Bbranes and supersymmetric quivers in 2d
HJE
Bbranes and supersymmetric quivers in 2d
Cyril Closset 0 1 2 5
Jirui Guo 0 1 2 3
Eric Sharpe 0 1 2 4
Topological Strings
0 850 West Campus Drive, Virginia Tech , Blacksburg, VA, 24061 U.S.A
1 220 Handan Road, Shanghai , 200433 China
2 Geneva 23 , CH1211 Switzerland
3 Department of Physics, Fudan University
4 Department of Physics MC 0435
5 Theory Department , CERN
We study 2d N = (0; 2) supersymmetric quiver gauge theories that describe the lowenergy dynamics of D1branes at CalabiYau fourfold (CY4) singularities. On general grounds, the holomorphic sector of these theories  matter content and (classical) superpotential interactions  should be fully captured by the topological Bmodel on the CY4. By studying a number of examples, we con rm this expectation and esh out the dictionary between Bbrane category and supersymmetric quiver: the matter content of the supersymmetric quiver is encoded in morphisms between Bbranes (that is, Ext groups of coherent sheaves), while the superpotential interactions are encoded in the A1 algebra satis ed by the morphisms. This provides us with a derivation of the supersymmetric quiver directly from the CY4 geometry. We also suggest a relation between triality of N = (0; 2) gauge theories and certain mutations of exceptional collections of sheaves. 0d N = 1 supersymmetric quivers, corresponding to Dinstantons probing CY5 singularities, can be discussed similarly.
Dbranes; Field Theories in Lower Dimensions; Supersymmetry and Duality

2.2
D1brane on C
4
2.3
Triality acting on N = (0; 2) supersymmetric quivers
Triality from mutation  a conjecture
4
Dinstanton quivers and gauged matrix models
D( 1)brane on C
5
N = 1 gauged matrix model from Bbranes at a CY5 singularity
4.2.1
4.2.2
4.3.1
4.3.2
1 Introduction 2
D1brane quivers and 2d N = (0; 2) theories
2.1
N = (0; 2) quiver gauge theory from Bbranes at a CY4 singularity
From Bbranes to quiver
{ i {
58
59
60
eld theories can be engineered on systems of branes in
string theory. The string theory embedding often provides us with an elegant geometric
understanding of eld theory phenomena. In particular, rich classes of eld theories, the
supersymmetric quiver gauge theories, can be engineered by considering parallel Dpbranes
at the tip of a conical local CalabiYau (CY) nfold Xn, with p = 9
2n, in type IIB
string theory. One obtains the following types of supersymmetric gauge theories in the
openstring sector:
6d N = (0; 1) quiver theories on D5branes at the tip of a CY2 cone.
4d N = 1 quiver theories on D3branes at the tip of a CY3 cone.
2d N = (0; 2) quiver theories on D1branes at the tip of a CY4 cone.
0d N = 1 quiver theories on Dinstantons at the tip of a CY5 cone.
All these quiver gauge theories consist of unitary gauge groups QI U(NI ) with matter elds
in adjoint and bifundamental representations.1 The 6d case corresponds to D5branes at
the tip of an ADE singularity C2= , and the quiver gauge theories are the corresponding
ADE quivers [1]. The 4d case has been thoroughly studied from various points of views
 see e.g. [2{15] for a very partial list. It is a special and important case because the
D3branes admit a smooth nearhorizon limit of the form AdS5
X5 [3, 4, 16] and the 4d
quiver gauge theories ow to nontrivial 4d N = 1 superconformal xed points.
The 2d and 0d cases had attracted less attention until more recently  see [17, 18]
for some early work. A recent breakthrough was the introduction of \brane brick
models" [19, 20], which gave an algorithm to determine the N = (0; 2) quiver gauge theory
1That is, elds XIJ in the fundamental of U(NI ) and in the antifundamental of U(NJ ).
{ 1 {
corresponding to D1branes probing a toric CY4 singularity, similarly to brane tiling
methods for D3branes at toric CY3 singularities [11{13]. The brane brick models were derived
using mirror symmetry in [21]. There are also hints that a similar structure exists for
D( 1)branes at toric CY5 singularities.2 Note that this line of work (and the present
paper) is only concerned with the classical structure of the N = (0; 2) gauge theory. In
a parallel development, there has been some important progress in our understanding of
twodimensional N = (0; 2) gauge theories as full edged quantum
eld theories [23{25].
Incidentally, it was discovered that the simplest SQCDlike N = (0; 2) theories enjoy a
beautiful triality [23]  an infrared \duality" of order three similar to Seiberg duality.
Triality also seems to be a generic property of D1branes quivers [26]. There has also been
some interesting recent work on engineering N = (0; 2) models from Ftheory [27{30]. See
also [31{42] for related works on quantum aspects of N = (0; 2) theories.
In this paper, we study 2d and 0d quivers from the point of view of Bbranes on the CY
nfold Xn. A Bbrane is simply a (halfBPS) Dbrane in the topological Bmodel on Xn.
The Bmodel is a gs = 0 limit of type II string theory which (somewhat trivially) captures
all 0 corrections. It can thus be used to accurately describe the local physics of branes
at a CalabiYau singularity. Since the Bmodel is independent of Kahler deformations, we
can use any convenient limit, such as, for instance, the large volume limit of a resolved
singularity, to study the quantities of interest. In this way, we loose a lot of important
information  for instance, we do not keep track of the central charges of the branes,
which determines their stability properties; yet, the Bmodel is su cient in order to extract
all the information about the holomorphic sector of the lowenergy open strings. That is,
we can read o the matter spectrum and the superpotential interactions of the lowenergy
quiver gauge theories on Dpbranes from the Bbranes alone.3
This approach was successfully carried out for D3branes at CY3 singularities [6{
10, 15].
What we present here is a straightforward extension of some of those earlier
works. It provides a string theory derivation of some brane brick models results, without
the need to rely on mirror symmetry. Our techniques are also more general, since they are
valid beyond the realm of toric geometry.4
Mathematically, a Bbrane E on Xn is an object in the (bounded) derived category of
coherent sheaves of Xn:
(1.1)
(1.2)
We can think of Bbranes E as coherent sheaves; more generally they are chain complexes of
coherent sheaves (up to certain equivalences). Given two Bbranes E , F , we may compute
their Ext groups:
2Very recently, these 2d and 0d quivers were also related to cluster algebras [22].
3An important caveat is that we need to be given a particular set of Bbranes, the \fractional branes",
4This is as a matter of principle. In this paper, all our examples will be toric geometries, somewhat by
happenstance, and also so that we can compare our results to the brane brick model literature.
E 2 Db(Xn) :
ExtiXn (E ; F ) ;
{ 2 {
which are the morphisms in the derived category. Physically, they encode the lowenergy
modes of the open strings stretched between the Dbranes E and F [43{46].
to [47, 48] for comprehensive reviews of the derived category approach to Bbranes.
We refer
Dbranes quivers from Bbranes.
Consider a D(9 2n)brane transverse to the
CalabiYau singularity Xn. Away from the singularity, the brane is locally in
at space.
From the point of view of Xn, it is a pointlike brane, which is described by a skyscraper
sheaf Op at a point p 2 Xn. When at the singularity, it is expected that Op \fractionates"
into marginally stable constituents. The resulting \fractional branes" fEI g realize a gauge
group:
Y U(NI )
I
(1.3)
HJEP02(18)5
on their worldvolume in the transverse directions. There are also massless open strings
connecting the fractional branes among themselves, which realize bifundamental (or
adjoint) matter elds XIJ . In this way, the lowenergy open string sector at the singularity
is described by a supersymmetric quiver gauge theory : to each fractional brane EI , we
associate a node in the quiver, denoted by eI . The matter elds corresponds to various
quiver arrows connecting the nodes: eI
the matter elds, which we will discuss below.
In all cases, the fractional branes are such that:
! eJ . There are also interaction terms among
These Ext0 = Hom groups are identi ed with vector multiplets in d = 10
2n dimensions;
a single vector multiplet is assigned to each node eI , realizing the gauge group U(NI ). The
other Exti groups (with i = 1;
gauge groups.
Ext0Xn (EI ; EJ ) = IJ :
(1.4)
; n
1) correspond to matter elds charged under the
We should probably emphasize that, in this paper, we will be mostly interested in
the supersymmetric quiver as an abstract algebraic object, consisting of nodes, arrows and
relations. The assignment of particular gauge groups U(NI ) is part of the data of a quiver
representation, and the gauge group ranks can vary depending on the physical setup (that
is, which Dbranes are we using to probe the singularity). In other words, our concept of
supersymmetric quiver can encode many di erent supersymmetric theories with the same
structure but distinct gauge groups.5
A crucial property of Ext groups on a CalabiYau variety Xn is the Serre duality
relation:
ExtiXn (EI ; EJ ) = ExtnXni(EJ ; EI ) ;
i = 0;
; n :
(1.5)
This corresponds to the CPT symmetry of the ddimensional quiver quantum eld theory.
Generalizing some relatively wellknown results for D3branes, it is natural to propose the
following identi cation of Ext groups with supersymmetry multiplets in various dimensions:
5Not all unitary gauge groups are allowed, however. Gauge anomalies provide strong constraints on the
allowed quiver ranks.
{ 3 {
where XIJ are 6d N
= (0; 1) hypermultiplets in the bifundamental representation of
U(nI )
U(nJ ). Note that the quiver link eI  eJ is unoriented since the
hypermultiplet is nonchiral  this corresponds to the Serre duality Ext1(EI ; EJ ) = Ext1(EJ ; EI ) on
X2. In this case, X2 must be an ADE singularity while the supersymmetric quivers are
extended Dynkin diagrams.
D3brane quivers. For D3branes on X3, we have:
Ext1X3 (EJ ; EI )
,
eI
! eJ
,
XIJ ;
where XIJ are 4d N = 1 chiral multiplets in the bifundamental of U(nI )
U(nJ ), or in the
adjoint of U(nI ) if I = J . The arrows are oriented. Therefore, such quiver gauge theories
are generally chiral theories. More precisely, we denote by:
1
dIJ
dim Ext1X3 (EJ ; EI )
the number of arrows from eI to eJ in the 4d N = 1 quiver. D3brane quivers are \ordinary"
quivers (with relations), consisting of nodes and arrows, of the type most studied by both
physicists and mathematicians.
D1brane quivers.
D1branes on X4 lead to the richer structure of 2d N
= (0; 2)
quiver gauge theories. Those quivers have two distinct types of arrows, corresponding to
(0; 2) chiral multiplets XIJ and (0; 2) fermi multiplets IJ , respectively. We propose the
identi cation:
,
,
,
,
,
eI
! eJ
,
XIJ ;
IJ :
XIJ ;
IJ :
Note that the Ext2X4 (EI ; EJ ) = Ext2X4 (EJ ; EI ) by Serre duality. Thus the second type of
arrow is unoriented. This corresponds to the selfduality of the fermi multiplet in such
theories. We also de ne:
1
dIJ
dim Ext1X4 (EJ ; EI ) ;
2
dIJ
dim Ext2X4 (EJ ; EI ) :
Here dI1J is the number of chiral multiplets from eI to eJ (in bifundamental representations
2 2
if I 6= J or adjoint representation if I = J ). Similarly, dIJ = dJI denotes the number
of bifundamental fermi multiplets if I 6= J , while 12 dI2J is the number of adjoint fermi
multiplets if I = J .
