Non-perturbative K3 orientifolds with NS–NS B-flux

29 July 1999

Physics Letters B 459 Ž1999. 497–506

Non-perturbative K3 orientifolds with NS–NS B-flux Zurab Kakushadze

1

Jefferson Laboratory of Physics, HarÕard UniÕersity, Cambridge, MA 02138, USA Department of Physics, Northeastern UniÕersity, Boston, MA 02115, USA Enrico Fermi Institute, UniÕersity of Chicago, Chicago, IL 60637, USA Received 6 May 1999; received in revised form 1 June 1999 Editor: M. Cveticˇ

Abstract We consider non-perturbative six dimensional N s 1 space-time supersymmetric orientifolds of Type IIB on K3 with non-trivial NS–NS B-flux. All of these models are non-perturbative in both orientifold and heterotic pictures. Thus, some states in such compactifications arise in ‘‘twisted’’ open string sectors which lack world-sheet description in terms of D-branes. We also discuss their dual F-theory compactifications on certain Voisin–Borcea orbifolds. In particular, the explicit construction of non-perturbative K3 orientifolds with NS–NS B-flux gives additional evidence for the conjectured extension of Nikulin’s classification in the context of Voisin–Borcea orbifolds. q 1999 Published by Elsevier Science B.V. All rights reserved.

In the recent years various six Žsee, e.g., w1–4x. and four Žsee, e.g., w5–12x. dimensional orientifold vacua have been constructed. In many cases the world-sheet approach to orientifolds is adequate and gives rise to consistent anomaly free vacua in six and four dimensions. However, there are cases where the perturbative orientifold description misses certain non-perturbative sectors giving rise to massless states w11,13,14x. In certain cases this inadequacy results in obvious inconsistencies such as lack of tadpole and anomaly cancellation. Examples of such cases were discussed in w9–11x. In other cases, however, the issue is more subtle as the non-perturbative states arise in anomaly free combinations, so that they are easier to miss.

1

E-mail: [email protected]

Recently such non-perturbative orientifolds have been studied in detail in w13,14x. In particular, six and four dimensional N s 1 supersymmetric nonperturbative orientifolds have been constructed in w13,14x using Type I-heterotic duality and the map between Type IIB orientifolds and F-theory as guiding principles. The discussion in w13,14x was Žmainly. confined to orientifolds with trivial NS–NS B-flux. The purpose of this note is to extend the results of w13,14x to compactifications with non-trivial NS–NS B-field turned on. In particular, here we will focus on six dimensional N s 1 supersymmetric non-perturbative orientifolds of Type IIB on K3 with nontrivial NS–NS B-flux. The origin of non-perturbative states in orientifold compactifications can be understood as follows. Thus, in the K3 orbifold examples of w3x the orientifold projection is not V , which we will use to

0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 7 2 2 - 4

498

Z. Kakushadzer Physics Letters B 459 (1999) 497–506

denote that in the smooth K3 case, but rather V J X , where J X maps the g twisted sector to its conjugate gy1 twisted sector Žassuming g 2 / 1. w15x. Geometrically this can be viewed as a permutation of two P 1 ’s associated with each fixed point of the orbifold w11x. ŽMore precisely, these P 1 ’s correspond to the orbifold blow-ups.. This is different from the orientifold projection in the smooth case where Žafter blowing up. the orientifold projection does not permute the two P 1 ’s. In the case of the V J X projection the ‘‘twisted’’ open string sectors corresponding to the orientifold elements V J X g are absent w16,11x. However, if the orientifold projection is V , then the ‘‘twisted’’ open string sectors corresponding to the orientifold elements V g are present w11x. In fact, these states are non-perturbative from the orientifold viewpoint and are required for gravitational anomaly cancellation in six dimensions. This can be explicitly seen from the construction of non-perturbative K3 orientifolds without the B-flux in w13,14x. The effect of turning on non-trivial NS–NS B-flux in the perturbatiÕe Žthat is, ‘‘untwisted’’. open string sectors is as follows. First, the rank of the gauge group is reduced by 2 b r2 w17x, where b is the rank of the matrix Bi j Ž i, j s 1,2,3,4 label the compact directions corresponding to K3.. Note that b is always an even integer, and in the case of K3 compactifications can take values b s 0,2,4. Second, the multiplicity of the 59 open string sector states Žif present. is no longer 1 Žas in the case without the B-field., but 2 b r2 w4x. Note that the B-field is quantized in half-integer units Žhere we are using the normalization where the B-field is defined modulo integer shifts. w17,4x. The effect of the NS–NS B-flux in the non-perturbatiÕe Žthat is, ‘‘twisted’’. open string sectors is a bit more non-trivial. The point here is that Type I-heterotic duality was an important guiding principle in determining the twisted open string sector spectrum of non-perturbative K3 orientifolds Že.g., in the case of the Z 3 orbifold limit of K3. without the B-flux w13x. In particular, in certain cases the corresponding heterotic dual is perturbative, which enables one to understand the non-perturbative sectors on the orientifold side. However, as was pointed out in w13x, once we turn on the B-field, the above property is no longer there - the heterotic duals in these cases are no longer perturbative. This makes

