Duality walls and defects in 5d \( \mathcal{N}=1 \) theories
Received: August
Published for SISSA by Springer
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c The Authors. 0 1
0 31 Caroline Street North , Waterloo, ON N2L 2Y5 Canada
1 Perimeter Institute for Theoretical Physics
We propose an explicit description of duality walls" which encode at low energy the global symmetry enhancement expected in the UV completion of certain dimensional gauge theories. The proposal is supported by explicit localization computations and implies that the instanton partition function of these theories satis es novel and unexpected integral equations.
Duality in Gauge Field Theories; Field Theories in Higher Dimensions; Su

Duality walls and defects in 5d
= 1 theories
3 Index calculations 4 Wilson loops
SU(N )N theories
Example: SU(N )N Nf =2 theories with Nf avors
Hemispheres with Wilson loop insertions
Example: SU(2) theories
Example: SU(3) theories
Duality and 't Hooft surfaces
Higgsing IN;N
Codimension 2 defects
Higgsing in the absence of a duality wall
Higgsing in the presence of a duality wall
Enhanced symmetry of SU(N + 1) theory
From Sp(N ) to exotic SU(N + 1)
Duality walls between Sp(N ) and SU(N + 1) theories
1 Introduction
Duality walls between SU(N ) gauge theories
Pure N = 1 SU(N )N gauge theory
Domain wall actions
SU(N )N Nf =2 SQCD with Nf < 2N
Duality walls for SU(N ) with Nf = 2N
Exceptional symmetries in SU(2) theories
A 5d Nekrasov's instanton partition function
A.1 SU(N ) partition function
A.2 Sp(N ) partition function
A.3 Hypermultiplets
B Partition functions of exotic SU(3) theory
B.1 Superconformal indices
Fivedimensional superconformal eld theories are a particularly rich subject of
investigation (see [1{5] for seminal work on the subject). The only constructions available for
these theories involve brane constructions, in particular quarterBPS webs of vebranes
in IIB string theory. Some of the
vedimensional SCFTs admit mass deformations to
vedimensional gauge theories, with the inverse gauge coupling playing the role of mass
deformation parameter. Several protected quantities in the
vedimensional SCFT are
computable directly from the lowenergy gaugetheory description [6].
More precisely, the space of mass deformations of the UV SCFT is usually decomposed
into chambers, which ow in the IR to distinctlooking gauge theories, or to the same gauge
theory but with di erent identi cations of the parameters. With some abuse of language,
these distinct IR theories may be thought of as being related by an \UV duality", in the
sense that protected calculations in these IR theories should match [7].
In such a situation, one may de ne the notion of \duality walls" between the di erent
IR theories [8]. These are halfBPS interfaces which we expect to arise from RG
starting from Januslike con gurations, where the mass deformation parameters vary
continuously in the UV, interpolating between two chambers. Duality walls between di erent
chambers should compose appropriately.
Furthermore, if we have some BPS defect in the UV SCFT, we have in principle
a distinct IR image of the defect in each chamber, each giving the same answer when
inserted in protected quantities. The duality walls should intertwine, in an appropriate
sense, between these images.
In this paper we propose candidate duality walls for a large class of quiver gauge
theories of unitary groups.1 The UV completion of these gauge theories has a conjectural
enhanced global symmetry whose Cartan generators are the instanton number symmetries
of the lowenergy gauge theory. The chambers in the space of real mass deformations
dual to these global symmetries are Weyl chambers and the duality walls generate Weyl
re ections relating di erent chambers.
The duality walls admit a Lagrangian description in the low energy gauge theory. The
fusion of interfaces reproduces the expected relations for the Weyl group generators thanks
to a beautiful collection of Seiberg dualities. This is the
rst nontrivial check of our
proposal. The second set of checks involve the computation of protected quantities.
The duality walls we propose give a direct physical interpretation to a somewhat
unfamiliar object: elliptic Fourier transforms (see [10] and references within). These are
invertible integral transformations whose kernel is built out of elliptic gamma functions. We
interpret the integral kernel as the superconformal index of the fourdimensional degrees
of freedom sitting at the duality interface and the integral transform as the action of the
duality interface on more general boundary conditions for the vedimensional gauge theory.
The integral identity which encodes the invertibility of the elliptic Fourier transform follows
from the corresponding Seiberg duality relations.
1Duality walls of the same kind, for 5d gauge theories endowed with a sixdimensional UV completion,
appeared rst in [9].
It follows directly from the localization formulae on the S4
S1 and the de nition
of a duality wall that the corresponding elliptic Fourier transform acting on the instanton
partition function of the gauge theory should give back the same partition function, up to
the Weyl re ection of the instanton fugacity. This is a surprising, counterintuitive integral
relation which should be satis ed by the instanton partition function. Amazingly, we nd
that this relation is indeed satis ed to any order in the instanton expansion we cared to
check. This is a very strong test of our proposal.
Experimentally, we
nd that this is the rst example of an in nite series of integral
identities, which control the duality symmetries of Wilson line operators. These relations
suggest how to assemble naive gauge theory Wilson line operators into objects which can
be expected to have an ancestor in the UV SCFT which is invariant under the full global
symmetry group.
We also identify a few boundary conditions and interfaces in the gauge theory which
transform covariantly under the action of the duality interface and could thus be good
candidates for symmetric defects in the UV SCFT. We brie y look at duality properties of
defects in codimension two and three as well.
Finally, we attempt to give a physical explanation to another instance of elliptic Fourier
transform which we found in the literature, which schematically appears to represent an
interface between an Sp(N ) and an SU(N + 1) gauge theories. We nd that the AC elliptic
Fourier transform maps the instanton partition function of an Sp(N ) gauge theory into the
instanton partition function of an exotic version of SU(N + 1) gauge theory with the same
After this work was completed, we received [11, 12] which have some overlap with the
number of avors.
last section of this paper.
Duality walls between SU(N ) gauge theories
Pure N = 1 SU(N )N gauge theory
Our rst and key example of duality wall encodes the UV symmetries of a pure
This gauge theory is expected to be a lowenergy description of a 5d SCFT with SU(2)
global symmetry, deformed by a real mass associated to the Cartan generator of SU(2).
In turn, the SCFT can be engineered by a BPS
vebrane web involving four semiin nite
external legs: two parallel NS5 branes, a ( 1; N ) and a ( 1;
N ) vebranes. The SU(2)
global symmetry is associated to the two parallel NS5 branes. See gure 1.
The mass deformation breaks SU(2) to a U(1) subgroup, which is identi ed with the
instanton U(1)in global symmetry of the SU(N ) gauge theory, whose current is the instanton
The Weyl symmetry acts as m !
m and the corresponding duality wall should relate
two copies of the same gauge theory, glued at the interface in such to preserve the
antidiagonal combination of the U(1)in instanton global symmetries on the two sides of the
The gauge theory is supported on the bundle of N parallel D5 branes. After removing the centre
of mass, the only nonnormalizable deformation is the separation m between the NS5 branes.
sides of an interface as open circles and the bifundamental matter as an arrow between them. The
extra baryonic coupling is denoted as a black dot over the arrow.
We propose the following setup: a domain wall de ned by Neumann b.c. for the
SU(N )N gauge theory on the two sides of the wall, together with a set of bifundamental
4d chiral multiplets q living at the wall, coupled to an extra chiral multiplet b by a 4d
W = b det q :
See gure 2 for a schematic depiction of the duality wall.
This system is rife with potential gauge, mixed and global anomalies at the interface,
which originate from the 4d degrees of freedom, from the boundary conditions of the 5d
gauge elds and and from anomaly in ow from the bulk ChernSimons couplings.
The cubic gauge anomaly cancels out beautifully: the bifundamental chiral multiplets
behave as N fundamental chiral multiplets for the gauge group on the right of the wall,
giving N units of cubic anomaly, which cancel against the anomaly in ow from the N units
of vedimensional ChernSimons coupling. Similarly, we get
N units of cubic anomaly
for the gauge group on the left of the wall, which also cancel against the anomaly in ow
from the N units of vedimensional ChernSimons coupling.
The bifundamental chiral multiplets also contribute to a mixed anomaly between the
bulk gauge elds and the baryonic U(1)B symmetry which rotates the bifundamental elds
involving the left gauge elds has the same sign and magnitude as the anomaly involving
gauge theory has N
avors and at low energy it glues the two 5d gauge groups together.
the right gauge elds. Both are the same as the anomaly which would be associated to a
single fundamental boundary chiral of charge 1.
We can make a nonanomalous U(1) global symmetry by combining U(1)B with U(1)in
from both sides of the wall. Under U(1) a boundary baryon operator will have the same
charge as an instanton particle on the left side of the wall, or an antiinstanton particle on
the right side of the wall. In particular, the proposed duality wall glues together U(1)in on
the two sides of the wall with opposite signs and thus has a chance to implement the Z2
We can also de ne a nonanomalous Rsymmetry by combining the Cartan generator
of the bulk SU(2)R symmetry and a boundary symmetry which gives charge 0 to the
bifundamentals, and thus charge 2 to b. The cancellation of the mixed gauge anomaly
proceeds as follows: the bulk gauge elds with Neumann b.c. contribute half as much as
4d SU(N ) gauge elds would contribute and thus the Rsymmetry assignment is the same
as for a 4d SQCD with Nf = N .
A neat check of this proposal is that two concatenated duality walls will annihilate
in the IR. Far in the IR, a pair of consecutive duality walls looks like a single interface
supporting fourdimensional SU(N ) gauge elds which arise from the compacti cation of
vedimensional SU(N )N gauge theory on the interval. Together with the quarks
associated to each duality wall, that gives us a fourdimensional S(N ) gauge theory with
N avors, deformed by a superpotential coupling
which sets to zero the two baryon operators det q and det q~.
