Root Bundles and Towards Exact Matter Spectra of F-theory MSSMs
Martin Bies, Mirjam Cveti?, Ron Donagi, Muyang Liu, Marielle Ong
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Root Bundles andTowards Exact Matter Spectra of F-theory MSSMs
Martin Bies , , Mirjam Cvetič , , , Ron Donagi , , Muyang Liu , Marielle Ong Department of Mathematics, University of Pennsylvania,Philadelphia, PA 19104-6396, USA Department of Physics and Astronomy, University of Pennsylvania,Philadelphia, PA 19104-6396, USA Center for Applied Mathematics and Theoretical Physics, University of Maribor,Maribor, Slovenia
Motivated by the appearance of fractional powers of line bundles in studies of vector-like spectra in 4d F-theory compactifications, we analyze the structure and origin ofthese bundles. Fractional powers of line bundles are also known as root bundles and canbe thought of as generalizations of spin bundles. We explain how these root bundles arelinked to inequivalent F-theory gauge potentials of a G -flux.While this observation is interesting in its own right, it is particularly valuable forF-theory Standard Model constructions. In aiming for MSSMs, it is desired to argue forthe absence of vector-like exotics. We work out the root bundle constraints on all mattercurves in the largest class of currently-known F-theory Standard Model constructionswithout chiral exotics and gauge coupling unification. On each matter curve, we conducta systematic “bottom”-analysis of all solutions to the root bundle constraints and allspin bundles. Thereby, we derive a lower bound for the number of combinations of rootbundles and spin bundles whose cohomologies satisfy the physical demand of absence ofvector-like pairs.On a technical level, this systematic study is achieved by a well-known diagrammaticdescription of root bundles on nodal curves. We extend this description by a countingprocedure, which determines the cohomologies of so-called limit root bundles on fullblow-ups of nodal curves. By use of deformation theory, these results constrain thevector-like spectra on the smooth matter curves in the actual F-theory geometry.1 ontents
1. Introduction 32. Root bundles in F-theory 6
3. Root bundles from limit roots 15
4. Limit root applications in F-theory 27
5. Conclusion and Outlook 34A. Fiber structure of F-theory Standard Models 38
A.1. Away from Matter Curves . . . . . . . . . . . . . . . . . . . . . . . . . . 39A.2. Over Matter Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39A.3. Over Yukawa Loci . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
B. Induced line bundles in F-theory Standard Models 48
B.1. G -flux and matter surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 48B.2. Computational strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . 50B.3. Root bundle constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . 55B.4. Limit roots in base space P . . . . . . . . . . . . . . . . . . . . . . . . . 572 . Introduction String theory elegantly couples gauge dynamics to gravity. This makes string theorya leading candidate for a unified theory of quantum gravity. As such, it must accountfor all aspects of our physical reality, especially the low energy particle physics that weobserve. As a first order approximation, one desires an explicit demonstration in whichone can actually obtain the particle spectrum of the Standard Model from string theory.In the past decades, enormous efforts have been undertaken to achieve this goal. Manyof these models concentrated on perturbative corners of string theory, such as the E × E heterotic string [1–8] or intersecting branes models in type II [9–15] (see also [16] andreferences therein). These perturbative models were among the first compactificationsfrom which the Standard Model gauge sector emerged with its chiral or, in the caseof [4, 5], even the vector-like spectrum. Unfortunately, these constructions are limiteddue to their perturbative nature in the string coupling, and they typically suffer fromchiral and vector-like exotic matter. Among these perturbative models, the first globallyconsistent MSSM constructions are [4,5] (see [17,18] for more details on the subtle globalconditions for slope-stability of vector bundles).The non-perturbative effects in string theory are elegantly described by F-theory [19–21]. As a non-perturbative extension of type IIB string theory, the framework of F-theorydescribes the gauge dynamics on 7-branes including their back-reactions (to all ordersin the string coupling) onto the compactification geometry B n . These back-reactionsare encoded in the geometry of an elliptically fibered Calabi-Yau space π : Y n +1 ։ B n .By studying this space Y n +1 with well-established tools of algebraic geometry, one canthen ensure the global consistency conditions of the physics in − n non-compactdimensions.An important characteristic of 4d N = 1 F theory compactifications (i.e., n = 3 ),which must match the particle physics that we observe, is the chiral fermionic spectrum.In F-theory, this spectrum is uniquely fixed by a background gauge flux, which in turnis most conveniently specified by the internal C profile in the dual M-theory geometry.The chiral spectrum then only depends on the flux G = dC ∈ H (2 , ( Y ) . By now,there exists an extensive toolbox for creating and enumerating the so-called primaryvertical subspace of G configurations [22–28]. The application of these tools led tothe construction of globally consistent chiral F-theory models [26, 28–30], which recentlyculminated in the largest class of explicit string vacua that realize the Standard Modelgauge group with their exact chiral spectrum and gauge coupling unification [31].However, these methods are insufficient to determine the exact vector-like spectrumof the chiral zero modes (i.e., not just the difference between chiral and anti-chiralfields). This is because the zero modes depend not only on the flux G , but also onthe flat directions of the potential C . The complete information is encoded in theso-called Deligne cohomology . In [32–34], methods for determining the exact vector-likespectra were put forward. This approach exploits the fact that (a subset of) the Delignecohomology can be parameterized by Chow classes. By use of this parameterization,one can extract line bundles L R that are defined on curves C R in B . In the dual IIBpicture, this can be interpreted as localization of gauge flux on matter curves, which3ifts some vector-like pairs on these curves. Explicitly, the zero modes are counted bythe sheaf cohomologies of L R and we have h ( C R , L R ) massless chiral and h ( C R , L R ) massless anti-chiral superfields in representation R on C R .Although this procedure works in theory for any compactification, technical limitationsarise in practical applications. Intuitively, one may think of the technical difficulties asreflections of the delicate complex structure dependence of the line bundles cohomologies.Even state-of-the-art algorithms such as [35–37] (see also [33, 34]) on supercomputersspecifically designed for such computations (such as Plesken at Siegen University ), canoftentimes not perform the necessary operations in realistic compactification geometries.For instance, the models studied in [32–34] focused on computationally simple geometriesas a result. While this led to a proof of principle, these models have unrealistically largenumbers of chiral fermions. Therefore — even though it is expected — it remains anopen question whether or not F-theory compactifications can actually realize effectivetheories that resemble the matter spectra of the Standard Model.Recently, [38] the complex structure dependence of line bundle cohomologies wasinvestigated . This analysis was inspired from the F-theory GUT models discussed in [34]and focused on simple geometries, in which the algorithms in [35] could generate a largedata set [39]. This data was analyzed by use of data science techniques, in particulardecision trees. A theoretical understanding of this data was achieved by supplementingthe data science results by Brill-Noether theory [40] (see [41] for a modern expositionand [42] for an earlier application of Brill-Noether theory in F-theory). These insightsled to a quantitative study of jumps of charged matter vector pairs as a function of thecomplex structure parameters of the matter curves.
Results
In globally consistent F-theory constructions with the exact chiral spectraof the Standard Model and gauge coupling unification [31], the vector-like spectra onthe low-genus matter curves are encoded in cohomologies of a line bundle, which areidentified with a fractional power of the canonical bundle. On high-genus curves, thesefractional powers of the canonical bundle are further modified by contributions fromYukawa points.In order to make sense of these fractional powers, we study the G -flux in more detail.The models in [31] consider a background G -flux, which not only leads to the exactchiral spectra but also satisfies global consistency conditions, such as the D3-tadpolecancelation and masslessness of the U (1) -gauge boson. We lift this very background G -flux to a gauge potential in the Deligne cohomology to identify the line bundles L R .This process requires an understanding of the intermediate Jacobian J ( Y ) , which labelsinequivalent gauge backgrounds. A naive analysis, which does not properly take theintermediate Jacobian into account, leads to the fractional line bundle powers mentionedabove. In past works [32–34], such scenarios were avoided as it is not immediately clearhow to interpret these fractional expressions. However, since these expressions appearubiquitously in compact models with realistic chiral indices, this work analyzes the originand meaning of these bundles in detail.The objects we are therefore interested in are fractional powers of line bundles, also4nown as root bundles. They may be thought of as generalizations of spin bundles.Similar to spin bundles, root bundles are far from unique. The mathematics of rootbundles indicates that we should think of the different root bundles as being induced frominequivalent gauge potentials for a given G -flux. While [32–34] has already anticipatedthat inequivalent gauge potentials for a given G -flux lead to different vector-like spectra,the root bundle interpretation allows one to test this expectation.In general, not all root bundles on the matter curves are induced from F-theory gaugepotentials in the Deligne cohomology H D ( b Y , Z (2)) . This mirrors the expectation thatonly some of the spin bundles on the matter curves are consistent with the F-theorygeometry b Y . This raises the interesting and important question of identifying whichroots and spin bundles are induced top-down. While this work does not answer thisquestion, we hope that it initiates and facilitates this analysis by providing a systematicapproach to all root bundles and spin bundles on the matter curves. In particular,we identify pairs of root bundles and spin bundles such that their tensor product is aline bundle whose cohomologies satisfy the physical demand of the presence/absence ofvector-like pairs.On a technical level, this requires a sufficient understanding of root bundles and theircohomologies on a matter curve C R . We gain this control from a deformation C R → C • R into a nodal curve. On the latter, root bundles are described in a diagrammatic way byso-called limit roots [43]. We extend these ideas to a counting procedure for the globalsections of limit roots, which we use to infer the cohomologies of root bundles on C R .This approach is demonstrated in the largest class of currently-known constructions ofglobally consistent F-theory Standard Models without chiral exotics and gauge couplingunification [31]. In one particular such geometry, we derive a lower bound to the numberof pairs of root bundles and spin bundles whose tensor product is a line bundle withoutvector-like exotics. Outline
In Section 2, we recall zero mode counting in F-theory and the appearance offractional line bundle powers. We explain that these fractional powers of line bundles,also known as root bundles, relate to inequivalent gauge potentials in F-theory. Theseideas are subsequently applied to the largest currently-known family of globallyconsistent F-theory Standard Model constructions without chiral exotics and gaugecoupling unification [31]. We explain how the background G -flux, which satisfies globalconsistency conditions such as the cancellation of the D3-tadpole and the masslessnessof the U (1) -gauge boson, induces root bundles on the matter curves. Details of thisderivation are summarized in appendix B. This derivation heavily relies on a detailedunderstanding of the elliptically fibered 4-fold F-theory geometry b Y , includingintersection numbers in the fiber over the Yukawa points. We supplement the earlierworks [26, 31, 44] with a complete list of all fiber intersection numbers in appendix A.In Section 3 we first summarize well-known results about root bundles before wedescribe the limit root constructions, which were originally introduced in [43]. We extendthe limit root constructions by a counting procedure for the global sections of limit rootson full blow-ups of nodal curves. In fortunate instances, this even provides a means to5xplore Brill-Noether theory of limit roots , which we demonstrate in an example inspiredfrom [45].Finally, we apply these ideas to globally consistent constructions of F-theory StandardModels without chiral exotics and gauge coupling unification [31] in Section 4. In anexplicit base space geometry, we deform the matter curves to nodal curves, constructlimit roots on these nodal curves, identify the number of global sections of these limitroots and finally use deformation theory to relate these counts to the cohomologies ofroot bundles in the actual F-theory geometry. Thereby, we explicitly prove the existenceof root bundle solutions without vector-like pairs. Technical details of the specific basegeometry and the limit root constructions are summarized in appendix B.4.
2. Root bundles in F-theory
Zero mode counting in F-theory
We consider an F-theory compactification to fourdimensions given by a singular, elliptically fibered 4-fold π : Y ։ B . We assumethat this fibration has a section s ∼ = B and admits a smooth, flat, crepant resolution b π : b Y ։ B . In such a compactification, the flux G ∈ H (2 , ( b Y ) is subject to thequantization condition [46] G + 12 c ( T b Y ) ∈ H (2 , Z ( b Y ) = H (2 , ( b Y ) ∩ H ( b Y , Z ) . (2.1)For simplicity, we focus on compactifications with even c ( T b Y ) , which holds true for F-theory compactifications on an elliptically fibered smooth Calabi-Yau 4-fold with glob-ally defined Weierstrass model [47]. Under the simplifying assumption that c ( T b Y ) iseven, Equation (2.1) requires that G ∈ H (2 , Z ( b Y ) . We will show an example of thisand the following root bundle analysis in the largest currently-known class of F-theoryStandard Model constructions with gauge coupling unification and no chiral exotics [31]in Section 2.2.Over the codimension-2 matter curves C R ⊆ B , the reducible fibers of b Y containa chain of P s. A state with weight w in the representation R corresponds to a linearcombination of these P ’s. By fibering this linear combination over the matter curve C R ,one obtains the matter surface S R . The chiral index of the massless matter localizedon the matter curve C R ⊆ B is then specified by [48–54] [23, 29, 55] as χ ( R ) = Z S R G . (2.2) For zero mode counting of half-integer quantized G -fluxes, see e.g. [32]. In general, a G -flux can induce different chiral indices and vector-like spectra on the different weightstates. In such instances, it makes sense to keep track of w and write S wR . However, in anticipationof [31], we focus on gauge invariant G -fluxes, which induce the same chiral index and vector-likespectra for all weight states. G -flux has been analyzed in [32–34]. We employthe short exact sequence → J ( b Y ) ι ֒ −→ H D ( b Y , Z (2)) b c −→→ H (2 , Z ( b Y ) → , (2.3)where there exists a surjection b c that maps the gauge potential A ∈ H D ( b Y , Z (2)) asan element of the Deligne cohomology group to its G -flux. The Deligne cohomologyclasses encode the full gauge background data. Therefore, they parallel the internal3-form potentials C in the dual M-theory picture in which G = dC . As long as C ′ − C is a closed 3-form, C ′ has the same field strength G as C . In F-theory, suchclosed 3-form potentials are encoded by the intermediate Jacobian. Put differently, twoinequivalent A ′ , A ∈ H D ( b Y , Z (2)) with b c ( A ′ ) = b c ( A ) = G differ by A ′ − A = ι ( B ) ,where B ∈ J ( b Y ) is an element of the intermediate Jacobian corresponding to a closedM-theory 3-form potential. The Deligne cohomology group H D ( b Y , Z (2)) is rather hard to handle in explicitcomputations. However, we can parametrize (at least a subset of) H D ( b Y , Z (2)) bythe Chow group CH ( b Y , Z ) . This is summarized in the commutative diagram hom ( b Y , Z ) CH ( b Y , Z ) H (2 , alg ( b Y ) 00 J ( b Y ) H D ( b Y , Z (2)) H (2 , Z ( b Y ) 0 b γ γ b c (2.4)Unless stated differently, the symbol A is reserved for an element A ∈ CH ( b Y , Z ) , bywhich we specify an F-theory gauge potential A = b γ ( A ) .In order to count the zero modes in representation R in the presence of such a gaugepotential b γ ( A ) ∈ H D ( b Y , Z (2)) , we consider the matter surface S R with ι S R : S R ֒ → b Y , π S R : S R ։ C R . (2.5)The cylinder map, which sends A ∈ CH ( b Y , Z ) to a class D R ( A ) ∈ Pic ( C R ) , is therestriction to S R followed by integration over the fibers to C R : D R ( A ) = π S R ∗ (cid:0) ι ∗ S R ( A ) (cid:1) ∈ Pic ( C R ) . (2.6)The matter spectrum is then determined in terms of sheaf cohomology groups: h ( C R , L R ( A )) ↔ chiral zero modes ,h ( C R , L R ( A )) ↔ anti-chiral zero modes , (2.7)where L R ( A ) = O C R ( D R ( A )) ⊗ O C R O spin C R , (2.8) Equivalently, different gauge potentials in H D ( b Y , Z (2)) differ by their Wilson lines [56, 57]. For more details, see [32] and references therein. O spin C R an appropriate spin bundle on C R . This is a refinement of Equation (2.2),since Riemann-Roch gives χ ( R ) = h ( C R , L R ( A )) − h ( C R , L R ( A )) = χ ( L R ( A )) = deg ( D R ( A )) = Z S R G . (2.9) Roots of F-theory gauge potentials