understanding twisted open string sectors in the presence of the B-flux more involved. Nonetheless, it turns out that there is a way around this difficulty. The key simplification here is due to the observation of w7,18x Žalso see w19,20x. that turning on the B-field in an orientifold compactification is equivalent to turning on non-commuting Wilson lines. Let us be more precise here. For the sake of definiteness let us consider a compactification on a two-torus T 2 . As was explained in detail in w18x, turning on Žquantized. B-filed of rank b s 2 on T 2 is equivalent to modding out by the following freely acting orbifold. Thus, let us turn on two Z 2 valued Wilson lines on T 2 - one corresponding to the a-cycle of T 2 , and the other one corresponding to the b-cycle. Each of these Wilson lines can be described as a freely acting Z 2 orbifold which amounts to shifting the T 2 lattice by a half-lattice shift along the corresponding cycle of T 2 . In the closed string sector the effect of non-zero B-field then corresponds to having non-trivial discrete torsion between the generators, call them g 1 respectively g 2 , of the two Z 2 subgroups. In the open string sector each Z 2 has a non-trivial action on the corresponding Chan-Paton charges. This action is described by the 16 = 16 Chan-Paton matrices 2 gg 1 respectively gg 2 . The effect of non-zero B-field in the open string sector then corresponds to having non-commuting Chan-Paton matrices: gg 1 gg 2 s ygg 2 gg 1. That is, even though the Žfreely acting. orbifold group is Z 2 m Z 2 Žwith discrete torsion., the corresponding embedding in the Chan-Paton charges is in terms of the non-Abelian dihedral group D4 Žsee w18x for details.. It is now clear how to deal with turning on the B-field in the twisted open string sectors. We start with the cases without the B-field, which have been understood in w13,14x, and then mod out by the above freely acting orbifold with the corresponding action on the Chan-Paton charges. This procedure is relatively straightforward to carry out, albeit there are some technical subtleties Žsuch as the fact that various relative signs in the corresponding projec-

2 Here we work with 16=16 Žrather than 32=32. Chan-Paton matrices for we choose not to count the orientifold images of the corresponding D-branes.

Z. Kakushadzer Physics Letters B 459 (1999) 497–506

tions require some care. one encounters in the process. One also must take into account not only the original twisted sectors modded out by the above action, but also the additional sectors corresponding to fixed points of the K3 orbifold group element Žwhich is a pure twist. accompanied by the Z 2 m Z 2 shifts. For the sake of brevity here we will skip Žmost of. the details of the derivation of the corresponding massless spectra, and simply summarize the results in various tables. We will, however, make various clarifying remarks as we discuss the corresponding models. Thus, consider Type IIB on M2 s T 4rZ N , where the generator g of Z N , N s 2,3,4,6, acts on the complex coordinates z 1 , z 2 on T 4 as follows: gz1 s v z 1 ,

gz 2 s vy1 z 2 ,

Ž 1.

where v ' expŽ2p irN .. This theory has N s 2 supersymmetry in six dimensions. Next, we consider the V orientifold of this theory. Note that, as we have already mentioned in the previous section, the V projection acts as in the smooth K3 case. This, in particular, implies that we must first blow up the orbifold singularities before orientifolding. After orientifolding the closed string sector contains the N s 1 supergravity multiplet in six dimensions plus the usual tensor supermultiplet. We also have some number n cH of closed string sector hypermultiplets plus some number n˜ T of extra tensor multiplets. Note that n cH q n˜ T s 20 Žthis follows from the fact that before orientifolding in the N s 1 language we have 20 hypermultiplets and 20 tensor multiplets associated with K3.. The precise values of n cH and n˜ T , however, depend on the order of the orbifold group N as well as the rank b of the B-field. Thus, in the Z 2 case the untwisted closed string sector gives rise to 4 hypermultiplets only. The Z 2 twisted closed string sector produces 16 hypermultiplets and no tensor multiplets for b s 0, 12 hypermultiplets and 4 tensor multiplets for b s 2, and 10 hypermultiplets and 6 tensor multiplets for b s 4. This can be seen as follows w4x. Consider a special point in the moduli space where the T 4 lattice is that of SO Ž8.. In this case the rank of the B-field b s 2. For this special lattice the Z 2 twist can be rewritten in terms of lattice shifts Žsee w4x for details., and in

