Geometric constraints in dual F-theory and heterotic string compactifications

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Citation Anderson, Lara B., and Washington Taylor. “Geometric Constraints in Dual F-Theory and Heterotic String Compactifications.” J. High Energ. Phys. 2014, no. 8 (August 2014).

As Published http://dx.doi.org/10.1007/JHEP08(2014)025

Publisher Springer-Verlag

Version Final published version

Citable link http://hdl.handle.net/1721.1/91263

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Detailed Terms http://creativecommons.org/licenses/by/4.0/ JHEP08(2014)025 Springer July 5, 2014 bundle. We May 27, 2014 : August 5, 2014 : 1 : P Accepted Received Published 10.1007/JHEP08(2014)025 doi: Published for SISSA by b [email protected] , . 3 and Washington Taylor a 1405.2074 The Authors. c F-Theory, Superstrings and Heterotic Strings, Superstring Vacua

We systematically analyze a broad class of dual heterotic and F-theory models , [email protected] E-mail: Department of Physics, Robeson Hall,850 West 0435, Campus Virginia Tech, Drive, Blacksburg,Center VA for 24061, Theoretical U.S.A. Physics, Department77 of Massachusetts Physics, Avenue, Massachusetts Cambridge, Institute MA of 02139, Technology, U.S.A. b a Open Access Article funded by SCOAP Keywords: ArXiv ePrint: a constraint on thelimits extent on to the way which in the(4962) which the F-theory gauge heterotic threefold group bundle bases can canmodels for decompose. be dual where We enhanced, the F-theory/heterotic explicitly common construct corresponding constructions all twofoldexamples in to base of the surface the subset is general of toric, results. and give both toric and non-toric fibered Calabi-Yau fourfold. Weof match stable these conditions vector with bundlesinto conditions on the for the conditions the heterotic under existence that side, which many and allowed such F-theory show bundles models that correspond canexceptional F-theory to be structure vector gives bundles constructed. groups, new on and the In insight for determine heterotic particular, side bundles a with we topological of find condition this that type. is only We satisfied show that in many cases the F-theory geometry imposes the two sides of thethe duality. heterotic Specifically, theory we give is afibered compactified complete with on classification a a of single models smoothF-theory section where Calabi-Yau model and threefold has carries that a smoothformulate is corresponding simple irreducible elliptically threefold conditions vector for base the bundles, that geometry has and on the the the F-theory form dual side of to a support an elliptically Abstract: that give four-dimensional supergravity theories, and compare the geometric constraints on Lara B. Anderson Geometric constraints in dualstring F-theory compactifications and heterotic JHEP08(2014)025 7 24 26 18 5 28 14 10 21 on divisors on curves 18 5 9 15 7 f, g f, g 28 8 36 15 20 9 37 32 27 – i – 22 21 34 23 8 1 4 31 34 29 20 1 6 5.3.1 Spectral5.3.2 covers Localized5.3.3 matter and spectral Limitations covers of spectral covers 4.2.1 Constraints4.2.2 on SU(2) and SU(3) Constraints enhancement from codimension two loci 4.1.3 F-theory4.1.4 bounds on twists Toric4.1.5 bounds on twists General4.1.6 constraints on geometry from General constraints on geometry from 4.1.1 General4.1.2 constraints from F-theory geometry Constraints for toric bases 2.2.5 Summary of 6D duality and relevance for 4D 2.2.1 Dual2.2.2 6D geometries Geometric2.2.3 conditions on vacua Moduli2.2.4 and the stable degeneration Constraints on limit bundles and gauge symmetry 1.2.1 Classification1.2.2 and enumeration of models Topological constraints on symmetries and spectra 5.2 Matter spectra5.3 in heterotic theories The spectral cover construction 5.1 The Bogomolov bound 4.2 Constraints on gauge enhancement 3.3 Heterotic/F-theory duals with toric base surfaces 4.1 Constraints on threefold base geometry 3.1 Geometry of3.2 heterotic/F-theory duality Possible base surfaces for smooth heterotic/F-theory duals 2.2 Heterotic/F-theory duality in six dimensions 1.3 Outline 2.1 Heterotic/F-theory duality in eight dimensions 1.1 Introduction 1.2 Overview of main results 5 Heterotic constraints 4 F-theory constraints 3 Heterotic/F-theory duality in four dimensions 2 Lessons from heterotic/F-theory duality in higher dimensions Contents 1 Introduction and overview JHEP08(2014)025 39 43 44 46 47 48 48 57 2 65 B ] of possibilities that 5 – 3 61 η 52 40 50 ]. Tremendous effort has been expended in 64 39 2 , 66 1 53 – 1 – with a codimension two singularity but no 3 61 F = 2 42 40 38 B i η 2 m P F 46 43 bundles over bundles over 4 2 3 1 1 P P gauge group 10.1 Detailed physics of10.2 smooth heterotic/F-theory Expanding dual heterotic/F-theory pairs duality 7.9 Generic G-flux that breaks gauge symmetry 7.4 dP 7.5 dP 7.6 dP 7.7 An example7.8 of an upper Examples and with lower non-trivial bound chiral on matter 7.1 7.2 7.3 An F-theory model over 6.1 Effective condition6.2 on Effective constraint6.3 and gauge groups F-theory constraints6.4 and SO(32) models Base-point free6.5 condition A note of caution: G-flux and F-theory) on different geometric spacesexploring [ the range of theoriesfor theories that in can higher be dimensions withmodels realized extended has through supersymmetry, the a such range tractable compactification. of scope,known possible While for constructions string theories seem in to four give dimensions rise with to minimal such supersymmetry a vast “landscape” [ 1.1 Introduction Since the early days of string theorytheories it has in been four known that and astring wide higher range theories dimensions of (and different can physical their be more realized recently by discovered higher-dimensional compactifying relatives ten-dimensional M-theory B A brief exploration of rigid bundles 1 Introduction and overview 10 Conclusions and open questions A Properties of elliptically fibered Calabi-Yau three- and fourfolds 8 Consequences for heterotic bundles 9 Enumeration of heterotic/F-theory dual pairs with toric bases 7 Examples 6 Equivalence of constraints JHEP08(2014)025 (1.1) 1 − n B 3 admits a more K −→ ] suggests that a = 1 theories that +1 8 n – +1 Y N n 6 Y : f , π + 1)-fold with section over a +1 ]. In this paper we analyze how n n 9 Y F-theory on – 2 – ]. The effective low-dimensional theories arising and the K3-fibered manifold = 1 supersymmetric theories with chiral matter ] ] for some recent systematic studies). The duality ⇔ 1 20 , − N 18 n 22 1 , – 19 B − 21 n 10 B E −→ n X : h , π n X is elliptically fibered over n X For those theories that have dual heterotic and F-theory constructions, the compacti- A major obstacle in any systematic attempt to classify the possible compactification Compactifications of heterotic string theory and F-theory provide two corners of the Nonetheless, it may be possible by analyzing specific string constructions to ascertain = 1 theories from string theory may represent only the tip of a much larger iceberg Heterotic on fication geometries take the form [ where detailed description as an elliptically-fibered Calabi-Yau ( threefolds and fourfolds isthe even manifolds finite, that much determine less thebackgrounds, how effective involving theories. dual to heterotic characterize There and is, all F-theoryCalabi-Yau compactifications however, the threefolds at on properties and elliptically least fibered fourfolds, one of class whereometries of the is number finite, of and topologically some distinct systematic string analysis ge- is possible. relevant for string phenomenology and give rise to the low-energygeometries physics we and see in effective nature. rent theories limitation for on either ourknown, the mathematical for heterotic example, understanding whether of string the the or number relevant of F-theory geometries. distinct is diffeomorphism classes It the of is cur- Calabi-Yau not from compactifications of both heteroticby string the theory background and geometry F-theory of arebility that highly the these constrained compact constraints dimensions. might becan Indeed, strong arise it enough (in is to any characterize an dimension) whichcompactifications attractive from effective possi- heterotic theories to or 4 F-theory dimensions, compactifications, used or in to the characterize case which of string geometries could be erotic string compactifications; some recentpossible work effective has theories sought that to can exploreory be and on enumerate obtained a the from smooth compactification Calabi-Yaubetween of (see smooth the [ heterotic heterotic compactifications the- andalso equivalent been 4D broadly F-theory explored; constructions see has e.g. [ as a step towards acan more arise systematic from understanding string of the theory. space of 4D string landscape where 4-dimensional and exceptional gauge symmetries arise naturally. There is a tremendous literature on het- some global constraints andification systematic of features string of the theory.pactifications and theories associated Recent that constraints work arise onsystematic from on 6D analysis compact- supergravity globally is theories possible classifying [ inment 6D of six some string/F-theory dimensions aspects and com- of may thegeometric provide space constraints tools of on 4D for compactifications two a [ general similar classes treat- of string compactifications to 4D are related, they impose on 4DN physics. It iscomposed suspected, of in compactifications fact, describednon-K¨ahlerand that by non-geometric the more compactifications. general known mathematical constructions objects of including 4D it is difficult to systematically study the set of allowed models and the constraints that JHEP08(2014)025 ] 3- 25 = 1 K N and are . In the 8 n E B ⊆ i H -flux. While a further associated with smooth G . For the most part, in 1 n there are in general many − B ) are connected on the F- n 1 1 B − − n n ), but in four dimensions there B 1 B P with structure groups , Higgsing/unHiggsing transitions in the 3 n X B . The classification of such dual theories can be – 3 – 1 . For any given 2) on − n , -flux is clearly desirable, a good understanding of n B G B . At the next level of detail, theories can be classified 1 = 1 − i n ( that characterize allowed F-theory geometries. Theories i B , which would be associated with singular small instanton V 2 n B B ] for a discussion of the different roles that heterotic 5-branes can play. on the F-theory side fixes some of the topology of the dual heterotic theory by adding to the Calabi-Yau geometry a pair 19 fibered over (including those with different n 8 1 B E n P B × 8 -flux on the F-theory side lifts some moduli and can give disjoint sets ]. We primarily focus in this paper on classifying theories at the level E G 24 1 – . In dimensions eight and six there is a unique 1 22 − ], for a more constructive argument in the context of Weierstrass models). In n 7 B which is itself n . The choice of B n Beginning on the heterotic side of the duality, it is known that the number of topo- In this work we focus on 4-dimensional effective theories arising from heterotic string In some cases, the dual geometries we consider may include heterotic 5-branes wrapping the elliptic B 1 described in the of holomorphic vector bundles SO(32) heterotic theory, only a single vector bundle is used.fiber of the For CY fixed threefold. bundle See [ topology logical types of smooth(see elliptically also fibered [ Calabi-Yau threefoldsa with section heterotic is dimensional finite reduction,field [ the configuration 10-dimensional over gauge the field Calabi-Yau and threefold the must vacuum be gauge taken into account. These are resolution. For bothgeometries the Calabi-Yau where threefold the andgauge elliptic fourfold bundle geometries, fibrations in we admit the considerwrapping heterotic a curves only theory in (single) is the section. smooth,configurations. base and that We there assume are that no the heterotic 5-branes the underlying geometric structure that wefirst focus step on in in a this paper systematic seems understanding to of be general an 4D important theory F-theory on models. a smoothfibered elliptically Calabi-Yau fibered fourfold Calabi-Yau that threefold admits and a F-theory compatible on elliptic a fibration dual and has a smooth theories in 4D, of string vacua associatedthis with paper compactifications we on concentrateindependent a on of the given features moduli that liftingunderstanding depend and of other only the issues on consequences associated of the with geometry of heterotic bundles, but notclasses all of (specifically, the it bundles). fixeseffective some theory, For components which a correspond of given the toheterotic choice deforming second of bundle along/tuning Chern pictures, moduli canand in modify correspondingly the the F-theory modify gauge and the group bundle of structure the group low-energy on effective the theory, heterotic side. For by the geometry ofdistinct the topological F-theory base types of with different bases theory side by tensionlesstransitions string [ transitions and onof the heterotic side by small instanton done at increasing levelsthe of geometry. complexity At by the coarsest includingthe level, successively base the more theories can information be about heterotic/F-theory classified dual by the pairs topological (respectively typeare a of many point distinct possible and bases base JHEP08(2014)025 ≤ ) there V ( 2 2 c , c 1 ≤ c corresponding to 3 B generalized del Pezzo . We demonstrate here . The number of distinct 2 . 2 B 3 B X ) (note that in the case exceptional bundles over any generalized del V ( 3 1 c ’s are toric. The direct enumeration P 2 B )), it is known that the moduli space bundle over 4 V ( 1 bundles) is a straightforward calculation, c P is restricted to be a 1 )). ] appeared, in which magnetized brane models were 2 P 5.2 29 B – 4 – must be built and here we restrict our attention 3 B , when the geometries are smooth on both the het- 3 0 mod 2, and the second Chern class is bounded as 0 (built as 3 ≡ that can act as bases of smooth dual heterotic/F-theory ’s can be enumerated for any 1 B 2 = 1 supersymmetry in 4D has only finitely many compo- 3 c B B N ]. 7 supports an elliptically fibered Calabi-Yau fourfold. Moreover, 3 B can be constructed as a ) and a total Chern class, 3 is toric. The powerful mathematical toolkit of toric geometry allows ) fibered threefold V B 1 2 P is no longer a topological invariant). To argue that the number of heterotic geometries B 3 c ]. Although these proofs of general finiteness are at present not constructive, compatible with ], of which there are a finite number of topologically distinct types. Over these 27 2 , 28 ] guarantee that for stable, hermitian bundles with fixed first and second Chern classes 26 [ = 1 string vacua and their effective theories, as well as understanding constraints on ) by heterotic anomaly cancelation (see eq. ( 3 3 27 , The general structure just detailed is illustrated clearly in the simple case of 4D models This class of dual heterotic/F-theory models thus represents a reasonable starting point As we discuss further in section N As this paper was being completed, the paper [ More precisely, the moduliFiniteness space of of Mumford the semi-stable number sheaves on of heterotic geometries here is established in two steps. First, the results 26 4 2 3 TX ( 2 are a finite numberstructure of groups, possible valuesis for finite the we must third furtherc Chern observe that class considered over smooth ellipticallymodels fibered in Calabi-Yau this threefolds class over was del shown Pezzo to bases, be and finite. the number of described in more detail. of [ 1.2 Overview of main results For the convenience ofpaper the that we reader, believe have we some summarize novelty, and here indicate where some in of the paper the these main results results are of the for simple and direct computationsgeneralized in del Pezzo this surfaces class ofCalabi-Yau threefolds examples. and fourfolds, While only there 16of are of these all hundreds of associated F-theorywhich bases we carry out in this paper as an example of the general theoretical framework. with which we can4D get a first footholdthe into effective theories the arising problem from of string classifying geometry. and characterizing where the base we show that this finitebranches set of of the modulidistinct space gauge of Weierstrass group models and oversurfaces matter any in contents specific (section is 6.5 finite of) by [ a similar argument to that for base surface [ bases a rational ( to the case where that there are a finitePezzo number surface of such topologically that distinct of bundles nents it seems possible to systematically construct at least one importanterotic class and of F-theory dual sides models. the base surface (specified by rank( JHEP08(2014)025 . 9 that there is finite, and ]. The details 2 independent of 4.1 38 B giving rise to smooth 2 ) structure groups the ]. This construction is is finite. We construct B N 2 ’s to a finite enumeration 39 for F-theory models with , . Thus, for these structure B 3 B 2 19 [ B ). Irreducible bundles can only ) or Sp( N N over any 3 B . Some of the most interesting aspects ) or Sp( 6 , and classify the generic associated effec- ] and Berglund and Mayr [ N 3 37 B . These results are quite general, and do not and smooth heterotic duals bundle – 5 – 2 ]). These results are described in section for F-theory models with smooth heterotic 1 over smooth bases 4.1.6 B P 36 3 -bundles over toric surfaces – 3 B 1 B 30 P spectral cover construction and section bundles 1 P , and write a simple set of topological conditions that characterize 4.1.5 -bundle bases 1 ’s we find 4962 threefolds P 4.1.3 2 B that can be built as bundle structure. 3 1 B P ’s in section 3 B used to build bundles withbe structure constructed group via SU( acondition spectral of cover base-point when freedom. thebase second We point Chern find freeness class that condition ofthe for can the assumption SU( be bundle of satisfies derived any a from particular method the of F-theory bundle geometry construction perspective, while the last itemfrom below the describes heterotic aspects point of of matter view. content thatHeterotic are clearest bundles and the base-pointOne free of condition the mostheterotic general compactification is methods the known for explicitly constructing bundles suitable for theories. This extends earlier workof of this Rajesh correspondence [ areof elaborated this in correspondence section duality arise than when the other. a Ingroup constraint particular, and is the bundle next better structure two that items understood are describe on currently constraints on understood one the most side gauge clearly of from the the F-theory 1.2.2 Topological constraintsMatching on geometric symmetries F-theory and constraints spectra andWe heterotic show bundle constraints that theretheory models is and a conditions for close the correspondence existence of between smooth, the slope-stable bundles geometric in constraints heterotic on F- theory bases elliptically fibered fourfolds forFor the F-theory 16 compactifications toric withtive smooth theories. heterotic These manifolds duals. systematically add studied to to the date dataset (see of [ Calabi-Yau fourfolds that have been depend on toric geometrybeyond or the any other specific conditions on theEnumeration F-theory of base models geometry with toric As a concrete example of the general classification results, we explicitly construct all F- follows from the factthe that number the of number possible ofexplicit “twists” generalized bounds of del on the the Pezzoproblem twist surfaces in that section reduce theallowed classification of Classification of duals As described at theare conclusion a of the finite previoussmooth number section, heterotic of we duals show in on section elliptically fibered Calabi-Yau threefolds with section. This 1.2.1 Classification and enumeration of models JHEP08(2014)025 . . . 