D7-Brane Motion from M-Theory Cycles and Obstructions in the Weak Coupling Limit
aa r X i v : . [ h e p - t h ] S e p HD-THEP-08-1 14 January 2008
D7-Brane Motion from M-Theory Cyclesand Obstructions in the Weak Coupling Limit
A. P. Braun a , A. Hebecker a and H. Triendl b a Institut f¨ur Theoretische Physik, Universit¨at Heidelberg, Philosophenweg 16 und 19D-69120 Heidelberg, Germany b II. Institut f¨ur Theoretische Physik der Universit¨at HamburgLuruper Chaussee 149, D-22761 Hamburg, Germany. ([email protected], [email protected], and [email protected])
Abstract
Motivated by the desire to do proper model building with D7-branes and fluxes, we studythe motion of D7-branes on a Calabi-Yau orientifold from the perspective of F-theory. Weconsider this approach promising since, by working effectively with an elliptically fibredM-theory compactification, the explicit positioning of D7-branes by (M-theory) fluxesis straightforward. The locations of D7-branes are encoded in the periods of certainM-theory cycles, which allows for a very explicit understanding of the moduli space ofD7-brane motion. The picture of moving D7-branes on a fixed underlying space relieson negligible backreaction, which can be ensured in Sen’s weak coupling limit. However,even in this limit we find certain ‘physics obstructions’ which reduce the freedom ofthe D7-brane motion as compared to the motion of holomorphic submanifolds in theorientifold background. These obstructions originate in the intersections of D7-branesand O7-planes, where the type IIB coupling cannot remain weak. We illustrate this effectfor D7-brane models on CP × CP (the Bianchi-Sagnotti-Gimon-Polchinski model) andon CP . Furthermore, in the simple example of 16 D7-branes and 4 O7-planes on CP (F-theory on K3), we obtain a completely explicit parameterization of the moduli spacein terms of periods of integral M-theory cycles. In the weak coupling limit, D7-branemotion factorizes from the geometric deformations of the base space. Introduction
During the last years, significant progress has been made in the understanding of string-theoretic inflation, moduli stabilization, supersymmetry breaking and the fine tuning ofthe cosmological constant using the flux discretuum. The most studied and arguablybest understood setting in this context is that of type IIB orientifolds with D3- and D7-branes [1] (which has close cousins in M-theory [2–4]). Given this situation, it is clearlydesirable to develop the tools for particle-phenomenology-oriented model building in thiscontext. One obvious path leading in this direction is the study of the motion of D7-branesin the compact space and their stabilization by fluxes [5–9]. The long-term goal must beto achieve sufficient control of D7-brane stabilization to allow for the engineering of thedesired gauge groups and matter content based on the D7-brane open string sector [10].This is a non-trivial task since the underlying Calabi-Yau geometry has to be sufficientlycomplicated to allow for the necessary enormous fine tuning of the cosmological constantmentioned above.In the present paper we report a modest step towards this goal in the simple setting ofthe 8-dimensional Vafa model, where 16 D7-branes move on T /Z [11]. This motion canbe viewed equivalently as the deformation of the complex structure of the dual F-theorycompactification on K3. The relevant moduli space of D7-brane motion has recentlybeen studied as part of the moduli space of K3 × K3 compactifications of F-theory to4 dimensions (see, e.g. [5, 6, 12]).One of our main results is the parameterization of the D7-brane motion on thecompact space in terms of periods which are explicitly defined using the standard integralhomology basis of K3. In other words, we explicitly understand the motion of the 16 D7-branes and of the background geometry in terms of shrinking or growing M-theory cyclesstretched between the branes or between the branes and the orientifold planes. In ouropinion, this is a crucial preliminary step if one wishes to stabilize specific D7-braneconfigurations using fluxes (which, in this context, are inherited from M-theory fluxesand depend on the integral homology of K3). Further important points which we discussin some detail in the following include the geometric implications of Sen’s weak couplinglimit [13], the issue of obstructions arising when D7-brane motion is viewed from thetype IIB (rather than F-theory) perspective, and the relevance of Sen’s construction ofthe double-cover Calabi-Yau [13] to type IIB models with branes at singularities [14, 15].Since the subsequent analysis is necessarily rather technical, we now give a detaileddiscussion of the organization of the paper, stating the main methods and results of eachsection.In Sect. 2, we begin with a discussion of the possible backreaction of D7-branes on theembedding space. This is of immediate concern to us since, in contrast to other D-branes,D7-branes have co-dimension 2 and can therefore potentially modify the surroundinggeometry significantly, even in the large volume limit [16–18]. However, following theanalysis of [13], it is possible to consider only configurations where Im τ ≫ ∼ / (Im τ )), while each O-plane has deficitangle π . The solutions that contain a single D7-brane only develop a deficit angle at largedistances away from the brane [16–18]. Sen’s weak coupling limit assumes a compactspace with proper charge cancellation between D-branes and O-planes, which allows for2he possibility that no deficit angle arises. One may say that, in contrast to other modelswith branes on Calabi-Yau space, the D7-brane case is special in that one has to takethe weak coupling limit more seriously than the large volume limit to be able to neglectbackreaction.In Sect. 3 we discuss obstructions to D7-brane motion. For this purpose, we have togo beyond our simple model with base space CP . Using the Weierstrass description of anelliptic fibration over CP or over CP × CP (which corresponds to the Bianchi-Sagnotti-Gimon-Polchinski model [19, 20]), it is easy to count the degrees of freedom of D7-branemotion. We find that the motion of D7-branes is strongly restricted as compared to thegeneral motion of holomorphic submanifolds analysed in [7, 21–23]. One intuitive way ofunderstanding these ‘physics obstructions’ is via the realization that D7-branes alwayshave to intersect the O7-plane in pairs or to be tangent to it at the intersection point. Weemphasize this issue since it serves as an important extra motivation for our approach viaM-theory cycles: If the moduli space is described from the perspective of the M-theorycomplex structure, such obstructions are automatically included and no extra constraintson the possible motion of holomorphic submanifolds need to be imposed.Section 4 is devoted to a brief review of the geometry of K3. This is central to ouranalysis as the moduli space of D7-branes on T /Z (or, equivalently, the motion of 16D7-branes and 4 O7-planes on CP ) is dual to the moduli space of M-theory on K3 inthe limit where the K3 is elliptically fibred and the volume of the fibre torus is sentto zero. In this language, the weak coupling limit corresponds to sending the complexstructure of the torus, which is equivalent to the type IIB axiodilaton τ , to i ∞ . Werecall that the relevant geometric freedom is encoded entirely in the complex structureof K3, which is characterized by the motion of the plane spanned by Re Ω and Im Ωin a 20-dimensional subspace of H ( K , R ). Alternatively, the same information can beencoded in two homogeneous polynomials defining a Weierstrass model and thus anelliptic fibration.In Sect. 5, we recall that at the positions of D7-branes the torus fibre degenerateswhile the total space remains non-singular. When two or more D7-branes coincide, asingularity of the total space develops, the analysis of which allows for a purely geomet-ric characterization of the resulting ADE gauge symmetry. For the simple case of twomerging branes, we show explicitly that a homologically non-trivial cycle of K3 with thetopology of a 2-sphere collapses [24, 25]. This collapsing cycle is the basic building blockwhich will allow us to parameterize the full moduli space of D7-brane motion in termsof the periods of such cycles in the remainder of the paper.In Sect. 6, we start developing the geometric picture of the D7-brane moduli space,which is one of our main objectives in the present paper. Our basic building block is the S cycle stretched between two D7-branes introduced in the previous section. Here, weconstruct this cycle from a somewhat different perspective: We draw a figure-8-shaped1-cycle in the base encircling the two branes and supplement it, at every point, with a1-cycle in the torus fibre. In this picture, it is easy to calculate the intersection numbersof such cycles connecting different D-brane pairs, taking into account also the presenceof O-planes (see the figures in this section). The Dynkin diagrams of the gauge groupsemerging when several branes coincide are directly visible in this geometric approach. Inparticular, the relative motion of four branes ‘belonging’ to one of the O-plane can befully described in terms of the above 2-brane cycles. The pattern of the corresponding3-cycles translates directly in the Dynkin diagram of SO(8). To obtain a global picture,we will have to supplement the cycles of these four SO(8) blocks by further cycles whichare capable of describing the relative position of these blocks.Before doing so we recall, in Sect. 7, the duality of F-theory on K3 to the E × E heterotic string on T . This is necessary since we want to relate the geometrically con-structed 2-cycles discussed above to the standard integral homology basis of K3, which isdirectly linked to the