these terms
lead to super-leading powers of $\lambda$ with respect to the supergravity
term; as argued in [78], they can be seen as finite string theory counterterms
which regulate the UV divergences of the AdS${}_{5}\,\times\,$S5 supergravity
loop expansion. Their presence is in fact crucial to restoring the
$SL(2,\mathbb{Z})$ invariance (S-duality) of the strong coupling expansion. At
weak coupling, their cancellation – required so that the perturbative
expansion proceeds only in integer powers of $\lambda$ – implies intriguing
inter-genera relations among the residues, see eq. (7.9) of [8]. (See also
[42].) The integration contour of the first term in brackets may be closed
either to the left to develop the strong coupling expansion (in negative
powers of $\tilde{\lambda}$); to the right to develop the weak-coupling
expansion (in positive powers of $\tilde{\lambda}$); or not deformed at all,
thus giving an expression which is exact in $\lambda$.

For example, at $\mathfrak{g}=0,1$ one finds

$\displaystyle\mathcal{G}_{p}^{(\mathfrak{g}=0)}(\lambda)=$
$\displaystyle\leavevmode\nobreak\ \frac{p-1}{2p}+\frac{1}{2\pi
i}\int_{\text{Re}\,s=\frac{1}{2}}ds\leavevmode\nobreak\ \frac{\pi}{\sin(\pi
s)}\,s(1-s)\,\Lambda(1-s)\,\tilde{\lambda}^{s-1}\,f_{p}^{(0)}(s)\leavevmode\nobreak\
,$ (5.20) $\displaystyle\mathcal{G}_{p}^{(\mathfrak{g}=1)}(\lambda)=$
$\displaystyle-\frac{p^{2}-1}{48}\sqrt{\lambda}-\frac{(p^{2}-1)(p^{2}-4)}{48}$
$\displaystyle+\frac{1}{2\pi
i}\int_{\text{Re}\,s=\frac{1}{2}}ds\leavevmode\nobreak\ \frac{\pi}{\sin(\pi
s)}\,s(1-s)\,\Lambda(1-s)\,\tilde{\lambda}^{s-1}\,f_{p}^{(1)}(s)\leavevmode\nobreak\
.$ (5.21)

where $f_{p}^{(0)}(s)$ and $f_{p}^{(1)}(s)$ were given in (5.3) and (5.6),
respectively. The first term in (5.3.1) is the ‘renormalization term’
discussed above, and descends from the second term in brackets in (5.19) for
$g=0$. At risk of repetition, we stress anew that this spectral representation
is exact in $\lambda$: no expansion has been made.

$\mathcal{G}_{p}^{(N)}(\tau)$ contains non-perturbative effects at both large
$\lambda$ and large $N$. This follows from the analysis in [8], where they
were determined for $p=2$, and the fact that the $p$-dependence of
$f_{p}^{(g)}(s)$ in (5.3) and (5.6) does not enter the large-order asymptotics
of the relevant series (for all genera). The effects at large $\lambda$ and
large $N$ will therefore come in positive even powers of the scales [8]

$\Lambda_{\lambda}\coloneqq\exp\left(-\sqrt{\lambda}\right)\qquad\text{and}\qquad\Lambda_{\lambda_{{\mathsf{S}}}}\coloneqq\exp\left(-\sqrt{\lambda_{{\mathsf{S}}}}\right)$
(5.22)

respectively, where $\lambda_{{\mathsf{S}}}:={16\pi^{2}N^{2}/\lambda}$ is an
S-dual ‘t Hooft coupling. Both corrections are present in the ‘t Hooft limit.
The $p$-dependence of the non-perturbative series, entering only in the
expansion coefficients, would be nice to examine in future work.

It is formally interesting to derive two alternative presentations of these
same objects $\mathcal{G}_{p}^{(\mathfrak{g})}(\lambda)$.

First, it is possible to show (see Appendix G.3) that the genus-$\mathfrak{g}$
correlator can be written entirely in terms of genus-$\mathfrak{g}$ spectral
overlaps as follows:

$\displaystyle\mathcal{G}_{p}^{(\mathfrak{g})}(\lambda)=$
$\displaystyle\leavevmode\nobreak\ \frac{1}{2\pi
i}\int_{\text{Re}\,s=\mathfrak{g}+\frac{1}{2}+\epsilon}ds\frac{\pi}{\sin(\pi
s)}s(1-s)\Lambda(1-s)\tilde{\lambda}^{s-1}f_{p}^{(\mathfrak{g})}(s)$
$\displaystyle-\sum_{k=2}^{\mathfrak{g}}(-1)^{k+1}\tilde{\lambda}^{k-1}k(k-1)\Lambda(1-k)f_{{p}}^{(\mathfrak{g})}(k)$
(5.23)

This rewriting involves shifting the contour of (5.19) and making use of
identities relating different genera – namely, eq. (7.9) of [8] – that are
required by consistency of the weak-coupling expansion in integer powers of
$\lambda$. The novelty of this expression is that the integrand involves only
the genus-$\mathfrak{g}$ overlap, unlike what (5.19) seems to give. Note the
absence of the second line for $\mathfrak{g}<2$.

