Global F-theory Models: Instantons and Gauge Dynamics
PPreprint typeset in JHEP style - HYPER VERSION
UPR-1216-TNSF-KITP-10-035
Global F-theory Models: Instantons and GaugeDynamics
Mirjam Cvetiˇc, I˜naki Garc´ıa-Etxebarria and James Halverson
Department of Physics and Astronomy, University of Pennsylvania,Philadelphia, PA 19104-6396, USAandKavli Institute for Theoretical Physics, Kohn Hall,UCSB, Santa Barbara, CA 93106, USAE-mail: [email protected] , [email protected] , [email protected] Abstract:
We elucidate certain aspects of F-theory gauge dynamics, due to quantumsplitting of certain brane stacks, which are absent in the Type IIB limit. We also providea working implementation of an algorithm for computing cohomology of line bundles onarbitrary toric varieties. This should be of general use for studying the physics of globalType IIB and F-theory models, in particular for the explicit counting of zero modes forrigid F-theory instantons which contribute to charged matter couplings. We illustrate thediscussion by constructing and analyzing in detail a compact F-theory GUT model in whicha D-brane instanton generates the top Yukawa coupling non-perturbatively. a r X i v : . [ h e p - t h ] J a n ontents
1. Introduction 12. F-theory Gauge Dynamics 3
3. ˇCech Cohomology of Line Bundles over Toric Varieties 6 dP
4. Example: a
10 10 5 H Yukawa Coupling in F-theory 13
5. Conclusions 23
1. Introduction
F-theory [1] provides a promising framework for studies of string vacua with potentiallyrealistic particle physics. It combines many of the nice features of type IIB, particularlylocalization of gauge degrees of freedom on D-branes, with some of the nice features ofheterotic models, such as the natural appearance of exceptional groups.Most effort so far has centered on the classical aspects of F-theory models, i.e. couplingsthat can be computed as wave function overlaps. This perturbative sector is already enoughfor constructing appealing and phenomenologically promising models (there is a rapidlygrowing literature on the topic, starting with [8, 9, 10, 11]). Nevertheless, non-perturbativecorrections to this picture can in some instances be the dominant contribution, and modifythe picture substantially.Classical examples come from gaugino condensation or euclidean instantons wrappingisolated cycles. Under favorable conditions [12], they can generate a superpotential for theK¨ahler modulus associated with the cycle. This effect has very important applications for For recent efforts within the Type IIA intersecting D-brane framework, see [2, 3] for a review and[4, 5, 6, 7] for a systematic study of local realistic MSSM quivers. – 1 –oduli stabilization in IIB and F-theory [13, 14], and it is an essential ingredient of manysemi-realistic type IIB scenarios [15].Another important effect coming from D-brane instantons has been greatly clarifiedin the type II context in recent years. Whenever a D-brane instanton intersects a D-branestack, there are some zero modes in the instanton worldvolume that are charged underthe gauge symmetry on the D-brane stack. Integration over these charged zero modes cangenerate F-term couplings for matter fields living on the D-brane stack [16, 17, 18, 19].In the type II context these charged instantons solve a long-standing difficulty in con-structing realistic models: brane stacks have U (1) factors that survive perturbatively, andgenerally forbid certain important couplings in the MSSM lagrangian. Typical examplesare the top-quark Yukawa couplings in SU (5) GUT models, and the µ term. ChargedD-brane instantons do not necessarily respect perturbative U (1) symmetries, and thus cangenerate these couplings (see [20] for a recent review).A very attractive feature of F-theory compactifications is that, due to their closerelation with exceptional groups, these U (1) factors can be absent, and thus the couplingswhich are problematic from the type II point of view can be obtained perturbatively.Nevertheless, one may still investigate the effect of non-perturbative effects in F-theory.The motivations are many: instanton effects can be naturally suppressed, depending onthe volume of the cycle wrapped by the instanton, a feature that can be quite convenientwhenever one desires to obtain a hierarchy. One may also want to try a hybrid approach,building a good model in the better understood IIB context with some couplings comingfrom euclidean instantons and then uplifting to F-theory to improve some aspects of themodel. Finally, and perhaps most importantly, non-perturbative effects will be there inany case, and one must be able to understand how they affect the model at hand.With this motivation in mind, in this paper we discuss in detail the F-theory uplift ofa particular global type IIB model in which D-brane instantons are known to play a crucialrole in generating MSSM couplings, in particular the top-quark Yukawa coupling. We usethis model as a prototype illustrating the two major themes of this paper: the quantumsplitting of classical brane stacks, and our implementation of line bundle cohomology com-putations using the ˇCech complex. These are issues that are important for any realisticF-theoretical model building, and we are explicit about their resolution. Before going intoa complete discussion, let us briefly review both issues in turn.The first issue, quantum brane splitting, arises when uplifting interesting weakly cou-pled IIB models to F-theory. It can easily happen, as in our particular example, thatbrane stacks that make perfect sense in IIB split into sub-stacks when uplifted to F-theory.This is due to non-perturbative D( −
1) instantons that start modifying the geometry assoon as g s is non-vanishing. A convenient way of studying this problem is using D3 braneprobes, for which the D( −
1) effects appear as ordinary gauge instantons, and the ambientgeometry appears as the geometry of the Coulomb branch of moduli space. The effects ofD( −
1) instantons on the geometry can thus be understood using Seiberg-Witten theory.We do this in section 2, explaining in which cases the quantum modification occurs, andin which it does not.Line bundle cohomology computations arise naturally when analyzing neutral instan-– 2 –on zero modes, which in F-theory are counted by the cohomology groups of the trivialsheaf on the worldvolume of the instanton [12]. The framework in which we work is toricgeometry , and accordingly our Calabi-Yau fourfold Y will be a complete intersection ina six (complex) dimensional ambient space X Σ (cid:48)(cid:48) , specified by toric data, with instantonswrapping divisors of Y . Sheaf cohomology on the instanton can then be computed by firstcomputing the cohomology of the toric ˇCech complex (twisted by L ) on X Σ (cid:48)(cid:48) , and thenusing the Koszul complex to project down results to divisors of Y . We have collected andreviewed the relevant mathematical background in section 3. The algorithm for computingˇCech cohomology that we review there, while straightforward, quickly becomes intractableif done by hand. Luckily, it is not hard to instruct a computer to do it, and we providea working implementation that should be useful for doing general computations of linebundle cohomology on arbitrary toric varieties.Previous works on charged instantons in F-theory include [27], which lays down partof the framework required for studying charged instantons, and [28, 29], which proposeto use euclidean D-branes to implement local F-theory models for GMSB (although onehas to be careful in determining which instantons can be responsible for supersymmetrybreaking [30, 31]).This paper is organized as follows. In section 2 we analyze the quantum splittingof classical brane stacks, determining in which cases it occurs. In section 3 we discussthe mathematical background required for computing zero modes of instantons in thetoric context, and provide the link to our implementation of the algorithm for computingˇCech line bundle cohomology on toric varieties. In section 4 we illustrate these generalconsiderations in a particular example with interesting phenomenological features.While we were writing our results, we became aware of the work [32], which providesan efficient algorithm and computer implementation for computing line bundle cohomologyon toric varieties, and thus overlaps with our discussion in section 3. We thank the authorsof that work for sharing their insights.