499

the latter language it is straightforward to see that 12 fixed points are symmetric under the action of V , whereas 4 of them are antisymmetric. The above conclusion for the b s 4 case then follows by considering the four-torus of the factorized form T 2 m T 2 and turning on rank 2 B-field in each T 2 . For each T 2 the symmetric and antisymmetric fixed points can be deduced from the b s 2 case discussed above as follows. First consider the case where we have T 2 m T 2 with rank 2 B-field in the first T 2 and no B-field in the second T 2 . Then the second T 2 gives 4 fixed points which are symmetric under V . Since we know that we must have total of 12 symmetric and 4 antisymmetric fixed points for b s 2, it then follows that the first T 2 gives 3 symmetric and 1 antisymmetric fixed points. Now it is clear that in the case where both T 2 ’s have rank 2 B-field turned on, there are total 10 symmetric and 6 antisymmetric fixed points. Next, let us consider other Z N cases with N s 3,4,6. In the untwisted sector we have 2 hypermultiplets and no tensor multiplets. For N s 4,6 we have the Z 2 twisted sector. As was discussed in w4x, after the appropriate projections, in the Z 2 sector we have: 10 symmetric and no antisymmetric fixed points for N s 4,b s 0; 8 symmetric and 2 antisymmetric fixed points for N s 4,b s 2; 7 symmetric and 3 antisymmetric fixed points for N s 4,b s 4; 6 symmetric and no antisymmetric fixed points for N s 6 and all three values of b s 0,2,4. Finally, in the Z N twisted sectors with N s 3,4,6 the fixed points are always symmetric under V . This can be seen, say, for the Z 3 case by going to the special point in the moduli space where T 4 factorizes as T 2 m T 2 with each T 2 corresponding to the SUŽ3. lattice. In this case each T 2 has rank 2 B-field turned on. For the SUŽ3. lattice the Z 3 twist can be rewritten in terms of the corresponding Z 3 valued shift, and then it is not difficult to see that all three fixed points in each T 2 are symmetric under V . Putting all of the above together, we conclude that for the non-perturbative V orientifolds of Type IIB on K3 we have: n cH s 20, n˜ T s 0 for N s 2,b s 0; n cH s 16, n˜ T s 4 for N s 2,b s 2; n cH s 14, n˜ T s 6 for N s 2,b s 4; n cH s 20, n˜ T s 0 for N s 3,6,b s 0,2,4; n cH s 20, n˜ T s 0 for N s 4,b s 0; n cH s 18, n˜ T s 2 for N s 4,b s 2; n cH s 17, n˜ T s 3 for N s 4,b s 4. This is reflected in the corresponding tables.

Z. Kakushadzer Physics Letters B 459 (1999) 497–506

500

Let us now turn to the open string sector. In the Z 3 case the twisted Chan-Paton matrices are completely fixed by the tadpole cancellation conditions for each value of b. The corresponding perturbative spectra can be found in w4x. As to the non-perturbative Z 3 twisted 99 open string sector, in the b s 0 case it was given in w13x. To obtain the correspond-

ing non-perturbative sector states for b s 2,4, we mod out by the freely acting orbifold action discussed above. The massless spectra of the Z 3 nonperturbative orientifolds with various values of b are summarized in Table 1. In the Z N cases with N s 2,4,6 the tadpole cancellation conditions do not uniquely fix the twisted

Table 1 The massless spectra of the non-perturbative Type IIB orientifolds on T 4rZ N for N s 3,4 with the NS–NS B-field of rank b. The semi-colon in the column ‘‘Charged hypermultiplets’’ separates 99 and 55 representations. The subscript ‘‘U ’’ indicates that the corresponding Ž‘‘untwisted’’. state is perturbative from the orientifold viewpoint. The subscript ‘‘T ’’ indicates that the corresponding Ž‘‘twisted’’. state is non-perturbative from the orientifold viewpoint. The UŽ1. charges are not shown, and by ‘‘neutral’’ hypermultiplets we mean that the corresponding states are not charged under the non-Abelian subgroups Model

b

Gauge group

Charged hypermultiplets

Neutral hypermultiplets

Extra tensor multiplets

Z3

0

wUŽ8. m SO Ž16.x99

Ž28,1.U Ž8,16.U 9 = Ž28,1.T

20

0

2

UŽ8. 99

36 U 9 = 28 T

20

0

4

SO Ž8. 99

9 = 28 T

20

0

0

wUŽ8. m UŽ8.x99 m wUŽ8. m UŽ8.x 55

Ž28,1;1,1.U Ž1,28;1,1.U Ž8,8;1,1.U Ž28,1;1,1.T Ž1,28;1,1.T same as above with 99 l 55 Ž8,1;8,1.U Ž1,8;1,8.U

20

0

2

wUŽ4. m UŽ4.x99 m wUŽ4. m UŽ4.x 55

Ž6,1;1,1.U Ž1,6;1,1.U Ž4,4;1,1.U Ž10,1;1,1.T 3 = Ž6,1;1,1.T Ž1,10;1,1.T 3 = Ž1,6;1,1.T same as above with 99 l 55 2 = Ž4,1;4,1.U 2 = Ž1,4;1,4.U

