#### Quadrality for supersymmetric matrix models

Received: March
Quadrality for supersymmetric matrix models
Sebastian Franco 1 2 4 6 8 9 10
Sangmin Lee 1 2 3 4 5 7 9 10
Rak-Kyeong Seong 0 1 2 4 9 10
Cumrun Vafag 1 2 4 9 10
0 Department of Physics and Astronomy, Uppsala University
1 Seoul 08826 , Korea
2 365 Fifth Avenue , New York NY 10016 , U.S.A
3 Department of Physics and Astronomy, Seoul National University
4 160 Convent Avenue , New York, NY 10031 , U.S.A
5 College of Liberal Studies, Seoul National University
6 The Graduate School and University Center, The City University of New York , USA
7 Center for Theoretical Physics, Seoul National University
8 Physics Department, The City College of the CUNY
9 Cambridge , MA 02138 , U.S.A
10 SE-751 08 Uppsala , Sweden
We introduce a new duality for N = 1 supersymmetric gauged matrix models. This 0d duality is an order 4 symmetry, namely an equivalence between four di erent theories, hence we call it Quadrality. Our proposal is motivated by mirror symmetry, but is not restricted to theories with a D-brane realization and holds for general N = 1 matrix models. We present various checks of the proposal, including the matching of: global symmetries, anomalies, deformations and the chiral ring. We also consider quivers and the corresponding quadrality networks. Finally, we initiate the study of matrix models that arise on the worldvolume of D(-1)-branes probing toric Calabi-Yau 5-folds.
Brane Dynamics in Gauge Theories; D-branes; Supersymmetric Gauge The-
1 Introduction 2 Supersymmetric gauged matrix models
Mirror symmetry and D-branes at toric singularities
Quadrality from mirror symmetry
Local geometry for order n dualities
Conclusions
B Review: C3=Z3
A The Hilbert series for C3=Z3, C4=Z4 and C5=Z5
{ 1 {
Quadrality
Checks
The quadrality dual
The meaning of quadrality in 0d
Abelian avor anomalies
Periodicity
Deformations
Chiral ring
Quadrality networks
D-brane theories
C
5
Local CP4
Local (CP1)4
The chiral ring and the probed CY5
Towards quadrality from compacti cation
4
5
6
7
8
9
2.1
2.2
2.3
2.4
3.1
3.2
3.3
3.4
4.1
4.2
5.1
5.2
5.3
5.4
7.1
7.2
7.3
7.4
Introduction
Duality, the equivalence between seemingly di erent theories, is one of the most fascinating
phenomena in quantum
eld theory. Some of the best-understood examples of dualities
involve supersymmetry, since it provides an enhanced control of the theories.
Seiberg duality, an equivalence between two 4d N = 1 gauge theories in the IR limit, is
the prototypical example of a supersymmetric duality [1]. More recently, Gadde, Gukov and
Putrov (GGP) discovered that 2d N = (0; 2) gauge theories exhibit triality [2]. This is an IR
equivalence, analogous to Seiberg duality, but of order 3. This means that triality relates
three di erent theories and that the original theory is recovered after three consecutive
transformations. The existence of triality makes it clear that the space of dualities is far
richer than naively suspected and that, in general, it involves order n dualities.
The realization of quantum
eld theories in terms of branes is an extremely fruitful
approach for understanding and uncovering dualities. In a recent paper [3], building on [4],
we showed that mirror symmetry provides a geometric uni cation of dualities in di erent
dimensions. Mirror symmetry naturally explains why (10
2n)-dimensional quantum
theories, which are associated with CY n-folds, exhibit duality symmetries of order n
eld
1.
Our work not only explained 4d Seiberg duality and 2d GGP triality in a uni ed framework,
but also led us to conjecture an order 4 duality for N = 1 gauged matrix models in 0d,
which we called quadrality.1 The main goal of this paper is to introduce quadrality and to
perform various checks of the proposal.
Even though we use mirror symmetry to motivate quadrality, our claim is much
stronger: we postulate it is a property of general N = 1 gauged matrix models. Quadrality
applies to matrix models with potentials, which are ubiquitous in physics. N = 1
supersymmetry can be regarded as an appropriate dressing by fermions that allows for additional
control of the theory.
The organization of this paper is as follows. Section section 2 introduces the basics
of N = 1 gauged matrix models. Section section 3 motivates quadrality by considering
the 0d N = 1 theories on D(-1)-branes probing toric Calabi-Yau (CY) 5-folds and mirror
symmetry. Section section 4 introduces quadrality for general N = 1 gauged matrix models
and elaborates on its physical meaning. Various checks of the proposal are presented in
section section 5. Section section 6 discusses how theories connected by sequences of
quadrality transformations can be organized into quadrality networks. Section section 7
considers theories on the worldvolume of D(-1)-branes probing toric CY 5-folds. We also
show, in the local CP4 example, how the chiral ring of the gauge theory reproduces the
coordinate ring of the probed geometry and remains invariant under quadrality. Section
section 8 discusses the possible derivation of quadrality via compacti cation. We present
our conclusions in section section 9. In two appendices we review the application of Hilbert
series to the chiral rings of gauge theories on D-branes.
2
In this paper we study N = 1 supersymmetric gauged matrix models. This section presents
the basic properties of these theories. Despite the fact that the theories carry only one
1We can regard matrix models as quantum eld theories in 0d.
{ 2 {
HJEP07(21)53
supercharge Q, some useful concepts in supersymmetric theories such as holomorphy and
R-symmetry are still valid. The main reason is that in matrix models, unlike in quantum
mechanics or QFT, we do not have to consider Hermitian conjugates of operators. In the
matrix integral, all Fermi elds are regarded as independent holomorphic variables without
Hermitian conjugates.
The basic multiplets of the supersymmetric matrix models are as follows:
1. Gaugino multiplet V :
2. Chiral multiplet Xi:
3. Fermi multiplet a:
[Q; i] = i ;
fQ; ig = 0 ; [Q; i] = 0 :
fQ; ag = Ga ; [Q; Ga] = 0 ;
fQ; a
g = G ; [Q; Ga] = 0 :
a
Each line presents the components of a supermultiplet and how they transform under the
action of the supercharge. We have adopted a notation that resembles that of 2d (0; 2)
gauge theories.2 The superalgebra Q2 = 0 holds trivially in all multiplets. As far as the
superalgebra is concerned ( ; ) are independent of
and ( ; G) are independent of ( ; G).
However, in view of the structure of interaction terms, it is convenient to regard ( ; ; )
as a single multiplet and ( ; G; ; G) as another one.
We now describe the general structure of N = 1 gauged matrix models, focusing on
the abelian case. The non-abelian extension is straightforward. At least locally in eld
space, both the D-term and J -term contributions are Q-exact, namely
fQ;
g = D ; [Q; D ] = 0 :
Expanding in components and integrating out auxiliary elds, we obtain
SD =
'
X
X
1
2
2
1 D2 + D
X q i i i
i
X q i i i
2
i
t
t
i
X q i i i ;
X q i i i
i
2This resemblance may cause some confusion. See section 2.2 for precise relations.
{ 3 {
We have xed the overall sign of SD and SJ such that their bosonic terms are positive
de nite when
and
satisfy the reality condition ( i) =
i. This condition is necessary
in order for the matrix integral R D( ; ; ; )e S( ; ; ; ) to converge. In contrast, as
mentioned earlier, we do not impose any reality condition for fermions.
HJEP07(21)53
2.1
Novel interaction terms
N = 1 matrix models allow for yet another type of interaction, which we call H-terms:
F
X
1
2 a;b=1
SH =
Hab( ) a b ;
fQ; SH g =
X Hab( )Ja( ) b :
a;b
F
X Hab
b=1
Jb = 0 for each a :
where Hab =
Hba are antiholomorphic functions that depend exclusively on the 's.
Unlike SD and SJ , SH is not locally exact. Using the fact that fQ; ag = Ga and that
Ga = Ja( ) on-shell, we obtain
Since every a is independent, SH is supersymmetric if and only if
SJ =
'
a
X
a
Ga(Ja( )
Ga) + GaJ a( ) + X
Ja( )J a( ) + X
i
a
(2.6)
(2.7)
(2.8)
(2.9)
(2.10)
(2.11)
J -term mass:
H-term mass:
J
= m
X ;
H 1; 2 = m :
X =
=
+
+
;
G :
X =
;
Superspace notation and products.
With 0d N = 1 SUSY, it is possible to introduce
a Grassmann coordinate
and package the di erent components into super elds. For the
matter supermultiplets we have
We refer to this condition as the H-constraint.
Mass terms. The interactions that we just described give rise to two types of mass terms
as summarized in gure 1: a chiral-Fermi mass via a J -term and a Fermi-Fermi mass via
an H-term. They take the following form
The basic distinction between them resides in the commutation properties of their lowest
components. Let us now consider products of super elds.
{ 4 {
Let us now discuss how 2d (0; 2) gauge theories are dimensionally reduced down to 0d N = 2
matrix models, which we express in 0d N = 1 language. This class of theories provides
a concrete illustration of the structures that we introduced in the previous section. Upon
dimensional reduction (followed by a Wick rotation), every multiplet of the 2d (0; 2) gauge
theory splits into two di erent 0d N = 1 multiplets as follows:
{ 5 {
1. 2d gauge ! 0d gaugino + 0d chiral.
2. 2d chiral ! 0d chiral + 0d Fermi.
3. 2d Fermi ! a pair of 0d Fermi's.
Let us denote the 2d gauge, chiral and Fermi multiplets
The 0d eld content is
A ( = 1; : : : ; G2d) ;
I (I = 1; : : : ; C2d) ;
A ; A (A; A = 1; : : : ; F2d) :
fXig = fXI ; Y g ;
f ag = f A; A
; I g :
Despite using a similar notation for both 2d and 0d super elds, the distinction and
correspondence between them should be clear from their subindices. From (2.17), it follows
that the number of 0d multiplets of each type is given by
G0d = G2d ;
C0d = C2d + G2d ;
F0d = 2F2d + C2d :
In the special case of 2d theories on D1-branes probing toric CY 4-folds, the relation
C2d
G2d
F2d = 0 holds [5]. The 0d matrix models obtained by dimensionally reducing
such theories then satisfy
F0d
3C0d + 5G0d = 0 :
Returning to general 2d (0; 2) theories, we can derive the interaction terms of the 0d
theory from those of the 2d parent. The J -terms are
0d
a
J a
J
A
A
A =
A
E
A = EA
2d
I=( )
Y
X
X
Y
In the last column, we used the standard quiver notation for bifundamental and adjoint
elds.3 Below we will use this notation whenever it is helpful. We also adopted a convention
in which a barred superscript is the same as an unbarred subscript and vice versa.
