$\displaystyle\begin{split}\widehat{\mathcal{G}}_{p}^{(N|\text{max})}(\tau)&=\frac{p}{4}-\frac{3p}{16}N^{-\frac{3}{2}}\widetilde{E}_{\frac{3}{2}}(\tau)-\frac{p(p-4)}{8}N^{-2}+\frac{45p}{2^{8}}N^{-\frac{5}{2}}\widetilde{E}_{\frac{5}{2}}(\tau)\\\\[3.0pt]
&\quad+\frac{15p}{2^{15}}N^{-\frac{7}{2}}\Big{(}(128p-676)\,\widetilde{E}_{\frac{3}{2}}(\tau)+315\widetilde{E}_{\frac{7}{2}}(\tau)\Big{)}+\frac{p(p^{2}-6p+11)}{12}N^{-4}\\\\[3.0pt]
&\quad+\frac{315p}{2^{18}}N^{-\frac{9}{2}}\Big{(}(64p+4)\,\widetilde{E}_{\frac{5}{2}}(\tau)+315\widetilde{E}_{\frac{9}{2}}(\tau)\Big{)}\\\\[3.0pt]
&\quad-\frac{p}{2^{27}}N^{-\frac{11}{2}}\Big{(}32(143360p^{2}-866400p+1961161)\,\widetilde{E}_{\frac{3}{2}}(\tau)\\\\[3.0pt]
&\quad\qquad\qquad-56700(576p-1009)\,\widetilde{E}_{\frac{7}{2}}(\tau)-245581875\widetilde{E}_{\frac{11}{2}}(\tau)\Big{)}+O(N^{-6})\,.\end{split}$
(6.26)

This takes a remarkably simple form considering that this describes an
infinite family of integrated correlators. Compared to the corresponding
expansion of the (unnormalised) $p=2$ correlator, as given in e.g. eq. (5.67)
of [7], one notes that also higher-order even powers of $1/N$ are present,
which comes from the non-trivial large $N$ expansion of the ensemble averages
$\langle\widehat{\mathcal{G}}_{p}^{(N|\text{max})}\rangle$, cf. (6.7).

##  7 General $p$ ansatz

There is unlikely to be a simple closed-form expression for generic integrated
correlators for arbitrary multi-trace insertions $(i,j)$ at arbitrary $p$,
unlike for the maximal-trace correlators considered in Section 6. However, our
computations provide a good deal of insight into their general structure. In
particular, the difficulties in obtaining general higher $p$ results appear to
be of computational nature only.414141To be clear, generic integrated
correlator computations encounter two main technical complications. The first
is the degeneracy in the space of half-BPS operators and the associated mixing
of chiral operators on the sphere: as explained in Section 2.2, to obtain
correlators on $\mathbb{R}^{4}$ from the localisation relation (2.14) one
first needs to solve the mixing problem on $S^{4}$. As a consequence of the
growing number of multi-trace operators, the size of the corresponding mixing
matrices increases rapidly with $p$. The second is the presence of multi-trace
operators which leads to higher-point insertions in the matrix-model
integrals. This complicates the evaluation of their expectation values
considerably. As described in Appendix B.3 this amounts to the computation of
higher-index $N$-dependent coefficients, which one has to perform on a case-
by-case basis. As such, we do not expect any qualitative changes or new
features to arise when considering correlators beyond the explicit cases
considered so far.

With some confidence afforded by the structural uniformity of our results, we
now make some conjectural assertions about the form of generic integrated
correlators. The ansatz is

$\boxed{\mathcal{G}_{p}^{(N|i,j)}(\tau)={1\over 2}g_{p}^{(N|i,j)}(0)+{1\over
4\pi i}\int_{{\rm Re\,}s={1\over 2}}ds\,{\pi\over\sin(\pi
s)}s(1-s)(2s-1)^{2}\,g_{p}^{(N|i,j)}(s)\,E^{*}_{s}(\tau)\,,}$ (7.1)

where the function $g_{p}^{(N|i,j)}(s)$ is polynomial in $s$ with rational
coefficients. Let us explicate this ansatz in further detail:

  * •

We expect that the Eisenstein overlaps continue to be of the form

$\\{\mathcal{G}_{p}^{(N|i,j)},E_{s}\\}={\pi\over\sin(\pi
s)}s(1-s)f_{p}^{(N|i,j)}(s)\,,$ (7.2)

with the structure

$\displaystyle
f_{p}^{(N|i,j)}(s)=(2s-1)^{2}\,g_{p}^{(N|i,j)}(s)\,,\quad\text{where}\quad\text{deg}\left[g_{p}^{(N|i,j)}(s)\right]=2N+2\left\lfloor\frac{p}{2}\right\rfloor-6\,.$
(7.3)

The reflection-symmetric polynomials $g_{p}^{(N|i,j)}(s)$ have the indicated
degree, with expansion coefficients rational in $N$ and $p$. Consequently, 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.

  * •

In Section 3.3, we gave robust evidence that the cusp form overlaps vanish for
$\mathcal{G}_{p}^{(N|i,j)}(\tau)$ by matching the form (7.1) to explicit
calculations of instanton effects using localisation. We conjecture that this
remarkable property extends to general $p$ and all trace-structures $(i,j)$:

$(\mathcal{G}_{p}^{(N|i,j)},\phi_{n})=0\quad\forall\leavevmode\nobreak\
(N|i,j),\,p,\,n>0\,.$ (7.4)

  * •

In accord with our remarks above, the normalised ensemble averages will be
rational functions of $N$ and $p$. This is clearly visible in the maximal-
trace case, see (6.15).

The individual averages computed in Section 4 do not seem to exhibit any
particular pattern, beyond their rationality and some uniformity in the large
$N$ limit. On the other hand, the ensemble averages
$\langle\mathcal{G}_{p}^{(N)}\rangle$ of correlators of SPO’s do exhibit nice
patterns, leading us to conjecture their general form

$\displaystyle\langle\mathcal{G}_{p}^{(N)}\rangle=Q_{p}(N)\,(N-p+1)_{p-1}\,,\qquad(\text{for
SPO's})\,,$ (7.5)

for some rational function $Q_{p}(N)$.424242The fact that $Q_{p}(N)$ is
rational essentially follows from the rational $N$-dependence of the
coefficients which appear in the weak-coupling expansion obtained from the
matrix model integrals. See Appendix B. The Pochhammer factor ensures the
vanishing of $\langle\mathcal{G}_{p}^{(N)}\rangle$ for $N=1,2,\ldots,p-1$. By
considering the large $N$ expansion of (7.5), we can further constrain the
function $Q_{p}(N)$:

    1. (i)

At leading order in large $N$, the SPO’s reduce to single-trace operators and
hence we should recover the large $N$ limit of
$\langle\mathcal{G}_{p}^{(N|1,1)}\rangle$ given in (4.6). Therefore, we have

$\displaystyle Q_{p}(N)\leavevmode\nobreak\
\xrightarrow[]{\leavevmode\nobreak\ N\to\infty\leavevmode\nobreak\
}\leavevmode\nobreak\ \frac{p-1}{2p}\,N+\ldots.$ (7.6)

    2. (ii)

To be consistent with the general large $N$ expansion of the ensemble average,
cf. (F.1), $Q_{p}(N)$ must admit an expansion containing only integer powers
of $1/N$, i.e.

$\displaystyle
Q_{p}(N)=\frac{p-1}{2p}\,N+\sum_{k=0}^{\infty}\alpha_{k}(p)\,N^{-k}\,,$ (7.7)

with coefficients $\alpha_{k}(p)$ which we expect to be rational functions of
$p$.

  * •

All recursion relations derived herein determine $f_{p}^{(N|i,j)}(s)$ for all
$N$ in terms of lower $p$ spectral overlaps, together with the trivial initial
condition $f_{p}^{(N=1|i,j)}(s)=0$. Ultimately, everything is recursively
determined from the $p=2$ spectral overlap at $N=2$, given by
$f_{2}^{(2)}(s)=(2s-1)^{2}$. As such, the $p=2$ integrated correlator in the
$SU(2)$ theory (together with the knowledge of the recursion relations
themselves!) determines the higher-charge correlators. We expect that this
determinism will extend to all $p$.

  * •

The functional form of the recursion relations themselves is less clear. For
$p=4,5$, the recursion had the general structure

$\displaystyle\begin{split}f_{p}^{(N+1|i,j)}(s)-f_{p}^{(N|i,j)}(s)&=\Big{[}W_{p}^{(0)}(N)-W_{p}^{(1)}(N)\,s(1-s)\Big{]}\,f_{2}^{(N+1)}(s)\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
+\sum_{q=2}^{p-1}W_{p}^{(q)}(N)\,f_{q}^{(N|i,j)}(s)+W_{p}^{(p)}(N)\,f_{2}^{(N-1)}(s)\,,\end{split}$
(7.8)

for some functions $W_{p}^{(n)}(N)$, rational in $N$, which also depend on the
precise trace-structure $(i,j)$.434343As we have pointed out a few times by
now, in cases where $f_{p}^{(N|i,j)}(s)$ vanishes for $N=2$, the consistency
of such a recursion relation requires the functions $W_{p}^{(0)}(N)$ and
$W_{p}^{(1)}(N)$ to vanish at $N=1$. However, in light of the growing degree
with $p$ of the overlaps $f_{p}^{(N|i,j)}(s)$ – see equation (7.3) – the above
structure cannot hold for $p\geq 6$. To account for the higher degree
polynomials requires some additional terms on the RHS of (7.8). This could
potentially include terms with higher powers of $s(1-s)$, or higher integer
shifts of $N$ of the form $f_{2}^{(N+n)}(s)$. As such, it would be necessary
to consider further explicit examples with $p\geq 6$ to determine the most
general structure of these powerful recursion relations.

##  8 Future directions

The integrated correlators $\mathcal{G}_{p}^{(N|i,j)}(\tau)$ are remarkable
observables: they are non-trivial functions of the complexified gauge coupling
$\tau$, and yet tractable enough not only to be exactly determined, but to be
determined simply as polynomials in an appropriate functional basis. This
basis is, moreover, one in which S-duality invariance is manifest. This link
suggests that there may be lessons in the offing for how to better understand
$\tau$-dependence of $\mathcal{N}=4$ SYM observables at large.

These properties raise many conceptual and fundamental questions. We close by
discussing some of these, but we first describe some more concrete open
problems. There are several computations that are ripe for completion, which
include the following:

  * •

Our formulas allow the explicit determination of non-perturbative corrections,
both in $1/\lambda$ and $1/N$, to $\mathcal{G}_{p}^{(N|i,j)}(\tau)$ in the
large $N$ limit. This was analyzed previously for the $p=2$ case, where
corrections in powers of the two scales (5.22) – identified with fundamental
and D-string worldsheet instantons in AdS${}_{5}\,\times\,$S5, respectively –
are easily identified from the $SL(2,\mathbb{Z})$ spectral integral, the two
being related by S-duality. It would be interesting to understand the
$p$-dependence of these corrections. We also note that the analysis of [8] did
not compute all non-perturbative corrections in $N$, in particular leaving the
sum over genera to future study; this is an interesting problem because the
genus sum can in principle generate higher-order (e.g. “doubly”) non-
perturbative effects.

  * •

Another set of computations pertains to the genus-one correlators
$\langle{\cal O}_{2}{\cal O}_{2}{\cal O}_{p}{\cal O}_{p}\rangle$, both
integrated and unintegrated. On the integrated side, we have made a prediction
in (5.15) for a bulk computation of the finite one-loop correction to
$\mathcal{G}_{p}^{(N)}(\tau)$ in AdS${}_{5}\,\times\,$S5 supergravity. This
would be satisfying to see verified in the bulk. Moreover, as noted in
footnote 28, this prediction (in combination with the flat space limit) can be
used to fix the undetermined ambiguities in the unintegrated $\langle{\cal
O}_{2}{\cal O}_{2}{\cal O}_{p}{\cal O}_{p}\rangle$ correlator at one-loop
level from [34, 35], another worthwhile computation.

  * •

The ensemble averages of the (SPO|SPO) correlators through $p=5$ were seen in
(4.9)-(4.10) to obey an intriguing pattern. Can a closed-form expression be
derived, or perhaps inferred from more data points, for generic $p$?

  * •

The $SL(2,\mathbb{Z})$ spectral decomposition faciliates the study of the
ensemble statistics of observables over the $\mathcal{N}=4$ conformal manifold
$\mathcal{M}$. The variance of the integrated correlators over $\mathcal{M}$
follows immediately from their explicit spectral overlaps and a formula
derived in [8]:

$\mathcal{V}\left(\mathcal{G}_{p}^{(N|i,j)}\right)={\rm{vol}}^{-1}({\cal
F})\left({1\over 4\pi i}\int_{{\rm Re\,}s={1\over
2}}ds\,\left|{\pi\over\sin\pi
s}s(1-s)\Lambda(s)f_{p}^{(N|i,j)}(s)\right|^{2}\right)$ (8.1)

This was studied at both finite and large $N$ in [8] for the $p=2$ case. It
would be interesting to ask how the general $p$ integrated correlator
statistics behave as well.