18

2

4

wUŽ2. m UŽ2.x99 m wUŽ2. m UŽ2.x 55

4 = Ž1,1;1,1.U Ž2,2;1,1.U 6 = Ž3,1;1,1.T 10 = Ž1,1;1,1.T 6 = Ž1,3;1,1.T 10 = Ž1,1;1,1.T same as above with 99 l 55 4 = Ž2,1;2,1.U 4 = Ž1,2;1,2.U

17

3

Z4

Z. Kakushadzer Physics Letters B 459 (1999) 497–506

Chan-Paton matrices. More precisely, in the Z 2 case with b s 0 we have a unique Žup to equivalent representations. solution for g R , where R is the generator of Z 2 :

g R s diag Ž iI8 ,y iI8 . ,

Ž 2.

where In denotes the unit n = n matrix. ŽNote that the action of the orbifold group on the D9- and

501

D5-branes is similar.. However, as pointed out in w14x, for b / 0 we have two solutions:

g R s diag Ž iI2 3y b r2 ,y iI2 3y b r2 . ,

Ž 3.

g R s diag Ž I2 3y b r2 ,y I2 3y b r2 . ,

Ž 4.

where the first solution corresponds to the case without vector structure w21,22x, whereas the second

Table 2 The massless spectra of the non-perturbative Type IIB orientifolds on T 4rZ 6 with the NS–NS B field of rank b. The semi-colon in the column ‘‘Charged hypermultiplets’’ separates 99 and 55 representations. The subscript ‘‘U ’’ indicates that the corresponding Ž‘‘untwisted’’. state is perturbative from the orientifold viewpoint. The subscript ‘‘T ’’ indicates that the corresponding Ž‘‘twisted’’. state is non-perturbative from the orientifold viewpoint. The UŽ1. charges are not shown, and by ‘‘neutral’’ hypermultiplets we mean that the corresponding states are not charged under the non-Abelian subgroups. Note that 16 is a reducible representation of SUŽ4. Model

b

Gauge group

Charged hypermultiplets

Neutral hypermultiplets

Extra tensor multiplets

Z6

0

wUŽ4. m UŽ4. m UŽ8.x99 m wUŽ4. m UŽ4. m UŽ8.x 55

Ž6,1,1;1,1,1.U Ž1,6,1;1,1,1.U Ž4,1,8;1,1,1.U Ž1,4,8;1,1,1.U same as above with 99 l 55 Ž4,1,1;4,1,1.U Ž1,4,1;1,4,1.U Ž1,1,8;1,1,8.U 5 = Ž6,1,1;1,1,1.T 5 = Ž1,6,1;1,1,1.T 4 = Ž4,4,1;1,1,1.T Ž1,1,1;6,1,1.T Ž1,1,1;1,6,1.T Ž4,1,1;4,1,1.T Ž1,4,1;1,4,1.T

20

0

2

wUŽ4. m UŽ4.x99 m wUŽ4. m UŽ4.x55

Ž6,1;1,1.U Ž1,6;1,1.U Ž4,4;1,1.U same as above with 99 l 55 2 = Ž4,1;4,1.U 2 = Ž1,4;1,4.U 5 = Ž6,1;1,1.T 5 = Ž1,6;1,1.T 4 = Ž4,4;1,1.T Ž1,1;6,1.T Ž1,1;1,6.T Ž4,1;4,1.T Ž1,4;1,4.T

20

0

4

UŽ4. 99 m UŽ4.55

2 = Ž6;1.U 2 = Ž1;6.U 4 = Ž4;4.U 10 = Ž6;1.T 4 = Ž16;1.T 2 = Ž1;6.T 2 = Ž4;4.T

20

0

Z. Kakushadzer Physics Letters B 459 (1999) 497–506

502

solution corresponds to the case with vector structure w14x. In the Z 2 case there are no non-perturbative twisted open string sectors, and the corresponding massless spectra for various values of b can be found in w4x for the cases without vector structure, and in w13x for the cases with vector structure. Here we will therefore focus on the Z 4 and Z 6 cases with

and without vector structure. In the cases without vector structure the corresponding twisted massless spectra can be worked out by starting from the b s 0 case and modding out by the freely acting orbifold action discussed above. The resulting massless spectra for the Z 4 orbifold models are given in Table 1. In Table 2 we summarize the massless spectra for the Z 6 orbifold models without vector structure.

