Exact relations between M2-brane theories with and without Orientifolds

We study partition functions of low-energy effective theories of M2-branes, whose type IIB brane constructions include orientifolds. We mainly focus on circular quiver superconformal Chern-Simons theory on $S^3$, whose gauge group is $O(2N+1)\times USp(2N)\times \cdots \times O(2N+1)\times USp(2N)$. This theory is the natural generalization of the $\mathcal{N}=5$ ABJM theory with the gauge group $O(2N+1)_{2k} \times USp(2N)_{-k}$. We find that the partition function of this type of theory has a simple relation to the one of the M2-brane theory without the orientifolds, whose gauge group is $U(N)\times \cdots \times U(N)$. By using this relation, we determine an exact form of the grand partition function of the $O(2N+1)_{2} \times USp(2N)_{-1}$ ABJM theory, where its supersymmetry is expected to be enhanced to $\mathcal{N}=6$. As another interesting application, we discuss that our result gives a natural physical interpretation of a relation between the grand partition functions of the $U(N+1)_4 \times U(N)_{-4}$ ABJ theory and $U(N)_2 \times U(N)_{-2}$ ABJM theory, recently conjectured by Grassi-Hatsuda-Mari\~no. We also argue that partition functions of $\hat{A}_3$ quiver theories have representations in terms of an ideal Fermi gas systems associated with $\hat{D}$-type quiver theories and this leads an interesting relation between certain $U(N)$ and $USp(2N)$ supersymmetric gauge theories.


