Probing non-perturbative effects in M-theory on orientifolds

Using holography, we study non-perturbative effects in M-theory on orientifolds from the analysis of the S^3 partition functions of dual field theories. We consider the S^3 partition functions of N=4 Yang-Mills theory with O(n) gauge symmetry coupled to one (anti)symmetric and N_f fundamental hypermultiplets from the Fermi gas approach. In addition to the worldsheet instanton and membrane instanton corrections to the grand potential, which are also present in the U(n) Yang-Mills case, we find that there exist"half instanton"corrections coming from the effect of orientifold plane.


Introduction
In the last few years, we have witnessed a tremendous progress in our understanding of non-perturbative effects in M-theory. In particular, in the case of N = 6 U (N ) k × U (N ) −k ABJM theory, which is holographically dual to M-theory on AdS 4 × S 7 /Z k , we have a complete understanding of non-perturbative corrections thanks to the relation to topological string on local P 1 × P 1 [1] (see [2] for a review). Especially, the grand partition functions of ABJM theory at k = 1, 2, 4 are completely determined in closed forms [3,4]. However, for lower supersymmetric theories we still do not have a detailed understanding of non-perturbative effects in M-theory. d = 3 N = 4 theories are particularly interesting since these theories are sometimes related by mirror symmetry exchanging the Higgs branches and Coulomb branches [5]. Such N = 4 theories naturally appear as the worldvolume theories on M2-branes probing ALE singularities. For instance, N M2-branes probing A N f −1 ALE singularities have two descriptions related by mirror symmetry: U (N ) Yang-Mills theory with one adjoint and N f fundamental hypermultiplets, and a A N f −1 quiver gauge theory [5,6]. The former theory appears as a worldvolume theory on N D2-branes in the presence of N f D6-branes, and the latter description comes from the M-theory lift of D6-branes as Taub-NUT space. In the large N limit, these theories are holographically dual to M-theory on AdS 4 × S 7 /Z N f , and we can study non-perturbative effects in this background from the S 3 partition function of the field theory side. In [7][8][9], the S 3 partition function of these theories, known as the N f matrix model, are studied using the Fermi gas formalism [10]. It turned out that the instanton corrections in the N f matrix model are quite different from those in the ABJM theory and have a very intricate structure [8]. Instanton corrections in more general N = 4 quiver gauge theories are also studied in [11][12][13], but it is fair to say that we are far from the complete picture.
In this paper, we will study a natural generalization of N f matrix model: S 3 partition functions of d = 3 N = 4 O(n) or U Sp(n) Yang-Mills theories with N f fundamental and one (anti)symmetric hypermultiplets, first studied in [9] using the Fermi gas approach. In the Type IIA brane constructions, such models appear as worldvolume theories on N D2-branes in the presence of N f D6-branes and a orientifold plane. We will denote the model of gauge group G with N f fundamental and one symmetric (or anti-symmetric) hypermultiplets as G + S (or G + A), respectively.
By mirror symmetry, the U Sp(2N ) + A model is dual to a D N f quiver gauge theory 1 [5,6], which can be interpreted as the worldvolume theory on M2-branes probing the D N f ALE singularity. In the large N limit, this theory is holographically dual to M-theory on AdS 4 × S 7 /Γ D N f where Γ D N f is the dihedral subgroup of SU (2). This opens up an avenue to study M-theory on orientifolds from the analysis of S 3 partition functions of dual field theories. In particular, we can study the effects of orientifold plane in the M-theoretic regime where the string coupling g s of Type IIA theory becomes large. Orientifolds in Type IIB theory can be described in F-theory, while the strong coupling behavior of Type IIA orientifolds is still poorly understood. Our work is a first step towards the understanding of non-perturbative effects in M-theory on orientifolds 2 .
We find that the U Sp(n) + A(or S) model is related to the O(n) + A(or S) model by a shift of N f , hence it is sufficient to consider the O(n) case only. For the O(n) + A model we find that there are three types of instantons: worldsheet instantons, membrane instantons, and "half instantons". The first two types have direct analogues in the N f matrix model, while the last type is a new one coming from the effect of orientifold plane. In the Fermi gas picture, orientifolding corresponds to the reflection of fermion coordinate x → −x, and the "half instantons" can be naturally identified as the contribution of the twisted sector of this reflection. We find that the sign of this contribution depends on the parity (−1) n of the gauge group O(n). On the other hand, we could not find a clear picture of the instanton corrections in the O(n) + S model. This paper is organized as follows: In section 2, we first review the Fermi gas formalism of the S 3 partition functions of G + A or G + S models [9]. Then we explain our algorithm to compute the partition functions of these models exactly.
In section 3, we determine the coefficients C, B and A in the perturbative part of grand potential (3.1). The results are summarized in Table 2.
In section 4, we study the non-perturbative corrections to the grand potential using our data of exact partition functions. For the O(n) + A model, we find the first few coefficients of instanton corrections as a function of N f . We also comment on the instanton corrections in the O(n) + S model.
In section 5, we compute the WKB expansion of grand potential using the density matrix operator in [14], and reproduce the coefficients C, B and A for the O(2N + 1) + A model.
In section 6, using the different form of operator in [9], we compute the WKB expansion of the "twisted spectral trace" defined in (6.14). We argue that this contribution is related to the effect of orientifold plane.
Finally, we conclude in section 7. Additionally, we have two Appendices A and B. In Appendix A, we summarize the non-perturbative part of grand potential J np (µ) for various (half-)integer N f , determined from our data of exact partition functions. In Appendix B, we explain the derivation of the Wigner transform in (5.13).