The H-terms of the 0d theory are
Y )
A=( )
:
(2.21)
The H-constraint (2.9) for
I is automatically satis ed due to the trace condition of the
2d theory
X(JAH
AI
+ JAH
AI ) =
A
X
A
J A
X
!
= 0 : (2.22)
3Of course there can be more than one eld for every pair of subindices. We leave this possibility implicit
in order to simplify the notation.
(2.16)
(2.17)
(2.18)
(2.19)
(2.20)
HJEP07(21)53
The H-constraints for ( A; A) schematically read as follows
X HAI JI + X H
ABJB
I
B
!
A=( )
2E
( Y
X
X
Y ) ( Y
gure 2 shows how the two partial sums cancel against each other.
2.3
Gauge/global symmetries and anomalies
A 0d theory contains no derivatives. So, we cannot gauge a symmetry by turning an
ordinary derivative into a covariant derivative. It may seem unclear whether/how we can
distinguish a gauge symmetry from a global symmetry.
Guided by dimensional reduction from higher dimensions, we distinguish gauge
symmetries from global symmetries in three ways. First, we assign a gaugino multiplet for each
factor in the gauge symmetry. There is no multiplet associated to the global symmetry.
Second, we require that all observables (more on this below) be gauge invariant, but may
be charged under global symmetry. Third, we require that the \anomaly" for the gauge
symmetry vanishes, but allow for non-vanishing global anomaly. By anomaly we mean any
non-invariance of the integration measure under the symmetry action.
In the presence of an anomaly, gauge or global, the integration over the symmetry
orbit forces the partition function (i.e. the expectation value of the identity operator) to
vanish. Since all observables are gauge invariant, a gauge anomaly implies that the theory is
completely trivial. In contrast, a global anomaly can be cancelled by computing correlation
functions of charged observables.
In 0d, anomalies are linear. We will use the matching of abelian
in the spirit of 't Hooft anomaly matching in higher dimensions, when we argue for the
equivalence of several theories. The integration measure should carry a de nite charge, i.e.
anomaly, under global symmetries. If two matrix models are dual, as a necessary condition,
their global anomaly should be the same.
The superalgebra Q2 = 0 suggests that the observables of the theory are cohomology classes
of Q. We begin with the ring of all gauge invariant products of elementary elds and study
the cohomology of Q. The resulting observables,
(2.24)
still form a ring. By a slight abuse of language, we may call it the \chiral ring".
Suppose we enumerate all possible Oi's. In principle, solving the theory completely
means calculating arbitrary correlation functions among them,
Z
hOi1 Oi2
Oin i =
D( elds)Oi1 Oi2
Oin e S :
Here, the classical action, the integral measure and the observables are all independently
gauge invariant. As for the global symmetries, we leave the possibility that the integral
measure has some abelian anomaly, which can be cancelled by the global charges of the
observables.
In summary, if we want to establish duality between two theories, we should check the
following things: (1) global anomaly, (2) spectrum of observables (including their global
charge and R-charge), (3) correlation functions.
3
Geometric motivation for quadrality
In this section we use the geometric engineering of 0d N = 1 theories in terms of
D(-1)branes probing toric CY 5-folds to motivate a new duality. This duality turns out to be of
order 4, so we will refer to it as quadrality.
3.1
D-branes probing Calabi-Yau manifolds
The (10
2n)-dimensional gauge theories that live on the worldvolume of D(9
2n)branes probing toric CY n-folds have been studied in great detail for n = 3 and 4. For
toric CYs, the connection between gauge theory and geometry is considerably simpli ed
by T-dual brane con gurations: brane tilings for CY3 [6, 7] and brane brick models for
CY4 [5, 8, 9]. It is natural to continue with this sequence as summarized in table 1 and
consider n = 5, which corresponds to D(-1)-branes probing toric CY 5-folds. The theories
on their worldvolume are 0d N = 1 gauged matrix models of the type discussed in section
section 2.
T-duality relates D(-1)-branes probing toric CY 5-folds to Type IIA con gurations
analogous to brane tilings and brane brick models, to which we refer as brane hyperbrick
models. A brane hyperbrick model consists of an NS5-brane wrapped over a holomorphic
{ 8 {
4d N = 1
whole con guration lives on a T 4. Very much like their lower dimensional cousins, we
expect brane hyperbrick models to provide valuable tools for connecting geometry to gauge
theory. However, they are beyond the scope of this paper and we leave their development
for future work.
3.2
Mirror symmetry and D-branes at toric singularities
In order to set up the stage for the discussion in the coming section, we present a lightning
review of the mirror approach to D-branes probing toric CY singularities. We refer the
reader to [3, 10, 11] for details.
in Z
complex W plane
A toric CYn M is speci ed by its toric diagram V , which is a convex set of points
n 1. Its mirror geometry [12, 13] is an n-fold W given by a double bration over the
with u; v 2 C and x 2 C ,
= 1; : : : ; n
1. P (x1; : : : ; xn 1) is the Newton polynomial
(3.1)
(3.2)
(3.3)
W = P (x1; : : : ; xn 1)
W = uv
P (x1; : : : ; xn 1) =
X c~v xv11 : : : xvnn 11 ;
~v2V
{ 9 {
where the c~v are complex coe cients and we sum over points ~v in the toric diagram. By
rescaling the x variables, it is possible to set n of the coe cients to 1.
The critical points of P are (x1; : : : ; xn 1) such that
P (x1; : : : ; xn 1)
(x1;:::;xn 1)
= 0
= P (x1; : : : ; xn 1). The number of critical points is
equal to the number of gauge nodes in the eld theory [10].
The two bers are a holomorphic (n
2) complex dimensional surface
W coming from
P (x1; : : : ; xn 1) and a C
bration associated with the uv piece. The resulting Sn 2
S1
bration over a straight vanishing path that stretches between W = 0 and W = W
hence
gives rise to an Sn.4
The S1 ber vanishes at W = 0. The structure of the gauge theory is determined by
how the surviving Sn 2's intersect on the vanishing locus W 1(0) : P (x ) = W = 0. The
4Vanishing paths can be curved. We refer the reader to [3] for a discussion of this possibility. This
subtlety does not a ect our conclusions.
associated to W0 . We indicate the type of avor contributed by each cycle: chiral (Cin and Cout)
or Fermi (Fin and Fout). (b) Corresponding quiver diagram. We refer to this theory as T .
geometry of the Sn 2's can be e ciently described using tomography, which was introduced
in [11] and further developed in [3]. The x -tomography is the projection of the Sn 2
spheres at W = 0 onto the x -plane. In summary, we can obtain a detailed description of
the con guration of Sn in the mirror geometry by combining the con guration of vanishing
paths on the W -plane with the x -tomographies,
= 1; : : : ; n.
The discussion in the coming section will build on ideas introduced in [3]. It will just
use basic properties of vanishing paths on the W -plane, which is a universal ingredient for
any CY dimension.
3.3
Quadrality from mirror symmetry
Following the discussion in the previous section, mirror symmetry maps D(-1)-branes
probing a toric CY5 to a collection of Euclidean D4-branes wrapping 5-cycles. Let us consider
some elementary features of this mirror con guration. Given a vanishing cycle C0, the
branes wrapping the cycles that intersect with it can be regarded as avor branes. These
avor cycles can be classi ed into four groups, depending on the type of matter
elds
they contribute to C0: fundamental and antifundamental chirals and fundamental and
antifundamental Fermis. Following the quiver representation of these
elds, we refer to
fundamental and antifundamental representations as in and out, respectively. Extending
the ideas introduced in [3], we expect that for any C0 the vanishing paths that are
associated to the four types of avors are organized cyclically on the W -plane as shown in gure
3. This conjecture is a natural generalization of what occurs for CY 3- and 4-folds and is
also based on the expected symmetry between quadrality and inverse quadrality.
An important lesson from [3, 4] is that mirror symmetry not only provides a geometric
uni cation of dualities for a
xed dimension, but also uni es seemingly di erent QFT
equivalences across dimensions.
gure 4.a and b show that the mirror realizations of 4d
them in terms of the same type of transformation.
Seiberg duality and 2d triality are basically identical. This observation naturally leads to
conjecture that the transformation in gure 4.c gives rise to an equivalence between N = 1
gauged matrix models that we call quadrality.
Let us denote Qi the D-brane charge vector for the cycle Ci. The intersection matrix
hCi; Cj i = hQi; Qj i of a CY5 is antisymmetric. The sign of hQi; Qj i determines the
orientation of the bifundamental(s) connecting nodes i and j. The distinction between chiral
and Fermi elds requires additional information regarding the intersection. Without loss
of generality, let us assume that the only non-vanishing intersections of cycles are
hQ1; Q0i = 1
hQ3; Q0i = 1
hQ2; Q0i =
hQ4; Q0i =
1
1
In our discussion, any multiplicities of di erent types of
avors are absorbed into the
ranks of the corresponding
avor nodes. Introducing more general intersection numbers
hQi; Q0i 6=
1 or splitting the avor nodes into collections of multiple nodes with di erent
brane charges is straightforward. Non-vanishing intersections between the
avor nodes
in the initial theory, corresponding to additional matter elds, can also be incorporated
without a ecting our analysis.
Quadrality on node 0 corresponds, as shown in gure 4, to shrinking the cycle C0 to
zero size and reemerging on the W -plane on the wedge between C1 and C2 with a reversed
orientation. The brane charges transform as follows:
Q00 =
Q0
Q02 = Q2
Q03 = Q3
Q04 = Q4
Q01 = Q1 + hQ1; Q0iQ0 = Q1 + Q0
(3.4)
(3.5)
by =2, while keeping the avor nodes xed.
Inverse quadrality is obtained by moving C0 in the opposite direction to the wedge between
C3 or C4, or by applying quadrality three times.
Remarkably, even without a more detailed analysis of the mirror, we can draw
important conclusions about what this transformation implies for quadrality.
Rank of the gauge group.