root lattice of E × E . In particular, we explicitly identify the partof the holomorphic 2-form Ω which corresponds to the two Wilson lines of the heterotictheory on T and thus determines the gauge symmetry at a given point in moduli space.In Sect. 8, we start with the specific form of Ω which realizes the breaking of E × E to SO(8) (corresponding to the choice of the two appropriate Wilson lines). The 2-cycles orthogonal to this particular Ω-plane generate the root lattice of SO(8) and canbe identified explicitly with our previous geometrically constructed 2-cycles of the fourSO(8) blocks. Thus, we are now able to express these 2-cycles in terms of the standardintegral homology basis of K3. Geometrically, this situation corresponds to a base spacewith the shape of a pillowcase (i.e. T /Z ) with one O-plane and four D-branes at eachcorner. The remaining four 2-cycles of K3, which are not shrunk, can be visualizedby drawing two independent 1-cycles on this pillowcase and multiplying each of themwith the two independent 1-cycles of the fibre torus. Thus, we are left with the task ofidentifying these geometrically defined cycles in terms of the standard homology basisof K3. The relevant space is defined as the orthogonal complement of the space of theSO(8) cycles which we have already identified. We achieve our goal in two steps: First,we consider the smaller (3-dimensional) subspace orthogonal to all SO(16) cycles (theirshrinking corresponds to moving all D-branes onto two O-planes and leaving the tworemaining O-planes ‘naked’). Second, we work out the intersection numbers with the S -shaped 2-cycles connecting D-branes from different SO(8) blocks. After taking theseconstraints into account, we are able to express the intuitive four cycles of the pillowcasein terms of the standard K3 homology basis.Finally, in Sect. 9, we harvest the results of our previous analysis by writing down aconveniently parameterized generic holomorphic 2-form Ω and interpreting its 18 inde-pendent periods explicitly as the 16 D-brane positions, the shape of the pillowcase, andthe shape of the fibre torus. Of course, the existence of such a parameterization of themoduli space of the type IIB superstring on T /Z is fairly obvious and has been used,e.g., in [5] and in the more detailed analysis of [6]. Our new point is the explicit map-ping between the periods and certain geometrically intuitive 2-cycles and, furthermore,the mapping between those 2-cycles and the standard integral homology basis of K3.We believe that this will be crucial for the future study of brane stabilization by fluxes(since those are quantized in terms of the corresponding integral cohomology) and forthe generalization to higher-dimensional situations.We end with a brief section describing our conclusions and perspectives on futurework. 4 Deficit angle of D7-branes
In this section we discuss the backreaction of D7-branes on the geometry. In general, D p -branes carry energy density (they correspond to black-hole solutions for the gravitationalbackground [26]). For p <
7, this deforms the geometry at finite distances, but the spaceremains asymptotically flat at infinity. Thus, backreaction can be avoided by consideringD-brane compactifications in the large volume limit.By contrast, objects with codimension two (such as cosmic strings or D7-branes)produce a deficit angle proportional to their energy density. Thus, a D7-brane in 10dimensions may in principle have a backreaction on the geometry which is felt at ar-bitrarily large distances. Let us first consider the effect of the energy density of a D7-brane. From the DBI-action it is easy to see that the gravitational energy density of aD7-brane is proportional to e φ = 1 / Im τ ≡ /τ . A D7-brane is charged under the axion C = Re τ ≡ τ , and supersymmetry constrains τ to be a holomorphic function of thecoordinates transversal to the D7-brane, see e.g. [18]. As will become apparent in thefollowing, this implies τ → i ∞ at the position of the D7-brane. Thus the energy densitydoes not couple to gravity due to the vanishing of the string coupling near the D7-brane.But there is another effect, first investigated in [16, 17], which is due to the couplingof D7-branes to the axiodilaton τ . This coupling produces a non-trivial τ backgroundaround their position whose energy deforms the geometry. In flat space this effect pro-duces a deficit angle around the position of the D7-brane at large distance. Let us havea closer look at the case of finite distance and weak coupling, which is important forF-theory constructions in the weak coupling limit.The relevant part of the type IIB supergravity action is Z d x √ g (cid:16) R + ∂ µ τ ∂ ν ¯ τ τ g µν (cid:17) . (1)The equations of motion (and supersymmetry) imply that τ is a holomorphic function ofthe coordinate z parameterizing the plane transversal to the brane, τ = τ ( z ). Because ofthe SL (2 , Z ) symmetry of IIB string theory acting on τ , it is helpful to use the modularfunction j ( τ ) instead of τ itself for the description of the dependence of τ on z . Thefunction j is a holomorphic bijection from the fundamental domain of SL (2 , Z ) onto theRiemann sphere and is invariant under SL (2 , Z ) transformations of τ (details can befound in [27]). Here we need the properties j ∼ e − π i τ for τ → i ∞ , (2) j (e π i / ) = 0 , (3) j (i) = 1 . (4)Since τ is holomorphic in z , the modular invariant function j ( τ ) depends holomorphicallyon z , and we can use the Laurent expansion of j in z . As we encircle the D-brane at z = 0, j must encircle the origin once in the opposite direction (cf. Eq. (2)). Thus j mustbe proportional to 1 /z and we can write j ( τ ) ≃ λz (5)for small z . Here λ is a modulus, the overall scaling of the axiodilaton.5rom (1) we also deduce Einstein’s equation R µν = 14 τ (cid:0) ∂ µ τ ∂ ν ¯ τ + ∂ ν τ ∂ µ ¯ τ (cid:1) . (6)Note that there is no term representing the energy density of the brane, as argued above.If we parameterize the plane orthogonal to the brane by z and write the metric as ds = η µν d x µ d x ν + ρ ( z, ¯ z )d z d¯ z , (7)we finally arrive at the equation ∂ ¯ ∂ ln ρ = ∂ ¯ ∂ ln τ , (8)which is solved by ρ = τ f ( z ) f ( z ). For the simplest case of a single D7-brane ininfinite 10-dimensional space, one might expect that both τ and ρ will not depend onthe angle in the complex plane because of radial symmetry. However, this is not the caseas we now explain.The simplest solution is given by declaring Eq. (5) to be exact. Then τ ( z ) mapsthe transverse plane precisely once to the fundamental domain of τ . Remember that thefundamental domain contains three singular points which are fixed points under some SL (2 , Z ) transformation. These points are τ invariant under τ → i ∞ Tτ = e π i / STτ = i S ,where S and T are the standard generators of SL (2 , Z ). From (2) and (3) we see thatthe first two of these points are mapped to | j | → ∞ and j = 0. Thus by (5) these pointsare at the position of the brane and at infinity, respectively. But from (4) we see that the S -monodromy point is somewhere at finite distance and (5) tells us that this point sitsat z = λ . Thus the phase of λ singles out a special direction and the radial symmetryis broken. Nevertheless, if | λ | is very large compared to the region we are interested in,there is still an approximate radial symmetry, as can be seen from (2). The monodromypoint at z = λ does not deform the region near the brane (which is mapped to large τ )and the limit | λ | → ∞ blows up the region where the radial symmetry is preserved. Wewill see later that this limit corresponds to the weak coupling limit of Sen [13].We now return to the generic case (where extra branes may be present and (5) is onlyapproximate) and use the assumption that | λ | is very large. As we approach a radiallysymmetric situation in this limit, we can neglect the angular derivatives in (8) and arriveat 1 r ∂∂r (cid:16) r ∂ ln ρ∂r (cid:17) = 1 r ∂∂r (cid:16) r ∂ ln τ ∂r (cid:17) , (9)where r is the radius in the ( z, ¯ z )-plane. The deficit angle is given by α = − π · r ∂ ln ρ∂r . (10) This is of course only true approximately if we consider a solution with many branes, so that (5) isthe Laurent expansion around the position of a single brane. ∂α∂r = − π · ∂∂r (cid:16) r ∂ ln τ ∂r (cid:17) . (11)As discussed before, there is no energy density at the position of the brane and thereforethe deficit angle is zero there. By integration we obtain α = − π · r ∂ ln τ ∂r + π · r ∂ ln τ ∂r (cid:12)(cid:12)(cid:12) r =0 . (12)Let us estimate the behavior near the brane, where τ → i ∞ . From (2) we see that τ ≃ i2 π ln j ≃ − i2 π ln zλ , (13)and therefore τ ≃ − π ln (cid:12)(cid:12) zλ (cid:12)(cid:12) . (14)Thus (12) can be evaluated using r ∂ ln τ ∂r ≃ ∂ ln(ln( r/ | λ | )) ∂ ln r = 1ln( r/ | λ | ) ≃ − πτ . (15)Since τ → ∞ for r →
0, the second term in (12) vanishes and we find α ≃ τ . (16)We see that for r ≪ | λ | this becomes small and therefore, in this limit, the deficit angleis small, too. Thus, we have derived quantitatively at which distances backreaction issmall in the weak coupling limit.Away from the D7-brane the analysis presented above breaks down. This is due tothe monodromy point at λ which destroys the radial symmetry of the configuration. Todetermine the deficit angle that emerges at distances that are much larger than | λ | , onehas to solve (8). The solution, and thus the physics, depends on the boundary conditionsthat are chosen. These are encoded in the shape of the function f ( z ), which in turn is de-termined by the symmetry that is required of the solution. The classic solution of [16,17]which argues for a deficit angle π/