Fermi gas formalism and exact computation of partition functions
We study the S 3 partition functions of N = 4 G + A and G + S models with G = O(n) or U Sp(n), considered previously in [9]. Such models naturally appear as worldvolume theories on D2-branes in the presence of N f D6-branes and a orientifold plane.
As discussed in [9], depending on the type of orientifold plane, we find the following models as worldvolume theories on D2-branes: • O(2N ) + A: N D2-branes and N f D6-branes with a O2 − plane.
In [9], it is found that the S 3 partition functions of above models can be written as a system of N fermions in one-dimension (x i ∈ R) (2.1) where the density matrix ρ(x, y) is given by The parameters a, b, c, d for each model are summarized in Table 1. fixed the overall normalization of Z(N, N f ) in such a way that Z(N = 0, N f ) = 1, which is a natural normalization in the Fermi gas formalism [10]. Note that our normalization of Z(N, N f ) is different from [9] 3 . As discussed in [10], to study the non-perturbative corrections, it is more convenient to consider the grand partition function by summing over N with fugacity e µ with some positive constant c, which is equivalent to a shift of chemical potential µ → µ + log c. However, there is no room for this change of normalization since a shift of chemical potential will spoil the absence of µ 2 term in the perturbative part of grand potential (3.1).
More physically, ρ is identified with the Hamiltonian H of the fermion system as In the following sections, we will study the large µ expansion of the grand potential J(µ) From (2.1) and Table 1, one can easily see that the partition functions of U Sp(2N ) theory and O(2N + 1) theory are related by a shift of N f (2.7) Therefore, for our purposes it is sufficient to consider the models with O(n) gauge group. We can compute the canonical partition function Z(N, N f ) at fixed N once we know the trace Tr ρ from = 1 to = N . Using the Tracy-Widom lemma [17], the th power of ρ can be systematically computed by constructing a sequence of functions φ (x) ( = 0, 1, 2, · · · ) Then Tr ρ is given by The integrals in (2.8) and (2.9) can be easily evaluated by rewriting them as contour integrals and picking up residues, as in the case of ABJM theory [18,19]. Using this algorithm, we have computed the exact values of partition functions Z(N, N f ) of our models for various integer N f and half-integer N f up to some high N = N max , where N max is about 20-30. 4 Note that for a physical theory N f should be an integer, but at the level of matrix model (2.1) we can consider analytic continuation of N f to arbitrary continuous values. Such analytic continuation in N f is implicitly assumed in what follows.
Before moving on, let us comment on some interesting relations between our models (2.1) and some other theories. First, by mirror symmetry of d = 3 N = 4 theories, the U Sp(2N ) + A model is dual to a D N f quiver gauge theory with one fundamental flavor node added [6]. The equivalence of the S 3 partition functions of these two theories can be shown by using the result of [14] 5 .
Second, we find a nontrivial relation between the U Sp(2N ) + S model with N f = 1 and the ABJ theory with gauge group U (N ) 4 where M = N 2 − N 1 denotes the difference of the rank of gauge group U (N 1 ) k × U (N 2 ) −k of ABJ theory. This relation (2.10) can be understood from the relation found in [4] Ξ ABJ (µ, k = 4, M = 1) = Ξ − ABJM (µ, k = 2), (2.11) where Ξ − ABJM (µ, k = 2) is the grand partition function of ABJM theory at k = 2 computed from the odd-part ρ − of density matrix (2.12) One can easily show that the density matrix of U Sp(2N ) + S model with N f = 1 and the odd-part ρ − of ABJM theory at k = 2 are equivalent, up to a rescaling x, y → 2x, 2y and a similarity transformation, hence the relation (2.10) follows. Finally, we also find the equivalence of the partition functions of U Sp(2N ) + A with N f = 3 and the U (N ) Yang-Mills theory with one adjoint and N f fundamental hypermul- (2.13) This is expected from the isomorphism D 3 = A 3 . This relation (2.13) is recently proved in [20] using the technique in [14].

