Fermi gas approach to general rank theories and quantum curves
aa r X i v : . [ h e p - t h ] J u l YITP-20-87
Fermi gas approach to general rank theoriesand quantum curves
Naotaka Kubo ∗ ∗ Center for Gravitational Physics, Yukawa Institute for Theoretical Physics,Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan
Abstract
It is known that matrix models computing the partition functions of three-dimensional N = 4 superconformal Chern-Simons theories described by circular quiver diagrams canbe written as the partition functions of ideal Fermi gases when all the nodes have equalranks. We extend this approach to rank deformed theories. The resulting matrix modelsfactorize into factors depending only on the relative ranks in addition to the Fermi gasfactors. We find that this factorization plays a critical role in showing the equality of thepartition functions of dual theories related by the Hanany-Witten transition. Further-more, we show that the inverses of the density matrices of the ideal Fermi gases can besimplified and regarded as quantum curves as in the case without rank deformations. Wealso comment on four nodes theories using our results. ∗ [email protected] ontents N = 4 Chern-Simons theories with four nodes 33
A.1 Determinant formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39A.2 N independent factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41A.3 Hanany-Witten transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42A.4 Similarity transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 B Complex delta functions 45C Proof for quantum curves 45
M2-branes are important objects in M-theory, and thus it is indispensable for studying M-theory to understand their non-perturbative properties. One of the most important progressin the study of M2-branes is the discovery of its worldvolume theory. M2-branes on C / Z k can2e obtained by taking T-duality and the M-theory lift to a brane configuration consisting ofD3-branes with one direction compactified, an NS5-brane and a (1 , k )5-brane in type IIB stringtheory. The worldvolume theory of this brane configuration is the N = 6 superconformal Chern-Simons theory with gauge group U ( N ) k × U ( N ) − k called ABJM theory [1]. The subscriptionsof the unitary groups denote their Chern-Simons levels.The partition function of the ABJM theory on S reduces to the ABJM matrix model usingthe localization technique [2]. Analysis in the ’t Hooft limit N → ∞ , λ = Nk : fixed , (1.1)which is the traditional way to study matrix models, however, is not suitable, because thegravitational dual of the ABJM theory is M-theory on AdS × S / Z k and thus M-theory reducesto type IIA string theory in AdS × CP with the k → ∞ limit.A new approach called Fermi gas formalism was developed in [3] to resolve this problem.They suggested to rewrite the ABJM matrix model as the partition function of an ideal Fermigas with a non-trivial one-particle density matrix b ρ ABJM = 12 cosh b p
12 cosh b q , (1.2)with Plank constant ~ = 2 πk . After this reformulation, we can analyze the small ~ regime, inother words, the small k regime in the WKB or semiclassical expansion because this systemis now the quantum mechanical system. This approach enables direct derivation of the N behavior of the free energy [3]. It was found that the Fermi gas formalism was the powerfultool also for calculating the exact values of the partition functions for various k and N [4–6],membrane instantons [3, 6, 7], the vacuum expectation values of Wilson loops [8–11] and otherphysical quantities [12] (see also [13, 14] for reviews). All these successes demonstrate that theFermi gas formalism is an exceedingly useful approach to study the M-theory regime.There are many generalizations of the ABJM theory. In this paper, we consider two stepsof generalizations. The first generalization is to deform the rank of one node, which is calledABJ theory [15, 16]. It was conjectured that the ABJ theories with gauge groups U ( N ) k × U ( N + M ) − k and U ( N + k − M ) k × U ( N ) − k were Seiberg-like dual theories [17], which is theconsequence of the Hanany-Witten transition [18] in type IIB string theory. To confirm thisconjecture, we should deal with rank deformed theories. However, it was more difficult to applythe Fermi gas formalism to the ABJ matrix model than the case of the ABJM matrix model.Nevertheless, in [19–21] , they succeeded to rewrite the ABJ matrix model associated with thegauge group U ( N ) k × U ( N + M ) − k as the partition function of an ideal Fermi gas with the Another formalism was developed in [22]. b ρ ABJ M = i M Q Mj =1 b q − πit M,j k
12 cosh b p Q Mj =1 b q − πit M,j k b q + iπM . (1.3)They also indicated that the density matrices of the dual theories were identical. This equalityleads to the equality of the partition functions of the dual theories (up to a phase factor).The inverse of the ABJM density matrix (1.2) is the Laurent polynomial of e b q and e b p .Laurent polynomials of these operators are called quantum curves. On the other hand, it is notobvious that the inverse of the ABJ density matrix (1.3) can also be regarded as a quantumcurve. Nevertheless, it was found in [23] that the inverse matrix also can be put in the form ofa quantum curve. This progress allowed us to discuss, for example, relations between the ABJtheory and the TS/ST correspondence [24] and between the ABJ theory and the q -deformedPainlev´e equations [25].As we explained above, the ABJ(M) theory is the worldvolume theory of the system contain-ing D3-branes with one direction compactified, a single NS5-brane and a single (1 , k )5-brane.The second generalization is to increase the number of 5-branes arbitrarily [26]. The worldvol-ume theories of these brane configurations are N = 4 superconformal Chern-Simons theoriesdescribed by circular quiver diagrams. The matrix models associated with these gauge theoriesalso can be written as the partition functions of ideal Fermi gases when all the nodes have equalranks, and much studied in the Fermi gas description [3, 27–32] as with the ABJ(M) theory.However, when the ranks are different, so far the Fermi gas formalism is applied only to specialcases such as theories with gauge groups consisting of four unitary groups [33, 34].On the other hand, there was recently progress in the quantum curves. It was revealed thatsome types of quantum curves obey the Weyl group symmetry of SO (10) and the exceptionalgroups [35, 36]. Furthermore, it was found in [34] by connecting the SO (10) quantum curveto the rank deformed theories with four nodes without using the Fermi gas formalism that theHanany-Witten transition is realized as a portion of the Weyl group symmetry of SO (10). Thisfact implies that quantum curves play an important role in understanding M2-brane physics.Given the above considerations, we expect that the Fermi gas description and quantumcurves have sufficient potential to study non-perturbative effects such as the Hanany-Wittentransition. Therefore, it is important to apply the Fermi gas formalism not only to theorieswithout any rank deformations but also to rank deformed theories and put the density matricesin the form of quantum curves. In this paper, we realize these ideas.The strategy we use in this paper to apply the Fermi gas formalism is to cut the integrand ofthe matrix model at positions where corresponding ranks of nodes are lowest and locally applythe Fermi gas formalism to each part of matrix model, which we call deformed factor (because4ach deformed factor corresponds to each lump of 5-brane factors with ranks deformed fromthe loewst rank). Looking at this procedure from a different angle, we can apply the Fermi gasformalism to all of the rank deformations constructed by the deformed factors. Therefore, thisprocedure can be applied to a wide class of rank deformations. This procedure first appearedin [34] though they computed the deformed factor associated with the brane configurationconsisting of only one NS5-brane and one (1 , k )5-brane. In this paper, we focus on braneconfigurations consisting of an NS5-brane at the center, any number of (1 , k )5-branes on bothside of the NS5-brane and D3-branes stretched between two 5-branes such that the numbermonotonically increases with approaching the NS5-brane (see figure 