Type II string theory on Calabi-Yau manifolds with torsion and non-Abelian discrete gauge ...

22 Pages·2017·0.35 MB·English

Published for SISSA by Springer
Received: May 15, 2017
Accepted: July 15, 2017
Published: July 26, 2017
Type II string theory on Calabi-Yau manifolds with
torsion and non-Abelian discrete gauge symmetries
J
H
E
P
Volker Braun,a Mirjam Cveti(cid:20)c,b;c;d Ron Donagic;b and Maximilian Poretschkinb
0
aElsenstrasse 35, 12435 Berlin, Germany
7
bDepartment of Physics and Astronomy, University of Pennsylvania,
(
209 S 33rd St, Philadelphia, PA 19104, U.S.A.
cDepartment of Mathematics, University of Pennsylvania, 2
209 S 33rd St, Philadelphia, PA 19104, U.S.A. 0
dCenter for Applied Mathematics and Theoretical Physics, University of Maribor,
1
Mladinska ulica 3, 2000 Maribor, Slovenia
7
E-mail: [email protected], [email protected],
[email protected], [email protected] )
1
Abstract: We provide the (cid:12)rst explicit example of Type IIB string theory compacti-
2
(cid:12)cation on a globally de(cid:12)ned Calabi-Yau threefold with torsion which results in a four-
9
dimensional e(cid:11)ective theory with a non-Abelian discrete gauge symmetry. Our example
is based on a particular Calabi-Yau manifold, the quotient of a product of three elliptic
curves by a (cid:12)xed point free action of Z (cid:2) Z . Its cohomology contains torsion classes
2 2
in various degrees. The main technical novelty is in determining the multiplicative struc-
ture of the (torsion part of) the cohomology ring, and in particular showing that the
cup product of second cohomology torsion elements goes non-trivially to the fourth co-
homology. This speci(cid:12)es a non-Abelian, Heisenberg-type discrete symmetry group of the
four-dimensional theory.
Keywords: String Field Theory, Conformal Field Models in String Theory, Discrete
Symmetries
ArXiv ePrint: 1702.08071
Open Access, (cid:13)c The Authors.
https://doi.org/10.1007/JHEP07(2017)129
Article funded by SCOAP3.
Contents
1 Introduction 1
2 Non-Abelian discrete symmetries in Type IIB 3
2.1 Generalizations 6
3 The Calabi-Yau manifold 7
4 Submanifolds 8
J
4.1 The special Lagrangian submanifold Y 9
4.2 The four-dimensional submanifold Y 11 H
0
4.3 Cohomology of the Calabi-Yau manifold 13 E
5 Cup product 13 P
5.1 Cup product on Y0 13 0
5.2 Naturality and the Calabi Yau manifold 14
7
6 Outlook 14 (
A Notebook 15 2
A.1 Constructing Y0 16 0
A.2 Cohomology groups: the fundamental region 17
1
A.3 Cohomology groups: four-torus 17
7
A.4 Cohomology groups of Y 18
0
)
A.5 Cup products 18
A.6 Alternate basis 20 1
2
9
1 Introduction
Discrete symmetries are an integral part of particle physics; they play a key r^ole in the
Standard Model physics, such as those to prevent nucleon and lepton number violating
processes, with R-symmetry being the most prominent one, as well as examples of Abelian
and non-Abelian discrete symmetries to address the (cid:13)avor hierarchy.
In string theory, discrete symmetries are expected to be gauge symmetries. In recent
years there has been a (cid:13)urry of activities in studies of discrete symmetries in string and F-
theory compacti(cid:12)cation, focusing on their geometric origin. In F-theory the primary focus
was on the origin of Abelian discrete symmetries, which arise from Calabi-Yau geometries
which are genus-one (cid:12)brations without section, in contrast to elliptic (cid:12)brations with sec-
tions. A natural object associated with these compacti(cid:12)cations is the Tate-Shafarevich
(TS) group which is a discrete Abelian group that organizes inequivalent genus-one ge-
ometries which share the same associated Jacobian (cid:12)bration. The TS group speci(cid:12)es the
Abelian discrete gauge symmetry of the F-theory compacti(cid:12)cation. The study of F-theory
{ 1 {
compacti(cid:12)cations with discrete gauge symmetries Z was initiated in [1] and followed-up
n
in [2{9]. Most of the past works primarily focused on Z gauge symmetry. However, new
2
insights into aspects of the TS group, and its relations to M-theory vacua in the case of
Z were addressed in [8]. Furthermore, there has been progress in elucidating the origin of
3
Abelian discrete symmetries via F-theory/Heterotic duality [9].
