Geometric engineering on flops of length two
HJE
Geometric engineering on ops of length two
Andres Collinucci 1 2 4 7 8 9 10 11
Marco Fazzi 1 2 4 5 6 8 9 10 11
Roberto Valandro 0 1 2 3 4 8 9 10 11
0 The Abdus Salam International Centre for Theoretical Physics
1 32000 Haifa , Israel
2 Campus Plaine C. P. 231, B1050 Bruxelles , Belgium
3 Dipartimento di Fisica, Universita di Trieste
4 Universite Libre de Bruxelles and International Solvay Institutes
5 Department of Physics , Technion
6 Department of Mathematics and Haifa Research Center for Theoretical Physics and Astrophysics
7 Service de Physique Theorique et Mathematique
8 op of length one. In this paper , we
9 Strada Costiera 11 , I34151 Trieste , Italy
10 Via Valerio 2, I34127 Trieste , Italy
11 University of Haifa , 31905 Haifa , Israel
Type IIA on the conifold is a prototype example for engineering QED with one charged hypermultiplet. The geometry admits a study the next generation of geometric engineering on singular geometries, namely length two such as Laufer's example, which we a ectionately think of as the conifold 2.0. Type IIA on the latter geometry gives QED with highercharge states. In type IIB, even a single D3probe gives rise to a nonabelian quiver gauge theory. We study this class of geometries explicitly by leveraging their quiver description, showing how to parametrize the exceptional curve, how to see the divisors intersecting the curve. With a view towards Ftheory applications, we show how these divisors contribute to the enhancement of the MordellWeil group of the local elliptic bration de ned by Laufer's example.
Dbranes; Di erential and Algebraic Geometry; FTheory; Brane Dynamics

1 Introduction
2
3
4
6
7
2.1
2.2
3.1
3.2
4.1
4.2
4.3
5.1
5.2
5.3
5
Weil divisors
Divisors from the quiver
Divisors as avor branes
Singular description
Highercharge states
Conclusions A
MorrisonPinkham's example
Warmup: the conifold threefold
The singularity and its matrix factorization
Noncommutative crepant resolution and the quiver
Laufer's example
The threefold op Matrix factorization and singular divisors
Noncommutative crepant resolution
Laufer as 4d N = 1 quiver gauge theory
Finding the exceptional curve
Following the op transition
1
Introduction
The study of D3branes at threefold singularities is by now a venerable subject, dating back
at least to [1{3]. The worldvolume theory can be inferred to be a fourdimensional N = 1
quiver gauge theory describing the fractional branes that probe the singularity of a
CalabiYau threefold, on which one has de ned a type IIB background. Powerful mathematics has
been developed to extract a quiver with superpotential describing the eld theory directly
from the singularity. These techniques are greatly simpli ed in favorable situations, namely
when the singularity is an orbifold or toric [4{6]. The quintessential example of the latter
class is the conifold C,
P
x
2
y
2
z t = 0
C4 ;
about which a great deal is known both in eld theory [7, 8] and in singularity theory [9].
{ 1 {
spective, namely the geometric engineering paradigm [10, 11]: one de nes type IIA on the
singularity, and considers the e ective eld theory arising from the supergravity zero modes
on this space, supplemented by the light degrees of freedom created by D2branes wrapping
the vanishing cycles. The case of the conifold produces a fourdimensional N = 2 SQED
with one charged hypermultiplet. The U(1) gauge group comes from the reduction of the
type IIA RamondRamond C3 form,1 and the charged hyper from a D2 and an antiD2
wrapped on the exceptional sphere.
However, when the singularity does not belong to either class mentioned above, it is
in general more di cult to \read o " the eld theory from it. One has to resort to the
where the summands are a specially selected subset of MCM modules over R.3 (A can
indeed be understood as a noncommutative enhancement of R = EndR R.)
1Although throughout the paper we will work on noncompact threefolds, one always expects that there
be a normalizable harmonic twoform on which to reduce C3. Alternatively, one can consider the singularity
as a patch of a compact threefold.
2We also require that the singularity be isolated, and that its coordinate ring R be Gorenstein. In physics
language, the latter condition guarantees that the resolved space of the singularity SpecR be a CalabiYau
threefold, which is necessary to preserve supersymmetry in four dimensions.
3In the conifold, for instance, there are two such possible modules, and only one of them is chosen.
{ 2 {
The advantage of the MF description is that, once appriopriate MF's of the
hypersurface are known, the task of computing a quiver with relations (i.e. the cyclic derivatives
of the superpotential, or Fterms) is completely algorithmic. This approach has been put
forward in [19], where many examples of singularities (wellknown in the mathematical
literature) were used as string theory backgrounds and tackled from the NCCR/MF point of
view. The authors of that paper focused in particular on socalled threedimensional simple
ops. These are singular algebraic varieties that admit two crepant resolutions which are
birationally isomorphic. The conifold, which is moreover toric, is the simplest example of
threefold op, and can be labeled by an integer ` = 1 known as length. For our purposes
it is enough to characterize this integer as half the size of the matrices in an MF of the
singularity.4 Indeed the conifold hypersurface equation admits a 2`
singularities are quite generic and moreover higherlength
ops make up a much richer
class of threefolds [22]. Thus, providing examples of D3brane probe theories on such
singularities appears as a very interesting challenge in its own right. In this paper, we will
focus on a simple op of length ` = 2 known as Laufer's example [23]. It is de ned as the
following hypersurface in C4:
x2 + y3 + wz2 + w2n+1y = 0 ;
n 2 N>0 :
We will study it thoroughly from the D3probe perspective, i.e. by exploiting the NCCR
technique, which will give us a quiver description of the threefold. However, we will also
analyze the eld theory that arises from the type IIA geometric engineering point of view.
The novelty will be that, in contrast to more familiar singularities with exceptional cycles
with normal bundle OP1 ( 1)
is characterized by OP1 ( 3)
OP1 (1). In
OP1 ( 1) (such as the conifold), this class of singularities
eld theory, we will see that this translates to
having hypers of charge one and also of charge two.
The goal of this paper is twofold: rstly, it is meant to be an exposition of the powerful
tools from NCCR's to study nontoric, nonorbifold singularities. We will see that we can
not only derive the appropriate quiver with relations for the aforementioned lengthtwo op,
but we can also recover the resolved threefold as the moduli space of stable representations
of the quiver. The peculiarity of this type of threefold is that the associated quiver gauge
theory is nonabelian even for a single probe brane.
Secondly, this will be a rst example where a op of length two is described
continuously. In [19] the
op transition could only be seen as a Z2 symmetry that exchanges
two MF's of the singular hypersurface, akin to the case of the conifold which admits two
(small) resolutions via twobytwo MF's. However, the question as to how one can see this
from the Kahler geometry perspective remained unanswered. How should one describe the
exceptional P1 of the resolution? Can we track it all the way to zero volume, and then see
how the opped curve starts to grow in the other Kahler cone? We will accomplish this
task explicitly.
4See [
20, 21
] for a formal de nition.
{ 3 {
Finally, by relying on the NCCR/MF techniques we will also study Weil divisors of the
singular geometry, by regarding them as avor nodes in the quiver, in the spirit of [24, 25].
This is of particular interest in the context of Ftheory or geometric engineering in
IIA/Mtheory, where such divisors induce extra U(1) gauge symmetries.
This paper is organized as follows. In section 2 we introduce the NCCR and MF
techniques and apply them to the wellstudied conifold singularity, i.e. the simplest example
of lengthone op. In section 3 we introduce our main casestudy, i.e. Laufer's lengthtwo
op, whose NCCR we present in section 4. (In appendix A we present yet another example
of a lengthtwo
op.) In section 5 we study classes of Weil divisors de ned on Laufer's
singularity, and we also provide an alternative perspective on them leveraging the
M/Ftheory duality. In section 6 we show that, in the geometric engineering context, these
geometries admit highercharge hypers. We brie y present our conclusions in section 7.
2
Warmup: the conifold threefold
To set the stage, we shall rst study the wellknown conifold singularity by relying on the
powerful NCCR techniques, which we will introduce as we go along in the presentation.
Given the general familiarity with the conifold example, the use of the NCCR might seem
like an overkill. However, it is instructive to revisit this old example in the less familiar
NCCR language, as a warmup for our main casestudy, Laufer's geometry.