1 in section 3.1). Thecorresponding deformed factors are (3.4).Our main result is that, after similarity transformations which we explain in section 2.2,(3.4) is transformed into (3.7), which is nothing but the Fermi gas description. The non-trivialfactorization appears in this expression, and this factorization displays its potential when westudy dualities. We find that we can divide the factors into three types of factors, namely Z (CS) k,M , Z (vec / mat) k,M,M ′ and the determinant of the density matrix, and they obey identities expected fromthe Hanany-Witten transition separately. In other words, the equality between dual theoriesholds for each type of factor. We also comment on the good/ugly/bad classification given in[37] by studying divergences appearing in these factors.The other main result is that the inverses of the density matrices (3.10) can be regardedas quantum curves as in (4.7). Comparing these quantum curves to the brane configurations,their relation can be visually understood. By using this result, we study specific N = 4 Chern-Simons theories with four unitary groups. First, we partially prove the relation between thesegauge theories and quantum curves. Second, we find out that a Weyl group symmetry obeyedby the quantum curves strongly depends on the form of the density matrices by revising interms of the density matrices.This paper is organized as follows. In section 2, first we briefly review brane configurationsin type IIB string theory and matrix models which the partition functions of the worldvolumetheories of these brane configurations reduce to using the localization technique. Second, weexplain our strategy to apply the Fermi gas formalism. Third, we review the Fermi gas formal-ism to simple examples. In section 3, we present new deformed factors we study in this paperand apply the Fermi gas formalism. We also prove the equality between the deformed factorsrelated by the Hanany-Witten transition. In section 4, we put the inverses of the density matri-ces we obtained in the previous section in the form of quantum curves. We comment on themfrom the point of view of the brane configurations and argue the Hanany-Witten transition. Insection 5, we study specific N = 4 Chern-Simons theories with four nodes. Our results providenew insights into these gauge theories. Finally, we conclude and discuss some future directionsin section 6. Appendix A contains various formulas and their proofs. Appendix B comments5n delta functions whose argument is a complex number. Appendix C provides proof for therelation between the density matrices and the quantum curves. In this section we review the Fermi gas formalism. First, we present brane configurations in typeIIB string theory which we deal with in this paper. Second, we briefly review matrix modelscomputing the partition functions of the worldvolume theories of the brane configurations.Third, we explain our strategy to apply the Fermi gas formalism to these matrix models.Finally, we review the way we rewrite the matrix models as the partition functions of idealFermi gases for simple examples including the ABJ theory.In this paper we use the following shorthand notations N Y n . . . = N Y n =1 . . . , N Y n 1) and N = ( N , N , N ). The corresponding relative description is h M • M ◦ M ◦i P N . (2.4)P of superscription emphasizes that x direction is periodic. On the other hand, when we focuson a divided part, we describe only that part and exclude P. We also write ⋆ instead of • and ◦ when we avoid to distinguish them.Next, we consider the N = 4 superconformal Chern-Simons theories described by circularquiver diagrams which are the three-dimensional worldvolume theories of D3-branes along 012directions. D3-branes on each interval gives rise to a vector multiplet of U ( N a ) group, whileeach 5-brane give rise to two bifundamental hypermultiplets. The Chern-Simons level on eachinterval is k a = k s a − s a − ) . (2.5)One important observable of these supersymmetric gauge theories is the partition functions.Localization technique [2] reduces the path integral computing the partition functions on roundthree-sphere to matrix models. These matrix models are parameterized by the Cartan subal-gebra α ( a ) n ( n = 1 , , . . . , N a ) of each U ( N a ) group: Z s k ( N ) = Y a (cid:18) N a ! Z d N a α ( a ) (cid:19) Z classic Z vector1 − loop Z hyper1 − loop . (2.6) Z classic captures the value of the action, while Z vector1 − loop and Z hyper1 − loop capture the 1-loop determi-nants of vector multiplets and hypermultiplets respectively. N a ! is the order of the Weyl groupof U ( N a ). Z classic contains the contributions from Chern-Simons terms. Explicitly, the contribution is e S ka ( N a ; α ( a ) ) , (2.7)7or each h ⋆N a ⋆ i , where S k ( N ; α ) = iπk N X n ( α n ) . (2.8)The contribution to Z vector1 − loop is N a Y n 1, respectively.In summary, what we will achieve in this paper is as follows. We study a special classof deformed factors (2.17) which we will explain in section 3.1. We first use the determinantformula (A.1), so that all elements become the inner product of h α | and | β i or constant vectors.We then eliminate all the Fresnel factors and transform the position eigenvectors according to(2.26), and the resulting deformed factors are denoted by e Z s k, M . We finally transform theminto the form (2.21). In this section we review the computation for deformed factors associated with the braneconfigurations consisting of less than three 5-branes, and we confirm our strategy explained inthe previous section. We first review one 5-brane case in section 2.3.1. The deformed factorconsisting of one NS5-brane and one (1 , k )5-brane is closely related to the ABJ theory, hencewe review this deformed factor from this viewpoint in section 2.3.2. The computation of the matrix models associated with the N = 4 Chern-Simons theoriesdescribed by circular quiver diagrams naturally reduces to the local computation for each 5-brane if all the nodes have equal ranks [3]. Therefore, in this case, it is enough to compute the12eformed factors of ( s a ) = ± 1. The corresponding brane configurations (2.16) are h • i N , h ◦ i N , (2.27)respectively, and the corresponding deformed factors (2.17) are Z ( ± k (cid:18) N ; α ~ , β ~ (cid:19) = Q Nn 12 cosh b p | β n i N × Nn,n . (2.29)We transform these deformed factors into e Z (+1) (cid:18) N ; α ~ , β ~ (cid:19) = ~ N det " h α n | e − i ~ b p 12 cosh b p e i ~ b p | β n i N × Nn,n , e Z ( − (cid:18) N ; α ~ , β ~ (cid:19) = ~ N det " h α n | e − i ~ b q e − i ~ b p 12 cosh b p e i ~ b q e i ~ b p | β n i N × Nn,n , (2.30)according to (2.26). This similarity transformation do not affect for s = +1 case. On theother hand, for s = − e − i ~ b p e − i ~ b q f ( b p ) e i ~ b q e i ~ b p = f ( b q ) , (2.31)we finally obtain e Z (+1) (cid:18) N ; α ~ , β ~ (cid:19) = ~ N det " h α n | 12 cosh b p | β n i N × Nn,n , e Z ( − (cid:18) N ; α ~ , β ~ (cid:19) = ~ N det " h α n | 12 cosh b q | β n i N × Nn,n . (2.32)This form is clearly (2.21). Therefore, we have succeeded in applying the Fermi gas formalism.Note that the above expression shows that b ρ (+1) = 12 cosh b p , b ρ ( − = 12 cosh b q . (2.33)13 .3.2 ABJ theory In [19–21], they applied the Fermi gas formalism to the ABJ matrix model with complicatedcomputations. After that, it was gradually realized that this type of computations can beperformed more systematically by using operator formalism [10, 33, 42–44]. In [34], by usingthis formalism, they computed the deformed factor consisting of one NS5-brane and one (1 , k )5-brane, which is closely related to the ABJ theory. In this section we review this computationfor the ease of understanding the main computation of this paper in section 3.2 where we willapply the Fermi gas formalism to more complicated deformed factors.The brane configuration of the ABJ theory is h • M ◦i P N . We restrict the number of D3-branes ending on an NS5-brane and a (1 , k )5-brane to M ≥ M ≤ k. (2.34)The part of the brane configuration needed for the ABJ theory is h • M ◦ i N , and correspond-ing deformed factor (2.17) is Z (+1 , − k,M (cid:18) N ; α ~ , β ~ (cid:19) = i { ( N + M ) − N } ( N + M )! Z d N + M ν ~ N + M e − i ~ P N + Mm ν m × Q Nn 12 cosh b p − iπM | ν m i i N × ( N + M ) n,m h ~ √ k hh πit M,j | ν m i i M × ( N + M ) j,m It was argued in [16] that the ABJ theory with M > k do not exist as N = 6 unitary theory. det (cid:18) h ~ h ν m | 12 cosh b p + iπM | β n i i ( N + M ) × N m,n h ~ √ k h ν m | − πit M,j ii i ( N + M ) × M m,j (cid:19) , (2.37)where t M,j = M + 12 − j. (2.38)We transform all position eigenvectors according to (2.26), namely, h α n | → h α n | e − i ~ b p , | ν m i → e i ~ b p | ν m i , h ν m | → h ν m | e − i ~ b p e − i ~ b q , | β n i → e i ~ b q e i ~ b p | β n i , (2.39)and we eliminate the Fresnel factors e − i ~ P N + Mm ν m . The result is denoted by e Z (+1 , − k,M . Asexplained in section 2.2, what we operated for ν m is just Z dν m e − i ~ ν m | ν m ih ν m | = Z dν m e i ~ b p | ν m ih ν m | e − i ~ b p e − i ~ b q , (2.40)and the ABJ matrix model (2.36) is expressed as Z (+1 , − k ( N, N + M ) = 1 N ! Z d N µ ~ N e Z (+1 , − k,M (cid:16) N ; µ ~ , µ ~ (cid:17) . (2.41)We apply (2.31) and the identity e − i ~ b p e − i ~ b q | p ii = 1 √ i e i ~ p | p i , hh p | e i ~ b q e i ~ b p = √ ie − i ~ p h p | , (2.42)to e Z (+1 , − k,M , so that we obtain e Z (+1 , − k,M (cid:18) N ; α ~ , β ~ (cid:19) = i { ( N + M ) − N } − M e iθ k,M ( N + M )! k − M Z d N + M ν × det h ~ h α n | 12 cosh b p − iπM | ν m i i N × ( N + M ) n,m h ~ √ k hh πit M,j | ν m i i M × ( N + M ) j,m × det (cid:18) h h ν m | 12 cosh b q + iπM | β n i i ( N + M ) × N m,n [ h ν m | − πit M,j i ] ( N + M ) × M m,j (cid:19) , (2.43)where θ k,M = − π k (cid:0) M − M (cid:1) . (2.44) The second identity is for later convenience. i ~ M Y j ( ± πit M,j ) = i θ k,M . (2.45)We use the identity N ! Z d N α det (cid:16) [ f m ( α n )] ( N + M ) × Nm,n (cid:2) f ′ mj (cid:3) ( N + M ) × Mm,j (cid:17) det (cid:16) [ g n ′ ( α n )] N × Nn,n ′ (cid:17) = Z d N α det (cid:16) [ f m ( α n )] ( N + M ) × Nm,n (cid:2) f ′ mj (cid:3) ( N + M ) × Mm,j (cid:17) Y n g n ( α n ) , (2.46)to diagonalize the second determinant, and then we again use the determinant formula (A.1).The result is e Z (+1 , − k,M (cid:18) N ; α ~ , β ~ (cid:19) = i { ( N + M ) − N } − M e iθ k,M k − M × Z d N + M ν Q Nn 12 cosh b q + iπM | β n i ! M Y j h ν N + j | − πit M,j i ! . (2.47)Fortunately, there are direct inner products of position operators, namely the delta functions,in the last line. Actually, it is straightforward to perform the integration over ν n (1 ≤ n ≤ N ).On the other hand, we should be careful to carry out the integration over ν N + j (1 ≤ j ≤ M )because the arguments of the delta functions are now complex number − πit M,j . We have toshift the integration contour from R to R − πit M,j so that we can use the property of the deltafunctions. If there are poles in the region where integration contour passes, we have to takeaccount of its residues. Appendix B provides more details . Nevertheless, we can also performthe remaining integration as usual because the restriction (2.34) ensures that the integrationcontour passes through no poles. We confirm this fact. The part of the function of ν N + j havingpoles and the region where this part has no poles are1 Q Nn α n − ν N + j k , No pole region: | Im ( ν N + j ) | < πk. (2.48) For later convenience, we write general form. In this case, f ′ mj = 0. Strictly speaking, we cannot use the argument in appendix B because the integrand of (2.47) do not convergeat ν N + j → ±∞ . This is because the deformed factors (2.17) are not convergent. Therefore, in this paper weshall give all Chern-Simons levels a small imaginary part (the sign is chosen such that the integral converges)and taking it to zero in the end. | πt M,j | ≤ π ( k − ν m , and the result is e Z (+1 , − k,M (cid:18) N ; α ~ , β ~ (cid:19) = i { ( N + M ) − N } − M e iθ k,M k − M × Q Nn 12 cosh ν n + iπM , (2.49)where ν m = β m (1 ≤ m ≤ N ) − πit M,m − N ( N + 1 ≤ m ≤ N + M ) . (2.50)We divide this into a phase factor depending on k , a phase factor Ω independent of k , Z whichis related to the N -independent factor C s k, M and O which is related to the density matrix: e Z (+1 , − k,M (cid:18) N ; α ~ , β ~ (cid:19) = e iθ k,M Ω Z O . (2.51)The explicit definitions areΩ = i { ( N + M ) − N } − M ,Z = k − M M Y j 12 cosh ν n + iπM . (2.52) Z is independent of α n and β n , while O depends on them. First, by substituting (2.50) into Z , we get Z = i − M ( M − k − M M Y j 12 cosh b p Q Mj b q − πit M,j k b q + iπM | β n i N × Nn,n . (2.57)Finally, by combining all of the above results, we obtain e Z (+1 , − k,M (cid:18) N ; α ~ , β ~ (cid:19) = e iθ k,M Z (CS) k,M ~ N det (cid:18)h h α n | b ρ (+1 , − M | β n i i N × Nn,n (cid:19) , (2.58)where b ρ (+1 , − M = b ρ ABJ M defined in (1.3). This form is clearly (2.21). Therefore we have accom-plished applying the Fermi gas formalism to the ABJ matrix model. As explained in section 2.2, what we will achieve in this paper is to transform the deformedfactors to the form of (2.21), and in this section we carry out this plan. In section 3.1, wepresent deformed factors we focus on in this paper and the result of the Fermi gas approach. Insection 3.2, we show its derivation. In section 3.3, we prove that the Hanany-Witten transitionholds on the level of deformed factors. We first explain the brane configurations in type IIB string theory which we focus on in thispaper (figure 1). These brane configurations involve an NS5-brane at the center, any number of(1 , k )5-branes on both sides of the NS5-brane and D3-branes stretched between two 5-branes.18 (cid:2)(cid:3)(cid:4)(cid:5)(cid:3) Figure 1: The brane configurations we consider in this paper. The blue line and the red linesrepresent an NS5-brane and (1 , k )5-branes, respectively. The black lines represent coincidentD3-branes, and each symbol above the D3-branes refers to the number of D3-branes on eachsegment.We parameterize these brane configurations as (cid:10) ◦ M q ◦ . . . ◦ M • M ◦ . . . ◦ M q ◦ (cid:11) . (3.1)We restrict the number of D3-branes stretched between an NS5-brane and a (1 , k )5-brane toless than k as is the case with ABJ theory. We also assume that the number of D3-branesmonotonically increases with approaching the NS5-brane at the center, and the growth of thenumber of the D3-branes near the NS5-brane is almost larger than that far from the NS5-brane. To give a rigorous description of the above statement, we define the difference betweenthe adjacent ranks: M a = M a − M a +1 , M a = M a − M a +1 , (3.2)where M q +1 = M q +1 = 0. These values satisfy the following restrictions:0 ≤ M a ≤ k, ≤ M a ≤ k (for all a and a ) ,M a + 1 ≥ M b , M a + 1 ≥ M b (cid:0) for all a < b and a < b (cid:1) ,M a + M a ≤ k + 1 (for all a and a ) . (3.3)We explain the physical meaning of these restrictions in the second and the third lines at the endof this section. Note that these brane configurations include the brane configuration appearedin section 2.3.2 as a special case.The deformed factors (2.17) corresponding to the part of brane configurations (3.1) are Z ( − q, +1 , − q ) k, M (cid:18) N ; α ~ , β ~ (cid:19) = i − n ( N + M ) − N o + { ( N + M ) − N } Q qa (cid:0) N + M a (cid:1) ! Q qa ( N + M a )! Z q Y a d N + M a µ ( a ) ~ N + M a ! q Y a d N + M a ν ( a ) ~ N + M a ! e S k (cid:18) N + M ; µ (1) ~ (cid:19) e S − k (cid:18) N + M ; ν (1) ~ (cid:19) q Y a Z (cid:18) N + M a +1 , N + M a ; µ ( a +1) ~ , µ ( a ) ~ (cid:19) × Z (cid:18) N + M , N + M ; µ (1) ~ , ν (1) ~ (cid:19) q Y a Z (cid:18) N + M a , N + M a +1 ; ν ( a ) ~ , ν ( a +1) ~ (cid:19) , (3.4)where M q +1 = M q +1 = 0, M = (cid:0) M q , . . . , M , M , . . . , M q (cid:1) , (3.5)and we introduced the following notation:( − q, +1 , − q ) = ( q z }| { − , . . . , − , +1 , q z }| { − , . . . , − . (3.6)Note that the order of the production Q qa is q = a, a − , . . . , Z ( − q, +1 , − q ) k, M according to the rule (2.26), and the result is denoted by e Z ( − q, +1 , − q ) k, M asexplained in section 2.2. We find that e Z ( − q, +1 , − q ) k, M (cid:18) N ; α ~ , β ~ (cid:19) = e i Θ ( − q, +1 , − q ) k, M q Y a Z (CS) k,M a ! q Y a Z (CS) k,M a ! × q − Y a q Y b = a +1 Z (vec) k,M a ,M b q Y a q Y a Z (mat) k,M a ,M a ! q − Y a q Y b = a +1 Z (vec) k,M a ,M b ! × ~ N det (cid:18)h h α n | b ρ ( − q, +1 , − q ) M | β n i i N × Nn,n (cid:19) . (3.7)The definition of each symbol is as follows. The phase symbol isΘ ( − q, +1 , − q ) k, M = 12 − q X a