Table 3 The massless spectra of the non-perturbative Type IIB orientifolds on T 4rZ N for N s 4,6 with the NS–NS B field of rank b. The semi-colon in the column ‘‘Charged hypermultiplets’’ separates 99 and 55 representations. The subscript ‘‘U ’’ indicates that the corresponding Ž‘‘untwisted’’. state is perturbative from the orientifold viewpoint. The subscript ‘‘T ’’ indicates that the corresponding Ž‘‘twisted’’. state is non-perturbative from the orientifold viewpoint. The UŽ1. charges are not shown, and by ‘‘neutral’’ hypermultiplets we mean that the corresponding states are not charged under the non-Abelian subgroups. Note that 6 is a reducible representation of SpŽ4. Žin our conventions the rank of SpŽ2 n. is n. Model

b

Gauge group

Charged hypermultiplets

Neutral hypermultiplets

Extra tensor multiplets

Z 4 , VS

2

wUŽ4. m SpŽ4. m SpŽ4.x99 m wUŽ4. m SpŽ4. m SpŽ4.x 55

Ž4,4,1;1,1,1.U Ž4,1,4;1,1,1.U Ž4,1,1;1,4,1.U Ž4,1,1;1,1,4.U 2 = Ž6,1,1;1,1,1.T Ž1,6,1;1,1,1.T Ž1,1,6;1,1,1.T Ž4,4,1;1,1,1.T Ž4,1,4;1,1,1.T same as above with 99 l 55

18

2

Z 4 , VS

4

wUŽ2. m SpŽ2. m SpŽ2.x99 m wUŽ2. m SpŽ2. m SpŽ2.x 55

Ž2,2,1;1,1,1.U Ž2,1,2;1,1,1.U 2 = Ž2,1,1;1,2,1.U 2 = Ž2,1,1;1,1,2.U 2 = Ž3,1,1;1,1,1.T Ž1,3,1;1,1,1.T Ž1,1,3;1,1,1.T 4 = Ž2,2,1;1,1,1.T 4 = Ž2,1,2;1,1,1.T same as above with 99 l 55 24 = Ž1,1,1;1,1,1.T

17

3

Z 6 , VS

4

w SpŽ4. m SpŽ4.x99 m w SpŽ4. m SpŽ4.x55

Ž4,4;1,1.U Ž1,1;4,4.U 2 = Ž4,1;4,1.U 2 = Ž1,4;1,4.U 5 = Ž6,1;1,1.T 5 = Ž1,6;1,1.T 4 = Ž4,4;1,1.T Ž1,1;6,1.T Ž1,1;1,6.T Ž4,1;4,1.T Ž1,4;1,4.T

20

0

Z. Kakushadzer Physics Letters B 459 (1999) 497–506

503

Note that all of these models satisfy the gravitational anomaly cancellation condition w23x

a Type IIA dual, the Hodge numbers of the corresponding Calabi–Yau three-fold are given by w26x:

n H y n V s 273 y 29nT ,

h1,1 s r Ž V . q nT q 2 ,

Ž 5.

where n H , n V and nT are the total numbers of hypermultiplets, vector multiplets and tensor multiplets, respectively. ŽNote that nT s n˜ T q 1.. Next, we turn to the Z 4 and Z 6 cases with vector structure. Here we must have b s 2 or 4. Note that the perturbative open string spectrum of the Z 6 model with b s 2 with vector structure is the same as that of the Z 6 model with b s 2 without vector structure w14x. In fact, it is not difficult to show that the non-perturbative twisted open string spectra are also the same in these two cases. The massless spectrum of the Z 6 case with b s 4 with vector structure is summarized in Table 3. Finally, the massless spectra of the Z 4 models with b s 2 and b s 4 with vector structure are also given in Table 3. Note that all of the models in Table 3 also satisfy the gravitational anomaly cancellation condition Ž5.. As to the Abelian anomalies present in models with UŽ1. factors, they are expected to be canceled via the generalized Green-Schwartz mechanism w24,21x. In the remainder of this note we would like to discuss F-theory duals of the above non-perturbative orientifolds with NS–NS B-flux. In order to understand these F-theory compactifications, we will take an indirect route. Let us further compactify a given non-perturbative K3 orientifold on T 2 . The resulting four dimensional model has N s 2 supersymmetry. Let us go to a generic point in the moduli space where the gauge group is maximally Higgsed Žso that the gauge group is either completely broken or consists of Abelian factors only. 3. Let r Ž V . be the number of open string vector multiplets after Higgsing. Then the total number of vector multiplets is given by r Ž V . q nT q 2, where nT is the number of tensor multiplets in six dimensions. Let H 0 be the number of hypermultiplets neutral with respect to the left-over Abelian gauge group Žif any.. Then, if we assume that the resulting four dimensional model has

3

More concretely, first we maximally Higgs using the hypermultiplet matter, and then use the adjoint scalars in the N s 2 vector multiplets to break any remaining non-Abelian group to its Cartan subalgebra w25x.

Ž 6. Ž 7.

h 2,1 s H 0 y 1 .

Using various dualities between Type IIA, heterotic, Type I and F-theory, it is not difficult to see that these Calabi–Yau three-folds must be elliptically fibered, and F-theory compactifications on these spaces should be dual to the original six dimensional orientifold models. In the table below we give the Hodge Žand Euler. numbers of the Calabi–Yau three-folds for the perturbative Žthat is, V J X . K3 orientifolds as well as non-perturbative Žthat is, V . K3 orientifolds with various values of b Žnote that in the Z 2 case the V J X and V orientifolds are equivalent as the action of J X is trivial.: Model

b

Ž h1,1,h2,1 .

x

Ž r,a.