Ontheotherhand,theoriginofnon-Abeliandiscretegaugesymmetriesinstringtheory
is less understood. An important progress in this direction was made [10] in the context of
TypeIIstringtheorycompacti(cid:12)cationtofour-dimensions,buildingonearlierworks[11{13]
that study Abelian discrete gauge symmetries in Type II string theory compacti(cid:12)cation
on Calabi-Yau threefolds with torsion. In the context of Type IIB string theory, a non- J
Abelian Heisenberg-type discrete symmetry is realized [10] on a Calabi-Yau threefold with H
torsion whose second cohomology torsion elements have a non-trivial cup product into
E
fourth cohomology torsion ones. This speci(cid:12)c approach therefore requires the study of
P
Calabi-Yau threefolds with torsion by determining torsion cohomology groups and their
cup products, which is technically challenging. 0
Thepurposeofthispaperistoprovidea(cid:12)rstexplicitconstructionofaTypeIIBstring 7
theory compacti(cid:12)cation on a Calabi-Yau manifold that exhibits a Heisenberg-type discrete (
symmetry.1 For that purpose we choose an example of a Calabi-Yau threefold X with
6 2
torsion. This particular Calabi-Yau threefold occurs in the classi(cid:12)cation of [15] as the (cid:12)rst
0
example of the free quotient of a torus T6 by a (cid:12)nite group action. It had occurred previ-
1
ously, in various contexts, in [16{18]. It is the quotient of a product of three elliptic curves
bya(cid:12)xedpointfreeactionofZ (cid:2)Z whichwedescribeexplicitlyin(3.1). Theintegercoho- 7
2 2
mology of ourCalabi-Yau manifold X6 is given in (4.20). This Calabi-Yau has a symmetric )
Hodge diamond, with h1;2 = h1;1 = 3. If it is also a self-mirror manifold, to the best of our
1
knowledge, this would be the (cid:12)rst example of a self-mirror Calabi-Yau threefold where the
2
two (a priori independent) torsion groups Torsion(H2(X ;Z)) = Torsion(H5(X ;Z)) =
6 6
Z2 (cid:8) Z3 and Torsion(H3(X ;Z)) = Torsion(H4(X ;Z)) = Z3 actually di(cid:11)er and are 9
4 2 6 6 2
both non-zero.
Thisexampleallowsustocarryoutthechallengingcomputationofthecupproductsfor
torsion coholomology elements explicitly, which in turn determine the Heisenberg discrete
symmetry of the four-dimensional theory. The key strategy is to relate X to a lower
6
dimensional (real) submanifold Y . Recalling that X is the quotient of a six-torus by a
0 6
group G = Z (cid:2)Z , we take Y to be the quotient of a four-dimensional subtorus of X
2 2 0 6
that happens to be invariant under G. It comes with an inclusion map i : Y ! X and
0 6
a projection map going the other way, hence the restriction i(cid:3) : H(cid:3)(X ;Z) ! H(cid:3)(Y ;Z) is
6 0
surjective, and in fact it exhibits H(cid:3)(Y ;Z) as a direct summand of H(cid:3)(X ;Z). The cup
0 6
product calculation on the smaller Y can be carried out explicitly. This could probably be
0
done by hand using elementary topology. Instead, we obtain it as part of a computational
scheme that gives us many additional useful facts about the manifolds involved, including
the integer cohomology of X , given in (4.20).
6
1Non-AbeliandiscretesymmetriesofTypeIIBstringtheoryon AdS (cid:2)S5=Z werestudiedin[14]from
5 3
the perspective of AdS/CFT correspondence.
