Strong coupling expansion of free energy and BPS Wilson loop in $\mathcal N=2$ superconformal models with fundamental hypermultiplets

As a continuation of the study (in arXiv:2102.07696 and arXiv:2104.12625) of strong-coupling expansion of non-planar corrections in $\mathcal N=2$ 4d superconformal models we consider two special theories with gauge groups $SU(N)$ and $Sp(2N)$. They contain $N$-independent numbers of hypermultiplets in rank 2 antisymmetric and fundamental representations and are planar-equivalent to the corresponding $\mathcal N=4$ SYM theories. These $\mathcal N=2$ theories can be realised on a system of $N$ D3-branes with a finite number of D7-branes and O7-plane; the dual string theories should be particular orientifolds of $AdS_5\times S^5$ superstring. Starting with the localization matrix model representation for the $\mathcal N=2$ partition function on $S^4$ we find exact differential relations between the $1/N$ terms in the corresponding free energy $F$ and the $\frac{1}{2}$-BPS Wilson loop expectation value $\langle\mathcal W\rangle$ and also compute their large 't Hooft coupling ($\lambda \gg 1$) expansions. The structure of these expansions is different from the previously studied models without fundamental hypermultiplets. In the more tractable $Sp(2N)$ case we find an exact resummed expression for the leading strong coupling terms at each order in the $1/N$ expansion. We also determine the exponentially suppressed at large $\lambda$ contributions to the non-planar corrections to $F$ and $\langle\mathcal W\rangle$ and comment on their resurgence properties. We discuss dual string theory interpretation of these strong coupling expansions.


Introduction and summary
An important problem in understanding detailed workings of AdS/CFT duality is to study 1{N corrections to superconformal gauge theory observables and their matching to string loop corrections. BPS Wilson loop in N " 4 super Yang-Mills theory provides a remarkable example when its expectation value xWy as a function of N and λ " g 2 YM N can be found exactly [1]. Expanding first at large N and then at large λ one finds in the SU pN q theory xWy N"4 SUpNq " e λ 8N p1´1{N q L p1q N´1`´λ 4N˘" N e where we expressed the result in terms of the string coupling and tension of the dual AdS 5ˆS 5 string theory (1. 2) As was argued in [2], the particular structure (1.1) of the small g s , large T expansion of xWy is indeed expected on the string-theory side and may apply also to other closely related theories with less supersymmetry. Indeed, the same expansion (1.1) was found recently for two special N " 2 4d superconformal models -SU pN qˆSU pN q "orbifold" [3] and SU pN q "orientifold" [4] that are planar-equivalent to N " 4 SYM theory. Here the localization approach [5,6] allows one to expresses the expectation value xWy in terms of a non-trivial matrix model integral. One is then able to extract the large λ behaviour of the leading non-planar 1{N correction, finding that it scales as λ 3{2 relative to the planar (i.e. N " 4 SYM) term, in agreement with (1.1).
The aim of the present paper is to consider two other (SU pN q and Spp2N q) examples of N " 2 "orientifold" superconformal models for which xWy can be also computed using the localization matrix model of [5] (see also [7,8,9]). These models are still planar-equivalent to N " 4 SYM but in contrast to the "orientifold" model studied in [4] (N " 2 vector multiplet coupled to hypermultiplets in symmetric and in antisymmetric SU pN q representation) will contain a finite (N -independent) number n F of hypermultiplets in the fundamental representation. The later are effectively related to the presence of (a finite number of) D7-branes in the dual string theory description and thus to a different type of the orbifold/orientifold of AdS 5ˆS 5 string theory than in the previous case of n F " 0 [10,11,12,13]. We shall find that here the structure of the large N , large λ expansion of the BPS Wilson loop expectation value xWy will be different from (1.1), raising an interesting question of how to explain this on the dual string theory side.

Review of N " 2 models
Let us first review 4d N " 2 superconformal gauge theories we are interested in. The condition of conformal invariance of an SU pN q model with a number of hypermultiplets in the adjoint, fundamental, rank-2 symmetric, and rank-2 antisymmetric representations is [14,15] SU pN q : β 1 " 2N´2N n Adj´nF´p N`2q n S´p N´2q n A " 0 . (1. 3) The non-zero number of adjoints can only be n Adj " 1 when we find the N " 4 SYM (n F " n A " n S " 0). For n Adj " 0 we get N " 2 superconformal models with n F " 2N´pN`2q n S´p N2 q n A . To have planar equivalence with N " 4 SYM (and thus a relatively simple AdS dual) the number n F should not depend on N . This implies that n S`nA " 2 and thus there are only two non-trivial solutions that we shall refer to as "SA" (symmetric+antisymmetric) and "FA" (fundamental+antisymmetric) models SU pN q : SA : pn F , n S , n A q " p0, 1, 1q , FA : pn F , n S , n A q " p4, 0, 2q . (1.4) Both N " 2 theories are dual to certain orbifold/orientifold projections of AdS 5ˆS 5 superstring [13] and for that reason we shall refer to them respectively as the "SA-orientifold" and the "FAorientifold". It is the SA-orientifold model that was discussed in [4] and here we shall study the FA-orientifold model. For completeness, let us recall that the 4d conformal anomaly a and c coefficients of an N " 2 superconformal model are determined by the free-theory values, i.e. in terms of the total number of the vector multiplets and hypermultiplets (counting also dimensions of their representations): a " 5 24 n v`1 24 n h , c " 1 6 n v`1 12 n h . The resulting explicit values are given below SU pN q a c N " 4 SYM Similarly, in the case of the Spp2N q gauge group the condition of conformal invariance of the N " 2 model containing the adjoint, fundamental and antisymmetric hypermultiplets reads [14] (cf. (1.3)) 1 Spp2N q : β 1 " 2N`2´p2N`2q n Adj´nF´p 2N´2q n A " 0 . (1.5) The Spp2N q N " 4 SYM theory corresponds to n Adj " 1, n F " n A " 0. For n Adj " 0 demanding planar equivalence to N " 4 SYM implies that n F should be independent of N and thus the only solution is the FA-orientifold model with n F " 4, n A " 1 Spp2N q : FA : pn F , n A q " p4, 1q . (1.6) The corresponding conformal anomaly coefficients are given below:

Summary of the results
Let us now summarise the main results of this paper starting with the SU pN q case. As in the case of the SA-orientifold [4] the structure of the localization matrix model implies that the leading 1{N corrections to the Wilson loop expectation value can be expressed in terms of the corresponding 1 In this paper we shall denote by Spp2N q the compact symplectic group U Spp2N q " U p2N q X Spp2N, Cq (sometimes also denoted as SppN q) so that Spp2q " SU p2q. The dimensions of its adjoint, fundamental and antisymmetric representations are, respectively, dim Adj " dimrSpp2N qs " N p2N`1q, dim F " 2N , dim A " N p2N´1q´1. Note while the groups Spp2N q and SOp2N q and their representations are formally related by N Ñ´N [16], the index of a representation that enters the 1-loop beta-function is always positive (i.e. its sign is changed at the same time with taking N Ñ´N ). Thus the conformal invariance condition is not invariant and has different solutions for the two groups. For example, the antisymmetric representation of Spp2N q is mapped to the symmetric traceless representation of SOp2N q with the index 2N`2 which is larger than the index of the adjoint SOp2N q representation 2N´2. Thus there are no SOp2N q conformal theories with hypermultiplets in the symmetric traceless representation [14].
corrections to the gauge theory free energy F pλ, N q "´log Z on 4-sphere. For that reason the main effort goes into the study for the large N expansion of F .
To recall, in the case of the SU pN q N " 4 SYM theory where the partition function Z is given by the Gaussian matrix model [1,5] one finds (after subtracting the "trivial" UV divergence in a particular scheme, see Appendix A) [17,18,19,20] SU pN q : where CpN q (given by (A. 6) or by log of Barnes function) does not depend on λ. 2 The large N expansion of the free energy of the N " 2 FA-orientifold model which is planar-equivalent to the N " 4 SYM may be represented as SU pN q : F pλq " F N"4 pλq`N F 1 pλq`F 2 pλq`Op 1 N q . (1.8) The F 1 term was absent in the case of the SA-orientifold in [4] (it is related to the presence of the fundamental hypermultiplets in the spectrum of this N " 2 model). F 1 admits an explicit integral representation in terms of Bessel functions (3.14) allowing to find its strong coupling expansion where A is the Gleisher's constant. 3 There is just a finite number of "polynomial" in large λ corrections and an infinite number of exponential e´p 2n`1q ? λ corrections reflecting the asymptotic nature of the strong coupling expansion (see (6.19); here we omit the λ´1 {4 prefactor of e´? λ ).
F 2 may be written as the sum of the two different contributions: a simpler one r F 2 which is related to F 1 by a differential relation and a more complicated oneF 2 which turns out to be the same as the leading 1{N 2 correction to F in the SA-orientifold case in [4] where p...q 1 " d dλ p...q. As a result, 4 where the values of f i were given in (1.10). The form of the exponential corrections in r F 2 follows from those in F 1 and the relation in (1.11), and similar corrections are expected inF 2 .
2 Since the large N expansion of CpN q contains log N term (cf. footnote 29), its comparison with string theory would require a non-perturbative definition of the latter. Given that F is scheme-dependent, CpN q may be in principle eliminated by changing the scheme (e.g. by redefining the matrix model measure). We shall ignore this λ-independent constant in what follows. 3 Note that log 2 in f1 originates from the Dirichlet η-function value ηp1q " ř 8 k"1 p´1q k´1 k " log 2 (see (6.3)). 4 The analysis in [4] showed that the leading large λ term inF2 is definitely λ 1{2 . The derivation of its coefficient k1 " 1 2π was based on partially heuristic analysis of the determinant of an infinite matrix, whose matrix elements admit an asymptotic expansion for large λ. A comparison with Padé resummation of the determinant revealed that 1 2π may actually be a lower estimate of the exact value of k1. This issue will not be relevant for the large λ expansion in the models considered here where r F2 is dominant overF2 at large coupling.
The large N expansion of the circular 1 2 -BPS Wilson loop expectation value in this N " 2 theory can be written as where W 0 and W 0,2 are the leading N " 4 SYM contributions following from (1.1) [21, 1] while W 1 and W 2 are the genuine N " 2 corrections. As we will show, they can be expressed in terms of the 1{N corrections F 1 and F 2 to the free energy (1.8) by the following remarkable differential relations (cf. (1.11)) (1.17) Using (1.9)-(1.14) in (1.17) and normalizing to the leading planar value we then find for the strong coupling expansions of W 1 and W 2 Like F 1 in (1.8), the W 1 term in (1.15) was absent in the case of the SA-orientifold in [4] (where there were no odd powers in 1{N series). Also, in the SA-orientifold case the expansion of W 2 {W 0 started with the k 1 λ 3{2 term that originated from theF 2 term in (1.13) in view of (1.17). The expressions (1.19), (1.20) also contain exponential corrections as follows from (1.9),(1.12) and (1.17). Similar results are found in the case of the Spp2N q FA-orientifold model (1.6) which is more tractable as the corresponding localization N " 2 matrix model is simpler than in the SU pN q case.
It turns out that the structure of the corresponding matrix model implies that F 1 , F 2 and F 3 can be expressed in terms of the function F 1 in (1.8) (and its integral r F 2 in (1.11)) that appeared in the SU pN q case Similar expressions in terms of derivatives of F 1 appear to exist also for higher F n terms in (1.21). 5 Here we shall use the same definition for λ as in the SU pN q case, i.e. λ " g 2 YM N (i.e. without extra factor of 2 as, e.g., in [22]).
Computing the strong-coupling expansion of F n we find that (cf. (1.8),(1.9),(1.12)) where ∆F pol stands for the polynomial in λ part of the strong coupling expansion. Note that log λ term in (1.25) receives contributions only at orders N 2 , N and N 0 while the λ´1 term appears only at order N .
Remarkably, the sum of the leading large λ terms in ∆F pol at each order in 1{N appears to have a closed log expression (f 1 " log 2 4π 2 as in (1.10)) (1.26) Combined with the N 2 log λ term in (1.25) the leading strong-coupling expression for F is then Spp2N q : xWy " xWy N"4`∆ xWy , As in the SU pN q case, one finds that the N " 2 corrections W 1 and W 2 are expressed in terms of F 1 " 2F 1 and F 2 as in (1.17) so that Comparing W 1 and W 0 with W 1 and W 0 in the SU pN q case in (1.16),(1.17) we conclude that their ratio is the same for any λ. The analog of the strong-coupling expansions in (1.19),(1.20) is 6 Similar relations between higher order 1{N terms F n in free energy (1.21) and W n in (1.28) are expected also in general, with the dominant large λ term in F n determining the strong coupling asymptotics of W n . In particular, This may be compared with similar exponentiation of the leading large λ terms in the N " 4 SYM case: as one finds from (1.1) in SU pN q case [1] and from (1.29) in the Spp2N q case (see Appendix C) SU pN q : where W 0 is given by (1.18). Note that the p1`λ 1{2 8N q prefactor that generates odd powers of 1{N in the expansion of xWy N"4 in Spp2N q case in (1.37) can be absorbed into e ? λ in W 0 by shifting N Ñ N`1 4 in the definition of λ " g 2 YM N (assuming one keeps only the leading large λ term at each order in 1{N ). 7