The transformation of the rank of the gauge group follows
from conservation of the total brane charge. Initially, we have
Since the ranks of the avor nodes remain constant, we have
QT =
X Ni Qi :
4
i=0
Conservation of brane charge QT = Q0T implies
Dual avors.
The avor vanishing paths divide the W -plane into four wedges.
Quadrality corresponds to moving the cycle associated to the dualized gauge group to a neighboring
wedge. This implies that, in the quiver, the transformation of avors is a
=2 rotation of
the corresponding arrows, while keeping the avor nodes xed. This is shown in gure 5
This is yet another manifestation of the geometric uni cation of QFT equivalences in
di erent dimensions The transformations of avors in Seiberg duality and triality are also
given by 2 =k rotations, with k = 2; 3, respectively [3, 4].
Mesons. The modi cation of Q1 in (3.5) leads to non-vanishing intersections between
node 1 and nodes 2, 3 and 4 in the dual. These new intersections give rise to mesons.
Furthermore,
hQ01; Q2i = hQ0; Q2i !
hQ01; Q3i = hQ0; Q3i !
hQ01; Q4i = hQ0; Q4i !
12 (Fermi)
31 (Fermi)
M14 (chiral)
(3.6)
(3.7)
(3.8)
(3.9)
The chirality and type of eld for each meson is thus determined by those of the original
avors connecting node 0 to the global symmetry nodes. We conclude the dual contains
the following mesons:
12 (Fermi), 31 (Fermi) and M14 (chiral). This expectation will be
con rmed by a eld theory analysis in section section 4.1.
The quiver. Summarizing our previous analysis, we conclude that the quiver diagram for
the quadrality dual of theory T , which we call T 0, is the one shown in gure 6. Throughout
the paper, we will often adopt a notation in which X and
indicate chiral and Fermi
avors, from the perspective of the dualized gauge group, respectively. Similarly, M and
correspond to chiral and Fermi mesons, i.e. gauge singlets.
Periodicity.
As we mentioned, it is possible to argue that quadrality is an order 4 duality
based on the fact that the avor vanishing paths divide the W -plane into four wedges. Here
we provide a more explicit proof of the periodicity. For all i; j = 0; : : : ; 4, we have
hQ0i000; Q0j000i = hQi + hQi; Q0iQ0; Qj + hQj ; Q0iQ0i
= hQi; Qj i + hQi; Q0ihQ0; Qj i + hQj ; Q0ihQi; Q0i + hQi; Q0ihQ0; Q0i (3.10)
= hQi; Qj i
where in the last line we used the antisymmetry of the intersection pairing.
3.4
Local geometry for order n dualities
The minimal local geometry that captures the order (n
1) duality of a (10
2n)dimensional theory (i.e. 4d duality, 2d triality and 0d quadrality) is local CPn. This is
the Cn=Zn orbifold with action (1; : : : ; 1) on the di erent complex planes. Local CPn gives
rise to n critical points and hence to n independent n-cycles. This number is precisely
what is required for studying a generic eld content in every dimension. One of the cycles
accounts for the gauge group and the remaining n
1 corresponds to nodes (which are
also gauged) for all possible types of avors: two in 4d (fundamental and antifundamental
chirals), three in 2d (fundamental and antifundamental chirals, and Fermi) and four in 0d
(fundamental and antifundamental chirals, and fundamental and antifundamental Fermis).
volume.
! Wi we can make the corresponding n-cycle, here C0, shrink to zero
The Newton polynomial coming for the toric diagram for local CPn contains, in general,
n + 1 terms. It is possible to rescale n of its coe cients to 1, leading to
P (x1; : : : ; xn) = x1 + : : : + xn +
(3.11)
1
x1 : : : xn
+
;
with
2 C. The parameter
does not a ect the critical points i
. However, it modi es
the corresponding critical values Wi by an overall shift. Equivalently, we can think that
the critical values Wi are xed and that the origin of the W -plane is shifted by . Thus,
this geometry has enough freedom for studying the limit in which any of the n gauge groups
go to in nite coupling. We can attain this by sending
CP5, which makes the volume of the corresponding cycle vanish.
! Wi as shown in gure 7 for
4
Quadrality
In the previous section we motivated quadrality using mirror symmetry for con gurations
of D(-1)-branes probing toric CY 5-folds. It is however natural to conjecture that, as it
occurs for Seiberg duality and triality, that quadrality applies to arbitrary N = 1 gauged
matrix models. The elementary quadrality transformation, which we now explain in more
detail, can be phrased in terms of the simple SQCD-like theory T that we introduced earlier.
For quick reference, gure 8 reproduces the quiver diagrams for T and its quadrality dual.
Let us discuss T
rst. The four avor nodes and the gauge node in the quiver
correspond to U(Ni) = SU(Ni)
U(1)(i) groups. All matter elds transform in bifundamental
representations, so the global diagonal combination of all of them, Pi4=0 Qi decouples.5
Without loss of generality, we can identify the global symmetry with SU(N1)
U(1)(3).6 It is straightforward to read the
transformation properties of the matter elds under the global symmetry group from the quiver.
Cancellation of the abelian gauge anomaly constrains the ranks of the avor nodes
to satisfy
discussed earlier.
N1
N2 + N3
N4 = 0 :
(4.1)
5From now on Qi refers to the charge under U(1)(i). It should not be confused with the D-brane charges
6U(1)(4) is not independent, and the corresponding charge is simply Q4 = (Q0 + Q1 + Q2 + Q3).
If more nodes are gauged, as in the D-branes examples considered in section section 7,
there will be additional matter elds that ensure the cancellation of all gauge anomalies.
4.1
The quadrality dual
We propose the quadrality dual T 0 is given by the quiver in gure 8, together with some
J - and H-terms that we discuss below. Since N1 + N3
N2
N4 = 0, the abelian gauge
anomaly still vanishes. The global symmetry of the dual is SU(N1)
U(1)(3), in agreement with the original theory.7
Let us explain the arguments that lead to this proposal.
Dual gauge group.
The dual theory has a U(N00 ) gauge symmetry, with
N00 = N1
N0:
(4.2)
This result was derived in section section 3 in the case of theories with a D-brane realization
from conservation of the total brane charge. We postulate it holds in general.
Dual avors. For theories arising on D-branes, we used mirror symmetry to derive the
transformation of avors summarized in gure 5. Once again, we propose this
transformation applies to general theories.
Mesons.
There are three types of mesons in T 0. They can be expressed as composites of
the elds in the original theory. In all cases they must contain X10 which, for D-branes, is
the chiral eld charged under the avor node whose brane charge changes under quadrality.
7When determining the global symmetry of a theory, it is necessary to take into account its J- and
H-terms, which can in principle break the naive symmetries preserved by the quiver. We discuss these
interactions below.
(4.3)
(4.4)
The mesons are given by
M14 = X10 X04
The types of elds obtained by taking these products nicely coincide with the ones
established using the brane intersections in (3.9). Notice that in order to form the gauge
invariant meson
31, it is necessary to conjugate X10. This is a novel feature of 0d, which
does not arise in 4d or 2d.
Dual avors-meson couplings.
As it occurs in Seiberg duality and triality, there are
new interaction terms coupling the mesons to the dual avors. These terms are the most
general ones allowed by the gauge and global symmetries.8 In this case, the couplings are
J 12 = X20X01
J 40 = X01M14
H 03; 31 = X01
!
!
!
Quiver loop
( 12X20X01)
( 40X01M14)
To simplify visualization of the interactions, on the right column we give the corresponding
loops in the quiver. Below we will explain how they are crucial for consistency of quadrality.
Interestingly, the appearance of a novel type of meson 31 =
30 X10 is correlated with
the existence of a new type of interactions in 0d, the H-terms.
4.2
The meaning of quadrality in 0d
In this section we would like to examine what we can mean by duality in the context of a
0d QFT. Typically when we have a Seiberg-like duality we start with two distinct theories
in the UV, and only in the IR the two theories will become identical, as they ow to the
same conformal xed point. In such situations, some aspects, such as the chiral ring, are
invariants of the ow and can be studied on either side of the duality even before the
ow. Checking this match has been one of the key evidences for Seiberg-like dualities.
But duality extends beyond the chiral sector and is expected that at the IR
xed point,
correlations function of any collection of elds, whether chiral or not, match on both sides,
with a suitable dictionary of how operators from one side map to the other.
In the case at hand, a robust check of our proposed quadrality is to verify that the
chiral rings match for each dual version. However this is not enough to claim equivalence of
two theories, as there are non-chiral operators in the theory. So one would like to have the
analog notion of \IR" in such theories, so that one could say that the IR of all sides agree
for all operators. However, there is di culty de ning the notion of IR
xed point in the
present context, because the dimension of space-time is 0 and so we have no such notion.
Instead we propose the following alternative. As a supersymmetric theory
ows to the IR
in d > 0 cases, chiral elds do not get renormalized (which we will loosely call \F-terms")
8This principle also holds for more complicated theories. We should always include all interactions
allowed by the gauge and global symmetries.
but non-chiral elds do get renormalized (the \D-terms"). So it is natural to de ne the
notion of IR in d = 0 theories by saying that there is a deformation of the \D-terms" in
the Lagrangian (i.e. Q-trivial additions to the Lagrangian) which leads the theory to have
the expected superconformal symmetry. So we need to address what is the superconformal
group in 0d.
It is natural to expect that the IR superconformal eld theory (SCFT) has a U(1)
R-symmetry.
Then, the superconformal group in 0d should have a bosonic conformal
symmetry given by SO(1; 1)
U(1)R. Let us call the generators of this group
and
R. Moreover we expect, as in higher dimensions, that the number of supersymmetries gets
doubled at the conformal point. Since we started with one nilpotent supercharge, let us call
it Q+, we should obtain another one, Q . We expect the following superconformal algebra:
Q+2 =Q 2 = 0 ;
Note that this symmetry algebra is at the level of elds in the theory and not the Hilbert
space, because this theory has no time dimensions. So we conjecture that there is a
distinguished
xed point where the above algebra is a symmetry of eld space and correlation
functions.9 Whether such a point is unique is not clear and requires further study. In this
paper we check the quadrality only by checking the chiral rings match on all sides (as is
the case with duality checks in higher dimension). This low dimension case allows us to
possibly be able to check a more detailed statement by including all operators! It would be
interesting to pursue this direction and see if one can precisely
x the Lagrangian at the
conformal xed point of these theories and prove the quadrality symmetry for correlation
functions of all operators.