{ 2 {
The cup product on Y turns out to have the property we want: there are torsion
0
classes in H2(Y ;Z) whose product does not vanish in H4(Y ;Z). Even though we do
0 0
not fully calculate the multiplicative structure of the cohomlogy ring of X, the fact that
H(cid:3)(Y ;Z) is a direct summand of H(cid:3)(X ;Z) plus knowing the multiplicative structure of
0 6
the cohomlogy ring of Y su(cid:14)ces to allow us to conclude that there are torsion classes in
0
H2(X ;Z) whose product does not vanish in H4(X ;Z), as claimed. In particular, Type
6 6
IIB compacti(cid:12)cation on such a Calabi-Yau manifold leads to a four-dimensional theory
with a Heisenberg-type discrete symmetry group.2
Insection2wesummarizetheresultsof[10]regardingTypeIIBstringtheorycompact-
i(cid:12)cations leading to four-dimensionall theories with Heisenberg-type discrete symmetries. J
Our Calabi-Yau manifold is described in section 3. The rest of the paper is devoted to
H
the calculation of the torsion homologies and cup products for this manifold. In section 4
E
we determine the torsion cohomology groups for the Calabi-Yau manifold and some of its
P
submanifolds, and in section 5 the speci(cid:12)c results for the cup products are obtained. The
outlook is given in section 6. In the appendix we display the notebook containing the main 0
calculations. A link is provided to a site containing the code and the worksheet, so that 7
the interested reader can verify the calculations directly.
(
2
2 Non-Abelian discrete symmetries in Type IIB
0
In this section, we review the construction of non-Abelian discrete symmetries arising in
1
Type IIB compacti(cid:12)cations on a Calabi-Yau manifold X . The analysis follows closely
6
7
that of [10]. This latter analysis is closely related to the study [11] of Abelian discrete
)
symmetries in Type IIB compacti(cid:12)cations.
In general, the homology of a threefold X has two independent torsion sub-groups 1
6
given by
2
Torsion(H (X ;Z)) ’ Torsion(H (X ;Z)); 9
1 6 4 6
Torsion(H (X ;Z)) ’ Torsion(H (X ;Z)); (2.1)
2 6 3 6
associated with the torsion one-cycles (and Poincar(cid:19)e dual torsion four-cycles) and torsion
two-cycles (and Poincar(cid:19)e dual torsion three-cycles), respectively.
In the following, we shall (cid:12)rst restrict our discussion to the case that
Torsion(H (X ;Z)) ’ Torsion(H (X ;Z)) = Z ;
1 6 4 6 k
Torsion(H (X ;Z)) ’ Torsion(H (X ;Z)) = Z : (2.2)
2 6 3 6 k0
The Poincar(cid:19)e dual cohomology groups which are needed for the dimensional reduction of
Ramond-Ramond (cid:12)elds are accordingly given as
Torsion(H5(X ;Z)) ’ Torsion(H2(X ;Z)) = Z ;
6 6 k
Torsion(H4(X ;Z)) ’ Torsion(H3(X ;Z)) = Z : (2.3)
6 6 k0
2For a related work in the context of F-theory, see [19] where the origin of the Heisenberg-type discrete
symmetry does not seem to have a weak coupling limit to Type IIB string theory. In the context of
Heterotic string compacti(cid:12)cations some aspects of torsion homologies were studied, e.g., in [20] for the
Schoen manifold.
{ 3 {
Let (cid:26) , (cid:12) , !~ , and (cid:16) represent the generators of the torsion cohomologies
2 3 4 5
Torsion(H2(X ;Z)), Torsion(H3(X ;Z)), Torsion(H4(X ;Z)) and Torsion(H5(X ;Z)),
6 6 6 6
respectively. They satisfy the following relations
(cid:14)(cid:13) = k(cid:26) ; (cid:14)(cid:26)~ = k(cid:16) ;
1 2 4 5
(cid:14)(cid:11) = k0!~ ; (cid:14)! = k0(cid:12) ; (2.4)
3 4 2 3
where (cid:13) , ! , (cid:11) and (cid:26)~ are one-, two-, three- and four- cochains on X , respectively,
1 2 3 4 6
belonging to corresponding H(cid:3)(X ;U(1)) cohomologies.3 The above elements satisfy:
6
Z Z Z Z
(cid:13) ^(cid:16) = (cid:26) ^(cid:26)~ = (cid:11) ^(cid:12) = ! ^!~ = 1: (2.5)
1 5 2 4 3 3 2 4 J
X6 X6 X6 X6
H
Here k(cid:0)1 and k0(cid:0)1 are the torsion linking numbers between dual torsion p- and (5(cid:0)p)-
cycles(p = 1;3). Note,eqs.(2.4)and(2.5)canbeobtainedfromexpressionsthatdetermine E
torsion linking numbers, cf., appendix C of [11]. P
Thecup-productoftwotorsionclassesisagainatorsionclass. Thustheproduct (cid:26) ^(cid:26)
2 2 0
is some multiple of the generator !~ of Torsion(H4(X ;Z)):
4 6
7
(cid:26) ^(cid:26) = M !~ ; M 2 Z: (2.6)
2 2 4 (
Thecoe(cid:14)cientM isaninvariantofthemanifoldX . Sometimesitvanishes,andsometimes 2
6
it does not. In this work we describe an example where it is non zero. By employing (2.4) 0
this cup-product integrates to (cid:26) ^(cid:13) = M0(cid:11) , where M0 2 Z and kM = k0M0.