Comments on dual string theory interpretation
Let us now discuss string theory interpretation of these strong-coupling expansions derived on the gauge theory side. The SU pN q FA-orientifold (i.e. the N " 2 SU pN q superconformal model with n F " 4 and n A " 2) may be engineered in flat-space type IIB superstring as a low-energy limit of the worldvolume theory on a stack of coincident N D3-branes in the presence of four D7-branes and one O7-plane (see [13] and references there). 8 Taking the large-N near-horizon limit of the underlying brane configuration one concludes that the dual string theory should be a projection AdS 5ˆS 15 , S 15 " S 5 {G ori , of the original AdS 5ˆS 5 type IIB theory [13]. Here Z 2,orb of G ori " Z 2,orbˆZ2,ori acts as ϕ 1 Ñ ϕ 1`π , ϕ 2 Ñ ϕ 2`π and Z 2,ori acts as ϕ 3 Ñ ϕ 3`π on the coordinates of S 5 with the metric ds 2 5 " dθ 2 1`c os 2 θ 1 pdθ 2 2`c os 2 θ 2 dϕ 2 1`s in 2 θ 2 dϕ 2 2 q`sin 2 θ 1 dϕ 2 3 . Similarly, the dual string theory for the Spp2N q FA-orientifold (i.e. the N " 2 Spp2N q superconformal model with n F " 4 and n A " 1) corresponds [10,11] to the near-horizon limit of N D3-branes with 8 D7-branes stuck on one O7-plane, i.e. is the type IIB superstring on AdS 5ˆS 15 , S 15 " S 5 {Z 2,ori (D7 is wrapped on AdS 5ˆS 3 where S 3 is fixed-point locus of Z 2,ori ). In both SU pN q and Spp2N q cases, the presence of D7-branes introduces the new D3-D7 open string sector (with massless modes being related to the fundamental hypermultiplets in the corresponding gauge theory). That means, in particular, that the dual string theory perturbation theory will involve both closed-string and open-string world-sheet topologies, i.e. corrections of both even and odd powers in g s , corresponding to even and odd powers of 1{N on the gauge theory side.
While in the SU pN q N " 2 model one expects contributions from only orientable surfaces (with topologies of 2-sphere with holes and handles) in the Spp2N q case there should be additional contributions with non-orientable crosscups (as is also suggested by the structure of the 1{N expansion of perturbative gauge theory diagrams, cf. [7]). In the Spp2N q N " 4 SYM case all odd-power 1{N contributions should come from crosscups [23], while in the Spp2N q N " 2 FA-orientifold model there should be additional contributions from world sheets with boundaries introduced due to the presence of D7-branes (and related to the presence of fundamental hypermultiplets on the gauge theory side), see also [24].
Accounting for the open string (type I, or disc) term in the dual string theory effective action that here may be interpreted as the D7-brane world-volume action allowed to give [25,26] the holographic interpretation of the order N term in the (super)conformal anomalies of the Spp2N q FA-orientifold (cf. table below eq. (1.6)).
The AdS/CFT duality suggests that the conformal gauge theory free energy F on S 4 should be matched with the string partition function Z str in AdS 5ˆS 15 . The leading 2-sphere topology contribution to the (properly defined) Z str is approximated by the type IIB supergravity action (plus α 1 -corrections). In particular, in the maximally supersymmetric N " 4 SU pN q SYM case one can match the leading N 2 term in the free energy F " 4a logpΛ rq`f 0 , a " 1 4 pN 2´1 q, with the leading term in the supergravity action proportional to the (IR divergent) volume of AdS 5 (reproducing, in particular, the conformal anomaly [27,28]). Here Λ is a UV cutoff, r is the radius of S 4 and f 0 is a regularization scheme dependent constant (cf. (A.2)). In the particular scheme selected by the localization matrix model representation for the gauge-theory partition function Z " e´F (with the λ-independent measure) one finds that F N"4 "´1 2 pN 2´1 q log λ. Then the leading N 2 term in F N"4 can be matched [19] with the on-shell value of the supergravity term in the string effective action in AdS 5ˆS 5 (assuming particular IR cutoff in the AdS 5 volume). 9 The subleading 1 2 log λ term should come from the 1-loop (torus) contribution to Z str , which is again proportional to the regularized AdS 5 volume and receives contributions only from short multiplets, i.e. is the same as the 1-loop supergravity correction [29].
The localization matrix model result for the large N , large λ expansion of the free energy of the SU pN q FA-orientifold model in (1.8)-(1.14) may be written as where k 1 2 " k 2`p3 , k 1 3 " k 3`p4 . The leading 1{N terms in the Spp2N q FA-orientifold case are similar (see (1.25),(1.26), (1.27)).
Let us note that in the SU pN q case the´2 log λ term in (1.38) has the coefficient 1 In the Spp2N q case the analog of this coefficient in (1.25) is 1 2 N 2`1 2 N´3 16 . Thus in both cases not only the N 2 term (as expected from the planar equivalence) 10 but also the order N term 9 On the AdS5 side the IR cutoff ℓ is measured in units of the AdS5 radius L and is related to the product of the radius r of S 4 and UV cutoff Λ as rΛ " Lℓ α 1 " ? λ ℓ L [19]. Then the regularized AdS5 volume (with power ℓ n divergences dropped) scales as log ℓ L Ñ´log ? λ`logpΛrq, suggesting that F " 4a logpΛrq`... Ñ´2a log λ`.... 10 In the case of the N " 4 SYM theory with the group Spp2N q which may be viewed as an orientifold projection of U p2N q theory and which is dual to type IIB string on AdS5ˆRP 5 [23] the presence of the O3-plane (carrying RR charge of 1 4 ) leads to the effective shift of N by 1 4 and thus to the expression L 4 " 4πgsp2N`1 2 qα 12 for the AdS radius. As a result, one reproduces both leading N 2 and N terms in the conformal anomaly from the on-shell value of the 10d supergravity action [26,22]. For example, the N " 4 Spp2N q SYM free energy in (1.22) may be written as F "´N 2 log λ´1 2 N log λ or as F "´1 4 " p2N`1 2 q 2´1 4 ‰ log λ. From the flat space perspective, the shift N Ñ N`1 4 may be equivalently attributed to the crosscup contributions (cf. [26]). One may also interpret the odd-power 1{N terms in the Wilson loop expectation value of the N " 4 Spp2N q theory [7] (see (7.35), (7.36)) as is the same as in the a-anomaly coefficients of the two theories (see the tables below eq. (1.4)). At the same time, the order N 0 coefficient of log λ in the Spp2N q case does not match the one in the conformal anomaly. This is not surprising: as discussed in Appendix A below, in contrast to what happens in the N " 4 SYM case, in the N " 2 theory cases there is no a priori reason why the log λ term in the strong-coupling limit of the free energy derived from the localization matrix model should have the conformal a-anomaly as its coefficient.
Rewriting (1.38) in terms of the dual string theory coupling and string tension as defined in (1.2) we get (renaming coefficients to absorb factors of 2 and π) 11 F pT, g s q The leading (2-sphere) term in the tree-level string theory effective action 1 ..q evaluated on the AdS 5ˆS 15 background is expected to match the 1 g 2 s term in (1.39) (after using, as in the N " 4 SYM case [19], the IR cutoff related to T in the AdS volume).
The 1 gs term in (1.39) should come from the disc contribution, and, in the Spp2N q case, also from the crosscup topology. In particular, one may expect the T 2 gs log T term to originate from the curvature squared term 1  ? g but this term should cancel against the orientifold (crosscup) contribution (cf. [30]), so that the leading term in the D7-brane action should be the above curvature-squared term. The order g 0 s terms in (1.39) should come from the closed-string (torus) and open-string (annulus or disc with crosscup) 1-loop corrections. Since the compact S 15 part of the background is not smooth (orbifold action has fixed points) they may originate from "localized" contributions (rather than "extensive" contributions proportional to the volume of AdS 5ˆS15 like terms in the local part of the string effective action).
The resummed expression for leading strong coupling terms in the free energy of the Spp2N q theory (1.25),(1.26) written in terms of the string coupling and string tension in coming from the crosscup contributions, but they can also be formally generated (at least in the large λ expansion) by shifting N Ñ N`1 4 in the semiclassical string tension prefactor e 2πT (2πT " ? λ " L 2 α 1 with g 2 YM " 2ˆ4πgs, λ " g 2 YM N ) of the even-power 1{N terms in (1.37) (we thank S. Giombi for a discussion of this issue). 11 In contrast to the N " 4 SYM case, in the N " 2 Spp2N q case we shall assume that N is not shifted in the definition of AdS5 radius and string tension and will also ignore possible extra factor of 2 in the relation between gs and g 2 YM .
Remarkably, the leading log term (dots stand for terms that are subleading in 1{T at each order in g s ) has non-trivial dependence only on the string coupling. The special´π 8gs term (that also depends only on g s ) should be a particular crosscup contribution. The exponential corrections should have a world-sheet instanton interpretation, i.e. should be related to world sheets wrapping compact S 2 parts of S 15 that are non-contractable and thus stable due to orbifolding (see also discussion in section 6. Expressed in terms of the string coupling and tension in (1.2) the leading strong coupling terms in (1.41) become The computation of xWy on the string side should proceed in a similar way as for the circular loop in the AdS 5ˆS 5 case [31,2] (the minimal surface ending on a circle at the boundary of AdS 5 is the same AdS 2 one). The crucial difference is the presence of a new open-string sector and thus extra "disc with holes" and also (in the Spp2N q case) "disc with crosscups" diagrams, in addition to the "disc with handles" ones. In the SU pN q case the structure of subleading terms in (1.41), (1.42) is different compared to the N " 4 SYM case in (1.1). In particular, the order g 0 s term in (1.42) should correspond to the annulus contribution (with one boundary with Dirichlet and one -with Neumann boundary conditions).
The prediction (1.35) for the resummation of the leading large λ terms in the Spp2N q theory is the following specification of (1.42) 2πT`1´l og 2 π g s˘‰`. .. , (1.43) where we used (1.18) and f 1 " log 2 4π 2 from (1.10). Note that the structure in the exponent that involves a function of 1`c g s is similar to the one of the first log term in the free energy in (1.40). The expression (1.43) may be compared with the leading-order one in the case of, e.g., SU pN q N " 4 SYM theory (1.36) (the Spp2N q result (1.37) is similar, cf. footnote 10) that should represent the sum of handle insertions on the disc [2]. Similarly, (1.43) should be summing up the leading crosscup insertions. Finally, let us note that the exact in λ differential relations like (1.17), (1.31) between the 1{N corrections to the free energy and the Wilson loop expectation value that we find from the localization matrix model representation on the gauge theory side appear to be very non-trivial on the dual string theory side where F and xWy are computed using quite different procedures. It would be interesting to uncover their string theory interpretation.
The rest of this paper is organized as follows. We shall first discuss the SU pN q case. In Section 2 we shall review the structure of the matrix model representation for the partition function of the N " 2 superconformal FA-orientifold theory. In Section 3 we shall find the explicit representations for the leading non-planar corrections F 1 and F 2 to its free energy.
In Section 4 we shall discuss the matrix model representation for the Wilson loop expectation value xWy and in Section 5 find the general relations between the 1{N terms in xWy and the free energy F . Section 6 will contain the results of the strong-coupling expansion of the 1{N terms in xWy and F . In particular, in Section 6.3 we shall discuss the structure of exponentially small e´n ? λ corrections to the leading non-planar term in F , their resurgence properties and comment on their possible string theory interpretation. Section 7 will be devoted to a similar analysis in the Spp2N q FA-orientifold model: matrix model representation, structure of 1{N corrections to the free energy and xWy and strong-coupling expansions. This case turns out be much simpler than the SU pN q one and we are able to determine the structure of the large λ asymptotics of free energy in rather explicit way.
In Appendix A we will review the general structure of the partition function of N " 2 models as described by the localization matrix model and explain how it encodes the information about the value of the conformal anomaly a-coefficient of the N " 2 model. Appendix B will contain some details of derivation of the strong-coupling expansion of F 1 using Mellin transform. In Appendix C we will discuss the relation between the 1{N coefficients in the Wilson loop and in the free energy in the case of the Spp2N q theory and their large λ asymptotics.

