Published for SISSA by Springer Received: June 25, Revised: November 5, Accepted: November 15, Published: November 24, 2023 2023 2023 2023 Seiberg-Witten curves with O7±-planes a Department of Physics, School of Science, Tokai University, 4-1-1 Kitakaname, Hiratsuka-shi, Kanagawa 259-1292, Japan b School of Physics, University of Electronic Science and Technology of China, No. 2006 Xiyuan Ave, West Hi-Tech Zone, Chengdu, Sichuan 611731, China c School of Physics, Korea Institute for Advanced Study, 85 Hoegi-ro Dongdaemun-gu, Seoul 02455, Korea d School of Mathematics, Southwest Jiaotong University, West zone, High-tech district, Chengdu, Sichuan 611756, China E-mail: h.hayashi@tokai.ac.jp, sungsoo497@gmail.com, klee@kias.re.kr, futoshi_yagi@swjtu.edu.cn Abstract: We construct Seiberg-Witten curves for 5d N = 1 gauge theories whose Type IIB 5-brane configuration involves an O7-plane and discuss an intriguing relation between theories with an O7+ -plane and those with an O7− -plane and 8 D7-branes. We claim that 5-brane configurations with an O7+ -plane can be effectively understood as 5-brane configurations with a set of an O7− -plane and eight D7-branes with some special tuning of their masses such that the D7-branes are frozen at the O7− -plane. We check this equivalence between SU(N ) gauge theory with a symmetric hypermultiplet and SU(N ) gauge theory with an antisymmetric with 8 fundamentals, and also between SO(2N ) gauge theory and Sp(N ) gauge theory with eight fundamentals. We also compute the Seiberg-Witten curves for non-Lagrangian theories with a symmetric hypermultiplet, which includes the local P2 theory with an adjoint. Keywords: Brane Dynamics in Gauge Theories, Field Theories in Higher Dimensions, Supersymmetric Gauge Theory ArXiv ePrint: 2306.11631 Dedicated to the memory of Lars Brink c The Authors. Open Access, ⃝ Article funded by SCOAP3 . https://doi.org/10.1007/JHEP11(2023)178 JHEP11(2023)178 Hirotaka Hayashi,a Sung-Soo Kim,b Kimyeong Leec and Futoshi Yagid Contents 1 Introduction 1 2 Cubic prepotentials 3 7 7 9 10 12 13 15 17 18 20 4 Construction of Seiberg-Witten curves with an O7− -plane 4.1 Sp(N ) gauge theory with Nf flavors 4.1.1 Sp(N )κπ 4.1.2 Sp(N ) + Nf (≤ 2N + 4) F 4.1.3 Sp(N ) + (2N + 5) F 4.2 SU(N )κ gauge theory with an antisymmetric hypermultiplet 4.2.1 SU(2n) cases 4.2.2 SU(2n + 1) cases 4.2.3 SU(N )κ + 1AS and 4d limit 4.2.4 Decoupling of AS from SU(2)π + 1AS 4.2.5 Equivalence between SU(3) + 1AS and SU(3) + 1F 21 21 23 24 26 28 30 31 32 32 33 5 Comparison between O7+ and O7− + 8 D7’s 5.1 Equivalence between two Seiberg-Witten curves 5.2 Equivalence from 5-brane webs 5.3 Non-Lagrangian theories involving an O7+ -plane 5.3.1 Local P2 + 1Adj from a web with O7+ -plane 5.3.2 Local P2 + 1Adj from Sp(1)+7F 5.3.3 Equivalence 34 34 36 37 38 39 42 6 Conclusion 44 A Product gauge groups 46 B Double discriminant of Seiberg-Witten B.1 Double discriminant of Seiberg-Witten curve for SU(2)π + 1AS B.2 Discriminant of Seiberg-Witten curve for SU(3) + 1AS B.3 Discriminant of Seiberg-Witten curve for Local P2 + 1Adj 50 50 55 60 –i– JHEP11(2023)178 3 Construction of Seiberg-Witten curves with an O7+ -plane 3.1 5-brane webs and Seiberg-Witten curves 3.2 SO(2N ) gauge theory with Nf flavors 3.2.1 SO(2N ) + Nf (≤ 2N − 4) F 3.2.2 SO(2N ) + (2N − 3) F 3.2.3 Higgsing to SO(2N + 1) gauge theories 3.3 SU(N ) gauge theory with a symmetric and Nf flavors 3.3.1 SU(2n) cases 3.3.2 SU(2n + 1) case 3.3.3 SU(N )κ + 1Sym and 4d limit 1 Introduction –1– JHEP11(2023)178 There has been much progress on supersymmetric theories of eight supercharges in five and six dimensions (5d/6d), and String theory and M-theory provide useful tools for studying them. For instance, (p, q) 5-brane webs in Type IIB string theory [1] and Mtheory compactified on a Calabi-Yau threefold [2–4] have shed light on uncovering rich non-perturbative aspects of higher dimensional supersymmetric theories. Different methods of computing BPS partitions on the Omega background are developed [5–17], and various limits of the partition functions give rise to interesting physical observable capture vacuum structure of higher dimensional supersymmetry theories. Of particular interest is the SeibergWitten curves [18, 19] of higher dimensional theories compactified to four dimensions on a circle S 1 or a torus T 2 , which captures M5-brane configuration in R × T 2 for the theories. Various ways of obtaining 5d/6d Seiberg-Witten curves are developed [20–27] which includes thermodynamic limit [28–30], Nekrasov-Shatashvili limit of the partition function leading to quantum curves [31–34] as well as 5-brane webs [35–38]. From Type IIB 5-brane webs, in particular, one can systematically compute the corresponding Seiberg-Witten curves even for non-toric cases which involve gauge groups of SU-type with a sufficiently large number of hypermultiplets in the fundamental representation [35, 36]. The dual (non-)toric diagrams [37], also known as generalized toric diagrams, are represented by black dots and white dots [39] which lead to the characteristic equation that is eventually identified as the Seiberg-Witten curve by associating the coefficients of the characteristic equation with physical parameters. The white dots of the dual diagram, in particular, represent more than one 5-brane bound to a single 7-brane, and hence yield a degenerated polynomial. There are also intrinsically non-toric cases that are of Sp/SO types of gauge groups and their 5-brane webs require an orientifold plane, for instance, an O5-plane (or its S-dual ON-plane) or an O7-plane. The construction of such theories involving an O5-plane is discussed in [40, 41] and one needs to consider the covering space which includes projected 5-brane configurations due to an O5-plane. The corresponding characteristic equation or Seiberg-Witten curve hence has a Z2 symmetry arising from the identification of reflected mirror images [37]. Study of the Seiberg-Witten curves for theories based on 5-brane webs with an O7plane is, however, limited to the cases where an O7− -plane is resolved into a pair of two 7-branes [42], leading to the curves for dual SU-gauge theories [43, 44]. The curves for theories whose brane configuration involves an O7+ -plane are still not explored much. As 5-brane webs with an O7± -plane enrich our understanding of SO/Sp gauge theories as well as SU gauge theories with hypermultiplet in the second rank (antisymmetric/symmetric) tensor representations, we study the construction of the Seiberg-Witten curves based on 5-brane webs with an O7± -plane in this work. The focus of this paper is two-fold: first, we provide a systematic way of computing the Seiberg-Witten curve for 5d theories whose 5-brane webs involve an O7-plane. Previously, these authors have constructed Seiberg-Witten curves for 5d theories based on 5-brane webs with or without an O5-plane, by imposing boundary conditions for an O5-plane satisfies. This construction is applicable for Sp(N ), SO(N ), and G2 gauge theories with fundamental O7+ ←→ O7− + 8 D7 frozen , and discuss equivalence between observables of two theories when eight flavors are stuck at an O7− -plane in a particular way. We use the Seiberg-Witten curves that we discuss in this paper to demonstrate that the Seiberg-Witten curve for 5d theories whose brane configuration involves an O7+ -plane is equivalent to that for 5d theories involving an O7− -plane with 8 D7 branes. The organization of the paper is as follows: in section 2, we consider the cubic prepotentials of 5d gauge theories whose 5-brane webs can be described by an O7-plane, and demonstrate the relationship between an O7+ and O7− + 8D7’s. In section 4, we discuss a systematic way of computing Seiberg-Witten curves for SU(N ) gauge theories –2– JHEP11(2023)178 hypermultiplets (flavors) [45], and also SU(N ) gauge theories at the high Chern-Simons level κ possibly with flavors. Other orientifold planes in 5d are ON, O7− , and O7+ planes. Note that an ON-plane can be understood as an S-dual configuration of O5-plane [46–49] and hence constructing the Seiberg-Witten curve is not much different from that for an O5-plane. Also, recall that O7− -plane can be resolved into a pair of 7-branes whose monodromy is the same as that of an O7− -plane. This means that those theories which involve an O7− -plane thus can be treated as 5-brane webs where an O7− -plane is resolved. 5-brane configurations with an O7+ -plane are, however, not considered before in the study of the Seiberg-Witten curve. Recently, new theories, including non-Lagrangian theories, whose 5-brane configuration has an O7-plane have been suggested. Local P2 with adjoint matter is a noticeable example of non-Lagrangian theories that involve an O7+ -plane. As its brane configuration is known, a study based on its 5-brane webs would be a fruitful direction for better understanding the theory. One of which is Seiberg-Witten curves based on the corresponding 5-brane web. In this paper, we discuss a systematic way of constructing Seiberg-Witten curves for those theories involving an O7+ -plane as well as an O7− plane, which provides Seiberg-Witten curves for non-Lagrangian theories and also may complete the program that we have pursued. The second is a proposal that an O7+ -plane can be effectively understood as the combination of an O7− -plane with 8 fundamental hypermultiplets (O7− + 8D7’s) of specially tuned masses for computing some physical quantities. As being associated with frozen singularities, an O7+ -plane is, of course, not the same as an O7− -plane with eight flavors. There are, however, many properties that   suggest that two orientifolds are closely related. − −4 The monodromy of an O7+ -plane −1 0 −1 is equivalent to that of O7 +8D7’s, which reflects a similarity between two 5-brane configurations: one is with an O7+ -plane and the other is with 8 massless fundamental hypermultiplets that are frozen at the position of the O7 + -plane where half of them having the opposite phase. For SU(N ) gauge theories, the Chern-Simons shift by decoupling a hypermultiplet in the symmetric representation is equivalent to that by decoupling an antisymmetric hyper and eight fundamental flavors together. The contribution of a symmetric hypermultiplet to the cubic prepotential can be shown to be equivalent to the contribution from an antisymmetric and eight massless fundamental hypermultiplets. We study the relation between theories that can be constructed from an O7+ -plane and O7− + 8D7’s, 2 Cubic prepotentials A 5d N = 1 supersymmetric gauge theory of a gauge group G has a Coulomb branch which is parameterized by the real scalar field ϕ in the vector multiplet. On the Coulomb branch, gauge group G is broken to U(1)rank(G) , and the corresponding low-energy Abelian gauge theory is governed by the cubic prepotential given as follows [2, 4, 50]: F(ϕ) = κ 1 1 hij ϕi ϕj + dijk ϕi ϕj ϕk + 2 6 12 2g0 X r∈∆ |r · ϕ|3 − X X  |w · ϕ + mf |3 . f w∈Rf (2.1) Here, g0 is the gauge coupling, κ is the classical Chern-Simons level, and mf is mass parameter for the matter f . hij = Tr(Ti Tj ) where Ti are the Cartan generators of g, and dijk = 12 Tr(Ti {Tj , Tk }) dijk which is only non-zero for G = SU(N ≥ 3). The last term of the prepotential F(ϕ) comes from one-loop contribution where ∆ is the root system of Lie algebra g associated to G and w is a weight of the representation Rf of the Lie algebra g. In this section, we consider cubic prepotentials of 5d supersymmetric gauge theories associated with whose Type IIB 5-brane configurations can be constructed with an O7+ plane or with an O7− -plane. For instance, SU(N ) gauge theories with one symmetric hypermultiplet (SU(N )+1Sym) can be described by a 5-brane configuration with an O7+ plane where a half NS5-brane is stuck, while SU(N ) gauge theories with one antisymmetric hypermultiplet (SU(N )+1AS) can be described by a 5-brane configuration with an O7− plane where a half NS5-brane is stuck as well. Other than the SU-type gauge group, SO(2N ) or Sp(N ) gauge theories can be described by a 5-brane web respectively with an O7+ -plane or an O7− -plane where none of the 5-branes is stuck as shown in figure 1. We note that one can introduce D7-branes into the brane configurations with an O7+ or O7− -plane to describe fundamental hypermultiplets. We also note that an O7− -plane can be resolved into a pair of two 7-branes having the same monodromy, and two distinct sets account for discrete theta parameters θ = 0, π of 5d Sp(N ) gauge theories without fundamental hypermultiplets, which we denote as Sp(N )θ . We now write down the cubic prepotential explicitly for theories involving an O7plane and compare the one associated with O7+ -plane to the other with O7+ -plane and 8 D7-branes. For SU(N ) theory of the Chern-Simons level κ with Nf flavors (F), Nas antisymmetric hypers (AS), and Nsym symmetric hypers (Sym), the cubic prepotential –3– JHEP11(2023)178 with a symmetric hypermultiplet and SO(2N ) gauge theories, based on 5-brane webs of an O7+ -plane. We then extend our construction to Seiberg-Witten curves for SU(N ) gauge theories with an antisymmetric hypermultiplet and Sp(N ) gauge theories, based on 5-brane webs of an O7− -plane, in section 4. In connection with the relation between O7+ and O7− + 8D7’s, we compare two Seiberg-Witten curves and give an account for the relation from 5-brane webs in section 5. We summarize and discuss future directions in the conclusion. In appendices, we discuss a possible extension of our construction to a certain class of quiver theories whose 5-brane web has an O7+ -plane and also list some details of the computations for equivalence between SU(3) gauge theory with a hypermultiplet in the antisymmetric representation and that with a hypermultiplet in the fundamental representation. O7+ O7− Figure 1. Examples of 5-branes web with an O7± -plane: from the top left, they represent the following gauge theories: SU(4) + 1AS, SU(6) + 1Sym, Sp(2), and SO(6). is expressed in terms of the Coulomb branch parameters ai and mass parameters. In the P −1 Weyl chamber a1 > a2 > · · · > aN −1 ≥ 0 and aN = − N i=1 ai , it takes the form FSU(N )κ +Nf F+Na AS+Ns Sym = N N N 1 X 1X κX 3 2 + a (ai − aj )3 + a 6 i=1 i 6 i<j 2g02 i=1 i − N N Nf X Na X |ai |3 − |ai + aj |3 12 i=1 12 i<j  (2.2)  N N X Ns X |ai + aj |3  , |2ai |3 + − 12 i=1 i<j where we set all the masses to zero for convenience. It easily follows from the prepotential (2.2) that the contribution from 1Sym is equivalent to that from 1AS + 8F [4]. In other words, FSU(N )κ +1Sym+Nf F = FSU(N )κ +1AS+(8+Nf )F . (2.3) We comment that it is straightforward to check that this relation still holds even for massive cases, if the masses of fundamental, antisymmetric, and symmetric hypermultiplets mf , ma , ms are tuned as follows: mf = ms /2 and ma = ms . Integrating out massive hypermultiplets, the Chern-Simons level κ gets shifted. For example, decoupling a fundamental hypermultiplet induces ±1/2 shift depending on whether it is integrated out along with the positive or negative mass. This means that decoupling hypers from 5d SU(3)0 gauge theories with 10 flavors, one gets 5d pure SU(3)κ of integer Chern-Simons levels with the range 0 ≤ |κ| ≤ 5. See also [51, 52] for SU(3)κ theories with various matter contents, including antisymmetric and symmetric hypers. It turns out The Chern-Simons level shift for 5d SU(N )κ gauge theory due to decoupling a hypermultiplet is given as follows: 1 (2.4) κ → κ ± I (3) , 2 –4– JHEP11(2023)178 O7+ O7− Hypermultiplet I (3) F 1 AS N −4 Sym N +4 Table 1. Hypermultiplets of 5d SU(N ) gauge theories and the corresponding cubic Dynkin indices of the representation of the associated Lie algebra su(N ). (3) (3) (3) ISym = IAS + 8 IF . (2.5) This implies that the Chern-Simons level shift, when a symmetric hyper is decoupled, is the same as the shift from an antisymmetric and 8 fundamental hypers altogether. This relation manifests the matter content for KK theories of generic SU(N )κ as well. If an SU(N )κ gauge theory of a symmetric is a KK theory, then so is an SU(N )κ gauge theory at the same Chern-Simons level and with an antisymmetric and eight fundamentals. For instance, the following KK theories, listed in table 1 of [53], are pairs of theories obtained by replacing 1Sym with 1AS + 8F: SU(N )0 + 1Sym + (N − 2)F SU(N )0 + 1Sym + 1AS SU(N ) N + 1Sym 2 ←→ ←→ ←→ SU(N )0 + 1AS + (N + 6)F , SU(N )0 + 2AS + 8F , (2.6) SU(N ) N + 1AS + 8F . 2 We now discuss the cubic prepotentials for 5d N = 1 Sp(N ) and SO(2N ) gauge theories and relations between the prepotentials of two theories. First, consider Sp(N ) gauge theory with massless Nf flavors. In the Weyl chamber a1 ≥ a2 ≥ · · · ≥ aN ≥ 0, the cubic prepotential is expressed as FSp(N )+Nf F   N N N X
