Complex eigenvalue instantons and the Fredholm determinant expansion in the
Gross-Witten-Wadia model

Dan Stefan Eniceicu,1 Raghu Mahajan,1 and Chitraang Murdia.2,3

1 Department of Physics, Stanford University, Stanford, CA 94305-4060, USA 2
Berkeley Center for Theoretical Physics, Department of Physics, University of
California, Berkeley, CA 94720, USA 3 Theoretical Physics Group, Lawrence
Berkeley National Laboratory, Berkeley, CA 94720, USA

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the leading nonperturbative corrections to the strong-coupling
(ungapped) phase of the Gross-Witten-Wadia (GWW) integral over unitary
matrices, to one-loop order. We compute these corrections directly in terms of
eigenvalue tunneling in a holomorphic presentation of the integral over
eigenvalues. The leading nonperturbative contribution to the partition
function comes from a pair of complex eigenvalue instantons. We show that
these are in fact “ghost instantons”, which are extrema of the one-eigenvalue
effective potential on the “unphysical sheet” of the spectral curve and have
been discussed in detail recently by Mariño, Schiappa, and Schwick. Further,
we discuss the relationship of these instantons to the Fredholm determinant
expansion of the unitary matrix integral, which has recently become an object
of interest in the computations of BPS indices of supersymmetric gauge
theories and black holes. We find that, after taking the ’t Hooft limit, the
first correction given by the Fredholm determinant expansion of the GWW
integral agrees precisely with the leading nonperturbative correction, to one-
loop order.

###### Contents

  1. 1 Introduction and setup
  2. 2 Complex eigenvalue instantons in the strong-coupling phase
    1. 2.1 The perturbative expansion in the strong-coupling phase
    2. 2.2 Instanton contributions in the strong-coupling phase
  3. 3 The Fredholm determinant expansion of the GWW integral
  4. 4 Summary and Discussion
  5. A Further details about eigenvalue instanton computations
    1. A.1 Some perturbative correlators
    2. A.2 Expectation value of $\det U$
    3. A.3 The putative one-eigenvalue instanton contribution to $Z(N,t)$
  6. B Ghost-instanton contribution from tunneling anti-eigenvalues
  7. C Unitary matrix integrals with general single-trace potentials
    1. C.1 Preliminaries
    2. C.2 Resolvents, effective potentials, and the eigenvalue distribution
    3. C.3 Leading-order two-instanton contributions
    4. C.4 The special case of the GWW matrix integral

##  1 Introduction and setup

In this short note, we study the Gross-Witten-Wadia (GWW) integral [1, 2, 3]

$\displaystyle
Z(N,t):=\int\frac{\mathrm{d}U}{\operatorname{vol}U(N)}\,\exp\left(\frac{N}{2t}\left(\operatorname{Tr}U+\operatorname{Tr}U^{-1}\right)\right)\,.$
(1.1)

The integral is over $N\times N$ unitary matrices $U$. The original papers [1,
2, 3] arrived at this integral via studies of 2D lattice gauge theory with
gauge group $U(N)$. We are interested in the ’t Hooft large-$N$ limit, where
the ’t Hooft coupling $t$ is held fixed as $N\to\infty$. For later use, we
will also define

$\displaystyle\tau:=\frac{N}{2t}\,,\quad Z_{N}(\tau):=Z(N,t)\,.$ (1.2)

With ’t Hooft scaling, this integral exhibits a third-order phase transition
at $t=1$ [1, 2, 3]. Performing the change of variables from the matrix $U$ to
its eigenvalues [4], we can rewrite $Z(N,t)$ as

$\displaystyle
Z(N,t)=\frac{1}{N!}\int\prod_{i=1}^{N}\frac{\mathrm{d}\theta_{i}}{2\pi}\,\prod_{j<k}|e^{\mathrm{i}\theta_{j}}-e^{\mathrm{i}\theta_{k}}|^{2}\,\exp\left(\frac{N}{t}\sum_{i=1}^{N}\cos\theta_{i}\right)\,.$
(1.3)

The basic observation is that when $t$ is small, the potential
$-\frac{1}{t}\cos\theta$ is steep and confines the eigenvalues to lie in a
small interval, symmetric around $\theta=0$. As $t$ increases, the potential
becomes weaker and the width of the eigenvalue distribution increases because
of the Vandermonde repulsion. At $t=1$, the width of the eigenvalue
distribution becomes $2\pi$ and the eigenvalues cover the whole circle. This
signals a phase transition to the ungapped phase for $t>1$. As $t\to\infty$,
the potential vanishes; this reduces the problem to the pure Haar measure over
the unitary group, and the eigenvalues are uniformly distributed over
$\theta$. In the strong-coupling phase ($t>1$), the perturbative approximation
to various quantities can be obtained by expanding the exponential in the
integrand $\exp(\tau(\operatorname{Tr}U+\operatorname{Tr}U^{-1}))$ in powers
of $\tau$, and using the Haar integrals computed in [5], without worrying
about finite-$N$ trace relations.

Let us say a few more words about this phase transition. Large-$N$ matrix
integrals are famously known to possess nonperturbative effects due to one-
eigenvalue instantons [6, 7, 8]. These are saddles of the matrix integral in
the large-$N$ limit where one eigenvalue is removed from the cut (i.e. the
original support of the eigenvalue density) and moved to an extremum of the
_effective_ potential (which is a combination of the bare potential and the
Vandermonde repulsion [4, 9]). In the gapped, weak-coupling phase ($t<1$),
there is an eigenvalue instanton at $\theta=\pi$. It is a very interesting
fact that the action of this instanton goes to zero continuously at the phase
transition point, thus becoming unsuppressed [10]. In the ungapped, strong-
coupling phase ($t>1$), the eigenvalue instantons are qualitatively different
and they are located at complex values of $\theta$ [11].

The GWW integral exhibits a double-scaling limit, in which one zooms into the
$t\approx 1$ region. In this double-scaling limit, the solution to the GWW
integral is described by the Hastings-McLeod solution [12] to the Painlevé-II
equation [10]. In this limit, it is also equal to the partition function of a
particular minimal superstring model [13].

Our specific interest is in the instanton contributions to $Z(N,t)$, in the
ungapped, or the strongly-coupled phase $t>1$, outside the double-scaling
limit (but still staying within the ’t Hooft limit). The genus expansion in
this phase only has a planar contribution; all the $1/N^{2}$ corrections to
the free energy vanish. However, instanton corrections still exist. This fact
was appreciated long ago [14] and was studied in detail by Mariño [10] using
transseries solutions to the “string equation” both in and away from the
double-scaling limit. The explicit form for the leading instanton correction
in the $t>1$ phase was written down in [15] using a relationship between the
partition function and $\left\langle\det U\right\rangle$. It reads

$\displaystyle\frac{Z(N,t)}{\exp(N^{2}/4t^{2})}$
$\displaystyle=1-e^{-2NS_{\text{strong}}(t)}\,\frac{1}{N}\frac{t}{8\pi(t^{2}-1)^{3/2}}\left(1+O(N^{-1})\right)+O(e^{-4NS_{\text{strong}}(t)})\,,$
(1.4) $\displaystyle S_{\text{strong}}(t)$
$\displaystyle=\operatorname{arccosh}(t)-\sqrt{1-t^{-2}}\,,\quad\quad(t>1)\,.$
(1.5)

The denominator on the left side is the perturbative contribution to $Z(N,t)$,
as reviewed below.

One of the goals of the present work is to give a derivation of the result
(1.4) via a direct analysis of the saddle points of the eigenvalue integral
(1.3). The lack of such a derivation until now constitutes a fundamental gap
in the literature given that the original one-eigenvalue instanton
computations in the early ’90s were done using this approach [6] which has
since been adopted as the standard technique for calculating nonperturbative
effects in matrix integrals. The hindrance people faced in adapting this
computation to the GWW integral is that the instanton action predicted by
standard eigenvalue tunneling has the wrong sign and would therefore
incorrectly suggest that the nonperturbative corrections should be
exponentially enhanced, rather than suppressed in the GWW integral [11]. We
overcome this hindrance by showing that the exponentially-suppressed
contribution in (1.4) is, in fact, attributed to the recently discovered
“ghost instantons”.111 Ghost instantons are nonperturbative configurations
with opposite actions compared to those of regular instantons. The existence
of these nonperturbative sectors was first predicted by studying the resurgent
properties of the differential equation governing the specific heat of the
$(2,3)$ minimal string [16] (see also [17]). They were subsequently studied in
[18, 19, 20, 21], and were successfully explained in the context of Hermitian
matrix integrals in terms of anti-eigenvalue tunneling in [22]. Finally, they
were given a physical interpretation in minimal string theory as ghost
D-branes in [23]. See [24, 25, 26, 27] for subsequent applications. As
explained in [22], in large-$N$ matrix integrals with a two-sheeted spectral
curve, each eigenvalue instanton with action $S$ should have a partner ghost
instanton, whose action is $-S$ and represents eigenvalue tunneling to the
unphysical sheet. It is very interesting that the simple and heavily studied
GWW unitary matrix integral contains a directly measurable ghost instanton
effect. We also convincingly show that the leading nonperturbative
contribution is a two-instanton effect. Furthermore, the present direct
eigenvalue/ghost instanton approach to deriving nonperturbative corrections to
the asymptotics of matrix integrals has the advantage of being easily
generalized to unitary matrix integrals with more complicated potentials,
whereas the differential equation approach of [15] is specific to the GWW
potential.

The second goal of this work is to relate eigenvalue instantons to the
Fredholm determinant expansion of $Z(N,t)$ [28, 29]. There is an exact formula
for $Z(N,t)$ in terms of a Toeplitz determinant [14]

$\displaystyle Z(N,t)=\det\bigg{[}I_{k-l}(N/t)\bigg{]}_{k,l=1,\ldots,N}\,.$
(1.6)

Here $I_{n}(x)$ denotes the $n$th Bessel-I function. The theorem of [28, 29]
relates this Toeplitz determinant to a Fredholm determinant. Recently, this
Fredholm determinant expansion has attracted interest because of its role in
supplying finite-$N$ corrections to the index of supersymmetric black holes in
AdS, and of BPS states in supersymmetric gauge theories [30]. The index is
computed by a unitary integral with a double-trace potential, which, after a
Hubbard-Stratonovich transformation, is related to a GWW-type integral with an
infinite number of single-trace terms in the potential.222These couplings are
controlled by a few chemical potentials that enter in the definition of the
symmetry-resolved index. The $m$th term in this expansion of the index is of
order $e^{-c_{1}mN}$, and is related to contributions to the index coming from
$m$ giant gravitons or wrapped D-branes [31, 32, 33, 34, 35, 36, 37, 38, 39,
40, 41].333The Fredholm determinant expansion is related to, but not exactly
the same as the giant-graviton expansion [42]. However, it was shown that it
is possible to extract the terms of the giant-graviton expansion from those of
the Fredholm determinant expansion at least in the single-fugacity case [43].
For important clarifications regarding the subtleties surrounding wall-
crossing in the multiple-fugacity case see [44]. This observation provides the
underlying intuition for why the Fredholm determinant expansion of the GWW
integral itself should be related to the eigenvalue instanton expansion, which
also supplies corrections of order $e^{-c_{2}N}$. We will show that this
intuition is precise. The Fredholm determinant expansion is usually written in
terms of the quantity $\tau$ (which is why we introduced the auxiliary
definitions (1.2)), and we will show that by taking the ’t Hooft limit, the
first nontrivial term in the Fredholm determinant series agrees with (1.4).

The structure of this paper is as follows. In section 2, we demonstrate that a
direct analysis of eigenvalue instantons in the strongly-coupled phase of the
GWW integral reproduces the result (1.4). We demonstrate how eigenvalue
instantons on the “unphysical sheet” arise in this context and are the ones
that actually contribute. In section 3, we demonstrate that the first
nontrivial term in the Fredholm determinant expansion of $Z(N,t)$ also
reproduces (1.4) in the ’t Hooft limit. In section 4, we summarize the results
and discuss some open problems. Appendix A.1 contains some results for the
perturbative correlators that we need. In appendix A.2, we analyze the
quantity $\left\langle\det U\right\rangle$ and directly compute the first
nonperturbative contribution to it using the integral over eigenvalues. The
contribution comes from a single ghost instanton. In appendix A.3, we analyze
the putative one-instanton contribution to the partition function and show how
it eventually becomes a two-instanton contribution, giving another derivation
of (1.4). In appendix B, we analyze the problem from the perspective of
tunneling anti-eigenvalues of an associated supermatrix integral. Finally, in
appendix C, we extend the analysis of the leading nonperturbative effects due
to regular eigenvalue instantons as well as ghost instantons to the case of
the strong-coupling phase of matrix integrals with general (higher-order)
single-trace potentials. For this more general class of potentials, the
differential equation technique of [15] is not available for the precise
computation of instanton corrections, and thus our techniques provide new
results in these cases.

##  2 Complex eigenvalue instantons in the strong-coupling phase

In this section, we will reproduce (1.4) by a direct analysis of the
eigenvalue instantons in the integral (1.1), or equivalently, in (1.3).
Previously, this result was obtained in [15] by an analysis involving the
expectation value of $\det U$ and a certain nonlinear differential equation
that it satisfies.

It will help us to cast (1.3) into a holomorphic form; see, for example,
Section 1.2.3 of [4]. Let $z_{j}:=e^{\mathrm{i}\theta_{j}}$ and note that

$\displaystyle|e^{\mathrm{i}\theta_{j}}-e^{\mathrm{i}\theta_{k}}|^{2}$
$\displaystyle=-\frac{1}{z_{j}z_{k}}(z_{j}-z_{k})^{2}\,,\quad\frac{\mathrm{d}\theta_{i}}{2\pi}=\frac{\mathrm{d}z_{i}}{2\pi\mathrm{i}z_{i}}\,.$
(2.1)

Introducing the potential

$\displaystyle V(z)=-\frac{1}{2}(z+z^{-1})\,,$ (2.2)

we can rewrite (1.3) as

$\displaystyle Z(N,t)$
$\displaystyle=(-1)^{\frac{1}{2}N(N-1)}\frac{1}{N!}\int\prod_{i=1}^{N}\frac{\mathrm{d}z_{i}}{2\pi\mathrm{i}z_{i}^{N}}\prod_{j<k}(z_{j}-z_{k})^{2}\exp\left(-\frac{N}{t}\sum_{i=1}^{N}V(z_{i})\right)$
(2.3)
$\displaystyle=(-1)^{\frac{1}{2}N(N-1)}\,\mathrm{i}^{-N}\frac{1}{N!}\int\prod_{i=1}^{N}\frac{\mathrm{d}z_{i}}{2\pi}\prod_{j<k}(z_{j}-z_{k})^{2}\exp\left(\frac{N}{2t}\sum_{i=1}^{N}(z_{i}+z_{i}^{-1}-2t\log
z_{i})\right)\,.$ (2.4)

The advantage of this last form is that the integral over eigenvalues looks
locally like one that would be obtained from a matrix integral over Hermitian
matrices. The cost we pay is a logarithmic term in the potential. So, in
principle, we can use the results of [45, 19, 46] expressing the one-loop
instanton contribution in terms of the perturbative data around the leading
one-cut saddle point. In fact, as mentioned in the introduction, we will need
the corresponding formulas for the ghost instantons developed in [22].
However, the presence of a logarithmic term in the potential can cause
ambiguities because of the multi-valuedness of the logarithm in the complex
plane. Furthermore, since the eigenvalue distribution in the strong-coupling
phase covers the entire unit circle, the spectral curve consists of two
disconnected sheets. Because of these differences, we will derive all the
needed formulas from scratch, and for this we will find it useful to use the
form of the integral in (2.3) in which the integrand does not have any branch
cuts.