2 1 3 1
Thesetorsionsubgroupsgiveaprioririsetothreenon-commutingdiscretecyclicgroups
7
in the e(cid:11)ective four-dimensional Type IIB action. This can be seen from the following
Kaluza-Klein reduction Ansatz for the Type IIB closed string sector Ramond-Ramond )
(RR) and Neveu-Schwarz-Neveu-Schwarz (NSNS) two-form (cid:12)elds C ;B , respectively and 1
2 2
RR four-form (cid:12)eld C :
4 2
B = b1^(cid:26) +A1^(cid:13) ; (2.7) 9
2 2 1
C = b2^(cid:26) +A2^(cid:13) ; (2.8)
2 2 1
C = b3!~ +A3^(cid:11) +V3^(cid:12) +c ^! ; (2.9)
4 4 3 3 2 2
where bi and Ai (i = 1;2;3) are the three axions and three U(1) one-form gauge potentials,
respectively. (One-formpotential V3 andtwo-formpotential c , arenotindependent(cid:12)elds,
2
due to the self-duality of the (cid:12)ve-form (cid:12)eld strength F in Type IIB supergravity.)
5
TheseAns(cid:127)atzeensurethattheKaluza-KleinreductionofTypeIIBsupergravityresults
inane(cid:11)ectivefour-dimensional(cid:12)eldtheorywiththreemassiveU(1)one-formgaugepoten-
tials Ai (= i = 1;2;3). (For further details, see section 4 of [10].) E.g., for B one obtains:
2
Z
dB ^(cid:3)dB (cid:0)! (2.10)
2 2
M10=M4(cid:2)X6
Z Z Z Z
(db1(cid:0)kA1)^(cid:3)(db1(cid:0)kA1) (cid:26) ^(cid:3)(cid:26) + (dA1)^(cid:3)(dA1) (cid:13) ^(cid:3)(cid:13) ;
2 2 1 1
M4 X6 M4 X6
3This approach is a simpli(cid:12)ed version designed to keep the expository part short and clear. A bet-
ter mathematical treatment would involve Cheeger-Simons cocycles or, more or less equivalently, Deligne
cohomology. These keep track of both the continuous and the discrete parameters.
{ 4 {
which results in a Stu(cid:127)ckelberg mass term for A1. The Stu(cid:127)ckelberg mass contributions for
all three gauge (cid:12)elds A1, A2, A3 in the e(cid:11)ective four-dimensional action is of the following
schematic form:
L (cid:27) G (cid:17)i(cid:17)(cid:22)j; (2.11)
ij (cid:22)
where
(cid:17)i = @ bi(cid:0)kAi ; i = 1;2;
(cid:22) (cid:22) (cid:22)
(cid:17)3 = @ b3(cid:0)k0A3 (cid:0)Mb2(@ b1(cid:0)kA1): (2.12)
(cid:22) (cid:22) (cid:22) (cid:22) (cid:22)
This four-dimensional action is therefore invariant under the following non-commuting dis-
J
crete gauge transformations:
H
Ai ! Ai +@ (cid:21)i; bi ! b1+k(cid:21)i; i = 1;2;
(cid:22) (cid:22) (cid:22) E
A3 ! A3 +@ (cid:21)3+M0k(cid:21)2A1 +M0b1@ (cid:21)2; b3 ! b3+Mkb1(cid:21)2+k0(cid:21)3; (2.13)
(cid:22) (cid:22) (cid:22) (cid:22) (cid:22) P
where M 2 Z, M0 2 Z and kM = k0M0. This corresponds to a set of non-commuting Z , 0
k
Zk, Zk0 factors as long as M 6= 0, resulting in a non-Abelian discrete gauge symmetry of 7
the four-dimensional action, speci(cid:12)ed by k, k0 and M.