θ k,M a + θ k, M −M + q X a θ k,M a ! , (3.8)where θ k,M appeared in the ABJ theory and is defined in (2.44). The other N -independentfactors are Z (CS) k,M = 1 k M M Y j 12 cos πk (cid:0) t M,j − t M,j (cid:1) , (3.9)where Q Mt M,j = t M ′ ,j ′ means that j runs from 1 to M except for the case when t M,j = t M ′ ,j ′ . Z (vec) k,M,M ′ comes from 1-loop contributions of vector multiplets (2.9) , while Z (mat) k,M,M comes from 1-loopcontributions of hypermultiplets (2.10) as we will show in the next section. These factors arepositive as can be seen from this definition. The density matrix is b ρ ( − q, +1 , − q ) M = i M −M q Y a Q M a j a b q − πit Ma,ja k b q − iπM a q Y a Q M a j a b q − πit Ma,ja k ! × 12 cosh b p q Y a Q M a j a b q − πit Ma,ja k b q + iπM a ! q Y a Q M a j a b q − πit Ma,ja k . (3.10)Note that e Z ( − q, +1 , − q ) k, M reduces to (2.58) when q = 0 and q = 1.We give some comments on the setup and the results. First, the role of s = ± s = ± s = ± N = 0, in which case the last line of (3.7), namelythe Fermi gas factor vanishes. These are the matrix models computing the partition functionsof N = 4 superconformal Chern-Simons theories described by linear quiver diagrams [26, 40,41], because Z ( − q, +1 , − q ) k, M are matrix models of these theories as explained below (2.17) and inthis case the following identity holds: Z ( − q, +1 , − q ) k, M ( N = 0) = e Z ( − q, +1 , − q ) k, M ( N = 0) . (3.11)Third, we comment on the positive N -independent factor Z (CS) k,M . It is clear by comparing thebrane configurations (3.1) and the result (3.7) that Z (CS) k,M appears for each stack of coincidentD3-branes stretched between the same NS5-brane and the same (1 , k )5-brane. This is thenatural result because it is known that the worldvolume theory of this brane configuration21namely, a stack of M coincident D3-branes stretched between an NS5-brane and a (1 , k )5-brane) is U ( M ) k pure Chern-Simons theory [45, 46] and Z (CS) k,M is the partition function of thesame theory on S [47, 48].Fourth, we comment on the other positive N -independent factors Z (vec) k,M,M ′ and Z (mat) k,M,M . First, Z (vec) k,M,M ′ appears through (A.6), and the left-hand side of (A.6) diverges when the parametersexceed the second line of the restriction (3.3), namely when M + 2 = M ′ (see also the argumentin the below (A.6)). Z (mat) k,M,M also diverges when the parameters exceed the third line of therestriction (3.3), namely when M + M = k + 2. These divergences do not disappear by theprocedure in footnote 6. It was argued in [49] that when a matrix model diverges even afterthis procedure, the corresponding gauge theory is “bad” in the context of [37]. Furthermore,the more explicit suggestion appeared in [50, 51] for the worldvolume theory of the braneconfiguration (cid:10) • M ◦ M • (cid:11) (the corresponding matrix model is the complex conjugate ofone of (cid:10) • M ◦ M • (cid:11) , which is the special case of (3.1), and especially the positive factors areequal). Z (mat) k,M,M appears in this case, and they revealed that the gauge theory is “good”, “ugly”or “bad” when the parameters satisfy M + M ≤ k , M + M = k +1 or M + M ≥ k +2, respectively.Furthermore, the restrictions in the second and the third lines of (3.3) are equivalent under theHanany-Witten transition as we will explain in section 3.3, and thus it is natural to expect thatthe physical origin of divergence of Z (vec) k,M,M ′ and Z (mat) k,M,M are the same. Therefore, we expectthat if any one of the restrictions in the second and the third lines of (3.3) are not satisfied,the corresponding gauge theory is “bad”.Fifth, the Hermitian conjugate of a density matrix associated with a brane configuration(3.1) is one associated with the inverse of the brane configuration, namely (cid:10) ◦ M q ◦ . . . ◦ M • M ◦ . . . ◦ M q ◦ (cid:11) . (3.12)Explicitly, the identity (cid:16)b ρ ( − q, +1 , − q ) M (cid:17) † = b ρ ( − q, +1 , − q ) M t , (3.13)holds, where M t = (cid:0) M q , . . . , M , M , . . . , M q (cid:1) . (3.14)Note that this identity comes from the identity (cid:26) Z ( − q, +1 , − q ) k, M (cid:18) N ; α ~ , β ~ (cid:19)(cid:27) ∗ = Z ( − q, +1 , − q ) k, M t (cid:18) N ; β ~ , α ~ (cid:19) . (3.15)22 .2 Derivation In this section we compute the general deformed factors (3.4) and transform them into (3.7).The general flow of computation is similar to the ABJ case in section 2.3.2, and thus we omitsimilar points accordingly.We start with using the determinant formula (A.1) to (3.4): Z ( − q, +1 , − q ) k, M (cid:18) N ; α ~ , β ~ (cid:19) = i − n ( N + M ) − N o + { ( N + M ) − N } Q qa (cid:0) N + M a (cid:1) ! Q qa ( N + M a )! Z q Y a d N + M a µ ( a ) ~ N + M a ! q Y a d N + M a ν ( a ) ~ N + M a ! × e i ~ P N + M m (cid:16) µ (1) m (cid:17) e − i ~ P N + M m (cid:16) ν (1) m (cid:17) × q Y a det (cid:20) ~ h µ ( a +1) m a +1 | 12 cosh b p − iπMa | µ ( a ) m a i (cid:21) ( N + M a +1 ) × ( N + M a ) m a +1 ,m a h ~ √ k hh πit M a ,j a | µ ( a ) m a i i M a × ( N + M a ) j a ,m a × Z (cid:18) N + M , N + M ; µ (1) ~ , ν (1) ~ (cid:19) × q Y a det (cid:18) h ~ h ν ( a ) m a | 12 cosh b p + iπMa | ν ( a +1) m a +1 i i ( N + M a ) × ( N + M a +1 ) m a ,m a +1 h ~ √ k h ν ( a ) m a | − πit M a ,j a ii i ( N + M a ) × Ma m a ,j a (cid:19) . (3.16)It should be emphasized that we have applied (A.1) also to Z (cid:0) N + M , N + M (cid:1) . As explainedin section 2.2, we eliminate all the Fresnel factors and transform Z ( − q, +1 , − q ) k, M into e Z ( − q, +1 , − q ) k, M ac-cording to (2.26). We further proceed with the computation in the same manner as section2.3.2. More explicitly, we first use the formulas (2.31) and (2.42) so that all momentum oper-ators and momentum eigenvectors except for those in Z (cid:0) N + M , N + M (cid:1) become positionoperators and position eigenvectors respectively. Second, we diagonalize all determinants exceptfor Z (cid:0) N + M , N + M (cid:1) using the formula (2.46). Finally, we restore Z (cid:0) N + M , N + M (cid:1) using the determinant formula (A.1) backward. After these computations we obtain e Z ( − q, +1 , − q ) k, M (cid:18) N ; α ~ , β ~ (cid:19) = i − n ( N + M ) − N o + { ( N + M ) − N } i M − M e i Θ ( − q, +1 , − q ) k, M k − M − M × Z q Y a d N + M a µ ( a ) ! q Y a d N + M a ν ( a ) ! q Y a N + M a +1 Y m a +1 h µ ( a +1) m a +1 | 12 cosh b q − iπM a | µ ( a ) m a +1 i M a Y j a h πit M a ,j a | µ ( a ) N + M a +1 + j a i × Q N + M m 12 cosh b q + iπM a | ν ( a +1) m a +1 i ! M a Y j a h ν ( a ) N + M a +1 + j a | − πit M a ,j a i !) . (3.17)All the integrations seem to be performed using the delta functions coming from inner prod-ucts of position eigenvectors. However, fake divergences appear with a naive computation .Therefore, we introduce small convergence factors into all constant position eigenvectors: h πit M a ,j a | = lim ǫ a → h πit M a ,j a + ǫ a | , | − πit M a ,j a i = lim ǫ a → | − πit M a ,j a + ǫ a i . (3.18)We have to consider poles to perform the integration because of the argument in appendix Bas with the ABJ theory. To come to the point, there are no poles in the region where integrationcontour passes due to the restriction (3.3), so that we can perform all the integration using theproperty of the delta functions as with the ABJ theory. First, we perform the integration over ν (1) m . We should be careful about the case when m = N + M + j (1 ≤ j ≤ M ). The partof the function of ν (1) N + M + j having poles and the region where this part has no poles is1 Q N + M m µ (1) m − ν (1) m k , No pole region: (cid:12)(cid:12) Im (cid:0) ν (1) m (cid:1)(cid:12)(cid:12) < πk. (3.19)Moreover, the restriction in the first line of (3.3) leads | πt M ,j | ≤ π ( k − ν (1) m , we perform theintegration over ν (2) m . In this case, there are two types of functions of ν (2) N + M + j (1 ≤ j ≤ M )having poles: 1 Q N + M m µ (1) m − ν (2) m k , No pole region: (cid:12)(cid:12) Im (cid:0) ν (2) m (cid:1)(cid:12)(cid:12) < πk, Q M j ν (2) m +2 πit M ,j k ν (2) m + iπM , No pole region: (cid:12)(cid:12) Im (cid:0) ν (2) m (cid:1)(cid:12)(cid:12) < π ( M + 1) . (3.20) This divergence already appeared in [33]. The reason this divergence appear is we used the determinantformula (A.1). | πt M ,j | ≤ π ( k − | πt M ,j | ≤ πM respectively, so that integration contour again passes through