5d/6d DE instantons from trivalent gluing of web diagrams

Journal of High Energy Physics, Jun 2017

We propose a new prescription for computing the Nekrasov partition functions of five-dimensional theories with eight supercharges realized by gauging non-perturbative flavor symmetries of three five-dimensional superconformal field theories. The topological vertex formalism gives a way to compute the partition functions of the matter theories with flavor instanton backgrounds, and the gauging is achieved by summing over Young diagrams. We apply the prescription to calculate the Nekrasov partition functions of various five-dimensional gauge theories such as SO(2N) gauge theories with or without hypermultiplets in the vector representation and also pure E 6, E 7, E 8 gauge theories. Furthermore, the technique can be applied to computations of the Nekrasov partition functions of five-dimensional theories which arise from circle compactifications of six-dimensional minimal superconformal field theories characterized by the gauge groups SU(3), SO(8), E 6, E 7, E 8. We exemplify our method by comparing some of the obtained partition functions with known results and find perfect agreement. We also present a prescription of extending the gluing rule to the refined topological vertex.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

https://link.springer.com/content/pdf/10.1007%2FJHEP06%282017%29078.pdf

5d/6d DE instantons from trivalent gluing of web diagrams

Received: March Published for SISSA by Springer Hirotaka Hayashi 0 1 2 5 6 7 Kantaro Ohmori 0 1 2 3 4 6 7 0 7-3-1 Hongo , Bunkyo-ku, Tokyo 113-0033 , Japan 1 1 Einstein Drive , Princeton, NJ 08540 , U.S.A 2 4-1-1 Kitakaname , Hiratsuka-shi, Kanagawa 259-1292 , Japan 3 School of Natural Sciences, Institute for Advanced Study 4 Department of Physics, Faculty of Science, The University of Tokyo 5 Department of Physics, School of Science, Tokai University 6 Open Access , c The Authors 7 sions, Topological Strings , F-Theory We propose a new prescription for computing the Nekrasov partition functions of ve-dimensional theories with eight supercharges realized by gauging non-perturbative avor symmetries of three ve-dimensional superconformal eld theories. The topological vertex formalism gives a way to compute the partition functions of the matter theories with avor instanton backgrounds, and the gauging is achieved by summing over Young diagrams. We apply the prescription to calculate the Nekrasov partition functions of various ve-dimensional gauge theories such as SO(2N ) gauge theories with or without hypermultiplets in the vector representation and also pure E6; E7; E8 gauge theories. Furthermore, the technique can be applied to computations of the Nekrasov partition functions of vedimensional theories which arise from circle compacti cations of six-dimensional minimal superconformal eld theories characterized by the gauge groups SU(3); SO(8); E6; E7; E8. We exemplify our method by comparing some of the obtained partition functions with known results and nd perfect agreement. We also present a prescription of extending the gluing rule to the re ned topological vertex. Conformal Field Models in String Theory; Field Theories in Higher Dimen- - DE instantons from trivalent gluing of web 1 Introduction 2 A dual description of 5d gauge theory with D; E-type gauge group 2.1 5d SO(2N + 4) gauge theory 2.2 5d pure E6; E7; E8 gauge theories 3 Gluing rule and 5d SO(2N + 4) gauge theory 3.2 5d pure SO(2N + 4) gauge theory Example: 5d pure SO(8) gauge theory 3.3 Adding avors Example: 5d SO(8) gauge theory with four avors 4 5d gauge theory with E-type gauge group 4.1 5d pure E6 gauge theory 4.2 5d pure E7 gauge theory 4.3 5d pure E8 gauge theory 5 A 5d description of non-Higgsable clusters 5.1 5.2 5.3 O( n) model with n = 6; 8; 12 Another non-Higgsable cluster 6 Re nement 7 Conclusion A 5d SO(2N + 3) gauge theory B Some formulae for computation B.1 Re ned topological vertex B.2 Nekrasov partition function B.3 Schur functions Re ned partition function of Db2(SU(2)) matter from 6.2 Examples: 5d pure SO(8) gauge theory and O( 4) model The (re ned) topological vertex is a powerful tool to compute the all genus topological string amplitudes for toric Calabi-Yau threefolds [1{4]. One can compute the full topological string partition function like a Feynman diagram-like method and it can yield the full list of the Gromov-Witten invariants and the Gopakumar-Vafa invariants of a toric CalabiYau threefold in principle. The topological string partition function also has a physical interpretation through string theory or M-theory. When we consider M-theory on a noncompact Calabi-Yau threefold with a compact base that is contractible, the low energy e ective eld theory gives rise to a ve-dimensional (5d) theory with eight supercharges which has a ultraviolet (UV) completion [5{8]. Then M2-branes wrapping various holomorphic curves in the Calabi-Yau threefold yield BPS particles in the 5d theory. Therefore, the curve counting for a non-compact Calabi-Yau threefold is equivalent to the counting of BPS particles of the 5d theory and this implies that the topological string partition function is equal to the Nekrasov partition function up to some extra factors. Indeed several checks of the equality have been done for example in [9{13] for 5d SU(N ) gauge theories with avors by utilizing the method of the topological vertex. Recently, the topological vertex formalism has been extended for computing the topological string partition functions of certain non-toric Calabi-Yau threefolds [14{16].1 The new method makes use of a Higgs prescription of the superconformal index in [19, 20].2 In fact, some non-toric Calabi-Yau threefold can be obtained from a topology changing transition or a Higgsing from a toric Calabi-Yau threefold. Then applying the Higgsing prescription for the topological string partition function of the \UV" Calabi-Yau threefold gives rise to the topological string partition function of the \infrared" (IR) non-toric Calabi-Yau threefold. This new technique enables us to compute the Nekrasov partition functions of the 5d rank one E7; E8 theories [14, 15], the 5d SU(N ) gauge theory with a hypermultiplet in the antisymmetric representation and also the 5d Sp(N ) gauge theory [25]. Furthermore, it has been also applied to the calculation of the Nekrasov partition functions of 5d theories which has a six-dimensional (6d) UV completion, and non-trivial checks with the elliptic genus of the 6d self-dual strings have been done in [26, 27]. Although the new method enlarges the space of non-compact Calabi-Yau threefolds to which we can apply the topological vertex, there is still a large class of non-compact Calabi-Yau threefolds to which we have not yet known how to apply the topological vertex. An interesting class of such Calabi-Yau threefolds is the ones which yield 5d gauge theories with a gauge group SO(2N ) or E6; E7; E8. In this paper, we propose a new technique which enables us to compute the Nekrasov partition functions of the 5d pure gauge theories with a gauge group SO(2N ) or E6; E7; E8 from the topological vertex. The new method utilizes a dual description of the 5d pure gauge theory with a gauge group of DE-type. In fact, it turns out that the dual description is given by gauging the diagonal part of avor 1There is also another vertex-like approach to compute the unre ned topological string amplitudes for some non-toric Calabi-Yau threefolds [17, 18]. 2In terms of geometry, the Higgsing corresponds to a topology changing transition and a similar technique has been also used in [21{24] in the context of the re ned version of the geometric transition. symmetries of three 5d theories. We call such a gauging trivalent gauging. We have often encountered the case of gauging the diagonal part of avor symmetries of two 5d theories from toric Calabi-Yau threefolds or equivalently 5-brane webs [28{30]. Gauging the avor symmetries of three 5d theories is a natural generalization but goes beyond the standard picture of 5-brane webs. The main aim of this paper is to formulate a novel method to compute the Nekrasov partition functions of 5d theories constructed by the trivalent gauging. The 5d theories coupled by the trivalent gauging may be considered as \matter" parts for the gauging. We indeed develop a way to compute the partition functions of the 5d theories as a \matter" contribution for the gauging from the topological vertex. Then, the trivalent gauging can be implemented by inserting the Nekrasov partition function of vector multiplets for the gauging and summing over Young diagrams. The prescription may be interpreted as a generalization of the gauging for the superconformal index in fourdimension [31{33]. However, the extension to the gauging for ve-dimensional partition functions is quite non-trivial compared with the four-dimensional case since we need to add instanton contributions which appear by the gauging. The new prescription of the trivalent gauging not only apply to the partition functions of 5d theories which have a 5d UV completion but also apply the partition functions of 5d theories which have a 6d UV completion. An interesting class of 6d superconformal eld are an important ingredient for the atomic classi cation of general 6d SCFTs [35{37]. When the non-Higgsable cluster has only one tensor multiplet then they are called 6d for some 6d minimal SCFTs with eight supercharges have been proposed in [38]. In fact, gauging avor symmetries of three or four 5d theories. Therefore, we can use the trivalent gauging method and it is possible to compute the Nekrasov partition functions of the 5d descriptions of some 6d minimal SCFTs on a circle. The Nekrasov partition function for a 5d theory with a 6d UV completion can be also interpreted as the sum of the elliptic genera of the self-dual strings in the 6d SCFT. We will give a non-trivial check between the elliptic genus computed in [39]. We will further propose a 5d description of the 6d minimal see a non-trivial