5
In this section we collect additional checks of the quadrality proposal.
avor anomalies provides a non-trivial check of the proposal. As
mentioned earlier, the non-abelian
avor anomalies trivially vanish. The following table
summarizes the anomalies.
1
2
T
N0
N0
N0
N0. However, as explained above, it is not really independent
and can be determined in terms of the other abelian anomalies.
9It is possible (and probably likely) that the symmetry is only a symmetry of the correlation functions
and not the Lagrangian itself, because the path-integral measure may not be invariant.
Interestingly, the matching of the U(1)(1) and U(1)(2) anomalies between the two
theories tests the existence of the non-conventional 31 meson. However, it does not establish
whether this eld is a chiral or a Fermi. If we did not have independent derivations of the
transformations of the gauge group rank based on conservation of brane charge, matching
of abelian
avor anomalies would simultaneously test the combination of the rank, avor
and mesons rules.
5.2
Periodicity
As explained in section section 3, mirror symmetry implies that after four consecutive
quadrality transformations we should return to the original theory. This sequence is shown
in
gure 9. At various steps we have integrated out chiral-Fermi and Fermi-Fermi pairs,
due to mass terms of the form shown in gure 1. For this to be possible, the detailed form
of the J - and H-terms coupling mesons to dual avors is crucial. The periodicity of the
sequence of quadralities is hence a non-trivial check of these couplings.
The rank of the gauge group evolves as follows
N000 = N2
N0000 = N3
N00000 = N4
N0
N1 + N0
N2 + N1
N3 + N2
N0
We see that the gauge anomaly constraint on the ranks of the avors (4.1) is crucial for
returning to the original rank of the gauge group.
5.3
Following Seiberg's seminal work on 4d N = 1 duality [1], deformations have become a
standard tool for testing a wide range of equivalences between eld theories. Here we will
show that similar arguments can be applied to quadrality. When testing the e ect of a
deformation in Seiberg duality, one only needs to consider the magnetic dual. Remarkably,
for quadrality we can, and actually must, study the e ect of any deformation on the
full quadrality sequence.
This is a general feature of order n dualities. The e ect of
deformations on the entire collection of dual theories provides an (n-1)-fold increase in the
number of constraints and consistency checks. N = 1 matrix models have a rich set of
possible deformations with which to test quadrality. In particular, we can introduce X
and
mass terms, which correspond to J - and H-terms respectively.
The original sequence.
The original quadrality sequence was discussed in section
section 5.2. It consists of four theories T , T 0, T 00 and T 000, corresponding to the four quivers
in
gure 9. Since the global symmetry is preserved, the ranks of the avor nodes remain
equal to (N1; N2; N3; N4) in all theories. The rank of the gauge group evolves according
to (5.1).
Possible deformations and deformed sequences. Let us consider the original theory
T , which is shown in
gure 8.
All bifundamental
avors are, generically, rectangular
matrices. We can use global and gauge symmetries to simplify them, such that all entries
are zero except for, at most, those in N0
N0 diagonal submatrices. There are three
possible mass deformations of this theory:
a) X10 02 mass: T can be deformed by introducing an X10 02 mass term, i.e.
J 02 = m 10 :
For simplicity, here and in the deformations that we discuss below, we assume the
rank of the mass matrix is 1. It is straightforward to extend our discussion to higher
rank masses. We call T~ the resulting theory, which still has the same quiver diagram
of T , but with reduced ranks for some of the global nodes, as we now explain. The
mass term clearly breaks the global symmetry down to (N~1; N~2; N~3; N~4) = (N1
1; N2
1; N3; N4).10 The gauge group is una ected so its rank is N~0 = N0.
We now take T~ as the new starting point. Acting with quadrality, we obtain a
deformed sequence of theories T~, T~0, T~00 and T~000. Once again, the quivers for the
deformed sequence are those in gure 9. The di erences with respect to the original
sequence are the following. The ranks of the avor nodes are (N1
1; N2
1; N3; N4)
for all the theories. In addition, using the initial N~i's in (5.1) we determine the
evolution of the rank of the gauge group is
~
T
N0
T~0
N00
1
T~00
N000
T~000
N0000
(5.2)
(5.3)
connected to a new, rank 1, global node.
30X04 mass: this deformation is very similar to the one we have just considered. It
breaks the global symmetry down to (N1; N2; N3
1; N4
1). The rank of the gauge
group is una ected in the initial theory and evolves as follows
c)
H 30; 02 = m. The global symmetry is reduced to (N1; N2
gauge group remains the same in the starting theory and then follows the sequence
It is interesting to notice that a common feature of the three deformations considered
above is that, for each of them, only one theory in the sequence gets higgsed.11
Connecting the original and the deformed sequences.
Alternatively, if the
quadrality proposal is correct, we should be able to obtain the deformed sequences by mapping
the original deformations of T to deformations of T 0, T 00 and T 000. As we will now explain,
it is possible to verify that this is indeed the case.
Before doing so, it is convenient to discuss how general deformations are mapped
under quadrality. In the context of N = 1 matrix models, we refer to any modi cation of
J - or H-terms as a deformation. Deformations modify the global symmetry of a theory,
while preserving its gauge symmetry and matter content. When moving to a dual theory
we must now include all interactions that are consistent with the new global symmetry.
The modi cation in the interactions of the dual theory is identi ed with the map of the
original deformation. Typically, we are interested in introducing new interaction terms
to the original theory, which reduces the global symmetry. As a result, the dual theory
admits new interaction terms that are the translation of the deformation. This prescription
is very general and is the one used when mapping deformations under dualities in other
dimensions. For example, it is precisely the approach one uses in the well-known case of
mapping superpotential deformations of 4d N = 1 theories under Seiberg duality.12
For concreteness, let us focus on case (a), namely on a rank 1 X10 02 mass term.
The other deformations can be understood using similar ideas. The rst dual theory, T 0,
11In an abuse of language, what we mean by higgsing is that some of the gauginos become massive, which
reduces the gauge symmetry.
12It is important to note that while it is relatively straightforward to map deformations under duality
with our prescription, this cannot be done by simply rewriting the component expansions of some terms
in the Lagrangian. Here, again, 4d Seiberg duality provides a familiar example. Deformations can be
easily implemented at the level of the superpotential, just replacing combinations of chiral elds that are
charged under the dualized gauge group by mesons. This process, however, cannot be implemented as a
reorganization of the component expansion of the Lagrangian.
the meson M21, which becomes massive by coupling to other elds and is integrated out.
contains a Fermi meson
12 = X10 02. The deformation of T given by (5.2) maps to an
additional constant contribution to J 12 as follows
(5.6)
(5.7)
(5.8)
(5.9)
T : J 02 = m 10
T 0 : J 12 = X20X01 + m ;
where the rst term in J 12 is the usual coupling between the Fermi meson
12 and the
chiral avors in T 0. The constant term in J 12 , where m is the original rank 1 mass matrix,
is precisely the new interaction that is allowed when the global symmetry is reduced to
(N1
1; N2
1; N3; N4).13
The action now contains the term
SJ = J 12 J 12 + : : : = j 20 01 + mj2 + : : : ;
which xes at a non-zero value an entry in each of the diagonalized
20 and 01. This, in
turn, reduces the rank of the gauge group to N0
1, in perfect agreement with (5.3).
Having correctly reproduced T~0 by mapping the deformation, let us now consider the
two remaining theories in the sequence. The
xed non-zero values for
20 and
01 in T 0
map to mass terms in both T 00 and T 000 that break the global symmetry down to (N1
1; N2
1; N3; N4), as expected. The two mass terms are:
T 00 ! T~00
T 000 ! T~000
J -term:
J 10 = m 02
H-term: H 20; 01 = m
We can understand these deformations as follows. When T 0 is dualized, a chiral meson
M21 = X20X01 is generated. This meson is not shown in phases T 00 and T 000 of gure 9
because it becomes massive and can be integrated out.
gure 10 shows where M21 would
be in T 00 and T 000. This picture is just intended as a visual reference; to reintroduce M21 in
these theories we should also integrate in the elds they paired with.
These theories contain the following interactions:
T 00
T 000
J -term:
J 10 =
H-term: H 20; 01 =
13This is the 0d analogue of the map, at the level of the superpotential, between a mass term for quarks
in 4d SQCD and a linear term for mesons in its Seiberg dual.
The non-zero values for 20 and 01 in T 0 that we discussed above, translate into a non-zero
value for
M21 . When plugged into (5.9), it produces the mass deformations in (5.8). In
addition, no
avor gets a non-zero value in either T 00 or T 000 so the gauge group is not
higgsed. This is in agreement with the fact that the ranks of the gauge group in T~00 and
T~000 are equal to those in the undeformed sequence.
In summary, we have correctly reproduced the entire deformed sequence. This
matching provides a rather non-trivial check of the quadrality proposal, which takes into account
all types of mesons and interaction terms.
As another check of quadrality, we compare the chiral ring of the four theories. For
simplicity, we identify a few elements of the chiral ring, focusing on operators that can be
expressed as super elds. We restrict to operators built out of a small number of matter
elds, leaving a complete enumeration for future work.
The chiral ring is given by cohomology classes of Q, i.e. it consists of gauge invariant
operators that are Q-closed but not Q-exact. The top components of super elds are, by
de nition, Q-exact. The only exception is X, which has a single component.14 We are thus
interested in the case in which only the lowest component of a gauge invariant super eld,
either elementary or a product, survives. Such a component then becomes an element of
the chiral ring. The chiral ring is determined on-shell which, among other things, requires
that Ga = Ja for all elementary Fermi elds.
Let us rst consider gauge invariant elementary elds. For quick reference, below we
summarize some of the super eld discussion of section section 2 and indicate whether the
lowest components of the operators are in the chiral ring.
Super eld
lowest
Q(lowest)
Chiral ring?
HJEP07(21)53
where we have separated the X and X contributions.
we obtain
We can repeat the exercise for products of two matter elds. Following (2.12)-(2.14),
Super eld
lowest
Chiral ring?
X
X
X1X2
X1X2
X1 2
X1 2
14As mentioned earlier, even though we sometimes consider X and X separately, we should keep in mind
that they combine into a single super eld.