For an F-theory model, we need an F-theorygauge potential, i.e. a class in the Deligne cohomology group H D ( b Y , Z (2)) . This willbe specified as b γ ( A ) for some “potential" A ∈ CH ( b Y , Z ) . We find that the geometrydetermines a class b γ ( A ′ ) ∈ H D ( b Y , Z (2)) and an integer ξ ∈ Z > such that A is subjectto the two constraints: γ ( A ) = G , ξ · b γ ( A ) ∼ b γ ( A ′ ) . (2.10)The condition γ ( A ) = G immediately follows from eq. (2.4) and it means that b γ ( A ) is an F-theory gauge potential for the given G -flux. We will illustrate with severalexamples below that the absence of chiral exotics in the F-theory Standard Models boilsdown to the second constraint. It is important to notice that the gauge potential A specified by the two conditions in eq. (2.10) is in general not unique. It is difficult to saymuch about solutions in the Chow group itself, but going to the bottom row in (2.4), wesee that the collection of all ξ -th roots of b γ ( A ′ ) (if non empty) is a coset of the group ofall ξ -th roots of . In particular, the number of solutions is ξ · dim C ( J ( b Y ) ) .All these solutions lead to the same chiral spectrum (2.2), since they all have the samedegree when restricted to the curves C R , hence the same index. However, they coulddiffer in their actual spectrum (2.7). This extra flexibility is the key tool that we intendto use to produce a desirable spectrum such as the MSSM. Roots on the matter curves
In theory, we could simply analyze the algebraic cycles A which satisfy eq. (2.10). However, as we will demonstrate momentarily, we can explicitlyconstruct A ′ ∈ CH ( b Y , Z ) . Therefore, we have a sufficient level of arithmetic controlover b γ ( A ′ ) and it is natural to ask how the induced divisors D R ( A ′ ) and D R ( A ) arerelated. The map D R : CH ( b Y , Z ) → CH ( C R , Z ) ∼ = Pic( C R ) , as defined in eq. (2.6),factors through b γ and a group homomorphism H D ( b Y , Z (2)) → Pic( C R ) . (2.11)Thus, it follows that ξ · D R ( A ) ∼ D R ( A ′ ) ∈ Pic( C R ) . (2.12)This means that the F-theory gauge potential A = b γ ( A ) induces a divisor D R ( A ) , whose ξ -th multiple is linearly equivalent to the divisor D R ( A ′ ) that is induced by the F-theorygauge potential A ′ = b γ ( A ′ ) ∈ H D ( b Y , Z (2)) . Such a divisor D R ( A ) is termed a ξ -th rootof D R ( A ′ ) . 8n general, ξ -th roots of D R ( A ′ ) do not exist. When they do, they are not unique.This is particularly well known for the case ξ = 2 and D R ( A ′ ) = K R , where the 2ndroots of the canonical bundle are the spin structures on C R . If C R is a curve of genus g , then it admits g spin structures (see e.g. [58, 59]). This easily extends to ξ > and D R ( A ′ ) = K R . While we will provide more details on root bundles in Section 3.1,it suffices to state here that ξ -th roots of D R ( A ′ ) do exist if and only if ξ divides deg ( D R ( A ′ )) . So on a genus g curve, there exist ξ g ξ -th roots of D R ( A ′ ) .In general, it cannot be expected that ξ -th roots of D R ( A ′ ) ∈ Pic( C R ) and ξ -th rootsof A ′ = b γ ( A ′ ) ∈ H D ( b Y , Z (2)) are one-to-one. Rather, only a subset of the ξ -th roots of D R ( A ′ ) will be realized from F-theory gauge potentials in H D ( b Y , Z (2)) . In this sense,the root bundle constraint in eq. (2.12) is necessary but not sufficient to conclude thatthe divisor D R ( A ) stems from an F-theory gauge potential.It is an interesting question to investigate which roots of D R ( A ′ ) ∈ Pic( C R ) areinduced from F-theory gauge potentials. While this work does not answer this question,we hope to initiate and facilitate this study by providing a systematic approach to all ξ -th roots of D R ( A ′ ) . In particular, we will provide a counting procedure, which allowsone to infer the cohomologies of some of these root bundles. This allows one to searchfor roots which satisfy the physical demand of the presence/absence of vector-like pairs.Before we show an example of these notions in the F-theory Standard Models [31], letus briefly comment on spin bundles O spin C R . Recall that Freed-Witten anomaly cancelationrequires spin c -structures on D -branes in perturbative IIB-compactifications [60]. Asexplained in [61], this extends to the demand of spin c -structures on gauge surfaces S ⊂ B in F-theory compactifications. Then, a choice of spin c -structure on N C R | S inducesa unique spin c -structure on C R [62]. Therefore, the question of which spin c -structuresare realized from the F-theory geometry b Y arises. While this is a fascinating question,we will not answer it in this work. Rather, we will systematically study all ξ -th roots of D R ( A ′ ) and all spin bundles on C R . Our goal is to identify combinations of root andspin bundles such that their tensor product is a line bundle whose cohomologies satisfythe physical demand of presence/absence of vector-like pairs. We will now exhibit an example of the root bundle analysis in the largest class ofcurrently-known globally consistent F-theory Standard Model constructions that supportgauge coupling unification and avoid chiral exotics [31]. Earlier geometric details can befound in the works [26, 44]. For convenience, we briefly summarize the geometry beforewe discuss the G -flux and its lifts.The analysis of the induced line bundles, i.e., evaluating Equation (2.6), is both tediousand lengthy. It makes use of the intersection numbers in the fibers over the mattercurves and Yukawa points. As an extension of the past works on this class of F-theorygeometries, we list exhaustive details of the fiber geometry in Appendix A. The necessaryintersection computations are detailed in Appendix B. The latter includes a sectionon topological intersection numbers of non-complete intersections, which we determine9igorously from the Euler characteristic of the structure sheaf of the intersection variety. b Y For a base 3-fold B , the resolved elliptic fibration b Y is ahypersurface in the space X = B × P F . The fiber ambient space P F is the toricsurface with the following toric diagram. In the accompanying table, we indicate its Z -graded Cox ring: we e ue e v u v w e e e e H 1 1 1 E -1 -1 1 E -1 -1 1 E -1 -1 1 E -1 -1 1 (2.13)Equivalently, the Stanley-Reisner ideal of P F is given by I SR ( P F ) = h e w, e e , e e , e v, e u, e e , e e , e v, wu, we , we , ve , uv, e u i . (2.14)Consider sections s i ∈ H ( B , K B ) . Then, in the space X , the resolved 4-fold b Y is thehypersurface V ( p F ) with p F = s e e e e u + s e e e e u v + s e e uv + s e e e u w + s e e e e uvw + s e vw . (2.15)It is instructive to note that (cid:8) e e e e u , e e e e u v, e e uv , e e e u w, e e e e uvw, e vw (cid:9) (2.16)is a basis of H (cid:0) P F , K P F (cid:1) . Since X = B × P F and s i ∈ H ( B , K B ) , it followsfrom the Künneth-formula that p F is a section of K X . Consequently, b Y is a smoothelliptically fibered Calabi-Yau 4-fold. Gauge group, matter curves and Yukawa points
Over V ( s ) = { s = 0 } ⊂ B ,the fibration b Y admits an SU (2) gauge enhancement. Similarly, there is an SU (3) enhancement over V ( s ) . The fibration b π : b Y ։ B admits two independent sections s = V ( v ) and s = V ( e ) . We call s = V ( v ) the zero section and employ theShioda map to associate a U (1) -gauge symmetry to s . Consequently, b Y admits an SU (3) × SU (2) × U (1) gauge symmetry with zero section s = V ( v ) .10e label the matter curves by the representations of SU (3) × SU (2) × U (1) in whichthe zero modes, localized on these curves, transform: C ( , ) / = V ( s , s ) , C ( , ) − / = V (cid:0) s , s s + s ( s s − s s ) (cid:1) , (2.17) C ( , ) − / = V ( s , s ) , C ( , ) / = V (cid:0) s , s s + s ( s s − s s ) (cid:1) , (2.18) C ( , ) = V ( s , s ) . (2.19)These curves intersect in the Yukawa loci Y = V ( s , s , s ) , Y = V ( s , s , s s − s s ) , Y = V ( s , s , s ) , (2.20) Y = V ( s , s , s ) , Y = V ( s , s , s ) , Y = V ( s , s , s ) . (2.21)We represent the intersections among the matter curves including the physically relevantself-intersections as follows: Y Y Y Y Y Y (1 , − / (3 , / (1 , (3 , / (3 , − / (2.22)The topological intersection number is K B at Y , Y , Y , Y and · K B at Y , Y . G -flux Let us identify the root bundles whose sections count the localized zero modes in thepresence of the (candidate) G -flux introduced in [31]. This flux is a base dependent linear combination of the U (1) -flux ω ∧ σ , where σ is the Shioda (1 , -form associatedto the divisor s = V ( e ) , and of the matter surface flux G ( , ) / on the curve C ( , ) / : G ( a, ω ) = a · G ( , ) / + ω ∧ σ ∈ H (2 , ( b Y ) . (2.23)The parameters a ∈ Q and ω ∈ π ∗ (cid:0) H (1 , ( B ) (cid:1) are subjected to flux quantization, D -tadpole cancelation, masslessness of the U (1) -gauge boson, and exactly three chiralfamilies on all matter curves. These conditions are solved by ω = 3 K B · K B , a = 15 K B . (2.24)11xplicitly, the resulting flux candidate can be expressed as (see Appendix B.1 for details) G = − K B · (cid:18) e ] ∧ [ e ]+ b π ∗ (cid:0) K B (cid:1) ∧ (cid:0) − e ] − e ] − e ] + b π ∗ (cid:0) K B (cid:1) − u ] + [ v ] (cid:1) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) b Y . (2.25)In this expression, [ e ] = γ ( V ( e )) ∈ H (1 , ( X ) is the image of the divisor V ( e ) ⊆ X under the cycle map γ . Also, we use the projection map b π : b Y ։ B . This G -fluxcandidate cancels the D3-tadpole and ensures the masslessness of a U (1) -gauge boson.We must verify that the G -flux candidate in eq. (2.25) satisfies the flux quantization G + c ( T b Y ) ∈ H (2 , Z ( b Y ) [46]. As a necessary check, in [31], the integrals of G + c ( T b Y ) over all matter surfaces S R and complete intersections of toric divisors were worked out.By employing the results in [47, 63], these were found to be integral. A sufficient checkfor eq. (2.25) to be properly quantized is computationally very demanding and currentlybeyond our arithmetic abilities. Therefore, the authors of [31] proceeded under theassumption that this candidate G -flux is properly quantized. We will also follow thisline of thought.Furthermore, we slightly extend this result. Namely, we integrate only c ( T b Y ) over thematter surfaces and over the complete intersections of toric divisors. By the reductiontechnique in [28], we can relate these integrals to intersection numbers in the base B .An explicit computation reveals that the only quantities which are not manifestly evenare Z B c ( B ) ∧ K B , Z B α ∧ ( c ( B ) + K B ) for all α ∈ H , ( B , Z ) , (2.26)where c ( B ) is the second Chern class of B . For smooth 3-folds B that appear asa base of a smooth elliptic Calabi–Yau 4-fold, it is known [47] that c ( B ) + K B isan even class. Furthermore, [63] states that R B c ( B ) · K B = 24 is even as well. Itthus follows that c ( T b Y ) passes the necessary conditions for being even. Likewise, we canintegrate the G -flux candidate eq. (2.25) over the matter surfaces and over the completeintersections of toric divisors. All of those are found to be integral. Since a sufficientcheck is currently beyond our arithmetic abilities, we proceed under the assumption that c ( T b Y ) is even and that the G -flux candidate eq. (2.25) is integral.It should be mentioned that the G -flux (candidate) eq. (2.25) was chosen so that theF-theory Standard Model vacua are stable, that is the D3-tadpole can be canceled. Thisrequires n D = χ (cid:0) T b Y (cid:1) − Z b Y G ∧ G ∈ Z ≥ . (2.27)Moreover, the masslessness condition for the U (1) -gauge boson was enforced: ∀ η ∈ H , ( B ) : Z Y G ∧ σ ∧ π ∗ η ! = 0 . (2.28)12ere, σ is the (1 , -form that relates to the so-called Shioda-divisor associated with the U (1) [64, 65]. We now discuss the zero modes in the presence of the flux in Equation (2.25). Asexplained in Section 2.1, we thus look for a lift to H D ( b Y , Z (2)) in the diagramEquation (2.4). For computational simplicity, we aim to parametrize such a lift as A = b γ ( A ) with A ∈ CH ( b Y , Z ) . To describe a candidate, we recall that the cycle map γ : CH ( b Y , Z ) → H , alg ( b Y ) is a ring homomorphism in which the intersection product in CH ∗ ( X , Z ) is compatible with the cup product in H ∗ ( X , C ) . By De Rham’s theoremand the Hodge decomposition, it follows that H k ( X , C ) ∼ = H kDR ( X , C ) = M p + q =2 k H p,q ( X ) . (2.29)The cup product in H ∗ ( X , C ) respects the grading, and restricts to the wedge productof ( p, q ) -forms. For any two divisors V ( r ) and V ( s ) on X , it therefore follows that γ ( V ( r, s )) = γ ( V ( r ) · V ( s )) = [ r ] ∧ [ s ] . This also shows that [ r ] ∧ [ s ] ∈ H , ( b Y ) is in theimage of γ . With this in mind, it is natural to consider A ′ = − · (cid:18) V ( e , e ) − V ( e , t ) − V ( e , t ) − V ( e , t )+ V ( t , t ) − V ( t , u ) + V ( t , v ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) b Y ∈ CH ( b Y , Z ) , (2.30)where t i ∈ H ( X , α ∗ ( K B )) and α : X = B × P F ։ B . Note that γ ( A ′ ) = K B · G .Therefore, the gauge potential A ′ = b γ ( A ′ ) would induce chiral exotics unless we “divide"it by ξ = K B . Hence, we are led to consider gauge potentials A = b γ ( A ) ∈ H D ( b Y , Z (2)) with γ ( A ) = G , ξ · b γ ( A ) ∼ b γ ( A ′ ) . (2.31)Hence, we can infer that the line bundles induced from A = b γ ( A ) are K B -th roots ofthe ones induced from A ′ = b γ ( A ′ ) . We can explicitly compute the latter from eq. (2.30)and eq. (2.6). As an example, let us consider the curve C ( , ) / . For this curve, we find(details in Appendix B) D ( , ) / ( A ′ ) = 3 · V ( t, s , s ) = 3 · K B (cid:12)(cid:12) C ( , )1 / , t ∈ H ( B , K B ) , (2.32)where the last equality follows from the adjunction formula. From this, we concludethat D ( , ) / ( A ) satisfies ξ · D ( , ) / ( A ) = K B · D ( , ) / ( A ) ∼ D ( , ) / ( A ′ ) = 3 · K B (cid:12)(cid:12) C ( , )1 / . (2.33)13he zero modes in the representation ( , ) / are counted by the tensor product of theline bundle associated with D ( , ) / ( A ) and a spin bundle. Let us emphasize again thatwe wish to provide a systematic study of all ξ -th roots of D ( , ) / ( A ′ ) and all the spinbundles on C ( , ) / . To this end, recall the defining property of spin bundles on C ( , ) / : · D spin ( , ) / ∼ K ( , ) / ∼ K B (cid:12)(cid:12) C ( , )1 / . (2.34)Consequently, we notice K B · (cid:16) D ( , ) / ( A ) + D spin ( , ) / (cid:17) ∼ · (cid:16) K B · D ( , ) / ( A ) (cid:17) + K B · (cid:16) · D spin ( , ) / (cid:17) ∼ (cid:16) K B (cid:17) · K B (cid:12)(cid:12) C ( , )1 / . (2.35)Expressed in line bundles, we thus conclude that P ⊗ K B ( , ) / ∼ (cid:18) K B (cid:12)(cid:12) C ( , )1 / (cid:19) ⊗ (cid:16) K B (cid:17) ∼ K ⊗ (cid:16) K B (cid:17) ( , ) / . (2.36)By repeating this computation for the other matter curves, one finds the following rootbundle constraints:curve root bundle constraint C ( , ) / = V ( s , s ) P ⊗ K B ( , ) / = K ⊗ (cid:16) K B (cid:17) ( , ) / C ( , ) − / = V ( s , P H ) P ⊗ K B ( , ) − / = K ⊗ (cid:16) K B (cid:17) ( , ) − / ⊗ O C ( , ) − / ( − · Y ) C ( , ) − / = V ( s , s ) P ⊗ K B ( , ) − / = K ⊗ (cid:16) K B (cid:17) ( , ) − / C ( , ) / = V ( s , P R ) P ⊗ K B ( , ) / = K ⊗ (cid:16) K B (cid:17) ( , ) / ⊗ O C ( , )1 / ( − · Y ) C ( , ) = V ( s , s ) P ⊗ K B ( , ) = K ⊗ (cid:16) K B (cid:17) ( , ) (2.37)In this table, we use P H = s s + s ( s s − s s ) and P R = s s + s ( s s − s s ) . Notethat the line bundles on the Higgs curve C ( , ) − / and the curve C ( , ) / depend on theYukawa points Y = V ( s , s , s ) and Y = V ( s , s , s ) (see Appendix B for details). Itmust also be noted that for two divisors D and E , D ∼ E ⇒ n · D ∼ n · E . (2.38)The converse is not true. This is why we do not cancel common factors. Finally, let uspoint out that the toric base spaces for these F-theory Standard Model constructionsmust satisfy K B ∈ { , , , } [31]. We provide an explicit list of the root bundlesconstraints for these values of K B in Appendix B.3. Inspired by
Ρίζα , which is greek for root, P refers to root bundles throughout this article. . Root bundles from limit roots In the previous section, we explained that root bundles feature prominently in vector-likespectra in F-theory. For the largest currently-known class of globally consistent F-theoryStandard Model constructions without chiral exotics and gauge coupling unification[31], we have worked out these bundle expressions explicitly and summarized them inAppendix B.3. In aiming for MSSM constructions, i.e., vacua without vector-like exotics,the cohomologies of root bundles beg to be investigated. Therefore, our goal is toconstruct roots whose number of global sections is exactly the amount required by thephysical considerations.Before we exhibit an example of this in Section 4, we first summarize well-known factsabout root bundles in general. In particular, we outline an argument for the existence ofsuch bundles on smooth, irreducible curves. From this argument, it can be extrapolatedthat explicit constructions of root bundles on smooth, irreducible curves – not to mentionan explicit count of their sections – are very challenging at best. Fortunately, we canemploy deformation theory to simplify the task. Namely, it is possible to relate rootbundles on smooth, irreducible curves to so-called limit roots on nodal curves. Thisfollows from the detailed study in [43]. For convenience to the reader, we summarize theessential steps in these limit root constructions before we extend these ideas. Namely, weprovide a simple counting procedure for the global sections of many limit root bundles.Our analysis in Section 4 will employ exactly this counting strategy in order to gaininsights into the vector-like spectra of F-theory Standard Models.