Note added in v3:
The exact values of the several leading coefficients k n in (1.13) were recently found analytically in [32] (and also using a refined numerical method in [33]). In particular, k 1 " 1 8 . The estimate of k 1 as 1 2π suggested in [4] was based on an approximate analytic treatment ofF 2 , i.e. was not rigorous (see comments in footnote 4).
2 Matrix model representation for N " 2 SU pN q theory Using supersymmetric localization, the partition function of an N " 2 gauge theory on a sphere S 4 of unit radius may be written as a matrix integral over the eigenvalues tmu N r"1 of a NˆN hermitian traceless matrix m [5] (see also Appendix A) The "interacting action" S int pmq that vanishes in the N " 4 theory is non-trivial for the N " 2 theories. We will neglect the instanton contribution since we are going to consider the 1{N expansion. In the case of the N " 2 model containing hypermultiplets in the fundamental, symmetric and antisymmetric representations of SU pN q (with numbers subject to the conformal invariance condition (1.3)) one finds (see e.g. [34]) where H is given in terms of the Barnes G-function 12 Hpxq " We will normalize the N " 2 partition function (2.1) to its N " 4 SYM value. After scaling the matrix m Ñ a according to the normalized partition function of the FA-orientifold in (1.4) (n F " 4, n S " 0, n A " 2q may be written as where ζ 2i`1 " ζp2i`1q are the Riemann ζ-function values. Z in (2.6) is related to the free energy as Expanding ∆F at large N we find that the leading N 2 term cancels due to planar equivalence 13 so that The order N term was absent in the case of the SA-orientifold [4] where n F " 0.
The weak coupling expansions of F 1 and F 2 are readily computed by doing the matrix model integrals in (2.6) (here we setλ " λ 8π 2 ) Note that the exponential prefactor in the r.h.s. of (2.4) cancels in Sint in superconformal models (with n F satisfying (1.3)). 13 Note, in particular, that at large N the number of hypers in 2 antisymmetric representations 2ˆN pN´1q 2 « N 2 is the same as in the adjoint representation N 2´1 « N 2 . (2.13) We shall see that as in the case of the SA-orientifold in [4], the large N expansion of the BPS Wilson loop expectation value can be expressed in terms of F , so it is important to study the latter first.
3 Explicit representation for free energy corrections F 1 and F 2 Following the same strategy as in [4] we can find the explicit representations of the leading and next-to-leading terms in the 1{N expansion of the free energy (2.11). To this aim, let us introduce the generating function Expanding in powers of the "sources" η i , χ i and evaluating the integrals over a gives log Xpη, χq "N`1 2 χ 1`5 8 χ 2`¨¨¨˘``3 16 η 2 1`1 5 where we assume summation over i, j " 1, ..., 8. The linear in χ terms in (3.3) have the following general form where the coefficient R i may be written as The infinite-dimensional matrices Q and r Q in (3.3) can be expressed in terms of the connected correlators of tr a n (see e.g. [35]; here xABy c " xABy´xAyxBy) xtr a 2k 1 tr a 2k 2 y c " N k 1`k2 2 k 1`k2 Γpk 1`1 2 qΓpk 2`1 2 q π pk 1`k2 qΓpk 1 qΓpk 2 q . (3.7) The matrix Q ij is same as the one that appeared in the case of the SA-orientifold in [4] Using (2.7), the leading terms in the large N expansion of the free energy ∆F in (2.11) may then be represented as where B i pλq and C ij pλq were defined in (2.8),(2.9). To compute (3.10) we may use that This leads to an explicit weak coupling expansion of the leading large N correction to the free energy: This weak coupling expansion is clearly convergent, with radius of convergence π 2 . It can be summed up into an integral representation using the identity: t 2i e t`1 . (3.13) This leads to the compact expression . (3.14) It is straightforward to check that the expansion of the Bessel J 1 function, combined with the identity (3.13), leads to the weak coupling expansion in (2.12) and (3.12). However, the integral representation (3.14) can also be used to analyze the strong coupling expansion, which is an asymptotic expansion, in contrast to the convergent weak coupling expansion (3.12). The strong coupling expansion is discussed below in Section 6. The next subleading correction to the free energy, the OpN 0 q term F 2 in (2.11), may be naturally split as whereF 2 comes from the Q ij η i η j part of (3.10) (i.e. depends on C ij and Q ij ). ThisF 2 part is identical to the one for the SA-orientifold found in [4] and can be written as The properties of the weak coupling and strong coupling expansions ofF 2 pλq have been studied in detail in [4]. The second term in (3.15), denoted r F 2 pλq, comes from the r Q ij χ i χ j part of (3.10) (cf.(3.11)) It can therefore be written as a double sum: where the function B i pλq was defined in (2.8) and the coefficients r Q ij in (3.9) and we explicitly indicated summation over i, j. Thus, the weak coupling series representation for r Note that r F 2 pλq is simpler thanF 2 pλq, being only quadratic in the zeta factors ζ 2k`1 , whileF 2 pλq involves sums over products of zetas to all orders. The weak-coupling expansion of the total F 2 pλq (3.15) of course agrees with the direct expansion of F 2 pλq at weak coupling in (2.13).
Remarkably, there is a direct differential relation between r F 2 pλq and F 1 pλq. Indeed, differentiating r F 2 pλq in (3.20) with respect to λ we observe that the double sum factorizes in terms of the second derivative of the product λ F 1 pλq with respect to λ, implying that Thus the form of r F 2 pλq is determined by that of F 1 pλq. Using (3.14) we then get also This integral representation also permits a direct access to the strong coupling expansion of r F 2 pλq.

