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
0 731 Hongo , Bunkyoku, Tokyo 1130033 , Japan
1 1 Einstein Drive , Princeton, NJ 08540 , U.S.A
2 411 Kitakaname , Hiratsukashi, Kanagawa 2591292 , 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 , FTheory
We propose a new prescription for computing the Nekrasov partition functions of vedimensional theories with eight supercharges realized by gauging nonperturbative avor symmetries of three vedimensional 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 vedimensional 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 sixdimensional 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; Etype 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 Etype 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 nonHiggsable clusters 5.1 5.2 5.3
O( n) model with n = 6; 8; 12
Another nonHiggsable 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 CalabiYau threefolds [1{4]. One can compute the full
topological string partition function like a Feynman diagramlike method and it can yield the full
list of the GromovWitten invariants and the GopakumarVafa invariants of a toric
CalabiYau threefold in principle. The topological string partition function also has a physical
interpretation through string theory or Mtheory. When we consider Mtheory on a
noncompact CalabiYau threefold with a compact base that is contractible, the low energy
e ective eld theory gives rise to a
vedimensional (5d) theory with eight supercharges
which has a ultraviolet (UV) completion [5{8]. Then M2branes wrapping various
holomorphic curves in the CalabiYau threefold yield BPS particles in the 5d theory. Therefore,
the curve counting for a noncompact CalabiYau 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 nontoric CalabiYau threefolds [14{16].1 The
new method makes use of a Higgs prescription of the superconformal index in [19, 20].2
In fact, some nontoric CalabiYau threefold can be obtained from a topology changing
transition or a Higgsing from a toric CalabiYau threefold. Then applying the Higgsing
prescription for the topological string partition function of the \UV" CalabiYau
threefold gives rise to the topological string partition function of the \infrared" (IR) nontoric
CalabiYau 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 sixdimensional (6d) UV completion, and nontrivial
checks with the elliptic genus of the 6d selfdual strings have been done in [26, 27].
Although the new method enlarges the space of noncompact CalabiYau threefolds
to which we can apply the topological vertex, there is still a large class of noncompact
CalabiYau threefolds to which we have not yet known how to apply the topological vertex.
An interesting class of such CalabiYau 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 DEtype. In
fact, it turns out that the dual description is given by gauging the diagonal part of avor
1There is also another vertexlike approach to compute the unre ned topological string amplitudes for
some nontoric CalabiYau 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 CalabiYau threefolds or equivalently 5brane webs [28{30]. Gauging the avor
symmetries of three 5d theories is a natural generalization but goes beyond the standard
picture of 5brane 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
vedimensional partition
functions is quite nontrivial compared with the fourdimensional 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 nonHiggsable 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 selfdual strings in the 6d SCFT. We will give a nontrivial check between the
elliptic genus computed in [39]. We will further propose a 5d description of the 6d minimal
see a nontrivial 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 Etype. 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
nontrivial 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 nonHiggsable 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; Etype gauge group
Fivedimensional gauge theories with eight supercharges can be realized by compactifying
Mtheory on a singular CalabiYau threefolds X3 [5{8]. When the CalabiYau threefold X3
has a Gtype surface singularity over a sphere CB, then the low energy e ective eld theory
from the Mtheory 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 CalabiYau 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 Gtype singularity. We denote the
consists of spheres alighted along the Dynkindiagram 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 M2brane wrapping a curve C in Fg yields a massive Wboson for the
of g in the 5d gauge theory. Therefore, the size of C is a Coulomb branch modulus.
On the other hand, an M2brane 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 twodimensional 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 noncompact CalabiYau threefold whose compact base is given by the complex
surface Sg. More generally, Mtheory on a noncompact CalabiYau 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 5brane
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 weblike 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 CalabiYau geometry Xe3
bration over the base CB. The
nonAbelian 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 ChernSimons (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) 5brane
theory. The slope of the (p; q) 5brane is pq in the twodimensional (x5; x6)space. In particular,
a horizontal line represents a horizontal line and a vertical line represents an NS5brane. Since
the structure of the 5brane only appears in the (x5; x6)plane, we only write the twodimensional
plane for depicting a 5brane.
on an orbifold C3=
where the orbifold action of
avor symmetry arises nonperturbatively 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 Mtheory
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 5brane web. A 5brane web is
a dual con guration of a certain CalabiYau threefold Xe3 [30]. The directions which the
5brane extend are summarized in table 1. It is also useful to introduce 7branes attached to
the ends of external 5branes in a 5brane web con guration to read o the avor symmetry
of a 5d theory realized on a 5brane web [46]. The 5brane 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 Sdual 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 Sdual picture, the avor symmetry of the Dbp(SU(2)) theory is realized
perturbatively as background gauge
eld on two D7branes attached to the ends of the
external 5branes 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) ArgyresDouglas theory.
p CS level. We have
p D5branes which lie in the horizontal direction. The parallel two external NS5branes imply the
nonperturbative SU(2)
avor symmetry. Right: the Sdual con guration to the 5brane web on
the left. Namely the 5brane web for the Dbp(SU(2)) theory.