###  2.1 The perturbative expansion in the strong-coupling phase

In this subsection, as a warm-up and review, we compute the perturbative
approximation to various important quantities. These ingredients will be used
in the next subsection to obtain the instanton correction to one-loop order.

In the strong-coupling phase, the perturbative expansions for various
quantities can be obtained by directly expanding the exponential in the
defining integral (1.1) and using the following result of Diaconis and
Shahshahani (Theorem 2 of [5]):

$\displaystyle\int\frac{\mathrm{d}U}{\operatorname{vol}U(N)}\,\prod_{j=1}^{k}(\operatorname{Tr}U^{j})^{a_{j}}(\operatorname{Tr}U^{-j})^{b_{j}}$
$\displaystyle=\prod_{j=1}^{k}j^{a_{j}}(a_{j})!\,\delta_{a_{j},b_{j}}\quad\quad\text{if
}N\geq\sum_{j=1}^{k}j\,a_{j}\,.$ (2.5)

The point is that at large $N$ and in the strong-coupling phase, the
perturbative quantities can be computed by ignoring the constraint
$N\geq\sum_{j=1}^{k}j\,a_{j}$ in this result.

Let us see an example of this result in action and compute the perturbative
approximation to $Z(N,t)$ itself, by directly expanding the exponential in
(1.1):

$\displaystyle
Z(N,t)=\int\frac{\mathrm{d}U}{\operatorname{vol}U(N)}\sum_{j=0}^{\infty}\frac{1}{j!}\left(\frac{N}{2t}\right)^{j}(\operatorname{Tr}U+\operatorname{Tr}U^{-1})^{j}\,.$
(2.6)

Because of the Kronecker deltas in (2.5), we will only get a nonzero
contribution if $j$ is even and, furthermore, when expanding
$(\operatorname{Tr}U+\operatorname{Tr}U^{-1})^{j}$ via the binomial theorem,
we only keep the middle term. Thus, the perturbative large-$N$ partition
function in the strong-coupling phase is

$\displaystyle Z^{(0)}(N,t)$
$\displaystyle=\int\frac{\mathrm{d}U}{\operatorname{vol}U(N)}\sum_{k=0}^{\infty}\frac{1}{(2k)!}\left(\frac{N}{2t}\right)^{2k}\frac{(2k)!}{(k!)^{2}}(\operatorname{Tr}U)^{k}(\operatorname{Tr}U^{-1})^{k}$
(2.7)
$\displaystyle=\sum_{k=0}^{\infty}\frac{1}{k!^{2}}\left(\frac{N}{2t}\right)^{2k}\times
k!=\exp\left(\frac{N^{2}}{4t^{2}}\,\right).$ (2.8)

We used the integral (2.5) in the second line. This result agrees with the
known planar free energy of the GWW model [1, 2, 3, 10].

Next, we consider the computation of the resolvent. We would like to define
the resolvent as
$R(z):=\frac{1}{N}\left\langle\operatorname{Tr}\frac{1}{z-U}\right\rangle$.
However, it is well-known that the eigenvalue distribution in the strong-
coupling phase of the GWW integral covers the whole circle and so, in fact, we
will have two resolvents:

$\displaystyle R^{+}(z)$
$\displaystyle:=\frac{1}{N}\left\langle\operatorname{Tr}\frac{1}{z-U}\right\rangle\,,\quad\quad\text{for
}|z|>1\,,\quad\text{and}$ (2.9) $\displaystyle R^{-}(z)$
$\displaystyle:=\frac{1}{N}\left\langle\operatorname{Tr}\frac{1}{z-U}\right\rangle\,,\quad\quad\text{for
}|z|<1\,.$ (2.10)

These are the defining equations, but $R^{+}(z)$ can be analytically continued
from the outside to the inside of the unit disk, and $R^{-}(z)$ can be
analytically continued from the inside to the outside of the unit disk.

For $R^{+}(z)$, by expanding $\frac{1}{z-U}=\sum_{k=1}^{\infty}z^{-k}U^{k-1}$
as appropriate for $|z|>1$, expanding the exponential in the GWW measure (1.1)
as in the computation of $Z(N,t)$, and using the result (2.5) we obtain

$\displaystyle R^{+}(z)=\frac{1}{z}+\frac{1}{z^{2}}\frac{1}{2t}\,.$ (2.11)

The asymptotic behavior $R^{+}(z)\to\frac{1}{z}$ as $z\to\infty$ is needed for
the correct normalization of the density of states. The computation of
$R^{-}(z)$ is similar, except we need to expand
$\frac{1}{z-U}=-\sum_{k=1}^{\infty}U^{-k}z^{k-1}$ as appropriate for $|z|<1$.
The result is

$\displaystyle R^{-}(z)=-\frac{1}{2t}\,.$ (2.12)

We note that as $t\to\infty$, these results reduce to the known resolvents for
the pure Haar ensemble.444See, for example, the discussion around Eq. (3.2.52)
of [4]. Furthermore, these results allow us to extract the density of states
via

$\displaystyle\rho(\theta)=\frac{R^{+}(z)-R^{-}(z)}{2\pi\mathrm{i}}\frac{\mathrm{d}z}{\mathrm{d}\theta}=\frac{1}{2\pi}\left(1+\frac{1}{t}\cos\theta\right)\,,\quad\theta\in[-\pi,\pi)\,.$
(2.13)

Again, this result for the density of states is well-known [1, 2, 3, 10]. Note
that this density does not make sense in the weak-coupling phase $t<1$ since
the expression would not be positive for all $\theta$.

###  2.2 Instanton contributions in the strong-coupling phase

Now that we know the resolvents and the density of states, we can compute the
derivative of the one-eigenvalue effective potential. The one-eigenvalue
effective potential is the sum of the bare external potential and a
logarithmic two-body repulsion term caused by the Vandermonde determinant,
$V_{\text{eff}}(z)=V(z)+t\log
z-\frac{2t}{N}\left\langle\operatorname{Tr}\log(z-U)\right\rangle$. We are
following the conventions of [10, 46]. The second term $t/z$ is due to the
logarithmic term in the bare potential appearing in (2.4). Because of the same
reasons as explained in the computation of the resolvent, in fact we need two
analytic functions $V_{\text{eff}}^{+}(z)$ and $V_{\text{eff}}^{-}(z)$ defined
outside and inside the unit circle, respectively. We have

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}V_{\text{eff}}^{+}(z)$
$\displaystyle=-\frac{1}{2}\left(1-z^{-2}\right)+\frac{t}{z}-2t\,R^{+}(z)=-\frac{1}{2}-\frac{1}{2z^{2}}-\frac{t}{z}\,,$
(2.14) $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}V_{\text{eff}}^{-}(z)$
$\displaystyle=-\frac{1}{2}\left(1-z^{-2}\right)+\frac{t}{z}-2t\,R^{-}(z)=\frac{1}{2}+\frac{1}{2z^{2}}+\frac{t}{z}\,.$
(2.15)

We note that the expression for
$\frac{\mathrm{d}}{\mathrm{d}z}V_{\text{eff}}^{-}(z)$ is simply the negative
of the expression for $\frac{\mathrm{d}}{\mathrm{d}z}V_{\text{eff}}^{+}(z)$.
The two common zeroes of these functions are the locations of the eigenvalue
instantons:

$\displaystyle z_{1}^{\star}=-t+\sqrt{t^{2}-1}\,,\quad
z_{2}^{\star}=-t-\sqrt{t^{2}-1}\,.$ (2.16)

We note that both $z_{1}^{\star}$ and $z_{2}^{\star}$ are real and negative,
with $z_{1}^{\star}$ being inside the unit circle and $z_{2}^{\star}$ being
outside the unit circle. These locations were also found numerically in [11].

The explicit expressions for $V_{\text{eff}}^{+}(z)$ and
$V_{\text{eff}}^{-}(z)$ are

$\displaystyle V_{\text{eff}}^{+}(z)$
$\displaystyle=-\frac{z}{2}+\frac{1}{2z}-t\log(-z),$ (2.17) $\displaystyle
V_{\text{eff}}^{-}(z)$ $\displaystyle=\frac{z}{2}-\frac{1}{2z}+t\log(-z),$
(2.18)

A plot of $V_{\text{eff}}^{+}(z)$ for $z<0$ is shown in figure 1.

Figure 1: A plot of $V_{\text{eff}}^{+}(z)$ (2.17) on the negative real axis
for $t=1.1$. It has a local maximum at $z_{1}^{\star}=-t+\sqrt{t^{2}-1}$ and a
local minimum at $z_{2}^{\star}=-t-\sqrt{t^{2}-1}$. The function
$V_{\text{eff}}^{+}(z)$ is initially defined outside the unit disk and then
analytically continued inside it. The values of $V_{\text{eff}}^{+}$ at the
local maximum and minimum are $\pm S_{\text{strong}}(t)$.

If we only looked at the original domain of definition of $R^{+}(z)$, which is
outside the unit disk, we might be tempted to compute the action of the
instanton as

$\displaystyle
V_{\text{eff}}^{+}(z_{2}^{\star})=\sqrt{t^{2}-1}-t\log\left(\sqrt{t^{2}-1}+t\right)<0.$
(2.19)

Since $V_{\text{eff}}^{+}(z_{2}^{\star})<0$, this will lead to an _enhanced_
contribution $\exp\left(-\frac{N}{t}V_{\text{eff}}^{+}(z_{2}^{\star})\right)$.
The fact the eigenvalue instanton in the strong-coupling phase seems to have
the wrong sign was noted earlier in [11]. However, the defining contour of
integration cannot be deformed to the corresponding steepest-descent contour,
which is along the real axis (since $V_{\text{eff}}^{+}(z)$ has a local
minimum along the real axis at $z_{2}^{\star}$).

Now, the main idea is that in order to obtain the instanton contribution in
(1.4), we should instead consider the contribution proportional to
$\exp\left(-\frac{N}{t}V_{\text{eff}}^{+}(z_{1}^{\star})\right)$. Note that
$z_{1}^{\star}$ is located inside the unit disk, whereas the original region
of definition of $R^{+}(z)$ in (2.9) is the exterior of the unit disk. In this
sense, the contribution can be labelled as a ghost-instanton since it
represents an eigenvalue tunneling to a location on the “unphysical sheet”
[22]. We note that

$\displaystyle\exp\left(-\frac{N}{t}V_{\text{eff}}^{+}(z_{1}^{\star})\right)$
$\displaystyle=\exp\left(-N\left(\log(t+\sqrt{t^{2}-1})-\sqrt{1-t^{-2}}\right)\right)=\exp(-NS_{\text{strong}}(t))\,.$
(2.20)

It is encouraging that the quantity $S_{\text{strong}}(t)$, defined in (1.5),
appears in this expression.

At this stage of the analysis, it is possible that we have contributions of
order $\exp(-NS_{\text{strong}}(t))$. It is a well known result from other
means of analysis that the leading correction is a two-instanton effect [10,
15, 47].555This is true even in the weak-coupling phase [10], although for a
qualitatively different reason. See also footnote 8. We show directly in
appendix A.3 that the contribution to the partition function coming from the
tunneling of a single eigenvalue vanishes. More precisely, if we try to pull
out one eigenvalue from the cut, we are forced to pull out a second one as
well in order to get a nonzero result. Thus, the putative one-instanton
contribution becomes a two-instanton contribution, and the approach of
appendix A.3 provides an alternative derivation of the result of this section.
However, we note that there are other observables that do receive
contributions from one-eigenvalue instantons. One such example is
$\left\langle\det U\right\rangle$ [48, 49, 47], which we analyze in appendix
A.2 using a direct instanton calculation in the eigenvalue integral.

Thus, let us move on to analyzing two-instanton contributions. Using the
methods of appendix A.3, it is possible to show that there is no contribution
of order $e^{-2NS_{\text{strong}}(t)}$ from a configuration where two
eigenvalues are placed at the same extremum of the effective potential. So we
consider the configuration where one eigenvalue tunnels to $z_{1}^{\star}$ and
another tunnels to $z_{2}^{\star}$. We denote this contribution by
$Z^{(1,1)}(N,t)$, which has the following explicit expression:

$Z^{(1,1)}(N,t)=\frac{1}{(N-2)!}(-1)^{N(N-1)/2}\int_{\mathcal{C}_{1}}\frac{\mathrm{d}z_{1}}{2\pi\mathrm{i}z_{1}^{N}}e^{-\frac{N}{t}V(z_{1})}\int_{\mathcal{C}_{2}}\frac{\mathrm{d}z_{2}}{2\pi\mathrm{i}z_{2}^{N}}e^{-\frac{N}{t}V(z_{2})}(z_{1}-z_{2})^{2}\\\
\times\int_{\mathcal{C}_{0}}\prod_{i=3}^{N}\frac{\mathrm{d}z_{i}}{2\pi\mathrm{i}z_{i}^{N}}e^{-\frac{N}{t}\sum_{i=3}^{N}V(z_{i})}\times\prod_{j=3}^{N}\big{[}(z_{1}-z_{j})^{2}(z_{2}-z_{j})^{2}\big{]}\times\prod_{3\leq
k<l\leq N}(z_{k}-z_{l})^{2}\,.$ (2.21)

Here, the tunnelled eigenvalues are denoted by $z_{1}$ and $z_{2}$ and these
variables are integrated along the steepest-descent contours of the respective
saddle points.