matching with the elliptic genus computation recently done in [40]. The organization of this paper is as follows. In section 2, we rst determine a dual description for the 5d SO(2N + 4) gauge theory with or without hypermultiplets in the vector representation and also for the 5d pure gauge theories with a gauge group of E-type. In section 3, we present a new technique to compute the topological string partition function from the trivalent gauging of 5d theories. We then apply the method to compute the Nekrasov partition function of the 5d SO(2N + 4) gauge theory with or without avors and perform non-trivial checks with known results. We then apply the trivalent gauging prescription for the partition functions of the pure E6; E7; E8 gauge theories in section 4. In section 5, the trivalent gauging method is applied to 5d descriptions for some minimal 6d SCFTs. We also support for it. We further comment on a 5d description of a non-Higgsable cluster theory with multiplet tensor multiplets in section 5.4. In section 6, we present a way to extend the prescription of the trivalent gauging to the re ned topological vertex formalism. We then conclude our work in section 7. In appendix A, we describe a relation between the SO(2N + 4) gauge theory and the SO(2N +3) gauge theory from a Higgsing, which provides a way to compute the Nekrasov partition function of the SO(2N +3) gauge theory from the SO(2N + 4) gauge theory. We nally summarize technical tools used in this paper in appendix B. This paper accompanies a Mathematica notebook which is available from the arXiv web site. The notebook performs some of the computations of topological vertices exhibited in section 3, section 4 and section 5, and the computation of the Hilbert series explained in appendix B. The notebook utilizes the Mathematica application LieART [41]. The notebook only provides calculations related to the unre ned limit. A dual description of 5d gauge theory with D; E-type gauge group Five-dimensional gauge theories with eight supercharges can be realized by compactifying M-theory on a singular Calabi-Yau threefolds X3 [5{8]. When the Calabi-Yau threefold X3 has a G-type surface singularity over a sphere CB, then the low energy e ective eld theory from the M-theory compacti cation yields a 5d pure gauge theory with a gauge group G. Here G is either AN = SU(N + 1); (N = 1; 2; ), DN+2 = SO(2N + 4); (N = 2; 3; E6; E7; E8. The resolution of the singularity means that the 5d gauge theory is on the Coulomb branch. The Calabi-Yau manifold Xe3 after the resolution contains a collection of spheres bered over the base sphere CB. The intersections among collection of the bered spheres form a shape of the Dykin diagram of the Lie algebra g (the Lie algebra of a Lie group G) corresponding to the resolution of the G-type singularity. We denote the consists of spheres alighted along the Dynkin-diagram of type g by Fg. Each sphere in Fg corresponds to a simple root of g and let a collection of spheres corresponding to a root be C . Then an M2-brane wrapping a curve C in Fg yields a massive W-boson for the of g in the 5d gauge theory. Therefore, the size of C is a Coulomb branch modulus. On the other hand, an M2-brane wrapping the base CB yields an instanton particle of the 5d gauge theory. The size of the base CB is then related to g2 coupling. We also denote a complex surface which is C where gY M is the 5d gauge bration over CB by S . From this construction it is clear that the gauge theory information is encoded in the complex two-dimensional space Sg which is given by the Fg bration over the base CB. The e ect of gravity may be neglected by taking a limit where the transverse direction to Sg is in nitely large. We will always take the eld theory limit and hence the background Xe3 is a non-compact Calabi-Yau threefold whose compact base is given by the complex surface Sg. More generally, M-theory on a non-compact Calabi-Yau manifold which is a the complex surface S is contractible then the 5d theory has a UV completion [5{8] and 3An uncommon convention for N is due to the construction of its dual theory in this section. the theory becomes a SCFT when the volume of S vanishes . We will restrict our attention to such a case in this paper. this case, we can use the powerful technique of toric geometry or a dual picture of 5-brane webs in type IIB string theory [28{30]. In this section, we will argue that the cases of picture, although we are not sure whether there exists any kind of brane construction which physically realizes that web-like picture. 5d SO(2N + 4) gauge theory Let us rst consider the case of G = DN+2; N = 2; 3; has the compact surface Sso(2N+4) which is a Fso(2N+4) . The Calabi-Yau geometry Xe3 bration over the base CB. The non-Abelian SO(2N + 4) gauge symmetry is recovered at the origin of the Coulomb branch moduli space which corresponds to the limit where the spheres forming the Fso(2N+4) ber shrink simultaneously over the base CB, recovering the DN+2 surface singularity over the base CB. It is possible to further shrink the base CB. Then the whole complex surface Sso(2N+4) shrinks to zero size and the gauge coupling become in nitely strong. This limit corresponds to the conformal limit where nonperturbative particles as well as perturbative particles become simultaneously massless, and therefore the 5d theory becomes a superconformal eld theory. In order to obtain a dual gauge theory description we consider a di erent order of shrinking of the surface Sso(2N+4). The ber Fg consists of N + 2 spheres whose shape is the Dynkin diagram of type DN+2. Among the N + 2 spheres, there is one special sphere Cg which intersect with adjacent three spheres. We then consider Cg as a base and shrink the other spheres including CB. Since CB is bered over Cg, the geometry develops an A1 singularity wrapping Cg after shrinking CB. Hence the theory has an SU(2) gauge symmetry. Furthermore, we have three singular points on Cg. Two of them originate from contracting a surface Ssu(2) which has a Fsu(2) ber over CB. The other singular point originates from contracting a surface Ssu(N) which has a Fsu(N) ber over CB. Since the singularities arise from shrinking the complex surfaces, each singular point yields a 5d SCFT and they are coupled by the SU(2) gauge symmetry associated to the A1 singularity over Cg. Hence each of the SCFTs should have an SU(2) avor symmetry and the diagonal part of the three SU(2) avor symmetries is gauged. Therefore, the dual description is realized by the SU(2) gauging of the three 5d SCFTs. We call the gauging trivalent gauging. Let us then see the three superconformal eld theories in detail. Two of them come from shrinking the complex surface Ssu(2). Hence, the 5d theory is a pure SU(2) gauge theory with its mass parameter turned on. The pure SU(2) gauge theory should have an SU(2) avor symmetry in UV which can be used for the SU(2) trivalent gauging. Hence, the discrete theta angle for the pure SU(2) gauge theory should be zero. The other SCFT comes from shrinking the complex surface Ssu(N). Therefore the 5d theory is a pure SU(N ) gauge theory. Since the pure SU(N ) gauge theory should have an SU(2) avor symmetry in UV again for the SU(2) trivalent gauging, the Chern-Simons (CS) level should be For each case of the pure SU(2) gauge theory and the pure SU(N ) gauge theory, the SU(2) (p; q) 5-brane theory. The slope of the (p; q) 5-brane is pq in the two-dimensional (x5; x6)-space. In particular, a horizontal line represents a horizontal line and a vertical line represents an NS5-brane. Since the structure of the 5-brane only appears in the (x5; x6)-plane, we only write the two-dimensional plane for depicting a 5-brane. on an orbifold C3= where the orbifold action of avor symmetry arises non-perturbatively in UV. To deal with the avor symmetry we should directly consider the UV superconformal eld theory of the pure SU(2) gauge theory and the pure SU(N ) N gauge theory,4 which we denote by Db2(SU(2)) and DbN (SU(N )) respectively. Here the notation Dbp(SU(2))5 implies a SCFT which arises from M-theory g = (!2; ! 1; ! 1 gp = (!2p; ! p; ! p) = (1; 1; 1) with !2p = 1 and p = 2; 3; of C3. Note that the orbifold action . The three components act on the three complex coordinates yields an A1 singularity, leading to an SU(2) avor symmetry. The Dbp(SU(2)) theory is then a rank (p 1) SCFT with an SU(2) avor symmetry. Therefore, it has p 1 Coulomb branch moduli and one mass parameter. In particular, Db2(SU(2)) theory is the yields the same SCFT as E1 theory in [45]. It is illustrative to describe the Dbp(SU(2)) theory by a 5-brane web. A 5-brane web is a dual con guration of a certain Calabi-Yau threefold Xe3 [30]. The directions which the 5brane extend are summarized in table 1. It is also useful to introduce 7-branes attached to the ends of external 5-branes in a 5-brane web con guration to read o the avor symmetry of a 5d theory realized on a 5-brane web [46]. The 5-brane web for the pure SU(p) p gauge theory is given in gure 1. To understand the SU(2) avor symmetry \perturbatively", it might help to take a S-dual of the web, which is also depicted in gure 1. The Sduality of the 5d theory is simply given by the 2 rotation of the web in the (x5; x6)-plane. Note that in the S-dual picture, the avor symmetry of the Dbp(SU(2)) theory is realized perturbatively as background gauge eld on two D7-branes attached to the ends of the external 5-branes extending in the right direction. However, we do not have internal D5branes and the theory does not admit a Lagrangian description. On the other hand the 4SU(N ) implies that an SU(N ) gauge theory with the CS level . 