Large N expansion of Wilson loops in the Gross-Witten-Wadia matrix model

Large N expansion of Wilson loops in the Gross-Witten-Wadia matrix model

[ \qquad [
Abstract

We study the large N expansion of winding Wilson loops in the off-critical regime of the Gross-Witten-Wadia (GWW) unitary matrix model. These have been recently considered in arXiv:1705.06542 and computed by numerical methods. We present various analytical algorithms for the precise computation of both the perturbative and instanton corrections to the Wilson loops. In the gapped phase of the GWW model we present the genus five expansion of the one-cut resolvent that captures all winding loops. Then, as a complementary tool, we apply the Periwal-Shevitz orthogonal polynomial recursion to the GWW model coupled to suitable sources and show how it generates all higher genus corrections to any specific loop with given winding. The method is extended to the treatment of instanton effects including higher order 1/N corrections. Several explicit examples are fully worked out and a general formula for the next-to-leading correction at general winding is provided. For the simplest cases, our calculation checks exact results from the Schwinger-Dyson equations, but the presented tools have a wider range of applicability.

a,c]Eleonora Alfinito b,c]Matteo Beccaria


Large N expansion of Wilson loops in the Gross-Witten-Wadia matrix model

 

  • Dipartimento di Ingegneria dell’Innovazione, Università del Salento, Via Arnesano, 73100 Lecce, Italy

  • Dipartimento di Matematica e Fisica Ennio De Giorgi,
    Università del Salento, Via Arnesano, 73100 Lecce, Italy

  • Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Lecce

E-mail: matteo.beccaria@le.infn.it

 


 


1 Introduction

The study of the 1/N expansion of matrix models is a topic of clear broad relevance [brezin1993large, Rossi:1996hs, akemann2011oxford, wang2013application]. Matrix models started as elementary building blocks in the large N analysis of “vector” models with O(N) flavour symmetry [Stanley:1968gx]. Their 1/N expansion appeared originally as an alternative perturbative approach to the determination of critical exponents [Ma:1973zu] or the starting point for a planar ’t Hooft expansion [tHooft:1973alw]. In some specific applications, the space-time dimensionality does not play a role and calculations are effectively reduced to matrix models, i.e. “single-link” group integrals. A well-known old example is 2d Yang-Mills theory on a lattice where, in the temporal gauge for the Wilson action, the partition function factorizes over the lattice links [Gross:1980he]. In a more recent setting, reduction to matrix models may be due to superconformal symmetry, as in the case of supersymmetric circular Wilson loop operators in the \mathcal{N}=4 SYM theory that may be reduced to a Gaussian matrix model, see e.g. [Erickson:2000af, Drukker:2000rr] and the \mathcal{N}=2 extension in [Pestun:2007rz]. In general, for theories admitting an AdS dual, the matrix model formulation allows for non-trivial tests of AdS/CFT, as in the case of BPS Wilson loops in higher representations in \mathcal{N}=4 SYM that have a dual description in terms of D-branes [Drukker:2005kx, Gomis:2006sb, Gomis:2006im, Hartnoll:2006is, Okuyama:2006jc, Yamaguchi:2007ps, Drukker:2006zk, Buchbinder:2014nia, Chen-Lin:2016kkk]. Other modern applications exploit matrix models as quantum gauge theories in zero dimension capturing some aspects of more realistic (or complicated) theories, like random surface models of quantum gravity or non critical and topological string on specific backgrounds [Marino:2004eq].

In this paper we reconsider the large N expansion of the Gross-Witten-Wadia (GWW) model [Gross:1980he, Wadia:2012fr]. This is the simplest unitary matrix model with Wilson action

S_{GWW}(U)=N\,g\,\text{tr}(U+U^{\dagger}), (1.1)

where g is the large N planar ’t Hooft coupling. The GWW model has a large N third order phase transition at g_{c}=1. 111For a renormalization group approach to the matrix model large N transition, see for instance [Brezin:1992yc, Higuchi:1994dv]. The critical point is captured by a double scaling limit [Periwal:1990gf, Periwal:1990qb] and is associated with type 0B theory in d=0 dimension, i.e. pure 2-d supergravity [Klebanov:2003wg]. It also plays an important role in the description of the adjoint unitary model which is used to discuss the non-perturbative aspects of the Hagedorn transition for tensionless IIB string theory in AdS [Liu:2004vy, AlvarezGaume:2005fv]. The transition in this context is holographically dual to the Hawking-Page transition on the bulk gravity side [Witten:1998zw].

The analysis of the GWW model away from the critical point is particularly interesting in the study of its non-perturbative multi-instanton corrections. These have been discussed in [Marino:2008ya] from the point of view of resurgence and trans-series analysis. Multi-instanton configurations in the GWW model may have an interpretation in terms of eigenvalue tunneling [shenker1991strength, David:1990sk], and admit a more general characterization as complex saddles of the partition function where the U eigenvalues real phases are continued to the complex plane and allowed to accumulate on non-trivial cuts off the unit circle [Buividovich:2015oju, Alvarez:2016rmo].

The GWW model has a gapless phase for g<g_{c} where the unitary matrix eigenvalue density is supported on the whole circle. In this phase, the (perturbative) higher genus corrections to the free energy vanish beyond genus zero, while non-perturbative corrections remains non trivial. Beyond the third order transition point, for g>g_{c}, the GWW model is in a different phase where a gap opens in the the eigenvalue density which is non zero on a coupling dependent interval. In this phase, the free energy has both perturbative and non-perturbative corrections and these are linked by resurgence [Marino:2008ya].

Recently, the winding Wilson loops

W_{k}=\frac{1}{N}\,\langle\text{tr}(U^{k})\rangle, (1.2)

have been analyzed in [Okuyama:2017pil] in both phases of the model. Series expansions in 1/N have been proposed for both the purely perturbative genus expansion as well as for the instanton parts of W_{k}. 222 The instanton corrections have an exponentially suppressed pre-factor \mathcal{C}\,\exp(-\ell\,N\,S(g))\, where S(g) is the instanton action and \ell is the instanton order. As discussed later, this is multiplied by a power series of 1/N corrections \mathcal{W}^{(\ell)}_{k,0}(g)+1/N\,\mathcal{W}^{(\ell)}_{k,1}(g)+1/N^{2}\,% \mathcal{W}^{(\ell)}_{k,2}(g)+\dots corrections, where \mathcal{W}^{(\ell)}_{k,n} are non-trivial functions of the coupling. Technically, these expansions have been obtained by a careful analysis of the numerical exact expressions of W_{k} at finite N and coupling. A numerical fitting procedure provided the coefficients of the 1/N expansion as rational functions of the coupling at genus three accuracy.