Wilson loop expectation value
The N " 2 vector multiplet of the N " 2 theories contains the gauge vector A µ , a complex scalar ϕ, and two Weyl fermions. The 1 2 -BPS Wilson loop depends only on the fields of the vector multiplet and is defined as where the contour x µ psq represents a circle of unit radius and the trace is taken in the fundamental representation. The expectation value of W may be computed in the matrix model as (cf. (2.6)) xWy " xtr e 2πm y " Its large N expansion may be written as where we separated the N " 4 SYM parts The leading terms in the weak-coupling expansions of the N " 2 parts W 1 and W 2 are found to be   Let us find the closed form of the series for the simpler W 1 term that is linear in ζ 2n`1 . W 1 gets contributions from the single-trace term in (2.7) that were absent in the case of the SA-orientifold in [4]. If we write S int in (2.7) as S 1`S2 where S 1 " ř 8 i"1 B i pλq tr`a ? Picking up the part linear in S 1 gives (4.9) Using (3.6), we then find which agrees with (4.6). 14 Using the identity (3.13) we can resum this double series expansion into an explicit integral representation It is straightforward to verify that the expansion of the Bessel functions, combined with the identity (3.13), leads to the weak coupling expansion in (4.6). A closed expression for W 2 pλq in (4.3) will be given in the next section after relating it to the corresponding terms in the free energy.

General relations between the 1{N terms in xWy and F
The coefficients W 1 and W 2 in the large N expansion (4.3) of the Wilson loop expectation value turn out to have close relation with the F 1 and F 2 in the free energy expansion (2.11) (see also Appendix C).
To relate W 1 to F 1 let us first write (4.10) as We notice that differentiating (5.1) over λ leads to the expression where the double sum factorizes. Using the expression for F 1 in (3.12) we then obtain This relation may be written as Using also the expression for W 0 in (4.4) we conclude that 14 Let us note that doing the sum over p for each n we obtain the exact form of the coefficients of all ζ2n`1 terms atching Eq. (3.29) of [34].
The term W 2 in (4.3) turns out to be related to F 2 in (2.11),(3.15) by This can be proved in the same way as in [4] 15 by expanding the Wilson loop factor to leading order, using the large N factorization of correlators and observing that the insertion of tr a 2 is the same as the insertion of the Gaussian "action" which, in turn, can be obtained by differentiating the matrix model integral over λ.
Using that in F 2 " r F 2`F2 and (3.21) we may represent (5.6) as In view of (5.5) the first term here is thus related to the square of dW 1 dλ .
6 Strong coupling expansions of the N " 2 SU pN q free energy and Wilson Loop In this section we present results for the large λ expansions of the terms F 1 pλq and F 2 pλq in the large N expansion (2.11) of the free energy. Using the relations (5.5),(5.6) these will also determine the expansion of the terms W 1 pλq and W 2 pλq in the large N expansion (4.3) of the Wilson loop.

Large λ expansion of F 1 and F 2
The large λ expansion of the first subleading large N correction F 1 pλq in (2.11) for the free energy can be derived in several different but complementary ways. The simplest way is to use the representation where ηp2i`1q is the value of the Dirichlet η-function. Then the expansion (3.12) for F 1 yields Expanding at large λ gives an expansion that can be evaluated using ζ-function regularization Using the η-function values ηp1q " log 2 , ηp´1q " 1 4 , η 1 p´1q "´1 4´l ogp2q 3`3 log A , ηp´2q " 0, ηp´3q "´1 8 , (6.4) where A is Glaisher's constant, we thus obtain the strong coupling expansion . (6.6) Here we indicated that there is only a finite number of power-law corrections: as will be discussed below in Section 6.3 and Appendix B, all further corrections turn out to be exponentially small as λ Ñ`8. An indication of this is that all higher order corrections in (6.3) have coefficients that are expressed in terms of η-function values that vanish.
The strong coupling expansion (6.5)-(6.6) for F 1 pλq can be also obtained from the integral representation (3.14) using the Mellin transform method (see Appendix B), or by expanding the e 2πt pe 2πt`1 q 2 " ř 8 n"1 p´1q n`1 n e´p 2n´1qπt factor in the integral representation (3.14) and integrating. TheF 2 part (3.16) of F 2 in (3.15) is same as in the SA-orientifold and thus [4] The strong coupling expansion of r F 2 in (3.20) may be derived directly from (3.21) using (6.5) 16 where f i have the values listed in (6.6). Notice that, as for F 1 pλq in (6.5), there is only a finite number of power law corrections, followed by exponentially suppressed terms, whose origin is discussed below in Section 6.3.

Large λ expansion of W 1 and W 2
Using the relations (5.5), (5.6), (5.7) allows us to find the strong coupling expansions of W 1 and W 2 from those of F 1 and F 2 . In particular, from (6.5) and the expansion of W 0 in (4.4) 128λ`. ..¯, (6.10) we find (dropping exponentially suppressed parts, cf. (6.5)) Comparing (6.7) and (6.8) we observe that r F 2 dominates overF 2 at the first two leading orders of expansion in λ " 1. As a result, the dominant contribution to W 2 comes from the first term in (5.7) where we used (6.5). The contribution to (6.12) coming fromF 2 term in (5.7) is so that in total where the values of f 1 , f 2 and k 1 are given in (6.6),(6.7). 16 Note that the value of the constant term p4 can not be deduced from the differential relation (3.21) and requires separate derivation using the method of Appendix B that gives p4 " 1 16`l og 2 12`l og π 16´3 4 log A.