7 5.3 7.8 cameral -bundles de- G . These limitations on gauge + 1), giving rise to 4D GUT 7 n E with a brief review of the dual- 2 with examples given in section ] can in principle provide construc- ]. In particular, in the dual F-theory 8 44 45 , 43 – 6 – and section 4.2 ) bundles. More general methods such as the cannot be enhanced to an N 6 or SU(5) symmetry [ E 6 E ] and other approaches to constructing general 42 – ) and Sp( 40 [ N . ] based on a theorem of Looijenga [ 9 , 19 . Examples of cases where this condition is violated in F-theory are described 7 6.4 In addition, we find that many F-theory models that should have smooth heterotic symmetry contain chiral matter. Examples of this type are described in section 6 1.3 Outline This paper is organizedity as between follows: heterotic we string beginon theory in the and section nature F-theory of in heterotic/F-theory dimensions duality, eight the and classification six. of models, We and focus constraints arise due to theG-flux heterotic is incorporated. geometry, with On the implicationstral heterotic for side cover these the construction correspond with dual to bundles structuretheories F-theory built with, group model via for given the when example, spec- by SU(2 geometries enumerated in this work,E we find that many examples of theories with generic enhancement are described in section Chiral matter The circumstances under which thestood low-energy on theory the has heterotic chiral side. matter are We better identify under- a class of situations in which chiral matter must tions SU(2) and SU(3) gauge groupsnot are be constrained in by a F-theory broken geometry phase sowith of that the an they structure SU(5) can- of gauge a group. codimension Incodimension one these singularity in two cases the the singularities F-theory restriction geometry. is related associatedfor In to other example, cases, in matter some fields cases constrain an enhancement — so that, Limitations on gauge enhancement Geometric constraints on the F-theorylow-energy side theory not given only a provide compactification a topology,the minimal but gauge gauge can group group also for can limit the be the enhanced extent over to a which given base geometry. For example, in many situa- has important consequences for thegroup problem breaking/enhancement. of vector The bundleThe deformations base-point corresponding and free symmetry F-theory condition conditionsection is and described circumstances in forin section its sections failure are described in cover construction scribed in [ tions of bundles withclassification these of situations more in general whichto bundles structure with exist groups. exceptional — structure groups Our thoughan are analysis in open expected gives many problem a in cases general geometry. explicitly describing The property the of properties base-point-freedom of and such its bundles violation is also in the moduli space. duals violate this base-pointbundles free having condition. exceptional orconstraints We SO(8) show for structure that SU( groups these and are thus all do not associated violate with the above groups base-point-freeness of (a part) of the second Chern class is required for all bundles JHEP08(2014)025 ] for 47 . Some ]. From 3 surface 10 53 K , 3 surface in is the unique ] (see [ 52 K 1 we summarize ]. For example, P 46 3 , ]. 51 = – , is dual in certain 21 1 1 22 and SO(32) heterotic , B P 48 8 , 21 E , → 46 × , 19 2 8 contains some examples. In Y 22 surfaces, glued together along E : 7 3 degenerates into a fiber-product of 9 K π heterotic theory, this duality can dP , 8 we compare the constraints on the 2 E Y 6 × 8 E – 7 – 3 surface K 3 that produce different non-Abelian symmetries limit corresponds to decomposing the to be large in the heterotic theory. The heterotic K 8 and a smooth heterotic dual construction. Finally, a 2 E 2 T is a point, and the F-theory base B × 0 8 B E describes the constraints on both the Calabi-Yau threefold and 5 base (corresponding to the positions of 7-branes in the language of . In the case of the 8D ] for details. 2 1 gives a more detailed description of the geometric constraints on the T P 48 — the so-called “stable degeneration limit” [ 4 we describe the results of the systematic enumeration of all smooth F- 5 (i.e. Wilson lines) can be matched to the symmetries arising from ADE 9 2 T we summarize the consequences of our study for heterotic bundle moduli spaces, 8 In the 8D stable degeneration limit, all the features of the two theories, including the volume modulus is mapped into a complex structure modulus of the For the limit which produces the SO(32) heterotic theory, the bundle over this point. Thus, there is a single moduli space of 8D models connected by 5 2 1 In this case the baseP manifold “Higgsing” type transitions that reduce or increase the size of the gauge grouprational by surfaces, de-tuning see [ bundles on degenerations of the ellipticover points fiber in of the Type IIB); these degenerationsthe were point classified of view mathematically of by the classification Kodaira of [ models and constraints, the 8D story is quite simple. into a (singular) fiber productan of elliptic two curve elliptically fibered moduli parameterizing the vacua,the can possible be gauge matched groups exactly arising [ from different configurations of the heterotic flat gauge (separate) limits of its parameterstring space theories to on the perturbative be understood most explicitlyrealized in by the taking weak coupling theT limit volume of of theF-theory. effective theory, Geometrically, which the is Beginning with the initiala formulation review), of the F-theorywindow duality in through of which 8 F-theory both dimensions theories withcompactified [ can the on be better heterotic an understood. string elliptically In has fibered 8 provided dimensions, an F-theory important technical details are relegated to appendices. 2 Lessons from heterotic/F-theory duality2.1 in higher Heterotic/F-theory dimensions duality in eight dimensions two sides and show when theyinformation are about equivalent and the when geometry onesection side of of the the duality other provides side. new and in Section section theory geometries with toric base brief summary of this work and associated open questions are given in section dimensional compactifications in theheterotic/F-theory remainder duality of in the four paper.interest. dimensions In and Section section describeF-theory the side, range and section of constructionsbundle of geometries on the heterotic side. In section on the effective theory, illustrating features and tools that are helpful in analyzing four- JHEP08(2014)025 ]. 3- 12, K 24 , 1 are ≤ ], cor- 3 fiber − 21 [ n K 22 1 , must have F can also be 21 G = 2 with ]. A complete P 2 8 n B F = that is elliptically 2 3 B Y 3 fibrations are taken ’s that admit a single 2 K , in which the 1 B − ), a heterotic theory on an n B ] consists of a finite family of is dual to F-theory on a 1.1 7 , where the Hirzebruch surface 1 n → ]. Codimension two singularities − . The base F 1 n 56 P is an elliptic fibration over a more +1 B → ’s corresponds to the mathematical n 2 3 Y 3 = Y B → Y : . As in ( 1 1 : n f B − π π X n : B ], in which curves of self-intersection ], though in this paper we describe aspects of h 19 – 8 – π 58 , , n . As in 8D, codimension one singularities in the 57 1 [ X B fibration over -fold 1 n ]) is reached. A systematic classification of F-theory bases P , which is a blow-up of a Hirzebruch surface [ 2 59 B ]). with the same base, 55 bundle over +1 1 n , there is a connected moduli space of elliptically fibered Calabi-Yau P Y 2 B ], and the more general “semi-toric” class of 60 minimal surface theory ]. The global description of the moduli space is much more complicated on ’s [ 61 2 + 1)-fold is itself a B ], reviewed in [ n , and the dual F-theory geometry is a Calabi-Yau threefold n 1 54 F P ([ In principle, heterotic/F-theory duality can be extended beyond the set of smooth dual In lower dimensions, heterotic/F-theory duality is understood by fibering the 8D du- = action [ = according to the intersection properties of effective divisors is given in [ 18 , or the Enriques surface [ , 1 ∗ 2 2 2 2 B enumeration of allowed basesfor is toric in principleC possible and has beenthe heterotic carried side, out where explicitly multiple coincident small instantons must be analyzed systemati- tions. The global spacesuch of moduli 6D spaces F-theory connectedthe compactifications by [ set tensionless of string type moduliframework transitions. of spaces associated The connectivity withblown of down distinct until aP minimal surface (in this case a Hirzebruch surface general 2-dimensional base responding to a morerealized general on the F-theory side, e.g.For after each a choice tensionless of string transitionthreefolds from describing a set of 6D theories connected by Higgsing and unHiggsing transi- non-simply laced groups whenencode monodromy matter is fields. present [ geometries by incorporating non-perturbativetheory. effects such In as this NS5-branes case, in the the dual heterotic F-theory geometry compactification space is aB K3 surface thatfibered is with elliptically section over fibered aB over Hirzebruch the surface, commonelliptic base fibration encode a gauge group in the F-theory picture, which in 6D can include 2.2 Heterotic/F-theory duality in2.2.1 six dimensions Dual 6D geometries For dual heterotic/F-theory compactifications to six dimensions, the perturbative heterotic elliptically fibered Calabi-Yau fibered ( is in turn ellipticallyto fibered as be described compatible, above. andprimarily The in both elliptic the are stable and degeneration chosenthe limit to duality [ that have are sections. true more This generally, independent duality of has this been limit. studied F-theory and heterotic sides,a the root only lattice constraint thatΓ is can be that embedded the into gauge the algebra unique signature (2,ality 18) over unimodular a lattice nontrivial shared base manifold or tuning moduli to modify the singularity structure of the elliptic fibration. On both the JHEP08(2014)025 and n > ] and D 22 12 yields , − 21 . n n < F − that are sub-bundles = 2 ]). On the heterotic V B 47 12 (originally described ], the degrees of freedom ≤ 56 n , ], that is, that they are slope- 1 22 , 21 with self-intersection ) with structure groups embedded into n 2 F ) must have vanishing first Chern class. However, ,V 8 1 . The constraint = E that cannot be resolved to yield a total space 2 V 2 ⊆ 2 B – 9 – B B H are traceless, all principal bundles 8 E (arising in the heterotic theory from the decomposition of the . On the F-theory side, this corresponds to the choice of n ) and to make useful statements about their properties. 2 V... ± 2 ,V ∧ 1 = 1 (minimal) supersymmetry in 6 dimensions. On the F- V, ,V ) are reducible, then their first Chern classes can be non-zero and satisfy 3 N 8 ) = 12 2 E , K 1 0 (mod 2). for the two-fold base V ( n ≡ 2 ]. Furthermore, the first Chern class of the principal bundles must F ) c . On the F-theory side this constraint follows from the fact that for ). We restrict attention in this paper to smooth heterotic/F-theory dual i 2 64 , V 1 – ( 1 V c 62 2.2.4 , each side of the duality encodes some nontrivial information about the geome- D with factor satisfy the Hermitian Yang-Mills equations [ bundle (i.e. that have structure group 0 (mod 2). 6 8 8 ]) gives a simple example of the type of geometric constraint that we explore later in ≡ E E The choice of a smooth heterotic/F-theory dual pair in 6D is determined by a single One of the most significant aspects of this duality is the way that the dual theories ) 3 surface and that the gauge bundles ( i More precisely, since the generators of -dimensional adjoint of 21 6 V K ( 1 the interpretation of these moduli in the two dualof an pictures. Asif described the in associated [ vector248 bundles c that is a Calabi-Yau manifold.on Thus, the in two this sides case give the rather same different analytic geometric constraint considerations 2.2.3 on the structure of Moduli the and allowedIt the theories. is stable worth degeneration briefly reviewing limit the moduli of heterotic/F-theory compactifications in 6 slope-stability of the vector bundles andrelates the heterotic the anomaly second cancellation Chern condition which gauge class bundles of the holomorphic12 tangent bundle the of existence K3 of witha an singularity those effective in of divisor any the in elliptic fibration over integer. In the heterotic theorythe vector this bundles appears as theHirzebruch choice surface of a fixed secondin Chern [ class for this paper for 4D compactifications. On the heterotic side this constraint follows from the a each (poly)stable [ vanish, theory side this appears asspace the condition is that a the Calabi-Yau totalcase elliptically threefold the fibered (more compactification resolved precisely,thought geometry the of is as manifold a a can singularside, be Calabi-Yau M-theory limit, this threefold; singular, as condition in reviewed the for corresponds which F-theory example to in description [ the can statement be that the compactification manifold is and effective theory of heterotichow compactifications the were used dual to Calabi-Yau threefold developspectra the geometry “dictionary” determines in of the the gauge F-theory symmetriesheterotic description. and backgrounds matter ( F-theory can in turn be usedrealize to the enumerate condition possible for geometries where no small instantons arise, which in2.2.2 6D limits us to Geometric conditionsEven on in vacua 6 try of its dual theory. In early explorations of F-theory [ cally (section JHEP08(2014)025 ) 6) g is the j > ] for a 6 in 4, K 66 base of the j < 0 (2.3) i > 1 , respectively P g ≥ 0 (2.2) ] and [ ) 3) fibered three- ≥ , which is embed- with J . On the F-theory ), where K 1 8 ) and 65 j 1 has structure group , I − K H E z f 2 6 − (6 V 56 ⊆ − , in n ( 1 (4 ), a Calabi-Yau threefold 6 1 O n H z 22 , -fibered threefolds. In this , ) 9 of 3 surface, while polynomials 3.3 21 K 1 ) (2.1) dP can be described in Weierstrass K and 4 2 G can be matched exactly to that 4 n − 1 , z ( i z F 1 Y )) + 20 (2.4) O z i ( V g 3) (see [ ( 0 and coefficients of + K ), of has structure group i x 1 f 8 such that 8 + 12 such that 12 + Y ) V End 2 , ≤ ≤ are (non-CY) 3 , z , and polynomials of high degree ( I J 1 4 in 1 2 K ], the heterotic/F-theory dictionary in 6D indi- , z – 10 – play a dual role in parameterizing the spectral V ( 1 ( encoded in the singularity structure of the elliptic 1 Y f k 1 22 h , i < G + , g and k ) 3 f 21 2 ) = ) n x 3 surface, Def( , i z 2 F ( where Y z with . The bundle K ) 19 = ( ( 2 j ) 2 1 i 1 i 2 , V − z Y 2 y − ; these explicit expressions give a local coordinate description on the F-theory side, the moduli encoded in the functions (6 h 3 2 (4 n K 2 n B . Parts of this moduli space in principle give a complete encod- B ∪ 8+ 12+ 2 = 0. Similarly, the second heterotic bundle , associated with the singularity structure of the F-theory elliptic 1 g f 2 B 1 i j 1 1 Y ) and the are sections of the line bundles z i z z G V . This correspondence can be made precise in the stable degeneration → I J =0 =0 i j X X f, g ∞ 3 Y = ) parameterize Weierstrass models for all elliptically (and ) = ) = 1 )) correspond to the moduli of the heterotic 2 2 ) are coordinates on z 2 2.3 , z , z 2.3 1 1 , z , and the resulting gauge group is the commutant ), where the polynomials over the base 1 z z 8 ), ( ( ( z 3 ), ( E g f 5.3 Y 2.2 As is now well understood [ ( 2.2 with commutant that is elliptically fibered over the Hirzebruch surface 3-fibration, giving rise to symmetry 2 3 2.2.4 Constraints onFor bundles a and gauge fixed symmetry topologyf, g of folds ing of each dual heterotic moduli space of sheaves with fixed total Chern class, denoted The correspondence between theof F-theory the moduli dual in heterotiction the bundles ( Weierstrass is model particularlycover and divisor transparent moduli on in the the heterotic side. spectral cover construc- limit, the infinitesimal deformationof space, the Def( bundles Def( modern treatment of this result in terms of limiting mixed Hodge structures). That is, side this corresponds to aK Calabi-Yau threefold with 7-branesfibration wrapping at the the point H fibration at limit, in which in ( of low degree (coefficientsparameterize one of of the heteroticparameterize bundles the other bundle ded in More abstractly, canonical class of the base of generic sections of these line bundles. cates that the “middle” polynomials (the coefficients of Y form as where ( easily confirmed in the toric description (reviewed in section JHEP08(2014)025 , 2 G as 3. c 3 K Y 3. In K effective D on = 0, fixed ω are generically 1 7 , over which the c n n F F ). → ] of K¨ahlerform on in 1 3 19 n Y as a Lie group. For example, has only one component for any bundle with that − : i bundles appearing in heterotic ω H π 4 F M in the 6-dimensional heterotic , 8 G E possible to bundles with smaller structure for a general discussion of the relevant i ⊆ V H the rank of 5.2 i , but also provides information about ] (see also figure H not 56 [ , 8 3 E Y 3, the threefolds ≥ – 11 – for a given vector bundle appearing in a representation of fixed rank ). The first of these is that for 2 n theories and matched exactly to complex structure , c . By tuning the complex structure to special loci in 1 3 D ) for , c Y 2 fixed subject to , c i 1 V , c (rank ω of fiber dimension 12, it is possible to deduce a number of facts about the dual i M H (rank encodes the unbroken symmetry ω corresponds to changing the gauge symmetry of the 6 ,... 