Let us look at root bundles on a smooth, complete Riemann surface (or curve) C ofgenus g . We focus on a line bundle L ∈ Pic( C ) and an integer n with n ≥ , and n | deg( L ) . We first recall the following definition. Definition 1 ( n -th root bundle) : An n -th root bundle of L is a line bundle P such that P n ∼ = L . Collectively, we denotethe n -th roots of P by Roots( n, L ) Equivalently, in the language of divisors, an n -th root D of a divisor e D is a divisorsuch that nD ∼ e D . The first important result about root bundles concerns theirexistence. While this seems to be a well-known fact, we were surprised to notice thatwell-established references, such as [66–68], do not give an explicit proof. Not only doesthe proof nicely illustrates the challenge in constructing root bundles on high genuscurves, it also allows us to easily understand why there are n g root bundles and whytheir differences are torsion divisors. For all these reasons, let us give a proof for theexistence of root bundles on smooth, complete Riemann surfaces. Proposition 2:
Let n ∈ Z with n ≥ . For every L ∈ Pic( C ) , there exists an n -th root bundle P of L ifand only if n | deg( L ) . 15 roof For the forward direction, if there exists an n -th root bundle P such that P n ∼ = L ,then n deg( P ) = deg( L ) and n | deg( L ) . Conversely, suppose that n | deg( L ) . Recall that J ( C ) is a complex torus of the form V / Λ , where V is a vector space of dimension g ,and Λ is a discrete subgroup of V of rank g . Denote the n -fold tensor product of linebundles by [ n ] : P nP .First, we will describe some properties of the map [ n ] : J ( C ) → J ( C ) . Observe thatits kernel is given by ker([ n ]) = { P ∈ V /
Λ : nP + Λ = Λ } = ((1 /n )Λ) / Λ ∼ = Λ /n Λ ∼ = ( Z /n Z ) g , (3.1)because Λ is a discrete subgroup of rank g . Hence, ker([ n ]) = [ n ] − (0) is finite, and hasdimension 0. Since J ( C ) is a complete variety, its image [ n ]( J ( C )) is a closed subvariety.For any a ∈ [ n ]( J ( C )) , the translation map t a : [ n ] − (0) → [ n ] − ( a ) , x x + a , (3.2)is an isomorphism. It follows from the dimension formula that n ] − (0) = dim[ n ] − ( a ) ≥ dim( J ( C )) − dim[ n ]( J ( C )) ≥ . (3.3)Hence, [ n ] : J ( C ) → J ( C ) is surjective.Now, consider the following commutative diagram, where deg is the degree map, and ( × n ) is the multiplication of integers by n . J ( C ) Pic( C ) Z J ( C ) Pic( C ) Z n ] [ n ] × n (3.4)Applying the Snake Lemma yields an exact sequence of the cokernels of the verticalmaps, i.e. J ( C ) / [ n ]( J ( C )) → Pic( C ) / [ n ](Pic( C )) ∼ = −→ Z /n Z → , (3.5)where the isomorphism is provided by the degree map. Since n | deg( L ) , we have that L ∈ [ n ](Pic( C )) . So, there exists P ∈ Pic( C ) such that P n ∼ = L, i.e. an n -th rootbundle. (cid:4) There are two important lessons that we can learn from this proposition. First, thereare n g n -th root bundles P of L . This is because Equation (3.2) is an isomorphism, andso, t deg( L ) : ( Z /n Z ) g ∼ = ker([ n ]) = [ n ] − (0) → [ n ] − (deg( L )) = Roots( n, L ) , (3.6)16s an isomorphism as well. If L = K C , then the n -th roots are called n -spin bundles [69].The second lesson concerns the difference between two root bundles. Any two n -throot bundles differ by an n -torsion line bundle, i.e. a line bundle M ∈ J ( C ) such that M n ∼ = O C .The Jacobian and the linear equivalence of divisors is well-understood for ellipticcurves E (see [66–68] for background). This allows us to exhibit examples of the abovenotions fairly explicitly. First, recall that E ∼ = J ( E ) ∼ = C / Λ and Λ = Z ⊕ Z · τ , where τ ∈ C is the complex structure modulus of the elliptic curve. We denote D ∈ Div( E ) by D = n X i =1 n i · ( p i ) , n i ∈ Z , p i ∈ E , (3.7)i.e. we place the points p i ∈ E in round brackets for notational clarity. Note that deg( D ) = P ni =1 n i and that a zero degree divisor satisfies D ∼ ⇔ n X i =1 n i · p i ! ∈ Λ . (3.8)From this, we can work out the divisor classes of the 2-torsion divisors in J ( E ) : D = [0] , D = (cid:20) − · (0) + 1 · (cid:18) (cid:19)(cid:21) , (3.9) D = h − · (0) + 1 · (cid:16) τ (cid:17)i , D = (cid:20) − · (0) + 1 · (cid:18) τ (cid:19)(cid:21) . (3.10)It follows ker([2]) ∼ = ( Z / Z ) , which we can intuitively collect in the following picture: ℑ ( z ) ℜ ( z ) D D D D τ (3.11)Note that { D , D , D , D } are exactly the four spin structures on E .In making contact with our physics applications, we should next investigate the sheafcohomologies of root bundles. Generally speaking, this is a very challenging task.However, on elliptic curves, the situation simplifies and we can achieve a completeclassification. Recall that any line bundle L ∈ Pic( E ) with deg( L ) = 0 is in the Kodairastable regime, i.e. we can infer its cohomologies from its degree. For deg( L ) = 0 we have h ( E, L ) = 1 ⇔ L ∼ = O E . (3.12)Therefore, it merely remains to study the cohomologies of roots P of a line bundle L ∈ Pic( E ) with deg( L ) = 0 . This is achieved by the following proposition.17 roposition 3: Let L ∈ Pic( E ) with deg( L ) = 0 and consider an integer n with n ≥ . Then,(i) L ∼ = O E : Exactly one n -th root P of L has h ( E, P ) = 1 , and the remaining n -throots Q have h ( E, Q ) = 0 .(ii) L = O E : All n -th roots P of L have h ( E, P ) = 0 . Proof (i) Since O E is an n -th root of itself, there is one n -th root P = O E with h ( E, P ) = 1 .Any other n -th root Q differs from P by a non-trivial n -torsion line bundle. Assuch, Q is non-trivial and has h ( E, Q ) = 0 .(ii) L is non-trivial. Hence, all n -th roots P are non-trivial and have h ( E, P ) = 0 . (cid:4) For example, among the P of a line bundle L ∈ Pic( E ) with deg( L ) = 0 ,at least have h ( E, P ) = 0 . We will make use of this simple result in Section 4.2.
For applications in F-theory, we wish to generalize Proposition 3 to matter curves C R with g > . Unfortunately on such curves, it is very hard to tell if a divisor is linearlyequivalent to zero. This is due to the current lack of practical understanding of theAbel-Jacobi map Div ( C R ) → J ( C R ) , whose kernel is exactly given by the (classesof) trivial divisors. This in turn makes it very challenging to identify n -torsion bundles,which forms a measure- subset of the Jacobian J ( C R ) . Consequently, it becomes almostimpossible to explicitly identify a single n -th root bundle P of a line bundle L on C R .To overcome this hurdle, we wonder if it is possible to simplify the matter curves C R .However, recall that the geometry of the matter curves is dictated by that of the ellipticfibration b π : b Y ։ B . Therefore, even though special, non-generic elliptic fibrations b Y may contain matter curves C R with simple geometries, it can be expected that suchfibrations lead to physically unwanted gauge enhancements.Therefore, in order to remain on physically solid grounds, we stick to the geometryof the matter curves C R as enforced by the generic fibration b Y . In this situation,there is still a way to improve our situation. Namely, suppose that ϕ : C simple R → C R is a deformation of a curve C simple R , whose simple geometry allows easy access to rootbundles and their cohomologies, into the actual physical matter curve C R ⊂ b Y . Then,we can wonder if the root bundles P simple R on C simple R approximate the roots P R on C R .In general, this sort of question leads to a deep discussion of deformation theory (seee.g. [70,71] for a modern exposition). In this work, we will not attempt to give a completeanswer. Rather, we make a special choice for C simple R . Inspired by [43], we focus on curves C simple R with singularities, which locally look like { x · y = 0 } , i.e. are nodes. On suchnodal curves C • R , roots P • R admit a description in terms of weighted diagrams [43]. Evenmore, there are exactly as many roots P • R as there are roots P R and we can, at least intheory, identify them with each other by tracing them along the deformation C • R → C R .18hat said, the next question is in what sense we can use the roots P • R on C • R toapproximate the cohomologies of the roots P R on the physical matter curve C R . To thisend, we first recall that refined section counting mechanisms exist for line bundles onsingular curves [38]. In exactly this spirit, we are able to extend the ideas from [43]. InSection 3.3 we will argue that it is often possible to count the number of global sectionsof roots P • R on a nodal curve C • R from simple combinatorics.It now remains to relate the cohomologies h i ( C • R , P • R ) to h i ( C R , P R ) . Since, the chiralindex is fixed from topology, it suffices to study how h ( C • R , P • R ) relates to h ( C R , P R ) .Since C • R is singular (and therefore non-generic) and C R expected to be smooth, atendency is known. This tendency goes by the name upper semi-continuity. It meansthat the number of global sections of P R must not increase when traced along C • R → C R to the root P R , i.e. h ( C R , P R ) ≤ h ( C • R , P • R ) . (3.13)It is a very interesting but also very challenging question to distinguish the roots P • R thatlose sections along C • R → C R from the roots with a constant number of sections. Whilewe hope to return to this question in the future, the physics applications in Section 4focus on a subset of roots, which do not lose sections. Namely, if χ ( P • R ) ≥ and h ( C • R , P • R ) = χ ( P • R ) , (3.14)then P • R cannot lose sections, since its numbers of sections is already minimal. Inparticular, it then holds h ( C R , P R ) = h ( C • R , P • R ) . In Proposition 2 we saw that n -th roots P of a line bundle L on a smooth curve C existif deg( L ) is divisible by n . This is not the case for reducible, nodal curves C • . Indeed,a root P • of a line bundle L • on such curves should restrict to a root on the irreduciblecomponents. However, even if n divides the degree of L • , it may not divide deg( L | Z ) forsome irreducible component Z of C • . This is elegantly circumvented by passing to limit n -th root bundles P ◦ on (partial) blow-ups C ◦ of C • , as originally introduced in [43].Just as every nodal curve C • can be described through its dual graph, these limit n -throot bundles P ◦ are determined by weighted graphs. This combinatorial data can beexploited to make the task of section-counting more tractable. For convenience to thereader, let us outline the important steps in these constructions before we explain thesection counting for limit roots. For more material on limit roots, we refer the readerto [72, 73] in which the pushforwards of these limits roots along the blow-up map, andtheir moduli are extensively studied. A point is a node if it has a neighborhood where the curve locally looks like { xy = 0 } in C . A nodal curve is a complete algebraic curve such that every point is either smooth,19r a node. Let C • be a connected (possibly reducible) nodal curve of arithmetic genus g . We associate to C • a dual graph Π C • in which(i) every vertex corresponds to an irreducible component C • i of C • ,(ii) every half-edge emanating from a vertex C • i is a node on C • i .If a node lies on both C • i and C • j , then the half-edges exiting from C • i and C • j jointogether to form an edge. For example, consider the Holiday lights – a nodal curve H • with 11 components given by: • a rational curve Γ with genus g (Γ) = 0 ,• 10 elliptic curves E , . . . , E with genus g ( E i ) = 1 .Each elliptic curve intersects no other curve except Γ , hence the name Holiday lights .Its dual graph can be visualized as follows: Γ E E E E E E E E E E (3.15)Each elliptic curve E i is represented by a green vertex, while the rational curve Γ isrepresented by the pink vertex.If π : C ◦ → C • is a blow-up of C • , then for every node n i ∈ C • , we denote theexceptional components by π − ( n i ) = E i ∼ = P . (3.16)Set C N = C ◦ \ ∪ i E i . Then, π | C N : C N → C • is the normalization of C • . For every node n i , the points in ( π | C N ) − ( n i ) = E i ∩ C N = { p i , q i } are called the exceptional nodes .From this point forward, we will consider blow-ups of C • on the full set of nodesunless stated otherwise. We will often refer to this setup as a full blow-up . More generalstatements exist for partial blow-ups, the details of which are fully treated in [43]. In all base spaces B of the globally consistent F-theory Standard Model constructions discussed inSection 2.2 and originally introduced in [31], the matter curves C R are contained in K3-surfaces.Motivated from [45], it stands to wonder if the matter curves C R admit a deformation into such a Holiday lights . Even more,
Holiday lights allow easy access to Brill-Noether theory of limit roots aswe will see momentarily. As such, they are very favorable nodal curves for our study. We hope toreturn to this question in the future. .3.2. Limit n -th roots and weighted graphs Let n be a positive integer, and L • be a line bundle on C • so that n | deg( L • ) . Denotethe full set of nodes by ∆ C • . Definition 4: A limit n -th root of L • associated to ∆ C • is a triple ( C ◦ , P ◦ , α ) consisting of:• the (full) blow-up π : C ◦ → C • ,• a line bundle P ◦ on C ◦ ,• a homomorphism α : ( P ◦ ) n → π ∗ ( L • ) ,satisfying the following properties:(i) deg( P ◦ | E i ) = 1 for every exceptional component E i ,(ii) α is an isomorphism at all points of C ◦ outside of the exceptional components.(iii) for every exceptional component E i of C ◦ , the orders of vanishing of α at theexceptional nodes p i and q i add up to n .We can also define limit n -th roots associated to a subset ∆ ⊆ ∆ C • in which the fullblow-up is replaced by the partial blow-up at ∆ , see [43] for details.Limit n -th roots over C • carry some combinatorial data, in the form of weightedgraphs, that takes into account the combinatorial aspects of the nodal curve C • .Conversely, these weighted graphs allow one to construct and recover limit n -th roots.Although the correspondence between limit n -th roots and these weighted graphs arenot one-to-one, it allows for a convenient parametrization of limit roots. First, let usintroduce the weighted graphs in question. Let e ∆ C • be the exceptional nodescorresponding to ∆ C • . Definition 5: A weighted graph associated to a limit n -th root ( C ◦ , P ◦ , α ) of L • is the dual graph Π C • endowed with weights assigned by the weight function w : e ∆ C • → { , . . . , n − } , (3.17)where w ( p i ) = u i and w ( q i ) = v i are the orders of the vanishing of α at p i and q i respectively.Such weighted graphs naturally satisfy two conditions:(A) w ( p i ) + w ( q i ) = u i + v i = n, (B) For every irreducible component C • i of C • , the sum of all weights assigned to thevertex corresponding to C • i is congruent to deg C • i L • (mod n ) .21e illustrate an example of a weighted graph by returning to the Holiday lights H • .We wish to find the limit 3rd roots of K H • . If C • i is a component of H • , then set k i = C • i ∩ ( H • \ C • i ) . Therefore, deg( K H • | C • i ) = 2 g ( C • i ) − k i , and the multi-degreeof K H • is (deg( K H • | Γ ) , deg( K H • | E ) , .., deg( K H • | E ) ) = (8 , , . . . , , (3.18)which has total degree is g ( H • ) − . So, the multi-degree of K H • is (16 , , . . . , . A weighted graph associated to the limit 3rd roots of K H • , as well as the multi-degreesof K H • , is given below. The labels inside the vertices are the multi-degrees of K H • , whilethe labels outside the vertices are the weights.2 1 212 1 212 1 212 1 21 212 1 (3.19)Given a weighted graph satisfying conditions A and B, we have a recipe for constructinglimit n -th roots of L • . Proposition 6:
Every weighted graph, whose underlying graph is Π C • , and whose weight function w : e ∆ C • → { , . . . , n − } satisfies conditions A and B, encodes a limit n -th root ( C ◦ , P ◦ , α ) of L . Moreover, this weighted graph coincides with the weighted graph associated to ( C ◦ , P ◦ , α ) of L • . Proof
Suppose we have a weighted graph satisfying the hypothesis of the proposition, andlet π | C N : C N → C • be the normalization of C • . Thanks to condition B, the line bundle ( π | C N ) ∗ ( L • ) − X p i ,q i ∈ e ∆ ( u i p i + v i q i ) (3.20)has on each irreducible component of C N degree divisible by n . Thus, on each irreduciblecomponent of C N it admits an n -th root. The collection formed from an n -th root oneach irreducible component is a line bundle P N ∈ Pic( C N ) . Let π : C ◦ → C • be thefull blow-up. Over each exceptional component, glue a degree one line bundle to P N to22btain a line bundle P ◦ ∈ Pic( C ◦ ) . Finally, define α : ( P ◦ ) n → π ∗ ( L • ) to be zero on theexceptional components, and α | C N : ( P N ) n = ( π | C N ) ∗ ( L • ) − X p i ,q i ∈ e ∆ ( u i p i + v i q i ) ֒ → ( π | C N ) ∗ ( L • ) (3.21)on C N . Then, ( C ◦ , P ◦ , α ) is the desired limit n -th root of L • associated to ∆ C • . (cid:4) The same statements follow when ∆ C • is replaced by a subset ∆ . In this case, thelimit roots associated to ∆ give rise to weighted subgraphs satisfying conditions A andB. Using the same procedure from Proposition 6, we can construct a limit root from aweighted subgraph.Every nodal curve C • and line bundle L • on C • has a total of n b (Π C • ) weightedsubgraphs satisfying conditions A and B, where b (Π C • ) = edges + connected components − vertices , (3.22)is the first Betti number of Π C • . We emphasize that this counts all of the weightedsubgraphs, whose edge sets coincide with subsets of ∆ C • . Curves, whose dual graphs aretrees, will have zero b , and thus, will only have one weighted graph. These curves aresaid to be of compact type. This is certainly the case for the Holiday lights H • . Here, b (Π H • ) = 0 and the weighted graph depicted in eq. (3.19) is the only possible weightedgraph for the 3rd limit roots of K H • .The correspondence between limit n -th roots and weighted graphs satisfying conditions A and B is not one-to-one. Indeed, the construction detailed in Proposition 6 involves achoice of a root P N of ( π | C N ) ∗ ( L • )( − P ( u i p i + v i q i )) . A careful count reveals that thereare n g limit n -th roots [43].We will apply the limit root construction in Proposition 6 to describe the limit 3rdroots of K H • on the Holiday lights H • . We proceed as follows:1. Blow-up all nodal singularities, and denote the exceptional component at the i -thnode by E i ∼ = P . This P touches E i at the exceptional node p i and Γ at q i .2. Let H N be the (full) normalization of H • , and consider the bundle ( π | H N ) ∗ ( K H • ) − X i =1 p i − X i =1 q i ! , (3.23)which has multi-degree (16 − , − , . . . , −
2) = (6 , , . . . , . This bundleadmits 3rd roots on H N , namely g ( E i ) = 9 roots on each elliptic curve E i and g (Γ) = 1 root on Γ . Hence, there are = 3 roots, and each has multi-degree (2 , , . . . , .3. Pick a 3rd root P N , and glue to it a degree one bundle over every E i . The resulting limit 3rd root P ◦ of K H • has multi-degree (2 , , . . . , , , . . . , over H ◦ , where deg( P ◦ | C • i ) = deg( P N | C • i ) , deg( P ◦ | E i ) = 1 . (3.24)23hese limit roots can be represented as follows:
200 0 0 000000 11 1 1 111111 (3.25)As before, the green vertices represent the elliptic curves E i , and the pink vertexrepresents Γ . The blue vertices represent the exceptional component E i , whichintersects E i and Γ . The multi-degrees of the limit root P ◦ is written inside thevertices. In particular, P ◦ restricts to a degree 1 line bundle over each exceptionalcomponent. Of ample importance for our analysis is the number of global sections of the limit roots.These arise from gluing sections on the irreducible components of the nodal curve acrossexceptional divisors, which is addressed in the next lemma.