Secondly, in [6] the genus-zero result
$\mathcal{G}_{p}^{(\mathfrak{g}=0)}(\lambda)$ was derived for all $\lambda$ as
an integral expression involving squared Bessel functions. In Appendix G, we
prove its equivalence to (5.20). Indeed, a “Bessel representation” at all
genera maybe derived by deforming the contour in (5.3.1) to the right,
yielding

$\displaystyle\mathcal{G}_{p}^{(\mathfrak{g})}(\lambda)=-\sum_{k=2}^{\infty}(-1)^{k+1}\tilde{\lambda}^{k-1}k(k-1)\Lambda(1-k)f_{{p}}^{(\mathfrak{g})}(k)\,.$
(5.24)

This is convergent for $|\lambda|\leq\pi^{2}$ for all finite $\mathfrak{g}$
and $p$. The expansion may be resummed using the integral identity

$\displaystyle\frac{2^{2s-2}}{\Gamma(2s)}\int_{0}^{\infty}dw\frac{w^{2s-1}}{\sinh^{2}(w)}=\zeta(2s-1)\,.$
(5.25)

The resulting expression is a one-dimensional integral of products of Bessel
functions. For example, we present such a result for
$\mathcal{G}_{3}^{(\mathfrak{g}=1)}(\lambda)$:

$\displaystyle\mathcal{G}_{3}^{(\mathfrak{g}=1)}(\lambda)=$
$\displaystyle-\int_{0}^{\infty}\frac{d\omega}{12\pi^{2}\lambda\omega\sinh^{2}\omega}\bigg{[}\left(7\lambda^{2}\omega^{4}+48\pi^{2}\lambda\omega^{2}-192\pi^{4}\right)J_{1}(\sqrt{\lambda}\omega/\pi)^{2}$
(5.26) $\displaystyle+$
$\displaystyle\lambda\omega^{2}\left(5\lambda\omega^{2}-48\pi^{2}\right)J_{0}(\sqrt{\lambda}\omega/\pi)^{2}+4\pi\sqrt{\lambda}\omega\left(48\pi^{2}-\lambda\omega^{2}\right)J_{1}(\sqrt{\lambda}\omega/\pi)J_{0}(\sqrt{\lambda}\omega/\pi)\bigg{]}.$

For the sake of completeness, we provide the results at genus one for
$p=2,4,5$ in Appendix G. Similar expressions for
$\mathcal{G}_{p}^{(\mathfrak{g})}(\lambda)$ for various $\mathfrak{g}$ and $p$
are easily obtained from (5.24).

####  5.3.2 Very strongly coupled limit

The VSC limit is the limit of $N\rightarrow\infty$ with $\tau$ fixed. In
contrast to the ’t Hooft limit, instanton contributions are not suppressed.
This limit is trivially extracted from the spectral decomposition (2.29) upon
inserting the large $N$ expansion of the overlaps (5.1) into (2.29), and
deforming the integration contour to develop the $1/N$ expansion. Summing over
residues in this way gives [8]

$\mathcal{G}_{p}^{(N)}(\tau)=\sum_{g=0}^{\infty}N^{p-2g}\left(\langle\\!\langle\mathcal{G}_{p}^{(g)}\rangle\\!\rangle-\sum_{m=1}^{\infty}N^{-m-\frac{1}{2}}\mathsf{R}_{m}^{(g)}E_{\frac{1}{2}+m}(\tau)\right),$
(5.27)

with residue function $\mathsf{R}_{m}^{(g)}$ defined as

$\mathsf{R}_{m}^{(g)}:=\underset{s=\frac{1}{2}+m}{\operatorname{Res}}\left[\frac{\pi}{\sin\pi
s}s(1-s)\Lambda(s)f_{{p}}^{(g)}(1-s)\right].$ (5.28)

The $\tau$-dependence appears only via Eisenstein series, and only down by
odd-half-integer powers of $1/N$.303030There can be terms down by powers of
$1/N^{2}$ as well, due to the ensemble average terms
$\langle\\!\langle\mathcal{G}_{p}^{(g)}\rangle\\!\rangle$, but these are
$\tau$-independent. Both of these features, like many others in this work,
follow from the relations $f_{\rm
np}^{(N)}(s)=(\mathcal{G}_{p}^{(N)},\phi_{n})=0$.