6, demands an SL (2 , Z ) invariant and non-singularmetric. The analysis of [18] argues that, due to its appearance in the definition of theKilling spinor, the function f ( z ) should be invariant under the monodromy transforma-tions of τ ( z ). This requirement introduces another z -dependent factor which leads to anasymptotic deficit angle 2 π/ T , ST and S in the complex plane.Without loss of generality, we can fix the T -monodromy point (i.e. the D7-brane) atzero and the ST -point at infinity. The definition of a deficit angle, which requires radialsymmetry, is possible at distances from the brane much smaller or much larger than thatof the S -monodromy point. In the first case the deficit angle is parametrically small, inthe second case it is 2 π/ π/
6. 7
Sen’s Weak Coupling Limit and its Consequencesfor D7-brane Motion
F-theory is defined by a Weierstrass model on some K¨ahler manifold [11]. The Weierstrassequation is y = x + f x + g , (17)where f and g are sections of the line bundles L ⊗ and L ⊗ respectively. The holomorphicline bundle L is defined by the first Chern class of the base space: c ( L ) = c ( B ) . (18)This equation is derived from the Calabi-Yau condition of F-theory , cf. [13].The brane positions are given by the zeros of the discriminant of the Weierstrassequation (17) ∆ = 4 f + 27 g . (19)Before going to the weak coupling limit, these objects are ( p, q ) branes which cannotbe all interpreted as D7 branes simultaneously. Their backreaction on the geometry isstrong [11]. Furthermore, the brane motion is constrained because the form of the homo-geneous polynomial in (19) is non-generic, i.e., the branes do not move independently.Let us discuss the weak coupling limit for F-theory compactifications, in which onecan formulate everything in terms of D7-branes and O7-planes. Following [13], we pa-rameterize f = Cη − h (20)and g = h ( Cη − h ) + C χ , (21)where C is a constant and η , h and χ are homogeneous polynomials of appropriatedegree (i.e. sections of L ⊗ n ). Note that f and g are still in the most general form if weparameterize them as above. The weak coupling limit now corresponds to C →
0. To seethis, consider the modular function j that describes the τ field : j ( τ ) = 4(24 f ) f + 27 g = 4(24) ( Cη − h ) ∆ . (22)The discriminant is given as ∆ = C ( − h )( η + 12 hχ ) (23)in the weak coupling limit. We observe that for C → | j | → ∞ everywhere awayfrom the zeros of h . Locally, this corresponds to the limit λ → ∞ in (5). Furthermore,four pairs of branes merge to form the O-planes at the positions where h = 0.The remaining branes are the D7-branes of this orientifold model. Their position isdefined by the equation η + 12 hχ = 0 . (24) From duality to M-theory we know that the compactification manifold of F-theory must be Calabi-Yau in order to preserve N = 1 supersymmetry. We have changed the normalization of j ( τ ) in order to agree with the physics convention [13]. L ⊗ . Weconclude that the D7-branes in an orientifold model do not move freely in general. Theseobstructions have, to our knowledge, so far not been investigated and are not includedin the common description of orientifold models. We want to clarify this point further inthe following.As an explicit example we consider the Weierstrass model on B ≡ CP × CP . Thereason why we take this example is that it has already been shown in [28] that the degreesof freedom of the Weierstrass model are in complete agreement with the CFT-descriptionof the T-dual orientifold model, the Bianchi-Sagnotti-Gimon-Polchinski model [19, 20].By counting the degrees of freedom we will show that this model has less degrees offreedom than a model of freely moving D-branes in the corresponding orientifold model.For this, we take the F-theory model described in [28], go to the weak coupling limit andrecombine all O7-planes into a single smooth O7-plane wrapped on the smooth base spaceof the Weierstrass model. The double-cover Calabi-Yau space is now easily constructed.Furthermore, we recombine all D7-branes into a single smooth D7-brane eliminating allD7-brane intersections. We first assume, following [7] that this D7-brane moves freely asa holomorphic submanifold (respecting, of course, the Z symmetry of the Calabi-Yau).This allows for a straightforward determination of the corresponding number of degreesof freedom from its homology. We will then compare this number with the degrees offreedom that are present in the actual F-theory model.Let us investigate the F-theory model in greater detail. We call the homogeneouscoordinates [ x : x ] for the first CP , and [ y : y ] for the second CP . By x and y we denote the generators of the second cohomology, where x corresponds to the cyclethat fills out the second CP and is pointlike in the first CP , and vice versa for y . In aproduct of complex projective spaces a section in a holomorphic line bundle correspondsto a homogeneous polynomial. We want to determine the degree of the homogeneouspolynomials that will be involved in the calculation. For this purpose, we calculate thefirst Chern class of the base space: c ( B ) = 2 x + 2 y . (25)From (18) we conclude that this is the first Chern class of the line bundle L . Thus,sections in L correspond to homogeneous polynomials of degree (2,2) and h , being asection in L ⊗ , is a homogeneous polynomial of degree (4 , ,
4) in two complex coordinates is irreducible. Thus, a generic h indeed describesone single O7-plane that wraps both CP s four times. Similarly, η is a homogeneouspolynomial of degree (8 , χ is of degree (12 , , CP s sixteen times. Note thatboth h and η + 12 hχ do not have any base locus and thus define smooth hypersurfacesin CP × CP by Bertini’s theorem, cf. [29].Equation (24) defines an analytic hypersurface S of complex dimension one, which isthe position of the D7-brane. This is just a Riemann surface, and in order to identify itstopology, it suffices to determine its Euler number. By the methods of [29, 30] it is easyto calculate the Euler number of a hypersurface defined by a homogeneous polynomial.The Euler characteristic is given as χ ( T ( S )) = Z S c ( T ( S )) , (26)9he first Chern class of T ( S ) is given in terms of the Chern classes of the normal bundleof S in B and the tangent bundle of B by the second adjunction formula: c ( T ( S )) = c ( T ( B )) − c ( N ( S )) . (27)The normal bundle of S in B is equivalent to the line bundle that defines S through oneof its sections . Putting everything together, we arrive at: χ ( T ( S )) = Z S c ( T ( S )) = Z S c ( T ( B )) − c ( N ( S ))= Z B ( c ( T ( B )) − c ( N ( S ))) ∧ c ( N ( S )) . (28)The Chern class of a line bundle on CP × CP that has sections which are homogeneouspolynomials of degree ( n, m ) is simply nx + my . Together with c ( T ( CP × CP )) = 2 x +2 y we find that χ ( T ( S )) = Z B ((2 − n ) x + (2 − m ) y ) ∧ ( nx + my ) = 2( n + m − nm ) . (29)Here we used the relations R x ∧ y = 1 and R x ∧ x = R y ∧ y = 0. By the simple relation χ ( S ) = 2 − g we can now compute the genus of S , and therefore the Hodge number h (1 , ( S ) to be h (1 , ( S ) = g = ( n − m − . (30)From [7] we know that the number of holomorphic 1-cycles of a D7-brane in theCalabi-Yau space that are odd under the orientifold action is equal to the number of itsvalid deformations . Above we computed the number of holomorphic 1-cycles that areeven under the orientifold action. To compute the number of holomorphic 1-cycles thatare odd under the orientifold projection, we construct the double cover of the brane. Thedouble cover of B is a hypersurface in a CP -fibration over the base B , which is definedby [13] ξ = h , (31)where ξ is the coordinate in one patch of CP . This introduces branch points at thelocation of the O-planes. To get back to the orientifold, one then has to mod out thesymmetry ξ → − ξ . This again gives the orientifold with the topology of B and withO7-planes at the zeros of h .We thus have to branch S over the intersection points of the O7-plane with theD7-brane. The double cover ˜ S of S is then formed by two copies of S that are joinedby a number of tubes that is half of the number of intersections. We can compute thenumber of intersections between the D7-brane and the O7-plane by using the cup productbetween the corresponding elements in cohomology: I ( D ,O = Z (4 x + 4 y ) ∧ ( nx + my ) = 4( n + m ) . (32) Note that this means that c ( N ( S )) defines a 2-form on B . One might be worried that setting m or n equal to zero, one can get a surface with g <
0. However,a homogeneous polynomial of degree ( n,
0) is reducible for n >
1, corresponding to a collection ofdisconnected surfaces. Note that in [7] this was shown for the case of a Calabi-Yau 3-fold compactification down to fourdimensions. Here we consider compactifications on K n -form then has one leg less, the degree of the relevant cohomology is reduced by one as well.
10o compute h (1 , of ˜ S , we simply compute its genus (which is equal to the number ofhandles of ˜ S ). Because we are considering the double cover, the genus of ˜ S is twice thegenus of S , plus a correction coming from intersections between the D7-brane and theO7-plane. As the intersections are pairwise connected by branch cuts which connect S to its image under the orientifold action, they introduce I ( D ,O − S .This is illustrated in Fig. 1. Thus h (1 , of ˜ S is given by h (1 , ( ˜ S ) = g ( ˜ S ) = 2 g ( S ) + 12 I ( D ,O − . (33)Now we can compute the number of holomorphic 1-cycles that are odd under the orien-tifold projection: h (1 , − ( ˜ S ) = h (1 , ( ˜ S ) − h (1 , ( S ) = g ( S ) + 12 I ( D ,O − n + 1)( m + 1) − . (34)Thus a freely moving D7-brane in an orientifold model, defined by a homogeneous poly-nomial of degree ( n, m ), has h (1 , − = ( n + 1)( m + 1) − n, m ).Figure 1: Illustration of the (double cover of ) a D7-brane intersecting an O7-plane infour points.