This fourdimensional theory has a wellknown lowenergy behaviour: it can be
debaryons B = det q, B~ = det q~, subject to a constraint
W = b det q + ~b det q~
BB~ = 2N :
Because of the bB + ~bB~ superpotential couplings, we can restrict ourselves to the locus
freedom required to Higgs the left and right
vedimensional theories back together, and
thus ow in the far IR back to a trivial interface. This is the expected behaviour for Z2
Domain wall actions
We should be able to use the domain walls to de ne a Z2 duality action on U(1)inpreserving
halfBPS boundary conditions for the SU(N )N
vedimensional gauge theory. As the
vedimensional gauge theories are IR free, we can describe most boundary conditions in terms
of their boundary degrees of freedom, which are in general some fourdimensional SCFTs
equipped with an SU(N ) and an U(1)in global symmetries with speci c cubic anomalies.
The exceptions are boundary conditions which (partially) break the gauge symmetry at
units of cubic 't Hooft anomaly, a U(1)@ global symmetry with a mixed 't Hooft anomaly
with the SU(N ) global symmetry equal to the contribution of a single fundamental chiral
eld of charge 1 and an Rsymmetry with a mixed 't Hooft anomaly with the SU(N ) global
symmetry equal to the contribution of N quarks of Rcharge 0. Such a theory can be used
to de ne a boundary condition for the 5d SU(N )N gauge theory which preserves a U(1)
symmetry, diagonal combination of U(1)in and U(1)@, and an Rsymmetry.
The action of the duality wall on this boundary condition gives a new theory B0 built
from B by adding N antifundamental chiral multiplets q of SU(N ), gauging the overall
the same type of mixed 't Hooft anomalies as we required for B (involving a new choice of
U(1)@ global symmetry).
In case of boundary conditions which break the gauge group to some subgroup H, we
can apply a similar transformation, which only gauges the H subgroup of SU(N ). For
example, the duality wall maps Dirichlet boundary conditions, which fully break the gauge
group at the boundary, to Neumann boundary conditions enriched by the set of N chiral
We can provide a more entertaining example: a selfdual boundary condition. We
de ne the boundary condition by coupling the vedimensional gauge elds to N +1 quarks
q0 and a single antiquark q~0. For future convenience, we also add N + 1 extra chiral
multiplets M coupled by the superpotential
Thus the boundary condition has an extra SU(N + 1)
U(1)e global symmetries de ned
at the boundary. The SU(N + 1) simply rotates q0 as antifundamentals and M as
fundamentals. The nonanomalous Rsymmetry assignments are akin to the ones for a 4d SQCD
with N + 1 avors.
The bulk instanton symmetry can be extended to a nonanomalous symmetry under
1=N . The remaining
non
1=N and on M with charge 1.
After acting with the duality interface, we nd at the boundary fourdimensional
SU(N ) gauge theory, with N + 1
avors given by the quarks q0 and antiquarks q and
q~0. The theory has a Seiberg dual description in the IR, involving the mesons and baryons
2It may be possible to consider a larger set of boundary conditions, involving singular boundary
conditions for the matter and gauge elds, akin to Nahm pole boundary conditions for maximally supersymmetric
gauge theories [13, 14].
tential coupling for the closed loop of three arrows.
det q antibaryon. The remaining qq0 mesons give N + 1 new fundamental chiral at the
boundary, the dual version of q0. The remaining antibaryons give one antifundamental
chiral, the dual version of q~0. The baryons give the dual version of M .
We should keep track of the Abelian global symmetries. The dual quarks have
instanand U(1)e charge
In order for the selfduality to be apparent, we should rede ne our instanton symmetry
1=(2N ), on M
charges, but leave U(1)e una ected. It is natural to conjecture that this boundary condition
descends from an SU(2)ininvariant boundary condition for the UV SCFT, equipped with
an extra SU(N + 1)
U(1)e global symmetry.
We can generalize that to a dualitycovariant interface IN;N0 between SU(N )N and
SU(N 0)N0 , coupled to three sets of fourdimensional chiral elds: N + N 0 fundamentals w
of SU(N ), N + N 0 antifundamentals u of SU(N 0) and a set of bifundamentals v of SU(N 0)
If we act with an SU(N )N duality interface, we obtain a fourdimensional SU(N )
gauge theory with N + N 0 avors, fundamentals w and antifundamentals v and q.
Applying Seiberg duality, we arrive to an SU(N 0) gauge theory with N + N 0 avors. The
original superpotential lifts the u
elds and the vw mesons. The b det q superpotential
maps to a similar b det q_ involving the Seibergdual quarks which transform under the
vedimensional SU(N 0)N0 gauge elds. The
nal result is identical as what one would
obtain by acting with the SU(N 0)N0 duality interface.
The dualitycovariant interfaces IN;N0 have interesting properties under composition.
Consider the composition of IN;N0 and IN0;N00 : it supports a fourdimensional SU(N 0) gauge
theory coupled to N + N 0 + N 00 avors, which include the N + N 0 antifundamentals u,
N 0 + N 00 fundamentals w0, bifundamentals v and v0. If we apply Seiberg duality, we nd a
new description of a composite interface, which is actually a modi cation of IN;N00 ! Indeed,
we nd an SU(N + N 00) gauge group which is coupled to the 5d degrees of freedom just as
the avor group of IN;N00 , and is furthermore coupled to N + N 0 fundamentals and N 0 + N 00
Nf =2 SQCD. The
gauge theory is supported on the bundle of N parallel D5 branes. After removing the centre of mass,
the nonnormalizable deformation are the separation m between the NS5 branes and the vertical
separation mf between the semiin nite D5 branes and the intersection of one of the NS5 branes and
the ( 1; N ) vebrane. The latter parameter is the overall mass parameter for the hypermultiplets.
We drew the resolved
vebrane web for positive and negative values of the overall hypermultiplet
mass. The former is closely related, but not identical to the gauge coupling or mass for U(1)in. It
is possible to argue that the instanton mass mi actually equals m + N2f mf . The standard IR gauge
theory description is valid for m > 0 and m + Nf mf > 0. When m becomes negative and we ip its
sign to go to a dual parameterization, we exchange the roles of the NS 5 branes and thus the role of
mf and the auxiliary parameter m0f = mf + mN . Alternatively, we can use mf +m0f as a parameter,
2
which remains invariant under duality.
antifundamentals with a superpotential coupling to (N + N 0)
(N 0 + N 00) mesons. This
is consistent with the dualitycovariance of the interface.
The interface IN;N0 clearly has an SU(N + N 0) global symmetry. We can also de ne
an U(1)e nonanomalous global symmetry, acting with charge 1 on v,
N
N+N0 on u. The second U(1)in global symmetry can be taken to act with charge 1 on w,
1 on u and charge N + N 0 on instantons on the two sides.
The IN;N dualitycovariant interface is particularly interesting. It supports a baryon
operator det v charged under U(1)e only. If we give it a vev, by a diagonal vev of v, we
Higgs together the gauge elds on the two sides of the interface and the superpotential
coupling gives a mass to u and w. We arrive to a trivial interface. Later on in section 5
we will use IN;N to study the duality properties of of 't Hooft surface defects.
SU(N )N Nf =2 SQCD with Nf < 2N
theories with Nf avors, with Nf < 2N . The SCFT can be engineered by a BPS vebrane
web involving Nf + 4 semiin nite external legs: two parallel NS5 branes, a ( 1; N ) and a
N ) vebranes, Nf D5 branes pointing to the left. The SU(2) global symmetry is
associated again to the two parallel NS5 branes, while the Nf D5 branes support an U(Nf )
global symmetry. The vebrane webs and mass parameters are depicted in gure 6.
As the gauge elds are IR free, we expect to be able to describe a typical
halfBPS boundary condition for such gauge theories in terms of an SU(N )preserving
boundary conditions for the
vedimensional hypermultiplets, with a weak gauging of the
dimensional SU(N ) symmetry. Of course, it is also possible to only preserve, and gauge, at
the boundary some smaller subgroup H of the vedimensional gauge group. An extreme
example would be to give Dirichlet boundary conditions to the gauge elds.
HalfBPS boundary conditions for vedimensional free hypermultiplets may yet be
strongly coupled. On general grounds [15], it is always possible, up to Dterms, to
describe such boundary conditions as deformations of simple boundary conditions which set
a Lagrangian half of the hypermultiplet scalars (which we can denote as \Y") to zero at
the boundary. The remaining hypers (which we can denote as \X") can be coupled to a
boundary theory B by a linear superpotential coupling
W = XO
involving some boundary operator O. This gives a boundary condition which we could
Conversely, if we are given some boundary condition BX for free hypermultiplets, we
can produce a fourdimensional theory B by putting the 5d hypers on a segment, with
free chiral multiplets
sides of the interface.
With these considerations in mind, we can evaluate the 't Hooft anomaly polynomial
must be exactly half of the 't Hooft anomaly polynomial for a fourdimensional free chiral
with the same quantum numbers as X.
Our proposal for the duality interface generalizes the interface for pure SU(N )N gauge
theory: we set to zero at the boundary the fundamental half X of the hypermultiplets
on the right of the wall and antifundamental Y 0 on the left of the wall, with a boundary
W = b det q + Tr X0qY :
The combination of gauge anomalies from q and the boundary condition for the
hypermultiplet precisely matches the desired bulk ChernSimons level N
Nf =2. We denote as
X the elds which transform as antifundamentals of U(Nf ). In particular, we give them
1 under the diagonal U(1)f global symmetry in U(Nf ).
