#### Mirror symmetry and loop operators

Received: July
Mirror symmetry and loop operators
k 0
YN sin[ 0
0 Waterloo , Ontario, N2L 2Y5 , Canada
1 Perimeter Institute for Theoretical Physics
2 Department of Mathematics, King's College London
Wilson loops in gauge theories pose a fundamental challenge for dualities. Wilson loops are labeled by a representation of the gauge group and should map under duality to loop operators labeled by the same data, yet generically, dual theories have completely di erent gauge groups. In this paper we resolve this conundrum for three dimensional mirror symmetry. We show that Wilson loops are exchanged under mirror symmetry with Vortex loop operators, whose microscopic de nition in terms of a supersymmetric quantum mechanics coupled to the theory encode in a non-trivial way a representation of the original gauge group, despite that the gauge groups of mirror theories can be radically di erent. Our predictions for the mirror map, which we derive guided by branes in string theory, are con rmed by the computation of the exact expectation value of Wilson and Vortex loop operators on the three-sphere.
The Strand; London WC2R 2LS; United Kingdom
1 Introduction
2 Loop operators in 3d N = 4 theories
Wilson loop operators
Vortex loop operators
Brane realization of Wilson loop operators
Mirror of Wilson loops from S-duality
Mirror symmetry and loop operators: examples
Wilson loops in the U(N
1) node of T [SU(N )]
The other T [SU(N )] loops
Loops and mirror map in circular quivers
4.5 Loops in SQCD with 2N quarks and its mirror
5 Loop operators in 3d N = 4 theories on S3
3 Brane realization of loop operators and mirror map
Wilson loops on S3
Vortex loops on S3
Exact partition function of 3d/1d theories on S3
S3 Partition function and Wilson loops
Matrix model computing Vortex loops
5.4 Loops in T [SU(N )] 5.4.1 5.4.2 5.4.3
T [SU(2)] loops
Wilson loops in T [SU(N )]
Vortex loops in T [SU(N )]
Mirror map
Loops in SQCD
5.6 Hopping duality
A Supersymmetry transformations on S3 and SQM embedding
B Evaluations of the SQM index
U(k) theory with N fundamental and M anti-fundamental chirals
B.2 U(k) theory with N fundamental, M anti-fundamental chirals and one
adB.3 Two-node quiver
Wilson loop is speci ed by a curve
of the gauge group G
Wilson loop operators [1] play a central role in our understanding of gauge theories. A
in spacetime and by the choice of a representation R
WR( ) = TrRP exp
In a certain sense, they are the most fundamental observables in gauge theories.
Wilson loops raise an immediate challenge to any conjectured duality, be it a eld
theory duality or a gauge/gravity duality: what is the dual description of Wilson loops?
Even the more qualitative question of how the choice of a representation R of the gauge
group G labeling a Wilson is encoded in the dual theory is challenging, and in general the
answer is unknown. Indeed, the gauge groups of theories participating in a
eld theory
duality can be drastically di erent, and in gauge/gravity dualities there is not even a Lie
group G in sight in the dual bulk theory.
One may even argue that the existence of Wilson loops actually introduces a
puzzle for dualities. While global symmetries and 't Hooft anomalies between dual theories
must match, gauge symmetries between dual theories need not. In a sharp sense, gauge
symmetries are not symmetries, but rather redundancies in our local description of
particles of helicity one. Nevertheless, the gauge group G is not void of important physical
information about the theory: Wilson loop operators are labeled by a representation R
of G. And therefore, it is in this vain, that the gauge group is \physical" and its elusive
representations must be found in the dual.
In this paper we identify the dual description of half-supersymmetric Wilson loop
operators in gauge theories related by three dimensional mirror symmetry [2], an infrared
IR to the same superconformal eld theory (SCFT):
SCFT
We nd that there is a rather intricate \mirror map" relating Wilson loop operators in one
theory to Vortex loop operators in the mirror:
The mirror map in non-abelian gauge theories is rather subtle. In abelian gauge theories it
follows from the mapping of the abelian global symmetries of the mirror dual theories [3{5].1
1Vortex loop operators were previously studied in [6]. For Vortex loops in pure Chern-Simons theory
gauge theories encoded by a quiver diagram of linear or circular topology (see gure 1).
We propose an explicit UV de nition of the Vortex loop operators exchanged with
Wilson loops under mirror symmetry. These Vortex loop operators are constructed by
coupling 1d N
that enter in the description of the mirror map of Wilson loop operators is encoded by the
quiver diagram of gure 2.2
representation as follows:3
Theory B | typically have completely di erent gauge groups GA and GB, we are able
to encode the choice of representation R of a Wilson loop in one theory in the precise
choice of the 1d quiver gauge theory in
gure 2 describing the mirror dual Vortex loop
operator. The 1d quivers in this class are characterized by the ranks of their gauge nodes,
the number of fundamental and anti-fundamental chirals in the last node and the presence
or not of an adjoint chiral in each node. In particular, to a quiver gauge theory with
R = Sn1
gauge node,
R = An1
Snp np 1 : for a quiver with one adjoint chiral in each
Anp np 1 : for a quiver with one adjoint chiral for U(ni)
for i = 1; : : : ; p
1, but no adjoint chiral in the U(np) node,
2Arrows encode bifundamental chiral multiplets of two gauge nodes (circles) or of a gauge and a avor
node (square). For the details of superpotential couplings see section 2.2.
3See sections 3.1 and 5.3.2 for the general dictionary.
where Ak and Sk denote the k-th antisymmetric and symmetric representations of U(N )L,
if p < 0, or U(N )R, if p > 0. A generic product of symmetric and antisymmetric
representations like R = Sn1
Snp np 1 is obtained by removing
an adjoint chiral multiplet in some nodes. We propose, but have less quantitative evidence
for, that an arbitrary irreducible representation R characterized by a Young diagram can
be obtained from the family of quivers in gure 2 by setting all but one of the FI parameters
A Vortex loop associated to a given gauge group factor with N fundamental
hypermulM ). A Vortex loop labeled by a
representation R and by the integer M is constructed by gauging avour symmetries of the
obtained in this way can be encoded by a mixed 3d/1d quiver diagram, see e.g. gure 3.
The mirror map between loop operators that we uncover is rather intricate and rich. In
particular, the map depends strongly on the choice of integer M labeling a Vortex loop. As
an illustrative example, the Vortex loop operator VM;R in T [SU(N )] de ned by the 3d/1d
gure 3 maps under mirror symmetry to the following combination of Wilson
loop operators in T [SU(N )] (this theory is self-mirror)5
representation R.
U(N ) avour symmetry of T [SU(N )].
6U(M ) is embedded as U(M )
where W U(M) is a Wilson loop in a representation Rbs of U(M ), WN;qs is an abelian avour
Wilson loop of charge qs and
M is the set of representations that appear in the
decomposition of R =
s2 M (qs; Rbs) under the embedding U(1)
1).6 With the
representation R in [8], where they describe M2-brane surface defects, which are indeed labeled by a
5As we shall see in section 4.3 the Vortex loop operator VN;R maps to a Wilson loop for the U(N
1), and the U(1) is the diagonal factor in
algorithm we put forward in this paper, the mirror map between loop operators in
arbitrary linear and circular quivers can be constructed, and we provide explicit representative
examples for both types of quivers.
The key insight that allows us to construct the explicit mirror map between loop
operators in linear and circular quivers is the identi cation of the brane realization of
Wilson and Vortex loop operators in the Type IIB Hanany-Witten construction of 3d
as S-duality in Type IIB string theory [9]. By understanding the detailed physics of branes
in string theory, the action of S-duality on our brane realization of loop operators allows
nd an explicit map between brane con gurations, which in turn yields the mirror
map between Wilson and Vortex loop operators.
Vortex loop on an S1
brane-based predictions.
We provide quantitative evidence for our mirror maps by computing the exact
expecThe computation of the expectation value of Vortex loops combines in a rather interesting
way the computations of the S3 partition function in [17] with the supersymmetric
quantum mechanics index of [18, 19]. The detailed matrix integral capturing the expectation
S3 to the bulk gauge theory. All our computations con rm our
Our understanding of the action of duality on Wilson loops in the context of three
dimensional mirror symmetry give us ideas and renewed con dence that this problem can
also be tackled in other interesting dualities, like four dimensional Seiberg duality [20],
where the gauge groups of the two dual theories are also di erent, and subject to the
puzzles raised at the beginning.9
The plan of the rest of the paper is as follows. In section 2 we study the classes of
de nition of the SCFT. This analysis leads to consider Wilson and Vortex loop operators.
forward a brane realization of Wilson and Vortex loop operators. We give at least two
di erent UV descriptions for each Vortex loop operator, distinguished in particular by
the gauging of the global symmetries of the 1d quiver gauge theory with bulk 3d
The explicit 3d/1d theories obtained this way can be encoded by mixed 3d/1d quiver
diagrams as in
gure 4. We then develop a brane-algorithm that allows us to use the
action of S-duality on brane con gurations to construct the mirror map for loop operators
gauge theory labeled by a linear or circular quiver. Section 4 presents the detailed mirror
brane construction was used in [10, 11] to provide the bulk holographic description of Wilson loops in an
arbitrary representation of SU(N ), following the dictionary put forward in [12, 13] for the fundamental
8The mapping of the S3 partition functions themselves between mirror dual theories was initiated in [14,
15] and proven in general for linear and circular unitary quiver theories in [16].
map for representative classes of gauge theories, including T [SU(N )], circular quivers with
equal ranks on all nodes and supersymmetric QCD (SQCD). In section 5 we consider loop
operators on S3 and write down a matrix model representation of the exact expectation
value of a Vortex loop operator in terms of the supersymmetric index of the SQM that
de nes it and the matrix model for the 3d theory. We perform explicit computations of
the expectation value of Wilson and Vortex loop operators and con rm our brane-based
predictions for the mirror map.10 We also show that the distinct UV de nitions of a given
Vortex loop operator (see
gure 4) give rise to the same operator in the IR, by showing
that the expectation value of the two UV de nitions coincide, and are related by hopping
duality [21]. Some technical details are relegated to the appendices.