2. F-theory Gauge Dynamics
Suppose that we are given a type IIB compactification with a stack of D7 branes. We areinterested on lifting this configuration to F-theory, and in particular on the fate of the stackof D7 branes. We will see that in some cases the brane stack splits, and the F-theoreticalpicture is qualitatively different. For concreteness, in most of the analysis we will take thegauge group on the stack to be SO (6), as that is the one that will make an appearancein our example in section 4, but the discussion generalizes easily and we give some relatedresults at the end of our analysis. The basic physics at play here is similar to that which smooths out the O − plane inF-theory [35, 36]. It can be elucidated by studying the world-volume dynamics of a D We recommend [21, 22, 23] for an introduction and [24, 25, 26] for a more thorough treatment. – 3 –rane probe close to the SO (6) stack [37]. The theory on the worldvolume of the D3 branehas a Coulomb branch, with a Coulomb branch parameter that can be identified with theposition of the D3 brane in the direction transverse to the SO (6) stack. Furthermore, theexact solution of the gauge theory on the probe can be described in terms of an ellipticfibration over the Coulomb branch (the Seiberg-Witten solution [38, 39]), which can beidentified with the F-theory geometry in which the probe moves [35, 36, 37].We construct the SO (6) theory by putting three D7 branes on top of an orientifold.Due to the orientifold, the worldvolume theory on the D3 brane probe is an N = 2 SU (2)theory, and due to the three A branes we have 3 massless quarks in the theory (at weakcoupling). The Seiberg-Witten curve for the theory with three massless flavors is given by[39]: y = x ( x − u ) + t Λ ( x − u ) (2.1)with u the coordinate on the Coulomb branch in the SU (2) theory, Λ the strong couplingscale of the theory, and t a constant which we could absorb in the definition of Λ . Theelliptic fibration (2.1) over the complex u plane degenerates over two points. There is adegeneration of order one at u = − t Λ /
4, and a degeneration of order 4 at u = 0. Thisalready explains why we could not obtain the SO (6) singularity above: the coupling Λ of the SU (2) theory is determined by the string coupling at the position of the D3. Inthe perturbative IIB limit this coupling is everywhere vanishing, so Λ →
0, and the twosingular points collide, enhancing the degeneration to order 5, as we expected from Tate’sclassification. Nevertheless, at finite string coupling this SO (6) factor decomposes into adegeneration of degree 4, and a degeneration of degree 1, separated by Λ .Let us try to understand the physics a bit better. In order to do this it is convenientto use the classification of ( p, q ) 7-branes described in [40, 41]. The ordinary D7-branes areof type (1 , A -type branes. An isolated O − plane splits into twocomponents in F-theory, which can be denoted as B and C , of ( p, q ) type ( − ,
1) and (0 , SO (6) stack can then be described as a CBAAA stack. Wewould like to understand how the five seven-branes in our stack split as we switch on a finitecoupling. In order to do this, it is convenient to consider the form of the Seiberg-Wittencurve for large (and equal) mass for the three flavors. It is given by [39]: y = x ( x − u ) −
164 Λ ( x − u ) − m Λ ( x − u ) + 14 m Λ x − m Λ (2.2)As one would expect, this mass deformation separates the branes into three stacks: threebranes are located at u = m + Λ m/
8, and the two remaining branes are located at: u ± = 1512 (cid:18) Λ − m ± (Λ + 64 m ) (cid:113) Λ + 64Λ m (cid:19) (2.3)For large mass we can identify the branes at u ± as the components of the orientifold, andthe stack of three branes as the three A branes. Let us now smoothly take the mass to0. For some intermediate value of the mass the stack of three branes collides with thedegeneration at u + , and the branes can have their ( p, q ) labels altered in the collision.After the collision the stack of three A branes must become magnetic monopoles (0 ,
1) (so– 4 –e recover a fourplet of monopoles in the massless regime at u = 0 [39]), while the braneat u + must become a ( − ,
2) dyon. This is indeed possible to achieve if we take the braneat u + to be the B brane, and we take the two brane stacks to circle around each otheronce as they collide. We are left with u − , which was a spectator in the whole process, andwhich we identify as the C brane. We have depicted this process in figure 1. Figure 1:
Schematic representation of the motion of the branes described in the text as we tunethe mass parameter from large values down to 0. The red dot on the left represents the C brane (oftype (0 , B brane (type ( − ,
1) before doing the monodromy), andthe black dot on the right represents the stack of three A branes (type (1 ,
0) before monodromy).These A branes become (0 ,
1) branes after the monodromy (shown as the dashed red dot), and the B brane becomes a ( − ,
2) dyon. We have indicated the branch cuts associated with the monodromiesaround each brane by the dotted line.
In more detail, the process goes as follows: recall (from [40], for example) that a braneof type ( r, s ) becomes a brane of type ( m, n ) upon crossing the branch cut associated witha ( p, q ) brane, with: (cid:32) mn (cid:33) = (cid:32) − pq p − q pq (cid:33) (cid:32) rs (cid:33) . (2.4)In figure 1, we have chosen conventions in in which this is the monodromy for crossingthe branch cut counterclockwise. Denoting a brane of type ( p, q ) as X ( p,q ) , the sequence ofcrossings in figure 1 is then: CBA → CA X (1 , → CX (1 , X , → CX , X ( − , = C X ( − , (2.5)So the local geometry is simply the one obtained from putting four C branes together,which gives an SU (4) theory. The same result can be obtained by studying the formof (2.1) close to u = 0.Let us briefly comment on what happens in various other interesting configurations.If we tried to uplift SO (4) stacks we would run into the same phenomenon. The theory tostudy now is N = 2 SU (2) with 2 massless flavors. In this case the Seiberg-Witten curveis known to degenerate at two points, both of degree 2 [39]. Using the same argumentsas above, we can argue that they correspond to a stack of two (1 ,
1) branes and a CC stack. There is again a collision of stacks as we take the mass from 0 to large values, whichchanges the (1 ,
1) stack into an AA stack, and the CC stack into a couple of neighboring B – 5 –nd C branes. Similarly, lifting a SO (2) stack splits the configuration into three separateddegenerations of types (1 , B and C .One can argue in a similar fashion about what happens for most of the other classicalgroups: U ( N ) stacks induce U (1) dynamics on the probe, so no splitting occurs at finitecoupling since the theory is abelian, and thus IR-free. Sp ( N ) stacks induce SO (2) dynamicson the probe, again non-confining. SO (8) stacks give rise to a N = 2 N f = 4 SU (2) theoryon the probe, which is conformal, so no IR deformation of the geometry occurs. Similarly, SO (2 n ) stacks with n > The analysis above also clarifies how some of the string junctions foundin [33] for the BPS states of N = 2 N f < A N f BC description of the flavor groupand the actual quantum configuration of D7 branes forces us to take the effects of SL (2 , Z )monodromy into account. It is not hard to see that the states in the classification of [33]that should become massless according to Seiberg-Witten theory, do indeed “untangle”due to the winding motion and the Hanany-Witten effect [34], and go from being involvedstring junctions to simple ( p, q )-strings. These ( p, q ) strings then become massless whenthe D3 collides with the D7 branes.