A consecutive pair of these conjectural duality walls can be analyzed just as in the pure
gauge theory case, as the boundary conditions prevent the vedimensional hypers on the
interval from contributing extra light fourdimensional elds. They can be integrated away
to give a Tr X00q~qY coupling. As the meson qq~ is identi es with the identity operator in the
group which goes through the interface as a strip. The dashed arrows indicate which half of the
bulk hypermultiplets survives at the wall. We include a superpotential coupling for the closed loop
of three arrows.
4d SU(N ) gauge theory has N
avors and at low energy it glues the two 5d gauge groups together.
The theory includes a quartic superpotential coupling which arises from integrating away the
hypermultiplets in the segment. In the IR, it glues together the hypermultiplets on the two sides of
IR, the interface ows to a trivial interface for both the gauge elds and the hypermultiplets,
up to Dterms. Thus the interface is a reasonable candidate for a duality wall.
Next, we can look carefully at the anomaly cancellation conditions. It is useful to
express the anomaly cancellation in terms of fugacities. If we ignore for a moment the
Rcharge and say that q has fugacity
1=N , X has fugacity x and X0 has fugacity x0, the
superpotential imposes x =
instanton fugacities on the right to ir =
x Nf =2 and i` =
1(x0) Nf =2.
We can recast the relation as a statement about one combination of bulk fugacity
being inverted by the interface,
inverted irxNf =2 2N = i`(x0)Nf =2 2N .
= irxNf =2 and
1 = i`(x0)Nf =2, and one being not
Although these relations may look unfamiliar, they can be understood in a
straight
As far as Rsymmetry is concerned, the bulk Rsymmetry only acts on the scalar elds
in the hypermultiplets, with charge 1. Thus we expect that assigning Rsymmetry 0 to q
and 2 to b will both satisfy anomaly cancellation and be compatible with the superpotential
It is straightforward to extend to SQCD the dualitycovariant boundary conditions
and interfaces proposed for pure SU(N ) gauge theory. We refer to gure 9 for the quiver
description of the IN;M interface and to gure 10 for the Seibergduality proof of
dualitycovariance. The composition of IN;M and IM;S can again be converted to a modi cation
tential coupling for the closed loops of three arrows.
After removing the centre of mass, the nonnormalizable deformation are the separation m between
the NS5 branes and the separation m~ between the ( 1; N ) vebranes. The vertical separation mf
between the semiin nite D5 branes and the intersection of one of the NS5 branes and the ( 1; N )
N mf , m0 = mi + N mf .
Duality walls for SU(N ) with Nf = 2N
The SU(N ) theory with 2N
avors is rather special: in the UV, two distinct Abelian global
symmetries are expected to be promoted to an SU(2). Essentially, they are the sum and
di erence of the instanton and baryonic U(1) isometries. Correspondingly, we will nd two
commuting duality walls. In the vebrane construction, the extra symmetry is due to two
sets of parallel vebranes. See gure 11.
The rst duality wall is de ned precisely as before, i.e. set to zero at the boundary the
fundamental half X of the hypermultiplets on the right of the wall and antifundamental
the closed loop of three arrows.
Y 0 on the left of the wall, with a boundary superpotential
W = b det q + Tr X0qY :
For the second wall, we replace q with a set of bifundamental elds q~ in the opposite
direction, and set to zero at the boundary the antifundamental half Y of the
hypermultiplets on the right of the wall and fundamental X0 on the left of the wall, with a boundary
W = ~b det q~ + Tr Xq~Y 0 :
Both walls implement Z2 symmetries: the composition of two walls of the same type
ows to the identity, and they re ect one of the two fugacities
= irxN or ~ = irx N
while leaving the other one xed.
We can consider the concatenation of the two walls. That gives us a 4d SU(N ) gauge
theory coupled to q, q~ and the surviving half of the bulk hypermultiplet in the interval. If
we pick one of the two possible orders of the composition, we nd
W = b det q + Tr X0qY + ~b det q~ + Tr X0q~Y 00
with X0 being a set of 2N fundamental chiral multiplets and q, q~ antifundamentals.
If we concatenate the walls in the opposite order, we nd
W = b det q + Tr X00qY 0 + ~b det q~ + Tr Xq~Y 0
with Y 0 being a set of 2N antifundamentals and q and q~ fundamentals of the 4d gauge
The two possibilities are precisely related by Seiberg duality! The mesons produced
by the duality implement the switch in the boundary conditions for the hypermultiplets,
and the baryons are remixed so that the b and ~b couplings match as well. Thus the two
duality walls commute, as expected.
The duality walls we considered can be de ned with minor changes in quiver gauge theories
where one or more nodes satisfy a balancing condition
= Nc
Nf =2. In the language
of vebranes, if the quiver is engineered by a sequence of D5 brane stacks stretched
between NS5 branes, the balancing condition insures that either the top pair of semiin nite
pairs are parallel. See gure 14 for an example.
theory with N
avors at the left node and M at the right node. The ve U(1) global symmetries
(two instanton symmetries and three hypermultiplet masses) are enhanced to U(1)2
SU(2) SU(3)
because of the two sets of parallel vebranes. The six mass deformations in the picture satisfy a
relation: m0 = m + M mf + N m0f .
A sequence of k balanced nodes is expected to be associated in the UV to an SU(k + 1)
global symmetry, enhancing a certain combination of the instanton and bifundamental
hypermultiplet charges for these nodes.
We want to understand the e ect of a duality wall for a node of the quiver on the
other nodes of the quiver, and gure out how the duality walls for di erent nodes match
We can de ne the duality wall at a balanced node as we did for a single gauge group,
leaving the other gauge groups and other hypermultiplets continuous at the interface.
As the X0 and Y
elds for a given node are charged under the gauge groups at nearby
nodes, but have di erent Abelian charges, in order for the corresponding symmetries to
remain nonanomalous, we need to correct these Abelian charges by the instanton charge
at the nearby nodes on either sides of the interface. In terms of instanton fugacities, that
means that the instanton fugacities at the nearby nodes will have to jump by the sum of the
fugacities of X0 and Y , i.e. the fugacity
of q. That makes sense: the duality wall permutes
two consecutive semiin nite branes and the instanton symmetries at the other nodes are
associated to the relative distance of nearby vebranes. If we permute two vebranes whose
distance is associated to the fugacity , the distances from other vebranes jump by plus
or minus that distance and the fugacities jump by factors of
Let's denote the domain walls associated to nodes a with positive balancing condition
as Da+, and the ones associated to nodes a with negative balancing condition as Da . If
It is easy to show that all Da+ commute with all the Da . It is more interesting to
show that each sequence of consecutive walls with the same sign satisfy the relations of a
rise to a fourdimensional SU(Na) gauge theory with Na + Na+1 avors. For example, the
left hand side gives
Seiberg duality appears to neatly exchange the interfaces corresponding to the two sides of
the permutation group relation, up to a small mismatch concerning the b0 det q0 coupling
for the intermediate interface in the composition: b0 appears to couple on the two sides
to two di erent operators with the same fugacities. The mismatch can likely be explained
away by the possibility of operator mixing under Seiberg duality.
Exceptional symmetries in SU(2) theories
The UV completion of SU(2) gauge theories with Nf avors is expected to have an enhanced
ENf +1 global symmetry. This can be understood as a combination of the general UV
enhancement for SU(N ) gauge theories and the enhancement of U(Nf ) to SO(2Nf ) due to
the fact that the fundamental representation of SU(2) is pseudoreal. Indeed, the SU(2)
enhancement involves a linear combination of U(1)in and the diagonal U(1) subgroup of
U(Nf ) and thus it combines nontrivially with the enhancement of U(Nf ) to SO(2Nf ).
Correspondingly, we can
nd continuously many versions of our basic duality wall,
each labelled by a choice of U(Nf ) subgroup in SO(2Nf ) and a splitting of the
hypermultiplet scalars into N \X" and N \Y" complex scalar elds. It is most useful to look at
domain walls which preserve a common Cartan subalgebra of the global symmetry group,
implementing Weyl re ections in the UV.
If we denote the bulk quarks as Qi, i = 1;
; 2Nf , we can consider duality walls for
which the X
elds consist of Nf
k quarks from the i = 1;
; Nf range and k quarks
from the i = Nf + 1;
overall fugacity of the X
; 2Nf range. If we denote as xa the fugacities of the quarks, the
elds will be de ned as xNf = Q
a2X xa. The domain walls invert
It is important to point out that not all splittings are simultaneously possible. There
are two disconnected classes of choices of X and Y
elds among the Qi, distinguished
by comparing the sign of their \orientation" dX1dY1dX2dY2
. Intuitively, in order to
interpolate between boundary conditions in di erent classes we need to add a single chiral
doublet at the boundary, which contributes one unit to the discrete Z2 gauge anomaly of
SU(2). Thus either boundary conditions with even k are simultaneously nonanomalous,
or boundary conditions with odd k are simultaneously nonanomalous, but not both.
Notice that SU(2) gauge theories have no continuous theta angle, but have a discrete
Z2valued theta angle. One unit of discrete Z2 gauge anomaly at the boundary can be
compensated by a shift of the bulk discrete theta angle. Thus we expect the two classes
(even k and odd k) of boundary conditions to be associated to the two di erent choices of
bulk theta angle. Thus we have 2Nf 1 basic domain walls.
In general, composing two such domain walls associated to splittings (X; Y ) and
(X0; Y 0) will give an interface supporting an 4d SU(2) gauge theory, with as many
chiral quarks as the number of bulk avors which belong to X and Y 0 (or equivalently X0 and
Y ). The relations in the Weyl group of ENf +1 must correspond to Seiberglike dualities in
the corresponding domain wall theories.
For reasons of space, we will only verify these for the simplest nontrivial example,
involving Q3 and Q4. Both preserve the same SU(2) subgroup of the SO(4) global group,
and mix the instanton symmetry with the other SU(2) subgroup to an SU(3).
