Spin(7)manifolds as generalized connected sums and 3d \( \mathcal{N}=1 \) theories
HJE
Spin(7)manifolds as generalized connected sums and
Andreas P. Braun 0 1
Sakura SchaferNameki 0 1
0 Woodstock Road , Oxford, OX2 6GG , U.K
1 Mathematical Institute, University of Oxford
Mtheory on compact eightmanifolds with Spin(7)holonomy is a framework for geometric engineering of 3d N = 1 gauge theories coupled to gravity. We propose a new construction of such Spin(7)manifolds, based on a generalized connected sum, where the building blocks are a CalabiYau fourfold and a G2holonomy manifold times a circle, respectively, which both asymptote to a CalabiYau threefold times a cylinder. The generalized connected sum construction is rst exempli ed for Joyce orbifolds, and is then used to construct examples of new compact manifolds with Spin(7)holonomy. In instances when there is a K3 bration of the Spin(7)manifold, we test the spectra using duality to heterotic on a T 3 bered G2holonomy manifold, which are shown to be precisely the recently discovered twistedconnected sum constructions.
Di erential and Algebraic Geometry; String Duality; MTheory

N
=
4
Spin(7)manifolds as generalized connected sums
The construction
Calibrating forms
Topology of GCS Spin(7)manifolds
GCS Spin(7)manifolds as quotients of CY4
A simple example 4.4 3d eld theory and sectors of enhanced supersymmetry
5
Mtheory on GCS Spin(7)/heterotic on TCS G2
5.3.1
5.3.2
5.3.3
5.4.1
5.4.2
5.5.1
5.5.2
5.5.3
5.5.4
Heterotic string theory on G2manifolds
GCS Spin(7)manifolds and Mtheory/heterotic duality
Duality for the building blocks
Geometric preparation: dual pairs of G2 and CY3
The acyl G2manifold and its dual acyl CY3
The acyl CY4 and its dual acyl CY3
5.4
First example of dual pairs: new Spin(7) and its dual G2manifold
5.5
Second example of dual pairs
The Mtheory on GCS Spin(7)manifold
The dual heterotic model on TCS G2manifold
The acyl CalabiYau fourfold Z+
The acyl G2manifold Z
Mtheory spectrum on the GCS Spin(7)manifold
The dual heterotic model
1 Introduction
2
3
2.1
2.2
3.1
3.2
3.3
3.4
4.1
4.2
4.3
4.5
4.6
5.1
5.2
5.3
Spin(7)holonomy, Mtheory and 3d N = 1 theories
Spin(7)holonomy
Mtheory on Spin(7)manifolds
A Spin(7) Joyce orbifold as a generalized connected sum
Setup and motivation
Joyce orbifolds: Spin(7) and G2holonomy
TCSdecomposition of the Joyce G2manifold
Uplift to GCSdecomposition of the Spin(7)manifold
6
Discussion and outlook
A TCSconstruction of G2manifolds
{ i {
Introduction
Geometric engineering of supersymmetric gauge theories is fairly wellunderstood in
theories with at least four real supercharges. Most prominently in recent years, Ftheory has
established a precise dictionary between geometric data (and
uxes) and 6d and 4d
theories with N = 1 supersymmetry, and dually, Mtheory of course has a wellestablished
dictionary between 3d and 5d theories and CalabiYau geometries. For 4d N = 1 the
realization in terms of Mtheory compati cations on G2holonomy manifolds is already much
less well understood, in particular due to the scarcity of compact geometries of this type.
Even less is known about Mtheory on Spin(7)holonomy compacti cations, which yield
theory, and we relate these constructions in certain instances to heterotic string theory
on G2holonomy manifolds. A second motivation is to study Spin(7) compacti cation
in the context of M/Fduality and uplifting these to supersymmetry breaking vacua in
Ftheory [1] based on the observations that 3d N = 1 supersymmetry can be related to
the circlereduction of a not necessarily supersymmetric theory in 4d [2, 3].
The construction that we propose, and refer to as generalized connected sum (GCS)
construction of Spin(7)manifolds is motivated by a recent development for compact
G2holonomy manifolds in Mathematics, where a large class of compact G2holonomy manifolds
were constructed using a twisted connected sum (TCS) construction [4{6]. This
construction relies on the decomposition of the G2holonomy manifold in terms of two
asymptotically cylindrical (acyl) CalabiYau threefolds  for a sketch see
gure 1. Following
these mathematical developments, there has been a resurgence in interest in the string and
Mtheory compacti cations, which for TCSmanifolds have been studied in [7{14]. For a
review of Mtheory on G2 and Spin(7)holonomy manifolds related to earlier results in the
'90s and early '00s see [15].
One of the nice features of TCS G2manifolds is that the connected sum gives rise to
a eld theoretic decomposition in terms higher supersymmetric subsectors [11, 13]: the
asymptotic regions that are acyl CalabiYau threefolds (times a circle), give rise to 4d
N = 2 subsectors, whereas the asymptotic neck region where the two building blocks are
glued together is by itself K3 times a cylinder times S1, and corresponds to a 4d N = 4
subsector, which is present in the decoupling limit (in nite neck limit). The theory breaks
to N = 1, when the asymptotic region is of nite size and the states of the N = 4 vector
multiplet become massive, leaving only an N = 1 massless vector. This observation may
be used to study nonperturbative corrections, e.g. M2brane instantons [16], which has
been initiated for TCSmanifolds in [7, 8, 14].
For Spin(7)manifolds the type of constructions have thus far has been rather
limited. There are (to our knowledge) two constructions of compact manifolds with
Spin(7)holonomy due to Joyce: either in terms of a Joyce orbifold T 8= [17] or a quotient by an
antiholomorphic involution of a CalabiYau fourfold (CY4) [18]. There are variations of
{ 1 {
+
HKR
K3
1 x Z \S0
S
S
3d N = 1 theory, which is crucially related to the singular loci of the CalabiYau quotient,
and not only the sector that is 3d N = 2.
In the present paper we provide a di erent approach to constructing Spin(7)manifolds,
which is closer in spirit to the TCS construction. We will motivate a generalized connected
sum (GCS)construction where the building blocks are an open asymptotically
cylindrical (acyl) CalabiYau fourfold (CY4) and a open asymptotically cylindrical (acyl)
G2holonomy manifold (times S1), which asymptote to a CalabiYau threefold (times a
cylinder/line). This is shown in gure 2. Explicitly, we construct the acyl CY4 as
CY3 ,! CY4 ! C ;
(1.1)
and the G2 = (CY3 R)=Z2, where Z2 acts as a re ection on the circle and antiholomorphic
involution on the CY3. In case the antiholomorphic involution acts freely, these are
socalled `barely G2manifolds' [16], which have holonomy group SU(3) o Z2.
Field theoretically we will show that each of the building blocks will give rise to a
3d N = 2 subsector, and the asymptotic region CY3
cylinder results in the limit of an
in nite neck region in a 3d N = 4 sector. At nite distance, some of the modes of the
vector multiplet of 3d N = 4 become massive and the theory is broken to 3d N = 1.
{ 2 {
HJEP06(218)3
+
Z =CY
2 x S1
this paper: the left hand building block is an the acyl CalabiYau fourfold building block. We
require this to be asymptotically approach a CalabiYau threefold times a cylinder. One way to
realize this is in terms of a CY3
bration over an open P1. The right hand building block is a circle
times a G2manifold, which asymptotes to a CY3
I times a circle. The simplest way to realize this
is in terms of G2 = (CY3
region is CY3 cylinder.
R)=Z2 where Z2 acts as an antiholomorphic involution. The asymptotic
We motivate this construction in section 3 by considering a simple Spin(7) Joyce
orbifold, T 8= , which is known to have a related G2 Joyce orbifold, where the precise
connection is through Mtheory on K3 to Heterotic on T 3 duality applied to this toroidal
setting [22]. It is known that such Joyce orbifolds have a TCSdecomposition [12], which
we apply to the G2 Joyce orbifold, and subsequently uplift the decomposition to the Spin(7)
Joyce orbifold. The TCSbuilding blocks then map precisely to an open CY4 and G2
respectively. This motivates our general construction of GCS Spin(7)manifolds, which will
S1,
be given in section 4.
One may ask whether these constructions in fact globally t together to (resolutions
of) CY4 quotients, like in Joyce's constructions in [18]. Whenever the acyl G2 manifold
is realized as (the resolution of) a quotient of X3
R, this is indeed the case and we
can write our GCS Spin(7)manifolds globally as resolutions of quotients of CY4 by an
antiholomorphic involution. Here, one starts with a CalabiYau fourfold
bered by a
CalabiYau threefold CY3 as in (1.1), and acts on it with an antiholomorphic involution
on the CY3 ber combined with an action on the base P1 given by zi $
zj , i 6= j, where
zi are the homogeneous coordinates. The xed locus of this on the base is a circle and the
quotient space is halfCY4, which is a CY3fold
bered over a disc, and in the vicinity of
the boundary circle we obtain (CY3
R)=Z2 which is a (barely) G2manifold, times an S1.
Pulling the base half sphere apart, gives the decomposition into the GCS building blocks.
This is sketched in
gure 3. The key di erence to the examples and constructions in [18]
is however that the singularities are either absent or occur over a locus of real dimension
at least 1. In Joyce's construction via CY4= the singularities are only pointlike and get
resolved using an ALEspace. It seems likely that the analog to this procedure in our
construction is gluing in the open, acyl G2manifold.
{ 3 {
HJEP06(218)3
bered over a disc, which is locally a CY4. Above the boundary circle
(dark green) the quotient acts nontrivially on the CY3
interval, shown in light green, which results
in a G2. Pulling the base apart, results in the GCSdecomposition of the resulting Spin(7)manifold.
The paper is structured as follows: we begin with some background material on
Spin(7)holonomy and Mtheory compacti cations to 3d N = 1 in section 2. In section 3 we
motivate our construction by rst considering a Joyce orbifold example. Section 4 contains
our proposal for the general construction of GCS Spin(7)manifolds. We also comment
on how this construction relates to and di ers from other known setups and provide an
example of a new Spin(7)manifold. For some examples of GCS constructions there is a
K3bration of the Spin(7)manifold, and we utilize this to apply Mtheory/heterotic duality
and construct the dual heterotic on G2 compacti cations in section 5, and match the
spectra of the theories. We conclude with a discussion and outlook in section 6. For reference
we include a brief summary of the TCSconstruction of G2manifolds in appendix A.
2
Spin(7)holonomy, Mtheory and 3d N
= 1 theories
As a preparation for this work and to introduce some notation, this section reviews some
basic aspects of manifolds with holonomy group Spin(7), see [23] for a detailed discussion.
2.1
Spin(7)holonomy
A compact orientable 8dimensional manifold Z which has a Ricci at metric g with
holonomy group contained in Spin(7) supports a closed, selfdual fourform
= g , which can
be expressed in local coordinates (in which g is the euclidean metric) as
= dx1234 + dx1256 + dx1278 + dx1357
dx1368
dx1458
dx1467
dx2358
dx2367
dx2457 + dx2468 + dx3456 + dx3478 + dx5678 :
(2.1)
Here, we use dxijkl as a shorthand for dxi ^ dxj ^ dxk ^ dxl. This fourform is in the
stabilizer of the action of the holonomy group Spin(7). Such a form de nes a Spin(7)
structure, which is called torsion free if
is closed and selfdual.
{ 4 {
Having this structure in place does not necessarily mean that the holonomy group is
exactly Spin(7) and, as long as Z is simply connected, we may discriminate between di erent
cases by computing the A^ genus, which tells us the number of covariantly constant spinors:
^
A
1
2
3
hol(g)
Spin(7)
SU(4)
Sp(2)
dimensions, we will only be interested in the case A^ = 1.
terested in compacti cations of Mtheory which preserve N = 1 supersymmetry in three
For eightmanifolds Z with holonomy contained in Spin(7) and A^(Z) = 1, a necessary
and su cient condition for the holonomy group to be all of Spin(7) is that Z is simply
connected, 1(Z) = 0 [17, 23]. This still allows for cases with nontrivial subgroups of
Spin(7), and we will see examples of such spaces later on. We will refer to manifolds Z
with A^(Z) = 1 and a metric g with hol(g)
Spin(7) as barely Spin(7)manifolds.