3. ˇCech Cohomology of Line Bundles over Toric Varieties
In section 4.3, we will need to perform a calculation of sheaf cohomology in order to showthe absence of the τ ˙ α mode for an O (1) instanton. In this section we explain in detailthe steps involved in calculating such cohomologies on toric varieties, and refer the readerto a code we have written which performs such computations. For the sake of brevity,we assume that the reader is familiar with the main concepts used in the study of toricgeometry, but highly recommend [21, 22, 23] for an introduction and [24, 25, 26] for athorough treatment. For more details on ˇCech cohomology on toric varieties, see Chapter9 of [25], which we follow closely here. In general, the calculation of the ˇCech cohomology groups ˇ H ( U , F ) for a sheaf F on X requires knowledge of an open cover U of X , determination of the p th ˇCech cochainsˇ C p ( U , F ), and determination of the differential maps d p , which are the maps between theˇCech cochains in the ˇCech complex0 → ˇ C ( U , F ) d −→ ˇ C ( U , F ) d −→ . . . d l −→ ˇ C l − ( U , F ) d l − −−−→ ˇ C l ( U , F ) → . . . (3.1) We reproduce the Kodaira classification in a form convenient for F-theory use in table 5, in section 4.4. The authors of [25] have kindly decided to provide recent copies of the book at the web address listedin the references, until it is completed and published by AMS. – 6 –e will define the differentials in section 3.2 below. The p th ˇCech cochains keep track oflocal sections, as can be seen from the definitionˇ C p ( U , F ) ≡ (cid:77) ( i ,...,i p ) ∈ [ l ] p F ( U σ i ∩ · · · ∩ U σ ip ) , l = |U | , (3.2)where ( i , . . . , i p ) ∈ [ l ] p is a ( p + 1)-tuple of elements in the set [ l ] ≡ { , . . . , l } , which hasthe ordering i < · · · < i p . As p increases, the sections become more and more local, andthe ˇCech complex can be viewed intuitively as encoding how increasingly local sections “fittogether”. Given this data and intuition, the p th ˇCech cohomology groups are defined tobe ˇ H p ( U , F ) ≡ ker ( d p ) im ( d p − ) , (3.3)as usual. After determining the structure of the p th ˇCech cochains and the differential maps d p , the ˇCech cohomology can be computed directly as the cohomology of the complex (3.1).In the generic case, however, the computation might be further complicated by notknowing, a priori, an open cover of X . Fortunately, in the case where X is a toric variety X Σ , the affine toric variety U σ associated with a cone σ is a patch on X Σ . Then there is anatural choice for an open cover, namely U ≡ { U σ } σ ∈ Σ max , l = | Σ max | , (3.4)where Σ max is the set of top-dimensional cones. Moreover, to determine the structure ofthe p th ˇCech cochain in general, we must know the structure of F ( U i ∩ · · · ∩ U i p ), whichrequires knowing how the opens in U intersect. Again, it is a fortunate property of toricvarieties that the intersection of two opens is encoded in the intersection of two cones. Forexample, if σ , σ ∈ Σ max and τ = σ ∩ σ is a common face, then U σ ∩ U σ = U τ . (3.5)Thus, for toric varieties, the relevant intersections of opens are known, and one can proceeddirectly to determining the structure of the ˇCech cochains.The cochains we are interested in are the ˇCech cochains of a sheaf O X Σ ( D ) on a toricvariety X Σ with the natural open cover U on the toric variety. On an open patch U σ associ-ated to some cone, not necessarily top-dimensional, O X Σ ( D )( U ) is an O X Σ -module finitelygenerated by the set of monomials on U σ of class [ D ] for the divisor D = (cid:80) ρ a ρ D ρ . Thisjust means that an arbitrary α ∈ O X Σ ( D )( U ) is a linear combination of these monomialswith coefficients that are functions on X Σ . The monomials are local sections on the patch,so we write ˇ C p ( U , O X Σ ( D )) = (cid:77) ( i ,...,i p ) ∈ [ l ] p H ( U σ i ∩ · · · ∩ U σ ip , O X Σ ( D )) . (3.6)Determining the local sections of class [ D ] is not difficult. Considering the fact that (cid:81) ρ x a ρ ρ has class [ D ] for x ρ the homogeneous coordinate associated with the one-dimensional cone ρ ∈ Σ(1), there is a monomial of class [ D ] for each m ∈ M , given by (cid:89) ρ x (cid:104) m,u ρ (cid:105) + a ρ ρ , (3.7)– 7 –here u ρ ∈ N is the vector associated with the one-dimensional cone ρ and (cid:104) ., . (cid:105) is the dotproduct. It is of class [ D ] due to the fact that we have multiplied by a gauge invariantproduct of homogeneous coordinates, (cid:81) ρ x (cid:104) m,u ρ (cid:105) ρ .Calculating the structure of the ˇCech cochains involves determining which of the mono-mials are well-defined on a given patch. For example, if a monomial has (cid:104) m, u β (cid:105) < − a β for β ∈ Σ(1), then the monomial is only well defined on patches where x β (cid:54) = 0. This behavioris captured in a simple way by the notion of “+” and “-” regions in the M lattice, wherethe former is the halfplane (cid:104) m, u β (cid:105) ≥ − a β and the latter is the halfplane (cid:104) m, u β (cid:105) < − a β .The M lattice is then partitioned by the set of lines (cid:104) m, u ρ (cid:105) = − a ρ ∀ ρ ∈ Σ(1), where eachpartition is a region in the M lattice categorized by a string of +’s and -’s, one for eachhomogeneous coordinate. For example, on P , a lattice point m in the region with sign“ − + + − − ” would have a corresponding monomial which is only well-defined on patcheswhere x , x , and x are non-zero. We will henceforth name such a region R − ++ −− , forthe sake of notation. How many lattice points are in this region, or whether it exists at all,is highly dependent on the divisor D .Given this intuition about local sections in terms of signed regions, we would like torelate them directly to patches U σ , since we are interested in expressions of the form (3.6).We define P σ = { m ∈ M R | (cid:104) m, u ρ (cid:105) ≥ − a ρ , ∀ ρ ∈ σ (1) } , (3.8)whose intersection with the M lattice contains all lattice points m whose correspondingmonomials are local sections of U σ . More precisely, H ( U σ i ∩ · · · ∩ U σ ip , O X Σ ( D )) = (cid:77) m ∈ P i ...ip ∩ M C · χ m , (3.9)where χ m and P i ...i p are shorthand for the monomial corresponding to m and P σ i ∩···∩ σ ip ,respectively. This identification makes sense in terms of patches, because if m ∈ P σ ∩ M ,then its corresponding monomial is guaranteed to have positive exponent for the homo-geneous coordinates x ρ for all one-dimensional cones ρ in σ . This is necessary to bewell-defined on U σ , since D ρ = { x ρ = 0 } ⊆ U σ , ∀ ρ ∈ σ (1). It is sufficient because x ρ (cid:54) = 0on U σ for every ρ / ∈ σ (1). One should note, of course, that a given P σ is generically theunion of multiple signed regions, and moreover that a given signed region might contributeto multiple P σ for different cones in the fan.Having the requisite tools for explicitly constructing the ˇCech cochains, it is straightfor-ward to compute the differentials , and one can then directly compute the ˇCech cohomologygroups ˇ H p ( U , O X Σ ( D )) ≡ ker ( d p ) im ( d p − ) . (3.10)The previous discussion was general but perhaps somewhat abstract. We now proceed toillustrate how to apply these ideas in a simple but non-trivial example, dP . As we willsee, the ˇCech complex gives a simple and systematic (albeit cumbersome, if done by hand)way to compute line bundle cohomology. We do not give the general definition now, because we think it is more illustrative to state it when wewill use it in the detailed dP example. – 8 – .2 Calculating ˇCech Cohomology on dP As a concrete non-trivial example, we calculate an example of ˇCech cohomology for a linebundle over the first del Pezzo surface, dP . The del Pezzo surfaces are P × P and theblow-up of P at n points, n = 0 , . . . ,
8, which are denoted dP n . The fan which specifies dP as a toric variety is given in figure 1, and it is easy to see that the removal of u , whichcorresponds to the exceptional divisor of the blow-up, leaves us with the fan for P . Hence,this is dP , also known as the first Hirzebruch surface F .Coords Vertices Q Q Divisor Class x u =(1,0) 1 0 Hy u =(0,1) 1 1 H + Ez u =(-1,-1) 1 0 Hw u =(0,-1) 0 1 E (cid:80) i [ D i ] 3 2 3 H + 2 E Table 1:
GLSM charges for dP . To fix notation, the homogeneous coordinates x , y , z , and w are associated to therays u , u , u , and u , respectively. For this example, we choose to calculate the ˇCechcohomology groups ˇ H p ( U , O dP ( D )) for the divisor D = 5 D x − D w . For this divisor, thereare four lines which divide the M lattice into signed regions, given by l : m x = − , l : m y = 0 l : m x + m y = 0 , l : m y = − , (3.11)which correspond to the rays u , u , u , and u , respectively. The partitioned M lattice isgiven in figure 2, where each region has been labeled with the appropriate sign accordingto the conventions discussed in the previous section. Figure 2:
Signed regions in the M lattice corresponding to O (5 D x − D w ) over dP . We havedenoted by m x and m y the coordinate axes of the M lattice. Calculationally, rather than considering which signed regions have monomials well-defined on the intersection of a particular set of opens, it is useful to instead consider– 9 –n which intersections of opens a particular monomial is well-defined. In the end, thisessentially corresponds to considering the cohomological contribution of each point in the M lattice. All m in a given signed region will have the same contribution. This is usefulsince each point in the M lattice contributes independently to the cohomology. In otherwords, there is a grading on cohomology which allows us to consider the contribution ofeach m ∈ M independently. We refer the reader to chapter 9 of [25] for more details.For this reason, we would like to categorize those P σ ’s which contain the m ’s cor-responding to monomials well defined on a particular intersection, as a union of signedregions. The result is P = (cid:91) R ++ •• P = (cid:91) R • ++ • P = (cid:91) R •• ++ P = (cid:91) R + •• + P = (cid:91) R • + •• P = (cid:91) R •••• P = (cid:91) R + ••• P = (cid:91) R •• + • P = (cid:91) R •••• P = (cid:91) R ••• + P = (cid:91) R •••• P = (cid:91) R •••• P = (cid:91) R •••• P = (cid:91) R •••• P = (cid:91) R •••• , (3.12)where a • simply means that the union includes both the + and the − in that placeholder,so that ∪ R + •• = R +++ ∪ R ++ − ∪ R + − + ∪ R + −− . This allows us to consider the contributionsof a particular m ∈ M to a ˇCech cochain as a vector where different entries correspond todifferent intersections of opens. Examples will come when we do the actual calculation.The only technical aspect which must still be specified before actually computing thekernels and images of the differentials d p is the definition and form of the differentialsthemselves. In general, they are maps from ˇ C p ( U , F ) to ˇ C p +1 ( U , F ) defined by( d p σ ) i ...i p +1 = p +1 (cid:88) k =0 ( − k σ i ... ˆ i l ...i p +1 | U i ∩···∩ U ip +1 , (3.13)where ˆ i k indicates that this index is removed. For a given set of indices ( i , . . . , i p +1 ), thisspecifies one component in an element of ˇ C p +1 ( U , F ). As an example, the definition (3.13)gives ( d σ ) = σ | U ∩ U ∩ U − σ | U ∩ U ∩ U + σ | U ∩ U ∩ U (3.14)for the case where l = 4, which is our case for dP . Each component in an element of aˇCech cochain is specified by a ( p + 1)-tuple of indices, where the components are orderedin a vector according to the natural ordering on [ l ] p . Thus, equation (3.14) correspondsprecisely to the third row in d , listed below.All components of d p σ can be determined this way, which allows us to write the maps– 10 –s matrices. The result in our particular case is0 → ˇ C ( U , O dP (5 D x − D w )) d = -1 1 0 0-1 0 1 00 -1 1 00 -1 1 00 -1 0 10 0 -1 0 −−−−−−−−−−−−−→ ˇ C ( U , O dP (5 D x − D w )) (3.15) d = −−−−−−−−−−−−−−−−−→ ˇ C ( U , O dP (5 D x − D w )) d = ( -1,1,-1,1 ) −−−−−−−−−→ ˇ C ( U , O dP (5 D x − D w )) → . . . Notice that the definition of the differential, seen as a linear map between vector spaces,does not require us to specify which monomial we are dealing with. This informationwill only enter in the definition of the vector spaces ˇ C • ( U , O dP (5 D x − D w )), specifyingwhich elements of the vector space are necessarily vanishing due to the monomial underconsideration not being well defined in the relevant patch.Now all of the pieces are in place for a direct computation of cohomology. We emphasizeagain that it is sufficient to consider the cohomology corresponding to a given m ∈ M ,and then sum over the contributions from each m . Moreover, since all m ’s in a givensigned region contribute to the overall cohomology in the same way, it is only necessary tocompute the cohomological contributions for each signed region and to then multiply thatcontribution by the number of points in that region. This implies that all cohomologicalcontributions from non-compact regions must be zero, since there are an infinite numberof points, and the cohomology is finite. This means that we only need to calculate thecontributions from points in R +++ − and R + − + − .Let us study first the monomials in R + − + − . From equation (3.12) and the naturalordering of ( p + 1)-tuples in [ l ] p , elements of the ˇCech cochains for a given m can bewritten · χ m ∈ ˇ C ( U , O dP (5 D x − D w )) , abcd · χ m ∈ ˇ C ( U , O dP (5 D x − D w )) , (3.16) efgh · χ m ∈ ˇ C ( U , O dP (5 D x − D w )) , (cid:16) i (cid:17) · χ m ∈ ˇ C ( U , O dP (5 D x − D w )) , where a, b, c, d, e, f, g, h, i ∈ C . One can then consider the action of the appropriate d p ’s onthe these elements, and it is a straightforward exercise in linear algebra to show that all ofthe kernels and images are the same except for im ( d ) = 0, ker ( d ) = C . Thus, for each m in this region, the contribution is ˇ h • m = (0 , , Figure 3:
The only signed regionwhich contributes to the cohomol-ogy. definition of signed regions that “+”’s are inclusive while“-”’s are exclusive. With this in mind, only the filleddots in figure 3 contribute. Thus, the contribution of thisregion to the cohomology is given byˇ h • R + − + − ( U , O dP (5 D x − D w )) = (0 , , . (3.17)A similar argument in the R +++ − region shows thatit does not contribute to the cohomology, and thus we conclude thatˇ h • ( U , O dP (5 D x − D w )) = (0 , , . (3.18) Once we know the cohomology of line bundles on the ambient space, we can use an exactsequence known as the Koszul complex to obtain the cohomology on subspaces of thisambient space. Let us start with the case of induced line bundles on divisors of A . Denotingas N ∗ the dual of the normal bundle of our surface X on A , we have that:0 → N ∗ → O A → O X → A is toric. In the case of a divisor D of A , N ∗ is theline bundle O ( − D ). Furthermore, in order to obtain information about the cohomology ofa line bundle L on X , we can tensor the whole short exact sequence above by L , we get:0 → O ( L − D ) → O A ( L ) → O X ( L ) → → H ( A , O ( L − D )) → H ( A , O ( L )) → H ( X, O ( L )) →→ H ( A , O ( L − D )) → H ( A , O ( L )) → H ( X, O ( L )) →→ . . . → H d ( A , O ( L − D )) → H d ( A , O ( L )) → H d ( X, O ( L )) → d is the dimension of the ambient space.From here we can read the dimensions of the cohomology groups. A couple of veryuseful facts are that we can always split any exact sequence0 → A → B → C → D → . . . (3.22)into two pieces: 0 → A → B → X → → X → C → D → . . . (3.24) We would like to acknowledge a number of useful discussions with L. Anderson on the contents of thissection. – 12 –nd that for any short exact sequence 0 → A → B → C → B ) = dim( A ) + dim( C ) , (3.25)which allows one to compute the dimensions of the cohomologies in a straightforwardmanner.For the case of a complete intersection of three divisors in the ambient space (our casein the main text), there is a useful general form for the Koszul complex, given by:0 → ∧ N ∗ → ∧ N ∗ → N ∗ → O A → O D ∩ D ∩ D → , (3.26)where N is the sum of the normal bundles of the divisors, N ≡ N D ⊕ N D ⊕ N D .After splitting this sequence into short exact sequences, one uses those sequences to arriveat a number of long exact sequences in cohomology, which make it straightforward tocompute the relevant groups. For the complete intersection of more hypersurfaces, theabove sequence extends as one might expect.
