Non-perturbative transitions among intersecting-brane vacua

DFTT 9/2011

CPHT-RR041.0511

LPT-ORSAY 11-41

ROM2F/2011/05

Abstract

We investigate the transmutation of D-branes into Abelian magnetic backgrounds on the world-volume of higher-dimensional branes, within the framework of global models with compact internal dimensions. The phenomenon, T-dual to brane recombination in the intersecting-brane picture, shares some similarities to inverse small-instanton transitions in non-compact spaces, though in this case the Abelian magnetic background is a consequence of the compactness of the internal manifold, and is not ascribed to a zero-size non-Abelian instanton growing to maximal size. We provide details of the transition in various supersymmetric orientifolds and non-supersymmetric tachyon-free vacua with Brane Supersymmetry Breaking, both from brane recombination and from a field theory Higgs mechanism viewpoints.

\setstretch

1.1

## 1 Introduction and summary of results

It is well known that D5 branes can be described as gauge instantons on the world-volume of D9 branes in the limit in which the instanton size, related to the vacuum expectation value of the D9-D5 states, is zero [1]. This suggests that, blowing up the small instanton size, it is possible to connect orientifold vacua with D9 and D5 branes to string vacua with internal magnetic field fluxes. The resulting phenomenon has been briefly discussed in the literature [2] in some supersymmetric examples [3, 4]. From a string theory perspective this transition can be elegantly described, in the T-dual framework of intersecting branes, in terms of “brane recombination”, namely in terms of a new configuration of homology cycles wrapped by the branes that respects the RR charge distribution. It has also been suggested that this phenomenon can be captured in the low-energy limit by a conventional Higgs mechanism, at least in some simple examples, where scalars living at the brane intersections condense. For a - configuration the corresponding vev’s were identified by Witten [1] to be related to the instanton size. As a result, in the type I set-up, the condensate interpolates between zero-size instantons, in the zero-vev limit, and constant non-Abelian magnetic field, in the large-vev limit corresponding to a “fat” gauge instanton of maximal size.

In the present paper, we address the problem of possible non-perturbative transitions among different orientifold vacua, and our results extend the discussion in the literature in two respects. On the one hand, by considering the low-energy field equations for the Abelian theory living on a D9 brane with a compact internal space, we show that an Abelian constant magnetic field is generated by the solution, that is naturally related to the normalisable constant zero-mode of the Laplacian in the compact space. We show that this magnetic field (and not the non-Abelian configuration generated by a standard “fat” instanton) is T-dual to the supersymmetric angle configuration in toroidal intersecting-brane models. We point out interesting analogies and differences between the two classical configurations. The Abelian configuration contains, in addition to the constant magnetic field, a singular part which exactly reproduces the singularity of a more conventional non-Abelian instanton configuration in the singular gauge and in the zero-size limit. This singularity is actually due to the zero-size limit, and it is blown-up by giving the instanton a finite-size, related to the vev of D9-D5 states. We analyse the corresponding singularity for the Abelian case by studying the effects of the vev and of the Dirac-Born-Infeld non-linearities. We could not find a simple way to resolve the singularity, although it is possible that higher-derivative corrections to the Dirac-Born-Infeld Action may provide the tool.

On the other hand, we extend the analysis of non-perturbative transitions to the class of non-supersymmetric vacua with Brane Supersymmetry Breaking. There is a large literature on non-supersymmetric string vacua built over the years, in the heterotic strings [5, 6, 7, 8, 9, 10, 11], type 0 orientifolds [12, 13], type II orientifolds with supersymmetry broken by compactification [14, 15, 16, 17], by magnetic fields [3, 18, 19, 20, 21, 22, 23, 24, 25], at the string scale [26, 27, 28, 29, 30] or by a combination of these effects [31, 32]. Although most of the non-supersymmetric vacua are manifestly unstable at the classical level due to the tachyonic states appearing in some regions of the moduli space some, most notably the models with “Brane Supersymmetry Breaking” [26, 27, 28, 29, 30] have no manifest classical instabilities. At tree-level, these models exhibit supersymmetry in the closed-string sector, while in the open-string sector supersymmetry is explicitly broken or, more precisely, non-linearly realised on stacks of anti-branes [33, 34]. Technically, the breaking of supersymmetry is ascribed to the simultaneous presence of branes and planes, where are exotic orientifold planes with positive tension and positive RR-charge. This system breaks supersymmetry in a way similar to brane-antibrane pairs, but with the notable difference of not introducing any tachyonic instability.