The dimension of the moduli space of Ricci at metrics on a Spin(7)manifold is given
by b4 + 1. Together with A^ = 1, this number is already determined by the Euler
characteristic and the two Betti numbers b2 and b3 by using (2.3)
b
4 =
3
9
b2 + b3 :
Calibrated submanifolds of Spin(7)manifolds must be of real dimension four and are
called Cayley submanifolds. The dimension of the moduli space of such a Cayley
submanAll of these manifolds have b1(Z) = 0 and the remaining independent Betti numbers are
related to the A^ genus by
HJEP06(218)3
1
2
1
2
{ 5 {
ifold N is [23, 24]
mN = (N )
(N )
N
N ;
1
2
1
2
so that Spin(7)manifolds cannot possibly be bered by calibrated fourtori.
where (N ) is the Hirzebruch signature and (N ) is the Euler characteristic. Note that
this expression evaluations to
mN = 4
N
N
for a Cayley submanifold which is a K3 surface, where (N ) = 16 and
(N ) = 24. It
hence seems sensible to assume there exist Spin(7)manifolds
bered by K3 surfaces over
a fourdimensional base. Using duality to heterotic strings, it is precisely the existence of
such
brations which we will conjecture and exploit. In contrast, note that for a Cayley
submanifold with the topology of a fourdimensional torus T 4, the same computation gives
mN =
N
N
Mtheory on a Spin(7)manifold gives rise to 3d N = 1 supersymmetric theory and the
spectrum is encoded in the topological data of the Spin(7)manifold as follows:
#
b3(Z)
b2(Z)
b4 (Z)
1
3d N = 1 Multiplet
Scalar
Vector
Scalar
Scalar
g
C3
C3
11d Origin
Pib=31 i(3)'i
Pib=21 !(2)v
PbI4=1( I(4))
Volume modulus
where !(2), (3) are a basis of harmonic two and threeforms, and I(4) are a basis of harmonic
antiselfdual fourforms. For 3d N = 1 the scalar multiplet has only a real scalar as its
bosonic component and the vector multiplet is just a 3d vector. As we can dualize the 3d
vectors to real scalars, compacti cations of Mtheory on Spin(7)manifolds hence give rise to
ns = b4 + 1 + b2 + b3 =
8 + 2b3
(Z)
3
NM2 =
(Z)
1
24
spacetime lling M2branes [25, 26]. As such M2branes can freely move on Z, they each
contribute a further 8 real degrees of freedom.
In the absence of G4 ux, the e ective action for a smooth Spin(7)manifold is a 3d
N = 1 eld theory with b2 abelian vectors, and scalars with the following kinetic terms
GIJ
Z
Z I ^ J(4) ;
(4)
M
Z
Z
!(2)
^ ?!(2) :
The theory for a smooth Spin(7)manifold is an abelian gauge theory. With singularities
this can enhance to nonabelian gauge symmetries. With G4 ux, additional ChernSimons
terms and scalar potential are generated  for an in depth discussion of the e ective
theory see [20, 27{33]. In this paper we will not consider uxes but focus on the geometric
constructions.
Of course it is interesting to consider this in the future and study the
e ects of these on supersymmetry breaking and potential obstructions to dualities, both
M/Fduality [
34
], where in particular 4d Poincare invariance could be broken, as well as
obstructions to Mtheory/weaklycoupled heterotic duality [35, 36].
3
3.1
A Spin(7) Joyce orbifold as a generalized connected sum
Setup and motivation
Our goal is to construct new classes of Spin(7)manifolds  which we have motivated from
various points of view in the introduction. The construction which we will end up with is
inspired by combining two ideas:
{ 6 {
'
I
1. The recent construction of G2holonomy manifolds as twisted connected sums (TCS),
with each building block an asymptotically cylindrical CalabiYau threefold. We
review this construction in appendix A.
2. Mtheory on K3/heterotic on T 3 duality.
Combining these ideas will lead us to consider a generalized connected sum (GCS)
construction, where  as we will show  the building blocks are acyl CY fourfolds and
G2manifolds, respectively. To motive this we start with a wellknown construction by
Joyce of both G2 and Spin(7)manifolds and a wellknown duality, between Mtheory and
heterotic strings, which will be discussed in more detail in section 5.2, between
Mtheory on K3
!
Heterotic on T 3
(3.1)
This duality in 7d is based on the agreement of the moduli spaces of the two compacti
cations
Mhet=T 3 = MM=K3 = [SO(3; Z)
SO(10; Z)] nSO(3; 19)= [SO(3)
SO(10)]
R+;
(3.2)
which is both the Narain moduli space of heterotic string theory on T 3 and the moduli
space of Einstein metrics on K3 [37] (for a recent exposition in the context of berwise
application see [12]). The string coupling on the heterotic side is matched with the volume
modulus of the K3. We can
ber this over a fourmanifold M4 in such a way that there is
a duality of 3d N = 1 theories
Mtheory on Spin(7)manifold Z8
Heterotic on G2manifold J7
(3.3)
where Z8 is K3 bered over M4 and J7 has T 3 bers. This duality has been tested in the
case when both manifolds are Joyce orbifolds in [22]. More generally, berwise application
of this duality will lead us to a correspondence between two realizations of a 3d N = 1
theory, in terms of a Spin(7)compacti cation of Mtheory, and a T 3 bered G2holonomy
manifold.1 It is this setup which will motivate our construction of GCS Spin(7)manifolds:
it is the analog of the TCSdecomposition for G2manifolds on the heterotic side, mapped
to Mtheory using the duality. In this way we obtain a dual pair of connected sums:
Mtheory on GCS Spin(7)manifold
Heterotic on TCS G2manifold
(3.4)
As a warmup we now show how this works for a simple G2 Joyce orbifold, which has
a TCSdecomposition and determine what this decomposition corresponds to on the
Mtheory side. This will give a rst hint as to what the general connected sum construction
1The existence of T 3 brations of G2 manifolds has also been conjectured in the context of mirror
will be for Spin(7)manifolds.
symmetry for G2 manifolds in [13].
!
!
{ 7 {
We will start the construction of a Spin(7)manifold Z8 by considering a Joyce orbifold
T 8= , where
= Z24, where each generator of the order two subgroups acts as follows [23]:
The entries 12
denote x !
x + 12 . The singular sets are locally given by
x1
+
1
2
x2
+
1
2
+
S : 4
S : 4
S : 2
S : 2
C2=Z2
C2=Z2
C2=Z2
, the singularities of which are modelled on C4=Z2. It was
shown by Joyce that the resolution of the orbifold in this way yields a Spin(7)holonomy
manifold.
From this, the Betti numbers are computed as follows. First note that the only even
classes in H (T 8) under the Z24 (besides H0(T 8) and H8(T 8)) are given by 14 classes in
H4(T 8). Resolving the 8 singularities of the form C2=Z2
T 4=
f 1g gives
whereas the 4 singularities of the form C2=Z2
T 4 yield
b
b
b
2
3
4
b
b
b
2
3
4
6
0
1
6
4
1
C2=Z2
T 4=
f 1
g C^2=Z2
T 4=
Hence each of these 12 singularities contributes b2 = 1 and b4 = 6 and only the second
4 contribute a nonzero b3 = 4. Finally, the resolution of the 64 singularities of the form
C4=Z22 contribute b4 = 1 and nothing else. Altogether we can compute
is the stabilizer of x6 and so
The notation T(32;6;8) indicates the threetorus along the coordinates x2; x6; x8. We now
need to identify the action of
and
on this space. Again, there will be two components,
which we denote by X0 and X1=4, located at x6 = 0; 1=4, which are given by
x6 : ( ; ; ) =
;
1
2
; +
x6
x6 + 12 , and thus x6 2 I = [0; 1=4]. The generator
X0 :
X1=4 :
>>:z31=4 = x6 + ix2
Each of these halves is an open, K3 bered CY threefold. To see this introduce the
coordinates
G2manifold as follows: consider the T 7 given by x1; x2; x3; x5; x6; x7; x8, and act with
Z2. This is a Joyce construction of G2manifolds by orbifolds as already observed
TCSdecomposition of the Joyce G2manifold
Our goal is to rst identify the TCS description of this Joyce G2orbifold and to then
lift this to a connected sum description, GCS, for the Spin(7)manifold. First note that
T 7=h ; ; i is bered over an interval
HJEP06(218)3
(3.10)
(3.11)
(3.12)
(3.13)
(3.14)
The K3s are along z1; z2, where
acts nontrivially. The remaining coordinates combine
with the interval coordinate x6. At x6 = 0(1=4) the x8(x2) circle pinches. A sketch is
shown in
gure 4. The holomorphic threeform is invariant under these actions, making
these open CY3.
3.4
Uplift to GCSdecomposition of the Spin(7)manifold
Adding back the circle along the x4 coordinate, as well as the additional orbifold generator
z
,
0
1 !
acts trivially on x6, so that the
bration over the interval remains intact, but
:
z10, so that the
(3;0) form is not invariant any longer. Let us now again de ne two
building blocks along the x6 interval, close to either boundary. The interesting observation
is that the two halves behave quite di erently:
rst consider x6 = 0. De ne another
complex coordinate with the action of
z
0
4
x2 + ix4
!
0
z4 :
{ 9 {
Z1=4, which are an open CalabiYau fourfold and an open G2manifold times S(18). These are glued
together along the x6 coordinate. Both asymptote to Xs times a cylinder, where Xs is the Schoen
CalabiYau threefold. In this particular example each building block contains a K3, which is also
present in the neck region of the Schoen CalabiYau.
Together with the action of
on zi0, we see that
(4;0) = dz10 ^ dz20 ^ dz30 ^ dz40 is invariant,
0
and we obtain a building block which is an open CalabiYau fourfold
At the other end of the x6 interval we have
where M7 will be shown to be a G2 holonomy manifold. The action of
is
From this we can de ne the G2form
3 = dx4 ^ !(1;1) + Re (3;0) ;
3 =
2
1 !(1;1)
^ !(1;1)
dx4 ^ Im
(3;0) ;
(3.15)
(3.16)
(3.17)
(3.18)
which is invariant, as follows from the action of
and the coordinates z1=4.
In summary we obtained a GCSconstruction of a Spin(7) Joyce orbifold Z, which has
two building blocks, an open CY4 and an open G2, W times a circle, respectively. The
geometry in the middle of the x6 internal is
Q = Xs
where the Xs is a CY3 along the directions 1; 2; 3; 4; 5; 7. Not too surprisingly, this is in fact
the Schoen CalabiYau threefold. The action along these coordinates, written in terms of
the complex coordinates of Z0 is
(see [12] for a detailed discussion of that). Inside Xs, there is a K3 surface along z10; z20.
On the other hand the CY4 Z0 is bered by Xs over x6; x8.
Note that we may also think about this Spin(7) orbifold as a quotient of a
CalabiYau fourfold X4 by an antiholomorphic involution. The action of ; ;
respects the
holomorphic coordinates zi0 on T 8 and leaves
0
(4;0) invariant, so that it produces a
CalabiYau orbifold. The full Spin(7) orbifold is then formed by acting with , which acts as an
antiholomorphic involution. We expect this structure to persist after resolution.