There are various generalizations of our work – e.g. extension to
$\mathcal{N}=4$ SYM with other gauge groups, analysis of other families of
integrated correlators $\mathcal{G}_{p}^{(N|i,j)}(\tau)$ that exist for all
$p$, integration of $\langle{\cal O}_{2}{\cal O}_{2}{\cal O}_{2}{\cal
O}_{2}\rangle$ against other measures [85, 60] – which may be worth pursuing.
Slightly further afield, it remains unknown whether there is an analogous
supersymmetric construction of integrated $\langle{\cal O}_{p}{\cal
O}_{p}{\cal O}_{p}{\cal O}_{p}\rangle$, or for that matter even $\langle{\cal
O}_{p}{\cal O}_{q}{\cal O}_{r}{\cal O}_{s}\rangle$, correlators. If so, new
ideas are needed to determine the supersymmetric integration measure and to
relate it to some as-yet-unknown generalization of the localization methods
based on $\mathcal{N}=2^{*}$ partition functions. It should also be possible
to extend the integrated correlator construction to SCFTs in two spacetime
dimensions using localisation techniques.

Most fascinating are various foundational questions about these observables,
centered around the question of why they exhibit such simplicity, and the
relation to manifest S-duality invariance.

Our results for $\mathcal{G}_{p}^{(N|i,j)}(\tau)$ are characterized by strong
uniformity in the half-BPS charge $p$. This calls for a deeper explanation. At
large $N$, an obvious possible touchstone is the 10-dimensional “hidden
conformal symmetry” that is known to govern at least some aspects of planar
correlators at weak and strong coupling [23, 86]. The origin and regime of
applicability of the hidden conformal symmetry itself are still unknown. It is
tempting to ask whether, and how, the integrated correlators
$\mathcal{G}_{p}^{(N|i,j)}(\tau)$ are organized in some way by this symmetry,
perhaps at large $N$ only. If so, one may wonder whether the integrated
correlators are subject to stronger constraints than the unintegrated ones.

Is there a natural bulk description of these objects besides as integrals over
boundary points? That point of view is rather clumsy from the bulk spacetime
perspective. The integrated correlators have an alternative formulation as
derivatives of the free energy deformed by sources, but that quantity breaks
the $\mathcal{N}=4$ superconformal symmetry. Perhaps there is a nicer picture
which preserves the full symmetries without using the unintegrated correlator
or free energy as an intermediary.

Of course, since the $\mathcal{G}_{p}^{(N|i,j)}(\tau)$ are soluble for all $N$
and $\tau$, we are really after their bulk string theory description. The
formulas herein for $\mathcal{G}_{p}^{(N|i,j)}(\tau)$ may be regarded as the
exact AdS${}_{5}\,\times\,$S5 quantum string theory answers. As emphasized in
the introduction, the integrated correlators seem to be picking out some
privileged subsector of the AdS${}_{5}\,\times\,$S5 four-point string
amplitude. Whether this is a useful perspective that can be leveraged to teach
us something more general about string theory, either in
AdS${}_{5}\,\times\,$S5 or in 10-dimensional flat space, is an open question.
It would obviously be very interesting to make this more explicit from the
worldsheet point of view, perhaps first in various simplifying limits.

The central role of $SL(2,\mathbb{Z})$ in our results underscores not only the
power, but also the practical utility, of S-duality. That the integrated
correlators are so streamlined in the $SL(2,\mathbb{Z})$ eigenbasis suggests
that the $SL(2,\mathbb{Z})$ spectral decomposition may be generally useful in
studying unprotected observables of $\mathcal{N}=4$ SYM such as operator
dimensions or OPE coefficents. This deserves further investigation in the
context of the superconformal bootstrap and integrability (and their
combination [87, 29, 88]). In the present case, that the
$\mathcal{G}_{p}^{(N|i,j)}(\tau)$ can be written down in such simple form for
finite $N$ and $\tau$ is partly444444Note that there do exist non-perturbative
physical observables that have computable nonzero cusp form overlaps. An
example is the torus partition function of two free bosons on a Narain
lattice, for all values of the moduli [89]. due to the absence of Maass cusp
forms in their $SL(2,\mathbb{Z})$ spectral decomposition. Why do the cusp form
overlaps vanish? What is the physical principle that tells us when an
$SL(2,\mathbb{Z})$-invariant observable in $\mathcal{N}=4$ SYM has vanishing
cusp form overlap? Observables with this property are “non-chaotic” in a
precise mathematical, if not (yet) physical, sense. It seems important to
bridge this gap. This may help understand whether the mathematical notion of
arithmeticity is a useful physical criterion in characterizing CFT
observables.454545We point out the recent work [90, 91] as another, possibly
related, arena for this question. For earlier work in a slightly different
context, see [92].

Indeed, the radical simplicity of the integrated correlators – exact for all
$N$ and $\tau$, with ingredients no more complicated than rational functions
of $N$ and $p$ – deserves to be understood in the deepest terms possible. The
fundamental origin of the recursion relations derived herein, generalizing the
$\mathcal{G}_{2}^{(N)}(\tau)$ recursion relation of [7], is a major open
question. We made some suggestions about their provenance in Section 6.1. Can
they be derived from a 6d (2,0) theory point of view? Can we bootstrap them
directly without the input of localization? And finally, what are these
equations telling us about unintegrated CFT data at finite $N$ and $\tau$, and
can we somehow extract these data by cleverly combining the whole family of
integrated correlators indexed by $p$?

The integrated correlators seem to be beautiful and rich observables, standing
out even in the wonderful world of $\mathcal{N}=4$ SYM. They are, in our view,
well-deserving of intensive further study.

## Acknowledgments

We thank Shai Chester, Scott Collier, Gabriele di Ubaldo, Bertrand Eynard and
Silviu Pufu for discussions. HR would like to acknowledge useful discussions
with participants of the Simons Summer Workshop organized at the Simons Center
for Geometry and Physics, Stony Brook. This research was supported by ERC
Starting Grants 853507 and 679278.

##  Appendix A Colour factors

Here we list the colour factors $R_{p}^{(i,j)}(N)$ used in the main text. For
the cases $p=2,3$, these are just given by the general formula (2.7). For
$p=4,5$ we have

$\displaystyle R_{4}^{(1,1)}(N)$
$\displaystyle=\frac{(N^{2}-1)(N^{4}-6N^{2}+18)}{4N^{2}}\leavevmode\nobreak\
,$ $\displaystyle\qquad R_{5}^{(1,1)}(N)$
$\displaystyle=\frac{(N^{2}-1)(N^{2}-4)(N^{4}+24)}{5N^{3}}\leavevmode\nobreak\
,$ $\displaystyle R_{4}^{(1,2)}(N)$
$\displaystyle=\frac{(N^{2}-1)(2N^{2}-3)}{2N}\leavevmode\nobreak\ ,$
$\displaystyle\qquad R_{5}^{(1,2)}(N)$
$\displaystyle=\frac{6(N^{2}-1)(N^{2}-4)(N^{2}-2)}{5N^{2}}\leavevmode\nobreak\
,$ (A.1) $\displaystyle R_{4}^{(2,2)}(N)$
$\displaystyle=\frac{(N^{4}-1)}{2}\leavevmode\nobreak\ ,$ $\displaystyle\qquad
R_{5}^{(2,2)}(N)$
$\displaystyle=\frac{6(N^{2}-1)(N^{2}-4)(N^{2}+5)}{25N}\leavevmode\nobreak\ .$

The colour-factor of the two-point function of maximal-trace operators ${\cal
O}_{p}^{(\text{max})}=T_{2,\ldots,2}$ has been derived in [50]. In our
conventions, for even $p$ it reads

$\displaystyle
R_{p}^{(\text{max})}(N)=\frac{2^{p}\,\big{(}\frac{p}{2}\big{)}!}{p^{2}}\,\bigg{(}\frac{N^{2}-1}{2}\bigg{)}_{\frac{p}{2}}\,.$
(A.2)

At large $N$, the counting simplifies. In particular, computation of colour
factors may be computed by large $N$ factorization. Single-trace colour
factors become

$R_{p}^{(1,1)}(N)\approx{N^{p}\over p}\,.$ (A.3)

This is especially useful for computation of colour factors involving multi-
trace operators $T_{p_{1},\cdots,p_{n}}$, defined in (2.2). The large $N$
colour factor of $T_{p_{1},\ldots,p_{n}}$ with $n$ distinct indices is

$\left({p_{1}\cdots p_{n}\over p}\right)^{2}R_{p_{1}}(N)\ldots
R_{p_{n}}(N)\Big{|}_{N\rightarrow\infty}\approx{p_{1}\cdots p_{n}\over
p^{2}}N^{p}\qquad(\text{for}\leavevmode\nobreak\ p_{1}\neq p_{2}\neq\cdots\neq
p_{n})\,,$ (A.4)

where $p=p_{1}+\cdots+p_{n}=p$. For example, taking $p_{1}=2$ and $p_{2}=3$
reproduces the large $N$ asymptotic of $R_{5}^{(2,2)}(N)$. For coincident
indices, extra combinatoric factors emerge. For example, the large $N$ colour
factor of $T_{q,\ldots,q}$ with $n$ identical indices is

$\left({q^{n}\over
nq}\right)^{2}n!\,(R_{q}(N))^{n}\Big{|}_{N\rightarrow\infty}\approx{q^{n}\over(nq)^{2}}n!\,N^{nq}\,.$
(A.5)

Taking $q=2$ and $n=p/2$ reproduces the large $N$ asymptotic of
$R_{p}^{(\text{max})}(N)$.

##  Appendix B Localization details

###  B.1 Solving operator mixing on $S^{4}$ and integrated two-point functions

In this Section we focus on computing the denominator of (2.14) for
$p=2,3,4,5$. Let us define

$\displaystyle
D_{p}^{(N|i,j)}\equiv\sum_{\mu,\nu}v^{i,\mu}_{p}\bar{v}^{j,\nu}_{p}\partial_{\tau_{\mu}^{\prime}}\partial_{\bar{\tau}_{\nu}^{\prime}}\log\mathcal{Z}_{N}(\tau,\tau_{A}^{\prime},m)\bigg{|}_{\tau^{\prime}_{A}=m=0}\,,$
(B.1)

where we recall that $\mu$ belongs to the index set (2.21). One of the main
problems is to determine the vectors $v^{i,\mu}_{p}$ which encode the mixing
of weight-$p$ chiral operators of species $i$, denoted by ${\cal
O}_{p}^{(i)}$, with lower-weight chiral operators. This is done through the
algorithm devised in [50]. On $S^{4}$, a chiral operator of weight $p$ will
generically mix with all other chiral operators of weight $p-2,\,p-4,\,\ldots$
through the non-trivial background fields. For a given $p$, we begin by
listing all such operators of dimension $\geq 2$.464646On $S^{4}$, operators
of even conformal dimension also mix with the identity operator. However,
there is no need for us to consider this in the unmixing problem because the
final results differ only by one-point functions on $\mathbb{R}^{4}$, which
vanish. The operators ${\cal O}_{2}$ and ${\cal O}_{3}$ do not mix with any
other operator on the sphere. The two chiral operators ${\cal O}_{4}^{(i)}$
mix with ${\cal O}_{2}$ and likewise the two chiral operators ${\cal
O}_{5}^{(i)}$ mix with ${\cal O}_{3}$ (recall the operator definitions in
(2.3)).

Next, we construct the matrix of connected two-point functions on $S^{4}$ for
a given chiral operator of weight $p$ with every other chiral operator of
weight $p-2,\,p-4,\,\ldots$ with which it mixes. We will list below all such
matrices for operators up to $p=5$. First, we introduce some short-hand
notation which will be convenient. In what follows we restrict our treatment
and notation to single- and double-trace operators only, but everything can be
straightforwardly generalized to arbitrary multi-trace insertions following
the prescription of [50].

Let us introduce the notation $O_{p}$ and $O_{p,q}$ for single- and double-
trace chiral operators, respectively, which couple to respective sources
$\tau^{\prime}_{p}$ and $\tau^{\prime}_{p,q}$. Two-point functions on $S^{4}$
are constructed via

$\begin{split}\langle
O_{p}\bar{O}_{q}\rangle:=\frac{\partial_{\tau_{p}^{\prime}}\partial_{\overline{\tau}_{q}^{\prime}}\mathcal{Z}_{N}}{\mathcal{Z}_{N}}\bigg{|}_{\tau^{\prime}_{A}=m=0},\leavevmode\nobreak\
\langle
O_{p,q}\bar{O}_{r}\rangle:=\frac{\partial_{\tau^{\prime}_{p,q}}\partial_{\overline{\tau}_{r}^{\prime}}\mathcal{Z}_{N}}{\mathcal{Z}_{N}}\bigg{|}_{\tau^{\prime}_{A}=m=0},\leavevmode\nobreak\
\langle
O_{p,q}\bar{O}_{r,s}\rangle:=\frac{\partial_{\tau^{\prime}_{p,q}}\partial_{\overline{\tau}^{\prime}_{r,s}}\mathcal{Z}_{N}}{\mathcal{Z}_{N}}\bigg{|}_{\tau^{\prime}_{A}=m=0}.\end{split}$
(B.2)

Note that $O_{p}={\cal O}_{p}^{(1)}$ is the weight-$p$ single-trace chiral
operator introduced in the main text, while $O_{p,q}$ is a single
representative of the set of weight-$(p+q)$ double-trace chiral operators
${\cal O}_{p+q}^{(j)}$ indexed by $j$. Let us also introduce a standard
notation for the connected two-point functions on $S^{4}$ (one-point functions
do not vanish on $S^{4}$), obtained by differentiation with respect to the
free energy:

$\begin{split}\langle
O_{p}\bar{O}_{q}\rangle_{c}&:=\partial_{\tau^{\prime}_{p}}\partial_{\bar{\tau}^{\prime}_{q}}\log\mathcal{Z}_{N}\big{|}_{\tau^{\prime}_{A}=m=0}\\\
&=\langle O_{p}\bar{O}_{q}\rangle-\langle
O_{p}\rangle\langle\bar{O}_{q}\rangle\,,\end{split}$ (B.3)

and likewise for the other two-point functions in (B.2).