SU(2) avor symmetry appears nonperturbatively in the pure SU(p) gauge theory since
it is associated to a symmetry on the two (0; 1) 7branes or the two NS5branes.
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 nonLagrangian
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 weblike 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 weblike
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 5brane 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 5brane 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 5brane web with an O5plane as
in the left gure in gure 3. The 5brane 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 5brane web for the SO(4) gauge theory with the two 5brane webs for the pure
SU(2) gauge theory as in gure 3. Then the weblike gure on the right in gure 3 may be
considered as an Sdual 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 5brane web of the pure SO(2N +4)
gauge theory, M1 + M2 hypermultiplets in the vector representation can be added by
introducing M1 avor 5branes on the left and M2 avor 5branes 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 5brane web of the pure SO(2N + 4) gauge theory using an
O5plane. The right gure is a weblike description by replacing the 5brane for the SO(4) gauge
theory part in the left gure with the two 5branes webs of the pure SU(2) gauge theory with no
discrete theta angle. Now the three 5brane 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 5brane webs are connected by the trivalent gluing.
gure 4 but it is still possible to write down a 5brane web by introducing a con guration
of 5branes jumping over other 5branes [49]. With the 5brane 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 weblike 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 5brane web, the application of the topological vertex to the 5brane web
gives rise to its Nekrasov partition function [9{13]. However, it is not possible to simply
apply the topological vertex to the weblike 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 nonperturbatively 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 Sdual 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 nontrivial
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 weblike 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 nontrivial 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 selfintersection 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 nonHiggsable 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 nonHiggsable cluster
theories which contain multiple tensor multiplets or more than one base curves [34, 35].
The 6d theories again have no
avor symmetry. The Ftheory 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 nonHiggsable clusters, the last entry in table 7 has an orbifold
description [38]. The Ftheory 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 Mtheory
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 5brane 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 nonHiggsable 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 eightdimensional 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 eightdimensional 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 5brane 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 wellde 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
oneinstanton 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 onestring 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 nontrivial 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; Etype gauge
groups. This method gives a new way to compute the Nekrasov partition functions and
one advantage of this technique is that the higherorder instanton partition functions can
be obtained systematically simply by summing over Young diagrams with more boxes. We
also performed nontrivial 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 Dtype gauge group and
may yield the Nekrasov partition function of a gauge theory with a Btype 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 onestring 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
onestring 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 oneinstanton partition function for the
pure SO(8) gauge theory and also the re ned onestring 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 oneinstanton order which can be computed from the general formula (B.16).
The higherinstanton 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 nonLagrangian 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 SungSoo 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 5brane 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 5brane web with an O5plane, 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 CalabiYau manifold X3 of the form
gtop is the topological string coupling constant, NgC is the genus g GromovWitten 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 CalabiYau threefolds, it can be also applied to certain nontoric
CalabiYau threefolds by making use of a Higgsing or topology changing transition from
a toric CalabiYau threefold [14{16, 25{27]. Here we summarize the rule for applying the
re ned topological vertex to a toric CalabiYau threefold or a dual 5brane 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 CalabiYau threefold by
a way which is similar to the method using Feynman diagrams. We rst decompose a toric
diagram or 5brane 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 5brane 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 5brane 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 nonpreferred direction then the weight has a form of
where the framing factor for the nonpreferred 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 nonpreferred 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
CalabiYau threefold X3 is related to the Nekrasov partition function of a 5d theory with
eight supercharges realized from Mtheory 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 5brane 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 CalabiYau 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 oneinstanton part has been
also known and it is given by [74{78]
Z1Ginst =
; 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
Z1Ginst =
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 kinstanton partition function can be computed from the index
of the onedimensional ADHM quantum mechanics whose moduli space is given by the
corresponding kinstanton 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 kinstanton 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 simplylaced, 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 socalled 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 hooklength 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 LittlewoodRichardson 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.
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