D( 1)brane quivers.
Finally, we may consider Dinstantons on X5, which results in
a quiver with two types of oriented arrows:
(1.6)
(1.7)
(1.8)
(1.9)
(1.10)
(1.11)
The corresponding N = 1 gauged matrix model contains two types of \matter" multiplets,
the chiral and fermi multiplets [49]. In this case, the quantities:
1
dIJ
dim Ext1X5 (EJ ; EI ) ;
2
dIJ
dim Ext2X5 (EJ ; EI ) ;
(1.12)
give the number of arrows of either types from eI to eJ . We will brie y discuss these
gauged matrix models in section 4.
Elusive fractional branes.
The above identi cations between Ext groups and
supersymmetric multiplets in Dbrane quivers are conjectures, that we may check in many explicit
computations. The practical usefulness of these identi cations rely on identifying the
fractional branes EI in the rst place, as distinguished objects in the Bbrane category on Xn.
To the best of our knowledge, this remains an open problem in general. In this note, we
will deal with simple examples where we can describe the fractional branes explicitly.
Interactions terms: product structure in the derived category. Importantly, the
Dbrane quivers have interactions terms, which are encoded in superpotentials in various
dimensions. On D5branes, the interactions are fully determined by supersymmetry, while
D3brane quivers have a nontrivial 4d N = 1 superpotential W (X). The 2d N = (0; 2)
theories have two types of \superpotential" interactions, encoded in holomorphic functions
J (X) and E(X) [50]. The 0d N = 1 matrix models also have two kinds of holomorphic
\superpotentials", distinct from the 2d superpotentials, denoted by F (X) and H(X) [49].6
These interactions terms can be recovered from the fractional branes by considering
the product structure between Ext groups. Let A denote the graded algebra Ext for a
given set of fractional branes, where the grading is by the degree of the Ext groups. (It is
also the ghost number of the Bmodel.) There exists multiproducts:
of degree 2 k, satisfying relations amongst themselves, that generate a minimal A1
structhe product obtained by composition. These multiproducts correspond to disk correlators
in the topological Bmodel.
It is known that the A1 structure encodes the 4d N = 1 superpotential of D3brane
quivers [10, 52]. Following the same methods, we will be able to derive the 2d N = (0; 2)
and 0d N = 1 quiver interactions.
This paper is organized as follows. In section 2, we discuss the construction of 2d
N = (0; 2) supersymmetric quiver gauge theories from the knowledge the Bbranes on a
CY fourfold. In section 3, we discuss triality of 2d N = (0; 2) quivers in this context, and
relate triality to mutations of exceptional collections of sheaves. In section 4, we discuss
the similar construction of 0d N = 1 quiver theories from Bbranes on a CY
vefold. A
few complementary points are discussed in appendices.
6What we call F term was called Jterm in [49]. We choose this notation in order to distinguish between
the 2d and 0d interactions.
7A minimal A1 structure is an A1 structure in which m1 = 0 [51].
{ 5 {
D1brane quivers and 2d N
= (0; 2) theories
Twodimensional gauge theories with N = (0; 2) supersymmetry are built out of three
types of supermultiplets: vector, chiral and fermi multiplets [50]. In WessZumino gauge,
the vector multiplet (V; Vz) contains a gauge eld A , leftmoving gaugini and an auxiliary
scalar D, transforming in the adjoint of the Lie algebra g = Lie(G), with G the gauge
group.
The charged matter elds consist of chiral multiplets
and fermi multiplets
 and
of their chargeconjugate multiplets, the antichiral multiplet e and the antifermi multiplet
e, respectively. They satisfy the halfBPS conditions:
the chiral multiplets . In components, the chiral super eld reads:
D+
= 0 ;
D+
= E( ) :
=
+
+
+
with
a complex scalar and
+ a rightmoving fermion. The fermi super eld is given by:
=
+G
+E ;
with
a leftmoving fermion an G an auxiliary eld. The chiral and fermi multiplets are
valued in some representations R
and R
of g, respectively. Consequently, the potential
E( ) is valued in R as well. The canonical kinetic Lagrangian for the matter elds is:
Lkin =
Z
d +d + i Dz
;
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
with J the conjugate potential for the antifermi multiplet. This Lagrangian is
supersymmetric provided that (2.5) is satis ed. The auxiliary
elds G, G can be integrated out,
which sets G = J and G = J . We then obtain the following Lagrangian for the fermi
multiplets:
Lfermi =
2i
+ EE + J J +
+ : (2.7)
{ 6 {
with Dz the gauge covariant derivative, and with the trace over g kept implicit. A standard
superYangMills term can also be constructed for the vector multiplet. To every fermi
multiplet , we also associate an \N = (0; 2) superpotential" J = J ( ) transforming in
the conjugate representation R , such that:
R
Z
with the trace over g, Tr : R
! C. The interaction Lagrangian reads:
Tr (EJ ) = 0 ;
LJ =
d +
J ( )
Z
d +
J ( ) ;
Note that there is a symmetry that exchanges fermi and antifermi multiplets:
$
;
E $ J ;
E $ J :
In the presence of several fermi multiplets in distinct irreducible representations, each fermi
multiplet can be \dualized" independently.8
(2.8)
(2.9)
! eJ
2.1
N = (0; 2) quiver gauge theory from Bbranes at a CY4 singularity
Systems of D1branes at CY4 singularities engineer a simple yet rich class of gauge theories
with product gauge group:
HJEP02(18)5
G = Y U(NI ) :
I
To each U(NI ) gauge group, one associates an N = (0; 2) vector multiplet, denoted by a
node eI in a quiver diagram. The matter elds in chiral multiplets are in bifundamental
representations NI
NJ between unitary gauge groups. To each chiral multiplet XIJ in the
fundamental of U(NI ) and antifundamental of U(NJ ), we associate a solid arrow eI
in the quiver diagram. The matter
elds in fermi multiplets are also in bifundamental
representations. To each bifundamental fermi multiplet
IJ , we associate the dashed link
eI    eJ in the quiver diagram. While
IJ denotes a fermi multiplet in the bifundamental
NI
NJ of U(NI )
U(NJ ), the associated link in the quiver is unoriented, re ecting
the fermi duality (2.8).9
We may also have chiral and fermi multiplets in the adjoint
representation of a single gauge group U(NI ), corresponding to a special case of the above
with I = J .
To each
IJ , one associates an Eterm and a J term. Given that IJ transform in
the bifundamental representation NI
NJ , by convention, the potential E IJ transforms
in NI
NI
NJ as well, while the potential J IJ transforms in the conjugate representation
NJ . In other words, E IJ is given by a direct sum of oriented paths p (counted with
complex coe cients) from eI to eJ in the quiver, travelled along chiral multiplet arrows,
and J IJ is similarly a direct sum of oriented paths pe from eJ to eI :
E IJ (X) =
J IJ (X) =
X
paths p
X
paths peecIpeJ XJL1 XL1L2
where the sum is over all possible paths p and pe of lengths k and ek, respectively. The
eld rede nitions) as part of the
de nition of the N = (0; 2) suepersymmetric quiver. They must be such that the
supersymmetry constraint (2.5) holds. This means that, for any closed loop P for chiral multiplets
8See [53] for a discussion of some subtleties in this symmetry in (0; 2) NLSMs.
9In practice, we still nd it convenient to write oriented dashed arrows for fermi multiplets, re ecting a
choice of representation for the fermi multiplets (that is, which is
and which is ). This is because such
a choice is needed to write down the o shell supersymmetric action.
{ 7 {
in the quiver, we must have:
X
p;p
e
p+pe=P
cp ecpe = 0 ;
8P ;
where the sum is over all pairs of quiver paths p : eI !
based at fermi multiplets IJ such that the closed path p + pe coincides with P .
! eJ and pe : eJ !
2.1.1
From Bbranes to quiver
Consider a D1brane probing a local CalabiYau fourfold singularity X4. Away from the
singularity, the D1brane is described in the Bbrane category as a skyscraper sheave Op
at a point p 2 X4. At the singularity, we expect that the D1brane fractionates into a
nite number n of mutuallystable components:
Op = E1
En :
The fractional branes EI , with I = 1;
; n, are a distinguished set in the derived category
of coherent sheaves on the local CY fourfold. If we normalize the central charge of the
D1brane to Z(Op) = 1, the fractional branes must be such that their central charge align
at a special smallvolume point  a \quiver point"  in the quantum Kahler moduli
space of X4, with Z(EI ) 2 R>0 and PI Z(EI ) = 1. In the case of an orbifold of at space,
X4 = C4= , the \quiver point" is the orbifold point, where perturbative string theory
is valid, and the fractional branes are in onetoone correspondence with the irreducible
representations of
[1]. We will not study stability issues at all in this work. We will only
assume that we may identify (or guess) a suitable set of fractional branes. In general, there
might be many allowable sets of fractional branes, some of which give the same quiver, and
some of which give di erent quivers. This last possibility should correspond to eld theory
dualities. We will comment on this point in section 3.
Given the fractional branes:
EI 2 Db(X4) ;
as objects in the Bbrane category, we may compute the morphisms between them. For EI
and EJ given as coherent sheaves on X4, the morphisms are elements of the Ext groups:
ExtiX4 (EI ; EJ ) ;
i = 0; 1; 2; 3; 4 :
These groups encode massless open strings stretched between fractional branes [45]. We
should have:
metric quiver:
Ext0X4 (EI ; EJ ) = Hom(EI ; EJ ) = IJ C ;
to obtain a physical quiver. This is because Ext0 is identi ed with the massless gauge eld
in the open string spectrum. In our setup, we identify Ext0(EI ; EJ ) with the N = (0; 2)
vector multiplet at the node eI of the quiver.
The degreeone Ext groups are identi ed with the chiral multiplets in the
supersym,
XIJ ;
{ 8 {
(2.11)
! eI
(2.12)
(2.13)
(2.14)
(2.15)
(2.16)
By Serre duality, we have Ext3X4 (EJ ; EI ) = Ext1X4 (EI ; EJ ), so that Ext3X4 (EI ; EJ ) is identi ed
with the antichiral multiplets XIJ . This identi cation of chiral multiplets with Ext1 is
wellknown in the case of fourdimensional N = 1 quivers associated to D3branes on a
CY threefold [
7, 44, 48, 54
]. The new ingredient on a CY fourfold is that we also have
independent degreetwo Ext groups, with:
Ext2X4 (EJ ; EI ) = Ext2X4 (EI ; EJ )
by Serre duality on X4. It is natural to identify these groups with the fermi multiplets IJ
in the N = (0; 2) quiver:
,
,
IJ :
The selfduality relation (2.17) for Ext2 correspond to the fact that fermi and antifermi
multiplet are indistinguishable. For each pair of distinct nodes I, J , we may pick the basis
of the Ext2 vector spaces:
f ; g 2 Ext2X4 (EJ ; EI ) ;
f ; g 2 Ext2X4 (EI ; EJ ) ;
where
and
correspond to fermi multiplets
JI , respectively, and such that Serre duality exchanges
with
, and
with . This choice of basis is completely conventiondependent, however.