2.1
The singularity and its matrix factorization
Consider the wellknown CalabiYau (CY henceforth) threefold C de ned by the
following equation:
Wconifold : x
2
y
2
tz = 0
C4 :
It has a pointlike singularity at the vanishing locus of the ideal (x; y; z; t) of the coordinate
ring R := C[x; y; z; t]=(x2
y
2
tz). A threefold will admit a small, Kahler, crepant
resolution provided there is a Weil (but nonCartier) divisor. In the conifold case there are
two independent Weil divisors, given by the (zero locus of the) following ideals:
(x + y; z) and (x
y; z) :
Each of them produces, upon blowup, a nonsingular threefold. We thus obtain two
threefolds X
related by a simple op X+ 99K X . This means the two nonsingular varieties are
birationally isomorphic away from a subvariety. In the case of simple ops, such subvariety
is an irreducible, smooth, rational curve, namely the exceptional P1 locus of the resolutions.
As explained in the introduction, given a hypersurface with de ning equation P =
0, a matrix factorization (MF) of it is a pair of square matrices ( ; ) of appropriate
dimensionality n such that
where 1 n n is the n
n identity matrix. The conifold admits two inequivalent, irreducible,
nontrivial MF's, namely ( ; ) and ( ; ) with
(2.1)
(2.2)
(2.3)
(2.4)
=
= P 1 n n ;
=
"
x
y
t x + y
z #
;
=
"
x + y
t
x
z
y
#
:
{ 4 {
in R 2, the vanishing locus is where s becomes parallel to the generators of (the rankone)
im
(i.e. the columns of ). This happens for
det
"
s1 x + y
s2
t
#
= 0 ;
det
"
s1
s2 x
z
y
#
= 0 ;
or in other words when
Hence we have a whole family jD+j of nonCartier divisors parametrized by (s1; s2). Notice
that for the special choice s1 = 0, we obtain the locus (x + y; z), that is one of the Weil
divisors mentioned in (2.2). On the other hand, the divisor (x
y; z) belongs to the family
jD j de ned by
The two pairs are ordered and inequivalent, i.e. they are not related by similarity
transformations. The two Weil divisors (2.2) are related to the two MF's, as we now explain.
First, notice that these matrices can be seen as maps from R 2 to R 2
. This allows to
construct two modules over R as follows [
17
]:
M := coker R 2
! R 2 ;
M _ := coker R 2
! R 2
phism [26]. They are rankone over the whole conifold threefold except at the singular
point, where they are ranktwo. As we will see, in the resolved space these pullback to
O(1) and O( 1), i.e they become line bundles. The associated divisors are given by the
locus where a generic section of the bundle vanishes. These loci can be detected already
in the singular space. Consider for instance the map
. The locus we are looking for is
where a given element of R 2 is inside im . Given a generic section
(2.6)
(2.7)
(2.8)
(2.9)
s =
s1!
s2
s = 0 :
s0 = 0 ;
{ 5 {
associated with the second MCM module (M _). One can check that the union of jD+j and
jD j is in the class of a Cartier divisor.
2.2
Noncommutative crepant resolution and the quiver
We will now put this knowledge of MCM modules over the conifold to use, by constructing
the NCCR of the singularity, and subsequently obtaining its (commutative) small resolution
as a quiver moduli space.
It is wellknown that one can associate a quiver with relations to the conifold
singularity. In string theory this can be seen as follows: one considers a stack of N D3branes
at the singular point. The probe theory is described by the quiver in gure 1 and comes
equipped with the (KlebanovWitten) superpotential [7]:
WKW =
The nodes of the quiver are the fractional brane gauge groups, and the arrows the chiral
multiplets charged under such groups. On the other hand, consider a crepant resolution X
of the conifold (either among X ). Then the bounded derived category of X is equivalent
to the bounded derived category of representations of the quiver described above. One can
extract the quiver with superpotential of a given threedimensional singularity by using Van
den Bergh's NCCR's [19]. These resolutions require the knowledge of the MCM modules
Mi of the singular ring R, whose category was shown be equivalent [
17
] to a particular
category of MF's of the equation de ning the singular threefold. Namely, once an MF of
the singularity equation Wconifold is known, we can construct for free its MCM modules
via (2.5).
As already mentioned, the basic idea [27] behind the NCCR is to replace the
coordinate ring R = EndR R describing the singular space with the noncommutative ring
and that it have nite global projective dimension, which essentially means that all
projective modules over A admit a
nite resolution. That is the homological counterpart of
smoothness. (We refer to [28] for a pedagogical introduction to these notions.)
Moreover the ring A, which is also an algebra over R, can be thought of as the
(noncommutative) path algebra of a quiver with relations. Hence, to each summand in A is
associated a vertex in the quiver, while the number of arrows from one vertex to another
is given by dim HomR(Mi; Mj ) (where Mi can be R too). Once a presentation of the
singularity as quiver with relations is known, one can construct a geometric resolution of the
singular threefold via the geometric invariant theory of King [29].
We now apply this procedure to the conifold singularity (2.1) as a rst casestudy. The
NCCR can be characterized by a single MCM module de ned by choosing one of the two
MF's. Let us take M = coker
in (2.5) for concreteness. The pair ( ; ) is a MF of the
singular space, i.e.
=
= (x2
y
2
tz)1 2` 2` ;
with ` = 1. As we have already mentioned elsewhere, we can de ne the integer ` to be the
length of the op, which is a numerical invariant that characterizes it [
20, 21
].
In this case the noncommutative ring is simply A = EndR(R
M ). It can be decom
posed in four pieces as follows:
A = HomR(R; R)
HomR(M; M )
HomR(R; M )
HomR(M; R)
= R
eR
= R
eM
= M
i
= M _
i
(2.11)
(2.12)
{ 6 {
where in the last line we have written down the corresponding generators. The relevant
morphisms are i and i, which satisfy the relations
1 i 2 =
2 i 1 ;
1 i 2 = 2 i 1 ; i = 1; 2 :
(2.13)
These can be derived by checking the de nition of M in terms of the MF [19]. However,
the result can be repackaged as Fterms by de ning a formal superpotential, which in this
case happens to be (through no coincidence) the KlebanovWitten one (2.10). eR and
eM are the multiplicative identities (actually idempotents) of the ring at node R and M
respectively. The quiver is depicted in
gure 2. In this language, Dbranes on the singular
space are described as complexes of right Amodules, i.e. objects in the bounded derived
category Db(modA). This makes sense because the bounded derived category of these
modules is equivalent to the bounded derived category of coherent sheaves Db(X ) on the
resolved space [
16
]. Moreover Amodules are equivalent to representations of the quiver.
In particular, by studying the moduli space of the quiver representations corresponding to
fractional D3branes, we can recover the conifold variety C.
A ( nitedimensional) quiver representation is de ned by associating a (complex)
vector space with each node of the quiver and a linear map with each arrow.5 eR and eM
are set to the identity matrix 1 . The representation corresponding to a single D3brane
is shown in
gure 3, and is characterized by d~ = (1; 1), collecting in a dimension vector
d~ the dimensions of the vector spaces at the two nodes. The D3brane splits into two
fractional branes, one on each node.
i and
i are complex scalar elds that transform
in the bifundamental (and antibifundamental) of the product gauge group U(1)
U(1).
Neglecting the decoupled diagonal U(1), we can say that these elds have charges 1 under
the relative U(1).
5We will also encounter in nitedimensional, but nitelygenerated, quiver representations. These
correspond to noncompact, or avor, branes.
{ 7 {
The moduli space is parametrized by all the possible values of the maps i; i modulo
the action of the relative U(1) gauge group. This naturally leads to the toric variety
toric description of the conifold space. When the parameter
is zero, we have the singular
conifold, while for
= 0 the space is resolved. The resolved phases correspond to
< 0. We also see that in the rst phase we will have the irrelevant ideal condition
( 1; 2) 6= (0; 0), while in the second phase we have ( 1; 2) 6= (0; 0). By irrelevant ideal
we mean the ideal associated with the excised locus, much like in the construction of P1
from C2 by excising its origin.
Stability. Let us see here how to obtain the irrelevant ideal, when we consider the
(resolved) space as the moduli space of a quiver representation. The quiver representation of
interest here has dimension vector d~ = (1; 1). In order to de ne the resolved ambient space,
we rst need to impose socalled stability conditions. The rst step requires assigning a
vector ~ to this quiver such that ~ d~ = 0, i.e. ~ = (
rescaling: ~+ := ( 1; 1) and ~ := (1; 1).