Loop operators in 3d N
= 4 theories
The bosonic generators comprise those in the SO(3; 2) ' Sp(4) conformal algebra and
those that generate the SU(2)C
SU(2)H R-symmetry of the SCFT. The supercharges in
OSp(4j4) transform in the (4; 2; 2) representation of SO(3; 2)
SU(2)H . A UV
subgroup of OSp(4j4) that closes into the isometries of at space.11 Mirror symmetry is
the statement that a pair of UV gauge theories ow in the IR to the same SCFT with the
roles of SU(2)C and SU(2)H exchanged:
10In [3, 4] hV i was computed using a disorder de nition of V for abelian theories, which we reproduce
from our SQM perspective and which we extend to arbitrary non-abelian gauge theories.
a straight line in
at space.12
conformal line defects in an N
There are two physically inequivalent classes of
superplicitly.14
tries. Superconformal line defects supported on a time-like line are invariant either
under an U(1; 1j2)W or U(1; 1j2)V subalgebra of OSp(4j4).13 Let us exhibit this more
exFirst, a straight time-like line in
at space preserves SU(1; 1)
SO(3; 2). The U(1; 1j2)W subalgebra is embedded as follows. Under the embedding of
U(1)C in SO(3; 2)
SU(2)H , the supercharges
generating OSp(4j4) decompose as (2; 2)++
in U(1; 1j2)W are (2; 2)++
. The supercharges
.15 For U(1; 1j2)V the analysis is identical with the
roles of SU(2)C and SU(2)H exchanged. A defect preserving U(1; 1j2)W is invariant under
SU(2)H while a defect preserving U(1; 1j2)V is invariant under SU(2)C
The main goal of this paper is to give the UV description of these two classes of line
operators/defects and to identity how the UV descriptions are mapped under mirror symmetry.
A line defect in a UV Lagrangian description of a SCFT can be made invariant under
four of the supercharges in the 3d N
SU(2)H R-symmetry of the UV theory and obey
fQ AA0 ; Q BB0 g = (
2. In Lorentzian signature and in this basis the
There are two inequivalent 1d N
by a line defect. We denote them by SQMW and SQMV . SQMW preserves U(1)C
and the supercharges Q11A0 and Q22B0 , which anticommute to the generator of translations
H along the defect
SQMV preserves SU(2)V
U(1)H and the preserved supercharges Q1A1 and Q2B2 obey
fQ11A0 ; Q22B0 g = A0B0 H :
fQ1A1; Q2B2g = ABH :
fQ+Q+g = H
fQ ; Q g = H
[J+; Q+] =
[J ; Q ] =
[J+; Q+] = Q+
[J ; Q ] = Q
12Operators supported on curves obtained by acting on a straight line by broken conformal generators
are also half-supersymmetric and preserve an isomorphic symmetry algebra. Under a broken conformal
symmetry a time-like line becomes a rectangular time-like hyperbola, a space-like line becomes a space-like
hyperbola (which includes a circle) and a null line remains a null line.
13U(1; 1j2)W and U(1; 1j2)V are the
xed locus of an involution of OSp(4j4).
14A very similar analysis holds for a space-like line defect. For a null line defect, which we do not consider
in this paper, the preserved symmetries are larger.
15The other choice (2; 2)+
(2; 2) + corresponds to the same line defect but with opposite orientation.
= Q112, Q
Q221 = Qy112 while for
SQMV Q
= Q212, Q
SQMW and RH
J12 for SQMV , where J12 is the U(1)? rotation generator transverse to
the defect and RC and RH are the Cartan generators of SU(2)C and SU(2)H respectively.16
Therefore, up to shifts by the \ avour" symmetry RC
J12 for SQMW and by RH
= RH
can de ned, one preserving SQMW and the other SQMV . These line defects ow in the
IR to superconformal line defects preserving U(1; 1j2)W and U(1; 1j2)V respectively. Our
analysis can be succinctly summarized by the following diagram:
UV
3d N = 4 Poincare
OSp(4j4)
U(1; 1j2)W
U(1; 1j2)V
4 theory is by gauging
avour symmetries of the SQM theory with 3d N
= 4 vector
multiplets. The gauging of the defect avour symmetries with bulk vector multiplets is
made supersymmetric by embedding the defect vector multiplet of the supersymmetry
the line defect. The embedding is found by identifying which combination of elds in the
higher dimensional vector multiplet transform as the elds of the defect vector multiplet
under the supersymmetry preserved by the defect. Replacing the defect vector multiplet
multiplet elds ensures that the coupling of 1d elds to 3d elds is supersymmetric under
defect and bulk matter multiplets may also be added when defect matter multiplets can
be embedded in bulk hypermultiplets. Such couplings gauge defect
avour symmetries
with bulk avour or gauge symmetries, depending on which symmetries of the bulk matter
multiplet are global and which are gauged.
We consider UV line defects invariant under SQMW and SQMV obtained by gauging
J12 and RH
J12 is also the commutant of U(1; 1j2)W and U(1; 1j2)V in OSp(4j4). They
appear respectively in the anticommutator of the U(1; 1j2)W and U(1; 1j2)V supercharges preserved by the
corresponding defect.
SQMW : 2d N = (0; 4) ! 1d N = 4 SQM
SQMV : 2d N = (2; 2) ! 1d N = 4 SQM
Superpotential couplings between defect and bulk elds also play an important role in the
construction of defects.
Wilson loop operators
operator, which is labeled by a representation R of the gauge group. It is given by
WR = TrRP exp
multiplet. This operator manifestly breaks the SU(2)C symmetry acting on the three
scalars ~ = ( 1; 2
SU(2)H . If the operator is supported on a straight line,18 it preserves the 1d N
= 4
SU(2)H R-symmetry. In
the IR a Wilson line operator ows to a conformal line operator in the SCFT preserving
U(1; 1j2)W .
vector multiplet as follows19
SQM theory living on the line with the bulk 3d N
= 4 gauge theory. This coupling
elds of the 1d N = (0; 4)
17The choice of scalar determines an embedding of U(1)C in SU(2)C.
18When the scalar couples to the loop with constant charge a circular Wilson loop is not supersymmetric
in the UV theory. See, however, discussion at the end of this subsection and of circular Wilson loops on S
in section 5.1.
a0 = A0
1d =
vector multiplet preserves U(1)C
SU(2)H R-symmetry and the SQMW algebra.20
as a coupling of a fermi multiplet with bulk
elds appeared in [10, 11], where the defect
eld theory was derived from brane intersections in string theory. Inspired by [10, 11], a
role in section 3, where we will use S-duality of Type IIB string theory to identify the
mirror of Wilson loop operators.
TrRP exp
i (A + i ) x_ d
and preserve two supercharges: QAH0
. These Wilson loop operators are in the
cohomology of the supercharges QAH0 of the Rozansky-Witten twisted theory [24]21 obtained
by twisting spatial rotations with SU(2)C . Half-supersymmetric Higgs branch operators
are also in the cohomology of this twisted theory.
Vortex loop operators
is the R-symmetry already present in 4d while SU(2)V emerges as an R-symmetry in the
The embedding of the bosonic elds in the 1d vector multiplet (a3; ~ 1d; d), where ~ 1d is a
a0 = A0
~ 1d = ~
d = D + F12 :
coupling of the fermi multiplet with the bulk through the embedding (2.7) breaks the R-symmetry down
loop operators in [23] are in the cohomology of a supercharge of the Langlands twist [25].
This embedding makes manifest that SU(2)V is preserved and that SU(2)H is broken down
to U(1)H , as it selects one of the auxiliary elds transforming as a triplet of SU(2)H in the
the SQMV algebra.
plets or background vector multiplets. Background vector multiplets for avour symmetries
theories.23 Gauging 1d
that 1d and 3d
avour symmetries with background 3d vector multiplets means
avour
symmetries are identi ed by SQMV -preserving defect cubic superpotential couplings between
defect chiral multiplets and bulk hypermultiplets24
W = q~Ii qaIQia ;
where the index I is a 1d gauge index. The indices i, a are simultaneously indices for
avour symmetries and indices for either 3d
avour or gauge symmetries.