This corresponds exactly to the freedom (2.8) of labelling fermi and antifermi multiplets
in the supersymmetric eld theory.
For I = J , Ext2(EI ; EI ) is selfdual, and each pair of Serredual elements correspond to
a pair of fermi and antifermi multiplets
II , II in the adjoint representation of U(NI ).
As a simple consistency check of these identi cations between Ext groups and N =
(0; 2) super elds, it is interesting to look at the product variety X4 = X3 C, with X3 a CY
threefold singularity. This nonisolated singularity preserves N = (2; 2) supersymmetry in
twodimension, and the 2d quiver should simply be the dimensional reduction of the N = 1
supersymmetric quiver for D3branes on X3. Each 4d N = 1 vector multiplet decomposes
into one N = (0; 2) vector multiplet and one adjoint fermi multiplet, and each 4d N = 1
chiral multiplet decomposes into one N = (0; 2) chiral multiplet and one fermi multiplet.
In terms of Ext groups, this means that we should have:
Ext0X4 (EI ; EJ ) = Ext0X3 (EI ; EJ ) ;
Ext1X4 (EI ; EJ ) = Ext0X3 (EI ; EJ )
Ext2X4 (EI ; EJ ) = Ext1X3 (EI ; EJ )
Ext3X4 (EI ; EJ ) = Ext2X3 (EI ; EJ )
Ext4X4 (EI ; EJ ) = Ext3X3 (EI ; EJ ) :
Ext1X3 (EI ; EJ ) ;
Ext2X3 (EI ; EJ ) ;
Ext3X3 (EI ; EJ ) ;
This can be shown to be the case in general orbifolds C3=
C  see appendix A. Note
that (2.20) is consistent with Serre duality (1.5). One can similarly consider the
decomposition X4 = X2
C2, which preserves 2d N = (4; 4) supersymmetry.
{ 9 {
A comment on conventions.
To avoid any possible confusion, let us note that we are
using the physicist notation for the chiral multiplets in the N = (0; 2) superpotentials, and
the mathematical notation of composition when discussing elements of Ext . For instance,
we have:
x y. When talking about the fractional branes, we write these maps as:
y
x
E3 ! E2 ! E1 :
On the other hand, we have chosen the convention that Ext(EJ ; EI ) corresponds to the
chiral multiplet XIJ , so that the direction of the arrows in the quiver are ipped: a map
EJ ! EI corresponds to a quiver arrow eI ! eJ . In our example (2.21), denoting by X
and Y the chiral multiplets associated to the Ext group elements, we have:
where on the righthandside we associated a gauge group U(NI ) to each node eI . In these
conventions, we can write x y as the matrix product XY for the chiral multiplets.
Anomalyfree condition and quiver ranks.
Consider an N = (0; 2) quiver with nodes
feI g and gauge group (2.9). For each U(NI ) factor, the cancellation of the nonabelian
anomaly requires:
X
J6=I
dI1J + dI3J
2
dIJ NJ + 2NI
1 + dI1I
= 0 :
(2.24)
Here the rst sum is over the chiral and fermi multiplets in bifundamental representations,
while the second term denote the contribution from the vector multiplet (with dI0I = 1)
and from adjoint matter. Using Serre duality, this can be written as:
(2.21)
(2.22)
(2.23)
4
J i=0
X
X( 1)iNI dim ExtiX4 (EI ; EJ ) = 0 :
(2.25)
This condition imposes constraint on the allowed ranks NI in the quiver. If we consider
a single D1brane, the ranks NI should be xed from
rst principle; however, the explicit
dictionary between branecharge basis and quiverrank basis is not always known. The
anomalyfree condition then provides a strong constraint. The solutions to (2.25), as a
linear system for the positive integers NI , correspond to all stable Dbrane con gurations
at the singularity. In particular, the unique solution fNI g such that each NI is the smallest
possible positive integer is expected to correspond to a single D1brane. In the special case
of toric CalabiYau singularities, we know from [19, 21, 26] that there exists \toric quiver"
with equal ranks, NI = N , corresponding to N D1branes.
We should also mention that the abelian quadratic anomalies, from the U(1)I factors
in U(NI ), do not vanish in general. Instead, they should be cancelled by closed string
contributions a la GreenSchwarz [1, 17, 55].
2.1.2
To complete the determination of the N = (0; 2) supersymmetric quiver from the fractional
branes on X4, we need to discuss the E and J terms (2.10). It is convenient to package
them into a gaugeinvariant \(0; 2) superpotential" W de ned as:10
Here, the index I runs over all the fermi multiplets. This W can be computed by following
the methods of [10], which studied 4d N = 1 quiver theories on D3branes at CY3
singularities. On general ground, the superpotential coupling constants are encoded in open
string correlation functions. Those can be described in the language of A1 algebra  see
e.g. [56] and references therein.
An A1 algebra is a (graded) algebra A together with a set of multiplications mk :
such a way that there is an A1 map:11
for all integer n > 0. The rst relation states that (m1)
2 = 0, so one can think of
m1 : A ! A as a di erential. The Ext group elements between Bbranes, on the other
hand, generate a minimal A1 algebra, for which m1 = 0.
To compute the multiproducts on the Ext algebra, we proceed as follows. Given an
A1 algebra Ae, one de nes H (Ae) to be the cohomology of m1. If Ae has no multiplications
beyond m2, then it has been shown [57] that one can de ne an A1 structure on H (Ae) in
f : H (Ae) ! Ae ;
(2.28)
with f1 equal to a particular representation H (Ae) ,! Ae in which cohomology classes map
to (noncanonical) representatives in Ae, and such that m1 = 0 in the A1 algebra on H (Ae).
One can then use the consistency conditions satis ed by elements of an A1 map to solve
algebraically for f1
mk.
In terms of Bbranes, the algebra Ae is the algebra of complexes of coherent sheaves,
with chain maps between complexes. In that construction, m1 is essentially the BRST
charge Q of the Bmodel. The \physical" open string states then live in the cohomology
H (Ae), which gives us the derived category Db(X)  we refer to [48] for a pedagogical
discussion. We can identify the minimal A1 algebra A
H (Ae) with the Ext algebra we
are interested in.
Practically, in the examples discussed in this paper, each Bbrane will be a single
coherent sheaf, which can be represented in the derived category by a locallyfree resolution.
10This expression is only formal. The N = (0; 2) superpotential that appears in the gauge theory
Lagrangian is the usual Tr
I JI (X)), since superspace treats
and
asymmetrically. This formal W
rst appeared in [22]. It elegantly encodes the algebraic structure of the N = (0; 2) quiver relations. This
point is further discussed in appendix C.
11That is, a family of maps satisfying certain consistency conditions [57].
Consider a correlation function of r boundary vertex operators ai 2 A on the openstring
worldsheet. In the A1 language, this can be written as:
ha1
ari = ha1; mr 1(a2;
; ar)i ;
in terms of the higherproduct mr 1 and the pairing (2.29) [10]. Each Ext elements x 2 A
is dual to a \ eld" X in the supersymmetric quiver  see appendix C for further details.
In the case of a 2d N = (0; 2) quiver describing Bbranes on a CY4 geometry, we have the
Ext algebra:
; , as in (2.19). The coupling constants cJ and cE appearing as:
where the summands denote all Ext groups between the various fractional branes, of degree
0;
X, and by ;
; 4. Let us denote by x 2 A the Ext1 elements corresponding to the chiral multiplets
e 2 A the Ext2 elements corresponding to the fermi and antifermi multiplets
cJ Tr( X1
Xr) + cE Tr( X10
Xr00 )
in the superpotential (2.26) can be computed as the openstring correlators:
The Ext elements can then be represented by chain maps between resolutions, modulo
chain homotopies. The m2 products in A are given by chain map composition. The higher
products can then be computed by the procedure just described.
We elaborate on this procedure in appendix C.3, and we illustrate the computation of
the higher products, in a speci c example, in appendix D. All of the other examples below
will actually have mk = 0 for k > 2.
Open string correlators and A1 products.
Let A denote the Ext algebra associated
to a local CalabiYau nfold. There exists a natural \trace map" of degree
n, which we
denote by
: A ! C. Note that A is a graded algebra, with a of degree q if a 2 Extq.
Serre duality de nes a natural pairing of degree
n:
Explicit formula for the E and J terms.
We can now spell out the precise formula
for the coupling constants appearing in (2.10). Consider a fermi multiplet IJ
corresponding to
to
2 Ext2(EJ ; EI ), and the chargeconjugate antifermi multiplet
IJ corresponding
2 Ext2(EI ; EJ ). For each path p as in (2.10), we have the elements x 2 Ext1
corresponding to the chiral multiplets X. We thus have:
for the Eterm coe cients, and
One can show that:
and
for the J term coe cients. We can check this identi cation for a number of geometries
previously studied by independent techniques, and we nd perfect agreement.
Last but not least, we should note that, according to the dictionary (2.34){(2.35), the
Tr(EJ ) = 0 constraint (2.11) translates into a very nontrivial relation amongst products of
open string correlators. In appendix C, we give a general argument for why this constraint
will hold for E and J de ned by the A1 algebra as above. In addition, we will check,
in every example below, that the condition Tr(EJ ) = 0 indeed holds, thus providing
an additional consistency check on our computations. It would be interesting to also
understand the rstprinciple origin of this constraint in the CalabiYau fourfold geometry.
space, there is a single \fractional brane", the skyscraper sheaf Op, which corresponds to
a single transverse D1brane. Consider Op at the origin of C4, without loss of generality.
Ext0(Op; Op) = Ext4(Op; Op) = C ;
Ext1(Op; Op) = Ext3(Op; Op) = C4 ;
Ext2(Op; Op) = C6 :
From this result, we directly read o the N = (0; 2) supermultiplet content according to the
general rules. We have a single vector multiplet, 4 chiral multiplets and 3 fermi multiplets.
If there are N fractional branes at a point, all these
elds are in the adjoint of a U(N )
gauge group. This reproduces the eld content of maximally supersymmetric N = (8; 8)
YangMills theory in 2d, as expected. To compute the interaction terms, we will need to
describe the Ext algebra more explicitly.
2.2.1
An explicit basis for Ext (Op; Op)
The Ext algebra can be computed from the Koszul resolution of Op, which reads:
0 ! O ! O
D
where:12
A =
x y z w
0 y
0 1
0 C
w C
A
z
0
x
0
0
w
0
0
x
z 0 C
x
A
12Here and in the following, we denote a map M : Cn ! Cm by an m
maps corresponds to matrix multiplication (for instance, A B = AB).
n matrix, so that composition of
Let us present explicit expressions for the generators of Ext . We will use the notation:
Xji 2 Exti(Op; Op) ;
j = 1;
; dim Exti(Op; Op) :
(2.39)
Every Ext element can be represented by a chain map between two copies of the Koszul
resolution; the actual Ext element is given by the corresponding element in its cohomology,
by the de nition of Ext as a derived functor. First of all, Ext0(Op; Op) is spanned by the
single element:
Secondly, Ext1(Op; Op) is spanned by maps of the form:
A basis can be obtained by taking
O
?
1?
y
O
O
D
! O
! O
! O
?