The \ " category in (5.10) and (5.11) indicates operators that become elements of the chiral
ring if the G-components of the corresponding Fermi elds vanish. This is automatically
the case if such Fermis do not participate in any J -term oriented loop in the quiver.
Below we list operators in the chiral ring for the quadrality sequence shown in gure 9.
For convenience, we refer to the operators in terms of super elds, with the understanding
that their lowest components are the objects of interest.
O41
O31
O42
T
X04X10
30X10
X04 02
T 0
M 14
31
40X20
M 14 12
T 00
M 14
31
42
42
X03 01
34M 14
(5.12)
Matching T and T 0. In order to illustrate the main ideas that go into the matching,
it is instructive to discuss the correspondence between T and T 0 in detail. The rest of the
theories follow a similar logic. In particular the analysis of T 000 is identical to the one of
T 0 up to a re ection with respect to a vertical axis. Let us discuss the rows in (5.12) that
deserve special comments.
The operator O31 is given by 30 10 in T and by
31 in T 0. It may appear that T 0
also contains the operator
01 03 between this pair of nodes. However, the theory
has H 03; 31 =
01, which gives rise to the coupling
03
31 01. Since 31 does not
participate in any other loop in the quiver, the equation of motion for
31 forces
01 03 to vanish on-shell.
O42 corresponds to 04 02 in T . Interestingly, there are two operators in T 0 with
the right properties:
40 02 and
M14
12 . Which one should we use? First of all,
the two operators are not related by an equation of motion, so we cannot restrict to
just one of them. Remarkably, for Q-closedness it is necessary to consider a linear
combination of them: 40 02
M14 12 . We have
fQ; 40 02
M14 12 g = G40 20
M14 G 12
'
14 10 20
14 10 20 = 0 :
(5.13)
Both
40 and
12 participate in J -term loops, so their respective G-components do
not automatically vanish, but they compensate when combined. A similar
explanation applies to O31 in T 000.
Additional comments. Further things that work nicely for all the theories are:
Elementary singlet Fermis. Notice that the three theories other than T have singlet
Fermi elds:
12 in T 0, 23 in T 00 and
34 in T 000. They are not in the chiral ring,
because they all participate in J -term loops and hence their G-components do not
vanish. This is good, since there are no matching operators in the other theories,
most notably in T .
node, node 1, of a 2-gauge/6- avor quiver.
X
Fermis. There are two types of composite Fermi elds, which are of the general
forms X
and X . In (5.11) we argued that the X
operators are not in the chiral
ring. Indeed, such operators do not match between di erent theories. A simple
example is X20 03 in T 0, which does not have a counterpart in T .
Chiral rings can be fairly non-trivial in theories with multiple matter elds and gauge
groups, even if restricted to special subsectors. This point is illustrated in section 7.3,
where we compute all the chiral ring operators that only consist of chiral elds for two
quadrality dual theories on D(-1)-branes probing local CP4.
6
Quadrality networks
Theories connected by sequences of quadrality transformations can be nicely organized in
terms of quadrality networks. Analogous constructions exist for Seiberg duality [4, 14, 15]
and triality [2, 9]. These networks become particularly interesting for theories with multiple
gauge groups.
One simple generalization of the simple SQCD-like theory considered in previous
sections consists of merging several copies of the basic 1-gauge/4- avor quiver to make up
n-gauge/(2n + 2)- avor quivers. An n = 2 example with the quadrality action on gauge
node 1 is shown in gure 11.
The quadrality actions on the two gauge nodes do not commute, so all possible
combinations of quadrality lead to a network of quivers. It turns out that there are 44 quivers at
n = 2. The quivers can be divided into four types according to the type of eld connecting
the two gauge nodes. Quivers of the same type may have di erent meson contents.
gure
12 shows the complete quadrality network of n = 2 quivers. In the gure, the labels A, B,
C, D denote the four types of quivers as in gure 11. The blue and pink arrows indicate
quadrality transformations on node 1 and 2, respectively. The length-4 closed oriented
loops consisting of arrows of a given color correspond to four consecutive quadrality
transformations on the same node. As it occurs for Seiberg duality and triality, more general
quivers can lead to in nite quadrality networks.
7
D-brane theories
probing toric CY 5-folds.
7.1
C
5
In this section we study N = 1 matrix models arising on the worldvolume of D(-1)-branes
Let us rst consider D(-1)-branes in at 10d spacetime, i.e. C5. This theory is often called
the Type IIB matrix model and has been proposed as a nonperturbative formulation of type
IIB string theory [16{18]. The models presented in this and the coming section illustrate
how the general structures discussed in section section 2 arise in D-brane constructions.
Bosons and fermions are decomposed in terms of the SU(5)
SO(10) global
symmetry as follows
: 10v !
: 16s !
m(51) +
(15=2) +
m(5 1)
mn(101=2) +
m(5 3=2)
Here m and n are indices in the fundamental representation of SU(5). We choose the
convention where complex conjugation exchanges superscripts and subscripts. Contraction
of conjugate indices (AmBm) implies summation.
The 16 supercharges decompose as
Q : 16c ! 1 5=2 + 10 1=2 + 53=2 :
HJEP07(21)53
We will only use the singlet supercharge.
The on-shell form of the action of the matrix model can be split into three parts:
(7.1)
(7.2)
(7.3)
(7.4)
(7.5)
(7.6)
(7.7)
(7.8)
(7.9)
The D-term is
The J -term is
The H-term is
1
2
1
2
S = SD + SJ + SH :
SD = Tr
where "mnpqr is the SU(5) invariant tensor.
The supersymmetry variation is
m =
m ;
m = 0 ;
m = 0 ;
mn = [ m; n] ;
= [ m
; m] :
It is easy to show that SD, SJ and SH are separately invariant under a supersymmetry
transformation.
We can introduce independent coupling constants for the three terms.
Ratios among the three couplings are xed only if we turn on non-singlet supercharges
from (7.2).
Unlike SD or SJ , the supersymmetry of SH does not rely on a cancellation between a
purely bosonic term and a fermion bilinear term. Explicitly,
8( SH ) =
mnpqr Tr [ m; n] pq r
mn[ p; q] r
=
mnpqr Tr
mn[[ p; q]; r] = 0 :
In the last step, we used the Jacobi identity.
section section 2. We simply note that
It is straightforward to rephrase this theory in the (mostly) o -shell formalism of
J mn( ) =
[ m
; n] ;
Jmn( ) = [ m; n] ;
H
mn;pq( ) = "mnpqr r :
Let us now consider D(-1)-branes probing local CP4, namely the C5=Z5 orbifold with
action (1; 1; 1; 1; 1). This geometry is the simplest toric CY5 that can be studied using
mirror symmetry and, as discussed in section section 3.4, it is the minimal local geometry
realizing quadrality.
The corresponding gauge theory is obtained from the one for C
5 presented in the
previous section by standard orbifold techniques [
19, 20
]. Its quiver diagram is shown in gure
13. We have chiral multiplets Xim;i+1, Fermi multiplets
i+2;i and gaugino multiplets i;i.
mn
The on-shell supersymmetry variation is a natural generalization of (7.7):
m
i;i+1 =
m
i;i+1 ;
m
i;i+1 = 0 ;
i+1;i = 0 ;
m
im+n2;i = ( im+2;i+1 in+1;i
i+2;i+1 i+1;i) ;
n m
i;i = ( im;i+1 m
i+1;i
m
i;i 1 m
i 1;i) :
where the quiver node indices are de ned (mod 5). The absolute-value-square of a complex
matrix is de ned as jAj2 = AAy. Once again, SD, SJ and SH are separately invariant under
a supersymmetry transformation.
The terms in the action are
SD =
SJ =
SH =
5
X Tr
i=1
5
X Tr
i=1
1
2
1
8
i=1
jXim;i+1 m
i;i 1Xim 1;ij2 + i;i( im;i 1 m
m
i 1;i
m
i;i+1 m
i+1;i) ;
m
n
2 j i 1;i i;i+1
n m mn n
i 1;i i;i+1j2 + i+1;i 1( im 1;i i;i+1
n m
i 1;i i;i+1) ;
i+2;i i;i 2 i 2;i+2 ;
mn pq r
(7.10)
(7.11)
For theories on D(-1)-branes, the chiral ring operators consisting exclusively of chiral elds
should reproduce the coordinate ring of the probed CY5. More precisely, following the
discussion in section section 5.4, the operators we are interested in are products of
components of chiral elds. For simplicity, since we are focusing only on chiral elds, we
can drop the conjugation in our discussion. Using local CP4 as an example, in this section
we show that the CY5 can be recovered from two di erent quadrality phases. This fact
simultaneously demonstrates the connection of the chiral ring to the probed CY5 and the
invariance of this particular sector under quadrality.
The original theory. Let us rst consider the theory presented in section section 7.2.
For simplicity, let us focus on a single D(-1)-brane, namely we set N = 1 in the quiver
of gure 13. For generic N , we can diagonalize all elds and the full answer is the N th
symmetric product of the N = 1 result. We can construct the following 55 gauge invariant
operators
M m1m2m3m4m5 =
1m21 2m32 3m43 4m54 5m15 ;
mi = 1; : : : ; 5 :
(7.12)
In terms of the SU(5) global symmetry, they decompose as follows
5
5
5
5
5 = 1 + 4(24) + 5(75) + 6(126) + 1260 + 5(1750) + 4(224)
= [0; 0; 0; 0] + 4[1; 0; 0; 1] + 5[0; 1; 1; 0] + 6[2; 0; 1; 0] + [5; 0; 0; 0] (7.13)
+5[1; 2; 0; 0] + 4[3; 1; 0; 0] ;
where, in order to distinguish representations with equal dimension we also provided the
corresponding Dynkin labels.
Vanishing of the bosonic potential in (7.11) gives rise to the following relations
i 1;i i;i+1
n m
i 1;i i;i+1 = 0 ;
(7.14)
which fully symmetrize the indices in M m1m2m3m4m5 .