It is widely believed that all non-supersymmetric string vacua are unstable towards decaying into supersymmetric ones. When classical instabilities are present, the decay is clearly due to tachyon condensation [35, 36] that indeed interpolates between different unstable non-supersymmetric orientifold vacua [37]. In the Brane Supersymmetry Breaking case, however, it can only occur through non-perturbative effects. The least understood and most intriguing cases are the ones in which the candidate supersymmetric vacua have completely different geometry, O-plane contents and different twisted sectors when compared to their non-supersymmetric parents. The only known proposal in this direction [38] makes explicit use of S-duality, and thus the real dynamics of the transition cannot be followed within a perturbative CFT set-up. We have nothing to add about these cases in the present work. There are however other examples in which the non-supersymmetric classically stable model and the candidate supersymmetric vacuum have the same geometry and differ only in the configuration of branes. The fate of such vacua is part of the subject of the present paper. We argue that the magnetic transmutation, T-dual to brane recombination discussed in the literature, is responsible for the transition among non-supersymmetric vacua as well. Whereas in the supersymmetric case the states acquiring vev’s are flat directions, in non-supersymmetric examples we argue that positive masses are generated for these states, that makes the transition truly non-perturbative in nature. We present explicit examples built upon the orientifold of the type IIB [30] in six dimensions and upon the orientifolds of type IIA superstring in four dimensions, and discuss the relation of supersymmetric vacua or of configurations with Brane Supersymmetry Breaking and discrete torsion [30] with magnetised vacua in six [3] and four [39, 40, 41, 42, 43] dimensions.

The paper is organised as follows. In section 2 we discuss the transition among vacua as a brane recombination process, for models in six dimensions built upon the orbifold, and in four dimensions built upon the type IIB orientifold with discrete torsion. We address the problem both for supersymmetric and Brane Supersymmetry Breaking vacua. The description of the transition in terms of brane recombination is shown to work nicely in all examples. In section 3 we analyse the same problem from a low-energy viewpoint, in terms of a field theory Higgs mechanism triggered by non-vanishing vev’s of open-string states living at the intersection of different stacks of branes. We perform a case analysis for six-dimensional examples and show that when supersymmetry is preserved the Higgs mechanism exactly captures the dynamics by yielding the correct massless spectrum of the corresponding intersecting brane models. On the contrary, when supersymmetry is broken some states ought to get masses from Yukawa (for fermions) and quartic-order (for scalars) couplings that we argue to exist. In four dimensions, however, the models we study involve anti-branes that are not standard instantons of the gauge theory living on the D9 branes, rather they should be interpreted as new stringy non-perturbative configurations. As a consequence, at low energy the Higgs mechanism does not accurately captures the dynamics of the transition, whereas brane recombination does, as shown in section 2. In section 4 we perform an analysis of the quantum stability of the Brane Supersymmetry Breaking models, by computing the vacuum energy as a function of the anti-brane positions and/or Wilson lines and by computing the quantum induced masses for the states whose vev’s trigger the recombination process. We argue that a positive mass for states is generated, that suggests a non-perturbative transition associated to a real tunnelling phenomenon through a potential barrier. Section 5 contains a detailed field-theory analysis in a superfield formalism which shows explicitly the connection between the vev of the Higgs field and the emergence of the Abelian magnetic field. We also compare properties of our Abelian classical solution in a compact space with the conventional SU(2) instanton solution. Appendix A collects some partition functions that are used in the paper, while in appendix B more examples of six dimensional models are reported, both supersymmetric and with Brane Supersymmetry Breaking.

## 2 Brane recombination

In this section we study the possible connections among different string vacua, supersymmetric and not, as a result of brane recombination. In particular, we shall show that all string vacua, at least in a given orbifold construction, live in the same moduli space and are connected one to the other by the process of brane recombination. For simplicity we shall concentrate on some interesting prototype examples in and based on and orientifolds, with and without supersymmetry. The complete one-loop amplitudes are given in the appendices, where additional examples are also discussed. In the following we shall discuss in detail few cases, and for simplicity we shall adopt a more geometrical language, in terms of homology cycles wrapped by the branes.