For example, applying the above formulae to the $p=3$ case using the genus
expansion (5.6) of its spectral overlap through $g=2$ yields the following VSC
expansion:

$\displaystyle\mathcal{G}_{3}^{(N)}(\tau)=\frac{N^{3}}{3}-\frac{1}{2}N^{\frac{3}{2}}\widetilde{E}_{\frac{3}{2}}(\tau)-\frac{5N}{6}+\frac{45}{2^{5}}N^{\frac{1}{2}}\widetilde{E}_{\frac{5}{2}}(\tau)+25N^{-\frac{1}{2}}\left(\frac{31\widetilde{E}_{\frac{3}{2}}(\tau)}{2^{10}}-\frac{189\widetilde{E}_{\frac{7}{2}}(\tau)}{2^{12}}\right)$
$\displaystyle-$ $\displaystyle
225N^{-\frac{3}{2}}\left(\frac{25\widetilde{E}_{\frac{5}{2}}(\tau)}{2^{13}}+\frac{147\widetilde{E}_{\frac{9}{2}}(\tau)}{2^{15}}\right)+N^{-\frac{5}{2}}\left(\frac{10517\widetilde{E}_{\frac{3}{2}}(\tau)}{2^{19}}-\frac{2159325\widetilde{E}_{\frac{7}{2}}(\tau)}{2^{22}}-\frac{49116375\widetilde{E}_{\frac{11}{2}}(\tau)}{2^{24}}\right)$
$\displaystyle+$ $\displaystyle
45N^{-\frac{7}{2}}\left(\frac{26699\widetilde{E}_{\frac{5}{2}}(\tau)}{2^{24}}-\frac{215355\widetilde{E}_{\frac{9}{2}}(\tau)}{2^{23}}-\frac{93648555\widetilde{E}_{\frac{13}{2}}(\tau)}{2^{28}}\right)+O(N^{-\frac{9}{2}})\,,$
(5.29)

where

$\displaystyle\widetilde{E}_{s}(\tau):=\frac{2\Lambda(s)}{\Gamma(s)}\,E_{s}(\tau)\,.$
(5.30)

##  6 Results II: Integrated maximal-trace correlators for all $p$

So far, we have mostly restricted ourselves to integrated correlators with
$p\leq 5$. Recall that our approach was to obtain the weak-coupling expansion
of $\mathcal{G}_{p}^{(N|i,j)}(\tau)$ using the localisation relation (2.14),
from which we then inferred the spectral overlaps. When trying to generalise
the localisation computation to higher values of $p$ for most choices of
$(i,j)$, one is met with some computational expense, to be elaborated upon in
Section 7.

However, there exists a special class of integrated correlators, defined for
arbitrary $p$, which we now determine exactly: the so-called maximal-trace
correlators.

The key point is that, as shown in [50], one can recursively construct a basis
of operators on the sphere such that certain families of operators become
orthogonal to each other. The simplest such family of operators with this
property is given by maximal-trace operators, introduced earlier in (3.29),
which for even $p$ are given by

${\cal O}_{p}^{(\text{max})}\equiv T_{2,\ldots,2}\,.$ (6.1)

They mix only among themselves, leading to an effective reduction in the size
of mixing matrices.313131 The same reasoning applies also to the odd $p$
family of maximal-trace operators,
$\\{T_{3},\,T_{2,3},\,T_{2,2,3},\,T_{2,2,2,3},\,\ldots\\}$, but in what
follows we restrict ourselves to the even $p$ case. The solution to the sphere
mixing is given in terms of the vectors $v_{p}^{j}$, recall the discussion
around (2.21). For the first few cases of $p$ we find

$\displaystyle\begin{split}v_{2}^{j}&=\Big{\\{}1\quad 0\quad 0\quad 0\quad
0\leavevmode\nobreak\ \leavevmode\nobreak\ \cdots\Big{\\}}\,,\\\
v_{4}^{j}&=\Big{\\{}-i\tfrac{N^{2}+1}{2y}\quad 1\quad 0\quad 0\quad
0\leavevmode\nobreak\ \leavevmode\nobreak\ \cdots\Big{\\}}\,,\\\
v_{6}^{j}&=\Big{\\{}-\tfrac{3(N^{2}+1)(N^{2}+3)}{16y}\quad-i\tfrac{3(N^{2}+3)}{4y}\quad
1\quad 0\quad 0\leavevmode\nobreak\ \leavevmode\nobreak\
\cdots\Big{\\}}\,.\end{split}$ (6.2)

The above vectors admit the closed form expression

$\displaystyle
v_{p}^{j}=\binom{\frac{p}{2}}{j}\,\frac{\Big{(}\frac{N^{2}-1}{2}+j\Big{)}{}_{\frac{p}{2}-j}}{(2iy)^{\frac{p}{2}-j}}\,,$
(6.3)

where the label $j=1,2,3\ldots$ now runs only over the subset of maximal-trace
operators ${\cal
O}_{p}^{(\text{max})}\in\\{T_{2},\,T_{2,2},\,T_{2,2,2},\ldots\\}$ due to
decoupling from all other operators on the sphere.