If we concatenate the two walls, the intermediate SU(2) 4d gauge group will be coupled
to three avors, i.e. the six doublets q, q~, Q1, Q2. In the IR, they will ow to a set of 15
mesons. Two of them will be lifted by b and ~b and eight simply ip the boundary condition
on the left and right hypermultiplets so that we are left with Q1 and Q2 at both boundaries.
The remaining ones give a set of bifundamental elds between the left and right gauge
groups and a neutral singlet. The Pfa an superpotential involving the 15 mesons couples
the singlet to the determinant of the bifundamental eld and couples the bifundamental
to the boundary values of the hypermultiplet.
nal result is again a duality wall, combined with a permutation of the Q1, Q2
quarks with the Q3, Q4 quarks on one side of the wall. If we denote the two original duality
walls as D1 and D2, and the trivial duality wall permuting the two sets of quarks as D3,
we nd the relations
D1D2 = D2D3 = D3D1 ;
D2D1 = D3D2 = D1D3
which agree well with the properties of the three permutations in S3, the Weyl group
Index calculations
In this section, we consider the superconformal index (SCI) and the hemisphere index of
a 5d SCFT at the UV
xed point. The superconformal index is a trace over the BPS
operators in the CFT on RD, or over the BPS states on a sphere SD 1 times R via the
I(wa; q; p; q) = Tr( 1)F pj1+Rqj2+R Y waFa qk :
j1; j2 and R are the Cartan generators of the SO(5)
SU(2)R bosonic algebra and p; q
are their fugacities. Fa are the Cartans of the global symmetries visible in the classical
Lagrangian and wa are the corresponding fugacities. k is the instanton number and its
fugacity is q. This index can also be considered as a twisted partition function on S1
which was computed in [6, 17] using supersymmetric localization.
The hemisphere index is the supersymmetric partition function on an half of the sphere
S4 times S1 with a speci c boundary condition of the D4. We can also interpret it as
an index counting the BPS states on S1
R4 with Omega deformation, introduced in [18].
e 2. Roughly speaking, this index is an half of the superconformal index and thus the full
sphere index (or SCI) can be reconstructed by gluing two hemisphere indices. We will now
use these indices to test our duality proposal.
SU(N )N theories
Let us begin by pure SU(N )N gauge theories. The hemisphere index with Dirichlet b.c. is
The \gauge fugacity" zi becomes here the fugacity of the boundary global symmetry. ZiNnst
is the singular instanton contribution localized at the center of the hemisphere.
The gauge theory on the full sphere can be recovered from two hemispheres with
Dirichlet boundary conditions by gauging the diagonal SU(N ) boundary global symmetry.
So the full sphere index can be written as
IN ( ; p; q) = hIIN jIIN i
IIN (zi; ; p; q)IIN (zi; ; p; q) : (3.3)
QiN6=j (zi=zj)
i6=j
The integrand includes the contribution of the 4d gauge multiplet, with IV
being the contribution of the Cartan elements. The integration measure is simply 2dziz .
The overline indicates a certain operation of \complex conjugation", which inverts all
gauge/ avor fugacities.
Other boundary conditions or interfaces can be obtained from Dirichlet boundary
conditions by adding boundary/interface degrees of freedom and gauging the appropriate
diagonal boundary global symmetries. For example, if I4d
N;M (zi; zi0; p; q) is the
superconformal index of some interface degrees of freedom for an interface between SU(N ) and SU(M )
gauge theories, the sphere index in the presence of the interface becomes3
IIN (zi; ; p; q)IN4d;M (zi; zi0; p; q)IIM (zi0; ; p; q) :
QiN6=j (zi=zj ) QiN6=j (zi0=zj0 )
Hemisphere indices, or sphere indices with an interface insertion, can be thought of as
counting the number of boundary or interface local operators in protected representations
of the superconformal group.
Before going on, we should spend a few words on how to compute the correct instanton
contribution ZiNnst to the localization formula. The partition function is computed by
equivariant localization on the moduli space of instantons. The instanton moduli spaces have
singularities, whose regularization can be thought of as a choice of UV completion for the
theory. The standard regularization for unitary gauge group is the resolution/deformation
produced by a noncommutative background, or by turning on FI parameters in the ADHM
quantum mechanics [19, 20].
In principle, the standard regularization may not be the correct one to make contact
with the partition function of a given UV SCFT. For SCFTs associated to (p; q) vebrane
webs, the standard regularization is expected to be almost OK [21, 22]: the correct
instanton partition function is conjectured to be same as the standard instanton partition
function up to some overall correction factor, independent of gauge fugacities and precisely
associated to the global symmetry enhancement of the UV SCFT: each pair of parallel
( 1; q) semiin nite vebranes contributes a factor of4
Zextra( ; p; q) = PE
to the correction factor, where
is the fugacity for the global symmetry associated to
the mass parameter corresponding to the separation between the parallel ( 1; q)
semiin nite vebranes. This correction factor has been extensively tested against the expected
global symmetry enhancement of the superconformal indices. It appears to account for
the decoupling of the massive Wbosons living on the sixdimensional worldvolume of the
semiin nite vebranes.
3One can bring the 4d index under the conjugation. The inversion of fugacities can be understood as the
di erence in sign which appears when matching 5d and 4d fugacities for left or right boundary conditions.
4PE[f ] denotes the plethystic exponent of singleletter index f .
The standard instanton partition function computed by using equivariant localization
of [18, 23] result takes the following contour integral form
ZQNM(zi; q; p; q) =
PIk=1 I Zvec( I ; zi; p; q) ;
is known that the integral should be performed by using the Je reyKirwan (JK) method,
which is rst introduced in [24] and later derived in [25] for 2d elliptic genus calculations.
See [26{28] for applications to 1d quantum mechanics and a detailed discussion of contour
integrals. See also appendix A for details on instanton partition functions.
The correction factor from the parallel semiin nite NS5branes is
Zextra(q; p; q) = PE
Let us leave a few comment on this correction factor. This factor can also be read o from
the residues R
1 at in nity I =
1. R
1 are associated to the noncompact Coulomb
branch parametrized by vevs I of the scalar elds in the vector multiplet. In fact, the above
contour integral contains the contribution from the degrees of freedom along this Coulomb
branch and it is somehow encoded in the R
. The extra contribution is roughly an `half'
. The residue at the in nity is in general given by a sum of several rational
functions of p; q. The `half' here means that we take only an half of them such that it
satis es two requirements: when we add it to the standard instanton partition function, 1)
the full instanton partition function becomes invariant under inverting x
2) it starts with positive powers of x in x expansion. The second requirement follows from
the fact that the BPS states captured by the instanton partition function have positive
charges under the SU(2) associated to x. This half then gives the extra contribution from
the Coulomb branch and it also coincides with the correction factor (3.7). We will see
similar correction factors in the other examples below.
Since the Coulomb branch of the ADHM quantum mechanics dose not belong to the
instanton physics of the 5d QFT, we should remove its contribution to obtain a genuine 5d
partition function. So the correct instanton partition function of the 5d SCFT is expected
QiN;j=1 ( 1=N zi=zj0 ) ;
with q =
in this case.
ZiNnst(zi; ; p; q) = ZQNM(zi; ; p; q)=Zextra( ; p; q) ;
At this point, we are ready to study the duality interface. The easiest way to do so
is to look at the boundary condition obtained by acting with the duality interface on a
Dirichlet boundary, i.e. the dual of Dirichlet boundary conditions. This consists of the
duality interface degrees of freedom coupled to a single SU(N )N gauge theory, with the
second SU(N ) global symmetry left ungauged. More general con gurations can be obtained
immediately by gauging that SU(N ) global symmetry.
The 4d superconformal index of the duality interface degrees of freedom is simply
where zi and zi0 are the fugacities for the gauge group on the left and right of the wall. The
numerator factor comes from the bifundamental chiral multiplet q and the denominator
is from the singlet chiral multiplet b. The anomalyfree U(1) symmetry, which is a linear
combination of U(1)in instanton symmetry and U(1)B baryonic symmetry, rotates the
of b is precisely the inverse of the contribution of a chiral multiplet with the same Rcharge
and fugacity as B.
Thus the hemisphere index for dual Dirichlet boundary conditions is:
2 izi0 ( ) QiN6=j (zi0=zj0 )
If we have identi ed the correct duality interface, the hemisphere index for dual
Dirichlet b.c. should actually coincide with the hemisphere index for Dirichlet b.c., up to a
reection of U(1)in instanton charges, i.e. an inversion of the instanton fugacity
This motivates us to propose the following relation:
= 1+
which is checked up to x5 order.
D^ IIN (zi; ; p; q) = IIN (zi;
This is a highly nontrivial relation. The instanton partition function in the hemisphere
index on the right side of the wall has a natural expansion by positive powers of the
instanton fugacity . On the other hand, the instanton partition function on the left side
of the wall is expanded by the negative powers of . This relation is a very stringent test
of our conjectural duality wall.
We can test this conjectural relation for small N and the rst few orders in the power
series expansion in p; q. We
nd that the relation holds with a particular choice of the
integral contours. The contour should be chosen by the condition: jpj; jqj
keeping the contour to be on a unit circle. One can then check the duality relation order
by order in the series expansion of x
For SU(2) case, one nds
D^ IIN=2(
1) = IIN=2( )
= 1 +
pp=q and r
SU(N)(z) are the characters of dimension r representations of SU(N )
symmetry. We have actually checked this relation up to x7 order.
Similarly, for SU(3)3 case, one nds
D^ IIN=3(
1) = IIN=3( )
where j0i is the ground state tensored by the broken current supermultiplet. These states
carry U(1) gauge charges (N
2)N; 0; +(N
2)N respectively. Among these three states,
the rst and the third states carry appropriate SU(2)R charge for being a current multiplet.