Through $p=5$, we thus have the following mixing matrices:

  * •

$p=2$: There is just one single-trace operator which does not mix with any
other operator. So there is nothing to unmix here and we have $M_{2}=\langle
O_{2}\bar{O}_{2}\rangle_{c}$.

  * •

$p=3$: Again there is nothing to unmix, and we have $M_{3}=\langle
O_{3}\bar{O}_{3}\rangle_{c}.$

  * •

$p=4$: There are two mixing matrices to consider here since operators of
weight four have degeneracy two. For the single-trace weight-four operator
${\cal O}_{4}^{(1)}$ we denote the mixing matrix by $M_{4}^{(1)}$ whereas for
the double-trace weight-four operator ${\cal O}_{4}^{(2)}$, we denote the
corresponding mixing matrix by $M_{4}^{(2)}$. These are

$M_{4}^{(1)}=\left(\begin{array}[]{cc}\langle
O_{2}\bar{O}_{2}\rangle_{c}&\langle O_{2}\bar{O}_{4}\rangle_{c}\\\\[5.0pt]
\langle O_{4}\bar{O}_{2}\rangle_{c}&\langle
O_{4}\bar{O}_{4}\rangle_{c}\end{array}\right),\qquad
M_{4}^{(2)}=\left(\begin{array}[]{cc}\langle
O_{2}\bar{O}_{2}\rangle_{c}&\langle O_{2}\bar{O}_{2,2}\rangle_{c}\\\\[5.0pt]
\langle O_{2,2}\bar{O}_{2}\rangle_{c}&\langle
O_{2,2}\bar{O}_{2,2}\rangle_{c}\end{array}\right).$ (B.4)

  * •

$p=5$: Again, there are two mixing matrices to consider here since operators
of weight five have degeneracy two. For the single-trace weight-five operator
${\cal O}_{5}^{(1)}$ we denote the mixing matrix by $M_{5}^{(1)}$ whereas for
the double-trace weight-five operator ${\cal O}_{5}^{(2)}$, we denote the
corresponding mixing matrix by $M_{5}^{(2)}$. These are

$M_{5}^{(1)}=\left(\begin{array}[]{cc}\langle
O_{3}\bar{O}_{3}\rangle_{c}&\langle O_{3}\bar{O}_{5}\rangle_{c}\\\\[5.0pt]
\langle O_{5}\bar{O}_{3}\rangle_{c}&\langle
O_{5}\bar{O}_{5}\rangle_{c}\end{array}\right),\qquad
M_{5}^{(2)}=\left(\begin{array}[]{cc}\langle
O_{3}\bar{O}_{3}\rangle_{c}&\langle O_{3}\bar{O}_{2,3}\rangle_{c}\\\\[5.0pt]
\langle O_{2,3}\bar{O}_{3}\rangle_{c}&\langle
O_{2,3}\bar{O}_{2,3}\rangle_{c}\end{array}\right).$ (B.5)

We explictly compute
$M_{2},M_{3},M_{4}^{(1)},M_{4}^{(2)},M_{5}^{(1)},M_{5}^{(2)}$ as a function of
$N$ and $\tau$ by evaluating the matrix integrals using methods outlined in
Section B.3.

The next step is Gram-Schmidt orthogonalization of the matrices $M_{p}^{(i)}$.
As mentioned above there is nothing to unmix in the $p=2,3$ cases. In
particular this gives $\partial_{\tau_{3}}=\partial_{\tau_{3}^{\prime}}$. For
$p=4,5$, the Gram-Schmidt process gives the new orthogonal vectors
$\partial/\partial{\tau_{4^{(j)}}},\partial/\partial{\tau_{5^{(j)}}}$ (which
brings insertions of chiral operators ${\cal O}_{4}^{(j)}$ and ${\cal
O}_{5}^{(j)}$, respectively, in correlation functions on $\mathbb{R}^{4}$):

$\displaystyle\frac{\partial}{\partial\tau_{4^{(j)}}}=\frac{\partial}{\partial\tau^{\prime}_{4^{(j)}}}+v_{4}^{j,2}\frac{\partial}{\partial{\tau}}\,,\qquad\frac{\partial}{\partial\tau_{5^{(j)}}}=\frac{\partial}{\partial\tau^{\prime}_{5^{(j)}}}+v_{5}^{j,3}\frac{\partial}{\partial\tau_{3}}\,,$
(B.6)

where $j=1,2$ corresponds to single- and double-trace operators, respectively.
We find the following results for the $v$-vectors:

$\displaystyle v_{4}^{1,2}$ $\displaystyle=-\frac{\langle
O_{4}\bar{O}_{2}\rangle_{c}}{\langle
O_{2}\bar{O}_{2}\rangle_{c}}=-\frac{\left(2N^{2}-3\right)}{2yN}\,,$
$\displaystyle\qquad v_{5}^{1,3}$ $\displaystyle=-\frac{\langle
O_{5}\bar{O}_{3}\rangle_{c}}{\langle
O_{3}\bar{O}_{3}\rangle_{c}}=-\frac{5\left(N^{2}-2\right)}{4yN}\,,$ (B.7)
$\displaystyle v_{4}^{2,2}$ $\displaystyle=-\frac{\langle
O_{2,2}\bar{O}_{2}\rangle_{c}}{\langle
O_{2}\bar{O}_{2}\rangle_{c}}=-i\frac{\left(N^{2}+1\right)}{2y}\,,$
$\displaystyle\qquad v_{5}^{2,3}$ $\displaystyle=-\frac{\langle
O_{2,3}\bar{O}_{3}\rangle_{c}}{\langle
O_{3}\bar{O}_{3}\rangle_{c}}=-i\frac{N^{2}+5}{4y}\,.$

This data is sufficient to solve the sphere mixing problem for $p=2,3,4,5$ and
therefore allows to compute both the numerator and denominator of (2.14).

As for the denominator, assembling everything, we find the following results
for the matrix of integrated two-point functions $D_{p}^{(N|i,j)}$ on
$\mathbb{R}^{4}$ as defined in (B.1):

$\displaystyle\begin{split}D_{2}^{(N)}&=\frac{N^{2}-1}{2^{3}y^{2}}\,,\qquad\qquad
D_{3}^{(N)}=\frac{3(N^{2}-4)(N^{2}-1)}{2^{6}y^{3}N}\,,\\\\[5.0pt]
D_{4}^{(N)}&=\left(\begin{array}[]{cc}\frac{\left(N^{2}-1\right)\left(N^{4}-6N^{2}+18\right)}{2^{6}y^{4}N^{2}}&-i\frac{\left(N^{2}-1\right)\left(2N^{2}-3\right)}{2^{5}y^{4}N}\\\\[10.0pt]
i\frac{\left(N^{2}-1\right)\left(2N^{2}-3\right)}{2^{5}y^{4}N}&\frac{N^{4}-1}{2^{5}y^{4}}\\\
\end{array}\right),\\\\[5.0pt]
D_{5}^{(N)}&=\left(\begin{array}[]{cc}\frac{5\left(N^{2}-4\right)\left(N^{2}-1\right)\left(N^{4}+24\right)}{2^{10}y^{5}N^{3}}&-i\frac{15\left(N^{2}-4\right)\left(N^{2}-2\right)\left(N^{2}-1\right)}{2^{9}y^{5}N^{2}}\\\\[10.0pt]
i\frac{15\left(N^{2}-4\right)\left(N^{2}-2\right)\left(N^{2}-1\right)}{2^{9}y^{5}N^{2}}&\frac{3\left(N^{2}-4\right)\left(N^{2}-1\right)\left(N^{2}+5\right)}{2^{9}y^{5}N}\\\
\end{array}\right).\end{split}$ (B.8)

Note that the matrices $D_{p}^{(N|i,j)}$ are all positive definite for
physical values of $N$, as they should be for two-point functions. We can
further diagonalize $D_{4}^{(N|i,j)}$ and $D_{5}^{(N|i,j)}$ in a
$\tau,\bar{\tau}$ independent single-particle operator basis (discussed in
Section 2.1) but for our current purpose we do not need such an expression. It
is straightforward to check that the $N$-dependence of these results are
precisely those given by computing the flat space two-point function using
Wick contractions in the free $\mathcal{N}=4$ SYM with gauge group $SU(N)$.

###  B.2 Integrated four-point functions

In this Section, we compute the numerator of (2.14) for $p=2,3,4,5$ in the
zero instanton sector.

$\displaystyle
C_{p}^{(N|i,j)}\equiv\sum_{\mu,\nu}v^{i,\mu}_{p}\bar{v}^{j,\nu}_{p}\partial_{\tau_{\mu}^{\prime}}\partial_{\bar{\tau}_{\nu}^{\prime}}\partial_{m}^{2}\log\mathcal{Z}_{N}(\tau,\tau_{A}^{\prime},m)\Big{|}_{\tau^{\prime}_{A}=m=0}\leavevmode\nobreak\
,$ (B.9)

where we recall that $\mu$ belongs to the index set (2.21). We first note that
$\partial_{\tau_{\mu}^{\prime}}\partial_{\bar{\tau}_{\nu}^{\prime}}\partial_{m}^{2}\log\mathcal{Z}_{N}$
gives the following five terms that compute the connected integrated
correlator

$\displaystyle\partial_{\tau_{\mu}^{\prime}}\partial_{\bar{\tau}_{\nu}^{\prime}}\partial_{m}^{2}\log\mathcal{Z}=$
$\displaystyle\frac{\partial_{\tau_{\mu}^{\prime}}\partial_{\bar{\tau}_{\nu}^{\prime}}\partial_{m}^{2}\mathcal{Z}}{\mathcal{Z}}-\left(\frac{\partial_{\tau_{\mu}^{\prime}}\partial_{\bar{\tau}_{\nu}^{\prime}}\mathcal{Z}}{\mathcal{Z}}\frac{\partial_{m}^{2}\mathcal{Z}}{\mathcal{Z}}+\frac{\partial_{\tau_{\mu}^{\prime}}\mathcal{Z}}{\mathcal{Z}}\frac{\partial_{\bar{\tau}_{\nu}^{\prime}}\partial_{m}^{2}\mathcal{Z}}{\mathcal{Z}}+\frac{\partial_{\bar{\tau}_{\nu}^{\prime}}\mathcal{Z}}{\mathcal{Z}}\frac{\partial_{\tau_{\mu}^{\prime}}\partial_{m}^{2}\mathcal{Z}}{\mathcal{Z}}\right)+2\frac{\partial_{\tau_{\mu}^{\prime}}\mathcal{Z}}{\mathcal{Z}}\frac{\partial_{\bar{\tau}_{\nu}^{\prime}}\mathcal{Z}}{\mathcal{Z}}\frac{\partial_{m}^{2}\mathcal{Z}}{\mathcal{Z}}$
(B.10)

Single mass derivatives of the $\mathcal{N}=2^{*}$ partition function do not
appear above since those terms vanish for vanishing mass. At vanishing sources
and with $\mu,\nu$ refering to single-trace species $p_{1},q_{1}$
respectively, the above derivatives give the following insertions in the
matrix model integral:

$\displaystyle\begin{split}\partial_{\tau_{p}^{\prime}}&\to
i\pi^{p/2}\sum_{i=1}^{N}a_{i}^{p}\leavevmode\nobreak\
,\qquad\partial_{\bar{\tau}_{q}^{\prime}}\to-i\pi^{q/2}\sum_{i=1}^{N}a_{i}^{q}\\\
\partial^{2}_{m}&\to-4\sum_{i<j}\sum_{\ell=1}^{\infty}(-1)^{\ell}(2\ell+1)\left(a_{ij}\right)^{2\ell}\zeta(2\ell+1)\end{split}$
(B.11)

For $\mu,\nu$ refering to multi-trace species, as explained in Section 2.2,
the precise insertion depends upon the species under consideration. The first
line in (B.11) is easy to see from (2.2). On the other hand, insertions from
mass derivatives resulting from (2.2) are complicated. A pragmatic approach to
evaluate them is in the large $y$ expansion. One way to do this is to expand
the quantity

$\displaystyle\partial_{m}^{2}\left[\frac{1}{H(m)^{N}}\prod_{i<j}\frac{H^{2}(a_{ij})}{H(a_{ij}+m)H(a_{ij}-m)}\right]_{m=0}$
(B.12)

in small $a_{i}$. This expansion is represented by the $\ell$ sum in (B.11).
Performing the matrix integrals term-by-term in the $\ell$ sum gives an
asymptotic expansion for the integrated correlators at large $y$. So computing
$\partial_{\tau_{\mu}^{\prime}}\partial_{\bar{\tau}_{\nu}^{\prime}}\partial_{m}^{2}\log\mathcal{Z}$
boils down to computing the matrix integral (2.20) with insertions dictated by
the derivatives in (B.11) and their multi-trace generalization. For $p\leq 5$,
the most general such insertion involves computing the following474747For
correlators involving just single-trace operators, (B.13) is still the most
general insertion for generic $p$, but when multi-trace operators are
involved, there can be several more insertions of the type
$\sum_{i=1}^{N}a_{i}^{q}$.