In general, an orientifold of type II superstrings compactified on suitable Calabi-Yau manifolds involves an action of the world-sheet parity , possibly combined with some (order-two) automorphism of the internal manifold. In the following, we shall consider the anti-holomorphic involution , , so that the effective orientifold projection involves , where the left-handed space-time fermionic index is needed in order that be order-two on the whole string spectrum.

As is well known, the fixed locus of defines the location of the orientifold planes, that extend through the whole non-compact Minkowski space and wrap additional -cycles of the internal Calabi-Yau space. Given our choice of the anti-holomorphic involution, the fixed locus identifies a special Lagrangian submanifold of , and thus their introduction preserves some amount of supersymmetries. As usual, the consistency of the construction, i.e. R-R tadpole cancellation, requires the introduction of suitable number of D-branes that, aside from expanding in the non-compact dimensions, wrap suitable internal -cycles . The open-string excitations thus comprise unitary gauge groups and chiral matter in (anti-)symmetric and bi-fundamental representations, with multiplicities given by the intersection numbers of pairs of branes. We shall not review here the details of the construction, but refer the interested reader to the vast literature on the subject [44]. In the following we shall study specific examples related to six and four dimensional orbifold compactifications.

### 2.1 Six-dimensional models

The six-dimensional models we want to study are all based on the orbifold compactification of type IIB superstring. For simplicity we shall assume that the factorises into the product , both with purely imaginary complex structure , . The analysis can be easily extended to the case where the complex structure of the ’s has a quantised non-vanishing real part. We denote by the complex coordinate on the , where each entry identifies the position on the -th factor. The is generated by the single element that reverts all coordinates: .

To define the homology of is it convenient to start from that of the covering . If one denotes by and the canonical horizontal and vertical one-cycles of each rectangular , with intersection form given by

 ai∘aj=0,bi∘bj=0,ai∘bj=δij, (2.1)

the homology of is clearly given by suitable combinations of the and cycles. In particular is spanned by

 ¯πi=(a1⊗a2,b1⊗b2,a1⊗b2,b1⊗a2,a1⊗b1,a2⊗b2), (2.2)

with intersection form

 ¯I=σ1⊗diag(1,−1,−1). (2.3)

Modding-out the torus by the action implies that the cycles are not invariant under the . Therefore, one is bound to construct new cycles that correspond to invariant orbits of the orbifold group. In the case at hand one finds that the invariant two-cycles are

 πi=(1+g)¯πi. (2.4)

Moreover, when computing the intersection form, one has to take into account that the is a double cover of the orbifold, and therefore

 Iij=12πi∘πj=2¯Iij. (2.5)

These cycles, however, do not span the lattice since the resolved orbifold singularities introduce new two-cycles , , called exceptional or collapsed cycles, associated to the ’s localised at the 16 fixed points with coordinates , where . Their intersection form

 Ixy,vw=−2δxvδyw, (2.6)

is given by the Cartan matrix of the Lie algebra , while . Although the Kummer basis does not span entirely , it is actually a convenient choice to discuss the orbifold . A generic two-cycle, can thus be written as

 Πa=6∑i=1ciaπi+4∑x,y=1ϵxyaexy, (2.7)

where the and the are suitable coefficients. It is convenient to refer to

 ^Πa=6∑i=1ciaπi (2.8)

as the bulk cycle, since it is inherited from the covering and exists at a generic point on the orbifold. Notice that in the literature it is customary to refer to bulk cycles if the coefficients are suitable integers. We find, however, more appropriate to use this term for the component in the homology inherited by the covering torus, independently of the coefficients .

#### 2.1.1 Supersymmetric vacua

We have now all the ingredients to discuss the orientifold of type IIB superstring compactified on the orbifold.

Let us start discussing the supersymmetric case, where the orientifold projection employed is precisely the previously introduced. The fixed locus of is given by the and two-cycles that, taking into account a correct normalisation of the RR charge, identify the cycles

 ΠO=2π1,ΠO′=2π2, (2.9)

wrapped by the O7 and planes, respectively.