Accordingly, only the 1260 =
[5; 0; 0; 0], i.e. the totally symmetric 5-index representation of SU(5), survives from the
gauge invariant operators in (7.13). We call them the generators of the chiral ring and
label them Msm1m2m3m4m5 , with 1
m1
m2
m3
m4
m5
The generators satisfy rst order quadratic relations. First, note that
1260
1260 = 1004 + 2574 + 38500 + 4125 + 3150 + 11760
= [10; 0; 0; 0] + [8; 1; 0; 0] + [6; 2; 0; 0] + [4; 3; 0; 0]
(7.15)
+[2; 4; 0; 0] + [0; 5; 0; 0] :
The J -term relations (7.14) imply that in a product Msm1m2m3m4m5 Msn1n2n3n4n5 any pair
of indices (mi; ni) can be swapped leaving the product invariant. Using this, it is possible
to show that the generators obey 7000 quadratic relations, which transform in the 38500 +
3150 representation of SU(5). They can be written explicitly as follows
38500 = [6; 2; 0; 0] : Rpijqkrlsmtun = Msijkv1v2 Mslmnw1w2 v1w1pqr v2w2stu = 0 ;
3150 = [2; 4; 0; 0] : Tpi1jp2p3q1q2q3r1r2r3s1s2s3 = Msim1m2m3m4 Msjn1n2n3n4
(7.16)
{ 28 {
HJEP07(21)53
The variety formed by the generators subject to their rst order relations is not a complete
intersection. The plethystic logarithm of the Hilbert series g(t; xi; C5=Z5) of the variety is
thus not nite and takes the form
PL[g(t; xi; C5=Z5)] = [5; 0; 0; 0]t
([6; 2; 0; 0] + [2; 4; 0; 0])t2 + : : : ;
(7.17)
where t is a fugacity counting the degree in terms of the generators Msm1m2m3m4m5 . The
previous analysis is in precise agreement with the computation of the Hilbert series directly
from the toric geometry of C5=Z5, as explained in appendix section A. We hence conclude
the chiral ring reproduces the coordinate ring of the probed CY5. The SU(5) charges of
the generators transforming in the 1260 representation form a convex polytope in Z4. This
polytope | the lattice of generators | is the dual of the toric diagram of C5=Z5. Such
relation between the generators and the polytope dual to the toric diagram is described in
detail for the simpler example of C3=Z3 in appendix section B.
The quadrality dual. Let us now consider the theory obtained by acting with quadrality
on node 1. Quadrality generates the following mesons:
M52 = X51 X12 = 5
5 = 15 + 10
54 = X51 14 = 5
35 =
31 X51 = 10
10 = 5 + 45
5 = 10 + 40
where we express them as composites of the elds in the original theory and indicate in
blue the elds that become massive and are integrated out. The masses for elds in the
rst two lines are J -terms, while the one for the last line is an H-term. The resulting quiver
diagram is shown in
gure 14. We do not write the J - and H-terms explicitly; they can
be easily determined from the quiver and global symmetry. This theory is just one in an
in nite web of quadrality dual theories, analogous to the web of 4d Seiberg dual theories
on D3-branes probing local CP2 [4, 14].
Chiral elds can be labeled using fundamental and antifundamental SU(5) indices as
follows
i5j2 : 15
2k3 : 5
l34 : 5
4m1n : 10
( 15)p : 5
(7.19)
We restrict to i
j and m < n to match the 15 (symmetric) and 10 (antisymmetric)
representations, respectively.
Once again, let us focus on the case of a single D(-1)-brane, namely N = 1. Proceeding
as before, we construct gauge invariant operators using chiral elds, which transform under
SU(5) as follows
ij k l
52 23 34 4m1n( 15)p = 15
5
5
10
5
= 1 + 6(24) + 7(75) + 10(126) + 2(126) + 1260 + 7(1750)+1750
+ 3(200) + 6(224) + 7000 + 4(1024) + 3(10500)
+ 2(1701) + 1750 :
(7.20)
where the U(4) color indices at node 1 are properly contracted. For brevity, we do not
provide the Dynkin labels of the representations.
HJEP07(21)53
The next step is to determine which of the representations in (7.20) survive once J
terms are taken into account. This can be conveniently done in steps, as we explain below.
The key idea is the following. In this theory, all J -terms correspond to cubic loops in
the quiver, involving one Fermi and two chiral elds. This means that the J -terms are
quadratic in chiral elds. For each Fermi eld
R transforming in some representation R
of SU(5), the condition J R = 0 sets the R representation in the corresponding product of
scalar elds to zero. The J -terms for the
ve types of Fermi elds in the theory end up
eliminating several of the operators in (7.20).
It is instructive to discuss in detail how to obtain the generators. First consider the
product
34 j4k1 = 5
i
10 = 10 + 40 :
These two chiral elds, appropriately contracted, form J 13 . Since 13 transforms in the 10,
we know that the 10 is eliminated from the product so we are left with
jk
i
34 41jJ=0 = 40.
Taking the product with ( 15)l, we then have
34 j4k1 ( 15)ljJ=0
i
40
5 = 10 + 15 + 175 ;
where the inclusion sign indicates that we still have not considered all the relevant J -terms.
J 54 = 0 removes the 45 in the product j4k1 ( 15)l which, in turn, leaves us with
This can be conveniently written as
34 j4k1 ( 15)ljJ=0 = 15 = [2; 0; 0; 0] :
i
M(iAj) = i
34 j4m1 ( 15)m = i34Zj ;
with i and j symmetrized and Zj =
j4m1( 15)m.
(7.21)
(7.22)
(7.23)
(7.24)
Next, let us consider the product
J 35 = 0 eliminates the 40, resulting in
which is symmetric in the three indices.
i5j2 2k3 = 15
5 = 35 + 40 :
M(iBjk) =
i5j2 2k3jJ=0 = 35 = [3; 0; 0] ;
Combining (7.24) and (7.26), we conclude the surviving gauge invariants are
Mfsm1m2m3m4m5 = M(mA1)m2 M(mB3)m4m5 =
m1m2 2m33 3m44 Zm5 ;
52
which transform in the 1260 = [5; 0; 0; 0] representation due to the symmetrization of
all indices that follows from the remaining J -terms. This is in full agreement with the
generators of the original theory. Furthermore, it is possible to verify explicitly that the
generators satisfy the same 7000 quadratic relations of the original theory (7.16).
In summary, with the matching of generators and their rst order relations, we conclude
that the chiral rings of the two dual theories reproduce the coordinate ring of C5=Z5.
The matrix models on D(-1)-branes probing toric CY 5-folds have, in general, an in nite
number of dual phases connected by quadrality. Some of these phases are described by the
brane hyperbrick models discussed in section section 3.1, equivalently by periodic quivers
on T 4. We refer to them as toric phases. This is a straightforward generalization of the
concept of toric phases in 4d and 2d.
A toric node is a node in a toric phase whose dualization also results in a toric phase.15
It is natural to ask what is the structure of a minimal, i.e. with a minimum number of
elds, toric node for N = 1 matrix models.16
In every dimension, minimal toric nodes involve two bifundamental elds of each
possible type. In the corresponding periodic quivers, the two
elds in each of these pairs
emanate from the toric node in opposite directions. In 4d, such a node involves two
incoming and two outgoing chirals. This con guration maps to a square face in the brane
tiling [6]. Similarly, a minimal toric node in 2d has two incoming chirals, two outgoing
chirals and two Fermis. This translates into a cube in the brane brick model [9]. Finally, a
minimal toric node in 0d contains two incoming chirals, two outgoing chirals, two incoming
Fermis and two outgoing Fermis. It maps to a hypercube in the brane hyperbrick model.
Can we nd a relatively simple toric CY 5-fold that: a) contains at least a minimal toric
node and b) has more than one toric phase? For 4d gauge theories, a standard example of
a CY3 with these properties is local CP1
CP1, also known as F0 [6]. The analogous CY4
for 2d theories is local CP1
CP1
CP1, i.e. Q1;1;1=Z2 [9]. It is then natural to conjecture
15Notice that it is possible for a toric phase not to have any toric node.
16We emphasize minimality because we expect multiple possibilities for toric nodes. While toric nodes
have a unique structure in 4d theories, this is no longer the case in 2d. It is reasonable to anticipate a
similar behavior in 0d.
(7.25)
(7.26)
(7.27)
HJEP07(21)53
CP1 does the job for matrix models. Below we present some
evidence supporting this claim.
Consider the following choice of coe cients in the Newton polynomial of (CP1)4:
1
x
1
y
P (x; y; z; w) =
x +
+ i y +
+ 0:9(1 + i) z +
+ 0:9( 1 + i) w +
w
(7.28)
Following the mirror symmetry analysis reviewed in section section 3.2, gure 15.a shows
the resulting 16 critical values on the W -plane and the associated vanishing paths. Focusing
on the y = +1 subset, shown in
gure 15.b, we recognize the con guration of vanishing
paths for phase B of the Q1;1;1=Z2 brane brick model [9].17 This is a phase that indeed
contains minimal toric nodes, i.e. cubic brane bricks. By symmetry, the y =
1 and x =
subsets give rise to almost identical con gurations. The z = +1 vanishing paths are shown
in gure 15.c. They also correspond to phase B of Q1;1;1=Z2. Due to symmetry, the same
is true for z =
1 and w =
1. These observations, combined, suggest that it is very
plausible that the con guration in gure 15 corresponds to a simple phase with minimal
toric nodes. A detailed study of the matrix model(s) associated to this geometry would
be extremely interesting. It is however beyond the scope of this paper and we leave it for
future work.
8
Towards quadrality from compacti cation
Dualities in lower dimensional quantum eld theories can often be derived from the ones for
higher dimensional theories via compacti cation. For IR dualities, the interplay between
the low energy limit and the zero compacti cation size limit can be subtle and sometimes
the two limits do not commute.
This approach has been successfully exploited for deriving dualities in 3d and 2d SUSY
theories starting from 4d Seiberg duality. In [21, 22], dualities of 3d N = 2 theories were
17Here we are making a comparison to a CY 4-fold. We say that the con guration of vanishing paths of
the CY5 is equal to the one of the CY4, when they coincide on the W -plane.
theories were studied in [23, 24]. The resulting 2d theories depend on how the theory is
coupled to background
elds, which maps to a choice of R-symmetry. It is possible to
show that the reduction of Seiberg duality gives rise to triality by simultaneously picking
an appropriate choice of R-charges and turning on an FI term.
The index of the 4d theories plays a central role in the aforementioned studies. The
superconformal S3
S1 index becomes the S3 partition function of the 3d theory [25{27] and
the S2
T2 partition function becomes the elliptic genus in 2d. Agreement of the indices in
the compacti ed theories provides substantial support for the lower dimensional dualities.