1?
y
4
4
D
D
O
y
?
?
4
D
C
! O
! O
! O
! O
?
1?
y
6
6
C
C
?
?
y
4
6
C
B
! O
! O
! O
! O
?
1?
y
4
4
B
B
?
?
y
6
4
B
A
4
?
?
y
! O
! O
A
A
! O
?
1?
y
! O
A
! O
A
0
C ;
A
0
C ;
A
0
C
A
1
and demanding the diagram be anticommutative. For example, when
= (1; 0; 0; 0)t, we
can take
B
0 0 1 0 0 1
B 0 0 1 0 C
= BBBB 00 00 00 01 CCC
C
C ;
0 0 0 0
0 0 0 1 0 0 0 1
= BBB 00 00 00 00 10 01 CC
Similarly, Ext2(Op; Op) is spanned by maps of the form:
=
A
! O
6
?
!?
y
As before, we can choose ' to be one of the unit column vectors with six entries, and then
make the diagram commutative. For example,
' = BBBB 10 CCC
0 1 0 0 0 1
A
Ext3(Op;Op) is spanned by maps of the form:
is one of the unit vectors with four entries and
is such that A
=
D. For
example, when
= (0;0;0;1)t,
= ( 1;0;0;0). Finally, Ext4(Op;Op) is spanned by:
O
D! O4 C
! O6 B ! O4 A
! O
2.2.2 Multiplication of maps
The multiplication rule can be determined by composing these maps. For example, X31 X21
is computed by:
D
! O4 C
! O6 B
! O4 A
! O
D
! O4 C
! O6 B
! O4 A
! O
D
! O4 C
! O6 B
! O4 A
! O
?
21?y
?
31?y
?
21?y
with:
O
D
! O4 C
! O6 B
! O4 A
0 0 0 0 0 1
A
C ; 21 = BBB 00 00 00 01 00 00 CCC ; 21 = 0 0 1 0 ;
O
?
12?y
?
31?y
O
?
13?y
B 0 0 0 0 C
0 0 0 0
O
?
1?
y
?
21?y
?
31?y
0 0 1
0
31 = BBB 01 CC
C ; 31 = BBBB 00
A
0
C
1 0 0 C
Proceeding in this way, we nd the multiplication rules:
m2(Xi1; Xj1) = Xi1 Xj1 = BB
0
B
B
B
B
0
X12
X22
X42
X12
0
X32
X52
The product X1 Xj1 is given by the matrix element ij in (2.40). One can also compute
i
the products:
X62 X12 =
X52 X22 = X42 X32 = X14 ;
which commute. All other products between degreetwo maps vanish. This shows that the
Serre dual of X12; X22; X33 are X62;
X52; X42 respectively.
One can also show that the higher products vanish in this case  that is, mk = 0 if
k > 2. Therefore, any nonzero correlation function can be reduced to one of the following:
hX12 X31 X41i = 1 ;
hX32 X11 X41i = 1 ;
hX52 X11 X31i =
1 ;
hX22 X21 X41i =
hX42 X21 X31i = 1 ;
hX62 X11 X21i = 1 :
1 ;
2.2.3
The C4 quiver: N = (8; 8) SYM
on C
4 has the
eld content of N
From (2.36){(2.37), we see that the N = (0; 2) gauge theory corresponding to D1branes
= (8; 8) SYM. We can also verify that the product
structure encoded in (2.41) reproduces the correct supersymmetric interactions. In N =
(0; 2) notation, this theory consists of four chiral multiplets, denoted
and three fermi multiplets a (a = 1; 2; 3), with the E and J terms:
and
a (a = 1; 2; 3),
This is reproduced by our computation, with the identi cations:
for the chiral multiplets, and
Ea = [ ; a] ;
J a = abc b c :
=
X41 ;
a = (X31 ;
X21 ; X11) ;
a = (X62 ;
X52 ; X42) ;
a = (X12 ; X22 ; X32)
(2.40)
(2.41)
(2.42)
(2.43)
(2.44)
for the fermi multiplets, as one can easily check using (2.34){(2.35), and
Ea =
J
a =
X
ij
X
ij
h a; m2( i; j )i i j ;
h a; m2( i; j )i i j ;
Ext0(E0; E0) = C ;
Ext0(E1; E1) = C ;
Ext0(E0; E1) = 0 ;
Ext0(E1; E0) = 0 ;
Ext1(E0; E0) = 0 ;
Ext1(E1; E1) = 0 ;
Ext1(E0; E1) = C4 ;
Ext1(E1; E0) = C4 ;
Ext2(E0; E0) = C6 ;
Ext2(E1; E1) = C6 ;
Ext2(E0; E1) = 0 ;
Ext2(E1; E0) = 0 ;
for f ig the set of all chiral super elds  here, by abuse of notation, we identi ed the
quiver elds with the corresponding Ext elements in the openstring correlators. Note that
the condition (2.5) is satis ed, Tr(EaJ a) = 0. The interaction terms (2.42) display an
SU(3) avory symmetry. Onshell, there is a larger SU(4) avor symmetry, with ( a; a)
sitting in the 6 of SU(4). It will often be the case that the avor symmetry displayed by the
N = (0; 2) quiver is smaller than the symmetry expected from the CY4 geometry. Those
larger geometric symmetries can be thought to arise in the infrared of the gauge theory, as
accidental symmetries [20].
The next simplest class of examples are supersymmetric orbifolds of at space. Consider
the CY4 singularity C4= , with
brane EI for each irreducible representation I of
line bundle O with the corresponding
given by:
a discrete subgroup of SU(4). There exists one fractional
[46]. We also denote by I the trivial
equivariant structure. The fractional branes are
EI = I
Op ;
with Op the skyscraper sheaf supported at the origin. In the following, we consider a few
examples with
abelian, for simplicity.
2.3.1
Consider C4=Z2, where the generator of Z2 acts on the C4 coordinates (x; y; z; w) as:
We have two fractional branes:
(x; y; z; w) 7! ( x; y; z; w) :
E0 = 0
Op ;
E1 = 1
Op :
for the trivial and nontrivial representation of Z2, respectively. The dimensions of the Ext
groups can be computed following the methods of [46]. We have:
(2.45)
(2.46)
(2.47)
(2.48)
a at each nodes, and two sets of four chiral multiplets, Ai = (Aa; A4) and
Bi = (Ba; B4), in bifundamental representations. The quiver arrows have multiplicities equal to
the number of distinct chiral or fermi multiplets.
with the higher Ext groups determined by Serre duality. We can also recover this spectrum
from the results of section 2.2. Let us replace X in (2.39) by a, b, c, d according to the
following diagram:
HJEP02(18)5
c
7 E0 l
a
b
, E1
w
d ;
which encodes all possible Ext groups. From the Koszul resolution (2.38) and the fact that
the maps A; B; C; D are all odd under Z2, we see that the superscript of a and b can only
take values 1; 3, while the superscript of c and d can only take values 0; 2; 4, in agreement
with (2.48). This gives us the N = (0; 2) quiver indicated in gure 1.
The Bmodel correlation functions can be read o from (2.41). The N = (0; 2)
superpotential immediately follows. Let
100; 200; 300; 111; 211; 311 denote the fermi
superelds corresponding to c24; c25; c26; d24; d25; d26, respectively. Note that they are Serre dual to
c23; c22; c21; d23; d22; d21. Let us also denote the chiral super elds corresponding to aj1; bj1 by
Aj ; Bj . We then have, for instance:
and so on and so forth. It is convenient to introduce the notation:
J 1 =
00
E 100 =
0a0 ;
X
i;j
X
i;j
1a1 ;
c24bi1aj1
c23bi1aj1
BiAj = B2A3
B3A2 ;
BiAj = B1A4
B4A1 ;
Ai = (Aa; A4) ;
Bi = (Ba; B4) ;
with the index a = 1; 2; 3, to emphasize an SU(3) avor symmetry. The interaction terms
are given by:
J 0a0 = abcBbAc ;
E 0a0 = BaA4
B4Aa ;
J 1a1 = AaB4
E 1a1 = abcAbBc :
A4Ba ;
This satis es Tr(EJ ) = 0, and it is in perfect agreement with the results of [19]. Note that,
while the Lagrangian of the theory only has an SU(3)
U(1) global symmetry, the E and
J terms of either node, taken together, t into the 6 of SU(4), while the elds Ai and Bi
each sit in the 4 of SU(4). This is the sign of an enhanced global symmetry in the infrared
of the gauge theory, which can also be seen in the geometry.
(2.49)
(2.50)
(2.51)
=
+ F ;
F = F ( ) :
Here, the N = 1 superpotential F is an holomorphic function of the bosons
in chiral
multiplets.
Given the chiral multiplets i and fermi multiplets
a, one can write the
supersymmetric action:
as complex conjugate in the matrix integral, while there is a single fermion . The second
type of multiplet is the fermi multiplet , with a single fermionic component , such that:
SF =
Z
d F a( ) a = F a( )Fa( ) +
(4.3)
Another quadratic action in the fermions can be written in terms of an holomorphic
potential Hab( ) =
Hba( ):
SH = Hab( ) a b :
This is supersymmetric provided that HabFb = 0. The third type of sypersymmetry
multiplet is the gaugino multiplet, which implements a gauge constraint on
eld space. The
gaugino multplet V consists of two components, the fermion
and the real boson D, with:
V =
+ D :
Given a theory of chiral and fermi multiplets with some nontrivial Lie group symmetry, we
can gauge a subgroup G (with Lie algebra g) of that symmetry by introducing an gvalued
gaugino multiplet, with the action:
(4.2)
(4.4)
(4.5)
(4.6)
(4.7)
(4.8)
with
acting on
in the appropriate representation, and an overall trace over the gauge
group is implicit. Here is a 0d FayetIliopoulos parameter. Integrating out D, we obtain:
Sgauge =
2
i
:
where
(schematically), which is the moment map (minus the \level" ) of the
G action on the bosonic eld space.
4.1
N = 1 gauged matrix model from Bbranes at a CY5 singularity
Dinstantons at CY5 singularities engineer precisely such gauged matrix models with gauge
group Q
I U(NI ). For each node eI in the 0d N = 1 quiver, we have a U(NI ) gaugino
multiplet. The matter elds are either chiral or fermi multiplets, in adjoint or bifundamental
representations. We have thus a quiver with two type of oriented arrows: eI ! eJ for
chiral multiplets XIJ , and eI 99K eJ for fermi multiplets IJ . Finally, we also have the F
and Htype interaction terms. To each fermi multiplet
IJ , we associate the element FIJ ,
a direct sum over oriented paths p from eI to eJ , of length k:
Sgauge =
Z
d
1
2
D
i
+ i
;
X
FIJ (X) =
cIpJ XIK1 XK1K2
similarly to (2.10), with given coe cients cIpJ . In addition, to every pair of fermi multiplets
KL, we associate the Hterm action SHIJ;KL , which is a sum over closed loops p
from eI back to itself, which includes both IJ and
KL, in addition to chiral multiplets X:
X
paths pe e
SHIJ;KL =
cIJ;KL Tr( IJ XJM1
p
XMk 1K
KL XLN1
XNk0 1I ) :
Note that the closed path pe has length k + k0 + 2, including the two fermions.