(or i) is a 3d avour index, the superpotential breaks the (otherwise independent) avour
symmetries acting on chiral multiplets qa (or q~i) and hypermultiplets Qia to the diagonal
mass to the 1d chiral multiplets and 3d hypermultiplets that are acted on by the preserved
diagonal avour symmetry group.
Wilson loops can be encoded in a standard quiver diagram shown in gure 2.26
An adjoint chiral multiplet may be added to any U(ni) gauge group factor, an
option which we denote by a dashed line. Each adjoint chiral multiplet is coupled to the
neighbouring bifundamental chiral multiplets through a cubic superpotential, while nodes
without an adjoint chiral multiplet have an associated quartic superpotential coupling the
corresponding bifundamental chiral multiplets.
a combined 3d/1d quiver diagram (analogous 4d/2d quivers have appeared in [21] (see
also [8])).The quiver diagram makes explicit the 1d avour symmetries which are gauged
with bulk dynamical gauge elds and the avour symmetries which are identi ed with 3d
avour symmetries, as shown in gure 5. We use the mixed circle and square notation of [8]
to denote the 1d avour symmetries that are gauged with dynamical 3d vector multiplets.
This 3d/1d quiver also assigns a defect cubic superpotential coupling between 1d chiral
multiplets and 3d hypermultiplets for each triangle that can be formed with these elds.
22The choice of auxiliary eld determines an embedding of U(1)H in SU(2)H.
embedding (see appendix A), which we denote by Q, allows one to write supersymmetric couplings between
25When one of the indices is a 3d gauge index the superpotential indeed enforces the gauging of 1d avour
defect chiral multiplets and bulk hypermultiplets.
symmetries with a 3d dynamical vector multiplet.
vector mutiplets (dynamical or weakly gauged).
Demanding that the UV supersymmetric 3d/1d Lagrangian coupling 1d chiral
multiF12 = g2
is the moment map for the avour symmetry acting on the 1d chiral multiplets
that is gauged with the bulk (dynamical) vector multiplet and g is its 3d gauge coupling.
Therefore, in the semiclassical UV description, defect elds induce a singular Vortex
con guration on the 3d gauge elds. This justi es our use of the subscript V to describe
this class of line defects, which we refer as Vortex line defects/operators. These UV Vortex
line defects ow in the IR to conformal line operators in the SCFT preserving U(1; 1j2)V .
an arbitrary curve
in R3 have been constructed in (2.8), it should be possible to construct
would preserve two supercharges: QCA
coupling of the 1d N
= 4 SQM to the bulk 3d N
= 4 theory. Such a Vortex loop
A0 . These Vortex loop operators are
in the cohomology of the supercharges QCA of the other version of the Rozansky-Witten
twisted theory, obtained by twisting spatial rotations with SU(2)H . Half-supersymmetric
Coulomb branch operators, that is monopole operators, are also in the cohomology of this
twisted theory.
Given two UV mirror theories that ow in the IR to the same SCFT, we can construct
both classes of line operators in each of the UV theories. How are line operators mapped
under mirror symmetry? Since mirror symmetry exchanges SU(2)C with SU(2)H in dual
mirror theories, Wilson line operators of one theory are mapped to Vortex line operators
in the mirror and viceversa. This can be represented by the following diagram:
Our immediate goal is to come up with an algorithm that yields the duality map
between Wilson and Vortex loop operators in mirror dual theories.
Brane realization of loop operators and mirror map
gauge theories of [9] and recall how mirror symmetry gets realized as S-duality in string
theory. Central to the main goal of this paper is the brane realization of both types of
line defects discussed in the previous section that we put forward in this section. We then
devise an explicit algorithm using branes in string theory to identify the map between loop
operators in mirror dual theories.
energy limit of brane con gurations in Type IIB string theory [9]. This consists of an array
of D3, D5 and NS5 branes oriented as shown in table 1.27
The gauge theory associated to a brane con guration is constructed by assigning:
A U(N ) vector multiplet to N D3-branes suspended between two NS5-branes
A hypermultiplet in the fundamental representation of U(N ) to a D5-brane
intersecting N D3-branes stretched between two NS5-branes
A hypermultiplet in the bifundamental representation of U(N1)
U(N2) to an
NS5brane with N1 D3-branes ending on its left and N2 branes ending on its right
Depending on whether the x3 coordinate takes values on the line or is circle valued,
linear topology or circular topology: linear and circular quiver diagrams respectively. The
quiver diagrams for linear and circular quiver theories are presented in gure 1. The general
brane con guration realizing a linear quiver theory is shown in gure 6. For a circular quiver
the x3 direction is periodic and there are extra D3-branes stretched between the rst and
last NS5-branes.28
27For more details of these brane constructions see [9, 29{31].
28The number inside a circle denotes the rank of a gauge group factor. The number inside a
rectangle denotes the number of hypermultiplets in the fundamental representation of the gauge group factor
corresponding to the circle to which attaches. A line between two circles represents a bifundamental
hypermultiplet of the two gauge group factors connected by the line.
group factor U(Nc) has a number of fundamental hypermultiplets Nf obeying Nf
2. The second condition is automatically obeyed by linear quivers obeying the
rst condition, but it is an extra requirement for circular quivers. When these conditions are
satis ed the gauge group of the quiver can be completely Higgsed [29, 32] and there are no
monopole operators hitting the unitarity bound [29].29 Mirror symmetry is the statement
The SU(2)C
SU(2)H R-symmetry of such an irreducible SCFT coincides with the
SU(2)H R-symmetry of the UV gauge theory. In the brane construction the
Rsymmetry is realized geometrically as spacetime rotations: SU(2)C rotates x789 and SU(2)H
and the Higgs branch.30 These are hyperkahler manifolds invariant under SU(2)H and
SU(2)C and acted on by a group of isometries GC and GH respectively. GH is manifest
in the UV de nition of the SCFT and is realized as the avour symmetry acting on the
hypermultiplets, while only the Cartan subalgebra of GC is manifest in the UV. Each U(1)
gauge group factor gives rise to a manifest U(1) global symmetry, known as a
topological symmetry, which acts on the Coulomb branch. The abelian symmetry acting on the
Coulomb branch can be enhanced to a non-abelian GC symmetry when conserved currents
associated to the roots of GC can be constructed with monopoles operators.31 The
nontrivial, irreducible SCFT sits at the intersection of the Higgs and Coulomb branch where
the R-symmetry is enhanced to SU(2)C
SU(2)H . The IR SCFT inherits a GC
GH
global symmetry. In the brane realization, the Coulomb branch corresponds to the motion
of D3-branes along x789 while the Higgs branch to the motion of D3-branes along x456.32
mirror symmetry [9], whereby two di erent UV gauge theories ow to the same nontrivial
SCFT in the IR with the roles of SU(2)C and SU(2)H exchanged. Mirror symmetry is
realized as S-duality in Type IIB string theory combined with a spacetime rotation that
29When Nf < 2Nc or PiP^=1 Mi
30Mixed branches can emerge at submanifolds of the Higgs and Coulomb branch.
31GC maps to the avour symmetry acting on the hypermultiplets of the mirror theory.
2 is required for complete Higgsing in
circular quivers. Indeed, unless there are two D5-branes, D3-branes segments cannot be detached from the
2 for circular quivers the IR theory is believed to contain a decoupled
M nodes
sends x456 to x789 and x789 to
x456, which exchanges SU(2)C with SU(2)H . The
combined transformation, which we will refer as S-duality for brevity, maps the class of brane
con gurations we have discussed to itself. Given the brane con guration corresponding to
low energy dynamics of the S-dual brane con guration. The mirror UV gauge theory can
be read by rearranging the branes along the x3 direction, possibly using Hanany-Witten
moves [9] involving the creation/annihilation of a D3-brane when an NS5-brane crosses a
D5-brane, to bring the S-dual brane con guration to a con guration where the low energy
gauge theory can be read using the rules summarized above. This transformation preserves
the type of quiver, and thus the mirror of a linear quiver is a linear quiver and the mirror of
a circular quiver is a circular quiver.33 Examples of mirror-dual pairs of quivers are given
in gure 7.
= 4 SCFT in
at space admits canonical relevant deformations preserving
symmetries GC
GH acting on the Coulomb and Higgs branches of the SCFT. In a UV
realization of the SCFT, these deformations couple to a triplet of mass and FI parameters,
which transform in the (3; 1) and (1; 3) of SU(2)C
SU(2)H . Mass and FI deformations are
obtained by deforming the UV theory with supersymmetric background vector multiplets
in the Cartan of GH and supersymmetric background twisted vector multiplets34 in the
Cartan of GC respectively. In the brane realization, these parameters are represented by
the positions of ve-branes. The position of the i-th D5-brane along x789 corresponds to
a mass deformation m~i while the position of the i-th NS5-brane along x456 corresponds to
an FI parameter ~i. Mass and FI parameters are exchanged between mirror dual theories.