Coming back to our example, we see that a freely moving D7-brane on CP × CP corresponds to a homogeneous polynomial P of degree (16 , P = η + 12 hχ (35)in the weak coupling limit. Here h is fixed by the position of the O-plane. Let us countthe degrees of freedom contained in this expression: η is of degree (8 , χ is of degree (12 ,
12) and thus has 169 degrees of freedom.Furthermore, we can always factor out one complex number from this equation so thatwe have to substract one degree of freedom. Finally, there is some redundancy thatarises through polynomials K = h α that are both of the form η and 12 hχ . As α isa polynomial of degree (4 , , CP × CP .11efore discussing the local nature of these ‘physics obstructions’ (which are verydifferent from certain ‘mathematical obstructions’ restricting the motion of holomorphicsubmanifolds in specific geometries [7, 22, 31, 32]), we want to briefly review a secondexample. We take the base to be CP , so that there are 6 O7-planes and 24 D7-branes inthe weak coupling limit. We denote the only 2-cycle of CP , which is a CP , by x . Afterrecombination we have one D7-brane that wraps the cycle 24 x and one O7-plane on thecycle 6 x . From x · x = 1 we conclude that they intersect 144 times. Now we can repeatcounting the degrees of freedom contained in an unconstrained polynomial of degree 24,which is 324, and compare it to a polynomial of the form (35). For CP , h is of degree 6, χ is of degree 18 and η is of degree 12. By the same arguments as above we find that 252degrees of freedom are contained in (35) for CP . The two numbers differ by 72, whichis again half of the number of intersections between D7-branes and O7-planes.It is then natural to expect that the obstructed D7-brane deformations are relatedto the intersections between D7-branes and O7-planes. Indeed, the smallness of the cou-pling cannot be maintained in the vicinity of O7-planes. Thus, our argument that thebackreaction of D7-branes on the geometry is weak breaks down and we have no rightto expect that D7-branes move freely as holomorphic submanifolds at the intersectionswith O7-planes.At the level of F-theory, the above physical type-IIB-argument is reflected in the non-generic form of the relevant polynomials in the weak coupling limit. To see this explicitly,let us investigate (35) in the vicinity of an intersection point. We parameterize theneighborhood of this point by complex coordinates z and w . Without loss of generality,we take h = w (i.e. the O7-plane is at w = 0) and assume that the intersection is at z = w = 0. This means that P = ( η +12 hχ ) vanishes at z = w = 0 and, since we alreadyknow that h vanishes at this point, we conclude that η ( z = 0 , w = 0) = 0. Expanding η and χ around the intersection point, η ( z, w ) = m z + m w + . . . and χ ( z, w ) = n + n z + n w + . . . , we find at leading order P = m z + 12 n w + · · · = 0 . (36)In the generic case n = 0, this is the complex version of a parabola ‘touching’ the O-planewith its vertex. Thus, we are dealing with a double intersection point. In the specialcase n = 0, our leading-order P is reducible and we are dealing with two D7-branesintersecting each other and the O7-plane at the same point. The former generic casehence results from the recombination of this D7-D7-brane intersection. In both cases,we have a double intersection point. In other words, the constraint corresponds to therequirement that all intersections between the D7-branes and O7-planes must be doubleintersection points. There are two easy ways to count the number of degrees of freedomremoved by this constraint: On the one hand, demanding pairwise coincidence of the 2 n intersection points (each of which would account for one complex degree of freedom fora freely moving holomorphic submanifold) removes n degrees of freedom. On the otherhand, at each of the n double intersection points the coefficient of the term ∼ z in (36)must vanish, which also removes n complex degrees of freedom. Note that zw is subdominant w.r.t. to w , which is not true for z . This is also clear from the fact that, if we were to introduce by hand a term ∼ z in (36), ourintersection point would split into two.
12o summarize, we have again arrived at the conclusion that 2 n intersections betweena D7-brane and an O7-plane remove n of the degrees of freedom of the D7-brane motion.In particular, we have now shown that these ‘physics obstructions’ have a local reason:In the weak coupling limit, the O7-plane allows only for double intersection points. Atthe local level, our findings are easily transferred to compactifications to 4 dimensions:The O7-plane D7-brane intersections are now complex curves rather than points. Ateach point of such a curve, we can consider the transverse compact space, which is againcomplex-2-dimensional. In this space, we can perform the same local analysis as aboveand conclude that double intersection points are required.We end this section with a comment on an interesting application of F-theory andelliptic fibrations which may be useful for type-IIB model building on the basis of localCalabi-Yau constructions [14, 15]: Consider a non-compact K¨ahler manifold with SU(3)holonomy (a local Calabi-Yau). Such spaces play an important role in attempts to con-struct Standard-like models from branes at singularities. We will now sketch a genericprocedure allowing us to embed them in a compact Calabi-Yau.Assuming that the non-compact K¨ahler manifold is given as a toric variety, it isclearly always possible to make it compact by adding appropriate cones. Furthermore,this can always be done in such a way that the resulting compact K¨ahler manifold B hasa positive first Chern class. With the first Chern class we can associate a line-bundle L and a divisor. Positivity of the Chern class implies that this divisor is effective, i.e.,the line bundle L has sections without poles (the zero locus of such a section defines thedivisor). Now we can wrap an O7-plane (with four D7-branes on top of it) twice along theabove effective divisor. This corresponds to the orientifold limit of a consistent F-theorymodel. Indeed, as explained at the beginning of this section, we can define a Weierstrassmodel based on the bundle L on our compact K¨ahler manifold B . Since c ( L ) = c ( B ), wehave constructed an elliptically fibred Calabi-Yau 4-fold and hence a consistent F-theorymodel. The O7-plane, defined by the zero locus of h , is wrapped twice along the divisorsince h is a section of L ⊗ . At this point, we have already realized our local Calabi-Yau aspart of a compact type IIB model. It is intuitively clear (although a better mathematicalunderstanding would be desirable) that the O7-plane can be chosen in such a way thatit does not interfere with the compact cycles of the original local Calabi-Yau. Indeed,the Calabi-Yau condition has been violated by making the original model compact. Thisviolation is measured by the effective divisor associated with L . This divisor has thereforeno need to pass through the region where the original compact cycles (relevant for localCalabi-Yau model building) are localized. We can even go one step further and separatethe two O7-planes lying on top of the divisor of L . Subsequently, we can recombine themat possible intersection points, thereby arriving at a single smooth O-plane. Constructingthe double cover of the base branched along this O-plane, we obtain a compact Calabi-Yau (without O-plane) the orientifolding of which takes us back to the above F-theory This means that some of the 1-cycles that would be present in the double cover of a genericholomorphic submanifold are collapsed in the double cover of a D7-brane in the weak coupling limit.These are the 1-cycles that wind around two branch points. The fan of a toric Calabi-Yau is spanned by one-dimensional cones that are generated by vectorsending on a single hyperplane H . If we add a one-dimensional cone in the direction opposite to thenormal vector n H of H , we end up with a (in general) non-compact K¨ahler manifold of positive firstChern class. To make this space compact, we appropriately enlarge the fan. As this can always be donesuch that all the one-dimensional cones that are added are generated by vectors that end on H , we donot have to change the first Chern class. K In two complex dimensions there is, up to diffeomorphisms, just one compact Calabi-Yaumanifold: K
3. We will only collect the facts that we need; for a comprehensive reviewsee e.g. [33].The Hodge diamond of K . z α which are the integrals of theholomorphic 2-form Ω over integral 2-cycles. z α ≡ Z γ α Ω = Z K η α ∧ Ω ≡ η α · Ω . (38)Here η α are the Poincar´e-dual 2-forms corresponding the 2-cycles γ α . The real K¨ahlerform J can be decomposed in a basis of 2-forms in a similar way. Together, Ω and J specify a point in the moduli space of K3. They have to fulfill the constraintsΩ · Ω = 0 , J · Ω = 0 , Ω · ¯Ω > , J · J > . (39)Parameterizing Ω and J by 3 real forms x i , such that Ω = x + i x and J ∼ x , theconstraints translate to x i · x j = 0 for i = j (40)and x = x = x > . (41)The symmetry between the three real 2-forms x i is related to the fact that K S of complex structures on K H ( K , Z ). It can be shown [33] that with this natural scalar product, H ( K , Z ) is an even self-dual lattice of signature (3 , H ( K , Z ) such that the innerproduct forms the matrix U ⊕ U ⊕ U ⊕ − E ⊕ − E (42)where U = (cid:18) (cid:19) , (43)and E denotes the Cartan matrix of E . Choosing a point in the moduli space of K R , equipped with the inner14roduct (42). This space-like three-plane is spanned by the three vectors x i fulfilling theconditions (40) and (41) .The Picard group, defined asPic( X ) ≡ H , ( X ) ∩ H ( X, Z ) , (44)is given by the intersection of the lattice H ( X, Z ) with the codimension-two surfaceorthogonal to the real and imaginary parts of Ω. The dimension of Pic( X ), also calledPicard number, counts the number of algebraic curves and vanishes for a generic K K K T fiber and a section, the latter being equivalent to the base CP . Thus the space orthogonal to the plane defining the complex structure has a two-dimensional intersection with the lattice H ( K , Z ), which fixes two complex structuremoduli . One can show that the two vectors in the lattice corresponding to the base andthe fiber form one of the U factors in (42). Thus, Ω has to be orthogonal to the subspacecorresponding to this U factor. The precise position of J , which lies completely in this U factor, is fixed by the requirement that the fibre volume goes to zero in the F-theorylimit. The only remaining freedom is in the complex structure, which is now defined bya space-like two-plane in R , with the inner product U ⊕ U ⊕ − E ⊕ − E . (45)Any vector in the lattice of integral cycles of an elliptically fibred K D = p i e i + p j e j + q I E I , (46)where i, j run from one to two and I, J from 1 to 16. The p i as well as the p i are allintegers. The E ⊕ lattice is spanned by q I fulfilling P I =1 .. q I = 2 Z , P I =9 .. q I = 2 Z . Ineach of the two E blocks, the coefficients furthermore have to be all integer or all half-integer [34]. The only non-vanishing inner products among the vectors in this expansionare E I · E J = − δ IJ e i · e j = δ ij . (47)There are 18 complex structure deformations left in the elliptically fibred case: Ωmay be expanded in twenty two-forms, which leads to 20 complex coefficients. However,there is still the possibility of an arbitrary rescaling of Ω by one complex number, as wellas the complex constraint Ω · Ω = 0, so that we find an 18-dimensional complex structuremoduli space.This number can easily be compared to the moduli of K CP [11], defined by the Weierstrass equation y = x + f ( a, b ) x + g ( a, b ) (48) We identify H ( K , R ) and H ( K , R ) here and below. This behavior is a specialty of K h , = b . a : b ] are the homogeneous coordinates of CP and f and g are homogeneouspolynomials of degree 8 and 12 respectively. They are determined by 9 + 13 = 22 param-eters. There is an SL (2 , C ) symmetry acting on the homogeneous coordinates of the base CP and an overall rescaling of (48). This reduces the independent number of parametersto 18 [11]. From the perspective of F-theory compactified on K