$\displaystyle\bigg{<}\left(\sum_{i<j}(a_{i}-a_{j})^{2\ell}\right)\left(\sum_{i=1}^{N}a_{i}^{p}\right)\left(\sum_{i=1}^{N}a_{i}^{q}\right)\bigg{>}$
(B.13)

where the expectation value is computed in the Gaussian ensemble (2.20) whose
details we present in Section B.3. Putting together various ingredients we
find the following results for $C_{p}^{(N|i,j)}$ (in the expressions below the
matrix elements $D_{p}^{(N|1,1)}$ were computed in (B.8))

$\bullet$ $p=2$:

$\displaystyle
C_{2}^{(N)}=D_{2}^{(N)}\bigg{(}\frac{12N\zeta(3)}{y\pi}-\frac{75N^{2}\zeta(5)}{y^{2}\pi^{2}}+\frac{735N^{3}\zeta(7)}{2y^{3}\pi^{3}}-\frac{945N^{2}\left(7N^{2}+2\right)\zeta(9)}{4y^{4}\pi^{4}}$
$\displaystyle+\frac{114345N^{3}\left(N^{2}+1\right)\zeta(11)}{16y^{5}\pi^{5}}-\frac{351351N^{2}\left(11N^{4}+25N^{2}+4\right)\zeta(13)}{128y^{6}\pi^{6}}+\cdots\bigg{)}$
(B.14a)

$\bullet$ $p=3$:

$\displaystyle
C_{3}^{(N)}=D_{3}^{(N)}\bigg{(}\frac{18N\zeta(3)}{y\pi}-\frac{90N^{2}\zeta(5)}{y^{2}\pi^{2}}+\frac{735N^{3}\zeta(7)}{2y^{3}\pi^{3}}-\frac{2835N^{2}\left(2N^{2}+1\right)\zeta(9)}{4y^{4}\pi^{4}}$
$\displaystyle+\frac{114345N^{3}\left(3N^{2}+5\right)\zeta(11)}{64y^{5}\pi^{5}}-\frac{117117N^{2}\left(11N^{4}+40N^{2}+9\right)\zeta(13)}{64y^{6}\pi^{6}}+\cdots\bigg{)}$
(B.14b)

$\bullet$ $p=4$:

$\displaystyle C_{4}^{(N|1,1)}$
$\displaystyle=D_{4}^{(N|1,1)}\bigg{(}\frac{24N\zeta(3)}{y\pi}-\frac{60N^{2}\left(2N^{4}-6N^{2}+27\right)\zeta(5)}{\left(N^{4}-6N^{2}+18\right)y^{2}\pi^{2}}+\frac{105N^{3}\left(10N^{4}+3N^{2}+108\right)\zeta(7)}{2\left(N^{4}-6N^{2}+18\right)y^{3}\pi^{3}}$
$\displaystyle-\frac{945N^{2}\left(19N^{6}+83N^{4}+168N^{2}+144\right)\zeta(9)}{8\left(N^{4}-6N^{2}+18\right)y^{4}\pi^{4}}+\cdots\bigg{)}$
(B.14c) $\displaystyle C_{4}^{(N|1,2)}$
$\displaystyle=D_{4}^{(N|1,2)}\bigg{(}\frac{24N\zeta(3)}{y\pi}-\frac{30N^{2}\left(11N^{2}-9\right)\zeta(5)}{\left(2N^{2}-3\right)y^{2}\pi^{2}}+\frac{3675N^{5}\zeta(7)}{2\left(2N^{2}-3\right)y^{3}\pi^{3}}$
$\displaystyle-\frac{945N^{2}\left(40N^{4}+53N^{2}-12\right)\zeta(9)}{4\left(2N^{2}-3\right)y^{4}\pi^{4}}+\frac{114345N^{5}\left(13N^{2}+43\right)\zeta(11)}{32\left(2N^{2}-3\right)y^{5}\pi^{5}}+\cdots\bigg{)}$
(B.14d) $\displaystyle C_{4}^{(N|2,2)}$
$\displaystyle=D_{4}^{(N|2,2)}\bigg{(}\frac{24N\zeta(3)}{y\pi}-\frac{150N^{2}\left(N^{2}+3\right)\zeta(5)}{\left(N^{2}+1\right)y^{2}\pi^{2}}+\frac{735N^{3}\left(N^{2}+6\right)\zeta(7)}{\left(N^{2}+1\right)y^{3}\pi^{3}}$
$\displaystyle-\frac{945N^{2}\left(7N^{4}+72N^{2}+20\right)\zeta(9)}{2\left(N^{2}+1\right)y^{4}\pi^{4}}+\frac{114345N^{3}\left(N^{2}+15\right)\zeta(11)}{8y^{5}\pi^{5}}+\cdots\bigg{)}$
(B.14e)

$\bullet$ $p=5$:

$\displaystyle C_{5}^{(N|1,1)}$
$\displaystyle=D_{5}^{(N|1,1)}\bigg{(}\frac{30N\zeta(3)}{y\pi}-\frac{150N^{2}\left(N^{4}+6N^{2}+12\right)\zeta(5)}{\left(N^{4}+24\right)y^{2}\pi^{2}}+\frac{525N^{3}\left(5N^{4}+57N^{2}+48\right)\zeta(7)}{4\left(N^{4}+24\right)y^{3}\pi^{3}}$
$\displaystyle-\frac{1575N^{2}\left(7N^{6}+122N^{4}+126N^{2}+72\right)\zeta(9)}{4\left(N^{4}+24\right)y^{4}\pi^{4}}+\cdots\bigg{)}$
(B.14f) $\displaystyle C_{5}^{(N|1,2)}$
$\displaystyle=D_{5}^{(N|1,2)}\bigg{(}\frac{30N\zeta(3)}{y\pi}-\frac{30N^{2}\left(6N^{2}-5\right)\zeta(5)}{\left(N^{2}-2\right)y^{2}\pi^{2}}+\frac{105N^{3}\left(17N^{2}+8\right)\zeta(7)}{2\left(N^{2}-2\right)y^{3}\pi^{3}}$
$\displaystyle-\frac{945N^{2}\left(35N^{4}+87N^{2}-20\right)\zeta(9)}{8\left(N^{2}-2\right)y^{4}\pi^{4}}+\frac{38115N^{3}\left(31N^{4}+167N^{2}+18\right)\zeta(11)}{64\left(N^{2}-2\right)y^{5}\pi^{5}}+\cdots\bigg{)}$
(B.14g) $\displaystyle C_{5}^{(N|2,2)}$
$\displaystyle=D_{5}^{(N|2,2)}\bigg{(}\frac{30N\zeta(3)}{y\pi}-\frac{15N^{2}\left(11N^{2}+97\right)\zeta(5)}{\left(N^{2}+5\right)y^{2}\pi^{2}}+\frac{735N^{3}\left(N^{2}+14\right)\zeta(7)}{\left(N^{2}+5\right)y^{3}\pi^{3}}$
$\displaystyle-\frac{945N^{2}\left(13N^{4}+268N^{2}+133\right)\zeta(9)}{4\left(N^{2}+5\right)y^{4}\pi^{4}}+\frac{114345N^{3}\left(7N^{4}+200N^{2}+321\right)\zeta(11)}{64\left(N^{2}+5\right)y^{5}\pi^{5}}+\cdots\bigg{)}$
(B.14h)

Having computed these results, we remark that while for $p=2$ and $3$, where
the non-planar corrections start at four-loop order, for higher $p$, non-
planarity sets in at lower loop orders. Specifically, for the matrix of
$p=4,5$ correlators, we see that the non-planar corrections already appear at
two loops which is in agreement with known results in the literature [9, 12].

With these results, we now have all the ingredients to assemble the full
matrix of integrated correlators $\mathcal{G}_{p}^{(N|i,j)}(\tau)$ defined in
(2.14) for $p=2,3,4,5$ at finite $N$ in the zero-instanton sector in a weak-
coupling expansion.

###  B.3 Expectation values in the Gaussian (special) unitary ensemble

Here we demonstrate how we evaluate expectation values of the type (B.13) in
the Gaussian special unitary ensemble (2.20). We found [93, 94, 95, 96] as
useful references for the material presented in this section.

One of the technical complications in evaluating such expectation values is to
properly deal with the delta function constraint on the eigenvalues in (2.20).
When the insertions are just differences of eigenvalues then such a constraint
simply gives an $N$-dependent prefactor with the expectation values to be
evaluated in the usual Gaussian unitary ensemble. However, when the insertions
are not differences of eigenvalues (like in (B.13)), a systematic way to
handle the constraint is to use the integral representation of the delta
function, $\delta(\sum
a_{i})=\int\frac{d\mu}{2\pi}e^{i\mu\sum_{i}a_{i}}$.484848Alternatively, one
can use the physical Coulomb branch parameters $\boldsymbol{a}_{i}$
$\displaystyle\boldsymbol{a}_{i}=a_{i}-\frac{1}{N}\sum_{j=1}^{N}a_{j}\,,$
(B.15) which makes all insertions into functions of differences $a_{ij}$ that
can then be evaluated in the $U(N)$ ensemble. After doing a linear change of
variables

$\displaystyle a_{i}-\frac{i\mu}{2\alpha}=z_{i}$ (B.16)

and completing the squares in the exponent, we get that (B.13) evaluates to
(where for convenience we defined $\alpha:=2\pi y$)

$\displaystyle\begin{split}\eqref{insertion1}=&\int d^{N}z\leavevmode\nobreak\
e^{-\alpha\sum_{i=1}^{N}z_{i}^{2}}\prod_{i<j}(z_{ij})^{2}\sum_{i<j}\left(z_{ij}\right)^{2\ell}\int\frac{d\mu}{2\pi}e^{-\frac{\mu^{2}N}{4\alpha}}\sum_{k=1}^{N}\left[a_{k}(z_{k})\right]^{p}\sum_{k^{\prime}=1}^{N}\left[a_{k^{\prime}}(z_{k^{\prime}})\right]^{q}\\\
\equiv&\bigg{<}\sum_{i<j}\left(z_{ij}\right)^{2\ell}\int\frac{d\mu}{2\pi}e^{-\frac{\mu^{2}N}{4\alpha}}\sum_{k=1}^{N}\left[a_{k}(z_{k})\right]^{p}\sum_{k^{\prime}=1}^{N}\left[a_{k^{\prime}}(z_{k^{\prime}})\right]^{q}\bigg{>}\end{split}$
(B.17)

The expectation value in the second line is now understood to be in the usual
Gaussian unitary ensemble where the eigenvalues $z_{i}$ are unconstrained. The
sums appearing above can be opened up and simplified to the sum of following
five expectation values

$\displaystyle\begin{split}\eqref{temp0}=&\leavevmode\nobreak\
N(N-1)\bigg{<}z_{12}^{2\ell}\int\frac{d\mu}{2\pi}e^{-\frac{\mu^{2}N}{4\alpha}}a_{1}^{p+q}\bigg{>}+N(N-1)\bigg{<}z_{12}^{2\ell}\int\frac{d\mu}{2\pi}e^{-\frac{\mu^{2}N}{4\alpha}}a_{1}^{p}a_{2}^{q}\bigg{>}\\\
&+\frac{1}{2}N(N-1)(N-2)\bigg{<}z_{12}^{2\ell}\int\frac{d\mu}{2\pi}e^{-\frac{\mu^{2}N}{4\alpha}}a_{3}^{p+q}\bigg{>}\\\
&+N(N-1)(N-2)\bigg{<}z_{12}^{2\ell}\int\frac{d\mu}{2\pi}e^{-\frac{\mu^{2}N}{4\alpha}}(a_{1}^{p}a_{3}^{q}+a_{1}^{q}a_{3}^{p})\bigg{>}\\\
&+\frac{1}{2}N(N-1)(N-2)(N-3)\bigg{<}z_{12}^{2\ell}\int\frac{d\mu}{2\pi}e^{-\frac{\mu^{2}N}{4\alpha}}a_{3}^{p}a_{4}^{q}\bigg{>}\end{split}$
(B.18)

The relation (B.16) between the old ($a_{i}$) and the new ($z_{i}$)
eigenvalues is left implicit to avoid clutter in the expression above. Because
of the $\mu$ integral and the fact that $a_{i}$ depends on $\mu$ through
(B.16), it is easiest to handle these expectation values by first providing
sources to them and performing the $\mu$ integral, i.e., we first do the
following replacement above

$\displaystyle a^{k}\to\partial_{\gamma}^{k}e^{\gamma
a}\Big{|}_{\gamma=0}\leavevmode\nobreak\ ,\qquad
z_{12}^{2\ell}\to\partial_{\beta}^{2\ell}e^{\beta z_{12}}\Big{|}_{\beta=0}$
(B.19)

and then do the $\mu$ integrals to get

$\displaystyle\begin{split}\eqref{tempC1}=&\leavevmode\nobreak\
\sqrt{\frac{\alpha}{\pi
N}}\,\partial_{\beta}^{2\ell}\bigg{[}N(N-1)\partial_{\gamma_{1}}^{p+q}\bigg{<}e^{(\gamma_{1}+\beta)z_{1}-\beta
z_{2}}\bigg{>}e^{-\frac{\gamma_{1}^{2}}{4\alpha
N}}+N(N-1)\partial_{\gamma_{1}}^{p}\partial_{\gamma_{2}}^{q}\bigg{<}e^{(\gamma_{1}+\beta)z_{1}+(\gamma_{2}-\beta)z_{2}}\bigg{>}e^{-\frac{\left(\gamma_{1}+\gamma_{2}\right)^{2}}{4\alpha
N}}\\\
&+\frac{1}{2}N(N-1)(N-2)\partial_{\gamma_{3}}^{p+q}\bigg{<}e^{\beta(z_{1}-z_{2})+\gamma_{3}z_{3}}\bigg{>}e^{-\frac{\gamma_{3}^{2}}{4\alpha
N}}\\\
&+N(N-1)(N-2)\partial_{\gamma}^{p}\partial_{\delta}^{q}\bigg{<}\left(e^{(\gamma+\beta)z_{1}-\beta
z_{2}+\delta z_{3}}+e^{(\delta+\beta)z_{1}-\beta z_{2}+\gamma
z_{3}}\right)\bigg{>}e^{-\frac{\left(\gamma_{1}+\gamma_{3}\right)^{2}}{4\alpha
N}}\\\
&+\frac{1}{2}N(N-1)(N-2)(N-3)\partial_{\gamma_{3}}^{p}\partial_{\gamma_{4}}^{q}\bigg{<}e^{\beta
z_{1}-\beta
z_{2}+\gamma_{3}z_{3}+\gamma_{4}z_{4}}\bigg{>}e^{-\frac{\left(\gamma_{3}+\gamma_{4}\right)^{2}}{4\alpha
N}}\bigg{]}_{\beta=\gamma_{i}=0}\leavevmode\nobreak\ .\end{split}$ (B.20)

We see that the problem now boils down to computing expectation values of the
type

$\displaystyle\bigg{<}e^{\sum_{i=1}^{k}\gamma_{i}z_{i}}\bigg{>}$ (B.21)

in the Gaussian unitary ensemble.