The other simplifying feature is that, since ${\cal O}_{p}^{(\text{max})}$ is
built from products of $T_{2}$ only, one only needs to act with repeated
$\partial_{\tau}$ and $\partial_{\bar{\tau}}$ derivatives in order to bring
down insertions of these operators in the localisation computation. As such,
no new matrix integrals have to be performed when considering correlators of
these maximal-trace operators.

In what follows, we will denote the integrated correlators $\langle{\cal
O}_{2}{\cal O}_{2}{\cal O}_{p}^{(\text{max})}{\cal
O}_{p}^{(\text{max})}\rangle$ for $p$ even by
$\mathcal{G}_{p}^{(N|\text{max})}(\tau)$:

$\displaystyle\mathcal{G}_{p}^{(N|\text{max})}(\tau):=-\frac{2}{\pi}\int_{0}^{\infty}dr\int_{0}^{\pi}d\theta\,\frac{r^{3}\sin^{2}\theta}{u^{2}}\,\mathcal{H}^{(N|\text{max})}_{p}(u,v;\tau)\Big{|}_{u=1+r^{2}-2r\cos\theta,\,v=r^{2}}\,.$
(6.4)

following the notation of (2.10) and (2.13).323232 Note that for $p=2,4$,
$\mathcal{G}_{p}^{(N|\text{max})}(\tau)$ reduces to the previously discussed
cases $\mathcal{G}_{2}^{(N)}(\tau)$ and $\mathcal{G}_{4}^{(N|2,2)}(\tau)$,
respectively. The corresponding colour-factor appearing in the localisation
relation (2.14) is recorded in (A.2). We also introduce a hatted notation to
denote normalization by $R_{p}^{(\text{max})}(N)$, e.g.

$\widehat{\mathcal{G}}_{p}^{(N|\text{max})}(\tau):=\mathcal{G}_{p}^{(N|\text{max})}(\tau)\,R_{p}^{(\text{max})}(N)^{-1}\,,$
(6.5)

and likewise for other quantities.

Let us present, then elaborate upon, the result:

${\widehat{\mathcal{G}}_{p}^{(N|\text{max})}(\tau)=\langle\widehat{\mathcal{G}}_{p}^{(N|\text{max})}\rangle+{1\over
4\pi i}\int_{{\rm Re\,}s={1\over 2}}ds\,{\pi\over\sin(\pi
s)}s(1-s)(2s-1)^{2}\,\widehat{g}_{p}^{(N|{\rm max})}(s)\,E^{*}_{s}(\tau)}$
(6.6)

The ensemble average is

$\boxed{\langle\widehat{\mathcal{G}}_{p}^{(N|\text{max})}\rangle=\langle\mathcal{G}_{2}^{(N)}\rangle\,\Big{(}H_{\frac{N^{2}+p-3}{2}}-H_{\frac{N^{2}-3}{2}}\Big{)}}$
(6.7)

where $H_{n}$ denotes the harmonic number, and the overlaps obey the
polynomial recursion

$\displaystyle p(N^{2}+p-3)\,\widehat{g}_{p}^{(N|\text{max})}(s)$
$\displaystyle=2\Big{[}\big{(}(p-1)\,N^{2}+p^{2}-5p+5\big{)}-2\,s(1-s)\Big{]}\,\widehat{g}_{p-2}^{(N|\text{max})}(s)$
(6.8) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
-(p-2)(N^{2}+p-5)\,\widehat{g}_{p-4}^{(N|\text{max})}(s)+4g_{2}^{(N)}(s)\,,$

Recall that $\langle\mathcal{G}_{2}^{(N)}\rangle=N(N-1)/4$.

We first address the spectral overlaps. These are deduced as usual from the
weak-coupling expansion obtained from localisation.333333 As mentioned
earlier, one can verify that the weak-coupling expansion agrees with the
explicit results from perturbation theory upon integration. For the maximal-
trace correlators, we have performed this check to two-loop order and up to
$p=6$, see Appendix E.  The usual function $f_{p}^{(N|\text{max})}(s)$
parameterizing the Eisenstein overlap à la (2.38) is found to be of the now-
familiar form