5The notation of Dbp(SU(2)) has been introduced in [38] as a 5d uplift of the 4d Dp(SU(2)) theory [43, 44] which is equivalent to the 4d (A1; Dp) Argyres-Douglas theory. p CS level. We have p D5-branes which lie in the horizontal direction. The parallel two external NS5-branes imply the non-perturbative SU(2) avor symmetry. Right: the S-dual con guration to the 5-brane web on the left. Namely the 5-brane web for the Dbp(SU(2)) theory. SU(2) avor symmetry appears non-perturbatively in the pure SU(p) gauge theory since it is associated to a symmetry on the two (0; 1) 7-branes or the two NS5-branes. In summary, when we regard Cg as the base manifold, the geometry gives rise to the following 5d theory Db2(SU(2)) DbN (SU(2)) The SU(2) in the center of (2.3) implies the SU(2) trivalent gauging which couple the two Db2(SU(2)) theories and the DbN (SU(2)) theory by the diagonal gauging of their SU(2) avor symmetries. We argue that this is a dual description of the pure SO(2N + 4) gauge theory. One can check that the number of the moduli and the parameters of one theory match with those of the other theory. The pure SO(2N + 4) gauge theory has N + 2 Coulomb branch moduli and one mass parameter corresponding to the gauge coupling. The dual theory (2.3) has (N parameter from the gauging coupling of the SU(2) trivalent gauging in (2.3). This duality between SO(2N + 2) gauge theory and the SU(2) gauge theory (2.3) with non-Lagrangian matter is a generalization of base- ber dualities between 5d SU(N ) linear quiver gauge theories [29, 47] as well as 4d theories [48]. It is also possible to write a web-like picture for the dual theory (2.3). Noting that the Dbp(SU(2)) theory is given by the web in the right gure of gure 1, we can write a web-like picture for the theory (2.3) as in gure 2. Due to the trivalent gauging, it is not possible to write the diagram in gure 2 as a proper 5-brane web on a plane. Verifying that this picture somewhat makes sense is the main purpose of this paper. In particular, what to do with the \trivalent SU(2) gauging" in the picture is going to be given in the next section. Note that the lengths between the parallel horizontal legs for the three 5-brane webs are the size of CB and hence they should be equal to each other. We need to impose this condition for the partition function computation in the later sections. In the dual picture, the size gauge theory. The prescription for the \trivalent SU(2) gauging" is going to be given in the next section. Three webs actually does not live in the same plane, and thus do not cross each other in the cases we will deal with in this paper. of CB becomes the Coulomb branch modulus of the SU(2) trivalent gauging. In terms of the web diagram, the trivalent gauging may be thought of as trivalent gluing of the three webs which give rise to the Db2(SU(2)); Db2(SU(2)) and DbN (SU(2)) theories. We will use the terminology of trivalent gauging and trivalent gluing interchangeably in this paper. We can further support the dual description (2.3) in another manner. SO(2N + 4) gauge theory can be also realized by a 5-brane web with an O5-plane as in the left gure in gure 3. The 5-brane web con guration can be thought of as connecting a pure SU(N ) gauge theory with the CS level N with a pure SO(4) gauge theory by su(2), we may replace the 5-brane web for the SO(4) gauge theory with the two 5-brane webs for the pure SU(2) gauge theory as in gure 3. Then the web-like gure on the right in gure 3 may be considered as an S-dual con guration of the web in gure 2. This understanding also provides us with a way to introduce hypermultiplets in the vector representation of SO(2N +4). Starting from the 5-brane web of the pure SO(2N +4) gauge theory, M1 + M2 hypermultiplets in the vector representation can be added by introducing M1 avor 5-branes on the left and M2 avor 5-branes on the right as in gure 4. We here assume M1 N + 1 and M2 N + 1 and also M1 + M2 2N + 1. In fact the SO(2N + 4) gauge theory with Nf hypermultiplets in the vector representation has a 5d UV completion when Nf 2N + 1 [49].6 In the case when the number of avors saturates O5− gluing. The left gure represents a 5-brane web of the pure SO(2N + 4) gauge theory using an O5-plane. The right gure is a web-like description by replacing the 5-brane for the SO(4) gauge theory part in the left gure with the two 5-branes webs of the pure SU(2) gauge theory with no discrete theta angle. Now the three 5-brane webs are connected by the trivalent gluing. SU(2) × SU(2) O5− vector representation. multiplets in the vector representation by replacing the web for the SO(4) part with the two webs for the pure SU(2). The three 5-brane webs are connected by the trivalent gluing. gure 4 but it is still possible to write down a 5-brane web by introducing a con guration of 5-branes jumping over other 5-branes [49]. With the 5-brane web picture in gure 4, one can again apply the replacement of the web of the SO(4) gauge theory with the two webs of the pure SU(2) gauge theory as in gure 5. A dual picture may be obtained by simply rotating the web in gure 6. By denoting the web on the left part in gure 6 by DbNM1;M2 (SU(2)), a 5d theory which is dual to the 5d SO(2N + 4) gauge theory Figure 6. A web-like diagram which is obtained by rotating web in gure 5 by 2 . with M1 + M2 hypermultiplets in the vector representation is given by DbNM1;M2 (SU(2)) Db2(SU(2)) Here DbNM1;M2 (SU(2)) is the 5d rank (N 1) SCFT with an SU(2) SU(M1 + M2) SU(M1 + M2) 5d pure E6; E7; E8 gauge theories and E8. For each case, there is again one sphere Cg in the ber Fg which intersects with three adjacent spheres. We may consider Cg as a base and shrink the other spheres including CB. Then the shrinking of CB yields again an A1 singularity over Cg, leading to an SU(2) gauge symmetry. Cg has three singular points and each point gives rise to a certain 5d SCFT, depending on G = E6; E7 or E8. other two singularities originate from shrinking a surface Ssu(3). Repeating the same argument in section 2.1, the former yields the Db2(SU(2)) theory and the latter gives rise to the Db3(SU(2)) theory. Therefore, a dual description of the pure E6 gauge theory is given by the trivalent gauging of the Db2(SU(2)) theory and the two Db3(SU(2)) theories, namely Db2(SU(2)) Db3(SU(2)) : the gauge coupling of the SU(2) trivalent gauging. These numbers agrees with the numbers of the Coulomb branch moduli and the mass parameter of the pure E6 gauge theory. the Db2(SU(2)) theory, the Dc3(SU(2)) theory and the Db4(SU(2)) theory. Hence a dual description of the pure E7 gauge theory is Db2(SU(2)) from the gauge coupling of the SU(2) gauging. The numbers again agree with the numbers of the Coulomb branch moduli and the mass parameter of the pure E7 theory. Db3(SU(2)) and the Db5(SU(2)) theory. Then a dual picture of the pure E8 gauge theory is Db2(SU(2)) parameter. The numbers completely agrees with the eight Coulomb branch moduli and the one mass parameter of the pure E8 gauge theory. Gluing rule and 5d SO(2N + 4) gauge theory Having identi ed the dual gauge theory descriptions (2.3){(2.7) for the gauge theories with to compute their Nekrasov partition functions. The main tool is the topological vertex formalism [1{4], whose basic formulae are summarized in appendix B.1. When a 5d theory is realized on a 5-brane web, the application of the topological vertex to the 5-brane web gives rise to its Nekrasov partition function [9{13]. However, it is not possible to simply apply the topological vertex to the web-like descriptions of the theories (2.3){(2.7) due to the existence of the trivalent gauging of three 5d theories. In this section we propose a new technique which enables us to apply the topological vertex formalism to the trivalent gauging of three 5d theories. The result will come in the form of double expansion of instanton fugacity and the Coulomb branch parameter corresponding to the trivalent node of the Dynkin diagram of the gauge group, and that is compared with result from the localization computations up to some orders of those two expanding parameters. In this section, we focus on unre ned partition functions, and postpone the re ned cases to section 6. 4); E6; E7; E8 came from the duality frame which involves the SU(2) gauging of the diagonal part of SU(2) avor symmetries of three SCFTs. Although each SCFT is a UV SCFT of a gauge theory, the gauged SU(2) symmetry emerges non-perturbatively at UV, so we cannot have a Lagrangian description of the duality frame, and thus we need to develop a new way to compute the partition function of such a theory. The central idea is regarding those SCFTs as \SU(2) matter", although they do not have a Lagrangian description where the SU(2) symmetry is manifest. Recall that the Nekrasov partition function [51, 52] for an SU(2) gauge theory with hypermultiplets looks X Qjg j+j j Zh;yper(QB; Qm)ZS;U(2) vector(QB); are Young diagrams, Qg, Qm, QB are associated to the instanton fugacity, a mass parameter and Coulomb branch parameter, respectively.7 Zh;yper(QB; Qm) is the contribution from the hypermultiplets, and ZS;U(2) vector(QB) is that from the SU(2) vector multiplets. What we need now is a generalization of Zh;yper to the partition function of a general SCFT with an SU(2) avor symmetry. The pair of Young diagrams ( ; ) labels the xed points of the U(1) action in the U(2) instanton moduli space. Then, Zh;yper is the partition function of hypermultiplets with SU(2) background with the nontrivial instanton con guration labeled by ( ; ). Therefore, this concept is manifestly generalized into a general SCFT T , and we denote the partition function with the avor instanton background and avor fugacity QB by ZT; (QB). Then, the partition function of the trivalent SU(2) gauging of T1, T2 and T3 can be obtained by X Qjg j+j j ZT;1 (QB)ZT;2 (QB)ZT;3 (QB)ZS;U(2) vector(QB): This is similar to the gauging formula for 4d index [31{33], One might worry about the validity of this formula, since the formula (3.1) comes from the U(N ) instanton, and therefore it is not clear that the formula can be generalized into gauging of SCFTs with only SU(2) avor. Here we just go ahead, and it will turn out this prescription almost works. However, we occasionally need to subtract \extra factors" similar to what is discussed in subsection B.1 when the theory have avor symmetries as we will see in subsection 3.3. The next task is understanding