The Joyce orbifold (3.5) has several generalizations, which can be parametrized in
terms of the following data:
(3.19)
(3.20)
(3.21)
(3.22)
(3.23)
where the shift vectors can take the following values
Case I is the orbifold studied above. Instead of the bration over the interval (3.10), in this
general setting, we consider pulling these orbifolds apart along x3, with the interval x3 2
[0; 1=4]. There is an asymptotic middleregion, where the space is Xs = T(61;2;5;6;7;8)=h ; i,
which in fact is again the Schoen CalabiYau threefold for all of these Joyce orbifolds. The
asymptotic manifolds at the
and G2manifold M7, respectively, given by
xed points x3 = 0; 14 are again open a CalabiYau fourfold
x3
+
+
d3
x4
+
+
+
x5
+
c5
d5
(1; 1; 1; 1) ;
(1; 0; 1; 0) ;
(1; 1; 1; 0) ;
(1; 0; 1; 0) ;
x6
+
c6
+
1
2
1
2
1
2
1
2
d =
d =
d =
d =
x7
+
+
d7
x8
+
+
+
(0; 1; 1; 1)
(0; 1; 1; 1)
(0; 1; 1; 0)
(0; 1; 1; 0) :
I :
II :
III :
IV :
x1
+
c1
d1
x2
c2
+
+
1
2
1
2
1
2
1
2
c =
c =
c =
c =
The analysis will follow very much along the lines of the rst example we studied and we
will instead now move to generalize this construction beyond Joyce orbifolds.
We have seen that the Joyce orbifold Z8 (3.5) has a natural decomposition in terms
of a GCSconstruction, where one building block is an open CalabiYau fourfold, and the
other is a G2manifold times a circle, where both geometries asymptote to a CalabiYau
threefold Xs times a cylinder. With this example in mind, we now turn to providing a
generalization of this construction in the next section.
4
Spin(7)manifolds as generalized connected sums
In the discussion of the last section, we were led to think about Spin(7) Joyce orbifolds
as being glued from two eightmanifolds: a noncompact CalabiYau fourfold and the
product of a noncompact G2manifold with a circle. These two halves are glued along
their overlap, which is the product of a CalabiYau threefold with an open cylinder I
S
1
 see gure 2. We are now going to propose a generalization of this procedure in which both
the two building blocks Z
and the resulting Spin(7)manifold Z are no longer (resolutions
of) orbifolds. To start this discussion, let us rst propose the following de nitions:
De nition. An asymptotically cylindrical (acyl) CalabiYau fourfold Z+ is a noncompact
algebraic fourfold, which admits a Ricci at metric, is simply connected, and is di
eomorphic (as a real manifold) to the product of a cylinder I
S1 and a compact CalabiYau
threefold X3 outside of a compact submanifold
. The Ricci at metric of Z+
exponentially asymptotes to the Ricci at metric on the product I
S
1
X3 = X4 n .
Asymptotically CalabiYau manifolds were discussed in [38]. Similar to the explicit
construction of acyl threefolds in [5, 6], we expect to able to construct Z+ by excising a
ber X3 from a compact fourfold Z~+ with c1(Z~+) = [X3], which is bered by CalabiYau
threefolds. This implies that we can think of producing Z+ by appropriately cutting a
compact, CY3 bered CalabiYau fourfold in half, or via toric methods as in [9]. Likewise
De nition. An asymptotically CalabiYau (acyl) G2manifold Z is a noncompact
manifold of G2 holonomy, which is di eomorphic to the product of an interval I with a compact
CalabiYau threefold X3 outside of a compact submanifold of Z , and the Ricci at metric
on Z
exponentially asymptotes to the Ricci at metric on the product manifold I
X3.
Such manifolds were discussed in [39]. Note that such G2manifolds are easily
constructed as (resolutions of) orbifolds (X3
Rt)=Z2 with the Z2 acting as t !
t and as
an antiholomorphic involution on the CalabiYau threefold X3. The quotient by such
involutions can be thought of as a CalabiYau threefold X3
bered over an halfopen
interval which undergoes a degeneration at one end. In particular, it is not surprising if the
Ricci at metric on such a resolution of (X3
R)=Z2 asymptotes to the Ricci at metric
on X3
R far away from the origin of R. A particularly simple case is given by freely
acting antiholomorphic involutions. In this case no resolution is required, and the Ricci
at metric on (X3
R)=Z2 is simply the quotient of the Ricci at metric on X3
R. The
simplicity of this examples comes at a price, however, as the holonomy group of such acyl
G2manifolds is not the full G2 [39, 40], but only SU(3) o Z2, so that we can call them
acyl barely G2manifolds. Such acyl barely G2manifolds in particular have a nontrivial
fundamental group Z2.
We are now ready to propose our construction of what we will call generalized
conZ
Z
nected sum (GCS) Spin(7)manifolds. Take an acyl CalabiYau fourfold Z+ with
asymptotic neck region isomorphic to It+
1
S +
X3+ for t+ = 0
with asymptotic neck region isomorphic to It
X3 for t
l and a acyl G2manifold
= 0
l. Then Z+ and
1 can be glued as topological manifolds to a manifold Z by identifying the neck
regions It
S
1
X3 such that
t+ = l
+ =
t
:
X3+ = X3
(4.1)
for a biholomorphic map . If the Ricci at metrics on Z
asymptote to the Ricci at
metrics on X3 , this takes us close to a Ricci at metric on Z. We conjecture that for long
enough neck regions (l large enough) there exists a Ricci at metric g associated with a
torsion free Spin(7) structure on Z, which is found by a small perturbation of the Ricci at
metrics on Z+ and Z
S1.
If Z
are both simply connected, it follows from the Seifertvan Kampen theorem that
Z is a simply connected eightmanifold. This means that the holonomy group of Z must
be equal to Spin(7) (and not a subgroup) if Z has a torsionfree Spin(7) structure.
Before exploring the consequences of our proposed construction further, let us brie y
discuss the mathematical work needed to put our proposal on rm ground. The work [38{40]
on acyl CalabiYau fourfolds and acyl G2manifolds and their deformation theory, together
with clear criteria when the asymptotic CalabiYau threefolds X3 allow a biholomorphic
map , should clarify under which circumstances a gluing can be found for a given pair of
such manifolds. In our examples, we can easily nd such di eomorphisms by realizing X3
as hypersurfaces in toric varieties, so that a di eomorphic pair can be simply constructed by
writing down identical algebraic equations. The crucial task is then to show the existence of
a Ricci at metric with holonomy Spin(7) on the resulting topological manifold Z. By
making the neck regions very long, we expect that the torsion introduced when gluing Z+ and
Z
S1 can be made su ciently small for a torsionfree Spin(7) structure to exist nearby.
As we shall discuss in more detail in section 4.5, there are instances of GCS
Spin(7)manifolds which can globally be realized as (resolutions) of quotients of CalabiYau
fourfolds. Quotients by free actions, where no resolution is needed, or situations in which
resolutions with Ricci at metrics can be constructed provide nontrivial examples of our
construction in which their existence is proven by other means.
4.2
Both the acyl CalabiYau fourfold Z+ and the product of the acyl G2manifold Z
with
a circle have a torsion free Spin(7) structure. In this section we show how the
identication (4.1) between the asymptotic neck regions produces a globally de ned Spin(7)
structure.
Let us denote the Kahler form and holomorphic fourform of Z+ by !+ and
+. The
Cayley fourform de ning a torsionfree Spin(7) structure on Z+ is then given by
In the asymptotic neck region, we can introduce coordinates t+ and + and decompose the
SU(4) structure as
+ = Re + +
!+ ^ !+ :
(4.2)
3;+ are the Kahler form and holomorphic threeform on X3+. This means
that the Spin(7) structure in the neck region is
1
2
1
2
which is noting but the map (4.1) proposed earlier.
t+ =
+ =
3;+ =
!3;+ = !3;
t
3;
(4.3)
(4.4)
(4.5)
(4.6)
(4.8)
+ = d + ^ Re 3;+
dt+ ^ Im
3;+ +
!3;+ ^ !3;+ + d + ^ dt+ ^ !3;+ :
Similarly, the Spin(7) structure on Z
S
1 is given by
in terms of the G2 structure '
on Z . In the neck region, this is further decomposed as
+ = d
^ '
+ '
'
'
=
=
dt ^ !3; + Re 3;
1
2 !3; ^ !3; + dt ^ Im
3;
Spin(7) structure on the neck region of Z
S1 can be written as
in terms of the Kahler form and holomorphic threeform on X3 . We hence nd that the
=
d
^ dt ^ !3; + d
^ Re 3; +
1
2 !3; ^ !3; + dt ^ Im
3; :
(4.7)
This means that the two Spin(7) structure
are consistently glued together under a
di eomorphism which identi es the neck regions as
The easiest topological number to determine for the GCS Spin(7)manifold Z is given by
the Euler characteristic (Z). As it is additive, we have
(Z) = (Z+) + (Z
S1)
Z+ \ (Z
S1) = (Z+) ;
(4.9)
where we have used the fact that the Euler characteristic vanishes for any manifold with
an S1 factor.
Two copies Z+, Z+0 of the acyl CalabiYau fourfold Z+ can be glued to a compact
CalabiYau fourfold X4 = Z+ [ Z+0 such that Z+ \ Z+0 = I
threefold X3. It follows that
(X4) = 2 (Z+) = 2 (Z). As the Euler characteristic of
any CalabiYau fourfold is divisible by six [41], this implies that (Z) is divisible by 3, so
S
1
X3 for a CalabiYau
that b4 in (2.4) is always an integer.
As we have de ned an acyl CalabiYau fourfold to be simply connected, but left room
for the possibility of an acyl G2manifold to have a nontrivial fundamental group, the
Seifertvan Kampen theorem tells us that Z is simply connected if an only if Z
is simply
connected. If Z
is not simply connected, Z can have a nontrivial fundamental group,
which signals that the holonomy group of Z is smaller than Spin(7) [17, 23].
The easiest way to construct acyl G2manifolds is by a free quotient X3
R by Z2,
in which case the fundamental group of Z
is not trivial but equal to Z2. This works as
follows. Consider two points identi ed by the antiholomorphic involution on X3 over the
origin of R. Any path connecting two such points becomes a closed loop in the quotient and,
as the involution acts freely, cannot be homotopic to a point. If X3 is simply connected,
all such loops are homotopic, so that 1((X3
R)=Z2) = Z2, which implies that Z
only
has holonomy group SU(3) o Z2, i.e. is a barely G2 manifold. On Z, such loops give rise
to a nontrivial fundamental group, which is Z2 as well, so that Z does not have the full
holonomy group Spin(7). From the point of view of physics, compacti cation on such
`barely' Spin(7)manifolds does not give rise to extended supersymmetry as there is still
only one covariantly constant spinor. In fact, it is not hard to see that Z has holonomy
group SU(4) o Z2 in this case: the holonomy group SU(4) of Z+ and the holonomy group
SU(3) o Z2 of Z
share a common SU(3), the holonomy group of X3.
We can compute the cohomology groups of Z in terms of the cohomology groups of Z
and the pullbacks to X3 by using the MayerVietoris exact sequence, which implies that
where
can be expressed in terms of the restriction maps
Hi(Z; Z) = ker i
coker i 1
;
i : Hi(Z+; Z)
Hi(Z
S1; Z) ! Hi(X3
S1; Z)
+i : Hi(Z+; Z) ! Hi(X3
i : Hi(Z ; Z) ! Hi(X3; Z) :
S1; Z)
Note that both i and i 1 feature in i due to the product S1.
(4.10)
(4.11)
(4.12)
Under a few assumptions, which are met in the examples to be discussed later, we can
now explicitly work out the various contributions to H (Z). First of all, let us assume that
the acyl CalabiYau fourfold Z+ can be constructed from an algebraic fourfold Z~ which is
bered by CalabiYau threefolds by excising a
ber X3. Furthermore, let us assume that
the images of +2 and
+4 are surjective and that b3(Z+) = b5(Z+) = 0.