Perturbative part
In this section, we consider the large µ expansion of the grand potential (2.6), which takes the following form J(µ) = J pert (µ) + J np (µ), Here J pert (µ) in (3.1) is called the perturbative pert of grand potential. On the other hand, J np (µ) in (3.1) represents the non-perturbative corrections which are exponentially suppressed in the large µ limit. We will study J np (µ) in the next section.
In the large N limit, the free energy F = − log Z(N, N f ) is approximated by the Legendre transform of J pert (µ) 2) 5 We are grateful to Masazumi Honda for discussion on this point.
where µ * is the saddle point value of the chemical potential Since the free energy on S 3 is a nice measure of the degrees of freedom in d = 3 theories [21], (3.2) implies that the degrees of freedom of our models scale as N 3/2 , which is the expected behavior of M2-brane theories [22]. We would like to determine the coefficients C, B and A in (3.1) as a function of N f . The coefficient C is already found in [9] from the analysis of the classical Fermi surface. The coefficient B is a bit difficult since B receives a correction in the semi-classical WKB expansion (small-expansion). The coefficient A is much harder to determine since A receives corrections from all orders in the WKB expansion.
To circumvent this problem, we determine the coefficients B and A by matching our exact values of Z(N, N f ) and the perturbative partition function Z pert (N, N f ) given by the Airy function [10,23] where C is a contour in the µ-plane from e − πi 3 ∞ to e πi 3 ∞, and Z np (N, N f ) denotes the nonperturbative corrections coming from J np (µ). When N becomes large, the non-perturbative corrections Z np (N, N f ) are highly suppressed, so we can approximate the partition function by its perturbative part Z pert (N, N f ). By comparing the exact values of Z(N, N f ) and Z pert (N, N f ) in (3.4), we find the coefficients B and A for various models, which are summarized in Table 2. As pointed out in [14], the computation of B in [9] has an error, log 2 Table 2. The coefficients C, B and A in J pert (µ). C is already found in [9]. and our results of B are different from [9]. A c (k) in Table 2 is the constant term in the grand potential of U (N ) k × U (N ) −k ABJM theory, which is closely related to a certain resummation of the constant map contribution of topological string [8,24,25]   Table 2.
For integer k, A c (k) can be written in a closed form In particular, A c (1) appearing in Table 2 is given by In Figure 1, we show the plot of free energy for the O(n) + A models. As we can see, the exact values of free energy at integer N exhibit a nice agreement with the perturbative free energy (3.4) if we use the coefficients C, B and A in Table 2. We also find a similar agreement for the O(n) + S models. Let us explain in more detail how we found the results in Table 2. The coefficient B can be found easily by matching the Airy function (3.4) with the exact values of Z(N, N f ), since the N f -dependence of B is relatively simple. On the other hand, the constant A is a complicated function of N f . To find the constant A as a function of N f , first we estimated the numerical values of A by  (3.9) for N f = 1, 2, · · · , 9, while the solid curve is the plot of A in Table 2. for N as large as possible. In practice, we set N = N max in (3.9) where N max is the maximal value of N that the exact values of Z(N, N f ) is available. In this way, we obtained the constant A for various values of N f 's. Then, assuming that A is written as a linear combination of A c (k) with some k's, we fixed the coefficients of this linear combination 6 , and finally we arrived at the expressions of A in Table 2. In Figure 2, we show the plot of constant A for the O(2N + 1) + A model as an example. We can clearly see a nice agreement between the numerical values of A at integer N f estimated by using (3.9) and our proposal of A in Table 2. We also find a similar agreement for the other models in Table 2.
In section 5, we will derive the coefficients B and A of the O(2N + 1) + A model from the WKB expansion. For other models, we could not find a systematic method to compute B and A.
From Table 2, we find the following interesting relations between the O(2N ) models and the O(2N + 1) models (3.10) In section 6, we will argue that the right hand side of these relations can be naturally interpreted as the contributions of orientifold plane.