These examples show that the relation between duality and compacti cation is rather
nontrivial. Issues that need to be carefully addressed include the ratio between the UV
cuto
and the compacti cation radius, the di erence between compact and non-compact
scalars in 3d, appearance of extra terms in the superpotential due to nonperturbative e ects
and the need for summing over ux sectors that might correspond to defect operators.
It is natural to ask whether 0d quadrality can be derived from Seiberg duality or triality
along similar lines. The SUSY partition function of the higher dimensional theory on the
compacti cation manifold should be compared with the matrix integral of the 0d theory.
There are various alternatives for compactifying 4d N = 1 down to 0d N = 1, such as
S
2
T2 and S
2
S2. If, instead, we try to connect 2d (0; 2) to 0d N = 1, we run into
a puzzle. There is no known SUSY preserving S2 compacti cation of 2d (0; 2). On the
other hand, T2 leads to too much SUSY in 0d. It might be possible to compactify on a
more general Riemann surface, including punctures if necessary. Typically, higher genus
surfaces give rise to relatively complicated theories in lower dimensions that re ect the
geometric data.
Before addressing quadrality, it may be instructive to consider theories with extended
supersymmetry, relating 2d (2; 2) to 0d N = 2. We can investigate whether the S2 partition
function of the 2d (2; 2) theory has a 0d N = 2 interpretation. Along similar lines, [23]
studied compacti cations of 4d N = 2 to 2d (2; 2).
It would be extremely interesting to provide a eld theoretic derivation of quadrality
by obtaining it from a higher dimensional duality via compacti cation. This is however
beyond the scope of this paper. We plan to revisit this question in the future.
9
We introduced quadrality, a new order 4 duality that applies to N = 1 supersymmetric
gauged matrix models. Our proposal follows naturally from mirror symmetry, which
provides a uni ed framework that puts 4d Seiberg duality, 2d GGP triality and quadrality on
an equal footing. We expect that quadrality is not restricted to theories with a D-brane
realization and holds for general N
= 1 matrix models.
We performed various checks
of the proposal, including the matching of: global symmetries, abelian
avor anomalies,
deformations and the chiral ring. The chiral ring was computed in detail for a pair of
quadrality dual theories on D(-1)-branes probing local CP4, for which we showed not only
that it is the same in the two theories but that it reproduces the coordinate ring of the
CY5 singularity.
We also initiated the study of various aspects of the matrix models that arise on the
worldvolume of D(-1)-branes probing toric CY 5-folds.
There are various natural directions for future investigation. First, dualities are
powerful tools for elucidating the dynamics of quantum
eld theories in di erent dimensions.
The application of Seiberg duality to map the phase space of 4d N = 1 SQCD is a prime
example. It would be worth studying what quadrality can teach us about the dynamics
of matrix models. It would also be interesting to establish whether it is possible to derive
quadrality from a higher dimensional duality through compacti cation. Below we discuss
two additional open questions.
Evaluating the integral.
Conceptually, the most explicit way to verify quadrality would
be to compute the matrix integrals for the four dual theories and show that they are equal.
This would be analogous to the computation of the elliptic genus to verify 2d triality [2, 28].
For triality the three dual theories ow to the same SCFT in the IR [29]. The elliptic
genus is invariant under the RG
ow, making it possible to probe the SCFT from the
UV gauge theory. In 0d, the usual notion of RG
ow does not exist, so determining the
conditions under which the matrix integrals should agree becomes more subtle.
The 2d elliptic genus can be re ned by turning on
avor fugacities. In the case of
non-compact target spaces, the fugacities regulate divergent contributions. In 0d, since
there are no background gauge elds, it is not clear how to turn on fugacities.
Due to these issues, evaluating the integrals and comparing between di erent
quadrality phases is not straightforward. We hope to revisit these questions in future work.
M-theory lift.
Mirror symmetry relates D(-1)-branes probing a CY 5-fold to Euclidean
D4-branes wrapping 5-cycles in the mirror CY5. It would be interesting to determine the
M-theory lift of this con guration. The ED4-branes become EM5-branes wrapping the
original 5-cycles times the M-theory circle.
Wick-rotating and decompactifying the
Mtheory circle, we arrive at an M-theory con guration with physical M5-branes wrapping
5-cycles. The result is a supersymmetric quantum mechanics. The original 0d matrix model
can be reinterpreted as the dimensional reduction of the M-theory quantum mechanics.
The situation is somewhat analogous, but not equivalent, to the relation between the
Dinstanton matrix model and D0 quantum mechanics. In particular, the D(-1) matrix model
contains information on the Witten index of D0 quantum mechanics [30]. In the D0/D(-1)
connection, an explicit Lagrangian description is available on both sides. In contrast, in
the relation between the IIA matrix model and the M-theory quantum mechanics we are
considering, the Lagrangian is only known for the former but not for the latter. The precise
nature of the quantum mechanics of wrapped M5-branes is an interesting open problem.
More generally, it would also be interesting to determine whether some new duality
for supersymmetric quantum mechanics can be inferred from quadrality.
Acknowledgments
We would like to thank P. Putrov, M. Romo, N. Seiberg, E. Witten and S.-T. Yau for useful
and enjoyable discussions. We are also grateful to D. Ghim for collaboration on related
topics. We gratefully acknowledge support from the Simons Center for Geometry and
Physics, Stony Brook University, where some of the research for this paper was performed
during the 2016 Simons Summer Workshop. The work of S. F. is supported by the U.S.
National Science Foundation grant PHY-1518967 and by a PSC-CUNY award. The work
of S. L. was supported by Samsung Science and Technology Foundation under Project
Number SSTBA1402-08. The work of S. L. was also performed in part at the Institute for
Advanced Study supported by the IBM Einstein Fellowship of the Institute for Advanced
Study, and at the Aspen Center for Physics supported by National Science Foundation
grant PHY-1066293. The work of R.-K. S. is supported by the ERC STG grant 639220
\Curved SUSY". The work of C.V. is supported in part by NSF grant PHY-1067976.
A
The Hilbert series for C3=Z3, C4=Z4 and C5=Z5
The Hilbert series [31, 32] is a powerful tool for enumerating operators in a chiral ring and
for studying its geometric structure. It is formally de ned in algebraic geometry as the
generating function
g(t; R) =
1
n=0
X dim(Rn)tn ;
(A.1)
where R is an algebraic quotient ring and Rn is a component of R of degree n 2 N. The
fugacity t counts the degree of the component Rn. For multi-graded rings with
compoP1
n=0 dim(R~n)t1n1 : : : tknk , where t1; : : : ; tk are the fugacities of the grading.
nents R~n and grading ~n = (n1; : : : ; nk), the Hilbert series takes the form g(t1; : : : ; tk; R) =
Hilbert series from toric geometry.
When the chiral ring is a toric variety, its Hilbert
series can be derived directly from the toric diagram [31, 33]. For a toric CY n-fold, the
toric diagram is an n
1 dimensional convex polytope that admits at least one triangulation
in terms of (n
1)-simplices. From a triangulation, one can construct a dual web diagram.
For Calabi-Yau 3-folds, these are the so-called (p; q)-webs [34{36]. gure 16 shows the toric
diagrams and dual web diagrams for C3=Z3 and C4=Z4.
The Hilbert series of the toric variety X then can be de ned from a triangulation of
the toric diagram as follows
g(t1; : : : ; tn; X) =
~t ~v(i;j)) ;
(A.2)
where ~t ~v(i;j) = Qan tvaa(i;j). The index i = 1; : : : ; r runs over the simplices making up the
triangulation while j = 1; : : : ; n runs over the faces of each simplex. The vector ~v(i; j) is
the n-dimensional outer normal to the face of the fan associated to face j of simplex i.18
18Notice that for a CY n-fold, these normal vectors are n-dimensional, while we said that the toric and
web diagrams are (n
1)-dimensional. More precisely, the toric diagram of a CY n-fold lives on a hyperplane
in n-dimensions at distance 1 from the origin. This fact allows a trivial projection to (n
1)-dimension. In
the examples below, we reincorporate the nth coordinate in order to determine the normal vectors.
r
X
n
Y(1
i=1 j=1
Hilbert series for C3=Z3, C4=Z4, and C5=Z5.
toric diagrams for these three geometries are
The coordinates for the points in the
C3=Z3 : (1; 0; 1); (0; 1; 1); ( 1; 1; 1); (0; 0; 1)
C4=Z4 : (1; 0; 0; 1); (0; 1; 0; 1); (0; 0; 1; 1); ( 1; 1; 1; 1); (0; 0; 0; 1)
C4=Z4 : (1; 0; 0; 0; 1); (0; 1; 0; 0; 1); (0; 0; 1; 0; 1); (0; 0; 0; 1; 1); ( 1; 1; 1; 1; 1);
(0; 0; 0; 0; 1)
Notice that we have included the nth coordinate, which will become important in the
discussion that follows. Labeling the points in the toric diagrams from 1; : : : ; n + 1 in the
order they are listed in (A.3), then the unique triangulations of these diagrams can be
summarized in terms of the points in the toric diagram as follows
C3=Z3 : ff1; 2; 4g; f1; 3; 4g f2; 3; 4gg
C4=Z4 : ff1; 2; 3; 5g; f1; 2; 4; 5g; f1; 3; 4; 5g; f2; 3; 4; 5gg
C4=Z4 : ff1; 2; 3; 4; 6g; f1; 2; 3; 5; 6g; f1; 2; 4; 5; 6g; f1; 3; 4; 5; 6g; f2; 3; 4; 5; 6gg
(A.4)
The corresponding Hilbert series are
g(t; C3=Z3) =
g(t; C4=Z4) =
g(t; C5=Z5) =
1 + 7t + t2
t)3 ;
1 + 31t + 31t2 + t3
(1
1 + 121t + 381t2 + 121t3 + t4
(1
t)5
:
(A.3)
(A.5)
The plethystic logarithms [31, 32] of the Hilbert series are
PL[g(t; C3=Z3)] = 10t
PL[g(t; C4=Z4)] = 35t
PL[g(t; C5=Z5)] = 126t
27t2 + 105t3
465t2 + 8960t3
540t4 + : : : ;
201376t4 + : : : ;
7000t2 + 544500t3
48095250t4 + : : : :
Note that none of these plethystic logarithms has a
nite expansion, indicating that
the three toric varieties are not complete intersections.