(
Altogether there are three generators T ;T ;T associated with the discrete symme-
1 2 3
2
try groups Z ;Z ;Z , respectively. The important fact to note is that these generators
k k k0
0
T ;T ;T do not commute, provided that there is a non-trivial cup-product (2.6).
1 2 3
These discrete gauge symmetries of the e(cid:11)ective four-dimensional action lead to the 1
following discrete symmetry operations on a four-dimensional state (x), with charges 7
(q ;q ;q ) under (Z ; Z ;Z ). From a ten-dimensional perspective this state corresponds
1 2 3 k k k0 )
to a bound state of q fundamental strings and q D1-branes wrapping the torsional one-
1 2
1
cycle in Torsion(H (X ;Z)) = Z , and q D3-branes wrapping the torsional three-cycle in
1 6 k 3
Torsion(H (X ;Z)) = Z .4 The state transforms under T ;T ;T generators: 2
3 6 k0 1 2 3
9
T~ : (x) (cid:0)! e2(cid:25)ik(cid:0)1q1 (x);
1
T~ : (x) (cid:0)! e2(cid:25)ik(cid:0)1q2U (x);
2
T~ : (x) (cid:0)! e2(cid:25)ik0(cid:0)1q3 (x); (2.14)
3
where the charge rede(cid:12)nition matrix U is of the form:
0 1 0 10 1
q 1 0 M0 q
1 1
Bq C 7! B0 1 0 CBq C ; kM = k0M0: (2.15)
@ 2A @ A@ 2A
q 0 0 1 q
3 3
4General (q ;q )-strings, which source B and C gauge potentials, wrap torsional one-cycles and re-
1 2 2 2
sult in a Z (cid:2)Z symmetry at the level of four-dimensional particle states. D3-branes, which source
k k
C4 gauge potential, wrap torsional three-cycles and result in an additional Zk0 symmetry factor. When
thecorrespondingtorsionalclasseshavenon-trivialrelations,asspelledoutabove,thisAbelianstructureis
promotedtoanon-Abelianone. ThisphenomenonisanalogoustotheHanany-Wittene(cid:11)ect[21]wherefour-
dimensionalstringsemergefromNS5-branesandD5-braneswrappingtorsionalfour-cycles,andD3-branes
wrapping torsional two-cycles. The non-Abelian discrete symmetry emerges when torsional four-cycles in-
tersect non-trivially along a torsional two-cycle; crossing of four-dimensional strings, associated with NS5-
and D5-branes, leads to the creation of strings associated with D3-branes wrapping torsional two-cycles.
{ 5 {
These transformations can be derived from the ten-dimensional perspective by performing
discrete shift transformations on corresponding gauge potentials, supporting p-branes, and
deducing the induced charges after the discrete shifts in the ten-dimensional Chern-Simons
terms of the e(cid:11)ective action. (Again, for further details see [10].)
Thus, one observes that
T~ T~ = T~MT~ T~ (2.16)
1 2 3 2 1
resulting in a non-commuting discrete gauge symmetry, a Heisenberg discrete symmetry
group (Z (cid:2) Z ) o Z , speci(cid:12)ed by k, k0 and M. In special cases, say, when k = k0,
k k0 k
M = 1 the non-Abelian discrete gauge symmetry is given by (Z (cid:2)Z )oZ and further
k k k
J
specializations of k reduce to, e.g., Dih for k = 2 and (cid:1)(27) for k = 3.
4
H
2.1 Generalizations
E
It is straightforward to generalize this analysis to the case when the second torsion coho- P
mologies have multiple discrete factors. (For further details, see [19], section 2.)