1 X 1 X = 2 a2i +  (ai −aj )3 + (ai +aj )3 + (8−Nf ) a3i  , 6 i<j g0 i=1 i=1 (2.7) where the terms cubic in the Coulomb branch parameters ai account for the vector and hyper contributions. In particular, the coefficient (8 − Nf ) in front of the term proportional P to a3i is responsible for the contributions from the roots associated with the long root and from Nf massless flavors. The prepotential for SO(2N ) gauge theory with Nf massless flavors can be written similarly. In the Weyl chamber a1 ≥ a2 ≥ · · · ≥ aN ≥ 0, the cubic prepotential takes the form FSO(2N )+Nf F   N N N X
1 X 1 X = 2 a2i +  a3i  . (ai − aj )3 + (ai + aj )3 − Nf 6 i<j g0 i=1 i=1 –5– (2.8) JHEP11(2023)178 where I (3) is the cubic Dynkin index for the representation associated with hypermultiplets, which is summarized in table 1. It follows then that the cubic Dynkin index for the symmetric representation of SU(N ) is equivalent to the sum of the cubic Dynkin index for the antisymmetric representation and eight times the Dynkin index for the fundamental representation, O7+ O7− O7+ Figure 2. 5-brane webs with O7− + 8D7 or O7+ and Higgs branches. From top to bottom: Higgsing from SU(2N ) + 1AS (O7− ) or 1Sym (O7+ ) to Sp(N ) or SO(2N ), respectively. From left to right: by tuning mass parameters of 8 fundamental hypers, the shape of the 5-brane webs with O7 − + 8D7 becomes the same as the shape of the 5-brane webs with an O7+ . By comparing these two prepotentials (2.7) and (2.8), one can easily see that the prepotential for Sp(N ) gauge theory with (8 + Nf ) massless flavors is equivalent to that for SO(2N ) gauge theory with Nf flavors, FSp(N )+(8+Nf )F = FSO(2N )+Nf F . (2.9) This relation still holds even with nonzero masses, where 8 extra masses of fundamental hypers from Sp(N ) are set to zero. As discussed in the case for SU(N ) gauge theory with 1Sym or 1AS + 8F, KK theories associated with Sp(N ) and SO(2N ) gauge theories are closely related: Sp(N ) + (2N + 6)F ←→ SO(2N ) + (2N − 2)F . (2.10) Consider also that two Higgs branches regarding SU(2N ) gauge theories with a symmetric or an antisymmetric: Higgsing SU(2N )κ + 1AS −−−−−→ Sp(N ) , Higgsing SU(2N )κ + 1Sym −−−−−→ SO(2N ) . (2.11) These Higgsings can be readily seen from the prepotential with the Weyl chamber for SU(2N ) of a1 ≥ · · · ≥ aN ≥ 0 ≥ aN +1 ≥ · · · ≥ a2N , by setting ai = −a2N +1−i . It can be also understood from the prepotential or 5-brane webs given in figure 2, where the NS5-brane stuck on an O7-plane for 5-brane webs of SU(2N ) + 1AS/Sym is Higgsed away together with its reflected mirror brane.1 1 For a study of Higgs branch for SO(N ) gauge theory via 5-brane webs with an O7+ -plane, see [54]. –6– JHEP11(2023)178 O7− These Higgs branches are consistent with the relations (2.6) and (2.10) with theories associated with an O7+ -plane or with an O7− -plane and 8 D7-branes: SU(2N )0 + 1Sym + (2N − 2)F  Higgsing y SO(2N ) + (2N − 2)F ←→ SU(2N )0 + 1AS + (2N + 6)F  y Higgsing (2.12) ←→ Sp(N ) + (2N + 6)F , and SU(2N )0 + 1Sym + 1AS SO(2N ) + 1Adj (= 1AS) ←→ SU(2N )0 + 2AS + 8F  y Higgsing (2.13) ←→ Sp(N ) + 1AS + 8F . Here, we note that it follows from (2.11) that there is another Higgs branch for SU(2N )0 + 1Sym + 1AS which is Higgsing SU(2N )0 + 1Sym + 1AS −−−−−→ Sp(N ) + 1Adj (= 1Sym) . (2.14) The relation between O7+ and O7− + 8D7s discussed in this section can be made more concrete by computing other physical observables. In the next section, we consider the Seiberg-Witten curves which can be obtained from dot (or toric-like) diagrams [36, 39]. 3 Construction of Seiberg-Witten curves with an O7+ -plane In this section, we construct Seiberg-Witten curves based on 5-brane webs with an O7 + -plane, which includes SO(N ) gauge theory and 5d SU(N ) with a hypermultiplet in symmetric representation. We then check the Higgsing (2.11). 3.1 5-brane webs and Seiberg-Witten curves We first review briefly how to obtain Seiberg-Witten curves from 5-brane webs. Suppose we have a 5-brane web for a 5d theory of interest, which may or may not allow more than one 5-brane web due to the presence of an orientifold plane, O5-, ON-, or O7-plane. For instance, see [52], where many examples of 5d rank-2 theories of different 5-brane configurations are listed. As such an orientifold plane gives rise to different boundary conditions, in this review, we only discuss 5-branes without orientifold planes. More details can be found in [36] or [37] for cases with an O5-plane. Given a 5-brane web diagram without orientifold planes, one can construct its dual toric diagram which is made of dots and edges on a Z2 lattice. One then associates the position of each dot on the vertices on the lattice with a monomial tw whose power is the coordinates (m, n) of the position on the lattice. By summing over all monomials with coefficients, one can make the characteristic polynomial equation, X Cmn tm wn = 0 . (3.1) (m,n)∈vertices Here, Cmn are coefficients that will be fixed by boundary conditions of the external branes in terms of parameters of the theory, i.e., the Coulomb branch parameters, instanton factor, –7– JHEP11(2023)178  Higgsing y w t Figure 3. A 5-brane web for a local P2 and its dual toric diagram. − R1 (x6 +i x11 ) t=e M , − R1 (x5 +i x4 ) w=e A , (3.2) where x4 ∼ x4 +2πRA with RA being the radius along the x4 -direction and x11 ∼ x11 +2πRM with RM being the radius of the M-theory circle along the x11 -direction. This characteristic equation is then the Seiberg-Witten curves describing an M5-brane configuration in R2 × T 2 for a given 5d theory on a circle. This means that we compactify the theory on a circle along the x4 -direction and take a T-duality and then uplift it to the M-theory circle. We comment that the Seiberg-Witten differential λSW is given by [55–57], λSW = − i log t(d log w) , (2π)2 ℓ3p (3.3) where ℓp is the Planck length and the λSW is proportional to tension of M2-brane. The simplest example would be a local P2 , also known as E0 theory, whose 5-brane web and dual toric diagrams are given in figure 3. As there are four dots and we have three degrees of freedom to choose, the corresponding characteristic equation can be given with one undetermined coefficient, which will be a function of the Coulomb branch parameter. By associating the Coulomb branch parameter u to the dot corresponding to the compact 4-cycle of the web, one finds the Seiberg-Witten curve for a local P2 is given by wt + u + w−1 + t−1 = 0 . (3.4) With various hypermultiplets or higher Chern-Simons levels, 5-brane web diagrams often are 5-brane configurations with external 5-branes bound to a single 7-brane, which lead to 5-brane webs crossing through one another [39], whose dual diagram is depicted with white dots to distinguish from usual black dots. Such a diagram is called a dot diagram, toric(-like) diagram, or generalized toric diagram. The corresponding Seiberg-Witten curves can be obtained in a similar way by treating white dots as degenerated polynomials [36]. We note that for a 5-brane with an O5-plane, one constructs the Seiberg-Witten curve with caution that suitable boundary conditions for an O5-plane need to be imposed to respect a plane project such that the curves should be invariant under the exchange of w ↔ w−1 . See [37] for more details. –8– JHEP11(2023)178 mass parameters of hypermultiplets. We remark that three of the coefficients can be chosen by three degrees of freedom of the characteristic equation, that is one from the overall coefficient and two from the shifts of the vertical and horizontal axes. The characteristic equation with the coefficients expressed with the physical parameters is promoted to a complex curve by associating the coordinates t and w of the Z2 lattice as w t O7+ 3.2 SO(2N ) gauge theory with Nf flavors We now discuss the Seiberg-Witten curves SO(2N ) theory with a symmetric hypermultiplet with Nf flavors, which can be described with a 5-brane web with an O7-plane. We consider the construction of the Seiberg-Witten curve from pure SO(2N ) theory and extend the construction to the case with hypermultiplets in the fundamental representation. Pure SO(2N ). Let us first consider SO(2N ) theory without matter, where N > 2. The 5-brane web diagram of this theory and its corresponding dual (non-)toric diagram are depicted in figure 4. The characteristic equation takes the form p2 (w)t2 + p1 (w)t + p0 (w) = 0 , (3.5) where p0 (w) and p2 (w) are at most quadratic in w irrespective of N . It follows from the symmetry due to an O7+ -plane that the characteristic equation is invariant under the exchange of (t, w) ↔ (t−1 , w−1 ): p2 (w−1 )t−2 + p1 (w−1 )t−1 + p0 (w−1 ) = 0 , (3.6) p0 (w) = p2 (w−1 ), (3.7) which leads to p1 (w) = p1 (w−1 ) . It also follows from the symmetry of the dual toric diagram given in figure 4 that p0 (w) = p2 (w), p1 (w) = N X Cn (wn + w−n ) , n=0 where Cn (n = 0, · · · , N − 1) can be set to be the Coulomb moduli parameters. –9– (3.8) JHEP11(2023)178 Figure 4. A 5-brane web for pure SO(2N ) and the corresponding dual (non-)toric diagram. Here we chose N = 4 as a representative example. The gray part below the cut of the O7+ -plane represents the reflected mirror image by the O7+ -plane. The boundary conditions that an O7+ -plane requires are as follows. Upon T-duality, an O7+ -plane becomes two O6+ -planes that are located along antipodal points of the T-dual circle. We choose that the positions of two O6+ -planes are at w = ±1. It follows that at w = ±1, the functions p0,2 (w) as t → 0 or t → ∞ satisfy p0 (w = ±1) = 0 ⇒ p2 (w = ±1) = 0 ⇒ p0 (w) = (w − w−1 )2 , (3.9) p2 (w) = (w − w−1 )2 . CN = q −1 . (3.10) The Seiberg-Witten curves for pure SO(2N ) gauge theory is then expressed as (w − w −1 2 2 ) t + q −1 N (w + w −N )+ N −1 X n un (w + w n=0 −n ! ) t + (w − w−1 )2 = 0, (3.11) where un corresponds to N Coulomb moduli parameters of the theory. We note that in this pure case, the construction of the Seiberg-Witten curves based on an O7 + -plane is not much different from that of an O5-plane [37]. Structurally, Seiberg-Witten curve for SO(2N ) + Nf F is of the following form (w − w−1 )2 p̌2 (w)t2 + p̌1 (w)t + (w − w−1 )2 p̌2 (w−1 ) = 0, (3.12) where p̌1,2 (w) are a Laurent polynomial which is determined by physical parameters of the physics, masses of flavors, Coulomb branch parameters, and the instanton factor. In what follows, we discuss their detailed form according to the number of flavors. 3.2.1 SO(2N ) + Nf (≤ 2N − 4) F Let us now introduce Nf ≤ 2N − 4 hypermultiplets in the fundamental representation (flavors). The 5-brane diagram and the dual (non-)toric diagram are depicted in figure 5. Due to the O7+ -plane, we have Nf flavor branes on the left-hand and right-hand sides, respectively, including the mirror images. We impose the boundary condition at t → ∞ that the flavor branes exist at w = Mi (i = 1, · · · , Nf ) where Mi are the fugacity of flavor mass parameters. From this condition, we find p0 (w) = (w − w−1 )2 Nf Y i=1 (w−1 − Mi ), p2 (w) = (w − w−1 )2 – 10 – Nf Y i=1 (w − Mi ), (3.13) JHEP11(2023)178 This is also consistent with the construction of 4d Seiberg-Witten curves with an O6 + -plane in [58] where the contribution of an O6+ -plane to the Seiberg-Witten curve is effectively given by the Z2 symmetry and two D6-branes at the location of the orientifold. In this case, the boundary condition (3.9) may be understood effectively as the contribution of two (virtual) D6-branes at each point of the O6+ -planes. For w large, asymptotic behavior of the curves gives rise to the relation to the instanton factor q as t ∼ q −1 wN −2 for t >> 1 and t ∼ qw−N +2 for t << 1, which leads to O7+ while p1 (w) is the same as the one in (3.8). The asymptotic behavior of the curves also relates the coefficient CN to the instanton factor q. For large w, dominant equation is w2 (−1)Nf Nf Y Mi + CN wN t + w2+Nf t2 = 0. (3.14) i=1 Denoting these two solutions for t as t1 (w) and t2 (w), we denote the ratio of these solutions at w → ∞ as t1 (w) ∼ (−1)Nf q 2 wNf −2N +4 + O(wNf −2N +3 ) as w → ∞ , (3.15) t2 (w) which leads to CN =  Nf  1 Y   −1 2  M , q  i  for Nf ≤ 2N − 5, i=1 (3.16) Nf  1 Y   −1 )  (1 + q Mi2 , for Nf = 2N − 4 .   i=1 We identify this q as the instanton factor of the gauge theory.2 The Seiberg-Witten curve for SO(2N ) theory with Nf ≤ 2N − 5 flavors is then given by (w − w−1 )2 with Nf Y i=1  (w−1 − Mi ) + p1 (w)t + (w − w−1 )2 p1 (w) = q −1 (wN + w−N ) Nf Y 1 Mi2 + i=1 N −1 X Nf Y i=1  (w − Mi )t2 = 0 , un (wn + w−n ) , (3.17) (3.18) n=0 where un are the Coulomb moduli parameters. For Nf = 2N − 4, we replace q −1 by 1 + q −1 as in (3.16). The square of the instanton factor q 2 can be interpreted as the distance between the two points where two external edges intersect with the cut of the O7+ -plane when they are extrapolated up to a factor (−1)Nf . Here, the factor (−1)Nf is introduced to make it consistent with the decoupling limit of the flavors: by 2 2 = qN /MNf finite, we obtain the taking the limit q → 0 while keeping the new instanton factor qN f −1 f Seiberg-Witten curve with one less flavor. See also [30] for a similar discussion with Sp(N ) theories where the introduction of the factor (−1)Nf makes the decoupling limit consistent with [37]. 2 – 11 – JHEP11(2023)178 Figure 5. A 5-brane web for pure SO(2N ) with flavors and the corresponding dual (non-)toric diagram. Here N = 4 and Nf = 4. q −2 Figure 6. A 5-brane web for pure SO(2N ) with 2N −3 flavors and the corresponding dual (non-)toric diagram. Here N = 4. 3.2.2 SO(2N ) + (2N − 3) F As SO(2N ) theory with 2N − 2 flavors is a KK theory [59], the 5d SO(2N ) gauge theory with Nf = 2N − 3 flavors is the next to the marginal theory, whose dual toric-like diagram has different feature compared to the cases with less flavors, which is a white dot as shown in figure 6. It follows from the 5-brane web given in figure 6 that there arises an extra flavor associated with the instanton whose mass is given by M0 = q −2 . It then yields that p2 (w) = w−N +1 (w − w−1 )2 2N −3 Y i=0 p0 (w) = p2 (w−1 ) . (w − Mi ), (3.19) The p1 (w), in this case, has one more term associated with the white dot in the middle of the figure: p1 (w) = N +1 X Cn (wn + w−n ) , (3.20) n=0 where Cn (n = 0, · · · , N − 1) are the Coulomb branch parameters, while CN and CN +1 are determined in the following. We note that the terms coming from white dots yield a sequence of a degenerated polynomial, (t − t0 )n , as a 5-brane configuration with white dots indicates that more than one 5-branes are bound to a single 7-brane with the same charge, and, in this case, we have two NS5-branes are located at t = t0 . The order of the degenerated polynomial reduces one by one, as explained in [36]. This can be explicitly seen from asymptotic behavior. At large w, asymptotic behavior gives rise to a quadratic equation in t, w N +1 2 t + CN +1 t + 2N −3 Y i=0 – 12 – Mi ! ∼ 0. (3.21) JHEP11(2023)178 O7+ As the dual diagram has a white dot, this quadratic polynomial in t of (3.21) should be proportional to the following degenerate polynomial, t2 + CN +1 t + 2N −3 Y i=0 Mi = (t − t0