D7 branes are free to wrap a generic two-cycle on the . The Fourier coefficients ’s and ’s have now a physical interpretation in terms of charges of untwisted and twisted RR forms, respectively. In order to make contact with the perturbative CFT construction reported in appendix A, it is useful to limit our attention to factorisable bulk two-cycles, that are obtained by combining two one-cycles, one on each , and to express then Fourier coefficients in terms of the number of times the branes wraps these one-cycles. To be specific, we write

 ^Πa=4∑i=1ciaπi=2⨂i=1(miaai+niabi), (2.10)

with and co-prime integers. As a result, a bulk cycle is entirely specified by the integers , and

 (c1a,c2a,c3a,c4a)=12(m1am2a,n1an2a,m1an2a,n1am2a). (2.11)

The Fourier coefficients associated to the collapsed cycles identify the fixed point crossed by the cycle associated to the D7 brane, and are related to the orbifold action on the Chan-Paton coefficients. With a suitable normalisation

 ϵxya={±12iif(zx1,zy2)∈^Πa,0otherwise, (2.12)

and define the charge with respect to the twisted RR six-form potentials.

Although, by construction, is invariant under the orbifold action, generically it will not be so under , and thus on the orientifold one has to introduce also image branes wrapping the cycle . The Fourier coefficients of are completely determined once we define the action of on the basis two-cycles:

 ~Ω⋅(π1,π2,π3,π4,π5,π6)=(π1,π2,−π3,−π4,−π5,−π6),~Ω⋅exy=−exy. (2.13)

As a result, on the orbifold, the invariant configuration only wraps bulk cycles, and this is consistent with the perturbative string description where the twisted tadpoles are identically vanishing. Moreover, has only components along and , consistently with the geometry of the orientifold planes.

At this point, one has all the ingredients to derive the light (chiral) spectrum of a vacuum involving a given number of D7 branes. A vacuum configuration is then fully determined by specifying the wrapping numbers of the -th stack of D7 branes together with the coordinate of (at least) one fixed point crossed by them. In the following we shall assume that all branes pass through the fixed point at the origin of the .

As a result, each D7 brane supports a unitary gauge group so that

 GCP=∏aU(Na). (2.14)

Open strings living at the intersection of a and brane are in the bi-fundamental representation and come in families, where now

 Iab=Πa∘Πb=122∏i=1(mianib−mibnia)−2∑x,yϵxyaϵxyb. (2.15)

Open strings living at the intersection of a and brane are in the bi-fundamental representation and come in families, where now

 Ia¯b=Πa∘Π¯b=122∏i=1(mianib+mibnia)+2∑x,yϵxyaϵxyb. (2.16)

Finally, any time a brane intersects its image one has chiral matter in the symmetric and anti-symmetric representations. In particular, the number of (anti-)symmetric representations is

 (2.17)

where we have used the property that, for any D7 brane,

 ∑x,y2ϵxya=0 mod 4i,and4∑x,yϵxyaϵxya=−4. (2.18)

Although a generic configuration of D7 branes is not supersymmetric, if their relative angles with respect to the O planes satisfy the familiar condition , for each stack of branes, then some amount of supersymmetry is preserved. The condition on the angles is equivalent to the requirement that the two-cycles wrapped by the branes be actually a special Lagrangian submanifold calibrated by , where here is the holomorphic -form of the internal manifold. In the T-dual version in terms of magnetised background, the supersymmetry conditions translates into the self-duality of the internal flux, [3].

If we compare the expressions (2.15), (2.16) and (2.18) with the spectrum worked-out in appendix A and summarised in table A.1, we can identify

 Sab=−4∑x,yϵxyaϵxyb, (2.19)

i.e. the number of mutual intersections between branes and coinciding with the orbifold fixed points is precisely given by minus four times the inner product of the Fourier coefficients relative to the collapsed cycles.