0
Let us focus on the following speci(cid:12)c examples, which shall be relevant for the rest of
7
our analysis. The torsion cohomologies are chosen to be:
(
Torsion(H5(X ;Z)) ’ Torsion(H2(X ;Z)) = Z (cid:2)Z ;
6 6 k1 k2 2
Torsion(H4(X ;Z)) ’ Torsion(H3(X ;Z)) = Z ; (2.17)
6 6 k3 0
andthenontrivialcupproductofthegenerators(cid:26)(i) 2 Z (i = 1;2)isofthefollowingform: 1
2 ki
7
(1) (2)
(cid:26) ^(cid:26) = M!~ ; (2.18)
2 2 4 )
where !~ is the generator of Torsion(H4(X ;Z)) = Z . 1
4 6 k3
These generators satisfy the following relations:5 2
(i) (i) 9
(cid:14)(cid:13) = k (cid:26) ; (cid:14)(cid:26)~ = k (cid:16) ; i = 1;2;
1 i 2 4(i) i 5(i)
(cid:14)(cid:11) = k !~ ; (cid:14)! = k (cid:12) ; (2.19)
3 3 4 2 3 3
where (cid:12) , and (cid:16) (i = 1;2) represent the generators of the torsion cohomologies
3 5(i)
Torsion(H4(X ;Z)) and Torsion(H5(X ;Z)), respectively, and (cid:13) , ! , (cid:11) and (cid:26)~ are
6 6 1(i) 2 3 4(i)
respectiveone-,two-,three-andfour-cochains,elementsofthecorrespondingH(cid:3)(X ;U(1))
6
cohomologies, that satisfy:
Z Z Z Z
(cid:13)(i)^(cid:16) = (cid:26)(i)^(cid:26)~ = (cid:14)i; (cid:11) ^(cid:12) = ! ^!~ = 1: (2.20)
1 5(j) 2 4(j) j 3 3 2 4
X6 X6 X6 X6
Kaluza-Klein Ans(cid:127)atze for B , C and C gauge potentials, parallel those of (2.9):
2 2 4
B = b1 ^(cid:26)(i)+A1 ^(cid:13)(i); (2.21)
2 (i) 2 (i) 1
C = b2 ^(cid:26)(i)+A2 ^(cid:13)(i); (2.22)
2 (i) 2 (i) 1
C = b3!~ +A3^(cid:11) +V3^(cid:12) +c ^! : (2.23)
4 4 3 3 2 2
5Forthesakeofsimplicity,wechosespecializedrelations;fortheanalysisofmoregeneralcases,see[19].
{ 6 {
In the four-dimensional e(cid:11)ective action there are (cid:12)ve massive U(1) gauge (cid:12)elds A1 , A2
(i) (i)
and A3, and (cid:12)ve associated axions b1 , b2 , and b3, respectively. (Again, V3 and c are not
(i) (i) 2
independent (cid:12)elds, due to the self-duality of F .)
5
The Stu(cid:127)ckelberg mass contributions for to the e(cid:11)ective action again takes the
schematic form:
L (cid:27) G (cid:17)I(cid:17)(cid:22)J(cid:3); (2.24)
IJ(cid:3) (cid:22)
where (cid:17)I, complexi(cid:12)ed four-vectors, which take the following form:
(cid:22)
(cid:16) (cid:17)
(cid:17) = @ b2 (cid:0)(cid:28)@ b1 +k A2 (cid:0)(cid:28)A1 ; i = 1;2;
(cid:22)(i) (cid:22) (i) (cid:22) (i) i (cid:22)(i) (cid:22)(i)
J
(cid:16) (cid:17)
(cid:17)3 = @ b3+k A3 (cid:0)M b2 (cid:0)(cid:28)b1 k A1 ; (2.25)
(cid:22) (cid:22) 3 (cid:22) (1) (1) 2 (cid:22)(2) H
and (cid:28) = C +ie(cid:0)(cid:30) denotes the complexi(cid:12)ed string coupling of Type IIB string theory. This E
0
structure results in the discrete gauge invariance of the e(cid:11)ective four-dimensional action, P
which corresponds to the Heisenberg discrete symmetry speci(cid:12)ed by k , k , k and M. For
1 2 3 0
further details see [10] section 2 and [19], section 3.