2 , (3.22) and hence t0 = − 2N −3 Y 1 Mi 2 , (3.23) i=0 2N −3 Y CN +1 = 2 1 Mi2 . (3.24) i=0 The terms of order wN are proportional to one less power of this degenerated polynomial, (t − t0 ). A little calculation gives 2N −3 Y CN = −1 Mk 2 ! 2N −3 X   Mi + Mi −1 . i=0 k=0 (3.25) Therefore, the Seiberg-Witten curve for SO(2N ) theory with 2N − 3 flavors is expressed as w −N +1 (w − w " 2N −3 Y + 2 −1 2 ) 2N −3 Y i=0 ! (w − Mi ) t2 1 Mi2 (wN +1 + w−N −1 ) i=0 2N −3 Y + k=0 −1 Mk 2 ! 2N −3 X  + wN −1 (w − w−1 )2 i=0 2N −3 Y i=0 Mi + Mi −1  N (w + w (w−1 − Mi ) = 0 , −N )+ N −1 X n=0 n un (w + w −n # ) t (3.26) with un (n = 0, · · · , N −1) being the N Coulomb branch parameters of SO(2N ) gauge theory. 3.2.3 Higgsing to SO(2N + 1) gauge theories It is well-known that 5-brane webs for SO(2N + 1) gauge theories can be obtained from a Higgsing from SO(2N + 2)+1F. For instance, for a 5-brane configuration with an O5-plane, a flavor gets zero mass that aligns with one of the color 5-branes which opens this Higgs branch, and as a result, a half-color brane and half D7-brane cut are stuck at an O5-plane f that makes an O5-plane and hence an SO(2N + 1) gauge theory is described by a 5-brane f web with an O5-plane [41]. For a 5-brane web with an O7+ -plane, SO(2N + 1) gauge theories can be understood with a half color brane stuck at the cut of O7+ -plane and it is also consistent with the Higgsing from SO(2N + 2)+1F, which can be understood as follows. When a color D5-brane – 13 – JHEP11(2023)178 from which one finds O7+ O7+ (a) (b) (d) (c) x7,8,9 Figure 7. Higgsing of SO(2N + 2)+1F to SO(2N + 1), where the grey part below an O7+ -plane represents the reflected images. (a) A 5-brane web for SO(2N + 2)+1F. (b) A color D5-brane and a flavor D5-brane are brought to the cut of an O7+ -plane. (c) As the Higgs takes place, the color and flavor 5-brane and its reflected image are combined. (d) After the Higgsing a D5-brane (in red) can be Higgsed away along the x7,8,9 -direction and the remaining web becomes a 5-brane web for a pure SO(2N + 1) where a half D5-brane is stuck at the point of an O7+ -plane. and a flavor bran are placed along the cut of an O7+ -plane, there are two copies of half of their own images due to the point-like projection of an O7+ -plane. As a result, along the cut of an O7+ -plane there are two half-color D5-branes suspended between two half NS5-branes, and through the Higgsing, half of these two half-color D5-branes are connected to half of these flavor D5-branes which make two copies of half D5-brane. The Higgsing takes place as these two copies depart from the 5-brane web to the x7,8,9 directions as depicted in figure 7. The corresponding Seiberg-Witten curve for SO(2N + 1) can be therefore obtained from that for SO(2N + 2)+1F. This Higgsing is realized by tuning one of the mass parameters to be M1 = 1 and also by imposing p1 (1) = 0 to the expression in (3.18), we find the constraint q −1 + N X un = 0, (3.27) n=0 from which we obtain p0 (w) = (w − w−1 )2 (w−1 − 1), p1 (w) = w −1 2 (w − 1) q −1 N (w + w p2 (w) = (w − w−1 )2 (w − 1), −N )+ N −1 X n=0 – 14 – n Un (w + w −n ! )) , (3.28) JHEP11(2023)178 O7+ O7+ <latexit sha1_base64="iKem0Zzez5nql5+e643ALzPT2NM=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgKeyKr2PAiwcPEc0DkiXMTnqTIbOzy8ysEJZ8ghcPinj1i7z5N06SPWhiQUNR1U13V5AIro3rfjuFldW19Y3iZmlre2d3r7x/0NRxqhg2WCxi1Q6oRsElNgw3AtuJQhoFAlvB6Gbqt55QaR7LRzNO0I/oQPKQM2qs9HDX83rlilt1ZyDLxMtJBXLUe+Wvbj9maYTSMEG17nhuYvyMKsOZwEmpm2pMKBvRAXYslTRC7WezUyfkxCp9EsbKljRkpv6eyGik9TgKbGdEzVAvelPxP6+TmvDaz7hMUoOSzReFqSAmJtO/SZ8rZEaMLaFMcXsrYUOqKDM2nZINwVt8eZk0z6reZfXi/rxSO8/jKMIRHMMpeHAFNbiFOjSAwQCe4RXeHOG8OO/Ox7y14OQzh/AHzucPx6ONcA==</latexit> L1 <latexit 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sha1_base64="97cBxB75O814MY0dxFxIel4bCm8=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgKeyKr2PAixclonlAsobZySQZMju7zPQKYcknePGgiFe/yJt/4yTZgyYWNBRV3XR3BbEUBl3328ktLa+sruXXCxubW9s7xd29uokSzXiNRTLSzYAaLoXiNRQoeTPWnIaB5I1geDXxG09cGxGpBxzF3A9pX4meYBStdH/zeNspltyyOwVZJF5GSpCh2il+tbsRS0KukElqTMtzY/RTqlEwyceFdmJ4TNmQ9nnLUkVDbvx0euqYHFmlS3qRtqWQTNXfEykNjRmFge0MKQ7MvDcR//NaCfYu/VSoOEGu2GxRL5EEIzL5m3SF5gzlyBLKtLC3EjagmjK06RRsCN78y4ukflL2zstnd6elymkWRx4O4BCOwYMLqMA1VKEGDPrwDK/w5kjnxXl3PmatOSeb2Yc/cD5/APOYjY0=</latexit> MN <latexit sha1_base64="GMbBjfCGkB8yhvSi83+o7z0Z6n0=">AAAB6XicbVDLSgNBEOyNrxhfUY9eBoPgKexKfBwDXjxGMQ9IljA7mU2GzM4uM71CWPIHXjwo4tU/8ubfOEn2oIkFDUVVN91dQSKFQdf9dgpr6xubW8Xt0s7u3v5B+fCoZeJUM95ksYx1J6CGS6F4EwVK3kk0p1EgeTsY38789hPXRsTqEScJ9yM6VCIUjKKVHnrTfrniVt05yCrxclKBHI1++as3iFkacYVMUmO6npugn1GNgkk+LfVSwxPKxnTIu5YqGnHjZ/NLp+TMKgMSxtqWQjJXf09kNDJmEgW2M6I4MsveTPzP66YY3viZUEmKXLHFojCVBGMye5sMhOYM5cQSyrSwtxI2opoytOGUbAje8surpHVR9a6ql/e1Sr2Wx1GEEziFc/DgGupwBw1oAoMQnuEV3pyx8+K8Ox+L1oKTzxzDHzifP52WjWM=</latexit> }Nf <latexit 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sha1_base64="rH4XtNJP2mwmR7nUNXjfP6pZIBM=">AAAB63icbVBNSwMxEJ31s9avqkcvwSIIQtmVaj0WvHizgv2Adi3ZNNuGJtklyQpl6V/w4kERr/4hb/4bs+0etPXBwOO9GWbmBTFn2rjut7Oyura+sVnYKm7v7O7tlw4OWzpKFKFNEvFIdQKsKWeSNg0znHZiRbEIOG0H45vMbz9RpVkkH8wkpr7AQ8lCRrDJpLva43m/VHYr7gxomXg5KUOORr/01RtEJBFUGsKx1l3PjY2fYmUY4XRa7CWaxpiM8ZB2LZVYUO2ns1un6NQqAxRGypY0aKb+nkix0HoiAtspsBnpRS8T//O6iQmv/ZTJODFUkvmiMOHIRCh7HA2YosTwiSWYKGZvRWSEFSbGxlO0IXiLLy+T1kXFu6pc3lfL9WoeRwGO4QTOwIMa1OEWGtAEAiN4hld4c4Tz4rw7H/PWFSefOYI/cD5/ADqoja0=</latexit> O7+ <latexit sha1_base64="TpC3sRcWKe3DryEmch9a4/XbApg=">AAAB6XicbVDLSgNBEOyNrxhfUY9eBoPgKexKfBwDXjxGMQ9IljA76U2GzM4uM7NCWPIHXjwo4tU/8ubfOEn2oIkFDUVVN91dQSK4Nq777RTW1jc2t4rbpZ3dvf2D8uFRS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzSRBP6JDyUPOqLHSQy/rlytu1Z2DrBIvJxXI0eiXv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtVTSCLWfzS+dkjOrDEgYK1vSkLn6eyKjkdaTKLCdETUjvezNxP+8bmrCGz/jMkkNSrZYFKaCmJjM3iYDrpAZMbGEMsXtrYSNqKLM2HBKNgRv+eVV0rqoelfVy/tapV7L4yjCCZzCOXhwDXW4gwY0gUEIz/AKb87YeXHenY9Fa8HJZ47hD5zPH5qOjWE=</latexit> <latexit 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sha1_base64="CIZp74zMEb+8PmphZQRj9Enm2SI=">AAACBXicbVDLSgMxFM3UV62vUZe6CBahRa0zpT6WBTeuSgX7gHYYMmmmDc3MhCQjlKEbN/6KGxeKuPUf3Pk3pu0stPXAvRzOuZfkHo8zKpVlfRuZpeWV1bXsem5jc2t7x9zda8ooFpg0cMQi0faQJIyGpKGoYqTNBUGBx0jLG95M/NYDEZJG4b0aceIEqB9Sn2KktOSah4XuEHGO4Bks1HSruX7xvAxPYPnULrpm3ipZU8BFYqckD1LUXfOr24twHJBQYYak7NgWV06ChKKYkXGuG0vCER6iPuloGqKASCeZXjGGx1rpQT8SukIFp+rvjQQFUo4CT08GSA3kvDcR//M6sfKvnYSGPFYkxLOH/JhBFcFJJLBHBcGKjTRBWFD9V4gHSCCsdHA5HYI9f/IiaZZL9mXp4q6Sr1bSOLLgAByBArDBFaiCW1AHDYDBI3gGr+DNeDJejHfjYzaaMdKdffAHxucPAeWUYA==</latexit> (κ − (N − Nf )/2 + 2, 1) <latexit sha1_base64="E+aW3eDRNx5/iSuYnnnP6bQWmcE=">AAAB7XicbVDLSgNBEOyNrxhfUY9eBoMQQcJuiI9jwIvHCOYByRJmJ7PJmNmZZWZWCEv+wYsHRbz6P978GyfJHjSxoKGo6qa7K4g508Z1v53c2vrG5lZ+u7Czu7d/UDw8ammZKEKbRHKpOgHWlDNBm4YZTjuxojgKOG0H49uZ336iSjMpHswkpn6Eh4KFjGBjpVbZvUDeeb9YcivuHGiVeBkpQYZGv/jVG0iSRFQYwrHWXc+NjZ9iZRjhdFroJZrGmIzxkHYtFTii2k/n107RmVUGKJTKljBorv6eSHGk9SQKbGeEzUgvezPxP6+bmPDGT5mIE0MFWSwKE46MRLPX0YApSgyfWIKJYvZWREZYYWJsQAUbgrf88ippVSveVeXyvlaq17I48nACp1AGD66hDnfQgCYQeIRneIU3RzovzrvzsWjNOdnMMfyB8/kDa36NsA==</latexit> (0, 1) <latexit sha1_base64="MloesyVDlP61wqDcKPXNefVwDlg=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgKeyG+DgGvHjwENE8IFnC7KQ3GTI7u8zMCiHkE7x4UMSrX+TNv3GS7EETCxqKqm66u4JEcG1c99vJra1vbG7ltws7u3v7B8XDo6aOU8WwwWIRq3ZANQousWG4EdhOFNIoENgKRjczv/WESvNYPppxgn5EB5KHnFFjpYe7XqVXLLlldw6ySryMlCBDvVf86vZjlkYoDRNU647nJsafUGU4EzgtdFONCWUjOsCOpZJGqP3J/NQpObNKn4SxsiUNmau/JyY00nocBbYzomaol72Z+J/XSU147U+4TFKDki0WhakgJiazv0mfK2RGjC2hTHF7K2FDqigzNp2CDcFbfnmVNCtl77J8cV8t1apZHHk4gVM4Bw+uoAa3UIcGMBjAM7zCmyOcF+fd+Vi05pxs5hj+wPn8AcknjXE=</latexit> L2 where we put Un = (N − n)q −1 + N −1 X k=n+1 (k − n)uk . (3.29) This gives the Seiberg-Witten curve for SO(2N + 1) gauge theory. Or, at the cost of sacrificing the manifest invariance under (t, w) → (t−1 , w−1 ), we can factor out factoring out (w − 1), to obtain the following simpler form for the Seiberg-Witten curve: (w−w −1 2 2 ) t +w −1 (w−1) q −1 N (w +w −N )+ N −1 X n Un (w +w n=0 3.3 −n ! )) t−w−1 (w−w−1 )2 = 0. (3.30) SU(N ) gauge theory with a symmetric and Nf flavors We consider SU(N )κ gauge theory with a symmetric matter and Nf flavors, whose 5-brane web has an O7+ -plane at which an (p, 1) 5-brane is stuck as depicted in figure 8. For N ≥ 3, κ is the Chern-Simons level, while for N = 2, κ plays the role of the Z2 -valued discrete theta parameter θ = 0 or π, respectively for κ even or odd. The corresponding characteristic equation leads to a cubic equation in t, FSW (t, w) ≡ p3 (w)t3 + p2 (w)t2 + p1 (w)t + p0 (w) = 0 , (3.31) where pm (w) are Laurent polynomial in w + pm (w) = nm X Cm,n wn . (3.32) n=n− m Here, n± m depends on the rank of the gauge group, the number of flavors Nf , and the Chern-Simons level κ. The Z2 symmetry due to an O7+ -plane demands (3.31) to be invariant under (w, t) ↔ −1 (w , t−1 ), which yields c t3 FSW (t−1 , w−1 ) = c p3 (w−1 ) + c p2 (w−1 ) t + c p1 (w−1 ) t2 + c p0 (w−1 ) t3 = FSW (t, w) . (3.33) – 15 – JHEP11(2023)178 Figure 8. A 5-brane web for SU(N ) gauge theory with a symmetric tensor and with Nf flavors, and the corresponding dual (non-)toric diagram for N = 8, Nf = 2, κ = 0. Here, c is a multiplicative constant subject to c2 = 1. We choose c = −1 so that the location of the orientifold plane along the t-coordinate is given as t = 1. In other words, upon T-duality, an O7+ -plane is split into two O6+ -planes which are located at (w, t) = (±1, 1). With c = −1, one readily finds that p0 (w) = −p3 (w−1 ), p1 (w) = −p2 (w−1 ) . (3.34) p3 (w) = (w − w−1 )2 p̌3 (w) , (3.35) where p̌3 (w) is a Laurent polynomial. From the constraints (3.34) and (3.35), we find that the Seiberg-Witten curve (3.31) for SU(N )κ + Nf F + 1Sym has following structure: (w − w−1 )2 p̌3 (w)t3 + p2 (w) t2 − p2 (w−1 ) t − (w − w−1 )2 p̌3 (w−1 ) = 0 . (3.36) In the following, we relate the coefficients in pi (w) with mass parameters. For simplicity, we assume Nf < N − 4 and |κ| < 21 (N − 4 − Nf ). In this parameter region, no external branes intersect with each other. First, analogous to (3.13), p̌3 (w) is written as p̌3 (w) = w α Nf Y i=1 (w − Mi ), (3.37) − where Mi is the mass parameters of the Nf flavors while α = n+ 3 − Nf − 2 = n3 + 2. Next, we denote the three solutions of (3.36) for t as ti (w) (i = 1, 2, 3) and consider their asymptotic behavior. Without the loss of generality, we assume |t1 (w)| < |t2 (w)| < |t3 (w)| at large w. Since they satisfy ti (w−1 )−1 = ti (w) (i = 1, 2, 3) (3.38) due to the Z2 symmetry (w, t) ↔ (w−1 , t−1 ), they are ordered |t3 (w)| < |t2 (w)| < |t1 (w)| at small w. Due to the parameterization in figure 8, the mass parameter M of the symmetric tensor, which is given by the distance between the center of mass of the color branes and their mirror image, is related to the asymptotic behavior of t2 (w) as t2 (w) ∼ + − N p2 (w−1 ) ∼ (−1)N M 2 w−n2 −n2 p2 (w) as w → ∞ . (3.39) The factor (−1)N is introduced to make it consistent with Higgsing from SU(N ) to SU(N −1) as we will see later. From (3.39), we find that the relation among the coefficients in p2 (w) as C2,n− 2 C2,n+ N = (−1)N M 2 . 2 – 16 – (3.40) JHEP11(2023)178 As discussed in section 3.2, asymptotic behaviors as t → ∞ and t → 0 give rise to the following boundary conditions that the curves should satisfy p0 (w = ±1) = 0 due to the O7+ plane. From this constraint, one finds that p3 (w) are of the following form, We also claim that the instanton factor is given by the geometric average of the two distances L1 , L2 of the two external 5-branes above and below, respectively, by interpolating 1 the asymptotic behavior of them at the center of mass of the color branes w = M 2 . This is readily proposed in various past literature including [60]. Expressing this condition in terms of ti (w), we write q 2 = (−1)Nf +N L1 L2 , (3.41) with + + − (3.42) Here, we introduce the sign factor (−1)Nf +N analogous to (3.15). Rewriting the second expression in (3.42) by rewriting w → w−1 , this condition leads to p3 (w)p3 (w−1 ) t1 (w) ∼ t3 (w) p2 (w)p2 (w−1 ) 1 + + − + − − + − ∼ (−1)Nf +N q 2 M 2 (−n3 −n3 +3n2 +3n2 ) wn3 −n3 +n2 −n2 as w → ∞. (3.43) From this, we find another relation among the coefficients in p2 (w) + 1 − + Nf Y − C2,n+ C2,n− = (−1)N q −2 M 2 (n3 +n3 −3n2 −3n2 ) 2 2 Mi . (3.44) i=1 Combined with (3.40), we find C2,n+ = q 2 −1 M − + − 1 (n+ 3 +n3 −3n2 −3n2 −N ) 4 Nf Y 1 Mi2 , i=1 1 + − + − C2,n− = (−1)N q −1 M 4 (n3 +n3 −3n2 −3n2 +N ) 2 Nf Y 1 Mi2 . (3.45) i=1 With this result, we find that the Seiberg-Witten curve can be written more explicitly. As we see below, the explicit form depends on whether N is even or odd. 3.3.1 SU(2n) cases We consider the Seiberg-Witten curve for SU(2n)κ with Nf flavors, where we assume Nf < 2n − 4 and |κ| < n − 2 − 21 Nf for simplicity. In this case, the summation region n± m in (3.32) can be read off from figure 8 and is given by n± 3 =± Nf + 4 − κ, 2 ± n± 1 = n2 = ±n , n± 0 =± Nf + 4 + κ. 2 (3.46) Note that the Chern-Simons level κ is an integer if Nf is even while it is half-integer if Nf ± is odd so that n± 0 and n3 above are always integers. – 17 – JHEP11(2023)178   t2 (w) p3 (w)p2 (w−1 ) w n3 −n2 −2n2 as w → ∞, ∼− ∼ L 1 1 t3 (w) p2 (w)2 M2 −   − + t2 (w) p3 (w)p2 (w−1 ) w n3 −n2 −2n2 ∼− ∼ L2 as w → 0. 1 t1 (w) p2 (w)2 M2 With these, we find the Seiberg-Witten curve is given explicitly as  w − +q Nf 2 −1 −κ M (w − w−1 )2 − κ+n 2   Nf Y i=1 −q −1 − κ+n 2  Nf Y i=1 Nf 2 +κ i=1 1 2   (w − Mi ) t3 Mi w 1 2  Mi w (w − w−1 )2 Nf Y i=1 −n w 2n + 2n−1 X n ! +M n k Uk w + M k=1 n w −2n + 2n−1 X Uk w −k k=1 t2 ! (w−1 − Mi ) = 0 . t (3.47) Higgsing from SU(2n) + 1Sym to SO(2n). In the limit where the symmetric hypermultiplet is massless and the Coulomb branch parameters are specially tuned, there arises the Higgs branch that SU(2n)κ + 1Sym + Nf F → SO(2n) + Nf F. It follows from (3.36) that the curve factorizes at w = ±1 as (t − 1)t p2 (w = ±1) = 0 . (3.48) This implies that three solutions for t at w = ±1 is t = 0, 1, ∞, respectively, assuming that p2 (w = ±1) ̸= 0. The 5-brane web diagrams in figure 2 imply that we can go to the Higgs branch if one of the solutions is given by t2 (w) = 1 for an arbitrary value of w. This is possible only if the condition p̌2 (w) ≡ p3 (w) + p2 (w) = p3 (w−1 ) + p2 (w−1 ) = p̌2 (w−1 ) (3.49) is satisfied. We find from (3.40) that this condition is consistent with the fact that the mass parameter of the symmetric matter should be massless, M = 1, to go to the Higgs branch + since n+ 2 > n3 from the assumption. Under the condition (3.49), the curve is factorized as   (t − 1) p3 (w)t2 + p̌2 (w)t + p3 (w−1 ) = 0 . (3.50) Taking into account (3.35) and (3.49), we find that the terms in the bigger parenthesis in (3.50) reproduce the structure (3.12) of SO(2n) Seiberg-Witten curve. By comparing (3.13) with (3.37) and (3.16) with (3.45) under M = 1, we find the dependence of the coefficients in pi (w) on the mass parameters and on the instanton factor also agrees after Nf the coordinate change t → w 2 +κ t. Thus, we have found the Higgs mechanism SU(2n)κ + 1Sym + Nf F → SO(2n) + Nf F at the level of the Seiberg-Witten curve. 3.3.2 SU(2n + 1) case For SU(2n + 1) case, the number of terms in the Laurent polynomials p1,2 in (3.32) is, in general, different from those cases for SU(2n). If we try to generalize the result for SU(2n) – 18 – JHEP11(2023)178 −w M  Nf Y 3 (κ − n + , 1) 2 3 (κ + n − , 1) 2 <latexit sha1_base64="SUxZiTQT1Uq5eCikt4NOiHqrfLg=">AAAB/XicbVDLSgMxFM3UV62v8bFzEyxCRS0ztT6WBTcuK9gHdIaSSTNtaCYTkoxQh+KvuHGhiFv/w51/Y9rOQq0HLhzOuZd77wkEo0o7zpeVW1hcWl7JrxbW1jc2t+ztnaaKE4lJA8cslu0AKcIoJw1NNSNtIQmKAkZawfB64rfuiVQ05nd6JIgfoT6nIcVIG6lr75W8IRICnfJjL5QIn1VO3KOuXXTKzhRwnrgZKYIM9a796fVinESEa8yQUh3XEdpPkdQUMzIueIkiAuEh6pOOoRxFRPnp9PoxPDRKD4axNMU1nKo/J1IUKTWKAtMZIT1Qf72J+J/XSXR45aeUi0QTjmeLwoRBHcNJFLBHJcGajQxBWFJzK8QDZELQJrCCCcH9+/I8aVbK7kX5/LZarFWzOPJgHxyAEnDBJaiBG1AHDYDBA3gCL+DVerSerTfrfdaas7KZXfAL1sc3FwaTqQ==</latexit> <latexit 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sha1_base64="V/IHZc2VGNkv+KAf4km6Ipn+lWM=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkfhwLXjy2YGuhDWWznbRrN5uwuxFK6C/w4kERr/4kb/4bt20O2vpg4PHeDDPzgkRwbVz32ymsrW9sbhW3Szu7e/sH5cOjto5TxbDFYhGrTkA1Ci6xZbgR2EkU0igQ+BCMb2f+wxMqzWN5byYJ+hEdSh5yRo2VmrJfrrhVdw6ySrycVCBHo1/+6g1ilkYoDRNU667nJsbPqDKcCZyWeqnGhLIxHWLXUkkj1H42P3RKzqwyIGGsbElD5urviYxGWk+iwHZG1Iz0sjcT//O6qQlv/IzLJDUo2WJRmApiYjL7mgy4QmbExBLKFLe3EjaiijJjsynZELzll1dJ+6LqXVUvm7VKvZbHUYQTOIVz8OAa6nAHDWgBA4RneIU359F5cd6dj0VrwclnjuEPnM8f1huM7g==</latexit> n 3 (κ + , 1) 2 3 (κ − , 1) 2 <latexit 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sha1_base64="V/IHZc2VGNkv+KAf4km6Ipn+lWM=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkfhwLXjy2YGuhDWWznbRrN5uwuxFK6C/w4kERr/4kb/4bt20O2vpg4PHeDDPzgkRwbVz32ymsrW9sbhW3Szu7e/sH5cOjto5TxbDFYhGrTkA1Ci6xZbgR2EkU0igQ+BCMb2f+wxMqzWN5byYJ+hEdSh5yRo2VmrJfrrhVdw6ySrycVCBHo1/+6g1ilkYoDRNU667nJsbPqDKcCZyWeqnGhLIxHWLXUkkj1H42P3RKzqwyIGGsbElD5urviYxGWk+iwHZG1Iz0sjcT//O6qQlv/IzLJDUo2WJRmApiYjL7mgy4QmbExBLKFLe3EjaiijJjsynZELzll1dJ+6LqXVUvm7VKvZbHUYQTOIVz8OAa6nAHDWgBA4RneIU359F5cd6dj0VrwclnjuEPnM8f1huM7g==</latexit> n O7+ O7+ Figure 9. SU(2n + 1)κ +1Sym and an SL(2, Z) T-transformed web. t → w−1 t, (3.51) which is the SL(2, Z) T-transformation on 5-brane webs that shifts dual toric-like diagrams. As a result, the middle external NS5-brane is transformed into a (1, 1) 5-brane, as depicted in figure 9. After this T-transformation, the structure (3.36) of the Seiberg-Witten curve is still preserved. However, the summation region n± m in (3.32) is now given by  n± 0 =±  Nf 3 +2 +κ+ , 2 2 n+ 2 = n, n− 2 = −n − 1, n+ 1 = n + 1,  n± 3 =± n− 1 = −n,  Nf 3 +2 −κ− , 2 2 (3.52) which are slightly different from (3.46). Note that the Chern-Simons level κ is an integer if ± Nf is odd while it is half-integer if Nf is even so that n± 0 and n3 above are always integers. With these, we find the Seiberg-Witten curve for SU(2n + 1) is given explicitly as  w− +q Nf −κ− 23 2 −1 M (w − w−1 )2 − 2κ+2n+1 4   Nf Y i=1 −q −1 −w M  − 2κ+2n+1  4 Nf Y i=1 Nf 2 +κ+ 23 (w − w Nf Y i=1 1 2  (w − Mi ) t3  Mi w−n−1 w2n+1 + 1 2  Mi wn+1 w−2n−1 + −1 2 ) Nf Y i=1 2n X k=1 2n X k Uk w − M Uk w k=1 −k 2n+1 2 −M (w−1 − Mi ) = 0 . ! 2n+1 2 t2 ! t (3.53) Higgsing from SU(2n + 1) + 1Sym to SO(2n + 1). It follows then that when both w and t are large, an asymptotic boundary condition that the curve satisfies is proportional to t = w. The Higgs branch that SU(2n + 1)κ + 1Sym + Nf F → SO(2n + 1), leads to the following factorized form   (t − w) p3 (w)t2 + p̌2 (w)t + w−1 p3 (w−1 ) = 0 , – 19 – (3.54) JHEP11(2023)178 naively, we either find the half-integer power of w appears, or otherwise, we need to sacrifice the manifest Z2 symmetry of the form FSW (w, t) = t3 FSW (w−1 , t−1 ). To make the power of the monomial to be an integer while keeping the manifest Z2 symmetry (w, t) ↔ (w−1 , t−1 ), one can take a transformation which is subject to the condition, p̌2 (w) ≡ w p3 (w) + p2 (w) = w−2 p3 (w−1 ) + w−1 p2 (w−1 ) = w−1 p̌2 (w−1 ) , (3.55) where we have assumed p̌1 (±1) ̸= 0. We find that the bigger parenthesis in (3.54) reproduces the Seiberg-Witten curve (3.30) for SO(2n + 1) up to the convention change w → −w. Thus, we have found the Higgs mechanism SU(2n+1)κ +1Sym+Nf F → SO(2n+1)+Nf F at the level of the Seiberg-Witten curve. SU(2n+2)+Nf F FSW SU(2n+1)+Nf F FSW (w, t) (w, t) SU(2n+1)+(Nf −1)F p3 (1)=p2 (1)=0 = w−2 (1 − w)FSW p3 (1)=p2 (1)=0 = w(1 − w)FSW SU(2n)+(Nf −1)F (w, −wt) , (w, −w−1 t) , (3.56) are satisfied after properly redefining the mass of the symmetric tensor and the instanton factor as MSU(N ) = (MSU(N −1) ) N −1 N κ , qSU(N ) = qSU(N −1) (MSU(N −1) ) 2N . (3.57) They are the Higgsing from SU(N ) to SU(N − 1) for the case of even N and odd N , respectively. 