; ). This gives us two choices up to
Before we can proceed, we should de ne the notion of subrepresentations, and what
it means for a representation to be destabilized by a subrepresentation. Physically this
corresponds to our brane system decaying into a nonsupersymmetric con guration. King's
work [29] then tells us how to insure that our d~ = (1; 1) representation is stable in the
appropriate way, and guarantees that stability is equivalent to satisfying Dterm conditions
in the gauge theory.
~
One can show in detail how to exclude destabilizing subrepresentations for each of
the phases ~ . However, there is an easier and cleaner way of getting the answer [30{
32]. Given our twonode quiver with a chosen ~ = (
; ), one of the two nodes, say
the left one, will have a negative ~component, and the other one will have a positive
component. Then, calling Vleft and Vright the vector spaces associated to the two nodes,
the semistable representations are de ned by the requirement that the space of paths from
the \negative" to the \positive" node, i.e. from Vleft to Vright, must fully generate the space
Hom(Vleft; Vright) of the underlying vector spaces. For the phase ~
+ = ( 1; 1), this means
that the i generate the right C, i.e. that they are not both zero. For the opposite phase,
= (1; 1), we must impose analogously that 1 and 2 are not both zero, so that they
generate Hom(Vright; Vleft). We have then found the two irrelevant ideal conditions for the
two phases mentioned above. This trick will be used extensively in further sections.
The exceptional curve.
The exceptional locus is a single, irreducible P1 curve. Let us
explicitly see this. Given a choice of ~ = ~ , the moduli space of d~ = (1; 1) representations
corresponds to a resolution X
of the singularity C we started with. Therefore, there exists
a blowdown map
: X
! C : ( 1; 2; 1; 2) 7! (x; y; z; t) ;
(2.15)
{ 8 {
the toric C corresponding to the complexi cation of the relative U(1)
U(1)
U(1) of
the quiver. Namely:
x + y =
These four paths generate all gaugeinvariant functions of i; i.
Here we are interested in the ber over the origin,
such that all gaugeinvariants vanish. In the ~
1; 2 parametrize a P1. In the ~
1(0). This means that we want
+ phase this means that
phase, the roles of i and i
op. Let us brie y see how the relative U(1) gauge theory
Dterm in (2.14) allows us to follow continuously the op transition undergone by the
For
> 0 we see that j 1j2 = j 2j2 = 0 and j 1j2 + j 2j2 =
gives a nite Kahler
size twosphere. As we let
have j 1j
2 = j 2j2 = 0, and j 1j2 + j 2j2 =
opposite orientation.
! 0, the size also goes to zero. Then, when
< 0 we
gives the size of a di erent sphere with
3
Laufer's example
In this section we shall consider a class of pointlike threefold singularities that generalize
the conifold studied before.6 The generalization stems from the fact that such singularities
are examples of lengthtwo ops, as opposed to the lengthone case.7
3.1
The threefold op
Consider the following equation in C7 [
21
]:
Wuniv : x2 + uy2 + 2vyz + wz2 + (uw
v2)t2 = 0 :
(3.1)
It describes a singular hypersurface with two small resolutions Wuniv. This sixfold is called
the universal op of length two. By de nition, any threefold singularity Wthreefold that
has a crepant resolution with a lengthtwo curve as exceptional locus admits a morphism
Wthreefold is the pullback of Wuniv ! Wuniv (with Wu+niv 99K Wuniv).
into (3.1). More precisely, given the map Wthreefold ! Wuniv, the resolution Wthreefold !
Old examples of lengthtwo
ops were provided by Laufer [23] and
MorrisonPinkham [34]; these were later put into standard form by Reid [22]. In [
21
] it is described
how to derive all lengthtwo examples from the universal op, and a 2`
2` = 4
4 MF of
the latter is given.
6This has no relation to the socalled generalized conifold, often seen in the topological string literature.
7Just as the lengthone conifold threefold can be seen as a family of complex deformations of the A1
twofold singularity over a complex plane [33], the lengthtwo Laufer's case can be seen as a family of
deformed D4 singularities over C [
21
].
{ 9 {
Let us now describe in more detail the class of singular opping geometries studied
by Laufer. Let X be a rational, singular CY threefold. Let X ! X be a small resolution
and C = P1 the exceptional locus; let us call N the normal bundle to C in X . Then N
must be a sum of line bundles N = L1
L2, with c(det N ) = c1(L1) + c1(L2). By the
adjunction formula, T X = T C
N , one deduces that c1(L1) + c1(L2) =
RC c1(C) = 2. De ning the Chern numbers (n1; n2) :=
RC c1(L1 ; RC c1(L2) , we see that
c1(C), where
n1 + n2 =
2. In order to have an isolated singularity, it turns out that we can only have
(n1; n2) = ( 1; 1), ( 2; 0), and ( 3; 1), and this exhausts all possibilities [23]. ( 1; 1)
corresponds to the conifold op, ( 2; 0) to the socalled Reid's pagoda (i.e. x2n
y
2
tz =
0
C
4 with n
2), and ( 3; 1) is the case of interest to this paper.
origin of C4, and its de ning equation reads
Indeed Laufer showed [23] that, in the ( 3; 1) case, X can be written as a hypersurface
inside C4, and moreover we have a family of such singularities labeled by an odd integer
k = 2n + 1
3 (that is n 2 N>0). The hypersurface has an isolated singularity at the
WLaufer : x2 + y3 + wz2 + w2n+1y = 0
C4 :
As predicted in [
19, 21
] explicitly showed that the above hypersurface is a specialization
of (3.1), obtained via the restriction
t ! wn ;
u ! y ;
v ! 0 :
3.2
Matrix factorization and singular divisors
Like in the conifold case, with each resolved phase is associated an MF of Laufer's
threefold (3.2), i.e. the pairs ( L; L) and ( L; L) satisfying
L
L =
L
L = WLaufer 1 4 4 :
Since the corresponding MF is known for the universal op of length two, the
restriction (3.3) can be used to produce the two matrices for Laufer [19]:
L := 6
2
x
2
6 y
4
6 wz
wn+1y
y
x
wn+1
wz
z
wny
x
y
2
wn3
7 ;
L := 6
2
6
6
4
x
y
2
wz
wn+1y
y
x
wn+1
wz
z
x
2
y
wny z 7
wn3
7 :
y7
5
x
Notice that L = 2x 1 4 4
through the exact sequence
L. The matrix
L de nes an MCM Rmodule M = coker L
0
R 4
L
R 4
M
0 :
As done for the conifold case in (2.7), we can extract families of Weil divisors directly
from the MF (3.4). In the resolved space the ranktwo module M = coker L becomes
a ranktwo vector bundle. The divisors we are looking for are then Poincare dual to the
rst Chern class of the bundle. The class can be determined as the locus where two of
(3.2)
(3.3)
(3.4)
(3.5)
(3.6)
the bundle's (generic) sections become parallel. This locus can be identi ed already in the
singular space, by requiring that two sections of M be proportional to each other. To do
this we use the isomorphism between coker
L and im
L: when the domain of the map
L is restricted to be coker L, the map is bijective (this is valid on generic points of the
Laufer threefold).8 Hence, the locus where two sections of coker
L are parallel is the same
as the locus where two sections of im
L are parallel. Since the image im
L is generated
by the columns of
L, we can choose two columns of
L and
nd the locus where these
become parallel. Take e.g. the last two columns of
L. The locus we are looking for is
given by
z
2
6wny
64 x
y
2
wn3
rank 6
1 :
z2 + w2ny ;
yz + wnx ; xz + wny2 ; (3.2) :
yz + xwn ;
xy + wn+1z ; y2 + w2n+1 ; (3.2) ;
When all the twobytwo minors vanish we obtain the vanishing locus of the following ideal:
On the other hand, one can make a di erent choice, e.g. the second and last columns give
while the rst and last give
xz + wny2 ; xy + zwn+1 ; x2 + w2n+1y ; (3.2) :
In fact, taking generic combinations of columns and requiring them to be parallel gives a
whole family of Weil divisors. We will explore this further at the end of section 5.