Non-perturbative corrections
In this section, we study the non-perturbative part J np (µ) of grand potential. To find the coefficients in J np (µ, N f ) we follow the procedure in [19]. First we expand e Jnp(µ) as where g w,n are some µ-independent coefficients. Then the non-perturbative part Z np (N, N f ) of partition function is written as a sum of Airy functions and their derivatives 3) The first two types have natural analogues in the N f matrix model [7,8]. On the other hand, the last one in (4.3) has no counterpart in the N f matrix model, hence it is natural to interpret it as the effect of orientifold plane. Following [7,8], let us call the first two types in (4.3) worldsheet instantons and membrane instantons, respectively. For the last type in (4.3), we will call them "half instantons". Note that the weight of worldsheet instanton in the N f matrix model is e −4µ/N f , which is related to the worldsheet instanton in our case (4.3) by a rescaling N f → 2N f .

Worldsheet instanton
We conjecture that the worldsheet instanton corrections are given by is very similar to the worldsheet instantons in the N f matrix model [8] and those in the (1, q)-model at k = 2 [13]. We can check this conjecture (4.4) in the same way as in [8]. We first notice that for N f > 4 the worldsheet 2-instanton factor e −4µ/N f is larger than the factor e −µ of half instanton in (4.3). Thus, the non-perturbative part of canonical partition function has the following expansion where Z WS and Z WS are the contributions of worldsheet 1-instanton and worldsheet 2instanton to the canonical partition function. Now let us consider the following quantity where µ * is given by (3.3). If our conjecture of worldsheet instantons (4.4) is correct, δ should be exponentially small in the large N limit. In Figure 3, we plot the quantity δ for N f = 5, 7, 8, 9 in the O(2N + 1) + A model. As we can see in Figure 3, δ indeed decays exponentially as N becomes large. We have also checked this behavior for the O(2N ) + A model. We should also mention that we have performed a similar checks for other types of instantons studied below.
Half instanton From the results in Appendix A.1 and A.2, we conjecture that the half instantons are given by where ε is a sign depending on the parity of n of the gauge group O(n) ε = (−1) n .   We should stress that our conjecture of half instantons (4.7) is consistent with all results in Appendix A.1 and A.2 in a very non-trivial way.
Bound states The results in Appendix A.1 and A.2 suggest that there are various types of bound state contributions. The existence of such bound state contributions are first observed in ABJM theory [19,26]. Let us denote the bound state of -worldsheet instanton, m-membrane instantons, and n-half instantons as J ( ,m,n) (µ, N f ). Namely, For instance, there seems to exist a bound state of 1-worldsheet instanton e −2µ/N f and 1-half instanton e −µ . From the coefficient of O(e −2µ/N f −µ ) term for N f = 3, 1/2, 3/2, 5/2 in Appendix A.1 and A.2, we conjecture There should also exist a bound state J (0,1,1) of 1-membrane instanton e −2µ and 1-half instanton e −µ in order to reproduce the finite term of N f = 1 at order O(e −3µ ) Note that J (1,0,1) in (4.12) has a pole at N f = 1, while the 3-half instanton J (0,0,3) in (4.7) is regular at N f = 1. For the equation (4.13) to make sense, the bound state of membrane instanton and half instanton J (0,1,1) should cancel the pole coming from J (1,0,1) in (4.12). This pole cancellation mechanism, first discovered in the ABJM theory [19], gives a constraint for the possible form of the coefficient of J (0,1,1) . But this condition alone is not strong enough to determine J (0,1,1) .