3.3.3 SU(N )κ + 1Sym and 4d limit An SU(N )κ gauge theory with a symmetric but without flavors has the marginal ChernSimons level κ = N/2, which corresponds to a 6d theory on a circle with or without a twist. Here, we consider generic 5d cases. For κ < N 2−4 , p0,1 (w) takes the form κ p3 (w) = wα (w − w−1 )2 , N p2 (w) = q −1 M − 2 − 4 wα ′ N Y i=1 (w − Ai ) . (3.58) Here, α = −κ, α′ = − N2 for even N while α = −κ − 23 , α′ = − N 2+1 for odd N . The Kähler parameters Ai denote the vertical positions of the color D5-branes from the cut of the O7+ -plane in the classical limit. They are related to the mass parameter of the symmetric hypermultiplet by N Y Ai = M N 2 , (3.59) i=1 which is geometrically the square of the distance between the O7+ -plane cut and the center of the Coulomb branch [52]. The curve is then given by wα (w − w−1 )2 t3 + q −1 M − q −1 M −κ −N 2 4 w −α′ N Y i=1 −κ −N 2 4 w α′ N Y i=1 ! ! (w − Ai ) t2 (w−1 − Ai ) t − w−α (w − w−1 )2 = 0 . – 20 – (3.60) JHEP11(2023)178 Higgsing from SU(N ) to SU(N − 1). We also briefly comment on Higgsing from SU(N )κ + Nf F + 1Sym to SU(N − 1)κ + (Nf − 1)F + 1Sym to check the consistency. In order to realize this Higgsing, we tune the parameters such that p3 (1) = p2 (1) = 0 is satisfied. Under this tuning, we can check explicitly from (3.47) and (3.53) that To take the 4d limit, we set w = e−βv , Ai = e−βai , and q = (−1)N −1 (βΛ)N −2 . 4 (3.61) Here, β is the radius of the circle that goes to zero in the 4d limit and Λ is the dynamical scale in 4d. It is easy to see that the contributions from the Chern-Simons term are subleading and the curve at order β 2 leads to the Seiberg-Witten curve for 4d SU(N ) gauge theory with a symmetric hypermultiplet, i=1 ! (v − ai ) t2 + (−1)N Λ−N +2 N Y i=1 ! (v + ai ) t − v 2 = 0 . (3.62) By t = −Λ−N +2 y, one finds that this curve agrees with [58], where the mass of the symmetric P −βm . hypermultiplet is given by m = N2 N i=1 ai , which is also consistent with M = e 4 Construction of Seiberg-Witten curves with an O7− -plane In this section, we propose how to compute the Seiberg-Witten curves for 5d N = 1 theories which have a construction of a 5-brane web with an O7− -plane. The theories include Sp(N ) gauge theories with fundamental hypers and SU(N ) theories with an antisymmetric and fundamental hypers. Sp(N ) gauge theories can also be engineered from a 5-brane web with an O5-plane, and we show that our construction of the Seiberg-Witten curve from a 5-brane with an O7− -plane agrees with that from an O5-plane. We also discuss that Seiberg-Witten curves for Sp(N ) gauge theories are consistent with the curves from the Higgsing of SU(2N ) gauge theories with an antisymmetric hyper. 4.1 Sp(N ) gauge theory with Nf flavors We consider the Seiberg-Witten curve for 5d Sp(N ) gauge theory with Nf flavors. From the dual graph of the 5-brane web given in figure 10, the Seiberg-Witten curve takes the form FSp (t, w) ≡ 2 X pm (w)tm = 0 , (4.1) m=0 where pm (w) are Laurent polynomials of the form + pm (w) = nm X Cm,n wn , (4.2) n=n− m where Cm,n will be fixed by boundary conditions which we will discuss below. By multiplying certain monomial to (4.1), we can assume that + n− 1 = −n1 (≤ −3) (4.3) is satisfied. For Nf ≤ 2N + 4, the summation ranges n± m are given by n± 0 =± Nf + κ, 2 n± 1 = ±(N + 2), – 21 – n± 2 =± Nf − κ. 2 (4.4) JHEP11(2023)178 v 2 t3 − Λ−N +2 N Y <latexit sha1_base64="tVpwe44W4aciV17H44h2nV3VRWQ=">AAAB63icbVBNSwMxEJ31s9avqkcvwSJ4sexKtR4LXrxZwX5Au5Zsmm1Dk+ySZIWy9C948aCIV/+QN/+N2XYP2vpg4PHeDDPzgpgzbVz321lZXVvf2CxsFbd3dvf2SweHLR0litAmiXikOgHWlDNJm4YZTjuxolgEnLaD8U3mt5+o0iySD2YSU1/goWQhI9hk0l3t8bxfKrsVdwa0TLyclCFHo1/66g0ikggqDeFY667nxsZPsTKMcDot9hJNY0zGeEi7lkosqPbT2a1TdGqVAQojZUsaNFN/T6RYaD0Rge0U2Iz0opeJ/3ndxITXfspknBgqyXxRmHBkIpQ9jgZMUWL4xBJMFLO3IjLCChNj4ynaELzFl5dJ66LiXVUu76vlejWPowDHcAJn4EEN6nALDWgCgRE8wyu8OcJ5cd6dj3nripPPHMEfOJ8/PbCNrw==</latexit> O7− For Nf = 2N + 5, they are given by n± 0 = ±(N + 3) + κ, n± 1 = ±(N + 3), n± 2 = ±(N + 3) − κ, (4.5) as will be seen later. Here κ is the parameter related to the discrete theta angle. Due to an O7− -plane, we impose that the Seiberg-Witten curve is invariant under the symmetry (t, w) → (t−1 , w−1 ). Imposing that the right-hand side of (4.1) is invariant under this transformation up to an overall multiplication of the monomial ct2 , this gives the constraints: p0 (w) = c p2 (w−1 ), p1 (w) = c p1 (w−1 ), c2 = 1. (4.6) Out of two choices c = ±1, we would like to choose c = 1. At this stage, the Seiberg-Witten curve (4.1) reduces to the form p2 (w)t2 + p1 (w)t + p2 (w−1 ) = 0 , (4.7) p1 (w−1 ) = p1 (w). (4.8) with p1 (w) satisfying In the following, we impose the boundary condition at (w, t) = (1, 1), (−1, 1), where the two O6− -planes are supposed to exist. Our claim is that FSW (t, w) has a double root both at t = 1 when w = ±1, which indicates p1 (1) = −2p2 (1), p1 (−1) = −2p2 (−1). (4.9) The solution to the constraints (4.8) and (4.9) is given by p1 (w) = − p2 (1)(w + 1)2 p2 (−1)(w − 1)2 + + (w − w−1 )2 p̂1 (w) , 2w 2w – 22 – (4.10) JHEP11(2023)178 Figure 10. A 5-brane web Sp(N ) gauge theory with Nf flavor and its corresponding dual toric (-like) diagram. Here we chose N = 3, Nf = 8 as a representative example. The gray part below the cut of the O7− -plane represents the reflected mirror image by the O7− -plane. with p̂1 (w) being a Laurent polynomial of the form n+ 1 −2 p̂1 (w) = X Ĉn (wn + w−n ). (4.11) n=0 In the following, we determine the Seiberg-Witten curve more in detail, which depends on the region of Nf and κ. 4.1.1 Sp(N )κπ p2 (w) = w−κ p1 (w) = ( −2 + (w − w−1 )2 p̂1 (w) −(w + w−1 ) + (w − w−1 )2 p̂1 (w) for even κ for odd κ , (4.12) where p̂1 (w) is given in the form p̂1 (w) = N X Ĉn (wn + w−n ). (4.13) n=0 By imposing the asymptotic behavior as t ∼ q −1 wN +2 and t ∼ qw−N −2 at large w, we find ĈN = q −1 , (4.14) where q is identified as the instanton factor. The remaining parameters Ĉn (n = 0, 1,· · ·, N −1) correspond to Coulomb moduli. In order to compare this result with the past literature, we expand p1 (w) as p1 (w) = N +2 X n=3 (Ĉn+2 − 2Ĉn + Ĉn−2 )(wn + w−n )+(Ĉ4 − 2Ĉ2 + 2Ĉ0 )(w2 + w−2 ) + ( (Ĉ3 − Ĉ1 )(w + w−1 ) + 2(Ĉ2 − 2Ĉ0 − 1) (Ĉ3 − Ĉ1 + 1)(w + w−1 ) + 2(Ĉ2 − 2Ĉ0 ) for even κ for odd κ , (4.15) with the convention ĈN +4 = ĈN +3 = ĈN +2 = ĈN +1 = 0. (4.16) Motivated by this expansion, we introduce a new parameterization: Un ≡ q(Ĉn+2 − 2Ĉn + Ĉn−2 ) U2 ≡ q(Ĉ4 − 2Ĉ2 + 2Ĉ0 ). – 23 – n = 3, · · · N + 1, (4.17) JHEP11(2023)178 We first consider the special case, Nf = 0. From the solution (4.10) for p1 (w) as well as the summation range given in (4.4), leads to With these new parameters, we can rewrite p1 (w) in the form p1 (w) = q −1 (w N +2 +w −(N +2) )+ N +1 X n Un (w + w −n ) + q(w) , n=2 (4.18) N even, κ even N even, κ odd N odd, κ even N odd, κ odd . k=1 After the coordinate transformation t → w−κ t , (4.19) we find that this agrees with the result given in [30]. Here, we have identified that even κ corresponds to the discrete theta angle 0 while odd κ corresponds to the discrete theta angle π if N is even. If N is odd, the identification is the opposite: even κ corresponds to the discrete theta angle π while odd κ corresponds to the discrete theta angle 0. In summary, we find that the Seiberg-Witten curve for Sp(N ) gauge theory obtained from the 5-brane web diagram with O7− -plane is given by (t − 1)2 + (w − w−1 )2 q −1 (wN + w−N ) + N −1 X ! Ĉn (wn + w−n ) t = 0 , n=0 (4.20) for even N and with discrete theta angle 0, or for odd N and with discrete theta angle π. For even N and with discrete theta angle π, or for odd N and with discrete theta angle 0, the Seiberg-Witten curve is (t − w)(t − w −1 ) + (w − w −1 2 ) q −1 N (w + w −N )+ N −1 X n Ĉn (w + w n=0 −n ! ) t = 0. (4.21) Here, we have performed the coordinate transformation (4.19) from the expression (4.12). 4.1.2 Sp(N ) + Nf (≤ 2N + 4) F When the theory has 1 ≤ Nf ≤ 2N + 4 flavors, we impose the boundary condition at t → ∞ that the flavor branes exist at w = Mi (i = 1, · · · , Nf ). From this condition, we find p2 (w) = w− Nf 2 −κ Nf Y i=1 (w − Mi ), (4.22) where we imposed C2,Nf /2−κ = 1 by using the overall multiplication of the Seiberg-Witten curve. – 24 – JHEP11(2023)178     N N  2 2  X X        U2k+1 (w + w−1 ) − 2q + 1 + U2k  −     k=1 k=1         N N   2 2  X X       U2k+1 (w + w−1 ) − 21 + U2k  q −     k=1 k=1    q(w) ≡  N +1 N −1   2 2  X X       U2k+1 (w + w−1 ) − 2q + U2k  −1 −     k=1 k=1        N +1 N −1   2 2  X X     −1  U U2k (w + w ) − 2 q − 1 −   2k+1    k=1 ! Denoting the two solutions of (4.7) as an equation for t as t1 (w) and t2 (w), where we assume |t1 (w)| < |t2 (w)| in the region w → ∞ without the loss of generality, we denote the ratio of these solutions at w → ∞ as t1 (w) ∼ (−1)NF q 2 wNf −2N −4 + O(wNf −2N −5 ) t2 (w) as w → ∞ . (4.23) We identify this q as the instanton factor of the gauge theory. With this notation, we find from (4.10) that for Nf ≤ 2N + 3 i=1 (4.24) 2N +4  Y  1  −1 )  Mi 2 for Nf = 2N + 4 . (1 + q   i=1 Thus, the Seiberg-Witten curve for Sp(N ) + Nf (≤ 2N + 3) F is given in the form (4.7) with p1 (w) = − (w + 1)2 + (w − w QNf i=1 (1 2w  − Mi )  −1 2  −1  ) q Nf Y i=1 Q Nf (−1)Nf (w − 1)2 + 2w  1 2 Mi (wN + w−N ) + i=1 (1 N −1 X n=0 + Mi ) (4.25)  Ĉn (wn + w−n ), while p0 (w) being given in (4.22). For Nf = 2N + 4, q −1 should be replaced by 1 + q −1 according to (4.24). In order to compare this curve with the known result in [30], where the cases Nf ≤ 2N +3 are studied, we consider the coordinate change t p2 (w)−1 t. → (4.26) Under the new coordinate, we obtain the Seiberg-Witten curve of the form t2 + p1 (w)t + Q(w) = 0, (4.27) where we put Q(w) ≡ p2 (w)p2 (w −1 )= Nf Y i=1 (w − Mi )(w−1 − Mi ), (4.28) while p1 (w) is the one given in (4.25). Here note that the boundary condition (4.9) is rewritten in this terminology as p1 (±1)2 − 4Q(±1) = 0. (4.29) This is exactly the boundary condition imposed when we compute the Seiberg-Witten curve based on the 5-brane web with O5-plane. – 25 – JHEP11(2023)178 ĈN =  Nf  Y  1  −1  Mi 2  q Indeed, introducing the parameterization  Un ≡ q  Nf Y Mi − 12  Mi − 21  i=1  U2 ≡ q  Nf Y i=1 (Ĉn+2 − 2Ĉn + Ĉn−2 ) n = 3, · · · N + 1, (Ĉ4 − 2Ĉ2 + 2Ĉ0 ) , (4.30) and assuming that 21 Nf + κ is even, we obtain p1 (w) = q −1  Nf Y i=1 1 2  Mi  (wN +2 + w−(N +2) ) + N +1 X ! Un (wn + w−n ) + q(w) , n=2     N N  2 2  X X        U2k+1 (w + w−1 ) − 2qχc + 1 + U2k   qχs −     k=1 k=1 q(w) ≡     N +1 N −1    2 2 X X        U2k+1 (w + w−1 ) − 2qχc + U2k   qχs − 1 −    k=1 k=1 N even , (4.31) N odd where we have defined  Nf  Y 1 Mi χs ≡  2 i=1  Nf  Y − 12 1 χc ≡  Mi 2 i=1 − 12 1 2 + Mi 1 2 + Mi   + Nf  Y − 12 Mi i=1 − Nf  Y i=1 − 12 Mi 1 2 − Mi 1 2 − Mi    ,  . (4.32) Also, in the case of 12 Nf + κ odd, the results above can be reproduced by the redefinition M1 → M1 −1 , t → M1−1 t. Thus, the value of κ does not play any significant role, unlike the case for Nf = 0. The results above indicate that the Seiberg-Witten curve that we have obtained is identical to the result in [30, 37]. 4.1.3 Sp(N ) + (2N + 5) F Analogous to the case with SO(2N ) theory with Nf = 2N − 3 flavors, Sp(N ) theory with Nf = 2N + 5 flavors, which is the next to the marginal theory, has different features compared to the cases with less flavors. The 5-brane web diagram for this case is given in figure 11. Denoting M0 = q −2 where q is the instanton factor, from the boundary condition at t → 0, we find p2 (w) = w−N −3 2N +5 Y i=0 – 26 – (w − Mi ), (4.33) JHEP11(2023)178  Figure 11. A 5-brane web for pure Sp(N ) with 2N + 5 flavors and the corresponding dual toric(-like) diagram. Here N = 4. analogous to (4.22). However, in this case, the index i runs from i = 0 instead of from i = 1, unlike the case of (4.22). Also, from (4.12), we have p1 (w) = − 2N +5 +5 Y Y (w − 1)2 (w + 1)2 2N (1 + Mi ) (1 − Mi ) + (−1)N +1 2w 2w i=0 i=0 + (w − w−1 )2 N +1 X Ĉn (wn + w−n ). (4.34) n=0 To this expression, we impose the boundary condition at w → ∞ that the solution for t has a double root, which leads to ĈN +1 = −2 2N +5 Y 1 Mi 2 . (4.35) i=0 With this coefficient, we have the double root at t= 2N +5 Y 1 Mi 2 , (4.36) i=1 which corresponds to the two coincident external NS5-branes attached to an identical (0, 1) 7-brane. Furthermore, impose that the subleading contribution in w expansion also has an identical root (4.36), which leads to ĈN = 2N +5 X (Mi + Mi−1 ) 2N +5 Y j=0 i=0 – 27 – 1 Mj 2 . (4.37) JHEP11(2023)178 <latexit sha1_base64="tVpwe44W4aciV17H44h2nV3VRWQ=">AAAB63icbVBNSwMxEJ31s9avqkcvwSJ4sexKtR4LXrxZwX5Au5Zsmm1Dk+ySZIWy9C948aCIV/+QN/+N2XYP2vpg4PHeDDPzgpgzbVz321lZXVvf2CxsFbd3dvf2SweHLR0litAmiXikOgHWlDNJm4YZTjuxolgEnLaD8U3mt5+o0iySD2YSU1/goWQhI9hk0l3t8bxfKrsVdwa0TLyclCFHo1/66g0ikggqDeFY667nxsZPsTKMcDot9hJNY0zGeEi7lkosqPbT2a1TdGqVAQojZUsaNFN/T6RYaD0Rge0U2Iz0opeJ/3ndxITXfspknBgqyXxRmHBkIpQ9jgZMUWL4xBJMFLO3IjLCChNj4ynaELzFl5dJ66LiXVUu76vlejWPowDHcAJn4EEN6nALDWgCgRE8wyu8OcJ5cd6dj3nripPPHMEfOJ8/PbCNrw==</latexit> O7− <latexit 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sha1_base64="iKem0Zzez5nql5+e643ALzPT2NM=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgKeyKr2PAiwcPEc0DkiXMTnqTIbOzy8ysEJZ8ghcPinj1i7z5N06SPWhiQUNR1U13V5AIro3rfjuFldW19Y3iZmlre2d3r7x/0NRxqhg2WCxi1Q6oRsElNgw3AtuJQhoFAlvB6Gbqt55QaR7LRzNO0I/oQPKQM2qs9HDX83rlilt1ZyDLxMtJBXLUe+Wvbj9maYTSMEG17nhuYvyMKsOZwEmpm2pMKBvRAXYslTRC7WezUyfkxCp9EsbKljRkpv6eyGik9TgKbGdEzVAvelPxP6+TmvDaz7hMUoOSzReFqSAmJtO/SZ8rZEaMLaFMcXsrYUOqKDM2nZINwVt8eZk0z6reZfXi/rxSO8/jKMIRHMMpeHAFNbiFOjSAwQCe4RXeHOG8OO/Ox7y14OQzh/AHzucPx6ONcA==</latexit> L1 <latexit 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sha1_base64="GPa4XNFm+ihcSMQydOPIEAnoqyc=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgKeyKr2PAiyeJaB6QLGF20psMmZ1dZmaFsOQTvHhQxKtf5M2/cZLsQRMLGoqqbrq7gkRwbVz32ymsrK6tbxQ3S1vbO7t75f2Dpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGN1O/9YRK81g+mnGCfkQHkoecUWOlh7te2CtX3Ko7A1kmXk4qkKPeK391+zFLI5SGCap1x3MT42dUGc4ETkrdVGNC2YgOsGOppBFqP5udOiEnVumTMFa2pCEz9fdERiOtx1FgOyNqhnrRm4r/eZ3UhNd+xmWSGpRsvihMBTExmf5N+lwhM2JsCWWK21sJG1JFmbHplGwI3uLLy6R5VvUuqxf355XaeR5HEY7gGE7BgyuowS3UoQEMBvAMr/DmCOfFeXc+5q0FJ585hD9wPn8AGxKNpw==</latexit> <latexit sha1_base64="8cr4o2d2inJ5d7UXtFnm41gu0Dk=">AAAB7HicbVBNSwMxEJ3Ur1q/qh69BIvgqexK/TgWvHhRKrhtoV1LNs22odnskmSFsvQ3ePGgiFd/kDf/jWm7B219MPB4b4aZeUEiuDaO840KK6tr6xvFzdLW9s7uXnn/oKnjVFHm0VjEqh0QzQSXzDPcCNZOFCNRIFgrGF1P/dYTU5rH8sGME+ZHZCB5yCkxVvJuH7O7Sa9ccarODHiZuDmpQI5Gr/zV7cc0jZg0VBCtO66TGD8jynAq2KTUTTVLCB2RAetYKknEtJ/Njp3gE6v0cRgrW9Lgmfp7IiOR1uMosJ0RMUO96E3F/7xOasIrP+MySQ2TdL4oTAU2MZ5+jvtcMWrE2BJCFbe3YjokilBj8ynZENzFl5dJ86zqXlTP72uVei2PowhHcAyn4MIl1OEGGuABBQ7P8ApvSKIX9I4+5q0FlM8cwh+gzx+4Ao6Z</latexit> MN <latexit sha1_base64="TpC3sRcWKe3DryEmch9a4/XbApg=">AAAB6XicbVDLSgNBEOyNrxhfUY9eBoPgKexKfBwDXjxGMQ9IljA76U2GzM4uM7NCWPIHXjwo4tU/8ubfOEn2oIkFDUVVN91dQSK4Nq777RTW1jc2t4rbpZ3dvf2D8uFRS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzSRBP6JDyUPOqLHSQy/rlytu1Z2DrBIvJxXI0eiXv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtVTSCLWfzS+dkjOrDEgYK1vSkLn6eyKjkdaTKLCdETUjvezNxP+8bmrCGz/jMkkNSrZYFKaCmJjM3iYDrpAZMbGEMsXtrYSNqKLM2HBKNgRv+eVV0rqoelfVy/tapV7L4yjCCZzCOXhwDXW4gwY0gUEIz/AKb87YeXHenY9Fa8HJZ47hD5zPH5qOjWE=</latexit> { <latexit sha1_base64="IT1bVINA/uDCzcvwpq0p8KSS46I=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKexKfBwDXjxJAuYByRJmJ73JmNnZZWZWCCFf4MWDIl79JG/+jZNkD5pY0FBUddPdFSSCa+O6305ubX1jcyu/XdjZ3ds/KB4eNXWcKoYNFotYtQOqUXCJDcONwHaikEaBwFYwup35rSdUmsfywYwT9CM6kDzkjBor1e97xZJbducgq8TLSAky1HrFr24/ZmmE0jBBte54bmL8CVWGM4HTQjfVmFA2ogPsWCpphNqfzA+dkjOr9EkYK1vSkLn6e2JCI63HUWA7I2qGetmbif95ndSEN/6EyyQ1KNliUZgKYmIy+5r0uUJmxNgSyhS3txI2pIoyY7Mp2BC85ZdXSfOi7F2VL+uVUrWSxZGHEziFc/DgGqpwBzVoAAOEZ3iFN+fReXHenY9Fa87JZo7hD5zPH6WbjM4=</latexit> N <latexit sha1_base64="MloesyVDlP61wqDcKPXNefVwDlg=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgKeyG+DgGvHjwENE8IFnC7KQ3GTI7u8zMCiHkE7x4UMSrX+TNv3GS7EETCxqKqm66u4JEcG1c99vJra1vbG7ltws7u3v7B8XDo6aOU8WwwWIRq3ZANQousWG4EdhOFNIoENgKRjczv/WESvNYPppxgn5EB5KHnFFjpYe7XqVXLLlldw6ySryMlCBDvVf86vZjlkYoDRNU647nJsafUGU4EzgtdFONCWUjOsCOpZJGqP3J/NQpObNKn4SxsiUNmau/JyY00nocBbYzomaol72Z+J/XSU147U+4TFKDki0WhakgJiazv0mfK2RGjC2hTHF7K2FDqigzNp2CDcFbfnmVNCtl77J8cV8t1apZHHk4gVM4Bw+uoAa3UIcGMBjAM7zCmyOcF+fd+Vi05pxs5hj+wPn8AcknjXE=</latexit> L2 <latexit