$\displaystyle
f_{p}^{(N|\text{max})}(s)=(2s-1)^{2}\,g_{p}^{(N|\text{max})}(s)\,,\quad\text{where}\quad\text{deg}\left[g_{p}^{(N|\text{max})}(s)\right]=2N+p-6$
(6.9)

where the polynomials $g_{p}^{(N|\text{max})}(s)$ are even and of indicated
degree. Considering many cases of different $N$ and $p$ leads to
(6.8).343434As we have observed in previous cases, for low values of $p$ the
above formula neatly degenerates in a consistent way. This can be most easily
seen when considering the recursion relation for the unnormalised overlaps
$g_{p}^{(N|\text{max})}(s)$, i.e. after reinstating the normalisation factors
$R_{p}^{(max)}(N)$ in the recursion (6.8): for $p=2$, the equation is
trivially satisfied. For $p=4$, the first term in the second line vanishes, as
it should since $g_{0}^{(N|\text{max})}(s)=0$: $p=0$ corresponds to an
external identity operator, for which the connected correlator vanishes.
Moreover, the other coefficients combine to correctly recover relation (3.13).
Finally, for $p=6$, the two terms in the second line of (6.8) degenerate and
in particular their coefficients combine to give the correct recursion
relation consistent with the data. The central property here is that there are
no shifts in $N$: in contrast to the previously-studied cases, this is a
recursion relation in $p$ alone, with $g_{2}^{(N)}(s)$ as the initial
condition (together with the trivial condition $g_{0}^{(N|\text{max})}(s)=0$).
Indeed, the solution of the recursion relation for general $p$ and finite $N$
is simply proportional to the $p=2$ solution:

$\displaystyle{\widehat{g}_{p}^{(N|\text{max})}(s)=F_{p}(N,s)\,\widehat{g}_{2}^{(N)}(s)\,.}$
(6.10)

The functions $F_{p}(N,s)$ have the structure

$\displaystyle
F_{p}(N,s)=\frac{2^{-\frac{p}{2}}}{\Big{(}\frac{N^{2}+1}{2}\Big{)}{}_{\frac{p}{2}-1}}\,\sum_{m=0}^{\infty}\frac{2^{m+1}}{\Gamma(m+2)^{2}}\,\Big{(}\frac{p-2m}{2}\Big{)}_{m+1}\,{h}_{p}^{(m)}(s)\,N^{p-2-2m}\,,$
(6.11)

where the complete $s$-dependence is captured by the $h_{p}^{(m)}(s)$, which
turn out to be polynomials of order $2m$. Since $p\in 2\mathbb{Z}_{+}$, the
infinite sum over $m$ truncates at $m=p/2$. For some low values of $m$, the
polynomials $h_{p}^{(m)}(s)$ read

$\displaystyle\begin{split}h_{p}^{(0)}(s)&=1\,,\\\
h_{p}^{(1)}(s)&=s^{2}-s+p-4\,,\\\
h_{p}^{(2)}(s)&=s^{4}-2s^{3}+\tfrac{1}{4}(9p-28)s^{2}-\tfrac{1}{4}(9p-32)s+\tfrac{3}{8}(3p^{2}-26p+44)\,,\\\
h_{p}^{(3)}(s)&=s^{6}-3s^{5}+(4p-9)s^{4}-(8p-23)s^{3}+\tfrac{1}{2}(9p^{2}-62p+76)s^{2}\\\
&\quad-\tfrac{1}{2}(9p^{2}-70p+100)s+\tfrac{3}{2}(p-4)(p^{2}-10p+12)\,.\end{split}$
(6.12)

More data can be generated straightforwardly from the recursion relation
(6.8). The reflection symmetry of $\widehat{g}_{p}^{(N|\text{max})}(s)$
implies that $F_{p}(N,s)$ and the individual polynomials $h_{p}^{(m)}(s)$ also
have this symmetry:

$\displaystyle F_{p}(N,s)=F_{p}(N,1-s)\,,\qquad
h_{p}^{(m)}(s)=h_{p}^{(m)}(1-s)\,.$ (6.13)

As is evident from (6.6), the cusp form overlap vanishes:

$(\mathcal{G}_{p}^{(N|\text{max})},\phi_{n})=0\quad\forall\leavevmode\nobreak\
n>0\,.$ (6.14)

This assertion follows from the computations discussed in Section 3.3: for
many values of $p$, we have directly checked to the first few perturbative
orders in $1/y$ around one instanton that (6.6) agrees with the first-
principles localisation computation.

Finally, we turn to the ensemble averages
$\langle\widehat{\mathcal{G}}_{p}^{(N|\text{max})}\rangle$. For general $p$ we
have found an explicit formula (6.7), not just a recursion relation. Since
$p\in 2\mathbb{Z}_{+}$, an equivalent representation of (6.7) for the ensemble
averages is given by353535It may be worth noting that the unnormalised
average, $\langle\mathcal{G}_{p}^{(N|\text{max})}\rangle$, is actually a
polynomial in $N$ of total degree $p$. This follows from the explicit formula
(A.2) for $R_{p}^{(\text{max})}(N)$, whereupon the Pochhammer factor and the
sum combine to give rise to non-trivial polynomials in $N$ of degree $p-2$
containing only even powers of $N$.