As an example, the original vacuum of [45, 46] involves two stacks of 16 branes each, with wrapping numbers

 ^ΠD7∼(m1,n1;m2,n2)=(1,0;1,0),^ΠD7′∼(m1,n1;m2,n2)=(0,1;0,1). (2.20)

Pairs of stacks of D7 branes can recombine on the . Although the dynamics of this process cannot be described by the underlying perturbative CFT, the rules for analysing the recombination process are well established, and essentially amount to RR charge conservation [47]. For the orientifold this is particularly simple, since the invariant combinations of branes carry a charge only with respect to the and cycles. In fact, one finds that the recombination of and brane yields branes

 (2.21)

with , and wrapping numbers determined by the conditions

 m1cm2c=NaNcm1am2a+NbNcm1bm2b,n1cn2c=NaNcn1an2a+NbNcn1bn2b. (2.22)

The amazing result is that all supersymmetric orientifold vacua, with the same closed-string spectrum, are all in the same moduli space, and thus are connected to the model of [45, 46] via the recombination of suitable numbers of branes. This connection is expected to hold also in the case of compactifications on orbifolds of skew tori (with quantised real component of the complex structure [48]), though we have not analysed it in detail.

Let us consider in fact the complete recombination of the D7 and branes. The resulting configuration involves a single stack of 16 branes with wrapping numbers thus yielding a gauge group together with 4 hypermultiplets in the anti-symmetric representation. Similarly, one could recombine only 4 D7 branes together with the 16 ones, so that the resulting configuration contains for instance two different stacks of branes of the type

 12 D7 branes with:(m1,n1;m2,n2)=(1,0;1,0),4 D7′′ branes with:(m1,n1;m2,n2)=(1,2;1,2). (2.23)

The gauge group is now with hypermultiplets in the representations . This is precisely the magnetised vacuum without D5 branes found in [3].

In the T-dual picture in terms of magnetised branes the recombination process has the suggestive interpretation of brane transmutation, i.e. some D5 branes dilute into a constant magnetic background on the world-volume of the D9 branes. Alternatively, the process is somewhat dual to the small instanton transitions of [1], where now the instanton becomes fatter and fatter and invades the whole compact space. We shall give evidence to this picture in section 5.

In the field theory limit, the recombination of branes and is described in terms of the Higgsing of massless scalars in the bi-fundamental , and we shall review it in the next section.

#### 2.1.2 Brane Supersymmetry Breaking vacua

Brane Supersymmetry Breaking [27] is an interesting deformation of the supersymmetric orientifold, where the world-sheet parity is further dressed with an involution of the internal manifold that affects the projection of the twisted sector, . In this way, the Klein-bottle projection (anti-)symmetrises the (NS-NS) R-R twisted sector and therefore the closed-string spectrum, still supersymmetric, comprises 17 tensor multiplets together with 4 hypermultiplets aside, of course, of the gravitational supermultiplet. From a geometrical perspective, the involution affects the nature of the orientifold planes that now are and , i.e. the O-planes wrapping the cycle have positive NS-NS tension and positive R-R charge. As a result, they have a different orientation with respect to the more conventional , so that , while as before.

An additional difference with respect to the previous supersymmetric case, is that the presence of also affects the components of a D7 brane with respect to the collapsed cycles, and the behaviour of the latters under :

 ϵxya={±12if(zx1,zy2)∈^Πa,0otherwise, (2.24)

and

 ^Ω⋅(π1,π2,π3,π4,π5,π6)=(π1,π2,−π3,−π4,−π5,−π6),^Ω⋅exy=+exy. (2.25)

The choice of the coefficients reflects the orbifold action on the Chan-Paton labels, that is now real [27]. As a result of eq. (2.25), the invariant combination

 Πa+Π¯a=2c1aπ1+2c2aπ2+2∑x,yϵxyaexy (2.26)

has now components not only along the bulk and cycles, but also along the exceptional ones. This implies that additional conditions have to be imposed on a consistent configuration of D7 branes, resulting in additional tadpoles for the twisted RR six-form potentials. Of course, this is in agreement with the perturbative CFT description, that is summarised in appendix A.