how to compute such a partition function ZS;CFT with a nontrivial avor background. Note again that in our case the avor emerges nonperturbatively, and therefore methods relying on Lagrangian descriptions cannot be utilized. This is where the topological vertex helps. To be inspired, let us rewrite (3.1) using the topological vertex. The web diagram representing an SU(2) gauge theory with one fundamental hypermultiplet can be depicted as X Qjg j+j jf ; QB also for moduli and parameters of a 5d theory. where f ; is the framing factor: f ; (q) = f t1(q)f t (q) means the summation over a pair of Young diagrams ( ; ) assigned to the indicated internal edges. This summation over ; can be directly identi ed with that in (3.1) [9{ 13].8 Decoupling the hypermultiplet, the partition function reduces to the that of the pure SU(2) gauge theory and it is given by X Qjg j+j j ZS;U(2) vector(QB) = QB X Qjg j+j jf ; obtaining the equation Then, equating (3.1) and (3.3) gives aa ZS;U(2) vector(QB) = f ; Zh;yper(QB; Qm) = This equation tells us that assigning nontrivial Young diagrams to parallel external edges representing the SU(2) avor symmetry almost realizes the avor background labeled by those Young diagrams, but the division by the factor ZH;alf(QB) = QB QB ; = @ 11=2 is needed. This factor is the square root of ZS;U(2) vector, and thus we call this factor a contribution from a \half" vector. Noe let us apply this division for determining the partition function of the Db2(SU(2)) 8This is also true for re ned case, if one is careful about the preferred direction. See section 6. the SU(2) avor instanton background, we assign Young diagrams to the parallel external legs. Then the consideration (3.7) motivate us to declare that the partition function for the Db2(SU(2)) is given by ZDb;2(SU(2))(QB; Q) = ZbDb;2(SU(2))(QB; Q)=ZH;alf(QB) where ZbDb;2(SU(2))(Q) is the quantity computed by the topological vertex with nontrivial Young diagrams ; on the external edges, with Coulomb branch parameter Q. When which removes the constitutions coming from decoupled strings bridging the parallel 5branes. (3.9) is a natural generalization of that. In general, if a SCFT T with an SU(2) avor symmetry can be engineered by a web diagram which make the avor symmetry manifest, we claim that then the partition function ZT; with instanton avor background can be computed by the topological vertex in the same matter, namely the ratio of the naive topological vertex computation ZbT; and ZH;alf.9 In particular, a generalization to the partition function of the Dbp(SU(2)) matter is obvious. Let us check that (3.9) actually works. For that, we consider a limit of Coulomb branch parameters of the pure SO(8) gauge theory which gives an SU(3) gauge theory. In the dual frame (2.3), two of Db2(SU(2)) decouples in this limit, and thus we get a dual description of the SU(3) gauge theory. From (3.9), the partition function of this dual description is X Qjg j+j jZDb;2(SU(2))(QB; Q)ZSU(2) vector(QB) X Qjg j+j jf ; 9If the web of the SCFT T contains other manifest avor symmetries, then the partition function should be further divided by extra factors corresponding to those symmetries. The resulting web diagram is in fact nothing but the S-dual web for the pure SU(3) 1 gauge theory. Note that Q; Qg corresponds to the two Coulomb branch parameters of SU(3), and QB is the related to the gauge coupling of SU(3). Therefore the parameters Qg; QB exchanges their roles under the duality between the SU(3)1 description and (3.10). Now we can write down a prescription for partition functions for gauge theories dealt with in the previous section. For simplicity, here we explicitly state the pure SO(8) case. Let us denote the Coulomb branch parameters corresponding to edge nodes by Q1; Q 1; Q 2 that corresponding to the center node by Qg, and the parameter associated to the instanton counting by QB. From (2.3) and (3.9), the partition function is ZSO(8) = X Qjg j+j jZDb;2(SU(2))(QB; Q1)ZDb;2(SU(2)) (QB; Q 1)ZbDb;2(SU(2))(QB; Q 2) ZH;alf(QB) (QB; Q 1)ZDb;2(SU(2))(QB; Q 2)ZS;U(2) vector(QB) X Qjg j+j jZbDb;2(SU(2))(QB; Q1)ZbDb;2(SU(2)) X Qjg j+j jZbDb;2(SU(2))(QB; Q1)ZbDb;2(SU(2)) QB dual description (2.3). In the latter part of this paper we are going to make non-trivial checks of (3.12) and its generalizations by explicitly calculating the righthand side and comparing the result with eld theory computations. 5d pure SO(2N + 4) gauge theory We then move onto the explicit computation of the Nekrasov partition function of the pure SO(2N + 4) gauge theory, making use of the trivalent gluing rule obtained in section 3.1. Its dual theory is described by the trivalent gauging as in (2.3). Namely, it is realized by X Qjg j+j jf ; QB; Qi for i = 1; the trivalent SU(2) gauging of the diagonal part of the three SU(2) avor symmetries of the DbN (SU(2)) and the two Db2(SU(2)) theories. The web-like description of the 5d theory which is dual to the pure SO(2N + 4) gauge theory was given in gure 2. We then apply the gluing rule as well as the topological vertex to the web diagram. For that we rst compute the partition function of the \DbN (SU(2)) matter" part with non-trivial Young diagrams on the parallel external legs representing the SU(2) instanton background. To compute the partition function of the DbN (SU(2)) matter system, we assign Young diagrams f ag = f 1; ; N g. f ag = f 1; ; N 1g, f ag = f 1; Kahler parameters QB; fQag = fQ1; ; QN 1g to the lines in the web for the DbN (SU(2)) gure 7. By using the techniques in appendix B.1, the application of the (unre ned) topological vertex to the web in gure (7) yields ZbDb;N (SU(2))(QB; fQag) = f ag f ag f ag a=1 0 = ; 0 = N = N = ?. Note that we chose the last su xes of the topological vertices as the Young diagrams assigned to the vertical lines in the web in gure 7. The choice is useful for the comparison with the Nekrasov partition function from the localization method since then (3.13) is expanded by QB which is eventually related to the instanton fugacity of the pure SO(2N +4) gauge theory. A straightforward computation f ag a=1 H(Q) = I 1; 2 (Q) = Q = where s is the Schur function and q is the specialization of its arguments, both of which are brie y reviewed in appendix B.3. We introduced the notations signs in (3.16) are taken in the same order. As discussed in section 3.1, the partition function of (3.14) is not the one for the DbN (SU(2)) matter but one needs to divide it by the contribution of a \half" of the vector multiplets of (3.8), and its explicit partition function is ZH;alf(QB) = = q 21 jj jj2+ 12 jj tjj2 X QjBjqjj tjj2 Ze (q)Ze t (q)s (q ZbDb;N (SU(2))(QB; fQag) = q 21 jj jj2+ 12 jj tjj2 Therefore, the partition function of the DbN (SU(2)) matter is nally given by ZDb;N (SU(2))(QB; fQag) = One might worry that the contribution of the DbN (SU(2)) matter may be di erent when one rotates the diagram in and puts Young diagrams on the parallel external legs with an orientation outward. When we consider the usual quadrivalent SU(2) gauging, we glue such a web with the web in gure 7. However, it turns out that the partition function (3.19) does not change after the rotation with the opposite orientation of the arrows for ; . Therefore we may use the partition function (3.19) both for the gluing from the left and the right. Due to this symmetric property, it is possible to use (3.19) even for the trivalent gauging. Then as described in section 3.1, our proposal is that the partition function of the pure SO(2N + 4) gauge theory can be computed by treating the partition function (3.19) as a matter contribution for the SU(2) gauging. After inserting also the Nekrasov partition function of the SU(2) vector multiplets, we obtain ZSO(2N+4)(QB; Qg; fQag; Q 1; Q 2) = X Qjg j+j jZS;U(2) vector(QB)ZDb;2(SU(2))(Q 1) ZDb;2(SU(2))(Q 2)ZDb;N (SU(2))(QB; fQag); (3.20) where ZS;U(2) vector(QB) is the contribution from the SU(2) vector multiplets ZS;U(2) vector(QB) = X QjBjqjj tjj2 Ze (q)Ze t (q)s (q In the dual picture QB corresponds to the Coulomb branch modulus of the SU(2) frame, Qg is rather related to one of the Coulomb branch moduli of the pure SO(2N + 4) gauge theory and QB is related to the instanton fugacity of SO(2N + 4). It is possible to determine the precise relations between the Kahler parameters QB, Q 2; Q 1; Qg; fQag and the Coulomb branch moduli and the instanton fugacity of the pure SO(2N + 4) gauge theory. Let Cf be the curve whose Kahler parameter is Qf for f = 1. The N + 2 curves Cf ; f = 1 form the ber whose shape is the Dynkin diagram of so(2N + 4). Therefore, they are associated to the simple roots of the Lie algebra so(2N + 4) and we can parameterize Qi = e (aN i aN i+1); Q 1 = e (aN+1 aN+2); i = 1; Qg = e (aN aN+1); Q 2 = e (aN+1+aN+2); where ai; i = 1; ; N + 2 are the Coulomb branch moduli of the pure SO(2N + 4) gauge One the other hand, the instanton fugacity uSO(2N+4) is related to the size of the base uSO(2N+4) = QBh(Q 2; Q 1; Qg; fQag); where h is a certain monomial of arguments. In order to x the factor h, let us see the intersection numbers between the curves Ci; i = 1; B and the surface Sf which has the Cf bration over CB where f = 1. Due to the Dynkin diagram structure of the ber Fso(2N+4), the intersection matrix between Cf and Sf0 for f; f 0 = 1 forms the negative of the Cartan matrix of the so(2N + 4) Lie algebra. Furthermore, CB intersects only with Sg with the intersection curves Ci; i = 2. The intersection numbers are summarized as in table 2. In other words, the intersection numbers imply the Coulomb branch moduli dependence for the Kahler parameter. Since the instanton fugacity does not depend on the Coulomb branch moduli, the factor h(Q 2; Q 1; Qg; fQag) in (3.23) should be chosen so that uso(2N+4) does not depend on the Coulomb branch moduli or equivalently the corresponding curve has the zero intersection number with any surface Sf ; f = 1. This uniquely xes the factor h(Q 2; Q 1; Qg; fQag) and the instanton fugacity is given by uso(2N+4) = QBQg 2N Q 1N Q 2N Y Therefore, we conjecture that