$\displaystyle\langle\widehat{\mathcal{G}}_{p}^{(N|\text{max})}\rangle$
$\displaystyle={N(N-1)\over 2}\leavevmode\nobreak\
\sum_{m=0}^{\frac{p-2}{2}}\frac{1}{N^{2}+2m-1}\,.$ (6.15)

At leading order in large $N$ (with finite $p$),

$\langle\widehat{\mathcal{G}}_{p}^{(N|\text{max})}\rangle\leavevmode\nobreak\
\xrightarrow[]{\leavevmode\nobreak\ N\rightarrow\infty\leavevmode\nobreak\
}\leavevmode\nobreak\ {p\over 4}\,.$ (6.16)

For $p=2$ and $p=4$, we recover equations (4.6) and the lower-right entry of
(4.5), respectively, after multiplying by $R_{p}^{(\rm max)}(N)$ at large
$N$.363636We also observe that the linear scaling in $p$ matches the large $p$
limit of the classical, unintegrated $\langle pppp\rangle$ correlator studied
in [79], a different, but related, observable.

###  6.1 Comments and connections to $\mathcal{N}=2$ extremal correlators

The maximal-trace family of integrated correlators,
$\widehat{\mathcal{G}}_{p}^{(N|\text{max})}(\tau)$, has been determined above
for all $N$ and arbitrary $p$. Compared to the general trace structures
studied earlier in this work, its solution is especially clean. It also has
interesting connections to various other quantities studied in the literature.

First, note that $\mathcal{G}_{p}^{(N|\text{max})}(\tau)$ may be thought of as
a generalization of the so-called “extremal correlators” studied in 4d
$\mathcal{N}=2$ SCFTs. Let us summarize the latter. Extremal correlators
involve several chiral operators $O_{m}$ and a single anti-chiral operator
$\overline{O}_{n}$. The most well-studied extremal correlators are two-point
functions,

$G_{2n}(\tau):=\langle
O_{n}(0)\overline{O}_{n}(\infty)\rangle_{\mathbb{R}^{4}}$ (6.17)

These depend non-holomorphically on the complexified gauge coupling $\tau$,
and may be determined by localisation. In an $\mathcal{N}=2$ SCFT of arbitrary
rank, one may take $O_{n}$ to be $n$-fold composites of the weight-two chiral
primary, what we called ${\cal O}_{2}$. Such an $O_{n}$ is precisely our
family ${\cal O}_{p}^{(\rm max)}$ defined in (6.1), with the identification
$2n=p$. The correlators $G_{2n}(\tau)$ solve differential recursion relations
in $n$ [64, 47]. In rank-one $\mathcal{N}=2$ SCFTs, such as $SU(2)$ SQCD with
four flavors, the $G_{2n}(\tau)$ obey semi-infinite Toda equations relating
different values of $n$:373737In the special case of $\mathcal{N}=4$ SYM, this
may be solved for $G_{2n}(\tau)$ [50], and matches the color factor
$R_{p}^{(\rm max)}(N)$ defined in (A.2), up to a $p^{2}$ factor due to
normalization differences (see footnote 7). In this case, $G_{2n}(\tau)\propto
y^{-2n}$, so the RHS is manifestly $\tau$-independent.

$\Delta_{\tau}\log G_{2n}(\tau)=-4y^{2}\left({G_{2n+2}(\tau)\over
G_{2n}(\tau)}-{G_{2n}(\tau)\over G_{2n-2}(\tau)}-{G_{2}(\tau)}\right)$ (6.18)

At higher rank, differential relations are believed to exist, but to take a
more involved form that couples different families of operators besides merely
composites of ${\cal O}_{2}$ [50].

We can now state the connections between $G_{2n}(\tau)$ and
$\mathcal{G}_{p}^{(N|\text{max})}(\tau)$ with $2n=p$. Both are correlators
involving the composite operators ${\cal O}_{p}^{(\rm max)}$. While the former
are two-point functions, the latter are four-point functions, but integrated
over positions, leaving only $\tau$-dependence. Strikingly, both obey a
similar form of differential recursion: as a consequence of (6.14), the
recursion (6.8) uplifts to a differential relation for
$\widehat{\mathcal{G}}_{p}^{(N|\text{max})}(\tau)$,

$\begin{split}\,\Delta_{\tau}\,\widehat{\mathcal{G}}_{p-2}^{(N|\text{max})}(\tau)=&-\frac{p}{4}(N^{2}+p-3)\,\widehat{\mathcal{G}}_{p}^{(N|\text{max})}(\tau)+\frac{1}{2}\big{(}(p-1)\,N^{2}+p^{2}-5p+5\big{)}\widehat{\mathcal{G}}_{p-2}^{(N|\text{max})}(\tau)\\\
&-\frac{1}{4}(p-2)(N^{2}+p-5)\,\widehat{\mathcal{G}}_{p-4}^{(N|\text{max})}(\tau)+\mathcal{G}_{2}^{(N)}(\tau)\,.\end{split}$
(6.19)