Introduction
In a couple of years, there is remarkable progress in understanding non-perturbative effects in M-theory through gauge/gravity duality. Most important tools in this progress are the supersymmetry localization [1,2] and Fermi gas approach [3]. These are applied to partition functions in a class of low-energy effective theories of N M2-branes on S 3 and it has turned out that the partition functions are described by an ideal Fermi gas system: ρ(x j , x σ(j) ), (1.1) where ρ plays an role of the density matrix in the Fermi gas system. Thanks to these techniques, now we know detailed structures of the non-perturbative effects in M-theory on AdS 4 × S 7 /Z k [4,5,6,7], which is dual to the 3d N = 6 superconformal Chern-Simons (CS) theory known as the ABJ(M) theory [8,9] via AdS/CFT correspondence (see also important earlier works [10,11,12,13,14,15,16,17,18,19,20]). On the other hand, we still do not have detailed understanding of the non-perturbative effects "beyond ABJ(M) theory", namely more general M2-brane theories with less super-symmetry 1 (SUSY). For instance, it is unclear whether many attractive features found in the ABJ(M) theory such as the Airy functional behavior [13,3], pole cancellation [17,4] and correspondence to topological string [10,16,4] are universal for general M2-brane theories or accidental for the ABJ(M) theory. While the Airy functional behavior has been found for a broad class of M2-brane theories [3,25,26,27,28] and seems universal [29] (see also [30]), the other features have been found in few examples. This problem has been addressed in special cases of Imamura-Kimura type theory [31], whose type IIB brane construction consists of NS5-branes and (1, k)-5 branes connected by N D3-branes. Especially the orbifold ABJM theory and (p, q) model [32] have been studied well in [26,33,34,22,23,24]. Alsô D-type quiver theories [27,28] and O or USp gauge theories with single node [26,35] have been studied (see also [36]). In order to understand the non-perturbative effects in more detail, it is very important to investigate the non-perturbative effects in various theories of M2-branes.
In this paper we consider a generalization along a different direction. We study partition functions of low-energy effective theories of M2-branes on S 3 , whose type IIB brane constructions include orientifolds. We mainly focus on 3d superconformal CS theory of circular quiver type with the gauge group 2 O(2N + 1) × USp(2N) × · · · × O(2N + 1) × USp(2N). This theory is a natural generalization of the O(2N + 1) 2k × USp(2N) −k ABJM theory with N = 5 SUSY [38,9]. We show that the S 3 partition function of this type of theory is also described by an ideal Fermi gas system as in (1.1) and its density matrix ρ O(2N +1)×U Sp(2N ) takes the following form where Ξ (±) [µ] denotes the grand canonical partition function defined by ρ (±) . This relation implies that we can obtain non-perturbative information on the orientifold theory from the non-orientifold theory.
Here we present two interesting applications of our main result (1.5). One of them is to determine an exact form of the grand partition function of the O(2N + 1) 2k × USp(2N) −k ABJM theory with k = 1, whose SUSY is expected to be enhanced to N = 6 from N = 5. This is achieved by combining our result (1.5) with recent results of [39,40] and we obtain Here Ξ U (N ) 1 ×U (N ) −1 is the grand partition function of the U(N) 1 × U(N) −1 ABJM theory, whose exact form is conjectured as [40] where several definitions will be given in sec. 2.2. The other application of (1.5) is to give a natural physical interpretation of a mysterious relation recently conjectured by Grassi-Hatsuda-Mariño [39]. They conjectured a relation between the grand partition functions of the U(N + 1) 4 × U(N) −4 ABJ theory and U(N) 2 × U(N) −2 ABJM theory as (1.8) This should be compared with our result (1.5) for the N = 5 ABJM theory: Combining (1.8) with (1.9), we find Remarkably this relation is indeed equivalent to the conjecture in [9]. The N = 5 ABJM theory is expected to be low energy effective theories of N M2-branes probing C 4 /D k with the binary dihedral groupD k defined in (2.26). Since C 4 /D k for k = 1 is C 4 /Z 4 , moduli of the O(2N + 1) 2 × USp(2N) −1 ABJM theory become the same as the one of the U(N + M) k × U(N) −k ABJ(M) with k = 4. Therefore the work [9] conjectured that the O(2N + 1) 2 × USp(2N) −1 ABJM theory has the enhanced N = 6 SUSY and equivalent to the U(N + 1) 4 × U(N) −4 ABJ theory 3 : We also discuss that partition functions ofÂ 3 quiver theories have representations in terms of ideal Fermi gas systems associated withD-type quivers 5 and this leads an interesting relation between certain U(N) and USp(2N) SUSY gauge theories with single node. The U(N) gauge theory under consideration is N = 4 vector multiplet with one adjoint hyper multiplet and N f fundamental hyper multiplets, while the USp(2N) gauge theory is N = 4 vector multiplet with one anti-symmetric hyper multiplet and N f -fundamental hyper multiples. Regarding these theories, the work [35] has proposed the equivalence This relation is expected from 3d mirror symmetry 6 [42,43]. It is known that the U(N) and USp(2N) theories are equivalent toÂ N f −1 andD N f quiver theories without CS terms, where only one of the vector multiplets is coupled to one fundamental hyper multiplet. Sincê A 3 =D 3 , (1.12) should hold via the 3d mirror symmetries. In appendix we explicitly prove this relation by using the technique in [27,28]. This paper is organized as follows. In sec. 2, we consider the N = 5 ABJM theory with the gauge group O(2N + 1) 2k × USp(2N) −k . In sec. 3, we generalize our analysis in sec. 2 to more general quiver gauge theories. We also identify quantum mechanical operators in ideal Fermi gas systems naturally corresponding to orientifolds in type IIB brane constructions.
As interesting examples, we deal with orientifold projections of the (p, q) model and orbifold ABJM theory. Section 4 is devoted to conclusion and discussions. In appendix, we explicitly prove the equivalence (1.12). 3 In order to fix the value of M , we should compare not only the moduli but also discrete torsion [9]. 4 This statement has been partially checked by using superconformal index [41]. 5 The papers [27,28] have written partition functions ofD-type quiver theories in terms of ideal Fermi gas systems. it is unclear to us whether their derivation includesD 3 case. However their derivation apparently seems to considerD n≥4 and it is unclear to us whether their derivation includes theD 3 case or not. Hence we explicitly prove this for theD 3 case. Even if [27,28] did not prove it for theD 3 case, our derivation is not essentially new. 6 We thank Kazumi Okuyama for useful discussions on this point.
In this section we consider the N = 5 ABJM theory with the gauge group O(2N + 1) 2k × USp(2N) −k . We will generalize our analysis in this section to more general theory in next section.

Orientifold ABJM theory as a Fermi gas
Thanks to the localization [2], the partition function of the O(2N +1) 2k ×USp(2N) −k ABJM theory on S 3 can be written as 7 (see tab. 1 for detail) with k ′ = 2k. This equation tells us that the partition function of the N = 5 ABJM theory is described by the ideal Fermi gas system with the density matrix ρ N =5ABJM (x, y). We regard ρ N =5ABJM (x, y) as the matrix element of a quantum mechanical operator as in [3], The operatorρ N =5ABJM is defined aŝ By using the operator equations 8 e (2.10) Performing the similarity transformation we obtain the following highly simplified expression Recalling thatρ for the N = 6 ABJM theory with the gauge group U(N) 2k × U(N) −2k is given by 9ρ (2.14) This indicates that the density matrix operatorρ N =5ABJM of the N = 5 ABJM theory is the projection of the one of the N = 6 ABJM theory. Since the N = 5 ABJM theory is the orientifold projection of the N = 6 ABJM theory, presumably the operation of (1 −R)/2 tô ρ N =6ABJM corresponds to the orientifold projection. It is interesting if one can understand this relation more precisely.