8 i M E V = 0 that change the rank. A change in the rank (and hence structure group ⊆ ). In the F-theory geometry, this gauge symmetry can be varied by i n 8 8 V there is a maximum structure group ), that are stable with respect to a chosen K¨ahlerform , E H 2 n 2 ) refers to a fixed F 2 c , c ⊆ 1 has rank 4 as a Lie group, one relevant moduli space for , c i 1 , c 4 G F , since While the gauge symmetry of a 6D theory can often be made smaller or larger by The moduli space structure on the F-theory side is matched non-trivially not only to rank, c (rank To avoid confusion it should be noted here that the rank appearing in the definition of the moduli space It is important to note that while heterotic/F-theory duality is believed to hold for the full moduli ( determined by the decomposition of the adjoint of ω ) of the bundles 8 7 ω i i spaces of stable sheaves that are stable forM the appropriate “adiabatic” choiceH [ although theories would consist ofbundles, rank ranks 26 and group vector representations. bundles. See section Chern class topology space of the thefor two theories, suitably the weakly explicit coupled “dictionary” regions between of degrees parameter of space. freedom is On only the well heterotic understood side, this corresponds to moduli is a 6-dimensional gaugehypermultiplets symmetry associated that to cannot thethis be complex corresponds Higgsed to structure the away moduli.elliptic presence by fibration of giving must In a become vevs the singular divisorgauge to and F-theory of group has the geometry, self-intersection factor. a Kodaira On type the associated heterotic with side, a this nontrivial implies that for certain values of the second Higgsing or un-Higgsing throughthere de-tuning are and also tuninggroup constraints can moduli on be in both brokensingular the or sides (though F-theory expanded. that they picture, For can admit a restrict smooth the resolution). extent to This means which that a in gauge these cases there groups. In the reverse process,a in cases symmetry, where there complex is structure a deformationsby simple exist to realization charged break in hypermultiplets. the effective Completehave theory been “chains” as determined of a for direct these “Higgsing” thedeformations breaking/enhancement of dual of patterns 6 the corresponding threefolds theory ( changing the complex structuremoduli of space, it is possible toover augment divisors the Kodaira in singularity types thethis of base, the corresponds elliptic on which fibration the enhances heterotic the side gauge to group specializing symmetry of the 6D theory; — corresponding to adescribed connected above. deformation space of a dual Calabi-Yau threefold the irreducibility of deformations of H this context, nontrivial featuresmatched. of Considering the only F-theory heterotic onbruch and the elliptically bases F-theory fibered geometries threefolds with canmoduli the be 13 spaces Hirze- exactly of sheaves and the structure group M JHEP08(2014)025 × 4) , 0 r, ( ω = SU(3) , which is M C H ]. 22 is SU(2), for any of self-intersection V C 7 6 4 F E E 2 (we will sometimes for symmetry [ 6)) can occur. Note also that in L L L L L L III IV 7 L L 9 8 7 r > 2 12 11 10 H H H 2 3 H H H H H E G N > SO SO SO ). In F-theory this follows from SO SO SO SU SU N ). In particular, in any 6D F-theory 1 L L L L L 6 5 4 3 2 H H H H H SU SU SU SU SU must vanish to degrees 2, 2 on – 12 – f, g L L L 4 3 2 H H H Sp Sp SO SU(2) with non-standard commutants (e.g. , singularity associated with a curve or SO(12) in figure L 3 H 7 ) = 4, the maximal structure group of IV to SU(3) E SU V 8 ( E 2 c L 2 H SU . The top row above the horizontal blue dashed line corresponds to an alterna- 3. It follows then that the moduli space of stable sheaves, 2 is singular with a generic, non-Higgsable having nonzero degrees of vanishing lie to the right of the vertical red dashed line, K ]. Figure depicts Higgsing possibilities based on heterotic bundles with structure ,I 8 3 I F 56 f, g , which match with dual F-theory models. F-theory gauge groups from Kodaira sin- 8 → E 3 . The possible Higgsing/Enhancement chains for smooth heterotic/F-theory dual pairs; ⊂ Y to type : H π A fact that is perhaps not generally well appreciated is that there are also cases in 6D for upper SU(3)), generically associated with matter in the adjoint representation, which on the 3, the group cannot be enhanced to any larger SU( 2 up/rightward paths towards construction where the low-energy theory hasarising a from generic a (non-Higgsable) gauge Kodaira group− type SU(3) the fact that the Weierstrass coefficients brevity refer to such athat moduli space as “empty”). This corresponds in F-theory to thewhere there fact are nontrivial constraints onby the “unHiggsing” ways the in generic which model a in gauge a group given can component be of enhanced the moduli space (i.e. possible the heterotic theory some Higgsing chains lead to product gauge groups, as discussed further in text. theory. For example, if such bundle on contains no locally free sheaves (i.e. smooth bundles) for zontal dashed lines correspond toIV,III transitions between different Kodaira typestive in Higgsing F-theory sequence from from type G F-theory side involves wrapping on higherthat genus do curves not for have 6D heterotic models. duals, further Note unHiggsing that (e.g. in to F-theory SU( models Figure 1 modified from [ group gularities with such gauge groups can beunHiggsed forced to from anything the left geometry of (geometrically the “non-Higgsable”) line. and The cannot SU(3)’s be and SU(2)’s connected near the bottom by hori- JHEP08(2014)025 ) 4 F 9 ]. For in 69 – in SO(10) , however, 0 are rigid (or SO(12)) 67 ⊕ O , a heterotic m , if we begin 7 1 26 1 E ⊕ O m > (located on the , corresponding are particularly V 8 n 1 16 → ) matter in the ad- F E V with N 27 in a heterotic model. = is + V 2 8 3 m 12 can have a structure 10 B Y E − V > , but the group cannot be ⊂ ) → i ... bundle V bundle over K3 with instanton G ( 27 6 2 6 ← . V c E E 4.2 2. This condition corresponds on the ) models containing any representation SO(7) N ← n > SU(4) via . While in 6D, those F-theory models with 2 G ... → . 3 – 13 – X ← symmetry is unbroken. In these cases, no smooth 8 E SU(3), for example. In the latter case, explicit compu- -fold antisymmetric tensor representation [ × k SU(5) 2 . The divisors of self-intersection (i.e., the reduction of the G for this case, it is possible to move upwards (un-Higgsing) ← m 2 ). This corresponds on the heterotic side to no structure group , the obstacles to gauge group enhancement can be stronger = n ± G IV type singularity for 1 F 6= 0 this can only happen on one side, so only one gauge group H dual to smooth heterotic models, the gauge group factors live on → n . This is also true for SU( m or 2 A m 6 B 12, in which cases the generic symmetry of F of self-intersection +2. An SU(3) gauge group with an adjoint results, SU(3). Another interesting feature that can arise in this context is the = 0 in E is trivial and full 1 ˜ × S z = ,... 2 H G H = 9 path in figure n denotes the trivial line bundle over there are irreducible curves of genus one that have twice the divisor class of the 2 O F some Constraints on heterotic bundles dual to F-theory models on The set of possible Higgsing/unHiggsing chains in dual pairs of 6D heterotic/F-theory symmetry can be broken to SU(3) in several distinct ways, depending on whether the Here 9 8 strong for patch containing at all. That is, on the heterotic sidegroup to such the as fact thatappearance only of bundles product with gaugeOn groups the as F-theory one side thisclass of corresponds with the again positive factors to self-intersection, now a given gauge as group a on sum a of multiple two of irreducible the parts. divisor ple, in irreducible curve corresponding to an SU(3) factormutant. in Note the that dual for heteroticcan model have with adjoint the matter non-standard representations com- and lie on the top line of figure other than the fundamentalF-theory and models on divisors of self-intersection and cannot support acan higher be genus taken curve. with higher The multiplicity, divisors giving of a self-intersection higher + genus Riemann surface. For exam- commutant is tation of the branching rulessmaller generically groups along gives the matter Higgsing injoint chain. the representation is adjoint In only representation 6D possible F-theory forin when models, the the the SU( gauge F-theory group base arises on a curve of higher genus dimensions there are ato number gauge of group constraints enhancement of in this more type. detail We in explore section themodels 4D contains a constraints number ofE other interesting features. As depicted in figure with the lower rightand SU(3) to the rightunHiggsed along to the the right-hand path path smooth SU(4) heterotic duals have gaugealong groups that can alwaysfor be more enhanced to complicated 6D F-theory models (without smooth heterotic duals), and in four heterotic side to a constraintnumber on 9 the can extent be to deformed whichthe to the enhancement a SU(3) bundle withis reduced structure possible, group. while In theis the enhancement case not. at SU(3) Phrased hand, differently, in terms of the Higgsing chains given in figure incompatible with an JHEP08(2014)025 , 21 ) on an 2 4 we note ,V 1 < V ]. 2 ]. 71 , 79 < c 70 3 [ (not locally free) with K 3 and Donaldson-Thomas K base of sheaves 1 3 fibered Calabi-Yau four-fold. P 3. Thus for 0 K , 2 arises. Physically, in the heterotic , ω = 1 M 2 c 3, and explore a number of new features, K 3. On the F-theory side this corresponds to the – 14 – , must contain a point where the elliptic fibration is 2 , m ]). F = 1 72 2 c ]. On the heterotic side this blowing up corresponds to the ) and b) the existence of a “maximal” rank/structure group 8 2 , c 1 is non-empty but contains only sheaves and no smooth bundles. 3 with 11 curve in ]), no systematic tools exist for constructing the moduli spaces ω r, c 3. In six dimensions, essentially all of the information that can be − K ( correspond to point-like instantons (NS/M5 branes) on the K3 [ 78 ω M K – 10 M 73 10 or ) or the corresponding Donaldson-Thomas Invariants [ 3 − , 3 (see for example [ , c 9 2 , can be independently determined using known mathematics to study the K − 2 , c c 1 , c In the following sections we use heterotic/F-theory duality to develop analogous state- In this paper, we ask many of the questions described above in the context of 4D (rank More precisely, skypscraper sheaves supported over points in the ω ], in the cases above with “instanton number” 10 topology (see e.g. [ M ments to those made above for bundles on heterotic/F-theory duality, which relates the modulielliptically fibered space Calabi-Yau of threefold vector to bundles theIn ( moduli this of context, a however, the informationfar obtained fewer is made mathematical more techniques significantsheaves by are the on known fact Calabi-Yau that for threefolds. determining Indeed, the aside moduli from space a of handful stable of examples with special for a given heterotic geometry. Both of theliterature facts, in a) and the b) study wereinvariants previously of on known moduli in the spaces mathematics of stable sheaves on encodes a deepfibered and Calabi-Yau non-trivial threefolds correspondence (and theirof between stable stable the sheaves degeneration over moduli limits) andinferred space from the of the moduli F-theory elliptically space geometrythe about irreducibility the of heterotic bundle moduli space, including a) moduli space of non-perturbative vacua from thematched heterotic to side the of geometric the duality F-theory which description, could be though we do not2.2.5 pursue this further here. Summary of 6DTo duality summarize and this relevance review for 4D of the 6D story, in 6 dimensions heterotic/F-theory duality in the vector bundle) thattheory a these compact sheaves moduli space 22 that the moduli space In principle, this line of development could be used to develop a full description of the 6D that we focus on here;them we in encounter 4D detail. analogues F-theory of geometry theseof and situations heterotic/F-theory small later duality instantons but thus and lead do to not sheafythe the study degenerations structure inclusion in of the the heterotic physicaldle/sheaf picture, theory moduli making is spaces. it in Mathematically, clear agreement anydles attempt that with alone to results the construct in mathematical a a notion non-compact moduli space. of space bun- It of is bun- only by including sheaves (i.e., degenerations fact that the so singular that the point inture the of base a must be Calabi-Yau threefold blown [ upshrinking for of the an total instanton space to tothe a have desired the point; struc- topology. in These these cases cases go there outside are the smooth heterotic/F-theory dual paradigm vector bundles exist on JHEP08(2014)025 T = in ) ] of 3 , we 9 (3.1) (3.3) = 0, ], we R . B ] that A 2 9 19 + 3.3 B heterotic Σ 8 · (1) on each E − O × 8 E bounds (based on . In section . Following [ 4 Y is Poincar´edual to the 3.2 lower 0 . For the ω we have the classes that satisfy Σ 3 . 2 (3.2) 3 3 X ) , B B 8 and E 2 ). Heterotic/F-theory duality is = 1 of 2 B ]. We begin with a general abstract × B 8 T ( 80 . On , 1 E , 2 c ( ) + , i i B 19 (SO(32)) (3.4) ]. It was shown more generally in [ ζ − 19 + = 12 0 T, O ⊕ L , = Σ . The F-theory compactification space 2 ( ω ) 2 η ± 2 P – 15 – , and the curvatures split as (see appendix ∧ + B ) B 2 i + was identified by Friedman, Morgan, and Witten , as well as constraints tying the matter spectrum 2 = ( that is elliptically fibered with a single section over V η 1 ) = 0 in cohomology. 1 , and then characterize the possible compactification 3 B 3 H c T η ( ⊕ T B = 1 X 1 c 2 3.1 and Σ i + = 2 V F and R − as a projectivization of a sum of two line bundles T = 6 ( i (1) is a bundle that restricts to the usual Tr 2 η 3 is dual to a heterotic model on , R O 1 B bundle over 1 3 30 η B 1 P = 1 supergravity theories [ bundle N ), where 1 L P ( 1 that is a c 3 = are (pullbacks of) 2-forms and 4-forms on B i , the heterotic bundles are smooth and irreducible and there are no additional is a general line bundle on the base 2 ,T , ζ i B L η (1)) )) on the bundle structure group An F-theory model on fiber. There are sections Σ O V ( ( 1 1 2 in the dimensionally reducedlimit supergravity or action, type independent of ofthe the bundle axion-curvature stable squared construction. terms degeneration in Forof the the the 4D SO(32) bundle supergravity heterotic action on constrains the string, the F-theory the twisting side analysis to [ satisfy possible when This correspondence between for bundles in the stablethis degeneration correspondence limit in follows [ directly from the structure of axion-curvature squared terms where section. The Bianchi identity gives where c P corresponding to the relation theory, the bundle decomposes as a threefold turn acts as acan base construct the for an elliptically-fibered Calabi-Yau fourfold 3.1 Geometry ofWe heterotic/F-theory focus duality on the bestis understood on class a of smooth dualities, Calabi-Yau threefold a in which base the heterotic compactification 5-branes wrapping curves in the base. In the dual F-theory compactification, we consider for four-dimensional formulation of the duality in section geometries that are smoothgive on a both more sides detailed of description the of duality constructions in that section involve a toric base surface c of the theories to topology. 3 Heterotic/F-theory duality inWe four now briefly dimensions review the duality between heterotic string and F-theory compactifications unique to the 4D theory. These include deriving strong upper and JHEP08(2014)025 , ) 2 3 = H K B 6 H × − 1 ( = 4 over H O , g T ) = supports an . When such + K ⇒ 3 H 4 T ) allows for the B − 3 . These features ( fX 4 B = 4 ) is Y O and + ], such models can n . For any given base ], and included into ( 3.3 3 2 + 82 are determined by the 4 81 B X Σ i F , G = − -flux on 2 G ); these components together Y , corresponding to singularity fixed by ( 2 2 3.2 , G 1 defined by , and in some cases distinguished η × in ( ]). The F-theory fourfold geometry 3 1 ]. Note that the Kodaira singularity 1 i B 19 ζ G bundle (i.e., knowing 83 , ,[ 1 = P ]. As described in [ A 56 , G 82 22 – 16 – . The singularity structure of the F-theory ellip- i ) except for i G V as listed in table ( . The minimal group factors 2 ) (appendix is 3 c 2 8 B B f, g ( E 2 in a completely generic elliptic fibration over ]. There can only be a smooth heterotic dual when the only c with one section and smooth total space, not all topological with the topological data in + 3 of the dual F-theory 2 + i 22 Σ , 2 X , B , which are as before sections of line bundles H T ) g − 21 2 B ); for this we must consider in addition ( i 1 and V c ( f 3 c = 11 2 )). ζ and the twist 1 2 P + is the canonical class of B 1 = ζ K 1 A central piece of information on which we focus in our analysis is the generic gauge For elliptic fibrations Note that we assume that the elliptic fibration on the heterotic side has precisely one B ( , there is therefore a minimal gauge group 1 3 c by monodromy around the singulardictates divisor the [ physical gaugediffer algebra by only; a differentare discrete theories somewhat factor may cavalier that about have thethe gauge does distinction reader groups not between should that gauge affect keep algebra the in and algebra. mind gauge group, that but In in most much cases of the this only paper structure we fixed by the local on the heterotic side over where the commutant of tic fibration is determinedparameterized in by terms ofwhere a Weierstrass model minimal degrees of vanishing of 4D supergravity theory [ nonabelian gauge group factors are associatedB with the divisors Σ structures present over Σ a gauge group is present, it implies that the largest possible structure group for a bundle group in the low-energy 4D supergravity theorydual corresponding pair. to a given On heterotic/F-theory theelliptic F-theory fibration side, that the maycation threefold have of singularities base such along singularities certain indicates the divisors. presence of The nonabelian Kodaira gauge classifi- group factors in the also does not fix each correspond to one undetermined parametermany on of the the heterotic consequences side. thatbundles It can — is be discussed interesting derived later that from inin F-theory this these paper for two — the parameters are structure on largely of the independent heterotic of heterotic any side. possible freedom features of the heteroticthe bundle base are determined byidentification the of dual all F-theory components geometry. of satisfy Knowing kind of F-theory dual. Weside, leave this elliptic interesting question fibrations for without furtherthe work. a moduli On global the space section F-theory be of were understood Weierstrass explored from models in Higgsing infrom [ abelian Higgsing [ nonabelian U(1) symmetries, symmetries, allF-theory which of and which can should heterotic in have descriptions, a turn though clear be parallel we understood between do the not pursue this here. 6D case where the SO(32)2 heterotic theory is dual to F-theory on section. It isnontrivial possible Mordell-Weil that group) fibrations — with or with more multisections than — one may independent admit section some (i.e., more general with for a dual SO(32) heterotic compactification to exist. This generalizes the corresponding JHEP08(2014)025 , 4 is 7 + A 0 z so f = = 2, the f )) then the g b to have an + otherwise. The a deg , 4 f, g f + ( ) the algebra is b = 2 X arising in 4D supergrav- of interest, where + )( f i b D G aX − 4 f otherwise. Similarly, when , 2 + X 7 9 g i 2 2 e 2 , 7 )( , so e 3 vanish to various degrees, and the H a X 6 su su , e 8 . If this cubic can be algebraically trivial su )( − 3 a so g f, g . For more general F-theory models ], with for example a generic gauge X , including the effects of monodromy. 