A practical way to evaluate these expectation values is as follows. Using the
orthogonal Hermite polynomials [97], it is straightforward to show that (B.21)
can be written as the following sum over a certain determinant

$\displaystyle\bigg{<}e^{\sum_{i=1}^{k}\gamma_{i}z_{i}}\bigg{>}$
$\displaystyle=\frac{(N-k)!}{N!}\frac{G(N+2)\pi^{N/2}}{2^{N(N-1)/2}\alpha^{N^{2}/2}}\sum_{a_{1},\cdots,a_{k}=0}^{N-1}{\rm
det\,}_{i,j}\leavevmode\nobreak\ Q_{a_{i},a_{j}}(\gamma_{i})$ (B.22)

where

$\displaystyle{Q}_{m,n}(\beta)$
$\displaystyle:=e^{\frac{\beta^{2}}{4\alpha}}\left(\frac{\beta}{2\alpha}\right)^{n-m}L_{m}^{n-m}\left(-\frac{\beta^{2}}{2\alpha}\right)\leavevmode\nobreak\
.$ (B.23)

Here $L_{m}^{n}(x)$ are the associated Laguerre polynomials. We record here
the following orthogonality relation for Hermite polynomials
$H_{n}(x)=(-1)^{n}e^{x^{2}}\left(d^{n}/dx^{n}\right)e^{-x^{2}}$

$\displaystyle
P_{i}(x):=\frac{1}{\left(4\alpha\right)^{i/2}}H_{i}(\sqrt{\alpha}x)\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\
\int_{-\infty}^{\infty}dx\leavevmode\nobreak\ e^{-\alpha
x^{2}}P_{j}(x)P_{k}(x)=\delta_{jk}\frac{\Gamma(k+1)\sqrt{\pi}}{2^{k}\alpha^{k+\frac{1}{2}}}$
(B.24)

and the identity

$\displaystyle\int dae^{-\alpha a^{2}+\beta
a}P_{m}(a)P_{n}(a)=\frac{\sqrt{\pi}}{2^{n}}m!e^{\frac{\beta^{2}}{4\alpha}}\alpha^{-n-\frac{1}{2}}\beta^{n-m}L_{m}^{n-m}\left(-\frac{\beta^{2}}{2\alpha}\right)\leavevmode\nobreak\
.$ (B.25)

which are necessary to show the equality in (B.22). We note that $Q$ enjoys a
summation identity

$\displaystyle\sum_{m=0}^{N-1}{Q}_{m,m}(\beta)=e^{\frac{\beta^{2}}{4\alpha}}\sum_{m=0}^{N-1}L_{m}\left(-\frac{\beta^{2}}{2\alpha}\right)=e^{\frac{\beta^{2}}{4\alpha}}L_{N-1}^{1}\left(-\frac{\beta^{2}}{2\alpha}\right)=\frac{2\alpha}{\beta}{Q}_{N-1,N}(\beta)$
(B.26)

which allows us to compute (B.21) for $k=1$ in closed form as

$\displaystyle\big{<}e^{\gamma z_{1}}\big{>}=\frac{2\alpha}{\gamma
N}{Q}_{N-1,N}(\gamma)\frac{G(N+2)\pi^{N/2}}{2^{N(N-1)/2}\alpha^{N^{2}/2}}\leavevmode\nobreak\
.$ (B.27)

A convenient way to represent the sum over the determinant in (B.22) is to
think of it as sum over products of traces of products of matrices
${Q}_{a_{i},a_{j}}$. Working out the traces one finds that the result takes
the form of an exponential times an even polynomial in $\gamma_{i}$ :

$\displaystyle\sum_{a_{1},\cdots,a_{k}=0}^{N-1}{\rm
det\,}_{i,j}\leavevmode\nobreak\
Q_{a_{i},a_{j}}(\gamma_{i})=e^{\sum_{i=1}^{k}\frac{\gamma_{i}^{2}}{4\alpha}}\sum_{i_{1},i_{2}\cdots
i_{k}=0}^{\infty}C^{(k)}_{i_{1},i_{2}\cdots
i_{k}}(N)\left(\frac{\gamma_{1}}{\sqrt{2\alpha}}\right)^{i_{1}}\left(\frac{\gamma_{2}}{\sqrt{2\alpha}}\right)^{i_{2}}\cdots\left(\frac{\gamma_{k}}{\sqrt{2\alpha}}\right)^{i_{k}}$
(B.28)

where the coefficients $C^{(k)}_{i_{1},i_{2}\cdots i_{k}}(N)$ are functions of
$N$ and completely symmetric in the indices $i_{1},i_{2}\cdots i_{k}$. We do
not know them in closed form as a function of $N$ for generic indices but
nevertheless can determine for a specific set of indices, which is all that we
really need for calculations.

The relation (B.28) formally determines (B.22) and therefore (B.20). Working
out the relevant derivatives in (B.20) and with the knowledge of
$C^{(k)}_{i_{1},i_{2}\cdots i_{k}}(N)$ we can straightforwardly determine
(B.13) for given $\ell,p,q$.

##  Appendix C Recursion relations for spectral overlaps involving SPO’s

One possible representation of the recursion relation for the spectral
overlaps of the $p=4$ correlator with SPO’s can be written in the form

$\displaystyle\begin{split}f_{4}^{(N+1)}(s)&=\frac{N-1}{4(N+1)^{2}(N^{2}+2N+2)^{2}}\bigg{[}P_{1}(N)-(N-1)(N^{2}-4)^{2}s(1-s)\bigg{]}\,f_{2}^{(N+1)}(s)\\\\[3.0pt]
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+f_{4}^{(N|1,1)}(s)-\frac{2\left(2N^{2}+4N-1\right)}{(N+1)\left(N^{2}+2N+2\right)}\,f_{4}^{(N|1,2)}(s)\\\\[3.0pt]
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\frac{3\left(2N^{2}+2N+3\right)}{N(N+1)}\,f_{3}^{(N)}(s)-\frac{P_{2}(N)}{2N^{2}(N+1)\left(N^{2}+2N+2\right)}\,f_{2}^{(N)}(s)\\\\[3.0pt]
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\frac{N\left(N^{3}+11N^{2}+20N-2\right)}{4\left(N^{2}+2N+2\right)}\,f_{2}^{(N-1)}(s)\,,\end{split}$
(C.1)

where the polynomials $P_{i}(N)$ are given by

$\displaystyle\begin{split}P_{1}(N)&=3N^{7}+10N^{6}+37N^{5}+70N^{4}+8N^{3}-216N^{2}-316N-96\,,\\\
P_{2}(N)&=N^{7}+N^{6}-16N^{5}+72N^{3}-16N^{2}-120N-72\,.\end{split}$ (C.2)

Note that one can further manipulate the RHS of (C.1) to repackage the terms
$f_{4}^{(N|1,1)}(s)$ and $f_{4}^{(N|1,2)}(s)$ into the SPO overlap
$f_{4}^{(N)}(s)$ by using the various relations given in the main text.
However, the result of this rearrangement results in even more complicated
$N$-dependent rational coefficients.

Similarly, putting together the recursion relations for $f_{5}^{(N|i,j)}(s)$
from the main text to obtain the $p=5$ recursion of SPO’s yields

$\displaystyle\begin{split}f_{5}^{(N+1)}(s)&=\frac{2(N-1)}{(N+1)^{3}(N^{2}+2N+6)^{2}}\bigg{[}P_{3}(N)-(N-3)^{2}(N^{2}-4)^{2}s(1-s)\bigg{]}\,f_{2}^{(N+1)}(s)\\\\[3.0pt]
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+f_{5}^{(N|1,1)}(s)-\frac{10(N^{2}+2N-1)}{(N+1)(N^{2}+2N+6)}\,f_{5}^{(N|1,2)}(s)\\\\[3.0pt]
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\frac{8(N^{2}+N+2)}{N(N+1)}\,f_{4}^{(N|1,1)}(s)-\frac{48(N+2)(N^{2}+2N-1)}{N(N+1)(N^{2}+2N+6)}\,f_{4}^{(N|1,2)}(s)\\\\[3.0pt]
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\frac{3\,P_{4}(N)}{N^{2}(N+1)^{2}(N^{2}+2N+6)^{2}}\,f_{3}^{(N)}(s)-\frac{2\,P_{5}(N)}{N^{3}(N+1)^{2}(N^{2}+2N+6)^{2}}\,f_{2}^{(N)}(s)\\\\[3.0pt]
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+4N\,f_{2}^{(N-1)}(s)\,,\end{split}$ (C.3)

with polynomials

$\displaystyle P_{3}(N)$
$\displaystyle=3N^{8}-2N^{7}+90N^{6}+376N^{5}+447N^{4}-86N^{3}+180N^{2}+1152N+432\,,$
$\displaystyle P_{4}(N)$
$\displaystyle=2N^{9}-N^{8}+16N^{7}-123N^{6}-504N^{5}-950N^{4}-1356N^{3}-1296N^{2}-1584N-864\,,$
$\displaystyle P_{5}(N)$
$\displaystyle=N^{11}-12N^{10}-54N^{9}+158N^{8}+439N^{7}-566N^{6}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-1838N^{5}-1632N^{4}-312N^{3}-576N^{2}-3168N-1728\,.$ (C.4)

##  Appendix D One-instanton results

Here we provide results for the one-instanton contribution to the integrated
correlators $\mathcal{G}_{p}^{(i,j)}(N;\tau)$ for $p=2,4,5$ in a weak-coupling
expansion. The result for $p=3$ was already presented in the main text in
Section 3.3.

For $p=2$ the results were previously computed in [76, 7] which in our
conventions are

$\displaystyle\mathcal{G}_{2}^{(N)}(\tau)\Big{|}_{\text{1-inst}}=e^{2\pi
i\tau}\frac{R_{2}}{4}\left(-\frac{6N\Gamma\left(N-\frac{1}{2}\right)}{\sqrt{\pi}\Gamma(N+2)}-\frac{9(N-4)N\Gamma\left(N-\frac{3}{2}\right)}{8\sqrt{\pi}y\pi\Gamma(N+2)}+\frac{45N\Gamma\left(N-\frac{5}{2}\right)}{256\sqrt{\pi}y^{2}\pi^{2}\Gamma(N)}+\cdots\right)$
(D.1)

For $p=4$, the one-instanton weak-coupling expansion of the $(1,1)$ component
has the following form:

$\displaystyle\mathcal{G}_{4}^{(N|1,1)}(\tau)\Big{|}_{\text{1-inst}}=e^{2\pi
i\tau}\frac{R_{4}^{(1,1)}}{4}\frac{1}{N^{4}-6N^{2}+18}\sum_{n=0}^{\infty}\frac{\Gamma\left(N-\frac{3}{2}-n\right)}{\sqrt{\pi}\Gamma(N+2)(\pi
y)^{n}}h_{4,n}^{(1,1)}(N)$ (D.2)

where $h_{4,n}^{(1,1)}(N)$ are polynomials in $N$ of degree $6+n$. For first
few values of $n$ these are

$\displaystyle\begin{split}h_{4,0}^{(1,1)}(N)&=-\frac{3N}{4}\left(80N^{5}-360N^{4}+243N^{3}+648N^{2}-432N-432\right)\\\
h_{4,1}^{(1,1)}(N)&=-\frac{9N}{64}\left(80N^{6}-1720N^{5}+8803N^{4}-14157N^{3}-1392N^{2}+12528N+2880\right)\\\
h_{4,2}^{(1,1)}(N)&=\frac{45}{2^{11}}N^{2}(N-2)\left(80N^{5}+1640N^{4}-32037N^{3}+138231N^{2}-150228N-80136\right)\\\
h_{4,3}^{(1,1)}(N)&=-\frac{315}{2^{15}}N^{2}\bigg{(}80N^{7}-40N^{6}+4923N^{5}-144186N^{4}+936765N^{3}-2308230N^{2}+1935792N+7776\bigg{)}\end{split}$
(D.3)