Interestingly, notice that Laufer's geometry (3.2) can also be thought of as (a patch of)
2
an elliptic bration over a noncompact C(w;z) base. It is in fact described by the Weierstrass
model (after a trivial rede nition y 7!
y)
x2 = y3 + f (w; z) y + g(w; z) ;
f := w2n+1 ;
g :=
wz2 :
(3.11)
The discriminant is
4f 3 + 27g2 = w2(4w6n+1 + 27z4). We notice that the 7brane
locus splits into one with
ber type II and one with
ber type I1. At the intersection of
the two loci, where the elliptic bration is singular, the ber type is I0 . This would navely
correspond to a D4 enhancement. However, we know that the resolved
ber over this locus
only has one P1. This can be blamed on the fact that the Kodaira classi cation is only
reliable for elliptically
bered K3's.
However, this might be a hint that there is a Tbrane e ect at play, that is breaking
this D4 enhancement in a nonconventional way. An analogous situation was observed in
an SU(5) Ftheory setting in [35]: there, the Kodaira table navely predicted an E6type
8The space im
L is twodimensional (when
L is applied on R 4); moreover when
L acts on elements
of the twodimensional space im
L it gives zero, hence im L
ker L, which is also twodimensional.
Therefore im L = ker L. Hence, inside coker L, the kernel of L is empty and the map is invertible.
(3.7)
(3.8)
(3.9)
(3.10)
ber enhancement over a codimension three locus, but the ber type turned out to be
entirely di erent (not even of ADE type). The puzzle was resolved by [
36
] who realized
that the E6 was partially broken due to a Tbrane, or monodromic e ect.
This elliptic bration has an extra (rational) section given, in the patch w 6= 0, by
x =
z
3
w3n ;
y =
z
2
w2n :
(3.12)
In Ftheory compacti cations, extra sections of the elliptic bration correspond to massless
U(1) gauge symmetries in the lowerdimensional e ective theory.
The class of elliptic brations with one rational section has been studied by Morrison and Park in [37]. Laufer's
Locally, we can see that the extra section (3.12) is equivalent to the divisor (3.8).9 It is
then just one of the possible Weil divisors associated with the MF (3.5). We thus see how
the MF is related to the massless U(1)'s in Ftheory. This observation, as well as the study
of the local Ftheory model provided by (3.2) and its generalizations, will be the subject
of a companion paper [
38
].
4
Noncommutative crepant resolution
In this section we will use the NCCR technique to extract the exceptional lengthtwo
P1 locus of Laufer's singular threefold.
Moreover, we will identify the class of divisors
mentioned above in terms of quiver variables.
We will use the generalization to three
dimensions [27] of classic results by ArtinVerdier [
39
] on ADE twofold singularities to
characterize the divisors as rst Chern class of a vector bundle associated with an MCM
module. Like in the conifold example, there are two inequivalent resolutions, associated
with the two inequivalent MF's of Laufer's threefold and correspondingly with two MCM
modules. For the NCCR we pick up one of the two MCM modules and we construct the
noncommutative ring A as follows:
Without loss of generality, let us consider Laufer's example (3.2) with n = 1:
M = coker L ;
This geometry will be our main casestudy throughout the rest of the paper. The
noncommutative resolution is described by the twonode quiver depicted in
gure 4. The quiver
has a path algebra, whereby linear combinations of arrows can be taken, and the product of
two arrows is given by their concatenation, reading from right to left.10 The path algebra
must be supplemented with the following relations:
RL :
(b2 + dc)d = 0 ; c(b2 + dc) = 0 ;
ab + ba = 0 ;
a2 + bdc + dcb + b3 = 0 : (4.3)
9Modulo a sign due to the rede nition of y.
10If the product does not correspond to a logical concatenation, then it is zero. For instance, in this case,
d a = 0, whereas a d 6= 0.
Like for the conifold, this data can be physically interpreted as the algebra of open strings
attached to fractional branes that wrap the vanishing cycle of the geometry.
The maps (arrows of the quiver) a; b; c; d can be expressed in terms of the MF data, as
explained in [19]. In particular, any homomorphism
: R ! M lifts to a homomorphism
R 4
R 4
c
q
L
L
R
^
R 4
R 4
^
R
R
M
M
R
where ^
^ +
L q, for any q : R ! R 4
.
follows:
Conversely, any map
: M ! R is expressible via its uplift to a map ^ : R 4
! R as
(4.4)
(4.5)
(4.6)
(4.7)
where we must impose the condition
L = 0.
and codomain are either R or R 4:11
We can then follow [19] and give the morphisms a; b; c; d in terms of maps whose domain
a = 6
2 0 1 0 0
3
6 y 0 0 07
64 0 0 0 17
5
7 ;
From this, we can deduce relations between quiver loops (i.e. gaugeinvariants) and a ne
C4 coordinates. First, note that the following paths from left to right generate a vector
11We always consider cases where the quiver has two nodes, with the nontrivial node given by an MCM
module M = coker , with
an n
n matrix that factorizes the hypersurface equation, i.e. ( ; ) is a
MF. The arrow from R to M are the morphisms from R to M : these are generated by n
1 matrices with
one entry equal to 1 and the other to 0. The morphisms from M to R are generated by the rows of
[19].
There are also additional endomorphisms of R and M which sometimes need to be added. Typically, there
are relations among these morphisms that allow to reduce the number of generators, leaving a smaller set
of relations (this is what happens in Laufer's case).
space:
side node:12
The a ne ambient C4 coordinates can be recovered in terms of loops on the lefthand
cabd = x ;
cbd = y ;
cad = z ;
cd = w :
Other useful relations also hold, e.g.
Their usefulness will become clear in section 5.3.
Laufer as 4d N = 1 quiver gauge theory
Having described the NCCR, which amounts to the path algebra of a quiver with relations,
we will now extract the geometricallyresolved space from that data. In principle, we could
already extract the hypersurface equation (4.2) from the path algebra: the coordinate
ring of the hypersurface is simply the center of the noncommutative ring (path algebra)
However, we will now use a more powerful technique that will not only allow us to
reproduce the singular geometry, but will also give us its explicit resolutions in both its
phases, and show us the op transition. Mathematically, we will de ne a nitedimensional
representation of the quiver. This means that we replace the nodes with complex vector
spaces, the arrows become linear maps between them, and the relations in (4.3) must still be
imposed. In our case, the appropriate representation will have dimension vector d~ = (1; 2),
that is dR = 1; dM = 2 on the lefthand side and righthand side nodes, respectively. These
dimensions correspond to the ranks of the MCM modules, respectively.
Physically, we are studying the N = 1 quiver gauge theory that arises from probing
the singularity with a spacetime lling D3brane. The theory has gauge group U(1)
U(2),
and is depicted again in gure 5. The superpotential has been derived in [19], and reads:
The corresponding Fterm relations are the following:
WL = dcb2 +
cdcd + a2b +
1
2
1 b4 :
4
HomR(R; R) = HomR(M; M ) = R.
(4.8)
(4.9)
(4.10)
(4.11)
(4.12)
(4.13a)
(4.13b)
(4.13c)
(4.13d)
c
For this theory, a and b are adjoint U(2) elds, i.e. twobytwo matrices. d and c are
bifundamentals, s.t. d is a twobyone matrix, and c a onebytwo matrix. The theory has
a (classical) moduli space parametrized by the gaugeinvariant combinations of these four
elds. However, it can also be described by using coordinates of the quotient space
C12 ha; b; c; di = (C
GL2(C)) ;
where C is the complexi ed U(1) gauge group, and GL2(C) is the complexi ed U(2)
gauge group. Note though, that just as projective spaces are quotients of an appropriately
punctured complex vector space, so must this C12 be punctured. More precisely, we must
exclude some algebraically closed subspaces before we can de ne the quotient. This per se
does not insure that the result be nonsingular, but it at least enforces that the resulting
space be Hausdor . In gauge theory terms, this amounts to imposing Dterm constraints
of the following form:
for each node v, whereby one must also impose
X
a: !v
a ay
X
a:v!
ay a = v 1 Nv Nv ;
X
v
v dim Vv = 0 ;
dim Vv being the dimension of the vector space at the vnode (i.e. the entry dv of d~).
Labeling our nodes v = R; M , this translates to the following conditions:
ccy
dyd = R ;
ddy
cyc + [a; ay] + [b; by] =
M 1 2 2 :
Here, we must impose R +2 M = 0. Once we have properly excised the bad loci, and taken
the quotient by the gauge group action, the moduli space we are left with (after imposing
the relations (4.13a){(4.13d)) is expected to be the CY threefold probed by the D3brane.