In this paper, we discuss the analytical calculation of these expansions for the winding Wilson loops. We first recall some exact results that are consequences of the loop equations of the model. These are an important benchmark for the proposed methods whose range of applicability goes beyond the Wilson action. We then focus on the (perturbative) genus expansion in the gapped phase. To this aim, we follow three different approaches. First, we compute the higher genus one-cut resolvent according to the methods developed in [Ambjorn:1992gw] for the hermitian matrix models and exploiting the results of [Mizoguchi:2004ne] to map the resolvent to the GWW model. A systematic calculation provides explicit results at genus five. This allows to write down the genus expansion of all W_{k} at order 1/N^{10}. The calculation is somewhat straightforward and can be used to provide useful checks of other approaches.

As a second technique, we compute W_{k} by coupling the GWW model to auxiliary sources \bm{\rho}

S_{GWW}(U;\bm{\rho})=N\,g\,\text{tr}(U+U^{\dagger})+\sum_{k=2}^{\infty}\rho_{k% }\,\text{tr}(U^{k}+(U^{\dagger})^{k}). (1.3)

Derivatives of the free energy with respect to the sources \rho_{k} compute W_{k}. The free energy associated with the extended action (1.3) is computed by the orthogonal polynomial methods developed for hermitian matrix models in [Bessis:1979is, Bessis:1980ss, DiFrancesco:1993cyw] and extended to unitary matrix models in [Goldschmidt:1979hq, Periwal:1990qb, Periwal:1990gf]. Each source \rho_{k} is associated with an increasingly involved recursion relation for the orthogonal polynomial coefficients. This can be solved perturbatively at large N. We clarify some technical aspect of the procedure and perfectly reproduce the results for W_{k} obtained by the resolvent approach. These calculations extend the results of [Okuyama:2017pil] and confirm them up to some discrepancies that are important for reconciliation with old exact results relating W_{1} and W_{2} at finite coupling and N.

Finally, we discuss a third (practical) approach that is based on some peculiarities of the large coupling expansion of the finite N expressions of W_{k}. Guided by an educated guess for the structure of the genus expansion we can provide, with very modest effort, quite long genus expansions for the winding Wilson loops. This is somewhat interesting because it is cumbersome to extend the resolvent method at high orders in 1/N. This is not a difficulty for the orthogonal polynomial method, but in that case one has to deal with a recursion relation whose complexity increases with k. Instead, the proposed analytic bootstrap of finite N data can treat with minor effort with higher k and even for more complicated observables like Wilson loops in general small representations.

As a final result, according to the ideas of [Marino:2008ya], we also discuss the use of the orthogonal polynomial method for the GWW model coupled to sources to compute the non perturbative instanton corrections to the winding Wilson loop. In particular, we clarify various technical issues and provide explicit examples in the gapless phase.

The plan of the paper is the following. In Sec. (2), we briefly recall some basic facts about the GWW model and its large N expansion. Sec. (3) is devoted to the analysis of the high genus expansion in the gapless phase. In more details, in Sec. (3.1) we present the one-cut resolvent at genus five. In Sec. (3.2), we discuss how to apply the Periwal-Shevitz recursion method to the GWW model coupled to suitable sources in order to compute W_{k}. In Sec. (3.3), we reproduce and extend the previous results by an analytic bootstrap procedure that exploits a simple combination of educated insight and (small) finite N data. Finally, in Sec. (LABEL:sec:instanton), the methods presented in Sec. (3.2) are used for the precise calculation of large N effects in the instanton contribution to W_{k} in the gapless phase.

2 The Gross-Witten-Wadia model

The partition function of the GWW model, at finite matrix dimension N and coupling g, admits the following exact expression [Wadia:2012fr, Bars:1979xb]333We shall follow the notation of [Okuyama:2017pil] to simplify comparison.

Z(N,g)=\int_{U(N)}dU\,\exp\bigg{[}\frac{N\,g}{2}\,\text{tr}(U+U^{\dagger})% \bigg{]}=\det\left[I_{n-m}(N\,g)\right]_{n,m=1,\dots,N}, (2.1)

where I_{\nu}(x) is the modified Bessel function of the first kind. The observables we are going to study are the winding Wilson loops whose definition and exact expression read

W_{k}(N,g)=\frac{1}{N}\,\langle\text{tr}(U^{k})\rangle=\frac{1}{N}\,\text{tr}(% M_{0}^{-1}\,M_{k}),\qquad(M_{k})_{n,m}=I_{k+n-m}(N\,g). (2.2)

We remark that the (standard) Wilson loop W_{1} can be computed directly from the free energy (2.1) according to the obvious relation

W_{1}(N,g)=\frac{1}{N^{2}}\partial_{g}\,\log Z(N,g). (2.3)

In particular, from the known expansion of the free energy, this implies that

\lim_{N\to\infty}W_{1}(N,g)=\begin{cases}\frac{g}{2},&g<1,\\ 1-\frac{1}{2\,g},&g>1.\end{cases} (2.4)

At fixed g and large N, the winding Wilson loops can be computed in terms of the asymptotic distribution \rho(\theta) of the eigenvalues e^{i\theta} of the U(N) matrix [Gross:1980he]. This gives the result

\lim_{N\to\infty}W_{k}(N,g)=\begin{cases}0,&g<1,\\ \frac{1}{k-1}\left(1-\frac{1}{g}\right)^{2}\,P_{k-2}^{(1,2)}\left(1-\frac{2}{g% }\right),&g>1,\end{cases} (2.5)

where P^{(a,b)}_{n}(z) are Jacobi polynomials. Notice that the second derivative of W_{k} has a finite jump at the transition point g=1. The point g=1 separates an ungapped phase (g<1) where the large N eigenvalue distribution \rho(\theta) is supported on the whole circle, and a gapped phase (g>1) where a gap opens in the distribution \rho(\theta).

2.1 Loop equations for winding Wilson loops

An alternative approach to the large N limit is based on the relation between the GWW model and Yang-Mills theory in two dimensions in the lattice Wilson formulation. One can show that the higher winding Wilson loops obey an algebraic equation at fixed g and large N. This follows from the Migdal-Makeenko loop equation [Makeenko:1979pb, Migdal:1984gj] that, after using large N factorization, leads to the following generating function for w_{n}\equiv W_{n}(N\to\infty) [Paffuti:1980cs]

\displaystyle\sum_{n=0}^{\infty}w_{n}\,x^{n} \displaystyle=\frac{g}{4x}\,\sqrt{\left(\frac{2x}{g}+x^{2}+1\right)^{2}-\frac{% 4x^{2}(g-2w_{1})}{g}}+\frac{g\left(x^{2}-1\right)+2x}{4x}
\displaystyle=1+w_{1}\,x+\left(1-\frac{2\,w_{1}}{g}\right)\,x^{2}+\left[-\frac% {2}{g}+w_{1}\,\left(1+\frac{4}{g^{2}}\right)-\frac{2\,w_{1}^{2}}{g}\right]\,x^% {3}+\dots\,. (2.6)

Starting with (2.4), the relations (2.1) are equivalent to (2.5). However, it is useful to remark that the relation linking w_{2} to w_{1} is special being exact even at finite N [Wadia:1980rb, Friedan:1980tu] 444For a given N, this can be easily checked explicitly from the exact expressions in (2.2).