We start by considering $|z_{1}|>1$ and $|z_{2}|<1$ and rewrite the above
expression as

$Z^{(1,1)}(N,t)=\frac{1}{(N-2)!}(-1)^{N(N-1)/2}\int_{\mathcal{C}_{1}}\frac{\mathrm{d}z_{1}}{2\pi\mathrm{i}z_{1}^{N}}e^{-\frac{N}{t}V(z_{1})}\int_{\mathcal{C}_{2}}\frac{\mathrm{d}z_{2}}{2\pi\mathrm{i}z_{2}^{N}}e^{-\frac{N}{t}V(z_{2})}(z_{1}-z_{2})^{2}\\\
\times
z_{1}^{2(N-2)}\int_{\mathcal{C}_{0}}\prod_{i=3}^{N}\frac{\mathrm{d}z_{i}}{2\pi\mathrm{i}z_{i}^{N-2}}e^{-\frac{N}{t}\sum_{i=3}^{N}V(z_{i})}\times\prod_{j=3}^{N}\big{[}(1-z_{j}/z_{1})^{2}(1-z_{2}/z_{j})^{2}\big{]}\times\prod_{3\leq
k<l\leq N}(z_{k}-z_{l})^{2}\,.$ (2.22)

This expression will be useful since it contains the factors $(1-z_{j}/z_{1})$
and $(1-z_{2}/z_{j})$ which are such that $|z_{j}/z_{1}|<1$ and
$|z_{2}/z_{j}|<1$. We can now perform the $z_{i}$ integrals, which is like
computing a specific correlator in the GWW matrix integral of rank $N-2$ and
with $t\to t-2t/N$ [50, 45]:

$\begin{split}Z^{(1,1)}(N,t)&=-\int_{\mathcal{C}_{1}}\frac{\mathrm{d}z_{1}}{2\pi\mathrm{i}}\int_{\mathcal{C}_{2}}\frac{\mathrm{d}z_{2}}{2\pi\mathrm{i}}e^{-\frac{N}{t}\big{[}V(z_{1})+V(z_{2})\big{]}}\times\frac{z_{1}^{N-4}(z_{1}-z_{2})^{2}}{z_{2}^{N}}\\\
&\hskip 54.2025pt\times Z^{(0)}(N-2,t-2t/N)\left\langle
e^{\operatorname{Tr}\log\big{[}(1-z_{1}^{-1}U)^{2}(1-z_{2}U^{-1})^{2}\big{]}}\right\rangle_{(N-2,t-2t/N)}\\\
&=-\int_{\mathcal{C}_{1}}\frac{\mathrm{d}z_{1}}{2\pi\mathrm{i}}\int_{\mathcal{C}_{2}}\frac{\mathrm{d}z_{2}}{2\pi\mathrm{i}}e^{-\frac{N}{t}\big{[}V(z_{1})+V(z_{2})\big{]}}\times\frac{z_{1}^{N-4}(z_{1}-z_{2})^{2}}{z_{2}^{N}}\\\
&\hskip 54.2025pt\times
Z^{(0)}(N,t)\exp\left[-\frac{N}{t}\frac{1}{z_{1}}-\frac{N}{t}z_{2}-4\log(1-z_{2}/z_{1})\right]\,.\end{split}$
(2.23)

Here we have used $Z^{(0)}(N-2,t-2t/N)=e^{N^{2}/4t^{2}}=Z^{(0)}(N,t)$. We have
also used the result for the required correlator given in equation (A.9), the
details of which are presented in appendix A.1. We can express this result in
terms of the effective potentials (2.17) and (2.18) as

$\frac{Z^{(1,1)}(N,t)}{Z^{(0)}(N,t)}=-\int_{\mathcal{C}_{1}}\frac{\mathrm{d}z_{1}}{2\pi\mathrm{i}}\int_{\mathcal{C}_{2}}\frac{\mathrm{d}z_{2}}{2\pi\mathrm{i}}e^{-\frac{N}{t}\big{[}V_{\text{eff}}^{+}(z_{1})+V_{\text{eff}}^{-}(z_{2})\big{]}}\frac{1}{(z_{1}-z_{2})^{2}}.$
(2.24)

Let us now be a bit more explicit about the contours. The contour
$\mathcal{C}_{1}$ passes through $z_{1}^{\star}$ and $\mathcal{C}_{2}$ passes
through $z_{2}^{\star}$. Note that $|z_{1}^{\star}|<1$ and so this lies
outside the domain from which we originally started analyzing the $z_{1}$
integral. This is the reason why these eigenvalue instantons are referred to
as “ghost instantons.” A similar comment applies for $z_{2}$. The steepest-
descent contours at these extrema are along the purely imaginary direction.

The integrals can be evaluated using the saddle-point approximation to get the
final answer

$\begin{split}\frac{Z^{(1,1)}(N,t)}{Z^{(0)}(N,t)}&=-\frac{t}{2\pi
N}\frac{1}{|\partial^{2}V_{\text{eff}}^{+}(z_{1}^{\star})|^{1/2}|\partial^{2}V_{\text{eff}}^{-}(z_{2}^{\star})|^{1/2}}\frac{1}{(z_{1}^{\star}-z_{2}^{\star})^{2}}e^{-2NS_{\text{strong}}(t)}\\\
&=-\frac{t}{8\pi N(t^{2}-1)^{3/2}}e^{-2NS_{\text{strong}}(t)}\,.\end{split}$
(2.25)

Thus, we have managed to precisely reproduce the result (1.4) for the leading
instanton correction to the GWW partition function in the strong-coupling
phase.

The analysis in this section derived the leading nonperturbative contribution
to the GWW integral by tunneling eigenvalues to the unphysical sheet of the
spectral curve. In [22], it was realized that, at least in Hermitian matrix
integrals, such ghost instantons can be equivalently realized by tunneling
“negative-charge eigenvalues” to the corresponding locations on the _physical_
sheet. This leads to a very interesting description of ghost instantons in
terms of saddles of an associated supermatrix integral. We explore this
description for the GWW matrix integral in appendix B and reproduce equation
(2.25) from this perspective.

##  3 The Fredholm determinant expansion of the GWW integral

The GWW integral (1.1) equals the Toeplitz determinant (1.6) [14]. The so-
called Fredholm determinant expansion of such Toeplitz determinants is given
by the convergent expansion [29, 28]

$\frac{Z_{N}(\tau)}{Z_{\infty}(\tau)}=\sum_{m=0}^{\infty}(-1)^{m}\sum_{\genfrac{}{}{0.0pt}{}{\\{r_{i}\\}:N\leq
r_{1}<\cdots
r_{m}}{r_{i}\in\mathbb{Z}+1/2}}\det\begin{pmatrix}\widetilde{K}(r_{1},r_{1};\tau)&\widetilde{K}(r_{1},r_{2};\tau)&\dots&\widetilde{K}(r_{1},r_{m};\tau)\\\
\vdots&\vdots&\ddots&\vdots\\\
\widetilde{K}(r_{m},r_{1};\tau)&\widetilde{K}(r_{m},r_{2};\tau)&\cdots&\widetilde{K}(r_{m},r_{m};\tau)\end{pmatrix}\,.$
(3.1)

The quantity $\widetilde{K}(r,s;\tau)$ is given by the generating function

$\displaystyle\sum_{r,s\in\mathbb{Z}+\frac{1}{2}}\widetilde{K}(r,s;\tau)z^{r}\,w^{-s}:=\frac{\mathcal{J}(z;\tau)}{\mathcal{J}(w;\tau)}\frac{\sqrt{zw}}{z-w}\,,\quad\text{with
}|w|<|z|\,,$ (3.2)

and with $\mathcal{J}(z;\tau)$ defined as666We have specialized the definition
of $\mathcal{J}(z;\tau)$ to the GWW integral. The more general expression can
be found in [29, 28, 30].

$\displaystyle\mathcal{J}(z;\tau):=\exp\left(\tau(z-z^{-1})\right)=\sum_{n\in\mathbb{Z}}z^{n}J_{n}(2\tau)\,.$
(3.3)

Here $J_{n}$ denotes the Bessel-J function, and the last equality is the well
known generating function of the Bessel-J functions.

This Fredholm determinant expansion was used in [30] to give a convergent
“giant graviton” expansion for unitary matrix integrals that arise in the
computation of supersymmetric indices in gauge theories. Since giant gravitons
are related to wrapped D-branes, and D-instantons are related to eigenvalue
instantons, this suggests that the giant-graviton corrections to the index are
related to eigenvalue instantons.777We thank Steve Shenker for suggesting this
connection. The unitary matrix integrals arising in the computation of the
supersymmetric index differ from the GWW model in that they contain double-
trace terms in the action (of the form
$\operatorname{Tr}U\operatorname{Tr}U^{-1}$). While we have not succeeded in
finding an eigenvalue instanton interpretation for the giant graviton
expansion of these matrix integrals, we show in this section that for the GWW
model, which has a single trace action, the first Fredholm-determinant
correction in the ’t Hooft limit is precisely equal to the leading instanton
correction to the partition function (1.4).

As mentioned in the introduction, we always work in the strongly-coupled phase
of the GWW integral, $t>1$. This is because the denominator on the left side
of (3.1) refers to a limit where $\tau$ (and not $t$) is held fixed as
$N\to\infty$. Recall from equation (1.2) that $\tau=\frac{N}{2t}$. At this
stage of the analysis, it is possible that (3.1) is not relevant for the ’t
Hooft scaled model (one would need to take $t$ of order $N$), but we will see
that the $m=1$ term agrees with the first eigenvalue-instanton contribution in
the ’t Hooft limit for all $t>1$.

We can simplify the ratio of $\mathcal{J}$’s appearing on the right side of
(3.2) as

$\displaystyle\frac{\mathcal{J}(z;\tau)}{\mathcal{J}(w;\tau)}$
$\displaystyle=\exp(\tau(z-z^{-1}))\exp(\tau(w^{-1}-w))$ (3.4)
$\displaystyle=\sum_{n,m\in\mathbb{Z}}z^{n}w^{-m}J_{n}(2\tau)J_{m}(2\tau)\,,$
(3.5)

and get

$\displaystyle\frac{\mathcal{J}(z;\tau)}{\mathcal{J}(w;\tau)}\frac{\sqrt{zw}}{z-w}$
$\displaystyle=\sum_{n,m\in\mathbb{Z}}z^{n-\frac{1}{2}}w^{-m+\frac{1}{2}}J_{n}(2\tau)J_{m}(2\tau)\left(1+\frac{w}{z}+\frac{w^{2}}{z^{2}}+\ldots\right)\,.$
(3.6)

Comparing the left side of (3.2) and the right side of this last equation for
$r=s$, we find

$\displaystyle\widetilde{K}(r,r;\tau)=\sum_{n=r+\frac{1}{2}}^{\infty}J_{n}(2\tau)^{2}\,,\quad
r\in\mathbb{Z}+\frac{1}{2}\,.$ (3.7)

We will only be interested in the $m=1$ term in (3.1), which becomes

$\displaystyle\frac{Z_{N}(\tau)}{Z_{\infty}(\tau)}$
$\displaystyle=1-\sum_{r=N+\frac{1}{2}}^{\infty}\widetilde{K}(r,r;\tau)+(\text{terms
with $m\geq 2$})$ (3.8)
$\displaystyle=1-\sum_{k=1}^{\infty}k\,J_{N+k}(2\tau)^{2}+(\text{terms with
$m\geq 2$})\,.$ (3.9)

Note that the index of the Bessel functions appearing in the final expression
is large, greater than $N$. We would like this expression to reproduce (1.4),
including the action of the instanton and the one-loop determinant.

In order to do so, we need to work in the ’t Hooft limit, and thus consider
$\tau$ of order $N$. In this regime, both the index and the argument of the
Bessel-J functions appearing in (3.9) are large. Therefore, Debye’s expansion
of the Bessel-J function [51] is valid:

$\displaystyle
J_{N}\left(\frac{N}{\cosh\alpha}\right)=e^{-N(\alpha-\tanh\alpha)}\,(2\pi
N\tanh\alpha)^{-\frac{1}{2}}\left(1+O\left(\frac{1}{N}\right)\right)\,.$
(3.10)

The right side of this expression looks exactly like a saddle-point
contribution. Comparing this to (3.10) is encouraging. The on-shell action
seems to be just right, after setting $\cosh\alpha=t$:

$\displaystyle\alpha-\tanh\alpha=\operatorname{arccosh}(t)-\sqrt{1-t^{-2}}=S_{\text{strong}}(t)\,.$
(3.11)

However, we need to do some more work to get the correct one-loop factor. We
will get this from the infinite sum over the integer $k$ in (3.9), and the
results will match.

Let us delve into the details of the one-loop prefactor. Define $\alpha_{k}$
and $\beta_{k}$ using

$\displaystyle\frac{N}{t}=\frac{N+k}{\cosh\alpha_{k}}=(N+k)\,\beta_{k}\,.$
(3.12)

Now using Debye’s expansion, we get

$J_{N+k}\left(\frac{N}{t}\right)^{2}\approx\frac{1}{2\pi(N+k)\sqrt{1-\beta_{k}^{2}}}\left(\frac{\beta_{k}\,e^{\sqrt{1-\beta_{k}^{2}}}}{1+\sqrt{1-\beta_{k}^{2}}}\right)^{2N+2k}\,.$
(3.13)

When summing over $k$, we need to be careful about taking

$\beta_{k}=\frac{N/t}{N+k}\approx\frac{1}{t}\left(1-\frac{k}{N}\right).$
(3.14)

Since the right side of (3.13) involves a large exponent, corrections of order
$1/N$ inside the parentheses can yield order-one results. Note that the
approximation (3.14) only makes sense for $k\ll N$, but we will use it to do
the sum in (3.9). The corrections to this approximation become very important
at $k\sim N$, but then the corresponding terms become comparable to the
contribution from the second nontrivial term in the Fredholm determinant
expansion, and we are not working to that accuracy.

Being careful about the $1/N$ corrections to $\beta_{k}$ discussed above, we
obtain

$\displaystyle\frac{J_{N+k}(N/t)^{2}}{J_{N}(N/t)^{2}}$
$\displaystyle\approx\left(\frac{1}{t+\sqrt{t^{2}-1}}\right)^{2k}.$ (3.15)

Noting also from (3.10) that

$\displaystyle J_{N}\left(N/t\right)^{2}$ $\displaystyle\approx
e^{-2NS_{\text{strong}}(t)}\cdot\frac{1}{2\pi N\sqrt{1-1/t^{2}}}\,,$ (3.16)

we compute the contribution of one giant graviton (3.9) as

$\displaystyle\frac{Z(N,t)}{e^{N^{2}/4t^{2}}}-1$
$\displaystyle=-\sum_{k=1}^{\infty}k\,J_{N+k}(N/t)^{2}$ (3.17)
$\displaystyle\approx-e^{-2NS_{\text{strong}}(t)}\cdot\frac{1}{2\pi
N\sqrt{1-1/t^{2}}}\sum_{k=1}^{\infty}k\left(\frac{1}{t+\sqrt{t^{2}-1}}\right)^{2k}$
(3.18)
$\displaystyle=-e^{-2NS_{\text{strong}}(t)}\,\frac{1}{N}\,\frac{t}{8\pi(t^{2}-1)^{3/2}}\,.$
(3.19)

The result (3.19) agrees precisely with our target (1.4).