Remarks
1. The representation (2.14) ofρ N =5ABJM gives the matrix element (2.15) This gives the following representation of the partition function , (2.16) where we have rescaled as x → 2x. Let us compare this with the partition function of the USp(2N) gauge theory with N = 4 vector multiplet, one symmetric hyper multiplet and N f -fundamental hyper multiples 10 (called USp + S theory in [26]): 9 Note that the definition of in (2.7) is slightly different from the one usually used in Fermi gas systems associated with U (N ) CS theories. 10 When we go to the last line from the second line, we have used sinh 2 µ j = 4 sinh 2 µj 2 cosh 2 µj 2 . (2.17) Comparing this with (2.16), we easily see that the N = 5 ABJM theory with k = 1 agrees with 11 the USp + S theory with N f = 1: It is interesting if one can understand this relation by the brane constructions. Note that this result is essentially the same as the recent result in [35], which has shown the equivalence between the grand partition function of the USp + S theory with N f = 1 and Ξ (−) part of the U(N) 2 × U(N) −2 ABJM theory. Because of (2.14), our result is equivalent to this result: 2. When we identify the quantum mechanical operator (2.8) associated with ρ N =5ABJM (x, y), we could use the following identity once or twice instead of (2.9), Then the partition function Z N =5ABJM is described by different representations ofρ. If we use this identity and (2.9) just once by once, then we find  11 We can also compare this with the O(2N + 1) gauge theory with N = 4 vector multiplet, one symmetric hyper multiplet and N f -fundamental hyper multiples (O(2N + 1) + S theory). Because of To summarize, we have four different representations ofρ to describe the same partition function Z N =5ABJM : where we can freely choose "+" or "−" at every "f ± (1 ±R)f ± " and f ± is given by (2.23) In this paper we always choose "−" since taking "−" seems technically simpler.
In terms of (1.6), we can explicitly write the exact form of the grand partition function of the O(2N + 1) 2 × USp(2N) −1 ABJM theory.

Grassi-Hatsuda-Mariño exact functional relation from geometry
Grassi, Hatsuda and Mariño conjectured the relation (1.8) on the grand canonical partition function of the ABJ theory [39]: . Physical interpretation of this relation has been unclear and therefore this relation has been considered as accidental. Now we give a physical interpretation on this relation. Let us compare this result with our result (1.5): This relation is equivalent to the conjecture in [9]. The O(2N + 1) 2k × USp(2N) −k ABJM theory is expected to be low energy effective theories of N M2-branes probing C 4 /D k with the binary dihedral groupD k , whose action to the complex coordinate (z 1 , z 2 , z 3 , z 4 ) of C 4 is Since C 4 /D k for k = 1 is C 4 /Z 4 , the moduli of the O(2N + 1) 2 × USp(2N) −1 ABJM theory become the same as the one of the U(N + M) 4 × U(N) −4 ABJ(M) theory. Therefore the work [9] conjectured that the O(2N + 1) 2 × USp(2N) −1 ABJM theory has N = 6 SUSY and equivalent to the U(N + 1) 4 × U(N) −4 ABJ theory (see [41] for the test by superconformal index): If we assume this, then our result (1.

Generalization
In this section we generalize our analysis in sec. 2 to a class of CS theory, which is circular quiver with the gauge group [O(2N + 1) × USp(2N)] r and bi-fundamental chiral multiplets one by one between nearest neighboring pairs of the gauge groups.

Fermi gas formalism
Let us consider the circular quiver CS theory with the gauge group O(2N+1) fundamental hyper multiplets, respectively. We parametrize the CS levels k a , k ′ a as k a = kn a , k ′ a = kn ′ a with rational numbers n a and n ′ a . Applying the localization, the partition function becomes [2] By similar arguments to sec. 2.1, we rewrite the partition function as Here the function ρ(x, y) is defined by whereρ U (N ) is the density matrix operator associated with the non-orientifold theory, which is obtained by the replacement O(2N + 1), USp(2N) → U(N) in the orientifold theories. This relation shows thatρ for the orientifold theory is the projection of the one without the orientifolds.