the partition function (3.20) yields the Nekrasov partition function of the pure SO(2N + 4) gauge theory after inserting the gauge theory parameters given by the relations (3.22) and (3.24).10 Example: 5d pure SO(8) gauge theory Let us explicitly compute the partition function (3.20) obtained from the SU(2) trivalent ZSO(8)(QB; Qg; Q1; Q 1; Q 2) = X Qjg j+j jZS;U(2) vector(QB)ZDb;2(SU(2))(Q 1) ZDb;2(SU(2))(Q 2)ZDb;2(SU(2))(QB; fQ1g); 10In this paper, we ignore the perturbative partition function from vector multiplets in the Cartan subalgebra of a gauge group G. The contribution cannot be captured from the topological vertex calculation but it can be easily recovered by the general formula ZCartan = H(1)rank(G); where rank(G) is the rank of the gauge group G. Eq. (5.39) should agree with the SU(3) Nekrasov partition function given by ZSNUek(3) = ZSPUer(t3) 1 + ukSU(3)ZSInUs(t3);k ; ZSInUs(t3);k = Eij(s) = ai 1`i(s) + 2(aj(s) + 1): moduli related to Q1; Q2; Q3 by ZSPUer(t3) is the perturbative part of the SU(3) partition function and e a1 = Qe13 Qe23 ; e a2 = Qe1 3 Qe23 : ZSPUer(t3) = H(Qe1)2H(Qe2)2H(Qe1Qe2)2: We checked that eq. (5.39) agrees with the Nekrasov partition function (5.41) of the pure SU(3) gauge theory by identifying the instanton fugacity uSU(3) as uSU(3) = until the order of Q3BQe21Qe22 and Q2BQe31Qe22. We are now ready to apply the op transition to the partition function (5.33). We assume that the same prescription for the op transition apply for the trivalent \SU(1)" gauging. We conjecture that the partition function of the 5d theory (5.33) after the op transition is given by ZOo(p 3)(QB; Qe1; Qe2; Qe3) = ZO( 3)(QB1; Qe1QB; Qe2QB; Qe3QB) The partition function (5.46) can be directly compared with the elliptic genus (5.34). The Kahler parameters Qe1; Qe2; Qe3 form the a ne Dynkin diagram of su(3) and we can for example choose Qe1; Qe2 for the simple roots of the su(3) corresponding to the 6d SU(3) symmetry. Then a map between Qm1; Qm2; Qm3 and Qe1; Qe2 is which can be written by complex structure modulus of the torus is Qe1 = Qm1Qm12; Qe2 = Qm2Qm13; Qm1 = Qe13 Qe23 ; Qm2 = Qe1 3 Qe23 ; Q = Qe1Qe2Qe3: 12 (2; 8; 1) + 12 (1; 8; 2) the negative of the self-intersection numbers of the base spheres. By using the maps (5.48) and (5.49), we checked that (5.46) agrees with implies that the string fugacity is given by Qs = QB Another non-Higgsable cluster equivalently one P orbifold description of (T 2 C2)= So far we have focused on the O( n) models which contain only one tensor multiplet or with the orbifold action given by (5.1), leading to its 5d description after a circle compacti cation. There are still another non-Higgsable cluster theories which contain multiple tensor multiplets or more than one base curves [34, 35]. The 6d theories again have no avor symmetry. The F-theory geometry has a compact base which is an elliptic bration over a collection of spheres given in table 7. They are also important ingredients for constructing 6d SCFTs. Among the three non-Higgsable clusters, the last entry in table 7 has an orbifold description [38]. The F-theory geometry is (T 2 C2)= with the orbifold action g = (! 6; !; !5); one Z2 xed point and two Z4 xed points. Then we consider a 5d description of this 6d theory. We can simply consider M-theory singularity and the 5d theory has an SU(2) gauge symmetry. Around the Z2 xed point, the geometry becomes C3= 0 with the action g0 = g2 = (! 12; !2; !10) = (! 4; !2; !2) = (!0 2; !0; !0); is an orbifold C3= is the Db2(SU(2)) theory at the Z2 xed point. Around the Z4 xed point, the geometry with the orbifold action (5.51). It is possible to write a 5-brane web corresponding to the orbifold geometry and it is depicted in gure 22. The 5d theory has an SU(2) avor symmetry with three Coulomb branch moduli. We denote the 5d theory with the orbifold acby Db (SU(2)), Therefore, the 5d theory for the non-Higgsable cluster is Db2(SU(2)) Db (SU(2)) Db (SU(2)) The 5d theory is again given by the SU(2) trivalent gauging. Let us see whether the numbers of 5d gauge theory parameters agrees with the expectation from 6d. The number of vector multiplets in the Cartan subalgebra in 6d is should have an eight-dimensional Coulomb branch moduli space. In 5d, Db2(SU(2)) theory has one Coulomb branch modulus and two Db (SU(2)) theories have 2 3 = 6 Coulomb branch moduli. By adding one Coulomb branch modulus from the trivalent SU(2) gauging, the 5d theory has an eight-dimensional Coulomb branch moduli space which agrees with the expectation. Since the 6d theory has no avor symmetry, the 5d theory should have only one mass parameter, Indeed the 5d theory (5.53) has one mass parameter coming from the instanton fugacity of the SU(2) trivalent gauging. Re nement So far we have considered the unre ned partition function where the two parameters 1; 2 are set to 1 = 2. In this section, we extend the rule for the trivalent SU(2) gluing to the re ned topological vertex formalism. Instead of performing the calculation in full generality, we will focus on a speci c example of the pure SO(8) gauge theory and describe how the trivalent SU(2) gauging can be generalized to the re ned case. The application to other cases will be carried out in a similar manner in principle. Re ned partition function of Db 2(SU(2)) matter from In order to perform the computation for the trivalent SU(2) gauging for the re ned case, we rst need to determine the re ned partition function of the DbN (SU(2)) matter correspondThe Kahler parameters satisfy Qm1 Q = Qm2 Q0. ing to the web in gure 7. Similarly to the topological vertex formalism, we assign the re ned topological vertex which is labeled by three Young diagrams corresponding to three legs at each vertex of a 5-brane web. However the role of the three legs is not symmetric and we assign t, q and a preferred direction for each leg. Furthermore, when one glues a leg with t(or q) with another leg, then the another leg should be labeled by q(or t). Let us rst think about the case when we choose the vertical directions in gure 7 for the preferred direction, then the gluing leg in the horizontal direction should be labeled by t or q. In order to have the consistent gluing for the re ned topological vertex, one needs to label t or q in a di erent way for the horizontal legs in the web for the other DbN (SU(2)) matter. When we glue two DbN (SU(2)) matter system then this gluing rule causes no problem. However when we consider the trivalent gluing with three DbN (SU(2)) matter system, then it is di cult to glue three webs consistently with the gluing rule for the re ned topological vertex. This problem can be avoided when we choose the horizontal direction in gure 7 for the preferred direction. This is also conceptually plausible. The equation (3.6) which we relied on can be generalized to the re ned case only when the preferred direction is taken to be horizontal. However another problem arises since some vertex does not have a leg in the preferred direction and we cannot apply the re ned topological vertex to such a vertex. In fact, there is a way to solve the second problem by using a op transition. To see that we focus on the case of the Db2(SU(2)) matter which we will use for the computation of the re ned partition function of the pure SO(8) gauge theory. Although we cannot apply the re ned topological vertex to the web for the Db2(SU(2)) matter with the horizontal direction chosen for the preferred direction, we can rst apply the re ned topological vertex to a di erent but a related to web in gure 23. From the web in gure 23, we can perform a op transition with respect to the curves whose Kahler parameters are Qm1 and Qm2 as in gure 24. Then we obtain a web on the right in gure 24. From the right web in gure 24, one can send Qm1 ; Qm2 ! 0, giving rise to a web in between the webs in gure 24, the Kahler parameters are related by gure 23. From the comparison Qm1 = QE11; Q = QF QE1 ; Qm2 = QE21; Q0 = QF QE2 : The same trick has been used to obtain the re ned partition function for the DbP2 theory [69]. We then rst compute the re ned partition function for the web in gure 23. The application of the re ned topological vertex to the web in gure 23 yields Zeb ; (QB; Q; Q0; Qm1 ; Qm2 ) = C 2t? t2 (t; q)C? 2 t2 (t; q)( QBQQ0)j 2jfe 2t (t; q) C? t1 2 (q; t)( Q)j 1jC 1t 1 t (t; q)( QB)j 1jfe 1t (t; q) C 1 1 t (t; q)( Q0)j 1jC t1? 2 (q; t)( Qm1 )j 2j( Qm2 )j 2j; where QQm1 = Q0Qm2 . After a calculation, we get s 1= (t q )s 1= (t q QjB1j+j 2j( 1)j 1j+j 1jQj 1j+j 2jQ0j 1j+j 2j )s 1= 0 (t q )s t1= 0 (t q X( Q)j jq 21 jj jj t 2 jj tjj2 Ze (t; q)Ze t (q; t)s 1 (t q 2 1 In order to apply the op transition in gure 24, we use a similar trick which we used in section 5.3. The insertion of (6.1) into (6.3) gives )s 1= 0 (t q )s t1= 0 (t q s 1= (t q )s 1= (t q ZRt1C2 (QE11; t; q)ZR1C2 (QE21; q; t)QjE11j+j 2jQjE21j+j 2j: Z op(t; q) = Qli!m0 Zconifold(Q) ZRC(Q 1; t; q) where [F (Q)]Q0 implies that we take the zeroth order of Q from F (Q). Therefore, applying the limit QE1 ; QE2 ! 