Similarly for $p=4$ and the $(1,2)$ component of the correlator, we have

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

where $h_{4,n}^{(1,2)}(N)$ are polynomials in $N$ of degree $4+n$. For the
first few values of $n$ these are

$\displaystyle\begin{split}h_{4,0}^{(1,2)}(N)&=-\frac{3N}{4}\left(52N^{3}-153N^{2}+72N+72\right)\\\
h_{4,1}^{(1,2)}(N)&=-\frac{9N}{64}\left(52N^{4}-713N^{3}+2387N^{2}-2088N-480\right)\\\
h_{4,2}^{(1,2)}(N)&=\frac{45}{2^{11}}(N-2)N^{2}\left(52N^{3}+499N^{2}-6357N+13356\right)\\\
h_{4,3}^{(1,2)}(N)&=-\frac{315}{2^{15}}N^{2}(N+1)\left(52N^{4}-321N^{3}+411N^{2}+594N-1296\right)\end{split}$
(D.5)

Finally, the $(2,2)$ component of the integrated correlator for $p=4$ was
presented in (3.30). Consistently, we note that

$\displaystyle
4\mathcal{G}_{4}^{(2|1,1)}(\tau)\Big{|}_{\text{1-inst}}=2\mathcal{G}_{4}^{(2|1,2)}(\tau)\Big{|}_{\text{1-inst}}=\mathcal{G}_{4}^{(2|2,2)}(\tau)\Big{|}_{\text{1-inst}}$
$\displaystyle
4\mathcal{G}_{4}^{(3|1,1)}(\tau)\Big{|}_{\text{1-inst}}=2\mathcal{G}_{4}^{(3|1,2)}(\tau)\Big{|}_{\text{1-inst}}=\mathcal{G}_{4}^{(3|2,2)}(\tau)\Big{|}_{\text{1-inst}}$
(D.6)

which should be the case since (as observed for the correlator in the zero
instanton sector in Section 3.2) $2T_{4}=T_{2,2}$ for $N=2,3$.

Likewise at $p=5$ we have the following results. For the $(1,1)$ component, we
have the following form for the one-instanton weak-coupling expansion

$\displaystyle\mathcal{G}_{5}^{(N|1,1)}(\tau)\Big{|}_{\text{1-inst}}=e^{2\pi
i\tau}\frac{R_{5}^{(1,1)}}{4}\frac{1}{N^{4}+24}\sum_{n=0}^{\infty}\frac{\Gamma\left(N-\frac{3}{2}-n\right)}{\sqrt{\pi}\Gamma(N+3)(\pi
y)^{n}}h_{5,n}^{(1,1)}(N)$ (D.7)

where $h_{5,n}^{(1,1)}(N)$ are polynomials in $N$ of degree $7+n$. For the
first few values of $n$ these are

$\displaystyle\begin{split}h_{5,0}^{(1,1)}(N)&=-\frac{15N}{2}\left(16N^{6}-76N^{5}+42N^{4}+243N^{3}-48N^{2}-336N-144\right)\\\
h_{5,1}^{(1,1)}(N)&=-\frac{45N}{32}\left(16N^{7}-492N^{6}+3094N^{5}-5549N^{4}-3333N^{3}+9744N^{2}+7056N+960\right)\\\
h_{5,2}^{(1,1)}(N)&=\frac{225}{2^{10}}N^{2}\left(16N^{7}+580N^{6}-17098N^{5}+121349N^{4}-299895N^{3}+96582N^{2}+338592N+80304\right)\\\
h_{5,3}^{(1,1)}(N)&=-\frac{1575}{2^{14}}N^{2}(N-3)\bigg{(}16N^{7}+180N^{6}+5462N^{5}-145329N^{4}+824655N^{3}\\\
&\qquad\qquad\qquad\qquad\qquad-1221822N^{2}-645648N-864\bigg{)}\end{split}$
(D.8)

For the $(1,2)$ component, the form of the correlator is

$\displaystyle\mathcal{G}_{5}^{(N|1,2)}(\tau)\Big{|}_{\text{1-inst}}=\mathcal{G}_{5}^{(N|2,1)}(\tau)\Big{|}_{\text{1-inst}}=e^{2\pi
i\tau}\frac{R_{5}^{(1,2)}}{4}\frac{1}{N^{2}-2}\sum_{n=0}^{\infty}\frac{\Gamma\left(N-\frac{3}{2}-n\right)}{\sqrt{\pi}\Gamma(N+3)(\pi
y)^{n}}h_{5,n}^{(1,2)}(N)$ (D.9)

where $h_{5,n}^{(1,2)}(N)$ are polynomials in $N$ of degree $5+n$. For the
first few values of $n$ these are

$\displaystyle\begin{split}h_{5,0}^{(1,2)}(N)&=-\frac{3N}{2}\left(32N^{4}-84N^{3}-31N^{2}+140N+60\right)\\\
h_{5,1}^{(1,2)}(N)&=-\frac{9N}{32}\left(32N^{5}-644N^{4}+2589N^{3}-1865N^{2}-2940N-400\right)\\\
h_{5,2}^{(1,2)}(N)&=\frac{45}{2^{10}}N^{2}\left(32N^{5}+684N^{4}-14427N^{3}+66091N^{2}-78290N-33460\right)\\\
h_{5,3}^{(1,2)}(N)&=-\frac{315}{2^{14}}N^{2}(N-3)\left(32N^{5}+156N^{4}+3213N^{3}-57321N^{2}+161710N+360\right)\leavevmode\nobreak\
.\end{split}$ (D.10)

And finally for the $(2,2)$ component, the form of the correlator is

$\displaystyle\mathcal{G}_{5}^{(N|2,2)}(\tau)\Big{|}_{\text{1-inst}}=e^{2\pi
i\tau}\frac{R_{5}^{(2,2)}}{4}\frac{1}{N^{2}+5}\sum_{n=0}^{\infty}\frac{\Gamma\left(N-\frac{3}{2}-n\right)}{\sqrt{\pi}\Gamma(N+3)(\pi
y)^{n}}h_{5,n}^{(2,2)}(N)$ (D.11)

where $h_{5,n}^{(2,2)}(N)$ are polynomials in $N$ of degree $5+n$. For the
first few values of $n$ these are

$\displaystyle\begin{split}h_{5,0}^{(2,2)}(N)&=-15N(2N+1)\left(N^{3}-N^{2}+4N-15\right)\\\
h_{5,1}^{(2,2)}(N)&=-\frac{45N}{16}\left(2N^{5}-19N^{4}+37N^{3}-180N^{2}+492N+100\right)\\\
h_{5,2}^{(2,2)}(N)&=\frac{225}{2^{9}}N^{2}\left(2N^{5}+13N^{4}-154N^{3}+541N^{2}-2647N+5719\right)\\\
h_{5,3}^{(2,2)}(N)&=-\frac{1575}{2^{13}}N^{2}(N-3)(N+1)(N+2)\left(2N^{3}-9N^{2}+10N+18\right)\end{split}$
(D.12)

Again, we note that the one-instanton results for the different components of
the correlators are consistent with each other for low values of $N$

$\displaystyle\left(\frac{6}{5}\right)^{2}\mathcal{G}_{5}^{(3|1,1)}(\tau)\Big{|}_{\text{1-inst}}=\left(\frac{6}{5}\right)\mathcal{G}_{5}^{(3|1,2)}(\tau)\Big{|}_{\text{1-inst}}=\mathcal{G}_{5}^{(3|2,2)}(\tau)\Big{|}_{\text{1-inst}}$
$\displaystyle\left(\frac{6}{5}\right)^{2}\mathcal{G}_{5}^{(4|1,1)}(\tau)\Big{|}_{\text{1-inst}}=\left(\frac{6}{5}\right)\mathcal{G}_{5}^{(4|1,2)}(\tau)\Big{|}_{\text{1-inst}}=\mathcal{G}_{5}^{(4|2,2)}(\tau)\Big{|}_{\text{1-inst}}$
(D.13)

which should be the case since $6T_{5}=5T_{2,3}$ for $N=3,4$.

##  Appendix E Matching to two-loop perturbation theory

In this appendix we demonstrate that the weak-coupling expansion of the
integrated correlators $\mathcal{G}_{p}^{(N|i,j)}(\tau)$ obtained from the
localisation computations matches the explicit integration of $\langle{\cal
O}_{2}{\cal O}_{2}{\cal O}_{p}^{(i)}{\cal O}_{p}^{(j)}\rangle$ up to two-loop
order in perturbation theory. For the $p=2$ case this has already been
established in [7],494949 See also the recent work [41] for a comparison up to
four-loop order and interesting connections to periods of Feynman graphs.  and
here we extend this analysis to higher values of $p$.

In [9], the family of (unintegrated) $\langle{\cal O}_{2}{\cal O}_{2}{\cal
O}_{p}^{(i)}{\cal O}_{p}^{(j)}\rangle$ correlators has been computed using
traditional Feynman diagram methods to order $g_{\text{YM}}^{4}$, with
explicit data for the colour factors of different trace structures of the
external operators ${\cal O}_{p}^{(i)}$ given up to $p=6$. To make use of the
wealth of data contained in reference [9], we first need to make contact with
our conventions and recast their results into a manifestly $SO(6)$ covariant
formulation. After applying a crossing transformation to bring the correlator
into the ‘22pp’ orientation, its full $y$-dependence is easily restored by
simply uplifting the bosonic propagator factors $x_{ij}^{2}$ to
superpropagators, i.e. by mapping $1/x_{ij}^{2}\mapsto g_{ij}$. We then pull
out the prefactor $g_{12}^{2}g_{34}^{p}$ and restore the factor of
$\mathcal{I}$ to bring their results into the $SO(6)$ covariant form (2.10),
where the interacting part $\mathcal{H}_{p}^{(i,j)}$ of the correlator admits
the weak-coupling expansion

$\displaystyle\mathcal{H}_{p}^{(i,j)}(u,v;\tau)=g_{\text{YM}}^{2}\,\mathcal{H}_{p,\,\text{one-
loop}}^{(i,j)}(u,v)+g_{\text{YM}}^{4}\,\mathcal{H}_{p,\,\text{two-
loop}}^{(i,j)}(u,v)+O(g_{\text{YM}}^{6})\,.$ (E.1)

The one- and two-loop contributions read

$\displaystyle\begin{split}\mathcal{H}_{p,\,\text{one-
loop}}^{(i,j)}(u,v)&=-2\,C_{1,p}^{(i,j)}(N)\cdot\frac{u}{v}\Phi^{(1)}(u,v)\,,\\\
\mathcal{H}_{p,\,\text{two-
loop}}^{(i,j)}(u,v)&=N\,\frac{u}{v}\cdot\Bigg{(}\frac{1}{4}\Big{[}(1+v-u)\,C_{1,p}^{(i,j)}(N)+\frac{2u}{N}\,C_{2,p}^{(i,j)}(N)\Big{]}\big{(}\Phi^{(1)}(u,v)\big{)}^{2}\\\
&\qquad\qquad+C_{1,p}^{(i,j)}(N)\,\Big{[}\Phi^{(2)}(u,v)+\frac{1}{v}\Phi^{(2)}(\tfrac{1}{v},\tfrac{u}{v})\Big{]}\\\
&\qquad\qquad+\frac{1}{u}\Phi^{(2)}(\tfrac{1}{u},\tfrac{v}{u})\Big{[}C_{1,p}^{(i,j)}(N)+\frac{p-2}{N}\,C_{3,p}^{(i,j)}(N)\Big{]}\Bigg{)}\,,\end{split}$
(E.2)

where
$\Phi^{(l)}(u,v)=-\frac{1}{x-\bar{x}}\phi^{(l)}(x^{\prime},\bar{x}^{\prime})$
with $\phi^{(l)}(x,\bar{x})$ being the well-known $l$-loop ladder integrals.
The $C_{n,p}^{(i,j)}(N)$ with $n=1,2,3$ are certain colour-factors encoding
the $N$-dependence of the one- and two-loop correlators. Their precise form
depends on the trace structure of the external operators ${\cal O}_{p}^{(i)}$
and ${\cal O}_{p}^{(j)}$ and is given on a case-by-case basis in [9] (for
$p=3$ see their eqs. (44)-(46), for $p=4$ see Table 2, and for $p=5,6$ see
Tables 6 and 7, respectively).505050In order to account for some differences
in operator normalisations and conventions for Wick contractions, we simply
compare their colour-factors of the free correlator and rescale their
$C_{n,p}^{(i,j)}(N)$ such that the free theory answer matches ours, amounting
to some factors of $4\pi^{2}$ and $p$.

As a final step, we need to perform the integration over Euclidean space given
by

$\displaystyle
I_{2}[f(u,v)]=-\frac{2}{\pi}\int_{0}^{\infty}dr\int_{0}^{\pi}d\theta\,\frac{r^{3}\sin^{2}\theta}{u^{2}}\,f(u,v)|_{u=1+r^{2}-2r\cos\theta,\,v=r^{2}}\,,$
(E.3)

which defines the integrated correlator, recall (2.13). This amounts to
integrating the various orientations of the one- and two-loop ladder integrals
$\Phi^{(l)}(u,v)$ present in (E.2). As shown in [7], one has515151 Note that
we have a relative factor of 4 difference in the definition of the integral
$I_{2}$ compared to reference [7], namely
$I_{2}[f(u,v)]|_{\text{here}}=\frac{1}{4}I_{2}[f(u,v)]|_{\text{there}}$.