Exponentially suppressed corrections at large λ
The leading large N correction to the free energy F 1 pλq has, in addition to the "perturbative" terms in (6.5), also exponentially suppressed corrections in the large λ limit. These can be computed directly from the integral representation (3.14). It is actually slightly simpler to begin with the combination d 2 dλ 2 pλF 1 q which appears in the relation to W 1 as in (5.4). From the integral representation (3.14) we deduce that Both these expressions are exact, but the first expression in terms of Bessel J-functions is well suited to a small λ expansion, while the second expression in terms of Bessel K-functions is well suited to a large λ expansion. As λ Ñ`8 each Bessel K-function in (6.15) is given by the exponentially small factor e´p 2n`1q ? λ , multiplied by an asymptotic series in 1 ? λ . Thus we obtain an expansion in the form of an "instanton sum", with each exponential multiplied by a "fluctuation expansion" in inverse powers of ? λ: The reconstruction of F 1 pλq from this expansion requires two integrations, and the integration constants are easily fixed by the comparison with (6.5),(6.6). As a result, we find that F 1 in (6.5) may be represented as Here F pol 1 is the "polynomial" in λ " 1 part, with a finite number of nonzero coefficients f j as in (6.5)-(6.6), and F exp 1 is the exponentially small contribution given by . (6.18) Here the sum over n looks like an "instanton" expansion: for each n and k the incomplete Γfunction terms in (6.18) are proportional to e´p 2n`1q ? λ when λ Ñ`8. Using the expansions of these Γ-functions we find explicitly that For each n, the fluctuation series is factorially divergent, but it is resurgent in the sense that the large l behaviour is encoded in the low l terms. To see this explicitly, let us define the "fluctuation" coefficients from (6.19): . (6.20) The first few low-order values of c l are given by At large order, l Ñ 8, these coefficients are alternating in sign and factorially divergent, and including the subleading corrections the large order behaviour can be written as: Notice that the numerators of the subleading corrections correspond precisely to the low order coefficients in (6.21). The powers of 2 correspond to the difference between the two Bessel function saddles (e´x vs. e`x) whose ratio is e´2 x . Thus we see that the subleading corrections to the large-order growth of the fluctuation coefficients are directly encoded in the low-order fluctuation coefficients. This behaviour in (6.22) is the typical low-order/large-order resurgence relation [36,37,38]. These resurgence properties are inherited from the large argument expansion of the Bessel function term in square brackets in the r.h.s. of (6.15). Furthermore, this resurgent behaviour of F 1 pλq is inherited by the exponentially small corrections to the Wilson loop ratio W 1 pλq{W 0 pλq in (6.11), due to the expression (5.5) relating W 1 pλq to F 1 pλq. Similar exponential terms will appear in the strong coupling expansion of F 2 and W 2 and also in the corresponding terms in the Spp2N q theory case discussed in the next section.
The exponential e´c ? λ corrections found here in the 1{N term in N " 2 free energy are generally expected in observables in conformal gauge theory with an AdS string dual. The perturbative expansion (in inverse string tension) in 2d string sigma model is expected to be asymptotic and such corrections may have a world-sheet theory origin (which may be different in different observables). Similar terms appear, e.g., in the N " 4 SYM theory in the large λ expansion of the cusp anomalous dimension (see [39,40] and also [41,42] for their relation to resurgence).
One may conjecture that the e´p 2k`1q ? λ terms in F 1 have a string instanton interpretation in terms of world sheets wrapping part of the compact internal space S 15 that has fixed points under the orientifold/orbifold action on S 5 (see discussion in the Introduction).
It is useful to compare this with what happens in the case of the Wilson loop expectation in N " 4 SYM theory (see (1.1),(4.3),(4.4)). The large λ expansion of the Bessel I 1 function in W 0 in (4.4) leads to just two exponential terms in (6.10), with the subleading one being imaginary (the same pattern is found also for higher 1{N terms in xWy in (1.1)). While the leading e ? λ term in (6.10) represents the expansion near the minimal AdS 2 surface embedded in AdS 5 , the second term may be interpreted 17 [43,44] as the contribution of an unstable surface wrapping S 2 of S 5 . 18 Note that higher order terms " e´n ? λ do not appear, as multiple wrappings would correspond to multiply wrapped Wilson loop.
In contrast, in the case of F 1 pλq in the N " 2 theory we get an infinite series of exponential terms as here multiple wrappings should be allowed 19 and they have real coefficients as the corresponding world-sheet solutions should be stable due to orbifolding of S 5 . Note that the appearance of the imaginary term in the formal large λ expansion of W 0 is related to fact that the asymptotic expansion of the Bessel I 1 function about the dominant e ? λ term is non Borel summable: the coefficients of the expansion about e ? λ are factorially divergent and nonalternating in sign and then the naive Borel summation integral has an imaginary contribution, and this must be cancelled against the ie´? λ term as total W 0 should be real. At the same time, the exponentially small factors e´p 2k`1q ? λ in F 1 are multiplied by asymptotic series that are Borel summable (note that the c l coefficients in (6.20) are factorially divergent but alternate in sign) and therefore, one finds only real exponentially suppressed contributions.
In view of the relation (5.5) between W 1 and F 1 and the expansion of W 0 in (6.10) the resulting expression for the 1{N correction W 1 to the Wilson loop in the N " 2 theory will thus contain two different sources of the subleading exponential corrections since Thus, d dλ W 1 has a trans-series expansion involving an overall e ? λ factor, multiplied by even powers of e´? λ . These alternate between being real and imaginary, 20 in such a way that the full transseries is well-defined and real (as W 1 should be when λ is real and positive). The same structure also survives the λ-integration that gives W 1 . The resurgence properties of this final trans-series for W 1 would be interesting to study in more detail. 21 7 N " 2 superconformal Spp2N q theory Let us now repeat similar analysis in the case of the FA-orientifold model (1.6) with the gauge group Spp2N q.

Matrix model formulation
The structure of the matrix model here is the same as in (2.1). For the model with n Adj , n A and n F expressed in terms of them using the finiteness condition (1.5) the interacting action in (2.1) reads [9] (cf. (2.7) and also Appendix A) To recall, the 1{N correction F1 should be given by string path integral over surfaces of disc topology with free boundary. 20 From the string theory point of view, W1 comes from contributions of world sheets with annulus topology (with one boundary being fixed by the Wilson loop circle and the other being free). Then the argument about stability of all wrappings of subspace in S 15 (given above for F1 case) should no longer apply. 21 An alternative approach is to start directly with the integral representation for W1 in (4.11) and perform the large λ expansion, getting both perturbative and non-perturbative contributions.
where the matrix a is in the 2N -dimensional fundamental representation of Spp2N q. The expression (7.1) greatly simplifies for the FA-orientifold where n Adj " 0, n A " 1 (and n F " 4): only the singletrace term survives so that (cf. (2.7)) 22 where B i is same as in (2.8).
Perturbative calculations are most efficiently performed by the same methods as in [45] in the SU pN q case. The matrix model variable is written in a basis of spp2N q generators in the fundamental representation with the following normalization Then the matrix model measure is simply Integration is done with respect to the Gaussian weight e´t r a 2 (cf. (2.6)), i.e. it reduces to repeated Wick contractions using xa r a s y " δ rs and the Spp2N q fusion/fission relations [46,47] tr pT a M 1 T a M 2 q " tr pT a M 1 q tr pT a M 2 q " 1 4 tr pM 1 M 2 q´1 4 p´1q n 2 tr pM 1 M 2 q, (7.6) where M 1 and M 2 are products of generators, n 2 is the number of factors in M 2 , and M 2 is the product in reverse order. In particular, one finds the following useful correlators 23 xtr a 2n y " N n`1 2 1`n Γp 1 2`n q ?