3, 17 of these 18 parame-ters describe the locations of D7-branes and O-planes on the base of the fibration, CP .The remaining parameter describes the complex structure of the fiber and correspondsto the axiodilaton τ . At the same time, these 18 parameters describe the variation ofthe complex structure of the K
3, so that one can interpret D-brane moduli as complexstructure moduli of an elliptically fibred higher-dimensional space.
As D7-branes are characterized by a degeneration of the elliptic fibre one may wonderwhether the total space, in our case K
3, is still smooth. Its defining equation shows that K K f + 27 g (49)determine the points of the base where the roots of the Weierstrass equation degenerate.Let us examine this further. The Weierstrass equation (48) can always be written in theform y = ( x − x )( x − x )( x − x ) . (50)It is easy to show that in this notation∆ = ( x − x ) ( x − x ) ( x − x ) . (51)At a point where the fibre degenerates, two of the x i coincide so that, adjusting thenormalization for convenience, (48) reads locally y = ( x − x ) . (52)By a change of variables this is equivalent to xy = 0, representing an A singularity of thefibre. To see what happens to the whole space we have to keep the dependence on the basecoordinates. Let us deform away from the degenerate point by shifting x → x ± = x ± δ .This means that now y = ( x − x + )( x − x − ) = ( x − x ) + δ . (53)The quadratic difference of the now indegenerate roots is given by( x + − x − ) = 4 δ . (54) Note that the term quadratic in x is absent in the canonical form (48). This corresponds to choosingthe origin of our coordinate system such that the three roots x i sum up to zero. δ ∼ ∆ . (55)Since we also want to see what happens to the full space, we reintroduce the the depen-dence on the base coordinates, ∆ = ∆( a, b ), and write (53) as y = ( x − x ) + ∆( a, b ) . (56)Without loss of generality we assume a = 0 and use a as an inhomogeneous coordi-nate. Near the singularity, where ∆ = ( a − a ) n , the Weierstrass model then reads y = ( x − x ) + ( a − a ) n (57)which is clearly singular if n is greater than one. By a change of variables this is againequivalent to yx = ( a − a ) n . (58)Thus, simple roots ( n = 1) of ∆ do not lead to any singularity of the whole space, it ismerely the fibration structure that becomes singular.We have seen that the K X ≡ { x, y, a | x + y + a = ǫ } . (59)We have shifted x and a for simplicity. Here a is an affine coordinate on the base and ǫ resolves the singularity by moving the two branes away from each other.As the situation is somewhat analogous to the conifold case [35], we will perform asimilar analysis: We first note that ǫ can always be chosen to be real by redefining thecoordinates. Next, we collect x, y, a in a complex vector with real part ξ and imaginarypart η . The hypersurface (59) may then be described by the two real equations ξ − η = ǫ, ξ · η = 0 . (60)We can understand the topology of X by considering its intersection with a set of 5-spheres in R given by ξ + η = t , t > ǫ : ξ = t ǫ , η = t − ǫ , ξ · η = 0 . (61)If we assume for a moment that ǫ = 0, the equations above describe two S s of equalsize for every t that are subject to an extra constraint. If we take the first S to beunconstraint, the second and third equation describe the intersection of another S witha hyperplane. Thus we have an S bundle over S for every finite t . This bundle shrinksto zero size when t approaches zero so that we reach the tip of the cone. Furthermore,17he bundle is clearly non-trivial since the hyperplane intersecting the second S rotatesas one moves along the first S .Let us now allow for a non-zero ǫ , so that X is no longer singular. The fibre S stillshrinks to zero size at t = ǫ , but the base S remains at a finite size. This is the 2-cyclethat emerged when resolving the singularity. We now want to show that the S at thetip of the resolved cone is indeed a non-trivial cycle. This is equivalent to showing thatthe bundle that is present for t > ǫ does not have a global section, i.e., the S at t = ǫ cannot be moved to larger values of t as a whole. Consider a small deformation of this S , parameterized by a real function f ( ξ ): ξ = ǫ + f ( ξ ) . (62)This means that we have moved the S at t = ǫ in the t -direction by 2 f ( ξ ), i.e. t f ( ξ ) + ǫ . (63)The equations for the S fibration over this deformed S read η = f ( ξ ) (64) η · ξ = 0 . (65)Deforming the S means choosing a continuous function η ( ξ ) subject to these equations.Since the set of all planes defined by (65) can be viewed as the tangent bundle of an S , we can view η ( ξ ) as a vector field on S . As we know that such vector fields haveto vanish in at least two points, we conclude that f ( ξ ) has to vanish for two values of ξ . We do not only learn that the S at the tip of the cone cannot be moved away, butalso that the modulus of its self-intersection number is two . This number is expected,as one can show that any cycle of K f , g and ∆ vanish [36]. For the whole K
3, we canuse the ordinary ADE classification of the arising quotient singularities [33], which isequivalent to the classification of simply laced Lie algebras. The intersection pattern ofthe cycles that emerge in resolving the singularity is precisely the Dynkin diagram ofthe corresponding Lie algebra. This ‘accidental’ match is of course expected from theF-theory point of view which tells us that singularities are associated with gauge groups.The correspondence between the singularity type of the whole space and the singularitytype of the fibre is more complicated when F-theory is compactified on a manifold withmore than two complex dimensions [37–40]. This can also be anticipated from the factthat gauge groups which are not simply laced appear only in IIB orientifolds with morethan one complex dimensions. Note that this viewpoint also works for the classic conifold example [35], which is a cone whosebase is an S bundle over S . As the Euler characteristic of S vanishes, the same arguments as beforegive the well-known result that the S at the tip of the resolved conifold can be entirely moved into thebase. Geometric picture of the moduli space
In this section we want to gain a more intuitive understanding of the cycles that areresponsible for the brane movement. For this we picture the elliptically fibred K3 locallyas the complex plane (in which branes are sitting) to which a torus has been attachedat every point. We will construct the relevant 2-cycles geometrically. We have alreadyseen that the cycles in question shrink to zero size when we move the branes on topof each other, so that these cycles should be correlated with the distance between thebranes. Remember that D7-branes have a non-trivial monodromy acting on the complexstructure of the fibre as τ → τ + 1, which has T = (cid:18) (cid:19) (66)as its corresponding SL (2 , Z ) matrix. Similarly, O7-planes have a monodromy of − T ,where the minus sign indicates an involution of the torus, meaning that the complexcoordinate z of the torus goes to − z . Thus 1-cycles in the fibre change orientation whenthey are moved around an O-plane.If we want to describe a 2-cycle between two D-branes, it is clear that it must haveone leg in the base and one in the fibre to be distinct from the 2-cycles describing thefibre and the base. Now consider the 1-cycle being vertically stretched in the torus . Ifwe transport this cycle once around a D-brane and come back to the same point, thiscycle becomes diagonally stretched because of the T-monodromy. If we then encircleanother brane in the opposite direction, the 1-cycle returns to its original form, so thatit can be identified with the original 1-cycle. This way to construct a closed 2-cycle wasalready mentioned in [24]. This 2-cycle cannot be contracted to a point since it cannotcross the brane positions because of the monodromy in the fibre. The form of the 2-cycleis illustrated in Fig. 2. We emphasize that to get a non-trivial cycle, its part in the fibretorus has to have a vertical component. a bc Brane 1 Brane 2
Figure 2:
The cycle that measures the distance between two D-branes. Starting with acycle in the (0 , direction of the fibre torus at point a , this cycle is tilted to ( − , at b . Because we surround the second brane in the opposite way, the cycle in the fibre isuntilted again so it can close with the one we started from. Next we want to compute the self-intersection number. In order to do this, we considera homologous cycle and compute the number of intersections with the original one. By Of course this notion depends on the SL (2 , Z )-frame we consider, but anyway we construct the cyclein the frame where the branes are D-branes. In the end the constructed cycle will be independent of thechoice of frame. S that wraps the fibre in the horizontal direction, so that it shrinks to a point atthe brane positions. Thus it is topologically a sphere, which fits with the self-intersectionnumber of − BA Figure 3:
The self-intersection number of a cycle between two D-branes. As shown in thepicture, we may choose the fibre part of both cycles to be (0 , at A , so that they do notintersect at this point. At B however, one of the two is tilted to (1 , , whereas the otherhas undergone a monodromy transforming it to ( − , . Thus the two surfaces meet twicein point B . = Figure 4:
The loop between two D-branes can be collapsed to a line by pulling it ontothe D-branes and annihilating the vertical components in the fibre. All that remains is acycle which goes from one brane to the other while staying horizontal in the fibre all thetime.