7
Thus, in order to determine the Heisenberg discrete group of Type IIB string compact-
(
i(cid:12)cations on a Calabi-Yau threefold with torsion, the plan is to identify second cohomology
torsion classes and to determine their non-trivial cup products. As explained in the intro- 2
duction, we proceed to relate the Calabi-Yau threefold X with torsion to a simpler space 0
6
Y , a submanifold, where the cup product is under control. In particular, we exhibit a
0 1
torsion class t in the second cohomology H2(X ;Z) whose restriction to Y is non-zero and
6 0 7
squares to a non- zero class on the auxiliary Y . Functoriality of this cup product then
0
)
(cid:12)xes the rest. In this paper we apply this strategy the example of the Calabi-Yau threefold
X , de(cid:12)ned in the section below, and explicit calculations are derived in the subsequent 1
6
two sections 4 and 5. 2
9
3 The Calabi-Yau manifold
Our Calabi-Yau threefold X,6 will be the quotient of a six-torus (in fact the product of
three elliptic curves) by a (cid:12)nite group action. The (cid:12)rst and best known example of such
a quotient was studied by Vafa and Witten in [22]. Let E = C=(Z+(cid:28) Z) 3 z , be three
i i i
elliptic curves, i = 0;1;2. Their product admits an action of the group G = Z (cid:2) Z ,
2 2
generated by the transformations;
g0 : (z ;z ;z ) 7!(cid:0)z ; (cid:0)z ; (cid:0)z (cid:1);
1 0 1 2 0 1 2
(3.1)
g0 : (z ;z ;z ) 7!(cid:0)(cid:0)z ; z ; (cid:0)z (cid:1):
2 0 1 2 0 1 2
Vafa and Witten then consider a crepant resolution of this quotient.
Various other quotients, by di(cid:11)erent group actions, were considered by Oguiso and
Sakurai [18] in the process of studying the collection of all Calabi-Yau manifolds with an
in(cid:12)nite fundamental group. They obtained a partial classi(cid:12)cation, which has very recently
6For simplicity, in the rest of the paper we shall omit the subscript 6 for a Calabi-Yau threefold, i.e.
X !X.
6
{ 7 {
been completed in [23]. All actions of the basic group G = Z (cid:2)Z and of all its Abelian
2 2
extensions were classi(cid:12)ed in [15]. (Those actions of Abelian extensions that specialize to
the Vafa-Witten action on G had been classi(cid:12)ed earlier, in [24].) It turns out that, up
to obvious equivalences, there are four such actions of G, cf. [15], Lemma 1.2.2. They
all have the same linearization, so they di(cid:11)er only in the shifts. Exactly one of these G
actions is (cid:12)xed-point free: the (cid:12)xed-point free G = Z (cid:2) Z action is generated by the
2 2
transformations:
g : (z ;z ;z ) 7!(cid:0)z + 1; (cid:0)z ; (cid:0)z (cid:1);
1 0 1 2 0 2 1 2 (3.2)
g : (z ;z ;z ) 7!(cid:0)(cid:0)z ; z + 1; (cid:0)z + 1(cid:1):
2 0 1 2 0 1 2 2 2
This modi(cid:12)es the Vafa-Witten action by adding some non-trivial shifts. It is these shifts J
that make the action (cid:12)xed-point free, and therefore the quotient: H
E
X = (E (cid:2)E (cid:2)E )=G (3.3)
0 1 2
P
is a manifold, with no need for a resolution. This particular quotient is described in
0
example 2.17 of [18], where it is attributed to Igusa [16], page 678, and to Ueno [17],
7
Example [16.16]. Each of g ;g and g := g (cid:14)g preserves the constant holomorphic (3;0)-
1 2 3 1 2
formonE (cid:2)E (cid:2)E , sothisformdescendstoanowherevanishingholomorphic(3;0)-form (
0 1 2
on X. On the other hand, the gi project out any constant one-forms, so the quotient X 2
of (3.3) is a proper Calabi-Yau threefold in the sense that compacti(cid:12)cation preserves only
0
the minimal amount of supersymmetry. We note that its holonomy group is G, and its
1
fundamental group is easily seen to be the semidirect product
7
(cid:25) (X) = Z6oG; (3.4)
1 )
cf. [15], section 1.5 and table 1 on page 10. Finally, the only invariant (1;1)-forms are 1
dz ^dz(cid:22), i = 0;1;2, leading to the Hodge diamond
i i 2
1 9
0 0
0 3 0
hpq(cid:0)X(cid:1) = 1 3 3 1: (3.5)
0 3 0
0 0
1
For further details see [15]. In this work we will analyze the cohomology ring of this
Calabi-Yau manifold X of (3.3). In [15] it was noted (Lemma 1.7.1) that this is one of four
topologically inequivalent free quotients (the others are by various abelian extensions of
G). All are Calabi-Yau manifolds with Hodge numbers (3;3), and it seems plausible that
similar calculations can be carried out for each of these (cid:12)xed-point free actions. We will
not pursue these other manifolds in the present work.