Lemma 7:
Let p , p be two distinct points on P , and a , a ∈ C . For every p ∈ P \ { p , p } , thereexists a unique section s ∈ H ( P , O P ( p )) such that s ( p ) = a and s ( p ) = a . Proof
Endow P with its standard open cover { U , U } . Let z ∈ U and w ∈ U be localcoordinates so that w = z in U ∩ U . Since
PGL(2) acts transitively on P , we mayassume that p = 0 , p = 1 , p = ∞ , (3.26)without loss of generality. The desired section s is given by s | U ( z ) = ( a − a ) z + a , s | U = ( a − a ) + a w . (3.27)It remains to show uniqueness. Recall that every section t ∈ Γ( P , O P ( p )) is given bytwo analytic functions t = t | U ∈ Γ( U , O P ( p )) and t = t | U ∈ Γ( U , O P ( p )) suchthat over U ∩ U , t ( z ) = zt ( w ) = zt (1 /z ) . (3.28)24f t ( z ) = P k ≥ α k z k and t ( w ) = P k ≥ β k w k , the above implies that X k ≥ α k z k = z X k ≥ β k w k ! = z X k ≥ β k z − k ! = X k ≥ β k z − k . (3.29)It follows that α k = β k = 0 for k > , which leaves α = β and α = β . As such, everysection t is given by t ( z ) = α z + α and t = α + α w. If t also satisfies t (0) = a and t (1) = a , then within the chart U containing , ∈ P ,a = t (0) = α , a = t (1) = α + α = α + a . (3.30)Thus, t ( z ) = ( a − a ) z + a and t ( w ) = ( a − a ) + a w , which coincides with s . (cid:4) By virtue of the above lemma, there is a unique way of gluing a local section over anexceptional component to local sections over the irreducible components at each end.This leads us to the following corollary.
Corollary 8:
Let C • be a connected nodal curve with irreducible components C • , . . . , C • k . Let L • bea line bundle on C • , and n be an integer with n ≥ and n | deg( L • ) . For any limit n -throot ( C ◦ , P ◦ , α ) of L • , h ( C ◦ , P ◦ ) = k X i =1 h ( C • i , P ◦ | C • i ) . (3.31) Proof
Let two irreducible components C • i and C • j intersect an exceptional component E ∼ = P at p i ∈ C • i and p j ∈ C • j respectively. Set Y = C • i ∪ E ∪ C • j . Then, we have h ( Y, P ◦ ) ≥ h ( C • i , P ◦ | C • i ) + h ( C • j , P ◦ | C • j ) + h ( E , P ◦ | E ) − h ( C • i ∩ E , P ◦ | C • i ∩E ) − h ( C • j ∩ E , P ◦ | C • j ∩E ) ≥ h ( C • i , P ◦ | C • i ) + h ( C j , P ◦ | C • j ) + 2 − − ≥ h ( C • i , P ◦ | C • i ) + h ( C • j , P ◦ | C • j ) . (3.32)It remains to prove equality. Recall that the number of independent conditions met at p i and p j is at most 2 – the number of intersection points on E . The previous lemmashowed that there are exactly two independent conditions; one at each p i and p j . Thus,equality holds. Since any two irreducible components of C • either intersect a commonexceptional component, or they do not in the full blow-up, the result follows. (cid:4) Let us apply these results to the
Holiday lights H • , and count the global sections of thelimit 3rd roots of K H • . Recall that H • is the union of a rational curve Γ , and 10 ellipticcurves E i . Also, the limit 3rd root P ◦ of K H • has multi-degree (2 , , . . . , , , . . . , .Since Γ is rational, h (Γ , P ◦ | Γ ) = h ( P , O P (2)) = 3 . By the above results, we have h ( H ◦ , P ◦ ) = h (Γ , P ◦ | Γ ) + X i =1 h ( E i , P ◦ | E i ) = 3 + (cid:26) (cid:27) + · · · + (cid:26) (cid:27) (3.33)25he last term in the above expression means or . This refers to the two cases describedin Proposition 3 in which P ◦ | E i is either non-trivial or trivial.This example highlights the general fact that a line bundle of degree d over a smoothcurve can have different h ’s. Since counting the global sections of a limit root isequivalent to counting its local sections over the smooth irreducible components, weaddress the effect of this phenomenon on section-counting in the following corollary. Corollary 9:
Let C • be a connected nodal curve with irreducible components C • , . . . , C • k . Let L • bea line bundle on C • , and n be an integer with n ≥ and n | deg( L • ) . For any limit n -throot ( C ◦ , P ◦ , α ) of L • , k X i =1 min h ( C • i , P ◦ | C • i ) ≤ h ( C ◦ , P ◦ ) ≤ k X i =1 max h ( C • i , P ◦ | C • i ) , (3.34)where for each i , the minimum and maximum are taken over all line bundles of degree deg( P ◦ | C • i ) over C • i .In the example of the Holiday lights H • , min P ◦ | Γ ∈ Pic (Γ) h (Γ , P ◦ | Γ ) = 3 , max P ◦ | Γ ∈ Pic (Γ) h (Γ , P ◦ | Γ ) = 3 , (3.35) min P ◦ | Ei ∈ J ( E i ) h ( E i , P ◦ | E i ) = 0 , max P ◦ | Ei ∈ J ( E i ) h ( E i , P ◦ | E i ) = 1 , (3.36)for i = 1 , ..., . Hence, ≤ h ( H ◦ , P ◦ ) ≤ . Let
Roots( n, L • ) ◦ be the set of limit n -th roots of L • on C • . In a broader sense, wewish to understand the map, h ( C ◦ , · ) : Roots( n, L • ) ◦ → N ∪ { } , P ◦ h ( C ◦ , P ◦ ) , (3.37)For curves of compact type, every limit root comes from one weighted graph, and isconstructed over the full blow-up. In this case, the global sections of the limit root arefully determined by the local sections over the irreducible components. Hence, we cancompute | h ( C ◦ , · ) − ( a ) | for every a ∈ N ∪ { } , i.e. the number of limit n -th roots with h = a . We illustrate this with the Holiday lights , which is a curve of compact type.Denote the number of elliptic curves on which P ◦ | E i is non-trivial by N i . Then, thenumber N P ◦ ( h ) of limit 3rd roots with specific h are as follows:26 i
10 9 8 7 6 5 4 3 2 1 0 N P ◦ (3) 1 (cid:0) (cid:1) · (cid:0) (cid:1) · (cid:0) (cid:1) · (cid:0) (cid:1) · (cid:0) (cid:1) · (cid:0) (cid:1) · (cid:0) (cid:1) · (cid:0) (cid:1) · (cid:0) (cid:1) · (cid:0) (cid:1) · N P ◦ (4) 1 (cid:0) (cid:1) · (cid:0) (cid:1) · (cid:0) (cid:1) · (cid:0) (cid:1) · (cid:0) (cid:1) · (cid:0) (cid:1) · (cid:0) (cid:1) · (cid:0) (cid:1) · (cid:0) (cid:1) · N P ◦ (5) 1 (cid:0) (cid:1) · (cid:0) (cid:1) · (cid:0) (cid:1) · (cid:0) (cid:1) · (cid:0) (cid:1) · (cid:0) (cid:1) · (cid:0) (cid:1) · (cid:0) (cid:1) · N P ◦ (6) 1 (cid:0) (cid:1) · (cid:0) (cid:1) · (cid:0) (cid:1) · (cid:0) (cid:1) · (cid:0) (cid:1) · (cid:0) (cid:1) · (cid:0) (cid:1) · N P ◦ (7) 1 (cid:0) (cid:1) · (cid:0) (cid:1) · (cid:0) (cid:1) · (cid:0) (cid:1) · (cid:0) (cid:1) · (cid:0) (cid:1) · N P ◦ (8) 1 (cid:0) (cid:1) · (cid:0) (cid:1) · (cid:0) (cid:1) · (cid:0) (cid:1) · (cid:0) (cid:1) · N P ◦ (9) 1 (cid:0) (cid:1) · (cid:0) (cid:1) · (cid:0) (cid:1) · (cid:0) (cid:1) · N P ◦ (10) 1 (cid:0) (cid:1) · (cid:0) (cid:1) · (cid:0) (cid:1) · N P ◦ (11) 1 (cid:0) (cid:1) · (cid:0) (cid:1) · N P ◦ (12) 1 (cid:0) (cid:1) · N P ◦ (13) 1 Factor This table says that for N i = 10 , we find N P ◦ (3) = 1 · limit 3rd roots P ◦ with h = 3 .Similarly, for N i = 4 , we find N P ◦ (3) = (cid:0) (cid:1) · · limit 3rd roots P ◦ with h = 6 . Forease of presentation, the overall factors are collected at the bottom of this table.We would like to generalize this section-counting of limit roots for all curves, whichmay have multiple weighted subgraphs. Complications arise when counting the globalsections of limits roots over partial blow-ups; namely, it is unclear what h of a limit rootis over a singular component of the curve, i.e., the component containing a singularitythat has not been blown up. Although we will not discuss this direction in this paper,it presents an interesting problem which we hope to revisit in the future.
4. Limit root applications in F-theory
After the detailed exposition of root bundles and limit roots in the previous section,we now wish to apply these techniques to F-theory. We first outline how limit rootscan be used to provide an explicit and oftentimes constructive argument for the absenceof certain vector-like exotics. We demonstrate these ideas in one particular geometryamong the largest class of currently-known globally consistent F-theory Standard Modelswithout chiral exotics and gauge coupling unification [31]. We will argue that there aresolutions without vector-like exotics in the representations C ( , ) / , C ( , ) − / , C ( , ) / and C ( , ) . Let us look at an F-theory compactification to 4-dimensions on a space Y , which admitsa smooth, flat, crepant resolution b Y . As explained in Section 2.1, root bundles appearnaturally in such settings when studying vector-like spectra. We found that the geometrydetermines a class A ′ = b γ ( A ′ ) ∈ H D ( b Y , Z (2)) for some A ′ ∈ CH ( b Y , Z ) , and an integer ξ ∈ Z > such that A is subject to the two constraints: γ ( A ) = G , ξ · b γ ( A ) ∼ b γ ( A ′ ) . (4.1)27he condition γ ( A ) = G immediately follows from eq. (2.4) and it means that A = b γ ( A ) is an F-theory gauge potential for the given G -flux. The second condition ensures theabsence of chiral exotics in the F-theory Standard Models [31]. It follows that the linebundle on the matter curve C R satisfies P R = O C R ( D R ( A )) ⊗ O C R O C R (cid:16) D spin C R (cid:17) , (4.2)where D R ( A ) and D spin C R are solutions to the root bundle constraints ξ · D R ( A ) ∼ D R ( A ′ ) , · D spin C R ∼ K R . (4.3)Recall from Section 3 that these root bundle constraints have many solutions. In general,it cannot be expected that all solutions are realized from roots in H D ( b Y , Z (2)) andspin c -structures on the gauge surfaces. We reserve a detailed study of this interestingand challenging question for future works. In this article, we study all the ξ -th roots of D R ( A ′ ) and all of the spin divisors D spin C R systematically. Our goal is to identify roots P R subject to the physical demand of absence/presence of vector-like pairs. In future works,we hope to identify which of these desired roots stem from F-theory gauge potentials in H D ( b Y , Z (2)) .At special loci of the complex structure moduli space, massive vector-like pairs canbe rendered massless. Mathematically, this is reflected in the fact that deformations ofa line bundle can have higher cohomologies. For example, if we assume χ ( P R ) ≥ , thenwe could have:Geometry of curves C R ( h ( C R , P R ) , h ( C R , P R )) Generic ( χ ( P R ) , Less generic ( χ ( P R ) + 1 , ≡ ( χ ( P R ) , ⊕ (1 , Even less generic ( χ ( P R ) + 2 , ≡ ( χ ( P R ) , ⊕ (2 , ... ...In [38], such cohomology jumps have been analyzed in large detail. In particular, it wasexplained that even on generic curves, line bundles with the same chiral index need nothave the same cohomologies. This classic observation goes by the name of Brill-Noethertheory [40] (see also [42] for another application of Brill-Noether theory to F-theory).This observation in particular applies to root bundles. In Section 3.1, we have explainedthat of the four spin structures on an elliptic curve, one has h ( E, O spin E ) = 1 and theother three have vanishing number of global sections. This is a special instance of theresults in [58,59], which show that all odd spin structures have odd number of zero modes,while the remaining even spin structures have even number of zero modes. Generallyspeaking, different roots P R will have different numbers of zero modes.That said, our task is to construct root bundles P R with the cohomologies that arephysically desired. For simplicity, let us assume χ ( P R ) ≥ . Inspired by physics, weshould then distinguish the generic case h ( P R ) = χ ( P R ) and the non-generic case h ( P R ) > χ ( P R ) . The former corresponds to the absence of exotic vector-like pairs,28hile the latter most prominently features on the Higgs curve in F-theory StandardModel constructions. In the latter case, for MSSM constructions, one wishes to achieve h ( P R ) = χ ( P R ) + 1 so that the additional vector-like pair describes a Higgs field.We approach the task of constructing such physically desired root bundles P R by firstconsidering a deformation C R → C • R , where C • R is a nodal curve. Therefore, P R → P • R becomes a root bundle on the nodal curve C • R . We focus on roots P • R , which we candescribe by limit roots P ◦ R on the full blow-up C ◦ R of C • R . For those limit roots, wecan employ the technology described in Section 3 in order to identify h ( C ◦ R , P ◦ R ) . Thisenables us to identify roots P ◦ R with h ( C ◦ R , P ◦ R ) = χ ( P ◦ ) + δ from simple combinatorics,where δ ∈ Z ≥ is the physically desired offset.The pushforward of limit roots P ◦ R along the blow-up map π : C ◦ R → C • R preservesthe number of global sections, i.e. h ( C ◦ R , P ◦ R ) = h ( C • R , P • R ) . We have thus identifiedthe roots on C • which have the physically desirable cohomologies. In theory, we cantrace those roots P • R along the deformation C • R → C R to find roots P R on the originalcurve C R . Crucially though, such a deformation can change the number of sections(see e.g. [38]). For the deformation C • R → C R , which turns a nodal (i.e. singular andthus non-generic) curve into a smooth, irreducible curve, it is known that the number ofsections is an upper semi-continuous function. This means that the number of sectionseither remains constant or decreases as we trace P • R to P R on C R : h ( C R , P R ) ≤ h ( C • R , P • R ) = χ ( P R ) + δ . (4.4)The natural question is thus to look for roots P • R for which equality holds. This happensin the generic case, i.e. the case δ = 0 . This is because the number of sections is thenalready minimal on C • R and thus, it must remain constant along the deformation to C • R : h ( C R , P R ) = h ( C • R , P • R ) = χ ( P R ) . (4.5)The upshot of this strategy, which we summarize in Figure 1, is that we can providea lower bound to the number of roots P R without vector-like exotics by studying thecombinatorics of limit roots on the full blow-up C ◦ R of the nodal curve C • R . In aiming for F-theory MSSMs, the non-generic case δ = 1 is also fairly importantfor the Higgs curve. While it is not hard to construct limit roots on C ◦ ( , ) − / withexactly 4 sections, the corresponding roots P • ( , ) − / satisfy h ( C • ( , ) − / , P • ( , ) − / ) = 4 ,which is larger than the minimal value χ ( P ( , ) − / ) = 3 . Since the number of sectionsis non-minimal, we cannot conclude from upper semi-continuity that the number ofsections remains constant. Rather, we expect some of those roots P • ( , ) − / to lose asection when traced to C ( , ) − / . Currently, we do not know a sufficient discriminatingproperty that allows us to identify the roots P • ( , ) − / for which the number of sectionsremains constant. We reserve this interesting mathematical question for future work. Recall that at least one of these roots stems from an F-theory gauge potential in CH ( b Y , Z ) . eformationUpper SC h remains 3Matter curve C R Nodal curve C • R P R Limit rootsPushforward h ( P ◦ ) = h ( P • ) Blow-up curve C ◦ R P • R P ◦ R Figure 1.: Roots P R with h ( C R , P R ) = 3 from roots P • R on a nodal curve C • R and limit roots P ◦ R on its blow-up C ◦ R . We now continue the analysis initiated in Section 2.2, where we summarized the geometryof the largest currently-known class of globally consistent F-theory Standard Modelswithout chiral exotics and gauge coupling unification [31]. The chiral index on all fivematter curves C ( , ) / = V ( s , s ) , C ( , ) − / = V (cid:0) s , s s + s ( s s − s s ) (cid:1) , (4.6) C ( , ) − / = V ( s , s ) , C ( , ) / = V (cid:0) s , s s + s ( s s − s s ) (cid:1) , (4.7) C ( , ) = V ( s , s ) , (4.8)is thus exactly three. We worked out the root bundle constraints (c.f. Appendix B.3).In aiming for an MSSM construction, which comes with exactly one Higgs pair, thevector-like spectrum is subject to the demand h ( C ( , ) − / , P ( , ) − / ) . As explainedabove, since χ ( P ( , ) − / ) , our current technology does not allow us to tend tothis case. However, we can address the absence of vector-like exotics on the remainingmatter curves in a constructive way. That is, we can construct solutions to the constraint h ( C ( , ) / , P ( , ) / ) = h ( C ( , ) − / , P ( , ) − / )= h ( C ( , ) / , P ( , ) / ) = h ( C ( , ) , P ( , ) ) . (4.9)To outline these steps, let us first look at the quark-doublet curve C ( , ) / = V ( s , s ) ,where s , s are generic sections of K B . To make our construction explicit, let us focuson base spaces B with K B = 18 . It then follows from Appendix B.3 that we aretrying to argue for the existence of root bundles that satisfy P ⊗ , ) / ∼ K ⊗ , ) / , h ( C ( , ) / , P ( , ) / ) = 3 . (4.10)For this, it suffices to argue that root bundles with the following properties exist P ⊗ , ) / ∼ K ⊗ , ) / , h ( C ( , / ) , P ( , ) / ) = 3 . (4.11)We achieve a proof of existence by studying a deformation C ( , ) / → C • ( , ) / . Letus work with a concrete base geometry, we opt for the toric base space B = P with K B = 18 , whose details are summarized in Appendix B.4.30o describe the deformation C ( , ) / → C • ( , ) / , we first notice that s is a polynomialin the homogeneous coordinates { x i } ≤ i ≤ of P . Since s is a section of K P , itcontains the monomial Q i =1 x i . This allows us to consider the deformation V ( s , s ) = C ( , ) / → C • ( , ) / = V Y i =1 x i , s ! . (4.12)Since we assume generic s , C • ( , ) / is manifestly nodal in the K3-surface V ( s ) and thetechniques of Section 3 apply. To this end, we first identify the dual graph of C • ( , ) / ,which has irreducible components:curve equation genus deg (cid:18) · K C • ( , )1 / (cid:12)(cid:12)(cid:12) C i (cid:19) C V ( x , s ) 1 6 C V ( x , s ) 1 6 C V ( x , s ) 0 12 C V ( x , s ) 0 12 C V ( x , s ) 0 0 n C ( i )8 o ≤ i ≤ V ( x , x − α i x ) 0 0 n C ( i )10 o ≤ i ≤ V ( x , x − α i x ) 0 0 (4.13)For convenience, we list the degree of · K C • ( , )1 / on all irreducible components sinceEquation (4.11) instructs us to construct third roots of this bundle. By taking theStanley-Reisner ideal of P into account (see Appendix B.4), one finds the dual graphof C • ( , ) / : C C C C C C ( i )8 C ( i )10 (4.14)We mark the P s in pink and the elliptic curves in green. This diagram is easily extendedto a weighted diagram, which encodes a 3rd root of · K C • ( , )1 / . This involves placingweights w i ∈ { , } subject to the following two rules (cf. Section 3.3): For any toric base space B with homogeneous coordinates x i , Q i x i is a section of K B ∼ P i [ x i ] .