$\displaystyle
I_{2}\Big{[}\frac{u}{v}\Phi^{(l)}(u,v)\Big{]}=I_{2}\Big{[}\frac{u}{v}\Phi^{(l_{1})}(u,v)\Phi^{(l_{2})}(u,v)\Big{]}=-\frac{1}{2}\binom{2l+2}{l+1}\,\zeta(2l+1),\quad\text{for
}l_{1}+l_{2}=l\,.$ (E.4)

The one-loop contribution is thus straightforwardly integrated by specialising
the above formula to $l=1$, which yields

$\displaystyle I_{2}\Big{[}\frac{u}{v}\Phi^{(1)}(u,v)\Big{]}=-3\zeta(3)\,.$
(E.5)

On the other hand, upon closer inspection of the two-loop correlator in (E.2)
we note that six different position space structures are present, given by
$\big{(}\Phi^{(1)}(u,v)\big{)}^{2}$ with different prefactors as well as the
three orientations of $\Phi^{(2)}(u,v)$. Importantly, we find that all of the
six different structures individually integrate to the same constant obtained
by setting $l=2$ in (E.4), and one has

$\displaystyle\begin{split}I_{2}\Big{[}\frac{u}{v}\,\big{(}\Phi^{(1)}(u,v)\big{)}^{2}\Big{]}&=I_{2}\Big{[}u\,\big{(}\Phi^{(1)}(u,v)\big{)}^{2}\Big{]}=I_{2}\Big{[}\frac{u^{2}}{v}\,\big{(}\Phi^{(1)}(u,v)\big{)}^{2}\Big{]}\\\
=I_{2}\Big{[}\frac{u}{v}\,\Phi^{(2)}(u,v)\Big{]}&=I_{2}\Big{[}\frac{u}{v^{2}}\,\Phi^{(2)}(\tfrac{1}{v},\tfrac{u}{v})\Big{]}=I_{2}\Big{[}\frac{1}{v}\,\Phi^{(2)}(\tfrac{1}{u},\tfrac{v}{u})\Big{]}=-10\zeta(5)\,,\end{split}$
(E.6)

which essentially follows from the fact that these structures are related by
crossing transformations and thus become indistinguishable from each other
after performing the integration.

With the integrals (E.5) and (E.6) in place, one can easily assemble the one-
and two-loop contributions to the integrated correlator (as mentioned above,
to account for different conventions we have rescaled the colour-factors
$C_{n,p}^{(i,j)}(N)$ from [9] so that the free theory correlators match, see
also footnote 50). We have then compared this with the first two orders of the
$1/y$ expansion obtained from the localisation integral (2.14), finding
perfect agreement for the cases $p=3,4,5$ for all trace configurations
$(i,j)$, as well as for the $p=6$ maximal-trace correlator!

The agreement we find provides further evidence that the localisation relation
(2.14) derived in [6] not only computes integrated correlators of single-trace
operators, but can be extended to multi-trace operators and thus allows us to
access the entire set of integrated correlators
$\mathcal{G}_{p}^{(N|i,j)}(\tau)$. In particular, it shows that our
prescription to incorporate such higher-trace insertions in the matrix model
as explained around (2.21) is indeed correct.

Furthermore, this matching also indicates that we have correctly solved the
operator mixing problem on $S^{4}$, which we performed explicitly up to $p=5$
in Appendix B.1 and for the maximal-trace correlators for all even $p$ in
Section 6, thus extending the results of [6] beyond the single-trace sector
and to finite $N$.

##  Appendix F Consistency check: Ensemble average vs. genus expansion

One may perform a consistency check of our results by demonstrating equality
between the large $N$ expansion of the ensemble averages
$\langle\mathcal{G}_{p}^{(N)}\rangle$ and the genus expansion of the spectral
overlaps $f_{p}(s)$. This is instructive because the match involves not just
the leading large $N$ limit of $\langle\mathcal{G}_{p}^{(N)}\rangle$ at every
genus, but the subleading terms as well.

It follows from [8], eq. (10.3), that the large $N$ expansion of
$\langle\mathcal{G}_{p}^{(N)}\rangle$ may be reconstructed from the genus
expansion of the spectral overlaps as

$\displaystyle\langle\mathcal{G}_{p}^{(N)}\rangle=\frac{1}{2}\sum_{g=0}^{\infty}N^{p-2g}\Big{[}f_{p}^{(g)}(1)+N^{-1}f_{p}^{(g)}(0)\Big{]},$
(F.1)

Note that, in terms of the quantity
$\langle\\!\langle\mathcal{G}_{p}^{(g)}\rangle\\!\rangle$ defined earlier,
$\langle\\!\langle\mathcal{G}_{p}^{(g)}\rangle\\!\rangle={1\over
2}f_{p}^{(g)}(1)$. Equation (F.1) implies in turn that the $1/N$ expansion
truncates at genus-$g^{*}$ if $f_{p}^{(g)}(1)=f_{p}^{(g)}(0)=0$ for all
$g>g^{*}$.

As one can see from the polynomials $n^{(g)}_{p}(s)$ given in the previous
section, this truncation holds for $p=2$ and $p=3$ with thresholds $g^{*}=0$
and $g^{*}=1$, respectively. For example, specialising (F.1) to the $p=2$ case
and recalling the genus-zero overlap from (5.2), one has

$\displaystyle\langle\mathcal{G}_{2}^{(N)}\rangle=\frac{1}{2}\Big{(}N^{2}f_{2}^{(0)}(1)+Nf_{2}^{(0)}(0)\Big{)}=\frac{N(N-1)}{4},$
(F.2)

which indeed reproduces the correct $p=2$ ensemble average using only the
genus-zero contribution $f_{2}^{(0)}(s)$. A similar check exists for
$\langle\mathcal{G}_{3}^{(N)}\rangle$ given in (4.2).

On the other hand, $\langle\mathcal{G}_{4}^{(N)}\rangle$ and
$\langle\mathcal{G}_{5}^{(N)}\rangle$ in (4.8) have polynomial $N$-dependent
denominators, giving rise to a $1/N$ expansion which does not truncate. This
is consistent with the higher-genus spectral overlaps given in the previous
section (cf. (5.6)), which do not vanish at $s=0,1$. Nevertheless, one can
still verify that (F.1) correctly reproduces the finite $N$ averages (4.8)
term by term in a $1/N$ expansion, which we have checked up to genus
four.525252We note that the averaged correlators
$\langle\mathcal{G}_{p}^{(N|i,j)}\rangle$, i.e. those not projected onto the
SPO basis, do admit such a truncation, as seen in equations (4.3)-(4.4). Why
these truncations exist is not clear to us.

##  Appendix G More on the ’t Hooft limit

In this appendix we give several details on the ’t Hooft limit and its
interplay with the $SL(2,\mathbb{Z})$ spectral decomposition which were
skipped in the main text.

###  G.1 Genus-zero spectral overlap for general $p$

First we derive the genus-zero, generic $p$ spectral overlap (5.3). The idea
is to make use of the known genus-zero correlator for generic $p$, recast it
into the spectral form and read off the overlap. The genus-zero correlator is
given by

$\displaystyle\mathcal{G}_{p}^{(\mathfrak{g}=0)}(\lambda)=\int_{0}^{\infty}d\omega\leavevmode\nobreak\
\omega\,\frac{J_{1}\big{(}\frac{\sqrt{\lambda}}{\pi}\omega\big{)}^{2}-J_{p}\big{(}\frac{\sqrt{\lambda}}{\pi}\omega\big{)}^{2}}{\sinh^{2}(\omega)}\,.$
(G.1)

which was first obtained in [6] through large $N$ matrix model techniques. We
want to write the above expression into the spectral form in (5.20), which we
rewrite more compactly below

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

Note that the integration contour is now anchored just beyond the $s=1$ pole
that comes from $\Lambda(1-s)$. This form absorbs the first term in (5.20)
into the $s$ integral. Next, we make use of the following Mellin-Barnes
representation for the Bessel function:

$\displaystyle J_{m}(x)J_{n}(x)=\frac{1}{2\pi
i}\int_{t_{0}-i\infty}^{t_{0}+i\infty}dt\frac{\Gamma(-t)\Gamma(m+n+2t+1)}{\Gamma(m+t+1)\Gamma(n+t+1)\Gamma(m+n+t+1)}\left(\frac{x}{2}\right)^{m+n+2t}\leavevmode\nobreak\
,$ (G.3)

which holds for $x>0$. The contour running parallel to the imaginary axis is
anchored at a point $t_{0}$ such that the all poles of $\Gamma(-t)$ and all
poles of $\Gamma(m+n+2t+1)$ are on opposite sides of $t_{0}$. Making use of
the identity

$\int_{0}^{\infty}dx\frac{x^{\alpha}}{\sinh^{2}x}=\left(\frac{1}{2}\right)^{\alpha-1}\Gamma(\alpha+1)\zeta(\alpha)\leavevmode\nobreak\
,$ (G.4)

we have

$\displaystyle\int_{0}^{\infty}dx\frac{x^{\alpha}y^{\alpha-1}}{\sinh^{2}x}J_{m}(xy)J_{n}(xy)=-\frac{1}{2\pi
i}$
$\displaystyle\int_{t_{0}-i\infty}^{t_{0}+i\infty}dt\left(\frac{\Gamma(m+n+2t+1)}{\Gamma(t+1)\Gamma(m+t+1)\Gamma(n+t+1)\Gamma(m+n+t+1)}\right)$
$\displaystyle y^{m+n+2t+\alpha-1}\frac{\pi}{\sin(\pi t)}$
$\displaystyle\left(\frac{1}{2}\right)^{2m+2n+4t+\alpha-1}\Gamma(m+n+2t+\alpha+1)\zeta(m+n+2t+\alpha)\leavevmode\nobreak\
.$ (G.5)

Plugging this representation into (G.1), combining the two Mellin-Barnes
integrals in the resulting expression into one by a change of variables and
making use of the functional equation for the Riemann zeta function,

$\displaystyle\zeta(s)=2^{s}\pi^{s-1}\zeta(1-s)\sin\left(\frac{\pi
s}{2}\right)\Gamma(1-s)\,,$ (G.6)

we find the spectral form (G.2) from which the spectral overlap
$f_{p}^{(0)}(s)$ is extracted.

###  G.2 Genus-one correlators for various values of $p$

Following the same strategy as in the previous subsection, we record Bessel
representations for genus-one SPO correlators
$\mathcal{G}_{p}^{(\mathfrak{g}=1)}(\lambda)$ for $p\leq 5$, and for the
maximal trace family
$\widehat{\mathcal{G}}_{p}^{(\mathfrak{g}=1|\text{max})}(\lambda)$ for all
$p$.

We start with the SPOs. The result for $p=3$ was presented in (5.26). Upon
performing the $k$ sum in (5.24) by making use of (5.25) we find

$\displaystyle\mathcal{G}_{2}^{(\mathfrak{g}=1)}(\lambda)=$
$\displaystyle\int_{0}^{\infty}d\omega\frac{-\lambda\omega^{3}}{48\pi^{3}\sinh^{2}\omega}\bigg{[}12\pi
J_{0}\big{(}\tfrac{\sqrt{\lambda}}{\pi}\omega\big{)}{}^{2}+2\sqrt{\lambda}\omega
J_{1}\big{(}\tfrac{\sqrt{\lambda}}{\pi}\omega\big{)}J_{0}\big{(}\tfrac{\sqrt{\lambda}}{\pi}\omega\big{)}+5\pi
J_{1}\big{(}\tfrac{\sqrt{\lambda}}{\pi}\omega\big{)}{}^{2}\bigg{]}$ (G.7)
$\displaystyle\mathcal{G}_{4}^{(\mathfrak{g}=1)}(\lambda)=$
$\displaystyle-\int_{0}^{\infty}\frac{d\omega}{48\pi^{3}\lambda^{2}\omega^{3}\sinh^{2}\omega}\bigg{[}6\pi\lambda\omega^{2}\left(7\lambda^{2}\omega^{4}+152\pi^{2}\lambda\omega^{2}-2688\pi^{4}\right)J_{0}\big{(}\tfrac{\sqrt{\lambda}}{\pi}\omega\big{)}{}^{2}$
$\displaystyle+2\sqrt{\lambda}\omega\left(\lambda^{3}\omega^{6}+156\pi^{2}\lambda^{2}\omega^{4}-5856\pi^{4}\lambda\omega^{2}+32256\pi^{6}\right)J_{0}\big{(}\tfrac{\sqrt{\lambda}}{\pi}\omega\big{)}J_{1}\big{(}\tfrac{\sqrt{\lambda}}{\pi}\omega\big{)}$
(G.8)
$\displaystyle+\pi\left(47\lambda^{3}\omega^{6}-1080\pi^{2}\lambda^{2}\omega^{4}+19776\pi^{4}\lambda\omega^{2}-64512\pi^{6}\right)J_{1}\big{(}\tfrac{\sqrt{\lambda}}{\pi}\omega\big{)}{}^{2}\bigg{]}$
$\displaystyle\mathcal{G}_{5}^{(\mathfrak{g}=1)}(\lambda)=$
$\displaystyle-\int_{0}^{\infty}\frac{d\omega}{4\pi^{2}\lambda^{3}\omega^{5}\sinh^{2}\omega}\bigg{[}\lambda\omega^{2}\left(5\lambda^{3}\omega^{6}-416\pi^{2}\lambda^{2}\omega^{4}+16512\pi^{4}\lambda\omega^{2}-147456\pi^{6}\right)J_{0}\big{(}\tfrac{\sqrt{\lambda}}{\pi}\omega\big{)}{}^{2}$
$\displaystyle+4\pi\sqrt{\lambda}\omega\left(-5\lambda^{3}\omega^{6}+1712\pi^{2}\lambda^{2}\omega^{4}-34944\pi^{4}\lambda\omega^{2}+147456\pi^{6}\right)J_{0}\big{(}\tfrac{\sqrt{\lambda}}{\pi}\omega\big{)}J_{1}\big{(}\tfrac{\sqrt{\lambda}}{\pi}\omega\big{)}$
$\displaystyle+\left(7\lambda^{4}\omega^{8}+424\pi^{2}\lambda^{3}\omega^{6}-21248\pi^{4}\lambda^{2}\omega^{4}+213504\pi^{6}\lambda\omega^{2}-589824\pi^{8}\right)J_{1}\big{(}\tfrac{\sqrt{\lambda}}{\pi}\omega\big{)}{}^{2}\bigg{]}$
(G.9)

Correlators at higher-genus for all $p$ can be obtained in principle in a
similar way.