##  4 Summary and Discussion

In this paper we have analyzed the leading nonperturbative correction to the
Gross-Witten-Wadia integral (1.1) in the strong-coupling phase, $t>1$. In the
past, the result (1.4) was derived using techniques such as orthogonal
polynomials and by studying differential equations satisfied by various
observables [10, 15, 47].

In section 2, we showed that it is possible to derive (1.4), which is a two-
eigenvalue instanton effect, by expressing the GWW integral as a holomorphic
integral over its eigenvalues (2.3) and directly analyzing the instanton
effects in this integral. The integral expression for the two-eigenvalue
instanton contribution involving the effective potentials is given in (2.24),
correct to one-loop order. We also showed in appendix A.3 that there are no
one-eigenvalue instanton contributions to the partition function, and in
appendix A.2 that there are other observables such as $\left\langle\det
U\right\rangle$ that do receive contributions from one-eigenvalue instantons.
All these computations were done via a direct analysis of the integrals (1.1)
and (2.3), and involve eigenvalue tunneling to complex locations on the
unphysical sheet. Such instantons are called ghost instantons and their action
has the opposite sign compared to their partners on the physical sheet.

The GWW integral admits a double-scaling limit, and in this limit, it is dual
to the minimal superstring [13]. An interesting future goal is to reproduce
the precise nonperturbative corrections from ZZ-instantons in the minimal
superstring, generalizing previous work in the $c=1$ bosonic string [52, 53],
the $c<1$ bosonic string [54, 46] and the $\hat{c}=1$ type 0B string [55].
However, we note that the results of the present paper predict a very
surprising organization of the asymptotics of the minimal superstring
partition function. Usually, in perturbative string theory, the partition
function is described by the exponential of a perturbative genus expansion
associated to closed surfaces, potentially supplemented by nonperturbative
D-brane effects. In contrast to the usual case, in the minimal superstring
dual to the strong-coupling phase of the GWW model, the perturbative
contribution (sum over closed surfaces) to the partition function is entirely
absent. The leading asymptotics are instead associated to the exponential of
disk and cylinder diagrams for a pair of ghost ZZ branes. We are currently
investigating this problem [56].

In section 3, we showed that the first nontrivial term in the Fredholm
determinant expansion of the Toeplitz determinant (1.6), in the ’t Hooft limit
and to one-loop order, precisely reduces to the result (1.4). We only analyzed
the strong-coupling phase of the GWW model ($t>1$), but the weak-coupling
phase ($t<1$) also has eigenvalue instanton effects [10]. An open problem is
whether these can also be related to the Fredholm determinant expansion.

Our motivation for studying this connection was to provide an eigenvalue
instanton interpretation for the giant-graviton expansion studied in the
context of supersymmetric index computations in [30]. The unitary matrix
integrals in [30] have double trace terms in the action, and these are related
to matrix integrals with a single-trace action (like the GWW model) via a
Hubbard-Stratonovich transform. We leave it for future work to study the fate
of the eigenvalue instantons after doing a Hubbard-Stratonovich transform.
However, we are tempted to make some preliminary remarks about this problem.

For this, we consider a simple integral with the action containing a single
double-trace term [57].

$\displaystyle
Z(N,a_{1}):=\int\frac{\mathrm{d}U}{\operatorname{vol}U(N)}\,\exp\left(a_{1}\operatorname{Tr}U\operatorname{Tr}U^{-1}\right)\,.$
(4.1)

This model has a phase transition at $a_{1}=1$. We work in the phase with
$a_{1}<1$ for which the eigenvalue density is ungapped i.e., supported on the
entire unit circle. We first write this integral as Hubbard-Stratonovich
transform of the GWW integral and then use the Fredholm determinant expansion
of the GWW integral. Directly, we can use Theorem 5.2 in [30], specialized to
the present case, to conclude that

$\displaystyle
Z(N,a_{1})=\frac{1}{1-a_{1}}\left(1-\frac{a_{1}^{N+1}}{(N+1)!}+O(a_{1}^{N+2})\right)\,.$
(4.2)

Note that the leading correction is of order $e^{-N\log N}$, which is
parametrically smaller than what is expected from eigenvalue instantons (which
should be of the form $e^{-cN}$).

Now, the unitary matrix integrals that show up in the computation of
supersymmetric indices have an infinite number of double-trace couplings
turned on. It is possible that models with an infinite number of terms behave
qualitatively differently than this simple “$a_{1}$-model.” We leave it for
future work to explore this further.

#### Acknowledgments.

We would like to thank Abhijit Gadde, Ohad Mamroud, Sameer Murthy, Steve
Shenker, Douglas Stanford, and Mykhaylo Usatyuk for discussions. D.S.E. is
supported by the Shoucheng Zhang Graduate Fellowship. C.M. is supported in
part by the U.S. Department of Energy, Office of Science, Office of High
Energy Physics under QuantISED Award DE-SC0019380 and contract DE-
AC02-05CH11231.

##  Appendix A Further details about eigenvalue instanton computations

###  A.1 Some perturbative correlators

In this subsection, we explain the computation of $\left\langle
e^{\operatorname{Tr}\log\big{[}(1-z_{1}^{-1}U)^{2}(1-z_{2}U^{-1})^{2}\big{]}}\right\rangle_{(N,t)}$
that is needed in (2.23). From now on, we will suppress the $(N,t)$ subscript
on the expectation value. Also recall that we would like to define this
correlator initially for $|z_{1}|>1$ and $|z_{2}|<1$, and then analytically
continue the expression obtained.

Using the identity $\left\langle e^{X}\right\rangle=\exp\left[\left\langle
X\right\rangle+\frac{1}{2}\left\langle XX\right\rangle_{c}+\ldots\right]$, we
have

$\left\langle
e^{\operatorname{Tr}\log\big{[}(1-z_{1}^{-1}U)^{2}(1-z_{2}U^{-1})^{2}\big{]}}\right\rangle\approx\exp\Big{[}2\left\langle\operatorname{Tr}\log(1-z_{1}^{-1}U)\right\rangle+2\left\langle\operatorname{Tr}\log(1-z_{2}U^{-1})\right\rangle\\\
+2\left\langle\operatorname{Tr}\log(1-z_{1}^{-1}U)\operatorname{Tr}\log(1-z_{1}^{-1}U)\right\rangle_{c}+2\left\langle\operatorname{Tr}\log(1-z_{2}U^{-1})\operatorname{Tr}\log(1-z_{2}U^{-1})\right\rangle_{c}\\\
+4\left\langle\operatorname{Tr}\log(1-z_{1}^{-1}U)\operatorname{Tr}\log(1-z_{2}U^{-1})\right\rangle_{c}\Big{]}\,.$
(A.1)

Let us now evaluate the various correlators present on the right side. The
starting point for these computations is the single-trace correlator

$\left\langle\operatorname{Tr}U^{j}\right\rangle=\begin{cases}N\,,&j=0\,,\\\
\frac{N}{2t}\,,&j=\pm 1\,,\\\ 0\,,&\text{otherwise,}\end{cases}$ (A.2)

and the double-trace connected correlator

$\left\langle\operatorname{Tr}U^{j}\operatorname{Tr}U^{k}\right\rangle_{c}=|j|\,\delta_{j,-k}\,.$
(A.3)

These are evaluated by expanding the exponential in the GWW measure (1.1) as
in the computation of $Z^{(0)}(N,t)$ in (2.8), and using the result (2.5).

The result (A.2) allows us to compute the single-trace correlators appearing
in (A.1) by doing a Taylor series expansion

$\displaystyle\left\langle\operatorname{Tr}\log(1-zU^{-1})\right\rangle$
$\displaystyle=-\sum_{j=1}^{\infty}\frac{1}{j}z^{j}\left\langle\operatorname{Tr}U^{-j}\right\rangle=-\frac{N}{2t}z\,,$
$\displaystyle|z|<1\,,$ (A.4)
$\displaystyle\left\langle\operatorname{Tr}\log(1-z^{-1}U)\right\rangle$
$\displaystyle=-\sum_{j=1}^{\infty}\frac{1}{j}z^{-j}\left\langle\operatorname{Tr}U^{j}\right\rangle=-\frac{N}{2t}\frac{1}{z}\,,$
$\displaystyle|z|>1\,.$ (A.5)

Using the result (A.3), we see that two of the double-trace connected
correlators appearing in (A.1) vanish:

$\displaystyle\left\langle\operatorname{Tr}\log(1-z_{1}U^{-1})\operatorname{Tr}\log(1-z_{2}U^{-1})\right\rangle_{c}$
$\displaystyle=\sum_{j,k=1}^{\infty}\frac{1}{jk}z_{1}^{j}z_{2}^{k}\left\langle\operatorname{Tr}U^{-j}\operatorname{Tr}U^{-k}\right\rangle_{c}=0\,,$
$\displaystyle|z_{1}|,|z_{2}|<1\,,$ (A.6)
$\displaystyle\left\langle\operatorname{Tr}\log(1-z_{1}^{-1}U)\operatorname{Tr}\log(1-z_{2}^{-1}U)\right\rangle_{c}$
$\displaystyle=\sum_{j,k=1}^{\infty}\frac{1}{jk}z_{1}^{-j}z_{2}^{-k}\left\langle\operatorname{Tr}U^{j}\operatorname{Tr}U^{k}\right\rangle_{c}=0\,,$
$\displaystyle|z_{1}|,|z_{2}|>1\,,$ (A.7)

while the third one is

$\displaystyle\left\langle\operatorname{Tr}\log(1-z_{1}^{-1}U)\operatorname{Tr}\log(1-z_{2}U^{-1})\right\rangle_{c}$
$\displaystyle=\sum_{j,k=1}^{\infty}\frac{1}{jk}z_{2}^{j}z_{1}^{-k}\left\langle\operatorname{Tr}U^{-j}\operatorname{Tr}U^{k}\right\rangle_{c}$
$\displaystyle=-\log(1-z_{2}/z_{1})\,,\quad\quad\quad|z_{1}|>1\,,|z_{2}|<1\,.$
(A.8)

Substituting the results (A.4)-(A.8) into (A.1), we get the final result

$\left\langle
e^{\operatorname{Tr}\log\big{[}(1-z_{1}^{-1}U)^{2}(1-z_{2}U^{-1})^{2}\big{]}}\right\rangle_{(N,t)}\approx\exp\left[-\frac{N}{t}\frac{1}{z_{1}}-\frac{N}{t}z_{2}-4\log(1-z_{2}/z_{1})\right]\,.$
(A.9)

This is the correlator that we needed for the computation of $Z^{(1,1)}(N,t)$
in (2.23).

###  A.2 Expectation value of $\det U$

Even though there is no one-instanton contribution to the partition function,
there are other observables that do receive contributions from single
instantons. One such quantity is $\left\langle\det U\right\rangle$, which we
analyze in this appendix. This was also discussed in [47] from the perspective
of orthogonal polynomials. Our goal is to redo the computation directly in
terms of a one-eigenvalue instanton in the integral (2.3).

It is known that the quantity $\left\langle\det U\right\rangle$ vanishes to
all orders perturbation theory in the strong-coupling phase of the GWW model
[15, 49, 48]. As usual, one way to see this is to series expand the
exponential in the defining integral (1.1) and then expand
$(\operatorname{Tr}U+\operatorname{Tr}U^{-1})^{n}$ via the binomial theorem. A
generic term will look like

$\left(\frac{N}{2t}\right)^{n}\frac{1}{k!(n-k)!}\int\frac{\mathrm{d}U}{\operatorname{vol}U(N)}\,\det
U(\operatorname{Tr}U)^{k}(\operatorname{Tr}U^{-1})^{n-k}\,,$ (A.10)

with $0\leq k\leq n$. Making a change of variables $U\to
e^{\mathrm{i}\alpha}U$, we see that we must have $N=n-2k$ in order to get a
nonzero result. In particular, this means that $n\geq N$. The first nonzero
term in the series (with $n=N,k=0$) is thus exponentially small in $N$. See
also the discussion below (A.13).

In this subsection, we show that the leading contribution to $\left\langle\det
U\right\rangle$ in the strong-coupling phase comes from a one-eigenvalue
instanton. Pulling out one eigenvalue from the cut, and placing it at some
position $z_{1}$ outside the unit circle, we have the expression

$\displaystyle\left\langle\det U\right\rangle_{(N,t)}^{(1)}$
$\displaystyle=\frac{1}{Z^{(0)}(N,t)}\times\frac{1}{(N-1)!}(-1)^{N(N-1)/2}\int_{\mathcal{C}_{1}}\frac{\mathrm{d}z_{1}}{2\pi\mathrm{i}z_{1}^{N}}e^{-\frac{N}{t}V(z_{1})}$
$\displaystyle\hskip
36.135pt\times\int_{\mathcal{C}_{0}}\prod_{i=2}^{N}\frac{\mathrm{d}z_{i}}{2\pi\mathrm{i}z_{i}^{N}}e^{-\frac{N}{t}\sum_{i=2}^{N}V(z_{i})}\times\prod_{j=2}^{N}(z_{1}-z_{j})^{2}\times\prod_{2\leq
j<k\leq N}(z_{j}-z_{k})^{2}\times\prod_{j=1}^{N}z_{j}$
$\displaystyle=\frac{1}{Z^{(0)}(N,t)}\times\frac{1}{(N-1)!}(-1)^{N(N-1)/2}\int_{\mathcal{C}_{1}}\frac{\mathrm{d}z_{1}}{2\pi\mathrm{i}z_{1}^{N-1}}e^{-\frac{N}{t}V(z_{1})}$
$\displaystyle\hskip
36.135pt\times\int_{\mathcal{C}_{0}}\prod_{i=2}^{N}\frac{\mathrm{d}z_{i}}{2\pi\mathrm{i}z_{i}^{N-1}}e^{-\frac{N}{t}\sum_{i=2}^{N}V(z_{i})}\times\prod_{j=2}^{N}\big{[}z_{1}^{2}(1-z_{j}/z_{1})^{2}\big{]}\times\prod_{2\leq
j<k\leq N}(z_{j}-z_{k})^{2}$
$\displaystyle=(-1)^{N-1}\int_{\mathcal{C}_{1}}\frac{\mathrm{d}z_{1}}{2\pi\mathrm{i}z_{1}^{N-1}}e^{-\frac{N}{t}V(z_{1})}\times
z_{1}^{2(N-1)}\times\left\langle
e^{2\operatorname{Tr}\log(1-z_{1}^{-1}U)}\right\rangle_{(N-1,t-\frac{t}{N})}$
$\displaystyle\approx(-1)^{N-1}\int_{\mathcal{C}_{1}}\frac{\mathrm{d}z_{1}}{2\pi\mathrm{i}}e^{-\frac{N}{t}V(z_{1})}\times
z_{1}^{(N-1)}\times e^{-\frac{N}{t}\frac{1}{z_{1}}}$
$\displaystyle=-\int_{\mathcal{C}_{1}}\frac{\mathrm{d}z_{1}}{2\pi\mathrm{i}z_{1}}e^{-\frac{N}{t}V_{\text{eff}}^{+}(z_{1})}.$
(A.11)

We used the fact that
$Z^{(0)}\left(N-1,t-\frac{t}{N}\right)=Z^{(0)}\left(N,t\right)$, We have used
the results (A.5) and (A.7) to get the expression in the second-to-last line.
In the last line, we have used (2.17).