4. Example: a
10 10 5 H Yukawa Coupling in F-theory
In this section we will illustrate the previous considerations in a particular example ofphenomenological interest. The discussion is organized as follows: in section 4.1, we presentthe geometric data for a IIB GUT orientifold compactification on a Calabi-Yau manifoldrealized as a hypersurface in a toric variety. This IIB background has an euclidean D3-braneinstanton generating a 10 10 5 H coupling. In section 4.2, we lift this IIB model to F-theoryby specifying an elliptically fibered Calabi-Yau fourfold as a complete intersection in a six-dimensional toric variety. In section 4.3, we study the lifted (now M5) instanton generatingthe 10 10 5 H coupling. By an explicit line bundle cohomology computation we show theabsence of fermionic zero modes that would make the contribution of the instanton to thesuperpotential vanish. In section 4.4, we present the precise form of the Tate sections andshow that many features of the gauge D O SO (6) gauge stack cannot beobtained at nonzero g s . For the sake of reference, in this section we present the geometric data relevant for theF-theory lift of the manifold M (dP ) , henceforth called X , which is a Calabi-Yau threefoldhypersurface in a four-dimensional toric variety. It was presented in [44] as a suitable– 13 –anifold for SU (5) GUT model building in IIB orientifold compactifications. It exhibitsmany desirable features, including the generation of the 10 10 5 H Yukawa coupling via aeuclidean D X Σ are given in table 2.As required by the Calabi-Yau condition, the hypersurface X has divisor class equal tothe anticanonical class of the ambient toric variety, that is (cid:80) i [ D i ]. In addition to thisCoords / Vertices Q Q Q Q Divisor Class x = (1 , , ,
0) 3 0 0 0 3 Mx = (0 , , ,
0) 2 0 0 0 2 Mx = (0 , , ,
0) 0 1 0 0 Nx = (0 , , ,
1) 0 0 1 0 Ox = ( − , − , − , −
1) 0 1 1 -1 N + O − Px = ( − , − , ,
0) 1 -1 -1 -1 M − N − O − Px = ( − , − , − ,
0) 0 0 -1 1 − O + Px = ( − , − , , −
1) 0 -1 0 1 − N + P (cid:80) i [ D i ] 6 0 0 0 6 M Table 2:
GLSM charges for X Σ , the ambient toric variety on the IIB side whose Calabi-Yau hyper-surface is the threefold X. We have chosen the basis of linearly inequivalent divisors ( M, N, O, P )to have charges (1 , , , , (0 , , , , (0 , , , , (0 , , ,
1) under the indicated C ∗ gauge symmetries Q i of the GLSM. We have also indicated next to each field x i the corresponding one-dimensionalgenerator of the fan. information, the orientifold involution σ is taken to be σ : x (cid:55)→ − x , (4.1)under which the divisors D and D are fixed. This identifies them as O O
7] = [ D ] + [ D ]. Furthermore, via projective equivalences it can be seen that the points x = x = x = 0 and x = x = x = 0 are fixed points of the σ -action, and thus are thelocations of O The intersection ring on the base can be computed using standard techniques of alge-braic geometry. In order to talk sensibly about intersections we need to give a triangulationof the fan, or equivalently the Stanley-Reisner ideal (loosely speaking, the set of monomi-als in which not all terms can vanish simultaneously). We choose the following simplicialtriangulation: [[1 , , , , [1 , , , , [1 , , , , [1 , , , , [1 , , , , [1 , , , , [1 , , , , [1 , , , , [1 , , , , [1 , , , , [2 , , , , [2 , , , , [2 , , , , [2 , , , , [2 , , , D i is the vanishing locus of the homogeneous coordinate x i , which can be written in terms of thegenerators of the divisor group, as in table 2. As noted in [44], the points x = x = x = 0 and x = x = x = 0 are also fixed under the involution.However, the monomials x x and x x are in the Stanley-Reisner ideal, and thus these points are not in X Σ . This is equivalent to them being in the set Z Σ in the homogeneous coordinate construction of thistoric variety, X Σ = ( C − Z Σ ) /G. – 14 –here the integers refer to generators of the fan (so n stands for x n ). The correspondingStanley-Reisner ideal is SRI = { x x , x x , x x , x x , x x , x x x } , (4.3)which can be seen directly from the triangulation. For example, since there is no cone inthe triangulation with both 3 and 5, we know that x x is in the Stanley-Reisner ideal.In fact, this variety was analyzed in [44] using a different base of divisors. We can justtake the result quoted there and change the base to our M, N, O, P cycles. The change ofbase is the following: M = 12 D = 3 D + D + 2 D + 2 D N = D = D + D O = D = D + D P = N + O − D = D + D + D (4.4)Here we have defined the basic divisors D i as the ones given by x i = 0, and we have usedthe linear equivalences of divisors: D = 9 D + 3 D + 6 D + 6 D D = 6 D + 2 D + 4 D + 4 D D = D + D D = D + D (4.5)Using this change of basis, and the intersection form given in [44]: I X = D (7 D − D − D − D − D D − D D − D D + D D + D D ) (4.6)we can easily obtain the triple intersection form in terms of M, N, O, P . The result is: I X = M (7 M + 2 M N + 2
M O + 3
M P + N O + N P + OP + P P ) (4.7)
In this section we present the lift of the IIB orientifold model to F-theory, where thegeometry is that of an elliptically fibered Calabi-Yau fourfold Y of the form T (cid:44) −→ Y − (cid:16) X/σ. (4.8)We follow the prescription of [45, 46, 47], which is generalizable to many lifts of IIB orien-tifolds, and discuss the details of this particular lift. We construct first the base of the We also list the linear equivalence relation for D , although it is not necessary for our calculationsabove. There has also been great progress recently in constructing semi-realistic global models directly inF-theory [48, 49, 50, 51, 52]. – 15 –lliptic fibration
X/σ as a hypersurface in a new ambient toric variety X Σ (cid:48) , with homoge-neous coordinates whose GLSM charges have been changed relative to their counterpartsin X Σ to account for modding out by the orientifold action. We then determine the divisorclass of the hypersurface X/σ and use it to calculate the canonical bundle of the base, K X/σ , which is crucial in determining the precise form of the Tate sections a n . Next, hav-ing relevant knowledge of the fourfold base, we construct a six-dimensional toric variety X Σ (cid:48)(cid:48) , in which the Calabi-Yau fourfold Y is a complete intersection of two hypersurfaces,one for the base and one for the fiber. The GLSM charges for the homogeneous coordinatesof the base carry over from the toric variety X Σ (cid:48) , and we show how to determine the GLSMcharges for the fiber-related coordinates x , y , and z from the Weierstrass equation. Wealso briefly mention how one could arrive at the ambient toric variety of the fourfold X Σ (cid:48)(cid:48) without explicitly constructing the intermediate toric variety X Σ (cid:48) . The Base of Y Since we would like to stay in the framework of toric geometry, we will start by constructinga toric ambient space for the base. Specifically, the Calabi-Yau threefold X on the IIB sideis a hypersurface in the toric variety X Σ , so that one can construct the base of the fourfoldby modding out by the orientifold action, giving a new toric ambient space X Σ (cid:48) , and bymapping the hypersurface constraints appropriately. This requires a map from X Σ to X Σ (cid:48) which is 2-to-1 away from the O x , x , x , x , x , x , x , x ) (cid:55)→ ( x , x , x , x , x , x , x , x ) (4.9) ≡ (˜ x , ˜ x , ˜ x , ˜ x , ˜ x , ˜ x , ˜ x , ˜ x ) , where the latter are the homogeneous coordinates of X Σ (cid:48) . The effect of such a map is asimple doubling of the GLSM charges of ˜ x and ˜ x relative to x and x , while the chargesof the other ˜ x i are left unchanged. This is sufficient to determine the toric data of X Σ (cid:48) presented in table 3.Coords/Vertices Q Q Q Q Divisor Class˜ x = (1 , , ,
0) 3 0 0 0 3 I ˜ x = (0 , , ,
0) 2 0 0 0 2 I ˜ x = (0 , , ,
0) 0 2 0 0 2 J ˜ x = (0 , , ,
1) 0 0 1 0 K ˜ x = ( − , − , − , −
1) 0 1 1 -1 J + K − L ˜ x = ( − , − , ,
0) 1 -1 -1 -1 I − J − K − L ˜ x = ( − , − , − ,
0) 0 0 -2 2 − K + 2 L ˜ x = ( − , − , , −
1) 0 -1 0 1 − J + L (cid:80) i [ ˜ D i ] 6 1 -1 1 6 I + J − K + L Table 3:
GLSM Charges for X Σ (cid:48) , the four-dimensional ambient toric variety for the base X/σ ofthe elliptic fibration on the F-theory side. We have indicated the generators of the fan.