As noted above, we assume that we can construct Z
as (a resolution) of the quotient
(X3
Rt)=Z2 by an antiholomorphic involution. The resolution must be such that it
preserves a metric of holonomy G2 on Z
and the property of Z
being asymptotically
cylindrical. We will denote the numbers of even/odd classes of X3 under this Z2 by bie
and bio, respectively. Finally, the kernels of the restriction maps i and their ranks are
abbreviated as
N i
ker i
n
i
jN i j :
With this notation and under the assumptions we have made, (4.10) implies that
b1(Z) = 0
b2(Z) = n2+ + n2 + be2
b3(Z) = n
2 + n3
b5(Z) = n
4 + n5
b6(Z) = n6+ + n5 + bo4
b7(Z) = 0 :
b4(Z) = n
3 + n4 + n4+ + bo2 + bo3 + be3 + be4
(4.13)
(4.14)
This can be seen as follows. For H1(Z), the unique class in H1(X3
S1) is in the image
of 1 and solely originates from H1(Z
coker( 0) = 0 implies b1(Z) = 0.
S1). Hence j ker( 1)j = 0, which together with
Let us now consider b2(Z). As the unique class in H1(Z
S1) is in the image of 1
we nd that coker( 1) = 0 so that H2(Z) = ker( 2) follows. In turn, ker( 2) has three
di erent contributions:
ker( 2) = ker( +2)
ker( 2 )
im( +2) \ im( 2 ) :
(4.15)
As +2 is surjective by assumption, the last term is simply given be2 and we recover the
expression in (4.14).
The computation for b3(Z) is made particularly simple by our assumptions.
As
contribution to b3(Z) is from ker( 2 )
ker( 3 ) = N 2
N 3 .
j coker( 2
)j = 0, it only receives a contribution from ker( 3). As b3(Z+) = 0, the only
For b4(Z), all potential contributions are nontrivial. First of all, there is a contribution
n4+ from ker +4, as well as n3 + n4 from ker 3 + ker 4 . Furthermore, there are bo2 classes
in coker( 3), which correspond to threeforms with one leg on the product S1 of the cylinder
region, and bo3 classes in coker( 3) which correspond to threeforms purely on X3. Finally,
there is a contribution
(im +4) \ (im 4 )
(im 3 ) ;
(4.16)
which simply has dimension be4 + be3 as +4 is surjective. Together, all of these contributions
give the expression for b4(Z) in (4.14).
The computation of b5(Z) again only receives a contribution from ker 5 as coker 4
is empty by assumption.
Furthermore,
5 is only nontrivial on Z
S1, so that
j ker 5j = n
The computation of b6(Z) is similar to b2(Z), it receives a term bo4 from coker( 5), as
well as terms n5 and n6+ from ker( 6). Note that N 6 = 0 as Z is a G2 manifold.
Finally, coker( 6) = 0 and ker( 7) = 0 gives b7(Z) = 0.
The various contributions n
i and hie; hio are not all independent and their relations
make Poincare duality for (4.14) manifest. For antiholomorphic involutions of a
CalabiYau threefold X3, the Lefschetz xedpoint theorem combined with Poincare duality
implies that bo2 = be4 and be2 = bo4. Furthermore, Poincare duality on the compact G2 manifold
(X3
S1)=Z2 implies that n2 = n
5 and n3 = n4 . Together with n2+ = n6+, these relations
imply that bi(Z) = b8 i(Z) in (4.14).
The number of deformations of the Ricci at metric is given by b4 + 1 for manifolds
of Spin(7)holonomy. Using (2.4), this can be written as
(4.17)
(4.18)
(4.19)
which becomes
1
3
1
3
b4 + 1 =
8 +
2
b2 + b3 + b4 ;
b4 + 1 =
8 + n3 +
2 + n4+
n2+ + 2bo2
be2 + be3 + bo3
in our GCSconstruction. The Euler characteristic of Z is given by
(Z) = 2 + 2b2(X3) + b3(X3) + n4+ + 2n2+ ;
and only depends on the data of Z+, as expected.
Even though the topology of the resolutions of Joyce's Spin(7)orbifolds discussed in
section 3 is most conveniently computed otherwise, one can also use their decomposition
as GCS Spin(7)manifolds together with the relations of this sections to nd their topology
from resolutions of two building blocks Z .
4.4
3d
eld theory and sectors of enhanced supersymmetry
In compacti cations of MTheory on a smooth Spin(7)manifold, the low energy e ective
theory at the classical level gives a 3d N = 1 supergravity theory with nv = b2(Z) massless
U(1) vector multiplets and nr = b3(Z)+b4 (Z)+1 massless uncharged real scalar multiplets.
As the Spin(7)manifolds considered here are glued from pieces with holonomy SU(4) and
G2, respectively, we expect to nd subsectors of enhanced supersymmetry in our spectrum,
similar to the observations made in [11] regarding TCS G2 manifolds, to arise from localized
forms in the building blocks.
Consider rst the multiplets arising from localized forms on Z
S1. By (4.14), each
twoform in N 2 gives rise to both a twoform and a threeform on Z, which combine
to form the bosonic eld content of a 3d N
= 2 vector multiplet. Furthermore, each
threeform in N 3 gives rise to a real scalar due to its appearance in the formula for b3(Z)
in (4.14). As for compacti cations of MTheory on a G2manifold times a circle, the moduli
associated to these threeforms pair up with deformations of the metric to form 3d N = 2
chiral multiplets: this can be seen from (4.18), which shows that each threeform in N 3
corresponds to an antiselfdual fourform, i.e. a deformation of the metric.
Let us now turn to multiplets which arise from localized forms on Z+. Each twoform
of Z+ in N+2 gives rise to a 3d N = 1 vector multiplet due to its contribution to b2(Z).
As for compacti cations of MTheory on CalabiYau fourfolds, we expect this degree of
freedom to pair up with a real scalar to the bosonic degrees of freedom of a 3d N = 2
vector multiplet. This degree of freedom must originate from an antiselfdual fourform
in N+4 . Furthermore, we expect the remaining antiselfdual fourforms in N+4 to appear
pairwise, so as to combine into 3d N = 2 chiral multiplets.
To see how this comes about, we need to exploit the fact that Z+ and its
compactication Z~+ are Kahler manifolds and carry a Lefschetz SU(2) action. The upshot is that
the antiselfdual fourforms on Z~+ are given by the fourforms in H3;1(Z~+)
H1;3(Z~+),
together with fourforms of the type !k ^ J , where J is the Kahler form on Z~+ and the !k
are (1; 1) forms on Z~+ such that !k ^ J 3 = 0. As every twoform in N+2 gives a fourform
in N+4 upon wedging with J , we can hence associate an antiselfdual fourform in N+4 on
Z+ to every element of N+2 (modulo the (1; 1) forms in the image of +2). The remaining
antiselfdual fourforms in N+4 then have to appear pairwise, as they must correspond to
H3;1(Z~+)
H1;3(Z~+). As we can think about the forms in N+4 as being localized on the
acyl CalabiYau fourfold Z+ far away from the gluing region, (anti) selfduality on Z+ will
imply (anti) selfduality of the corresponding forms on Z.
Note that this argument precisely re ects how these degrees of freedom originate in
Physics. For twoforms in N+2 and selfdual fourforms in N+4 , there are deformations of the
Ricci at metric of Z+ (Kahler and complex structure deformations), which do not alter
the cylindrical region in which Z+ is glued to Z
S1. As we expect the Ricci at metric
of Z to be well approximated by the Ricci at metrics of Z+ far away from the gluing
region, these become deformations of the Spin(7)manifold Z if we stretch the neck region
su ciently long. The metric deformations associated with N+2 and N+4 , together with the
3d vectors originating from the C eld on N+2 , only see the geometry of a CalabiYau
fourfold and hence give rise to a subsector with 3d N = 2 supersymmetry. As the property
of forms being (anti) selfdual is a topological constraint, the counting and identi cations
we have performed will persists throughout the moduli space of Z, so that the subsectors
with enhanced N = 2 supersymmetry in 3d will persist. We have summarized the result
of this discussion in table 1.
The remaining moduli of MTheory on the GCS Spin(7)manifold Z do not appear
in multiplets with enhanced supersymmetry, as they are associated with the gluing along
X3
S1. These moduli are, however, in onetoone correspondence with the geometric
deformations of the CalabiYau threefold X3 in the neck region, i.e. Kahler and complex
structure deformations of X3. While compacti cation of MTheory on X3 S
1 S1 results in
a theory with N = 4 supersymmetry in 3d, these multiplets are truncated to N = 1 for
MTheory compacti ed on Z. Similar to the case of TCS G2 manifolds [11], the eigenvalues
unHJEP06(218)3
N = 2 vector
N = 2 chiral
N = 4 vector
N = 2 vector
N = 2 chiral
Origin
N+2
h3;1(Z~+)
b2(X3), b3(X3)
N 2
N 3
multiple from the neck region is only massless in the in nite neck limit and away from this becomes
der the Laplace operator of the appropriate nonharmonic forms needed to lift these N = 1
multiplets to N = 4 multiplets go to zero in the limit in which the neck region is stretched.
GCS Spin(7)manifolds as quotients of CY4
Our construction has a natural connection to Spin(7)manifolds constructed as quotients
of CalabiYau fourfolds by antiholomorphic involutions, however we will see that there
are key di erence to the previous constructions in [18, 20]. Consider a compact CalabiYau
fourfold X4, which is bered by CalabiYau threefolds over a base P
coordinates [z1 : z2] and let z = z1=z2. Let us denote the ber over a point p of the P1 base
by X3(p). For appropriate brations, we may then tune the complex structure (de ning
1 with homogeneous
equation) of X4 such that
For appropriate antiholomorphic involutions
construct an antiholomorphic involution
acting on X3(p) for all p, we may then
X3(z) = X3(1=z) :
=
(z1 ! z2)
z2 ! z1
1
Sf
(X3
R)= :
acting in X4. This will produce a singular Spin(7)manifold Zs = X4= , and we will
assume that it can be resolved into a smooth Spin(7)manifold Z.
In the P1 base the xed locus of this involution will be given by the circle Sf1 = fjzj2 = 1g
and the quotient e ectively truncates the base P1 to a disc with boundary. We may make
this disc very large and furthermore con ne all of the singular
bers of the
bration of
X3(p) near the origin. If the bration of X3(p) over the xed Sf1 is trivial, the bers over
Sf1 all become identical and we may simply denote them by X3. Cutting along a circle of
xed radius now produces one half near the boundary of the disc which may be described as
In the limit in which the disc is very large, a resolution of Zs to Z is equivalent to a
resolution of (X3
R)= to a smooth acyl G2 manifold Z . The other half of Z near the origin
of the disc does not require resolution and becomes an acyl CalabiYau fourfold Z+ which
(4.20)
(4.21)
(4.22)
asymptotes to X3. In the middle region, the two spaces Z+ and Z
the product of an interval and X3
given above. An illustration is given in gure 3.
Sf1 . This is precisely the gluing construction we have
Sf1 are glued along
Let us highlight two aspects of the topology of Z which are immediately recovered
from this point of view. The resolution of Zs to Z only introduces new cycles sitting over
a circle, so that the topological Euler characteristic of Z is the same as that of Zs. But
the topological Euler characteristic of Zs is simply given by half of the Euler characteristic
of X4. In fact, we may think of X4 as being glued from two copies of the acyl
CalabiYau fourfold Z+, which implies that the Euler characteristic of Z is equal to that of Z+,
which is the same result found from our gluing construction earlier. Furthermore, we have
observed that we produce a barely Spin(7)manifold with fundamental group Z2, whenever
Z is constructed from a free quotient of X3
R by . This means that we may write Z
as the free quotient X4= , so that we immediately recover 1(Z) = Z2.
As we have explained, a GCS Spin(7)manifold can be described as a resolution of a
quotient of a CalabiYau fourfold by an antiholomorphic involution whenever the acyl
G2 manifold Z
can be described as a resolution of (X3
R)= for an antiholomorphic
involution
. In the absence of a di erent construction for acyl G2 manifolds, our GCS
construction is hence equivalent to forming quotients of a speci c class of CalabiYau
fourfolds. In this context, it implies that a resolution of singularities of the quotient by
is
already captured by resolving quotients (X3
R)= . Furthermore, it gives a completely
new point of view on such geometries; it shows how to distinguish subsectors of enhanced
supersymmetry and, as we shall see, allows to construct dual heterotic backgrounds on TCS
G2 manifolds. Finally, the realization that many GCS Spin(7)manifolds are simply
(resolutions of) quotients X4= in fact proves that our construction produces eightdimensional
manifolds with Ricci at metrics with holonomy group Spin(7) (or SU(4) o Z2).