Furthermore, the coe cients
in (A.6) are sums of dimensions of irreducible representations of SU(3), SU(4) and SU(5),
respectively.
This computation con rms that the chiral ring discussed in section 7.3 indeed
corresponds to the coordinate ring of C5=Z5.
The
rst term in the plethystic logarithm
corresponds to the 126 generators of the toric variety transforming in the [5; 0; 0; 0]
representation of SU(5). The second term in the expansion indicates that these 126 generators
satisfy 7000 quadratic relations, which transform in the [6; 2; 0; 0]+[2; 4; 0; 0] representations
of SU(5).
B
Review: C3=Z3
While the D-brane constructions of this paper focus on D(-1)-branes probing toric CY
5-folds, it is enlightening to review the case of D3-branes probing C3=Z3 in further detail.
This example is useful because it exhibits many of the concepts that apply to the CY5
case in a considerably simpler context. For N D3-branes, the worldvolume theory is a
4d N = 1 supersymmetric gauge theory with the quiver diagram shown in gure 17 and
superpotential
The global symmetry of the theory is SU(3)
U(1)R.
The Hilbert series for a single D3-brane has been computed in [37]. It takes the form
W = ijk i12 j23 3k1 :
g(t; x1; x2; C3=Z3) =
1
X[3n; 0]tn ;
n=0
(B.1)
(B.2)
generators form the convex polygon dual to the toric diagram.
where [3n; 0] is the character of the SU(3) representation, with the entries being Dynkin
labels of the representation. t is the fugacity for the U(1)R symmetry. The plethystic
PL[g(t; x1; x2; C3=Z3)] = [3; 0]t
[2; 2]t2 + ([1; 1] + [1; 4] + [2; 2] + [4; 1])t3 + : : : : (B.3)
This matches the result obtained from toric geometry in (A.6). There are 10 generators
which transform in the [3; 0] representation of SU(3) satisfying 27 relations transforming in
the [2; 2] representation of SU(3). The vacuum moduli space is not a complete intersection
and for the abelian theory it is precisely C3=Z3.
The 10 generators can be written in terms of the chiral bifundamental elds
as follows,
where one sets i
j
k such that M ijk transform in the [3; 0] representation of SU(3).
This follows from the fact that the F -terms associated to the superpotential in (B.1),
i
ab
(B.4)
(B.5)
(B.6)
(B.7)
M ijk = i12 j23 3k1 ;
= ijk jbc cka = 0 :
following quotient variety,
M ijk=hRmij n = 0i :
fully symmetrize the indices of M ijk.
The plethystic logarithm (B.3) indicates that there are quadratic relations between the
generators M ijk transforming in the [2; 2] representation of SU(3). We can identify these
Rmij n =
M ik1k2 M jl1l2 k1l1m k2l2n = 0 :
Knowing the generators and their rst order relations, C3=Z3 can be expressed as the
The SU(3) charges of the generators, which correspond to the exponents of x1nx x2ny in the
character of [3; 0],
x
x
1
x
2
1 +
1
1
x
2
3 + 1 ;
can be plotted on a Z2 lattice. It is possible to transform the lattice points associated to
2nx; nx
ny), which is an SL(2; Z)
transformation and a rescaling. The resulting points form a lattice triangle, which is the
dual re exive polygon of the toric diagram of C3=Z3 as illustrated in gure 18 [37].
Open Access.
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.
[INSPIRE].
(2017) 106 [arXiv:1609.01723] [INSPIRE].
[1] N. Seiberg, Electric-magnetic duality in supersymmetric nonAbelian gauge theories, Nucl.
Phys. B 435 (1995) 129 [hep-th/9411149] [INSPIRE].
[2] A. Gadde, S. Gukov and P. Putrov, (0; 2) trialities, JHEP 03 (2014) 076 [arXiv:1310.0818]
[3] S. Franco, S. Lee, R.-K. Seong and C. Vafa, Brane brick models in the mirror, JHEP 02
[4] F. Cachazo, B. Fiol, K.A. Intriligator, S. Katz and C. Vafa, A geometric uni cation of
dualities, Nucl. Phys. B 628 (2002) 3 [hep-th/0110028] [INSPIRE].
[5] S. Franco, D. Ghim, S. Lee, R.-K. Seong and D. Yokoyama, 2d (0; 2) quiver gauge theories
and D-branes, JHEP 09 (2015) 072 [arXiv:1506.03818] [INSPIRE].
[6] S. Franco, A. Hanany, K.D. Kennaway, D. Vegh and B. Wecht, Brane dimers and quiver
gauge theories, JHEP 01 (2006) 096 [hep-th/0504110] [INSPIRE].
[7] S. Franco, A. Hanany, D. Martelli, J. Sparks, D. Vegh and B. Wecht, Gauge theories from
toric geometry and brane tilings, JHEP 01 (2006) 128 [hep-th/0505211] [INSPIRE].
[8] S. Franco, S. Lee and R.-K. Seong, Brane brick models, toric Calabi-yau 4-folds and 2d (0; 2)
quivers, JHEP 02 (2016) 047 [arXiv:1510.01744] [INSPIRE].
[9] S. Franco, S. Lee and R.-K. Seong, Brane brick models and 2d (0; 2) triality, JHEP 05
(2016) 020 [arXiv:1602.01834] [INSPIRE].
[10] B. Feng, Y.-H. He, K.D. Kennaway and C. Vafa, Dimer models from mirror symmetry and
quivering amoebae, Adv. Theor. Math. Phys. 12 (2008) 489 [hep-th/0511287] [INSPIRE].
[11] M. Futaki and K. Ueda, Tropical Coamoeba and torus-equivariant homological mirror
symmetry for the projective space, Commun. Math. Phys. 332 (2014) 53 [INSPIRE].
[12] K. Hori and C. Vafa, Mirror symmetry, hep-th/0002222 [INSPIRE].
[13] K. Hori, A. Iqbal and C. Vafa, D-branes and mirror symmetry, hep-th/0005247 [INSPIRE].
[14] S. Franco, A. Hanany, Y.-H. He and P. Kazakopoulos, Duality walls, duality trees and
fractional branes, hep-th/0306092 [INSPIRE].
D 70 (2004) 046006 [hep-th/0402120] [INSPIRE].
[16] N. Ishibashi, H. Kawai, Y. Kitazawa and A. Tsuchiya, A large-N reduced model as
[17] M. Fukuma, H. Kawai, Y. Kitazawa and A. Tsuchiya, String eld theory from IIB matrix
[18] H. Aoki, S. Iso, H. Kawai, Y. Kitazawa and T. Tada, Space-time structures from IIB matrix
HJEP07(21)53
B 506 (1997) 84 [hep-th/9704151] [INSPIRE].
07 (2013) 149 [arXiv:1305.3924] [INSPIRE].
[20] M.R. Douglas, B.R. Greene and D.R. Morrison, Orbifold resolution by D-branes, Nucl. Phys.
[21] O. Aharony, S.S. Razamat, N. Seiberg and B. Willett, 3d dualities from 4d dualities, JHEP
[22] O. Aharony, S.S. Razamat, N. Seiberg and B. Willett, 3d dualities from 4d dualities for
orthogonal groups, JHEP 08 (2013) 099 [arXiv:1307.0511] [INSPIRE].
[23] M. Honda and Y. Yoshida, Supersymmetric index on T 2
S2 and elliptic genus,
arXiv:1504.04355 [INSPIRE].
11 (2015) 163 [arXiv:1506.08795] [INSPIRE].
JHEP 09 (2011) 133 [arXiv:1104.4482] [INSPIRE].
003 [arXiv:1104.2592] [INSPIRE].
[24] A. Gadde, S.S. Razamat and B. Willett, On the reduction of 4d N = 1 theories on S2, JHEP
[25] Y. Imamura, Relation between the 4d superconformal index and the S3 partition function,
[26] A. Gadde and W. Yan, Reducing the 4d index to the S3 partition function, JHEP 12 (2012)
[27] F.A.H. Dolan, V.P. Spiridonov and G.S. Vartanov, From 4d superconformal indices to 3d
partition functions, Phys. Lett. B 704 (2011) 234 [arXiv:1104.1787] [INSPIRE].
[28] F. Benini and B. Le Floch, Supersymmetric localization in two dimensions,
arXiv:1608.02955 [INSPIRE].
arXiv:1404.5314 [INSPIRE].
046007 [hep-th/9804123] [INSPIRE].
[29] A. Gadde, S. Gukov and P. Putrov, Exact solutions of 2d supersymmetric gauge theories,
[30] M.B. Green and M. Gutperle, D instanton partition functions, Phys. Rev. D 58 (1998)
[31] S. Benvenuti, B. Feng, A. Hanany and Y.-H. He, Counting BPS operators in gauge theories:
quivers, syzygies and plethystics, JHEP 11 (2007) 050 [hep-th/0608050] [INSPIRE].
[32] B. Feng, A. Hanany and Y.-H. He, Counting gauge invariants: the plethystic program, JHEP
03 (2007) 090 [hep-th/0701063] [INSPIRE].
[33] D. Martelli, J. Sparks and S.-T. Yau, Sasaki-Einstein manifolds and volume minimisation,
Commun. Math. Phys. 280 (2008) 611 [hep-th/0603021] [INSPIRE].
[34] O. Aharony and A. Hanany, Branes, superpotentials and superconformal xed points, Nucl.
Phys. B 504 (1997) 239 [hep-th/9704170] [INSPIRE].
and grid diagrams, JHEP 01 (1998) 002 [hep-th/9710116] [INSPIRE].
[hep-th/9711013] [INSPIRE].
695 [arXiv:1201.2614] [INSPIRE].
superstring , Nucl. Phys. B 498 ( 1997 ) 467 [ hep -th/9612115] [INSPIRE].
model , Prog. Theor. Phys . 99 ( 1998 ) 713 [ hep -th/9802085] [INSPIRE].
[19] M.R. Douglas and G.W. Moore , D-branes, quivers and ALE instantons , hep-th/9603167 [35] O. Aharony , A. Hanany and B. Kol , Webs of (p; q) ve-branes, ve-dimensional eld theories [36] N.C. Leung and C. Vafa , Branes and toric geometry, Adv. Theor. Math. Phys. 2 ( 1998 ) 91 [37] A. Hanany and R.-K. Seong , Brane tilings and re exive polygons , Fortsch. Phys . 60 ( 2012 )