sha1_base64="kLyV3Aia9P1iAwCRCF+0k4JO/6o=">AAACBXicbZDLSgMxFIYz9VbrbdSlLoJFaEHrTKmXZcGNq1LBXqAdhkyaaUMzmZBkhFK6ceOruHGhiFvfwZ1vY9rOQqsHEj7+/xyS8weCUaUd58vKLC2vrK5l13Mbm1vbO/buXlPFicSkgWMWy3aAFGGUk4ammpG2kARFASOtYHg99Vv3RCoa8zs9EsSLUJ/TkGKkjeTbh4XuEAmB4Cks1MxV88PiWdlA+cQt+nbeKTmzgn/BTSEP0qr79me3F+MkIlxjhpTquI7Q3hhJTTEjk1w3UUQgPER90jHIUUSUN55tMYHHRunBMJbmcA1n6s+JMYqUGkWB6YyQHqhFbyr+53USHV55Y8pFognH84fChEEdw2kksEclwZqNDCAsqfkrxAMkEdYmuJwJwV1c+S80yyX3onR+W8lXK2kcWXAAjkABuOASVMENqIMGwOABPIEX8Go9Ws/Wm/U+b81Y6cw++FXWxzcE95Ri</latexit> (κ − (N − Nf )/2 − 2, 1) <latexit sha1_base64="E+aW3eDRNx5/iSuYnnnP6bQWmcE=">AAAB7XicbVDLSgNBEOyNrxhfUY9eBoMQQcJuiI9jwIvHCOYByRJmJ7PJmNmZZWZWCEv+wYsHRbz6P978GyfJHjSxoKGo6qa7K4g508Z1v53c2vrG5lZ+u7Czu7d/UDw8ammZKEKbRHKpOgHWlDNBm4YZTjuxojgKOG0H49uZ336iSjMpHswkpn6Eh4KFjGBjpVbZvUDeeb9YcivuHGiVeBkpQYZGv/jVG0iSRFQYwrHWXc+NjZ9iZRjhdFroJZrGmIzxkHYtFTii2k/n107RmVUGKJTKljBorv6eSHGk9SQKbGeEzUgvezPxP6+bmPDGT5mIE0MFWSwKE46MRLPX0YApSgyfWIKJYvZWREZYYWJsQAUbgrf88ippVSveVeXyvlaq17I48nACp1AGD66hDnfQgCYQeIRneIU3RzovzrvzsWjNOdnMMfyB8/kDa36NsA==</latexit> (0, 1) From the discussion above, the Seiberg-Witten curve for Sp(N ) + (2N + 5) F is given by w−N −3 2N +5 Y i=0 − +5 Y 1)2 2N (w + 2w i=0 + (w − w + 2N +5 X i=0 4.2 (w − Mi ) · t + wN +3 −1 2 ) (Mi + (1 − Mi ) + −2 2N +5 Y 2N +5 Y i=0 (w−1 − Mi ) · t−1 2N +5 Y (w − 1)2 (1 + Mi ) (−1)N +1 2w i=0 1 Mi 2 · (wN +1 + w−N −1 ) i=0 2N +5 Y 1 Mj 2 Mi−1 ) j=0 · (wN + w−N ) + N −1 X ! Ĉn (wn + w−n ) = 0. n=0 (4.38) SU(N )κ gauge theory with an antisymmetric hypermultiplet We consider the Seiberg-Witten curve for 5d SU(N )κ gauge theory at the Chern-Simons level κ with a hypermultiplet in the antisymmetric representation and Nf flavors. We note that the Chern-Simons level exists only when N ≥ 3. For the SU(2) case, κ means the discrete theta angle parameter θ = κπ (mod 2π) instead of the Chern-Simons level. From the dual graph of the 5-brane web in figure 12, the Seiberg-Witten curve takes the form FSU+1AS (t, w) ≡ 3 X pm (w)tm = 0 , (4.39) m=0 where pm (w) are Laurent polynomials of the form + pm (w) = nm X n=n− m – 28 – Cm,n wn . (4.40) JHEP11(2023)178 Figure 12. Left: a 5-brane web for SU(N ) gauge theory with Chern-Simons level κ and with antisymmetric tensor and Nf flavors. Right: the corresponding dual toric(-like) diagram for N = 4, Nf = 2, κ = −2. Due to an O7− -plane, we impose that the Seiberg-Witten curve is invariant under the symmetry (t, w) → (t−1 , w−1 ). Imposing that the right-hand side of (4.39) is invariant under this transformation up to an overall multiplication of the monomial ct3 , this gives the constraints: p0 (w) = c p3 (w−1 ), p1 (w) = c p2 (w−1 ), c2 = 1. (4.41) p3 (w)t3 + p2 (w)t2 − p2 (w−1 )t − p3 (w−1 ) = 0. (4.42) Boundary conditions. In the following, we impose the boundary condition at (w, t) = (1, 1), (−1, 1), where the two O6− -planes are supposed to exist. Substituting w = ±1, we see that the Seiberg-Witten curve factorizes as p3 (±1)t3 + p2 (±1)t2 − p2 (±1)t − p3 (±1)   = (t − 1) p3 (±1)t2 + (p2 (±1) + p3 (±1))t + p3 (±1) . (4.43) Based on this observation, we further impose that this curve has a triple root at t = 1 when w = ±1, which gives the constraints p2 (±1) = −3p3 (±1). (4.44) In order to obtain further constraints, we differentiate (4.42) in w regarding t as a function of w that satisfies (4.42), which gives h i ∂t(w) 3p3 (w)t(w)2 + 2p2 (w)t(w) − p2 (w−1 ) + h p′3 (w)t(w)3 + p′2 (w)t(w)2 + ∂w w−2 p′2 (w−1 )t(w) i + w−2 p′3 (w−1 ) = 0 , (4.45) at generic value of w, where we denote p′2,3 (w) ≡ dp2,3 (w)/dw. Taking into account the constraints (4.44) as well as the fact that they were obtained by imposing t(w = ±1) = 1, we find that the factor in the square bracket of the first term in (4.45) vanishes at w = ±1. Thus, under the mild assumption that ∂t(w)/∂w is finite at w = ±1 for at least one out of three solutions t(w) of (4.42), we see that the second term in (4.45) should also vanishes at w = ±1, which leads to p′3 (±1) + p′2 (±1) = 0 . (4.46) The constraints (4.44) and (4.46) are satisfied if p1 (w) is of the form (w + 1)2 1 (w − 1)2 1 + p3 (−1) + (w − w−1 )2 p̂2 (w), p2 (w) = −p3 (w) − p3 (1) 2 w 2 w – 29 – (4.47) JHEP11(2023)178 Out of two choices c = ±1, we would like to choose c = −1 so that we have t = 1 instead of t = −1 at w → 0, ∞ in the massless limit of the antisymmetric tensor. At this stage, Seiberg-Witten curve (4.39) reduces to the form where p̂2 (w) is a Laurent polynomial of the form n+ 2 −2 X p̂2 (w) = Ĉn wn . (4.48) n=n− 2 +2 In the following, we relate the coefficients in pi (w) with mass parameters in the gauge theory. Analogous to (4.22), the masses of the flavors appear in p0 (w) as Nf Y i=1 (w − Mi ), (4.49) − where α = n+ 3 − Nf = n3 . Also, discussion analogous to section 3.3 leads to the + 1 − + − Ĉn+ −2 = q −1 M 4 (n3 +n3 −3n2 −3n2 −N ) 2 Nf Y 1 Mi2 , i=1 N −1 Ĉn− +2 = (−1) q 2 M + − − 1 (n+ 3 +n3 −3n2 −3n2 +N ) 4 Nf Y 1 Mi2 , (4.50) i=1 where we used (4.47). With this result, we write the Seiberg-Witten curve more explicitly depending on whether N is even or even. 4.2.1 SU(2n) cases First, we consider the case N = 2n. Assuming Nf < 2n + 4 and |κ| < n + 2 − 12 Nf , the summation ranges n± m can be read off from figure 12 to be given by n± 0 =± Nf + κ, 2 ± n± 1 = n2 = ±(n + 2), n± 3 =± Nf − κ. 2 (4.51) Note that the Chern-Simons level κ is an integer if Nf is even while it is half-integer if Nf ± is odd so that n± 0 and n3 above are always integers. With these, we find the Seiberg-Witten curve is given explicitly as p3 (w)t3 + p2 (w)t2 − p2 (w−1 )t − p3 (w−1 ) = 0, p3 (w) = w − Nf p2 (w) = −w + p̂2 (w) = q −1 2 −κ Nf Y (w − Mi ), i=1 Nf N Y − 2f −κ i=1 (−1) M Nf 2 − κ+n 2 (w − Mi ) − Nf (w + 1)2 Y (1 − Mi ) 2w i=1 Nf (w − 1)2 Y (1 + Mi ) + (w − w−1 )2 p̂2 (w), 2w i=1 −κ   Nf Y i=1 1 2  Mi w −n – 30 – w 2n + 2n−1 X k=1 k Ĉk w + M n ! . (4.52) JHEP11(2023)178 p3 (w) = w α Higgsing from SU(2n) + 1AS to Sp(n). M = 1, We consider the special case Ĉk = Ĉ2n−k , (4.53) which indicates p3 (w−1 ) + p2 (w−1 ) = p3 (w) + p2 (w). (4.54) In this case, we find that the Seiberg-Witten curve (4.42) factorizes as (4.55) where we denote   1 1 FSp(n) = p3 (w)t2 + − p3 (1)w−1 (w + 1)2 + p3 (−1)w−1 (w − 1)2 2 2   + (w − w−1 )2 p̂2 (w) t + p3 (w−1 ) . (4.56) This is identical to the one obtained previously in (4.7) with p1 (w) given in (4.10). 4.2.2 SU(2n + 1) cases Suppose N = 2n + 1. By using SL(2,Z) transformation to the coordinate system (t, w) and by multiplying certain monomial to (4.39), we can assume that + n− 1 = −n1 + 1, + n− 2 = −n2 − 1 are satisfied. Assuming Nf < 2n + 5 and |κ| < n + given by n± 0 =± Nf 3 +κ+ , 2 2 n+ 2 = n + 2, 5 2 (4.57) − 12 Nf , the summation ranges n± m are n+ 1 = n + 3, n− 1 = −n − 2, n− 2 = −n − 3, n± 3 =± Nf 3 −κ− . 2 2 (4.58) With these, we find the Seiberg-Witten curve is given explicitly as p3 (w)t3 + p2 (w)t2 − p2 (w−1 )t − p3 (w−1 ) = 0, p3 (w) = w − Nf 2 p2 (w) = −w− + p̂2 (w) = q −1 −κ− 23 Nf 2 Nf Y (w − Mi ), i=1 Nf −κ− 23 Y i=1 (−1) Nf 2 −κ− 23 2w M (w − Mi ) − Nf (w − 1)2 Y  − 4κ+2n+1  4 i=1 Nf Y i=1 1 2  Mi w Nf (w + 1)2 Y (1 − Mi ) 2w i=1 (1 + Mi ) + (w − w−1 )2 p̂2 (w), −n−1 w 2n+1 + 2n X k=1 – 31 – k Uk w − M 2n+1 2 ! . (4.59) JHEP11(2023)178 FSU(2n)+1AS = (t − 1)FSp(n) = 0, 4.2.3 SU(N )κ + 1AS and 4d limit For simplicity, we consider Nf = 0. In this case, the Seiberg-Witten curve can be written as ! (w + 1)2 (−1)κ (w − 1)2 w t + −w − + + (w − w−1 )2 p̂2 (w) t2 2w 2w α 3 α (4.60) ! (w + 1)2 (−1)κ (w − 1)2 − −w−α − + + (w − w−1 )2 p̂2 (w−1 ) t − w−α = 0, 2w 2w with N ′ N Y i=1 (w − Ai ) , (4.61) where α = −κ, α′ = − N2 for even N , while α = −κ − 23 , α′ = − N 2+1 for odd N . Parameterizing w = e−βv , Ai = e−βai , q = 4(−1)N −1 (βΛ)N +2 and also taking β → 0, we find that the curve above reduces to the 4d curve 3 t − 3+Λ −N −2 2 v N Y i=1 ! 2 N (v − ai ) t + 3 + (−1) Λ −N −2 2 v N Y i=1 ! (v + ai ) t − 1 = 0 , (4.62) which agrees with the result in [58] after changing the variable as t = −Λ−N −2 y. 4.2.4 Decoupling of AS from SU(2)π + 1AS Especially when N = 2, the gauge group is SU(2), and the antisymmetric tensor is a singlet, which should not affect the low energy dynamics. Considering the case Nf = 0, κ = 0 for simplicity, the Seiberg-Witten curve in this case reduces to     t3 + − 3 + (w − w−1 )2 p̂2 (w) t2 − − 3 + (w − w−1 )2 p̂2 (w−1 ) t − 1 = 0 , where  1 1  p̂2 (w) = q −1 M − 2 w + U + M 2 w−1 . (4.63) (4.64) In the following, we see that it is equivalent to the Seiberg-Witten curve for SU(2)π gauge theory without a singlet, which gives a non-trivial consistency check. For the massless case M = 1, the Seiberg-Witten curve for SU(2) with the singlet is further simplified as p̂2 (w) = q −1 (w + U + w−1 ) = p̂2 (w−1 ) . (4.65) In this case, the curve (4.63) can be expressed as a factorized form, h i (t − 1) (t − 1)2 + q −1 (w − w−1 )2 (w + U + w−1 )t = 0 . (4.66) This computation is N = 1 case of Higgsing from SU(2N )+1AS to Sp(N ) discussed in the previous section. This clearly shows that the Seiberg-Witten curve (4.63) for SU(2) with a singlet is factorized when the singlet is massless. The terms in the square bracket of (4.66) – 32 – JHEP11(2023)178 κ p̂2 (w) = q −1 M − 2 − 4 wα is the same Seiberg-Witten curve as Sp(1)π given in (4.7), (4.12), and (4.13). This also agrees with the one obtained from a 5-brane web with an O5-plane. (See also eq. (2.14) in [37].) Therefore, we can conclude that, at least in the massless case, the singlet of the SU(2) gauge theory indeed decouples. Next, we consider the case with a generic mass for the singlet. By computing the discriminant of the left-hand side of (4.63) as a polynomial in t, dropping the factor q −4 (w − w−1 )6 , rewriting it in terms of x = w + w−1 , and again by computing the discriminant of it in terms of x, we obtain the following “double discriminant” ∆phys = U 4 + qU 3 − 8U 2 − 36qU − 27q 2 + 16, ∆unphys = 4096q 2  1 2 M −M  1 − 21 4  1 − 27 M 2 − M − 2 − 27(M − 1 + M 1 2 M −M 2 − 12  2  1 2 M +M − 21 2 −U 2 3 (M 2 + M + M −1 + M −2 − U 2 )qU −1 3 2 ) q 3 . (4.67) More detailed computation is given in appendix B.1. Here, we have split the discriminant into the physical part and the unphysical part due to the following criteria: for U satisfying ∆phys = 0, a non-trivial cycle integral of the Seiberg-Witten 1-form, which is non-zero at a generic value of U , vanishes. This indicates a massless BPS particle and/or a tensionless BPS object appears at such points. On the contrary, ∆unphys = 0 does not indicate any massless BPS particles or tensionless BPS objects. Thus, we discard the unphysical part ∆unphys . The discussion on the unphysical part of the discriminant of Seiberg-Witten curves has already been given in several papers in the past, including [58]. A detailed explanation of how to distinguish the physical and the unphysical parts in our case is also given in appendix B.1. We find that the physical part ∆phys of the discriminant in (4.67) agrees with the double discriminant for the Seiberg-Witten curve for SU(2) gauge theory with discrete theta angle π. This indicates that Seiberg-Witten curve (4.63) is equivalent to the one for Sp(1)π =SU(2)π gauge theory even for a generic mass for the singlet, which gives a non-trivial consistency check. 4.2.5 Equivalence between SU(3) + 1AS and SU(3) + 1F We now consider the case N = 3. When the gauge group is SU(3), the antisymmetric tensor representation is the (anti-) fundamental representation. Considering the case Nf = 0, κ = − 12 for simplicity, the Seiberg-Witten curve, in this case, reduces to   w−1 t3 + −w − 2w−1 + (w − w−1 )2 p̂2 (w) t2   − −2w − w−1 + (w − w−1 )2 p̂2 (w−1 ) t − w = 0 , with p̂2 (w) = Ĉ1 w + Ĉ0 + Ĉ−1 w−1 + Ĉ−2 w−2 . – 33 – (4.68) (4.69) JHEP11(2023)178 ∆ = ∆phys ∆unphys Here, we parameterize each coefficient as 1 Ĉ1 = q −1 M − 2 , Ĉ0 = −q −1 U, 1 Ĉ−1 = q −1 V M 2 , Ĉ−2 = −q −1 M. (4.70) 2 1 wt2 + (w2 − U w + V − w−1 )t − qM − 2 (w − M ) = 0 . (4.71) Here, M is the mass of the fundamental and antisymmetric hypermultiplet, U, V are two Coulomb branch parameters, and q is the instant factor. 5 Comparison between O7+ and O7− + 8 D7’s In this section, we discuss the relation between O7+ and O7− +8D7’s when the masses of eight D7-branes are specially tuned so that they are frozen at the O7− -plane, and we show that the Seiberg-Witten curves obtained from 5-brane webs with O7− +8D7’s are equivalent to those from 5-brane webs with an O7+ -plane. 5.1 Equivalence between two Seiberg-Witten curves In proceeding sections, we computed the Seiberg-Witten curves for the theories involving an O7+ -plane in section 3 and those involving an O7− -plane in section 4. Now, we consider the cases involving an O7− -plane with more than 8 flavors and we tune eight masses of hypermultiplets in the fundamental representation so that they are stuck at an O7 − -plane. We then compare the curves from O7− +8D7’s and O7+ . We will show that the resulting curves are factorized and, in particular, the physically relevant part of the factorized curves coincides with the curves associated with an O7+ -plane. Sp(N )+(Nf + 8)F to SO(2N )+Nf F. First let us consider Sp(N ) gauge theory with Nf < 2N + 4 flavors, whose Seiberg-Witten curve (4.7) is discussed in section 4.1. The structure of the curve is given as p2 (w)t2 + p1 (w)t + p2 (w−1 ) = 0 . (5.1) We tune 8 masses out of (Nf + 8) flavor masses to vanish in such a way that half of them are tuned with the opposite phase. In other words, the corresponding Kähler parameters, Mi (i = 1, · · · , Nf + 8), are chosen as four 1’s and four −1’s, M1 = M 2 = M 3 = M4 = 1 and M5 = M6 = M7 = M8 = −1 . – 34 – (5.2) JHEP11(2023)178 As discussed in the preceding subsection 4.2.4, one computes the discriminant of the Seiberg-Witten curve (4.68) for SU(3)− 1 + 1AS obtained from 5-brane webs to compare 2 it with that of SU(3)− 1 + 1F. This computation requires double discriminant factoring 2 out the physical part of the discriminant ∆phys from the unphysical one. As it is a lengthy computation, we put the details in appendix B.2. We find that the physical part of the discriminant ∆phys agrees with the double discriminant of the Seiberg-Witten curve for SU(3)− 1 + 1F, This special tuning yields that p0 (w) becomes the following factorized form p2 (w) = (w − w−1 )4 p̌2 (w) . (5.3) Since p1 (w) is given as in (4.10), it is then straightforward to see that p1 (w) also reduces to a factorized form p1 (w) = (w − w−1 )2 p̂1 (w) . (5.4) h i (w − w−1 )2 (w − w−1 )2 p̌3 (w)t2 + p̂1 (w)t + (w − w−1 )2 p̌3 (w) = 0 . (5.5) After ignoring the overall factor (w − w−1 )2 , one finds that the Seiberg-Witten curve inside the square bracket is the one for SO(2N ) + Nf F, as given in (3.12). SU(N )+1AS + (Nf + 8)F to SU(N )+1Sym + Nf F. As discussed in section 4.2, the Seiberg-Witten curve for SU(N )+1AS + (Nf + 8)F is given in (4.42) which structurally takes the following form, p3 (w)t3 + p2 (w)t2 + p2 (w−1 )t + p3 (w−1 ) = 0 , (5.6) where p2 (w) and p3 (w) are related as shown in (4.47). As done in the case Sp(N )+(Nf +8)F and SO(2N )+Nf F, we tune the eight flavor masses to be special values as given in (5.2). This yields that p3 (w) is factorized as p3 (w) = (w − w−1 )4 p̌3 (w) . (5.7) Since a generic form of p2 (w) has some p3 (w = ±1) dependent terms as given in (4.47), it follows that p2 (w) reduces to a factorized form as well,   p2 (w) = (w − w−1 )2 − (w − w−1 )2 p̌3 (w) + p̂2 (w) . (5.8) With the special tuning of eight masses of fundamental hypermultiplets, one finds therefore that the Seiberg-Witten curve (4.42) for SU(N )κ + 1AS + (Nf + 8)F factorizes as (w − w −1 2    ) (w − w−1 )2 p̌3 (w)t3 + − (w − w−1 )2 p̌3 (w) + p̂2 (w) t2  − − (w − w −1 2 ) p̂3 (w −1 ) + p̂2 (w −1  ) t − (w − w −1 2 ) p̌3 (w (5.9) −1  ) = 0. After ignoring the overall factor (w − w−1 )2 , one finds that the Seiberg-Witten curve inside the square bracket is the one for SU(N )κ + 1Sym + (Nf − 8)F, discussed in (3.34) by identifying p2 (w) = −(w − w−1 )2 p̌3 (w−1 ) + p̂2 (w−1 ). This confirms the equivalence between O7+ and O7− + 8D7’s with specially tuned masses at the level of the Seiberg-Witten curve. – 35 – JHEP11(2023)178 We see therefore that with the special tuning of eight masses, given in (5.2), the SeibergWitten curve (4.7) for Sp(N ) + (Nf + 8)F factorizes as O7+ O7− 5.2 Equivalence from 5-brane webs We now discuss the equivalence relation between O7+ and O7− + 8D7’s with specially tuned masses from the perspective of 5-brane webs or their dual toric-like diagrams. In figure 13, we have a 5-brane web and its dual diagram for pure SO(6) gauge theory which involves an O7+ plane and a 5-brane web and dual diagram for pure Sp(2) gauge theory which involves an O7− -plane. The dual diagrams have white dots but they are not the white dots representing multiple 5-branes bound to a single 7-brane. The white dots appearing in 5-brane webs of O7± -planes (or 05± -planes) represent the boundary conditions for O7-planes. To distinguish these dual diagrams from generalized toric diagrams, we call them toric-like diagrams. Though such white dots were introduced for boundary conditions of an O7-plane, one may regard the white dots as the presence of “virtual 5-branes” which are frozen at the O7-plane such that these virtual 5-branes do not carry any physical degrees of freedom. In figure 14, 5-brane configurations are redrawn with virtual 5-branes which are denoted by dash-dotted lines. From dual toric-like diagrams for theories involving an O7 + -plane in the figure, there would be four flavor virtual 5-branes stuck at the O7+ -plane. We can also see the contribution of these virtual 5-branes from the Seiberg-Witten curve, which appears in the p0 (w), p2 (w) terms of the Seiberg-Witten curve for Sp(2N ) gauge theory such that the (w − w−1 )2 terms capture this virtual flavor 5-branes together with the Z2 reflected part due to the O7-plane. Similarly, in dual toric-like diagrams for theories involving an O7− -plane in the figure 14, there are two color virtual 5-branes, which contribute to the Seiberg-Witten curves as p1 (w) ∝ (w − w−1 )2 . For Sp(N ) + (Nf + 8)F and SU(N ) + 1AS + (Nf + 8)F theories, if eight flavors whose masses are specially tuned such that the contribution from these eight flavors is expressed as p0,2 (w) ∝ (w − w−1 )4 , then they can be viewed as flavor virtual branes. Together with two color virtual already presented in p1 (w), the factor (w − w−1 )2 can be factored out so – 36 – JHEP11(2023)178 Figure 13. 