As in the case of the partition function itself, even though we started by
considering $z_{1}$ to be outside the unit circle, the relevant saddle point
lies inside the unit circle. The defining integration contour is such that we
need to pick the orientation of the steepest-descent contour as being
vertically _upwards_ through $z_{1}^{\star}$. Doing a saddle point evaluation
at $z_{1}=z_{1}^{\star}$, we get

$\left\langle\det
U\right\rangle_{(N,t)}^{(1)}\approx\frac{1}{2\pi(-z_{1}^{\star})}e^{-\frac{N}{t}V_{\text{eff}}^{+}(z_{1}^{\star})}\sqrt{\frac{2\pi
t}{N|\partial^{2}V_{\text{eff}}^{+}(z_{1}^{\star})|}}=e^{-NS_{\text{strong}}(t)}\frac{1}{\sqrt{N}}\,\frac{\sqrt{t/2\pi}}{(t^{2}-1)^{1/4}}.$
(A.12)

This is indeed the known leading contribution to
$\left\langle\det{U}\right\rangle$ in the strong-coupling phase of the GWW
model [15].

We can do a quick consistency check of this result, which we have derived from
saddle-point analysis of the integral over eigenvalues. There is a known
relationship between $\left\langle\det U\right\rangle_{N,t}$ and the partition
function [10, 15]

$\displaystyle\left\langle\det
U\right\rangle_{(N,t)}^{2}=1-\frac{Z(N+1,t+\frac{t}{N})\,Z(N-1,t-\frac{t}{N})}{Z(N,t)^{2}}\,.$
(A.13)

First note that we can use (A.13) and the result
$Z^{(0)}(N,t)=\exp(N^{2}/4t^{2})$ to show that $\left\langle\det
U\right\rangle$ vanishes to all orders in perturbation theory (when $t>1$).
Second, it is straightforward to plug in (A.12) on the left side and (1.4) on
the right side and verify the equality (A.13).

###  A.3 The putative one-eigenvalue instanton contribution to $Z(N,t)$

In this subsection, we show that there are no one-eigenvalue instanton
contributions (of order $e^{-NS_{\text{strong}}(t)}$) to $Z(N,t)$ in the
strong-coupling phase. If we try to pull out one eigenvalue from the cut, we
shall see that we are forced to pull out a second one as well in order to get
a nonzero result.

This result is known from previous analyses [10, 47, 15] via other methods. In
fact, it is also true in the weak-coupling phase that the leading instanton
correction to the perturbation series for the partition function is a two-
instanton effect [10].888In the weak-coupling phase, the eigenvalue instanton
is a maximum of the effective potential that lies on the unit circle, which is
the original integration contour. There is a general argument that such saddle
points cannot contribute to the integral. A quick way to see this is that it
would yield a purely imaginary contribution to the partition function, which
is ruled out since the partition function is purely real. In an analysis using
the Borel transform, this is attributed to median resummation, or the
cancellation of nonperturbative ambiguities [10]. Our goal is to see this
explicitly via an analysis of the eigenvalue instantons in the integral (2.3).

Pulling out just one eigenvalue $z_{1}$ and placing it outside the unit
circle, the putative one-eigenvalue instanton contribution to $Z(N,t)$ is
given by

$Z^{(1)}(N,t)=\frac{1}{(N-1)!}(-1)^{N(N-1)/2}\int_{\mathcal{C}_{1}}\frac{\mathrm{d}z_{1}}{2\pi\mathrm{i}z_{1}^{N}}e^{-\frac{N}{t}V(z_{1})}\\\
\times\int_{\mathcal{C}_{0}}\prod_{i=2}^{N}\frac{\mathrm{d}z_{i}}{2\pi\mathrm{i}z_{i}^{N}}e^{-\frac{N}{t}\sum_{i=2}^{N}V(z_{i})}\times\prod_{j=2}^{N}(z_{1}-z_{j})^{2}\times\prod_{2\leq
j<k\leq N}(z_{j}-z_{k})^{2}.$ (A.14)

We can rewrite the above expression as

$Z^{(1)}(N,t)=\frac{1}{(N-1)!}(-1)^{N(N-1)/2}\int_{\mathcal{C}_{1}}\frac{\mathrm{d}z_{1}}{2\pi\mathrm{i}}\,z_{1}^{N-2}e^{-\frac{N}{t}V(z_{1})}\\\
\times\int_{\mathcal{C}_{0}}\prod_{i=2}^{N}\frac{\mathrm{d}z_{i}}{2\pi\mathrm{i}z_{i}^{N-1}}e^{-\frac{N}{t}\sum_{i=2}^{N}V(z_{i})}\times\prod_{j=2}^{N}\big{[}z_{j}^{-1}(1-z_{j}/z_{1})^{2}\big{]}\times\prod_{2\leq
j<k\leq N}(z_{j}-z_{k})^{2}.$ (A.15)

Now we do the $z_{2},z_{3},\ldots,z_{N}$ integrals to get

$Z^{(1)}(N,t)=(-1)^{N-1}\int_{\mathcal{C}_{1}}\frac{\mathrm{d}z_{1}}{2\pi\mathrm{i}}z_{1}^{N-2}e^{-\frac{N}{t}V(z_{1})}\times
Z^{(0)}\left(N-1,t-\frac{t}{N}\right)\left\langle\det
U^{-1}\det(1-z_{1}^{-1}U)^{2}\right\rangle_{(N-1,t-t/N)}.$ (A.16)

Using the fact that
$Z^{(0)}\left(N-1,t-\frac{t}{N}\right)=Z^{(0)}\left(N,t\right)$, we get

$\displaystyle\frac{Z^{(1)}(N,t)}{Z^{(0)}(N,t)}=(-1)^{N-1}\int_{\mathcal{C}_{1}}\frac{\mathrm{d}z_{1}}{2\pi\mathrm{i}}z_{1}^{N-2}e^{-\frac{N}{t}V(z_{1})}\left\langle\det
U^{-1}\det(1-z_{1}^{-1}U)^{2}\right\rangle_{(N-1,t-t/N)}.$ (A.17)

Similar to $\left\langle\det U\right\rangle$, the expectation value
$\left\langle\det U^{-1}\det(1-z^{-1}U)^{2}\right\rangle$ vanishes at the
perturbative order. We can show this by expanding the second term as a power
series in $z^{-1}$

$\begin{split}\left\langle\det
U^{-1}\det(1-z^{-1}U)^{2}\right\rangle_{(N,t)}&=\sum_{k=0}^{\infty}\frac{2^{k}}{k!}\left\langle\det
U^{-1}\left(-\sum_{j=1}^{\infty}\frac{z^{-j}}{j}\operatorname{Tr}U^{j}\right)^{k}\right\rangle_{(N,t)}\,.\end{split}$
(A.18)

Note that the coefficients of $z^{-m}$ vanish for $m>2N$, since
$\det(1-z^{-1}U)$ is a polynomial in $z^{-1}$ of degree $N$. Using an argument
similar to the one for $\left\langle\det U\right\rangle$ in appendix A.2, we
can prove that this quantity is exponentially small in $N$. Thus, the
perturbative contribution to $\left\langle\det
U^{-1}\det(1-z^{-1}U)^{2}\right\rangle$ vanishes, and we now look for
eigenvalue instanton contributions to this quantity.

The leading contribution to $\left\langle\det
U^{-1}\det(1-z^{-1}U)^{2}\right\rangle$ comes from a one-eigenvalue instanton,
just like $\left\langle\det U\right\rangle$. Plugging this one-eigenvalue
instanton contribution back into (A.17) will result in an expression identical
to (2.24), which is a two-instanton effect. Thus, this method yields an
alternative derivation of the result (1.4).

Pulling out one eigenvalue from the cut, and placing it at some position
$z_{1}$ inside the unit circle, we have the expression

$\displaystyle\left\langle\det
U^{-1}\det(1-z^{-1}U)^{2}\right\rangle^{(1)}_{(N,t)}=\frac{1}{Z^{(0)}(N,t)}\times\frac{(-1)^{N(N-1)/2}}{(N-1)!}\int_{\mathcal{C}_{2}}\frac{\mathrm{d}z_{1}}{2\pi\mathrm{i}z_{1}^{N+1}}e^{-\frac{N}{t}V(z_{1})}(1-z_{1}/z)^{2}$
$\displaystyle\hskip
14.45377pt\times\int_{\mathcal{C}_{0}}\prod_{i=2}^{N}\frac{\mathrm{d}z_{i}}{2\pi\mathrm{i}z_{i}^{N-1}}e^{-\frac{N}{t}\sum_{i=2}^{N}V(z_{i})}\times\prod_{j=2}^{N}\big{[}(1-z_{1}/z_{j})^{2}(1-z_{j}/z)^{2}\big{]}\times\prod_{2\leq
j<k\leq N}(z_{j}-z_{k})^{2}$
$\displaystyle=(-1)^{N-1}\int_{\mathcal{C}_{2}}\frac{\mathrm{d}z_{1}}{2\pi\mathrm{i}z_{1}^{N+1}}e^{-\frac{N}{t}V(z_{1})}(1-z_{1}/z)^{2}\times\left\langle
e^{2\operatorname{Tr}\log(1-z^{-1}U)+2\operatorname{Tr}\log(1-z_{1}U^{-1})}\right\rangle_{(N-1,t-\frac{t}{N})}$
$\displaystyle\approx(-1)^{N-1}\int_{\mathcal{C}_{2}}\frac{\mathrm{d}z_{1}}{2\pi\mathrm{i}z_{1}^{N+1}}e^{-\frac{N}{t}V(z_{1})}(1-z_{1}/z)^{2}\times\frac{e^{-\frac{N}{t}\frac{1}{z}-\frac{N}{t}z_{1}}}{(1-z_{1}/z)^{4}}$
$\displaystyle=-\int_{\mathcal{C}_{2}}\frac{\mathrm{d}z_{1}}{2\pi\mathrm{i}}e^{-\frac{N}{t}V_{\text{eff}}^{-}(z_{1})}e^{-\frac{N}{t}\frac{1}{z}}\frac{1}{z_{1}(1-z_{1}/z)^{2}}\,.$
(A.19)

We have used (A.4), (A.5) and (A.8) to get to the second-to-last line.

In (A.17), we need $\left\langle\det
U^{-1}\det(1-z^{-1}U)^{2}\right\rangle^{(1)}_{(N-1,t-t/N)}$, with shifted
values of $N$ and $t$. Note that $V_{\text{eff}}^{-}(z_{1})$ given in (2.18)
has an explicit dependence on $t$. The required shift $t\to t-t/N$ nicely
absorbs the $1/z_{1}$ factor present in the integrand in (A.19), so we get

$\left\langle\det
U^{-1}\det(1-z^{-1}U)^{2}\right\rangle_{(N-1,t-t/N)}^{(1)}=\int_{\mathcal{C}_{2}}\frac{\mathrm{d}z_{2}}{2\pi\mathrm{i}}e^{-\frac{N}{t}V_{\text{eff}}^{-}(z_{2})}e^{-\frac{N}{t}\frac{1}{z}}\frac{1}{(1-z_{2}/z)^{2}}\,,$
(A.20)

where we have relabeled the integration variable as $z_{2}$. Substituting this
into (A.17), we get

$\displaystyle Z^{(1,1)}(N,t)$
$\displaystyle=(-1)^{N-1}\int_{\mathcal{C}_{1}}\frac{\mathrm{d}z_{1}}{2\pi\mathrm{i}}z_{1}^{N-2}e^{-\frac{N}{t}V(z_{1})}\int_{\mathcal{C}_{2}}\frac{\mathrm{d}z_{2}}{2\pi\mathrm{i}}e^{-\frac{N}{t}V_{\text{eff}}^{-}(z_{2})}e^{-\frac{N}{t}\frac{1}{z_{1}}}\frac{1}{(1-z_{2}/z_{1})^{2}}$
$\displaystyle=-\int_{\mathcal{C}_{1}}\frac{\mathrm{d}z_{1}}{2\pi\mathrm{i}}\int_{\mathcal{C}_{2}}\frac{\mathrm{d}z_{2}}{2\pi\mathrm{i}}e^{-\frac{N}{t}V_{\text{eff}}^{+}(z_{1})-\frac{N}{t}V_{\text{eff}}^{-}(z_{2})}\frac{1}{(z_{1}-z_{2})^{2}}\,.$
(A.21)

This is now a two-eigenvalue instanton contribution, which we have indicated
by changing $Z^{(1)}(N,t)$ to $Z^{(1,1)}(N,t)$. This equation is exactly
(2.24) and thus gives us an alternative derivation of (1.4), the leading
nonperturbative correction to $Z(N,t)$ in the strong-coupling phase.

##  Appendix B Ghost-instanton contribution from tunneling anti-eigenvalues

A key insight of [22] is the fact that the ghost-instanton contributions to a
matrix integral can be described in two equivalent ways: either as the
tunneling of regular eigenvalues to the unphysical sheet of the spectral
curve, or by upgrading the matrix integral to a supermatrix integral,999We
will exclusively use the “physical supermatrix integral.” The physical
supermatrix integral is defined explicitly in terms of an integral over the
eigenvalues of the supermatrix, and should be contrasted with the “ordinary
supermatrix integral,” which plays no role here. See [22] and [58] for details
and equation (B) for the precise definition we will use. and tunneling a
corresponding number of negative-charge eigenvalues to the physical sheet. The
equivalence between these two pictures is summarized by equations (3.13) and
(3.21) of [22] for the single-anti-eigenvalue contributions and for the
multiple-anti-eigenvalue contributions, respectively.