no poles.The similar argument holds for the other ν ( a ) m a . Therefore, we can perform the integration over ν ( a ) m a in the order a = 1 , , . . . , q . We can also carry out the integration over µ ( a ) m a in the order a = 1 , , . . . , q in the similar manner. The only difference is that there are factors1 Q N + M m µ (1) m − ν (1) m k , (3.21)in (3.17) which now contain imaginary constant − πit M a ,j a instead of real number ν (1) m . Theregion where this function has no poles is q Y a Q M a j a µ ( a ) ma +2 πit Ma,ja k , No pole region: (cid:12)(cid:12)(cid:12) Im (cid:16) µ ( a ) m a (cid:17)(cid:12)(cid:12)(cid:12) < π ( k + 1 − M a ) . (3.22)Moreover, the restriction in the third line of (3.3) leads (cid:12)(cid:12)(cid:12) πt M a ,j a (cid:12)(cid:12)(cid:12) ≤ π ( k − M a ), so that wecan perform the integration over µ ( a ) m a . From the above, we can carry out all the integration: e Z ( − q, +1 , − q ) k, M (cid:18) N ; α ~ , β ~ (cid:19) = i − n ( N + M ) − N o + { ( N + M ) − N } i M − M e i Θ ( − q, +1 , − q ) k, M k − M − M × q Y a lim ǫ a → ! q Y a lim ǫ a → ! q Y a N + M a +1 Y m a +1 12 cosh µ ma +1 − iπM a × Q N + M m 12 cosh ν ma +1 + iπM a , (3.23)where µ m = α m (1 ≤ m ≤ N )2 πit M q ,m − N + ǫ q (cid:0) N + 1 ≤ m ≤ N + M q (cid:1) πit M q − ,m − N −M q + ǫ q − (cid:0) N + M q + 1 ≤ m ≤ N + M q − (cid:1) ...2 πit M ,m − N −M + ǫ (cid:0) N + M + 1 ≤ m ≤ N + M (cid:1) , (3.24)25nd ν m = β m (1 ≤ m ≤ N ) − πit M q ,m − N + ǫ q ( N + 1 ≤ m ≤ N + M q ) − πit M q − ,m − N −M q + ǫ q − ( N + M q + 1 ≤ m ≤ N + M q − )... − πit M ,m − N −M + ǫ ( N + M + 1 ≤ m ≤ N + M ) . (3.25)To proceed the computation, we divide this into a phase factor depending on k , a phasefactor Ω independent of k , Z which is related to the N -independent factor C s k, M and O whichis related to the density matrix: e Z (+1 , − k,M (cid:18) N ; α ~ , β ~ (cid:19) = e i Θ ( − q, +1 , − q ) k, M Ω Z O . (3.26)The explicit definitions areΩ = i − n ( N + M ) − N o + { ( N + M ) − N } + M − M ,Z = k − M − M q Y a lim ǫ a → ! q Y a lim ǫ a → ! q − Y a M a +1 Y j a +1 12 cosh µ N + ja +1 − iπM a × Q M j 12 cosh ν N + ja +1 + iπM a , O = q Y a N Y n 12 cosh µ n − iπM a × Q Nn Q N + M m = n +1 µ n − µ m k Q Nn Q N + M m = n +1 ν n − ν m k Q ( m,m ) / ∈ ( N + j,N + j ) 2 cosh µ m − ν m k q Y a N Y n 12 cosh ν n + iπM a , (3.27)where Q ( m,m ) / ∈ ( N + j,N + j ) means that m and m run the range where m and m are not largerthan N simultaneously. In other words, they run the following three ranges: ≤ m ≤ N, ≤ m ≤ N ≤ m ≤ N, N + 1 ≤ m ≤ N + M N + 1 ≤ m ≤ N + M , ≤ m ≤ N . (3.28) Z is independent of α n and β n , while O depends on them. We start with Z and especiallyfocus on the factors independent of µ n (in other words, the factors depending only on ν n ). Wealso put k − M and the limit operation ǫ a → 0, and we denote by I it. By substituting (3.25)26nd using the following identities M Y j 12 cosh ν ma +1 + iπM a = q Y a i − M a ( M a − Z (CS) k,M a ! q Y a lim ǫ a → ! q − Y a q Y b = a +1 M b Y j b Q M a j a − πi ( t Mb,jb − t Ma,ja ) + ǫ b − ǫ a k − πit Mb,jb + iπM a + ǫ b . (3.30)We now take the limit ǫ a → a = 1 , , . . . , q . The limitfor a = 1 is trivial. On the other hand, we should use the identity (A.6) to take the limit for a = 2 , , . . . , q . We then get I = i − M ( M − q Y a Z (CS) k,M a ! q − Y a q Y b = a +1 Z (vec) k,M a ,M b ! . (3.31)Next, we calculate the factors depending only on µ n with k − M and lim ǫ a → . The process isthe same with the above, and the result is k − M q Y a lim ǫ a → ! M Y j 12 cosh µ ma +1 − iπM a = i M ( M − ) q Y a Z (CS) k,M a ! q − Y a q Y b = a +1 Z (vec) k,M a ,M b . (3.32)27inally, we calculate the factors depending on both µ n and ν n :1 Q M j Q M j µ N + j − ν N + j k = q Y a q Y a Z (mat) k,M a ,M a . (3.33)In summary, Z can be written as Z = i M ( M − ) − M ( M − q Y a Z (CS) k,M a ! q Y a Z (CS) k,M a ! × q − Y a q Y b = a +1 Z (vec) k,M a ,M b q Y a q Y a Z (mat) k,M a ,M a ! q − Y a q Y b = a +1 Z (vec) k,M a ,M b ! . (3.34)Next, we combine Ω and the phase appeared in the above expression: i M ( M − ) − M ( M − Ω = i N ( M −M ) . (3.35)Finally, we combine this phase and O . By substituting (3.24) and (3.25) and using the identities N + M Y m f ( µ m ) = N Y n f ( α n ) ! q Y a M a Y j a f (cid:16) πit M a ,j a (cid:17) , N + M Y m f ( ν m ) = N Y n f ( β n ) ! q Y a M a Y j a f ( − πit M a ,j a ) ! , (3.36)we obtain i N ( M −M ) O = N Y n i M −M q Y a Q M a j a α n − πit Ma,ja k α n − iπM a q Y a Q M a j a α n − πit Ma,ja k ! × Q Nn The quantum curves at first naturally appeared as the inverses of the density matrices associatedwith the N = 4 Chern-Simons theories described by circular quiver diagrams without any rankdeformations [3, 30]. In this situation, namely M = , the density matrices are the productof (2.33) following (2.23) because we can cut the whole matrix models at all gauge nodes anddecompose it completely as explained in section 2.2. Therefore, the inverses of the densitymatrices are the products of (cid:16) b Q + b Q − (cid:17) and (cid:16) b P + b P − (cid:17) , where b Q = e b q , b P = e b p . (4.1)Thus they are Laurent polynomials of b Q and b P .On the other hand, it is highly nontrivial that whether the inverses of rank deformed densitymatrices, which are denoted by b H sM = ( b ρ sM ) − , (4.2)can be written as the specific form which can be termed the quantum curve or not. The progressfor the ABJ theory gave a positive answer [23] (see also appendix A.3 in [34]) b H (+1 , − M = b Q − M b P + b Q M b P − , (4.3)for the ABJ density matrix (1.3), where b Q M = e iπM b Q + e − iπM b Q − . (4.4)In the next section we show that the inverses of the density matrices (3.10), which we found inthis paper, are also the quantum curves. In this section, we relax the restrictions against the relative ranks (3.3). We assume only that M a and M a are non-negative integers. For the right side and the left side of the labels of therelative ranks M = (cid:0) M q , . . . , M , M , . . . , M q (cid:1) , (4.5)we define the product of b Q M operator respectively: b Q M R = q Y a b Q M a , b Q M L = q Y a b Q M a . (4.6)31he inverses of the density matrices (3.10) are b H ( − q, +1 , − q ) M = b Q − M R b P b Q M L + b Q M R b P − b Q − M L . (4.7)We prove this identity in appendix C.The above result means that the quantum curves associated with the brane configurations(3.1) are (4.7). This is natural in the following sense. As explained above, a quantum curve asso-ciated with a brane configuration without rank deformations can be obtained by replacing eachNS5-brane and each (1 , k )5-brane in the brane configuration to (cid:16) b P + b P − (cid:17) and (cid:16) b Q + b Q − (cid:17) respectively in reverse order (since quantum curves are the inverses of density matrices). Rankdeformations break this factorization. However, for the ABJ theory case (4.3), if we considerthe term containing b P and the term containing b P − separately, the similar replacing rulecan be applied. That is, the brane configuration of the ABJ theory is h • M ◦i P N , and theABJ quantum curve has the structure that b Q ± M is located on the left side of b P ± . Our setup(3.1) contains the larger number of (1 , k )5-branes on both sides of the NS5-brane. (4.7) meansthat we can also apply a similar replacing rule to these brane configurations. That is, thecorresponding quantum curves (4.7) have the structure that the same number of b Q ± M as the(1 , k )5-branes are located on both sides of b P ± .The concept of order of operators in the above argument seems to be meaningless becausetwo b Q ± M are commutative and also we can exchange b Q ± M and b P ± using the identity b P α b Q M = b Q M − αk b P α , (4.8)where we used b P α b Q β = e − i ~ αβ b Q β b P α . However, conversely, we can consider that this impliesdualities in the gauge theory side. In fact, the exchange of b Q ± M and b P ± is related to theHanany-Witten transition as explained in the next paragraph. We also give a discussion abouta “Hanany-Witten move of two (1 , k )5-branes” in section 6, which is related to the exchangeof two b Q ± M .We finally comment on the Hanany-Witten transition in terms of the quantum curves. Thequantum curves associated with the dual theories are identical since the corresponding