6 − + F ± 21 [ 2 X X g 2 2 , m 4 f is a perfect square, and 7 2 f su i F 7 8 , , + 4 e e G so are those that can be forced to arise from around the divisor 6 3 su g 3 , e . Since the precise conditions that determine 8 1 su X D – 17 – so ··· + when g 2 6 z e 2 g deg over the divisors Σ is a perfect square, and + z f 2 f, g 1 g g 12 2 2 2 3 3 3 4 4 5 5 ]. In general, the gauge group in a particular model may be larger deg + 8 0 ], this can be determined by performing an expansion g when = 3 is somewhat more complicated; in this case, the algebra depends 83 = 3 , if it factorizes into the form ( , g 8 su 56 = 4 the algebra is , g so deg g , ··· over the curve of self-intersection -6 in + deg = 2 6 , 2 E f z . The gauge algebra summands associated with group factors 2 = 3 f f In a number of places we will need to know the precise gauge algebra associated with The only gauge factors listed in table + z 1 deg case deg on the factorization properties offactorized the cubic into a product ofgauge three algebra terms is to the form ( As described in [ f an algebraic coordinate thatthe vanishes monodromy on arehere expressed a differently succinct ingauge summary various of algebra places the is in possible the situations. literature, we When collect deg singularity that arises in theof generic models elliptic with fibration different over gaugethe that groups minimal base. can gauge be While group tuned a for over wide each each variety base. base, we aregiven focused degrees here of on vanishing of “non-Higgsable” matter [ than the minimum groupF-theory dictated base by threefolds it the is structure possiblesingularity to of corresponding tune to the the a base. Weierstrass coefficients gaugedoes factor For not arise example, SU(5) as over over an certain some automatic divisors, consequence though of this the group geometry of any base corresponding to a 6D F-theory models overgroup Hirzebruch of surfaces (i.e., those without smooth heteroticgroups, duals) some there involving can multiple beof gauge more structures group complicated arising minimal factors gauge in — maximally in Higgsed analogy 6D to F-theory the constructions, the which general can set contain singularity structure of the F-theory geometry is the gaugethe algebra. structure of the F-theoryin base cases threefold, with independent of a the smooth choice heterotic of dual. elliptic fibration, This is analogous to the generic gauge groups for Table 1 ity theory from divisorsassociated on structure which group Weierstrass factors parameters for dual heterotic bundles. JHEP08(2014)025 1 3 ]. 3, P ). − 1 84 [ ≤ × P ], the n 1 n × ˆ 60 P . E 1 ≤ 4 = P 2 or above, 0 = − F 0 generalized del F with 0 . In principle, all 2 n B 3 or below consist of dP − that only have effective 2 B , matching with the expectation 2 curves having an intersection ]. Over each of the possible base . We use these conditions in the 9 points, the bases 8 ) that admit an SO(32) heterotic 2 giving different geometries on the 9, there are generalized del Pezzo − 2 g ≤ so 86 bundle over a base surface with , B T n ( 1 1 85 P c . Our restriction to models with smooth < n < 6.3 = 2 1 7.3 ]. When the base contains a curve of self- T n, 8 2. These 16 bases are listed in table over which the elliptic fibration geometry is a on the F-theory side — is fixed uniquely for any − – 18 – 2 with a set of 2 curves, corresponding to generalized del Pezzo 3 2 curve), and the broader class of blown up at B − B can be described using toric geometry. In [ n − 2 2 P dP B ]. For each with a 28 0 F 2 [ 2 or above. with a smooth heterotic dual can be classified by determining , given by − − 3 n that contain no curves of self-intersection B , and 11 other toric generalized del Pezzo surfaces with various 2 2 dP B F for each del Pezzo and generalized del Pezzo (and for T — and hence the base and = 9; these surfaces are classified in [ T 0 surfaces 3 or below, the total space of the elliptic fibration becomes singular, and in n F − there are a wide range of possible twists 2 B (which is a limit of surfaces, which are in general described as limits of del Pezzo surfaces containing = 9, the classification is slightly more complicated. There are 279 rational elliptic 2 F The set of bases It is also worth noting that in the cases n del Pezzo . The corresponding minimal gauge group is always 2 complete set of toriconly bases interested in for those elliptic caseswhich threefold where restricts fibrations all us was effective to curves enumerated.the only have self-intersection surfaces 16 Here possible we bases: are combinations of the curves del of Pezzo self-intersection surfaces heterotic side on thepossible Calabi-Yau F-theory describing bases an ellipticall fibration allowed over twists 3.3 Heterotic/F-theory dualsA with particularly toric simple base class surfaces of bases For surfaces with different combinations of surfaces with surfaces F-theory side. Each such geometry will correspond to a different class of bundles on the and Pezzo curves of self-intersection surfaces corresponding tostructure limits corresponding of to any proper subgraph of the extended Dynkin diagram an example of such aheterotic model duals is means given that incurves we section of limit self-intersection our analysis here to bases the intersection general the heterotic theory acquiresspecial an cases enhanced where gauge an group.curves F-theory can construction Note apparently that on exist there a withoutno are an smooth extra some heterotic nonabelian dual gauge in group, such cases though as there the is dual still Calabi-Yau geometry would be singular; 3.2 Possible baseThere surfaces are for only smooth a heterotic/F-theorysmooth limited duals Calabi-Yau class on of the bases heteroticelliptically fibered side. Calabi-Yau threefold The exists set canstructure of be classified complex of according base to effective surfaces the divisors intersection over which on an the base [ analysis in several places in the remainder of thedual, paper. the twist B from the heterotic side. This is shown in section and if it does not factorize algebraically, the algebra is JHEP08(2014)025 2 a B D used (3.9) (3.6) (3.8) (3.13) (3.10) (3.11) (3.12) 3 + 4 for bundle. directly Z k 1 2 = P B = 1), and the bundle over 3 r ± 1 N , P 0 , (0 1). The coordinates , , 0) . , i 6. 1 , ) in this basis. Any toric D = (0 , and write i 4 ± ) } i , that defines the k , y y . +1 . i . This describes a base with (0 k . For convenience we will use | ≤ T } x 2 + 1, are 3D rays that project } ≤ , + 3 − v c T ], we characterize the base α | Z α i i k 0) , ∀ ∀ x | = ( , , , = b ≤ ≤ 0 88 i | 4 6 , − , , v 2 1 a . | 1 , which is the trivial i , , . . . , k i 0) and 87 , a N (1 1 | are in one-to-one correspondence with ± ) 0 , ≤ i D P D (3.5) (3.7) is a coordinate choice on 1) i ∈ i i f i ≥ − 0 i ≥ − , t X x y 0) 0) ∓ × i , α ∈ { α , , , a +1 +1 = (1 − − 1 with = 0 1 0 k , y k α w i P 0 , , , 3 a k k v =1 =1 x i i , w m, w X X m, w Z × D h 0 h 1 – 19 – : and : ∈ w ∼ = = (1 = ( = (0 = (0 +1 P =0 , . . . , v , ∗ , and ) ∗ 3 3 a k X i 0 0 0 } of the dual lattice whose inner product with all of ± ± v N = +1 w +1 N s w D k ∗ k can be written as 3 ∈ ∈ ) = + 1 = w parameterizing the twist D 2 N a, b, c B 2 i B m m B t = +3 { ( { 1 ,k are associated with c = = M are the basis vectors ( , . . . , k +2 g = 0 k G F , giving α ∈ 2 a w w ) 1, then the monomials in v ∈ { K − − , a } a, b, c can then be described in terms of a 3D toric fan are the triplets ( 2 , . . . , r = ( B , the rays f, g , . . . , k m 1 = 0 1 . We can choose a basis in which P k correspond to the divisors Σ , α × ∈ { α ± 1 i is not less than -4, ) = w s 2 P α , with the third component B For a toric F-theory base, we can compute the (anti)canonical class of The toric language gives a simple description of the monomials available in the Weier- Following standard methods in toric geometry [ w a ( = bundle over are v 1 , 3 0 1 1 We have then associated with the rays As a simple example,F for the case monomials in from the toric description. There are two equivalence relations on the set of divisors Similarly, the monomials in strass description of the F-theoryB model. If the set ofthe 1D elements rays describing the toricthe threefold to The vanishing of the thirdto component eliminate of two redundant degreesindices of freedom in thethe twist total number of rays generating the 3D toric fan. where by the toric fan,h consisting of vectors of the remaining vectors defining P JHEP08(2014)025 , 3 0 A B for , we 2 (4.1) (3.14) B D < 4.1.5 · then , A 4.1.6 0 singularity ] to identify . Combined , it is called 8 8 ± e Q D < · ∈ . We begin (sec- for some relevant 2 . In section i D γ B , which give a set of 3 7.8 are that there are no B 4.1.4 3 B . effective. This means in par- -bundle threefold bases in the , we use the general conditions X we describe constraints on the 1 bundle over P 1 4.1 has a negative intersection P T, with 4.1.3 effective) S + . This result was used in [ associated with specific divisors in the X 2 X as a D ( + K T 3 D B − ]. = − X, 19 A – 20 – ). In section + on a surface by repeated applications of the preceding rule i = 2Σ having negative self-intersection i A we describe further constraints on the gauge group D 3 γ i D 4.1.2 γ K describing 4.2 i − T ) also follows straightforwardly from the definition of ) must vanish on X . While in higher dimensions the Zariski decomposition can A ( = A 3.14 A , meaning that of , this gives a set of conditions that are necessary, but not suffi- ∈ O T s and twist 2 . As discussed below for more general F-theory geometries with a with multiplicity B i i D i D t P ) and the adjunction formula [ as a component = 3.1 ) with a brief overview of the general F-theory constraints, which are easy to make D T In the analysis in this section we repeatedly use a basic result from algebraic geometry, Similarly, we have 4.1.1 Zariski decomposition of rigid divisors When such a decompositionthe is carried out over the rational numbers with an irreducible effective divisor contains ticular that any section the “non-Higgsable clusters” that classify6D the F-theory intersection compactifications. structure of More base generally, an surfaces effective divisor may contain a number sufficient conditions for an acceptablethree F-theory singularities model, and G-flux subject that toaspects we issues do of not from G-flux address codimension for here 4D (see section F-theory compactifications). which states that if an effective divisor base. The toric descriptionwe of derive these a simple constraints set iswith of given the necessary in constraints conditions section on associatedcient, with for the the divisors existence of Σ include a more good general, nonlocal F-theory conditions compactification associated geometry. with In curves section in in the Kodaira classification.the These conditions base can geometry be describedtion in terms of constraintsexplicit on in the toric contextto (section derive a set of local constraints on the twist through “unHiggsing” by moving on the Calabi-Yau moduli4.1 space. Constraints onThe threefold base basic geometry conditions oncodimension one an or F-theory codimension threefold two base loci geometry with singularities worse than the 4 F-theory constraints In this section weclass describe of the models geometric with constraintsthreefold smooth on geometry, heterotic and duals. in In section of section the corresponding 4D supergravity theory and the extent to which it can be enhanced where heterotic dual, the formulagiven ( in ( JHEP08(2014)025 , on D ∩ (4.2) ± 2, and f, g is that − 3 . base locus B = Σ = , when the C . D 7.8 · 3.1 + 6 D i 12 then the model α , vanish on a divisor in m, w 2 of the form h g ]. , known as the 3 i 91 ∈G B 12 on a point then the point D = 4 or 6 on any irreducible 6 on any curve, or the curve m i , , + 27 γ n 3 . When the degrees are nonzero, can refer either to the divisor in ]. We focus in this paper on the f α The degrees of vanishing of for = 0. D 89 D = min reach 8 g D 11 there must be a Calabi-Yau fourfold α · 3 D < n f, g X B , while ]. As described in section deg 3 B 46 is a curve of self-intersection , are given by , – 21 – α D 22 effective, w + 4 , 6 on any divisor i , 21 X α [ ∆) reach or exceed (4, 6, 12) on a divisor, the fibration cannot vanish to orders 4 3 B m, w f, g, with h f, g for the curve in X pulled-back from a corresponding divisor in the base surface and the discriminant ∆ = 4 ∈F m 3 C + B f, g D = min = 2 f 6 on a codimension three locus but do not exceed 8 . Similarly, A , α D using the explicit description of the Weierstrass monomials as elements of D associated with the ray 3 α B deg , depending on context. D 2 4, then that admit good F-theory models. The constraint on codimension three loci is a divisor on B − 3 must admit a section of vanishing degree D B = is straightforward. For example, if 3 In a number of places in this section we focus on curves in Related constraints in the toric language of “tops” were described in [ D or in . We will generally use A , the corresponding 4D supergravity theory gets a nonabelian gauge group contribution · 11 nK 2 3 3 These are easily computed fordegrees any cannot given both base reach and or divisor. exceed 4 For a good F-theory base, these The F-theory constraints describedtheory above bases are particularly simplethe to dual lattice, describe as for describedthe toric in divisor F- the previous section. and matter content in thewith low-energy codimension theory. 3 We include singularities,future therefore work. leaving in the Codimension our three resolution analysis singularities models of are the discussed status further4.1.2 in of section these models Constraints to for toric bases (points) on the base ismust less be clear; blown if up thevanishing for degrees reach of a 4 good Calabi-Yaumay resolution. be On problematic the yet otherconstraints cannot hand, associated be if with blown the up codimension degrees directly of one [ and two loci, associated with gauge groups becomes too singular to− admit a Calabi-Yau resolution.effective divisor Thus, a constraintwould on need to beof blown up, the giving singular abases elliptic different base fibration structure, to for exist. a Calabi-Yau resolution This provides a strong set of constraints on that is ellipticallyWeierstrass fibered coefficients over B depending upon the Kodaira typeWhen of the vanishing the degrees corresponding of singularity ( in the elliptic fibration. B 4.1.1 General constraints fromWe begin F-theory with geometry a generalFor statement of a the good F-theory constraints F-theory that model hold for to any exist geometry. on a base of A where B be more subtle, for surfaces the computation of the terms JHEP08(2014)025 . . 2 2 D 1 − are B (4.7) (4.4) (4.5) (4.6) (4.3) k = . g . . Such 1 D D F , where · defined by and ) ) +12 is a section = D i k and to order − D 2 ∩ k β f 3.1 f − B − w + 2 + α K on Σ , and k + , according to table ). Thus, m, w α h + = 2 matter transforming defined in ( D ], any higher genus divisor ∈G )Σ on Σ 3 bundle over N 8 k m B 1 k . are given by − P D, k < n . − 12 β 8 · z a z D (12 with non-positive self-intersection ]), or may in some cases represent that is of the form Σ 8 = min 12 T 3 ) respectively. We can use this fact = 0. Consider then the intersection ) ∩ f T. g 3 ) derived above in the toric context B g T − 90 k 6 on any curve or the F-theory base . We consider the various possibilities ) D α B β , 2 + D − k ··· ··· D − · D 2 B in vanishes to order 3.14 ∩ Σ , in analogy to the 6D situation. In some compatible with any particular base 2 − + n + α K , using the fact that ( k β ( k C 2 2 K 2 T D , with (6 z z − − z − B k ,D 2 2 − 3 g − α f g , we have = deg 2 7.2 and to order 8 in K gives a singular elliptic fibration that cannot be D + + – 22 – ∞ K 6 D 2 − 6 z z · D − , can be determined by imposing the condition that = 1 1 B = 2Σ ( locally in a region around the locus Σ ) − 2, where f g ( z 3 T T O . Note that, as shown in [ − +8 ) O + on Σ + 2 K i k f, g 0 0 = β B k g f − − w is at on the toric curve ). Similarly, D n = = · + + + ) and ( α g T f D f, g − )Σ ) k k 2 m, w − − h nK is a section of (4 . In this case, the relation ( (8 − ∈F k ( − over any base 3 g m − 2 T − = 0, where Σ K vanishes to order 4 Σ z k k − , and = min ( z + − k f O f 3 β is a rational curve. We focus on divisors 0, since determining conditions on these divisors is sufficient to bound the total K D 4 on Σ ∩ ≤ D α − k D ( We begin with the case A set of necessary conditions on should not vanish to orders 4, 6 on any curve D − O · deg number of twists of negative self-intersection in theresolved base to a Calabi-Yau, so it is sufficient to restrict attention to rational curves f, g is associated with an irreducible effective divisordepending on upon the self-intersectionwhen of D of 12 sections of to determine constraints on the possible twists the coordinate The term described in section holds more generally from the adjunction formula applied to Writing the Weierstrass functions singular, as occurs inmatter 6D charged compactifications under (see U(1) e.g. gauge [ factors. 4.1.3 F-theory boundsWe on now specialize twists to the class of F-theory geometries that have smooth heterotic duals as 1, 2 along a givenunder curve gauge they groups generally carried indicate on the thetoric presence divisors 4D of cases, however,a there situation are is no described suchcodimension explicitly nonabelian two in gauge singularities section groups. may simply An represent example cusps of where the such discriminant becomes The degrees of vanishing of Again, these degrees cannotmust both be reach blown up or along exceed that curve 4 to give a new base. When these degrees reach or exceed they indicate the presence of a generic gauge algebra factor on JHEP08(2014)025 6 = − 6. 1) , . 2 +1, − as a 2 , 2. A k +1 = 4 | ≤ i (4.11) (4.12) (4.10) B nK bundle − ]. The D w = D 1 , though − i · 87 = (0 = P [ D i T . i v | must vanish k < n respectively, D D · k 0) k , = 0 or , g 1 D − 2 curve in form a connected i k − 6 ) contains ). In the 3D toric f − by considering the 2 0) and , )). Associated with T , i k, ) for all B ) 3.6 v k . (If k 3.8 − is = (0 − +1 − = (1 i i n )–( n ˜ 1 = 1, so we have ( n w ( , v − ( − i i 3.5 − 1 for any D v → − to degrees 4, 6, with a similar , v · need not vanish on 1 ) , associated with the extension 2 i . Thus, the twist must satisfy 2 − 5 | ≤ (4.8) − i . g (4.9) C , t Σ +1 K v D . i 12 6 1 1 nK · − ∩ , t − − so that i T , D | = 0 | ≤ | ≤ | ≤ 2 , t 1 D D D N D at least to order 4 = 0, so − · · · i · = (0 t i 2, so T T T D D T | | | · − , these reduce the problem of identifying all in the third dimension ( ) .) 1 is smooth, the third ray has the form T k – 23 – +1 , w , so we have a bound in this case of i 7 : 2 : 2 − . This means that the sections − D > + to vanish on 0) · B 1. In this case, 2 is the self-intersection of the divisor n , w ≥ = 0 : = 1 : ≥ , i in the base having self-intersection − 0 T K n n n n n , 6 associated with a given base ray f, g . , w = D k < n − i 1 − ≥ 2 in this paper. 