This quiver structure naturally arises from open strings between fractional D(
1)branes at a CY5 singularity, where each node eI corresponds to a fractional brane EI . As
The nonvanishing Ext0 elements are identi ed with the gaugino multiplets. The
degreeone Ext groups are identi ed with chiral multiplets:
e
(4.9)
(4.10)
(4.11)
(4.12)
(4.14)
(4.15)
,
eI
! eJ
,
,
XIJ ;
IJ :
in bifundamental (if I 6= J ) or adjoint (if I = J ) representations. Similarly, the degreetwo
Ext groups are identi ed with the fermi multiplets:
Ext2X5 (EI ; EJ ).
By Serre duality, we also have Ext4X5 (EJ ; EI ) = Ext1X5 (EI ; EJ ) and Ext3X5 (EJ ; EI ) =
Interaction terms.
The F terms (4.3) and Hterms (4.4) also arise naturally in the
Bmodel. As discussed in section 2.1.2, the Extgroup generators satisfy an A1 algebra with
multiproducts mk. Consider a fermi multiplet IJ corresponding to
2 Ext2(EJ ; EI ), and
let us denote by
2 Ext3(EI ; EJ ) the Serre dual generator. For each path p as in (4.8), we
have the elements x 2 Ext1 corresponding to the chiral multiplets X. We propose that:
for the F term coe cients in (4.8). Similarly, consider the fermi multiplets
2 Ext2(EL; EK ), respectively. We propose that
the Hterm coe cients in (4.9) are given by:
p
e
cIJ;KL =
D
xJM1
xMk 1K
xLN1
xNk0 1I
E
=
m2( ; mk(xJM1 ;
e
; xMk 1K ; ; xLN1 ;
; xNk0 1I )) ;
with ek = k + k0 + 1. We will check this prescription in some examples below. Note that
this corresponds exactly to computing the formal 0d N = 1 superpotential:
which was recently introduced in [22].
4.2
We can work out the very simplest case, a D( 1) brane on X5 = C5, exactly like in
section 2.2. Consider the skyscraper sheaf Op at the origin of C5. We have:
Ext0(Op; Op) = Ext5(Op; Op) = C ;
Ext1(Op; Op) = Ext4(Op; Op) = C5 ;
Ext2(Op; Op) = Ext3(Op; Op) = C10 :
Using the above dictionary to N = 1 super elds, this reproduces the expected eld content
of the maximallysupersymmetric N = 16 matrix model, as we will review below.
Proceeding as before, the Koszul resolution of Op on C5 reads:
0 ! O ! O
E
! O
! O
10 B
! O
! O ! Op ! 0;
(4.16)
(4.17)
0 u 1
B
B
w C
y CA
x
(4.18)
w 00 CCCC ; E = BBB z C :
C
C
Similarly to section 2.2, we choose as bases of the Ext groups the commutative diagrams
composition. The products m2(Xi1; Xj1) = X1 Xj1 are given by:
i
whose leftmost nonzero vertical map has 1 at an entry and 0 elsewhere. We denote them
by Xi, following the same conventions. The multiplication rule is again determined by
j
0
x 0
0
0
u
0
0
0
0
0
x
X11
X21
X31
X41
X51
0
0
0
w
0
y
0
z 0
0 y
0
0
0
0
u
z
0
0
X11
0
X12
X22
X42
X72
X21
X12
0
X32
X52
X82
X31
X22
X32
0
X62
X92
0 C
C
C
u C
w CA
z
w 0 C
0 w
B
B
B
B 0
BB y
B
B
B
B
B 0
0
z 0
x 0
u
0
y
0
0
0
0
0
u
z
0
u
0
0
y
0
0
z
w 0
x 0
x 0
u C
C
C
A
w
0
0
u
z
0
0
x
u C
w CC
z C
A
y
X41
X42
X52
X62
0
X120
X51
X72
X82
X92
X120
0
products m2(Xi1; Xj2) mapping Ext1
Ext2 to Ext3, according to:
The elements in this table are the products of the elements in the rst column multiplied
by elements in the rst row. (For example, X11
X21 =
X12.) Similarly, we have nonzero
X12 X22 X32 X42
X52 X62 X72
X92 X120
X31 X13
We also nd the following Serre dual elements to Xi2:
Using the multiplication rule (and the cyclic property of the openstring correlators), we see
that any nonzero correlation function can be computed in terms the following
hX3X1X1itype correlators:
and the following hX2X2X1itype correlators:
hX13X51X41i = 1 ;
hX43X51X11i = 1 ;
hX73X41X11i = 1 ;
hX130X21X11i = 1 ;
hX32X120X11i = 1 ;
hX22X120X21i = 1 ;
hX12X120X31i = 1 ;
hX12X92X41i = 1 ;
hX12X62X51i = 1 ;
hX23X51X31i = 1 ; ; hX33X51X21i = 1 ;
hX53X41X31i = 1 ;
hX83X31X21i = 1 ;
hX63X41X21i = 1 ;
hX93X31X11i = 1 ;
hX52X92X11i = 1 ;
hX42X92X21i = 1 ;
hX42X82X31i = 1 ;
hX22X82X41i = 1 ;
hX22X52X51i = 1 ;
hX62X82X11i = 1 ;
hX72X62X21i = 1 ;
hX52X72X31i = 1 ;
hX32X72X41i = 1 ;
hX32X42X51i = 1:
(4.19)
(4.20)
(4.21)
(4.22)
HJEP02(18)5
4.2.2 The C5 quiver: N = 16 SYM
Consider the N = 16 supersymmetric GMM with gauge group U(N), corresponding to N
D( 1)branes in at space. Its eld content can be deduced from dimensional reduction
of 2d N = (8; 8) SYM in section 2.2.3. In N = 1 language, we have a single U(N) gaugino
multiplet, 5 chiral multiplets in the adjoint representation, and 10 fermi multiplets in the
adjoint representation. It is convenient to denote the chiral and fermi multiplets by
n and
mn =
nm, with n = 1;
5, since n and nm transform in the 5 and 10 of an SU(5)
avor symmetry. This spectrum is reproduced by the Ext groups above. We identify the
elds with the Ext elements according to Xn1 =
n, n = 1;
; 5, and:
X12 = 21 ;
X62 = 43 ;
X22 = 31 ;
X72 = 51 ;
X32 = 32 ;
X82 = 52 ;
X42 = 41 ;
X92 = 53 ;
X52 = 42 ;
X120 = 54 :
The interaction terms are determined by the F  and Hterms [49]:
Fmn =
m n
n m ;
Hmn;pq =
mnpqr r :
HJEP02(18)5
One can check that the openstring correlators (4.21){(4.22) precisely reproduce these
interactions. Note that, to check that the Hterm:
SH =
4
1 mnpqr Tr( mn pq r)
is supersymmetric, we need to use the Jacobi identity for U(N ). This is equivalent to the
nontrivial condition HabFb = 0 mentioned above, which must always be realized by the
Bbrane correlators.
Given the above results for C5, we can easily study various N = 1preserving orbifolds
Consider for instance C5=Z5, where Z5 acts as:
(x; y; z; w; t) 7! (!x; !y; !z; !w; !t) ;
! = e 5
2 i
on the C5 coordinates. We have ve fractional branes denoted by Ei, i = 0;
weights for the sheaves in the Koszul resolution of Ei are given by:
i + 1
BB i + 2 CC
BB i + 3 CC
BB i + 3 CC
B i + 3 C
C
A
i + 3
! BB i + 1 CC D! BBB ii ++ 22 CCCC !C BBB ii ++ 33 CCCC !B BBB i + 4 CCC !A
E B
C B B
i :
0 i + 4 1
B i + 4 C
(4.23)
(4.24)
(4.25)
(4.26)
; 4. The
(a) C5=Z5(1; 1; 1; 1; 1) quiver.
(b) C5=Z3(1; 1; 1; 1; 2) quiver.
chiral and fermi multiplet arrows indicate their multiplicities.
We then nd the spectrum:
Ext[0C5=Z5](Ei; Ej) =
Ext[1C5=Z5](Ei; Ej) =
Ext[2C5=Z5](Ei; Ej) =
0
0
0
( SpanCfX10g if j
i mod 5 ;
otherwise,
( SpanCfX11; X21; X31; X41; X51g if j + 1
i mod 5 ;
otherwise,
( SpanCfX12; X22; X32; X42; X52; X62; X72; X82; X92; X120g if j + 2
i mod 5 ;
otherwise.
The higher Ext groups are obtained by Serre duality. The correlation functions can be
read o from (4.21){(4.22). Let us introduce the chiral multiplets:
In : eI
with I an integer mod 5, m; n = 1;
; 5, and
quiver is shown in gure 10a. The interaction terms are:
I
mn =
Inm. The gauged matrix model
F Imn =
m n
I I+1
In Im+1 ;
H Imn; Ip+q2 = mnpqr Ir 1
:
Note the obvious SU(5) avor symmetry. This quiver was discussed in [49, 67, 68].
4.3.2
As a last example, consider the C5=Z3 orbifold:
(x; y; z; w; t) 7! (!x; !y; !z; !w; !2t) ;
! = e 3 :
2 i
(4.27)
(4.28)
(4.29)
We have three fractional branes Ei, i = 0; 1; 2. The weights for the sheaves in the Koszul
resolution of Ei are:
i + 1
0
8> 0
The spectrum consists of:
Ext[0C5=Z3](Ei; Ej ) =
SpanCfX10g if j
i mod 3 ;
BB i + 2 CC
i + 1
i
! BB i + 1 CC D! BBB ii ++ 22 CCCC !C BBB ii ++ 11 CCCC !B BBB i + 2 CCC !A
E B
C B B
i :
Ext[1C5=Z3](Ei; Ej ) = <
SpanCfX21; X31; X41; X51g if j + 1
Ext[2C5=Z3](Ei; Ej ) = <
>: SpanCfX11g
8> SpanCfX12; X22; X42; X72g
>: SpanCfX32; X52; X62; X82; X92; X120g if j + 2
if j
i mod 3 ;
if j + 2
i mod 3 ;
i mod 3 ;
if j
if j + 1
i mod 3 ;
i mod 3 ;
i mod 3 :
(4.30)
(4.31)
(4.32)
(4.33)
The corresponding 0d N = 1 quiver is shown in gure 10b. The correlation functions can
be read o from (4.21){(4.22). Taking advantage of the residual SU(4) avor symmetry,
let us introduce the chiral multiplets:
with a = 1;
4, and I an integer mod 3. Similarly, we de ne the fermi multiplets:
AI : eI
X!11 eI 1 ;
BIa : eI
X1a+1
interaction terms read:
mn =
nm are de ned as in (4.23). In this notation, the
I
F ab = BIaBIb+1
BIbBIa+1 ;
H Ia; Ibc =
abcdBId 1
F Ia =
AI BIa 1
H Iab; Ic 1
= abcdBId 1
Many more N = 1 matrix models can be worked out in this way. It would also be instructive
to study fractional branes on local Fano fourfold varieties, such as the resolution of the
C5=Z5(1; 1; 1; 1; 1) to Tot(O( 5) ! P4). We leave this and many other related questions
for future work.