densitymatrices are identical (3.45). Of course, this identity can be checked directly. Using (4.8), weobtain b H ( − q, +1 , − q ) M origin = b Q − M originR b P b Q M b Q M HWL + b Q M originR b P − b Q − M b Q − M HWL = b Q − M originR b Q − ( k − M ) b P b Q M HWL + b Q M originR b Q k − M b P − b Q − M HWL = b H ( − ( q − , +1 , − ( q +1)) M HW . (4.9)32his identity reveals that the appearance of k D3-branes in the Hanany-Witten move is trans-lated into the noncommutativity between the position operator and the momentum operator. N = 4 Chern-Simons theories with four nodes So far we studied the rank deformed gauge theories from three viewpoints, namely the view-points of brane configurations, density matrices and quantum curves. In this section we studythe brane configurations consisting of two NS5-branes and two (1 , k )5-branes using these results.Concretely, we focus on s = { +1 , +1 , − , − } and s = { +1 , − , +1 , − } . The worldvolumetheories of these brane configurations are the N = 4 Chern-Simons theories described by thecircular quiver diagrams with the gauge groupsU ( N ) k × U ( N ) × U ( N ) − k × U ( N ) , U ( N ) k × U ( N ) − k × U ( N ) k × U ( N ) − k . (5.1)The first and the second theories are called (2 , 2) theory and (1 , , , 1) theory, respectively.Note that they are expected to share the same quantum curve since they are related by theHanany-Witten transition.The reason why we focus on these gauge theories is that motivated by the discovery ofsymmetries of these gauge theories by combining the Fermi gas formalism and numerical studies[31, 33, 52], a “symmetry” of the associated quantum curve has been studied [35]. Theydefined that quantum curves are symmetric if they are equal up to similarity transformations.This definition is natural because the values of the matrix models (2.14) are invariant undersimilarity transformations. They revealed that the associated quantum curve obeys the Weylgroup symmetry of SO (10).It was found in [34] that this discovery led to two important results. First, they foundthe explicit relations between the brane configurations associated with the (2 , 2) theory andthe (1 , , , 1) theory and the associated quantum curve by using the symmetries in both sides,namely the Hanany-Witten transition and the SL (2 , Z ) transformation in type IIB string theoryand the Weyl group symmetry in the quantum curve, instead of using the Fermi gas approach.Second, in the quantum curve side, they found new symmetry which cannot be generated bythe Hanany-Witten transition and the SL (2 , Z ) transformation. In this section we review thesetwo points and comment on them using our results.33 .1 Quantum curves from Fermi gas approach In this section at first we review the conjecture in [34] which predicts the explicit relationbetween the matrix model of the (2 , 2) theory and a quantum curve, and then we partiallyprove this conjecture by using our results. We label the ranks of the (2 , 2) theory as D f M + f M • f M + 2 f M • f M + f M + f M ◦ f M ◦ E P N . (5.2)They conjectured that the corresponding quantum curve is b Hα = b Q b P + (1 + e e ) b P + e e b Q − b P + (cid:0) e e − (cid:1) b Q + Eα + e − e ( e + e ) b Q − + e e − b Q b P − + e − e (cid:0) e + e − (cid:1) b P − + e − e b Q − b P − , (5.3)where e i = e πi f M i . α and E are not determined in the conjecture because these coefficients donot affect to the symmetry of the quantum curve. The direct derivation of the conjecture byusing the Fermi gas approach is also important in this meaning because the Fermi gas approachdetermines also these coefficients. Note that the coefficients of b Q +1 and b P +1 are adjusted bythe similarity transformations generated by b Q and b P , which multiply each terms by constantsaccording to b P α b Q β = e − i ~ αβ b Q β b P α .So far we focused on the (2 , 2) theory. Nevertheless, this conjecture also treats the (1 , , , , 2) theory and the (1 , , , 1) theoryand the quantum curves are expected to obey the Hanany-Witten transition (this is the casefor our setup (4.9)). Therefore, it is enough to study the (2 , 2) theory.In this paper we related the brane configurations to the quantum curves by using the Fermigas approach. We now compare this relation to the conjecture (5.3) and show that these areconsistent. Unfortunately, we cannot deal with all the rank deformations with the techniqueswe developed in this paper. The non-trivial brane configurations we can deal with are h M • • M ◦ ◦i P N , h • • M + M ◦ M ◦i P N , (5.4)where M and M are non-negative integers. Note that we can also deal with the s = ± f M i can be negative and half-integer. They indicated that it is necessary to fix one interval between two 5-branes for relating brane configurationsand quantum curves. In this statement we implicitly put this “reference” on the left most D3-branes of thebrane configuration in (5.2). b H (+1 , − M (cid:16) b H (+1 , − M (cid:17) † , (5.5)where we took account of (3.13). This quantum curve has been calculated in [34] because thecomponents are only the ABJ quantum curves, and they confirmed that the result is consistentwith the conjecture.Next, we consider the second line of (5.4). The corresponding quantum curve is b H (+1 , − M ,M ) b H (+1) . (5.6)We substitute (4.7), take the similarity transformation b P − M k [ · ] b P M k (this operation changes b Q to e iπM b Q ) and multiply it by m m − where m i = e iπM i : m m − b P − M k b H (+1 , − M ,M ) b H (+1) b P M k = n(cid:16) b Q + b Q − (cid:17) (cid:16) b Q + m m − b Q − (cid:17) b P + (cid:16) m m b Q + b Q − (cid:17) (cid:16) b Q + m − b Q − (cid:17) b P − o × (cid:16) b P + b P − (cid:17) . (5.7)Now we return to the conjecture (5.3). The brane configuration we focus on now is (cid:16) f M , f M , f M (cid:17) = (cid:18) M + 12 M , M , − M (cid:19) , (5.8)(and N = − M + M ) in the notation of (5.2), so that the conjectured quantum curve is b Hα = b Q b P + (cid:0) m m − (cid:1) b P + m m − b Q − b P + (1 + m m ) b Q + Eα + m − (cid:0) m + m − (cid:1) b Q − + m m b Q b P − + m − ( m + m ) b P − + m − b Q − b P − . (5.9)These two quantum curves are exactly the same. Moreover, we find that α = m − m and35 = 2 m − m + 2 m m − . In [34], they indicated that the two quantum curves associated with h N • N ◦ N • N ◦i P , h N • N ◦ N • N ◦i P , (5.10)are identical up to the similarity transformations based on the conjecture (5.3) (see (3.17) in[34]). This symmetry would be interesting because this symmetry cannot be generated by theHanany-Witten transition and the SL (2 , Z ) transformation. In this paper, we call this typeof symmetry non-trivial symmetry. It would be natural to ask when the non-trivial symmetryarises. In this section, we try to answer this question by considering the non-trivial symmetryin terms of the density matrices. We focus on the special cases (cid:10) • M ◦ • M ◦ (cid:11) P N , b H (+1 , − M b H (+1 , − M , (cid:10) • ◦ M • M ◦ (cid:11) P N , b H ( − , +1 , − M,M b H (+1) , (5.11)to use our results. We wrote the corresponding quantum curves on the right side. The aboveexplanation did not clarify the meaning of the non-trivial symmetry. Therefore, we explain thispoint at first. In [35], they found that the two quantum curves are equal up to the similaritytransformation: b G − b H (+1 , − M b H (+1 , − M b G = b H ( − , +1 , − M,M b H (+1) , (5.12)where b G satisfies b G − b P b G = b Q − − M − k b P b Q M + k . (5.13)This similarity transformation is non-trivial because we can only consider the similarity trans-formations which transform a polynomial to another polynomial, while this similarity trans-formation seems not to satisfy this restriction at first sight. It is for this reason we call thesymmetry generated by b G non-trivial symmetry. In what follows we study the non-trivialsymmetry using our results.We first realize that b G is equal to b G M defined in (A.19). To understand the reason b G M appears, we consider the inverse of (5.12) and then get the identity between the density matrices: b G − M b ρ (+1 , − M b ρ (+1 , − M b G M = b ρ (+1) b ρ ( − , +1 , − M,M . (5.14)36his identity is trivial because of (3.10), namely b ρ (+1 , − M b ρ (+1 , − M = i M Q Mj b q − πit M,j k 12 cosh b p Q Mj b q − πit M,j k b q + iπM × i M Q Mj b q − πit M,j k 12 cosh b p Q Mj b q − πit M,j k