A nonzero choice for the ~ = ( R; M ) vector corresponds to resolving the singular space.
Let us apply the same trick used to determine the stability of the d~ = (1; 1) quiver
representation of the conifold. In Laufer's case, the quiver representation of interest will
have dimension vector d~ = (1; 2). Again there are two choices of a ~vector satisfying
(4.14)
(4.15)
(4.16)
(4.17a)
(4.17b)
~ d~ = 0 up to rescaling: ~+ := ( 2; 1) and ~ := (2; 1). We will study both separately,
and then see how to make a smooth
op transition from one to the other. Like for the
conifold, the semistable representations are de ned by the requirement that the space of
paths from Vleft to Vright must fully generate the space Hom(Vleft; Vright) of the underlying
vector spaces. For the phase ~+ = ( 2; 1), we must then impose that
Paths(R; M ) = hd; ad; bd; abdi = C2 :
In order for this to be true, we must require that (d; ad; bd) not all be collinear, when viewed
as column twovectors. For the opposite stability condition, ~
= (2; 1), we must impose
HJEP04(218)9
Paths(M; R) = hc; ca; cb; cabi = Hom(C2; C) = C2 :
This means that (c; ca; cb) should not all be collinear as row twovectors.
We can summarize the situation by constructing the two following irrelevant ideals
corresponding to the two resolution phases:
Therefore, there exists a blowdown map
(4.18)
(4.19)
(4.20)
(4.21)
(4.22)
(4.23)
(4.24)
In the phase ~+ (
~ ), the elements of I+ (I ) cannot vanish simultaneously. We will later
see that these ideals are made of the homogeneous coordinates of the exceptional P1 in
each phase.
4.2
Finding the exceptional curve
In this section, we will nd the
ber of the resolution in either phase, and see that it
corresponds to a P1. Before doing so, let us brie y explain some generalities.
Given a choice of ~ = ~ , the moduli space of d~ = (1; 2) representations corresponds
to a resolution X
of the singular hypersurface WLaufer we started with (with n = 1), i.e.
WLaufer : x2 + y3 + wz2 + w3y = 0
C4 :
~
~
+ = ( 2; 1) : I+ := (ad ^ d; bd ^ d) ;
= (2; 1) : I := (ca ^ c; cb ^ c) :
: X
! WLaufer : (a; b; c; d) 7! (x; y; z; w)
where the coordinates of C4 are functions of the quiver variables that are invariant under
the C
GL2(C) group. Put compactly:
(x; y; z; w) 2 R[a; b; c; d]C
GL2(C) :
These invariants were derived in [19] and are reported in (4.10). The important point is
that they are made as traces of gaugeinvariant loops. Now, we are interested in the ber of
the origin
1(0). This means that we want the locus in X
such that gaugeinvariants are
traceless. These are traces of either numbers or twobytwo matrices, and all their positive
powers. Therefore, all loops must correspond to nilpotent maps, respecting the stability
conditions [31, 32].
+ = ( 2; 1). Let us start with the phase ~+ = ( 2; 1). The most basic loops
based on the lefthand node are:
R : cd ; canbmd
8n; m ;
and products thereof ;
(4.25)
where anbm means any combination of (integer) powers of those arrows. Requiring their
nilpotency means setting them to zero, since they are given by onebyone matrices. On
the righthand node we have:
M :
dc ;
danbmc ;
anbm ;
8n; m ;
and products thereof :
(4.26)
Note that the rst two sets of loops are automatically nilpotent if we set cd = 0, as required
by the nilpotency of the lefthand side loops.
Taking both sets of variables together, we see that the ber is given by the
following ideal:
cd; a2; b2; cad; cbd; cabd :
Remember that, in this phase, ad; bd and d cannot be collinear.
Hence the ideal (cd; cad; cbd) immediately implies c = 0. All in all, we can simplify the answer to
IP1+ := c; a2; b2; ab + ba :
basis that allows us to \see" the P
1 more directly. Let us illustrate this.
Given this ideal, and the irrelevant ideal I+ = (ad ^ d; bd ^ d), we can choose a convenient
From the irrelevant ideal, we have that a and b cannot vanish simultaneously. First,
we would like to prove that a and b are proportional. Say, without loss of generality, that
a 6= 0 (the considerations we will make are symmetric in a $ b). Then it must have a
nontrivial onedimensional kernel, generated by a vector va. The relation ab + ba tells us
then that ab va = 0. From this we deduce one of two possibilities:
bva / va. This is impossible, since a nilpotent matrix cannot have nonzero eigenvalues.
bva = 0. This implies that ker a
ker b.
Similarly, we can prove that ker b
ker a, which implies that ker a = ker b. Now, the
nilpotency of a and b gives us the following picture
im a
im b
ker a
=
ker b
from which we infer that either im a = im b, or one of the two variables is zero. In either
case, the conclusion is that a / b. Now we can use the SL2(C)
a basis for C2 such that ad and bd / (1; 0)t and d / (0; 1)t. Using a combination of the
left C and the two C inside GL2(C), we can x d = (0; 1). This leaves one C that acts
GL2(C) symmetry to x
(4.27)
(4.28)
(4.29)
on the pair (ad; bd) via rescaling. Summarizing, we have the following Ansatz for the most
general solution to F and Dterms:
a =
"
0
with an action of the following C subgroup of GL2(C):
C : (a; b) 7!
"
0
#
Clearly, this turns (ad; bd) ^ d into homogeneous coordinates:
(ad; bd) ^ d = ( ; ) 7! ( ;
) :
The fact that this pair is exactly the irrelevant ideal makes this into an actual P1[ : ].
= (2;
1). In this phase, c; ca and cb cannot all be collinear. Hence, the ideal
(cd; cad; cbd) implies d = 0. The curve is given by
On this side, the Ansatz for the P1 is equally simple to nd, and turns out to be the
The gauge group is broken to the following C :
The homogeneous coordinates of the P1 are then
a =
"
0
#
"
0
(ca; cb) ^ c = ( ; ) ;
(4.30)
(4.31)
(4.32)
(4.33)
(4.34)
(4.35)
(4.36)
(4.37)
(4.38)
HJEP04(218)9
which constitute the irrelevant ideal for this phase, and transform as expected.
4.3
Following the op transition
In the previous two sections we showed how the resolved space looks by imposing two
opposite stability conditions, ~ = ( 2; 1) and (2; 1). In both cases, we see a P1. Hence,
the op transition has to do with negating the stability parameter. However, we would like
to follow this transition continuously through the singularity, see a twosphere shrink to
zero size, and a new one grow. In order to do this it is best to use the Dterm constraints
dyd
ccy = 2 ;
ddy
cyc + [a; ay] + [b; by] = 1 2 2 :
Let us make the following overarching Ansatz, which interpolates between (4.30) and (4.35):
"
0
a =
;
b =
;
c =
It satis es the Fterm constraints (4.3). Plugging the above Ansatz into the Dterm
constraints, we get the following equations:
instead of ~stability. For reading convenience, we repeat these here. Taking ~ = ( 2 ; ),
we have:
(4.39)
(4.40)
(4.42)
(4.43)
(4.44)
(4.45)
p
2j j2
j j
2
j j2 = ;
j j2 + j j2 = j j2 + j j2 :
"
j j2 + j j
2
2j j
2
0
j j
2
j j2 = ;
0
j j
2
j j
2
#
> 0. We can rewrite this system as follows:
Now we see the transition continuously. Starting at
> 0, our solution at the level of
Fterms for the twosphere dictates
= 0. Hence, we see that j j2 = , which implies
that j j2 + j j2 = , signaling a nite Kahler size for the exceptional sphere. As we send
! 0, the size goes to zero. After transitioning to
imposes
= 0. Now we have j j2 =
, and hence j j2 + j j2 =
a di erent sphere of nite size.
< 0, our solution for the sphere
, which corresponds to
5
Weil divisors
In this section we will analyze in detail the Weil divisors associated with the small
resolutions. Already for the conifold, we may see that they can be detected both in the singular
phase and in the resolved phase.