0 to (6.5), we obtain ZbL; (QB; QF ) = q 21 (jj tjj2+jj tjj2)Ze t (t; q)Ze t (t; q) s 1= (t q )s 1= (t q )s 1= 0 (t q )s t1= 0 (t q For the re ned partition function of the Db2(SU(2)) matter, one needs to divide the re ned partition function by a \half" of the partition function of the SU(2) vector multiplets ZH;alf;L(QB) = q 21 (jj tjj2+jj tjj2)Ze t (t; q)Ze t (t; q) Then we consider the quantity G (Q; t; q) = Zconifold(Q) = ZRC(Q; t; q) Zconifold(Q) In fact, G (Q) is a polynomial of degree j j + j j in Q [66, 68]. Therefore, the following limit is well-de ned Ge (t; q) = lim G (Q 1; t; q)Qj j+j j: By using the op invariance for the partition function of the resolved conifold (6.7), the limit for ZRC(Q; t; q) can be taken as Therefore, the re ned version of the Db2(SU(2)) matter contribution is given by ZDb;2(SU(2));ref(QB; QF ) = In order to treat (6.12) for the Db2(SU(2)) matter, we check whether the web diagram which is given by the rotation compared to gure 23 but with the opposite direction for the arrows of ; yields the same partition function. We then compute the partition function for the web in gure 25 and apply the limit (6.1). By following the same steps, we obtain the partition function ZbR; (QB; QF ) = t 21 (jj jj2+jj jj2)Ze (q; t)Ze (q; t) s 1= (t q )s 1= (t q )s 1= 0 (t q )s t1= 0 (t q SU(2) theory gives Dividing (6.13) by the another half of the partition function of the vector multiplet for the Z0 Db;2(SU(2));ref(QB; QF ) = ZH;alf;R(QB) = t 21 (jj jj2+jj jj2)Ze (q; t)Ze (q; t) It is not clear whether (6.12) is equal to (6.14) but we checked that they are indeed equal Therefore, we can use (6.12) for the re ned partition function of the Db2(SU(2)) matter. Examples: 5d pure SO(8) gauge theory and O( 4) model In the previous subsection, we computed the re ned version of the partition function of the Db2(SU(2)) matter. In this section we apply the trivalent SU(2) gauging for the re ned partition function and obtain the Nekrasov partition functions of the pure SO(8) gauge theory and the 5d theory from the O( 4) model on a circle. theory is given by theory is given by Pure SO(8) gauge theory. A 5d dual description of the pure SO(8) gauge theory is ZSU(2)(QB; Qg) = X Qjg j+j jZS;U(2) vector,ref(QB); Hence, we propose that the re ned Nekrasov partition function of the pure SO(8) gauge ZSO(8)(QB; Qg; Q1; Q2; Q3) = X Qjg j+j jZS;U(2) vector,ref(QB)ZDb;2(SU(2));ref(QB; fQ1g) ZDb;2(SU(2));ref(QB; fQ2g)ZDb;2(SU(2));ref(QB; fQ3g); (6.18) where the Kahler parameters are related to the gauge theory parameters by (3.22) partition function of the pure SO(8) gauge theory until the order of Q31Q23Q33Qg3 for the one-instanton part. O( 4) model. We can also make use of the re ned Db2(SU(2)) matter contribution to compute the Nekrasov partition function of the 5d theory (5.3) which arises from a circle compacti cation of the O( 4) model. In this case, we gauge four re ned partition functions of the Db2(SU(2)) matter and the full partition function is given by ZO( 4)(QB; Qg; Q1; Q2; Q3; Q4) = X Qjg j+j jZS;U(2) vector,ref(QB) ZDb;2(SU(2));ref(QB; fQ1g)ZDb;2(SU(2));ref(QB; fQ2g) ZDb;2(SU(2));ref(QB; fQ3g)ZDb;2(SU(2));ref(QB; fQ4g): order of Q21Q22Q32Q42Qg for the one-string part. We checked that (6.19) agrees with the elliptic genus (5.6) of the O( 4) model until the In this paper, we have proposed a novel method to compute the topological string partition functions/Nekrasov partition functions of 5d theories constructed by the trivalent gluing/gauging. A dual description of 5d pure gauge theories with a gauge group of D; Etype is given by the SU(2) trivalent gauging of three 5d DbN (SU(2)) matter theories. We have proposed a way to apply the topological vertex formalism to the trivalent gauging and successfully calculated their Nekrasov partition functions. We rst computed the partition function of the 5d DbN (SU(2)) theory with non-trivial avor instanton backgrounds, which can be used for a matter contribution for the SU(2) gauging. Then, combining the DbN (SU(2)) matter contributions with the partition function of the SU(2) vector multiplets yields the Nekrasov partition functions of the 5d pure gauge theories of D; E-type gauge groups. This method gives a new way to compute the Nekrasov partition functions and one advantage of this technique is that the higher-order instanton partition functions can be obtained systematically simply by summing over Young diagrams with more boxes. We also performed non-trivial checks with the known results of the SO(8) gauge theory with or without avors and also the pure E6; E7; E8 gauge theories up to some order of the gluing parameters. Moreover, we will see in appendix A that applying a Higgsing prescription to the Nekrasov partition function of a gauge theory with a D-type gauge group and may yield the Nekrasov partition function of a gauge theory with a B-type gauge group. Therefore, with the Higgsing procedure as well as the trivalent gluing method, it is now possible to compute the Nekrasov partition functions of 5d pure gauge theories with a ABCDE gauge group from the topological vertex. Another application of the trivalent gluing method is that we can also compute the Nekrasov partition functions of 5d theories which have a 6d UV completion. In particular three 5d DbN (SU(2)) matter theories. We applied the trivalent gauging method for the Nekrasov partition function with the elliptic genus of the one-string calculated in [39]. We also proposed a 5d description of the O( 3) model and calculated its Nekrasov partition function. Remarkably, we found perfect agreement with the elliptic genus result of the one-string in [40] up to some orders. In every case, the computation for higher instantons can be achieved very systematically and the trivalent gauging method provides a powerful tool to compute their elliptic genera. We also determine a 5d description of another nonHiggsable cluster theory and the 5d theory can be again described by the SU(2) gauging of three 5d theories. Most of the computation in this paper have been done in the unre ned limit. We also argued that it is possible to extend the computation for the re ned topological vertex when we choose the preferred direction to the gluing direction. Indeed we have checked that the trivalent gluing prescription works for the re ned one-instanton partition function for the pure SO(8) gauge theory and also the re ned one-string elliptic genus of the O( 4) model. We expect that the re ned calculation can be generalized to other cases. As for the comparison with the exceptional instantons of 5d theories, we restrict the check to the one-instanton order which can be computed from the general formula (B.16). The higher-instanton partition functions of the exceptional gauge groups have been calculated in [19, 70{72]. However, a direct comparison of the results obtained in this paper with the results in [19, 70{72] may not be straightforward since the explicit expressions in the literature seems not to be compatible with the unre ned limit. It would be interesting to extend the computation for the Nekrasov partition function of the exceptional gauge groups to the re ned one by using the technique in section 6. Similarly the unre ned limit also prevented us from comparing the results with computations from other ing to extend the Nekrasov partition function computation for the 5d descriptions of the O( 6); O( 8); O( 12) models to the re ned partition function computation and perform checks with the results in [61{63]. We expect that our trivalent gauging method has vast applications. In this paper we only consider vector matter of the SO(2N + 4) gauge group. It will be interesting to generalize our method to include matter in di erent representations. Furthermore, our method is applicable to any SU(N ) gluing of possibly non-Lagrangian matter. Finding more dualities among 5d/6d theories like what we argued in section 2 and computing Nekrasov partition functions would be fruitful. We would like to thank Sung-Soo Kim, Kimyeong Lee, Mastato Taki, Futoshi Yagi, and Yuji Tachikawa for useful discussion and conversation. We would also like to thank the RIKEN workshop of the Progress in Mathematical Understanding of Supersymmetric Theories during a part of this work. H.H. would like to thank Korea Institute for Advanced Study for hospitality. K.O. is partially supported by the Programs for Leading Graduate Schools, MEXT, Japan, via the Advanced Leading Graduate Course for Photon Science and by JSPS Research Fellowship for Young Scientists. K.O. gratefully acknowledges support from the Institute for Advanced Study. 5d SO(2N + 3) gauge theory In section 3.3, we have computed the partition function of the SO(2N + 4) gauge theory branch of the SO(2N + 4) gauge theory with M1 + M2 avors, it is possible to realize a 5d SO(2N + 3) gauge theory with Nf 1 avors in the far infrared. A 5-brane web picture for the Higgsing has been presented in [73]. Therefore, one can apply the Higgsing prescription for the Nekrasov partition function of the SO(2N +4) gauge theory with Nf avors to obtain the Nekrasov partition function of the SO(2N + 3) gauge theory with Nf From the Higgsing procedure of the 5-brane web with an O5-plane, the Higgsing from the SO(2N + 4) gauge theory with Nf avors to the SO(2N + 3) gauge theory with Nf avors may be achieved by setting one mass parameter and also one Coulomb branch modulus to be zero. We can for example choose aN+2 = mNf = 0: ; N + 2 and the mass parameters by mi; i = 1; In fact, the tuning condition (A.1) can be directly applied to the Nekrasov partition function of the SO(2N + 4) gauge theory with Nf avors. A similar Higgsing prescription has been used to compute the Nekrasov partition function of the rank one E7 theory [14] and also the rank one E8 theory [15]. In the re ned case, the tuning is not as simple as (A.1) but the parameters are xed to be qt 21 or t 2 q we can directly use the tuning condition of (A.1). Let us see how the condition (A.1) works for the perturbative part. The perturbative partition function of the SO(2N + 4) gauge theory with Nf avors can be written as12 . However, in the unre ned case, ZSPOer(t2N+4);Nf = H(1)N+2 i=1 f=1 Inserting the condition (A.1) into (A.3) yields ZSPOer(t2N+4);Nf jEq. (A.1) = H(1)N i=1 f=1 = H(1) 1ZSPOer(t2N+3);Nf 1 up to op transitions (A.2). Therefore, the perturbative partition function of the SO(2N + 3) gauge theory with Nf 1 avors is reproduced except for the factor H(1) 1 which can be understood as a singlet contribution o the Higgs vacuum. When one includes the instanton partition function, a natural expectation is that ZSO(2N+4);Nf jEq. (A.1) = H(1) 1ZSO(2N+3):Nf 1 2; Nf = 2 by using the localization result (B.17). Assuming that (A.5) is correct, it is then possible to compute the Nekrasov partition function of the 5d SO(2N + 3) gauge theory with avors by combining (3.42) and (A.1) from the relation (A.5). Some formulae for computation In this appendix, we collect formulae which we have used for the calculation of the (re ned) topological vertex as well as the Nekrasov partition function in this paper. 