9 − + 1 or 0 respectively in the obvious fashion to respect the 6 . This is less than or equal to D ≤ i D = (1 there are twists ≤ must vanish on · k w D 1 vanish to degrees 4, 6 on k n · − to degree +1 D i i with smooth heterotic dual geometries to a finite enumeration , g T ˜ w ) k D , v 3 f, g shows that f k i bundle over the sequence of rays B → + = 6, then ( , v 1 − 6 when 0 ) 1 P , n + 1 with 1 − D ( i i − · this intersection is negative, and ( i v − . As promised in section = 4 , 2, then , t T 2 0 n 0 over the divisor vanish on n B 1 or , ≥ 6, which would force i = k t ), where the integer − D , g i · D we can perform a linear transformation taking +1 D > k = (1 · i f 2 for any rational curve · D > T 3 ) 1 · − N T T i n, t ) T . If w − k These bounds provide strong constraints on the twists that are allowed for a Similar reasoning shows that analogous constraints hold for curves of self-intersection Now, consider the case , D < D 1 − · − n lattice Since we are assuming( that the base we replace cyclic ordering of rays). We(note can that choose this a is basis a for the different three choice of 2D basis rays thanof that the used corresponding in 3D ( rays smooth heterotic duals in section 4.1.4 Toric boundsThe on bounds twists on twists can beon seen explicitly the in twist the toric context.local geometry We can of identify the the bounds over any base smooth F-theory bases problem, since the curves ofset. negative or We summarize 0 the self-intersection results in of the a base complete enumeration of all twists over toric bases with We only need the results for 0 and for curvestwist of over more any negative rational self-intersection; curve the of complete (non-positive) set self-intersection of constraints for the ( when constraint with the opposite(For sign example, for if Σ all other component for on which would mean that T similar argument for Σ When JHEP08(2014)025 . . , , , 6 6 i 2 )– ± , , v B B B 2 4.8 = 4 + 2 (4.13) (4.14) (4.15) (4.16) (4.17) (4.18) (4.21) c i ≥ − . B ˜ t = 0. For c . Inequal- 2 ) becomes i ˜ t G n , with 0 this implies B < 4.18 F + bundle over they will also be . c 1 c < f, g i a , to avoid having a P m nt = 2, ( to determine bounds 12 when ) in the toric context for ) becomes 3 − n ∗ do not have degrees 4 3 B. ≤ B 4.11 i N +1 for any ˜ t ). The final constraint im- i 4.18 t f, g ∈ ≥ − 2 0. Then we have the follow- ) becomes )–( ) + c b, c , and since i ≥ c 1 ˜ t 4.8 i i 4.17 ˜ t − = 0, ( ˜ t on divisors i + t = 1 are shown in figure a, b, c n for a solution of all the inequalities i , to reproduce the conditions ( in the base B ˜ t B = nb c < (4.19) n , . B i f, g (4.20) = ( 6. Finally, for B D c > B ˜ t − 2 12 6 1 parameterizing the ∩ ≥ − a ≤ m ≥ − ≤ − = 2 ≤ ). For associated directly with the divisors Σ ± T i a b c b i | ≤ | ≤ | ≤ ˜ t n ˜ t i i i ˜ ˜ ˜ T t t t | | | where 4.18 = Σ → → → → → − , ) C ) becomes 2 )–( – 24 – i ˜ t 2 : = 6) for B B B B B n, 4.17 4.15 = 0 : = 1 : ≥ − 0 we need . Let us assume that − ), we need B of the twist ) must be satisfied for all monomials in n n n ( , ± < 1 , which combined with ( s D g i i ≥ − i ≥ − i ≥ − i ≥ − comes from the condition that − c > · 4.15 B i = 6, so we have the constraint + 1 4.17 − − i i ˜ − ˜ w +1 − t s T i i w B = ( + 2 ˜ ˜ w w + c m, determines the nontrivial part of the twist around the ray m, s i i ) takes h h +1 ˜ t i ˜ ) and ( i m, m, w . But this means that there must be at least one integral point sat- ˜ h h w D ≤ − · 4.12 . Therefore, we need only ask whether there can exist any solutions m, s b 4.17 → T and since B ) to inequalities ( + ∃h i − +1 = w i i = w ˜ t < B plane within which any solutions must lie (figure we used curves of the form B, b, c a ) 0 from ( c , which combined with ( − - c b ) becomes B − c < 4.1.3 + 2 1)( ) on the other hand, need only be satisfied by at least one point 4.18 is maximally negative, so if there are any solutions in c i − a ˜ t ). As in the general situation described in the previous subsection, the strongest con- i 1. To summarize, we have reproduced the constraints ( 4.17 ˜ t ≤ ≤ = 1, ( b 4.11 i ˜ 4.1.5 General constraintsIn on section geometry from on the individual components We now describe more general constraints on The constraints on coefficients in to exist. Thisn is weakest for so ( 2 t poses a condition that restricts thewhen solutions within this triangle. Thisacceptable constraint for is weakest of the form ( and since All of these inequalities exceptity ( ( 4, 6 singularity on isfying all of these inequalities.angle in The the second through fourth inequalities define a simple tri- straint on the twist component on the curves associated with ing constraints on the monomials associated with The parameter and can be constrained geometrically depending( upon linear transformation ( JHEP08(2014)025 ) − 2; 6. T . − − and ) ). If ± k − = on Σ = T − cannot 8 B n − e (4 , − − 2 is nonzero 7 − f, g e 2 5 K = parameterized g 6 K . This follows a is not effective 2 4 g − ( L B since − 2 T ( O - T , − O 5 b 4 H is not effective then 2 on the divisors Σ = yz K L T ) (4.22) c 6 2 , = 0 corresponds to Σ − b − f, g H K z 2 6 K − 6 of the discriminant locus that c − 1)( is a section of = 4) must be nonzero. The circled k respectively. Thus, we need only 5 − , g B g k ( 2 n f f K 6 + ( − , nt 2 − K = 1 over a curve of self-intersection 2 4 in coordinates where i ˜ t K 4 – 25 – − 6 . On the other hand, if yz 12 − − 2, since + c ≥ = 0. Furthermore, if £ ) = n , the components b T 5 2 6 g ).) − + correspond to weakest conditions, which hold at c 2 L 6.4 0 4.1.3 > K , b 6 0 b, c are sections of H on the base cannot give further constraints on ) respectively. In particular, − are associated with sections of the line bundles 6 ( 6 T . This point is relevant in ruling out gauge algebra factors − n - D ) i , g k 4 ³ D f c − on Σ + 6, or the corresponding region for k (6 − to higher degree than they do on the corresponding curve in b > c 2 cannot be effective for is effective, as is any positive combination of effective divisors. Similarly, if is effective, then this line bundle admits a section, so generically K D 2) corresponds to the monomial 2 6 . Constraints imposed on dual monomials in the Weierstrass function in the toric description of a twist T nt − K − , 6 ( − − does not vanish to degree 6 on Σ 2 2 − O g As described in section a, b, c K K = 0 corresponds to 6 6 then there are no− sections and and vanish to degree and − and Note that the divisors vanish on from the fact that consider constraints associated with the degrees of vanishing of For a smooth F-theoryboundary geometry, at at leastpoint one (5 monomial in they shaded region (notunder including these the conditions (section Figure 2 by the depicted constraints on JHEP08(2014)025 is in )– T . To D 4.8 2 4.1.3 (4.26) (4.23) (4.24) (4.25) + ∩ B 2 must be − in K 4 6 ≤ D k = Σ − 2, where the . , g − C 2 ≤ . = k 2 1 since f since the vanishing B . In general, “nonlo- ≥ D j 2 , · to be well-behaved on is effective if and only ) B m ,D 2 i D T is effective where the sign is changed K D f, g − . + − T 2 + − 4)( K . 6 2 on curves − − K are not sufficient to guarantee that without divisors of self-intersection a divisor in 3 m 2 D . f, g D · B 2 T + (6 K ⇒ − must be nonvanishing. But by a parallel 6 is effective = +1, then by the analysis of section with 4 mt − ≤ T , along with the local twist conditions ( D k D − . This means that one of = · cannot be effective for − 2 = +1 ∩ , g . A parallel argument shows that D T T – 26 – 3 2 vanish up to but not including degrees (4, 5). − K − D ≤ mt 4 4.1.5 K k k · 4 f − − , g = Σ k 2 − ,T f 2 C K ) = 4 − bundle over a base T 1 − in the base with self-intersection = ). Note, however, that there are no further constraints asso- − P conditions that must be satisfied for any F-theory compactifi- 2 D D is not a constant and must vanish on some curves · K do not vanish to degrees 4, 6 on Σ 4.11 both vanish on 6 5 5 D g − )–( : both vanish to degrees 4, 6 on the associated curve ( f, g , g 3 that is a m 4.8 D f 3 necessary ∃ then f, g B 2 K 6 − is not effective then 6= at least one other coefficient T T 3 do not vanish to degrees 4, 6 on Σ − on all equations. we know that nonvanishing. Again, a parallel argument to the above means that in this case If there is anyassociated component curve of the twist is In this case all coefficients of If avoid having B argument to the above this means that One possibility is that ), give a set of 2 Any good F-theory base geometry must satisfy these conditions and must fit into one From these considerations we can distinguish several possible configurations of allowed are well-behaved on all curves T K f, g 6 B) C) A) 4.11 f, g cal” effects from other divisorsthe can local limit conditions ( the rangeciated of with allowed curves twists formed more from stringentlyon than the such curves intersection cannot of be two divisors greater than at the corresponding points in The conditions (A-C) derived in( section cations on a space below -2. Thesethe conditions preceding are subsection not, are however,divisors, both sufficient. the local necessary While constraints and on the twist sufficient components conditions for derived in of the 3 categories (A-C).for A similar set of conditions hold for Σ 4.1.6 General constraints on geometry from geometries on the F-theory side where again the lastif term is effective.effective This if shows and that only if − JHEP08(2014)025 ) ) 6, = N 1 n, k 4.1 ≤ ) are D (4.28) (4.29) · D · 1 4.11 T D = 1, if this )–( 2 from which it D 0 for in the base. · 4.8 , with ) ), which satisfies . T 2 ≥ k D 3 T without -2 curves, ) = B − D ,D 2 k · 1 1 . This can be seen by and ( n B − ( T − D D · Σ n − . , there is at least one to be well-behaved on all (4.27) T ( 2 ∩ 0 0 ≤ − 2 4.1.5 B − ) = 6 < < 2 f, g η ,D in ) ) 2 D − k k · D nK for Σ − 5 , − − . In some situations, the vanishing − 6 ∩ γ T 2 n n 1 ( Q ( ( B , X, has a decomposition of the form ( = ( D ) − − k + T Q = = ) 2 − ) ) seems to allow k 1 2 n on all divisors and curves when the base is ( − D D ηD · · 4.10 n + ( k < n T T is effective, 1 of the form f, g – 27 – ) ) ≤ − − that contains two divisors k k ≤ ) T 2 Q 2 γD γ − − − 2 B 2 nK = n n ( ( − ) of K − Q ) is that for each divisor 6 − − η . = 4.1 6 and 0 − − k , 5 but are not constants will generically have points where = = , so ( − 1 2 n,k , = 4 n D D Q 0 · · n ≥ ≥ Q Q 2 ) with no -2 curves, when the base has -2 curves the more general as long as 0 would vanish to degrees 4, 6 on = 1. While the bound ( D k, γ F · 2 D − ) so that also impose constraints that limit the extent to which gauge group factors D f, g or · n ,X 1 2 0 ≥ (and all divisors Σ P f, g ,D ≥ . This condition is automatically satisfied for any base η k < n ± 2 1 vanish to degrees 4 Σ 3; in the second type, constraints are associated with codimension two loci in the − D ∩ < k · is effective and has sections that do not vanish on any -1 curves = D f, g D 5 2 , X γ N > 6 In formulating these conditions, note again that we have not considered potential prob- This shows that while the conditions described in section A general statement of the sufficient conditions on As an example of a further constraint following from the interaction between twists vanish to degrees 4, 6, which are of codimension 3 in the full base D · = 4 or 6, Q 2 n the dual heterotic picture.type, We generic consider SU(2) two and SU(3) specific gaugewith types groups are of constrained such from constraints. being enhancedF-theory In to picture. the SU( first In bothof cases, sufficient matter the to restriction represent on a enhancement Higgsed is phase related of to a the theory absence with higher symmetry. The constraints describedtheory so compactification, far and for limit anycannot the given be geometry broken possible without impose changing geometries a theconditions minimum F-theory that on base gauge can group that bein used the effective for supergravity theory F- larger can be gauge enhanced groups. by “unHiggsing” These matter fields constraints to give form nontrivial limitations on bundle structure in These other issues are deferredwhich to future work. In specific,f, g however, note that any curve on 4.2 Constraints on gauge enhancement del Pezzo (or conditions stated above must be included to give alems set with codimension of 3 sufficient singularities conditions. orF-theory G-flux, compactification which even may when make a the geometry sufficient unsuitable conditions for discussed here are satisfied. with since for a -1 curve so sufficient for a model to have acceptable with follows that curves ( We then have a decomposition ( on different divisors, consider aD base were to hold then considering (as usual, for JHEP08(2014)025 ) 3 12 N su , 6 , or cannot ) gauge 2 N su contains a 3 anywhere 3 F = N > singularity where 2 B N I ∆ are forced to vanish ) with N f, g, ∆ necessarily vanish to degrees 2, to a given Kodaira singularity type do not vanish on a given divisor but ∆ f, g, D ∆ on a codimension two locus to 4 g is such that by type III or type IV singularities in the 3 3 and SU(2) or SU(3) can be realized when . In this case, the base B f, g, and , 3 D ∆ on a locus of codimension 2. This occurs 2 F f , ) by tuning the Weierstrass moduli to get an – 28 – N f, g, 3 on which on a divisor − G ) where N Σ = · since this can only happen from a type 4 (Type III and IV singularities in the Kodaira classification). D , singularity in the Kodaira classification). Another realization of 2 , N . This means that for many theories with generic SU(2) or SU(3) I 4, giving rise to a non-Higgsable SU(2) or SU(3) gauge group. In 3 on ∆ vanish to degrees 1 , D 2 , (an , this corresponds in the heterotic dual picture to a constraint on how . f, g, N > realization of SU( 7 N 7 1 singularities, their physical properties are rather different. When 3 or 2 − , 3 N 2 I , A and singularity. This is clear in the low-energy 6D theory, as there is no matter charged 2 I 1 ∆ vanish to degrees 2 A similar constraint occurs for a wide range of 4D F-theory compactifications. In many As a concrete example of where this constraint is relevant, we begin by considering Note that while the bounds considered here on gauge enhancement are a consequence do not vanish on − N In another class ofis situations, limited the by extent the towhen degrees which the of a enhancement vanishing gauge of of group theautomatically raises factor factor the can degrees of be vanishingor of enhanced beyond. such cases these gauge groupsin cannot the be moduli enhanced space, toby though higher tuning SU( the moduli. SU(2) gaugeappear We factors in describe can section generally some be specific enhanced examples to where SU(3) 4.2.2 these kinds of Constraints constraints from codimension two loci discuss in section the associated bundle over K3 can be decomposed. 4D compactifications the structure ofto the degree base 1 divisor Σ with self-intersection Σ 2, 4. The resulting SU(3)be gauge enhanced group in to the SU(4)A corresponding or 6D supergravity any theory higherunder SU( the SU(3), such as would arise under Higgsing from a larger gauge group. As we f, g gauge symmetries there is in principlesymmetry no (particularly branch no of SU(5) the theory gauge with symmetry). enhanced SU( the 6D case of F-theory compactification on While type III and IVtype singularities are in one sense simplygauge special algebras limit are points forced on to theF-theory exist loci geometry, on of there a is divisor gauge no group way with to tune the Weierstrass moduli to realize any SU( One unusual feature oftwo the separate gauge ways in groups F-theory, SU(2) associateda and with standard two SU(3) different is Kodaira that singularities.vanishes they There to is can order be realizedSU(2) in arises when f, g, of the geometry, it isgroup, possible that as in discussed some casesdescribed further G-flux in in may this decrease section. later the size sections. of the4.2.1 This gauge would not Constraints affect on SU(2) the and upper SU(3) bounds enhancement JHEP08(2014)025 . 4 8 = to F E 2 3.1 C · in f, g 2 i vanish on ,C H 2 of − contains two i z, w = 2 G 1 B , from which the . In some cases, 6 C ) · = 1 heterotic vacua. 1 on the F-theory side 3.1 C 8 z, w N ) is a non-perturbative E ( by increasing the degrees O ). In such situations an 5.2 or (i.e., degenerations of the = 0 (see the discussion in 4 = 0 (5.2) 7 + on a codimension two locus. 2 1 F 5.3 c 3 B eff ,E ] w or 6 2 f, g 6 E W bz heterotic string gives rise to a num- E 8 = . We will be interested here in these given in section ) + [ E 2 2 g η × V ,V , 8 ( 1 4 factor to the commutant 2 E ) V 8 ch E z, w − 6 vanishing of ( ) , 0 (mod 2) (5.1) – 29 – 1 O V ≡ ( + 2 ) 2 2 , are 4D compactifications of F-theory that have het- 4, doing this raises the degrees of vanishing of ch V , leading to a point in the base that must be blown up. , 6 ( with self and mutual intersections 2 1 − azw 2 C c ) · 3 to 3 ), the definition of 1 = ,C ≡ ) of vector bundles on a Calabi-Yau threefold with structure , and there is matter charged under both groups, as described 1 2 2 C 2 ) factor can be tuned to an f 3.3 C 1 TX g C 2 ( ,V V 2 ⊕ 1 ( G 2 1 V on 2, which break each c ch , , a necessary condition for the existence of spinors; in the case of su g i V , the compactification of the = 1 1 and i , f 8 E denotes the total class of effective curves wrapped by 5-branes. In this work, ). Henceforth in this paper we focus attention on irreducible bundles and = 1. In such a situation, the 6D theory has a non-Higgsable gauge group with ⊆ eff 2 ] i C = 0. The second constraint on the second Chern characters of the bundles is the H W respectively we have · 2.2.2 6= 0 will not affect ( 1 1 2 c ]. While naively the We consider a pair ( While this kind of enhancement constraint only arises for 6D F-theory models that do In six dimensions, constraints of this type only appear for F-theory constructions eff 8 ] ,C ,C 6 on the intersection point 3 1 , W contribution arising fromwhere NS5-branes [ (equivalently M5we branes, will in not heterotic includebundle M-theory), corresponding 5-branes to wrapping sheaves supported curves[ over in curves) the and as base a result, any possible term class of the bundles irreducible principal bundles, thissection reduces to thetake condition familiar 10D anomaly cancellationin condition the we context of have heterotic/F-theory already duality. encountered The in last section term in ( The first of these conditions is equivalent to the vanishing of the second Steifel-Whitney that (i.e., Chern classes)bounds of on the topology, vector as well bundles as in the conditions for supersymmetric groups The bundles must satisfy the topological constraints 5 Heterotic constraints As noted in section ber of consistency constraints, linking the topological data of the Calabi-Yau threefold with not have heterotic duals, inwhich 4D we the discuss issue further is in mucherotic more duals section violating general. the base-point One free keysymmetry condition class (see can section of have an examples, obstruction toas enhancement such to an an enhancement would lead to a 4 of vanishing of 4 Writing the Weierstrass coefficients explicitly, inC a coordinate system where above conclusions follow directly. without smooth heteroticeffective duals. irreducible divisors A− simple example isLie when algebra summands the base in [ JHEP08(2014)025 , V V ] ]. = 0 1 (5.4) (5.5) of b 94 ¯ a – H F 92 b ¯ a g associated to F = 1 4D heterotic ) if and only if ), Thanks, however, V = 0, is by definition N 5.3 ( 12 2 , 0 F < rk ) = F 0 , ( 2 with respect to a given K¨ahler ω . ], it is possible to translate this F that solves ( ∧ = 0 (5.3) 64 < rk b ¯ ω A – a F ∧ 62 b ¯ ) a , and the field strength ) F ( b ¯ V 1 ) for all sub-sheaves, and “poly-stable” if slope stable a , with 0 ( c g . Regardless of the structure group V 3 V i ( ) = 0 (5.6) X ∀ µ < µ , g Z ) V ) i – 30 – ( ) ≤ F ⊂ V ). The consequences of the condition µ F ( F ) 3 = 0 ( 1 ( µ µ b ¯ F X ¯ a ( rk F µ admits a connection ) = if = V ) = is defined to be V ( . ab 13 F µ V ( ω F µ to satisfy the well-known “Hermitian-Yang-Mills” equations [ i V stable and . A bundle One of our goals in this work is to try to use heterotic/F-theory duality ) if for all sub-sheaves i 3 V ] for some recent results in the math/physics literature on bundle moduli spaces on 14 X ( are not known analytically except in very special cases. 