Z2 , V

0 2 4 0 2 4 0 2 4 0 2 4 0 2 4 0 2 4 0 2 4

Ž3,243. Ž7,127. Ž9,69. Ž20,14. Ž16,10. Ž16,10. Ž7,127. Ž9,69. Ž10,40. Ž9,69. Ž9,69. Ž9,69. Ž11,59. Ž7,55. Ž7,55. Ž3,243. Ž5,185. Ž6,156. Ž3,243. Ž3,243. Ž3,243.

y480 y240 y120 12 12 12 y240 y120 y60 y120 y120 y120 y96 y96 y96 y480 y360 y300 y480 y480 y480

Ž2,4. Ž6,8. Ž8,10. Ž11,9. Ž11,11. Ž11,11. Ž6,8. Ž8,10. Ž9,11. Ž6,8. Ž6,8. Ž6,8. Ž2,0. Ž2,2. Ž2,2. Ž2,4. Ž4,6. Ž5,7. Ž2,4. Ž2,4. Ž2,4.

Z3 , V J X Z4 , V J X Z6 , V J X Z3 , V Z4 , V Z6 , V

Note that the above table covers the cases with and without vector structure alike. In the fourth column we have displayed the Euler number x s 2Ž h1,1 y h2,1 .. The meaning of the last column will become clear in a moment. All of the above Calabi–Yau three-folds correspond to Voisin–Borcea orbifolds w27,28x Žfor a

Z. Kakushadzer Physics Letters B 459 (1999) 497–506

504

physicists discussion, see, e.g., w29,30x.. Here we will therefore briefly review some facts about Voisin–Borcea orbifolds. Thus, let W 2 be a K3 surface Žwhich is not necessarily an orbifold of T 4 . which admits an involution J such that it reverses the sign of the holomorphic two-form V 2 on W 2 . Consider the following quotient: Y3 s Ž T 2 m W 2 . rY ,

Ž 8.

where Y s  1,S4 f Z 2 , and S acts as Sz 0 s yz 0 on T 2 Ž z 0 being a complex coordinate on T 2 ., and as J on W 2 . This quotient, known as a Voisin–Borcea orbifold, is a Calabi–Yau three-fold with SUŽ3. holonomy which is elliptically fibered over the base B2 s W 2rB, where B s  1, J 4 f Z 2 . Nikulin gave a classification w31x of possible involutions of K3 surfaces in terms of three invariants Ž r,a, d .. The result of this classification is plotted in Fig. 1 Žwhich has been borrowed from w11x. according to the values of r and a. The open and closed circles correspond to the cases with d s 0 and d s 1, respectively. ŽThe cases denoted by ‘‘m’’ are outside of Nikulin’s classification, and we will discuss them shortly.. In the case Ž r,a, d . s Ž10,10,0. the base B2 is an Enriques surface, and the corresponding Y3 has Hodge numbers Ž h1,1,h2,1 . s Ž11,11.. In all the other cases the Hodge numbers are given by: h1,1 s 5 q 3r y 2 a , h

2,1

s 65 y 3r y 2 a .

Ž 9. Ž 10 .

For Ž r,a, d . s Ž10,10,0. the Z 2 twist S is freely acting Žthat is, it has no fixed points.. For Ž r,a, d . s

Fig. 1. Open circles and dots represent the original Voisin–Borcea orbifolds. The line of m’s corresponds to the extension discussed in the text.

Ž10,8,0. the fixed point set of S consists of two curves of genus 1. The base B2 in this case is P 2 blown up at 9 points. In all the other cases the fixed point set of S consists of one curve of genus g plus k rational curves, where g s 12 Ž 22 y r y a . , Ž 11 . k s 21 Ž r y a . . Ž 12 . Note that except for the cases with a s 22 y r, r s 11, . . . ,20, the mirror pair of Y3 is given by the Voisin–Borcea orbifold Y˜3 with r˜ s 20 y r, a˜ s a. Under the mirror transform we have: g˜ s f, f˜s g, where f ' k q 1. In the cases a s 22 y r, r s 11, . . . ,20, the mirror would have to have r˜ s 20 y r and a˜ s a s r˜ q 2, where r˜ s 0, . . . ,9. We have depicted these cases in Fig. 1 using the ‘‘m’’ symbol. In particular, we have plotted cases with a s r q 2, r s 0, . . . ,10. The Hodge numbers for these cases are still given by Ž9. and Ž10. Žwhich follows from their definition as mirror pairs of the cases with a s 22 y r, r s 11, . . . ,20.. ŽThis is true for a s r q 2, r s 0, . . . ,9. Extrapolation to r s 10 is motivated by the fact that in this case we get Ž h1,1 ,h2,1 . s Ž11,11. which is the same as for Ž r,a, d . s Ž10,10,0... In w11x it was argued that these Voisin–Borcea orbifolds also exist, albeit they are singular. In fact, some of them can be constructed explicitly Žsee w11x for details.. For instance, the case with r s 2,a s 4 has Hodge numbers Ž h1,1,h2,1 . s Ž3,51., which correspond to the singular Calabi–Yau realized as the Z 2 m Z 2 orbifold with discrete torsion w32,33x. As we have already mentioned, here we would like to briefly review F-theory compactifications on Voisin–Borcea orbifolds Žwhich correspond to N s 1 supersymmetric vacua in six dimensions.. Thus, consider F-theory on Y3 with Ž r,a, d . / Ž10,10,0. or Ž10,8,0.. This gives rise to the following massless spectrum in six dimensions. The number of tensor multiplets is T s r y 1. The number of neutral hypermultiplets is H s 22 y r. The gauge group is SO Ž8. m SO Ž8. k . There are g adjoint hypermultiplets of the first SO Ž8.. There are no hypermultiplets charged under the other k SO Ž8.’s. Under mirror symmetry 4 g and f s k q 1 are interchanged. Thus, 4