In [18] Joyce proposed another construction of Spin(7)manifolds base on nonfree
quotients of CalabiYau fourfolds with orbifold singularities was given. Crucially, the
resulting singular Spin(7) orbifolds are taken to only have isolated singularities here, which
can be resolved by gluing in appropriate ALE spaces. Even though similar in spirit, the
antiholomorphic involution appearing here are either free or have xed loci of real dimension
at least one, so these constructions are distinct. It seems likely, however, that there are
examples for which the resolutions can be described by gluing in a acyl G2 manifold times
a circle, so that there will be many Spin(7)manifolds which can be found using both
4.6
A simple example
We now construct a new Spin(7)holonomy manifold using the GCS construction. There
will be further examples in the next section, which are geared toward the application of
Mtheory/Heterotic duality.
Consider a smooth anticanonical hypersurface in weighted projective space P11114.
The resulting space is a CalabiYau threefold X1;149 with Hodge numbers h1;1 = 1 and
h2;1 = 149. We now show how to construct an acyl G2manifold Z
which asymptotes
to X1;149
R and an acyl CalabiYau fourfold Z+ which asymptotes to X1;149
S
1
R,
and then glue Z+ and Z
S1 to a Spin(7)manifold Z as described above. Let us rst
describe the acyl G2manifold Z . The CalabiYau threefold X1;149 admits a freely acting
antiholomorphic involution, which is obvious by choosing a Fermat type hypersurface
where [x1 : x2 : x3 : x4 : x5] are the homogeneous coordinates of P11114. Letting xi ! xi
then acts as a
xedpoint free antiholomorphic involution, as the above equation has no
solutions purely over the reals. The odd/even Hodge numbers are
he2 = 0 ;
ho2 = 1 ;
he3 = ho3 = 149 ;
and (X1;149
R)=Z2 is an acyl G2manifold. Note that ni = 0 for all i as no resolution of
singularities is required on Z .
To nd an acyl CalabiYau fourfold, we consider an algebraic fourfold Z~+ which is a
bration of X1;149 over P1 and has c1(Z~+) = [X1;149]. Such a Z~+ can easily be constructed
by as the zero locus of a generic section of O(8; 1) over P11114
h1;1(Z~+) = 3, which means that n2+ = 1. Furthermore, the resulting acyl CalabiYau
P1. Such a fourfold has
fourfold Z+ = Z~+ n X1;149 has (Z+) =
(Z~+)
(X1;149) = 1680 =
(Z).
Note that we may choose an element of the algebraic family Z~+ such that the ber over
some chosen point in the base P1 is described by (4.23), so that Z+ manifestly asymptotes
to a CalabiYau threefold which is isomorphic to the CalabiYau threefold in the neck
region of Z . For the resulting GCS Spin(7)manifold constructed from this choice of Z+
and Z , it now follows that
(4.23)
(4.24)
HJEP06(218)3
b2(Z) = 1
b3(Z) = 0
b4 (Z) = 551
b4+(Z) = 1125 :
(4.25)
Using (2.4), this is most e ciently calculated by using (Z) = 1680. One can also work
out that n4+ = 1374, which reproduces b4(Z) = 1676 using (4.14). Due to the free action
of the antiholomorphic involution on X1;149, this example produces a barely
Spin(7)manifold. By considering more general antiholomorphic involutions and resolving the
resulting singularities, similar examples with full holonomy Spin(7) can be constructed
from this procedure.
Hodge numbers of X4 are
In agreement with our general discussion in section 4.5, Z may also be found as the
free quotient of a CalabiYau fourfold X4 realized as a hypersurface in P11114
P1. The
h1;1(X4) = 4
h2;1(X4) = 0
h3;1(X4) = 548
h2;2(X4) = 2252
(4.26)
and (X4) = 3360. The free involution
acts as
xi ! xi
z1 ! z2
z2 ! z1 ;
(4.27)
where [z1 : z2] are homogeneous coordinates on the base P1 and xi are the homogeneous
coordinates on P11114. We may choose a smooth invariant hypersurface on which
acts
without xed points as
4
i
(z12 + a5z1z2 + z22)x52 + X(z12 + aiz1z2 + z22)xi8 = 0 ;
(4.28)
for a set of real parameters ai. A little thought reveals that a onedimensional subspace of
the forms in H1;1(X4) is even under , which together with (X4) = 3360 and b3(X4) = 0
implies (4.25).
HJEP06(218)3
5
Mtheory on GCS Spin(7)/heterotic on TCS G2
In this section we construct a new class of GCS Spin(7) manifolds Z following our general
discussion in section 4, where the building blocks are an open acyl CalabiYau fourfold Z+
and an open acyl G2manifold Z
(times a circle), which are both asymptote to a cylinder
times the same CalabiYau threefold X3. The additional input in this section is that we
will consider these in the context of the duality to Heterotic strings theory on G2manifolds,
which we show to be TCSmanifolds, with building blocks X .
Our strategy in nding dual compacti cations will be similar to the one we used in [12],
i.e. using
berwise duality between Mtheory and heterotic string theory, we will identify
dual pairs of building blocks, which can then be glued to
nd the dual Spin(7) and
G2manifolds. As this requires to carefully dissect geometries on both sides, the following
discussion will unfortunately be rather technical. After motivating our construction for
a dual pair of geometries, we will summarize our results in section 5.4 before checking
the spectra. In our rst example, the E8
E8 gauge symmetry is broken by a (quotient
of a) Wilson line, and we can easily match the light degrees of freedom of Mtheory to
those of the dual heterotic string theory. In general, such a match is much more di cult,
as it requires to determine the number of moduli associated with a vector bundle on a
G2manifold. We present a second closely related example in which we use the proposed
duality to instead predict the number of bundle moduli on the G2 manifold.
We begin with a brief summary of heterotic G2systems in section 5.1 and a discussion
of the duality in section 5.2, before then constructing examples of dual pairs.
5.1
Heterotic string theory on G2manifolds
Before we begin with a detailed discussion of dual pairs, we should explain how to construct
3d N = 1 theories from heterotic string theory on a G2holonomy manifold together with
a vector bundle. Here we will discuss brie y heterotic compacti cations on G2holonomy
manifolds.
A compact orientable 7dimensional manifold J which has a Ricci at metric g with
holonomy group contained in G2 supports a closed threeform
, the Hodge dual of which
is closed as well, which can be expressed in local coordinates (in which g is the euclidean
= dx123 + dx145 + dx167 + dx246
dx257
dx347
dx356 :
(5.1)
This threeform and its Hodge dual are preserved by the action of G2. Besides having a
Ricci at metric, G2manifolds support a single covariantly constant spinor.
In contrast to Spin(7)manifolds, a necessary and su cient condition for the holonomy
group of a manifold J with a Ricci at metric g and hol(g)
G2,to be equal to G2 is
that the fundamental group of J is
nite rather than trivial. We will refer to manifolds
which support a single covariantly constant spinor, but have holonomy group SU(3) o Z2 as
barely G2manifolds. The only nontrivial Betti numbers of a G2manifold are b2 and b3 as
b1 = 0, and the moduli space of Ricci at metrics for a G2manifold has real dimension b3.
For G2manifolds, calibrated submanifolds can have dimension three (associative cycles), in
which case they are calibrated by , oder dimension four (coassociative cycles), in which
case they are calibrated by
.
Let us now discuss compacti cation of heterotic E8
E8 strings on G2manifolds.
Earlier studies of this in the context of resolutions of Joyce orbifolds T 7=
have appeared
in [22, 42, 43]. The most in depth analysis of the conditions that such a compacti cation
needs to satisfy were derived in [44] and are given in terms of a heterotic G2system. This is
comprised of a G2holonomy manifold and its tangent bundle (J; T J ) as well as an E8
E8vector bundle with connection (V; A). The curvatures R and F of T J and V satisfy
Furthermore there can be NS5branes wrapped on associative threecycles, subject to the
anomaly condition
R ^
= 0 ;
F ^
= 0 :
0
4
(Tr F ^ F
Tr R ^ R) = dH ;
(5.2)
(5.3)
where dH =[NS5] the Poincare dual of the homology class corresponding to associatives
wrapped by NS5branes.
5.2
GCS Spin(7)manifolds and Mtheory/heterotic duality
Up to this point, our construction of GCS Spin(7)manifold was largely motivated for
constructions of new geometries. An alternative motivation is to realize that these give
rise to Mtheory duals of heterotic strings on TCS G2manifolds. A review of the TCS
construction is given in appendix A. Let us hence consider heterotic strings compacti ed
on a manifold of G2 holonomy J , together with an appropriately chosen vector bundle,
giving an e ective 3d theory with N = 1 supersymmetry. Fiberwise duality to Mtheory
is possible if J admits a calibrated
bration by threetori T 3, which are replaced by K3
surfaces in the dual compacti cation of Mtheory. As 3d compacti cations of Mtheory with
N = 1 supersymmetry are found upon compacti cation on manifolds of Spin(7) holonomy,
we expect the resulting geometry to be in this class.
We already discussed the 7d duality between M/K3 and Het/T 3 in section 3.1. We
now apply this
berwise to construct dual pairs of TCS and GCS constructions, in cases
when there is a K3 bration on the GCS Spin(7)manifold and dually on the heterotic side
there is a T 3 bered TCS G2manifold. In this section we make some initial observations
about the general features of these compacti cations.
If J is a TCS G2manifold, mirror symmetry for Type II strings implies the existence
of a calibrated T 3 bration, which realizes a mirror map in the spirit of
StromingerYauZaslow (SYZ) [45] via three Tdualities [13]. Such a
bration exists if one of the two
building blocks, say X+, used in the TCSconstruction of J is bered by K3 surfaces S+,
which in turn admit an bration by an elliptic curve E
E ,! S+ ! P1 :
(5.4)
E
1
Sb; becomes the SYZ
ber of X .
This implies that X+ admits an elliptic bration with ber E as well, and the T 3 bers of
J restricted to E
1
Se;+ on X+
Se1;+. The Donaldson matching (A.1) then maps E to a
sLag ber of the K3surface S , and Se;+ to Sb1; . Going away from the neck region of X ,
1
Next let us consider how this maps under Heterotic/Mtheory duality:
HJEP06(218)3
X+: we replace the T 3 bration given by E
1
Se;+ on X+
Se1;+, with a K3 surface
S+. Then only two out of the three forms of the hyperKahler structure of S+ have
a nontrivial bration over the base. In particular, we may choose to described the
resulting K3
bration algebraically and end up replacing the product of the acyl
1
CalabiYau threefold X+ with Se;+ by an acyl CalabiYau fourfold Z+.
X : here, the SYZ ber of X is bered such that application of Heterotic/Mtheory
duality leads to a K3 bration in which all three (1,1)forms of the hyperKahler
structure have a nontrivial variation over the base. As in the 4d N
= 1 duality
between heterotic strings and Mtheory, we replace a CalabiYau threefold by a
G2manifold. In the present setup, this means replacing an acyl CalabiYau threefold X
with an acyl G2manifold Z . In the cylinder region of J , where J is di eomorphic
to a product S
simply nd X3
1
S
S
1
S
1
I for a K3surface S with a calibrated T 2 bration,2 we
I by application of Heterotic/Mtheory duality.