The chiral open-string spectrum is still captured by the intersection form of the two-cycles wrapped by the branes. A generic brane still supports a unitary gauge group , while open strings stretched between intersecting and branes come in bi-fundamental representations with a degeneracy given by

 Iab=Πa∘Πb=12[2∏i=1(mianib−mibnia)−4∑x,yϵxyaϵxyb], (2.27)

open strings stretched between intersecting and branes come in bi-fundamental representations with a degeneracy given by

 Ia¯b=Πa∘Π¯b=12[2∏i=1(mianib+mibnia)−4∑x,yϵxyaϵxyb], (2.28)

and, finally, any time a brane intersect its image chiral fermions carry a (anti-)symmetric representation with a degeneracy given by

 (2.29)

Comparison with table 2 allows us to identify

 ϵaϵbSab=−4∑x,yϵxyaϵxyb, (2.30)

where, in the CFT analysis, counts, as usual, the number of mutual intersections between branes and coinciding with the orbifold fixed points, while takes into account the action of the orbifold on the Chan-Paton degeneracies. Note also that, since the are now real, . Special care is needed when a D7 brane wraps the same cycle as an orientifold plane, since now gauge groups become orthogonal or symplectic depending on whether the O-plane is an or an one.

Since supersymmetry is explicitly broken in the open-string sector, charged scalars do not carry the same representations as their “would be” fermionic superpartners. Although the geometrical description adopted in this section is not suited for determining their spectrum, the perturbative CFT analysis of appendix A gives complete information about their representations and multiplicities, and the complete spectrum is summarised in table A.2.

Also in this case with Brane Supersymmetry Breaking we conjecture that all vacua, with the same closed-string spectrum, are connected to the original model of [27] via the recombination of suitable numbers of branes, though they might be separated by an energy barrier as argued in section 4.

Before we show the connection by working out some explicit examples, one should note that the rules for brane recombination (2.22) are now changed since they have to take into account also the conservation of the twisted charges, that are now non-vanishing. Let us suppose we recombine one and one brane

 (Πa+Π¯a)∪(Πb+Π¯b)=(m1am2a+m1bm2b)π1+(n1an2a+n1bn2b)π2+2∑x,y(ϵxya+ϵxyb)exy. (2.31)

This cannot yield automatically a factorisable brane since, in general

 2∑x,y(ϵxya+ϵxyb)≠4, (2.32)

that is to say the associated cycle crosses more or less fixed points than the canonical four touched upon by a factorisable cycle. As a result, in order to allow the recombination of two or more branes into a factorisable one the additional condition

 2∑x,y∑aϵxya=0 mod 4 (2.33)

must hold. In this case the bulk cycle of the recombined branes is identified by the wrapping numbers , with

 m1cm2c=∑am1am2a,n1cn2c=∑an1an2a. (2.34)

Let us discuss now some simple examples where the various recombinations of branes in the original model [27] yield new and old vacua with Brane Supersymmetry Breaking. The model involves eight copies of physical branes wrapping the following cycles:

 Π±D7=^ΠD7±12(e11+e12+e21+e22),Π±¯¯¯¯¯¯D7′=^Π¯¯¯¯D7′±12(e11+e13+e31+e33), (2.35)

where and , with the minus sign implying that the branes are actually anti-branes, and thus their cycle has an opposite orientation. The sign in front of the exceptional cycles reflects, as already stated, the different orientifold action on the associated Chan-Paton labels, and thus the various twisted RR charges.

In this case, one needs to recombine at least three different stacks of branes in order to get a factorisable one. For instance, one could recombine222The additional factor 2 takes into account the fact that the and branes wrap the same cycle as their orientifold images. Here and in the following we shall omit to indicate the orientifold images, though they are tacitly present in the identification of the invariant cycle.

 (4×2Π−D7)∪(8×2Π+¯¯¯¯¯¯D7′)∪(8×2Π−¯¯¯¯¯¯D7′) (2.36)

to yield a new vacuum with eight horizontal D7 branes of type , four horizontal D7 branes of type and four oblique branes wrapping the cycle

 ΠD7o=12(π1−4π2+2π3−2π4)−12(e11+e12+e21+e22). (2.37)

This configuration clearly satisfies both the untwisted and the twisted tadpole conditions, and reproduces the Brane Supersymmetry Breaking vacuum without D5 antibranes of [3] and with anti-self-dual magnetic background, with gauge group .