The goal of this appendix is to reproduce the leading ghost-instanton
contribution to the GWW matrix integral in the strong-coupling phase in the
supermatrix picture. This complements the derivation in section 2 of the same
result, equation (2.25), which was obtained by tunneling regular eigenvalues
to the opposite sheet. Since we will be interested in the contribution from
one negative-charge-eigenvalue tunneling to $z_{1}^{\star}$ and one negative-
charge-eigenvalue tunneling to $z_{2}^{\star}$, the starting point will be a
unitary $(N+2|2)\times(N+2|2)$ supermatrix integral, defined in terms of the
integral over its eigenvalues101010In the following, the notation $\bar{z}$ is
used to denote a negative-charge-eigenvalue of the supermatrix. It does not
denote complex conjugation. as:

$\displaystyle Z\left(N+2|2\right)$
$\displaystyle:=\frac{1}{(N+2)!\;2!}\text{PV}\int_{\mathcal{C}_{0}}\prod_{i=1}^{N+2}\frac{\mathrm{d}z_{i}}{2\pi\mathrm{i}z_{i}}\int_{\bar{\mathcal{C}}_{0}}\prod_{j=1}^{2}\frac{\mathrm{d}\bar{z}_{j}}{2\pi\mathrm{i}\bar{z}_{j}}\;\Delta_{N+2|2}(z,\bar{z})^{2}\times$
$\displaystyle\hskip
108.405pt\times\exp\left(-\frac{N}{t}\sum_{i=1}^{N+2}V\left(z_{i}\right)+\frac{N}{t}\sum_{j=1}^{2}V\left(\bar{z}_{j}\right)\right),$
(B.1)

where we define the super-Vandermonde determinant

$\Delta_{N+2|2}(z,\bar{z})=\frac{\left|\bar{z}_{1}-\bar{z}_{2}\right|\prod_{1\leq
i<j\leq
N+2}\left|z_{i}-z_{j}\right|}{\prod_{i=1}^{N+2}\left|\bar{z}_{1}-z_{i}\right|\left|\bar{z}_{2}-z_{i}\right|}\,,$
(B.2)

and the PV denotes a particular regularization prescription required to render
the integral finite [22]. In the Hermitian case, this prescription corresponds
to the Cauchy principal value (see [22] for details). In the unitary case, a
definition was provided in [43]. Since we are solely interested in using the
supermatrix integral description to obtain nonperturbative corrections to an
ordinary matrix integral, the choice of prescription does not play a role, so
we will not attempt to specify it in the present unitary case. We can use the
identity

$|z_{j}-z_{k}|^{2}=-\frac{1}{z_{j}z_{k}}(z_{j}-z_{k})^{2}\,,$ (B.3)

which is valid on the unit circle, to cast the supermatrix integral in a
holomorphic form

$Z\left(N+2|2\right)=-\frac{(-1)^{(N+2)(N+1)/2}}{(N+2)!\;2!}\text{PV}\int_{\mathcal{C}_{0}}\prod_{i=1}^{N+2}\frac{\mathrm{d}z_{i}}{2\pi\mathrm{i}z_{i}^{N}}\int_{\bar{\mathcal{C}}_{0}}\prod_{j=1}^{2}\frac{\bar{z}_{j}^{N}\mathrm{d}\bar{z}_{j}}{2\pi\mathrm{i}}\times\\\
\times\frac{\left(\bar{z}_{1}-\bar{z}_{2}\right)^{2}\prod_{1\leq i<j\leq
N+2}\left(z_{i}-z_{j}\right)^{2}}{\prod_{i=1}^{N+2}\left(\bar{z}_{1}-z_{i}\right)^{2}\left(\bar{z}_{2}-z_{i}\right)^{2}}\times\exp\left(-\frac{N}{t}\sum_{i=1}^{N+2}V\left(z_{i}\right)+\frac{N}{t}\sum_{j=1}^{2}V\left(\bar{z}_{j}\right)\right)\,.$
(B.4)

The leading ghost-instanton correction to the $N\times N$ GWW matrix integral
will be obtained from the above expression after modifying the contours of
integration for the two anti-eigenvalues to pass through $z_{1}^{\star}$ and
$z_{2}^{\star}$, respectively. Since there are two ways to choose which anti-
eigenvalue tunnels to which contour, amounting to an overall factor of two,
the leading contribution is

$Z^{(1,1)}(N,t)=-\frac{(-1)^{(N+2)(N+1)/2}}{(N+2)!}\text{PV}\int_{\bar{\mathcal{C}}_{1}^{\star}}\frac{\bar{z}_{1}^{N}\mathrm{d}\bar{z}_{1}}{2\pi\mathrm{i}}\int_{\bar{\mathcal{C}}_{2}^{\star}}\frac{\bar{z}_{2}^{N}\mathrm{d}\bar{z}_{2}}{2\pi\mathrm{i}}\left(\bar{z}_{1}-\bar{z}_{2}\right)^{2}\exp\left(\frac{N}{t}\sum_{j=1}^{2}V\left(\bar{z}_{j}\right)\right)\times\\\
\times\int_{\mathcal{C}_{0}}\prod_{i=1}^{N+2}\frac{\mathrm{d}z_{i}}{2\pi\mathrm{i}z_{i}^{N}}\;\frac{\prod_{1\leq
i<j\leq
N+2}\left(z_{i}-z_{j}\right)^{2}}{\prod_{i=1}^{N+2}\left(\bar{z}_{1}-z_{i}\right)^{2}\left(\bar{z}_{2}-z_{i}\right)^{2}}\exp\left(-\frac{N}{t}\sum_{i=1}^{N+2}V\left(z_{i}\right)\right).$
(B.5)

Writing $U=\text{diag}\left(z_{i}\right)_{i=1,\dots,N+2}$, and noting that

$\displaystyle\prod_{i=1}^{N+2}\left(\bar{z}_{1}-z_{i}\right)^{2}$
$\displaystyle=\det U^{2}\det\left(1-\bar{z}_{1}U^{-1}\right)^{2},$ (B.6)
$\displaystyle\prod_{i=1}^{N+2}\left(\bar{z}_{2}-z_{i}\right)^{2}$
$\displaystyle=\bar{z}_{2}^{2(N+2)}\det\left(1-\bar{z}_{2}^{-1}U\right)^{2},$
(B.7)

we find

$Z^{(1,1)}(N,t)=-\frac{(-1)^{(N+2)(N+1)/2}}{(N+2)!}\text{PV}\int_{\bar{\mathcal{C}}_{1}^{\star}}\frac{\bar{z}_{1}^{N}\mathrm{d}\bar{z}_{1}}{2\pi\mathrm{i}}\int_{\bar{\mathcal{C}}_{2}^{\star}}\frac{\bar{z}_{2}^{N}\mathrm{d}\bar{z}_{2}}{2\pi\mathrm{i}}\left(\bar{z}_{1}-\bar{z}_{2}\right)^{2}\exp\left(\frac{N}{t}\sum_{j=1}^{2}V\left(\bar{z}_{j}\right)\right)\times\\\
\times\int_{\mathcal{C}_{0}}\prod_{i=1}^{N+2}\frac{\mathrm{d}z_{i}}{2\pi\mathrm{i}z_{i}^{N+2}}\;\frac{\prod_{1\leq
i<j\leq
N+2}\left(z_{i}-z_{j}\right)^{2}}{\det\left(1-\bar{z}_{1}U^{-1}\right)^{2}\cdot\bar{z}_{2}^{2(N+2)}\det\left(1-\bar{z}_{2}^{-1}U\right)^{2}}\exp\left(-\frac{N}{t}\sum_{i=1}^{N+2}V\left(z_{i}\right)\right).$
(B.8)

More compactly, we have

$Z^{(1,1)}(N,t)=-\text{PV}\int_{\bar{\mathcal{C}}_{1}^{\star}}\frac{\mathrm{d}\bar{z}_{1}}{2\pi\mathrm{i}}\int_{\bar{\mathcal{C}}_{2}^{\star}}\frac{\mathrm{d}\bar{z}_{2}}{2\pi\mathrm{i}}\;\bar{z}_{1}^{N}\bar{z}_{2}^{-N-4}\left(\bar{z}_{1}-\bar{z}_{2}\right)^{2}\exp\left(\frac{N}{t}\sum_{j=1}^{2}V\left(\bar{z}_{j}\right)\right)\times\\\
\times
Z^{(0)}\left(N+2,t+2t/N\right)\left\langle\frac{1}{\det\left(1-\bar{z}_{1}U^{-1}\right)^{2}\det\left(1-\bar{z}_{2}^{-1}U\right)^{2}}\right\rangle_{(N+2,t+2t/N)}.$
(B.9)

We can also express the inverse determinants as exponentials of traces, and
use the relations derived in appendix A to obtain:

$\displaystyle\left\langle\frac{1}{\det\left(1-\bar{z}_{1}U^{-1}\right)^{2}\det\left(1-\bar{z}_{2}^{-1}U\right)^{2}}\right\rangle$
$\displaystyle=\left\langle\exp\left[-2\operatorname{Tr}\log\left(1-\bar{z}_{1}U^{-1}\right)-2\operatorname{Tr}\log\left(1-\bar{z}_{2}^{-1}U\right)\right]\right\rangle$
$\displaystyle\approx\exp\Big{[}-2\left\langle\operatorname{Tr}\log\left(1-\bar{z}_{1}U^{-1}\right)+\operatorname{Tr}\log\left(1-\bar{z}_{2}^{-1}U\right)\right\rangle+4\left\langle\operatorname{Tr}\log\left(1-\bar{z}_{1}U^{-1}\right)\operatorname{Tr}\log\left(1-\bar{z}_{2}^{-1}U\right)\right\rangle_{c}\Big{]}$
$\displaystyle=\exp\left[\frac{N}{t}\left(\bar{z}_{1}+\frac{1}{\bar{z}_{2}}\right)-4\log\left(1-\frac{\bar{z}_{1}}{\bar{z}_{2}}\right)\right],$
(B.10)

where, in the second line, we used the identity $\left\langle
e^{X}\right\rangle=\exp\left[\left\langle
X\right\rangle+\frac{1}{2}\left\langle XX\right\rangle_{c}+\ldots\right]$. The
previous equation, along with the expressions for the effective potential
inside and outside the unit circle, which we reproduce here for convenience,

$\displaystyle V_{\text{eff}}^{+}\left(\bar{z}_{2}\right)$
$\displaystyle=-\frac{\bar{z}_{2}}{2}+\frac{1}{2\bar{z}_{2}}-t\log(-\bar{z}_{2}),$
(B.11) $\displaystyle V_{\text{eff}}^{-}\left(\bar{z}_{1}\right)$
$\displaystyle=\frac{\bar{z}_{1}}{2}-\frac{1}{2\bar{z}_{1}}+t\log(-\bar{z}_{1}),$
(B.12)

and the fact that $Z^{(0)}(N+2,t+2t/N)=Z^{(0)}(N,t)$ allow us to
write:111111Note that the regularization prescription no longer plays a role
since the contours for $\bar{z}_{1}$ and $\bar{z}_{2}$ do not intersect.

$\frac{Z^{(1,1)}(N,t)}{Z^{(0)}(N,t)}=-\int_{\bar{\mathcal{C}}_{1}^{\star}}\frac{\mathrm{d}\bar{z}_{1}}{2\pi\mathrm{i}}\int_{\bar{\mathcal{C}}_{2}^{\star}}\frac{\mathrm{d}\bar{z}_{2}}{2\pi\mathrm{i}}\;\frac{1}{\left(\bar{z}_{1}-\bar{z}_{2}\right)^{2}}e^{\frac{N}{t}\left[V_{\text{eff}}^{-}\left(\bar{z}_{1}\right)+V_{\text{eff}}^{+}\left(\bar{z}_{2}\right)\right]}.$
(B.13)

This expression is the analog of equation (2.24) which was obtained in the
description where the ghost instanton contributions are obtained by tunneling
regular eigenvalues to the unphysical sheet. We can evaluate the integrals
using the saddle-point approximation around
$\left(\bar{z}_{1},\bar{z}_{2}\right)=\left(z_{1}^{\star},z_{2}^{\star}\right)$.
Note that since both
$\partial_{z}^{2}V_{\text{eff}}^{-}\left(z_{1}^{\star}\right)$ and
$\partial_{z}^{2}V_{\text{eff}}^{+}\left(z_{2}^{\star}\right)$ are positive,
the contours for $\delta\bar{z}_{i}=\bar{z}_{i}-z_{i}^{\star}$ will be along
the imaginary axis, so each integral will contribute an extra factor of
$\mathrm{i}$. Recalling that

$\frac{N}{t}V_{\text{eff}}^{-}\left(z_{1}^{\star}\right)=\frac{N}{t}V_{\text{eff}}^{+}\left(z_{2}^{\star}\right)=-NS_{\text{strong}}(t),$
(B.14)

we obtain

$\displaystyle\frac{Z^{(1,1)}(N,t)}{Z^{(0)}(N,t)}$
$\displaystyle\approx-\frac{t}{2\pi
N}\frac{1}{\left|\partial^{2}V_{\text{eff}}^{-}\left(z_{1}^{\star}\right)\right|^{1/2}\left|\partial^{2}V_{\text{eff}}^{+}\left(z_{2}^{\star}\right)\right|^{1/2}}\frac{1}{\left(z_{1}^{\star}-z_{2}^{\star}\right)^{2}}e^{-2NS_{\text{strong}}(t)}$
(B.15) $\displaystyle=-\frac{t}{8\pi
N\left(t^{2}-1\right)^{3/2}}e^{-2NS_{\text{strong}}(t)}.$ (B.16)

This agrees with the result obtained in section 2 and provides a derivation of
the leading nonperturbative contribution to the GWW integral in the strong-
coupling phase realized via anti-eigenvalue tunneling on the physical sheet.