12Note that the op invariance of the partition function of the resolved conifold implies H(Q) = H(Q 1): We always make use of (A.2) to compare the perturbative partition functions from the topological vertex with the perturbative partition function from the localization result. Namely we check the equality between the two perturbative partition functions up to the op transitions (A.2). Fg = with the preferred direction. The preferred direction is denoted by jj. Re ned topological vertex The topological vertex is a powerful tool to compute the all genus topological string amplitude [1, 2] for a Calabi-Yau manifold X3 of the form gtop is the topological string coupling constant, NgC is the genus g Gromov-Witten invariant vertex is parameterized by the topological string coupling and it is possible to further generalize it to the re ned topological vertex by introducing two parameters q; t corresponding the application to toric Calabi-Yau threefolds, it can be also applied to certain non-toric Calabi-Yau threefolds by making use of a Higgsing or topology changing transition from a toric Calabi-Yau threefold [14{16, 25{27]. Here we summarize the rule for applying the re ned topological vertex to a toric Calabi-Yau threefold or a dual 5-brane web. The re ned topological vertex formalism provides us with a method to compute the all genus topological string amplitude on a background of a toric Calabi-Yau threefold by a way which is similar to the method using Feynman diagrams. We rst decompose a toric diagram or 5-brane web into trivalent vertices with three legs. We assign a Young diagram to each leg with some orientation. When the leg is an external leg, then we assign a trivial Young diagram on it. We also need to choose a preferred direction in the diagram and one leg of the re ned topological vertex should be in the preferred direction. We then assign t; q for the other two legs of the vertex. The t; q assignment should be compatible with the gluing rule which we will mention below. Let ; ; be three Young diagrams. When the three legs of a vertex is labeled by a pair of (t; ), (q; ) and with the preferred direction a (i; j) = jt as in gure 26, we assign to the vertex of a 5-brane web the re ned topological vertex s t= (t q )s = (t Here we also de ned Ze (t; q) = Y l (i; j) = i Then we need to glue the vertices for going back to the original 5-brane web. For each gluing of two legs, the assigned Young diagram on one leg should be transposed compared to the Young diagram on the other leg. Then the gluing is done by summing over a Young associated to the two legs with a weight. When we glue along the preferred direction then the weight takes a form of where the framing factor for the preferred direction is When we glue along the non-preferred direction then the weight has a form of where the framing factor for the non-preferred direction is f (t; q) = ( 1)j jt jj 2tjj2 q jj 2jj2 : Kahler parameter for a curve associated to the glued internal line. When we glue along the non-preferred direction, we need to connect a leg on which q is assigned with a leg on which t is assigned. By assigning re ned topological vertex (B.3) for each vertex and also the weights (B.6) or (B.8), the topological string partition function is given by summing all the assigned Young diagrams. The rules for the unre ned version can be obtained simply by setting t = q. An important point is that the topological string partition function for a certain local Calabi-Yau threefold X3 is related to the Nekrasov partition function of a 5d theory with eight supercharges realized from M-theory compacti cation on Xe3 or equivalently on a 5brane web dual to Xe3 [9{13]. In fact, it turns out the topological string partition function calculated from the re ned topological vertex contains contributions that are not present in the Nekrasov partition function and one needs to extract that factor [14, 53{55]. The factor is related to the contribution from strings between parallel external legs. Therefore the factor can be read o from a 5-brane web and for example the extra factor from a web in gure 28 is given by Zextra = We call such a factor extra factor. Therefore, the Nekrasov partition function of a 5d theory can be computed by the topological string partition function of the corresponding Calabi-Yau threefold by dividing it by the extra factor, ZNek = Note that the re ned topological vertex computation does not include the perturbative contribution from vector multiplets in the Cartan subalgebra but it can be easily recovered since it has a general form ZCartan = for a gauge group G. Nekrasov partition function In this section we summarize the result of the Nekrasov partition function for some 5d gauge theories with eight supercharges. For a gauge with a gauge group G, the perturbative partition function of the vector multiplets is given by ZvPeecrt = ZCartan + is a set of positive roots and a = (a1; ; arankG) represents the Coulomb branch moduli in the Cartan subalgebra. The perturbative partition function of hypermultiplets in the representation r is ZhPyeprt = i;j=1 w2r where w is a weight of the representation r. Note that the comparison using the perturbative partition functions (B.13) and (B.14) is done up to op transitions. For the pure gauge theory with a gauge group G, the general result for the one-instanton part has been also known and it is given by [74{78] Z1G-inst = ; are roots of the Lie algebra g, h_ is the dual Coxeter number,13 l is a set of long the expression after putting all the terms over a common denominator takes a form Z1G-inst = 2 + ( 1)1+ne(e + ) a(e(h_ 2) a + 1) Q number of positive roots which satisfy _ + is again a set of positive roots and e = P . n stands for the Next we turn to the result of the instanton partition function from the localization technique [51, 52, 79{82]. The k-instanton partition function can be computed from the index of the one-dimensional ADHM quantum mechanics whose moduli space is given by the corresponding k-instanton moduli space. We here quote the result of the instanton partition function for the SO(N ) gauge theory with hypermultiplets in the vector representation. The k-instanton partition function for the SO(N ) gauge theory with Nf hypermultiplets in the vector representation is given by a contour integral over the dual Sp(k) gauge Zk{inst = 14In the case when G is simply-laced, l is a set of all the roots. for N = 2n + 1. Here the notation 2 sinh( x y) means 2 sinh( x y) = 2 sinh(x + y)2 sinh( x + y)2 sinh( x y). ai; i = 1; ; n are the Coulomb branch hypermultiplets in the vector representation and it is given by Zhyp = Y Y 2 sinh I=1 f=1 Kirwan residue rule [82]. Schur functions Here we summarize the formulas on Schur functions which is needed to perform the topological vertex computations. Schur polynomials s (x1; ; xn) with nite variables can be 1 I<J k 2 sinh QIk=1 Qin=1 2 sinh I 2J +2 + QIk=1 2 sinh ( 1 I<J k 2 sinh I J + 2 Zvec = for N = 2n and Zvec = SO(N ). More concretely, jWkj = 2kk!. Finally jWkj is the order of the Weyl group of the Sp(k) which is the dual gauge group of The contour integral (B.17) can be systematically evaluated by so-called the Je ery 1 I<J k 2 sinh I 2J +2 + QIk=1 2 sinh ( QIk=1 2 sinh I2+ + QIk=1 Qin=1 2 sinh 1 I<J k 2 sinh I J + 2 = 1 ; L is a integer partition. Schur polynomials have a scaling property In particular, we often use principal specialization of Schur function, de ned by ; xn) = (A )ij = ; axn) = aj js (x1; s (q ) = s (q1=2; q3=2; q5=2; Stanley's hook-length formula [83] says s (q ) = q jj2jj Y where u runs through boxes of the Young diagram , and hook(u) is a(u) + `(u) + 1. The important point is that the righthand side is nite product and thus this formula is exact with respect to q. This formula is the reason why we can compute partition functions from topological vertices exactly with respect to the exponentiated We also encounter Schur functions with arguments like = ( 1; use of the formula s (x; y) = ) = s (q1=2 1; q3=2 2; qL0=2 L0 ; q(L0+1)=2 ; L0) is another partition. To compute this function explicitly, we make where x; y are sets of variables and c ; are Littlewood-Richardson coe cients. Set x to be the rst L0 variables of (B.26) and y to be the remaining, and use (B.22) for the former and (B.25), (B.24) for the latter. Using (B.27) repeatedly, we can also compute Schur We also encounter two variants of Schur functions, which are skew Schur functions s = (x) = X c ; s (x); s (xjy) = and super Schur functions is not included in . = ?, and 0 when A Mathematica implementation which automates computations of Schur functions like (B.28) and those generalization to skew and super Schur functions is available online at https://github.com/kantohm11/SchurFs. In the main part of this paper, we used the following formulas [84] Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. hep-th/0207114 [INSPIRE]. Phys. 254 (2005) 425 [hep-th/0305132] [INSPIRE]. [hep-th/0701156] [INSPIRE]. [hep-th/9603150] [INSPIRE]. D-branes, JHEP 05 (2005) 039 [hep-th/0502061] [INSPIRE]. eld theories, Nucl. Phys. B 483 (1997) 229 [hep-th/9609070] [INSPIRE]. theory, Nucl. Phys. B 497 (1997) 155 [hep-th/9609071] [INSPIRE]. theories and degenerations of Calabi-Yau spaces, Nucl. Phys. B 497 (1997) 56 [hep-th/9702198] [INSPIRE]. Math. Phys. 7 (2003) 457 [hep-th/0212279] [INSPIRE]. Theor. Math. Phys. 10 (2006) 1 [hep-th/0306032] [INSPIRE]. [hep-th/0310235] [INSPIRE]. genera, JHEP 03 (2008) 069 [hep-th/0310272] [INSPIRE]. [arXiv:0710.1776] [INSPIRE]. JHEP 06 (2014) 014 [arXiv:1310.3854] [INSPIRE]. (2015) 093 [arXiv:1409.0571] [INSPIRE]. Commun. Math. Phys. 265 (2006) 201 [hep-th/0505192] [INSPIRE]. surface defects, JHEP 01 (2013) 022 [arXiv:1207.3577] [INSPIRE]. Math. Phys. 98 (2011) 225 [arXiv:1006.0977] [INSPIRE]. [arXiv:1007.2524] [INSPIRE]. Math. Phys. 333 (2015) 187 [arXiv:1105.5117] [INSPIRE]. arXiv:1210.2733 [INSPIRE]. 03 (2017) 112 [arXiv:1609.07381] [INSPIRE]. 