Acknowledgments
We would like to thank P. Aspinwall, S. Franco, D. Ghim, C. Herzog, S. Katz, W. Lerche,
I. Melnikov, T. Pantev, and R.K. Seong for useful conversations and comments. E.S. was
partially supported by NSF grants PHY1417410 and PHY1720321.
A
Dimensional reductions
Fourfolds versus threefolds. Let X be a CalabiYau orbifold [Xc=G] of complex
dimension 3, with a set of fractional branes fEig supported at a point p 2 Xc, a xed point
of the Gaction. Let N3 denote the normal bundle Np=X .
Let us build another CalabiYau orbifold Y = C
X, which again has an isomorphic
set of fractional branes fEig, supported at x
0
f g
p 2 C
X, of codimension four. Let N
denote the normal bundle to x in Y , and 0 the structure sheaf with trivial Gequivariant
Ext0Y (Ei; Ej ) = H0(x; i
j )
G
Ext1Y (Ei; Ej ) = H0(x; i
Ext2Y (Ei; Ej ) = H0(x; i
Ext3Y (Ei; Ej ) = H0(x; i
Ext4Y (Ei; Ej ) = H0(x; i
= Ext0X (Ei; Ej ) ;
= Ext0X (Ei; Ej )
= Ext1X (Ei; Ej )
= Ext2X (Ei; Ej )
= Ext3X (Ei; Ej ) :
j
j
j
j
j
j
(N3
^2N3))G
(^2N3
^3N3))G
^3N3)
G
This directly con rms (2.20) in the case of an orbifold singularity. We conjecture that it
holds more generally.
Fourfolds versus twofolds. Similarly, we may consider X a CalabiYau orbifold [Xc=G]
of complex dimension 2, with a set of fractional branes fEig supported at a point p 2 Xc, a
xed point of the Gaction. Let NX denote the normal bundle Np=X . Let us build another
CalabiYau orbifold Y = C
2
X, which again has an isomorphic set of fractional branes
fEig, supported at x
f(0; 0)g p 2 C
X, of codimension four. Let N denote the normal
bundle to x in Y , and 0 the structure sheaf with trivial Gequivariant structure. Then,
2
0
N =
NX ) 2
^2NX ) 2 = NX
NX2
^2NX ;
(^2NX ) 2
;
^2NX = ^2NX :
j
j
j )
G
( 02
NX ))G
Ext0X (Ei; Ej )
Ext1X (Ei; Ej )
Ext1X (Ei; Ej )
Ext2X (Ei; Ej ) ;
Ext2X (Ei; Ej )
j
(NX
^2NX )G
(NX ) 2
^2NX ))G
(^2NX ) 2))G
This decomposition corresponds to the dimensional reduction of a 6d N = 1 quiver theory
(or, equivalently, of a 4d N = 2 theory) to 2d, giving rise to an N = (4; 4) quiver theory.
Each N = (4; 4) vector multiplet splits into one N = (2; 2) vector multiplet, two chiral
multiplets and one fermi multiplet. Each N = (4; 4) hypermultiplet splits into two chiral
and two fermi multiplets. This is precisely the decomposition seen here.
B
Fractional D3branes on a local P
2
Consider the wellknown case of fractional D3branes on the CalabiYau threefold:
Xe3 = Tot(O( 3) ! P2) ;
(B.1)
which is a crepant resolution of the orbifold singularity X3 = C3=Z3. The corresponding 4d
N = 1 quiver gauge theory is very well studied  see e.g. [2, 3, 58, 69]. In this appendix,
we review this 4d N = 1 quiver using the Bbrane language. This will help to illustrate,
in a more familiar context, the tools that we similarly use to study D1brane quivers.
B.1
Fractional branes and supersymmetric quivers
Let us discuss two particular sets of fractional branes. Below, we will see how they are
related by mutation of exceptional collections, providing a geometric realization of Seiberg
duality [58].
rst set of fractional branes: theory (I)
Fractional branes on the resolution (B.1) can be constructed from the data of a strongly
exceptional collection on P2, as in section 3.3. Let us rst consider the exceptional collection:
EI = f
2(2) ;
The corresponding three fractional branes on Xe3 are:
E0 = i O ;
E1 = i
E2 = i
(B.2)
(B.3)
where i is the inclusion from P2 into Xe3.
Let z0; z1; z2 be the homogeneous coordinates of P2 and Ui be the open set in which
zi 6= 0. Denote the local coordinates in Ui by (xi; yi) and the coordinate of the
O( 3) in Ui by wi. We have w1 = x30w0; w2 = y03w0 = y13w1. In the following we will take
ber of
Koszul resolutions:
0 ! O(k + 3) w0
(k + 3) w0
! O(k) ! i OP2 (k) ! 0
(k) ! i
P2 (k) ! 0 :
It is straightforward to compute the Ext groups themselves. The Ext1 quiver reads:
Ext1(i O( 1)[2]; i (1)[1]) is generated by ci 2 C0(Hom1(i O( 1)[2]; i (1)[1])):
A basis of the Ext groups can be chosen as follows:
Ext1(i O; i O( 1)[2]) is generated by ai 2 C2(Hom 1(i O; i O( 1)[2])):
O(2)
a1 =
1
x0y0
! O( 1)
1
x20y0
;
a3 =
1
x0y02
:
Ext1(i (1)[1]; i O) is generated by bi 2 C0(Hom1(i (1)[1]; i O)):
i O o
a
3
9
i (1)[1]
3
c
! O
(1)
O
(1)
b1 : (x0;y0)?y
( x0; y0)?y
b2 : ( 1;0)??y
c1 :
(4)
! O( 1)
y0 ??
x0 y
(1)
c2 :
b3 : (0; 1)?
c3 :
(4)
?
y
O(3)
O(2)
1 ??
0 y
(4)
(1)
O
! O( 1)
10 ??y
(1)
(a) C3=Z3 quiver.
(b) A Seiberg dual quiver.
multiplicity 3, while the arrow M has multiplicity 6.
The generator of Ext3(i O; i O) can be chosen to be t 2 C2(X; Hom1(i O; i O)) with
One can then compute:
tjU0 =
1
x0y0
:
(m2(ai; m2(bj ; ck))) = ijk :
Note that there is a GL(3) symmetry inherited from P2, and a corresponding SU(3) avor
symmetry in the N = 1 gauge theory.
The N = 1 quiver gauge theory is the one shown in
gure 11a, with a gauge group
U(N )
U(N )
U(N ). The bifundamental chiral multiplets Ai; Bi; Ci correspond to the
Ext1 elements ai; bi; ci, and the product structure (B.4) leads to the N = 1 superpotential:
W = Tr( ijkAiBj Ck) :
This quiver can also be obtained by orbifold projection from 4d N = 4 theory [2, 3].
B.1.2
A second set of fractional branes: theory (II)
Consider another strongly exceptional collection on P2:
EII = fO( 1) ; O ; O(1)g :
The corresponding fractional branes are:
E0 = i O[1] ;
E1 = i O(1) ;
E2 = i O( 1)[2] :
We repeat the same analysis as before. The Ext1 quiver reads:
(B.4)
(B.5)
(B.6)
(B.7)
The corresponding N = 1 quiver is shown in gure 11b.
x
a0
3
f
3
b0
6
d0
/ i O(1)
b01 :
O(2)
O(3)
O(3)
O(4)
Ext1(i O[1]; i O(1)) is generated by b0i 2 C0(Hom1(i O[1]; i O(1))):
1?
y
O
O
! O(1)
a02 :
b02 :
O(2)
d01 =
d04 =
1
1
x0y0
x20y02
O(2)
O(3)
O(3)
! O( 1)
A0i = ai ;
B0i = bi ;
O
O
! O(1)
! O(1)
a03 :
b03 :
1
1
;
d03 =
d06 =
1
1
x20y0
x0y03
;
:
0d01 d03 d021
Mij = BBBd03 d05 d04CC :
C
Ext1(i O(1); i O( 1)[2]) is generated by d0n 2 C2(Hom 1(i O(1); i O( 1)[2])):
The eld theory is shown in gure 11b. The elds A0i, B0i are both in the 3 of the SU(3)
avor symmetry, while the elds Mij = Mji span the 6 of SU(3). They are identi ed with
the Ext1 elements according to:
O(2)
O(3)
O(3)
O(4)
O
O
y
! O(1)
(B.8)
(B.9)
One can then derive the superpotential:
be U(2N )
duality.
B.2
Seiberg duality as mutation
W = A0iMij A0j :
Moreover, due to the nonabelian anomalycancellation condition, the gauge group must
U(N )
U(N ). This is also what is obtained from the usual rules of Seiberg
The two N = 1 quiver theories of gure 11 are related by a Seiberg duality on node e0.
Consider for instance the \Theory (I)". A Seiberg duality at node e0 reverses the arrows
Ai and Bj while generating the new mesons Mfij , with the identi cation Mfij = AiBj across
the duality. The superpotential dual to (B.5) reads:
W = ijkMfij Ck + A0iMfij B0j :
(B.10)
This contains a mass term for Ci and the antisymmetric part of Mfij . Integrating those
elds out, we are left with \Theory (II)", including the superpotential (B.9).
Similarly, if we start from Theory (II) and perform a Seiberg duality at node e0, we ip
the arrows A0i, B0j , and generate the dual mesons N ij = A0iB0j , with the superpotential:
W = Mij N ji + BiN ij Aj :
(B.11)
Integrating out the massive elds  Mij and the symmetric part of N ij  we recover
Theory (I) and (B.5), with the identi cation N ij =
ijkCk.
Mutation of exceptional collection. It was proposed in [58] that Seiberg duality could
be realized as mutation on exceptional collections of sheaves. Start with Theory (II) and
the corresponding exceptional collection EII (B.6). Using the left mutation:
LOO(1) =
1(1)
(B.12)
(B.13)
(C.1)
(C.2)
(C.3)
(C.4)
on P2, we see that a left mutation of the collection EII at the second sheaf precisely gives
the collection EI in (B.2):
fO( 1) ; O ; O(1)g
fO( 1) ;
Therefore, the Seiberg duality at node e0 of Theory (I) is indeed realized by a mutation
of the underlying sheaves. This observation has been generalized to a number of other
cases [8].
C
A1 structure and N
= (0; 2) quiver
In this appendix, we discuss the A1 structure of the Ext algebra, and how it is related to
the structure of the N = (0; 2) quiver. This discussion is a straightforward generalization of
a similar discussion for 4d N = 1 quivers by Aspinwall and Katz [10]. See also [51, 68, 70].
C.1
An algebraic preliminary
Let V be a graded vector space, and let T (V ) be the associated graded tensor algebra:
Let d be an derivative operator of degree 1 acting on T (V ), satisfying the graded
Leibniz rule:
d(A
B) = dA
B + ( 1)jAjA
dB ;
with A; B 2 T (V ), and jAj denoting the degree of A. We also require that:
Using the Leibniz rule, the action of d on T (V ) is determined by its action on V itself. Let
us decompose d as:
d V = d1 + d2 +
with
dk : V ! V k
:
T (V ) =
1
M V n
:
n=1
d2 = 0 :
denote the corresponding map of degree
A as being the dual of V [1]:
together with the multiproducts:
Let V [1] denote the vector space V with all degrees decreased by one, and let s : V ! V [1]
1. Given this data, we can de ne an A1 algebra
given by the dual of the map s k dk s 1 : V [1] ! V [1] k. The nilpotency condition (C.3)
is equivalent to the following A1 relation on the multiproducts:
X
( 1)r+stmn+1 s(1 r
ms
1 t) = 0 ;
8n > 0 ;
where the sum is over all r; t
0, s > 0, such that r + s + t = n [51].