##  Appendix C Unitary matrix integrals with general single-trace potentials

The goal of this appendix is to extend the calculation of the leading (ghost)
instanton contributions in the ungapped (strong-coupling) phase to the case of
a matrix integral with a general single-trace potential:

$Z\left(N,t_{k}^{\pm}\right):=\int\frac{\mathrm{d}U}{\operatorname{vol}U(N)}\exp\left(\sum_{k=1}^{K}\left(\frac{t_{k}^{+}}{k}\operatorname{Tr}\left(U^{k}\right)+\frac{t_{k}^{-}}{k}\operatorname{Tr}\left(U^{-k}\right)\right)\right).$
(C.1)

Here, the integral is over $N\times N$ unitary matrices $U$, the parameters
$t_{k}^{\pm}$ are complex conjugates of one another,
$t_{k}^{+}=\left(t_{k}^{-}\right)^{\dagger}$, and $K$ is an arbitrary positive
integer. To guarantee that the general integral indeed admits an ungapped
phase, where the distribution of eigenvalues in the large-$N$ limit covers the
entire unit circle, we will impose the sufficient condition (see later
discussion)

$\frac{2}{N}\sum_{k=1}^{K}|t_{k}^{+}|<1\,.$ (C.2)

The GWW integral is a special case where $K=1$ and

$t_{1}^{+}=t_{1}^{-}=\frac{N}{2t}\,.$ (C.3)

###  C.1 Preliminaries

Before discussing instanton corrections to the large-$N$ limit of this matrix
integral, we will first note some useful results similar to those listed in
appendix A.1. Using the theorem of Diaconis and Shahshahani, equation (2.5),
which we recall here for convenience,

$\displaystyle\int\frac{\mathrm{d}U}{\operatorname{vol}U(N)}\,\prod_{j=1}^{k}(\operatorname{Tr}U^{j})^{a_{j}}(\operatorname{Tr}U^{-j})^{b_{j}}$
$\displaystyle=\prod_{j=1}^{k}j^{a_{j}}(a_{j})!\,\delta_{a_{j},b_{j}}\quad\quad\text{if
}N\geq\sum_{j=1}^{k}j\,a_{j}\,.$ (C.4)

we note that the large-$N$ limit of the matrix integral (C.1) evaluates to

$\displaystyle Z_{\infty}\left(t_{k}^{\pm}\right)$
$\displaystyle=\int\frac{\mathrm{d}U}{\operatorname{vol}U(N)}\prod_{k=1}^{K}\left[\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\frac{1}{m!\,n!}\left(\frac{t_{k}^{+}}{k}\right)^{m}\left(\frac{t_{k}^{-}}{k}\right)^{n}\left(\operatorname{Tr}U^{k}\right)^{m}\left(\operatorname{Tr}U^{-k}\right)^{n}\right]$
$\displaystyle=\prod_{k=1}^{K}\left[\sum_{m=0}^{\infty}\frac{1}{m!}\left(\frac{t_{k}^{+}t_{k}^{-}}{k}\right)^{m}\right]$
$\displaystyle=\prod_{k=1}^{K}\exp\left(\frac{t_{k}^{+}t_{k}^{-}}{k}\right).$
(C.5)

The correlator we will need for the instanton computation is the same as the
one in equation (A.1) which we recall here for convenience:

$\left\langle
e^{\operatorname{Tr}\log\big{[}(1-z_{1}^{-1}U)^{2}(1-z_{2}U^{-1})^{2}\big{]}}\right\rangle\approx\exp\Big{[}2\left\langle\operatorname{Tr}\log(1-z_{1}^{-1}U)\right\rangle+2\left\langle\operatorname{Tr}\log(1-z_{2}U^{-1})\right\rangle\\\
+2\left\langle\operatorname{Tr}\log(1-z_{1}^{-1}U)\operatorname{Tr}\log(1-z_{1}^{-1}U)\right\rangle_{c}+2\left\langle\operatorname{Tr}\log(1-z_{2}U^{-1})\operatorname{Tr}\log(1-z_{2}U^{-1})\right\rangle_{c}\\\
+4\left\langle\operatorname{Tr}\log(1-z_{1}^{-1}U)\operatorname{Tr}\log(1-z_{2}U^{-1})\right\rangle_{c}\Big{]}\,.$
(C.6)

Similar to equation (A.2), the single-trace large-$N$ correlator evaluates to

$\left\langle\operatorname{Tr}U^{j}\right\rangle=\begin{cases}N\,,&j=0\,,\\\
t_{j}^{-}\,,&1\leq j\leq K\,,\\\ t_{|j|}^{+}\,,&-K\leq j\leq-1,\\\
0\,,&\text{otherwise.}\end{cases}$ (C.7)

This correlator can be obtained immediately by taking derivatives of equations
(C.1) and (C.5) with respect to the corresponding parameters $t_{j}^{\pm}$.
Similarly, one can obtain the double-trace connected correlator

$\left\langle\operatorname{Tr}U^{j}\operatorname{Tr}U^{k}\right\rangle_{c}=|j|\,\delta_{j,-k}\,,$
(C.8)

by noting that for $j,k\geq 1$,

$\displaystyle\left\langle\operatorname{Tr}U^{j}\operatorname{Tr}U^{k}\right\rangle_{c}$
$\displaystyle=\frac{jk}{Z_{\infty}}\partial_{t_{j}^{+}}\partial_{t_{k}^{+}}Z_{\infty}-\left(\frac{j}{Z_{\infty}}\partial_{t_{j}^{+}}Z_{\infty}\right)\left(\frac{k}{Z_{\infty}}\partial_{t_{k}^{+}}Z_{\infty}\right)=0\,,$
(C.9)
$\displaystyle\left\langle\operatorname{Tr}U^{-j}\operatorname{Tr}U^{-k}\right\rangle_{c}$
$\displaystyle=\frac{jk}{Z_{\infty}}\partial_{t_{j}^{-}}\partial_{t_{k}^{-}}Z_{\infty}-\left(\frac{j}{Z_{\infty}}\partial_{t_{j}^{-}}Z_{\infty}\right)\left(\frac{k}{Z_{\infty}}\partial_{t_{k}^{-}}Z_{\infty}\right)=0\,,$
(C.10)
$\displaystyle\left\langle\operatorname{Tr}U^{j}\operatorname{Tr}U^{-k}\right\rangle_{c}$
$\displaystyle=\frac{jk}{Z_{\infty}}\partial_{t_{j}^{+}}\partial_{t_{k}^{-}}Z_{\infty}-\left(\frac{j}{Z_{\infty}}\partial_{t_{j}^{+}}Z_{\infty}\right)\left(\frac{k}{Z_{\infty}}\partial_{t_{k}^{-}}Z_{\infty}\right)=j\,\delta_{j,k}\,.$
(C.11)

The single-trace correlator expression allows us to compute the single-log
correlators

$\displaystyle\left\langle\operatorname{Tr}\log(1-zU^{-1})\right\rangle$
$\displaystyle=-\sum_{j=1}^{\infty}\frac{1}{j}z^{j}\left\langle\operatorname{Tr}U^{-j}\right\rangle=-\sum_{j=1}^{K}\frac{1}{j}z^{j}t_{j}^{+}\,,$
$\displaystyle|z|<1\,,$ (C.12)
$\displaystyle\left\langle\operatorname{Tr}\log(1-z^{-1}U)\right\rangle$
$\displaystyle=-\sum_{j=1}^{\infty}\frac{1}{j}z^{-j}\left\langle\operatorname{Tr}U^{j}\right\rangle=-\sum_{j=1}^{K}\frac{1}{j}z^{-j}t_{j}^{-}\,,$
$\displaystyle|z|>1\,.$ (C.13)

The double-trace correlator allows us to compute the double-log connected
correlators,

$\displaystyle\left\langle\operatorname{Tr}\log(1-z_{1}U^{-1})\operatorname{Tr}\log(1-z_{2}U^{-1})\right\rangle_{c}$
$\displaystyle=\sum_{j,k=1}^{\infty}\frac{1}{jk}z_{1}^{j}z_{2}^{k}\left\langle\operatorname{Tr}U^{-j}\operatorname{Tr}U^{-k}\right\rangle_{c}=0\,,$
$\displaystyle|z_{1}|,|z_{2}|<1\,,$ (C.14)
$\displaystyle\left\langle\operatorname{Tr}\log(1-z_{1}^{-1}U)\operatorname{Tr}\log(1-z_{2}^{-1}U)\right\rangle_{c}$
$\displaystyle=\sum_{j,k=1}^{\infty}\frac{1}{jk}z_{1}^{-j}z_{2}^{-k}\left\langle\operatorname{Tr}U^{j}\operatorname{Tr}U^{k}\right\rangle_{c}=0\,,$
$\displaystyle|z_{1}|,|z_{2}|>1\,,$ (C.15)
$\displaystyle\left\langle\operatorname{Tr}\log(1-z_{1}^{-1}U)\operatorname{Tr}\log(1-z_{2}U^{-1})\right\rangle_{c}$
$\displaystyle=\sum_{j,k=1}^{\infty}\frac{1}{jk}z_{2}^{j}z_{1}^{-k}\left\langle\operatorname{Tr}U^{-j}\operatorname{Tr}U^{k}\right\rangle_{c}$
$\displaystyle=-\log(1-z_{2}/z_{1})\,,\quad\quad\quad|z_{1}|>1\,,\;|z_{2}|<1\,.$
(C.16)

Substituting the results (C.12)-(C.16) into (C.6), we get the following final
expression for the correlator we will need:

$\left\langle
e^{\operatorname{Tr}\log\big{[}(1-z_{1}^{-1}U)^{2}(1-z_{2}U^{-1})^{2}\big{]}}\right\rangle\approx\exp\left[-2\sum_{j=1}^{K}\frac{1}{j}z_{1}^{-j}t_{j}^{-}-2\sum_{j=1}^{K}\frac{1}{j}z_{2}^{j}t_{j}^{+}-4\log\left(1-z_{2}/z_{1}\right)\right]\,.$
(C.17)

###  C.2 Resolvents, effective potentials, and the eigenvalue distribution

We can now evaluate the resolvents, the effective potentials inside and
outside the unit circle, as well as the eigenvalue distribution in the
ungapped phase of the general matrix integral (C.1). As in the case of the GWW
integral, there will be two resolvents corresponding to the outside and inside
of the unit circle, respectively,

$\displaystyle R^{+}(z)$
$\displaystyle:=\frac{1}{N}\left\langle\operatorname{Tr}\frac{1}{z-U}\right\rangle\,,\quad\quad\text{for
}|z|>1\,,\quad\text{and}$ (C.18) $\displaystyle R^{-}(z)$
$\displaystyle:=\frac{1}{N}\left\langle\operatorname{Tr}\frac{1}{z-U}\right\rangle\,,\quad\quad\text{for
}|z|<1\,.$ (C.19)

We can evaluate these just as before, this time using (C.7),

$\displaystyle R^{+}(z)$
$\displaystyle=\frac{1}{N}\sum_{k=1}^{\infty}z^{-k}\left\langle\operatorname{Tr}U^{k-1}\right\rangle=\frac{1}{z}+\frac{1}{N}\sum_{k=1}^{K}t_{k}^{-}z^{-k-1}\,,$
(C.20) $\displaystyle R^{-}(z)$
$\displaystyle=-\frac{1}{N}\sum_{k=1}^{\infty}z^{k-1}\left\langle\operatorname{Tr}U^{-k}\right\rangle=-\frac{1}{N}\sum_{k=1}^{K}t_{k}^{+}z^{k-1}\,.$
(C.21)

The density of states in the general matrix integral (C.1) will be given by

$\rho(\theta)=\frac{R^{+}(z)-R^{-}(z)}{2\pi\mathrm{i}}\frac{\mathrm{d}z}{\mathrm{d}\theta}=\frac{1}{2\pi}\left(1+\frac{1}{N}\sum_{k=1}^{K}\left(t_{k}^{+}e^{ik\theta}+t_{k}^{-}e^{-ik\theta}\right)\right)\,.$
(C.22)

We can check that the condition (C.2) ensures the eigenvalue density is non-
negative. Writing

$t_{k}^{\pm}=\frac{N}{2q_{k}}e^{\pm i\varphi_{k}}\,,$ (C.23)

where $q_{k}>0$ and $\varphi_{k}\in[0,2\pi)$, the condition (C.2) takes the
form

$\sum_{k=1}^{K}\frac{1}{q_{k}}<1\;,$ (C.24)

and the eigenvalue density takes the form

$\rho(\theta)=\frac{1}{2\pi}\left(1+\sum_{k=1}^{K}\frac{1}{q_{k}}\cos\left(k\theta+\varphi_{k}\right)\right)\,.$
(C.25)

Thus, $\rho(\theta)>0$ for all $\theta\in[0,2\pi)$.

We can now evaluate the derivative of the effective potential inside and
outside the unit circle. For convenience, we will modify the conventions used
in section 2 slightly. We adapt the holomorphic form of the matrix integral,
equations (2.3) and (2.4), to the present case:

$\displaystyle Z\left(N,t_{k}^{\pm}\right)$
$\displaystyle=(-1)^{\frac{1}{2}N(N-1)}\frac{1}{N!}\int\prod_{i=1}^{N}\frac{\mathrm{d}z_{i}}{2\pi\mathrm{i}z_{i}^{N}}\prod_{j<k}(z_{j}-z_{k})^{2}\exp\left(-\sum_{i=1}^{N}V(z_{i})\right)$
(C.26)
$\displaystyle=(-1)^{\frac{1}{2}N(N-1)}\,\mathrm{i}^{-N}\frac{1}{N!}\int\prod_{i=1}^{N}\frac{\mathrm{d}z_{i}}{2\pi}\prod_{j<k}(z_{j}-z_{k})^{2}\exp\left(-N\sum_{i=1}^{N}\left(\frac{1}{N}V\left(z_{i}\right)+\log
z_{i}\right)\right)\,,$ (C.27)

where

$V(z)=-\sum_{k=1}^{K}\left(\frac{t_{k}^{+}}{k}z^{k}+\frac{t_{k}^{-}}{k}z^{-k}\right)$
(C.28)

The derivative of the effective potential is then obtained as

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}V_{\text{eff}}^{+}(z)$
$\displaystyle=\frac{1}{N}V^{\prime}\left(z\right)+\frac{1}{z}-2R^{+}(z)$
$\displaystyle=-\frac{1}{N}\sum_{k=1}^{K}\left(t_{k}^{+}z^{k-1}-t_{k}^{-}z^{-k-1}\right)+\frac{1}{z}-2\left(\frac{1}{z}+\frac{1}{N}\sum_{k=1}^{K}t_{k}^{-}z^{-k-1}\right)$
$\displaystyle=-\frac{1}{z}-\frac{1}{N}\sum_{k=1}^{K}\left(t_{k}^{+}z^{k-1}+t_{k}^{-}z^{-k-1}\right)\,,$
(C.29) $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}V_{\text{eff}}^{-}(z)$
$\displaystyle=\frac{1}{N}V^{\prime}(z)+\frac{1}{z}-2R^{-}(z)$
$\displaystyle=-\frac{1}{N}\sum_{k=1}^{K}\left(t_{k}^{+}z^{k-1}-t_{k}^{-}z^{-k-1}\right)+\frac{1}{z}+\frac{2}{N}\sum_{k=1}^{K}t_{k}^{+}z^{k-1}$
$\displaystyle=\frac{1}{z}+\frac{1}{N}\sum_{k=1}^{K}\left(t_{k}^{+}z^{k-1}+t_{k}^{-}z^{-k-1}\right)\,.$
(C.30)

As in the GWW case, we see that the analytic continuations of the derivatives
of the effective potential are additive inverses of each other. Furthermore,
upon integrating,

$\displaystyle V_{\text{eff}}^{+}(z)$ $\displaystyle=-\log
z-\frac{1}{N}\sum_{k=1}^{K}\left(\frac{t_{k}^{+}}{k}z^{k}-\frac{t_{k}^{-}}{k}z^{-k}\right)\,,$
(C.31) $\displaystyle V_{\text{eff}}^{-}(z)$ $\displaystyle=\log
z+\frac{1}{N}\sum_{k=1}^{K}\left(\frac{t_{k}^{+}}{k}z^{k}-\frac{t_{k}^{-}}{k}z^{-k}\right)\,,$
(C.32)

we note that

$V_{\text{eff}}^{-}\left(1/z^{\dagger}\right)=\left(V_{\text{eff}}^{+}(z)\right)^{\dagger}\,,$
(C.33)

where the dagger symbol denotes complex conjugation. The eigenvalue instanton
locations are given by the common zeros of (C.29) and (C.30),

$N+\sum_{k=1}^{K}\left(t_{k}^{+}\left(z^{\star}\right)^{k}+t_{k}^{-}\left(z^{\star}\right)^{-k}\right)=0\,.$
(C.34)

This equation can be written as a polynomial equation of degree $2K$ and
therefore has $2K$ complex solutions. However, an important observation is
that if a complex number $z^{\star}$ is a zero of (C.34) then
$1/\left(z^{\star}\right)^{\dagger}$ is also a zero of it. This can be seen
immediately by taking the complex conjugate of (C.34). Thus, equation (C.34)
has $K$ pairs of zeroes, $z^{\star}$ and $1/\left(z^{\star}\right)^{\dagger}$,
where one element of each pair is inside the unit circle and the other is
outside. Note that there are no solutions on the unit circle itself since the
left side of equation (C.34) is proportional to the eigenvalue density (C.22),
which is nowhere vanishing due to condition (C.2).