W_{2}(N,g)=1-\frac{2}{g}\,W_{1}(N,g), (2.7)

and is thus an important constraint on the computations. Actually, the exact relation (2.7) may be generalized to higher winding loops by inspecting the loop equations at specific low windings. Using, for instance, the results of [Green:1980bg] it is easy to prove the next two exact relations (W_{k}\equiv W_{k}(N,g))

\displaystyle W_{3} \displaystyle=-\frac{2}{g}+\left(1+\frac{4}{g^{2}}+\frac{2}{g^{2}\,N^{2}}% \right)\,W_{1}-\frac{2}{g}\,W_{1}^{2}-\frac{2}{g\,N^{2}}\,\partial_{g}W_{1},
\displaystyle W_{4} \displaystyle=1+\frac{4}{g^{2}}+\frac{8}{g^{2}N^{2}}+\left(-\frac{8}{g}-\frac{% 8}{g^{3}}-\frac{28}{g^{3}N^{2}}\right)\,W_{1}+\frac{12}{g^{2}}\,W_{1}^{2}+% \frac{12}{g^{2}N^{2}}\,\partial_{g}W_{1}, (2.8)

that are additional constraints holding at finite N and g. Going at higher winding involves more complicated correlators. For instance, the loop equation for W_{5} links it to \langle\text{tr}(U^{2})^{2}\rangle. Actually, as we explained in the Introduction, we are going to use results like (2.7) and (2.1) as a test in the GWW model of more general tools that, in principle, may be used for unitary models with actions different from the simplest Wilson one.

2.2 Structure of 1/N and non-perturbative corrections

The free energy of the GWW model admits the large N genus expansion

\displaystyle F(N,g) \displaystyle=\log Z(N,g)=F^{\rm pert}(N,g)+F^{\rm inst}(N,g),
\displaystyle F^{\rm pert}(N,g) \displaystyle=\sum_{\ell\geq 0}N^{2-2\,\ell}\,F_{\ell}(g), (2.9)

where F^{\rm inst}(N,g) are instanton contributions exponentially suppressed at large N. The genus expansion coefficients F_{\ell}(g) are trivial in the ungapped phase

F_{0}(g)=\frac{g^{2}}{2},\qquad F_{\ell\geq 1}(g)=0. (2.10)

Instead, in the gapped phase, all coefficients F_{\ell}(g) are non zero. They start with

F_{0}(g)=g-\frac{1}{2}\,\log g-\frac{3}{4},\quad F_{1}(g)=\zeta^{\prime}(-1)-% \frac{1}{12}\,\log N-\frac{1}{8}\log(1-1/g), (2.11)

and, for \ell\geq 2, may be written in the general form

F_{\ell\geq 2}(g)=\frac{B_{2\,\ell}}{2\,\ell\,(2\,\ell-2)}+\frac{1}{(g-1)^{3\,% \ell-3}}\sum_{n=0}^{\ell-2}c_{n}^{(\ell)}\,g^{n}, (2.12)

where the coefficients c_{n}^{(\ell)} may be systematically computed at higher genus by solving the pre-string equation governing the partition function [Goldschmidt:1979hq, Periwal:1990qb, Periwal:1990gf] as discussed in details in [Marino:2008ya]. The non perturbative part F^{\rm inst}(g) in (2.2) is non trivial in both phases. Its detailed structure as well as the relation with resummation of the genus expansions have been fully elucidated in [Marino:2008ya].

In the case of winding Wilson loops, the recent results of [Okuyama:2017pil] suggest that we can write again a decomposition similar to (2.2)

\displaystyle W_{k}(N,g) \displaystyle=W_{k}^{\rm pert}(N,g)+W_{k}^{\rm inst}(g),
\displaystyle W_{k}^{\rm pert}(N,g) \displaystyle=\sum_{\ell\geq 0}N^{2-2\ell}W_{k,\ell}(g), (2.13)

where the genus expansion correcting (2.4) and (2.5) is non trivial in the gapped phase. The instanton contribution is present in both phases and takes the general form

W_{k}^{\rm inst}(N,g)=\sum_{\ell=2}^{\infty}\frac{e^{-\ell\,N\,S(g)}}{4\,\pi\,% N^{2}}\sum_{n=0}^{\infty}\frac{\mathcal{W}^{(\ell)}_{k,n}(g)}{N^{n}}, (2.14)

where the instanton action is

S(g)=\cosh^{-1}(1/g)-\sqrt{1-g^{2}}, (2.15)

and the exponential prefactor comes together with an infinite perturbative tail factor associated with the coefficients \mathcal{W}_{k,n}^{(\ell)}.

In the following sections, we shall first focus on the gapped phase and discuss how to derive analytically the perturbative contributions W_{k,\ell}(g) in (2.2) at high genus. Later, we shall address the computation of the coefficients \mathcal{W}_{k,n}^{(\ell)} in the non-perturbative contribution focusing on the ungapped case where this is the only non trivial correction to the winding Wilson loops at large N.

3 High genus expansion in the gapped phase

In order to clarify the motivations of our analysis, let us start by remarking that from the relation (2.3) and the known expansion of the free energy, we obtain the exact expansion of W_{1}^{\rm pert} in the gapped phase

W_{1}^{\rm pert}=1-\frac{1}{2\,g}-\frac{1}{N^{2}}\,\frac{1}{8\,(g-1)\,g}-\frac% {1}{N^{4}}\,\frac{9}{128\,(g-1)^{4}}-\frac{1}{N^{6}}\,\frac{9\,(25\,g+17)}{102% 4\,(g-1)^{7}}+\dots. (3.1)

For the winding loops, the results of [Okuyama:2017pil] – denoted below by a tilde – are based on a numerical fitting procedure and in the case of W_{2} and W_{3} they read

\displaystyle\widetilde{W}_{2}^{\rm pert} \displaystyle=\frac{(g-1)^{2}}{g^{2}}+\frac{1}{N^{2}}\,\frac{1}{4\,(g-1)\,g^{2% }}+\frac{1}{N^{4}}\,\frac{9}{64\,(g-1)^{4}\,g}+\frac{1}{N^{6}}\,\frac{451\,g^{% 2}+297\,g+23}{1024\,(g-1)^{7}\,g^{2}}+\dots,
\displaystyle\widetilde{W}_{3}^{\rm pert} \displaystyle=\frac{(g-1)^{2}(2\,g-5)}{2\,g^{3}}+\frac{1}{N^{2}}\,\frac{10-28% \,g+15\,g^{2}}{8\,(g-1)\,g^{3}}+\frac{1}{N^{4}}\,\frac{3\,(20-90\,g+96\,g^{2}-% 35\,g^{3})}{128\,(g-1)^{4}\,g^{3}}+\dots. (3.2)

Similar expansions at genus two are also provided for the higher winding Wilson loops. The expression \widetilde{W}_{2} is not compatible with the exact relation (2.7). Indeed, that relation implies, using the expansion (3.1), the result

W_{2}^{\rm pert}=\frac{(g-1)^{2}}{g^{2}}+\frac{1}{N^{2}}\,\frac{1}{4\,(g-1)\,g% ^{2}}+\frac{1}{N^{4}}\,\frac{9}{64\,(g-1)^{4}\,g}+\frac{1}{N^{6}}\,\frac{9\,(1% 7+25\,g)}{512\,(g-1)^{7}\,g}+\dots, (3.3)

that differs from the first of (3) at the N^{-6} level. For this reason, it seems important to compute analytically the expansions of W_{k}^{\rm pert} in order to test the numerical fitting in [Okuyama:2017pil]. An analysis of the instanton corrections in the gapped phase will be addressed later in Sec. (LABEL:sec:instanton).