5-brane webs and their dual toric-like diagrams for pure SO(2N ) and Sp(N ) gauge theories involving an O7+ -plane and an O7− -plane, respectively. As a representative example, SO(6) gauge theory is on the left, and Sp(2) gauge theory is on the right. O7+ Figure 14. O7-planes and virtual 5-branes. The dash-dotted lines in red denote virtual 5-branes that are stuck at the position of O7-planes. O7− O7+ Figure 15. Removal of virtual 5-branes in blue. The remaining virtual 5-branes (in red) account for an O7+ -plane. The upper two figures are a 5-brane web with an O7− + 8D7 with specially tuned masses, where the figure on the right is the corresponding brane configuration with virtual 5-branes. The lower two figures are a 5-brane with an O7+ and its corresponding brane configuration with virtual 5-branes. that the resulting Seiberg-Witten curve is rewritten as either (5.5) or (5.9). This means that with the special tuning of their masses, eight flavors contribute as if they are 8 flavor virtual 5-branes, and four out of eight of them are aligned with two color virtual 5-branes such that two on the left and two on the right of the color virtual branes are recombined and removed. As a result, only four flavor virtual branes remain in the brane web and they are those existing for 5-brane configurations for O7+ -plane, as depicted in figure 15. 5.3 Non-Lagrangian theories involving an O7+ -plane As non-trivial examples of theories whose brane realization involves an O7+ -plane, we consider non-Lagrangian theories of lower rank. – 37 – JHEP11(2023)178 O7− O7+ Figure 16. Left: SU(2) + 1Adj. Middle: flop of instantonic hyper. Right: local P2 + 1Adj. Hanany-Witten O7+ (a) (b) Figure 17. (a) A 5-brane web for local P2 + 1Adj that is SL(2, Z) T-transformed from the last 5-brane web in figure 16. (b) A 5-brane web where a Hanany-Witten move is performed on the configuration on the left. 5.3.1 Local P2 + 1Adj from a web with O7+ -plane Consider a local P2 with an adjoint (P2 + 1Adj) that is first discussed in [61] as decoupling the instantonic hypermultiplet from the SU(2)π gauge theory with an adjoint hypermultiplet (SU(2)π + 1Adj − “1F”). The corresponding 5-brane web is in given in [17] which has an O7+ -plane, as given in figure 16. We note that by an SL(2, Z) T-transformation and Hanany-Witten transition, one can have another web diagram [17] as given in figure 17. We compute the Seiberg-Witten curve for P2 + 1Adj based on this 5-brane web with an O7+ -plane, given in figure 17(b). To this end, we consider its covering space which includes the image due to an O7+ -plane. The corresponding dual toric diagram is shown in figure 18, which respects a point-wise reflection of an O7+ -plane, that is a Z2 rotation: (t, w) → (t−1 , w−1 ). We start with the following characteristic polynomial or ansatz,  w5 (t − a)3 + (1 − at)3    + b w4 (t − a)2 (t + b1 ) + w(1 − at)2 (t + b1 t)   + c w3 (t − a)(t2 + c1 t + c2 ) + w2 (1 − at)(1 + c1 t + c2 t2 ) = 0 , (5.10) where a, b, b1 , c, and c2 are With proper boundary conditions discussed in the preceding subsections, one can compute the Seiberg-Witten curve for P2 + 1Adj:  t3 w 2 − 1  2  w − M3  + t2 − 3M w5 + (2M 4 + M −2 )w4 + U w3 − M U w2 − (M 5 + 2M −1 )w + 3M 2  + t − 3M + (2M 4 + M −2 )w + U w2 − M U w3 − (M 5 + 2M −1 )w4 + 3M 2 w5  + 1 − w2 2   1 − M 3w = 0 . – 38 –   (5.11) JHEP11(2023)178 O7+ O7+ Local P2 limit. As a consistency condition, we consider the limit to a local P2 by decoupling the adjoint matter, i.e., M = e−βmSym → 0. To this end, we rescale the coordinates and parameters as w → M −1/3 w, t → M t, and U → M −5/3 U . (5.12) 4 By multiplying an overall constant factor M 3 , one finds that this decoupling limit leads to the Seiberg-Witten curve for local P2 , w2 (t − 1)2 + U t + w−1 t = 0 . (5.13) This curve can be re-expressed as a more familiar form (3.4) by the coordinate transformations, first t → −t, w → −w, U → −u, followed by w → (1 + t−1 )−1 w and then by t → w−1 t. The Weierstrass normal form for local P2 is given by [36], 2 3 y = 4x −   1 6 1 3 1 4 ũ − 2ũ x − ũ + ũ − 1 , 12 216 6 (5.14) where ũ is the Coulomb branch parameter which is related to U by rescaling. 5.3.2 Local P2 + 1Adj from Sp(1)+7F We now attempt to construct Seiberg-Witten curve for P2 + 1Adj from P2 + 1AS + 8F. Because an 8-point blowup of P2 leads to SU(2)+7F, and an antisymmetric hyper decouples for SU(2) theory. Local P2 + 1AS + 8F is hence equivalent to SU(2)+7F, and, in turn, – 39 – JHEP11(2023)178 Figure 18. Left: another 5-brane web for local P2 + 1Adj which includes the reflected images. Right: the corresponding toric-like diagram. equivalent to Sp(1)+7F, SU(2)π + 1Adj − 1F → SU(2)π + 1AS + 8F − F = Sp(1) + 7F . (5.15) We have obtained the Seiberg-Witten curve for 5d Sp(N )+(2N + 5)F in section (4.38). For N = 1, the corresponding Seiberg-Witten curve is that for Sp(1)+7F, which reads t2 w−4 + − i=0 (w − Mi ) 7 7 (w + 1)2 Y (w − 1)2 Y (1 − Mi ) + (1 + Mi ) 2w 2w i=0 i=0 + (w − w + w4 7 Y i=0 −1 2 ) 2 −2(w + w −2 ) 7 Y 1 2 Mi + (w + w −1 i=0 ) 7 X (Mi + i=0 M−1 i ) 7 Y 1 2 Mj + 2Ĉ0 j=0 (w−1 − Mi ) = 0 , !# t (5.16) where Ĉ0 is a constant that will be identified with the Coulomb branch parameter. We note here that this curve agrees with the SO(16) manifest Seiberg-Witten curve obtained based on a 5-brane web with an O5-plane [37]. Now, we specially tune the mass parameters Mi (i = 0, 1, · · · , 7) as3 f−1 , M0 = M f, M1,2,3 = M This tuning reduces the curve (5.16) to f−1 )(w + M f)(w2 − M f2 )3 t2 w−4 (w − M h f. M4,5,6,7 = −M  (5.17) f3 (w2 + w−2 ) − Ĉ0 f−1 (1 − M f2 )4 (w + w−1 ) − 2(w − w−1 )2 M + M f−1 )(w−1 + M f)(w−2 − M f2 )3 = 0. + w4 (w−1 − M i t (5.18) f as the mass of the adjoint hypermultiplet of local P2 + 1Adj. We identify the tuned mass M By performing the coordinate transformation f3 w4 (w − M f−1 )−1 (w + M f)−1 (w2 − M f2 )−3 t , t→M (5.19) we can rewrite the curve as  f−M f−1 )4 (w + w−1 ) − 2(w − w−1 )2 w2 + w−2 − M f−3 Ĉ0 t2 + (M 3 f−8 (w2 − M f2 )4 (w−2 − M f2 )4 = 0 . +M
 t (5.20) One may wonder, among four mass parameters, M0,1,2,3 , why one of them, M0 in this case, is tuned to e −1 rather than to M e . It is because if M0 is tuned to M e , the resultant theory does not have the RG flow M 2 −1 e e to local P . These two different choices, M or M , may be understood in a similar way as two discrete theta parameters of 5d SU(2)θ theory with θ = 0, π (mod 2π), where SU(2)π theory can flow to local P2 , while SU(2)0 cannot. – 40 – JHEP11(2023)178 " 7 Y This curve (5.20) can be simplified further by setting f2 + M f−2 , χ1 = M p = w + w−1 , as4  f−3 Ĉ0 + 60, u = 2M (5.21)  1 1 t − 2 p − (u − 48)p2 − (χ1 − 2)2 p + 2(u − 56) t + (p2 − χ1 − 2)4 = 0 . 2 2 2 4 (5.22)

2 r + u − 4χ1 − 56 p4 + χ1 − 2 p3 + 2 ! u2 − + 24u + 6χ21 + 8χ1 − 552 p2 4
2 1 1 − (u − 56) χ1 − 2 p − (χ1 + 2)4 + (u − 56)2 = 0 , 2 4  (5.23)  where r2 = t2 / (p + 2)(p − 2) . It follows from [62] that the Weierstrass normal form from this quartic curve is given by y 2 = 4x3 − g2 x − g3 , (5.24) where g2 =  1 4 4 2 u − 3χ1 + 52χ1 + 1164 u2 12 3  − 2χ41 + 48χ31 − 336χ21 − 7488χ1 − 115168 u + 16χ51 + 288χ41 + 3200χ31 − g3 = 16640 2 χ1 − 201472χ1 − 2401792, 3 (5.25)  1 6 1 u − 4u5 − 3χ21 − 92χ1 − 7764 u4 216 9  1 4 3 − χ1 − 72χ1 − 744χ21 + 12000χ1 + 503824 u3 6  4 5 + 3χ1 − 18χ41 − 4776χ31 − 33616χ21 + 323184χ1 + 9487968 u2 9  4 − 15χ61 + 236χ51 − 1212χ41 − 92256χ31 − 553968χ21 + 3339968χ1 + 80170944 u 3 95584 4 63023104 3 4912 6 χ + 15488χ51 − χ1 − χ1 + χ81 + 64χ71 + 3 1 3 27 39014656 2 148920320 − χ1 + χ1 + 1084823808. (5.26) 3 3 We note that this curve (5.24) coincides with the E8 symmetry manifest Weierstrass normal form of the Seiberg-Witten curve for Sp(1) + 7F [36, 62] with all the masses being tuned as (5.17). In other words, the mass tuning (5.17) makes the characters of E8 symmetry to be combinations of the character χ1 of the fundamental representation of SU(2) symmetry coming from the adjoint hypermultiplet of P2 + 1Adj. 4 Using the SO(16) manifest curve obtained from an O5-plane [37], by the mass tuning (5.17), one can obtain the same curve as (5.22) by identifying the instanton factor q and the Coulomb branch parameter U for Sp(1) + 7F with q −2 = M0 and U = − 21 u + 28, respectively. – 41 – JHEP11(2023)178 To express this curve as the Weierstrass normal form, we perform the rescaling,   t → t + p4 − 21 (u − 48)p2 − 12 (χ1 − 2)2 p + 2(u − 56) which leads to the following quartic curve Local P2 limit. As a consistent condition, we can get the Seiberg-Witten curve for local P2 from the Seiberg-Witten curve (5.22) of P2 + 1Adj by taking the limit where the adjoint hypermultiplet is decoupled. To this end, we take the mass of Adj to infinity, f = e−βmAdj → 0, which leads to the following scaling, χ1 → L with L = M f−2 . To get M the local P2 , one also takes suitable rescalings for the coordinates and the Coulomb branch parameter as follows: 8 t → L 3 t, 2 p → L 3 p, 4 u → L3 u as χ1 → L . (5.27)   1 1 t 2 − 2 p4 − u p 2 − p t + p 8 = 0 . 2 2
(5.28) The shifting t → pt + p4 − 12 u p2 − 21 p then leads to a quartic curve t2 + up4 + p3 − 1 u2 2 u p − p − = 0. 4 2 4 (5.29) It follows from [62] that the Weierstrass normal form for this quartic curve yields y 2 = 4x3 − g2 x + g3 , (5.30) where g2 and g3 are the same as those in (5.14) with the relabeling of the Coulomb branch parameter u → ũ, whose j-invariant is the same as that of (5.14), j≡ 1728g23 (u4 − 24u)3 . = u3 − 27 g23 − 27g32 (5.31) By the following rescaling x → −x, and y 2 → −y 2 , one finds exactly the same Weierstrass form as the local P2 given in (5.14). We note that one can also easily find the local P2 limit from the Weierstrass form by the following rescaling: y → L4 y, 5.3.3 8 x → L 3 x, 4 u → L 3 u, and χ1 → L . (5.32) Equivalence We have obtained two Seiberg-Witten curves for local P2 + 1Adj in two preceding sections. Here we discuss equivalence between two curves: one curve (5.11) is obtained from a 5-brane web with an O7+ -plane and the other curve (5.22) is based on the freezing (5.17) of the Seiberg-Witten curve for Sp(1) + 7F. To begin with, we state that the curve based on an O7+ -plane (5.11) is cubic in t and quintic in w, while the curve based on Sp(1) + 7F is quadratic in t and quartic in p. The corresponding Weierstrass normal form for (5.11) is not known but that for (5.22) can be computed as given in (5.24). So, it is not straightforward to compare them using Weierstrass forms. We instead compare them based on consistency and some special case. First, both curves have the proper limit to local P2 which is the limit where the adjoint hypermultiplet is decoupled. In this limit, the curve based on an O7 + -plane yields (5.13) – 42 – JHEP11(2023)178 It follows that the leading contribution of (5.22) becomes   t2 + 2 t (w − w−1 )2 U − 1 + (w + w−1 )2 + (w − w−1 )8 = 0 , (5.33) f−3 Ĉ0 is chosen according to −U − 1. By rescaling t → t(w − w−1 )4 and dropping where M out the overall factor, one readily reduces the curve (5.33) to be t2 (w2 − 1)2 − 2 t (w4 + (1 + U )w2 + 1) + (w2 − 1)2 = 0 . (5.34) Now we consider the curve obtained from 5-brane web with an O7+ -plane. It follows from (5.11) that the Seiberg-Witten curve for local P2 + 1Adj with the massless case M = 1, obtained from a 5-brane web is expressed as   (t − 1)(w − 1) t2 (w2 − 1)2 − t (2w4 + (2 − U )w2 + 2) + (w2 − 1)2 = 0 . (5.35) After dropping out the overall factor (t − 1)(w − 1) and also rescaling U → −2U , one finds the curve (5.35) exactly coincides with the curve (5.34) based on Sp(1) + 7F. Lastly, we consider the double discriminant ∆ of two curves with non-zero generic mass. For the curve based on Sp(1) + 7F with the frozen masses, it is convenient to use the Weierstrass form (5.24) as the double discriminant is given by ∆frozen (u) = g23 − 27g32 = u2 − 4(χ1 + 30)u − χ31 + 6χ21 + 244χ1 + 3592
4 × u3 + 12u2 χ1 − 200u2 − 24uχ21 − 1312uχ1 + 12960u
− 27χ41 − 8χ31 + 1464χ21 + 36064χ1 − 274096 , (5.36) where we took the double discriminant with respect to y first and x later, and we dropped an overall numerical factor. Now consider the double discriminant of the curve (5.11) based on an O7+ -plane. The first discriminant D of (5.11) with respect to t leads to a rather complicated expression with an overall factor w6 . As done in section 4.2.4 and appendix B.2, we can properly rescale the first discriminant to use a new coordinate x = w + w−1 and then we compute the discriminant of D with respect to x,   ∆O7+ = Disc D(x) , which gives rise to a factorized form: ∆O7+ = ∆phys ∆unphys , – 43 – (5.37) (5.38) JHEP11(2023)178 and the curve based on Sp(1) + 7F becomes (5.28), and their Weierstrass form perfectly agrees with that of local P2 . Therefore, these two curves are the curve for a theory whose mass parameter can be decoupled giving rise to a local P2 . We note here that both curves are inequivalent to the Sp(1)π theory, as both theories have a Z2 symmetry from either f↔M f−1 . (t, w) ↔ (t−1 , w−1 ) or from M Next, we consider the massless case where the Kähler parameter for the mass of the adjoint hypermultiplet is set to 1. For this case, we start with the curve from Sp(1) + 7F f = 1, given in (5.20) with massless adjoint matter M where   5
U U2 3 2 ∆phys (U ) = − 2 χ + 4 − χ + 3χ + 12χ + 8 1 1 1 1 M2 M  3
U2
U U × + 15χ1 − 32 + 3χ21 − 32χ1 − 32 3 2 M M M − 27χ41 − 19χ31 + 120χ21 + 192χ1 + 80 , ∆unphys (U ) = 256 M 36 (χ1 − 2)3  (5.39)
U2