Having deduced the GLSM charges for the homogeneous coordinates ˜ x i in X Σ (cid:48) , wemust also deduce the divisor class of X/σ . To this end, the divisor class of X in X Σ is– 16 – i [ D i ] = 6 I . Monomials of this divisor class in X Σ get mapped to monomials of basecoordinates in X Σ (cid:48) via the map (4.9), from which we can read off the divisor class of X/σ in X Σ (cid:48) . For example, from x x x x (cid:55)→ ˜ x ˜ x ˜ x ˜ x (4.10)we see that X/σ has class 6 I . From this, the anticanonical bundle of the base can becomputed from the adjunction formula to be K X/σ = c ( T X/σ ) = (cid:80) i ˜ D i − I = J − K + L .Thus, we see that X/σ is not Calabi-Yau.At this point, one could explicitly construct the Tate form of the elliptic fibration,since it is specified by sections a n ∈ H ( X/σ, K − nX/σ ), and we have calculated the divisorclass of the anticanonical bundle. This method was employed in [47] and was fruitful inexamining the gauge enhancements associated with fiber degenerations. However, since weare interested in counting instanton zero modes via cohomologies of a divisor wrapped bya vertical M Y as acomplete intersection in a toric ambient space. In doing so, we will be able to apply thealgorithm descibed in section 3 in a straightforward manner. The Elliptically-Fibered Fourfold Y
The process of constructing the ambient toric variety X Σ (cid:48)(cid:48) of the fourfold Y is fairly intu-itive, as one might expect, and essentially amounts to appropriately adding homogeneouscoordinates for the fiber. In addition, since we wish to realize the fourfold as a completeintersection Y ≡ { P X/σ = 0 } ∩ { P T = 0 } , (4.11)we must specify the divisor class of the polynomials P X/σ and P T . The polynomial P T isusually chosen to be in either the Weierstrass form or (equivalently) the Tate form for anelliptic curve. For ease in determining the relevant GLSM data, we will use the Weierstrassform in this section, but will later move to the Tate form to make the determination ofgauge enhancements more tractable. There is no technical difference, of course, since theTate sections determine f and g . We merely choose one or the other based on what iseasiest for the particular task at hand.Beginning with the base, the GLSM charges for the homogeneous coordinates in X Σ (cid:48) carry over directly to X Σ (cid:48)(cid:48) , with the addition of the fact that they are uncharged underthe GLSM charge associated with the fiber, Q . We immediately know that [ P X/σ ] = 6 I ,since it is must have the same divisor class as X/σ in X Σ (cid:48) .In addition to the polynomial P X/σ , we must take into account a polynomial P T corresponding to the elliptic fiber. As mentioned above, in this section we choose the form P T ≡ y − x − f xz − gz , (4.12)the vanishing locus of which gives an elliptic curve in Weierstrass form, where f and g areglobal sections f ∈ H ( X/σ, K − X/σ ) and g ∈ H ( X/σ, K − X/σ ). In the case where f and g aremerely complex numbers, rather than sections, the Weierstrass equation can be consideredto be a degree six hypersurface in P , , . This gives the charges Q of x , y , and z under the– 17 –rojective scaling associated only to the fiber coordinates. Moreover, from homogeneity ofthe Weierstrass equation, the classes [ D x ] and [ D y ] can be determined as2[ D y ] = 3[ D x ] = [ g ] + 6[ D z ] , (4.13)where we use [ g ] = 6[ K X/σ ] = 6 J − K + 6 L . In addition, since we have two equationsand three unknowns, we choose [ D z ] = M , so that it does not transform under projectivescalings of the base. This is sufficient to determine the toric data of X Σ (cid:48)(cid:48) presented intable 4.Coords / Vertices Q Q Q Q Q Divisor Class˜ x = (1 , , , , ,
0) 3 0 0 0 0 3 I ˜ x = (0 , , , , ,
0) 2 0 0 0 0 2 I ˜ x = (0 , , , , ,
0) 0 2 0 0 0 2 J ˜ x = (0 , , , , ,
0) 0 0 1 0 0 K ˜ x = (0 , , , , ,
0) 0 1 1 -1 0 J + K − L ˜ x = ( − , − , , , ,
0) 1 -1 -1 -1 0 I − J − K − L ˜ x = (6 , , , , ,
0) 0 0 -2 2 0 − K + 2 L ˜ x = ( − , − , , − , ,
0) 0 -1 0 1 0 − J + Lx = (0 , , , , ,
3) 0 2 -2 2 2 2 J − K + 2 L + 2 My = ( − , − , − , − , − , −
2) 0 3 -3 3 3 3 J − K + 3 L + 3 Mz = (9 , , , , ,
0) 0 0 0 0 1 M (cid:80) i [ D i ] 6 6 -6 6 6 6 I + 6 J − K + 6 L + 6M Table 4:
GLSM Charges for X Σ (cid:48)(cid:48) , the six-dimensional ambient toric variety for the ellipticallyfibered Calabi-Yau fourfold Y, which is a complete intersection of two hypersurfaces. We haveindicated the generators of the fan. The reader should note, though, that the intermediate step of constructing the toricambient space X Σ (cid:48) of the base is not really necessary, since the GLSM charges of homoge-neous coordinates in X Σ (cid:48) are a subset of the GLSM charges of homogeneous coordinatesin X Σ (cid:48)(cid:48) and one can easily deduce the charges of y via the Calabi-Yau condition and theadjunction formula. This yields c ( T Y ) = c ( T X Σ (cid:48)(cid:48) ) − N P X/σ − N P T (4.14)= (cid:88) i [ D i ] − I − D y ] = 6 J − K + 6 L + 6 M − D y ] = 0 , with Poincar´e duality implied. In the same way, one could determine the charges of x , andagain one could choose z to only be charged under Q . It is then possible to read off theclass of f and g , or equivalently the Tate sections a n , from the homogeneity of P T , withoutever explicitly calculating the anticanonical bundle. Of course, these different viewpointsare all closely tied together, and the method one uses is a matter of preference. So far, we have focused on discussing the F-theory model which will have non-perturbativecorrections to the 10 10 5 H Yukawa coupling, but we have not yet discussed in detail the– 18 –roperties of the instanton which generates the coupling. The reason for this is that mostknown properties of euclidean branes in F-theory are known only from the properties ofthe relevant instanton in the IIB model.Charged zero modes in particular, which are the ones ultimately responsible for gen-erating the Yukawa coupling, are still poorly understood from a purely F-theoretical pointof view. What we have in mind when making this statement is the description of F-theoryas M-theory with vanishing fiber. The properties of charged zero modes on the euclideanM5 are not well understood. Nevertheless, F-theory is also IIB at strong coupling, and insimple situations like ours the description in terms of euclidean D3 branes is still expectedto be mostly correct. See, for example, [27] for a recent paper which takes this viewpoint,obtaining a number of rules for the spectrum of charged modes (these agree with the onesobtained in IIB, except in the case where exceptional degenerations of the fiber appear).What this means, in practice, is that the known computation of the superpotentialcoupling is isomorphic to the one done in IIB, except for the issue of saturation of