In Mtheory geometric engineering, these divisors play an important role. The
singularities we are studying can be obtained from a smooth hypersurface CY space by restricting
its complex structure (specializing the de ning equation). In this process, the threefold can
gain new codimensionone submanifolds. In the singular space these are Weil divisors, that
in the resolved phase become honest Cartier divisors. In Mtheory, abelian gauge
symmetries emanate from the reduction of the supergravity C3form along harmonic, normalizable
twoforms that are Poincare dual to the new divisors. In Ftheory compacti cations not
all the divisors will correspond to abelian gauge symmetries. However, these extra divisors
are the natural objects to use in order to produce massless abelian gauge bosons in the
e ective theory. As we have seen in (3.12), in this case the elliptic
bration develops an
extra (rational) section that belongs to the family of Weil divisors; this is the condition for
the Ftheory to have an abelian gauge symmetry [37, 40].
In this section, we will introduce a new way of detecting such divisors in algebraic
varieties that admit small resolutions. The conifold is an obvious case: it admits two
but that are separately only Weil. In either resolution phase ~ , the divisor D
families of Weil divisors jD j whose union jD+ + D j is in the class of a Cartier divisor,
will
intersect the exceptional P1 at a point, and D
will actually contain it as the total space
of O( 1)P1 over it.
We will explore how to carry out this analysis for Laufer's example with n = 1 in this
section. We will nd that, here too, there are two divisors D
such that one intersects the
P1 and the other contains it in one phase, and vice versa in the other phase. From the
quiver perspective, we will actually recover the extra divisor, and the whole linear system
in which it moves, as opposed to only the representative corresponding to the extra section.
Divisors from the quiver
Divisor in phase ~ = ( 2; 1). Let us start by
which is the one that intersects the exceptional P1 at a point in the phase ~
+ = ( 2; 1).
The idea is to construct a line bundle such that the zero locus of its sections intersects the
exceptional P1 once. In Mtheory, this will mean that we have a U(1) gauge group, and
matter with charge one under it, given by a membrane wrapping the sphere.
nding the family of divisors jD+j,
The construction is straightforward. First, note that our resolved space X+ comes
equipped with a tautological bundle of the form
Vtaut = LR
VM ;
where LR is a line bundle whose structure group is the left C , and VM is a ranktwo vector
bundle with structure group identi ed with GL2(C). The arrows in the quiver correspond
to sections of these bundles as follows:
d 2 (L_R
VM ) ;
c 2 (LR
VM_ ) ;
a; b 2 (End(VM )) :
There is an ambiguity that allows us to twist Vtaut into Vtaut
bundle. This is equivalent to the statement that an overall U(1) gauge group decouples
L
~, where L~ is any line
from the quiver theory. We can gauge x this by tensoring with L_R, such that now
Vtaut = O
V ;
(5.1)
(5.2)
(5.3)
(5.4)
where the rst summand is the trivial bundle (i.e. the structure sheaf of X+), and the second
a ranktwo vector bundle. Now, we have the following assignments to the various arrows:
d 2 (V) ;
c 2 (V_) ;
a; b 2 (End(V)) :
ArtinVerdier theory [
39
], and its generalization to smallresolved threefolds by Van Den
Bergh [
18, 27
], tells us how to construct a line bundle L such that c1(L) = c1(Vtaut). In
(5.5)
(5.6)
(5.7)
(5.8)
(5.9)
(5.10)
(5.11)
This is easily con rmed by looking at our Ansatz (4.30), which we repeat for convenience:
"
0
#
!
0
1
a =
; b =
; c = 0 :
Since ab = 0 on the P1, VjP1 is generated by the sections hd; ad; bdi. Given our Ansatz,
these take the following form:
ad =
; bd =
;
d =
0
!
!
0
1
:
These are clearly sections of O(1)
O.
Finally, given the generators (5.7) of the line bundle L, our claim is that the U(1)
corresponds to a family of divisors of the form
D+ = c1(ad ^ d) + c2(bd ^ d) + c3(abd ^ d) :
(5.12)
O
d
V
^ d
L
0 :
This line bundle clearly satis es c1(L) = c1(V). From here, we see that we can generate all
sections of L as
(L) = had ^ d; bd ^ d; abd ^ di :
Note that, on the locus (4.28) of the P1, the third generator abd ^ d is identically vanishing.
Comparing the remaining sections with the irrelevant ideal I+ in (4.20), we conclude that
be
2VjP1 = O(a + b). Therefore,
The irrelevant ideal we have excised imposes that V be generated by its sections. This
means that its restriction to the P
1 must decompose into a sum of line bundles of
nonnegative degree: VjP1 = O(a)
O(b). The line bundle of equal rst Chern class must then
L = OP1 (1) :
VjP1 = O(1)
O :
those references it is proven that the vector bundle V in (5.3) must occur in an exact
sequence of the form
O
V
L
0 ;
d : O ! V
where the divisor associated to L intersect the exceptional P1 once. To see that such a
sequence must exist is simple. Pick a generic section of V, say d. This de nes the rst map
. Since d is nowhere vanishing, the rank of this map is always one. Therefore,
the cokernel of the map must be a line bundle. De ne L to be that cokernel. The exterior
product ^ d de nes an explicit map ^ d : V ! L such that (5.5) is an exact sequence:
ΦD
ΦU
a
b
q1,2
1
p˜1,2
q˜
2
p
jD j, intersecting the P1 at a point in the
listing the sections of the dual bundle V_:
1).
Now we would like to describe the family of divisors
phase. The logic is the same. We start by
(V_) = hc; ca; cb; cabi :
Pick a particular section, say c, and construct the exact sequence:
0
O
c
V
L
0 :
In the phase ~ , c is nowhere vanishing, so the cokernel is a bona de line bundle. From
this, we can construct the generic D
divisor:
D
= c1(ca ^ c) + c2(cb ^ c) + c3(cab ^ c) :
To understand the fact that jD+ + D j is a family of Cartier divisors, simply note that
the product D+ D
is a section of L
L
_ = O. Hence, such a section can be deformed
by an arbitrary constant, thereby making the corresponding divisor miss the exceptional
locus completely. This means that, under the blowdown map
in (4.23), this divisor
can escape the singularity, and is therefore Cartier.
5.2
Divisors as avor branes
We now show how to obtain the divisor (5.12) by relying exclusively on the quiver gauge
description of the singularity. We will add avor nodes to the quiver, which will correspond
to noncompact D7branes along either Weil divisor. Naturally, the pointlike D3brane
(obtained from the (1; 2)representation of the quiver), will have a avor D3D7 spectrum.
We will then de ne the Weil divisors as the loci in the moduli space of the quiver where
some of this matter becomes massless.
c
1
(5.13)
(5.14)
(5.15)
HJEP04(218)9
those of the dual bundle V
written as follows:
Consider the avored version of Laufer's quiver depicted in
gure 6. The squares
correspond to avor D7branes, whereas round nodes to fractional D3branes. The arrows
connecting color to
avor branes are D3D7 states. Note, that we must have two q elds
and two p~ elds in order to make the avor branes anomaly free.
The arrows connecting the boxes are D7D7 states. The superpotential (4.12) will be
modi ed by additional terms describing the interaction of these new states. Let us call
si the sections (4.18) of the vector bundle V:
(V) = fsigi4=1 := fd; ad; bd; abd; g, and s~i
_ (i.e. (5.13)). The avored superpotential can be schematically
WLv = WL + Mn p p~n + Mfm q~ qm + K
Up q~ + Dmnqmp~n D ;
(5.16)
(5.17)
HJEP04(218)9
where
two D7branes.
Mn
i
Cnsi
and
Mfm
C~mi s~i
and where
= 1; 2 is a U(2) index, m; n = 1; 2, i = 1; : : : ; 4, and Cni; C~mi; K; Dmn are
numerical coe cients. The elds
U;D describe the states between the two D7branes. Let
us consider the situation when their vev vanish, that means no recombination between the
We want to see what happens to the D3D7 states when we move the D3brane away
from the singularity, that means when we give a generic vev to the sections si and s~i
(satisfying the Fterms (4.13a){(4.13d)). The masses of these states are described by the
2
2 matrices M and Mf in (5.17). At a generic point away from the singularity, both M
and Mf have maximal rank and all the D3D7 states become massive. This means that the
D3brane is not on top of the D7brane. On the other hand, when one of the two matrices
has rank one (i.e. either det M = 0 or det Mf = 0), a vector like pair becomes massless: we
interpret these states as the D3D7 states that become massless when the D3brane is on
top of one of the two D7branes.