96 1 , , 1 V with 95 i H , V 78 ∈ ] for a discussion of such heterotic 5-branes and G-flux). i ω 19 L It is poly-stable bundles that we must consider in the context of For a supersymmetric vacuum, the vanishing of the 10-dimensional gaugino variation For recent progress in solvingNote these that equations all via poly-stableSee numeric bundles approximations, [ are see automatically for semi-stable, example but [ the converse does not hold. = slope poly-stable 12 13 14 threefolds and what properties characterize the associated heterotic effective theories. Calabi-Yau threefolds in examples with special topology. Calabi-Yau vacua, and although the studyand of ongoing such subject bundles and in their algebraichow moduli geometry, to spaces at fully is present classify a very and rich little enumerateYau the is threefolds. moduli known space in of general stableto about bundles understand (sheaves) on as Calabi- much as possible about which stable bundles can exist on Calabi-Yau vector bundles describing a goodand heterotic satisfy vacuum must be holomorphic, slope poly-stable for the physical K¨ahlerform where for any sheaf, A bundle is calledV semi-stable is form the background Ricci-flat Calabi-Yauthe metric bundle to the powerful Donaldson-Uhlenbeck-Yau theoremproblem [ in differential geometrytheorem, into a one holomorphic bundle in algebraic geometry. According to the DUY The first half of thesethe conditions, condition namely that the the vanishingholomorphic vector of bundle functions is over holomorphic theare (i.e., base not that so its easy transitionposed to functions state, a are however; significant solving challenge this to partial the differential construction equation of has supersymmetric historically heterotic vacua, since (see [ requires each bundle non-perturbative effects in the form of 5-branes wrapping the elliptic fibers may be present JHEP08(2014)025 . 3 2 2 B X B (5.8) ) we can )) on the 3 )) and the 2 X A.9 ( B in the explicit 1 ( , . (with section). 1 1 is slope (poly-) , ω 0 1 2 h forms on the base V B ≥ ) = 0, in which case } ) , . . . h , as in ( 0 2 V → ( , ω 1 2 A 3 c both for the form on = 1 { ∧ = 1 X bundle 0 0 (5.7) η k a ) = 1 + : ω 3 ≥ N and X ∧ )) is an ample divisor on π ( 6 2 ω ) } (5.12) 1 ˆ , η ) (5.9) Z 1 B (with . Which form is used should be ∧ 1 )) ( ) , ( ζ 3 ∗ h
1 ( k 1 u , B ) (5.11) ) ∗ ( π { 1 ) (5.10) ω base g π k V t ω ( base ( + 2 1 + ) + base 0. ∗ c ω = ζ 4 0 ω ( , . . . h ( ˆ ( . ω 1) ≥ ∗ Z ∗ ω ∗ 2 ) — on ˆ π Mπ ∧ π ω X − B η = 1 ) ) ( ∧ Mπ + ∧ + u ∗ η N 0 ( α 0 ( ) ( 0 π – 31 – + ( ∗ ω f ω V 0 ( π − ( α aω + ω 2 ) ∧ ω c 3 = α ) + ( V = 3 ) = ˆ b )) ( X ω X 2 ζ ω V ( R ( base = = ∗ 2 ω Nc 2 π c 2 ˆ ∧ Y + base η 3 0 , we have ω ( X ∗ ω Z A π ∧ ∧ ) ) are pullbacks of, respectively, η 0 . Substituting this form for the K¨ahlermoduli into the Bogomolov ζ ( ( ∗ ∗ M π π Mω ( are coordinates on the base are coordinates on the elliptic fiber (described as a degree six hypersurface ] for a review), which states that if a rank } X } = u Z ˆ 97 Z , then { 3 ) and ˆ Y, X η ( is a constant and ˆ ∗ X, ) and a π { Let us consider the consequences of the Bogomolov bound for a bundle over an Thus far our discussion of the consistency conditions on heterotic vacua has been gen- 231 . (Note that in other sections we often use the notation e.g. P 2 for some constant bound, the constraint becomes where Without loss of generality, we can scale the K¨ahlerform to set B and for the pullbackapparent form in any — given technically equation frombasis context.) of Expanding appendix the form section K¨ahler defined as above.expand Recalling the the second Chern geometric class identities of in the bundle appendix as where where in eral. We restrict our attentionnamely now to smooth, those elliptically threefolds thatMore fibered can specifically, give Calabi-Yau we rise restrict threefolds, to our F-theory(i.e., consideration duals, to the the Mordell-Weil case in groupmanifold which of can there is sections be a put is single in trivial section Weierstrass and form as For simplicity, let us consider firstthe the Bogomolov case bound of reduces vector to bundles with There are a numbermorphic of vector constraints bundle. that Onebound” slope-stability of (see places the [ on most thestable important with topology of respect of these to a isCY a holo- the 3-fold choice so-called of “Bogomolov K¨ahlerform 5.1 The Bogomolov bound JHEP08(2014)025 ) 3 1 X  5.13 (5.13) on the M i associated H ). η V , the r.h.s. is . All matter 3 8 X denotes a set of E = SU(3) over } ) H A , r ), 0 (5.14) ). Note that in ( A 2 ) (5.17) R ≥ inside of 8 A.5 ( B 2 , i { ] for a standard review): , 1 B 1 ) (5.16) H K A . . · 2 and with structure group base , it is necessary that we evaluate 2 , r from the Bogomolov bound, the ) + ( η B ω i B A 3 ¯ 3 of the base then, by definition, this (see [ G · V η , + R i -form on η in ( ζ } H 27 base 2 A + , 0 (5.15) ω × M ) = 2 i { ≥ ) + ( G base base 3 , ω 1) + ω · , base ) ∧ 27 – 32 – ω G · η but not sufficient for the stability of ( Mη ), is large). This limit is achieved by taking , the commutant of η ∗ i and the possible states in the theory are determined ) + ( π = G 6 1 (Ad( A.6 ∧ , ω E 0 to → ∧ 78 ω = has positive intersection with the K¨ahlerform of the base ( ) 8 3 is an effective divisor η necessary . Thus, the Bogomolov bound becomes the condition G E X V 2 → ( 248 η Z ), it is possible to derive consistency conditions on 2 B c K 3 . For example, the presence of a bundle with · 248 5.11 X H η Z , as given in ( ) is that 3 under the direct product × = X 8 0 G ) must be examined in a case-by-case manner. That is, given a choice of E sufficiently large, it is clear that the dominant constraint from the Bogo- ω 5.14 ∧ factor down to M 0 5.14 8 ) represents the adjoint representation of ω E G ∧ ] in which the volume of the elliptic fiber is small compared to that of the base ) while the l.h.s. is computed by integrating over the threefold ) viewed now as the coefficient of the η ). For 19 ( ζ ∗ 5.14 π There is one limit, however, that is of particular interest in heterotic/F-theory duality. In general, to extract consistency conditions on 5.14 . In the adiabatic limit, it is impossible for the Bogomolov bound to be satisfied unless and a K¨ahlerform in ( 2 where Ad( representations of breaks one by the decomposition of the 10-dimensional gaugerepresentation field, of that is from the decomposition of the 248-dimensional 5.2 Matter spectraAs in described heterotic above, theories theCalabi-Yau presence threefold of breaks ain vector the bundle low-energy effective theory arises under dimensional reduction from components B If this is taken tois hold simply for the any ample condition K¨ahlerform that In order to take thethis stable expression with degeneration not limit just of anylimit” K¨ahlerform, but section one [ chosen in the appropriate(and “Adiabatic the volume of in ( molov bound in ( condition in ( ζ to the underlying bundlethat being the Bogomolov slope-stable bound (though is once again it is important to recall (with and ( computed in terms of intersection products on the base and Moreover, we recall that by the triple intersection numbers in ( JHEP08(2014)025 ) A r (5.19) (5.21) (5.18) ), etc). ) and as V ∨ ( ), the third V ( ) is written in X,V 3 ( c m 1 2 . n < rk − 5.21 -singlet fields. of matter in the 4- 3 6 ) h E ) = ... V type multiplets minus the number ( ) = X,V 3 ( etc. 3 27 . ) = 0, h , X,V V )) ) example. − ( ( ∨ ) = 0, with 1 ) V 6 m ) = Ch c ( V E h 0 V n )) ) = 0 (5.20) ], ∧ X,V X,V ∨ V ) ( ( End ) 98 ( 1 ∨ 2 0 X, ,V h h ( 3 X, 0 ( gauge boson, the low energy theory can ). X X,V 1 H End ) + X,V ( of these massless scalar fields it is necessary ) + ( 8 6 ( 0 1 1 E ,H h X, H h ) ( Multiplicity 1 X,V – 33 – representations, as well as X,V V ( ( h 2 1 1 ) = ∧ and the condition h h 27 ) to the , 3 − ,V V V − 3 ] for further details). In heterotic theories, the exact ( X 1 ) = 1 theory is determined by the Atiyah-Singer index multiplicity 27 27 0 ( ) is fact determined by the index of the fundamental X and 99 1 Field ( ) = 0 N V 27 ,H X,V H ) ( V,... . Matter multiplicities for the 0 p Ind( h ,V and End 3 of the V,S ¯ 3 X ) = ( m 1 ) = 0 this is further given by Ind( 15 V Table 2 ∧ , V H ( ∨ 1 Ind( c V is the rank 3 vector bundle valued in the fundamental of SU(3) (and hence theory, to find the theory given above, or the number of families in a heterotic Standard Model. 6 V 6 E example above this leads to the bundle-valued cohomology groups shown in E ] as the alternating sum -bundle. The chiral asymmetry in all other representations (i.e., for the induced 6 98 H E where 2 ’s in the The chiral index While the decomposition above is sufficient to determine the is associated to the Physically, the chiral index in a heterotic compactification counts the number of generations minus the 27 ∨ 15 vector bundles, representation given above (see [ number of anti-generations of chiralof particles. For example, the number of In the case that Chern character. Finally, itterms of should the be vector noted bundleprincipal that associated to the the index fundamental given representation in of the ( underlying (as well asFinally, the it induced should be representations;a recalled i.e., result, that the by chiral Serre index duality can [ be expressed simply as the difference, theorem [ But by slope stability of the bundle For the table V to count the number of bundle-valuedunder 1-forms dimensional on reduction, the Calabi-Yau the manifold.by zero-modes More the of precisely, dimensions the of 4-dimensionalover vector the theory bundle-valued Calabi-Yau are cohomology threefold, determined groups such as (in the representation That is, incontain addition charged to matter in the the adjoint-valued dimensional JHEP08(2014)025 ) V ( and ∗ ), let π , with p 3 ). Then 5.8 7.8 R X σ , U → V -fold cover of the ) can be matched ). As in ( N G 5.9 bundles, N )) (5.22) that is an V ( 3 . On any open set ∗ = 0 defines the section X V π p is given by ˆ Z ) (5.23) ) (5.24) ,R S ] of η 2 is defined as in ( ( ,V ) ∗ 98 B η ( π U r ( 1 ]. This is the well-known “spectral cover” H ] + − and m σ π 42 [ 2 ( = is a divisor in , p r – 34 – B N M + S 19 H p ) independent of any particular method of bundle . The class of → 2 ] = S 3 1 map (in fact a full functor on the category of coherent S ,V ) = [ 1 ) which (in the case of simply-laced U → X − 2 V by the pre-sheaf : ). Moreover, for some bundles with these structure groups, ) where B 2 S X,V π N ( B ,L m S H -th derived push forward [ p ) or Sp(2 ] provides a 1 N ]. 40 [ to a pair ( 41 is a rank-1 sheaf on 16 , S ) is the V 39 L V ( ∗ π is the zero section of and be the coordinates of the elliptic fiber (where p 2 σ R ˆ Z B For heterotic theories defined over elliptically fibered Calabi-Yau threefolds, the The precise conditions for stability and consistency of spectral cover bundles will be discussed further ˆ Y, 16 ˆ where X, in the next section. which are slope-stable in the adiabaticMukai transform region described in the previoussheaves) section, from the Fourier- base construction [ 5.3.1 Spectral covers The spectral coverstructure construction group can SU( be used to build rank general, and throughoutthe this holomorphic work vector we bundlesconstruction. attempt ( as It will, farconstructing however, as be vector possible useful bundles to in onduality certain keep elliptically is examples fibered our particularly to manifolds study well appeal in of understood to which [ one heterotic/F-theory method of geometry. We explore this localizedin matter the and context the of chiral the index spectral further cover in construction5.3 section of vector bundles below. The spectralThus cover far construction our discussion of heterotic compactifications and constraints has been completely This formalism allows for a very(supported precise over notion loci of in “localized” the matterexactly in base the to heterotic the theory localized matter associated to 7-brane intersections in the dual F-theory a decomposition where can locally be represented on work we use the simple structurespectrum of heterotic and matter chiral index to extract arising useful from information F-theory about compactifications the on Calabi-Yau fourfolds. bundle-valued cohomology groupsof defined the above base/fiber have geometry. a Using the simple techniques decomposition of Leray in spectral terms sequences, there exists massless matter spectrum is frequently easier to compute than in F-theory, and in this JHEP08(2014)025 k is , g 2 k f K (5.25) (5.26) − such that 0 . This can M above. The 2 ) is effective 2 S B ) (5.27) B L 2 ( 1 is base-point free B ( can be represented η 17 1 Nc c ]. if , S ∗ S − ). However, the converse π )) ] also for the analogous 18 ] 2 η 100 , ...  19 19 19 ⊕ ). It is straightforward to jK are sections of line bundles ... 2 Nλ Z 1 [ j + V , + a ]. 3 + η )) and  3 ⊕ η is not effective. In view of the ( ] (see [ spectral covers are stable in the to be [ X 1 2 − 1 ( M O η 4.1 V 19 4 S N  , spectral cover ˆ 2 L is irreducible Z H + is an effective class in B ˆ play a dual role as coefficients Y S ( η , the associated vector bundle under Fourier- to be nonvanishing, and since 3 odd [ η 0 3 j ∗ S a 0 a ), the π X H a N + ), but also the rank-1 sheaf N  irreducible 2 λ − for )) = 5.24 − N η ˆ Y ˆ – 35 – ( Z 1 2 3 is SU( in ( ˆ − 2  X 2 N V η and its topology after Fourier-Mukai transform, it is ⊗ O a + ˆ X j ]. However, the condition that the cover is irreducible 2 σ V + ⊗ B N a  39 N can be nonvanishing if , λ ,K ˆ , but for now it should simply be noted that this condition Z j 2 ] that for a spectral cover bundle there exists some value 0 19 be an effective class in + a . It can be argued that B a ( 6.4 η 1 2 S 100 0 = must be effective for  H s η even and ) structure group). The coefficients N ∈ N j N a ) = 0 fixes the first Chern class of ) = ), the bundle associated to that spectral data is slope stable for the given region of S V ] that this is true if and only if for ( L is irreducible as an algebraic curve in 1 2 ( 2 N c 5.11 1 S B c ˆ 100 X , in ( N 0 to be slope stable. By construction, ] for examples. a 39 , ] for example). The condition of base point free-ness will be explored in further M V it is necessary that 19 101 ,  has no base locus in a decomposition of the type ( defining the stable region of moduli K¨ahler space. For Calabi-Yau threefolds however, the arguments 102 3 To fully determine the bundle There is a further condition that must be imposed in order for the spectral cover 0 71 Note that if More precisely, it is known [ M X M 18 17 i.e., K¨ahlermoduli space. For K¨ahlertwofolds the proofof of stability is constructiveare and not yields constructive an and explicit we value are restricted toMukai considering transform the limit willdoes be not indecomposable hold. (i.e.See not [ Some a reducible direct spectral sum covers can still correspond to indecomposable vector bundles. for detail for surfaces in section guarantees that there exist irreducible curves in the classnecessary [ to specify not onlycondition the that class bundle adiabatic region given above [ places another condition on ( (see [ be seen by notingeffective that no other coefficient Bogomolov condition in thebuilt to previous be section, slope-stable it in is the adiabatic clear region that of spectral K¨ahlermoduli space. cover bundles are that in the dualityin to the F-theory, F-theory thetwo Weierstrass coefficients sides model, of providing the ain duality. direct In map order for betweenshow the [ the spectral moduli cover on to be the an actual algebraic surface over the base which can locally be described as polynomial functions with appropriate degrees. Note as the zero set of the polynomial ending in construction for Sp(2 in the case that the structure group of JHEP08(2014)025 ] λ ). 19 Z ), a we , ] for ) are 2 (5.28) 6 2 B 5.29 103 B ( and the ( 2 1 σ , including H V Nc ) and ( − )) (5.31) i 2 η 5.28 B ] ( 1 ), ( and 103 ), and the parameter , . By the Leray spectral Nc i (see section (6.2) of [ 3 η N 5.27 2 − X 45 )) (5.32) , B η 2 ( , if ) must be an integral class in i B 39 S in ( ∧ V , S ⊂ = SU( 1 L η σ ( 19 1 H  Nc is spanned by the class c 1 4 ). S ∩ − S is odd V is even − η ). The Chern classes of a spectral cover ( , are [ ) may occur include Noether-Lefschetz ( 2 3 ∨ N λ · N c S λ ( 1 2 :  ,V 1 if if , λη ) mod 2 (5.29) 3 N 1 2 and 2 N h X , ) = B ) differ only by an even element of ( 1 2 . For a bundle ( is chosen subject to ( 2 1 N – 36 – + V 1 η h ( B c S 2 + )) = 2 3 ( ) 2 L 1 2 ≡ c m m, )+ ch B B ( η ( ( form 1 has structure group and degenerate (singular) spectral covers (see [ ,V 1 has a larger Picard group (i.e. more independent divisor 3 c ) = } = and S V Nc S 1 X V N λ , η ( 1 − 1 ) for such examples). In these more general situations, the − { h η 24 3 ( − 5.27 N ∧ and the integers η η − ) = ∧ is even it is clear that this integrality condition imposes V σ ) = 0, Ind( , the bundle ) is localized at the intersections V ∧ N ) holds when the cohomology of λσ 2 ( η 1 B c ) may not hold. We will see the need for such interesting possibilities (and 5.27 X,V ( ) = ) = 2 ) = 0 (5.30) . This condition arises from the fact that 1 is base-point free, and an 5.29 Z V V V S → i ( ( ( H : η 2 3 1 ). When , specified by ∈ ) spectral covers, at least some of the zero-modes of the theory have a simple c c c S Z V N , π m S ( The essential heterotic constraints on a bundle constructed via spectral covers can be Finally, with this data in hand it is possible to extract the full topology of 1 , 1 5.3.2 Localized matterIn and SU( spectral covers realization in terms ofsequence the arguments geometry outlined of above, the itmined spectral can by cover be shown that the matter in the theory deter- spectral cover bundle is guaranteedmoduli to space exist in and toexplore which be the heterotic/F-theory way slope that duality stable some (for isgiving of a information these well region about same understood). in constraints when K¨ahler independent appear these of in In constraints the a section must dual specific fourfold be method geometry, true of bundle based construction. on topological data Note that since simply encapsulated by the effective, constraint ( their F-theory duals) in later sections. the chiral index, Ind( bundle pullback of the cohomologythere can in be the situations in base.classes). which While Examples this whereloci is this in expected the increase complex to in some structure be generalizations of true of generically, ( H where “mod 2” indicatesThe that relation ( is either integer or half-integer depending on where where JHEP08(2014)025 ) )) to η ... ) or ( + V 5 N ), the z theory. 