We also need to impose the U(1) gauge invariance.
Therefore, the instanton operator provides a broken current supermultiplet when the
classical CSlevel satis es
N = 0 :
This supports the U(1)in ! SU(2) global symmetry enhancement of the SU(N ) N gauge
theory at the UV
We now consider SU(N ) gauge theory with fundamental hypermultiplets. The Nf
fundamental hypermultiplets induce on the instanton moduli space Nf complex fermionic
zero modes carrying the
avor charges and U(1) gauge charge N
2. The quantization
leads to Nf raising operators Ca; a = 1;
; Nf and they act on the states as
which may signal the symmetry enhancement of the UV CFT.
If we impose the standard bound j j
Nf =2 as in [36], one nds that r should
be 0 or Nf and the broken current multiplet exists only when
r = 0 :
Nf =2) ;
r = Nf :
= N
Nf =2 :
Nf . These states have U(1) gauge charge (N
Nf =2) and avor
charges. We can construct the instanton operators by tensoring these states with the above
gaugino contribution and imposing U(1) gauge invariance. Then one can see that there
exist candidate broken current supermultiplets having zero U(1) gauge charge when
We nd that the broken current multiplets may exist if r
n. The states
n can survive when
while the states with r
n can survive when
symmetry and carries the baryoninc U(1)B
Nf =2 or Nf =2, respectively.
U(1)in global symmetry will be enhanced as expected to
at the UV
xed point by the instantonic conserved currents,
Thus the SU(Nf )
bound by n:
symmetry is enhanced to SU(Nf )
If we relax the bound on , though, other possibilities occur. Suppose we violate the
Nf =2 = 0 ;
Nf =2 ;
= N + r
Nf =2 ;
Nf =2 :
These states provide candidate broken current multiplets in the rank r antisymmetric
representation of the SU(Nf ) avor group.
There is no symmetry group whose adjoint representation is decomposed into irreps
involving any rank r > 2 antisymmetric representation of a subgroup. Thus we expect
theories with n > 2 to be truly incompatible with an UV completion. The constraint (7.9)
of the 5d CFTs in [11, 38, 39]. A similar analysis has been done in [12].
For r = 1 (or r = Nf
= N + 1
Nf =2 (or
1 + Nf =2)
the candidate broken currents transform in the (anti)fundamental representation of the
avor symmetry with the U(1)B charge
Nf =2 + 1 (or Nf =2
an UV CFT may exist with enhanced global symmetry SU(Nf + 1)
U(1). The current
multiplet of the SU(Nf + 1) is in the adjoint representation which is decomposed by current
multiplets in the adjoint and a fundamental and an antifundamental representation of the
subgroup SU(Nf ). The fundamental and antifundamental current multiplets are generated
by following the above procedure in the instanton background. In particular, when
it gives a current multiplet which is a singlet under the SU(Nf ) avor symmetry. Thus in
this case we have a bigger symmetry enhancement to SU(2N + 2)
SU(2). Furthermore,
= 12
= 0 and Nf = 2N + 2, both states r = 1 and r = Nf
1 survive and provide
two broken current multiplets in the fundamental and antifundamental representations.
Therefore the symmetry of the UV CFT may be enhanced to SU(2N + 4).
Similarly, the instanton state with r = 2 (or r = Nf
2) generates the broken current
multiplet in the antisymmetric representation of the SU(Nf ) when
= N + 2
Nf =2 (or
U(1) at the UV
xed point. When
= 1 (or
1) and Nf = 2N + 2, one
is singlet under the SU(Nf ). So the enhanced symmetry of the UV
xed point becomes
SO(4N + 4)
= 12 (or
) and Nf = 2N + 3, two states with
r = 2 and r = Nf
1 (or r = 1 and r = Nf
2) can provide current multiplets in the
antisymmetric and the fundamental representations of the SU(Nf ) with di erent U(1)B
N + 12 and N + 12 respectively. So the enhanced global symmetry of the UV
CFT is SO(4N + 8). Lastly, when
= 0 and Nf = 2N + 4, two instanton states with
2 survive and they provide current multiplets in the rank 2 and
r = 2 and r = Nf
2 antisymmetric representation of the avor symmetry. It has been conjectured
is expected to be UV complete and has a 6d
xed point. The corresponding 6d theory is
the (DN+2; DN+2) minimal conformal matter theory [40, 41].
The discussion in this subsection strongly supports the duality proposed in this section.
Following the fermion zero mode analysis above, the SU(N + 1) gauge theory with the
CS= N +3
2N + 2 and SO(2Nf )
SU(2) when Nf = 2N + 3, which is the same as the
expected UV global symmetry of the dual Sp(N ) gauge theory.
SU(Nf + 1)
SU(Nf + 1)
SU(Nf + 2)
SU(N ) (N+2 Nf =2)
SO(2Nf + 2)
From Sp(N ) to exotic SU(N + 1)
We rst discuss the superconformal index and the instanton partition function of Sp(N )
gauge theory. The superconformal index of the Sp(N ) gauge theory with Nf fundamental
avors takes the form
ISNp;Nf (wa;qSp;p;q) =
I YN
i=1 2 izi
QiN=1QaN=f1(ppqzi =wa;p;q)1
Zk=1 =
The function ZSNp;;Ninfst is the instanton partition function of Sp(N ) gauge theory, which
can be computed using localization of the path integral on the instanton moduli space
given in [30, 42]. The 5d Sp(N ) instanton partition functions are studied in great detail
in [6, 26]. The results are summarized in appendix A.
The Sp(N ) gauge theory has O(k) dual gauge group in the ADHM quantum mechanics.
At each instanton sector we will compute two partition functions Zk+ and Z
and O(k) , respectively,
Zk ( ; m; 1;2) =
with k = 2n +
= 0 for odd N + Nf and
for even N + Nf while choosing the same
mass signs for all matter elds for notational convenience.
The k instanton partition function can be written as
ZSkp(odd)( ; m; 1;2) =
ZSkp(even)( ; m; 1;2) =
given by sum of two partition functions
Zk+=1 =
p3=2q3=2 QaN=f1 wa 1=2( 1 + wa)
q) QiN=1(1
for O(1)+ and O(1) , respectively.
p3=2q3=2 QaN=f1 wa 1=2(1 + wa)
q) QiN=1(1 + ppqzi )
For higher instantons, we need to evaluate the contour integral over O(k) Coulomb
branch parameters using the JKresidue prescription. For example, the 2instanton
partition function has a contour integral over one variable 1 for O(2)+ sector, whereas has no
integral for O(2) sector. The JKprescription tells us that the poles we should pick up are
i + + = 0 ;
2 1 + 1 = 0 ;
2 1 + 2 = 0 ;
mod 2 ) : (7.16)
ZSNp;;Ninfs=t8 = ZSNp;;NQfM=8=ZSNpf;e=x8tra ;
where ZSp;QM is the standard instanton partition function before removing the extra factor.
Next, we need to assemble the instanton partition function and 1loop determinants
into the hemisphere partition function for Dirichlet boundary conditions:
IISNp;Nf (zi; wa; qSp; p; q) =
QiN=1
QaN=f1(ppqzi =wa; p; q)1
The hemisphere index for the SU(N + 1) theory is similarly de ned and takes the form
IISNU+1;Nf (zi; wa; qSU; p; q) =
QiN6=+j 1(pqzi=zj ; p; q)1
QiN=+11 QaN=f1(ppqzi=wa; p; q)1
at in nity I =
therefore de ned as
The sum over the JKresidues plus the O(2) contribution gives the full 2instanton
parmechanics associated to a classical noncompact Coulomb branch. The partition function
involves an extra contribution coming from this continuum which should be removed to
obtain the correct QFT partition function.
nd that the extra contribution takes
ZSNpf;e=x8tra = PE
The halfintegral coe cient in the letter index obviously shows that this is coming from
a continuum. This correction factor can also be obtained by taking a half of the residue
1 in the integral formula. The QFT instanton partition function is
with an apriory unknown instanton contribution ZSNU+;i1n;sNt f .
The degrees of freedom on the duality wall have the 4d index contribution
QiN=+11 QjN=1 (
where z0 and z are the fugacities for the bulk SU(N + 1) and Sp(N ) gauge groups. To
couple this to the 5d index, we need to multiply the 4d Sp(N ) vector multiplet contribution
and integrate the Sp(N ) gauge fugacities z. The result is given by
D^ IISNp;Nf =
where wa is the fugacity for U(Nf )
SO(2Nf ) avor symmetry and
(C)(z; z0; ) =
Our conjecture is that the duality action D^ on the hemisphere index of the Sp(N )
gauge theory converts it into the hemisphere index of the SU(N + 1) gauge theory in the
other side of the wall. So the following relation is expected to hold
D^ IISNp;Nf (zi; wa; qSp; p; q) = IISNU+1;Nf (zi0; wa0; qSU; p; q) :
In this relation, the fugacities for the global symmetry in two sides of the wall should be
identi ed as
wa =
1=2wa0 ;
qSp = (N+1)=2 Y(wa) 1=2 ;
qSU =
1 Y(wa0) 1=2 :
The rst relation comes from the the constraint of the 4d superpotential.
mined the second and the third relations experimentally from the duality relations (7.24)
and (7.29), but they agree with the relations expected from cancellation of the mixed
't Hooft anomalies for the duality wall.
The simplest example would be the duality action between Sp(2) and SU(3) gauge
theories with Nf
avors. To evaluate the integral in (7.24) and see the duality relation,
we should choose a particular contour. We take the contour to be along a unit circle while
D^ IIS2p;0(zi; qSp)
= 1 +
SU(3) is the SU(3) character of the dimension r irrep with fugacities zi. We checked
that the right hand side agrees with the perturbative part of the SU(3) hemisphere index
and admits an expansion in nonnegative powers of qSU, up to the order x5.