This was written in terms of the central charge and $n=p/2$ in (1). Despite
obvious differences, this equation is intriguingly similar to the decoupled
semi-infinite Toda chain equations obeyed by the extremal two-point functions
$G_{2n}(\tau)$ in the $SU(2)$ theory [47, 48, 49, 50].383838Dividing both
sides by $G_{2}(\tau)$, the RHS becomes reminiscent of a connected four-point
function. Note, though, that (6.19) holds for all $N$! Evidently, the extra
power of $\mathcal{N}=4$ supersymmetry makes the integrated four-point
function even more soluble than its $\mathcal{N}=2$ extremal counterparts.
Altogether, this suggests that one may think of the Laplace difference
equations for integrated $\mathcal{N}=4$ SYM correlators in general, and
(6.19) in particular, in the same spirit as the Toda equations (6.18), and
that they may be derivable by some generalization of the $tt^{*}$ method that
led to (6.18).

The other point we wish to make is the following: the large $p$ limit of
$\mathcal{G}_{p}^{(N|\text{max})}(\tau)$ is a large charge limit [80, 51].
Since $\mathcal{G}_{p}^{(N|\text{max})}(\tau)$ is determined explicitly for
all $p$, this regime is easy to access, given the simplicity of the recursion
(6.8). For example, together with the spectral decomposition (6.6), it is
quick to see that $\mathcal{G}_{p\gg 1}^{(N|\text{max})}(\tau)$ limit admits a
‘t Hooft-like expansion with $g_{\rm YM}^{2}\,p$ fixed, for any finite $N$.
This generalizes a main result of [52], which established the ‘t Hooft-like
limit of the extremal correlators (6.17) in rank-one SCFTs at large charge
(also previously studied in [81, 82, 83]). Non-perturbative effects in large
$p$ or large $\lambda$ are also straightforward to determine along the lines
of [8], again using the explicit recursion above. Moreover, there are several
interesting regimes of $p$ and $N$ scaling together. Work in these directions
is ongoing [84].

###  6.2 Large $N$ expansion

Let us proceed to develop the large $N$, fixed $p$ expansion of the
(normalised) maximal-trace overlaps $\widehat{g}_{p}^{(N|\text{max})}(s)$.

In light of (6.10), the large $N$ expansion follows from that of the functions
$F_{p}(N,s)$ and $\widehat{g}_{2}^{(N)}$. The large $N$ expansion of the
latter393939Recall that
$\widehat{g}_{2}^{(N)}(s)=2/\big{(}(2s-1)^{2}(N^{2}-1)\big{)}\,f_{2}^{(N)}(s)$.
has been discussed in [7, 8], with the genus-$g$ overlaps $f_{2}^{(g)}(s)$
being of the general form depicted in (5.6).

On the other hand, the genuinely new information about the $p$-dependence of
the maximal-trace overlaps lies entirely in the functions $F_{p}(N,s)$. For
some fixed value of $p$, we note that both the sum in the numerator as well as
the Pochhammer factor in the denominator of (6.11) give rise to even
polynomials in $N$ of degree $p-2$. Therefore, the large $N$ expansion of
$F_{p}(N,s)$ starts at order $N^{0}$ and proceeds in even powers of $1/N$.
Explicitly, for the first few orders we find the expansion

$\displaystyle\begin{split}F_{p}(N,s)&=\frac{p}{2}+\frac{p(p-2)}{8}(s+1)(s-2)\,N^{-2}\\\
&\quad+\frac{p(p-2)}{72}(s+1)(s-2)\big{[}(p-4)s(s-1)-3(2p-5)\big{]}\,N^{-4}\\\
&\quad+\frac{p(p-2)}{1152}(s+1)(s-2)\big{[}(p-4)(p-6)s^{3}(s-2)-(p-4)(17p-38)s^{2}\\\
&\quad\qquad\qquad\qquad\qquad\qquad\quad\quad+2(p-4)(9p-22)s+24(3p^{2}-14p+14)\big{]}\,N^{-6}\\\
&\quad+O(N^{-8})\,.\end{split}$ (6.20)

At order $N^{-2m}$ (with $m\geq 1$) we find an overall factor $(s+1)(s-2)$
times a degree $2m-2$ polynomial in $s$, whose coefficients depend only
polynomially on $p$. We have verified that this simple structure persists to
high order in $m$. Moreover, as a consequence of (6.13), every term in this
expansion is invariant under $s\mapsto 1-s$.