Membrane instanton
We do not have a direct information of the coefficient of membrane instantons in the results of Appendix A.1 and A.2. However, from the pole cancellation mechanism and the information of finite terms at N f = 1, 2, we can make a conjecture of membrane 1-instanton, as we will see below.
From the expression of the membrane instanton in the N f matrix model [13], it is natural to conjecture that the coefficient of 1-membrane instanton is proportional to 2µ + 1 (4.14) From (A.2) and (A.4), we observe that the 1-membrane instanton term is absent for half- Also, the result of N f matrix model in [8] suggests that b 1 (N f ) is given by a certain combination of gamma-functions. Furthermore, J M2 should reproduce the finite terms at WS + J half + J M2 should cancel those poles and give the finite terms in the right hand side of (4.16). From this pole cancellation condition and other conditions mentioned above, we conjecture that J (1) M2 is given by One can show that our conjecture (4.17) indeed reproduces the right hand side of (4. 16) and vanishes when N f is positive half-integer, as required. It would be nice to see if our conjecture (4.17) of 1-membrane instanton is correct or not, by computing it from the WKB expansion as in [8,13]. We conjecture that the worldsheet instanton corrections are given by (4.19) We have performed a similar check as in Figure 3 for our conjecture (4.19), and confirmed that (4.19) correctly reproduce the large N behavior of exact values Z(N, N f ) for various N f 's.
As for the membrane instantons, we were unable to determine their coefficient from the data in Appendix A.3 and A.4 alone. It would be interesting to study the structure of instanton corrections in this model further.

WKB expansion (I)
As discussed in [14], we can compute the coefficients C and B in the perturbative part of grand potential by formally introducing the Planck constant and performing the small expansion, although the physical theory corresponds to = 2π. Since the coefficients C and B receive corrections only up to order O( 0 ) and O( 2 ), respectively, we can fix C and B by computing the first two terms of WKB expansion and simply setting = 2π at the end. On the other hand, the coefficient A receives all order corrections in , hence it is not obvious if we can find the constant A in J pert (µ) from this formal WKB expansion. Nevertheless, as we will see below, at least for the O(2N + 1) + A model we can guess the all order expression of the WKB expansion of A, and by setting = 2π we find the constant A in a closed form. For models other than O(2N + 1) + A, we found difficulty in computing the leading (classical) term in the WKB expansion. Therefore, in this section we will focus on the O(2N + 1) + A model. Note that, the O(2N + 1) + A model is related to the U Sp(2N ) + A model by a shift of N f (2.7), which in turn is dual to a D-type quiver theory by mirror symmetry [6].