2015 (2015) 083B02 [arXiv:1504.03672] [INSPIRE]. JHEP 01 (2017) 093 [arXiv:1607.07786] [INSPIRE]. Phys. B 504 (1997) 239 [hep-th/9704170] [INSPIRE]. and grid diagrams, JHEP 01 (1998) 002 [hep-th/9710116] [INSPIRE]. [hep-th/9711013] [INSPIRE]. Nucl. Phys. B 747 (2006) 329 [hep-th/0510060] [INSPIRE]. conformal theories, Commun. Math. Phys. 275 (2007) 209 [hep-th/0510251] [INSPIRE]. polynomials, Commun. Math. Phys. 319 (2013) 147 [arXiv:1110.3740] [INSPIRE]. [35] J.J. Heckman, D.R. Morrison and C. Vafa, On the classi cation of 6D SCFTs and generalized ADE orbifolds, JHEP 05 (2014) 028 [Erratum ibid. 06 (2015) 017] [arXiv:1312.5746] [INSPIRE]. [36] M. Del Zotto, J.J. Heckman, A. Tomasiello and C. Vafa, 6d conformal matter, JHEP 02 [37] J.J. Heckman, D.R. Morrison, T. Rudelius and C. Vafa, Atomic classi cation of 6D SCFTs, Fortsch. Phys. 63 (2015) 468 [arXiv:1502.05405] [INSPIRE]. arXiv:1608.03919 [INSPIRE]. Phys. 63 (2015) 294 [arXiv:1412.3152] [INSPIRE]. minimal conformal matter, JHEP 08 (2015) 097 [arXiv:1505.04439] [INSPIRE]. JHEP 01 (2013) 191 [arXiv:1210.2886] [INSPIRE]. type Dp(G), JHEP 04 (2013) 153 [arXiv:1303.3149] [INSPIRE]. eld theories, nontrivial xed points and string dynamics, Phys. Lett. B 388 (1996) 753 [hep-th/9608111] [INSPIRE]. ve-dimensional En eld theories, JHEP 03 (1999) 006 [hep-th/9902179] [INSPIRE]. duality, JHEP 04 (2012) 105 [arXiv:1112.5228] [INSPIRE]. Phys. B 497 (1997) 173 [hep-th/9609239] [INSPIRE]. 163 [arXiv:1507.03860] [INSPIRE]. JHEP 10 (2016) 126 [arXiv:1512.08239] [INSPIRE]. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE]. 244 (2006) 525 [hep-th/0306238] [INSPIRE]. and the 5d superconformal index, JHEP 01 (2014) 079 [arXiv:1310.2150] [INSPIRE]. junctions, JHEP 01 (2014) 175 [arXiv:1310.3841] [INSPIRE]. duality in 5d supersymmetric gauge theory, JHEP 03 (2014) 112 [arXiv:1311.4199] [63] M. Del Zotto and G. Lockhart, On exceptional instanton strings, arXiv:1609.00310 three dimensional Sicilian theories, JHEP 09 (2014) 185 [arXiv:1403.2384] [INSPIRE]. [75] C.A. Keller, N. Mekareeya, J. Song and Y. Tachikawa, The ABCDEFG of instantons and [76] G. Zafrir, Instanton operators and symmetry enhancement in 5d supersymmetric USp, SO [77] M. Billo, M. Frau, F. Fucito, A. Lerda and J.F. Morales, S-duality and the prepotential in classical gauge groups, JHEP 05 (2004) 021 [hep-th/0404125] [INSPIRE]. [1] A. Iqbal , All genus topological string amplitudes and ve-brane webs as Feynman diagrams , [2] M. Aganagic , A. Klemm , M. Marin ~o and C. Vafa , The topological vertex, Commun . Math. [3] H. Awata and H. Kanno , Instanton counting, Macdonald functions and the moduli space of [4] A. Iqbal , C. Kozcaz and C. Vafa , The re ned topological vertex , JHEP 10 ( 2009 ) 069 [5] E. Witten , Phase transitions in M-theory and F-theory, Nucl . Phys . B 471 ( 1996 ) 195 [6] D.R. Morrison and N. Seiberg , Extremal transitions and ve-dimensional supersymmetric [7] M.R. Douglas , S.H. Katz and C. Vafa , Small instantons, del Pezzo surfaces and type-I' [8] K.A. Intriligator , D.R. Morrison and N. Seiberg , Five-dimensional supersymmetric gauge [9] A. Iqbal and A.-K. Kashani-Poor , Instanton counting and Chern-Simons theory, Adv . Theor. [10] A. Iqbal and A.-K. Kashani-Poor , SU(N ) geometries and topological string amplitudes , Adv. [11] T. Eguchi and H. Kanno , Topological strings and Nekrasov's formulas , JHEP 12 ( 2003 ) 006 [12] T.J. Hollowood , A. Iqbal and C. Vafa , Matrix models, geometric engineering and elliptic [13] M. Taki , Re ned topological vertex and instanton counting , JHEP 03 ( 2008 ) 048 [14] H. Hayashi , H.-C. Kim and T. Nishinaka , Topological strings and 5d TN partition functions , [15] H. Hayashi and G. Zoccarato , Exact partition functions of Higgsed 5d TN theories , JHEP 01 [16] H. Hayashi and G. Zoccarato , Topological vertex for Higgsed 5d TN theories , JHEP 09 [17] D.-E. Diaconescu , B. Florea and N. Saulina , A vertex formalism for local ruled surfaces , [18] D.-E. Diaconescu and B. Florea , The ruled vertex and nontoric del Pezzo surfaces , JHEP 12 [19] D. Gaiotto and S.S. Razamat , Exceptional indices, JHEP 05 ( 2012 ) 145 [arXiv:1203.5517] [20] D. Gaiotto , L. Rastelli and S.S. Razamat , Bootstrapping the superconformal index with [21] T. Dimofte , S. Gukov and L. Hollands , Vortex counting and Lagrangian 3-manifolds , Lett. [22] M. Taki , Surface operator, bubbling Calabi-Yau and AGT relation, JHEP 07 ( 2011 ) 047 [23] M. Aganagic and S. Shakirov , Knot homology and re ned Chern-Simons index , Commun. [24] M. Aganagic and S. Shakirov , Re ned Chern-Simons theory and topological string , [25] H. Hayashi and G. Zoccarato , Partition functions of web diagrams with an O7 -plane , JHEP [26] S.-S. Kim , M. Taki and F. Yagi , Tao probing the end of the world , Prog. Theor. Exp. Phys. [27] H. Hayashi , S.-S. Kim , K. Lee and F. Yagi , Equivalence of several descriptions for 6d SCFT, [28] O. Aharony and A. Hanany , Branes, superpotentials and superconformal xed points , Nucl. [29] O. Aharony , A. Hanany and B. Kol , Webs of (p; q) ve-branes, ve-dimensional eld theories [30] N.C. Leung and C. Vafa , Branes and toric geometry, Adv. Theor. Math. Phys. 2 ( 1998 ) 91 [31] C. Romelsberger , Counting chiral primaries in N = 1, D = 4 superconformal eld theories , [32] J. Kinney , J.M. Maldacena , S. Minwalla and S. Raju , An index for 4 dimensional super [33] A. Gadde , L. Rastelli , S.S. Razamat and W. Yan , Gauge theories and Macdonald [34] D.R. Morrison and W. Taylor , Classifying bases for 6D F-theory models, Central Eur . J. [39] B. Haghighat , A. Klemm , G. Lockhart and C. Vafa , Strings of minimal 6d SCFTs , Fortsch. [40] H.-C. Kim , S. Kim and J. Park, 6d strings from new chiral gauge theories , [41] R. Feger and T.W. Kephart , LieART | a Mathematica application for Lie algebras and representation theory, Comput . Phys. Commun . 192 ( 2015 ) 166 [arXiv:1206.6379] [42] H. Hayashi , S.-S. Kim , K. Lee , M. Taki and F. Yagi , A new 5d description of 6d D-type [43] S. Cecotti and M. Del Zotto , In nitely many N = 2 SCFT with ADE avor symmetry , [44] S. Cecotti , M. Del Zotto and S. Giacomelli , More on the N = 2 superconformal systems of [45] N. Seiberg , Five-dimensional SUSY [46] O. DeWolfe , A. Hanany , A. Iqbal and E. Katz , Five-branes, seven-branes and [47] L. Bao , E. Pomoni , M. Taki and F. Yagi , M 5- branes, toric diagrams and gauge theory [48] S.H. Katz , A. Klemm and C. Vafa , Geometric engineering of quantum eld theories , Nucl. [49] O. Bergman and G. Zafrir , 5d xed points from brane webs and O7-planes , JHEP 12 ( 2015 ) [50] H. Hayashi , S.-S. Kim , K. Lee , M. Taki and F. Yagi , More on 5d descriptions of 6d SCFTs, [51] N.A. Nekrasov , Seiberg-Witten prepotential from instanton counting , Adv. Theor. Math. [52] N. Nekrasov and A. Okounkov , Seiberg-Witten theory and random partitions, Prog . Math. [53] O. Bergman , D. Rodr guez-Gomez and G. Zafrir , Discrete [54] L. Bao , V. Mitev , E. Pomoni , M. Taki and F. Yagi , Non-Lagrangian theories from brane [55] O. Bergman , D. Rodr guez-Gomez and G. Zafrir , 5 - brane webs, symmetry enhancement and [56] D.R. Morrison and C. Vafa , Compacti cations of F-theory on Calabi-Yau threefolds . 1 , Nucl . [57] D.R. Morrison and C. Vafa , Compacti cations of F-theory on Calabi-Yau threefolds . 2 , Nucl . [58] B. Haghighat , A. Iqbal , C. Kozcaz , G. Lockhart and C. Vafa , M-strings, Commun. Math. [59] B. Haghighat , C. Kozcaz , G. Lockhart and C. Vafa , Orbifolds of M-strings, Phys. Rev. D 89 [60] C. Vafa , Evidence for F-theory, Nucl . Phys . B 469 ( 1996 ) 403 [hep-th /9602022] [INSPIRE]. [61] A. Gadde , S.S. Razamat and B. Willett , \ Lagrangian" for a non-Lagrangian eld theory with N = 2 supersymmetry , Phys. Rev. Lett . 115 ( 2015 ) 171604 [arXiv:1505.05834] [INSPIRE]. [62] P. Putrov , J. Song and W. Yan , ( 0 ; 4) dualities, JHEP 03 ( 2016 ) 185 [arXiv:1505.07110] [64] A. Iqbal and A.-K. Kashani-Poor , The vertex on a strip , Adv. Theor. Math. Phys. 10 ( 2006 ) [65] Y. Konishi and S. Minabe , Flop invariance of the topological vertex, Int . J. Math. 19 ( 2008 ) [66] S. Gukov , A. Iqbal , C. Kozcaz and C. Vafa , Link homologies and the re ned topological vertex, Commun . Math. Phys. 298 ( 2010 ) 757 [arXiv:0705.1368] [INSPIRE]. [67] M. Taki , Flop invariance of re ned topological vertex and link homologies , arXiv:0805. 0336 [68] H. Awata and H. Kanno , Macdonald operators and homological invariants of the colored Hopf link , J. Phys. A 44 ( 2011 ) 375201 [arXiv:0910.0083] [INSPIRE]. [69] A. Iqbal and C. Kozcaz , Re ned topological strings and toric Calabi-Yau threefolds , [70] C.A. Keller and J. Song , Counting exceptional instantons , JHEP 07 ( 2012 ) 085 [71] A. Hanany , N. Mekareeya and S.S. Razamat , Hilbert series for moduli spaces of two [72] S. Cremonesi , A. Hanany , N. Mekareeya and A. Za aroni, Coulomb branch Hilbert series and [73] G. Zafrir , Brane webs and O5-planes , JHEP 03 ( 2016 ) 109 [arXiv:1512.08114] [INSPIRE]. [74] S. Benvenuti , A. Hanany and N. Mekareeya , The Hilbert series of the one instanton moduli [78] M. Billo , M. Frau , F. Fucito , A. Lerda and J.F. Morales , S-duality and the prepotential of N = 2? theories (II): the non-simply laced algebras , JHEP 11 ( 2015 ) 026 [79] M. Marin~o and N. Wyllard , A note on instanton counting for N = 2 gauge theories with [80] N. Nekrasov and S. Shadchin , ABCD of instantons, Commun. Math. Phys. 252 ( 2004 ) 359 [81] S. Shadchin , On certain aspects of string theory/gauge theory correspondence , [82] C. Hwang , J. Kim , S. Kim and J. Park, General instanton counting and 5d SCFT , JHEP 07 ( 2015 ) 063 [Addendum ibid . 04 ( 2016 ) 094] [arXiv:1406.6793] [INSPIRE]. [83] R.P. Stanley , Theory and application of plane partitions . Part 2, Studies Appl. Math. 50 [84] I.G. Macdonald , Symmetric functions and Hall polynomials , Oxford University Press, Oxford


This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2FJHEP06%282017%29078.pdf

Hirotaka Hayashi, Kantaro Ohmori. 5d/6d DE instantons from trivalent gluing of web diagrams, Journal of High Energy Physics, 2017, 1-66, DOI: 10.1007/JHEP06(2017)078