With this expansion of $F_{p}(N,s)$ at hand, it is clear that the normalised
overlaps $\widehat{g}_{p}^{(N|\text{max})}(s)$ for general $p$ admit the large
$N$ expansion

$\displaystyle\widehat{g}_{p}^{(N|\text{max})}(s)=\sum_{g=0}^{\infty}N^{-2g}\Big{[}N^{s-1}\,\widehat{g}_{p}^{(g|\text{max})}(s)+(s\rightarrow
1-s)\Big{]}\,.$ (6.21)

Let us reiterate that the genus-$g$ overlaps
$\widehat{g}_{p}^{(g|\text{max})}(s)$ can be computed using (6.10) from the
expansion (6.20) of $F_{p}(N,s)$ together with the known genus-$g$ overlaps
$f_{2}^{(g)}(s)$ studied previously in [7, 8]. Importantly, since the
$F_{p}(N,s)$ are polynomial in $s$, the pole structure of
$\widehat{g}_{p}^{(g|\text{max})}(s)$ is identical to their $p=2$
counterparts, which we discussed in Section 5.1. As such, the general
structure of the expansion in the ’t Hooft or the very strongly coupled limit
remains unchanged, the only effect being that the coefficients acquire some
$p$-dependence. Let us illustrate this by explicitly developing the
corresponding large $N$ expansions.

####  6.2.1 ’t Hooft and very strongly coupled limits

We start by considering the ’t Hooft limit of the normalised correlators
$\widehat{\mathcal{G}}_{p}^{(N|\text{max})}(\tau)$, which admit the genus
expansion

$\displaystyle\widehat{\mathcal{G}}_{p}^{(N|\text{max})}(\tau)=\sum_{\mathfrak{g}=0}^{\infty}N^{-2\mathfrak{g}}\,\mathcal{G}_{p,0}^{(\mathfrak{g}|\text{max})}(\lambda)+(\text{non-
perturbative in $N$})\,.$ (6.22)

As mentioned previously, (anti-)instanton contributions are non-perturbatively
suppressed in $1/N$ and hence this is an expansion in the zero-instanton
sector only.404040Again, we will drop the ‘0’ subscript and it is understood
that we work perturbatively in $1/N$. Note that since we consider the
normalised correlator, there is a difference of $N^{p}$ in the overall power
of $N$ compared to (5.16). Since the genus-zero overlaps are directly related
to the $p=2$ overlaps by an overall factor of $p$, i.e.
$\widehat{f}_{p}^{(0|\text{max})}(s)=p\,f_{2}^{(0)}(s)$, it follows that the
$\mathfrak{g}=0$ term is simply given by

$\displaystyle\widehat{\mathcal{G}}_{p}^{(\mathfrak{g}=0|\text{max})}(\lambda)=p\,\mathcal{G}_{2}^{(\mathfrak{g}=0)}(\lambda)\,,$
(6.23)

where $\mathcal{G}_{2}^{(\mathfrak{g}=0)}(\lambda)$ was given in Section
5.3.1. At higher orders in $1/N$ such a factorisation no longer happens and
some polynomial dependence on $p$ will appear. For example at the next order,
for $\mathfrak{g}=1$, one has

$\displaystyle\begin{split}\widehat{\mathcal{G}}_{p}^{(\mathfrak{g}=1|\text{max})}(\lambda)&=-\frac{p}{16}\,\sqrt{\lambda}-\frac{p(p-4)}{8}+\frac{1}{2\pi
i}\int_{\text{Re}\,s=\frac{1}{2}}ds\leavevmode\nobreak\ \frac{\pi}{\sin(\pi
s)}\,s(1-s)\,\Lambda(1-s)\,\tilde{\lambda}^{s-1}\,\widehat{f}_{p}^{(1|\text{max})}(s)\,.\end{split}$
(6.24)

Like the SPO integrated correlators considered earlier, this admits a Bessel
representation, which we provide in Appendix G.2. Its strong coupling
expansion is

$\begin{split}\widehat{\mathcal{G}}_{p}^{(\mathfrak{g}=1|\text{max})}(\lambda\gg
1)&=\frac{p}{16}\bigg{[}-\sqrt{\lambda}-2(p-4)+\frac{15(32p-169)\,\zeta(3)}{32\lambda^{\frac{3}{2}}}+\frac{315(16p+1)\,\zeta(5)}{64\lambda^{\frac{5}{2}}}\\\
&\qquad\quad+\frac{14175(576p-1009)\,\zeta(7)}{8192\lambda^{\frac{7}{2}}}+\frac{1091475(80p-173)\,\zeta(9)}{4096\lambda^{\frac{9}{2}}}+O(\lambda^{-\frac{11}{2}})\bigg{]}.\end{split}$
(6.25)

Lastly, let us also consider the large $N$ expansion in the VSC limit.
Following the same steps as described in Section 5.3.2, we arrive at