Indeed, in the brane realization of mirror symmetry through S-duality the roles of the
NS5 and D5-branes are exchanged. The positions of the 5-branes in the x3 direction are
irrelevant in the infrared 3d SCFT. For instance, the separation between two consecutive
33The irreducibility condition of the IR SCFT is preserved under mirror symmetry, except for a circular
quiver with a single node, whose mirror dual has a single fundamental hypermultiplet.
34In twisted multiplets the roles of SU(2)C and SU(2)H are exchanged.
NS5-branes is inversely proportional to the coupling gY2M of the e ective low-energy 3d
SYM theory living on the D3-branes stretched between the two NS5-branes. In the deep
IR, where the Yang-Mills coupling diverges, the dependence on gY2M disappears.
Linear quivers that ow to irreducible, interacting SCFT's can be labeled by two
partitions of N |
and ^ | and are denoted by T^ [SU(N )] [29]. Circular quivers owing
to interacting SCFT's are labeled also by two partitions of N and a positive integer L, and
can be denoted by C^[SU(N ); L] [31].35 Under mirror symmetry
T^ [SU(N )] () T ^[SU(N )]
C^[SU(N ); L] () C ^[SU(N ); L] ;
and the role of the two partitions are exchanged. The Coulomb branch of these theories,
and by mirror symmetry the Higgs branch, describe the moduli space of monopoles in
the presence of Dirac monopole singularities for linear quivers and the moduli space of
instantons on a vector bundle over an ALE space for circular quivers.
In this paper we give a brane realization of both classes of loop operators discussed
in section 2 and put forward an algorithm that produces a map between loop operators of
mirror dual theories.
Brane realization of Wilson loop operators
A key ingredient in our derivation of the mirror map of loop operators is identifying a brane
realization of Wilson loop operators, which are labeled by a representation R of the gauge
quiver gauge theory admits a simple brane interpretation, obtained by enriching the setup
We start with the brane realization of a supersymmetric Wilson loop in the k-th
antisymmetric representation of a U(N ) gauge group factor in the quiver, which we denote
by Ak. Such an operator insertion is realized by adding k F1 strings stretched in the x9
direction ending at one end on the N D3-branes where the U(N ) gauge group is supported
and at the other end on a D5' brane, de ned as a D5-brane stretched in the x045678
directions.36
The brane con guration realizing such a Wilson loop is given in table 2.
The array of fundamental strings ends between the two NS5-branes over which the N
D3-branes are suspended. This brane setup is depicted in the example of gure 8-a.
This enriched brane con guration is supersymmetric: it preserves the SQMW
subal35In this paper we will not need the explicit mapping between the data of the quivers and , ^ and L,
but present it here for completeness. It is based on the linking numbers of the D5-branes li and NS5-branes
^lj, which obey Pk
Nj + Psk^=j Ms, where
36Adding the D5'-brane does not break any further symmetries beyond those broken by the F1-strings.
Table 2. Brane Realization of Wilson Loops in 3d N = 4 Gauge Theories.
of a Wilson loop in the Ak representation of the U(3) node of a linear quiver. Here 1
b) Brane con guration with k F1-strings ending on a D5-brane, realizing the insertion of a Wilson
loop in the Sk representation of the U(2) node of the same linear quiver.
open strings stretched between the D3 and the D5'-branes gives rise to 1d complex fermions
gauged Fermi multiplet
dt y [i@t + (A0 +
representation of U(N )
U(1), where U(N ) is the gauge group on the D3-brane segment
where the F1-strings end and the U(1) is the avour symmetry associated to the D5'-brane.
The fermions can be integrated out exactly and yield37
global anomaly for the U(1)
U(N ), since under large gauge transformations Z0 !
37Here we put the system on a circle of length .
Our brane realization of the Wilson loop, however, engineers a bare supersymmetric
Chernthe presence of the Chern-Simons term therefore yields
Z =
The presence of k F1-strings stretched between the D3 and D5'-branes is represented in
the gauge theory by the insertion of k creation operators for these fermions in the past and
k annihilation operators in the future. Physically, these operators insert a charged probe
into the gauge theory. Integrating out these fermions inserts a supersymmetric Wilson loop
operator in the k-th antisymmetric representation [10]
= e i mkTrAk U :
The weights of the k-th antisymmetric representation of U(N ) admit an elegant
description in the brane construction. We must distribute k F1-strings among N D3-branes
(all k F1-strings terminate at the other end on a single D5'-brane). To a pattern of k
F1-strings where kj strings end on the j-th D3-brane (see gure 9) we associate a set of N
+ kN , with kj
0 for all j. However,
not all positive integers kj are allowed. There can be at most one F1-string stretched
between a D3-brane and a D5'-brane. This is the so-called s-rule [9], and is a constraint
that follows from Pauli's exclusion principle [36]. Therefore, the allowed con gurations are
described by a collection of N non-negative integers fkj g with the constraint that kj
This set of con gurations is in one-to-one correspondence with the weights of the k-th
antisymmetric representation of U(N ), i.e. of Ak.
We now turn to a Wilson loop in the k-th symmetric representation of U(N ), which
we denote by Sk. Inserting a Wilson loop in the k-th symmetric representation is realized
by adding k F1 strings stretched in the x9 direction ending at one end on the N D3-branes
where the U(N ) gauge group is supported and at the other end on a D5-brane stretched in
the x012456 directions and localized in the x9 direction. The array of fundamental strings
ends between the two NS5-branes over which the N D3-branes are suspended. This setup
is illustrated in gure 8-b. In this case the charged probe particle inserted by the Wilson
loop can be though of as arising from a very heavy hypermultiplet, represented by adding a
D5-brane to the theory and then taking the D5-brane far away from the stack, thus giving
it a large mass and making the hypermultiplet elds nonrelativistic. Integrating out the
heavy hypermultiplet in the presence of k heavy insertions yields a supersymmetric Wilson
loop operator in the k-th symmetric representation [10, 11].39
38This coupling is obtained by inserting the ux produced by the D5'-brane on the non-abelian
ChernSimons term on the worldvolume of the D3-branes. This is T-dual to the haf-integral Chern-Simons term
discussed in [35].
a large mass to a hypermultiplet.
39In [11] the heavy charged particle was obtained by going to the Coulomb branch while here by giving
4, ending on each of the four
D3-branes, associated to a weight (k1; k2; k3; k4) of a representation of U(4).
The weights of the k-th symmetric representation of U(N ) admit an elegant description
in the brane construction. We again distribute k F1-strings among N D3-branes (all k
F1strings terminate at the other end on a single D5-brane). To a pattern of k F1-strings
where kj strings end on the j-th D3-brane (see gure 9) we associate a set of non-negative
integers fkj g obeying k = k1 + k2 +
+ kN , with kj
0 for all j. In this case an arbitrary
number of F1-strings can be stretched between the D5 and a D3-brane. Therefore the set
of con gurations is in one-to-one correspondence with the weights of the k-th symmetric
representation of U(N ), i.e. of Sk.
Our brane construction can be easily generalized to Wilson loops in the tensor product
a=1Sk(a)
d0
b=1 Al(b) . This requires considering F1-strings stretched between d D5-branes
and d0 D5'-branes and the N D3-branes that support the gauge group. For the above
mentioned representation k(a) F1-strings must emanate from the a-th D5-brane and l(b)
F1-strings from the b-th D5'-brane. Integrating out the massive charged particles produced
by this con guration yields a Wilson loop in the desired representation. Furthermore, the
set of allowed F1-string con gurations, labeled by fkj(1); : : : ; kj(d); lj(1); : : : ; lj
gj=1:::N with
2 N, lj(b)
2 f0; 1g and such that Pj kj
the weights in the representation R =
a=1Sk(a)
wj = Pd
a=1 kj
; wN ) in the orthogonal basis with
In order to describe Wilson loops in the above mentioned representation we have to
place the d D5 and d0 D5'-branes at di erent positions in the x3 and x9 directions. The
separation in the x9 direction is not essential at this stage but plays a role when we identify
the S-dual brane con guration and the mirror dual Vortex loop. The separation in the x3
direction is more crucial: if two D5-branes sit at the same x3 position, the pattern of
F1strings is more complicated since strings can now break and be stretched between the two
D5-branes preserving the same amount of supersymmetry. In this case we expect that the
brane con guration would insert a Wilson loop in an irreducible representation of U(N ),
as in [10, 11]. An irreducible representation R of U(N ) labeled by a Young diagram with
operations we can pull out of the integral the factors depending on the weight w,
V (p)(w) = ( 1)(p 1)wtot 1
2 Pjp jw (j)
1 Z
= ( 1)(p 1)wtot 1
2 Pjp jw (j) 1 Z
In the last step we have re-transformed the matrix model integrand into the ZT [SU(p)]
partition function, but with unshifted mass parameters. The resulting matrix model is
only the w dependent prefactor
V (p)(w) = ( 1)(p 1)wtot 1
2 Pjp jw (j) :
The full Vortex loop vev is obtained by summing over all the single weight contributions.