4 Submanifolds
Since the group action eq. (3.2) only ever changes the imaginary part of the coordinates
z by a sign, there are a number of G-invariant (real) submanifolds of the product, hence
i
{ 8 {
submanifolds of the quotient X, obtained by setting the imaginary part to zero. On the
covering space, these are sub-tori of E (cid:2)E (cid:2)E . After dividing out the group action, we
0 1 2
obtain the special Langrangian 3-manifold
Y ,!X; (x ; x ; x ) 7! (x ; x ; x ); (4.1)
0 1 2 0 1 2
three four-dimensional submanifolds
Y ,!X; (x ; x ; x ; y ) 7! (x +(cid:28) y ; x ; x );
0 0 1 2 0 0 0 0 1 2
Y ,!X; (x ; x ; x ; y ) 7! (x ; x +(cid:28) y ; x ); (4.2)
1 0 1 2 1 0 1 1 1 2
Y2 ,!X; (x0; x1; x2; y2) 7! (x0; x1; x2+(cid:28)2y2); J
and three 5-dimensional submanifolds H
E
Y ,!X; (x ; x ; x ; y ; y ) 7! (x +(cid:28)y ; x +(cid:28)y ; x );
01 0 1 2 0 1 0 0 1 1 2
P
Y ,!X; (x ; x ; x ; y ; y ) 7! (x +(cid:28)y ; x ; x +(cid:28)y ); (4.3)
02 0 1 2 0 2 0 0 1 2 2
0
Y ,!X; (x ; x ; x ; y ; y ) 7! (x ; x +(cid:28)y ; x +(cid:28)y ):
12 0 1 2 1 2 0 1 1 2 2
7
In addition to the submanifold embeddings, we note that there are also projection maps
(
X ! Y , X ! Y , and X ! Y by ignoring the imaginary part of one, two, and all three
ij i
complex coordinates. Therefore, these submanifolds are all retractions and the relative 2
cohomology long exact sequences split into 0
H(cid:3)(X;Z) ’H(cid:3)(Y;Z)(cid:8)H(cid:3)(X;Y;Z) 1
’H(cid:3)(Y ;Z)(cid:8)H(cid:3)(X;Y ;Z); 0 (cid:20) i < 3; (4.4) 7
i i
’H(cid:3)(Y ;Z)(cid:8)H(cid:3)(X;Y ;Z); 0 (cid:20) i < j < 3: )
ij ij
1
In the remainder of this section we now discuss the integral cohomology of these subma-
2
nifolds.
9
4.1 The special Lagrangian submanifold Y
Again, all constant 2-forms on the covering space torus are projected out by the G-action.
Hence, this is a rational homology sphere. Its fundamental group and Abelianization is
(cid:25) (Y) = Z3oG; H (Y) = (cid:25) =[(cid:25) ;(cid:25) ] = Z (cid:8)Z : (4.5)
1 1 1 1 1 4 4
To summarize, the integral cohomology is
8
>Z d = 3
>
>
>
><Z (cid:8)Z d = 2
Hd(Y;Z) = 4 4 (4.6)
>0 d = 1
>
>
>
>:Z d = 0
and, by degree count, there can be no non-trivial cup products.
To get some explicit handle on the torsion cohomology generators, let us write down
cochains in a cellular model. Our model of choice is going to be cubical cells with iden-
ti(cid:12)cations. First, by cubical cells we mean (hyper-)cubes in some Rn whose vertices are
{ 9 {