3.1 Genus expansion of the one-cut resolvent

The solution of the loop equations for all winding loops W_{k} at a given genus order may be achieved by resolvent techniques. The GWW model can be mapped to a non-polynomial hermitian matrix model according to the results of [Mizoguchi:2004ne]. In particular, the gapped phase of the GWW model is associated with a (symmetric) one-cut resolvent. For a general hermitian matrix model, there exists a systematic iterative scheme for the resolvent calculation with explicit results up to genus two [Ambjorn:1992gw].555 Further discussion and details can be found in [Akemann:2001st, Marino:2007te]. We briefly discuss the method and then apply to the GWW model at genus five. The resolvent of the GWW model is defined by the expectation value

G(t)=\frac{i}{N}\,\left\langle\text{tr}\frac{t+U}{t-U}\right\rangle, (3.4)

and the values of W_{k} can be read from its large t expansion. We shall exploit the relation

G(t)=(1+z^{2})\,\omega(z)-z,\qquad t=\frac{1+i\,z}{1-i\,z}, (3.5)

where \omega(z) is the resolvent of a hermitian matrix model in terms of M=M^{\dagger}

\omega(z)=\frac{1}{N}\left\langle\text{tr}\frac{1}{z-M}\right\rangle,\qquad U=% \frac{i-M}{i+M}. (3.6)

The relation between the GWW partition function and that of the hermitian matrix model is simply

\displaystyle\int dU\,e^{-N\,\text{tr}W(U)}=\int dM\,e^{-N\,\text{tr}V(M)},% \qquad V(z)=W(t)+\log(1+z^{2}). (3.7)

According to [Ambjorn:1992gw], we also introduce the 2-point (connected) resolvent

\omega(z,z^{\prime})=\left\langle\text{tr}\frac{1}{z-M}\,\frac{1}{z^{\prime}-M% }\right\rangle. (3.8)

Both \omega(z) and \omega(z,z^{\prime}) admit a genus expansion

\omega=\sum_{g\geq 0}N^{-2\,g}\,\omega_{g}. (3.9)

The first Makeenko loop equation for the hermitian matrix model reads [Makeenko:1991tb]

\oint\frac{d\zeta}{2\,\pi\,i}\frac{V^{\prime}(\zeta)}{z-\zeta}\,\omega(\zeta)=% \omega(z)^{2}+\frac{1}{N^{2}}\,\omega(z,z), (3.10)

where the integral is along a simple positive curve enclosing all singularities of \omega(\zeta). The genus expansion of (3.10) reads

\widehat{K}\,\omega_{g}(z)=\sum_{g^{\prime}=1}^{g-1}\omega_{g^{\prime}}(z)\,% \omega_{g-g^{\prime}}(z)+\frac{\delta\omega_{g-1}(z)}{\delta V}, (3.11)

where \widehat{K} is a suitable integral linear operator and the loop insertion derivative \delta/\delta V has a complicated but explicit expression that can be found in [Ambjorn:1992gw]. The iterative solution of (3.11) starts by expressing the resolvent as

\omega_{g}(z)=\sum_{n=1}^{3g-1}\bigg{[}A_{g}^{+(n)}\,\chi^{+(n)}(z)+A_{g}^{-(n% )}\,\chi^{-(n)}(z)\bigg{]}, (3.12)

where the functions \chi^{\pm(n)}(z) are explicit eigenfunctions of \widehat{K} that can be given explicitly and depend on the potential V through the so-called moments (k\geq 1)

M_{k}=\oint\frac{d\zeta}{2\pi\,i}\frac{V^{\prime}(\zeta)}{(\zeta-x)^{k+1/2}(% \zeta-y)^{1/2}},\quad J_{k}=\oint\frac{d\zeta}{2\pi\,i}\frac{V^{\prime}(\zeta)% }{(\zeta-x)^{1/2}(\zeta-y)^{k+1/2}}, (3.13)

where x, y are the real endpoints of the resolvent cut [y,x]\subset\mathbb{R}. Plugging (3.12) in (3.11), we can determine the constants A_{g}^{\pm(n)} and thus the resolvent. The lowest order results are

\displaystyle\omega_{0}(z) \displaystyle=\frac{A^{2}\,z\,\left(2\,g+1+z^{2}\right)-2\sqrt{A^{2}+1}\sqrt{z% ^{2}-A^{2}}}{A^{2}\left(z^{2}+1\right)^{2}},
\displaystyle\omega_{1}(z) \displaystyle=-\frac{A^{4}\sqrt{A^{2}+1}\left(A^{2}-2z^{2}-1\right)}{16\left(z% ^{2}-A^{2}\right)^{5/2}},
\displaystyle\omega_{2}(z) \displaystyle=\frac{A^{8}\,\sqrt{1+A^{2}}}{1024\,(z^{2}-A^{2})^{11/2}}\,\bigg{% [}36A^{10}-180A^{8}z^{2}+9A^{6}\left(41z^{4}+2z^{2}+1\right)
\displaystyle-2A^{4}\left(206z^{6}+57z^{4}+36z^{2}+5\right)+A^{2}\left(292z^{8% }+300z^{6}+285z^{4}+118z^{2}+21\right)
\displaystyle+12z^{2}\left(18z^{6}+34z^{4}+26z^{2}+7\right)\bigg{]}, (3.14)

where A=\frac{1}{\sqrt{g-1}} is the endpoint of the (symmetric) cut [y,x]\equiv[-A,A]. Inserting in (3.5) the genus expansion of the resolvent, see (3.9), and expanding at large t, one obtains the genus expansion of all winding Wilson loops. We pushed the calculation extending the list in (3.1) up to genus 5. 666The rather unwieldy expressions of the coefficients in (3.12) and of the resolvent are available under request. The standard Wilson loop W_{1} reads

\displaystyle W_{1}^{\rm pert} \displaystyle=1-\frac{1}{2\,g}-\frac{1}{N^{2}}\,\frac{1}{8\,(g-1)\,g}-\frac{1}% {N^{4}}\,\frac{9}{128\,(g-1)^{4}}
\displaystyle-\frac{1}{N^{6}}\,\frac{9\,(25\,g+17)}{1024\,(g-1)^{7}}-\frac{1}{% N^{8}}\frac{9\left(6125\,g^{2}+10750\,g+2381\right)}{32768\,(g-1)^{10}}
\displaystyle-\frac{1}{N^{10}}\,\frac{9\left(694575\,g^{3}+2160925\,g^{2}+1293% 325\,g+140927\right)}{262144(g-1)^{13}}+\dots. (3.15)