0 ⊗ O )). Note a 7 the data 2 η 5.25 E ( ] for some +( ⊗ 2 x all = SU( ) ,K ⊗ O ... 2 104 H , in exactly the same + N B 3 ( 2 20 ⊗ 2 z 0 B 2 a H ,K 2 +( ∈ 3 B ) x ( u 0 = ( (not just its topology) and )) etc., must be considered. 2 H 2 a y as expected. V ∈ V ( 8 0 ) ) is E u ( 19 . → 5.33 End N 2 7 a B E X, ( ) can be explicitly calculated. From the 1 = 0 (5.33) multiplets of the 4-dimensional ˆ -sheeted spectral cover associated to the ,H X 5.23 ) = 0 in 2 56 N a 2 V ) bundles arising in heterotic compactifications 2 a ]. , etc) as well. At present, it is not known how + – 37 – ) bundle, but other curves associated to other ∧ N 2 V 20 ) (i.e., the matter valued in the fundamental N p ) and ( ˆ Z X, ). With this data in hand, the restriction of 0 ( ) spectral cover of the form shown in ( a 1 V,S 5.22 X,V N H k ( 5.21 1 ∧ ) or Sp(2 H at the zero-locus of the section N σ ), the dual Weierstrass model to ( . = 0 there is an enhancement of 4.6 ), such as 2 7.8 a )–( 5.16 4.5 ) will be controlled by the zeros of -flux. We will return to the issue of chiral matter and heterotic/F-theory V in ( G 8 indicates the presence of charged matter in the 4D theory, the exact matter E 2 B of . This localized matter appears as the )) and perhaps more importantly, its applicability is limited even in these settings. 2 248 B N Finally, it is worth noting that although the presence of “matter curves” in the class More generally, given an SU( = 0] in Note that in the dual F-theory geometry matter curves appear in the shared base intersects the zero-section N 19 a That is, not all consistentcan SU( be represented by well-behaved spectral covers. way. In the notation of ( and on the vanishing locus Despite the fact that itthe constitutes dual one heterotic/F-theory of the landscape, mostand the studied care spectral and cover must best construction understood bebundle is corners taken moduli far of in from space generalizing general orspectral results the cover derived generic construction in dual isSp(2 this fourfold valid only context geometries. for to As special the pointed structure full groups out vector (i.e., above, the also a choice of duality again in section 5.3.3 Limitations of spectral covers spectrum with multiplicities cannotof be the bundle, determined including withoutof detailed fully course, properties specifying the of third aboth Chern the particular fiber class and ( baseperspective implicit of in F-theory, ( we expectthe this exact further (chiral) data matter spectrum to depends be crucially necessary, not since just it on is the known fourfold that geometry, but to construct all suchprogress associated in this spectral direction), coverslikewise though in be in full a the notion generality case of (see localized of [ matter some [ of[ these representations there will the low-energy 4D theory —of for the that other representations appearingFor in the these, decomposition wefundamental must representation consider of not theinduced vector SU( just bundles the (such as exact zero-mode spectrum along the curve localized matter countedrepresentation of by that this matter need not be chiral, and this is of course not the full matter spectrum of S over The exact multiplicity of these fields can be found by a Leray calculation to determine the for a review). For example, for an SU(2) spectral cover of the form JHEP08(2014)025 ]. 106 , summing 105 i E ) is N p is called “regular” ). E for some unknown bundle of the form )) = V 0 p 2 I 5.29 E | V ( M > M ) and ( Aut ] as the heterotic “Standard 71 1 (5.34) 5.28 ]. In particle, this will be true for generic fibers  107 V is defined with respect to the restricted ,M V bundle on the elliptic curve has an automor- α . N ] for some recent results on degenerate spectral ]. In fact, there are indications that this criteria B – 38 – Mω 101 110 + 0 of the bundle to each elliptic fiber ω p E | = ). See [ V ω 3 0). Unlike the previous one, this condition cannot be stated TX = → i V ] are not regular [ E 109 , → O 2 108 I ). For indecomposable stable bundles exists, compatible with the large volume, weakly-coupled limit that we require ) and the role of quantization conditions like ( → V 5.5 i E 7.9 phism group of the minimum possible dimension: dim( Regular (i.e., that the restricted rank Semistable The second condition of “regularity” appearing above was introduced by Atiyah in his For the first of these, the semi-stability of Since this has an impact on the heterotic/F-theory comparisons undertaken in this and is not constructive. The condition above is sufficient to guarantee that a stable 2. 1. 0 → O 6 Equivalence of constraints We now consider the relation between constraintson on bundle F-theory geometry constructions and on the the constraints heterotic side. dual geometry (includingEmbedding” in in such which well-knowncovers examples and [ dual F-theoryin geometry. detail, but The itgeometric regularity may constraints condition arising be will in relevant dual notsection in heterotic/F-theory be explaining pairs, including some explored exotic of here G-flux the (see unsolved questions regarding as a simple globalfor restriction generic on bundles. the Indeed, heteroticregularity it moduli criterion. is space significant For that and instance, manyconstruction is it good [ harder is heterotic known to bundles that characterize willcan most fail bundles sometimes the described be via consistently the violated monad in the context of perturbative heterotic/F-theory classification of semi-stable sheaves on ellipticif curves [ when it isbundles restricted (i.e., to every by elliptic theto fiber divisor-line zero) it rather bundle decomposes than correspondence,0 into a a a non-trivial set poly-stable extension sum of (for of points example, line on Atiyah’s region for for heterotic/F-theory dual pairs.moduli It space is is important certainly to notThis note, necessary point however, for that is the explored this consistency further limitation of in of heterotic appendix theories [ when the K¨ahlerclass is chosen to be where, as above, for threefoldsM the stability proof is based on slope in ( work, it istransform worth is briefly a well-behaved reviewing functor onconditions: some the moduli that of space the of these restriction sheaves, subject constraints to here. the following The Fourier-Mukai JHEP08(2014)025 . , ) 6 2 4 T 1 , f c c , ± 3 6 (6.1) 2 e − K = must be η 6 0 G − . With the g is effective. ± = 2( are effective. that carries a B 1 2 c and η − − 6 0 ) is effective that must be effective. to have a term of f 2 − A 1 . f B c and 1 η ( 2 c 1 1 5 η − = 2 ≥ Nc − t η 2 η for bundles with structure − = do not vanish to degrees 4 − η η 5) be effective (corresponding 2 / T 1 K c f, g − 6 we mean that are well-behaved on Σ 24 2 T − , where it was shown that must be necessary conditions on the B K − ± 4 + η f, g 2 ≥ correspond to gauge algebras η − 3, on the other hand, must be effective for K 2 is a necessary condition for the existence 4.1.5 A 6 g 1 = 5( T − c t or 2 N > − 5 = 3 2 = 0, or ≥ – 39 – 2 − is trivial and the structure group on the heterotic , su vanishing, so K 1 2 η − (or greater). This matches with the result found 4 ) must be effective and η G η f 1 K 2 − c 6 = are effective is a necessary condition for the existence 5 , where by su or − ]. Note that these conditions for bundles with structure 1 ± ± for a bundle to exist on the heterotic side was generalized c − g η η i 2 38 η η η , then ≥ ) groups with 7 E N = 6( T t − 6 , showing that this constraint is more general and independent of 1 − c 2 5.3 is equivalent to the heterotic condition that K = 6 + 6 ). For SU( η − 1 g or a lower term in or Σ SU(3) precisely agree with the condition that 4 , g − 3 , so , it is only necessary that = SU(2), so we conclude that 8 ≤ E , again in agreement with [ H ] the effective constraint on ]. Similarly, bundles with structure 3 In fact, from the F-theory side the constraint is significantly stronger. The constraint 38 38 was needed in section bundle construction. In the caseside where is to a nontrivial nonvanishing so of a bundle within structure [ algebra which have is effective. The resultstable for these andgroup the SU(2) other minimal gauge group types are shown in analysis in the previousgauge section. group no For larger example,degree than if we consider aThis divisor corresponds Σ ongroup the heterotic side to a constraint on 6.2 Effective constraint and gaugeIn groups [ to situations where thethe 4D F-theory gauge side. group can These be constraints seen can to be be readily restricted attained in by specific a ways generalization on of the on the F-theory side statesThis that constraint, either and theheterotic analogous side constraint for for the existence of any smooth bundle with the specified components of good heterotic dual. Thisbound matches is with necessary what in isstable known order degeneration of to the limit. satisfy heterotic thegeometry theory, The Bogomolov shows where that bound fact this this in that bound theany this on bundle adiabatic the bound construction, limit heterotic irrespective is side of of must necessary the of be for the more any stable general and degeneration good applies limit. F-theory to it follows that the conditionof that a good F-theory geometry. This is thus a necessary condition for the existence of a We begin by showingon that either the Σ F-theory conditionThis that follows directly from themust analysis be of effective section inidentification any F-theory geometry where 6.1 Effective condition on JHEP08(2014)025 2 − is a (6.3) 2 f bundle 2 = 1 ) so that 2 0. In the − P B g ( 1 ) is an effective c 2 . Both of these = 2 D < 3 B · B ( = 2 i D 1 has three complex O η · T ac 3 ) is base-point free on ) down to SO(8), just g formed as a 2 − i D ) = η 3 η + T /Z + B x 3 2 2 f − K 2 + K 3 6 x 1 1 1 1 1 1 of heterotic bundles, identified from − c c means that c c c c ( 5 3 2 9 4 3 2 1 24 14 0 (6.2) ]. H O = 0 ac ≥ ≥ ≥ ≥ 38 . ≥ ≥ ≥ η bound η η η ≥ 0 η = 1, since ( η η D η · ≥ there is a unique twist D ; ) · 2 2 4 . G f D g T 4 2 2 B , · , 8 7 6 1 F 8 2. e e e 3 K ± su . This matches the dual heterotic theory where a , – 40 – T ). Thus, the base-point free condition imposes 8 so 2 − su is a section of ± so K 4.9 = 1 3 6 6 2 . In this case the F-theory conditions are that g n 4 g − 2 2 3 f ) becomes ( , 7 8 , 1 GH e e B 8 6 su su . Let us examine the consequences of this condition for e 6.2 and of negative self-intersection such that 2 so , on a surface this simply means that there does not exist 3 B where B D is dual to the SO(32) heterotic theory on a generic elliptically O 4.1 n in 3 = 1, we have for certain structure groups − B η n D , for any given base ) = = 3 T 2 D for a given 4D gauge group · with − η 2 D D K 4 1. In particular, this means that any F-theory base − . For example, a heterotic bundle with structure group SU(2) is only possible when − ( 2 O B is a rational curve, so ( . Constraints on is effective. These results match those found in [ D 1 c For a curve 2 . As discussed in section 2 − This is automatically satisfiedno from additional ( conditionsintersection for F-theory bases associated with twists over curves of self- for all effective divisors curves satisfying where We now consider theF-theory conditions geometry. that On the theB heterotic heterotic base-point side, free theany condition constraint effective imposes is divisor on that (curve) F-theory picture this means that generic choice of bundle will breakas the in full SO(32) the (really 6D Spin case (32) fibered where Calabi-Yau the threefold F-theory over dual the of base heterotic on6.4 K3 is F-theory Base-point on an free elliptically condition F-theory on the resulting fibered Calabi-Yau threefold over section of are therefore simply complexroots numbers and so the resulting that gauge the algebra is cubic divisor on η 6.3 F-theory constraintsAs and discussed SO(32) in section models Table 3 F-theory bounds on JHEP08(2014)025 . . i ) 0 ), 1. B D D , f 1 is − 1 ∩ (6.8) (6.4) when 4.10 a, b, c , f ≥ − − 2 ± ≥ − 6. This a ), as well , , g = ( 3 . A similar g 2 curve, the 8 B, c m 4.16 e − 6 on Σ cannot vanish since , ≤ 1. From ( ) associated with ). As described )–( or B ± k , − f, g 7 1. 2. In this case, we − e − = B, b 4.12 4.14 , − b − to degrees 4 6 n − . Furthermore, if the D ( e = · = − + 2 − D T n D a 2 with an associated twist ∩ = +1 over a · = − , it is clear that this is not c < b 4 vanishing case the generic i − . A similar result follows for ≥ − ˜ t , T 8 2 D c e · = 0. Thus, the base-point free ) to ) to degree at least 1, or by T D 7 ) when = 4 or 6. D · b, c e B that satisfy the base-point free con- . k 2 curve has i + T B 6. In addition, there must be at least (6.7) η 1) − − , . , (6.5) (6.6) 0) n 2 0 would vanish to degrees 4 , ( 1) 0) gauge group on Σ 1 − , for = 4 , , ≥ , − 8 0 0 − 1 e B , , , f, g 2 vanish on D cannot be − = 6. From figure · 2 or − c > B – 41 – nK for then satisfy the inequalities ( with self-intersection T , f B = (0 = (1 = (0 = ( 7 3 − − e G B i ± 1 D cannot be a perfect square, since if it was then it b − , , f ˜ w contains no curves of self-intersection lower than − +1 s 4 − i i 4 F ), , this would mean that every monomial . Thus, in the non-base-point free cases where an F- 2 ˜ ˜ g b w w , g 6 and again 5 e B 1 in is a curve of self-intersection g ∗ = 4 and 4.17 + 2 D D a N B ∈ 4 then ≥ , ) , this condition means that in the 3 c ) becomes automatically gives rise to and not 2 4 6.2 f 3.1 B a, b, c 5 or higher, or they would vanish on Σ . From table , 8 e 1 for both = ( that satisfies ( = 1, we have a local set of rays in the toric fan as in ( 1. or . Assume that m i , this means that m − = 0, so ( ≥ − = +1. In this situation, ( 7 D 1, and this is ruled out for both values of to degree at least 2, etc. An immediate consequence is that e c = · D − D i · D ˜ t T · 4.1.5 4.1.3 2 = ), which becomes to degrees 3 T = K i − ˜ t − First, we can ask if it is possible to have a gauge group factor on Σ These conditions can be made more explicit in the toric context. Given a divisor We can now analyze the consequences for the gauge group on the divisors Σ Now consider the case of a curve B, c when 4.18 + = possible, however. The onlyb simultaneous solution in ( Thus, when the base-point freegauge condition algebra is summand violated associated through withΣ Σ All monomials as ( one monomial a summand would have theory construction is possible,consideration the holds generic for gauge the group generic cannot gauge be group on Σ with means that there cannotdegrees be of a vanishing generic arewould 3 vanish to degreeAs two on reviewed in section gauge group must be in section having in section vanish on on Σ This relation is onlycondition satisfied will with be both violated whenever signsthese the when twist are over the a only nonzero possibilities. the base-point free condition is violated. This corresponds to the condition (C) analyzed dition on the heterotic side, since have over a del Pezzo base JHEP08(2014)025 , ) y N ) = ] (or = . The 4 g − 112 a, b, c , is 2. There regarding 4 which must 1 z = ( 111 ≥ − 4 on Σ ), independent , m c N is 3 when the base-point 4 f, g f = 6 is ) or Sp( , given by B N c ignored G-flux is a perfect square, the gauge 4 , both associated with vanishing g 4 f . If , in a local expansion in coordinates we have − and 6 – ··· 6 e 3 + 5 . z can be seen most easily from the leading term 5 g 4 – 42 – 7.9 f . So this combination of vanishing degrees can be + 1. 2 . This is in good agreement with what is known of 4 − 2 z and in sections g and the degree of vanishing of 4 = 6 g η . Since the only allowed monomial at order e i T , section 4 ˜ t f = and + ’s that are generalized del Pezzo surfaces have well-defined , g . In this case the constraint for 1 7.8 2 8 c − B so when , = 6 4 + f 1 , on Σ η 6 e heterotic bundle with structure group SU( , 7 f, g e , any 8 ] will exist for the heterotic bundles (though explicit conditions on bundle e bundles over 42 4 for – 1 , , and otherwise it is 6 P 40 e ) structure group bundles is not possible when the base point free condition is 2), again as depicted in figure N − , 5 In some cases it may be possible for non-trivial G-flux (in the singular limit of the For the gauge groups associated with the more general exceptional algebras, it is The upshot of this analysis is that the only gauge algebras that are possible for Now let us consider the possible summands , 6 − fourfold geometry) to changethe the Weierstrass apparent equation. symmetry groupflux Although that in counterintuitive would from a be the smoothsingular inferred perspective M-theory limit from of and limit, Abelian is suchrecently G- expected symmetry-breaking been to explored by in occur flux the in can context a of be wide local generic F-theory range models in of as 4D the “T-branes” F-theory [ models. This has the results presented herebounds that on are structure basedbe groups purely taken and on into F-theory accountbundle geometry. moduli for space. In a Some deriving fullreferences general the are aspects description given of of in G-flux section in the 4D F-theory F-theory physics models, and and relevant dual heterotic heterotic duals, even when thebe base-point needed free for condition explicit is construction violated, though of new the6.5 tools appropriate may bundles on the A heterotic note side. To of close caution: this section, G-flux we return briefly to a caveat mentioned in section section struction [ topology are at present not astral well covers) understood and in as this a context result, asthe the they construction base-point are of free in a condition the sensible is case bundle. notmodels of necessarily spec- The on a analysis requirement of for this section suggests that all F-theory violated. Furthermore, these resultsbe demonstrate necessary that for the base-point-freeof condition the must spectral cover construction. expected that other bundle constructions such as the more general “cameral cover” con- the structure groupfree of condition is the violatedtheory, bundle namely are in the the commutantsheterotic/F-theory duality of dual in the these heterotic possible cases,and model since gauge Sp( the algebras when spectral of cover the construction the for base-point 4D SU( around the relevant divisor,algebra which is in this casewhich is is Σ not afree perfect condition square, fails on thesame gauge result algebra follows must for always Σ be degrees of 3 is a simultaneous solution( to the inequalities forrealized. this The value distinction of between in the Weierstrass coefficient JHEP08(2014)025 3 12 B has (7.1) ≤ 2 ] and ]. . The t is that B 2 bundles P ≤ 38 = 1. The 116 , 1 , tH P = vanishes on 30 H 2 = · 5 115 can break the g B T i H ζ that are with 3 that have a clear dual . B in which H ) i η 3 , t 1 B − fibration over , 1 1 ) for a curve of self-intersection P − corresponding to different twists in 4.11 = ( 3 . 