Another possibility would be to recombine all branes of the original model. This would identify the two-cycle , without any leg along the collapsed cycles. What kind of brane could this correspond to? Notice that in the original configurations all branes were crossing the orbifold fixed points, and in particular the fixed point at the origin of the with . As a result, also the recombined brane(s) should cross the origin, since the recombination process does not involve any deformation associated to brane displacements and/or Wilson lines. Moreover, in this vacuum all branes passing through the origin cross exactly four fixed points, and thus wrap four collapsed cycles. As a result, the complete recombination of all branes in the original model must result in, at least, two different stacks of branes wrapping the same bulk cycle but with opposite twisted charges. Given the Fourier coefficients one has the solution

 (8×2Π+D7)∪(8×2Π−D7)∪(8×2Π+¯¯¯¯¯¯D7′)∪(8×2Π−¯¯¯¯¯¯D7′)=8(Πd+Πd′), (2.38)

with

 (2.39)

The Chan-Paton gauge group is now with left-handed fermions in the adjoint representation and in four copies of the , and right handed fermions in two copies of the bi-fundamental representation . From the CFT analysis of appendix A one also finds 16 real scalars in the together with 8 real scalars in the , as summarised in table 2.1.

More examples can of course be studied, though we leave a detailed analysis to the interested reader.

### 2.2 Four-dimensional models

In four dimensions we focus our attention on a specific class of models based on the orbifold with discrete torsion. The naive orientifold construction with orthogonal D-branes is a generalisation of the Brane Supersymmetry Breaking construction with supersymmetry broken in the open-string sector [30]. However, it was shown in [41, 39, 40] that supersymmetric vacua exist also for this compactification and involve non-trivial angles and/or magnetic backgrounds. The natural question is then whether the two constructions are related by some (non-perturbative) effect, like brane recombination.

The group is generated by the elements

 g=(+,−,−),h=(−,−,+), (2.40)

while, as usual, , and the -th entry denotes the action of the orbifold on the -th factor in the factorisable . The presence of discrete torsion, i.e. a relative sign in the independent orbit in the twisted sector, exchanges the and Hodge numbers of the smooth Calabi-Yau manifold [49], and acts as mirror symmetry in the type II compactification.

The description of the homology of the space is similar to the six-dimensional case previously studied. is generated by the eight bulk cycles

 {¯π}i=(a1⊗a2⊗a3,b1⊗b2⊗b3,b1⊗b2⊗a3,a1⊗a2⊗b3,  b1⊗a2⊗b3,a1⊗b2⊗a3,a1⊗b2⊗b3,b1⊗a2⊗a3) (2.41)

inherited from the . These three-cycles are common both to the orbifolds with and without discrete torsion. In the former case, however, one has also to consider the exceptional cycles that are built by tensoring a collapsed two-cycle , localised at the fixed point of the -th element of the orbifold group, with a one-cycle of the spectator :

 αgxy=2a1⊗egxy,βgxy=2b1⊗egxy,αfxy=2a2⊗efxy,βfxy=2b2⊗efxy,αhxy=2a3⊗ehxy,βhxy=2b3⊗ehxy. (2.42)

Altogether, these generate whose dimension is indeed . Here and in the following we use the index to label both the -th element of the orbifold group and the -th component of the factorisable . These are in fact related since, for instance, leaves the first fixed, while is inert on the second one, and on the third .

The intersection form of these cycles can be straightforwardly derived from the one of the covering and from the intersection form of the exceptional cycles in . In particular, noting that under the action each bulk three-cycle has exactly three orbifold images

 πi=(1+g)(1+h)¯πi, (2.43)

and that collapsed cycles of different twisted sectors do not intersect, one finds the following non-vanishing entries of the intersection form:

 Iij=14πi∘πj=−4iσ2⊗14, (2.44)

and

 Iλκxy,vw=12αλxy∘βκvw=−4δλκδxvδyw, (2.45)

where is the Pauli matrix, is a identity matrix, and we have used the fact that, for a given twisted sector, , while . A generic three-cycle of can thus be written as

 Πa=8∑i=1ciaπi+∑λ=g,f,h4∑x,y=1(μxya,λαλxy+νxya,λβλxy). (2.46)

Also in this case it is customary to refer to

 ^Πa=8∑i=1ciaπi (2.47)

as the bulk cycle.