This expression agrees with (3.1) and we checked agreement of higher genus corrections with the pre-string expansion of the free energy. The resolvent prediction for the second Wilson loop is

\displaystyle W_{2}^{\rm pert} \displaystyle=\frac{(g-1)^{2}}{g^{2}}+\frac{1}{N^{2}}\,\frac{1}{4\,(g-1)\,g^{2% }}+\frac{1}{N^{4}}\,\frac{9}{64\,(g-1)^{4}\,g}
\displaystyle+\frac{1}{N^{6}}\,\frac{9(25\,g+17)}{512\,(g-1)^{7}\,g}+\frac{1}{% N^{8}}\,\frac{9\left(6125\,g^{2}+10750\,g+2381\right)}{16384\,(g-1)^{10}\,g}
\displaystyle+\frac{1}{N^{10}}\,\frac{9\left(694575\,g^{3}+2160925\,g^{2}+1293% 325\,g+140927\right)}{131072\,(g-1)^{13}\,g}+\dots. (3.16)

The exact relation (2.7) is satisfied, as it should. The next Wilson loop is 777 It is instructive to insert this expansion into the large N relation (2.1). As expected, the 1/N^{2} terms do not cancel due to non-factorization corrections to the loop equations.

\displaystyle W_{3}^{\rm pert} \displaystyle=\frac{(g-1)^{2}(2g-5)}{2g^{3}}+\frac{1}{N^{2}}\,\frac{15g^{2}-28% g+10}{8(g-1)g^{3}}-\frac{1}{N^{4}}\,\frac{3\left(35g^{3}-96g^{2}+90g-20\right)% }{128\,(g-1)^{4}g^{3}}
\displaystyle-\frac{1}{N^{6}}\,\frac{27\left(35g^{3}-85g^{2}+62g+30\right)}{10% 24(g-1)^{7}g^{2}}
\displaystyle-\frac{1}{N^{8}}\,\frac{9\left(17325g^{4}-27010g^{3}-11765g^{2}+6% 5748g+13470\right)}{32768(g-1)^{10}g^{2}}
\displaystyle-\frac{1}{N^{10}}\,\frac{9\left(1576575g^{5}-537075g^{4}-5526245g% ^{3}+8822345g^{2}+7716666g+816990\right)}{262144(g-1)^{13}g^{2}}+\dots. (3.17)

It is clear that similar results for any W_{k} can be written immediately, at this genus order. Examples are 888Notice that the 1/N^{4} contribution in W_{5} is different from [Okuyama:2017pil].

\displaystyle W_{4}^{\rm pert} \displaystyle=\frac{(g-1)^{2}\left(g^{2}-6g+7\right)}{g^{4}}+\frac{1}{N^{2}}% \frac{16g^{3}-70g^{2}+90g-35}{2(g-1)g^{4}}
\displaystyle+\frac{1}{N^{4}}\frac{226g^{3}-624g^{2}+561g-154}{32(g-1)^{4}g^{4% }}+\frac{1}{N^{6}}\,\frac{9\left(202g^{3}-470g^{2}+275g+35\right)}{256(g-1)^{7% }g^{3}}
\displaystyle+\frac{1}{N^{8}}\,\frac{9\left(32250g^{4}-42564g^{3}-37885g^{2}+5% 8950g+8505\right)}{8192(g-1)^{10}g^{3}}
\displaystyle+\frac{1}{N^{10}}\frac{9\left(2908150g^{5}+30850g^{4}-10788813g^{% 3}+5267705g^{2}+6286275g+585585\right)}{65536(g-1)^{13}g^{3}}+\dots,
\displaystyle W_{5}^{\rm pert} \displaystyle=\frac{(g-1)^{2}\left(2g^{3}-21g^{2}+56g-42\right)}{2g^{5}}+\frac% {1}{N^{2}}\frac{5\left(35g^{4}-260g^{3}+630g^{2}-616g+210\right)}{8(g-1)g^{5}}
\displaystyle+\frac{1}{N^{4}}\,\frac{6615g^{5}-32672g^{4}+60438g^{3}-52332g^{2% }+21098g-3192}{128(g-1)^{4}g^{5}}
\displaystyle-\frac{1}{N^{6}}\frac{45\left(693g^{6}-3923g^{5}+9670g^{4}-12250g% ^{3}+7910g^{2}-2394g+336\right)}{1024(g-1)^{7}g^{5}}
\displaystyle-\frac{1}{N^{8}}\frac{45\left(33033g^{6}-173082g^{5}+416379g^{4}-% 341620g^{3}-103320g^{2}+179172g+8694\right)}{32768(g-1)^{10}g^{4}}
\displaystyle-\frac{1}{N^{10}}\frac{45}{262144(g-1)^{13}g^{4}}\bigg{(}1756755g% ^{7}-7613879g^{6}+13944571g^{5}+15923913g^{4}
\displaystyle-51364240g^{3}+14031360g^{2}+16537374g+1073898\bigg{)}+\dots\ . (3.18)

Notice that the expansions for W_{3} and W_{4} are in full agreement with (2.1).

The advantage of the resolvent method is that the fixed genus corrections to all W_{k} are obtained altogether in one shot. However, pushing the calculation at higher genus becomes more and more cumbersome. Besides, in the ungapped phase, the use of this approach to compute the non-perturbative instanton corrections is quite involved, see for instance [Marino:2007te]. In the next section, we discuss a different complementary strategy. Its complexity increases with k, but – at fixed k – it easily generates the full 1/N expansion with minor effort. It will be later extended to capture instanton corrections following the ideas put forward in [Marino:2008ya].

As a final comment about the methods presented in this section, we remark that the 2-point generating function (3.8) is a byproduct of the resolvent calculation. This may be exploited to get expansions for other observables involving product of traces of powers of U, like those appearing in the study of Wilson loops in small representations.

3.2 The Periwal-Shevitz recursion for the GWW model coupled to sources

Let us change variables and introduce the (string) coupling g_{s}=1/(N\,g) to trade the matrix dimension N. We can consider the partition function

Z_{V}(N,g_{s})=\int dU\,\exp\bigg{[}\frac{1}{g_{s}}\,\text{tr}[V(U)+V(U^{% \dagger})]\bigg{]}, (3.19)

for a generic potential V(z). The idea is simply to couple the GWW model to suitable sources in order to generalize the relation (2.3) to higher winding. To this aim we consider

V(U)=\frac{1}{2}\,\bigg{(}U+\rho\,g_{s}\,U^{k}\bigg{)}, (3.20)

and extract the winding Wilson loop W_{k} from

W_{k}=\frac{1}{N}\,\lim_{\rho\to 0}\partial_{\rho}\log Z_{V}. (3.21)

The point is that (3.19) may be treated in full generality by using the method of orthogonal polynomials developed in [Bessis:1979is, Bessis:1980ss] and extended to unitary models in [Periwal:1990gf]. To remind the basic facts, one introduces monic polynomials p_{n}(z)=z^{n}+\cdots that are orthogonal with respect to the circle measure d\mu