Using (5.67) we obtain
hVR(p)i = ( 1)(p 1)jRj X e
2 Pjp jwj ;
representation R. Up to the overall sign factor, V (p) evaluates to a background Wilson
R
loop in the representation R of the U(p)
U(N )J subgroup of the topological symmetry
acting on the Coulomb branch. This is precisely the prediction (4.7) derived from the
brane picture.
We now turn to the evaluation of the vevs of the Vortex loops VM;R of the U(N
Note that the Vortex loop vev hVN;Ri, associated to the splitting of hypermultiplets
1 and reproduces our prediction (4.8).
In order to evaluate the VM;R vevs, we are going to consider their \right" SQM
realization, namely we choose to insert in the 3d matrix model the right SQM index (5.45) with
the additional background loop (5.49). Again we start by considering a single weight
contribution VM (w), with w = (w1;
; wN 1), and we simplify the matrix model by replacing
the factors irrelevant to the computation by R [
VM (w) = lim
k=1 QjN0 ch( j
k=1 QjN0 ch(mk
where j denote the eigenvalues of the U(N
convenience. The cancellation between numerator and denominator factors leads to
VM (w) = lim
1 Z
2 N wtot QiN<0j sh2( i
N 0! QkN=M+1 QjN0 ch( j
mk) QkM=1 QjN0 ch( j + iwjz
A generalized version of this formula, for N
j) QiNe<j sh(ei
j=1 ch( (j)
j=Ne+1
We compute this matrix model by using a generalized Cauchy determinant formula. Let
us remind the Cauchy determinant formula
We use this formula in the matrix model to replace the factor depending on the weight w,
= N 0
mj) j=1 ch( (j) + iw (j)z
mj) j=M+1
of (5.79), which just produces a sign. This is justi ed because these factors do not have
QkM=1 QjN=01 ch( j + iwjz
j=1 ch( (j) + iw (j)z
mj) j=M+1
wj)z) QiM<j sh(mi
We may now plug this result into the matrix model and pull out the sum over permutation
= lim
QiM<j sh(mi
N 0! QkN=M+1 QjN0 ch( j mk) QiN<0j sh( i
j +i(wi wj)z)
poles crossing the integration contours as z goes from 0 to 1. We obtain
= lim
QiM<j sh(mi
mj ) j=1 ch( (j) + iw (j)z
mj ) j=M+1
k=M+1 QjN0 ch( j
It is now possible to relabel the eigenvalues
j in each integral and recognize each
term in the sum as the same matrix model with shifted masses mj ! mj
iw (j)z, for
p. The shifted matrix model is simply the bare partition function Z, as can be
VM (w) = lim
k=M+1 QjN0 ch( j
QiM<j sh(mi
mj ) j=1 ch( j + iw (j)z
mj ) j=M+1
X ( 1)N0 PjM=1 w (j) 1
where wM; = (w (1); w (2);
; 0). Summing over the weights w of the
representation R and using (5.67) yields the nal result
where wM = (w1; w2;
; 0). To check mirror symmetry, it is useful to
reexpress the result in terms of Z, which is manifestly invariant under the exchange of mass
and FI parameters,
hVM;Ri = ( 1)(N+M 1)jRj X e
2 N PjN=M1+1 wj 1
We remind the reader that we did not keep track of the overall sign in the evaluation of
the SQM index, so that our evaluation of Vortex loops are only valid up to an overall sign.
This completes our evaluation of the matrix models computing the vevs of the
T [SU(N )] Vortex loop operators.
Mirror map
With all the computations out of the way, it remains to con rm the rest of the mirror map
predictions of section 4 between loop operators in T [SU(N )]. Let start with the simplest
case of the Vortex loop VN 1;R discussed in section 4.2. Its vev is given by (5.83) with
M = N
node given by (5.68),
hVN 1;Ri =
we observe that the two vevs are mapped, up to an irrelevant sign, under the exchange of
the masses and FI parameters mj $ j ,
This is precisely the prediction (4.4) of mirror symmetry obtained from the brane picture.
The more complicated mirror symmetry predictions of section 4.3 are also easily
checked with our explicit exact results. The formula (5.83) for the VM;R loops vevs
expresses a decomposition into contributions labeled by representations (q; Rb) appearing in
the decomposition of R under the subgroup U(1)
U(M ), where U(M ) is embedded as
1) and U(1) is embedded diagonally in U(N
1) ! U(1)
where we have used N
mass in the mirror dual T [SU(N )], shifted by an imaginary number to reabsorb ( 1)
facto a linear combination of W U(M) Wilson loops combined with
tors. This expresses the fact that the VM;R Vortex loop is mapped under mirror symmetry
avor Wilson loops. This
reproduces the mirror symmRebtry prediction (4.6) found by studying carefully the brane
realization of the loop operators and their mapping under S-duality, up to the imaginary
loop is curious and would deserve more investigation.
M denotes the set of representations (q; Rb) in the decomposition of R counted
with degeneracies. Equation (5.83) can be re-expressed as
hVM;Ri = ( 1)MjRj
The factor 1 P
the exchange of mass and FI parameters to the vev given in (5.68) of the U(M ) Wilson
wb2Rbs SbwbZ is labeled by a representation Rbs of U(M ) and is mapped under
loop labeled by a representation Rbs. We obtain the explicit mirror symmetry map
This completes successfully the checks of mirror symmetry for the T [SU(N )] theory.
We have found that T [SU(N )] Wilson loops and Vortex loops can be expressed as
operators acting on the partition function by imaginary shifts of the FI or mass parameters
(see also [50]). More generally, in all T^ [SU(N )] linear quiver theories it seems possible
to express Wilson loops / Vortex loops as operators acting on the partition function by
imaginary shifts of a \generalized" set of FI parameters/ mass parameters. (we will not
provide a proof of this result in this paper).
Loops in SQCD
As our nal example, we consider loops in the U(N ) theory with 2N fundamental and its
mirror dual theory, discussed in section 4.5. We focus on the prediction (4.15), relating the
Wilson loop WR of the U(N ) theory to the Vortex loop Ve1;R
other mirror maps (4.16) can be easily checked with the explicit matrix models computing
(N) of the mirror theory. The
the vevs of the operators, by computations essentially identical to those presented above
for the T [SU(N )] loops.
We denote 1
2 the FI parameter and ma, a = 1;
hypermultiplets in the U(N ) theory, ea
ea+1, a = 1;
me1; me2 the masses of fundamental hypermultiplets in the mirror theory. The matrix models
computing the S3 partition function of the U(N ) theory and its mirror dual are given by
; 2N , the masses of fundamental
1, the FI parameters and
This result will follow from our computations.
( 1; 2; ma) = ( me1; me2; ea) :
Z = e
Ze = ee
where we used to fact that the mirror theory can be decomposed into three pieces: two
T [SU(N )] theories (left and right parts of the quiver in gure 33-b) whose SU(N )
hypermultiplet avor symmetries are gauged with the U(N ) gauge symmetry of the central node
(which has two more hypermultiplets by itself). The matrix model is then a combination of
these three pieces and can be expressed using two T [SU(N )] partition functions with mass
parameters identi ed with the U(N ) node eigenvalues j and FI parameters as indicated.
We have also added background Chern-Simons terms given by
e = e2 i 2 P2aN=1 ma ;
ee = e2 i(me 1+me 2) P2aN=N+1 ea ;
which are unphysical, as they belong to the set of nite counterterms parametrizing
ambiguities of the partition function, but are useful to obtain partition functions which match
exactly under mirror symmetry, namely under the identi cation
= ( i)
sh sh(M1
analytically continued to
2 C. After simpli cations we obtain
W (w) =
N ! Q[ma] shN ( 1
YN e2 i( 1m (j)+ 2m 0(j)+N iwjm (j))
e2 i( 2m (j)+ 1m 0(j)+N iwjm 0(j)+N )
We now turn to the mirror dual operator which should be the Vortex loop Ve1;R
to the discussion in section 4.5. We compute its vev from its 3d/1d realization with the left
(N), according
SQM. The matrix model following from (5.48), with SQM factor (5.47) and background
loop (5.49) can be expressed after cancellations of factors of ch as
W (w) =
where we de ne Q[xa]
we use the identity
2 PjN wj j e2 i( 1 2) PjN j
1 a<b N sh(xa xb)sh(xa+N
xb+N ). To compute the integrals
; (5.96)
X Ve (w)
Ve (w) =
The matrix model computing the vev of a Wilson loop WR in the U(N ) theory is
hWRi =
W (w) =
2 PjN wj j
Using twice the Cauchy determinant formula (5.77) we nd
2 PjN e (j)wjz Y
Plugging the explicit values (5.51) leads to
Ve (w) =
lign, since these factors do not have poles as z goes from 0 to 1. The ratio of sh factors
then simpli es to ( 1)(N 1)wtot . The remaining integrals can be performed using (5.97)
Ve (w) =
YN e2 i(me 1 e (j)+me 2 e 0(j)+N iwj e (j))
e2 i(me 2 e (j)+me 1 e 0(j)+N iwj e 0(j)+N )
to the matrix models. This con rms the mirror symmetry prediction (4.15)
( me1; me2; ea), up to a sign, which was not carefully analyzed in the computations leading
phases e , ee to the matrix models.