neutralzero modes (¯ τ modes in particular). In this case there are more intrinsic ways of determiningthis spectrum. The most well known way is using Witten’s characterization of fermioniczero modes as elements of the cohomology of the structure sheaf of the divisor [12]. Inthe rest of this section we will use this representation, together with the result in [27] thatthe ¯ τ mode can be identified with an element of H , ( D ), to argue that the ¯ τ modes areprojected out in our context. Before going into that, we would also like to mention that onecan also understand the absence of dangerous neutral fermionic modes using the stronglycoupled IIB viewpoint [31], so we already know what the answer should be. Nevertheless,computing the cohomology is an instructive exercise, to which we now proceed.We study an M D = D X/σ ∩ D T ∩ D , whichis the intersection of the base, fiber, and D divisors in the ambient toric sixfold X Σ (cid:48)(cid:48) . Thepresence or absence of the τ ˙ α zero modes for this instanton are determined by the sheafcohomology group H ( D , O D ) , (4.15)which can be related to sheaf cohomologies on X Σ (cid:48)(cid:48) via Koszul sequences. For toric varieties,sheaf cohomology is equivalent to the ˇCech cohomology groups ˇ H p ( U , L ), where U is anopen cover and L is a line bundle on the toric variety. Thus, our task is to computeˇ H ( U , O D ) by calculating the ˇCech cohomology groups of various line bundles on X Σ (cid:48)(cid:48) .We can do this easily using our implementation of the algorithm in section 3.The divisor D is the intersection of three divisors of the sixfold, whose normal bundlesare given by N X/σ = O (6 I ) N T = O (6 J − K + 6 L + 6 M ) N D = O ( J + K − L ) . (4.16) And since in this case we have a weakly coupled limit of the system, we also know the answer from aCFT analysis in IIB [53, 54, 55, 56]. – 19 –ne can relate these objects on the ambient toric variety to the structure sheaf on D viathe Koszul sequence 0 → ∧ N ∗ → ∧ N ∗ → N ∗ → O X Σ (cid:48)(cid:48) → O D → , (4.17)where N ∗ is the dual of N ≡ N X/σ ⊕ N T ⊕ N D . For practical purposes, we split this intothree short exact sequences as0 → ∧ N ∗ → ∧ N ∗ → K → → K → N ∗ → K → → K → O X Σ (cid:48)(cid:48) → O D → , each of which gives a long exact sequence in cohomology, as outlined in section 3. Looking tothe parts of the long exact sequences relevant for the immediate calculation of H ( D , O D ),we have · · · → H ( X Σ (cid:48)(cid:48) , O X Σ (cid:48)(cid:48) ) → H ( D , O D ) → H ( X Σ (cid:48)(cid:48) , K ) → H ( X Σ (cid:48)(cid:48) , O X Σ (cid:48)(cid:48) ) → . . . · · · → H ( X Σ (cid:48)(cid:48) , N ∗ ) → H ( X Σ (cid:48)(cid:48) , K ) → H ( X Σ (cid:48)(cid:48) , K ) → . . . (4.19) · · · → H ( X Σ (cid:48)(cid:48) , ∧ N ∗ ) → H ( X Σ (cid:48)(cid:48) , K ) → H ( X Σ (cid:48)(cid:48) , ∧ N ∗ ) → . . . , one part for each short exact sequence. Calculating the cohomology of these line bundleson toric varieties, we arrive at the results H ( X Σ (cid:48)(cid:48) , ∧ N ∗ ) = 0 H ( X Σ (cid:48)(cid:48) , ∧ N ∗ ) = 0 H ( X Σ (cid:48)(cid:48) , N ∗ ) = 0 (4.20) H ( X Σ (cid:48)(cid:48) , O X Σ (cid:48)(cid:48) ) = 0 H ( X Σ (cid:48)(cid:48) , O X Σ (cid:48)(cid:48) ) = 0 , where Serre duality was useful for efficiently computing H ( X Σ (cid:48)(cid:48) , ∧ N ∗ ). Using theseresults, it is easy to see that H ( D , O D ) = 0 . (4.21)We see that the τ ˙ α is projected out for this instanton, as one might expect, since it is thelift of an O (1) instanton in IIB. Thus, since the τ ˙ α modes are projected out and the cycle D is rigid, we expect an M D to give a non-perturbative correction tothe 10 10 5 H Yukawa coupling.As a final word, there is a technical point that may be bothering the reader: theinstanton is on top of an Sp (2) stack of branes, and thus the fiber degenerates everywhereover its worldvolume. From this point of view, computing the cohomology of the relevantdivisor of the fourfold seems to not be well-defined. Nevertheless, with the definition thatwe have adopted here there are no issues, since cohomologies of line bundles on the ambienttoric space are always well-defined. This point was further explored and reinforced in [27],where it was tested that the relevant cohomology does not change under blow-ups of thegeometry that smooth out the degeneration of the fiber.– 20 – .4 The Tate Form for the Uplift In this section we discuss the details of the Tate form for the elliptic fiber in the fourfold Y . We construct the most general form of the sections a n of the Tate form in terms of thehomogeneous coordinates associated with the base and show that at a point in complexstructure moduli space the degenerations of the elliptic curve recover two of the three gaugegroups seen in the IIB limit. We show that the third group SO (6) is recovered only in Sen’sweak coupling limit. The results that we obtain in this example agree with, and illustrate,the general discussion in section 2.The fourfold Y , as mentioned, is an elliptic fibration over the base X/σ . The ellipticfiber is often cast in the Weierstrass form y = x + f xz + gz , where f ∈ H ( X/σ ; K − X/σ )and g ∈ H ( X/σ ; K − X/σ ) encode how the fiber varies over the base. Often more useful inpractice, however, is the Tate form y + a xyz + a yz = x + a x z + a xz + a z , (4.22)where a n ∈ H ( X/σ ; K − nX/σ ) instead encode the variation of the fiber over the base. Par-ticular combinations of the a n ’s are grouped into variables b = a + 4 a , b = a a + 2 a , b = a + 4 a (4.23)which are related to f and g by f = − ( b − b ) , g = − ( − b + 36 b b − b ) . (4.24)The discriminant, which encodes the locations of degeneration of the elliptic fiber, and thusthe 7-branes, takes the form∆ F = 4 f + 27 g = − b ( b b − b ) − b − b + 9 b b b . (4.25)The geometry of the base determines the explicit form of the sections a n and the discrimi-nant ∆ F , from which the singularities of the fiber, and thus the corresponding gauge groupsof the 7-branes, can be read off. The data relating the vanishing order of the sections a n and the discriminant ∆ F to the singularity type is reproduced in table 5.In our case we have a n ∈ H ( X/σ ; O ( n ( J − K + L )). Since they must be globalsections, the orders of vanishing of the homogeneous coordinates x i ∈ X Σ (cid:48) appearing inthe monomials must be positive. Thus, the divisors corresponding to the monomials mustbe effective : [ D ] = (cid:88) i n i [ D i ] = n ( J − K + L ) n i ≥ ∀ n i . (4.26)Satisfying this condition for the case at hand yields the result n = − n + n n = 2 n − n n = n = n = n = n = 0 , (4.27)which completely determines the allowed monomials in each section a n . Note that thisgives a = c ˜ x ˜ x a = c ˜ x ˜ x + c ˜ x ˜ x , (4.28)– 21 –ing. discr. gauge enhancement coefficient vanishing degreestype deg(∆) type group a a a a a I A SU (2) 0 0 1 1 2I ns3 s3 ns2 k k C k SP (2 k ) 0 0 k k k I s2 k k A k − SU (2 k ) 0 1 k k k I ns2 k +1 k + 1 [unconv.] 0 0 k + 1 k + 1 2 k + 1I s2 k +1 k + 1 A k SU (2 k + 1) 0 1 k k + 1 2 k + 1II 2 — 1 1 1 1 1III 3 A SU (2) 1 1 1 1 2IV ns s A SU (3) 1 1 1 2 3I ∗ ns0 G G ∗ ss0 B SO (7) 1 1 2 2 4I ∗ s0 D SO (8) 1 1 2 2 4I ∗ ns1 B SO (9) 1 1 2 3 4I ∗ s1 D SO (10) 1 1 2 3 5I ∗ ns2 B SO (11) 1 1 3 3 5I ∗ s2 D SO (12) 1 1 3 3 5I ∗ ns2 k − k + 3 B k SO (4 k + 1) 1 1 k k + 1 2 k I ∗ s2 k − k + 3 D k +1 SO (4 k + 2) 1 1 k k + 1 2 k + 1I ∗ ns2 k − k + 4 B k +1 SO (4 k + 3) 1 1 k + 1 k + 1 2 k + 1I ∗ s2 k − k + 4 D k +2 SO (4 k + 4) 1 1 k + 1 k + 1 2 k + 1IV ∗ ns F F ∗ s E E ∗ E E ∗ E E Table 5:
Refined Kodaira classification resulting from Tate’s algorithm [57], from [58]. In orderto distinguish the “semi-split” case I ∗ ss2 k from the “split” case I ∗ s2 k one has to work out a furtherfactorization condition which is part of the aforementioned algorithm, see § which leads to b = c ˜ x ˜ x + 4 c ˜ x ˜ x + 4 c ˜ x ˜ x . It can be shown that in Sen’s limit, theorientifold is located at b = 0, which in our case corresponds to O7-planes on the divisor[ O
7] = [ D ] + [ D ]. This is precisely the result of the simple analysis on the IIB side.– 22 –ontinuing this analysis for the sake of examining possible gauge enhancements gives a = c ˜ x ˜ x + c ˜ x ˜ x ˜ x a = c ˜ x ˜ x + c ˜ x ˜ x ˜ x + c ˜ x ˜ x (4.29) a = c ˜ x ˜ x + c ˜ x ˜ x ˜ x + c ˜ x ˜ x ˜ x + c ˜ x ˜ x . This is the most general form for the Tate sections in this model, from which the “minimal”gauge enhancements can be read off. For example, using table 5, it can be seen from theorder of vanishing along D that it has minimal gauge group G (a similar phenomenonwas found in [20]). Rather than constructing the most general allowed fibration for thismodel, however, we would like to reproduce as much of the IIB physics as possible in theF-theory lift. Moving to a point in complex structure moduli space where a = c ˜ x ˜ x a = 4 c c ˜ x ˜ x ˜ x a = a = a = 0 (4.30)∆ F = − c c ˜ x ˜ x ˜ x (27 c ˜ x ˜ x + c ˜ x ) , it is readily seen that the gauge groups along D and D are SO (10) and Sp (2), respectively,as is the case in IIB. However, recovering the factor of SO (6) along D requires taking c →
0, which sends ∆ F → g s →
0. This is precisely Sen’s limit[35]. This agrees beautifully with the discussion in section 2.
5. Conclusions
In this paper we have addressed a number of conceptual and technical issues which arisein the analysis of instanton effects in F-theory.We started in section 2 by explaining the reason for an obstruction that can appearwhen trying to lift certain stacks of branes in IIB to F-theory. This also allowed us tomake predictions about which IIB brane configurations are obstructed. In the process, wedescribed in some detail the behavior of D7 branes as we go from large to vanishing flavormasses.We continued in section 3 by discussing a way of computing sheaf bundle cohomologyon toric varieties. This is essential for showing that the τ ˙ α mode is projected out, whichrequires calculation of the cohomology group H ( D , O D ), where D is the fourfold divisorwhich the M dP in section 3.2. In 3.3, we discuss the Koszul sequence, whichgives a long exact sequence in cohomology which allows one to compute the cohomology H ( D , O D ) by knowing information about ˇCech cohomology of line bundles on the ambi-ent toric variety. Finally, in section 3.4, we provide some details about where to find ourready-to-use computer implementation of the algorithm.In section 4 we illustrated the considerations in the previous sections in a particularexample. We introduced in section 4.1 the Calabi-Yau threefold M ( dP ) , henceforth called– 23 – , as a Calabi-Yau hypersurface in a four-dimensional toric variety X Σ . In section 4.2,we performed the F-theory lift of the IIB orientifold compactification on X , following [45].For the sake of clarity, we presented the lift in two steps. First, we presented the fourfoldbase X/σ as a hypersurface in a four-dimensional toric variety X Σ (cid:48) by properly moddingout by the orientifold action σ . Next, we presented the uplifted Calabi-Yau fourfold Y asa complete intersection of two hypersurfaces in a six-dimensional toric variety X Σ (cid:48)(cid:48) , oneassociated to the fiber and one to the base.In section 4.3, we addressed the issue of instanton zero modes in the F-theory uplift ofthe IIB orientifold compactification on X . In the F-theory lift, we showed the absence ofthe fermionic τ ˙ α zero modes for a vertical M H Yukawa coupling.In section 4.4 we determined explicitly the Tate sections and discriminant for the F-theory lift, allowing us to see the location of seven branes as divisors in the base over whichthe fiber degenerates, as well as their associated gauge group. At a generic point in modulispace, this data determines the “minimal” gauge enhancements along the seven branes,but we showed a point in complex structure moduli space which recovers, in F-theory, theproper location of the orientifold and two of the three gauge seven branes seen on the IIBside. Interestingly, it is only in Sen’s IIB limit that the proper enhancement of the thirdgauge seven brane is obtained, in agreement with the general discussion of section 2.F-theory compactifications provide a rich field of study both for formal and phenomeno-logical questions. The way the results in this paper came to be nicely illustrates this con-nection: we set out to study a particular model with some nice phenomenological features,and we were driven to fascinating questions in Seiberg-Witten theory and algebraic geome-try. There is no doubt that there are still plenty of interesting phenomena to be elucidatedin the quest for fully realistic F-theory models.
Acknowledgments
We would like to acknowledge interesting discussions with Lara Anderson, Ralph Blumen-hagen, B.G. Chen, Andres Collinucci, Ron Donagi, Josh Guffin, Benjamin Jurke, JeffreyC.Y. Teo and Timo Weigand. We would also like to thank the authors of [32] for inform-ing us of their upcoming work prior to publication. We are also grateful to the editor ofJHEP, who suggested a rearrangement of the sections to increase the clarity of the expo-sition. We gratefully acknowledge the hospitality of the KITP during the Strings at theLHC and in the Early Universe program for providing a stimulating environment duringthe completion of this work. I.G.E. thanks N. Hasegawa for kind support and constantencouragement. This research was supported in part by the National Science Foundationunder Grant No. NSF PHY05-51164, DOE under grant DE-FG05-95ER40893-A020, NSFRTG grant DMS-0636606 and Fay R. and Eugene L. Langberg Chair.
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