\displaystyle\oint d\mu\,p_{n}(z)\,p_{m}(1/z)=h_{n}\,\delta_{nm},\qquad d\mu=% \frac{1}{2\,\pi\,i}\frac{dz}{z}\,\exp\bigg{[}\frac{1}{g_{s}}(V(z)+V(1/z))\bigg% {]}. (3.22)

Quite generally, it can be shown that the polynomials p_{n}(z) obey the functional recursion

p_{n+1}(z)=z\,p_{n}(z)+f_{n}\,z^{n}\,p_{n}(1/z),\qquad\frac{h_{n+1}}{h_{n}}=1-% f_{n}^{2}. (3.23)

The normalizations h_{n}, cf. (3.22), and their ratios r_{n}=h_{n}/h_{n-1}, determine the exact partition function in (3.19)

Z_{V}=\prod_{n=0}^{N-1}h_{n}=h_{0}^{N}\,\prod_{n=1}^{N}r_{n}^{N-n}. (3.24)

It is convenient to introduce t=1/g=N\,g_{s} and write the free energy as 999In some cases, it may be convenient to subtract a reference free energy. Here, we are interested in the linear in \rho part of F and this subtraction is not needed.

g_{s}^{2}\,F=\frac{t^{2}}{N}\,\log h_{0}+\frac{t^{2}}{N}\,\sum_{n=1}^{N}\bigg{% (}1-\frac{n}{N}\bigg{)}\,\log r_{n}. (3.25)

We remark that the quantity h_{0}=\oint d\mu is a non-trivial function of \rho. The other relevant quantities r_{n} obey a recursion that depends on V. Defining for convenience s_{n}^{2}=1-f_{n}^{2}=r_{n+1}, one can show that the following identity holds

g_{s}\,(n+1)\,\frac{f_{n}^{2}}{s_{n}}=\oint d\mu\,p_{n+1}(z)\,p_{n}(1/z)\,% \frac{d}{dz}\bigg{[}V(z)+V(1/z)\bigg{]}. (3.26)

As shown in [Periwal:1990qb], the r.h.s. of (3.26) may be evaluated in explicit form by means of the Migdal-Gross operatorial formalism [Gross:1989vs, Gross:1989aw]. The result is the remarkable formula

g_{s}\,(n+1)\,\frac{f_{n}^{2}}{s_{n}}=\oint\frac{du}{2\,\pi\,i}\,\mathcal{S}% \bigg{[}V^{\prime}(z_{n+1})-\frac{1}{z_{n+1}^{2}}\,V^{\prime}(1/z_{n+1})\bigg{% ]}, (3.27)

where, in the square bracket, we replace the non-commutative formal expansions

\displaystyle z_{m} \displaystyle=u\star s_{m}-f_{m-1}\star f_{m}-u^{-1}\star f_{m-2}\star s_{m-1}% \star f_{m}-u^{-2}\star f_{m-3}\star s_{m-2}\star s_{m-1}\star f_{m}-\cdots,
\displaystyle z_{m}^{-1} \displaystyle=u^{-1}\star s_{m-1}-f_{m}\star f_{m-1}-u\star f_{m+1}\star s_{m}% \star f_{m-1}-u^{2}\star f_{m+2}\star s_{m+1}\star s_{m}\star f_{m-1}-\cdots\ , (3.28)

and the \mathcal{S} operation brings the powers u^{k} to the left using the commutation relations

X\star s_{m}\star u^{k}\star Y=X\star u^{k}\star s_{m+k}\star Y,\qquad X\star f% _{m}\star u^{k}\star Y=X\star u^{k}\star f_{m+k}\star Y. (3.29)

After this procedure, we simply pick the residue in u and make commutative the \star-product. For the potentials V(z)=z^{k}/2 with k=1,2 we obtain the known recursions, see [Periwal:1990gf],

\displaystyle k \displaystyle=1,\quad g_{s}\,(n+1)\,f_{n}=\frac{1}{2}(1-f_{n}^{2})\,(f_{n-1}+f% _{n+1}),
\displaystyle k \displaystyle=2,\quad g_{s}\,(n+1)\,f_{n}=(1-f_{n}^{2})\,(-f_{n-2}f_{n-1}^{2}-% f_{n}f_{n-1}^{2}-2f_{n}f_{n+1}f_{n-1}-f_{n}f_{n+1}^{2}
\displaystyle+f_{n-2}-f_{n+1}^{2}f_{n+2}+f_{n+2}). (3.30)

Higher k may be worked out algorithmically according to the above rules. For instance, for k=3, relevant to the computation of W_{3}, we get the 26-term recursion

\displaystyle k \displaystyle=3,\quad g_{s}\,(n+1)\,f_{n}=\frac{3}{2}(1-f_{n}^{2})\,(f_{n-2}^{% 2}f_{n-1}^{3}+f_{n}^{2}f_{n-1}^{3}+2f_{n-2}f_{n}f_{n-1}^{3}+f_{n-3}f_{n-2}^{2}% f_{n-1}^{2}
\displaystyle-f_{n-3}f_{n-1}^{2}+3f_{n}^{2}f_{n+1}f_{n-1}^{2}+2f_{n-2}f_{n}f_{% n+1}f_{n-1}^{2}-f_{n+1}f_{n-1}^{2}-f_{n-2}^{2}f_{n-1}
\displaystyle+3f_{n}^{2}f_{n+1}^{2}f_{n-1}-f_{n+1}^{2}f_{n-1}-2f_{n-2}f_{n}f_{% n-1}+2f_{n}f_{n+1}^{2}f_{n+2}f_{n-1}-2f_{n}f_{n+2}f_{n-1}
\displaystyle+f_{n}^{2}f_{n+1}^{3}-f_{n-3}f_{n-2}^{2}+f_{n+1}^{3}f_{n+2}^{2}-f% _{n+1}f_{n+2}^{2}+f_{n-3}-2f_{n-2}f_{n}f_{n+1}+2f_{n}f_{n+1}^{3}f_{n+2}
\displaystyle-2f_{n}f_{n+1}f_{n+2}-f_{n+1}^{2}f_{n+3}+f_{n+1}^{2}f_{n+2}^{2}f_% {n+3}-f_{n+2}^{2}f_{n+3}+f_{n+3}). (3.31)

The longer k=4 case is reported in App. (LABEL:app:PS4).

3.2.1 Continuum limit and large N expansion

According to [Marino:2008ya], whose notation we adopt, we can make the following replacements at large N

g_{s}\,n\to z,\quad r_{n-1}\to R(z,g_{s}). (3.32)

For instance, the recursion for k=1 becomes the functional equation, cf. Eq. (4.18) of [Marino:2008ya],

z\,\sqrt{1-R(z,g_{s})}=\frac{1}{2}\,R(z,g_{s})\,\bigg{[}\sqrt{1-R(z+g_{s},g_{s% })}+\sqrt{1-R(z-g_{s},g_{s})}\bigg{]}. (3.33)

Higher k recursion as in (3.2) and (3.2) are treated in the same way and lead to structurally similar equations. An important remark is that in the potential (3.20) the additional \rho-dependent source term comes with an explicit g_{s} factor making a perturbative treatment (in both g_{s} and \rho) feasible.