Identification of operators corresponding to orientifolds
Here we identify quantum mechanical operators, which naturally correspond to the orientifolds 15 O3 ± in type IIB brane construction. First, it is known that D5-brane, NS5-brane and (1, k)-5 brane naturally correspond to 16 (see e.g. [44,45]) This is actually consistent withρ of N = 3 circular quiver CS theory with U(N) gauge group and SL(2, Z) symmetry in type IIB string. For example, let us consider the (p, q)-model, whose IIB brane construction consists of p NS5-branes and q (1, k)-5 branes connected by N D3-branes on a circle (see fig. 1). This theory is N = 3 circular quiver superconformal CS theory with the gauge group U(N) k × U(N) q−1 0 × U(N) p−1 −k andρ (p,q) associated with this theory isρ
As discussed in sec. 2.1,ρ for the N = 5 ABJM theory iŝ If we assume that this can be rewritten aŝ whereÔ O3 ± corresponds to O3 ± , then it is natural to identify 17 This identification is consistent for more general quiver gauge theories described in sec. 3.1.

Orientifold projection of (p, q)-model
As an interesting example, we consider orientifold projection of the (p, q)-model analyzed well in [26,33,34,22,23,24]. The (p, q)-model is the circular quiver theory with the 17 As mentioned in remark 2 of sec. 2.1, we have multiple representations ofρ to describe the same partition function. Then identifications of OÕ 3 ± are more generally Then let us consider a circular quiver theory with the brane construction (see fig. 3) Then correspondingρ iŝ This can be understood as the projection of the (2p, 2q)-model. We can also consider the orientifold projection of the (p, q)-model with odd p and q. For example suppose the brane construction This is the projection of the (1, 2m + 1)-model. [Right] Its orientifold projection.

Orientifold projection of orbifold ABJM theory
Next we consider the orientifold projection of the orbifold ABJM theory. Recalling that the brane construction of the orbifold ABJM theory is [(D3) − (NS5) − (D3) − (1, k)] r , let us take the following brane construction (see fig. 4) which gives the [O(2N +1) 2k ×USp(2N) −k ] r circular quiver superconformal CS theory. Then correspondingρ iŝ which is the projection of the orbifold ABJM theory. We can express the grand partition function of the orientifold projected orbifold ABJM theory in terms of the one of the N = 5 ABJM theory by using the argument in [21]. Namely, when the density matrix operatorρ satisfiesρ = (ρ ′ ) r , the grand partition function becomes (3.14) Multiplet One-loop determinant  6) and (1.7), we can also explicitly write the one of the orientifold projected orbifold ABJM theory with k = 1 in terms of (1.6).

Comments on O(2N ) × U Sp(2N ) type
In this section we give some comments on partition functions of By similar arguments to sec. 3.1, we find where (3.18) which is of course the same as the result of [37]. Next we consider operators corresponding to orientifolds O3 ± . Let us recall that the brane construction of the O (2N , we arrive at the following identification (3.20)