We hence nd the statement that Mtheory duals of heterotic strings on TCS G2manifolds
are compacti ed on Spin(7)manifolds, which allows a construction as proposed in our
GCSconstruction: an acyl CalabiYau fourfold Z+ and an acyl G2manifold Z
are glued along a cylinder times a CalabiYau threefold.
times an S1
In our discussion, we have so far ignored that the heterotic compacti cation comes
equipped with a vector bundle on J , and we need to identify what this is mapped to
on the dual Mtheory geometry. For X+, where only E varies nontrivially, replacing
X+
1
Se;+ by Z+ is simply a noncompact version of the 3d duality between heterotic
strings and Mtheory on CalabiYau fourfolds. We can hence apply the usual logic of how
bundles constructed from spectral covers are translated in Ftheory [46]. Restricting J
to X
Se1; , holomorphic vector bundles on a (noncompact) CalabiYau threefold get
translated to the geometry of a (noncompact) G2manifold, similar to discussed in [12].
Gluing Z+ and Z
S1 to a compact Spin(7)manifold Z then implies that the geometry
of Z determines a heterotic G2system on the G2manifold J . It is hence tempting to
construct such vector bundles on twisted connected sum G2manifolds by appropriately gluing
2Depending on a choice of complex structure, this is an elliptic or sLag bration.
holomorphic vector bundles on the building blocks X , see [47] for a concrete realization
of this idea. In section 5.5 we will use this logic to count the number of bundle moduli
on a TCS G2manifold and con rm a matching with the degrees of freedom of the dual
Mtheory compacti cation on a GCS Spin(7)manifold.
Duality for the building blocks
Before heading to the full construction of Mtheory/Spin(7) and Heterotic/G2 we rst
explore the dual pairs that are relevant for each building block. Recall:
GCS Spin(7)manifold Z is built out of an acyl CY4 building block Z
and an acyl
G2building block Z+.
TCS G2manifold J is build out of two acyl CY3, X .
We will now construct dual pairs for CY4 and G2 and correspondingly on the heterotic side
CY3 in sections 5.3.2 and 5.3.3. These will then be used as building blocks to construct
the GCS/TCS dual pairs. As a preparation we rst need to construct pairs of G2 and CY3
and duals for M/het duality.
Geometric preparation: dual pairs of G2 and CY3
In this section we discuss a wellsuited example for building the G2 building block of the
GCSmanifold, by consider Mtheory on G2manifold and the heterotic on CY3
compacti cation. Similar to the strategy used in [16], we can
nd such a pair by forming an
appropriate quotient of a dual pair of 4d N = 2 compacti cations, which are
M/X43;43
S1
!
Het/K3 Th2 :
(5.5)
(5.6)
On the heterotic side we choose a trivial vector bundle on the K3 surface and break E8
E8
to U(1)16 by introducing Wilson lines on an the Th2 factor. Tadpole cancellation then
necessitates the introduction of 24 NS5branes wrapped on T 2. Such a compacti cation
of the heterotic string theory on K3 Th2 is dual to Mtheory compacti ed on X43;43
S1,
where X43;43 is a CY3fold, chosen to be a
bration of K3surfaces with polarizing lattice
N = U
( E8)
( E8) over P1. The resulting CY3 is constructed as a hypersurface in a
toric variety A associated to the pair of re exive polytopes
0
= BBB
3 3
1 1
12 0 12 CA
= BBB
1 1
1 0
6 6
and has h1;1(X43;43) = h2;1(X43;43) = 43. Besides the 4d N = 2 supergravity multiplet
(which also contains a graviphoton), both of these compacti cations have 43 N = 2 vector
multiplets, 43 hypermultiplets and one universal hypermultiplet.
We can now quotient both theories in (5.5) by an involution to nd dual compacti
cations of Mtheory on a G2manifold and heterotic strings on a CalabiYau threefold with
a vector bundle. This truncates the light excitations to be described by a theory with 20
N = 1 U(1) vectors and 67 N = 1 chiral multiplets.
On the type IIA side we act with a free involution Zo2 to produce a G2manifold3 M as
M = (X43;43
S1u)=Zo2 ;
(5.7)
where Zo2 sends the coordinate u !
u and acts on X43;43 as a antiholomorphic involution.
This antiholomorphic involution is chosen such that it sends all of the homogeneous
coordinates of the ambient toric space A for X43;43 to their complex conjugate and furthermore
swaps any homogeneous coordinate of A associated with the lattice point (a; b; c; d) on
with the homogeneous coordinate associated with the lattice point (a; b; c; d) on
. Some
thought reveals that this is a symmetry of A and that one can choose the hypersurface
equation of X43;43 such that Zo2 becomes a symmetry of X43;43 (without enforcing
singularities) and acts as Zo2 :
!
, as appropriate for an antiholomorphic involution. There
are 47 divisors on X43;43, which descend from A by intersecting X43;43 with the vanishing
loci of the homogeneous coordinates of A, out of which 43 are linearly independent. There
are 21 pairs of homogeneous coordinates fzi ; i = 1
21g on which Zo2 acts as
HJEP06(218)3
Zo2 :
Zo2 :
z
i
!
z
i
wj ! wj :
and the remaining 5 homogeneous coordinates fwj ; j = 0
5g are acted on as
Such an involution acts freely, because a xed point would require that zi
to the C
action on the homogeneous coordinates of A. For our choice of zi
actions imply that xed points require zi+ = zi
= 0. For our choice, such intersection
vanish for any pair zi , so that there are no
xed points. Out of the four linear relations
among the divisors of A, only one involves the 42 divisors associated with fzi ; i = 1
21g
and the remaining three involve the fwj ; j = 0
combinations of such divisors and we nd that4
g
5 . Hence there are precisely 20 even
As Zo2 is an antiholomorphic involution, we also have that b3 (X43;43) = 44, from which
h1+;1(X43;43) = 20
h1;1(X43;43) = 23 :
b2(M ) = h1+;1(X3)
b3(M ) = h1;1(X3) + b3+(X3) = 67 ;
the P
1 base of the K3
why there are 8 + 12 = 20 even classes in b2(X3).
which gives the number of N = 1 U(1) vector and chiral multiplets for compacti cation of
MTheory on M .
3As the involution we consider acts freely, this produces only a `barely' G2manifold, the holonomy group
of which is only the subgroup SU(3) o Z2 of G2.
4Another way to see this is as follows: the threefold X43;43 is bered by elliptic K3 surfaces with two
II over P1 and this involution acts as the antipodal map of the base of this elliptic
bration (another way
to see why it acts freely) in particular identifying the two II
bers. It furthermore acts (nonfreely) on
bration and identi es the 24 reducible K3 bers on X3 pairwise. This reproduces
(5.8)
(5.9)
zi , up
the C
(5.10)
(5.11)
with Hodge numbers h1;1(X11;11) = h2;1(X11;11) = 11. Furthermore, it acts on the bundle
data by twisting the two Wilson lines such as to break E8
E8 to U(1)8. We can think
about the surviving bundle data as an extension of a line bundle associated with a Wilson
E8 to U(1)8, and there are eight complex degrees of freedom specifying
The orbifold furthermore identi es the 24 NS5branes on Th2 pairwise,6 so that the
number of 4d N = 1 U(1) vector and chiral multiplets on the heterotic side is given by
z !
On the dual heterotic side, the corresponding involution Z2h acts as a combination of
z (on the Th2 factor) together with the Enriques involution5 on the K3 surface to
produce a CalabiYau threefold
X11;11 = (K3
Th2)=Z2h
(5.12)
(5.13)
(5.14)
(5.15)
(5.16)
nv = 8 + 12
nc = 1 + 2 11 + 3 12 + 8
in perfect agreement with the dual Mtheory on the G2manifold M . With this preparation
in hand we can now study the duality for the building blocks.
The acyl G2manifold and its dual acyl CY3
We can turn X11;11 and M in the last subsection into a dual pair of an acyl CalabiYau
threefold and an acyl G2manifold by cutting each of them in half. Let us start with M
in (5.7). By cutting along the middle of the interval remaining of the circle S1u, M is turned
into an acyl (barely) G2manifold Z
which we may represent by
Z
= (X43;43
Ru) =Zo2 ;
where Zo2 acts as before on X43;43 and as u !
fold of Z
is X43;43 and there are b2+(X43;43) = 20 classes in the image of the restriction
map from H2(Z ; Q) to H2(X43;43; Q), as well as b3+(X43;43) = 44 classes in the image of
the restriction map from H3(Z ; Q) to H3(X43;43; Q). There are no elements in the kernels
u on Ru. The asymptotic CalabiYau
threeof these maps.
trivial) K3 bration
On the heterotic side, CalabiYau threefold X11;11 (5.12) carries a natural (almost
K3 ,! X11;11 ! P1h ;
where the base P1h is found as the quotient of T 2. Over four points on this P1h, the K3
h
ber is truncated to an Enriques surface, but is constant otherwise. Note that this way of
thinking reproduces the fact that the Euler characteristic of X11;11 vanishes from
(X11;11) = 24 (2
4) + 4 12 = 0 ;
5The Enriques involution is the unique xedpoint free nonsymplectic involution on a K3 surface and
is identi ed by the invariants (r; a; ) = (10; 10; 0) in the classi cation of [48].
6Equivalently, one can compute that ch2(X11;11) = 12[Th2].
where L(2) denotes a lattice L with its inner form rescaled by a factor of two. The
expression for N (X ) is nothing but the even sublattice of H2(K3; Z) under the Enriques
involution. Furthermore, X
may be compacti ed to a compact building block X~ in the
sense of [5] by gluing in a single K3 ber. It now follows from a computation as above that
(X~ ) = 24, which together with b2(X~ ) = 11 (and b1(X~ ) = 0 implies that b3(X~ ) = 0.
This is all of the data we will need in using X
as a building block for a G2manifold.
Note that we have indeed decomposed M and X11;11 by cutting along the same S1.
This becomes particularly clear, by noting that these dual compacti cations descend from
a compacti cation to
ve dimension, in which Mtheory is put on X43;43 and heterotic
string on K3 S1. The dual models we have constructed are then found by compactifying
both side on a further S1u and quotienting by Z2. It is the quotient of S1u, which becomes
one of the two circles on Th2, which is cut in half to produce Z
and X , respectively.
The acyl CY4 and its dual acyl CY3
We now construct a dual pair of an acyl CalabiYau fourfold Z+ (as a building block for
the Spin(7) GCS) and the heterotic dual acyl CalabiYau threefold X+ (as a building
block for the G2 TCS). As we want Z+ to asymptote to X43;43 so that we can glue Z+ with
Z
S1, we construct
(5.17)
(5.18)
(5.19)
(5.20)
so that the algebraic threefold Z~+ satis es
Z+ = Z~+ n X403;43 ;
X43;43 ,! Z+ ! P1 ;
~
c1(Z~+) = [X403;43] :
Note that we may also construct Z+ by cutting a CalabiYau fourfold X4, which is a
bration of X43;43 over P1 in two halves. This fourfold is a toric hypersurface de ned by
the pair
where we have excised the four bers which are Enriques surfaces from the base P1h in the
rst term and added them back in for the second. If we now cut P1h into two halves, each of
which is C and contains two of the Enriques bers over it, we produce an acyl CalabiYau
as the TCS G2 building block
K(X ) = ;
( E8(2)) ;
0
B
= BBB
B
0
= BBB
1 1
0 0
6 6
6 6
1 C
C
h1;1(X4) = 68 ;
h3;1(X4) = 292 ;
h2;2(X4) = 1484 ;
(5.21)
so that (X4) = 2208. As a generalization of [9] to fourfolds, Z~+ can also be constructed
from one of the two isomorphic tops in
corresponding to the bration by X43;43. Either
way, it follows that the Euler characteristic of Z+ is (Z+) = 1104, half of the Euler
characteristic of X4.
The restriction maps i : Hi(Z+; Q) ! Hi(X403;43
S1; Q) are fairly easy because all of
the divisors and none of the threecycle on X43;43 descend from Z+. Furthermore, there are
12 divisor classes on Z+ associated with reducible bers of the bration by X43;43. Hence
HJEP06(218)3
we conclude
j im( 2)j = 43 ;
j im( 3)j = 0 ;
j ker( 2)j = 12 ;
j ker( 3)j = 0 ;
j coker( 2)j = 0 ;
j coker( 3)j = 131 :
(5.22)
This will be all the data from Z+ we need in the following.