Now we want to determine the intersection number between different cycles andconsider a situation with three D7-branes. There is one cycle between the first two branesand one between the second and the third, each having self-intersection number −
2. FromFig. 5 it should be clear that they intersect exactly once. If one now compares the waythey intersect to the figure that was used to determine the self-intersection number, onesees that the two surfaces meet with one direction reversed, hence the orientation differsand we see that the mutual intersection number is +1. Thus we have shown that theintersection matrix of the N − N D-branes is minus theCartan Matrix of SU ( N ).We now want to analyze the cycles that arise in the presence of an O7-plane. Two D7-branes in the vicinity of an O7-plane can be linked by the type of cycle considered above.20 A Figure 5:
Mutual intersection of two cycles. Start by taking both cycles to have fibre part (0 , at B . The fact that we closed a circle around the D-brane tells us that one of thetwo has been tilted by one unit at A . Thus they meet precisely once. However, there are now two ways to connect the D-branes with each other: we can passthe O-plane on two different sides, as shown in Fig. 6. By the same argument as before,each of these cycles has self-intersection number −
2. To get their mutual intersectionnumber, it is important to remember the monodromy of the O-plane, which contains aninvolution of the torus fiber. Thus, the intersection on the right and the intersection onthe left, which differ by a loop around the O-plane, have opposite sign. As a result, theoverall intersection number vanishes.
D7D7 O7 plane
Figure 6:
Cycles that measure how D-branes can be pulled onto O-planes.
Now we have all the building blocks needed to discuss the gauge enhancement in theorientifold limit, which is SO (8) . We should find an SO (8) for each O7-plane with fourD7-branes on top of it. The cycles that are blown up when the four D7-branes move awayfrom the O7-plane are shown in Fig. 7. It is clear from the previous discussion that allcycles have self-intersection number − c intersects every other cycle preciselyonce. Thus, collecting the four cycles in a vector ( a, b, c, d ), we find the intersection form D = − − − − , (67)which is minus the Cartan matrix of SO (8). It is also easy to see that collapsing onlysome of the four cycles yields minus the Cartan matrices of the appropriate smaller gaugeenhancements. 21 c da Figure 7:
Four D-branes and an O-plane. The D-branes are displayed as circles and theO-plane as a cross. To simplify the picture we have drawn lines instead of loops. E × E E × E heterotic string. We focus on the pointsrelevant to the discussion in this paper.As M-theory compactified on S is dual to type IIA in 10 dimensions, we can relateM-theory to type IIB by compactifying on a further S and applying T-duality. Thus,M-theory on T corresponds to type IIB on S . The complexified type-IIB couplingconstant is given by the complex structure of the torus. Furthermore, taking the torusvolume to zero corresponds to sending the S radius on the type IIB side to infinity. Inother words, M-theory on T with vanishing volume gives type IIB in 10 dimensions. Onemay think of this as a T compactification of a 12d theory, which we take as our workingdefinition of F-theory. Considering the compactification of M-theory on an ellipticallyfibred manifold Y and using the above argument for every fibre, we arrive at type IIBon the base space. This can be re-expressed as an F-theory compactification on Y .One can now study the stabilization of D7-branes through fluxes by investigating thestabilization of a (complex) fourfold Y on the M-theory side and mapping the geometryto the D-brane positions [5, 6]. One can also translate the 4-form fluxes of M-theory to3- and 2-form fluxes in the IIB picture. In particular, we can consider 4-cycles built froma 2-cycle stretched between two D7-branes (see previous section) and a 2-cycle of theD7-brane. We then expect 2-form flux on D7-branes to arise from M-theory 4-form fluxon such cycles. This fits nicely with the fact that 2-form flux on D7-branes T-dualizesto an angle between two intersecting D6-branes, so that it is only defined relative to theflux on another D7-brane. Note that being in the F-theory limit (i.e. ensuring that Y iselliptically fibred and the fiber volume vanishes) also has to be realized by an appropriateflux choice on the M-theory side.The foundation of the duality between F-theory and the E × E heterotic string isthe duality between M-theory compactified on K T [41]. In the limit in which the fibre of K S decompactifies so that we end up with heterotic E × E on T [11, 25, 42–44].In this duality the complex structure of the base and the fibre of K T on the heterotic side. The preciserelation between these parameters has been worked out in [43]. The volume of the baseof K E × E thatis achieved by Wilson lines on the heterotic side appears in the form of deformations ofthe Weierstrass equation away from the E × E singularity on the F-theory side. Thisis equivalent to deforming the complex structure of K K E × E point byΩ E × E = e − ˜ U ˜ Se + ˜ U e + ˜ Se . (68)We have set the first coefficient to one by a global rescaling and used the fact thatΩ · Ω = 0. We can now add a component in the direction of the E factors to (68) inorder to break the gauge group.In the heterotic theory, the Wilson lines are two real vectors that are labelled bytheir direction in T : W I and W I . Let us normalize these Wilson lines such that theassociated gauge-theoretical twist is P = e − π i W I E I , (69)with the action E γ → P E γ P − = e − π i γ I W I E γ , (70)and analogously for W I . This means that, whenever there is a root vector γ such that γ · W and γ · W are integers, the corresponding root survives the Wilson line breaking.Let us define W I E I = W I E I + ˜ U W I E I and write the complex structure at a generalpoint in moduli space asΩ = e + ˜ U e + ˜ Se − (cid:18) ˜ U ˜ S + 12 ( W ) (cid:19) e + W I E I . (71)Here we denote ( W I E I + ˜ U W I E I ) simply by ( W ) . Note that Ω · Ω = 0 holds for allvalues of the parameters. If we find a surviving root γ = γ I E I in the E × E lattice, itsinner product with the complex structure is Ω · γ = − n − ˜ U m . This means there existsa cycle γ ′ ≡ γ I E I + ne + me with the property Ω · γ ′ = 0, implying that this cycle hasshrunk to zero size. Thus, extra massless states are present and a gauge enhancementarises, which is precisely what one expects from a surviving root on the heterotic side.We can conclude that, by (71), we have consistently identified the properly normalizedWilson lines W and W of the heterotic description with the appropriate degrees offreedom of the complex structure of K3 on the F-theory side. SO (8) singularity of K If we describe the possible deformations of K f and g , we see that singularities arise at specific points inmoduli space. We can also describe the deformations of K3 as deformations of the complexstructure, Ω. In this description we can also reach points in the moduli space wherethe K3 is singular. By comparing the singularities, we can map special values of thecomplex structure moduli to a special form of the Weierstrass model. This enables us todescribe the positions of the D-branes and O-planes, which are explicit in the polynomialdescription, by the values of the complex structure moduli of K H ( Z ) with self intersection − J . If we now consider the inner product on the sublattice spanned by such roots,we find minus the Cartan Matrix of some ADE group. This structure tells us the kindof singularity that has emerged [5, 33].We now want to discuss the cycles that shrink to produce an SO (8) singularity. Forthis we have to find a change of basis such that the D ⊕ in the inner product of thelattice H ( K , Z ) becomes obvious. The well-known Wilson lines breaking E × E downto SO (8) are W = (cid:18) , , , (cid:19) W = (cid:0) , , , (cid:1) . (72) W breaks E × E down to SO (16) × SO (16), W breaks this further down to SO (8) .Inserting this in (71), we find Ω at the SO (8) point:Ω SO (8) = e + ˜ U e + ˜ Se − (cid:16) ˜ U ˜ S − − ˜ U (cid:17) e + W I E I . (73)Note that setting ˜ S = 2 i and ˜ U = i reproduces the complex structure given in [5].The lattice vectors orthogonal to Ω SO (8) span the lattice D ⊕ . Using the expan-sion (46), their coefficients have to satisfy − (cid:16) ˜ U ˜ S − − ˜ U (cid:17) p + p − W I q I + ˜ Sp + ˜ U ( p − W I q I ) = 0 . (74)As we know that these lattice vectors must be orthogonal to the complex structure forevery value of ˜ U and ˜ S , we find the conditions p = 0 p − W I q I = 0 p = 0 p − W I q I = 0 . (75)These equations are solved by the following four groups of lattice vectors: A B C D E − E − E + E − e − E + E e + E − E E − E − E + E − E + E E − E − e − E − E e + E + E − E + E E − E E − E − E + E − E − E E + E (76)It is not hard to see that there are no mutual intersections between the four groups,and that the intersections within each group are given by the D matrix (67). This servesas an explicit check that (73) is indeed the correct holomorphic two-form of K SO (8) point.It should be clear that one can choose different linear combinations of the basisvectors in each block that still have the same inner product. This only means we candescribe the positions of the D-branes by a different combination of cycles, which are ofcourse linearly dependent on the cycles we have chosen before and span the same lattice.24 Figure 8:
The assignment between the geometrically constructed cycles between branesand the cycles of the table in the text. Note that the distribution of the cycles 1,3 and 4is ambiguous.