31. Along each edge: The sum of weights is .2. At each node: The sum of weights equals the degree in Equation (4.13) modulo .It is readily verified that the following weighted diagram satisfies these rules:1 22 1 2121 21 122 12 12 12 12 12 12 12 12 1212121 12 12 12 12 121 2
12 12606000000 0 0 0 0 0 0 (4.15)We then study the limit roots P ◦ ( , ) / on the full blow-up C ◦ ( , ) / of C • ( , ) / , which areencoded by this diagram. The degree of each such limit root P ◦ ( , ) / is as follows: − C C − C − − − − − − − − − − − − (4.16)Note that we denote the blow-up P s in blue and that, by construction of the limit roots,we consider a degree line bundle on each of these. It follows that the (total) degree32f each such limit root P ◦ ( , ) / is . This is expected from Equation (4.11) since itis equivalent to χ ( P ◦ ( , ) / ) = 3 . Here, we claim even more, namely that some of theselimit roots have exactly three global sections.To see this, recall from Section 3.4 that the number of global sections of a limitroot P ◦ ( , ) / is simply given by the sum of the sections on each irreducible componentof C • ( , ) / . Hence, we have to add the number of sections on the green and pinkcomponents in Equation (4.16). From the degrees, it follows that only C , C and C support a non-zero number of sections, namely h (cid:16) C , P ◦ ( , ) / (cid:17) = (cid:26) (cid:27) , h (cid:16) C , P ◦ ( , ) / (cid:17) = 1 , h (cid:16) C , P ◦ ( , ) / (cid:17) = 2 . (4.17)The notation for C reminds us of the fact that on an elliptic curve, a line bundle withvanishing degree can either have or global section. Moreover, recall from Section 3.3that the limit roots on C are actually the 3rd roots of a line bundle of vanishingdegree. In anticipation of this situation, we have already given a detailed exposition ofexactly those root bundles on elliptic curves in Section 3.1. In particular, it follows fromProposition 3 that at least of the C satisfy h ( C , P ◦ ( , ) / ) = 0 .Note that also C admits · different roots. However, in contrast to C , all ofthese roots are in the Kodaira stable regime and have exactly one section. Therefore,we conclude that we found at least · limit roots P ◦ ( , ) / with h (cid:16) C ◦ ( , ) / , P ◦ ( , ) / (cid:17) . (4.18)It therefore follows from our discussion in Section 4.1 that there are at least 72 solutionsto Equation (4.11) and consequently, also to Equation (4.10). Let us emphasize thatthis analysis does not guarantee that one of these 72 solutions stems from an F-theorygauge potential in H D ( b Y , Z (2)) . This top-down study is reserved for future work.Along exactly the same lines, we can argue that also C ( , ) − / = V ( s , s ) and thesinglet curve C ( , ) = V ( s , s ) admit at least 72 solutions to the root bundle constraintswith exactly three global sections. This leaves us to discuss the vector-like spectrum on C ( , ) / = V (cid:0) s , s s + s ( s s − s s ) (cid:1) . (4.19)On this curve we look for root bundles with (c.f. Appendix B.3) P ⊗ , ) / ∼ K B ⊗ , ) / ⊗ O C ( , )1 / ( − · Y ) , h ( C ( , ) / , P ( , ) / ) = 3 , (4.20)where Y = V ( s , s , s ) . For this, it suffices to find solutions to P ⊗ , ) / ∼ K B ⊗ , ) / ⊗ O C ( , )1 / ( − · Y ) , h ( C ( , ) / , P ( , ) / ) = 3 . (4.21)To this end we consider the deformation C ( , ) / → C • ( , ) / with C • ( , ) / = V ( s , s − s ) ∪ V ( s , s − s ) ∪ V ( s , s + s ) ≡ Q ∪ Q ∪ Q , (4.22)33hich is obtained from s → s − s , s → s − Y i =1 x i , s → Y i =1 x i , (4.23)and generic s , s , s . Therefore, C ( , ) / → C • ( , ) / turns this matter curve into threenodal curves, each of which looks like the curve C • ( , ) / that we discussed above. Fromthis point on, we can again employ the limit root techniques. On a technical level, theonly distinction to the constructions presented for C • ( , ) / is that we have to carefullytake into account the line bundle contributions from the Yukawa point Y . Also, theresulting weighted diagrams become very large since the nodal curve C • ( , ) / has 51irreducible components. For these reasons, it suffices to state that we can argue for atleast · solutions to Equation (4.21). Details are provided in Appendix B.4. Wereserve a detailed top-down study of which root bundles arise from an F-theory gaugepotential in H D ( b Y , Z (2)) for the future.
5. Conclusion and Outlook
This work is motivated by the frequent appearance of fractional powers of line bundleswhen studying vector-like spectra of globally consistent 4d F-theory Standard Modelswith three chiral families and gauge coupling unification [31]. In these models, thevector-like spectra on the low-genus matter curves are naively encoded in cohomologiesof a line bundle that is identified with a fractional power of the canonical bundle. Onhigh-genus curves, these fractional powers of the canonical bundle are further modifiedby contributions from Yukawa points. In order to understand these fractional bundles,we have analyzed their origin and nature.First, in section 2.1, we analyzed the origin of such fractional powers of line bundles.We recalled that the vector-like spectra are not specified by a G -flux, but rather byits associated gauge potential in the Deligne cohomology H D ( b Y , Z (2)) [32–34]. In fact,a given G -flux has many such gauge potentials. To see this, recall that in the dualM-theory picture, G = dC , where C is the internal M-theory 3-form potential. Anyother 3-form potential C ′ with closed C ′ − C still has G as its field strength. Suchclosed 3-form potentials are encoded by the intermediate Jacobian J ( b Y ) in the F-theorygeometry. While it is well-defined in theory, it can be very challenging to associate evena single gauge potential in H D ( b Y , Z (2)) to a given G -flux in practice. We were able totie the appearance of fractional powers of line bundles to exactly this challenge.For an F-theory model, we need an F-theory gauge potential, i.e., a class in theDeligne cohomology group A ∈ H D ( b Y , Z ) . This will be specified as b γ ( A ) for some“potential" A ∈ CH ( b Y , Z ) . We found that the geometry determines a class A ′ = b γ ( A ′ ) ∈ H D ( b Y , Z (2)) and an integer ξ ∈ Z > such that A is subject to thetwo constraints: γ ( A ) = G , ξ · b γ ( A ) ∼ b γ ( A ′ ) . (5.1)34he condition γ ( A ) = G immediately follows from eq. (2.4) and it means that A = b γ ( A ) is an F-theory gauge potential for the given G -flux. In the dual M-theory picture, itstates that the 3-form potential C satisfies dC = G . We illustrated with severalexamples that the absence of chiral exotics in the F-theory Standard Models [31] boilsdown to the second constraint.It is important to notice that the gauge potential A = b γ ( A ) specified by the twoconditions in eq. (5.1) is in general not unique. The collection of all ξ -th roots of b γ ( A ′ ) (if non-empty) is a coset of the group of all ξ -th roots of . In particular, the number ofsolutions is ξ · dim C ( J ( b Y ) ) . All these solutions lead to the same chiral spectrum (2.2) sincethey all have the same degree when restricted to the curves C R , and hence, the sameindex. However, they could differ in their actual spectrum (2.7). This extra flexibility isthe key tool that we intend to use to produce a desirable spectrum such as the MSSM.In theory, we could proceed by studying gauge potentials A = b γ ( A ) ∈ H D ( b Y , Z (2)) subject to eq. (5.1). However, in practice it seems more efficient to proceed with thealgebraic cycle A ′ , which we could construct explicitly in the largest currently-knownclass of globally consistent F-theory Standard Models without chiral exotics and gaugecoupling unification [31]. Hence, we have a sufficient level of arithmetic control over A ′ = b γ ( A ′ ) . In particular, we can identify the Z -Cartier divisor D R ( A ′ ) induced from A ′ on the matter curve C R . It follows that ξ · D R ( A ) ∼ D R ( A ′ ) . (5.2)Divisors D R ( A ) , which solve this equation for given D R ( A ′ ) and ξ , are called rootdivisors and their associated line bundles are root bundles . They exist if and only if ξ divides the degree of D R ( A ′ ) . Such root bundles are by no means unique. For example,spin bundles on a genus g matter curve C R are 2nd roots of the canonical bundle K R and there are g such roots. Similarly, on a genus g -curve, Equation (5.2) admits ξ g solutions (if they exist).It is well-known that not all spin bundles have the same number of global sections.Rather, roughly half of the spin bundles on a curve C R have an odd number of globalsections and the remaining ones have an even number [58, 59]. More generally, we cantherefore expect that the gauge potentials A = b γ ( A ) subject to eq. (5.1) lead to differentvector-like spectra. This mirrors the physical expectation that inequivalent F-theorygauge potentials — equivalently, in the dual M-theory picture, two 3-form potentials C and C ′ that differ by a closed 3-form — will in general lead to different vector-likespectra. This was anticipated e.g. in [32–34].In general, only a subset of the root divisors in eq. (5.2) are induced from F-theorygauge potentials in H D ( b Y , Z (2)) . While this work does not answer the importantquestion of which root divisors are induced from F-theory potentials, we hope that thiswork initiates and facilitates this study by providing a systematic analysis of all rootbundles and spin bundles on the matter curves. Our goal in this work was to identifycombinations of root bundles and spin bundles on the matter curves, such that theirglobal sections satisfy the physical demand of the presence/absence of vector-like pairs.While we expect that our techniques apply more generally, we have focused on thelargest currently-known class of globally consistent F-theory Standard Models with35ealistic chiral spectra [31], which emphasizes the genuine appearance of root bundlesin vector-like spectra of F-theory compactifications. It should be mentioned that thebackground G -flux in these F-theory Standard Models models does not only leadto realistic chiral spectra, but also allows cancelation of the D3-tadpole and ensuresmasslessness of the U (1) -gauge boson. We summarize the involved technical steps inthe derivation of these root bundle constraints in appendix B. This derivation heavilyrelies on a detailed understanding of the elliptically fibered 4-fold F-theory geometry b Y ,including intersection numbers in the fiber over the Yukawa points. We supplement theearlier works [26, 31, 44] by providing a complete list of all fiber intersection numbers inappendix A.Our approach to identifying root and spin bundles on the matter curves, whosecohomologies are physically desired for the presence/absence of vector-like pairs, isinspired by the work in [43], which gives a diagrammatic description of root bundleson nodal curves C • R . More explicitly, it relates these roots with so-called limit roots on (partial) blow-ups C ◦ R of C • R . We summarized these ideas in section 3, and thenintroduced counting procedures for the global sections. In order to fully appreciate thisfinding, recall from [38] that in general one will merely find a lower bound. The argumentthat we provide in this work is stronger – it provides an exact count of the global sectionsof limit roots on full blow-ups of C • R . This observation may be interesting in its ownright since it provides a combinatoric access to Brill-Noether theory of limit roots. Wedemonstrated this for a nodal curve – the Holiday lights H • . This curve is of compacttype and its only blow-up that is to be considered for the limit root is its full blow-up.Our approach then allowed us to identify exactly how many limit roots possess a certainnumber of global sections. It will be an interesting mathematical question to extendthese ideas to partial blow-ups. We reserve this analysis for future work.Given these insights on root bundles on nodal curves C • R , it remained to extractinformation on root bundles on actual matter curves C R in F-theory compactifications.As the latter are typically smooth, it is natural to wonder what we can say about (limit)roots when traced along a deformation C • R → C R . In particular, we can wonder ifthere are deformations of C R that are conducive for a more fruitful analysis. As wehave already mentioned, curves of compact type, such as the Holiday lights , are primecandidates. The lack of cycles in their dual graph limits the number of possible weightedgraphs, so much so that we have a complete understanding of the limit roots and theirglobal sections. In contrast, the dual graphs of the deformed matter curves C • R in explicitgeometries are more complex in which there are multiple weighted subgraphs, and limitroots over partial blow-ups. In particular, some singularities on the curve still remainin its partial blow-up, and it is therefore far more challenging to count the sections.It would be useful to compare these two examples in more depth and to determineexactly what features of the dual graph allow for better section-counting. One obviousfeature is the cyclomatic number, which happens to be the first Betti number of a graphwhen viewed as a 1-dimensional simplicial complex. Curves of compact type have zerocyclomatic number, and thus, are topologically simple. Subsequently, we can explorepossible ways of deforming C R to a nodal curve whose dual graph has these desirablefeatures. 