For a discussion of F-theory compactifications on mirror manifolds, see, e.g., w34x.

Z. Kakushadzer Physics Letters B 459 (1999) 497–506

the vector multiplets in the adjoint of SO Ž8. k are traded for g y 1 hypermultiplets in the adjoint of the first SO Ž8.. That is, gauge symmetry turns into global symmetry and Õice-Õersa. This can be pushed further to understand F-theory compactifications on Calabi–Yau three-folds with a s r q 2, r s 1, . . . ,10, which give the following spectra. The number of tensor multiplets is T s r y 1. There are H s 22 y r neutral hypermultiplets. In addition there are g s 10 y r hypermultiplets transforming as adjoints under a global SO Ž8. symmetry. There are no gauge bosons, however. It is not difficult to verify that this massless spectrum is free of gravitational anomalies in six dimensions. In fact, in w11x it was conjectured that all of these singular Calabi–Yau three-folds exist, and F-theory compactifications on these singular spaces are equivalent to F-theory compactifications on smooth Calabi–Yau three-folds according to the following relation w11x: F-theory on Y3 with Ž h1,1 ,h2,1 . s Ž r q 1,61 y 5r . is equivalent to

505

string backgrounds gives evidence for the conjectured extension of Nikulin’s classification. In fact, in our discussion of non-perturbatiÕe K3 orientifolds we have encountered four new cases: Ž hˆ 1,1 ,hˆ 2,1 . s Ž5,185.,Ž6,156., and Ž h1,1 , h 2,1 . s Ž11,59.,Ž7,55.. The first two cases correspond to the Yˆ3 three-folds with r s 4,5, respectively. Our construction of non-perturbative K3 orientifolds with the NS–NS B-flux therefore gives additional evidence for the above conjecture. As to the other two cases, they correspond to Voisin–Borcea orbifolds Žwithin Nikulin’s classification. with Ž r,a. s Ž2,0. Žin this case the base is the Hirzebruch surface F4 . respectively Ž r,a. s Ž2,2. Žin which case the base is either F0 or F1 .. Before we end this note, we would like to point out that, having understood non-perturbative K3 orientifolds with NS–NS B-flux, it is now straightforward to construct non-perturbative orientifolds on T 6rG orbifolds with SUŽ3. holonomy in the presence of NS–NS B-flux along the lines of w13,14x.

F-theory on Yˆ3

Acknowledgements

with Ž hˆ 1 ,1 ,hˆ 2,1 . s Ž r q 1,301 y 29r .

This work was supported in part by the grant NSF PHY-96-02074, and the DOE 1994 OJI award. I would also like to thank Albert and Ribena Yu for financial support.

= Ž r s 1, . . . ,9 . .

Ž 13.

Thus, for instance, for r s 2 we get Ž hˆ ,hˆ s Ž3,243., which is the elliptic Calabi–Yau given by the T 2 fibration over the base P 1 m P 1. In the above discussion of perturbatiÕe K3 orientifolds we have encountered Calabi–Yau threefolds with the Hodge numbers Ž hˆ 1,1 , hˆ 2,1 . s Ž3,243.,Ž7,127.,Ž9,69.,Ž10,40., and also Ž h1,1,h2,1 . s Ž20,14.,Ž16,10. Žsee the above table.. The last two cases are within Nikulin’s classification, and correspond to Ž r,a. s Ž11,9. and Ž11,11., respectively. The first four cases, however, are outside of Nikulin’s classification, and correspond to the above Calabi– Yau three-folds Yˆ3 Žwhose Hodge numbers are given by Ž hˆ 1,1,hˆ 2,1 . s Ž r q 1,301 y r .. with r s 2,6,8,9, respectively. The cases r s 6,8 were originally discussed in w35x. In fact, in w36x the Calabi–Yau threefold with Ž hˆ 1,1 ,hˆ 2,1 . s Ž7,127. Žcorresponding to r s 6. was explicitly constructed. Here we also have an additional case with r s 9. The fact that perturbative K3 orientifolds with NS–NS B-field are consistent 1,1

2,1 .