For general Nf
D^ IIS2p;Nf (zi;wa;qSp)
= 1+ S3U(3) U(Nf )x+
U(Nf ) is the U(Nf ) character with fugacities (wa0) 1 of a irrep labeled by a Young
Y
tableau Y . We have identi ed the parameters as (7.25). The perturbative part on the
right hand side agrees with that of the SU(3) theory and the other parts are expanded by
nonnegative powers of qSU. This relation has been checked at least up to x3 order.
In appendix B, we shall suggest a UV prescription of the instanton moduli space of our
exotic SU(3) theory with matter elds, whose partition function precisely reproduces the
right hand side. In addition, we will explicitly compute the superconformal index of this
SU(3) theory and show the desired global symmetry enhancement at the UV
One can also consider the generalization to higher rank gauge theories. Acting with
the duality wall on the hemisphere index of the Sp(3) theories, we obtain
= 1+ S4U(4) U(Nf )x+
3, by identifying the parameters as (7.25). We checked that the right hand side
agrees with the perturbative part of the SU(4) hemisphere index and admits an expansion
in nonnegative powers of qSU, at least up to the order x4.
Of course, the duality wall can also act in the opposite direction, from SU(N + 1)
where the 4d index of the boundary degrees of freedom involving the 4d vector multiplet
(A)(z0; z; ) =
QjN=+11 (
QiN6=+j 1 (zi0=zj0 ) QiN>+j 1 (
zi =zj0 )
The contour is chosen along the unit circle with an assumption x
parameters are matched as (7.25).
Of course, this follows from the CA and AC inversion formula introduced in [10]:
(C)(z; z0; )f (z) = f (x) ;
d z0 (A)(z0; z; )f (z0) = f (x) :
Note that the contours should be chosen along unit circles by assuming x
for the Atype integral, but by assuming x
< 1 for the Ctype integral as speci ed
In this subsection, we will study the properties of BPS Wilson loops under the conjectural
duality in the previous sections. We will focus on the simplest cases: fundamental Wilson
loops of the Sp(2) and SU(3) gauge theories meeting at the interface. The Wilson loops on
two sides of the wall are connected at the boundary by the bifundamental chiral multiplet
q. The chiral multiplet q has charge 12 under the nonanomalous U(1) global symmetry.
To cancel the global charge when it couples to the Wilson loops, we combine the gauge
Wilson loops with a
avor Wilson loop for the U(1) , with
from the similar argument in section 4. We will compute hemisphere indices and test this
avor charge 12 , which follows
duality property between two fundamental Wilson loops.
We rst compute the hemisphere indices with fundamental Wilson loops inserted at the
origin. We need to compute the instanton partition function in the presence of Wilson loops.
As explained in section 4, Wilson loops are represented by equivariant Chern characters
in the localization, and that for the fundamental Wilson loop is given in (A.13). Then the
localized partition function can be written in terms of the equivariant Chern characters as
For the Sp(N ) gauge theory, the equivariant Chern character for the fundamental
Wilson loop can be written, at kinstantons, as
Chf+und(e ; e ) =
Chfund(e ; e ) =
q)(pq) 1=2 X(e I + e I + ) ;
q)(pq) 1=2 X(e I + e I + ei
with k = 2n +
= 0 or 1. Here the superscripts
means those for O(k) sectors.
Then the 1instanton partition function can be written as
Wk+=1 =
Wk=1 =
(pq)3=2
PiN=1(e i + e i )
q) QiN=1(1
q)(pq) 1=2 QaN=f1 2 sinh m2a
(pq)3=2
PiN=1(e i + e i ) + (1
q)(pq) 1=2 QaN=f1 2 cos m2a
q) QiN=1(1 + ppqe i )
There could be extra instanton corrections to the Wilson loop index as we have seen in
section 4. For the cases in this section, however, we nd that there is no such corrections
up to certain order in x expansion.
Now we consider the duality wall action on the hemisphere index of the Sp(N ) theory
with the fundamental Wilson loop. We propose that the fundamental Wilson loop partition
function of the Sp(N ) theory is mapped to that of the SU(N + 1) theory after passing
through the duality wall as follows:
Sp(N);Nf (zi; wa; ) =
1=2Wfund
with the parameter identi cation in (7.25). The duality action D^ is de ned in the same way
as in (7.22), but the hemisphere indices IISNp;;NSUf in both sides are replaced by the Wilson
avor Wilson loop.
We compute the hemisphere indices of the Sp(2) gauge theories and test this duality.
= 3
+ S2U(2)(y) S3U(3)(z)x3+ S2U(2)(y) S3U(3)(z)2
D^ WfSupn(d2);0(zi; qSp)
1=2WfSuUnd(3);1(zi; qSU)
for Nf = 0, and
D^ WfSupn(d2);1(zi;w;qSp)
1=2WfSuUnd(3);1(zi;w0;qSU)
= S3U(3)(z)+ (w10) 1 SU(3)(z)2+(w10) 1=2qSU x+
3
+ (w10) 2 SU(3)(z)+(w10) 2 SU(3)(z)+(w10) 1 SU(2)(y) S3U(3)(z)2 x2
10 8 2
expansion in nonnegative powers of qSU and the perturbative part agrees with that of the
SU(3) gauge theory, up to x4 order. It also turns out that the right hand sides agree up to
x4 order with the hemisphere indices of the SU(3) theories with the fundamental Wilson
loop whose instanton partition functions are computed using the UV prescription given in
The research of DG and HK was supported by the Perimeter Institute for Theoretical
Research at Perimeter Institute is supported by the Government of Canada
through Industry Canada and by the Province of Ontario through the Ministry of
Economic Development and Innovation.
5d Nekrasov's instanton partition function
The moduli space of instantons has complicated singularities which are associated to one or
more instantons shrinking to zero size. In the context of vedimensional supersymmetric
gauge theories, these eld con gurations are outside the obvious regime of validity of the
gauge theory description of the theory. Correspondingly, the de nition of the gauge
theory instanton partition functions through equivariant localization on the instanton moduli
spaces requires a prescription of how to deal with the singularities, which will depend on a
choice of UV completion of the gauge theory.
It is very challenging to work directly on the singular moduli spaces. Even in the
absence of extra matter
elds this was done only recently [43] using the technology of
equivariant intersection cohomology. Extra matter elds, in the form of hypermultiplets
transforming in some representation of the gauge group, provide additional fermion zero
modes in the instanton background which are encoded into some appropriate characteristic
class inserted in the equivariant integral. The correct description of these characteristic
classes over the singular instanton moduli space is poorly understood.
The standard alternative to working with the singular moduli spaces, available for
classical groups only, is to employ the ADHM technology to provide a resolution of the
singularities in the instanton moduli space. The ADHM construction has a clear motivation
in terms of a string theory UV completion. It realizes the instantons as D0 branes in
presence of other brane systems which engineer the gauge theory itself.
It is important to realize that this is not obviously the same as the quantum
theory UV completion we are after, which should involve some 5d SCFT or perhaps a 6d
SCFT. Luckily, it appears that the answers computed by the ADHM construction can be
easily corrected to sensible eld theory answers, as long as the matter content of the gauge
theory does admit a reasonable string theory lift.
When that is not the case, it is not
obvious that a construction of the correct bundle of fermion zero modes will actually be
available in the ADHM description of the moduli space. We will encounter some of these
issues in the sections A.3 and B.
When the ADHM construction for a gauge group G exists, it can be described as
a one dimensional gauged linear sigma model of dual gauge group G^, called the ADHM
quantum mechanic (ADHM QM). The Higgs branch of this theory coincides with the
instanton moduli space. This theory has bosonic SU(2)1
SU(2)R symmetry and
4 real supercharges Q A_, where the SO(4) = SU(2)1
SU(2)2 corresponds to the spatial R4
rotation and the SU(2)R is the Rsymmetry in 5d. The indices
= 1; 2; _ = 1; 2; A = 1; 2
are the doublets of SU(2)1; SU(2)2; SU(2)R symmetries respectively. The ADHM QM
consists of the (0,4) hypermultiplets
) in adjoint rep;
) in fundamental rep
and the vector multiplet (At; ; A_). The bosonic elds in the hypermultiplets are called
In order to apply the ADHM construction to a vedimensional gauge theory we need
to nd within the ADHM quantum mechanics a construction of the bundle of fermionic
zero modes associated to the hypermultiplets. Concretely, that means adding extra elds
to the quantum mechanics which add the appropriate fermionic bundle on top of the Higgs
branch of the theory. If a string theory description of the gauge theory is available, one
can usually read o from it the required extra degrees of freedom.
If the instanton moduli space was not singular, it would be possible to derive simple
relationships between the characteristic classes in the equivariant integral associated to
hypermultiplets in di erent representations. If a string theory construction is not available for
some representation, one can try to guess an ADHM description for that representation by
imposing the same relationship on the the corresponding characteristic classes/equivariant
indices in the ADHM equivariant integral. Some equivariant indices for hypermultiplets in
simple representations are given in [30]. We will present below the equivariant indices and
partition functions for the hypermultiplets used in the main text and discuss the di culties
associated to this naive choice of UV completion.
The k instanton partition function takes the following integral expression
Zk( ; m; 1;2) =
1 I
PIk=1 I Zvec( ; ; 1;2) Y ZRa ( ; ; ma; 1;2) ;
where ZRka is the contribution from a hypermultiplet in Ra representation and ma is the
mass parameter. We will often use fugacities zi
ema . The vector multiplet
Zvec( ; ; 1;2) =
QIk6=J 2 sinh I 2 J QIk;J 2 sinh I J2+2 +
The instanton partition function takes the form of the instanton series expansion as
with an instanton counting parameter q. The Zk is the k instanton partition function. It
is the supersymmetric Witten index of the 1d ADHM QM. It also admits a path integral
representation. The supersymmetric localization was employed to evaluate this path
integral of the ADHM quantum mechanics in [18, 23]. See also [26{28] for 1d localization
calculations. We will now summarize some results.