Thus, like in the GWW case, the eigenvalue instantons come in pairs, whose
leading contribution we evaluate in the next subsection.

###  C.3 Leading-order two-instanton contributions

We are interested in obtaining the analog of (2.25) associated to the
eigenvalue instanton locations $z^{\star}$ and
$1/\left(z^{\star}\right)^{\dagger}$ for the matrix integral with a general
single-trace potential, (C.1). Our starting point is the analog of equation
(2.21),

$Z^{(1,1)}\left(N,t_{k}^{\pm}\right)=\frac{1}{(N-2)!}(-1)^{N(N-1)/2}\int_{\mathcal{C}_{1}}\frac{\mathrm{d}z_{1}}{2\pi\mathrm{i}z_{1}^{N}}e^{-V(z_{1})}\int_{\mathcal{C}_{2}}\frac{\mathrm{d}z_{2}}{2\pi\mathrm{i}z_{2}^{N}}e^{-V(z_{2})}(z_{1}-z_{2})^{2}\\\
\times\int_{\mathcal{C}_{0}}\prod_{i=3}^{N}\frac{\mathrm{d}z_{i}}{2\pi\mathrm{i}z_{i}^{N}}e^{-\sum_{i=3}^{N}V(z_{i})}\times\prod_{j=3}^{N}\big{[}(z_{1}-z_{j})^{2}(z_{2}-z_{j})^{2}\big{]}\times\prod_{3\leq
k<l\leq N}(z_{k}-z_{l})^{2}\,,$ (C.35)

where we once again start by considering $|z_{1}|>1$ and $|z_{2}|<1$. We now
follow the same steps laid out in equations (2.22) - (2.25). First rewrite the
previous equation as:

$Z^{(1,1)}\left(N,t_{k}^{\pm}\right)=\frac{1}{(N-2)!}(-1)^{N(N-1)/2}\int_{\mathcal{C}_{1}}\frac{\mathrm{d}z_{1}}{2\pi\mathrm{i}z_{1}^{N}}e^{-V(z_{1})}\int_{\mathcal{C}_{2}}\frac{\mathrm{d}z_{2}}{2\pi\mathrm{i}z_{2}^{N}}e^{-V(z_{2})}(z_{1}-z_{2})^{2}\\\
\times
z_{1}^{2(N-2)}\int_{\mathcal{C}_{0}}\prod_{i=3}^{N}\frac{\mathrm{d}z_{i}}{2\pi\mathrm{i}z_{i}^{N-2}}e^{-\sum_{i=3}^{N}V(z_{i})}\times\prod_{j=3}^{N}\big{[}(1-z_{j}/z_{1})^{2}(1-z_{2}/z_{j})^{2}\big{]}\times\prod_{3\leq
k<l\leq N}(z_{k}-z_{l})^{2}\,.$ (C.36)

This allows us to perform the integrals in $z_{3},\dots,z_{N}$ and express the
result as a correlator in the analog of (C.1):

$\begin{split}Z^{(1,1)}\left(N,t_{k}^{\pm}\right)&=-\int_{\mathcal{C}_{1}}\frac{\mathrm{d}z_{1}}{2\pi\mathrm{i}}\int_{\mathcal{C}_{2}}\frac{\mathrm{d}z_{2}}{2\pi\mathrm{i}}e^{-V(z_{1})-V(z_{2})}\times\frac{z_{1}^{N-4}(z_{1}-z_{2})^{2}}{z_{2}^{N}}\\\
&\hskip 54.2025pt\times Z^{(0)}\left(N-2,t_{k}^{\pm}\right)\left\langle
e^{\operatorname{Tr}\log\big{[}(1-z_{1}^{-1}U)^{2}(1-z_{2}U^{-1})^{2}\big{]}}\right\rangle_{(N-2,t_{k}^{\pm})}\\\
&=-\int_{\mathcal{C}_{1}}\frac{\mathrm{d}z_{1}}{2\pi\mathrm{i}}\int_{\mathcal{C}_{2}}\frac{\mathrm{d}z_{2}}{2\pi\mathrm{i}}e^{-V(z_{1})-V(z_{2})}\times\frac{z_{1}^{N-4}(z_{1}-z_{2})^{2}}{z_{2}^{N}}\\\
&\hskip 54.2025pt\times
Z^{(0)}\left(N,t_{k}^{\pm}\right)\exp\left[-2\sum_{k=1}^{K}\frac{1}{k}z_{1}^{-k}t_{k}^{-}-2\sum_{k=1}^{K}\frac{1}{k}z_{2}^{k}t_{k}^{+}-4\log\left(1-z_{2}/z_{1}\right)\right]\,,\end{split}$
(C.37)

which follows since in the large-$N$ limit,
$Z^{(0)}\left(N-2,t_{k}^{\pm}\right)=Z^{(0)}\left(N,t_{k}^{\pm}\right)$, and
using the expression for the correlator (C.17). The previous relation has a
simple form in terms of the effective potentials, which is analogous to
(2.24):

$\frac{Z^{(1,1)}\left(N,t_{k}^{\pm}\right)}{Z^{(0)}\left(N,t_{k}^{\pm}\right)}=-\int_{\mathcal{C}_{1}}\frac{\mathrm{d}z_{1}}{2\pi\mathrm{i}}\int_{\mathcal{C}_{2}}\frac{\mathrm{d}z_{2}}{2\pi\mathrm{i}}e^{-N\big{[}V_{\text{eff}}^{+}(z_{1})+V_{\text{eff}}^{-}(z_{2})\big{]}}\frac{1}{(z_{1}-z_{2})^{2}}\,.$
(C.38)

There are now two possibilities depending on whether we expect a regular
eigenvalue instanton or a ghost instanton contribution. These depend on
whether we choose $\mathcal{C}_{1}$ to pass through $z^{\star}$ and
$\mathcal{C}_{2}$ to pass through $1/\left(z^{\star}\right)^{\dagger}$ or vice
versa. For concreteness, let us assume without loss of generality that
$z^{\star}$ is the eigenvalue instanton location outside the unit circle and
$1/\left(z^{\star}\right)^{\dagger}$ is inside the unit circle.

Taking $\mathcal{C}_{1}$ to pass through $z^{\star}$ and $\mathcal{C}_{2}$ to
pass through $1/\left(z^{\star}\right)^{\dagger}$, we obtain the regular
eigenvalue instanton contribution:

$\frac{Z^{(1,1)}\left(N,t_{k}^{\pm}\right)}{Z^{(0)}\left(N,t_{k}^{\pm}\right)}=\frac{1}{2\pi
N}\frac{1}{\sqrt{\partial^{2}V_{\text{eff}}^{+}(z^{\star})}\sqrt{\partial^{2}V_{\text{eff}}^{-}\left(1/\left(z^{\star}\right)^{\dagger}\right)}}\frac{1}{\left(z^{\star}-1/\left(z^{\star}\right)^{\dagger}\right)^{2}}e^{-2NS_{\text{strong}}^{\text{regular}}}\,,$
(C.39)

where

$S_{\text{strong}}^{\text{regular}}:=\frac{1}{2}\left[V_{\text{eff}}^{+}\left(z^{\star}\right)+V_{\text{eff}}^{-}\left(1/\left(z^{\star}\right)^{\dagger}\right)\right]\,.$
(C.40)

Note that $S_{\text{strong}}^{\text{regular}}$ is real due to the relation
(C.33) between the effective potentials. The regular eigenvalue instanton pair
contribution is the one that determines the leading nonperturbative correction
associated to the pair
$\left(z^{\star},\,1/\left(z^{\star}\right)^{\dagger}\right)$ to the general
matrix integral (C.1) when $S_{\text{strong}}^{\text{regular}}>0$.

Taking $\mathcal{C}_{1}$ to pass through $1/\left(z^{\star}\right)^{\dagger}$
and $\mathcal{C}_{2}$ to pass through $z^{\star}$, we obtain the ghost
instanton contribution:

$\frac{Z^{(1,1)}\left(N,t_{k}^{\pm}\right)}{Z^{(0)}\left(N,t_{k}^{\pm}\right)}=\frac{1}{2\pi
N}\frac{1}{\sqrt{\partial^{2}V_{\text{eff}}^{+}\left(1/\left(z^{\star}\right)^{\dagger}\right)}\sqrt{\partial^{2}V_{\text{eff}}^{-}\left(z^{\star}\right)}}\frac{1}{\left(z^{\star}-1/\left(z^{\star}\right)^{\dagger}\right)^{2}}e^{-2NS_{\text{strong}}^{\text{ghost}}}\,,$
(C.41)

where

$S_{\text{strong}}^{\text{ghost}}:=\frac{1}{2}\left[V_{\text{eff}}^{+}\left(1/\left(z^{\star}\right)^{\dagger}\right)+V_{\text{eff}}^{-}\left(z^{\star}\right)\right]\,.$
(C.42)

Similarly, $S_{\text{strong}}^{\text{ghost}}$ is real due to relation (C.33)
between the effective potentials. Note also that instanton action
corresponding to the ghost instanton is the additive inverse of the instanton
action corresponding to the regular instanton,

$S_{\text{strong}}^{\text{ghost}}=-S_{\text{strong}}^{\text{regular}}.$ (C.43)

The ghost instanton pair contribution is the one that determines the leading
nonperturbative correction associated to the pair
$\left(z^{\star},\,1/\left(z^{\star}\right)^{\dagger}\right)$ to the general
matrix integral (C.1) when $S_{\text{strong}}^{\text{ghost}}>0$.

Note that the leading regular eigenvalue instanton contribution (C.39) differs
from the leading ghost instanton contribution (C.41) only in the sign of the
action $S_{\text{strong}}$ and an overall sign due to the relations between
the effective potentials.

This method also allows us to compute the contributions from other, more
general, instanton configurations. The defining contour of the matrix integral
tells us how to weigh these various contributions.

###  C.4 The special case of the GWW matrix integral

Finally, we show how the leading-order ghost-instanton contribution (C.41) in
the matrix integral with a general single-trace potential reduces to its
counterpart (2.25) in the GWW matrix integral. The GWW integral (1.1) is a
special case of the general matrix integral (C.1) where $K=1$ and
$t_{1}^{+}=t_{1}^{-}=\frac{N}{2t}$. The effective potentials (C.31) and (C.32)
reduce to

$\displaystyle V_{\text{eff}}^{+}(z)=-\log
z-\frac{1}{2t}\left(z-\frac{1}{z}\right)\,,$ (C.44) $\displaystyle
V_{\text{eff}}^{-}(z)=\log z+\frac{1}{2t}\left(z-\frac{1}{z}\right)\,,$ (C.45)

and the eigenvalue instanton location condition (C.34) reduces to

$1+\frac{1}{2t}\left(\left(z^{\star}\right)+\frac{1}{z^{\star}}\right)=0\,,$
(C.46)

which has solutions

$z^{\star}=-t-\sqrt{t^{2}-1}\,,\qquad\qquad
1/\left(z^{\star}\right)^{\dagger}=-t+\sqrt{t^{2}-1}\,.$ (C.47)

The regular instanton action (C.40) reduces to:

$S_{\text{strong}}^{\text{regular}}=-\left(\log\left(t+\sqrt{t^{2}-1}\right)-\sqrt{1-t^{-2}}\right)\,,$
(C.48)

while the ghost instanton action (C.42) reduces to:

$S_{\text{strong}}^{\text{ghost}}=+\left(\log\left(t+\sqrt{t^{2}-1}\right)-\sqrt{1-t^{-2}}\right)\,.$
(C.49)

Thus, the GWW matrix integral will have a leading ghost-instanton contribution
in the strong-coupling phase. Additionally,

$\displaystyle\partial^{2}V_{\text{eff}}^{+}\left(1/\left(z^{\star}\right)^{\dagger}\right)$
$\displaystyle=-\left(z^{\star}\right)^{2}\sqrt{1-t^{-2}}\,,$ (C.50)
$\displaystyle\partial^{2}V_{\text{eff}}^{-}\left(z^{\star}\right)$
$\displaystyle=-\frac{1}{\left(z^{\star}\right)^{2}}\sqrt{1-t^{-2}}\,,$ (C.51)
$\displaystyle\left(z^{\star}-1/\left(z^{\star}\right)^{\dagger}\right)^{2}$
$\displaystyle=4\left(t^{2}-1\right)$ (C.52)

Therefore, the leading nonperturbative contribution to the large-$N$ limit of
the GWW matrix integral in the strong-coupling phase will be a special case of
(C.41),

$\frac{Z^{\left(1,1\right)}\left(N,t_{k}^{\pm}\right)}{Z^{(0)}\left(N,t_{k}^{\pm}\right)}=-\frac{t}{8\pi
N\left(t^{2}-1\right)^{3/2}}e^{-2NS_{\text{strong}}^{\text{ghost}}}\,.$ (C.53)

This agrees with (2.25).

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