36n this work, we have focused on deformations C R → C • R which arise naturally bymodifying the defining polynomials in a concrete base geometry B . Most curves thatwe encountered in this way had planar dual graphs. Still, for the most involved mattercurve discussed in this article, the dual graph is non-planar. The subject of planarityraises many interesting questions and applications in graph theory [74–78]. However, thegeometric significance for a nodal curve to have a non-planar dual graph is not mentionedin the literature to our knowledge [79, 80]. It is possible that planarity does not playa role in the geometry of nodal curves. Indeed, the curve associated to the well-knownnon-planar graph K , is quite ordinary. Nevertheless, it would be useful to explore thisfeature as it raises the question of whether there are better ways to represent a givendual graph.For a physical application, we have studied vector-like spectra of F-theory StandardModels without chiral exotics in section 4. In aiming for MSSM constructions, we shouldwonder what we can say about the global sections of a root P • R as we trace it to a root P R along a deformation C • R → C R . In this work, we did not attempt to provide a completeanswer to this question. Rather, we recalled that a certain behavior of the cohomologiesalong such a deformation is known. This is called upper semi-continuity and it meansthat the number of global sections cannot increase when tracing a root P • R on C • R to aroot P R on C R . Put differently, h ( C R , P R ) ≤ h ( C • R , P • R ) . (5.3)For F-theory MSSM constructions, it is important to understand (limit) roots on theHiggs curve with h ( C ( , ) − / , P ( , ) − / ) = 4 = 1 + χ ( P ( , ) − / ) . While we can constructroots P • ( , ) − / with h ( C • ( , ) − / , P • ( , ) − / ) = 4 , upper semi-continuity does then notguarantee that P • ( , ) − / → P ( , ) − / along C • ( , ) − / → C ( , ) − / yields roots with 4global sections. Rather, the roots could lose sections along this transition (cf. [38]). Toour knowledge, a sufficient criterion that identifies the Higgs roots P • ( , ) − / that do notlose sections is currently unknown. However, given the physical significance of such acondition, we hope to return to this interesting question in the future.Even a subset of (limit) roots that do not lose sections along C • R → C R is valuable. Weidentified a family of such roots P • R . Namely, for a root with h ( C • R , P • R ) = χ ( P R ) ≥ ,it follows from upper semi-continuity that h ( C R , P R ) = h ( C • R , P • R ) . Any such rootthus satisfies h i ( C R , P R ) = ( χ ( P R ) , , which means it describes a zero mode spectrumon C R without vector-like pairs. For example, in the F-theory MSSM constructions, thisis a desired feature for the representations C ( , ) / , C ( , ) − / , C ( , ) / , C ( , ) for whichvector-like pairs are exotic, i.e. have thus far not been observed in particle accelerators.We have applied these techniques to a particular F-theory geometry among the largestcurrently-known class of globally consistent F-theory Standard Model constructionswithout chiral exotics and gauge coupling unification [31]. To this end, we worked withthe base space B = P . This 3-fold is one of the triangulations of the 39-th polyhedronof the Kreuzer-Skarke list [81], hence the name. In this space, we have explicitly deformedthe matter curves C R to nodal curves C • R . On those nodal curves, we could then easilyconstruct limit roots on the full blow-up C ◦ R of C • R which have exactly 3 sections. We37ollect details on the base space B = P and limit roots on the blow-up of a genus g = 82 matter curve in appendix B.4. In future works, we hope to investigate which ofthese desired root bundles are realized from F-theory gauge potentials in H D ( b Y , Z (2)) .To fully appreciate these findings, let us point out that this task cannot be performedwith state-of-the-art algorithms such as [35] unless one explicitly specifies the line bundledivisor in question. In past works [33, 34], such constructions were described. Acomputer model of such line bundles (by dualizing the corresponding ideal sheaf) requiresGröbner basis computations. Even by the use of state-of-the-art algorithms such as[82], the involved geometries resulted in excessively long runtimes and heavy memoryconsumption. By approaching root bundles from limit roots on full blow-ups, thesecomplications are circumvented at the cost of studying deformation theory.This work provides a constructive approach to identifying limit root bundles on fullblow-ups of a nodal curve with specific number of global sections. Since our approachis completely constructive, we anticipate a computer implementation which can find allsuch limit roots. For this, one would work out all of the weighted diagrams associatedto the dual graph of a nodal curve C • R , and then identify the limit roots with the desirednumber of global sections. In generalizing this approach even further, we anticipatea scan over many of the F-theory Standard model geometries in [31]. By employingstate-of-the-art data-science and machine learning techniques, it can be expected thatsuch a scan will lead to a more refined understanding of F-theory Standard Modelconstructions. We hope to return to this fascinating question in the near future. Acknowledgements
We thank Gavril Farkas, Iñaki García-Etxebarria, Ling Lin andClaire Voisin for valuable discussions. M.B., R.D. and M.O. are partially supported byNSF grant DMS 2001673 and by the Simons Foundation Collaboration grant
A. Fiber structure of F-theory Standard Models
In this section, we investigate the fiber structure of the resolved 4-fold with SU (3) × SU (2) × U (1) Y gauge symmetry as employed in the largest currently-knownclass of globally consistent F-theory Standard Models without chiral exotics and gaugecoupling unification [31]. We work out the intersection numbers in the fibers over genericpoints of the gauge divisors, matter curves and Yukawa loci. The knowledge of the fiberstructure determines the vector-like spectrum in this F-theory vacuum.38 .1. Away from Matter Curves A.1.1. SU (2) Gauge Divisor
This gauge divisor is V ( s ) . Here, the defining equation of P F factors as p F = e (cid:0) e e e e s u + e e e s u v + e e e s u w + e e e s uvw + s vw (cid:1) . (A.1)The Cartan divisors are therefore as follows D SU (2)0 = V (cid:0) e e e e s u + e e e s u v + e e e s u w + e e e s uvw + s vw , s (cid:1) ,D SU (2)1 = V ( e , s ) . (A.2)The intersection numbers in the fiber over a generic base point p ∈ V ( s ) are: D SU (2) i · D SU (2) j · ˆ π − ( p ) D SU (2)0 D SU (2)1 U (1) Y D SU (2)0 -2 2 0 D SU (2)1 A.1.2. SU (3) Gauge Divisor
This SU (3) gauge divisor V ( s ) relates to the Cartan divisors as follows: D SU (3)0 = V (cid:0) e e e e s u + e e e e s uv + e e s v + e e s uw + e e e s vw, s (cid:1) ,D SU (3)1 = V ( e , s ) , D SU (3)2 = V ( u, s ) . (A.4)The intersection numbers in the fiber over a generic base point p ∈ V ( s ) are: D SU (3) i · D SU (3) j · ˆ π − ( p ) D SU (3)0 D SU (3)1 D SU (3)3 U (1) Y D SU (3)0 -2 1 1 0 D SU (3)1 D SU (3)2 A.2. Over Matter Curves
Intersection Structure over C ( , ) / away from Yukawa Loci Over the matter curves, singularity enhancements occur. They are geometrically relatedto the presence of new P -fibrations, of which linear combinations eventually serve asmatter surfaces. Over C ( , ) / = V ( s , s ) the following P -fibrations are present: P (cid:0) ( , ) / (cid:1) = V ( s , s , e e e e s u + e e e s uv + e e s uw + e s vw ) , P (cid:0) ( , ) / (cid:1) = V ( s , s , e ) , P (cid:0) ( , ) / (cid:1) = V ( s , s , e ) , P (cid:0) ( , ) / (cid:1) = V ( s , s , u ) , P (cid:0) ( , ) / (cid:1) = V ( s , s , e ) . (A.6)39hese P -fibrations relate to restrictions of the SU (3) and SU (2) Cartan divisors:
Original Split components over C R D SU (2)0 P (cid:0) ( , ) / (cid:1) + P (cid:0) ( , ) / (cid:1) + P (cid:0) ( , ) / (cid:1) + P (cid:0) ( , ) / (cid:1) D SU (2)1 P (cid:0) ( , ) / (cid:1) D SU (3)0 P (cid:0) ( , ) / (cid:1) + P (cid:0) ( , ) / (cid:1) + P (cid:0) ( , ) / (cid:1) D SU (3)1 P (cid:0) ( , ) / (cid:1) D SU (3)2 P (cid:0) ( , ) / (cid:1) (A.7)Over p ∈ C ( , ) / which is not a Yukawa point, these P -fibrations intersect as follows: P (cid:0) ( , ) / (cid:1) P (cid:0) ( , ) / (cid:1) P (cid:0) ( , ) / (cid:1) P (cid:0) ( , ) / (cid:1) P (cid:0) ( , ) / (cid:1) P (cid:0) ( , ) / (cid:1) -2 1 0 0 1 P (cid:0) ( , ) / (cid:1) P (cid:0) ( , ) / (cid:1) P (cid:0) ( , ) / (cid:1) P (cid:0) ( , ) / (cid:1) (A.8)The intersection numbers between the P -fibrations over C ( , ) / and the pullbacks ofthe Cartan divisors are readily computed as follows: D SU (2)0 D SU (2)1 D SU (3)0 D SU (3)1 D SU (3)2 U (1) Y P (cid:0) ( , ) / (cid:1) -1 1 -1 1 0 -1/6 P (cid:0) ( , ) / (cid:1) P (cid:0) ( , ) / (cid:1) P (cid:0) ( , ) / (cid:1) P (cid:0) ( , ) / (cid:1) S ( a )( , ) / over C ( , ) / are linear combinations of the above P -fibrations.We use P to denote such a linear combination compactly. Explicitly, P is a list of themultiplicities with which these P -fibrations appear in the above order: P = (0 , , , , ↔ · P (cid:0) ( , ) / (cid:1) + 4 · P (cid:0) ( , ) / (cid:1) . We apply β to indicate the Cartan charges of such a linear combination, these notationswill also be adopted for the other matter curves. All that said, the matter surfaces over C ( , ) / take the following form: Label ~P β
Label ~P βS (1)( , ) / (0 , , , ,
0) (1) ⊗ (0 , S (4)( , ) / (0 , , , ,
0) ( − ⊗ (1 , − S (2)( , ) / (0 , , , ,
0) ( − ⊗ (0 , S (5)( , ) / (0 , , , ,
1) (1) ⊗ ( − , S (3)( , ) / (0 , , , ,
0) (1) ⊗ (1 , − S (6)( , ) / (0 , , , ,
1) ( − ⊗ ( − , (A.10)40 ntersection Structure over C ( , ) − / away from Yukawa Loci For convenience, we employ p Hi to denote the following polynomials: p H = s e e e u + s w , p H = s e e e u + s s e e e uv − s s vw + s s vw ,p H = s e e e u + s e e e uw + s w , p H = s s e e e u + s s w − s s w ,p H = s s + s s − s s s , p H = s s e e e u + s s e e e uv + s s vw ,p H = s e e e u + s s e e e uv − s s e e e uv + s s vw . (A.11)Over C ( , ) − / which is not a Yukawa point, the following P -fibrations are present: P (cid:0) ( , ) − / (cid:1) = V ( s , p H , p H , p H , p H ) , P (cid:0) ( , ) − / (cid:1) = V ( s , p H , p H , p H , p H , p H · e e e u + p H · v ) , P (cid:0) ( , ) − / (cid:1) = V ( s , p H , e ) . (A.12)Equivalently, P = V ( s , p H , p H ) arises from the analysis of a primary decomposition.The above P -fibrations relate to restrictions of the SU (2) Cartan divisors as follows:
Original Split components over C R Original Split components over C R D SU (2)0 P (cid:0) ( , ) − / (cid:1) + P (cid:0) ( , ) − / (cid:1) D SU (2)1 P (cid:0) ( , ) − / (cid:1) (A.13)Over p ∈ C ( , ) − / which is not a Yukawa point, these P -fibrations correspond to therepresentation state at the right column and intersect each other as follows: P (cid:0) ( , ) − / (cid:1) P (cid:0) ( , ) − / (cid:1) P (cid:0) ( , ) − / (cid:1) P (cid:0) ( , ) − / (cid:1) -2 1 1 P (cid:0) ( , ) − / (cid:1) P (cid:0) ( , ) − / (cid:1) ~P β Label ~P βS (1)( , ) − / (1 , ,
0) (1) S (2)( , ) − / (1 , ,
1) ( − (A.15) Intersection Structure over C ( , ) − / away from Yukawa Loci Over C ( , ) − / = V ( s , s ) the following P -fibrations are present: P (cid:0) ( , ) − / (cid:1) = V ( s , s , e e e s u + e e e e s uv + e e s v + e e s vw ) , P (cid:0) ( , ) − / (cid:1) = V ( s , s , u ) , P (cid:0) ( , ) − / (cid:1) = V ( s , s , e ) , P (cid:0) ( , ) − / (cid:1) = V ( s , s , e ) . (A.16)41hese P -fibrations relate to restrictions of the SU (3) Cartan divisors as follows:
Original Split components over C R Original Split components over C R D SU (3)0 P (cid:0) ( , ) − / (cid:1) + P (cid:0) ( , ) − / (cid:1) D SU (3)1 P (cid:0) ( , ) − / (cid:1) D SU (3)2 P (cid:0) ( , ) − / (cid:1) (A.17)Over p ∈ C ( , ) − / which is not a Yukawa point, these P -fibrations intersect as follows: P (cid:0) ( , ) − / (cid:1) P (cid:0) ( , ) − / (cid:1) P (cid:0) ( , ) − / (cid:1) P (cid:0) ( , ) − / (cid:1) P (cid:0) ( , ) − / (cid:1) -2 1 0 1 P (cid:0) ( , ) − / (cid:1) P (cid:0) ( , ) − / (cid:1) P (cid:0) ( , ) − / (cid:1) (A.18)The matter surfaces S ( a )( , ) − / take the following form: Label ~P β
Label ~P β
Label ~P βS (1)( , ) − / (0 , , ,
1) (1 , S (2)( , ) − / (0 , , ,
1) ( − , S (3)( , ) − / (0 , , ,
1) (0 , − (A.19) Intersection Structure over C ( , ) / away from Yukawa Loci Over C ( , ) / which is not a Yukawa point, the following P -fibrations are present: P (cid:0) ( , ) / (cid:1) = V ( s , s s − s s s + s s , s e e u + s e v,s s e e u − s s e v + s s e v, s e e u + s e e e uv + s e v ) , P (cid:0) ( , ) / (cid:1) = V ( s , s s − s s s + s s , s s e e e e u + s s e e v + s s e e w,s e e e e u + s e e e e uv + s e e uw + s e e v + s e e e vw,s s e e e e u − s s e e e e u − s s e e v − s e e w,s s e e e e u + s s e e v − s s e e v + s e e w ) , P (cid:0) ( , ) / (cid:1) = V ( s , s s + s s − s s s , u ) , P (cid:0) ( , ) / (cid:1) = V ( s , s s + s s − s s s , e ) . (A.20)Due to primary decomposition analysis, P (cid:0) ( , ) / (cid:1) can be rewritten as P (cid:0) ( , ) / (cid:1) = V ( s , us e e + vs e , s s e e u − s s e v + s s e v ) − V ( s , s , s ) . (A.21)The above P -fibrations relate to restrictions of the SU (3) Cartan divisors as follows:
Original Split components over C ( , ) / Original Split components over C ( , ) / D SU (3)0 P (cid:0) ( , ) / (cid:1) + P (cid:0) ( , ) / (cid:1) D SU (3)1 P (cid:0) ( , ) / (cid:1) D SU (3)2 P (cid:0) ( , ) / (cid:1) (A.22)42ver p ∈ C ( , ) / which is not a Yukawa point, these P -fibrations intersect as follows: P (cid:0) ( , ) / (cid:1) P (cid:0) ( , ) / (cid:1) P (cid:0) ( , ) / (cid:1) P (cid:0) ( , ) / (cid:1) P (cid:0) ( , ) / (cid:1) -2 1 0 1 P (cid:0) ( , ) / (cid:1) P (cid:0) ( , ) / (cid:1) P (cid:0) ( , ) / (cid:1) (A.23)The matter surfaces S ( a )( , ) / take the following form: Label ~P β
Label ~P β
Label ~P βS (1)( , ) / (1 , , ,
0) (1 , S (2)( , ) / (1 , , ,
1) ( − , S (3)( , ) / (1 , , ,
1) (0 , − (A.24) Intersection Structure over C ( , ) away from Yukawa Loci Over the singlet curve C ( , ) = V ( s , s ) the following two P -fibrations are present: P (( , ) ) = V (cid:0) s , s , e e e e s u + e e s uv + e e e e s uw + e s w (cid:1) , P (( , ) ) = V ( s , s , v ) . (A.25)These fibrations intersect as follows: P (( , ) ) P (( , ) ) U (1) Y P (( , ) ) -2 2 -1 P (( , ) ) P (( , ) ) as matter surface for the singlet state with q U (1) Y = 1 . A.3. Over Yukawa Loci