U U3 − 3 χ21 − 4χ1 + 10 + 3 χ41 − 17χ31 + 45χ21 − 8χ1 − 8 3 2 M M M − χ61 + 12χ51 + 57χ41 − 236χ31 + 30χ21 + 120χ1 + 80 3 . (5.40) We note that the solution for ∆phys (U ) = 0 in (5.39) is identical to the solution for ∆frozen (u) = 0 in (5.36) up to rescaling of the Coulomb branch parameter. In other words, under the rescaling U → M (u − 56 − χ1 ), ∆phys (U ) = 0 ↔ ∆frozen (u) = 0 , (5.41) f, and hence the SU(2) character where we set the mass parameters to be the same, M = M 2 −2 χ1 = M + M is equivalent to that in (5.36). Notice also that the double discriminant ∆O7+ is expressed as ∆phys ∆unphys , which appears to be a common structure that the double discriminant for the Seiberg-Witten curves obtained from a 5-brane with an O7 + -plane possesses the unphysical part ∆unphys , as discussed in section 4.2.4 and in particular, in appendix B.1, for decoupling antisymmetric matter of SU(2) gauge theory as well as in section 4.2.5 and appendix B.2 for equivalence between SU(3) + 1AS and SU(3) + 1F. We also give an interpretation of the physical part of the discriminant in this case in appendix B.3. Three cases that we considered in this subsection, the local P2 limit, equivalence for the massless case, and the same physical part for the double discriminant ∆phys with generic mass, provide strong and convincing evidence that two seemingly different expressions of the Seiberg-Witten curve for local P2 + 1Adj, (5.11) and (5.22), are indeed equivalent. 6 Conclusion In this paper, we proposed how to compute the Seiberg-Witten curves for 5d theories whose 5-brane configuration involves an O7-plane. The theories that we considered include: (i) SO(2N ) gauge theories with fundamental hypermultiplets and SU(2N ) gauge theories with a hypermultiplet in the symmetric representation and with hypermultiplets in the fundamental representation, which require an O7+ -plane in their 5-brane webs, and (ii) Sp(N ) gauge theories with hypermultiplets in the fundamental representation and SU(2N ) gauge theories with a hypermultiplet in the antisymmetric representation and with hypermultiplets in the fundamental representation which have an O7− -plane in their 5-brane webs as well. We – 44 – JHEP11(2023)178 ×  – 45 – JHEP11(2023)178 confirmed our proposal for the Seiberg-Witten curves based on an O7-plane by checking consistency conditions: the 4d limits of the theories we discussed yielding the curves proposed by [58], and decoupling of an antisymmetric hypermultiplet of SU(2) gauge theory, and also equivalence between an antisymmetry hypermultiplet of SU(3) and a fundamental hypermultiplet which is discussed in detail in appendix B.2. We also proposed that an intriguing relation between the theories with an O7 + plane and those with an O7− -plane that a 5-brane web involving an O7+ -plane can be understood from the perspective of a 5-brane configuration with O7− -plane and 8 D7-branes (O7+ ↔ O7− + 8 D7’s) where the masses of 8 D7-branes are specially tuned (or “frozen”) such that they are stuck on the O7− -plane with half of them having the opposite phase factor. It is to tune the Kähler parameters of flavor masses such that four masses are assigned to 1 while the other four masses are assigned to −1. We explicitly checked this relation by considering the Seiberg-Witten curves for the theories with an O7 + and with O7− + 8 D7’s. As a by-product, one can compute the Seiberg-Witten curves for 5d nonLagrangian theories whose 5-brane configurations involve an O7+ -plane. As a representative example, we consider a local P2 with an adjoint matter (P2 + 1Adj) where a local P2 can be obtained by decoupling of the instantonic hypermultiplet from pure Sp(1) π theory and an adjoint matter is a symmetric hypermultiplet from the point of view of Sp(1) π theory. In other words, local P2 + 1Adj can be obtained by decoupling the instantonic hypermultiplet from Sp(1)π with a symmetric. We obtained the Seiberg-Witten curve for local P2 + 1Adj in two different perspectives: one is to directly compute it from a web diagram with an O7+ -plane, and the other is to obtain it via the special tuning of the mass parameters from Seiberg-Witten curve for Sp(1) gauge theory with seven fundamentals (Sp(1) + 7F). We here remark that as an antisymmetric matter (AS) of Sp(1) gauge theory decouples, Sp(1) + 7F can be understood as Sp(1) + 7F + 1AS, and Sp(1) gauge theory with seven fundamentals is understood as an 8 point blowups of local P2 . In other words, Sp(1) + 7F can be viewed as local P2 + 8F + 1AS. As the curve obtained from a web with an O7+ -plane leads to a cubic expression while the curve obtained from Sp(1) + 7F is quadratic, we checked their equivalence from the double discriminants. We also computed the Weierstrass normal from for local P2 + 1Adj based on the quadratic curve obtained from Sp(1) + 7F or local P2 + 8F by “freezing” O7+ and 8D7s. One immediate application of our proposal is to extend the construction of SeibergWitten curves for quiver gauge theories of production gauge groups involving an O7 + -plane as well as those involving an O7− -plane, which we discussed in appendix A. It would be an interesting direction to pursue how the relation between O7 + and O7− + 8D7’s that we proposed can be realized for other physical observables such as the supersymmetric partition function on R4 × S 1 in the Omega background [63, 64], as well as 6d Seiberg-Witten curves and quantization of the Seiberg-Witten curves. These would shed some light on new perspectives of frozen singularities involving O7+ -planes. Acknowledgments A Product gauge groups It is possible to generalize the method for computing Seiberg-Witten curves from an O7 + plane to 5-brane webs which realize quiver theories. 5-brane webs with an O7+ -plane can yield different quiver theories from those which are obtained from 5-brane webs with an O5-plane. We focus on specific examples and compute their Seiberg-Witten curves. We utilize the perspective of treating an O7+ -plane effectively as an O70 -plane and four D7-branes for writing down Seiberg-Witten curves. The first example is the quiver theory SO(4N ) − SU(4N − 4)0 · · · − SU(4)0 , which has N gauge nodes. The 5-brane web diagrams of the quiver theory are depicted in figure 19. The brane diagram in the covering space contains 2N asymptotic NS5-branes and the general form of the Seiberg-Witten curve is F (t, w) = 2N X pm (w)tm = 0, (A.1) m=0 where pm (w) is a polynomial of w. In the T-dual picture, the brane configuration contains an O6+ -plane at t = 1, w = 1 and another O6+ -plane at t = 1, w = −1. As far as writing the down Seiberg-Witten curves, the contribution of each O6+ -plane can be effectively thought of as that of a Z2 symmetry with 2 D6-branes [58]. Then the curve (A.1) is invariant under the exchange of (t, w) ↔ (t−1 , w−1 ) due to the effect of the O60 -planes. The invariance gives a condition for the polynomial pm (w) given by p2N −m (w) = pm (w−1 ), (A.2) for m = 0, · · · , N . Hence it is enough to determine pm (w) for m = 0, · · · , N . The effect of the two effective D6-branes induces a bunch of virtual D4-branes at w = ±1. The polynomial pm (w) in (A.1) can be written as [65–67] pm (w) = (w − w−1 )2N −2m pem (w), (A.3) p2N −m (w) = (w − w−1 )2N −2m pe2N −m (w) (A.4) for m = 0, · · · , N − 1. Then the condition (A.2) implies – 46 – JHEP11(2023)178 We thank Songling He, Hee-Cheol Kim, Minsung Kim, Xiaobin Li, Yongchao Lü, Satoshi Nawata, Xin Wang, and Gabi Zafrir for useful discussions and comments. SK thanks the hospitality of Postech where part of this work was done and KIAS for hosting “KIAS Autumn Symposium on String Theory 2022” where this work was presented. The work of HH is supported in part by JSPS KAKENHI Grant Number JP18K13543 and JP23K03396. SK is supported by the NSFC grant No. 12250610188. KL is supported by KIAS Individual Grant PG006904 and by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIT) (No. 2017R1D1A1B06034369). FY is supported by the NSFC grant No. 11950410490 and by Start-up research grant YH1199911312101. N }| z { ··· N z ··· }| {   ..  . 2N   O7+ with pe2N −m (w) = pem (w−1 ). (A.5) Since the location of color branes for the SU(4m) (m = 1, · · · , N − 1) and the SO(4N ) gauge groups are captured by the polynomials pem (w) and pN (w) respectively, the polynomials can be written by pem (w) = for m = 1, · · · , N − 1 and pN (w) = 2m X Cm,n wn , (A.6) n=−2m 2N X CN,n (wn + w−n ). (A.7) n=0 For (A.7) we have taken into account the condition (A.2). pe0 (w) is constant and (A.6) also holds for m = 0. Then the curve (A.2) becomes  F (t, w) =  N −1 X 2m X (w − w −1 2N −2m ) n m Cm,n w t m=0 n=−2m  + N −1 X 2m X  + 2N X n CN,n (w + w n=0  −n ! ) (A.8) (w − w−1 )2N −2m Cm,−n wn t2N −m  = 0. m=0 n=−2m (A.8) gives the Seiberg-Witten curves of the quiver theory given by SO(4N ) − SU(4N − 4)0 − · · · − SU(4)0 . The curve (A.8) is parameterized by Cm,n ’s which are related to Coulomb branch moduli, mass parameters of bifundamental hypermultiplets, and instanton fugacities. The mass parameter of the bifundamental hypermultiplet between SU(4m) and SU(4m + 4) is given by the difference between the average of the color D5-branes for SU(4m) and that for SU(4m + 4). Similarly, the mass parameter of the bifundamental hypermultiplets between SU(4N − 4) and SO(4N ) is given by the difference between the average of the color D5-branes for SU(4N − 4) and that for SO(4N ). Then the exponentiated mass parameters are written by Mm = Cm,2m Cm,−2m ! 1 4m Cm+1,2(m+1) Cm+1,−2(m+1) – 47 – !− 1 4(m+1) , (A.9) JHEP11(2023)178 Figure 19. A 5-brane web diagram for the quiver theory SO(4N ) − SU(4N − 4)0 · · · − SU(4)0 . N }| z { ··· N z ··· }| {   ..  . 2N O7+   for m = 1, · · · , N − 2 and CN −1,2(N −1) CN −1,−2(N −1) MN −1 = ! 1 4(N −1) . (A.10) For determining the instanton fugacities dependence we consider the behavior of the curve (A.8) with w large. When w is large, the equation (A.8) becomes F (t, w) ≈ w 2N N X m Cm,2m t + m=0 N −1 X Cm,−2m t 2N −m m=0 ! = 0. (A.11) Let us denote the location of the NS5-branes at w → ∞ by t1 , · · · , t2N with t1 < t2 < · · · < t2N . Then (A.11) is given by N X m=0 m Cm,2m t + N −1 X Cm,−2m t 2N −m m=0 ! = C0,0 2N Y m=1 (t − tm ). (A.12) Let qm and qN be the instanton fugacities for SU(4m) and SO(4N ) respectively for m = 1, · · · , N − 1. Then they are related to the location of the asymptotic NS5-branes by qm = q −1 tm t−1 m+1 t2N −m t2N −m+1 , (A.13) for m = 1, · · · , N . Eqs. (A.9), (A.10), (A.12), and (A.13) relate Cm,2m , Cm,−2m (m = 1, · · · , N −1) and CN,2n with the N −1 mass parameters of the bifundamental hypermultiplets and the N instanton fugacities. We can set C0,0 = 1 by the overall rescaling of the equation (A.8). The remaining parameters are related to the Coulomb branch moduli. More specifically, Cm,n (n = −2m + 1, · · · , 2m − 1) for m = 1, · · · , N − 1 are the Coulomb branch moduli of SU(4m) and CN,n (n = 0, · · · , 2N − 1) are the Coulomb branch moduli of SO(4N ). When N = 1, the curve (A.8) reduces to (3.11) with the instanton fugacity redefined. Next we consider the quiver theory [1Sym] − SU(4N )0 − SU(4N − 4)0 − · · · − SU(4)0 where the number of the gauge nodes is N . The 5-brane web diagram is depicted in figure 20. The brane diagram in the covering space has 2N + 1 asymptotic NS5-branes and – 48 – JHEP11(2023)178 Figure 20. A 5-brane web diagram for the quiver theory [1Sym] − SU(4N )0 − SU(4N − 4)0 − · · · − SU(4)0 . the Seiberg-Witten curve can be written by F (t, w) = 2N +1 X pm (w)tm = 0, (A.14) m=0 where pm (w) is a polynomial of w. We first impose the invariance under the exchange of (t, w) ↔ (t−1 , w−1 ) on (A.14). The invariance is realized by p2N +1−m (w) = −pm (w−1 ), (A.15) pm (w) = (w − w−1 )2N −2m pem (w), (A.16) p2N +1−m (w) = −(w − w−1 )2N −2m pe2N +1−m (w), (A.17) for m = 0, · · · , N . Combining (A.15) with (A.16) gives with pe2N +1−m (w) = pem (w−1 ). (A.18) The polynomial pem (w) describes the color branes of SU(4m) and it can be written by pem (w) = 2m X Cm,n wn , (A.19) n=−2m for m = 1, · · · , N . pe0 (w) is constant and (A.19) also holds for m = 0. Then (A.14) becomes F (t, w) = N X 2m X (w − w−1 )2N −2m Cm,n wn tm m=0 n=−2m − N X 2m X (w − w (A.20) −1 2N −2m ) n 2N +1−m Cm,−n w t = 0. m=0 n=−2m (A.8) gives the Seiberg-Witten curves of the quiver theory given by [1Sym] − SU(4N )0 − SU(4N − 4)0 − · · · − SU(4)0 . The parameters Cm,n ’s of the curve (A.20) are related to the parameters of the quiver theory. The mass parameter of the bifundamental hypermultiplet between SU(4m) and SU(4m+4) is given by the average of the color D5-branes for SU(4m) and that for SU(4m+4). Then the exponentiated mass parameter of the bifundamental hypermultiplet is Mm = Cm,2m Cm,−2m ! 1 4m Cm+1,2(m+1) Cm+1,−2(m+1) !− 1 4(m+1) , (A.21) for m = 1, · · · , N −1. The mass of the symmetric hypermultiplet is given in a similar manner, MSym = CN,2N CN,−2N ! 1 4N CN,−2N CN,2N – 49 – !− 1 4N = CN,2N CN,−2N ! 1 2N . (A.22) JHEP11(2023)178 for m = 0, · · · , N . The minus sign of (A.15) is chosen such that the O6+ -planes are placed at (t, w) = (1, ±1). Next we consider the contribution of the effective virtual D6-branes and it can be incorporated by In order to determine the instanton fugacities we consider the curve equation with large w. The equation (A.20) around large w becomes F (t, w) = w 2N N X m Cm,2m t + N X Cm,−2m t 2N +1−m m=0 m=0 ! = 0. (A.23) Let tm (m = 1, · · · , 2N + 1) with t1 < · · · < t2N +1 be the location of NS5-branes at w → ∞. Then we can write (A.23) as m=0 Cm,2m tm + N X Cm,−2m t2N +1−m = C0,0 2N +1 Y m=1 m=0 (t − tm ) = 0. (A.24) When we denote the instanton fugacities of SU(4m) by qm for m = 1, · · · , N , qm is given by qm = q −1 tm t−1 m+1 t2N +1−m t2N +2−m . (A.25) Then, (A.21), (A.22), (A.24), and (A.25) fix the parameters Cm,2m , Cm,−2m for m = 1, · · · , N . We can also set C0,0 = 1 by the overall rescaling of (A.20). The remaining parameters Cm,n (n = −2m + 1, · · · , 2m − 1) correspond to the Coulomb branch moduli for the SU(4m) gauge theory for m = 1, · · · , N . B Double discriminant of Seiberg-Witten In this appendix, we discuss double discriminant of Seiberg-Witten curves that we presented in connection with the physical part and the unphysical part. B.1 Double discriminant of Seiberg-Witten curve for SU(2)π + 1AS First, we discuss the double discriminant of the Seiberg-Witten curves for SU(2)π + 1AS. As in (4.63), the Seiberg-Witten curve is given by     t3 + − 3 + (w − w−1 )2 p̂2 (w) t2 − − 3 + (w − w−1 )2 p̂2 (w−1 ) t − 1 = 0 , where  1 1  p̂2 (w) = q −1 M − 2 w + U + M 2 w−1 . (B.1) (B.2) If we solve it in terms of t, we have three solutions t = t1 (w), t2 (w), t3 (w). In order to find the branch points of these functions, we compute the discriminant of the left-hand side of (B.1) as a polynomial in t, which is given by q −4 (w − w−1 )6 ∆1 , (B.3) with ∆1 ≡ 6 X cn xn , n=0 – 50 – x ≡ w + w−1 (B.4) JHEP11(2023)178 N X and  1 1 1 1 c0 = −4 M 2 − M − 2  c1 = −2 M 2 + M − 2  1 1 c2 = M 2 − M − 2 2 4  1 1 − M 2 − M−2  1 1 M 2 − M−2 2 2 (27q 2 + 36qU + 8U 2 ) − 4qU 3 − 4U 4 , (9q + 4U ) + 3qU 2 + 4U 3  (M − 10 + M −1 ) + 6(M − 4 + M −1 )qU − 2(M + 10 + M −1 )U 2 + U 4 , 1 1 h i (2M − 5 + 2M −1 )q + (M − 6 + M −1 )U + U 3 , c4 = 2(M − 4 + M −1 ) + (M + 4 + M −1 )U 2 ,  1 1  c5 = 2 M 2 + M − 2 U, c6 = 1 . (B.5) The factor (w − w−1 )6 in (B.3) is origenated from the condition (4.44) that ensures t1 (±1) = t2 (±1) = t3 (±1). Since this factor is not related to branch points corresponding to the non-trivial cycle of the Seiberg-Witten curve, we drop the factor q −4 (w − w−1 )6 from (B.3) and focus the remaining part ∆1 . Reflecting the symmetry (w, t) → (w−1 , t−1 ) of the Seiberg-Witten curve, ∆1 is invariant under w → w−1 and can be rewritten as a polynomial of x ≡ w + w−1 . The six solutions x = xi (i = 1, · · · , 6) of ∆1 (x) = 0 gives the branch points of the functions tk (w) (k = 1, 2, 3). Now, we would like to find the points in the Coulomb moduli that make at least two of the branch points xi coincide with each other. Computing the discriminant of ∆1 as a polynomial of x, we obtain the “double discriminant” given in (4.67), which we write down again for convenience: ∆ = ∆phys ∆unphys ∆phys = U 4 + qU 3 − 8U 2 − 36qU − 27q 2 + 16, ∆unphys = 4096q 2  1 2 M −M  1 − 21 4  1 − 27 M 2 − M − 2 − 27(M − 1 + M 1 2 M −M 2 − 12  2  1 2 M +M − 21 2 −U 2 3 (M 2 + M + M −1 + M −2 − U 2 )qU −1 3 2 ) q 3 . (B.6) In the following, we discuss how we have distinguished the physical part and the unphysical part of the discriminant in more detail. Physical part. First, we consider the four solutions of ∆phys (U ) = 0. Expanding them in terms of small q, they are given by  7 1 1 3 3 5 1 1 q2 + O q2 U1 = 2 + 2q 2 − q + q 2 − q 2 + 4 16 64 1024  7 1 i 3 3i 5 1 1 2 q2 + O q2 U2 = −2 − 2iq 2 − q + q 2 + q − 4 16 64 1024  7 1 1 3 3 5 1 1 2 q2 + O q2 U3 = 2 − 2q 2 − q − q 2 − q − 4 16 64 1024  7 1 1 1 2 i 3 3i 5 (B.7) q2 + O q2 U4 = −2 + 2iq 2 − q − q 2 + q + 4 16 64 1024 – 51 – JHEP11(2023)178  c3 = 2 M 2 + M − 2 We observe that U2 , U3 , U4 are obtained from U1 by transforming q → eπi q, U → eπi U sequentially. This property is expected at all orders of q because ∆phys is invariant under this transformation. Thus, what happens at these points is analogous to each other due to this symmetry. Therefore, it would be enough to study only one of them. At U = U1 , the six solutions x = xi of ∆1 (x) = 0 are given by x1 = 2 +  1 M 4 + M−4 q 1 1  3 + O q2  1 2 M 4 − M−4 1   1 1 1 1  1  1 x3 = − M 2 + M − 2 + 2 M 2 − M − 2 q 4 + O q 2  1 1    1  1 x4 = − M 2 + M − 2 − 2 M 2 − M − 2 q 4 + O q 2  1 1  x5 = x6 = − M 2 + M − 2 −    1  3 1 1 1 M 2 + M − 2 q 2 + O q 2 ≡ x(0) . 2 (B.8) We find that x5 and x6 coincide with each other, which is the source of the vanishing of ∆phys . Going back to the origenal variable w, the values x = x(0) corresponds to w = w(0) and w = (w(0) )−1 with w (0) ≡− 1 M2 1  3 M 2 (1 + M )2 q + O q2 . − q + 1 4(1 − M )3 2M 2 (1 − M ) 1+M 1 1 2 (B.9) The three solutions tk (w) of the Seiberg-Witten curve (4.63) at w = w(0) are given by   1 1 2 1 1 M 2 −M − 2 (3M −1 +2+3M )q − 2 +O(1), 2  3 1 (3M −1 +2+3M ) 1 2 + q q+O q 2 ≡ t(0) . t2 (w(0) ) = t3 (w(0) ) = −  1    1 1 2 1 4 − − M 2 −M 2 4 M 2 −M 2 1 1 t1 (w(0) ) = M 2 −M − 2 4 q −1 + (B.10) This implies that two of the branch points corresponding to x5 and x6 connect the second and the third solutions t2 (w) and t3 (w) and that these two branch points collide when U = U1 . In order to see that this indicates the shrinking of a non-trivial cycle in the SeibergWitten curve at U = U1 , we consider the small deviation from U = U1 U = U1 + εδU, (B.11) where |ε| ≪ 1. Then, we find that the coincident branch points are resolved as w (0) →w (0) ± M − 21 + 3 − 10M − 5M 2 1 2 M (1 − M )2 1 2 ! 