Notice that the columns of M (Mf) are two generic sections of the bundle V (V_).
Hence det M = 0 (det Mf = 0) happens exactly when two sections of the bundle V (V_)
become parallel, i.e. on top of the Weil divisors studied in section 5.1.
We have then proven that the Weil divisors correspond to the locus where a D3D7
state becomes massless. To see this more concretely, take the locus det M = 0. By eld
rede nition one can set C1i si = d (and the second column of M is a linear combination
ad; bd; abd). The determinant of M then gives exactly the locus (5.12) obtained before.
Analogous considerations hold for the locus det Mf = 0.
5.3
Singular description
Having described the two families of divisors jD+j and jD j in the resolved geometries,
we would like to describe the image of this family under the blowdown map (4.23), in
order to gain direct access to it in the singular space. In Mtheory compacti ed on the
CY threefold, the U(1) gauge group associated with these divisors exists, whether we
resolve the singularity or not. Hence, the divisors we found must exist on the singular
space as Weil divisors. Our strategy is to construct singlets out of sections
D+ and
described in section 5.1, so that we can gain a description in terms of the a ne C
coordinates (x; y; z; w).
Our strategy will be slightly counterintuitive. In order to describe the family jD+j,
which are the divisors that intersect the P
1 at a point in the ~
+ phase, we will actually
rst describe it in the ~
phase, and vice versa for the other divisor family. Here is why. In
the ~
phase, we have an irrelevant ideal I
= (ca ^ c; cb ^ c). Incidentally, the generators
of this ideal are sections of L_, which is exactly what we need to create singlets from
D+ 2
(L). The fact that they form an irrelevant ideal means that, if we create the ideal
of products ( D+ I ), we will get an ideal whose zero locus must correspond to the zero
locus Z( D+ ) of D+ . For technical reasons, we will nd it more convenient to throw in
the singlets made by multiplying also by (cab ^ c). The ideal in the resolved space will have
the same zero locus; however, in the singular space, we will obtain a cleaner description of
the Weil divisor family. Explicitly, de ne the following singlet matrix:
Z( D+ ) = Z
D+ ca ^ c;
D+ cb ^ c;
D+ cab ^ c :
The ideal I+ can now be written entirely in terms of a ne C
4 coordinates by using the
relations in (4.10). First, let us use the general fact that, given sections si of V and s~i of
V_, we have the following identity:
For instance, we would have identities like
Let us setup a matrix of such relations:
(s ^ r)(s~ ^ r~) = (s s~)(r r~)
(s r~)(r s~) :
(ca ^ c)(ad ^ d) = (ca2d)(cd)
(cad)2 =
yw2
z2 :
(5.18)
(5.19)
(5.20)
h ad bd abd ^ d
i
ca2bd 3
cbabd 7
ca2b2d5
Plugging in (4.10) and (4.11), one obtains
Z( D+ ) =
2yw
x
w 6 x w2
4 y2 zw yw2
y2 3
zw7
5
0 cad 1
B cbd C
cabd
0
z1
cad cbd cabd :
(5.21)
z y
x :
(5.22)
Therefore, a generic section
D+ will have the following description as a Weil divisor:
Z( D+ ) ~k = 6
xw + zy
2
4
yw2
y2w
z
2
xz
w3
y
2
wy2
xz 3 0
k11
zw2 + xy 75 B@k2CA = 0 :
zw2 + xy
yw3
x
2
k3
(5.23)
We recover the locus (3.8) (with n = 1) by taking ~k = (1; 0; 0)t. Notice that this coincides
with the extra section (3.12) of the elliptic
bration given by the local Ftheory model
WLaufer = 0
P231[ y : x : 1]
respectively the loci (3.9) or (3.10).
C(w;z). By setting ~k = (0; 1; 0)t or ~k = (0; 0; 1)t we obtain
2
Let us write down the singular description of the Weil divisor corresponding to
D
by going to the + phase, multiplying by a matrix made from the irrelevant ideal in that
phase. There is nothing to calculate, we simply transpose the threebythree matrix we
just constructed. Therefore, a generic section
will have the following description as a
Weil divisor:
Z( D ) ~k = 6 xw + zy
2
4
yw2
2
y2w
xz zw2 + xy
D
2
w3
x
2
k3
(5.24)
This is the same family one would obtain from the MF ( L; L) and the corresponding
MCM module M _ = coker L, following the computations of section 3.2.
6
Highercharge states
Throughout the paper we have used the branes at singularities paradigm, whereby we
describe the quiver gauge theory arising from a spacetime lling D3brane that is pointlike
in the internal space. We will now switch to the socalled geometric engineering picture
in IIA, whereby we study the e ective eld theory that arises by reducing type IIA
supergravity on our threefold, supplemented by the Dparticles arising from various D2branes
wrapping the exceptional P1.
So far, Laufer's geometry seems to behave in perfect analogy with the conifold: it has
a vanishing P1 that can be blownup crepantly; the curve can be opped; the threefold
admits two Weil divisors, one of which cuts the curve at one point in one resolution phase,
and the other one cuts the curve in the other phase. It seems then that the only novelty
of this geometry is simply that it is more complicated to describe.
In this section, we will discover an important qualitative di erence: IIA on Laufer's
example admits hypers of higher charge under the U(1) generated by the Weil divisors.
This is in stark contrast to the conifold, which only admits a hyper of charge one.
Once again we will use the example of the conifold as a familiar reference, in order to set
the stage. In that case the two simple representations d~ = (1; 0) and d~ = (0; 1) correspond
to the objects OC (a D2brane) and OC ( 1)[1] (an antiD2 with
category Db(X ). In the case of Laufer's example, it was explained in [19] that d~ = (0; 1)
corresponds to OC ( 1)[1], whereas d~ = (1; 0) corresponds to some di erent (not locally
ux) respectively, in the
free) object S with support over the curve C. The following simple equation
d~ = (1; 2) = (1; 0) + 2 (0; 1)
(6.1)
shows us that, at the Ktheory level, the class of a point p 2 X (i.e. a pointlike D0brane)
is given by
[Op] = [S] + 2 [OC ( 1)[1]] = [S]
2 [OC ( 1)] , [S] = [Op] + 2 [OC ( 1)] :
(6.2)
From this, we can conclude that the d~ = (1; 0) representation must correspond to some
bound state of two D2branes. Hence, if we construct a hyper in the e ective theory by
taking a d~ = (0; 1) Dparticle (i.e. the antiD2 with
ux wrapped on P1) and its
corresponding antiparticle (i.e. the oppositelyoriented membrane corresponding to the object
shifted by [1]), and normalize its charge to one, then we can also build a hyper of charge
two from the d~ = (1; 0) representation and its corresponding antibrane. To summarize in
a succinct language:
(0; 1) ; (0; 1)[1]
(1; 0) ; (1; 0)[1]
hyper of charge one ;
hyper of charge two :
(6.3)
(6.4)
Constructing the antibranes of a given representation can be done either by passing to the
derived category of quiver representations Db(modA) or, in the resolved geometry, to the
derived category of coherent sheaves Db(X ).
Now, these two hypers are never simultaneously massless. Depending on the value of
the B eld, only one of them can be made massless at a time. The mass formula for a
Dparticle is given by the modulus jZj of its central charge in fourdimensional N = 1
language. In turn, the central charge is a function Z( ; B + iJ ) of its RamondRamond
charge vector , and of the complexi ed Kahler modulus of the threefold. It gets worldsheet
instanton corrections that are subleading at large volume, where the formula reduces to
the following:
Z
X3
^ e (B+iJ) + worldsheet instanton corrections :
In the case where the CY has no compact fourcycles (our case), the formula receives no
0 corrections (as we will show momentarily), and simpli es to the following integral:
Z
P1
Z =
B + iJ
F ;
where F is the worldvolume DBI ux on the membrane. Now, our (1; 0) and (0; 1) hypers
are branes that di er in rank, and in their DBI ux, by one unit of induced D0 charge.
Hence, by shifting the B eld accordingly, either one can be made massless.
As promised, let us brie y argue that this formula for the central charge is indeed
uncorrected. (This is a recurring folk theorem that we have been aware of for a long time.)