Free energy
The free energy of the Spp2N q FA-orientifold has the same structure of the 1{N expansion as in (2.11), i.e. after the subtraction of the N " 4 SYM free energy we have (see (1.22)) where we included two more terms, compared to (2.11). To get the explicit expressions for the terms F 1 pλq, F 2 pλq, and F 3 pλq we repeat the analysis in Section 3 (the computation of F 4 follows similar steps). In this case we need to consider the analog of the generating function (3.2) containing only χ-part Xpχq " ż Da e´t r a 2 e V pχ,aq , V pχ, aq " Evaluating the integrals gives Using (7.7)-(7.9), this may be written as and with R i and r Q ij being the same as in (3.5) and (3.9). The free energy ∆F in (7.10) is then obtained by acting on´log X with the operator expp´B i B Bχ i q and setting χ i Ñ 0. 24 This replaces χ i Ñ´B i (cf. (3.11)) and thus ∆F pλq " Equivalently, we just start with exp "´ř 8 i"1 Bipλq tr`a ?

N˘2
i`2 ‰ (cf. (7.2)), compute its expectation value expanding in powers of Bi terms using the connected correlators in (7.7)-(7.9) and then rewrite the result as e´∆ F . The F 1 term in (7.10) is then simply (7.17) where F 1 pλq is the corresponding SU pN q term in (3.12). Thus, F 1 pλq for the Spp2N q model also has an exact integral representation of the form in (3.14) multiplied by factor of 2.
For the F 2 term we obtain where r F 2 pλq is the same as in (3.19), (3.20), (3.21). We conclude that in this Spp2N q model the F 2 term is much simpler than in the SU pN q case in (3.15) -it does not contain the analog of theF 2 term (3.16). In (7.18), the first term is linear in the ζ 2n`1 -values, while the second is quadratic. The presence of this first term is related to the different structure of the large N expansion in (7.7) that contains the 1{N term which was absent in the SU pN q case. 25 Furthermore, since F 1 pλq has a simple integral representation (3.14), and r F 2 pλq is directly related to F 1 pλq as in (3.21), we see from (7.17) and (7.18) that in the Spp2N q model both F 1 pλq and F 2 pλq have explicit integral representations that permit precise analysis of both the convergent weak coupling expansion and the asymptotic strong coupling expansion. This carries over to the Wilson loop corrections, as discussed in the next subsections.
Finally, from (7.16) we conclude that the 1{N term F 3 pλq in (7.10) is given by Using that according to (2.8) we have B i " λ i`1 and also the relation in (3.21), the expression for F 3 may be written as (cf. (7.18)) where f 1 pλq " d dλ f pλq. 25 Note, for example, that xtr a 6 y " It is possible to generalize the above computation of F 3 to the case of the next terms F 4 and F 5 in (7.10). The analog of the last term in (7.20) with highest number of powers of derivatives over λ or of highest power in pλF 1 q 2 turns out to be (cf. (7.19),(7.8),(7.9),(7.15)) .. , (7.21) .. , (7.22) where we used that, as follows from (7.8),(7.9), These terms provide the dominant contributions in F 4 and F 5 at strong coupling: F 4 " λ 4 , F 5 " λ 5 (see below). Comparing the last term in (7.20) with (7.21) and (7.22) we observe a definite pattern for generalization of these leading terms 3 rpλF 1 q 2 s k`2¯`. .. , k " 0, 1, 2, ... . (7.24) Here the k " 0 case represents the second (2F 2 ) term in (7.18) given by the integral over λ, F 2 pλq " 1 2 pλF 1 q 1´ş dλ λ rpλF 1 q 2 s 2 . It is natural to expect that the full expressions for higher order 1{N corrections F n in the free energy in (7.10) will be expressed in terms of derivatives of F 1 pλq. The integral representation for F 1 (3.14) will then imply a similar representation not only for F 2 (cf. (7.18), (3.22)) and F 3 (7.20) but also for all F n .

Strong coupling expansion of free energy
Given the relations (7.17), (7.18) and (7.20) the strong coupling expansions of the free energy terms F 1 , F 2 and F 3 in (7.10) follow from the SU pN q results for F 1 and r F 2 in (6.5),(6.6) and (6.8) and the leading terms in F 4 and F 5 from (7.21),(7.22) Here Ope´? λ q stands for the corresponding exponentially suppressed corrections " λ´k {4 e´n ? λ that follow from the ones in F 1 in (6.17), (6.19). 26 26 While F1 has exponentials that are odd powers of e´? λ , F2 (that contains squares of derivatives of F1 and cross-terms, cf. (7.18)) has both even and odd powers of e´? λ . Similarly, for F3 in (7.20) one also finds both odd and even powers of e´? λ .
We observe that the leading large λ asymptotics of F n appears to be λ n . Note also that F 3 has no log λ term while the order λ´1 term appears only in F 1 . Assuming that all higher F n terms are expressed in in terms of derivatives of λF 1 as in (7.18),(7.20),(7.21), (7.22) the only log λ corrections will come from F 1 and F 2 , i.e. the coefficient of the log λ term in F receives contributions only from the N 2 , N and N 0 orders in the 1{N expansion while the λ´1 term in F is exactly captured by (7.25).
Including also the N " 4 SYM contribution in (1.22) the full expression for the free energy expanded at large λ may be written as .. , (7.32) where ∆F pol represents the polynomial in λ " 1 contributions with Fp λ N q being the sum of the leading λ n terms at each order in 1{N .
Remarkably, the coefficients in (7.32) suggest that F has the following exact form Fp λ N q " log`1`2f 1 λ N˘. (7.33) Using that according to (1.2) we have λ N " 4πg s we conclude that this leading order term expressed in terms of string parameters non-trivially depends just on string coupling (8πf 1 " 2 π log 2) This term should be summing the leading large string tension contributions from each order in string topological expansion The term´π 2 2 N λ "´π 8 1 gs in (7.30) should also have a special origin on the string side, coming from a particular crosscup or disc contribution not involving (in contrast to the 1 gs term in (7.34)) extra powers of string tension (and thus subleading compared to (7.34) at large T ).