The large N expansion of the free energy is achieved by standard methods, see the clean exposition in [Marino:2008ya]. We can rewrite (3.26) in the form

\displaystyle g_{s}^{2}\,F \displaystyle=\int_{0}^{t}dz\,(t-z)\,\log R(z,g_{s})+\sum_{p=1}^{\infty}g_{s}^% {2p}\,\frac{B_{2p}}{(2p)!}\frac{d^{2p-1}}{dz^{2p-1}}\left.\bigg{[}(t-z)\,\log R% (z,g_{s})\bigg{]}\right|_{z=0}^{z=t}
\displaystyle+\frac{t\,g_{s}}{2}\bigg{[}2\,\log h_{0}-\log R(0,g_{s})\bigg{]}. (3.34)

A consequence of (3.2.1) is

g_{s}^{2}\,F^{\prime\prime}(t)=\log R(t,g_{s})-\sum_{p=1}^{\infty}g_{s}^{2p}\,% \frac{B_{2p}}{2p\,(2p-2)!}\frac{d^{2p}}{dt^{2p}}\,\log R(t,g_{s}), (3.35)

which is equivalent to

F(t+g_{s})-2\,F(t)+F(t-g_{s})=\log R(t). (3.36)

Notice that (3.35) and (3.36) are weaker than (3.2.1) since they determine essentially F^{\prime\prime} and miss a linear contribution in t. In our context this is bad since the coefficients of this linear part may and do depend on \rho forcing us to use (3.2.1) with the non-trivial \rho dependent contribution from h_{0}. 101010In the later study of non-perturbative corrections this will not be an issue since an algebraic contribution to F cannot mix with a non-perturbative one.

In the gapped phase we are considering, the function R has a regular expansion in powers of g_{s} plus non perturbative contributions. The perturbative part 111111For ease of notation we do not emphasize that we are dealing with the perturbative part of R and use R\equiv R^{\rm pert}. is thus

R(z,g_{s})=\sum_{\ell=0}^{\infty}R_{\ell}(z)\,g_{s}^{\ell}. (3.37)

The coefficients R_{\ell} may be expanded and computed at first order in \rho

R_{\ell}(z)=R_{\ell,0}(z)+R_{\ell,1}(z)\,\rho+\mathcal{O}(\rho^{2}). (3.38)

The \rho independent part R_{\ell,0}(z) is well-known, see again [Marino:2008ya], and is non zero only for even \ell. The first terms read

\displaystyle R_{2,0}(z)=\frac{1}{8}\frac{z}{(1-z)^{2}},\quad R_{4,0}(z)=\frac% {9z(z+3)}{128(1-z)^{5}}\quad R_{6,0}(z)=\frac{9z\left(17z^{2}+152z+125\right)}% {1024(1-z)^{8}}, (3.39)

and so on. For the k=2 and k=3 Wilson loops we get respectively

\displaystyle R_{1,1}^{k=2}(z)=z(3z-2), \displaystyle R_{3,1}^{k=2}(z)=\frac{z^{2}(3z-5)}{8(z-1)^{3}},
\displaystyle R_{5,1}^{k=2}(z)=-\frac{9(z-3)z^{2}(3z+7)}{128(z-1)^{6}}, \displaystyle R_{7,1}^{k=2}(z)=\frac{27z^{2}\left(17z^{3}+39z^{2}-465z-375% \right)}{1024(z-1)^{9}},\dots (3.40)

and

\displaystyle R_{1,1}^{k=3}(z)=-\frac{3}{2}z\left(10z^{2}-12z+3\right),\quad R% _{3,1}^{k=3}(z)=-\frac{3z\left(30z^{3}-82z^{2}+75z-25\right)}{16(z-1)^{3}},
\displaystyle R_{5,1}^{k=3}(z)=\frac{3z\left(50z^{4}+212z^{3}-963z^{2}+766z-24% 5\right)}{256(z-1)^{6}},
\displaystyle R_{7,1}^{k=3}(z)=-\frac{27z\left(150z^{5}+826z^{4}-3483z^{3}-100% 5z^{2}+2105z-945\right)}{2048(z-1)^{9}},\dots (3.41)

The part of \log h_{0} that is linear in \rho is 121212We normalize h_{0} by its value at g_{s}=0.

\left.\partial_{\rho}\log h_{0}\right|_{\rho=0}=\frac{1}{2}\bigg{[}\frac{I_{k}% (1/g_{s})}{I_{0}(1/g_{s})}-1\bigg{]}, (3.42)

and may be easily expanded at small g_{s}. Replacing in the general formula (3.2.1) the explicit coefficients R_{\ell}(z) appearing in (3.37), we fully reproduce the genus 5 expansion obtained with the resolvent method. 131313We recall the small g_{s} expansion reconstruct the large N expansion. Besides, t=1/g in (3.2.1). The advantage of this procedure is that the functional equation (3.33) – or similar for higher k – may be easily and systematically expanded in g_{s} at any desired order by minor effort thus generating the full 1/N genus expansion.

3.3 Analytical bootstrap from finite N data

To conclude our discussion of the perturbative genus expansion in the gapped phase, we present an analytical bootstrap method that quite efficiently exploits finite N data to general long 1/N series for the winding Wilson loops W_{k}. This is a sort of educated trick whose great advantage is computational simplicity and flexibility. Indeed, it can be applied to more general observables like generic Wilson loops in higher “small” representations.

The long genus expansions that we have provided in, e.g. (3.1) and (3.1) , allow to identify strong regularities in how the coefficients W_{k,\ell}(g) in (2.2) depend on g. These regularities were first postulated in [Bessis:1980ss] for the hermitian matrix models and may be observed in the GWW model as well. As a warm-up, consider for instance the expansion of W_{1}^{\rm pert} in (3.1). It apparently takes the form

W_{1}^{\rm pert}=1-\frac{1}{2\,g}+\frac{1}{N^{2}}\,\frac{1}{8\,g\,(1-g)}+\sum_% {p=2}^{\infty}\frac{1}{N^{2p}}\frac{1}{(g-1)^{3p-2}}\,\sum_{n=0}^{p-2}c_{p,n}% \,g^{n}. (3.43)

Since we are in the gapped phase, it makes sense to expand at large g, and this gives the re-organized expansion