Let us now
nd the dual geometry for the heterotic string. As Z+ and X4 are bered
by elliptic K3 surfaces, this is fairly straightforward, as we simply need to apply the known
rules of heteroticFtheory duality. In particular, Z+ and X4 are
bered by elliptic K3
surfaces with Picard lattice U
( E8)2, so that the dual heterotic theory has no E8
E8
vector bundles turned on. Furthermore, Z+ and X4 are elliptic
brations over (blowups
of) C
P
1
P1. These blowups capture the presence of the NS5branes needed on the
heterotic side to satisfy the Bianchi identity. Hence the dual heterotic geometry dual to is
X4 given by an elliptically
bered CalabiYau X3;243 with base P
The threefold X3;243 is constructed from a pair of re exive polytopes with vertices
P1 times a circle S1.
0
= BBB
3 3 3
1 1 0
0 0 1
2 1
C ;
1
0
= BBB
1 1
0 6
0 0
1 1
1 1
0 6
0 6
6
C : (5.23)
and it follows that X3;243 has Hodge numbers (h1;1; h2;1) = (3; 243). The dual acyl CY
building block X+ of the acyl CalabiYau fourfold Z+ is constructed using by cutting
X3;243 along an S1 in the base of its K3 bration. Equivalently, we may glue in a K3 ber
to compactify X+ to a building block X~+ which can be constructed using the methods
of [9] from a pair of projecting tops
0
= BBB
3 3 3 C
1 1 0 CA
0 0 1
C ;
0
= BBB
1 1
0 6
0 0
1
1
6
6
1
1
6
6
1
1
6
0
1
1
6
0
1
1
1
6
6
1
1
6
6
1 1
1 C
0
C :
(5.24)
(5.25)
0
1
0
0
1
0
0
0
It follows that
N (X+) = U
K(X+) = ;
h2;1(X~+) = 112 :
Z = Z+ [ (Z
S1) ;
where Z+ \ (Z
S1) = X3
S1 = and X3 = X43;43. It immediately follows that
(Z+) = 1104. The remaining Betti numbers are found from the MayerVietoris
b2(Z) = j ker 2j + jcoker 1j = (20 + 12) + 0 = 32 ;
where we have used that 20 classes in h1;1(X43;43) are in the common image of the restriction
maps 2 : H2(Z+)
H2(Z
H2(X43;43) is 12dimensional. Furthermore,
S1) ! H2(X43;43
S1) and that the kernel of +2 : H2(Z+) !
b3(Z) = j ker 3j + jcoker 2j = 0 ;
as all of the classes in h1;1(X43;43) are in the image of +2, no class in b3(X43;43) is in the
image of +3 and the restriction maps 3 have no kernels.
We can now use (2.9) to nd
b4 (Z) = 327 ;
Note that there are Wilson lines breaking E8
E8 to U(1)16 on the product S1 on the
heterotic side, which are dual to the volumes of the divisors associated with the ( E8) 2
in the Picard lattice of the K3 bers of X4.
First example of dual pairs: new Spin(7) and its dual G2manifold
Above we have constructed a pair of an acyl CalabiYau fourfold Z+ and an acyl
G2manifold Z
which both asymptote to (a cylinder/an interval times) the same CalabiYau
threefold X43;43, together with the two dual acyl CalabiYau threefolds X+ and X
together with the corresponding bundle data on the heterotic side. We will now glue Z+
with Z
S1 to a GCS Spin(7)manifold Z and X
S1 to a TCS G2manifold J and
verify that the light elds in the e ective eld theories indeed agree.
The Mtheory on GCS Spin(7)manifold
As detailed in section 4, the Spin(7)manifold Z is constructed as
and it follows that there are nc = 360 real degrees of freedom originating from the metric
and threeform for compacti cations of Mtheory on Z. Out of these, 32 sit in 3d N = 1
vector multiplets. Furthermore, a consistent compacti cation requires the introduction
of 1104=24 = 46 M2 branes. Each of these contributes a further 8 real moduli from
moving on Z.
Alternatively, this Spin(7)manifold can also be obtained as a free quotient of the
CalabiYau fourfold X4 speci ed as a toric hypersurface by (5.20).
The dual heterotic model on TCS G2manifold
The dual heterotic side has various components: geometry, vector bundle and NS5branes.
We will determine all the dual data, and compare with the spectrum obtained in the
Mtheory compacti cation  and nd agreement.
(5.26)
(5.27)
(5.28)
(5.29)
Geometry.
Mtheory on Z is dual to heterotic on a G2manifold J which can be obtained
by berwise duality as
J = (X+
Se1;+) [ (X
Se1; ) ;
where X
are the acyl CalabiYau threefolds introduce in sections 5.3.3 and 5.3.2 above.
Recall that the lattices N (X ) are
E8(2) ;
h2;1(X ) we hence immediately nd [6]
and there is a matching such that N (X+) \ N (X ) = ;
. Together with K(X ) and
(5.30)
(5.31)
(5.32)
b2(J ) = K(X+) + K(X ) = 0
b2(J ) + b3(J ) = 23 + 2 (K(X+) + K(X )) + 2 h2;1(X+) + h2;1(X ) = 247 ;
giving b3(J ) = 247. We hence nd 247 real degrees of freedom from the metric of a heterotic
compacti cation on J and no moduli associated with the B eld.
Bundle data.
Recall from section 5.3.1 that the heterotic compacti cation on X11;11 =
X
dual to Mtheory on M = Z
has eight complex bundle degrees of freedom,
associated with a bundle which is an extension of appropriate Wilson lines on the double
cover of X11;11. The origin of the corresponding degrees of freedom in the dual Mtheory
suggests to associate 4 complex bundle degrees of freedom with each of the S1s in the
K3
Th2 double cover of X11;11. For X , one of these two S1s has disappeared, so that we
are left with 8 real bundle degrees of freedom there. In particular, these give the nontrivial
holonomies in E8
1
E8 along Sb; in the base C of the K3
G2manifold, these get mapped to Wilson lines on Se1;. Note that this is precisely what is
expected for the duality between Mtheory on the CalabiYau fourfold X4 and heterotic
strings on X3;243
Se1: the nonzero volumes of the divisors on Z+ associated with the
( E8) 2 in the Picard lattice of the K3 bers of Z+ are mapped to Wilson lines on Se1+ on
the heterotic side. We hence conclude that J carries a vector bundle V inherited from a
bration on X . In the TCS
bundle is captured by the volumes of the curves in ( E8 2) of the K3 bers of Z .
bundle on X
which breaks E8
E8 ! U(1)8 and has eight real moduli. The data of this
NS5branes.
Furthermore, our compacti cations of the heterotic string require the
addition of NS5branes for consistency as the gauge bundle V does not have a second Chern
character. We will infer the NS5branes which we need to include by exploiting piecewise
duality. In particular, we can think about a representative of the rst Pontrjagin class of
J as being glued from representatives of c2(X~ ).
The second Chern class of X~+ is
c2(X~+) = 46HH^+ + 46H+ + + 23H^+ + + 11 +2
= 46H+H^+ + 24H+ + + 12H^+ + ;
(5.33)
where H^+ is the divisor class corresponding to xing a point on the base of the K3 bration
on X~+ and H+ is the class corresponding to xing a point on the other P1 in the P1
P^1
base of the elliptic bration on X~+. The divisor class + is represented by the section of
this elliptic
bration and these classes satisfy the relation
that c1(X~+) = [H^+].
The 46 NS5branes on the curve HH^ are wrapped on the elliptic ber of X+ times
1
the auxiliary S+;e multiplying X+ in the construction of J . As such, they are wrapped
precisely on the conjectured
ber of the T 3
bration of J used for the duality and we
can associated them with the 46 M2 branes present on the Mtheory side. Let us con rm
this by counting the number of moduli. Each of these NS5branes has 4 real degrees of
freedom by displacing the elliptic
ber on X+. Furthermore, there is one real degree of
freedom from the scalar
in the tensor multiplet (the position along the 11th direction in
heterotic Mtheory) and three real degrees of freedom from the selfdual twoform B on
the b1(T 3) = b2(T 3) = 3 cycles of the T 3 they are wrapped on. This makes 8 degrees of
+( + + 2H+ + H^+) = 0. Note
freedom matching the counting for the M2 branes.
The 12 NS5branes on H^+ + are points on the P1 base of the K3 bration of X~+ so
that they are wrapped on a threemanifold with the topology S2 S1 in J . Within X+, each
such NS5brane has 2 real moduli associated with displacement,7 one modulus associated
with the worldvolume scalar , and one modulus from the selfdual twoform B. At least
in the Kovalev limit, we are hence led to associate 4 real moduli with each such NS5brane.
Finally, there are the 24 NS5branes in the class H+ + of X~+
Se1;+. On X+ = X~+ n S+0,
they are wrapped on the whole of the open base C
1
Se;+ of the K3 bration on X+
Se1;+.
Similarly, there are 12 NS5branes wrapped on a double cover the open base C
K3
bration on X
1
Se; as we have seen in section 5.3.1. Note that this sector of
NS51
Se; of the
branes becomes 24 copies of the T 2 factor times the interval in (X+
Se1;+) \ (X
Se1; ) =
I
T 2
K3. On the G2manifold, these NS5branes can hence be joined to form 12
irreducible NS5branes. This has several e ects. First of all, the relative positions of the
24 branes on X+
S1+;e, which are points on the K3 ber S+ of X+ , are pairwise xed
to be symmetric under the Enriques involution acting on S . Second, each such pair of
NS5branes only has two real moduli of deformation. Because they are wrapped on the
Cbase of the K3 bration on X+ and the elliptic bration of X+ is nontrivial there, they
cannot be displaced in the direction of the elliptic curve of X+. Third, the two branches of
each pair of NS5branes are swapped when encircling two special points in the base of the
K3 bration on X , so that we should think of each such pair of NS5branes as a single
NS5brane wrapped on a threemanifold L which is a double cover of S3 branched along
two unlinked S1s. We conclude that there are 12 NS5branes wrapped on threemanifolds
L inside J with two real deformations each. It is not hard to see that b1(L) = b2(L) = 1, so
that each of these NS5branes is associated with 4 real degrees of freedom: 2 real moduli of
displacement together with 2 real moduli from
and B adding up to 4 real moduli each.
Counting of degrees of freedom.
We are now ready to count the number of degrees of
freedom on the heterotic side. To do so, we can neglect the 46 NS5branes wrapped on the
T 3 ber of J , as these are mapped to M2 branes in Mtheory and we have already matched
7The associated holomorphic curves which lift to associatives are xed to lie on the section of the elliptic
bration of X+.
their degrees of freedom. Summarizing the di erent contributions discussed above, the
remaining light elds for heterotic strings on J are hence the 247 moduli of the Ricci at
metric on J , the 8 real degrees of freedom in the bundle V , 8 U(1) vectors which remain
massless in E8
E8, 24 NS5branes with 4 real moduli each, and the dilaton, giving a total
of 360 real degrees of freedom. Out of these, there are 32 3d N = 1 vector multiplets,
originating from the 8 surviving U(1)s together with the 24 NS5branes wrapped on
threecycles with b1 = 1. We have hence veri ed that the number of light elds between heterotic
string theory on J and Mtheory on Z precisely agrees.
Second example of dual pairs
Let us now make consider a variation of the previous example and work out the topology
of the Spin(7) associated with putting an E8
E8 vector bundle on J . To describe such a
situation, we intend to replace Z
and Z+ by an acyl G2manifolds and acyl CY4 originating
from a K3 bration with Picard lattice U instead of U
( E8) 2.