Wilson loop
The 1 2 -BPS Wilson loop is again defined as in (4.1). In the Spp2N q N " 4 SYM theory its expectation value (exact in N and λ defined still as λ " N g 2 YM ) is given by the sum of the Laguerre polynomials [7] The resulting 1{N expansion is The Laguerre polynomials in (7.35) are the basic ones, while in the SU pN q case in (1.1) we have the associated Laguerre polynomial arising from the sum in (7.35) without parity restriction on the index, i.e. from the identity L p1q N pxq " Then the N " 2 expectation value may be written as in (1.28) 28 (7.37) where the N " 4 parts W 0,n are given by (7.36) λq . (7.39) The relation between the genuine N " 2 parts W 1 and W 2 in (7.37) and the free energy terms in (7.10) is the same (up to factor of 1/2) as in SU pN q case in (5.5),(5.6) (see Appendix C) We thus find using (5.5) and (7.18) (cf. (5.7)) Like for the free energy in (7.20)-(7.22), these relations can be extended also to higher 1{N orders. Using (7.40),(7.41) we find for the strong-coupling expansion of the coefficients in (7.37) Note that like F n in free energy the Wilson loop coefficients W n have additional exponentially suppressed corrections " e´? λ at strong coupling, which are resurgent, and which follow directly from the exponentially suppressed corrections to F 1 pλq derived in Section 6.3. Similar relations between higher order 1{N terms F n in free energy (1.21) and W n in (1.28) are expected also in general, with the dominant large λ term in F n determining the strong coupling asymptotics of W n (see Appendix C). In particular,

A Partition function of N " 2 matrix model and conformal anomaly
Let us first recall that the conformal anomaly coefficients a and c in N " 2 superconformal models are not renormalized, i.e. are given just by their free-theory values found by summing up contributions of particular fields (see, e.g., [48]). In a model with n v vector multiplets and n h hypermultiplets one finds In particular, in the N " 4 SYM theory (n v " n h ) with group G we get a " 1 4 dim G. The free energy of a massless superconformal model on S 4 of radius r may be written aŝ where Λ is a UV cutoff, i.e. the r dependence is controlled by the a-coefficient. The free energy thus depends on a subtraction scheme and below we shall denote by F its regularized value. The localization matrix model expression for the partition function Z of N " 2 gauge theory on S 4 is [5] Z " e´F " ż Da e´8 In the N " 4 SYM case Z 1-loop paq " 1 and doing the Gaussian integral we get [17,18] (for G " SU pN q) where GpN`2q " ś N k"1 Γp1`kq is the Barnes G-function. 29 Setting r " 1 we conclude that in the subtraction scheme assumed in the localization approach F N"4 "´2a log λ (up to a λ-independent constant). This was noted in [19] and an AdS/CFT interpretation of this result was suggested.
One may wonder what happens in other N " 2 superconformal models, in particular, if the conformal anomaly a-coefficient is also encoded the log λ term of the large λ expansion of the free energy F on S 4 . For the models that are planar-equivalent to N " 4 SYM this is certainly the case at the leading N 2 order but as we shall see below this does not need to be be true at subleading orders in 1{N .
For an N " 2 model with a collection of hypermultiplets in representation R " 'R i of a group G with algebra g one finds [5] 30 29 Note that the large N expansion of CpN q may be written as CpN q " 1 4 N 2 " 3`4 logp4πq ‰´N " log N`logp2πq´1 ‰´5 12 log N`OpN 0 q. 30 We ignore the instanton contribution since it is exponentially suppressed in the 1{N expansion we are interested in here. Ẑ 1-loop coming from the ratio of 1-loop determinants on S 4 in a constant scalar a background does not depend on λ but does depend on r. Note that the product over roots here includes also the "massless" contributions of the zero roots corresponding to Cartan directions for which α¨a " 0 (same also applies to the product over weights in the case of the adjoint representation).
The regularized value ofẐ in (A.7) used in [5] was where Hpxq " Gp1`xq Gp1´xq is the product of the Barnes G-functions. Notice that here the contribution of the "massless" terms present in (A.7) is trivial as Hp0q " 1. As a result, the contribution of (A.8) to the log r term in F or to the conformal anomaly is trivial -the r dependence can be absorbed into the rescaling of the integration variable a in (A.3) and this the resulting Z will depend on r in the same way (A.5) as in the N " 4 SYM case.
To properly account for the conformal anomaly of the N " 2 model we need to go back to the original unregularized expression (A.7) and compute its dependence on the radius r. Rearranging (A.7) using that where µ stands for α¨a or w¨a, we conclude that the non-trivial dependence on r (that cannot be absorbed into a) is captured by the infinite product factor that can be defined using the standard Riemann ζ-function regularization as Redefining ra Ñ a to account for the dependence on r in the free action in (A.3) and in Z 1-loop pa rq we need also to include the contribution of the Gaussian measure or the N " 4 term in (A.5), so that the total r dependence of the N " 2 free energy is (cf. (A.2)) F " " dim G´1 6 pdim G´dim Rq ‰ log r`... " 4a log r`..., a " 5 24 dim G`1 24 dim R , (A. 12) in agreement with the general expression for the a-anomaly in (A.1).
We have thus shown that it is the "bare" expression for the matrix model integral (A.3) using (A.7) that correctly includes the conformal a-anomaly term in free energy. It is clear that the direct correlation between the dependence on r and on λ is a feature of only the Gaussian part of the integral in (A.3). In particular, the dependence of the N " 2 free energy on log λ beyond the leading planar limit need not be controlled by the a-anomaly coefficient as that happened in the N " 4 SYM case in (A.5).
Nevertheless, we have found (see discussion below (1.38)) that not only the order N 2 but also the order N coefficient of the log λ term in the large λ limit of the free energies of the SU pN q 31 Here we use that the total number of roots counting also the trivial Cartan ones is the same as dim G. and Spp2N q FA-orientifold theories computed in this paper do agree with the corresponding terms in the conformal a-anomalies. We suspect that the matching of the order N term should be also related to the fact that these models are planar-equivalent to N " 4 SYM theory.
B Derivation of large λ expansion of F 1 using Mellin transform In the main text, we computed the large λ expansion of F 1 using the approach described in (6.1)-(6.3). Here we shall compute the large λ expansion of F 1 given by the integral representation (3.14) by applying the Mellin transform method (see e.g. [49,50]). The first step is to rewrite (3.14) in the form of a Mellin convolution hpxq " pf ‹ gqpxq " The Mellin transform is r hpsq " Mrhspsq " ş 8 0 dx x s´1 hpxq " r f psq r gp1´sq. If α ă s ă β is the fundamental strip of analyticity of r hpsq, the asymptotic expansion of hpxq for x Ñ 8 is obtained from the poles of its Mellin transform in the region s ě β. In particular, the pole 1 ps´s 0 q n gives a term p´1q n pn´1q! 1 x s 0 log n´1 x in the asymptotic expansion of hpxq. Explicitly, let us first put (3.14) in the equivalent form 32 ?
xWy " xWy N"4`W 1`1 N W 2`1 N 2 W 3`O p 1 N 3 q , (C. 13) we see that derivatives of both F n and F n`1 terms in ∆F in (C.9) contribute to W n . In particular, B B k F 1 " 2R k contributes to the order N (planar) part of xWy while for W 1 we find (C.14) where p...q 1 " B λ p...q. Since B j " λ j`1 , differentiating W 1 over λ gives  pj`1qpj`2qR j B j "´λ 8 W 0 pλF 1 q 2 , (C. 15) where W 0 " 4 ? λ I 1 p ? λq as in (7.38). This demonstrates the relation in (7.40). Similarly one can show also that W 2 "´λ 2 8 W 0 F 1 2 . The example of W 2 suggests that the dominant at large λ term in W n comes from the dominant term in the corresponding F n . Indeed, from (C.9) and the expression for the dominant term in F 3 in (7.19) we get for the leading order large λ contribution This is indeed the leading at large λ term in the exact expression for W 2 in terms of F 1 " 2F 1 in (7.41).
Applying the same logic to find the large λ contribution in W 3 we use the expression for the dominant term in F 4 in (7.21) where c ijkm is given in (7.23