SU(N ) partition function
bulk 5d theory, one can also turn on a classical CS coupling
3. It induces a ChernSimons coupling in the 1d quantum mechanics [44, 45].
Zinst =
The hypermultiplet factor will be discussed later.
We still have the contour integral to be evaluated. The contour integral of the
instanton partition function should be performed using the Je reyKirwan method [26]. If the
hypermultiplet factor has only fermionic contributions, as our naive expectation from the
zero mode analysis in the 5d QFT, we need to take into account only the vector multiplet
factor. The JKprescription tells us that the residue sum of the following poles will give
the nal result.
i + + = 0 ;
J + 1 = 0 ;
J + 2 = 0 ;
with I > J . However, we will see that the hypermultiplets can introduce extra bosonic
degrees for the UV completion of their zero modes. Thus they can also provide nontrivial
JKpoles above the poles from the vector multiplet. We will discuss some examples below.
Sp(N ) partition function
Since the O(k) group has two disconnected components O(k)+ and O(k) , we will get two
partition functions Zk+ and Z
k at each instanton sector. The k instanton partition function
is then given by a sum of these two functions. In addition, the Sp(N ) gauge theory has a
angle associated with
4 (Sp(N )) = Z2 [3]. Two possible
parameters lead to
the following two di erent combinations [7, 46]:
ZSp = <8 21 (Zk+ + Zk ) ;
= 0
When the theory couples to more than one fundamental hypermultiplet, the
comes unphysical because it can be e ectively absorbed by ipping the sign of a single
mass of one fundamental matter.
The k instanton partition function takes the form
Zk ( ; m; 1;2) =
Yn d2 Ii Zvec( ; ; 1;2) Y ZRa ( ; ; ma; 1;2) ;
with k = 2n +
= 0 or 1. The Weyl factor is given by
=0 =
=1 =
=0 =
=1 =
The vector multiplet for O(k)+ sector gives the contribution
2+ + QiN=12sinh
2+ + QiN=12sinh
I 2 i+ + QIn=12sinh 2 I
I J +2 +
Zvec =
Zvec =
with k = 2n.
with k = 2n + 1 and
For O(k) sector, the vector multiplet contribution is
2+ + QiN=12cosh
+ +)QiN=12sinh(
i+ +) I=1
I=1 2sinh
2+ + QiN=12sinh
I 2 i+ + QIn=112sinh 2 I
A hypermultiplet develops fermion zero modes in the instanton background. The presence
of the fermion zero modes can be observed using an index theorem. Accordingly, it is
expected that the bulk hypermultiplets induce fermionic degrees on the instanton moduli
space. When we attempt to engineer an ADHM quantum mechanics description of these
fermionic zero modes on the Higgs branch, however, extra bosonic degrees of freedom are
in general required. Often these bosonic zero modes give rise to extra classical branches
of vacua in the ADHM quantum mechanics, or extra continuum contributions to the
spectrum, which may be spurious from the point of view of the 5d gauge theory. In string
theory constructions, they may describe D0 branes moving away from the brane system
which engineers the 5d gauge theory. These spurious branches of vacua must be carefully
subtracted from the nal answer.
We can give a few simple examples of this phenomenon. The instanton moduli space
of a 5d gauge theory with an adjoint hypermultiplet has a string theory embedding. The
instanton states can be interpreted as the D0/D4brane bound states in this case. The
1d gauge theory living on the D0branes is described by the ADHM quantum mechanics
with additional matter elds corresponding to the bulk adjoint hypermultipet. This theory
involves extra real 4 dimensional bosonic elds that parametrize the 4 transverse directions
to the D4branes in which the 5d gauge theory supports. The noncommutativity parameter
(or FI parameter) in the 1d QM generally make these directions massive. However, when
the commutativity is restored, these branches of vacua open up D0branes (or instantons)
can escape to in nity.
Similarly, the UV completion of instanton dynamics in Sp(N ) gauge theory with an
antisymmetric and fundamental hypermultiplets has extra bosonic degrees of freedom from
the hypermultiplets. Its string theory embedding is given by D0D4D8O8 brane
system [47]. The extra bosonic modes again parametrize the transverse directions to the
D4branes. In particular, the ADHM for this theory does not have noncommutative
deformation of the space. Hence the observables computed using this UV completion in general
involves extra contributions to be subtracted o . One can
nd examples in [26].
Next, we can describe our guess for the contribution of hypermultiplets in tensor powers
of the fundamental representation, based on the prescription given in [30]. If we could ignore
the singularities, the hypermultiplets introduce vector bundles on the instanton moduli
space, and the vector bundles are constructed by tensor products of an universal bundle E .
The tensor product structure of the vector bundle inherits that of the representation of the
5d hypermultiplet. We will now pretend that the same prescription can be applied to the
ADHMresolved moduli space of instantons. In [30], it was suggested that the equivariant
index for the hypermultiplet can be computed by taking tensor product of the equivariant
Chern character of the bundle E , which is given by [30, 48]
ChE (e ; e ; p; q) = fund(e i )
q)(pq) 1=2
where fund(e i ) and fund(e I ) denote the character of the fundamental representations of
the guage group G and the dual gauge group G^, respectively. For example, the equivariant
indices for the hypermultiplets in the fundamental, symmetric, antisymmetric and adjoint
representations are given by, respectively,
indfund(e ; e ; p; q) =
indsym(e ; e ; p; q) =
indanti(e ; e ; p; q) =
indadj(e ; e ; p; q) =
Zfund = Y 2 sinh
where the tensor product of the Chern character is de ned using the usual tensor product
ChE E (e ; e ; p; q) =
Ch^2E (e ; e ; p; q) =
ChE (e ; e ; p; q)2 + ChE (e2 ; e2 ; p2; q2)i ;
ChE E (e ; e ; p; q) = ChE (e ; e ; p; q)
The equivariant indices in other representations can be obtained in the similar manner.
The resulting index computed in this way contains terms independent of the fugacity e I
for G^. These terms amount to the perturbative contribution, so we will ignore them when
we compute the instanton partition function.
The contribution to the instanton partition function of the hypermultiplets can be
easily obtained using the relevant equivariant indices. There is a conversion rule for 5d
indR =
ZR = Y h2 sinh zi ini
Thus the plethystic exponential of the equivariant index yields the instanton partition
function contribution of the hypermultiplet. One can check that the contribution from
an adjoint hypermultiplet computed using this prescription agrees with that from the
localization of the ADHM quantum mechanics in [49].
Let us present explicit expressions for the hypermultiplets discussed in the main
context. For SU(N ) gauge theory, the fundamental hypermultiplet contribution is
with a mass parameter m. The antisymmetric hyper has the following contribution
Zasym =
QiN=1QIk=12sinh I + 2i m QIk>J 2sinh I + J 2 m
I J +m
I J +m
For Sp(N ) gauge theory, the fundamental representation has the contribution
for O(k)+, and
with k = 2n + 1, and
Zf+und = 2 sinh
Zfund = 2 cosh
Zfund = 2 sinh
Next, we can assemble a modi cation of the bare ADHM quantum mechanics which
would reproduce these modi cations to the equivariant integrand. The contribution for
the fundamental hypermultiplet implies that a fundamental matter induces a (0; 4) fermi
multiplet in fundamental representation of G^ in the ADHM QM. This agrees with our
expectation that the hypermultiplet develops fermion zero modes in the instanton
background. On the other hand, the contribution from the antisymmetric hyper has factors
in denominator as well as the factors in numerator. The numerator factors correspond to
a fermi multiplet in the bifundamental representation of G
fermi multiplets in the antisymmetric representation of G^. While, the denominator factors
corresponds to a pair of (0; 4) hypermultiplets in the symmetric representation of G^. This
means that the UV completion of the zero modes acquires nontrivial bosonic degrees which
G^ and a conjugate pair of
are not present in the zero mode analysis of the 5d QFT.
The computation of the 1d equivariant integral requires both an integrand and a choice
of integration contour/prescription. The latter, in a sense, can be used to include or exclude
the contribution of certain classical branches of vacua, by selecting which poles should be
picked by the contour integral. The standard prescription in 1d localization computations
is the JKprescription. To read the relevant poles from the JKprescription, we should know
the exact representations of the extra bosonic degrees under G^ rotation. However, although
the recipe given in [30] and in this section allows us to know the matter contents in the
ADHM QM, it yet has an ambiguity in the exact representations of the multiplets. More
precisely, it cannot distinguish a certain complex representation R and its conjugation,
prescriptions for it case by case in the main context.
'. Since we could not resolve this issue, we will give
Further spurious contributions included by the standard JKprescription have to be
removed on a casebycase basis. See [26] for few examples.
Partition functions of exotic SU(3) theory
In this appendix, we propose a prescription to compute the instanton partition functions
of the exotic SU(3) theories with matters. With these results, we compute the hemisphere
indices and then show that they agree with the hemisphere indices obtained in section 7.2
using the duality wall action on the Sp(2) hemisphere indices.
We are interested in the SU(3) SQCD with
= 5
Nf =2, which obviously violates
bound, the localization integral of the instanton partition function from the usual ADHM
are associated to the classical Coulomb branch of vacua in the ADHM quantum mechanics
and not to the to the instanton moduli space which is described by the Higgs branch.