\displaystyle W_{1}^{\rm pert} \displaystyle=1-\frac{1}{2g}-\frac{1}{8N^{2}}\frac{1}{g^{2}}-\frac{1}{8N^{2}}% \frac{1}{g^{3}}+\bigg{(}\frac{c_{2,0}}{N^{4}}-\frac{1}{8N^{2}}\bigg{)}\,\frac{% 1}{g^{4}}+\bigg{(}\frac{4c_{2,0}}{N^{4}}-\frac{1}{8N^{2}}\bigg{)}\,\frac{1}{g^% {5}}
\displaystyle+\bigg{(}\frac{c_{3,1}}{N^{6}}+\frac{10c_{2,0}}{N^{4}}-\frac{1}{8% N^{2}}\bigg{)}\,\frac{1}{g^{6}}+\bigg{(}\frac{c_{3,0}}{N^{6}}+\frac{7c_{3,1}}{% N^{6}}+\frac{20c_{2,0}}{N^{4}}-\frac{1}{8N^{2}}\bigg{)}\,\frac{1}{g^{7}}
\displaystyle+\bigg{(}\frac{c_{4,2}}{N^{8}}+\frac{7c_{3,0}}{N^{6}}+\frac{28c_{% 3,1}}{N^{6}}+\frac{35c_{2,0}}{N^{4}}-\frac{1}{8N^{2}}\bigg{)}\,\frac{1}{g^{8}}% +\dots. (3.44)

The simple observation is that each power of 1/g receives contributions from a finite number of terms coming from the 1/N expansion. This implies that it is possible to match them with the large g expansion of the exact Wilson loop at finite N. This is easily obtained by replacing in (2.1) or (2.2) the modified Bessel function by its asymptotic expansion at large argument

I_{\nu}(x)\sim\frac{e^{x}}{\sqrt{2\,\pi\,x}}\sum_{k=0}^{\infty}\frac{(\frac{1}% {2}+\nu)_{k}(\frac{1}{2}-\nu)_{k}}{k!}\,(2\,x)^{-k}. (3.45)

After this replacement, all exponential factors cancel and we get an (asymptotic) series in 1/g whose coefficients depend on N. Some explicit examples are

\displaystyle W_{1}^{(N=2)} \displaystyle=1-\frac{1}{2\,g}-\frac{1}{32\,g^{2}}-\frac{1}{32\,g^{3}}-\frac{7% 3}{2048\,g^{4}}-\frac{25}{512\,g^{5}}-\frac{5153}{65536\,g^{6}}-\frac{149}{102% 4\,g^{7}}-\frac{2550997}{8388608\,g^{8}}+\dots,
\displaystyle W_{1}^{(N=3)} \displaystyle=1-\frac{1}{2\,g}-\frac{1}{72\,g^{2}}-\frac{1}{72\,g^{3}}-\frac{1% 7}{1152\,g^{4}}-\frac{5}{288\,g^{5}}-\frac{1897}{82944\,g^{6}}-\frac{29}{864\,% g^{7}}-\frac{1299533}{23887872\,g^{8}}+\dots,
\displaystyle W_{1}^{(N=4)} \displaystyle=1-\frac{1}{2\,g}-\frac{1}{128\,g^{2}}-\frac{1}{128\,g^{3}}-\frac% {265}{32768\,g^{4}}-\frac{73}{8192\,g^{5}}-\frac{44513}{4194304\,g^{6}}-\frac{% 899}{65536\,g^{7}}+\dots, (3.46)

and so on. Of course, the calculation leading to (3.3) is easy and fully straightforward using any computer algebra system. Any of the above expansions may be compared with (3.3) and this gives immediately c_{2,0}=-9/128. In a similar way, the coefficient of 1/g^{6} gives c_{3,1}=-225/1024 and so on. It goes without saying that these values agree with the previous expansions. If needed, one may add the information from different N to obtain further constraints. To see a less trivial example, consider the expansion of W_{3}^{\rm pert}. From inspection of (3.1), our Ansatz is now

W_{3}^{\rm pert}=\frac{(g-1)^{2}(2g-5)}{2g^{3}}+\sum_{p=1}^{\infty}\frac{1}{N^% {2p}}\frac{1}{(g-1)^{3p-2}\,g^{3}}\sum_{n=0}^{p+1}d_{p,n}g^{n}. (3.47)

Expansion of (3.47) at large g reads

\displaystyle W_{3}^{\rm pert} \displaystyle=1-\frac{9}{2g}+\bigg{(}\frac{d_{1,2}}{N^{2}}+6\bigg{)}\frac{1}{g% ^{2}}+\bigg{(}\frac{d_{1,1}+d_{1,2}}{N^{2}}-\frac{5}{2}\bigg{)}\frac{1}{g^{3}}% +\bigg{(}\frac{d_{2,3}}{N^{4}}+\frac{d_{1,0}+d_{1,1}+d_{1,2}}{N^{2}}\bigg{)}% \frac{1}{g^{4}}
\displaystyle+\bigg{(}\frac{d_{2,2}+4d_{2,3}}{N^{4}}+\frac{d_{1,0}+d_{1,1}+d_{% 1,2}}{N^{2}}\bigg{)}\frac{1}{g^{5}}+
\displaystyle+\bigg{(}\frac{d_{3,4}}{N^{6}}+\frac{d_{2,1}+4d_{2,2}+10d_{2,3}}{% N^{4}}+\frac{d_{1,0}+d_{1,1}+d_{1,2}}{N^{2}}\bigg{)}\frac{1}{g^{6}}
\displaystyle+\bigg{(}\frac{d_{3,3}+7d_{3,4}}{N^{6}}+\frac{d_{2,0}+4d_{2,1}+10% d_{2,2}+20d_{2,3}}{N^{4}}+\frac{d_{1,0}+d_{1,1}+d_{1,2}}{N^{2}}\bigg{)}\frac{1% }{g^{7}}+\dots. (3.48)

Explicit expansion of W_{3} at (small !) N=2,3 gives

\displaystyle W_{3}^{(N=2)} \displaystyle=1-\frac{9}{2g}+\frac{207}{32g^{2}}-\frac{93}{32g^{3}}-\frac{297}% {2048g^{4}}-\frac{81}{512g^{5}}-\frac{12465}{65536g^{6}}-\frac{567}{2048g^{7}}% +\dots,
\displaystyle W_{3}^{(N=3)} \displaystyle=1-\frac{9}{2g}+\frac{149}{24g^{2}}-\frac{193}{72g^{3}}-\frac{179% }{3456g^{4}}-\frac{47}{864g^{5}}-\frac{545}{9216g^{6}}-\frac{61}{864g^{7}}+\dots. (3.49)

Again, from any of the two expansions we immediately obtain d_{1,2}=\frac{15}{8} from the 1/g^{2} term after comparing with (3.3). Then, the 1/g^{3} term gives d_{1,1}=-\frac{7}{2}. Going further, we see the need for combining different values of N. The 1/g^{4} term contains d_{2,3} and d_{1,0}. Taking N=2,3 we have

\left\{\begin{array}[]{ll}\frac{1}{4}\,d_{1,0}+\frac{1}{16}\,d_{2,3}-\frac{535% }{2048}=0,\\ \frac{1}{9}\,d_{1,0}+\frac{1}{81}\,d_{2,3}-\frac{445}{3456}=0,\cr\omit\span% \@@LTX@noalign{ }\omit\\ \end{array}
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