We notice that the partition functions Z and Ze can be found by setting to zero the weight
w and removing the normalization factors Z 1 and Ze 1 in the formula for W (w) and Ve (w)
respectively. We observe then that Z and Ze are exactly mapped under the identi cation
Hopping duality
We have been claiming several times that each Vortex loop can be realized (at least) by
two di erent 3d/1d defect theories, which are read from the brane realization by moving
the D1-branes to the closest NS5 on the left or on the right. The equivalence between the
two defect theories is called hopping duality, in analogy with [21] (see also [8]). We can
show that the S3 partition function of the two defect theories indeed match.
Consider a Vortex loop VM(N;R) in a certain 3d quiver theory, labeled by a representation
M ) of the K fundamental hypermultiplets
of that node. It is realized by a brane con guration of gure 38-a, with jRj D1-branes
ending on N D3-branes with K
M D5-branes on the left and M D5-branes on the right.
The associated left and right 3d/1d theories are shown in 38-b.
by (5.48) with the 1d index (5.45):
The matrix model associated to the right 3d/1d theory and computing hVM(N;R) i is given
hVM(N;R) iright = lim
: (5.102)
; (5.104)
a=1 ch(ma
] indicates the matrix model associated to the other nodes of the 3d quiver,
which play no role in the check of the hopping duality, l and r are the \FI parameters"
K-M M N
K-M M
branes which are not shown in the picture. b) Left and right 3d/1d quivers read from moving the
D1-branes to the NS5 on the left or on the right, related by the hopping duality.
associated to the left and right NS5-branes ( l
r is the FI parameter of the U(N ) node),
ma are the masses of the fundamental hypermultiplets and
kr are the eigenvalues
of the U(Nl) and U(Nr) nodes standing respectively on the left and on the right of the
U(N ) node in the quiver diagram. This simpli es to
hVM(N;R) iright = lim
1 Z dN
a=M+1 ch( j
QN
a=1 ch(ma
iwj z) QkN=r 1 ch( kr
: (5.105)
The meaning of the z ! 1 limit, as explained after equation (5.48), is to take the analytical
continuation of the matrix model computed with iz 2 R. We can thus perform the change
iwj z, leading to
hVM(N;R) iright = lim
1 Z dN e2 ( l r)zjRje2 i( l r) PjN j QiN<j sh2( i
j +iwj z) QK
a=M+1 ch(ma
a=1 ch(ma
j ) QkN=r 1 ch( kr
j + iwj z)
The analytical continuation z ! 1 can be taken directly in the integrand for the factors
in the numerator, because they do not have poles for z 2 C. The expression then
simpli es again and matches the matrix model computing hVM(N;R) i from the left 3d/1d theory
3d/1d theory. b) When the numbers of D3-branes on both sides of N5-branes are equal, namely
the ranks of adjacent nodes are equal, the D1-branes can be moved across the NS5-branes. The
Vortex loops they realize are all equivalent.
hVM(N;R) iright = lim
1 Z dN
j +iwj z) QaK=M+1 ch(ma
j +iwj z)
QaM=1 ch(ma
j ) QkN=r 1 ch( kr
= hVM(N;R) ileft ;
model associated to the left 3d/1d theory given by (5.48) with the 1d index (5.47), after
simpli cation of some factors of ch. Note that the additional background Wilson loop is
important to get a precise match.
The hopping duality also explains the equivalence of Vortex loops labeled by
representations of di erent nodes. This occurs for instance for the Vortex loops of circular quivers
with nodes of equal ranks described in section 4.4 (see (4.13), (4.14)).
Consider the quiver in gure 39-a with two adjacent nodes U(N1) and U(N2), with
K1 = 0 + (K1
K1 and K2 fundamental hypermultiplets respectively and assume N1
N2. The vortex
loop V0(;NR11), with R1 a representation of U(N1) and the subscript 0 indicating the splitting
0), can be computed from the matrix model associated to the right 3d/1d
hV0(;NR11)iright = lim
1 Z dN1 dN2
j=1 k=1 ch(ek
where we indicated only the matrix factors inserted by the 1d defect and part of sh factors
of the two nodes, which will play a role in the check of hopping duality. Moreover the
ch factors in the numerator of (5.45) have been canceled by the matrix factor of the
U(N2) bifundamental hypermultiplet. This Vortex loop is realized by the brane
con guration of gure 39-a with jR1j D1-branes ending on the N1 D3-branes supporting
1) node and standing to the right of the K1 D5-branes.
to the right of the NS5-brane without changing the loop operator inserted. When
D1branes stand on the right of the NS5-branes, they realize a Vortex loop VK( N2;2R)2 with R2 a
representation of U(N2) and the subscript K2 indicating the splitting of U(N2) fundamental
hypermultiplets K2 = K2 + (K2
actually the same as the right 3d/1d theory associated to V0(;NR11), except for one important
di erence, which is that the FI parameter of the terminating SQM node (the node with N1
fundamental and N2 anti-fundamental chiral multiplets) is positive for VK( N2;2R)2 and negative
for V0(;NR11). The matrix model computing hVK( N2;2R)2i from the left 3d/1d theory is given by
K2). The left 3d/1d theory associated to VK( N2;2R)2 is
hV0(;NR1)iright = lim
hVK( N2;2R)2ileft = lim
1 Z dN1 dN2
j=1 k=1 ch(ek
When N1 = N2
make the replacement QiN<j sh( i
j) ! ( 1)(N 1)jRj QiN<j sh( i
can be moved freely across the NS5. Let us consider this case rst. Starting from the matrix
j + i(wi wj)z) in the
integrand. This allows us to use the Cauchy determinant formula (5.77) with j ! j +iwj,
1 Z dN dN e Y sh( i
Relabeling j !
hV0(;NR1)iright = lim
1 Z dN dN e Y sh( i
= lim
( 1)(N 1)jRj e2 rjRj Y Y
j=1 k=1 ch(ek
= hVK( N2;2R) ileft ;
where we have used the Cauchy identity to obtain the second equality and we have again
puting hVK( N2;2R) i, with N2 = N .
This shows that the Vortex loops realized by D1-branes ending on the left or on the
right of an NS5 with equal numbers of D3s on both sides are equivalent. Combining this
property with the hopping duality between the left and right 3d/1d quiver realization of
a Vortex loop, we prove the equivalence of Vortex loops in circular quivers with nodes of
equal rank (4.13), (4.14). This is illustrated in gure 39-b.
When N1 > N2 the map of Vortex loops is more complicated. It can be found from
the brane picture in the same way as we found mirror maps between loop operators. We
consider the brane realization of V0(;NR11) with jR1j D1-strings ending on the N1 D3-branes
gure 39-a. As N1 > N2, the NS5 has a D3-spike on its left side. We then move the
D1-branes to the left, across the NS5-brane. Some D1-branes can be moved along the
D3spike and get attached to the NS5-brane, realizing avor Wilson loops for the U(1) global
the precise prediction from the brane picture is found to be
symmetry associated to the NS5-brane. The other D1-branes end on the N2 D3-branes on
the right of the NS5 and realize a Vortex loop VK( N2;2R)2. Recycling the ideas of section 3.2,
is the set of representations (qs; Rbs) appearing in the decomposition of R1 under
the subgroup U(1) U(N2)
U(N1 N2) U(N2)
U(N1), with U(1) embedded diagonally
into U(N1
N2). WNS5;q denotes the Wilson loop of charge q under the U(1) topological
symmetry associated to the NS5.
This map can be checked by explicit computations, using the generalized Cauchy
formula (5.78). It implies that the Vortex loops V0(;NR11) and VK( N2;2R)2 are redundant and that
in order to describe the mirror map with Wilson loops of the mirror theory, it is su cient
to consider only the loops V0(;NR11) or the loops VK( N2;2R)2. In general for each pair of consecutive
D5-branes in the brane picture, we need only to consider the Vortex loops realized with
D1branes placed between the two D5-branes, with a xed number of NS5-branes on their left
and on their right. This is the mirror statement to the fact that between two consecutive
NS5-branes, we need only consider Wilson loops realized with F1-strings placed between
the two NS5s, with a xed number of D5-branes on their left and on their right.
Acknowledgments
We would like to thank Kevin Costello, Nadav Drukker, Davide Gaiotto, Heeyeon Kim,
Hee-Cheol Kim, Bruno Le Floch, Stefano Cremonesi, R. Santamaria and E. Witten for
discussions. B.A. thanks Perimeter Institute for its generous hospitality during several
visits between 2013 and 2015. JG is grateful to the KITP for its warm hospitality during
the \New Methods in Nonperturbative Quantum Field Theory" Program in early 2014,
which was supported in part by the National Science Foundation under Grant No. NSF
PHY11-25915. This research was supported in part by Perimeter Institute for
Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada
through Industry Canada and by the Province of Ontario through the Ministry of
Research and Innovation. J.G. also acknowledges further support from an NSERC Discovery
Grant and from an ERA grant by the Province of Ontario. B.A. acknowledges support
by the ERC Starting Grant N. 304806, \The Gauge/Gravity Duality and Geometry in
String Theory".