The line of reasoning goes as follows. In general for, say, a onemodulus CY there are four
periods of the
3form of the mirror CY which, in some basis, would take the following
form (see [
41
] for an introduction):
0 = P0(z) ;
1
1 =
2 i 0 log(z) + P1(z) ;
2 = P2(z) log2(z) + P20(z) ;
3 = P3(z) log3(z) + P30(z) ;
(6.5)
(6.6)
(6.7)
(6.8)
(6.9)
(6.10)
where z is the complex structure modulus of the mirror CY, and the Pi(z) are polynomials
in z. Now, 0 can be argued to be the central charge of the mirror to the D0brane. The
quotient Q =
1= 0 has a monodromy Q ! Q + 1 around z = 0. At large volume, with
complex Kahler modulus t = B + iJ , one makes the following match with the mirror side
as follows:
The monodromy above is mapped to the monodromy around the large volume point as
B ! B + 1.
satisfy is the following:
If we had nontrivial four and sixcycles, then it would be nontrivial to solve for t in
terms of z. However given that in our case there is only a twocycle, the only relation to
which allows one to de ne a mirror map z(t), which would typically be a nonperturbative
series in 0. We can reabsorb the disturbing P1(z) piece by rede ning z ! z~ such that
Z
Z
Z
twocycle
fourcycle
sixcycle
t =
1= 0 ;
t2 =
t3 =
2= 0 ;
3= 0 :
Z
P1
1
2 i
t =
log(z) +
P1(z)
P0(z)
1
2 i
t =
log(z~) :
Z( ; B + iJ ) =
^ e t ;
Z =
Z
P1
t
Z
X3
F
(6.11)
(6.12)
(6.13)
(6.14)
(6.15)
From this, we conclude that the exact central charge for a (Btype) Dbrane is given by
which reduces to
in the case of a membrane. Clearly, given a value for F , there will be a point in moduli
space where Z = 0, giving rise to a massless state.
7
In this paper we have studied in full detail a prototype CY threefold singularity that admits
a op of length two. Such geometries were of course previously known in mathematics, but
only in terms of their algebraic properties.
The approach we used here is to examine the worldvolume theory of a D3brane
probing Laufer's singularity in type IIB string theory. We have studied the resolution of the
singular geometry by using its noncommutative crepant resolution, and subsequently its
presentation as a quiver representation, i.e. by using the quiver geometric invariant
theory method.
We have shown how to extract the exceptional P1 locus from the quiver representation,
and how to follow continuously the op transition the P1 undergoes by looking at the
fourdimensional N = 1 gauge theory Dterms. Moreover we have de ned and studied (both in
the singular and resolved phase) two families of Weil divisors of the geometry, which can
be given a natural interpretation as U(1) divisors if we de ne a local Ftheory model on
Laufer's example.
By relying on the geometric engineering perspective, we also showed that IIA
compacti ed on this class of geometries consists in a fourdimensional N
= 2 U(1) gauge
theory with at least one hypermultiplet of charge one, and one hypermultiplet of charge
two. Both hypers can become massless, but at di erent points in the complexi ed Kahler
We would now like to speculate on the possibility of having states of charge higher than
two. These could be created if several fractional branes form appropriate bound states.
In geometries like the conifold, these are usually ruled out. The main reason is the fact
that the normal bundle N to the exceptional P1 curve is strictly negative, and, since any
bound state requires giving a vev to its sections, the states are not supported. However,
ops of length two have N = O( 3)
O(1), which means it is de nitely conceivable to
form bound states. We leave this interesting question for future work.
Finally, let us brie y comment on the Ftheory interpretation of Laufer's example. We
have seen that the latter geometry is already in Weierstrass form. The family of noncompact
divisors we have found by our methods includes one divisor that can be regarded as an
extra section to that bration, implying the rank of the MordellWeil group is one. The
exceptional P1 represents a ber enhancement over a codimensiontwo locus in the base of
the bration, giving rise to a charged hyper in six dimensions.
Acknowledgments
We would like to thank R. Argurio, M. Del Zotto, I Garc aEtxebarria, J. J. Heckman,
H. Jockers, J. Karmazyn, D. R. Morrison, M. Reineke, and M. Wemyss for
enlightening discussions, and especially D.R.M. for collaboration on a related project. A.C. and
M.F. thank the Ban International Research Station \Geometry and Physics of Ftheory"
workshop for a stimulating environment and hospitality during the nal stages of this work.
A.C. is a Research Associate of the Fonds de la Recherche Scienti que F.N.R.S. (Belgium).
The work of A.C is partially supported by IISN  Belgium (convention 4.4503.15). The work
of M.F. is partially supported by the Israel Science Foundation under grant No. 1696/15
and 504/13, and by the ICORE Program of the Planning and Budgeting Committee.
The work of R.V. has been supported by the Programme \Rita Levi Montalcini for young
researchers" of the Italian Ministry of Research.
A
MorrisonPinkham's example
Another example of threefold op of length two was constructed by MorrisonPinkham [34].
The hypersurface singularity in C4 is given by:
WMP : x2 + y3 + wz2 + w3y
wy2
w4 = 0 ;
2 C :
(A.1)
MP = 6
6 (y
w)y
2
6
4
2
6
4
6 (y
w)w2 wz
y
x
w
y
x
w
w)y x
z
x
z
x
w 3
z7
y 75
7 ;
w3
7 ;
y 7
5
(A.2)
(A.3a)
(A.3b)
(A.4)
(A.6)
(A.7)
This is not a oneparameter family of singularities. There are only two distinct classes
thereof, for
= 0 and
6= 0 [19]. In particular it is easy to notice that the
= 0 class
is equivalent to the n = 1 case (4.2) of Laufer's example. The above threefold is a special
case of the universal threefold op (3.1), obtained via the restriction [19]
t !
w ;
u ! y
w ;
v ! 0 ;
as one can easily check. We also obtain an MF of (A.1) by applying (A.2) to the MF of
Wuniv provided in [19]. Explicitly:
WMP = dcb2 + 12 cdcd + a2b + 13 b3 + 14 b4 :
We will now show that the Ansatz (4.30) also holds in the MorrisonPinkham case, and
can be obtained directly from the ideal
cd; a2; b2; cad; cbd; cabd; (ab)2 ;
satisfying
tial [19]:
MP
MP =
MP
MP = WMP 1 4 4 :
The quiver with relations producing an NCCR of (A.1) was obtained in [19], and is
equivalent to Laufer's one (depicted in gure 4). The relations in the path algebra read instead
RMP : (b2 + dc)d = 0 ; c(b2 + dc) = 0 ; ab + ba = 0 ; a2 + bdc + dcb + b2 + b3 = 0 ; (A.5)
and can be obtained as the Fterms (i.e. cyclic derivatives) of the following
superpotenMP = 6
w)y
w)w z 7
w)w2
wz (y
w)y x
where we just added the last relation with respect to the ideal (4.27) (which is appropriate
for Laufer's example). In that case, this relation was implied by the Fterms once one
assumed the others. With the Fterms (A.5), we need to impose it by hand.
We will now show that the ideal (A.7) implies that the nilpotent matrices a and b are
proportional to each other. This is surely true if a or b are equal to zero. So we need
to prove it for a 6= 0 and b 6= 0. We rst show that if a nilpotent twobytwo matrix is
nonzero, then its kernel and its image are isomorphic. In fact, if the matrix m is nonzero
(m 6= 0) and nilpotent (m2 = 0), the dimension of ker m and im m are equal to one.
Moreover, since it is nilpotent, im m
ker m, implying im m = ker m as the two subspaces
are onedimensional.
Now, we take a, b twobytwo nonzero, nilpotent matrices. ab must also be nilpotent,
because of (A.7). It can be either zero or not.
If ab = 0, then im b
ker a, that again means im b = ker a. But im b = ker b and
im a = ker a. So, a and b have the same image and kernel. Hence a / b.
If ab 6= 0, then dim im(ab) = dim ker(ab) = 1. Moreover ker(ab)
ker b, that means
ker(ab) = ker b, and im(ab)
im a, that means im(ab) = im a. This implies that a
and b have the same image and kernel and then are proportional to each other.
This proves that we can always bring a; b; c; d in the form (4.30) (in the phase ~ = ~+)
or (4.35) (in the phase ~ = ~ ). Just as in Laufer's case, we could now set up an overarching
Ansatz interpolating between the two phases, and allowing us to follow the simple
lengthtwo op continuously.
Open Access.
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