Conclusion and Discussions
In this paper we have studied the partition functions of the low-energy effective theories of M2-branes, whose type IIB brane constructions include the orientifolds. We have mainly focused on the circular quiver superconformal CS theory on S 3 with the gauge group O(2N + 1) × USp(2N) × · · · × O(2N + 1) × USp(2N), which is the natural generalization of the O(2N + 1) 2k × USp(2N) −k N = 5 ABJM theory. We have found that the partition function of this type of theory have the simple relation (1.5) to the one of the M2-brane theories without the orientifolds with the gauge group U(N) × · · · × U(N). By using this relation and the recent results in [39,40], we have found the exact form (1.6) of the grand partition function of the O(2N + 1) 2 × USp(2N) −1 ABJM theory, where its SUSY is expected to be enhanced to N = 6 [9]. As another application, we discussed that our result gives the natural physical interpretation of the relation (1.8) conjectured by Grassi-Hatsuda-Mariño. We also argued in appendix that the partition function ofÂ 3 quiver theory has the representation (A.10) in terms of an ideal Fermi gas system ofD-type quiver theory and this leads the relation (1.12) between the U(N) and USp(2N) SUSY gauge theories. Our result (1.2), (3.4) shows that the density matrix operator for the orientifold theory is the projection of the non-orientifold theory by the operator (1 −R)/2. It is nice if we can understand this relation more precisely. Our result also implies that one can systematically study the partition function of the orientifold theory by using techniques developed in the studies of the non-orientifold theory. For instance the technique introduced in [35] allows us to compute WKB expansion of Tr(ρ ℓR ) systematically 19 in terms of information on Wigner transformation ofρ ℓ . It is interesting to determine non-perturbative effects in the orientifold theories by such techniques.
Recalling that the U(N) k × U(N) −k N = 6 ABJM theory is described by topological string on local P 1 × P 1 , this relation would imply that the O(2N + 1) 2k × USp(2N) −k ABJM theory is described by certain projection in the topological string. There should be a physical meaning of (1 −R)/2 in the context of the topological string.
Although we have found the physical interpretation of one of relations conjectured by Grassi-Hatsuda-Mariño [39], they also conjectured other relations among the grand partition functions of the ABJ(M) theory with specific values of the parameters: (4.1) Although these relations might be accidental coincidences, it would be illuminating if we can find some physical interpretations.
One of immediate extensions of our analysis is to consider the gauge group O (2N . Probably this can be done by combining the technique in [20,5] with the Cauchy determinant-like formula (2.2). If this is the case,ρ for the O(2N 1 + 1) 2k × USp(2N 2 ) −2k N = 5 ABJ(M) theory would be projection 20 of the one of the U(N 1 ) 2k × U(N 2 ) −2k N = 6 ABJ(M) theory by (1 −R)/2. Another interesting direction is to study other supersymmetric observables such as supersymmetric Wilson loops. Then the techniques established in [7] would be efficient. whose partition function is described by so-called N f -matrix model [33]: This matrix model has been studied well in [26,33,22,23,24]. The USp(2N) gauge theory is N = 4 vector multiplet with one anti-symmetric hyper multiplet and N f -fundamental hyper multiples and its partition function is which has been analyzed in [26,27,35]. Regarding these theories, the work [35] has proposed the following equivalence 21 This relation is expected from 3d mirror symmetry [42,43]. It is known that the U(N) and USp(2N) theories are equivalent toÂ N f −1 andD N f quiver theories without CS levels, where only one of the vector multiples is coupled to one fundamental hyper multiplet, respectively. SinceÂ 3 =D 3 , the equation (1.12) should hold. In this section we explicitly prove this relation by using the technique in [27,28].
Here we show that partition function ofÂ 3 quiver theory has a representation in terms of an ideal Fermi gas system ofD-type quiver theory. Although this may be already proven in [27,28], it is unclear to us whether their derivation includes our analysis in this section or not and therefore we explicitly prove it. First we precisely explain what we would like to prove. Suppose the SUSY CS theory withÂ n quiver, namely the circular quiver with the gauge group U(N) k 1 × · · · U(N) k n+1 , which is coupled to N (a) f fundamental hyper multiplets. The partition function of theÂ n quiver theory can be denoted by [2]   where µ It is known that one can rewrite the partition function of theÂ n theory as [3] ZÂ where ρÂ n (x, y) is the density matrix of the ideal Fermi gas system associated with the quantum mechanical operatorρÂ n (Q,P ) = Next let us consider theD L+2 quiver CS theory with the gauge group U(N) fig. 5) . The partition function of this theory is given by [2] It is also known that the partition function of theD L+2 quiver theory is described by an ideal Fermi gas system [27,28]: In this section we prove As mentioned above, this may be already proven in [27,28]. However their derivation apparently seems to take L ≥ 2, where at least one U(2N) node is present, and it is unclear to us whether their derivation includesD 3 (L = 1) case or not. Therefore we explicitly prove this relation. Now let us consider the A 3 quiver theory: Let us redefine the variables as inD 3 -quiver language: Then the partition function becomes where I, J = 1, · · · , 2N. By inserting to the integrand and using the Cauchy determinant formula, we find (A. 15) Below in this subsection we just repeat the argument of [27]. According to [27], we introduce R(j) = N + j, R(N + j) = j. (A. 16) Now we would like to rewrite the integral in terms of a kernel acting on set of N eigenvalues K(σ) among x J 's, which is dependent on the permutation σ. More precisely, we take K(σ) such that Rτ −1 Rτ (j) ∈ K(σ) for given j ∈ K(σ). Then we rewrite the partition function as where s(j) = 0 for j = 1, · · · , N 1 for j = N + 1, · · · , 2N . (A.18) Note that we can also write this as Averaging over these, we obtain 2 sinh A.2Â n → U (N ) + adj.
Suppose theÂ n quiver theories without CS terms, where only one of the U(N) vector multiples is coupled to one fundamental hyper multiplet. This theory is related to the U(N) gauge theory with N = 4 vector multiplet, one adjoint hyper multiplet and n+1 fundamental hyper multiplets. We can easily show this for the partition functions [47,3,45]. To be self contained, here we repeat its derivation. The density matrix operatorρ of theÂ n theory is A.3D n → U Sp + A We also review the proof of the 3d mirror symmetry between the partition functions on S 3 of theD n quiver and USp + A theories. The gauge group of theD n quiver theory consists of four U(N) nodes and (n − 3) U(2N) nodes, where one of U(N) nodes associates one fundamental hypermultiplet. The partition function of this theory is given by