Modding-out the compactification by implies that four different types of O6 planes are introduced at the fixed locus of , , and . In particular, given the action of the orbifold on the three-cycles and the fact that and , one obtains the following geometry of , , and planes with associated three-cycles

 Πo,ϵo=2ϵoπ1,Πg,ϵg=−2ϵgπ7,Πf,ϵf=−2ϵfπ5,Πh,ϵh=−2ϵhπ3, (2.48)

where we have properly normalised the cycles as pertained to O6 planes. The signs determine the type of orientifold plane , with positive associated to a conventional O-plane with negative tension and charge, while refers to an exotic plane with positive tension and charge. These signs are actually not arbitrary, but have to be correlated to the presence/absence of discrete torsion as [30]

 ϵ=ϵoϵgϵfϵh. (2.49)

In the following, we shall make the definite choice and as in [30, 39, 50], while the other options can be discussed in a similar fashion and require only minor modifications.

A generic D6-brane on the orbifold is thus identified by the Fourier coefficients of its associated three-cycle with respect to the basis in previously introduced. In particular, as in the six-dimensional case, these coefficients are uniquely specified by the wrapping numbers associated to the canonical one-cycles of the -th . As a result,333The signs in eq. (2.50) can be determined by the consistency with the CFT description of the vacuum configuration, and differ from those of [40] the similar minus signs appear in the definition of the cycle wrapped by the O-planes. However, from a direct comparison with the transverse-channel Klein-bottle partition function it seems to us that the definitions (2.48) and (2.50) are more appropriate.

 {ca}i=14(ma1ma2ma3,na1na2na3,na1na2ma3,ma1ma2na3,na1ma2na3,ma1na2ma3,ma1na2na3,na1ma2ma3), (2.50)

while

 μxya,λ=12maλϵxya,λ,νxya,λ=12naλϵxya,λ. (2.51)

The Fourier coefficients associated to the collapsed cycles identify the fixed points crossed by the bulk cycle associated to the D6 brane, and are related to the orbifold action on the Chan-Paton coefficients. So in the case at hand with , one has

 ϵxya,g={±12iif(∙,zx2,zy3)∈^Πa,0otherwise,ϵxya,f={±12iif(zx1,∙,zy3)∈^Πa,0otherwise, (2.52)

while

 ϵxya,h={±12if(zx1,zy2,∙)∈^Πa,0otherwise. (2.53)

Generically, the cycle is not left invariant by , and thus one needs also introduce its image in order to build an invariant configuration. To this end, one has to use the previous action of on the canonical one-cycles of each , as well as its action on the collapsed ones

 ~Ω⋅αλxy=−ϵλαλxy,~Ω⋅βλxy=ϵλβλxy, (2.54)

where the signs are precisely those that determine the type of O-planes associated to the , and twists. Also in this case, it is useful to introduce the invariant combination , in order to determine the effective cycle wrapped by an invariant combination of branes.

The chiral spectrum can be derived as usual by computing the intersection numbers between different stacks of branes and between branes and orientifold planes [40], or by a direct CFT computation of the annulus and Möbius-strip amplitudes [50]. There are some subtleties here associated to the different nature of the orientifold planes. For instance, while D6 branes lying on planes yield unitary gauge groups, when lying on a the Chan-Paton coefficients become real and the gauge groups turns out to be symplectic. Moreover, when D6 branes wrap the same cycle as an O6-plane the twisted charge assignments of eqs. (2.52) and (2.53) have to be modified so to reflect the correct orbifold action on the Chan-Paton labels, that can be found, for instance in [51, 52]. In the following, we shall not be concerned with the complete spectrum but will mainly focus our attention on the gauge group.

Also for this four-dimensional example, different vacua with common closed-string sector, i.e. different solutions of the tadpole conditions, can be connected via brane recombination. In particular, it is quite amusing that the simplest vacuum with Brane Supersymmetry Breaking [50] is actually connected to the supersymmetric solutions of [39, 40, 53]. Let us show this connection in few illustrative examples.

The simplest solution of tadpole conditions is in terms of different stacks of D6 and branes that are parallel to the various O-planes. Their wrapping numbers are

 D6o=(1,0;1,0;1,0),D6f=(0,−1;1,0;0,1),D6g=(1,0;0,1;0,−1),D6h=(0,1;0,