References w1x G. Pradisi, A. Sagnotti, Phys. Lett. B 216 Ž1989. 59; M. Bianchi, A. Sagnotti, Phys. Lett. B 247 Ž1990. 517; Nucl. Phys. B 361 Ž1991. 539. w2x E.G. Gimon, J. Polchinski, Phys. Rev. D 54 Ž1996. 1667. w3x E.G. Gimon, C.V. Johnson, Nucl. Phys. B 477 Ž1996. 715; A. Dabholkar, J. Park, Nucl. Phys. B 477 Ž1996. 701. w4x Z. Kakushadze, G. Shiu, S.-H.H. Tye, Phys. Rev. D 58 Ž1998. 086001. w5x M. Berkooz, R.G. Leigh, Nucl. Phys. B 483 Ž1997. 187. w6x C. Angelantonj, M. Bianchi, G. Pradisi, A. Sagnotti, Ya.S. Stanev, Phys. Lett. B 385 Ž1996. 96. w7x Z. Kakushadze, Nucl. Phys. B 512 Ž1998. 221. w8x Z. Kakushadze, G. Shiu, Phys. Rev. D 56 Ž1997. 3686; Nucl. Phys. B 520 Ž1998. 75. w9x G. Zwart, Nucl. Phys. B 526 Ž1998. 378. w10x G. Aldazabal, A. Font, L.E. Ibanez, ´˜ G. Violero, Nucl. Phys. B 536 Ž1998. 29. w11x Z. Kakushadze, G. Shiu, S.-H.H. Tye, Nucl. Phys. B 533 Ž1998. 25.

506

Z. Kakushadzer Physics Letters B 459 (1999) 497–506

w12x Z. Kakushadze, Phys. Lett. B 434 Ž1998. 269; Nucl. Phys. B 535 Ž1998. 311; Phys. Rev. D 58 Ž1998. 101901. w13x Z. Kakushadze, hep-thr9904007. w14x Z. Kakushadze, hep-thr9904211. w15x J. Polchinski, Phys. Rev. D 55 Ž1997. 6423. w16x J.D. Blum, Nucl. Phys. B 486 Ž1997. 34. w17x M. Bianchi, G. Pradisi, A. Sagnotti, Nucl. Phys. B 376 Ž1992. 365. w18x Z. Kakushadze, Nucl. Phys. B 544 Ž1999. 265. w19x E. Witten, JHEP 9802 Ž1998. 006. w20x A. Sen, S. Sethi, Nucl. Phys. B 499 Ž1997. 45. w21x M. Berkooz, R.G. Leigh, J. Polchinski, J.H. Schwartz, N. Seiberg, E. Witten, Nucl. Phys. B 475 Ž1996. 115. w22x K. Intriligator, Nucl. Phys. B 496 Ž1997. 177; J.D. Blum, K. Intiligator, Nucl. Phys. B 506 Ž1997. 223. w23x M.B. Green, J.H. Schwartz, P.C. West, Nucl. Phys. B 254 Ž1985. 327; J. Erler, J. Math. Phys. 35 Ž1994. 1819. w24x A. Sagnotti, Phys. Lett. B 294 Ž1992. 196.

w25x S. Kachru, C. Vafa, Nucl. Phys. B 450 Ž1995. 69. w26x C. Vafa, Nucl. Phys. B 469 Ž1996. 285. w27x C. Voisin, in: A. Beauville et al. ŽEds.., Jornees ´ de Geome´ trie Algebrique d’Orsey, Asterisque, vol. 218, Soc. Math. ´ ´ France, 1993, p. 273. w28x C. Borcea, in: B.R. Greene, S.-T. Yau ŽEds.., Mirror Manifolds II, International Press and AMS, 1997, p. 717. w29x D.R. Morrison, C. Vafa, Nucl. Phys. B 473 Ž1996. 74; B 476 Ž1996. 437. w30x P.S. Aspinwall, Nucl. Phys. 460 Ž1996. 57. w31x V.V. Nikulin, in: Proc. ICM, Berkeley, 1986, p. 654. w32x A. Font, L.E. Ibanez, ´˜ F. Quevedo, Phys. Lett. B 217 Ž1989. 272. w33x C. Vafa, E. Witten, J. Geom. Phys. 15 Ž1995. 189. w34x E. Perevalov, G. Rajesh, Phys. Rev. Lett. 79 Ž1997. 2931. w35x E.G. Gimon, C.V. Johnson, Nucl. Phys. B 479 Ž1996. 285. w36x P. Berglund, E.G. Gimon, Nucl. Phys. B 525 Ž1998. 73.