WKB expansion in O(2N + 1) + A model
As discussed in [14], the density matrix ρ(x, y) in (2.2) for the O(2N + 1) + A model can be written as a matrix element x|ρ|y of the quantum mechanical operator ρ = e −H of the following form 7 Note that in [9] a different expression of operator ρ was used. We will consider the operator in [9] in the next section. One advantage of ρ D in (5.2) is that it has not only a reflection symmetry [ρ D , R] = 0, but also has a property that its trace with R insertion vanishes [14] Tr(ρ D R) = 0, ( = 1, 2, · · · ). (5.4) This implies that the grand partition function is written as Ξ(µ) = Det(1 + e µ ρ) = Det(1 + e µ ρ D ), (5.5) and the grand potential is given by As noticed in [27], the WKB expansion of the grand potential (5.6) is most easily obtained from the WKB expansion of the spectral trace Tr ρ s D . By analytically continuing Tr ρ s D from integer s to arbitrary complex s, the grand potential is written as a Mellin-Barnes type integral where the integration contour γ is parallel to the imaginary axis with 0 < (s) < 1. By picking up poles at positive integers s = ∈ Z >0 , we recover (5.6). On the other hand, deforming the contour in the direction (s) ≤ 0, we can find a large µ expansion of J(µ). The WKB expansion of spectral trace takes the following form The relation between the density matrix in (2.2) with a = c = 1, b = d = 0, and the operator in (5.2) can be shown by using the relation [14] sinh The leading term Z 0 (s) is given by the classical phase space integral, simply replacing the operators ( x, p) in (5.2) by classical commuting variables (X, P ) From the WKB expansion of the spectral trace (5.8), we can easily find the WKB expansion of grand potential by replacing s in D(s, ) by the µ-derivative ∂ µ , and acting it on the leading term J(µ) = D(∂ µ , )J 0 (µ), (5.10) where J 0 (µ) is the leading term in the WKB expansion of grand potential Now, let us move on to the computation of D n (s) in (5.8).
In many examples of d = 3 N = 4 theories [11,13], it turned out that D n (s) was a rational function of s. Therefore, it is natural to assume that this is also the case for our expansion (5.8). Then, the easiest way to determine D n (s) is to make an ansatz that D n (s) is a rational function of s, and fix the coefficients in the ansatz by matching the WKB expansion of Tr ρ D for integer from = 1 to some = max , where we choose max as the number of independent coefficients in the ansatz of D n (s). Once we determined D n (s) in this way, we can check the agreement of Tr ρ D and D n ( ) for > max .
To compute the WKB expansion of Tr ρ D for integer , it is convenient to use the Wigner transform of the operator ρ D . In general, the Wigner transform O W is defined by As explained in Appendix B, the Wigner transform of ρ D is given by Using the property of Wigner transformation the Wigner transform of the th power of ρ D is given by the star-product of (ρ D ) W 's Using the fact that the trace Tr ρ D is written as a classical phase space integral of (ρ D ) W 18) and plugging the WKB expansion of (ρ D ) W (5.16) into (5.18), finally we find the WKB expansion of the trace Tr ρ D .
Using the above method, we have computed D n (s) up to n = 13. We find that D n (s) has the following form D n (s) = p n (s) 96 n n j=1 (2(N f + 1)s + 2j − 1)(s + 2j − 1) , (5.19) where p n (s) is a (4n) th order polynomial of s. The first few terms are given by From this we can read off the general structure of p n (s) for n ≥ 2 n (N f ) is a (2n − 1) th order polynomial of N f . Note that p 1 (s) is an exception: p 1 (s) has a factor s 2 (1 − s) while p n (s) (n ≥ 2) has a factor s 3 (1 − s). As we will see in the next subsection, this is related to the difference of the corrections of C, B and A.

Perturbative part of O(2N + 1) + A from WKB expansion
By deforming the contour γ to the left half plane (s) ≤ 0 in (5.7), we can find the large µ expansion of the grand potential J(µ). It turns out that the perturbative part J pert (µ) comes from the pole at s = 0. The leading contribution of the WKB expansion reads (5.21) The -corrections can be computed systematically by applying the relation (5.10) to the perturbative part J pert (µ) = D(∂ µ , )J pert,(0) (µ). (5.22) Since J pert,(0) (µ) is a cubic polynomial in µ, the derivatives ∂ m µ with m ≥ 4 do not contribute to J pert (µ). By expanding D n (∂ µ ) up to ∂ 3 µ , we find We have checked this behavior up to n = 13 and we believe that this is true for all n.
From the expansion in (5.23), one can easily see that C and B receive corrections only up to O( 0 ) and O( 2 ), respectively. Acting the differential operator on the leading term J pert,(0) and setting = 2π, finally we arrive at the correct C and B of the O(2N + 1) + A model in Table 2 1 We can also determine the constant A by summing over all order corrections. From (5.21) and (5.23), one can easily see that the constant A is given by (5.25) By comparing this with the small k expansion of A c (k) [24] A c (k) = 2ζ(3) we find that A is written as Using the explicit values of A c (4) and A c (2) from (3.6) one can see that (5.27) correctly reproduces the constant A of the O(2N + 1) + A model in Table 2.