(C.5)
(C.6)
(C.7)
(C.8)
(C.9)
(C.10)
(C.11)
(C.12)
Ext algebra and N = (0; 2) quiver
In our physical setup, the vector space A is spanned by the various Exti groups (i =
; 4) among the fractional branes on a CY4 singularity. Schematically:
A = Ext0
Ext1
Ext2
Ext3
Ext4 :
The grading of A is given by the degree i of Exti. Any z 2 A of degree q is associated
to a local vortex operator in the Bmodel, with the degree identi ed with the ghost
number. Given z 2 A, let z(1) denote the corresponding oneform descendant. The oneform
operators can be used to deform the Bmodel according to [10, 70]:
S ! S + X Zi zi(1) :
i
The coupling Zi is identi ed with a \quiver eld" in the spacetime (D1brane) theory.
Note that Zi has degree 1
qi if zi has degree qi. The quiver elds are elements of the
vector space V , in the notation of subsection C.1.
Let us denote by z 2 A the Serre dual of z 2 A, with the Ext algebra A given by (C.8).
Let us then choose a basis of A according to:
fzig = fe0 ; x ; I ; I ; x ; e0g ;
C.2
0;
with:
e0 2 Ext0 ;
x
2 Ext1 ;
I ; I 2 Ext2 ;
x
2 Ext3 ;
e0 2 Ext4 :
As discussed in the main text, the choice of basis for Ext2 is arbitrary. Any given choice
introduces a distinction between the elements
and the Serre dual elements , which is a
matter of convention.
The dual vector space V spans the \quiver elds". We choose a basis of V :
fZig = fe ; X ; I ; I ; X ; eg ;
dual to (C.10). The element e correspond to the vector multiplets, while X
and
correspond to the chiral and fermi multiplets, respectively. Note the degrees:
degree:
e
1
X
0
I
1
X
In particular, the chiral multiplets have degree 0. Given this explicit basis of V , we de ne
a derivative d as follows. First, let us introduce the N = (0; 2) \superpotential":
with JI (X) and EI (X) some arbitrary functions of the chiral multiplets X . This W is an
arbitrary gaugeinvariant function of degree
1 that is independent of e, except that we
need to impose the constraint:
Let us also de ne the derivatives:
Tr(EI
J I ) = 0 :
by left derivation on W  that is, we use the cyclic property of the trace to write (C.13)
with X
is de ned as the sum of all
possible forms of W
de ne the degreeone derivative d on V as:24
with X
in front, with X
removed. Given the superpotential, we
d2X
= 0 ;
W = Tr
I
J I (X) +
EI (X) ;
(C.13)
(C.14)
(C.15)
(C.16)
(C.17)
(C.18)
de =
e
dX
= X
e ;
e
d I = EI (X)
d I = J I (X)
dX
de = X
e
X
e
I
I
X
X
X ;
e
e
e
e
+ X
I ;
I
e ;
I
X
I + I
I
e
e
e
e :
By direct computation, one can check that d2 = 0. The relations:
d2e = 0 ;
d2X
= 0 ;
d
2 I = 0 ;
d
2 I = 0
are obvious.25 The key relation is:
which holds true if and only if the nontrivial constraint (C.14) is satis ed. This is nothing
but the requirement that the N = (0; 2) superpotential be properly supersymmetric. Since
we explicitly displayed a nilpotent derivative d on the vector space V spanned by the quiver
24This is the analogue of equations (30) and (39) of [10].
25To check the last two relations, one uses that:
dF (X) = F (X) e e F (X) ;
for any degreezero holomorphic function F (X), which follows from the second line in (C.16).
elds, it follows from the general discussion above that the multiproducts mk acting on
the Ext vector space A satisfy the A1 relations (C.7). In this way, we see clearly that the
A1 relations on a CY4 are intimately related to the supersymmetry constraint (C.14).
We should also note that the di erential d de ned in (C.16) has:
d1 = 0 ;
(C.19)
mk = 0 for k
given by m2.
where dk is de ned as in (C.4), if and only if the potentials EI and J I do not contain any
linear terms in X . In such a case, we have m1 = 0 in the dual Ext algebra, which gives us
a minimal A1 structure. Linear terms in EI or J I are mass terms, and the corresponding
elds can always be integrated out, as discussed in examples in section 3. Therefore, (C.19)
always holds for the lowenergy quiver.
Similarly, we see from (C.4) that there exists nonzero higher products mk for k =
2;
; kmax, with kmax the highest order in the elds X
that appear in the potentials EI ,
J I . In the simplest case when EI , J I are all quadratic in the chiral multiplets, we have
3, and the A1 algebra reduces to an associative algebra with a product
C.3
General procedure to compute the higher products
Let us discuss in more detail the procedure to compute the higher products of the Ext
A1 algebra [10], which we outlined in section 2.1.2. Consider an A1 algebra Ae and the
A1 map:
Let the rst map:
f1 = i : H (Ae) ! Ae ;
(C.20)
(C.21)
reads:
be the inclusion map de ned by picking representatives of cohomology classes, and let
d = me1 : Ae ! Ae denote the di erential on Ae. The rst A1 constraint on the maps fk
i
m2( ; ) = i( ) i( ) + df2( ; ) :
We can compute i( ) i( ), and use the result to de ne m2( ; ) and f2( ; ). The next
A1 constraint is of the form:
i
m3( ; ; ) =
f2( ; m2( ; ))
f2(m2( ; ); ) + i( ) f2( ; )
f2( ; ) i( ) + df3( ; ; ):
Using the previouslycomputed m2 and f2, this expression allows us to compute m3 and
f3. Proceeding inductively in this fashion, one can construct mk and fk to any order k.
D
Higher products on a local P
1
P
1
In this appendix, we spell out the computation of the higher products on the local P1
P
1
geometry of section 2.5, using the procedure summarized in appendix C.3.
HJEP02(18)5
sition of the chain maps, one nds the products:
m2(b1; d1) = 10 ;
m2(a1; e1) = 10 ;
m2(b1; d2) = 20 ;
m2(a2; e1) = 20 ;
m2(b2; d1) = 30 ;
m2(a1; e2) = 30 ;
m2(b2; d2) = 40 ;
m2(a2; e2) = 40 :
It follows that f2(b; d) = f2(a; e) = 0. De ne the 1cochains
and as follows:
( )01 = ( )02 = ( )03 = ( )12 = ( )13 = 0 ; ( )23 = x 1u 1 ;
One can compute
where the chain maps are de ned by
d1 c1 = d 1 ;
d2 c2 = d 1 ;
d1 c2 = d 2 ;
d2 c2 = d 2 ;
1 =
1 =
0
0
) ;
) ;
2 =
2 =
0 !
0 !
( ; 0) ;
( ; 0) ;
f2(d1; c1) =
f2(d2; c1) =
1 ; f2(d1; c2) =
1 ; f2(d2; c2) =
c1 a1 = d 1 ; c2 a1 = d 2 ;
c1 a2 = d 1 ; c2 a2 = d 2 :
f2(c1; a1) =
f2(c1; a2) =
1 ; f2(c2; a1) =
1 ; f2(c2; a2) =
2 ;
2 :
2 ;
2 :
m3(d1; c1; a1) = 0 ;
m3(d1; c2; a1) = 0 ;
m3(d2; c1; a1) =
m3(d2; c2; a1) =
1 ;
2 ;
m3(d1; c1; a2) = 1 ;
m3(d1; c2; a2) = 2 ;
m3(d2; c1; a2) = 0 ;
m3(d2; c2; a2) = 0 :
between the corresponding complexes. This implies m2(d; c) = 0 and:
Thus, m2(c; a) = 0 and:
Plugging these results into the A1 map constraint:
im3(d; c; a) = f2(d; m2(c; a)) f2(m2(d; c); a) + d f2(c; a) f2(d; c) a + df3(d; c; a) ;
(D.2)
(D.3)
Plugging these results into:
f2(e1; c1) = 1 ; f2(e1; c2) = 2 ;
f2(e2; c1) = 1 ; f2(e2; c2) = 2 ;
f2(c1; b1) =
f2(c1; b2) =
1 ; f2(c2; b1) =
1 ; f2(c2; b2) =
2 ;
2 ;
1 =
1 =
0
0
2 =
2 =
0
0
( ; 0) ;
( ; 0) :
m3(e1; c1; b1) = 0 ;
m3(e1; c2; b1) = 0 ;
m3(e2; c1; b1) = 10 ;
m3(e2; c2; b1) = 20 ;
m3(e1; c1; b2) =
m3(e1; c2; b2) =
m3(e2; c1; b2) = 0 ;
m3(e2; c2; b2) = 0 :
0 ;
1
0 ;
2
Similarly, if we de ne
and
we get m2(e; c) = 0; m2(c; b) = 0 and
( )01 = ( )03 = ( )13 = ( )23 = 0 ; ( )02 = ( )12 = x 1u 1 ;
( )01 = ( )02 = ( )12 = ( )23 = 0 ; ( )03 = ( )13 =
x 1 ;
(D.4)
(D.5)
im3(e; c; b) = f2(e; m2(c; b)) f2(m2(e; c); b) + e f2(c; b) f2(e; c) b + df3(e; c; b) ;
This completes the computation of the threeproduct m3. As a consistency check, one can
verify that the m2 and m3 just computed satisfy the relevant A1 relations.
By the same procedure, we could also nd the m4 product, while the higher products
vanish. We can use various shortcuts to the correct answer, however. For instance, we can
impose Tr(EJ ) = 0 in the gauge theory quiver, which is equivalent to imposing the A1
relations. This leads to the result (2.87) for the m4 products amongst the Ext1 elements.
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[INSPIRE].
[1] M.R. Douglas and G.W. Moore, Dbranes, quivers and ALE instantons, hepth/9603167
[2] M.R. Douglas, B.R. Greene and D.R. Morrison, Orbifold resolution by Dbranes, Nucl. Phys.
B 506 (1997) 84 [hepth/9704151] [INSPIRE].
[3] S. Kachru and E. Silverstein, 4D conformal theories and strings on orbifolds, Phys. Rev.
Lett. 80 (1998) 4855 [hepth/9802183] [INSPIRE].
[4] I.R. Klebanov and E. Witten, Superconformal eld theory on threebranes at a CalabiYau
[5] B. Feng, A. Hanany and Y.H. He, Dbrane gauge theories from toric singularities and toric
duality, Nucl. Phys. B 595 (2001) 165 [hepth/0003085] [INSPIRE].
[6] M. Wijnholt, Large volume perspective on branes at singularities, Adv. Theor. Math. Phys. 7
(2003) 1117 [hepth/0212021] [INSPIRE].