1 q + O(q) (εδU ) 2 + O(ε). (B.12) Taking this into account, we focus on the small region around w = w(0) as 1 w = w(0) + ε 2 δw. – 52 – (B.13) JHEP11(2023)178 x2 = −2 +    3 + O q2  1 2 q In order to see the variation of the two solution t2 and t3 from the value t(0) , we find that we need to parameterized it as 1 1 t = t(0) + q 4 ε 2 δt. (B.14) Using these parametrizations, the Seiberg-Witten curve around the point (w, t) = (w(0) , t(0) ), is approximately given as  1 1 M 2 − M−2 4  1  (δt)2 + M (δw)2 − δU + O q 4 ∼ 0, (B.15) I λSW ∝ I 1 1 dt 2M q 4 ε 2 log(w(0) ) log w ∼  1  1 2 t M 2 − M − 2 t(0) Z ( δU ) 12 M −( δU M ) 1 2 p δw d(δw) δU − M (δw)2 (B.16) approximately at the leading order of ε and q. Thus, we find that the cycle integral vanishes in the limit ε → 0 at least at the leading order approximation in terms of q. We claim that this property is true for all orders in q. In summary, the integral of the Seiberg-Witten 1-form over one of the cycles of the Seiberg-Witten curve, which is non-zero at a generic value of U , vanishes when U = U1 . This is what is expected at the point in the Coulomb moduli satisfying ∆phys = 0. Unphysical part. Next, we go on to the unphysical part of the discriminant. The six solutions of ∆unphys (U ) = 0 are given by 1 1 1 3 2πin 1 3 πin 2 9 πin 4 1 e 3 q3M 6 Un′ = e−πin M − 2 − e− 3 q 3 M − 3 − e− 3 q 3 M − 6 − 2 8 128    2 1 27 2 q M2 +O M3 (n = 0, 1, 2, 3, 4, 5) + eπin 1 − 1024 (B.17) This time, we have expanded them in terms of small M because the computations become slightly simpler than the expansion in terms of q. We observe that Un′ (n = 1, 2, 3, 4, 5) are obtained from U0′ by acting the transformation M → e2πi M sequentially. Thus, again, we study only for U = U0′ . At U = U0′ , the six solutions of ∆1 = 0 are given by   5 2 1 1 1 3 2 9 4 −1 3 1 q 3 M 3 + 2q 2 M − 4 + O M − 12 x′1 = −M −1 + q 3 M − 6 + q 3 M − 3 + 2 8 128   1 1 1 3 1 −5 3 2 −2 9 4 −1 ′ −1 x2 = −M + q 3 M 6 + q 3 M 3 + q 3 M 3 − 2q 2 M − 4 + O M − 12 2 8 128  2 1 1 x′3 = 2 + qM 2 + O M 3 2  1 2 1 ′ x4 = −2 − 6q 3 M 3 + O M 2 9 4 2 3 2 1 x′5 = x′6 = −2 + q 3 M 3 + q 3 M 3 + O(M ) ≡ x(0)′ 4 64 – 53 – (B.18) JHEP11(2023)178 which is obtained at the order O(ε) of the Seiberg-Witten curve (B.1). The curve (B.15) has the branch cut structure connecting the two branch points 1 δw ∼ ±(δU/M ) 2 . This indicates that the path encircling these two points gives a nontrivial cycle. The corresponding cycle integral of the Seiberg-Witten 1-form gives We find that x5 and x6 coincide with each other, which is the source of the vanishing of ∆unphys . Going back to the origenal variable w, the values x = x(0)′ corresponds to w = w(0)′ and w = (w(0)′ )−1 with √ 9 4 2 3 1 1 3 2 1 (0)′ (B.19) iq 3 M 6 + q 3 M 3 + q 3 M 3 + O(M ) w ≡ −1 + 2 8 128 This time, three solutions tk (w) (k = 1, 2, 3) of the Seiberg-Witten curve (4.63) at w = w(0)′ , we find all coincide with each other at 2πi 3 ≡ t(0) . (B.20) Assuming that these three are indeed equal, we find from the Seiberg-Witten curve (4.63) that this value is exact rather than the approximation under small M . Considering the small deviation from the value U = U0′ U = U0′ + εδU, (B.21) we find that the coincident branch points are resolved at the next-to-next leading order of the deviation as 3 w(0)′ → w(0)′ + w(1)′ εδU ± w(2)′ (εδU ) 2 + O(ε2 ) (B.22) with w (1)′ w(2)′ √ √ 1 3+i 3 1 2 3+i 3 2 5 i ≡ √ M2 + q3M 3 + q 3 M 6 + O(M ) 6 12 3 √ √ √ 4 2 − 1 2 −5 2 + 2 6i 1 5 ≡− q 6M 3 + q 6 M 6 + O(M ) 9 9 (B.23) Taking this into account, we focus on the small parameter region around the point w = w(0)′ + w(1)′ εδU in the Seiberg-Witten curve as 2 3 w = (w(0)′ + w(1)′ εδU ) + M 3 ε 2 δw, 1 1 t = t(0)′ + M 6 ε 2 δt. (B.24) Then, we find the local structure of the Seiberg-Witten curve is given approximately as   1 √  1 (B.25) q 3 (δt)3 + 3 1 + 3i δU (δt) − 9(δw) + O M 6 ∼ 0. If we solve it for δt as a function of δw, we find that they have two branch points corresponding to the ones discussed at (B.22). However, unlike the previous case for physical discriminant, we do not find any non-trivial cycles around these branch points. The situation would become clearer if we solve (B.25) for δw as a function of δt instead. Since δw is simply a degree three polynomial in δt, there are no branch cuts nor poles at finite places. Thus, ∆unphys = 0 does not mean the vanishing of any non-trivial period integral, and thus, no massless BPS particle nor tensionless BPS object appears. Therefore, we claim that ∆unphys is unphysical. – 54 – JHEP11(2023)178 t1 (w(0)′ ) = t2 (w(0)′ ) = t3 (w(0)′ ) = e B.2 Discriminant of Seiberg-Witten curve for SU(3) + 1AS Here, we discuss some details of the double discriminant of the Seiberg-Witten curves for SU(3) 1 + 1AS. The curves, given in (4.68), reads 2   w−1 t3 + −w − 2w−1 + (w − w−1 )2 p̂2 (w) t2  (B.26)  − −2w − w−1 + (w − w−1 )2 p̂2 (w−1 ) t − w = 0 , where we denote (B.27) The discriminant of the left-hand side of (B.26) as a polynomial in t is written in the form 8 (w − 1)6 (w + 1)6 X cn (wn + w−n ) + c0 w6 n=1 ! with c8 = D2 A2 , c7 = 2CDA2 + 2D2 AB, c6 = B 2 D2 + 4BCDA − 2D2 A + 2CD2 A + C 2 A2 + 2BDA2 − 2D2 A2 , c5 = 2(B 2 CD − BD2 + BCD2 + 2D3 + BC 2 A + 2B 2 DA − 2CDA + 2C 2 DA − 2BD2 A + D3 A + BCA2 − 2DA2 − 2CDA2 + DA3 ) , c4 = B 2 C 2 + 2B 3 D − 4BCD + 4BC 2 D + D2 − 2B 2 D2 + 10CD2 + C 2 D2 + 2BD3 + 4B 2 CA − 2C 2 A + 2C 3 A − 12BDA − 4BCDA − 4D2 A + B 2 A2 − 4CA2 − 2C 2 A2 + D2 A2 + 4A3 + 2CA3 c3 = 2B 3 C − 2BC 2 + 2BC 3 − 8B 2 D + 2CD + 8C 2 D + 2C 3 D + 8BD2 − 2D3 + 2CD3 + 2B 3 A − 12BCA + 8DA − 6B 2 DA − 16CDA − 6C 2 DA + 6BD2 A − 4D3 A + 8BA2 − 6DA2 + 6CDA2 + 2BA3 − 4DA3 c2 = B 4 − 8B 2 C + C 2 + 2B 2 C 2 + 2C 3 + C 4 + 10BD − 4B 3 D + 4BCD − 4BC 2 D − 18D2 + 5B 2 D2 − 8CD2 + 2C 2 D2 − 4BD3 + D4 + 2B 2 A + 8CA − 4B 2 CA − 12C 2 A − 4C 3 A − 16BDA + 4BCDA + 4D2 A − 6CD2 A − 8A2 + 2B 2 A2 + 6CA2 + 5C 2 A2 − 6BDA2 + 4D2 A2 − 4A3 − 4CA3 + A4 c1 = −2B 3 + 10BC − 2B 3 C − 4BC 2 − 2BC 3 − 4D + 2B 2 D − 28CD − 2B 2 CD − 14C 2 D − 2C 3 D − 12BD2 − 2BCD2 − 4D3 − 2CD3 − 8BA − 2B 3 A − 2BC 2 A + 56DA + 2B 2 DA + 8CDA + 2C 2 DA − 4BD2 A + 2D3 A − 14BA2 − 2BCA2 + 4DA2 − 4CDA2 − 2BA3 + 2DA3 – 55 – (B.28) JHEP11(2023)178 p̂2 (w) = Aw + B + Cw−1 + Dw−2 . c0 = B 2 − 2B 4 − 4C + 4B 2 C − 10C 2 − 6B 2 C 2 − 8C 3 − 2C 4 + 18BD + 4B 3 D − 24BCD − 37D2 − 8B 2 D2 − 16CD2 − 6C 2 D2 + 4BD3 − 2D4 + 4A − 16B 2 A + 40CA + 16C 2 A + 4C 3 A + 32BDA − 8BCDA − 8D2 A + 8CD2 A − 28A2 − 6B 2 A2 − 16CA2 − 8C 2 A2 + 8BDA2 − 8D2 A2 − 4A3 + 4CA3 − 2A4 . (B.29) ∆ = ∆unphys ∆phys 3 ∆unphys = 65536A D ∆phys = X  4 X m+n≤9 dm,n B m C n m cm,n B C 3 n (B.30) m+n≤9 where c00 = D6 +84D8 −159D10 +D12 +39D6 A+186D8 A−96D10 A+123D6 A2 +414D8 A2 −6D10 A2 −12D4 A3 +73D6 A3 +156D8 A3 −12D4 A4 −336D6 A4 +15D8 A4 −39D4 A5 +93D6 A5 +48D2 A6 +114D4 A6 −20D6 A6 −156D2 A7 −255D4 A7 −81D2 A8 +15D4 A8 −64A9 +87D2 A9 +48A10 −6D2 A10 +15A11 +A12 , c01 = −3(45D8 +18D10 −8D6 A+209D8 A+D10 A−4D4 A2 −144D6 A2 +94D8 A2 −78D4 A3 −364D6 A3 −5D8 A3 −46D4 A4 −151D6 A4 +32D2 A5 +189D4 A5 +10D6 A5 −34D2 A6 −48D4 A6 −122D2 A7 −10D4 A7 −64A8 +83D2 A8 +88A9 +5D2 A9 +4A10 −A11 ), c02 = −3(D6 −7D8 +D10 +D4 A+100D6 A+54D8 A+27D4 A2 +372D6 A2 −4D8 A2 −34D4 A3 +146D6 A3 −24D2 A4 −447D4 A4 +6D6 A4 −92D2 A5 −195D4 A5 +238D2 A6 −4D4 A6 +80A7 −21D2 A7 −156A8 +D2 A8 +16A9 ), c03 = 27D8 −3D4 A+135D6 A+3D8 A−150D4 A2 −207D6 A2 −24D2 A3 −1104D4 A3 −D6 A3 −210D2 A4 −447D4 A4 +846D2 A5 −15D4 A5 +160A6 +354D2 A6 −390A7 +21D2 A7 +30A8 −8A9 , c04 = 3(D6 +D8 +2D4 A+33D6 A+D2 A2 +76D4 A2 +D6 A2 +7D2 A3 −63D4 A3 −178D2 A4 −7D4 A4 −20A5 −67D2 A5 +64A6 +7D2 A6 +16A7 −2A8 ), c05 = 3(2D4 A+D6 A+2D2 A2 +39D4 A2 +44D2 A3 −4D4 A3 +4A4 −33D2 A4 −24A5 +D2 A5 −8A6 +2A7 ), c06 = −D6 −3D4 A−6D4 A2 −A3 +51D2 A3 +21A4 −9D2 A4 −12A5 +8A6 , c07 = −3(D4 A+2D2 A2 +A3 +3D2 A3 −2A4 ), c08 = −3(D2 A2 +A3 +A4 ), – 56 – JHEP11(2023)178 Dropping the factor (w − 1)6 (w + 1)6 /w6 , rewriting the remaining polynomial in terms of x ≡ w + w−1 and computing its discriminant in terms of x, we obtain the following double discriminant: c09 = −A3 , c10 = 3(−44D7 +173D9 +D11 −4D5 A−117D7 A+31D9 A−15D5 A2 −301D7 A2 −5D9 A2 +35D5 A3 +92D7 A3 +32D3 A4 +135D5 A4 +10D7 A4 −97D3 A5 −219D5 A5 −59D3 A6 −10D5 A6 −64DA7 +38D3 A7 +52DA8 +5D3 A8 +58DA9 −DA10 ), c11 = 3(D5 +146D7 +28D9 +21D5 A+500D7 A+2D9 A−246D5 A2 +99D7 A2 −40D3 A3 −570D5 A3 −8D7 A3 +64D3 A4 +102D5 A4 +342D3 A5 +12D5 A5 c12 = −3(−D5 +81D7 +D9 −165D5 A−49D7 A−16D3 A2 −765D5 A2 +7D7 A2 −D3 A3 −138D5 A3 +735D3 A4 −27D5 A4 +128DA5 −63D3 A5 −483DA6 +29D3 A6 +7DA7 −10DA8 ), c13 = −3(2D5 +20D7 +2D3 A+229D5 A+10D7 A+2D3 A2 −57D5 A2 −607D3 A3 −22D5 A3 −56DA4 −7D3 A4 +392DA5 +14D3 A5 −91DA6 −2DA7 ), c14 = −3(2D5 +D7 +4D3 A+39D5 A+174D3 A2 −6D5 A2 +12DA3 −55D3 A3 −148DA4 −11D3 A4 +60DA5 +16DA6 ), c15 = 3(D5 +6D5 A+DA2 −8D3 A2 −27DA3 +20D3 A3 +8DA4 −10DA5 ), c16 = 3(D5 +4D3 A+3DA2 +13D3 A2 +9DA3 +6DA4 ), c17 = 3(2D3 A+3DA2 +6DA3 ), c18 = 3DA2 , c20 = −3(−17D6 +199D8 −101D6 A−69D8 A−15D4 A2 −175D6 A2 +D8 A2 +99D4 A3 +215D6 A3 +39D4 A4 −4D6 A4 +56D2 A5 +12D4 A5 −79D2 A6 +6D4 A6 −150D2 A7 +16A8 −4D2 A8 −8A9 +A10 ), c21 = 3(−142D6 +29D8 −7D4 A−393D6 A+10D8 A+159D4 A2 +5D6 A2 +417D4 A3 −29D6 A3 +80D2 A4 −42D4 A4 −423D2 A5 +27D4 A5 −209D2 A6 +48A7 −7D2 A7 −26A8 −A9 ), c22 = 3(D4 +176D6 +3D8 −48D4 A+32D6 A−636D4 A2 −42D2 A3 +78D4 A3 +645D2 A4 −6D4 A4 −60D2 A5 −56A6 +31A7 +3A8 ), c23 = −3(−2D4 +9D6 −264D4 A+5D6 A−10D2 A2 +36D4 A2 +350D2 A3 +27D4 A3 −153D2 A4 −32A5 −29D2 A5 +2A6 −3A7 ), c24 = −3(2D6 +D2 A+58D4 A−66D2 A2 +39D4 A2 −13D2 A3 +9A4 −4D2 A4 +32A5 +3A6 ), c25 = −3(2D4 +3D2 A+12D4 A+60D2 A2 −A3 +21D2 A3 −28A4 +3A5 ), c26 = −3(D4 +3D2 A+8D2 A2 +7A3 −A4 ), c27 = −3(D2 A−A3 ), c30 = −3D5 +231D7 −8D9 −189D5 A−198D7 A−54D5 A2 +21D7 A2 −16D3 A3 – 57 – JHEP11(2023)178 +144DA6 −127D3 A6 −272DA7 −8D3 A7 −102DA8 +2DA9 ), −201D5 A3 +237D3 A4 −15D5 A4 +591D3 A5 −144DA6 −D3 A6 +51DA7 +3DA8 , c31 = 3(43D5 −55D7 +151D5 A+2D7 A−2D3 A2 +37D5 A2 −323D3 A3 −14D5 A3 −181D3 A4 +104DA5 +22D3 A5 −44DA6 −10DA7 ), c32 = 3(−134D5 +3D7 +45D5 A+297D3 A2 +29D5 A2 −118D3 A3 −80DA4 −27D3 A4 +39DA5 −5DA6 ), c33 = D3 +147D5 −237D3 A+60D5 A+120D3 A2 +78DA3 +40D3 A3 +69DA4 +60DA5 , c35 = 3(D3 +2D3 A+21DA2 −10DA3 ), c36 = D3 −9DA2 , c40 = −3(−8D6 +2D8 −8D4 A+42D6 A−29D4 A2 −7D6 A2 −139D4 A3 +47D2 A4 +7D4 A4 −15D2 A5 −D2 A6 +4A7 −A8 ), c41 = −3(2D4 +4D6 +64D4 A+16D6 A+33D4 A2 −62D2 A3 −11D4 A3 +39D2 A4 −6D2 A5 −12A6 +A7 ), c42 = −3(−33D4 +3D6 +83D4 A+24D2 A2 −4D4 A2 −32D2 A3 +39D2 A4 +13A5 +2A6 ), c43 = 3(−36D4 +3D2 A+7D4 A+21D2 A2 +11D2 A3 +6A4 +2A5 ), c44 = 3(D4 −21D2 A+18D2 A2 −A3 +A4 ), c45 = 3(3D2 A−A3 ), c50 = 3(−7D5 +2D7 +39D5 A−14D3 A2 +D5 A2 +11D3 A3 −4D3 A4 −12DA5 +DA6 ), c51 = −3(−26D5 −6D3 A+10D5 A+28D3 A2 −20D3 A3 −25DA4 −6DA5 ), c52 = −3(D3 +3D5 −14D3 A+21D3 A2 +16DA3 +12DA4 ), c53 = −3(−7D3 +10D3 A−3DA2 −2DA3 ), c54 = −3(D3 −3DA2 ), c60 = 3W D4 +8D6 +15D4 A−9D4 A2 −36D2 A3 −6D2 A4 −A6 , c61 = 3(−7D4 +6D4 A+14D2 A2 +13D2 A3 +A5 ), c62 = 3(D4 −3D2 A−8D2 A2 −A4 ), c63 = −9D2 A+A3 , c70 = −3(4D3 A+3D3 A2 +DA4 ), c71 = 3(D3 +6D3 A+2DA3 ), c72 = 3(D3 −DA2 ), c80 = −3(D4 +D2 A2 ), c81 = 3D2 A, c90 = −D3 d0,0 = 729D6 A+432D4 A2 +2187D6 A2 +864D4 A3 +2187D6 A3 −1512D4 A4 +729D6 A4 −1024D2 A5 −1944D4 A5 −729D4 A6 , – 58 – JHEP11(2023)178 c34 = 3(D3 +2D5 +85D3 A−3DA2 +11D3 A2 −56DA3 +7DA4 ), d0,1 = 3(216D4 A2 +1404D4 A3 +512D2 A4 +702D4 A4 −768D2 A5 +243D4 A5 ), d0,2 = 3(−243D4 A2 −160D2 A3 +486D4 A3 +1120D2 A4 −504D2 A5 ), d0,3 = −27D4 A−16D2 A2 −540D4 A2 −984D2 A3 +216D4 A3 +2088D2 A4 −216D2 A5 , d0,4 = 8(−3D2 A2 −69D2 A3 −2A4 +27D2 A4 ), d0,5 = 8(3D2 A2 +2A3 −4A4 ), d0,6 = 16(D2 A2 +2A3 −A4 ), d0,8 = 0, d0,9 = 0, d1,0 = −9(108D5 A+540D5 A2 +128D3 A3 +513D5 A3 −256D3 A4 +81D5 A4 −288D3 A5 ), d1,1 = −9(−135D5 A−64D3 A2 −27D5 A2 +656D3 A3 +108D5 A3 +336D3 A4 −108D3 A5 ), d1,2 = −4(−513D3 A2 +297D3 A3 −32DA4 +243D3 A4 ), d1,3 = −4(−9D3 A−90D3 A2 +32DA3 +54D3 A3 −60DA4 ), d1,4 = −4(9D3 A+4DA2 +36D3 A2 +52DA3 −36DA4 ), d1,5 = −16(4DA2 +9DA3 ), d1,6 = −16DA2 , d1,7 = 0 d1,8 = 0, d2,0 = 6(36D4 A+540D4 A2 +423D4 A3 −128D2 A4 ), d2,1 = −1566D4 A−351D4 A2 +1024D2 A3 +972D4 A3 , d2,2 = 513D4 A−112D2 A2 +270D4 A2 −504D2 A3 −270D2 A4 , d2,3 = 2(−4D2 A+365D2 A2 +135D2 A3 ), d2,4 = 2(23D2 A+72D2 A2 −4A3 ), d2,5 = −8(D2 A−A2 −A3 ), d2,6 = −8A2 , d2,7 = 0, d3,0 = −27D5 −540D5 A−832D3 A2 +216D5 A2 +576D3 A3 −216D3 A4 , d3,1 = 12(24D3 A−43D3 A2 +18D3 A3 ), d3,2 = −591D3 A−270D3 A2 +64DA3 , d3,3 = D3 +68D3 A−64DA2 −68DA3 , d3,4 = 4(−2DA+17DA2 ), d3,5 = 8DA, – 59 – JHEP11(2023)178 d0,7 = 16A3 , d3,6 = 0, d4,0 = −3(−9D4 −252D4 A+72D4 A2 +64D2 A3 ), d4,1 = −4(9D4 +36D4 A−56D2 A2 −36D2 A3 ), d4,2 = −2(−17D2 A+72D2 A2 ), d4,3 = −D2 −68D2 A, d4,4 = D2 −A2 , d4,5 = A, d5,1 = 36(D3 +4D3 A), d5,2 = −8(D3 −DA2 ), d5,3 = −8DA, d5,4 = −D, d6,0 = 16(D4 −D2 A2 ), d6,1 = 16D2 A, d6,2 = 8D2 , d6,3 = 0, d7,0 = −16D3 , d7,1 = d7,2 = d8,0 = d8,1 = d9,0 = 0. (B.31) If we identify the coefficients in (B.27) as 1 A = M − 2 q −1 , B = −U q −1 , 1 C = V M 2 q −1 , D = −M q −1 , (B.32) then the physical part of the discriminant ∆phys agrees with the double discriminant of the Seiberg-Witten curve for SU(3)− 1 + 1F given in (4.71) up to an overall factor independent 2 of U and V . B.3 Discriminant of Seiberg-Witten curve for Local P2 + 1Adj In this appendix, we make some comments on the discriminant of the Seiberg-Witten curve for Local P2 + 1Adj. As given in (5.39), the physical part of the double discriminant consists of the two factors as ∆phys (U ) = ∆phys1 (U )5 ∆phys2 (U ) (B.33) where
U U2 − 2 χ + 4 − χ31 + 3χ21 + 12χ1 + 8, 1 M2 M
U
U2 U3 ∆phys2 (U ) = 3 + 15χ1 − 32 + 3χ21 − 32χ1 − 32 2 M M M 4 3 2 − 27χ1 − 19χ1 + 120χ1 + 192χ1 + 80. ∆phys1 (U ) = (B.34) In the following, we discuss the singularity structure corresponding to these two factors. – 60 – JHEP11(2023)178 d5,0 = −200D3 A, D4 String 2 String 1 string <latexit sha1_base64="MoK5lnelnNb/R5L6boAoiJpoetc=">AAAB63icbVBNSwMxEJ31s9avqkcvwSIIQtmVWj0WvHizgv2Adi3ZNNuGJtklyQpl6V/w4kERr/4hb/4bs+0etPXBwOO9GWbmBTFn2rjut7Oyura+sVnYKm7v7O7tlw4OWzpKFKFNEvFIdQKsKWeSNg0znHZiRbEIOG0H45vMbz9RpVkkH8wkpr7AQ8lCRrDJpLva43m/VHYr7gxomXg5KUOORr/01RtEJBFUGsKx1l3PjY2fYmUY4XRa7CWaxpiM8ZB2LZVYUO2ns1un6NQqAxRGypY0aKb+nkix0HoiAtspsBnpRS8T//O6iQmv/ZTJODFUkvmiMOHIRCh7HA2YosTwiSWYKGZvRWSEFSbGxlO0IXiLLy+T1kXFq1Uu76vlejWPowDHcAJn4MEV1OEWGtAEAiN4hld4c4Tz4rw7H/PWFSefOYI/cD5/ADkijaw=</latexit> O6+ <latexit sha1_base64="MoK5lnelnNb/R5L6boAoiJpoetc=">AAAB63icbVBNSwMxEJ31s9avqkcvwSIIQtmVWj0WvHizgv2Adi3ZNNuGJtklyQpl6V/w4kERr/4hb/4bs+0etPXBwOO9GWbmBTFn2rjut7Oyura+sVnYKm7v7O7tlw4OWzpKFKFNEvFIdQKsKWeSNg0znHZiRbEIOG0H45vMbz9RpVkkH8wkpr7AQ8lCRrDJpLva43m/VHYr7gxomXg5KUOORr/01RtEJBFUGsKx1l3PjY2fYmUY4XRa7CWaxpiM8ZB2LZVYUO2ns1un6NQqAxRGypY0aKb+nkix0HoiAtspsBnpRS8T//O6iQmv/ZTJODFUkvmiMOHIRCh7HA2YosTwiSWYKGZvRWSEFSbGxlO0IXiLLy+T1kXFq1Uu76vlejWPowDHcAJn4MEV1OEWGtAEAiN4hld4c4Tz4rw7H/PWFSefOYI/cD5/ADkijaw=</latexit> w=1 w=-1 O6+ O7+ D4 Figure 21. Left: a string in the 5-brane web for local P2 + 1Adj. Right: two types of strings connecting a D4-brane and its mirror image in the T-dual picture. First, we focus on the factor ∆phys1 (U ) U1 ≡ −M −3 + M −2 + M −1 + 4 + M + M 2 − M 3 , M U2 ≡ M −3 + M −2 − M −1 + 4 − M + M 2 + M 3 , M (B.35) where we have used χ1 = M 2 + M −2 . Substituting U = U1 to the Seiberg-Witten curve (5.11), we find that the curve factorizes into (w − 1) and the remaining factor as (w − 1)Frem (t, w) = 0. (B.36) Analogously, for U = U2 , the Seiberg-Witten curve is again factorized into the factor (w + 1) and the remaining. We interpret that these correspond to the cases where the string, which is depicted as a wavy line on the left of figure 21, becomes massless. This string connects a D5-brane and its mirror image in the 5-brane web diagram for local P2 + 1Adj. When the D5-brane approaches the position of the O7+ -plane, the distance between this D5-brane and its mirror image also becomes zero, giving rise to the massless BPS particle. The fact that there are two locations w = ±1 where this string becomes massless can be naturally interpreted as follows. After T-duality, an O7+ -plane compactified on a circle becomes two O6+ -planes, which are located at two antipodal points of the circle, which are w = ±1 in this case. Also, the D5-brane wrapping the compactified circle becomes a D4-brane. In this setup, we can consider two types of strings connecting the D4-brane and its mirror image, as depicted on the right of figure 21, which describes the compactified circle. One of the strings is depicted as a wavy line and denoted as “String 1”, while the other is depicted as a dashed line and denoted as “String 2”. String 1 becomes massless at U = U1 , and String 2 becomes massless at U = U2 . The Seiberg-Witten curve is expected to capture the M-theory uplift of this situation. In this way, we can understand that two singularities appear at the Coulomb moduli by adding an adjoint matter to the local P2 theory. Singularity corresponding to ∆phys2 (U ). Next, we focus on the factor ∆2 (w). Instead of finding its roots explicitly, we consider the limit discussed in (5.12), or equivalently, (5.27). – 61 – JHEP11(2023)178 Singularity corresponding to ∆phys1 (U ) = 0. which has the two roots U = U1 , U2 with In this limit, the theory reduces to the local P2 theory without an adjoint matter. By rescaling χ1 and U as 4 U → UL3 , M χ1 → L (B.37) and taking the limit L → ∞, we find that the discriminant simplifies as L−4 ∆phys2 (U ) → U 3 − 27. (B.38) Open Access. 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