The acyl CalabiYau fourfold Z+
Following the usual rules of heteroticMtheory (Ftheory) duality Z+ is now found as
one half of a compact CalabiYau fourfold X4 described by a generic Weierstrass elliptic
bration over P1
P
1
P1 and Z+ \ (Z
S1) = X3;243
S
1
I. Here, X3;243 has already
appeared in section 5.3.3 and X4 is found from the pair of re exive polytope:
0
B
= BBB
B
0
B
= BBB
B
1
0
0
0
0
2
1
0
0
0
0
1
0
0
0
1
1
0
0
0
3 3 3 3 C
0 1 0 0 CC
0 0 0 1 CA
1 0 1 0
C
1
1
1 1 C
C
6 6 CC ;
(5.34)
S
1
I and the
(5.36)
which implies
and (Z+) = 8784.
j im +2j = 3 ;
j im +3j = 0 ;
j ker +2j = 0
j ker +3j = 0
h2;1(X4) = 0 ;
h2;2(X4) = 11724 ;
(5.35)
and (X4) = 17568. We can write Z+ [ Z+ = X4 with Z+ \ Z+ = X3;243
data of Z+ relevant for its use in the construction of Z are
To construct Z , we rst consider M = Z
= X3;243
S1u=Z2. The threefold X3;243
was constructed as a toric hypersurface associated with the pair of re exive polytopes
shown in (5.23). Its homogeneous coordinates have the toric weights
Similar to our procedure in section 5.3.1 we can introduce an antiholomorphic involution
on X3;243 by letting8
y x w z1 z2 z^1 z^2
3 2 1
6 4 0
6 4 0
0
1
0
0
1
0
0
0
1
0
0
1
:
(y; x; w; z^1; z^2) ! (y; x; w; z^1; z^2)
(z1; z2) ! ( z2; z1) :
b2(Z) = b3(Z) = 0 :
b4 (Z) = 2919
This involution acts as the antipodal map on the P1 with homogeneous coordinates [z1; z2]
The resulting barely G2manifold M hence has Betti numbers
h1+;1(X3;243) = 0
h2+;1(X3;243) = 244
h1;1(X3;243) = 3
h2;1(X3;243) = 244
b2(M ) = 0 ;
b3(M ) = 247 :
We can now form an acyl G2manifold Z
and Z
= S1
X3;243. The restriction maps are
by cutting S1u in the middle, i.e. M = Z
j im 2 j = 0 ;
j im 3 j = 244 ;
j ker 2 j = 0
j ker 3 j = 0 :
5.5.3
Mtheory spectrum on the GCS Spin(7)manifold
From the acyl CalabiYau fourfold Z+ and the acyl G2manifold Z
S we can now form
a Spin(7)manifold Z = Z+ [ Z
S. It follows from the restriction maps that
Furthermore (Z) = 8784. Hence
and compacti cations of Mtheory on Z have 2920 real degrees of freedom from the metric
which all sit in real multiplets, as well 8784=24 = 366 spacetime lling M2branes.
Again, this Spin(7)manifold can also be obtained as a free quotient of the CalabiYau
fourfold X4 speci ed as a toric hypersurface by (5.34).
8In fact, X43;43 is connected to X3;243 by several singular transitions and the actions (5.8) and (5.9)
imply the one given here.
(5.37)
(5.38)
(5.39)
(5.40)
(5.41)
(5.42)
(5.43)
The dual heterotic compacti cation lives on the same G2manifold J as discussed in
section 5.4.2. Here, the metric contributes 247 real degrees of freedom. Now, however, we
choose to switch on a vector bundle V completely breaking E8
E8, such that V is at
on the T 3
bers of J for the duality to hold
berwise. It now follows from the proposed
duality that such a bundle must have
mV = 2920
1 = 2672
(5.44)
real moduli. Furthermore, the heterotic Bianchi identity in this context must force to
include 366 NS5branes wrapped on the T 3 ber of J .
We can reproduce the number of bundle moduli by studying the moduli of holomorphic
vector bundles on X
together with a gluing condition. For X , such bundles are inherited
from E8
E8 vector bundles on K3 symmetric under the Enriques involution. From the
point of view of the K3
bration on X , the bundle data are constant over the base C of
X . A holomorphic vector bundle on K3 has 4 112 real moduli (they all sit in
hypermultiplets for heterotic string theory on K3) and the restriction to be invariant imposes a
restriction
xing half, i.e. 224 real moduli of such a bundle.
For X+, the number of bundle moduli can be inferred as follows. Consider heterotic
strings on X3;243 = X+ [ X+. By a straightforward application of duality to Ftheory
(which is compacti ed on the manifold X4 discussed in section 5.5.1) immediately gives
that a generic vector E8
E8 model W , which is at on the elliptic
ber of X3;243, has
mW = 5344 real moduli [46, 49]. An E8
E8 vector bundle on X+ \ X+ = I
S
1
K3
has 4 112 real moduli, which gives the number of conditions when matching two generic
E8
E8 bundles W
on X . We hence expect
mW+ + mW
448 = mV :
(5.45)
As furthermore mW+ = mW , we nd that mW+ = 2896.
On the G2manifold J , we can hence construct a suitable vector bundle V by letting
V jX+
= W+. Such a bundle restricts to a vector bundle on S+ which then has to be
appropriately restricted to match V jX
Se1; . As we have seen VS is not a generic vector
bundle on a K3 surface, but there are 224 conditions arising at it must be symmetric under
the Enriques involution. Once this condition is met, VS
uniquely de nes V jX
Se1; . As
we have seen, the Enriques involution gives 224 real restrictions, so that we conclude that
mV = 2896
224 = 2672, which matches the expectation from the counting of degrees
of freedom on the Spin(7)manifold Z. Ignoring the 366 NS5branes wrapped on the T 3
ber of J , all of the moduli of this model sit in real multiplets, matching the result that
b2(Z) = 0 on the MTheory side.
6
Discussion and outlook
We proposed a new construction of eightmanifolds with Spin(7)holonomy, based on a
generalized connected sum (GCS), where two building blocks  a CalabiYau fourfold and
a G2holonomy manifold times S1  are glued together along an asymptotic region that is
a CalabiYau threefold times a cylinder. This construction is in part inspired by the recent
twisted connected sum (TCS) realization of G2holonomy manifolds, which has resulted in a
multitude of new examples of compact G2manifolds, and thereby a resurgence of interest in
the string/Mtheory context. Likewise the GCSconstruction that we propose, provides an
avenue to construct large classes of compact Spin(7)manifolds systematically. In particular
the construction of acyl CalabiYau threefolds using semiFano threefolds [5] or tops [9]
which is useful for TCSconstructions, has an obvious generalization to acyl CalabiYau
fourfolds, which is useful for expanding the set of examples of GCSconstructions.
We gave an alternative description of the GCS Spin(7)manifolds in terms of a quotient
by an antiholomorphic involution of a CalabiYau threefold
bered CalabiYau fourfold
in section 4.5, which is similar in spirit to the constructions of Joyce, however the key
di erence is that instead of gluing in ALEspaces at pointlike orbifold singularities, we
glue in G2manifolds with suitable asymptotics.
There is a multitude of future directions to consider:
1. M/Fduality for Spin(7): Mtheory on Spin(7) results 3d N = 1 theories, and we have
seen that there are subsectors of the e ective eld theory, which in the limit of in
nite asymptotic region enjoy enhanced supersymmetry, as discussed in section 4.4. It
would clearly be very interesting to apply M/Fduality to the GCS Spin(7)manifolds
and determine how the supersymmetry breaking in the 4d Ftheory vacuum is
realized [1]. Needless to say it is not di cult to construct GCSexamples that have an
elliptic bration and we will return to this shortly elsewhere. Again key for this will
be also to understand the uxes in the GCS Spin(7)manifolds.
2. M/Heteroticduality and heterotic G2systems: as was exempli ed in section 5, some
GCSconstructions have a K3 bration, so that Mtheory compacti cation on these
has a dual description in terms of Heterotic on G2manifolds with a T 3 bration with
vector bundle. We have seen that the GCSdecomposition of the Spin(7)manifold
maps in the dual G2compacti cation to a TCSdecomposition. Heterotic on
G2manifolds has been studied only very sparsely, and this approach may very well
provide further insight into the construction and moduli of vector bundles for
socalled heterotic G2systems. Furthermore, this duality also gives evidence for the
existence of associative T 3 brations on G2manifolds as conjecture from G2 mirror
symmetry in [13], and these may indeed be less rare as was proposed in [23] despite
the obstructions of associatives.
3. Nonabelian gauge groups: furthermore we expect, again through the duality to
heterotic, that the GCSSpin(7) manifolds can give rise to nonabelian gauge symmetries.
Locally, these will have a description in the form of an ADE singularities over Cayley
fourcycle (for a discussion of such noncompact examples see [29]). It would indeed
be interesting to develop this further and understand e.g. a Higgs bundle description
of the e ective theories and its relation to the compact Spin(7) manifolds  much
like what has been done in recent years for M/Ftheory on CY4.
4. Mirror symmetry: mirror symmetry for TCSG2 manifolds as studied in [10, 13] by
applying the mirror map to the CalabiYau threefold building blocks. It would be
interesting to see whether there is a similar way to study mirror symmetry for Spin(7)
manifolds (as proposed in [50]) by applying the mirror map to the building blocks.
Acknowledgments
We thank for Sebastjan Cizel, Magdalena Larfors, Xenia de la Ossa, James Sparks and
Yuuji Tanaka for discussions related to the present work. APB and SSN are supported
by the ERC Consolidator Grant 682608 \Higgs bundles: Supersymmetric Gauge Theories
and Geometry (HIGGSBNDL)".
A
TCSconstruction of G2manifolds
The TCS G2manifolds [4{6] have the topology of a K3 bration over an S3 and are
obtained by gluing two building blocks Z
that are algebraic threefolds, which are K3 bered
over S2. We will denote the generic K3 bers of each building block by S . Excising a
ber, the two building blocks asymptote to K3 S1
S1
I, where I is an interval. The
building blocks are not CalabiYau but have c1(Z ) = [ S ], which means that removing
a ber gives rise to acyl CalabiYau threefolds X . These are shown in
gure 1. Here we
summarize the salient points and introduce some notation that will be used in the main
text. We refer the reader for a more detailed exposition to [4{6, 12].
The TCSconstruction glues these two building blocks times an S1 together, with a
hyperKahler rotation (HKR), or Donaldson matching,
acting on the Kahler forms ! and
holomorphic (2,0)forms
as
N
T
= im(
= N ?
H2(S ; Z)
K(Z ) = ker(
)=[S ] :
8
Re (2;0)
Im
(2;0)
induces an isometry on the cohomologies of the bers: H2(S+; Z) = H2(S ; Z).
and an exchange of the circles in the base Se1;
1
= Sb; as shown in gure 1. In particular
What will be most important for this paper is how to determine the cohomologies of
the G2manifold in terms of data of the TCS gluing. For each building block we de ne the
restriction map on the second cohomology as
H2(Z ; Z)
H2(S ; Z) =
3;19 = U 3
( E8)2 ;
where we used the cohomology of the K3surfaces, where U is the hyperbolic lattice of
rank 2. The cohomology of the G2manifold, can then be written in terms of the following
lattices
(A.1)
(A.2)
(A.3)
Applying the MayerVietoris sequence to this problem gives the cohomology of the G2
H1(J; Z) = 0
H2(J; Z) = (N+ \ N )
H3(J; Z) = Z[S]
K(Z+)
K(Z )
H3(Z+)
3;19=(N+ + N )
H3(Z )
\ T+)
(N+ \ T )
K(Z+)
K(Z )
H4(J; Z) = H4(S)
(T+ \ T )
3;19=(N
+ T+)
3;19=(N+ + T )
H3(Z+)
H3(Z )
K(Z+)
K(Z )
H5(J; Z) =
3;19=(T+ + T )
K(Z+)
K(Z ) :
We will determine a similar relation for the Spin(7) GCSconstruction.
Open Access.
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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