Comment on the non-perturbative part of O(2N + 1) + A
In principle, we can also study the non-perturbative corrections J np (µ) from the WKB analysis in the previous subsection. J np (µ) comes from the poles on the negative real axis in the Mellin-Barnes representation (5.7). For instance, the leading term of spectral trace Z 0 (s) has poles at s = −1/2 and s = −1/N f , and their contribution to J np (µ) is given by (5.29) For these two terms, we find an all order expression of the coefficients. For the e −µ/N f term, we find (5.30) We have checked that this agrees with the WKB expansion up to n = 13. For the e −µ/2 term, we observe that the coefficient D(−1/2, ) is independent of N f 8 , hence it is simply given by setting N f = 2 in (5.30). From (5.30), one can see that the coefficients of e −µ/N f and e −µ/2 both vanish in the limit → 2π. This is consistent with our result in Appendix A.2 that there are no e −µ/N f and e −µ/2 terms in the grand potential of O(2N + 1) + A model. Note that (5.30) is essentially equal to a combination of q-gamma A similar expression of instanton coefficient has appeared in the (1, q)-model studied in [13].
It is more interesting to determine the coefficients of e −µ , e −2µ , or e −2µ/N f terms, which have non-vanishing contributions at = 2π. However, using our data of WKB expansion alone, we were unable to find an all order expression of those terms and set = 2π. It would be interesting to find those coefficients by computing the WKB expansion to more higher orders, or by other means.

WKB expansion (II)
In the previous section, we have considered the WKB expansion using ρ satisfying Tr(ρ R) = 0 [14]. Alternatively, as in [9] we can use ρ with Tr(ρ R) = 0. In this section, we will consider the WKB expansion using the operator in the latter case. Interestingly, as we will see below we find that the O(2N ) + A(or S) model and O(2N + 1) + A(or S) model can be thought of as a R = 1 and R = −1 subspace, respectively, of some bigger model. Namely the O(n) model corresponds to a projection to where ρ A and ρ S are given by In other words, this sign σ distinguishes the symmetric and anti-symmetric hypermultiplets. Then (6.4) can be written as One can easily show that the total grand potential for ρ = ρ σ 1+εR 2 can be decomposed as Note that (6.8) implies the following relation (6.10) The perturbative part of this relation is closely related to the difference of B and A found in (3.10). From the brane configuration in section 2, it is tempting to identify J R ρσ (µ) as the contribution of a half D2-brane stuck on the orientifold plane. In the rest of this section, we will consider the WKB expansion of this contribution.
Notice that Tr(OR) for some operator O can be obtained from the Wigner transform O W by simply setting X = P = 0 9 Thus the WKB expansion of Tr(ρ σ R) can be easily found from the WKB expansion of (ρ σ ) W , which can be systematically computed by using the method in the previous section. The Wigner transform of ρ σ is easily found to be (6.14) We will call Tr(ρ s σ R) in (6.14) the twisted spectral trace. The WKB expansion of J R ρσ (µ) can be found from the WKB expansion of twisted spectral trace Tr(ρ s σ R) The leading term Z R 0 (s) in (6.15) can be easily obtained as