[7] C.P. Herzog, Exceptional collections and del Pezzo gauge theories, JHEP 04 (2004) 069
[8] C.P. Herzog, Seiberg duality is an exceptional mutation, JHEP 08 (2004) 064
[9] P.S. Aspinwall and I.V. Melnikov, Dbranes on vanishing del Pezzo surfaces, JHEP 12
[hepth/0310262] [INSPIRE].
[hepth/0405118] [INSPIRE].
(2004) 042 [hepth/0405134] [INSPIRE].
Phys. 264 (2006) 227 [hepth/0412209] [INSPIRE].
[INSPIRE].
[10] P.S. Aspinwall and S.H. Katz, Computation of superpotentials for Dbranes, Commun. Math.
[11] A. Hanany and K.D. Kennaway, Dimer models and toric diagrams, hepth/0503149
[12] S. Franco, A. Hanany, K.D. Kennaway, D. Vegh and B. Wecht, Brane dimers and quiver
gauge theories, JHEP 01 (2006) 096 [hepth/0504110] [INSPIRE].
[13] 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 [hepth/0505211] [INSPIRE].
[14] 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 [hepth/0511287] [INSPIRE].
[15] C.P. Herzog and R.L. Karp, On the geometry of quiver gauge theories (Stacking exceptional
collections), Adv. Theor. Math. Phys. 13 (2009) 599 [hepth/0605177] [INSPIRE].
[16] D.R. Morrison and M.R. Plesser, Nonspherical horizons, I, Adv. Theor. Math. Phys. 3 (1999)
1 [hepth/9810201] [INSPIRE].
(1998) 161 [hepth/9707012] [INSPIRE].
[17] K. Mohri, Dbranes and quotient singularities of CalabiYau fourfolds, Nucl. Phys. B 521
(2017) 106 [arXiv:1609.01723] [INSPIRE].
[INSPIRE].
[18] H. GarciaCompean and A.M. Uranga, Brane box realization of chiral gauge theories in
twodimensions, Nucl. Phys. B 539 (1999) 329 [hepth/9806177] [INSPIRE].
[19] S. Franco, D. Ghim, S. Lee, R.K. Seong and D. Yokoyama, 2d (0; 2) Quiver Gauge Theories
and Dbranes, JHEP 09 (2015) 072 [arXiv:1506.03818] [INSPIRE].
[20] S. Franco, S. Lee and R.K. Seong, Brane Brick Models, Toric CalabiYau 4Folds and 2d
(0; 2) Quivers, JHEP 02 (2016) 047 [arXiv:1510.01744] [INSPIRE].
[21] S. Franco, S. Lee, R.K. Seong and C. Vafa, Brane Brick Models in the Mirror, JHEP 02
[22] S. Franco and G. Musiker, Higher Cluster Categories and QFT Dualities, arXiv:1711.01270
[23] A. Gadde, S. Gukov and P. Putrov, (0; 2) trialities, JHEP 03 (2014) 076 [arXiv:1310.0818]
arXiv:1404.5314 [INSPIRE].
[24] A. Gadde, S. Gukov and P. Putrov, Exact Solutions of 2d Supersymmetric Gauge Theories,
[25] A. Gadde, Holomorphy, triality and nonperturbative function in 2D supersymmetric QCD,
Phys. Rev. D 94 (2016) 025024 [arXiv:1506.07307] [INSPIRE].
[26] S. Franco, S. Lee and R.K. Seong, Brane brick models and 2d (0; 2) triality, JHEP 05
(2016) 020 [arXiv:1602.01834] [INSPIRE].
[arXiv:1601.02015] [INSPIRE].
[27] S. SchaferNameki and T. Weigand, Ftheory and 2d (0; 2) theories, JHEP 05 (2016) 059
GLSMs, Phys. Rev. D 96 (2017) 066015 [arXiv:1610.00718] [INSPIRE].
[29] C. Lawrie, S. SchaferNameki and T. Weigand, Chiral 2d theories from N = 4 SYM with
varying coupling, JHEP 04 (2017) 111 [arXiv:1612.05640] [INSPIRE].
[30] C. Lawrie, S. SchaferNameki and T. Weigand, The gravitational sector of 2d (0; 2) Ftheory
vacua, JHEP 05 (2017) 103 [arXiv:1612.06393] [INSPIRE].
[31] S.H. Katz and E. Sharpe, Notes on certain (0; 2) correlation functions, Commun. Math.
Phys. 262 (2006) 611 [hepth/0406226] [INSPIRE].
[32] E. Witten, Twodimensional models with (0; 2) supersymmetry: Perturbative aspects, Adv.
Theor. Math. Phys. 11 (2007) 1 [hepth/0504078] [INSPIRE].
[33] A. Adams, J. Distler and M. Ernebjerg, Topological heterotic rings, Adv. Theor. Math. Phys.
10 (2006) 657 [hepth/0506263] [INSPIRE].
[34] M.C. Tan, Twodimensional twisted models and the theory of chiral di erential operators,
Adv. Theor. Math. Phys. 10 (2006) 759 [hepth/0604179] [INSPIRE].
[35] J. McOrist and I.V. Melnikov, HalfTwisted Correlators from the Coulomb Branch, JHEP 04
(2008) 071 [arXiv:0712.3272] [INSPIRE].
[36] M.C. Tan and J. Yagi, Chiral Algebras of (0; 2) models: Beyond Perturbation Theory,
Lett. Math. Phys. 84 (2008) 257 [arXiv:0801.4782] [INSPIRE].
[37] R. Donagi, J. Gu n, S. Katz and E. Sharpe, A Mathematical Theory of Quantum Sheaf
Cohomology, Asian J. Math. 18 (2014) 387 [arXiv:1110.3751] [INSPIRE].
[38] R. Donagi, J. Gu n, S. Katz and E. Sharpe, Physical aspects of quantum sheaf cohomology
for deformations of tangent bundles of toric varieties, Adv. Theor. Math. Phys. 17 (2013)
1255 [arXiv:1110.3752] [INSPIRE].
[39] M. Dedushenko, Chiral algebras in LandauGinzburg models, arXiv:1511.04372 [INSPIRE].
[40] C. Closset, W. Gu, B. Jia and E. Sharpe, Localization of twisted N = (0; 2) gauged linear
models in two dimensions, JHEP 03 (2016) 070 [arXiv:1512.08058] [INSPIRE].
[41] J. Guo, Z. Lu and E. Sharpe, Quantum sheaf cohomology on Grassmannians, Commun.
Math. Phys. 352 (2017) 135 [arXiv:1512.08586] [INSPIRE].
[42] W. Gu and E. Sharpe, A proposal for (0; 2) mirrors of toric varieties, JHEP 11 (2017) 112
[arXiv:1707.05274] [INSPIRE].
2818 [hepth/0011017] [INSPIRE].
(2003) 263 [hepth/0212218] [INSPIRE].
[43] E.R. Sharpe, Dbranes, derived categories and Grothendieck groups, Nucl. Phys. B 561
[45] S.H. Katz and E. Sharpe, Dbranes, open string vertex operators and Ext groups, Adv. Theor.
HJEP02(18)5
[48] P.S. Aspinwall, Dbranes on CalabiYau manifolds, in porceedings of the Theoretical
Advanced Study Institute in Elementary Particle Physics (TASI 2003): Recent Trends in
pp. 1{152 [hepth/0403166] [INSPIRE].
[49] S. Franco, S. Lee, R.K. Seong and C. Vafa, Quadrality for Supersymmetric Matrix Models,
JHEP 07 (2017) 053 [arXiv:1612.06859] [INSPIRE].
[hepth/9301042] [INSPIRE].
[50] E. Witten, Phases of N = 2 theories in twodimensions, Nucl. Phys. B 403 (1993) 159
[51] B. Keller, Introduction to Ain nity algebras and modules, math.RA/9910179.
[52] P.S. Aspinwall and L.M. Fidkowski, Superpotentials for quiver gauge theories, JHEP 10
(2006) 047 [hepth/0506041] [INSPIRE].
(2009) 33 [hepth/0605005] [INSPIRE].
[54] D. Berenstein and M.R. Douglas, Seiberg duality for quiver gauge theories, hepth/0207027
[55] L.E. Iban~ez, R. Rabadan and A.M. Uranga, Anomalous U(1)'s in typeI and type IIB D = 4,
N = 1 string vacua, Nucl. Phys. B 542 (1999) 112 [hepth/9808139] [INSPIRE].
[56] M. Herbst, C.I. Lazaroiu and W. Lerche, Superpotentials, A1 relations and WDVV
equations for open topological strings, JHEP 02 (2005) 071 [hepth/0402110] [INSPIRE].
[57] T.V. Kadeishvili, The algebraic structure in the homology of an A1 algebra, Sobshch. Akad.
Nauk. Gruzin. SSR 108 (1982) 249.
[58] F. Cachazo, B. Fiol, K.A. Intriligator, S. Katz and C. Vafa, A Geometric uni cation of
dualities, Nucl. Phys. B 628 (2002) 3 [hepth/0110028] [INSPIRE].
[59] A. Hanany, C.P. Herzog and D. Vegh, Brane tilings and exceptional collections, JHEP 07
(2006) 001 [hepth/0602041] [INSPIRE].
[60] R. Bott, Homogeneous vector bundles, Annals Math. 66 (1957) 203.
[61] B. Kostant, Lie algebra cohomology and the generalized BorelWeil theorem, Annals Math.
74 (1961) 329.
[62] J. Guo, B. Jia and E. Sharpe, Chiral operators in twodimensional (0; 2) theories and a test
of triality, JHEP 06 (2015) 201 [arXiv:1501.00987] [INSPIRE].
JHEP 10 (2013) 121 [arXiv:1104.2853] [INSPIRE].
[63] K. Hori, Duality In TwoDimensional (2; 2) Supersymmetric NonAbelian Gauge Theories,
Series, Cambridge University Press, Cambridge U.K. (1990).
hepth/0006224 [INSPIRE].
HJEP02(18)5
CalabiYau, JHEP 09 (2005) 057 [hepth/0003263] [INSPIRE].
JHEP 08 (2000) 015 [hepth/9906200] [INSPIRE].
singularity , Nucl. Phys. B 536 ( 1998 ) 199 [ hep th/9807080] [INSPIRE].
[44] M.R. Douglas , Dbranes, categories and N = 1 supersymmetry , J. Math. Phys. 42 ( 2001 ) [46] S.H. Katz , T. Pantev and E. Sharpe , D branes, orbifolds and Ext groups, Nucl. Phys. B 673 [47] E. Sharpe, Lectures on Dbranes and sheaves , hepth/0307245 [INSPIRE].
[64] B. Jia , E. Sharpe and R. Wu , Notes on nonabelian (0; 2) theories and dualities , JHEP 08 [66] N. Ishibashi , H. Kawai , Y. Kitazawa and A. Tsuchiya , A LargeN reduced model as superstring , Nucl. Phys. B 498 ( 1997 ) 467 [ hep th/9612115] [INSPIRE]. [67] D.E. Diaconescu and M.R. Douglas , D branes on stringy CalabiYau manifolds , [68] M.R. Douglas , S. Govindarajan , T. Jayaraman and A. Tomasiello , Dbranes on CalabiYau [70] I. Brunner , M.R. Douglas , A.E. Lawrence and C. Romelsberger , D branes on the quintic,