Likewise, the result for the normalised genus-one maximal trace correlator
$\widehat{\mathcal{G}}_{p}^{(\mathfrak{g}=1|\text{max})}(\lambda)$ is

$\displaystyle\begin{split}\widehat{\mathcal{G}}_{p}^{(\mathfrak{g}=1|\text{max})}(\lambda)&=\int_{0}^{\infty}d\omega\leavevmode\nobreak\
\frac{p}{48\pi^{3}\lambda\omega\sinh^{2}\omega}\bigg{[}12\pi\lambda\omega^{2}\big{(}(p-3)\lambda
w^{2}-4\pi^{2}\big{)}J_{0}\big{(}\tfrac{\sqrt{\lambda}}{\pi}\omega\big{)}^{2}\\\\[3.0pt]
&\qquad\qquad\qquad-2\sqrt{\lambda}w\big{(}18\pi^{2}(p-2)\lambda\omega^{2}+\lambda^{2}w^{4}-96\pi^{4}\big{)}J_{0}\big{(}\tfrac{\sqrt{\lambda}}{\pi}\omega\big{)}J_{1}\big{(}\tfrac{\sqrt{\lambda}}{\pi}\omega\big{)}\\\\[3.0pt]
&\qquad\qquad\qquad-\pi\big{(}(12p-19)\lambda^{2}\omega^{4}-24\pi^{2}p\lambda\omega^{2}+192\pi^{4}\big{)}J_{1}\big{(}\tfrac{\sqrt{\lambda}}{\pi}\omega\big{)}^{2}\bigg{]}.\end{split}$
(G.10)

This is equivalent to (6.24).

###  G.3 Proof of (5.3.1)

Here we give the proof of (5.3.1).

Plugging (F.1) for the average in (5.19), the spectral decomposition of
integrated correlators in a genus expansion is given by

$\displaystyle\mathcal{G}_{p}^{(N)}(\lambda)=$
$\displaystyle\leavevmode\nobreak\
\frac{1}{2}\sum_{g=0}^{\infty}N^{p-2g}\Big{[}f_{p}^{(g)}(1)+N^{-1}f_{p}^{(g)}(0)\Big{]}$
(G.11) $\displaystyle+\frac{1}{2\pi
i}\int_{\text{Re}\,s=\frac{1}{2}}ds\frac{\pi}{\sin(\pi
s)}s(1-s)\sum_{g=0}^{\infty}N^{p-2g}\Big{[}\Lambda(1-s)\tilde{\lambda}^{s-1}+\Lambda(s)N^{2s-1}\tilde{\lambda}^{-s}\Big{]}f_{p}^{(g)}(s)\leavevmode\nobreak\
,$

We can absorb the first line into the second at the cost of introducing two
shifted contour integrals as follows:

$\displaystyle\mathcal{G}_{p}^{(N)}(\lambda)=$ $\displaystyle\frac{1}{2\pi
i}\int_{\text{Re}\,s=1+\epsilon}ds\frac{\pi}{\sin(\pi
s)}s(1-s)\sum_{g=0}^{\infty}N^{p-2g}\Lambda(1-s)\tilde{\lambda}^{s-1}f_{p}^{(g)}(s)$
(G.12) $\displaystyle+\frac{1}{2\pi
i}\int_{\text{Re}\,s=0-\epsilon}ds\frac{\pi}{\sin(\pi
s)}s(1-s)\sum_{m=0}^{\infty}N^{p-2m+2s-1}\Lambda(s)\tilde{\lambda}^{-s}f_{p}^{(m)}(s)\leavevmode\nobreak\
,$

where $\epsilon>0$ but infinitesimal. Noting that $f_{p}^{(m)}(s)$ has simple
poles at $s=-\frac{1}{2}-n$ for $n\in\mathbb{Z}_{\geq 0}$ and for all $m\geq
0$, commuting the $s$ integral with the genus expansion, the second line above
simplifies to the following sum over residues

$\displaystyle\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}N^{p-2-2(m+n)}\tilde{\lambda}^{n+\frac{1}{2}}\underset{s=-\frac{1}{2}-n}{\operatorname{Res}}\left[\frac{\pi}{\sin\pi
s}s(1-s)\Lambda(s)f_{{p}}^{(m)}(s)\right]$ (G.13)

We now introduce the $\mathfrak{g}$ variable,

$\displaystyle\mathfrak{g}:=m+n+1\,.$ (G.14)

Note that $\mathfrak{g}>n$. Rewriting the previous sum in terms of
$\mathfrak{g}$ and $n$ we get

$\displaystyle\sum_{\mathfrak{g}=1}^{\infty}N^{p-2\mathfrak{g}}\sum_{n=0}^{\mathfrak{g}-1}\tilde{\lambda}^{n+\frac{1}{2}}\underset{s=-\frac{1}{2}-n}{\operatorname{Res}}\left[\frac{\pi}{\sin\pi
s}s(1-s)\Lambda(s)f_{{p}}^{(\mathfrak{g}-n-1)}(s)\right]$ (G.15)

Note that we can extend the above sum to $\mathfrak{g}=0$, which vanishes
because of the $n$ sum. Therefore $\mathcal{G}_{p}^{(N)}(\lambda)$ now becomes

$\displaystyle\mathcal{G}_{p}^{(N)}(\lambda)=$
$\displaystyle\sum_{\mathfrak{g}=0}^{\infty}N^{p-2\mathfrak{g}}\bigg{(}\frac{1}{2\pi
i}\int_{\text{Re}\,s=1+\epsilon}ds\frac{\pi}{\sin(\pi
s)}s(1-s)\Lambda(1-s)\tilde{\lambda}^{s-1}f_{p}^{(\mathfrak{g})}(s)$ (G.16)
$\displaystyle+\sum_{n=0}^{\mathfrak{g}-1}\tilde{\lambda}^{n+\frac{1}{2}}\underset{s=-\frac{1}{2}-n}{\operatorname{Res}}\left[\frac{\pi}{\sin\pi
s}s(1-s)\Lambda(s)f_{{p}}^{(\mathfrak{g}-n-1)}(s)\right]\bigg{)}\leavevmode\nobreak\
,$

where we commuted the $s$ integral with the genus expansion and changed the
dummy summation index from $g$ to $\mathfrak{g}$ in the first line. Comparing
with (the perturbative part of) (5.16) we have

$\displaystyle\mathcal{G}_{p}^{(\mathfrak{g})}(\lambda)=$
$\displaystyle\frac{1}{2\pi
i}\int_{\text{Re}\,s=1+\epsilon}ds\frac{\pi}{\sin(\pi
s)}s(1-s)\Lambda(1-s)\tilde{\lambda}^{s-1}f_{p}^{(\mathfrak{g})}(s)$
$\displaystyle+\sum_{n=0}^{\mathfrak{g}-1}\tilde{\lambda}^{n+\frac{1}{2}}\underset{s=-\frac{1}{2}-n}{\operatorname{Res}}\left[\frac{\pi}{\sin\pi
s}s(1-s)\Lambda(s)f_{{p}}^{(\mathfrak{g}-n-1)}(s)\right]$ (G.17)

The above equation says that a genus-$\mathfrak{g}$ correlator includes such
renormalization terms from all genera $g<\mathfrak{g}$.

Next we ask, can we absorb the second line into the first by shifting the
integration contour? The answer is yes, but under certain conditions on the
overlaps.

First note that $f_{{p}}^{(\mathfrak{g})}(s)$ has a finite number of poles at
positive half integers,

$\displaystyle s=\frac{3}{2}+n\leavevmode\nobreak\ ,\qquad
n=0,1,2,...,\mathfrak{g}-1\leavevmode\nobreak\ ,$ (G.18)

as follows from (5.6). For $\mathfrak{g}=0$ there are no such poles. Therefore
we can shift the integration contour in the first line of (G.3) towards the
right over all the positive poles of $f_{{p}}^{(\mathfrak{g})}(s)$ mentioned
above. In doing so we encounter the positive integer poles of $\sin(\pi s)$.
Therefore we need to subtract the residue of these integer $\sin$ poles. When
all is said and done we land at the following expression for $\mathfrak{g}>0$:

$\displaystyle\mathcal{G}_{p}^{(\mathfrak{g}>0)}(\lambda)=$
$\displaystyle\frac{1}{2\pi
i}\int_{\text{Re}\,s=\mathfrak{g}+\frac{1}{2}+\epsilon}ds\frac{\pi}{\sin(\pi
s)}s(1-s)\Lambda(1-s)\tilde{\lambda}^{s-1}f_{p}^{(\mathfrak{g})}(s)$
$\displaystyle+\sum_{n=0}^{\mathfrak{g}-1}\tilde{\lambda}^{n+\frac{1}{2}}\underset{s=-\frac{1}{2}-n}{\operatorname{Res}}\left[\frac{\pi}{\sin\pi
s}s(1-s)\Lambda(s)f_{{p}}^{(\mathfrak{g}-n-1)}(s)\right]$
$\displaystyle-\sum_{n=0}^{\mathfrak{g}-1}\tilde{\lambda}^{n+\frac{1}{2}}\underset{s=\frac{3}{2}+n}{\operatorname{Res}}\left[\frac{\pi}{\sin\pi
s}s(1-s)\Lambda(1-s)f_{{p}}^{(\mathfrak{g})}(s)\right]$
$\displaystyle-\sum_{k=2}^{\mathfrak{g}}(-1)^{k+1}\tilde{\lambda}^{k-1}k(k-1)\Lambda(1-k)f_{{p}}^{(\mathfrak{g})}(k)$
(G.19)

Here we note the condition for consistency of the weak-coupling expansion,
which must not have half-integer powers of $\lambda$. This implies

$\displaystyle\underset{s=-\frac{1}{2}-n}{\operatorname{Res}}\left[\frac{\pi}{\sin\pi
s}s(1-s)\Lambda(s)f_{{p}}^{(\mathfrak{g}-n-1)}(s)\right]=\underset{s=\frac{3}{2}+n}{\operatorname{Res}}\left[\frac{\pi}{\sin\pi
s}s(1-s)\Lambda(1-s)f_{{p}}^{(\mathfrak{g})}(s)\right]$ (G.20)

which is precisely eq. (7.9) of [8]. Hence, this consistency condition implies
our final result, a spectral representation of the genus-$\mathfrak{g}$
correlator purely in terms of the genus-$\mathfrak{g}$ spectral overlap:

$\displaystyle\mathcal{G}_{p}^{(\mathfrak{g}>0)}(\lambda)=$
$\displaystyle\frac{1}{2\pi
i}\int_{\text{Re}\,s=\mathfrak{g}+\frac{1}{2}+\epsilon}ds\frac{\pi}{\sin(\pi
s)}s(1-s)\Lambda(1-s)\tilde{\lambda}^{s-1}f_{p}^{(\mathfrak{g})}(s)$
$\displaystyle-\sum_{k=2}^{\mathfrak{g}}(-1)^{k+1}\tilde{\lambda}^{k-1}k(k-1)\Lambda(1-k)f_{{p}}^{(\mathfrak{g})}(k)$
(G.21)

This is (5.3.1). At genus one, the second line is not present! However, for
higher genera, $f_{{p}}^{(\mathfrak{g})}(k)$ need not vanish for integer
$k\geq 2$. This is the penalty we pay for having a form like (G.3) for the
correlator $\mathcal{G}_{p}^{(\mathfrak{g})}(\lambda)$. Note that the above
expression is valid for $\mathfrak{g}>0$. For $\mathfrak{g}=0$, we instead
have

$\displaystyle\mathcal{G}_{p}^{(0)}(\lambda)=$ $\displaystyle\frac{1}{2\pi
i}\int_{\text{Re}\,s=1+\epsilon}ds\frac{\pi}{\sin(\pi
s)}s(1-s)\Lambda(1-s)\tilde{\lambda}^{s-1}f_{p}^{(0)}(s)$ (G.22)

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