Intersection Structure over Yukawa Locus Y Over the Yukawa point Y = V ( s , s , s ) the following P -fibrations are present: P ( Y ) = V ( s , s , s , e ) , P ( Y ) = V ( s , s , s , e ) , P ( Y ) = V ( s , s , s , e ) , P ( Y ) = V ( s , s , s , e ) , P ( Y ) = V ( s , s , s , u ) , P ( Y ) = V (cid:0) s , s , s , e e e s u + e e e s uv + s vw (cid:1) . (A.27)The intersection numbers in the fiber over Y are as follows: P ( Y ) P ( Y ) P ( Y ) P ( Y ) P ( Y ) P ( Y ) P ( Y ) -2 0 0 1 0 1 P ( Y ) P ( Y ) P ( Y ) P ( Y ) P ( Y ) (A.28)43estrictions of the fibrations over the matter curves relate to the P i ( Y ) as follows: Split P over C R Split P over Y Split P over C R Split P over Y P (cid:0) ( , ) / (cid:1) P ( Y ) + P ( Y ) P (cid:0) ( , ) − / (cid:1) P ( Y ) P (cid:0) ( , ) / (cid:1) P ( Y ) P (cid:0) ( , ) − / (cid:1) P ( Y ) P (cid:0) ( , ) / (cid:1) P ( Y ) P (cid:0) ( , ) − / (cid:1) P ( Y ) + P ( Y ) + P ( Y ) P (cid:0) ( , ) / (cid:1) P ( Y ) P (cid:0) ( , ) − / (cid:1) P ( Y ) P (cid:0) ( , ) / (cid:1) P ( Y ) P (cid:0) ( , ) − / (cid:1) P ( Y ) P (cid:0) ( , ) − / (cid:1) P i =1 P i ( Y ) P (cid:0) ( , ) − / (cid:1) P ( Y ) (A.29) Intersection Structure over Yukawa Locus Y Over the Yukawa point Y = V ( s , s , s s − s s ) the following P -fibrations are present: P ( Y ) = V (cid:0) s , s , s s − s s , s e e u + s e v, s e e u + s e v (cid:1) , P ( Y ) = V ( s , s , s s − s s , e ) , P ( Y ) = V ( s , s , s s − s s , e ) , P ( Y ) = V ( s , s , s s − s s , e ) , P ( Y ) = V ( s , s , s s − s s , u ) , P ( Y ) = V ( s , s , s s − s s , s e e e u + s w, s e e e u + s w ) . (A.30)The intersection numbers in the fiber over Y are as follows: P ( Y ) P ( Y ) P ( Y ) P ( Y ) P ( Y ) P ( Y ) P ( Y ) -2 0 1 0 0 1 P ( Y ) P ( Y ) P ( Y ) P ( Y ) P ( Y ) (A.31)Restrictions of the fibrations over the matter curves relate to the P i ( Y ) as follows: Split P over C R Split P over Y Split P over C R Split P over Y P (cid:0) ( , ) / (cid:1) P ( Y ) + P ( Y ) P (cid:0) ( , ) − / (cid:1) P ( Y ) + P i =2 P i ( Y ) P (cid:0) ( , ) / (cid:1) P ( Y ) P (cid:0) ( , ) − / (cid:1) P ( Y ) P (cid:0) ( , ) / (cid:1) P ( Y ) P (cid:0) ( , ) / (cid:1) P ( Y ) P (cid:0) ( , ) / (cid:1) P ( Y ) P (cid:0) ( , ) / (cid:1) P ( Y ) + P ( Y ) + P ( Y ) P (cid:0) ( , ) / (cid:1) P ( Y ) P (cid:0) ( , ) / (cid:1) P ( Y ) P (cid:0) ( , ) − / (cid:1) P ( Y ) P (cid:0) ( , ) / (cid:1) P ( Y ) (A.32) Intersection Structure over Yukawa Locus Y Over the Yukawa point Y = V ( s , s , s ) , we use A i to denote the reduced P -fibrationssuch that the following structure is presented: P ( Y ) = A ( Y ) = V ( s , s , s , e ) , P ( Y ) = A ( Y ) = V ( s , s , s , e ) , P ( Y ) = 2 A ( Y ) = V (cid:0) s , s , s , e (cid:1) , P ( Y ) = 2 A ( Y ) = V (cid:0) s , s , s , u (cid:1) , P ( Y ) = A ( Y ) = V (cid:0) s , s , s , us e e e e + vs e e + ws e e (cid:1) . (A.33)44estrictions of the fibrations over the matter curves relate to the P i ( Y ) as follows: Split P over C R Split P over Y Split P over C R Split P over Y P (cid:0) ( , ) / (cid:1) P i =2 A i ( Y ) P (cid:0) ( , ) / (cid:1) A ( Y ) + A ( Y ) + A ( Y ) P (cid:0) ( , ) / (cid:1) A ( Y ) P (cid:0) ( , ) / (cid:1) A ( Y ) P (cid:0) ( , ) / (cid:1) A ( Y ) P (cid:0) ( , ) / (cid:1) A ( Y ) P (cid:0) ( , ) / (cid:1) A ( Y ) P (cid:0) ( , ) / (cid:1) A ( Y ) P (cid:0) ( , ) / (cid:1) A ( Y ) (A.34)Their intersection numbers are slightly away from standard, namely A ( Y ) A ( Y ) A ( Y ) A ( Y ) A ( Y ) A ( Y ) -2 0 1 0 0 A ( Y ) A ( Y ) A ( Y ) A ( Y ) (A.35)The meaning of ( A ( Y )) = − becomes clear once we draw the associated diagram: A ( Y ) A ( Y ) A ( Y ) A ( Y ) A ( Y ) missing node N (A.36)Consequently, we see that the node N is missing and it holds P ( Y ) = 2 · N + N ,where N is the standard node that ordinarily appear instead of P ( Y ) . It follows (cid:0) P ( Y ) (cid:1) = 4 · N + 4 N N + N = 4 · ( −
2) + 4 · −
2) = − . (A.37)Likewise, A ( Y ) = N + · N leads to the half-integer intersection in Equation (A.35). Intersection Structure over Yukawa Locus Y Over the Yukawa point Y = V ( s , s , s ) the following P -fibrations are present: P ( Y ) = V ( s , s , s , e ) , P ( Y ) = V ( s , s , s , v ) , P ( Y ) = V (cid:0) s , s , s , s e e e u + s e e e uw + s w (cid:1) . (A.38)Restrictions of the fibrations over the matter curves relate to the P i ( Y ) as follows: Split P over C R Split P over Y Split P over C R Split P over Y P (cid:0) ( , ) / (cid:1) P ( Y ) P (( , ) ) P ( Y ) + P ( Y ) P (cid:0) ( , ) / (cid:1) P ( Y ) P (( , ) ) P ( Y ) P (cid:0) ( , ) / (cid:1) P ( Y ) (A.39)45he intersection numbers in the fiber over Y are as follows: P ( Y ) P ( Y ) P ( Y ) P ( Y ) -2 1 1 P ( Y ) P ( Y ) Intersection Structure over Yukawa Locus Y Over the Yukawa point Y = V ( s , s , s ) the following P -fibrations are present: P ( Y ) = V (cid:0) s , s , s , e , s e , u s e e e + uvs e e e e + v s e e + vws e e e (cid:1) = n · V ( s , s , s , e ) = n · A ( Y ) , P ( Y ) = V (cid:0) s , s , s , e (cid:1) = n · V ( s , s , s , e ) = n · A ( Y ) , P ( Y ) = V (cid:0) s , s , s , u (cid:1) = n · V ( s , s , s , u ) = n · A ( Y ) , P ( Y ) = V ( s , s , s , u s e e e + uvs e e e e + v s e e + vws e e ,u s e e + 2 u vs s e e e + u v s e e e + 2 u v s s e e e + 2 uv s s e e e + v s e , u s s e e + uvs s e e e + v s s e )= n · V (cid:0) s , s , s , u s e e + uvs e e e + v s e (cid:1) = n · A ( Y ) . (A.41)The total elliptic fiber over Y is given T ( Y ) = X i =0 P i ( Y ) = n · A ( Y ) + n · A ( Y ) + n · A ( Y ) + n · A ( Y ) . (A.42)Restrictions of the fibrations over the gauge divisor D SU (3) i are: D SU (3)0 | Y = n · A ( Y ) + n · A ( Y ) + n · A ( Y ) ,D SU (3)1 | Y = n · A ( Y ) ,D SU (3)2 | Y = n · A ( Y ) . (A.43)The total elliptic fiber can be obtained from the total torus over the gauge divisor D SU (3) i : T (cid:16) D SU (3) i | Y (cid:17) = ( n + n ) · A ( Y ) + n · A ( Y ) + n · A ( Y ) + n · A ( Y ) . (A.44)Since this must recover (Equation (A.42)), we conclude that n = n + n (A.45)Restrictions of the fibrations over the matter curves ( , ) / ) are as follows: P (cid:0) ( , ) / (cid:1) | Y = V (cid:0) s , s , s , u s e e + uvs e e e + v s e (cid:1) = A ( Y ) , P (cid:0) ( , ) / (cid:1) | Y = V (cid:0) s , s , s , u s e e + uvs e e e + v s e (cid:1) + P ( Y ) + V ( s , s , s , e , s e , u s e e e + uvs e e e e + v s e e + vws e e e )= A ( Y ) + n · A ( Y ) + n · A ( Y ) , P (cid:0) ( , ) / (cid:1) | Y = V (cid:0) s , s , s , e (cid:1) = n · A ( Y ) , P (cid:0) ( , ) / (cid:1) | Y = V (cid:0) s , s , s , u (cid:1) = n · A ( Y ) . (A.46)46he fibrations over the matter curves ( , ) / give the total elliptic fiber over Y as: T (cid:0) ( , ) / | Y (cid:1) = ( n + n ) · A ( Y ) + n · A ( Y ) + n · A ( Y ) + 2 · A ( Y ) . (A.47)Restrictions of the fibrations over the matter curves ( , ) − / are: P (cid:0) ( , ) − / (cid:1) | Y = V ( s , s , s , e , s e , u s e e e + uvs e e e e + v s e e + vws e e e )+ P ( Y ) = n · A ( Y ) + n · A ( Y ) , P (cid:0) ( , ) − / (cid:1) | Y = V (cid:0) s , s , s , u (cid:1) = n · A ( Y ) , P (cid:0) ( , ) − / (cid:1) | Y = V (cid:0) s , s , s , e (cid:1) = n · A ( Y ) , P (cid:0) ( , ) − / (cid:1) | Y = V (cid:0) s , s , s , e (cid:1) = n · A ( Y ) . (A.48)The fibrations over the matter curves ( , ) − / give the total elliptic fiber over Y as: T (cid:0) ( , ) − / | Y (cid:1) = ( n + n ) · A ( Y ) + n · A ( Y ) + n · A ( Y ) + n · A ( Y ) . (A.49)We conclude that the restriction from the two triplet matter curves to the Yukawa point Y preserve the elliptic fiber structure as presented in (Equation (A.42)) iff n = 2 .Before we continue our discussion of the factors n i , let us look at the intersectionnumbers among the A i ( Y ) as follows: A ( Y ) A ( Y ) A ( Y ) A ( Y ) A ( Y ) -2 1 1 2 A ( Y ) A ( Y ) A ( Y ) n i we can fix n = n = 2 intuitively. Then, by infering that − (cid:16) P (( , ) − / ) (cid:12)(cid:12) Y (cid:17) , we find n ∈ { , } . Intuitively, we discard n = 1 and pick n = 3 instead. By accepting all these steps above, we are then left to conclude n = 5 , n = n = n = 2 , n = 3 , n = 2 . (A.51)This finally, completes our understanding of the fiber structure over Y . Intersection Structure over Yukawa Locus Y Over the Yukawa point Y = V ( s , s , s ) the following P -fibrations are present: P ( Y ) = V ( s , s , s , e ) , P ( Y ) = V ( s , s , s , e ) , P ( Y ) = V ( s , s , s , u ) , P ( Y ) = V ( s , s , s , v ) , P ( Y ) = V (cid:0) s , s , s , s e e e e u + s e e v + s e e w (cid:1) . (A.52)Restrictions of the fibrations over the matter curves relate to the P i ( Y ) as follows: Split P over C R Split P over Y Split P over C R Split P over Y P (cid:0) ( , ) / (cid:1) P ( Y ) + P ( Y ) P (cid:0) ( , ) − / (cid:1) P ( Y ) P (cid:0) ( , ) / (cid:1) P ( Y ) P (cid:0) ( , ) − / (cid:1) P ( Y ) P (cid:0) ( , ) / (cid:1) P ( Y ) P (cid:0) ( , ) − / (cid:1) P ( Y ) P (cid:0) ( , ) / (cid:1) P ( Y ) P (cid:0) ( , ) − / (cid:1) P ( Y ) + P ( Y ) P (( , ) ) P i =0 P i ( Y ) + P ( Y ) P (( , ) ) P ( Y ) (A.53)47he intersection numbers in the fiber over Y are as follows: P ( Y ) P ( Y ) P ( Y ) P ( Y ) P ( Y ) P ( Y ) -2 1 1 0 0 P ( Y ) P ( Y ) P ( Y ) P ( Y ) (A.54) B. Induced line bundles in F-theory Standard Models
In this section, we give details on how we identify the root bundles in the largestcurrently-known class of globally consistent F-theory Standard Model constructionswithout chiral exotics and gauge coupling unification [31]. More details can be found inthe earlier works [26, 44]. We provide details on the employed G -flux in Appendix B.1.Subsequently, we outline our computational techniques in Appendix B.2 and summarizethe resulting root bundle constraints in Appendix B.3. Finally, we construct root bundlesolutions in compactifications over a particular 3-fold base space B in Appendix B.4. B.1. G -flux and matter surfaces U ( ) -flux We associate to the section s = V ( e ) a U (1) -flux. To this end, we employthe Shioda map to turn s into σ ∈ H (1 , ( b Y ) : σ = (cid:18) [ e ] − [ v ] − (cid:2)b π ∗ (cid:0) K B (cid:1)(cid:3) + 12 [ e ] + 13 [ e ] + 23 [ u ] (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ˆ Y . (B.1)In this expression, [ e ] = γ ( V ( e )) ∈ H (1 , ( b Y ) denotes the image of the divisor V ( e ) ⊆ X under the cycle map γ . Furthermore, recall that b π : b Y ։ B . The U (1) -fluxis then given by G U (1)4 ≡ ω ∧ σ ∈ H (2 , ( b Y ) , ω ∈ π ∗ ( H (1 , ( B )) . (B.2) Matter surface flux
To the matter surface S (1)( , ) / over the quark-doublet curve C ( , ) / (cf. Appendix A) one can associate a gauge invariant flux G ( , ) / ≡ h S (1)( , ) / i + 12 · (cid:2) P (( , ) / ) (cid:3) + 13 · (cid:2) P (( , ) / ) (cid:3) + 23 · (cid:2) P (( , ) / ) (cid:3) . (B.3) Total flux expression
One can now consider a linear combination of these fluxes G ( a, ω ) = a · G ( , ) / + ω ∧ σ ∈ H (2 , ( b Y ) . (B.4)48he parameters a ∈ Q and ω ∈ π ∗ (cid:0) H (1 , ( B ) (cid:1) are subject to flux quantization, D -tadpole cancelation, masslessness of the U (1) -gauge boson and exactly three chiralfamilies on all matter curves. These conditions are solved base-independently by ω = 3 K B · K B , a = 15 K B . (B.5)This leads to G = − K B · (cid:0) e ] ∧ [ e ] − K B ∧ (3[ e ] − e ] − e ] + K B − u + v ) (cid:1) . (B.6)For this G , it was verified in [31], that the integral over all matter surfaces S R andcomplete intersections of toric divisors is integral. This is a necessary condition, for thisalgebraic cycle to be integral. A sufficient check is computationally very demanding andcurrently beyond our arithmetic abilities. Therefore, we proceed under the assumptionthat this G -flux candidate eq. (2.25), is indeed integral and thus a proper G -flux.We next look at A ′ = − · (5 V ( e , e ) − V ( e , t ) − V ( e , t ) − V ( e , t )+ V ( t , t ) − V ( t , u ) + V ( t , v )) | b Y ∈ CH ( b Y , Z ) , (B.7)where t i ∈ H ( X , α ∗ ( K B )) and α : X = B × P F ։ B . Note that γ ( A ′ ) = K B · G .Therefore, this gauge potential would induce chiral exotics, unless we ”devide“ it by ξ = K B . Hence, we are led to consider gauge potentials A = b γ ( A ) ∈ H D ( b Y , Z (2)) with γ ( A ) = G , ξ · b γ ( A ) ∼ b γ ( A ′ ) . (B.8)Hence, we can infer that the line bundles induced from A = b γ ( A ) are K B -th roots ofthe ones induced from A ′ = b γ ( A ′ ) . The divisors D R ( A ′ ) are then K B -th roots of the D R ( A ′ ) . In the following, we outline the arithmetic identification of the divisors D R ( A ′ ) . Matter surfaces
As Equation (B.7) is gauge invariant, it suffices to focus on thefollowing matter surfaces (cf. Appendix A) S (1)( , ) / = V ( s , s , e ) , S (1)( , ) − / = V ( s , s , e ) , S (1)( , ) = V ( s , s , v ) ,S (1)( , ) − / = V ( s , s s + s s − s s s , s e e e u + s w,s s e e e u + s s w − s s w, s e e e u + s e e e uw + s w ) ,S (1)( , ) / = V ( s , s s − s s s + s s , s e e u + s e v,s s e e u − s s e v + s s e v, s e e u + s e e e uv + s e v ) . (B.9)49ote that S (1)( , ) − / and S (1)( , ) / are not complete intersections. In Appendix B.2.1,we explain how one can compute topological intersection numbers of cycles with them.Moreover, we can simplify the expressions for those two matter surfaces. S (1)( , ) − / = V ( s , us e e e + ws , u s e e e + uws e e e + w s ) ,S (1)( , ) / = V ( s , us e e + vs e , u s e e + uvs e e e + v s e ) − V ( s , s , s ) . (B.10)Therefore, we can express all matter surfaces as pullbacks from elements in CH ( X ) : S (1)( , ) / = V ( s , e ) − V ( e , e ) | ˆ Y ,S (1)( , ) − / = V ( s , p ) − V ( e , p ) − V ( s , v ) | ˆ Y ,S (1)( , ) − / = V ( s , e ) − V ( v, e ) | ˆ Y ,S (1)( , ) / = V ( s , q ) + V ( e , e ) − V ( e , q ) − V ( e , s ) − V ( u, q ) (cid:12)(cid:12) ˆ Y ,S (1)( , ) = V ( s , v ) − V ( v, w ) | ˆ Y , (B.11)where p = us e e e + ws and q = us e e + vs e . We exploit this in Appendix B.2.2to compute the actual intersection loci.Finally in Appendix B.2.3, we make use of the fact that we know that the mattersurfaces are particular P -fibrations over the matter curves. The matter surface fluxoriginates from the matter surface S ( , ) / . This allows us to derive the divisors D R ( A ′ ) intuitively from intersections in the fiber and intersections in the base. The former isfacilitated by knowledge of the intersection numbers listed in Appendix A. B.2. Computational strategies
B.2.1. Euler characteristic of structure sheaf of intersection varietyThe twisted cubic – a non-complete intersection
Let us start with a simple andinstructive example that involves a non-complete intersection. We consider P withhomogeneous coordinates [ x : y : z : w ] and focus on the hypersurface Y = V ( xw − yz ) .In this hypersurface Y , we consider the twisted cubic S = V ( xz − y , yw − z ) ∩ Y = V ( xz − y , yw − z , xw − yz ) ∼ = P , (B.12)and a union of two lines A = V ( x ) ∩ Y = V ( x, xw − yz ) = V ( x, y ) ∪ V ( x, z ) . (B.13)Crucially, note that S is not a complete intersection and cannot be expressed as anysort of pullback from P . In order to compute the topological intersection number S · A ,we notice that this intersection number coincides with the Euler characteristic of thestructure sheaf of the intersection variety V ( x, xz − y , yw − z , xw − yz ) .50et us denote the coordinate ring of P by R . Then an f.p. graded (left) R -module,which sheafifies to the structure sheaf in question, is given by R ( − ⊕ ⊕ R ( − (cid:16) xw − yz xz − y yw − z x (cid:17) T −−−−−−−−−−−−−−−−−−−−−−−−→ R ։ O S ·A → . (B.14)Denote this sequence by F M −−→ R ։ O S ·A → . A minimal free resolution is given by → F M −−→ F M −−→ F M −−→ R ։ O S ·A → , (B.15)where F = R ( − ⊕ , F = R ( − ⊕ and M = − wz y − y w x − y w yz − w − x xz − yw x y − xw , M = (cid:18) x − y − wx − y y − w − z (cid:19) . (B.16)The vector bundles e F i has the following sheaf cohomologies: O P ( − ⊕ O P ( − ⊕ O P ( − ⊕ ⊕ O P ( − e F ≡ e R ∼ = O P h h h h h i ( P , O S ·A ) = (3 , , , , i.e. S · A = χ ( O S ·A ) = 3 . Equivalently, we find V ( xw − yz, xz − y , yw − z , x ) = V ( x, y, z ) , (B.18)which allows us to conclude S · A = 3 · V ( x, y, z ) . Application to Higgs matter surface