We can make an assignments between the cycles in the table and the cycles constructedgeometrically as shown in Fig. 8.When we are at the SO (8) point, where 16 of the 20 cycles of K T /Z × T .The four cycles describing these deformations have to be orthogonal to all of the 16 branecycles. There are four cycles satisfying this requirement, e + W I E I e + W I E I e e , (77)and the torus cycles must be linear combinations of them. From the fibration perspective,the torus cycles are the cycles encircling two blocks (and thus two O-planes), so that themonodromy along the base part of those cycles is trivial. They can be either horizontalor vertical in the fibre, giving the four possibilities displayed in Fig. 9. Note that all ofthem have self-intersection zero and only those that wrap both fibre and base in differentdirections intersect twice. (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) or or ABDC
Figure 9:
The torus cycles have to encircle two blocks to ensure trivial monodromy alongthe basis part of the cycles. Note that the cycles which are orthogonal in the base and inthe fibre intersect twice because of the orientation change introduced in going around theO-plane.
To find out which linear combination of the forms in (77) gives which torus cycle, wewill consider a point in moduli space where the gauge symmetry is enhanced from SO (8) to SO (16) . At this point, there are 16 integral cycles orthogonal to Ω the intersectionmatrix of which is minus the Cartan matrix of SO (16) . Furthermore, we know thatthis situation corresponds to moving all D-branes onto two O7-planes. We will achieve25his leaving two of the four blocks untouched, while moving the D-branes from the otherblocks onto them. This means that we blow up one of the cycles in each of the blocks thatare moved, while collapsing two new cycles that sit in between the blocks. Doing thiswe find three independent linear combinations of the cycles in (77) that do not intersectany of the cycles that are shrunk.Before explicitly performing this computation, we choose a new basis that is equiv-alent to (77): α ≡ (cid:0) e + e + W I E I (cid:1) β ≡ e + e + W I E I ) e e . (78)In this basis we can write Ω at the SO (8) point asΩ SO (8) = 12 ( α + U e + Sβ − U Se ) . (79)We also have switched to a new parameterization in terms of U and S . They will turnout to be the complex structures of the base and the fibre torus.Let us now go to the SO (16) point by setting W I = 0 in (71). After switching againfrom ˜ S and ˜ U to S = ˜ U and U/ S − ˜ U , we findΩ SO (16) = e + U e + S β − U S e . (80)The 16 integral cycles that are orthogonal to Ω are: E F − e − E + E e + E − E − E + E E − E − E + E E − E − E + E E − E − E + E E − E − E + E E − E − E + E E − E E + E E + E (81)They are labelled as shown in Fig. 10. Note that this means that we have moved block A onto block C and block B onto block D . Figure 10:
The Dynkin diagram of SO (16) . One can check that, out of the basis displayed in (78), only α has a non-vanishingintersection with some of the 16 forms above, whereas all of them are orthogonal to26 , e , β . Thus, α is contained in the cycle that wraps the fibre in vertical direction andpasses in between A, B and
C, D (cf. Fig. 9). Furthermore, the non-zero intersectionbetween e and α tells us that e wraps the fibre horizontally (for this argument we used e · e = e · β = 0). Given these observations, it is natural to identify the four cycles(78) with the four cycles displayed in Fig. 9. More specifically, we now know that α isvertical in the fibre and passes in between A, B and
C, D , while e is horizontal in thefibre and passes in between A, C and
B, D . The four cycles characterize the shape of T /Z × T . Other possible assignments between the cycles of (78) and those displayedin Fig. 9 correspond to reparameterizations of the tori and are therefore equivalent toour choice.We now have to assign the cycles e and β to the two remaining cycles of Fig. 9. Forthis purpose, we will explicitly construct the cycles Z XY between the four SO (8) blocks.Since they can be drawn in the same way as the cycles between the D-branes, cf. Fig. 11,we see that all of them must have self-intersection number −
2. We also know that theirmutual intersections should be Z XY · Z Y Z = 1. From what we have learned so far, all ofthem should be either orthogonal to β and e or orthogonal to e and e . It is easy tocheck that the first case is realized by: Z AC = E − E − e Z AB = − ( e + e ) − E + E − E + E + e Z DB = E − E − e Z CD = e − e . (82)One can show that the second case is not possible. This can be seen from the followingargument:If we can find Z -cycles that are orthogonal to e and e , we can decompose them as Z XY = q I E I . (83)Note that the e i are now responsible for making the Z -cycles wind around the base torus,so that we do not loose any generality by omitting them in the decomposition above.Because of the constraint P q I = 2 Z , we can only have Z AB intersecting one of the cyclesin block A by putting q I = ± appropriately for I = 5 ..
8. By the structure of the lattice,(46), this forces us to also set q I = ± for I = 1 ..
4. It is clear that this will also makethis cycle intersect with one of the cycles in block C , contradicting one of its definingproperties. This means that we simply cannot construct the Z -cycles to be all orthogonalto e and e . AB Z AB Figure 11:
The cycles connecting the blocks. Note that all cycles in this picture are builtin the same way and thus all lie horizontally in the fibre.
We can now use the intersections of the Z -cycles with the complex structure at the SO (8) point, eq.(79), to measure their length and consistently distribute the four blocks27n the pillow. We find that Z AB · Ω SO (8) = − U/ Z AC · Ω SO (8) = − Z CD · Ω SO (8) = − U/ Z DB · Ω SO (8) = − . (84) Z BDCA
D BC A
ZDC Z Z AB U/21/2
Figure 12:
A schematic picture of the cycles between the blocks. We have drawn arrowsto indicate the different orientations. Note that the Z -cycles have intersection +1 withthe cycle they come from and − with the cycle they go to. Note that they sum up to atorus cycle, e , telling us which blocks are encircled. It is clear that we can add the two orthogonal cycles e and β to the Z -cycles, Z → Z + ne + mβ , without destroying their mutual intersection pattern. However, thischanges their length by n + U m . This means that we can make the Z -cycles wind aroundthe pillow n times in the real and m times in the imaginary direction. Calling the realdirection of the base (fibre) x ( x ′ ) and the imaginary direction of the base (fibre) y ( y ′ ),we can now make the identifications: e winds around x and x ′ e winds around x and y ′ α winds around y and y ′ β winds around y and x ′ . (85)Alternatively one can find the positions of e and α by computing their intersectionswith the Z -cycles: e · Z CD = e · Z AB = − , e · Z AC = e · Z DB = 0 ,α · Z AC = α · Z DB = − , α · Z CD = α · Z AB = 0 . (86)Note that these intersections change consistently when we let Z → Z + ne + mβ . Wedisplay the distribution of the torus cycles in Fig. 13.28 ee αβ Z Z
CA AB ZZ DBDC
ACD B
Figure 13:
The distribution of the torus cycles.