Comments on the non-perturbative corrections
By matching the WKB expansion of twisted spectral trace Tr(ρ s σ R), we find the first two non-perturbative corrections, coming from the poles at s = −1 and s = −2, in a closed form in . (6.19) It is tempting to identify these corrections as the "half instantons". For ρ S (σ = −1), the non-perturbative corrections in (6.19) vanish at = 2π, which is consistent with the absence of O(e −µ ) term in O(n) + S models.
On the other hand, for ρ A (σ = +1) the 1-instanton term has a pole at = 2π (6.20) This should be canceled by a term of order O(e −2πµ/ ), which is non-perturbative in and hence cannot be seen directly in the WKB expansion. As discussed in [27], the O(e −2πµ/ ) term might arise from a pole of the twisted spectral trace Tr(ρ s A R) at s = −2π/ . Our result of 1-instanton and 2-instanton in (6.19) suggests that Tr(ρ s A R) has a structure which has a pole at s = −2π/ . As in [27], using the Pade approximation, we have checked numerically that Tr(ρ s A R) has a pole very close to s = −2π/ . For the special value of N f = −1, which is not physical though, by matching the WKB expansion we find a closed form expression of the twisted spectral trace Tr(ρ s A R) which indeed has a pole at s = −2π/ , as expected. Although we do not have an analytic proof that Tr(ρ s A R) has a pole at s = −2π/ for general N f , we will assume that this is the case in the rest of this section.
We assume that Tr(ρ s A R) has a simple pole at s = −2π/ for general N f Then the contribution of pole at s = −2π/ to J R ρ A is given by This contribution has a pole at = 2π, and behaves in the limit → 2π as Thus, there is a possibility that the pole at = 2π cancels between (6.20) and (6.25). This pole cancellation occurs if r(1) is given by If we further assume then the total contribution correctly reproduces the 1-half instanton term in J half (4.7) . (6.28) For the N f = −1 case in (6.22), one can see that the residue at s = −2π/ indeed satisfies the above conditions It would be interesting to study the analytic structure of Tr(ρ s A R) for general N f and see if it indeed has a pole at s = −2π/ with the correct residue.

Conclusion
In this paper, we have studied non-perturbative effects in N = 4 O(n) Yang-Mills theories with N f fundamental and one (anti)symmetric hypermultiplets using the Fermi gas formalism for their S 3 partition functions. They are a natural generalization of N f matrix model and interesting in their own right since we can study the effects of orientifold plane in the strong coupling M-theoretic regime.
We determined the coefficients C, B and A in the perturbative part of grand potential as functions of N f in Table 2. We also studied instanton corrections to the grand potential using our exact values of canonical partition functions Z(N, N f ). We found that instanton corrections in the O(n) + A model and the O(n) + S model have slightly different structure. For the O(n) + A model, in addition to the worldsheet instantons (4.4) and membrane instantons (4.17), there are "half instanton" corrections coming from the effect of orientifold plane. We have argued that half instantons can be naturally identified as the contributions from the twisted spectral trace Tr(ρ s A R) in the Fermi gas picture. Also, we found a bound state of worldsheet instanton and half instanton (4.12). From the pole cancellation argument, there should also be a bound state of membrane instanton and half instanton as well.
It is interesting that the reflection R of one-dimensional Fermi gas system has a relation to the orientifolding in the spacetime. We find that the half instanton has a weight e −µ which is half of the weight of membrane instanton e −2µ . This type of half instanton corrections is also observed in other theories [4,16], and we believe that this is a general phenomenon in M-theory on orientifolds. It would be interesting to study the general structure of half instantons and clarify the precise relation to the Type IIA brane picture.
In the case of ABJM theory, the effect of bound state can be removed by introducing the effective chemical potential given by the quantum A-period [1,26]. It would be very interesting to see whether a similar redefinition of chemical potential works in our case. To study instanton corrections further, it is desirable to find a systematic method to compute the WKB expansion. In the case of N f matrix model, it has a natural one-parameter generalization of the model by introducing a Chern-Simons level k, and we can systematically study the WKB expansion around k = 0 [13]. It would be interesting to find a generalization of the O(n) + A(or S) models with Chern-Simons terms, along the lines of [14,15].

Acknowledgments
I am grateful to Yasuyuki Hatsuda, Masazumi Honda, and Marcos Marino for useful discussions. I would also like to thank the theory group in University of Geneva for hospitality.
A J np (µ, N f ) for various N f In this Appendix, we summarize the non-perturbative corrections for various (half-)integral N f , obtained from the data of exact values of Z(N, N f ).

A.1 O(2N ) + A
Here we summarize the non-perturbative corrections J np (µ, N f ) to the grand potential of the O(2N ) + A model.

A.2 O(2N + 1) + A
Here we summarize the non-perturbative corrections J np (µ, N f ) to the grand potential of the O(2N + 1) + A model.

A.3 O(2N ) + S
Here we summarize the non-perturbative corrections J np (µ, N f ) to the grand potential of the O (2N