We employ this technique as a consistencycheck on intersections with the non-complete matter surfaces. For instance, let us workout the topological intersection number of B = V ( e , e , p F ) with (cf. Equation (B.9)for p i ) S ≡ S (1)( , ) − / = V ( p , p , p , p , p ) . (B.19)in the elliptic fibration b Y over the base space B = P (cf. Appendix B.4). To constructthe structure sheaf of the variety V ( p , p , p , p , p , e , e ) , we model the coordinate ring51f X as R = Q [ s , s , s , s , s , s , u, v, w, e , e , e , e ] with Z -grading s s s s s s u v w e e e e K B E -1 -1 1 E -1 -1 1 E -1 -1 1 E -1 -1 1 (B.20)Then, an f.p. graded (left) R -module O S · B which sheafifies to O S · B is given by R ( − K B ) R ( − K B ) R ( − K B − H + E ) R ( − K B − H + E ) R ( − K B − H + 2 E ) R ( − E + E ) R ( − E ) T (cid:16) p p p p p e e p F (cid:17) T −−−−−−−−−−−−−−−−−−−−−−−−−−→ R ։ O S · B → . (B.21)Denote this sequence by F M −−→ F ։ O S · B → . A minimal free resolution is given by → F M −−→ F M −−→ F M −−→ F M −−→ F M −−→ F M −−→ F ։ O S · B → , (B.22)where rk( F ) = rk( F ) = 1 , rk( F ) = 6 , rk( F ) = 7 , rk( F ) = rk( F ) = 19 , rk( F ) = 25 . We compute the Euler characteristics of the e F i by computing their sheaf cohomologies.The latter is performed by use of the Künneth formula. Namely, since X = P × P F ,and H k ( X , L ) = H k ( P × P F , L ) = M i + j = k H i ( P , L ) ⊗ H j ( P F , L ) . (B.23)we can easily compute the cohomologies in question from line bundle cohomologies on P and P F . The Euler characteristics of the vector bundles e F i are χ ( e F ) = 1 , χ ( e F ) = − , χ ( e F ) = − , χ ( e F ) = 384 , (B.24) χ ( e F ) = 699 , χ ( e F ) = 266 , χ ( e F ) = 0 . (B.25)It follows that S · B = χ ( e O S · B ) = χ ( e F ) − χ ( e F ) + χ ( e F ) − χ ( e F ) + χ ( e F ) − χ ( e F ) + χ ( e F )= 1 − ( −
50) + ( − −
384 + 699 −
266 + 0 = 18 . (B.26) We could use the actual coordinate ring for the fibration over P . This ring has indeterminates andis Z -graded. As a consequence, the resulting computations take longer than the ones performedwith the simpler ring. Both lead to the same result. The twists of these free modules and the mapping matrices are huge. We therefore omit them here. .2.2. Line bundles from Chow ring of toric ambient space Let us repeat the intersection computation S · B by using S ≡ S (1)( , ) − / = V ( s , p ) − V ( e , p ) − V ( s , v ) | ˆ Y , (B.27)instead. Similarly, B = V ( e , e ) | ˆ Y . We define S ′ , T ′ ∈ CH ( X , Z ) via S ′ = V ( s , p ) − V ( e , p ) − V ( s , v ) , T ′ = V ( e , e ) . (B.28)Then, it follows S · b Y T = S ′ · X T ′ · X V ( p F ) . Explicitly, we find V ( s , p ) · X V ( e , e ) = V ( s , s w, e , e ) = V ( s , s , e , e ) ,V ( e , p ) · X V ( e , e ) = ∅ , V ( s , v ) · X V ( e , e ) = ∅ . (B.29)From a primary decomposition, we find h s , s , e , e , p F i = h e , e , s , s , s i . Note that e = e = 0 fixes all other homogeneous coordinates of P F . Hence π ∗ (cid:0) S · b Y T (cid:1) = V ( s , s , s ) . (B.30)If we consider B = P , then it follows from Equation (B.26) that V ( s , s , s ) must bea divisor of degree 18 on C ( , ) − / . Indeed, this is true because K P = 18 . It is not toohard to repeat this computation and find that A ′ in Equation (B.7) gives D ( , ) / ( A ′ ) = 3 · V ( K B , s , s ) , (B.31) D ( , ) − / ( A ′ ) = − (cid:2) V ( s , s , s ) − V ( K B , s , P H ) (cid:3) , (B.32) D ( , ) − / ( A ′ ) = 3 · V ( K B , s , s ) , (B.33) D ( , ) / ( A ′ ) = − (cid:2) V ( s , s , s ) − V ( K B , s , P R ) (cid:3) , (B.34) D ( , ) ( A ′ ) = 3 · V ( K B , s , s ) . (B.35)In this expression, we are using P H = s s + s ( s s − s s ) , P R = s s + s ( s s − s s ) . (B.36)By considering K B -th roots and adding spin bundles on the matter curves, one arrivesat the root bundle expressions summarized in Appendix B.3. B.2.3. Line bundles from fiber structure
Finally, let us present a third way to compute the induced line bundles. Even thoughthis approach is equivalent, it provides more intuition than the brute-force intersectioncomputations in CH ∗ ( X ) . To this end we make use of the genesis of the G -flux andthe fiber structure of b Y , which we outlined in Appendix A.53et us apply this strategy for the Higgs curve. We first recall that A ′ in Equation (B.7)can be thought of as A ′ = A ′ ( , ) / + A ′ U (1) = 15 · A ( , ) / + 3 · π ∗ (cid:0) K B (cid:1) · σ ∈ CH ( b Y , Z ) , (B.37)where (in abuse of notation) σ denotes the canonical lift of the 1-form associated to thesection s = V ( e ) via the Shioda map. On general grounds, it now follows that π ∗ ( S R · A ′ U (1) ) = q U (1) · K B (cid:12)(cid:12) C R . (B.38)For the Higgs curve, we have q U (1) = − / . Thus, π ∗ (cid:16) S ( , ) − / · A ′ U (1) (cid:17) = − · K B (cid:12)(cid:12) C ( , ) − / . (B.39)Note that (c.f. Equation (2.22)) C ( , ) − / · C ( , ) / = Y ∪ Y . Hence, the intersectionnumber of the Higgs matter surface and A ′ ( , ) / is found in the fiber over Y and Y : A ′ ( , ) / (cid:12)(cid:12)(cid:12) Y · S ( , ) − / (cid:12)(cid:12)(cid:12) Y = (1 / , / , , , / , · (0 , , , , ,
0) = − / , (B.40) A ′ ( , ) / (cid:12)(cid:12)(cid:12) Y · S ( , ) − / (cid:12)(cid:12)(cid:12) Y = (0 , / , / , , / , · (0 , , , , ,
1) = +1 / . (B.41)This implies D ( , ) − / (cid:16) A ′ ( , ) / (cid:17) = 15 · (cid:20) − Y + 12 Y (cid:21) . (B.42)We now use Y + Y = K (cid:12)(cid:12) C ( , ) − / (c.f. Equation (2.22)) to conclude that D ( , ) − / (cid:16) A ′ ( , ) / (cid:17) = 15 · (cid:20) − Y + 12 Y (cid:21) − · K (cid:12)(cid:12) C ( , ) − / (B.43) = 6 K (cid:12)(cid:12) C ( , ) − / − Y . Noting that Y = V ( s , s , s ) , and P H = s s + s ( s s − s s ) , we thus find D ( , ) − / (cid:16) A ′ ( , ) / (cid:17) = − (cid:2) V ( s , s , s ) − V ( K B , s , P H ) (cid:3) . (B.44)This is exactly the result that we found in Equation (B.32). Similarly, the line bundleexpressions found in Appendix B.2.2 for C ( , ) , C ( , ) − / , C ( , ) / can be verified byusing this strategy. For the quark-doublet curve, the situation is more involved since thematter surface flux is defined over this very matter curve so that self-intersections are tobe taken into account. Equivalently, we can give a quick argument by noting that thedivisor in question must be a linear combination of the Yukawa loci on C ( , ) / . Any ofthese Yukawa loci Y , Y , Y admits a pullback description: O C ( , )1 / ( Y ) ∼ = O C ( , )1 / ( Y ) ∼ = K B (cid:12)(cid:12) C ( , )1 / , O C ( , )1 / ( Y ) ∼ = 2 K B (cid:12)(cid:12) C ( , )1 / . (B.45)Therefore, the bundle must be of the form n · K B (cid:12)(cid:12) C ( , )1 / and the prefactor n is fixedby the chiral index. This gives D ( , ) / (cid:16) A ′ ( , ) / (cid:17) = 3 · K B (cid:12)(cid:12) C ( , )1 / .54 .3. Root bundle constraints By repeating the intersection computations, we obtain the root bundle constraints asfunctions of K B (c.f. Equation (2.37)). Since we analyze the case K B = 18 in moredetail momentarily, let us list the root bundles for such base spaces explicitly: curve g P d BN-theory C ( , ) / = V ( s , s ) P ⊗ , ) / = K B (cid:12)(cid:12) ⊗ C ( , )1 / h h ρ P ⊗ , ) − / = K B (cid:12)(cid:12) ⊗ C ( , ) − / ⊗ O C ( , ) − / ( − · Y ) h h ρC ( , ) − / = V ( s , s s + s ( s s − s s )) ... ... ...10 7 12 C ( , ) − / = V ( s , s ) P ⊗ , ) − / = K B (cid:12)(cid:12) ⊗ C ( , ) − / h h ρ P ⊗ , ) / = K B (cid:12)(cid:12) ⊗ C ( , )1 / ⊗ O C ( , )1 / ( − · Y ) h h ρC ( , ) / = V ( s , s s + s ( s s − s s )) ... ... ...10 7 12 C ( , ) = V ( s , s ) P ⊗ , ) = K B (cid:12)(cid:12) ⊗ C ( , )1 h h ρ The parameter ρ from Brill-Noether theory [40] provides a measure of how likely itis to find a degree d line bundle with certain number of global sections – the larger ρ is, the more likely such bundles exist. Notably, this parameter does not take the rootbundle constraints into account. See [38, 42] for an application of Brill-Noether theoryto F-theory and further explanations.For the considered F-theory Standard Model constructions, the toric base spaces mustsatisfy K B ∈ { , , , } [31]. Therefore, let us list the root bundle constraints forthese values of K B . Analogous to the above table, the root bundle constraints on C ( , ) − / , C ( , ) / , C ( , ) follow once the constraints on C ( , ) / , C ( , ) − / are known.For ease of presentation, we will merely list the constraints on C ( , ) / and C ( , ) − / :55 B curve g P d BN-theory C ( , ) / = V ( s , s ) P ⊗ , ) / = K B (cid:12)(cid:12) ⊗ C ( , )1 / h h ρ P ⊗ , ) − / = K B (cid:12)(cid:12) ⊗ C ( , ) − / ⊗ O C ( , ) − / ( − · Y ) h h ρC ( , ) − / = V ( s , s s + s ( s s − s s )) ... ... ...7 4 0 C ( , ) / = V ( s , s ) P ⊗ , ) / = K B (cid:12)(cid:12) ⊗ C ( , )1 / h h ρ P ⊗ , ) − / = K B (cid:12)(cid:12) ⊗ C ( , ) − / ⊗ O C ( , ) − / ( − · Y ) h h ρC ( , ) − / = V ( s , s s + s ( s s − s s )) ... ... ...8 5 6 C ( , ) / = V ( s , s ) P ⊗ , ) / = K B (cid:12)(cid:12) ⊗ C ( , )1 / h h ρ P ⊗ , ) − / = K B (cid:12)(cid:12) ⊗ C ( , ) − / ⊗ O C ( , ) − / ( − · Y ) h h ρC ( , ) − / = V ( s , s s + s ( s s − s s )) ... ... ...10 7 12 C ( , ) / = V ( s , s ) P ⊗ , ) / = K B (cid:12)(cid:12) ⊗ C ( , )1 / h h ρ P ⊗ , ) − / = K B (cid:12)(cid:12) ⊗ C ( , ) − / ⊗ O C ( , ) − / ( − · Y ) h h ρC ( , ) − / = V ( s , s s + s ( s s − s s )) ... ... ...13 10 6 .4. Limit roots in base space P We consider the smooth, complete toric 3-fold base P , whose Cox ring is Z -graded x x x x x x x x x x x I SR = h x x , x x , x x , x x , x x , x x , x x , x x , x x , x x ,x x , x x , x x , x x , x x , x x , x x , x x , x x , x x , x x , x x ,x x , x x , x x , x x , x x , x x i . (B.47) P is a particular triangulation of the 39-th polytope in the Kreuzer-Skarke list of toricthreefolds [81], hence the name. It follows that K P = 18 . Furthermore, for D i = V ( x i ) ,we find non-trivial topological intersection numbers D i D D D D i · K P D i ∈ { D , D , D , D } , we find D i · V ( s i ) · V ( s j ) = ∅ for any s i , s j ∈ H ( P , K P ) . Thedivisors D , D , D intersect the generic curve V ( s i , s j ) trivially but admit non-trivialintersections with non-generic curves.In Section 4.2, we discussed roots on the quark-doublet curve C ( , ) / . Here, weprovide details on the limit roots on C ( , ) / = V ( s , s s + s ( s s − s s )) . We use s → s − s , s → s − Y i =1 x i , s → Y i =1 x i , (B.49)and generic s , s , s to deform this curve into C • ( , ) / = V Y i =1 x i , s − s ! ∪ V Y i =1 x i , s − s ! ∪ V Y i =1 x i , s + s ! ≡ Q ∪ Q ∪ Q . (B.50)It is important to verify that this curve is nodal so that we can apply the limit roottechniques outlined in Section 3. A computationally favorable description is that a57oint p is a node if and only if the Jacobian matrices vanish identically at p but theHessian matrix does not [80]. Therefore, it is readily verified that for example, the points V ( x , s − s , s − s ) are indeed nodes.Consequently, we proceed to identify roots P • ( , ) / that solve the root bundle constraintin Appendix B.3 and admit exactly three sections. For this, it suffices to constructsolutions to (cid:16) P • ( , ) / (cid:17) ⊗ = K B (cid:12)(cid:12) ⊗ C • ( , )1 / ⊗ O C • ( , )1 / ( − · Y ) , h (cid:16) C • ( , ) / , P • ( , ) / (cid:17) = 3 , (B.51)where Y = V ( s , s , s ) . We notice that Y ∩ Q = Y ∩ Q = ∅ , which implies (cid:16) P • ( , ) / (cid:17) ⊗ (cid:12)(cid:12)(cid:12)(cid:12) Q = K B (cid:12)(cid:12) ⊗ Q , (cid:16) P • ( , ) / (cid:17) ⊗ (cid:12)(cid:12)(cid:12)(cid:12) Q = K B (cid:12)(cid:12) ⊗ Q ⊗ O Q ( − · Y ) = K B (cid:12)(cid:12) ⊗ Q ⊗ K B (cid:12)(cid:12) ⊗ ( − Q = K B (cid:12)(cid:12) ⊗ Q , (cid:16) P • ( , ) / (cid:17) ⊗ (cid:12)(cid:12)(cid:12)(cid:12) Q = K B (cid:12)(cid:12) ⊗ Q . (B.52)These observations allow us to draw a weighted graph, which encodes roots P • ( , ) / on C • ( , ) / . This graph is displayed in Figure 2.We find it important to mention that this graph is non-planar. This is remarkablebecause all other dual graphs considered in this work are planar. To our knowledge,there does not seem to be any result in the literature which suggests that the dualgraph of a nodal curve is necessarily planar. In fact, most of the literature, such as [79]and [80], only discuss examples of nodal curves with planar dual graphs. Although thereare well-known planarity criterion theorems, such as the Kuratowski’s theorem [83], weresorted to excessive computational checks to verify that C • ( , ) / has a non-planar dualgraph Figure 2. A more minimalistic example of this sort is the nodal curve whose dualgraph is K , . There are many interesting questions concerning planarity that arise ingraph theory, such as criterion theorems [74, 75], enumeration [76], and other variants ofplanarity [77, 78]. However, the significance of non-planarity for the geometry of nodalcurves is unknown. We hope to return to this interesting question in the future.Turning back to solving Equation (B.51), we note that the degrees of the roots encodedby Figure 2 are listed in Figure 3. In particular, the total degree is d = 84 , as expectedfor χ ( P • ( , ) / ) = 3 on this g = 82 curve. Recall that we identify the number of globalsections from Corollary 8, i.e. we add the number of sections on all curve componentsexcept the exceptional P s, which are colored in blue. Therefore, it suffices to focus onthe curves C Q , C Q , C Q , C Q , C Q , C Q and C Q . Each curve C Q and C Q admits36 roots whereas C Q only admits a unique root. These roots each have one section.It follows from Proposition 3 that of the roots on C Q , C Q , C Q and C Q , each curveadmits at least 35 roots which have no sections. We have thus found at least · solutions to Equation (B.51). In future works, we wish to investigate which of these rootbundles stem from F-theory gauge potentials in H D ( b Y , Z (2)) .58
142 1515 42513 33 33 33 33 33 33 33 33 33 33 33 3 33 33 3333 33 33555 A C
55 5 4 44 D
33 3 3 33 F
55 5 4 44 G
33 3 3 33 I K M
66 6633033000000 000000 A B
11 1 2 22 D
11 1 2 22 E
11 1 2 22 G
11 1 2 22 H K L
36 3618018000000 000000 C B
55 5 4 44 E
33 3 3 33 F
55 5 4 44 H
33 3 3 33 I M L
66 6633033000000 000000
Figure 2.: Weighted diagram of roots P • ( , ) / on C • ( , ) / which solve Equation (B.51). CDF GIK M − − C Q − C Q − − − − − − − − − − − − A BDE GHK L C Q − C Q − C Q C BEF HIM L − − C Q − C Q − − − − − − − − − − − − Figure 3.: Degrees of roots P • ( , ) / on C • ( , ) / encoded by Figure 2. Exceptional P s are indi-cated in blue and each carry a line bundle of degree d = 1 . eferences [1] P. Candelas, G.T. Horowitz, A. Strominger and E. Witten, VacuumConfigurations for Superstrings , Nucl. Phys.
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