b q + iπM ! , b ρ (+1) b ρ ( − , +1 , − M,M = 12 cosh b p ! i M − M Q Mj b q − πit M,j k b q − iπM Q Mj b q − πit M,j k × 12 cosh b p Q Mj b q − πit M,j k b q + iπM Q Mj b q − πit M,j k . (5.15)That is, b ρ (+1 , − M has the structure that (cid:16) b p (cid:17) − lies at the center and (cid:16)Q Mj b q − πit M,j k (cid:17) − and (cid:16) b q + iπM (cid:17) − × (cid:16)Q Mj b q − πit M,j k (cid:17) lie at the side, and b G M moves these of b ρ (+1 , − M into b ρ (+1 , − M . As a result, the length of one s = (+1 , − 1) decreases, namely b ρ (+1 , − M → b ρ (+1) ,and the length of another s = (+1 , − 1) increases, namely b ρ (+1 , − M → b ρ ( − , +1 , − M,M .Based on the above argument, we realize that the non-trivial symmetry exists when thequantum curves are product of two b ρ ( − q, +1 , − q ) M since b ρ ( − q, +1 , − q ) M also has the structure that (cid:16) b p (cid:17) − lies at the center and some (cid:16)Q Mj b q − πit M,j k (cid:17) − and some (cid:16) b q + iπM (cid:17) − × (cid:16)Q Mj b q − πit M,j k (cid:17) lie at the side.The above argument focuses only on the rank deformations which are the combination of two b ρ ( − q, +1 , − q ) M . However, for example, the non-trivial symmetry exists for general rank deformations(5.10) in the (1 , , , 1) theory. Therefore, we expect that our argument in this section can beapplied also for general rank deformations. Thus we expect that the presence or the absence ofthe non-trivial symmetry depends only on the number of the NS5-branes and the (1 , k )5-branes,and when the number of the NS5-branes (or the (1 , k )5-branes) is just two, the non-trivialsymmetry presents. In fact, the quantum curve associated with s = { +1 , − , − , − , − } obeys the Weyl group symmetry of E [35].Note that the positive N independent factors in (3.7) do not obey the non-trivial symmetry.For example, the factors associated with the first line of (5.11) consist only of the pure Chern-Simons factors Z (CS) k,M Z (CS) k,M , while the factors associated with the second line consist of Z (mat) k,M,M inaddition to the same pure Chern-Simons factors. Therefore, the non-trivial symmetry appearsonly in the partition functions of the ideal Fermi gases. In this meaning, it is essential for thenon-trivial symmetry that the form of quiver diagrams is not linear but circular.37 Conclusion and discussion In this paper we applied the Fermi gas formalism to the matrix models computing the partitionfunctions of N = 4 circular quiver Chern-Simons theories with various rank deformations. Ourmethod can be applied to a wide class of rank deformations since we provided the computa-tion method for deformed factors, which are parts of matrix models. After the computation,we found that the deformed factors factorize to the pure Chern-Simons factors, the other N independent factors depending on two relative ranks and the partition functions of ideal Fermigases. The equality between dual theories related by the Hanany-Witten transition holds foreach type of factor. We also found that the inverses of the density matrices can be rewritten asthe quantum curves. Using these results we studied specific N = 4 Chern-Simons theories withthe gauge group consisting of four unitary groups, and we especially focused on the non-trivialsymmetry generated by (A.19).The quantum curves obey this non-trivial symmetry, while the N independent factors donot obey it as explained in section 5.2. Interestingly, the same thing happens when we considerthe exchange of two M a or two M a in the left side notation of (3.2), namely difference notation.This is because the quantum curves (4.7) of before and after the exchange are completely equalas can be easily seen, while Z (vec) k,M,M ′ = Z (vec) k,M ′ ,M in general . We might be able to relate thissymmetry to a “Hanany-Witten move of two (1 , k )5-branes”. Though the argument developedin [18] cannot apply for the exchange of 5-branes of the same type, here we assume thatthe argument in section 3.3 also holds in this case. Then, since no D3-branes are created,the exchange of two adjacent (1 , k )5-branes in the brane configuration (3.1) has the effect ofexchanging two adjacent M a or two adjacent M a . In any case, it would be great to reveal thephysical origin of the symmetries obeyed by the quantum curves (or equivalently the densitymatrices) but not obeyed by the N independent factors.We shall list some further directions in the following. First, we performed the computationof the deformed factors under the restriction (3.3). The first line of this restriction, namely M a ≤ k and M a ≤ k are just for using the argument in appendix B from the technical point ofview. It is interesting to uncover the behavior of the matrix models in the Fermi gas descriptionbeyond the restriction.Second, as explained in the introduction, our main motivation to consider the Fermi gasformalism is the expectation that using the formalism we can calculate, for example, the mem-brane instantons and the exact values of the matrix models associated with the rank deformed N = 4 Chern-Simons theories as with the case of the ABJ(M) theory and the N = 4 Chern- The parameters of rank deformations on the left-hand side or the right-hand side of the inequality maynot satisfy the restriction in the second line of (3.3). However, we can calculate both sides when the differencebetween the two parameters is one. The result is Z (vec) k,M +1 ,M = 2 Z (vec) k,M,M +1 . P × P [10, 53]. It is interesting whether asimilar story exists in more general cases we studied.Third, in this paper we focused on the specific brane configurations (3.1), and thereforewe need to develop the computation technique to deal with more general brane configurations.Furthermore, there are various generalizations of the ABJM theory, and it is expected that theFermi gas formalism for these theories with rank deformations provide, for example, identitiesbetween the partition functions of dual theories. Therefore, we hope to apply the Fermi for-malism to, for example, mass deformed theories [41] (the result for the ABJM theory is in [54]),theories with gauge groups including Lie groups other than unitary groups (the result for thegauge group consisting of two groups is in [43]), theories including the fundamental matters [3,27], and the D-type quiver theories (the results for specific rank theories are in [55, 56]) andthe E-type quiver theories in the context of the ADE classification [57]. Acknowledgments We are grateful to Masazumi Honda, Seiji Terashima, Koji Umemoto, Shuichi Yokoyama andespecially Sanefumi Moriyama for valuable discussions and comments. This work was supportedby Grant-in-Aid for JSPS Fellows No.20J12263. A Formulas This appendix provides various formulas we use in our computations. A.1 Determinant formula In this section we show that the 1-loop contributions for each 5-brane (2.11) can be writtenas a determinant of a matrix all whose elements are inner products of vectors. Explicitly, thefollowing identities hold for non-negative integer N and M :39 Nn 12 cosh b p − iπM | β m i i N × ( N + M ) n,m h ~ √ k hh πit M,j | β m i i M × ( N + M ) j,m , Q N + Mm 12 cosh b p + iπM | β n i i ( N + M ) × Nm,n h ~ √ k h α m | − πit M,j ii i ( N + M ) × Mm,j (cid:19) , (A.1)where [ f a,b ] A × Ba,b denotes an A × B matrix whose ( a, b ) element is f a,b , and t M,j is defined in(2.38). We only derive the first identity in the rest of this section because the second identitycan be easily derived from the first identity.The starting point is the combination of the Cauchy determinant formula and the Vander-monde determinant formula [22, 58]: Q Nn 12 cosh πit M ′ ,j ′ + iπM + ǫ ) = i − MM ′ Z (vec) k,M,M ′ . (A.6)In the rest of this section, we prove this identityThe left-hand side of (A.6), which is denoted by I , is equal to I = i − MM ′ M ′ Y j ′ ( lim ǫ j ′ → M Y j πi ( t M ′ ,j ′ − t M,j ) + ǫ j ′ k ! e ǫj ′ + e − πi ( t M ′ ,j ′ + M ) e − ǫj ′ ) . (A.7)At this point, we consider separately the cases when M + M ′ is even or odd. First, we considerthe even case. In this case, zero appears in the denominator after taking the limit, because2 t M ′ ,j ′ + M is odd. However, the restriction M ′ ≤ M + 1 ensures the appearance of zero alsoin the numerator. Therefore, I = i − MM ′ M ′ Y j ′ M Y t M,j = t M ′ ,j ′ πi ( t M ′ ,j ′ − t M,j )2 k lim ǫ j ′ → ǫ j ′ k ǫ j ′ = i − MM ′ Z (vec) k,M,M ′ , (A.8)where Q Mt M,j = t M ′ ,j ′ means that j runs from 1 to M except for the case when t M,j = t M ′ ,j ′ . Wewould like to emphasize that I diverges if M + M ′ is even and M ′ ≥ M + 2. Second, we consider denominator of (3.37). The zeros at α n = 0 and β n = 0 cancel with zeros appeared in the numerator. In thismeaning, we do not mind this problem. Second, the residue of j = M +12 might be the half. However, this termvanishes as explained just below. Therefore, we also do not mind this problem. t M ′ ,j ′ + M is even, and weagain obtain (A.6). A.3 Hanany-Witten transition In this appendix we provide identities related to the Hanany-Witten transition. We assumethat non-negative integers k , M and M satisfy M ≤ k , M ≤ k and M + M ≤ k + 1. Theidentities related to the positive N -independent factors (3.9) are Z (CS) k,M = Z (CS) k,k − M ,Z (mat) k,M,M = Z (vec) k,k − M,M , (A.9)and related to the density matrices are i M Q Mj b q − πit M,j k 12 cosh b p Q Mj b q − πit M,j k b q + iπM = i − ( k − M ) Q k − Mj b q − πit k − M,j k b q − iπ ( k − M )2 12 cosh b p Q k − Mj b q − πit k − M,j k . (A.10)In the rest of this section, we prove these identities .We start with the well-known identity z k − k Y j (cid:16) z − e πik j (cid:17) . (A.11)By substituting z = e − k ( q − iπ ( M +1)) , we get i k ( − M q + iπM k Y j q − iπt M,j k . (A.12)We divide the production in the right-hand side into less than and not less than M + 1. Thelatter part becomes k Y j = M +1 q − iπt M,j k = i k − M k − M Y j =1 q − iπt k − M,j k . (A.13) The identity in the first line of (A.9) was proved in [49]. i − M q + iπM M Y j q − πit M,j k ! k − M Y j q − πit k − M,j k ! . (A.14)The combination of this identity and the identity obtained by changing M into k − M in thisidentity leads (A.10).Furthermore, by changing M into k − M in (A.14) we get i k − M Q k − Mj q − πit k − M,j k q + iπ ( k − M ) = M Y j 12 cosh q − πit M,j k . (A.15)By substituting q = 2 πit M,j + ǫ and multiplying j = 1 , , . . . , M we get i ( k − M ) M M Y j Q k − Mj πit M,j − πit k − M,j + ǫ k πit M,j + iπ ( k − M ) + ǫ = M Y j M Y j 12 cosh πit M,j − πit M,j − ǫ k . (A.16)We take the limit ǫ → (cid:12)(cid:12) t M,j − t M,j (cid:12)(cid:12) ≤ k − for the right-hand side, and we obtain the identity in the second line of (A.9).Finally, using (A.5) and (A.9) we get (cid:16) Z (CS) k,M (cid:17) = Z (vec) k,M,M = Z (mat) k,k − M,M = Z (mat) k,M,k − M = Z (vec) k,k − M,k − M = (cid:16) Z (CS) k,k − M (cid:17) . (A.17)Moreover, Z (CS) k,M is positive. Therefore, the identity in the first line of (A.9) holds. A.4 Similarity transformation In this section we prove the identity b G − M b P b G M = 1 b Q − M − k b P b Q M + k , b G M b P b G − M = b Q − M − k b P b Q M + k , (A.18)where b Q M is defined in (4.4), and b G M = M Y j 12 cosh b q − πit M,j k . (A.19)43he identity in the second line of (A.18) is the Hermitian conjugate of the first line. In the restof this section, we prove the first line.The quantum dilogarithm Φ b ( x ) plays an important role. This function can be expressedas [59–61] Φ b ( x ) = (cid:16) e π b ( x + i ( b + b − )); e πi b (cid:17) ∞ (cid:16) e π b − ( x − i ( b + b − ) ); e − πi b − (cid:17) ∞ , (A.20)where ( a ; q ) n = n − Y k =0 (cid:0) − aq k (cid:1) , (A.21)is the q-Pochhammer symbol. The quantum dilogarithm satisfiesΦ b (cid:0) x + i b (cid:1) Φ b (cid:0) x − i b (cid:1) = 11 + e π b x , Φ b (cid:0) x + i b − (cid:1) Φ b (cid:0) x − i b − (cid:1) = 11 + e π b − x . (A.22)We substitute x = q − πit M,j π √ k to the first line of (A.22) and multiply j = 1 , , . . . , M , then weobtain Φ √ k (cid:16) q π √ k + iM √ k (cid:17) Φ √ k (cid:16) q π √ k − iM √ k (cid:17) = e − M k q M Y j 12 cosh q − πit M,j k . (A.23)Moreover, using the second line of (A.22) and f ( b q ) b P = b P f ( b q + 2 πik ) we obtainΦ √ k (cid:16) b q π √ k − iM √ k (cid:17) Φ √ k (cid:16) b q π √ k + iM √ k (cid:17) b P Φ √ k (cid:16) b q π √ k + iM √ k (cid:17) Φ √ k (cid:16) b q π √ k − iM √ k (cid:17) = Φ √ k (cid:16) b q π √ k − iM √ k (cid:17) Φ √ k (cid:16) b q π √ k − iM √ k − i √ k (cid:17) b P Φ √ k (cid:16) b q π √ k + iM √ k (cid:17) Φ √ k (cid:16) b q π √ k + iM √ k + i √ k (cid:17) = 11 + e − iπ ( M + k ) b Q b P (cid:16) e iπ ( M + k ) b Q (cid:17) . (A.24)The combination of these two identities leads (A.18).44 Complex delta functions In this appendix we briefly comment on the integral Z R dxf ( x ) h x | a + ib i , (B.1)where a and b are real numbers. If f ( x ) = o (cid:0) e −| x | (cid:1) and f ( x ) has neither poles nor branchpoints in 0 < Im ( x ) < b or b < Im ( x ) < b > b < h x | a + ib i is considered to be the usual delta function. Concretely, we can perform the integration byshifting the integration contour from R to R + ib so that we can use the property of the deltafunction: Z R dxf ( x ) h x | a + ib i = 12 π Z R dpdxf ( x ) e ip ( x − a − ib ) = 12 π Z R dpdxf ( x + ib ) e ip ( x − a ) = f ( a + ib ) . (B.2)At the first line we inserted the identity operator R R dp | p iihh p | . This shows that we can calculatethe integral (B.1) regarding h x | a + ib i as the usual delta function as long as f ( x ) satisfies theconditions above. C Proof for quantum curves In this appendix we prove (4.7), where the left-hand side is defined in (4.2). We use the math-ematical induction. The base case is q = q = 0 case and is easily proven. In the induction step,we prove that if the identity (4.7) holds for b H ( − q, +1 , − q ) M , the identity also holds for b H ( − q, +1 , − ( q +1)) M ′ and b H ( − ( q +1) , +1 , − q ) M ′′ , where M = (cid:0) M q , . . . , M , M , . . . , M q (cid:1) , M ′ = (cid:0) M q , . . . , M , M + M, . . . , M q + M, M (cid:1) , M ′′ = (cid:0) M , M q + M , . . . , M + M , M , . . . , M q (cid:1) . (C.1)First, we show that if the identity (4.7) holds for b H ( − q, +1 , − q ) M , the identity also holds for b H ( − q, +1 , − ( q +1)) M ′ . This assumption and (4.3) lead (cid:26)b ρ ( − q, +1 , − q ) M (cid:16)b ρ (+1 , − M (cid:17) † (cid:27) − (cid:16) b H (+1 , − M (cid:17) † b H ( − q, +1 , − q ) M = b P b Q M + k b Q − M R + k b Q M L + (cid:16) b Q − M + k b Q − M R + k b Q M L + b Q M − k b Q M R − k b Q − M L (cid:17) + b P − b Q − M − k b Q M R − k b Q − M L , (C.2)where we used (4.8). We apply the similarity transformation generated by b G − M defined in(A.19), namely b G M [ · ] b G − M to the both side. The left-hand side becomes i M + M −M q Y a Q M a j b q − πit Ma,j k b q − iπM a q Y a Q M a t b q − πit Ma,j k ! × Q Mj b q − πit M,j k 12 cosh b p q Y a Q M a j b q − πit Ma,j k b q + iπM a ! Q Mj b q − πit M,j k b q + iπM × q Y a Q M a j b q − πit Ma,j k 12 cosh b p − = 2 cosh b p (cid:16)b ρ ( − q, +1 , − ( q +1)) M ′ (cid:17) − , (C.3)where we substituted (3.10), and the right-hand side becomes b Q − M − k b P b Q − M R + k b Q M L + (cid:16) b Q − M + k b Q − M R + k b Q M L + b Q M − k b Q M R − k b Q − M L (cid:17) + b Q M + k b P − b Q M R − k b Q − M L = b P b Q − M ′ R + k b Q M ′ L + (cid:16) b Q − M ′ R + k b Q M ′ L + b Q M ′ R − k b Q − M ′ L (cid:17) + b P − b Q M ′ R − k b Q − M ′ L = (cid:16) b P + b P − (cid:17) (cid:16) b Q − M ′ R b P b Q M ′ L + b Q M ′ R b P − b Q − M ′ L (cid:17) , (C.4)where we used (A.18). Therefore, we obtain b H ( − q, +1 , − ( q +1)) M ′ = b Q − M ′ R b P b Q M ′ L + b Q M ′ R b P − b Q − M ′ L . (C.5)Next, we show that if the identity (4.7) holds for b H ( − q, +1 , − q ) M , the identity also holds for b H ( − ( q +1) , +1 , − q ) M ′′ . The general flow of computation is similar to the above, and thus we omit thedetail. We start with (cid:16)b ρ (+1 , − M b ρ ( − q, +1 , − q ) M (cid:17) − = b H ( − q, +1 , − q ) M b H (+1 , − M b Q − M R b Q M L − k b Q − M − k b P + (cid:16) b Q − M R b Q M L − k b Q M − k + b Q M − k b Q M R − k b Q − M L (cid:17) + b Q M R b Q − M L + k b Q M − k b P − . (C.6)We apply the similarity transformation generated by b G M , namely b G − M [ · ] b G M to the both side.The left-hand side becomes (cid:16)b ρ ( − ( q +1) , +1 , − q ) M ′′ (cid:17) − b p , (C.7)and the right-hand side becomes (cid:16) b Q − M ′′ R b P b Q M ′′ L + b Q M ′′ R b P − b Q − M ′′ L (cid:17) (cid:16) b P + b P − (cid:17) . 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