At the orientifold point we can write the complex structure of T /Z × T asΩ T /Z × T = ( dx + U dy ) ∧ ( dx ′ + Sdy ′ )= dx ∧ dx ′ + Sdx ∧ dy ′ + U dy ∧ dx ′ + SU dy ∧ dy ′ . (87)In the above equation, U denotes the complex structure of the torus with unprimed coor-dinates, whereas S denotes the complex structure of the torus with primed coordinates.We have so far always switched freely between cycles and forms using the natural duality.We now make this identification explicit at the orientifold point: e = − dy ∧ dy ′ e = 2 dy ∧ dx ′ α = 2 dx ∧ dx ′ β = 2 dx ∧ dy ′ .Thus we have shown that the parameters U and S inΩ SO (8) = 12 ( α + U e + Sβ − U Se ) , (88)do indeed describe the complex structure of the base and the fibre torus.The findings of this section represent one consistent identification of the torus cyclesand the Z -cycles that connect the four blocks. It is possible to add appropriate linearcombinations of e , e , α and β without destroying the mutual intersections and theintersection pattern with the 16 cycles in the four SO (8) blocks. What singles out ourchoice is the form of Ω SO (8) in (73) as well as the SO (16) that was implicitly definedin (80). We have normalized the orientifold such that R T /Z dx ∧ dy = 1 / R T ∧ dx ′ ∧ dy ′ = 1. D-Brane positions from periods and the weak cou-pling limit revisited
In this section, we study deformations away from the SO (8) point. To achieve this, wehave to rotate the complex structure such that not all of the vectors spanning the D ⊕ lattice are orthogonal to it. In other words, we want to add terms proportional to theforms in (76) to Ω SO (8) . To do this, we switch to an orthogonal basis defined by˜ E = E + e , ˜ E I = E I , I = 2 .. , .. E = E + e , ˜ E J = E J + e / , J = 5 .. , .. . (89)As we will see, each ˜ E I is responsible for moving only one of the D-branes when we rotatethe complex structure toΩ = 12 (cid:16) α + U e + Sβ − (cid:0) U S − z (cid:1) e + 2 ˜ E I z I (cid:17) . (90)Here z denotes z I z I . Note that all of the ˜ E I are orthogonal to Ω SO (8) , so that we onlyhave to change the coefficient of e to maintain the constraint Ω · Ω = 0.We can use the information about the length of the blown-up cycles to computethe new positions of the branes. Let e.g. z = 0: This gives the first cycle of block C , C = − e − E + E , the length Ω · C = z , so that we move one brane away from theO-plane. As a result, the SO (8) at block C is broken down to SO (6). At the same time,the sizes of Z AC and Z CD are changed to Z AC · Ω = −
12 + z , Z CD · Ω = − U − z . (91)Thus we can move the brane from block C onto block A by choosing z = , or ontoblock D by choosing z = − U/
2. This can also be seen from the overall gauge groupwhich is SO (6) × SO (10) × SO (8) for these two values of z .As we have seen in the last paragraph, z controls the position of one of the fourD-branes located at block C , as compared to the position of the O-plane at block C . Ifwe let all of the z I be non-zero, we find the following values for the lengths of the cyclesin block C : C I C I · Ω C z − z C z − z C z − z C z + z (92)To determine the D-brane positions, it is important to note that the D-branes aremoving on a pillow, T /Z . We thus use a local coordinate system equivalent to C /Z . Itis centered at the position of the O-plane of block C at one of the corners of the pillow.The intersections of the cycles with Ω are line integrals along the base part of the cycles C I (see Fig. 14), multiplied by the line integral of their fibre part (which can be set to30 C C C CO−plane C
Figure 14:
The positions of D-branes on C /Z are measured by complex line integralsalong the cycles C I . As indicated in the picture, C /Z is obtained from the complex planeby gluing the upper part of the dashed line to its lower part. Due to the presence of theO-plane, the line integral along C has to be evaluated as indicated by the arrows. Using(92), one can see that the positions of the branes are given by the z I . unity locally). This is of importance for the length of C : due to the orientation flip in thefibre when surrounding the O-plane, we have to evaluate both parts of the line integralgoing from the O-plane to the D-branes to account for the extra minus sign. This isindicated by the arrows that are attached to the cycles in Fig. 14. It is then easy to seethat associating the z I with the positions of the D-branes yields the correct results. Notethat one achieves the same gauge enhancement for z = z and z = − z , because forboth values one of the C I is collapsed, cf. (92). Thus the D-branes labelled 3 and 4 haveto be at the same position in both cases, which fits with the fact that z I = − z I holdsdue to the Z action.By the same reasoning, the remaining z I give the positions of the other D-branesmeasured relative to the respective O-plane. For example, the moduli z to z give thepositions of the D-branes of block A (see (76)). We have also shown that we can connectthe four blocks by following the gauge enhancement that arises when we move a branefrom one block to another, cf. (91). This means that we can also easily connect thefour coordinate systems that are present at the position of each O-plane. We have nowachieved our goal of explicitly mapping the holomorphic 2-form Ω to the positions ofthe D-branes. For this we have used forms dual to integral cycles. These are the cyclesthat support the M-theory flux which can be used to stabilize the D-branes. By usingour results, it is possible to derive the positions of the D-branes from a given complexstructure (unless the solution is driven away from the weak coupling limit). We thus viewthis work as an important step towards the explicit positioning of D-branes by M-theoryflux.The geometric constructions of this article only make sense in the weak couplinglimit, in which the monodromies of the branes of F-theory are restricted to those of D7-branes and O7-planes. It is crucial that the positions of the D-branes and the shape of31he base torus factorize in the weak coupling limit, S → i ∞ . The shape of the base torusis measured by multiplying the cycles α , β , e and e with Ω. The result is independentof the positions of the branes in the weak coupling limit, as the only potential source ofinterference is the z in α · Ω =
U S − z , which is negligible as compared to U S . Thusthe branes can really be treated as moving on T /Z without backreaction in the weakcoupling limit.Certain gauge groups, although present in F-theory, do not show up in perturbativetype IIB orientifolds and thus cannot be seen in the weak coupling limit. The latticeof forms orthogonal to Ω only has the structure of gauge groups known from type IIBorientifold models when we let S → i ∞ in (71). This comes about as follows: Startingfrom the SO (8) point, we can cancel all terms proportional to W I E I in (71) when we areat finite coupling. In the limit S → i ∞ , the fact that β has S as its prefactor preventsthe cancellation of the term W I E I in Ω. The presence of this term ensures that onlyperturbatively known gauge enhancements arise.
10 Summary and Outlook
Our main points in this article were the implications of the weak coupling limit (whichis necessary if one wants to talk about D7-branes moving in a background Calabi-Yauorientifold) and the description of D7-brane motion on CP in terms of integral M-theorycycles.Regarding the weak coupling limit, we have rederived the fact that the deficit angle ofa D7-brane is zero in a domain whose size is controlled by the coupling. Furthermore, wehave pointed out ‘physics obstructions’ that arise in the weak coupling limit: The poly-nomial that describes the D7-brane has to take a special form in the weak coupling limit,so that some degrees of freedom are obstructed. We have counted the relevant degrees offreedom explicitly for F-theory on elliptically fibred spaces with base CP × CP (which isdual to the Bianchi-Sagnotti-Gimon-Polchinski model) and with base CP . From a localperspective, the obstructions arise at intersections between D7-branes and O7-planes.They demand that D7-branes always intersect O7-planes in double-intersection-points(in the fundamental domain of the orientifold model; not in the double-cover Calabi-Yau, where this would be a trivial statement). The obstructions imply that a D7-braneintersecting an O7-plane has fewer degrees of freedom than a corresponding holomor-phic submanifold. It would be very interesting to analyse the implications of this effectexplicitly in higher-dimensional models.The rest of this paper was devoted to an explicit discussion of the degrees of freedomof F-theory on K
3, corresponding to type IIB string theory compactified on an orientifoldof T . We were able to construct and visualize the cycles that control the motion of D7-branes on the base. Using these cycles, we have achieved a mapping between the positionsof the D-branes and the values of the complex structure moduli. This was done usingthe singularities that arise at special points in the moduli space.We consider our analysis an important preliminary step for the study of more realisticD7-brane models. First, realistic models should contain fluxes to stabilize the geometryand the D7-brane positions. Using a higher-dimensional generalization of our map be-tween cycles and D7-brane positions, it should be possible to determine explicitly the flux32tabilizing a desired D7-brane configuration. A first step in this direction, from whichone should gain valuable intuition, might be the reconsideration of F-theory on K × K S emerges.This S can be seen as the combination of a disc spanned in the tunnel connecting thetwo D7-branes and a horizontal cycle in the fibre torus.Although there exists a rich literature on elliptically fibred Calabi-Yau manifolds(see e.g. [39, 40, 45–49]), the direct visualization of the geometric objects achieved in thispaper becomes harder in higher dimensions. We hope that the use of toric geometrymethods will allow for a geometric understanding and simple combinatorial analysis insuch cases. Acknowledgments
We would like to thank Tae-Won Ha, Christoph L¨udeling, Dieter L¨ust, Michele Trapletti,and Roberto Valandro for useful comments and discussions.
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