Supersymmetry transformations on S3 and SQM embedding
and hypermultiplet on S3. By restricting to the four supercharges preserved by the SQMV
deformed algebra SU(1j1)l
SU(1j1)R in (5.19)(5.20), we work out the embedding of the
AA0 =
DI0 =
I =
2 AA0 ( I )AB BA0
6 AA0 ( I0 )A0 B0 D
i 2; 12; 21; D1 + iD2) = ( ; ; ; F ).
SQMV embedding. The four Killing spinors generating the deformed SQMV algebra
are (see section 5.2)
In the left-invariant frame, l and l are constant, while r and r have a spatial dependent
phase ei =L and e i =L respectively, where
We recall that the equations solved by the Killing spinors are
2 [0; 2 L] is the coordinate along the loop.
r l =
r r =
r l =
r r =
where L is the radius of S3. The supersymmetry transformations under the supercharges
in deformed superalgebra SQMV reduce to73
l =
l =
r =
r =
3 =
1 =
2 =
1 =
2 =
A =
= 1~2 + 2 1
2 F12 +
2 F12 +
2 F12 +
2 F12 +
fermions, with
= 1; 2.
vector multiplet (see for instance [18]) but in the presence of a background gauge
gauge eld a
The identi cation of the
R-symmetry, generated by J . The presence of the background
L1 a ects the covariant derivative of ; in the transformations above.
73We do not give the supersymmetry transformations of the auxiliary elds and transverse gauge elds,
1d vector multiplet : (v ; x3;
= 0
2 F12 +
1d chiral multiplet :
vector multiplet elds have vanishing U(1) charge, while the elds x1 + ix2 and
We now turn to the 3d N
These do not admit an o -shell formulation, so we provide the on-shell supersymmetry
A0 =
A = i
R-charge 1/2 transforming under complex conjugate representations of the gauge group.
Restricting to the same four supercharges generating deformed SQMV as above, we
1 = ( 1)1~2
( 1)1 = i~1D
( 2)1 = i 1D
( 1)2 =
( 2)2 =
2 = ( 1)2 1
i(D1 + iD2) 1~
3 in the supersymmetry transformations is identi ed with a real mass deformation with
chiral multiplets ( ; +;
complex parameter mF = Li for a
multiplets have GF charge qF =
( 1; ( 1)1; ( 2)1) 1 and ( 2; ( 1)2; ( 2
supersymmetry transformations arises from a superpotential coupling. In the o -shell
transformations it gets replaced with a complex auxiliary eld.
We note that the mass deformation obtained by giving a background to the GF avour
symmetry is not visible anywhere in the transformations (A.4), which means that the
multiplet comes with a U(1) charge.
adjoint chiral multiplet with bottom component
This allows us to identify this avor symmetry with
Moreover the background gauge eld a
tions (A.9) of the chiral multiplets with bottom components 1 and
2, which means that
these multiplets are not charged under J . We then have the identi cation
L1 for J is not visible in the
transformaThis identi cation of generators allows us to match the deformed SQMW supersymmetry
algebra with the SU(1j1)l
SU(1j1)r subalgebra preserved by the defect, as explained in
section 5.2.
Evaluations of the SQM index
In this appendix we compute the 1d N
for some quiver gauge theories with generic U(1)
R-symmetry background z and U(1)F
avor chemical potential
symmetry charges are summarized in table 6 and the speci c adjoint R-charges are given
in equation (5.43).
U(k) theory with N fundamental and M anti-fundamental chirals
chiral multiplets with real masses
j , R-charges r+ and GF
avor charge q+, and M
anti-fundamental chiral multiplets with masses ma, R-charges r and GF
avor charge q .
Mass parameters are in units of the inverse S1 radius. As explained in the main text the
take a negative FI parameter
< 0. This corresponds to the choice of FI parameter when
The partition function is given by
I = JK
k! sin( z)
I6=J sin[ (uI
I6=J sin[ (uI
where the integration contour is de ned so that it picks the residues at the poles from the
the de nition of JK
Res for
< 0, reviewed in the main text. This integral has poles
at uI = i^j , j = 1;
residue only when each u
; N , however, due to the sin[
uJ )] factors, we get a non-zero
I hits a di erent i^j , in particular we have non-zero residues
only when k
N . A non-vanishing residue at u
decomposition k = PN
k=1 kj , with k1;
; kN 2 f0; 1g, where kj = 1 if u
I = i^j for a
= fuI g1 I k is then associated to a
corresponding to permutations of the uI . The partition function is then expressed as
I =
k=1 kj = k
^j + i(ki
a=1 j=1 sh(^j
m^ a + i(kj
The partition function vanishes when k > N , consistent with the fact that there are no
supersymmetry vacua in that range.
Plugging the constraints r+ + r
= 2 and q+ + q
= 1=2 we obtain
j=1 a=1 ch( j
ma + ikj z)
This result matches the formula (5.45) giving the index as a sum over the weights of the
U(k) theory with N fundamental, M anti-fundamental chirals and one
adjoint chiral
chiral multiplets with real masses
j , R-charges r+ and GF
avor charge q+, M
antifundamental chiral multiplets with masses ma, R-charges r and GF
avor charge q and
avor charge qadj = 1.
parameters are in units of the inverse S1 radius. The charges obey the superpotential
constraints r
+ r+ = 2 and q
+ q+ = 1=2.
We introduce the complex parameters
i r2 z. Importantly we have a negative FI
In order to avoid higher order poles in the computation we keep Radj generic and only
set it to 2 at the end of the computation. The partition function or index is given by
I = JK
k! sin( z)
I6=J sin[ (uI
I=1 j=1
I=1 j=1
( uI + im^ a
sin[ ( uI + im^ a)]
I6=J sin[ (uI
( uI + im^ a
sin[ ( uI + im^ a)]
where the integration contour is de ned by picking the residues of the poles from the
\half" of the adjoint chiral multiplet factors Qk
I;J according to the JK
Res prescription
< 0. Concretely the integral is a sum over residues, each contribution corresponding
to a pole u
integers, k = PN
i=1 ki, ki
0, and picking uI ! ui;si = i^i
si R2adj z with si = 0;
The arrow ! indicates a mapping between the index I into the index (i; si). The residue
contribution coming from a given pole u
to permutations of the uI . The partition function is then given by a sum over the residue
I =
PiN=1 ki = k
The explicit evaluation of the residues leads to
= ( 1)(N+M)k Y
N kYi1 sh[^i ^j +i R2adj sz +iz] YN kYi1 sh[^i ^j +i R2adj (s+1)z iz] YN kYi1 sh[i R2adj (s+1)z iz]
i6=j
and N2 = k1 + k2.
where we have introduced a regulating mass parameter
for vanishing factors.75 Let us
sh[ ]
write I (z) = QiN=1 sh[ikiz+ ] . In the limit z ! 1 (at nite ) this factor become trivial.
Using the contraints r+ + r
= 2 and q+ + q
= 1=2, we obtain
i6=j
kj )z] Y Y
i=1 a=1 ch[ i
Here again the result matches the formula (5.45) giving the index as a sum over the weights
U(N2) quiver gauge theory with bifundamental chiral multiplets of
R-charges rb; re and GF -charges qb; qe. In addition we have N U(N2)-fundamental chiral
multiplets with real masses
j , R-charges r+ and GF -charge q+, M U(N2)-anti-fundamental
chiral multiplets with masses ma, R-charges r
and GF -charge q
and an adjoint chiral
1. The charges obey the super
potential constraints r
+ r+ = 2, q + q+ = 1=2, rb + r + Radj = 2 and qb + qe = 1. The FI
e
parameters are both taken negative 1; 2 < 0. We also assume N2 > N1. This setup
correN1 D1-branes ending on
a second extra NS5'-brane (see gure 40). The representation associated to this quiver is
R = Ak1
in order to avoid higher order poles in the computation we keep Radj generic and only set
75 can be introduced as a very small real mass for the adjoint chiral multiplet. In this case
appear in the other factors as well. We do not introduce it in the other factors since it would complicate
signi cantly the discussion, without changing the nal result in the 3d/1d matrix models.
it to 0 at the end. The partition function or index is given by
I =
I=1 j=1
(uI uJ + R2adj z z)] YN1 0 YN sin[
sin[ (vI i^j )]
( vI +im^ a z)]
sin[ ( vI +im^ a)]
The integration contour picks the residues at the poles selected by the JK-recipe as
explained in the main text. The recipe allows to take poles from the the fundamental chiral
mulitplet factors, as well as from \half" of the bifundamental and adjoint chiral multiplet
factors. However one can realize, for instance, that picking a pole u
I = u
the adjoint factor and a pole u
J = v
uI = v
K +q + 2rez z 1, where we have used the superpotential constraints, and in this case
I has an extra zero from the bifundamental factor, canceling the pole from the adjoint
factor. This kind of reasoning leads to the conclusion that we cannot take a pole from the
r
bz from the bifundamental factor, leads to
adjoint factor, as it yields a vanishing contribution.