Unfortunately, we do not know how to remove these spurious contributions when the
1. These poles
degree of the pole is higher than 1. In what follows, we will explain how to avoid having
higher degree poles at in nity by introducing `pseudo' hypermultiplets in the instanton
background. We will add two (or more) `pseudo' hypermultiplets and integrate them out
at the end. This will allow us to evaluate the instanton partition function without having
the problem of the higher degree poles at in nity.
Let us rst discuss the `pseudo' hypermultiplet and the ADHM quantum mechanics.
The `pseudo' hypermultiplet is simply the hypermultiplet in the antisymmetric
representation of the SU(3). It should be equivalent to the fundamental hypermultiplet for the SU(3)
gauge theory. This is indeed the case for the perturbative analysis. However, the
antisymmetric hypermultiplet a ects the ADHM quantum mechanics in a di erent way from
that of the fundamental hypermultiplet. Strictly speaking, the ADHM quantum mechanics
is designed for the U(N ) gauge theory since it involves singular U(1) instantons which is
regularized by introducing extra UV degrees of freedom. Therefore, fermion zero modes
from the antisymmetric hypermultiplet has a rather di erent UV completion than those
from the fundamental hypermultiplet in the ADHM QM.
The fermionic zero modes from the antisymmetric hypermultiplet provide many
nontrivial multiplets, not just fermi multiplets but possibly also hypermultiplets including
extra bosonic zero modes, in the ADHM QM as depicted in gure 16. The ADHM quantum
SU(2)R symmetry. See appendix A for details.
We then add a bifundamental chiral
fermion (black dashed arrow) of U(k) and SU(3) groups, and a (0; 4) fermi multiplet (blue
dashed arrow), which is a doublet under the SU(2)1 and in the antisymmetric representation
of U(k), and a hypermultiplet (red solid arrow) in the symmetric representation of U(k).
This is equivalent to add to the instanton moduli space a vector bundle given by the
antisymmetric product of the universal bundle in the fundamental representation.
We consider the SU(3) gauge theory with two `pseudo' hypermultiplets and Nf
fundamental hypermultiplets. The kinstanton partition function from the ADHM QM can be
where Zvkec; Zfkund; Zaksym are given in (A.5), (A.17), (A.18), respectively. We will set the
= 4
1, which is now controllable.
ZQM;k =
at in nity I =
We are essentially interested in the theory with
= 5
Nf =2 and without the `pseudo'
matters. This theory can be obtained by integrating out two `pseudo' hypermultiplets.
We will send their mass parameters ti to in nity. Then it will e ectively shift the bare
CSlevel by +1 and the low energy theory will have the CSlevel
= 5
Nf =2 as desired
for our exotic theory. To avoid the higher degree poles at in nity, we shall integrate out
the `pseudo' matters after evaluating the contour integrals. It thus allows us to compute
the instanton partition function of the exotic SU(3) theory without facing higher degree
poles at the in nity. This procedure can be interpreted as a UV prescription of the SU(3)
instanton moduli space at the exotic CSlevel. Here the `pseudo' hypermultiplets are used
as a UV regulator. We will restrict ourselves to the cases with Nf
consider the dual SCFT with Sp(2) gauge group.
8,5 for which we can
The contour integral will be evaluated using the JKresidue prescription. One then
notices that the `pseudo' matter contributions provide additional nonzero JKresidues. For
example, at one instanton, the JKresidues at the following poles are nonzero:
+ = 0 ;
Summing over all JKresidues including both from the vector multiplet and from the
`pseudo' hypermultiplets, we can compute the partition function with `pseudo' matters.
This is not quite our nal answer. To obtain the QFT partition function, we need to
strip o some overall factor associated to the extra bosonic
at directions introduced by
the `pseudo' hypermultiplets. We conjecture that the extra factor is given by
ZeNxftra;pseudo=PE qSUf Nf (wa; a;p;q) ;
f Nf =
p 1 2QaN=f1pwa
(1 p)(1 q)(1 pq 1= 2)(1 pq 2= 1)
+(pq)3=2( 1+ 2) 1+( 1 2) 1 U(Nf )(1=w)+( 1 2) 2 U(Nf )(1=w) ;
4 8
5One may notice that the integral has higher degree poles at in nity when Nf > 8. We may be able to
resolve this by introducing one more `pseudo' hypermultiplet, but we will not discuss these cases.
U(Nf ) is the character of the rank L antisymmetric irrep of the
L
Note that this extra factor is independent of the SU(3) gauge fugacities and thus it indeed
corresponds to the degrees of freedom decoupled from the 5d QFT. We have checked that,
after subtracting o this factor, the instanton partition function has no poles for a and is a
nite polynomial in 1 and 2, as expected, at 1instanton for all Nf and up to 2instantons
There is the usual correction factor coming from the continuum along the noncompact
1. We obtain
ZeNxftra;cont = PE
ZeNxftr=a8;cont = PE
qSU QaN=f1 pwa
1 Y pwa + pqp 1 2 Y pwa
The `correct' partition function can then be written as
ZiNnsft(zi; wa; a; qSU; p; q) = ZQNMf=ZeNxftra ;
where ZQM is the partition function of the ADHM QM evaluated with the JKprescription.
1,6 and take the leading contribution. By rescaling the instanton fugacity as qSU
We now integrate out the `pseudo' hypers. We will send their masses to in nity ta !
qSU, we will end up with the instanton partition function of the SU(3) theory with Nf
avors and the CSlevel
ZiNnsft(zi; wa; a; qSU=
Taking into account the extra factors carefully, we compute 1instanton partition
functions for Nf
8 and obtain
Zi3n;sNt;fk=1 =
Combining the 1loop determinant, we have checked that the hemisphere index of our exotic
SU(3) theory yields exactly the right hand side of the duality relation (7.27) between the
Sp(2) and SU(3) theories, in all examples at least up to x3 order. This result supports the
UV prescription of the exotic SU(3) theory in this section.
Similarly, we can compute the Wilson loop index of the exotic SU(3) theories using the
above UV prescription. An Wilson loop in a representation R inserts the corresponding
Nf =2.
equivariant Chern character into contour integral of the instanton partition function. At
kinstantons, the Wilson loop index before integrating out the pseudo hypers can be written as
WQM;k =
I=1 2 i
PIk=1 I Zvkec( ; )YZfkund( ;ma)YZaksym( ;ti);
where ChR( ; ) is the equivariant Chern character of the vector bundle in the
representation R. We will focus only on the Wilson loop in the fundamental representation whose
equivariant Chern character is given in (A.13). The contour integral is again evaluated
using the JKprescription. Since the Wilson loop insertion does not change the pole structure
of the integrand, we can pick up the same poles as before.
As we have seen above, the partition function involves the correction factors from
the Coulomb branch and the extra bosonic degrees of the `pseudo' matters given in (B.4)
and (B.3), which we should subtract o .
Due to the same reason as without Wilson
loops, we expect the correct Wilson loop index has no poles for the mass parameter a
of the `pseudo' matters. However, even after subtracting the correction factors in (B.4)
and (B.3), we notice that the Wilson loop index still has poles for a. We
nd that the
Wilson loop receives an additional correction when Nf > 0. For example, if we de ne new
Wilson loop indices taking the form
Wf3u;n0d(z; w; ; qSU) = WQM;fund
3;0
ZeNxftr=a0 ;
Wf3u;n1d(z; w; ; qSU) = WQM;fund
3;1
Wf3u;n2d(z; w; ; qSU) = WQM;fund
3;2
ZeNxftr=a1 + qSU (1
ZeNxftr=a2 + qSU
where II3;Nf is the bare hemisphere index without Wilson loops, these new indices have
no poles for a. We have checked this till 2instantons. Thus we suggest that the `correct'
Wilson loop index with `pseudo' matters should be this new index.
Let us integrate out the `pseudo' hypers by rescaling the instanton fugacity as qSU
! qSU and taking the limit ta !
1. It leads to the Wilson loop index of the exotic SU(3)
theory, given by
We have also checked that this Wilson loop index yields the results in (7.35) and (7.36)
obtained from the duality wall action on the dual Sp(2) hemisphere indices, up to x4 order.
Superconformal indices
Now, we compute the superconformal indices for the Sp(2) and SU(3) theories and check
the duality conjecture. Let us rst discuss the Sp(2) theories. The superconformal index
IS2p;Nf = 1+ 1+ SaOdj(2Nf ) x2+
SO(2Nf ) is the character of the r irrep of SO(2Nf ) symmetry with fugacities wa
and S denotes the conjugate spinor representation and S is the complex conjugation of
SO(2Nf )(wa2) denotes the fundamental character with fugacities wa2. For Nf = 8, we
fund
IS2p;Nf =8 = 1 +
SO(16) is the character of the rank 2 symmetric representation of SO(16). This
theory has an enhanced SU(2)
SO(16) global symmetry at the UV xed point. There are
additional BPS states at x2 order corresponding to the conserved currents with instanton
fugacity qSp and all BPS states properly arrange themselves to form representations of the
enhanced symmetry. Thus the result is consistent with the symmetry enhancement.
We now turn to the SU(3) theories. The superconformal index of the general SU(N )
SQCD can be written as
2 izi QiN=1
QiN6=j (zi=zj ; p; q)1
QaN=f1(ppqzi=wa; p; q)1
only enters in the instanton partition function.
= N + 2
Nf =2 which
For our SU(3) theories, the instanton partition functions are given in the previous
section, so the superconformal index computation is straightforward. We
nd that the
results perfectly agree with the indices of the dual Sp(2) theories computed in (B.11)
and (B.12), once we identify the fugacities of two dual theories as (7.25). This has been
checked at least up to x4 orders. This result provides a strong evidence for the duality
conjecture of the Sp(2) and SU(3) theories and also the symmetry enhancements of the
SU(3) theories at UV
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