A careful analysis along these lines leads to the following sets of poles (u ; v )
contributing to I:
With k1 = N1 and k2 = N2
N1, a pole fuI ; vJ g is characterized by a choice of
k1 =
k2 =
The explicit single-integral poles are given by
with ! denoting a mapping between the relevant indices. The range of the sj is such
that it correctly gives the N1 uI -poles and the N2 vI -poles.
The evaluation of the index gives
I =
PjN kj(1) = k1
PjN kj(2) = k2
I(k(1);k(2)) = F ( i; z)
a=1 j=1
sj=0
where we have used the superpotential constraints. F ( i; z) is a complicated expression
which will not be relevant for our analysis of mirror symmetry. It is expressed as a product
of the form
F ( i; z) = Y YN sin[
; i6=j
z + =2)]
i j + z + =2)]
where ; take real values. As in (B.10), there are terms which require a regularization
by a small mass deformation
and which evaluates to
I(k(1);k(2)) = F ( i; z)
a=1 j=1 ch[ j
This result reproduces correctly (5.45) as a sum over the weights of the representation
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
Lett. B 387 (1996) 513 [hep-th/9607207] [INSPIRE].
in 2+1 dimensions, JHEP 06 (2013) 099 [arXiv:1211.2861] [INSPIRE].
supersymmetric theories, JHEP 07 (2014) 137 [arXiv:1211.3409] [INSPIRE].
JHEP 03 (2009) 004 [arXiv:0810.4344] [INSPIRE].
gauge dynamics, Nucl. Phys. B 492 (1997) 152 [hep-th/9611230] [INSPIRE].
[INSPIRE].
arXiv:1407.1852 [INSPIRE].
[hep-th/0604007] [INSPIRE].
[15] S. Benvenuti and S. Pasquetti, 3D-partition functions on the sphere: exact evaluation and
mirror symmetry, JHEP 05 (2012) 099 [arXiv:1105.2551] [INSPIRE].
Chern-Simons Theories with Matter, JHEP 03 (2010) 089 [arXiv:0909.4559] [INSPIRE].
[21] A. Gadde and S. Gukov, 2d Index and Surface operators, JHEP 03 (2014) 080
[22] E.I. Buchbinder, J. Gomis and F. Passerini, Holographic gauge theories in background elds
and surface operators, JHEP 12 (2007) 101 [arXiv:0710.5170] [INSPIRE].
[INSPIRE].
[24] L. Rozansky and E. Witten, HyperKahler geometry and invariants of three manifolds, Selecta
[25] A. Kapustin and E. Witten, Electric-Magnetic Duality And The Geometric Langlands
[26] S. Gukov and E. Witten, Gauge Theory, Rami cation, And The Geometric Langlands
[27] N.R. Constable, J. Erdmenger, Z. Guralnik and I. Kirsch, Intersecting D-3 branes and
Superconformal Field Theories, JHEP 08 (2011) 087 [arXiv:1106.4253] [INSPIRE].
JHEP 12 (2012) 044 [arXiv:1210.2590] [INSPIRE].
06 (2011) 114 [arXiv:1105.0689] [INSPIRE].
[39] I.B. Samsonov and D. Sorokin, Super eld theories on S3 and their localization, JHEP 04
[44] N. Hama, K. Hosomichi and S. Lee, SUSY Gauge Theories on Squashed Three-Spheres,
[INSPIRE].
Unitarity and F-Maximization in Three-Dimensional Superconformal Theories, JHEP 10
Chern-Simons Contact Terms in Three Dimensions, JHEP 09 (2012) 091
[1] K.G. Wilson , Con nement of Quarks , Phys. Rev . D 10 ( 1974 ) 2445 [INSPIRE].
[2] K.A. Intriligator and N. Seiberg , Mirror symmetry in three-dimensional gauge theories , Phys.
[3] A. Kapustin , B. Willett and I. Yaakov , Exact results for supersymmetric abelian vortex loops [4] N. Drukker , T. Okuda and F. Passerini , Exact results for vortex loop operators in 3d [5] B. Assel , J. Gomis, and R.C. Santamaria, unpublished.
[6] N. Drukker , J. Gomis and D. Young , Vortex Loop Operators, M2 -branes and Holography, [7] G.W. Moore and N. Seiberg , Taming the Conformal Zoo , Phys. Lett . B 220 ( 1989 ) 422 [8] J. Gomis and B. Le Floch , M2-brane surface operators and gauge theory dualities in Toda , [9] A. Hanany and E. Witten , Type IIB superstrings, BPS monopoles and three-dimensional [10] J. Gomis and F. Passerini , Holographic Wilson Loops, JHEP 08 ( 2006 ) 074 [11] J. Gomis and F. Passerini , Wilson Loops as D3-branes , JHEP 01 ( 2007 ) 097 [12] J.M. Maldacena , Wilson loops in large-N eld theories , Phys. Rev. Lett . 80 ( 1998 ) 4859 [13] S.-J. Rey and J.-T. Yee , Macroscopic strings as heavy quarks in large-N gauge theory and anti-de Sitter supergravity , Eur. Phys. J. C 22 ( 2001 ) 379 [hep-th /9803001] [INSPIRE].
[14] A. Kapustin , B. Willett and I. Yaakov, Nonperturbative Tests of Three-Dimensional [16] B. Assel , Hanany-Witten e ect and SL(2; Z) dualities in matrix models , JHEP 10 ( 2014 ) 117 [17] A. Kapustin , B. Willett and I. Yaakov , Exact Results for Wilson Loops in Superconformal [18] K. Hori , H. Kim and P. Yi , Witten Index and Wall Crossing , JHEP 01 ( 2015 ) 124 [19] C. Cordova and S.-H. Shao , An Index Formula for Supersymmetric Quantum Mechanics, [20] N. Seiberg , Electric-magnetic duality in supersymmetric nonAbelian gauge theories , Nucl.
[23] K. Zarembo , Supersymmetric Wilson loops, Nucl. Phys . B 643 ( 2002 ) 157 [hep-th /0205160] [29] D. Gaiotto and E. Witten , S-duality of Boundary Conditions In N = 4 Super Yang-Mills Theory , Adv. Theor. Math. Phys. 13 ( 2009 ) 721 [arXiv:0807.3720] [INSPIRE].
[30] B. Assel , C. Bachas , J. Estes and J. Gomis , Holographic Duals of D = 3 N = 4 [31] B. Assel , C. Bachas , J. Estes and J. Gomis, IIB Duals of D = 3 N = 4 Circular Quivers , [32] J. de Boer , K. Hori , H. Ooguri and Y. Oz , Mirror symmetry in three-dimensional gauge theories, quivers and D-branes, Nucl . Phys . B 493 ( 1997 ) 101 [hep-th /9611063] [INSPIRE].
[33] N. Drukker and B. Fiol , All-genus calculation of Wilson loops using D-branes , JHEP 02 [34] S. Yamaguchi , Wilson loops of anti-symmetric representation and D5-branes , JHEP 05 [35] U. Danielsson , G. Ferretti and I.R. Klebanov , Creation of fundamental strings by crossing D-branes , Phys. Rev. Lett . 79 ( 1997 ) 1984 [hep-th /9705084] [INSPIRE].
[36] C.P. Bachas , M.B. Green and A. Schwimmer , ( 8 ; 0) quantum mechanics and symmetry enhancement in type-I' superstrings , JHEP 01 ( 1998 ) 006 [hep-th /9712086] [INSPIRE].
[37] C.G. Callan and J.M. Maldacena , Brane death and dynamics from the Born-Infeld action, Nucl . Phys . B 513 ( 1998 ) 198 [hep-th /9708147] [INSPIRE].
[38] H. Kim , S.-J. Lee and P. Yi , Mutation, Witten Index and Quiver Invariant , JHEP 07 ( 2015 ) [40] G. Festuccia and N. Seiberg , Rigid Supersymmetric Theories in Curved Superspace, JHEP [41] J. Gomis and N. Ishtiaque , Kahler potential and ambiguities in 4d N = 2 SCFTs , JHEP 04 [42] N. Hama , K. Hosomichi and S. Lee , Notes on SUSY Gauge Theories on Three-Sphere, JHEP [43] D.L. Ja eris , The Exact Superconformal R- Symmetry Extremizes Z , JHEP 05 ( 2012 ) 159 [45] E. Witten , Constraints on Supersymmetry Breaking, Nucl. Phys . B 202 ( 1982 ) 253 [46] L.C. Je rey and F.C. Kirwan, Localization for nonabelian group actions , Topology 34 ( 1995 ) [47] T. Nishioka , Y. Tachikawa and M. Yamazaki , 3d Partition Function as Overlap of [48] C. Closset , T.T. Dumitrescu , G. Festuccia , Z. Komargodski and N. Seiberg , Contact Terms , [49] C. Closset , T.T. Dumitrescu , G. Festuccia , Z. Komargodski and N. Seiberg , Comments on