2 B )–( , the geometry of the F-theory base are equivalent under reflection to those 7.9 , w to be a 4.8 t 3 0) 3.1 , B 0 in the 6D construction. Since the base , 11 – 43 – of the hyperplane class , 10 = (1 , 9 17 correspond to bases 1 = 0), and that there are valid models for 0 nH F ]. ]) and in the global context in both 4- [ ≤ . The general constraint on the twist , w t tH H 0) 101 , which must be blown up for a smooth threefold base, , 114 − ≤ , − 1 2 or below, the base-point free condition is never violated. 2 , = 3 − K 2 113 6 ), however, are not determined by the 4-fold geometry. On the 2 K = (0 − P − 0 are multiples locus Σ 3.2 and which illustrate various features discussed in the main part of w 2 8 2 e giving F-theory models with distinct smooth heterotic duals. P in ( = 18 ( B 2 i t P ζ are 2 . The twists 13 t effective). Models with negative B will not be visible in the F-theory geometry, and can only be seen in F-theory i tH bundles over . We thus have a total of 14 distinct bases ζ t − 1 ∼ P 2 18; this is the analogue of the constraints ( bundle over ) These features can be seen explicitly through a toric construction, where the parts of From the general classification of allowed F-theory models it follows that there is a In the context of the present investigation, we hope to explore the full moduli/vacuum The basic mechanism by which G-flux can break an apparent symmetry appearing i K 1 V 4 ( P | ≤ 2 − t value of a the fan from with positive some curves on the analogous to the Hirzebruch surfaces no curves of self-intersection The resulting 4D supergravity models have a range of gauge group factors according to the | +1. This class of F-theory models and theirvalid heterotic model duals with was described( in [ 7.1 We begin with theeffective simplest divisors example, on taking (anti-)canonical class is 7 Examples We examine some specificover examples various of bases F-theory modelsthe on paper. bases space of the dualprovide theories an including illustration G-flux of where inexotic the future G-flux, purely in work. geometric an criteria For example may now, appearing miss however, in solutions we section involving simply The components heterotic side, non-trivial bundlesgauge with group a just second as Chernin class effectively F-theory entirely as geometry. in bundlesc with The non-vanishing symmetry breakingwhen G-flux bundles is corresponding correctly incorporated. to the topology 6-dimensional compactifications [ from the geometric F-theorydual analysis in heterotic terms picture. of Asdetermines a discussed Weierstrass almost model in all section is of most the clear topology in the of the corresponding bundle on the heterotic side. equivalently “gluing data” [ JHEP08(2014)025 . , 8 ), ). 0 5. e H 2 2 , 2 bF 18. , g 6 on with K K 0 . For − , 2 6 f + 2 = 4 F − − . ≥ − t H, 8 6 t S, F aS e = = , − . There are 7 T = are effective, e tH , H, bF 6 gives T 4 T e base-point free. , + − + ± 4 = 12 ( f = 6 ( T 2 = , t b aS 2 to 0 for the first two K ± g 4 = 2 = 6. There is a single tH T ) − a is effective. K T k ≤ 8. Up to symmetries this 6 i η − − S ≤ (3) arise in the cases n · . (3) cannot be enhanced to an ( = su axis at a value less or equal to F T are qualitatively similar to the a, b − su ) ± z . 2 6. (In general, the condition that η 1 = m 4 ≤ this gives 108 + 1 = 109 distinct F and then focus on the case | F ≤ model with 8 a nK 1 T 3 8 − F − (2) or (2) and e − is a Hirzebruch surface. Some of these and t/ b su su | 2 + (2 + 0 , , as can be seen from the fact that and F 2 correspond to singular geometries on the B S − 12 0 vanish to degrees (2, 2). Both of these cases 3), so F = 2 ≤ do not automatically vanish to degrees 4 m > corresponds to the bundle decomposition where – 44 – 2 , t/ f, g ’s intersect the 18 in the toric picture since the plane spanned by F i 0 t K 2 on Σ , and we can again parameterize · , w ≤ , f, g ). For all other good bases . The general constraints on twists over curves of − F , and we can parameterize 2 at degree 6 can be described in any toric case as the t T . The general constraints on twists over curves of self- with T = 5, b ) The corresponding condition for Σ are effective, so F K − t m = 6 20 + 3 ]. The cases T F ↔ | − + 2 a S → − 30 12. There is a single a | ± S = 2 ≤ T = 1, and = 2 K T and 4 , the cone of effective divisors is generated by divisors = 2 m F 2 a, b T · − m F 2 1 give K F − ≤ K − ,S (8), parallel to the 6D model on = 18 ( − → − 12 so vanish to degrees 1 , b = 0 − T 12. Up to the sign symmetry of g F 0 is the number of instanton factors in each component of the gauge group, · = 12 | ≤ , there are a variety of gauge groups associated with the different twists. and ≥ b 2 a | t , we have f , we have , P intersects the third axis at (0 1 0 m, F 8 bundles over ± 3 F F = − 1 must all vanish as the corresponding divisors 2 , w P = 4, 1 = 2 | ≤ B g For For The unique model with an SO(32) heterotic dual is the model with On the heterotic side, the choice of Specific examples with generic gauge algebras t = 18 corresponds to the condition that a Thanks to L. Swanson for discussions on this point. S | , w 2 (5), though the algebra can be enhanced to the exceptional series · , not vanish on a divisor such as Σ 20 2 curves in the base. The cases − 1 1 model with so twists associated with validassociated F-theory with models. the different Again, twists, there and are all a models variety have of gauge groups gives 81 + 1 =over 82 distinct twists associated with valid F-theory models.There As in is the a models symmetryself-intersection under 0 and S symmetries under intersection 0 give For all other good bases models were also discussedmodels in described [ in the− previous section, andheterotic are side. always We base-point briefly free describeany Hirzebruch as the surface cases there are no with gauge algebra 7.2 Now consider F-theory models where the base For and are all non-effective. Similarly,correspond for to situations where thesu gauge algebra 6, as can be verified geometrically. η which must be nonnegative associated with the condition that vectors. The F-theory conditionΣ that w g condition that the plane spanning the Here we have used linear transformations to set the component of JHEP08(2014)025 , . − 1 T 1. 1. F F are − ≤ (7.8) (7.2) . a when = = = 2 are all S − · 2 S S contains T T = 2 T B · ) 3 b η k = on Σ B | − 4 a over f n 2 3 ( − B b − | , , and we parameterize 12 nK F − ≤ structure bundle on the bundles 7 F as F +4 . For the twist combination 1 · that violates the base-point . The general constraints on E S 1) − P − 7) gives vanishing degrees of T ± 2 , fails the base-point free condi- − F ). The models with = = 2 2 a T | on Σ 2 2 a K | model there are 10 base-point free − 6 K 3) codimension two singularity type − ) = (3 2 8 , − f, g − e . (7.7) (7.4) be effective the base-point free models K ) and a dual heterotic bundle structure ) 6 = a, b (7.5) (7.6) 1) has a negative intersection with 5 (7.3) T 2 , a , b − ± 0) 0) T g + (12 6 confirms that the cases described above 1 1 ≤ F , , , − exchanging Σ . An explicit computation of the monomials ) = − 0 1 0 − − S 2 a k , , , , , , ) T , which is not base-point free since − . This constrains 2 constrain 4 a K a, b ≤ η F 4 occurs for several T − . We have + 2 – 45 – = 24 ( − 8 2 , giving a (2 − = (0 = (0 = (1 = (0 = (0 n + → − F S − , b 0 1 2 3 2 +19 ± i ≥ − T = s + ( 4.1.2 w w w w α S K = (6 2 3 T = 12 B − nS a m, w − = 10 . h 2 = 2 T K T − 3 ) 1 all vanish on k < n k c +1, there is no vanishing of − 2 a − ≤ although there is no gauge group on Σ , g = 6 + 1, from which n 1 ( − = 2 model with a η , g 6, 0 Σ b − 8 0 , e ∩ = 2 2, , g 1 S b nK 8 up to symmetry, so along with the = 4 , f − , with a symmetry under n is parameterized by the integers 0 a < 2, 3, for a gauge algebra factor of ≤ 5), the vanishing degrees are 1, 2, so the gauge algebra contribution from Σ f , T a bF − + ≤ , so . This should correspond on the heterotic side to an k 2 aS 2 on Σ A similar analysis for the twist combination ( We consider explicitly the cases where We can describe these cases explicitly in toric language. The toric fan for Now we consider models with A situation mentioned in section have acceptable degrees of vanishing on all divisors and curves. ) = (2 su = a, b is Calabi-Yau described by the genericfree condition, elliptic with fibration over f, g in the dual latticeare that the satisfy only onesf, for g which the F-theory model is acceptabletion. and When that in( all these cases The twist the rays must be effective, and similar for so there are 14 non-base-point free configurations, with gauge groups up to have 0 configurations. For the non-base-pointwe free can configurations, choose up to the sign symmetry on effective for T twists over curves of self-intersection 0 and There is a single base-point free; from the constraint that where a codimension two singularityfrom arises codimension on one a singularities. curve AIn despite sample this the example absence case, of of this gaugen occurs groups > for the twist over the curve JHEP08(2014)025 , 2 E (7.9) , and − (7.10) 4 9) and f 1 , which , 2 E is − ,E − 1 , there is no is the proper H ) = (4 F 2 = in the base with F,E . In the singular = 0. The proper = a, b 2 F D D T E at two points, giving · 1 2 , we know this gives a P ’s. There is a single twist ,E 3 1 3 curve B − − ). 4.1.6 c = − 2 . These assertions can easily be . − b, E 2 associated with . For the twists ( , · Σ − can be constructed so that there is E 2 3 2 F ∩ 3 a, B using the conditions described in sec- E cE F − + 2 S ( = T 1 + 17 + = 1 1 E S ↔ 2 E ) · with a codimension two singularity but bE B + 2 1 = 9 3 + 1) in a basis where the intersection product is – 46 – E F F , η 0 a, b, c over , aF = = 3 3 (0 2 with = , B K 2 B , is constructed by blowing up and ( . T 0) 4) on the curve 2 2 , , − c ,E g 1 1 , ↔ , E (0 2 bundle . From the analysis in section b , , all elliptically fibered Calabi-Yau geometries over the base P 1 1) P on − 3.2 , 4.1.5 = 1. The cone of effective divisors is spanned by can have a smooth heterotic dual. In most cases, the absence of the 1 also has a simple toric presentation, but we use this more abstract , a H 2 3 − 2 T , F E = 3 · to degrees (2, 2) or greater. This means that no F-theory models on any = F 2 K B − 1). dP f, g = − ) vanish to degrees (3 . In this case the bundle has 1 , 4 and section 2 f 1 E are singular due to the Kodaira singularity over the over · f, g − 3 3 , no gauge group F F B We can now count the set of allowed twists This gives a number of explicit examples of F-theory constructions that violate the 4.1.3 = 11) the vanishing degrees are 3, 4, so the gauge algebra contribution on Σ , 2 there are symmetries under tion necessary and sufficient set of conditions for the set of allowed transform of If we parameterize the twist as with we can write asdiag(1 (1 formulation for the delbe Pezzo implemented examples outside the to toric illustrate context. how The the (anti-)canonical methods class of of dP this paper can 7.4 dP The second del Pezzoa surface, pair dP of exceptional divisors transform of the line passing through the two points is a third -1 curve values of the twist no extra nonabelian gaugedivisor group that factor. must carry a For gaugewhere example, group ( with factor, the though there twist ischecked a explicitly codimension using two the singularity monomials computed in the toric description. base heterotic dual is madefactor particularly in clear the by 4D the F-theoryheterotic appearance model dual of over theory an the this divisor additionalsingularity would in in gauge correspond the group Calabi-Yau to geometry. an It additional is interesting gauge to factor note, arising however, that at for the certain 7.3 An F-theory model over As discussed inB section vanishing of (5 the heterotic structure group is base-point free condition, wherebundle the having dual an heterotic exceptional model structure should group. nonetheless exist with a group of JHEP08(2014)025 . ) 3 − a 3 ↔ B b 2 | ≤ 1 i − F E E ≤ · , (7.11) (7.16) (7.18) (7.19) of the 2 T 3 − b | 6 b , − | D H , 2 + bundles 1 ) (7.15) 2 E b 2 = bundles over 1 · b b P − 2 1 T | + − P and under a, , H | 1 ,X 1 ’s are of type (B). , 1 3 b of type (B), follow- } 3 3 , b 3 = ( E , E 3 B 0 · − , 2 B 2 , T 2 − T ≥ | , , a 2 E 1 3 ) . a that are b E i = 1 c 3 at 3 generic points gives 3 ) is effective, which implies i (7.12) (7.13) (7.14) . E ∈ { − − ± 3 B 2 b 1 0) 0) 1) i 12 b , , , P H X ∓ 0 1 0 1) we have − , , , + (8 4 = + 1 0 0 | ≤ − 1 i 3 , , , − 1 , , a b , 1 E 3 distinct X X 2 ) b − b − b i , = (0 = (0 = (0 2 − 1 ∓ X ) is effective if ± b 1 2 3 2 3 − 4 b b E E E symmetry group there are 775 solutions of − , . To enumerate bases − − 6 1 − 6. Up to the symmetries listed above, there . , , . Blowing up + (8 b K 1) = 1 , i, j, k 1 curves 2 D 2 1) 1) 0) 6 9 8. The constraints that b 6 , a b − , F − − − − 3 , − – 47 – 1 ) | ≤ − ∓ , , b a 1 c a is again the proper transform of , 1 0 − | ≤ 2 4 = | ≤ 1 | − , , c − 3 k − b ± | − , 1 1 − b 2 T , , 1 b − b 8 0 − − a, (7.17) − − , , , − a, , , − 6 j ± = dP a b 1 | ≤ = (12 | and three 12 2 b b , it is sufficient to identify all twists , = ( | = (1 = (1 = (1 , − | ≤ 6 T B , = (3 i 3 − 1 2 3 b a | ≤ D | | ± a 12 = (12 a X X X K ,E | | ≤ 2 4.1.6 (2 = = c K T from the proper transforms of the lines connecting each pair of − | | 4 | ≤ i i ,E 2 − ↔ a ± 1 | . E − X E ) a E · of type (A), with | · 3 K 9 is similar to that for dP , b 4 3 − T 6 T ’s that satisfy these conditions, so the number of distinct | | − 3 B 1 − , T , this result agrees with the explicit enumeration done using toric methods that give good F-theory models is 472. This agrees with a direct analysis E 2 2 b | ≤ 2 b − − , − 1 of class (A). There are no -2 curves in the base, so all other H b 3 a | = dP − K = 2 6 a, 3 B ( − . As for dP 3 ,X , which maps = i 3 A simple enumeration shows thatall up these to conditions, the sodP a total of 776described distinct in possible section bases are satisfied and There is one base ing the analysis of section so that the local twist conditions In the toric pictureregular this hexagon. can A be divisor The seen (anti-) canonical as class the of 12-fold dihedral symmetry group There are symmetries under theX 6 permutations of the indices exceptional divisors E points. In a basis with intersection form diag(1 are 471 distinct over using toric methods, as described in section 7.5 dP The story for dP The constraints that are effective constrain 6 imply T JHEP08(2014)025 is )– 1), ) , H 0 4.8 which , (7.21) 6.5 0 , a, 0 2 , of type (B), ≤ . , and we can i 3 6 b = (0 B K i , the number of 6 4 2 P − 1). There are sym- B i, − = on the F-theory side ∀ , . fixed (7.20) 1 a ,...E 3 6 T , which complicates the 4 B − 5 0) ≤ , , . An explicit enumeration . The intersection form is E i 1 ,E | ≤ 0 3 b j 4 , are distinct; in terms of the i − B t − 0 , ,... } , 4 ) must satisfy the conditions 1 − 4 1 4 η E 0) i , , − t , , t , 3 0 − 3 , ,X − , 2 3 . These restrictions arise because of , t 1 12 , s = (0 4 2 . E | 1 = (3 -fibered base − X 4 Y , though there is no toric construction, , 1 1 , t , 2 − = 0 1 for which only one structure group 1 P K | 2 ↔ − ,E η − ij ) is effective iff i E 3 s, t , ∈ { 4 E X b − · ’s, as well as additional symmetries of the form = ( − 1 and dP i of the , = (1 T must be formed from a positive integral linear E 3 can be neither broken (Higgsed) or enhanced at – 48 – 3 | T ,E b T 12 − 13 D − , for which the nonzero intersection products are X , µ, ν, λ, σ X , effective, which along with the twist conditions ( H 2 , kl 6 b T ij ↔ X − | fibrations over dP = 2 ± , X 2 | ≤ 1 1 i b t K C ijkl P |  4 − | ) on divisors and curves as described in section 1 2 − = ’s. The twist = 1 when i ,E a, increases, however, the details of the calculation become more | i E 23 f, g = n λσ E = ( X · ˜ X ij , for example, there is an additional -1 curve from a conic passing that are · D T 5 ˜ ↔ | X 3 , , and for generalized del Pezzo surfaces, the analysis can be carried out ’s and 1 µν B i n ˜ ij E X E , X 1 = there are 4 exceptional divisors can be done efficiently and explicitly. − i 2 4 0 , a divisor = 4 ˜ B 4 X the analysis is again similar to dP µν n ˜ X ) give necessary and sufficient conditions for a good base · The consequences of having a twist For higher dP In dP For dP ’s over ’s, the symmetry group is simply the set of permutations on all 5 possible index values. µν 3 4.11 ˜ ˜ a degree of vanishing of ( that gives rise to such a restrictivegeometry condition to on the a gauge choice group of correspondspossible in partial (subject the bundle heterotic once topology again to the caveats arising from ignoring G-flux, see section 7.7 An exampleOne of of an the upper more novel and observationstries of lower there this bound study exist on is generic theany symmetries fact points that that in for the certain complexa fourfold structure geome- variety moduli of space features, of however all the failures of “enhancement” occur because of too-high complicated. For dP through all 5 blown upeffectiveness points, condition on divisors.twists In satisfying principle, the local however, for twistB conditions any is base finite, and the determination of the full set of ( shows that after taking accountfor of a symmetry total there of are 6977 6976 distinct bases in a similar fashion. As combination of As in the previousenumerate del all solutions Pezzo with examples, there is one solution with defining X X can be seen from the conditions that diag (1, -1, -1, -1, -1),metries with under (anti-)canonical arbitrary class permutations of the The full symmetry group is of order 5! = 120, and can be seen most clearly by re- For dP there are more symmetries,the and number the of effectiveness -1 condition curves is on increasingly the wall complicated of as and the 6 cone proper of transforms effective of divisors lines increases. 7.6 dP JHEP08(2014)025 2 1 3 to P G X = 0 S (7.23) (7.24) and is z ∩ bundles − F symmetry. for example, 4 + 13 SU(3) F 6 , while type according ) (7.22) 4 S E 4 defined by the , we cannot define ... CY 3 /F → = 7 6 6 structure group and B + 4 E − 6 smooth 8 F η z E 6 g indicates that a generic has + ) for 10) on the curve Σ is a perfect square then the 4 5 1 , bundle over the threefold Y : of the associated bundle from z SU(3), 4 V 5 and is of 4 5 2 . , g 7.24 g Y 4 3 → G F manifests a generic B + 2 3 4 4 G X symmetry is generic (i.e., cannot be z Y , has generic 4 4 1 + g η F ⊕ O + + ( ∨ ¯ 3 3 ) inside of X V + ) − of the appropriate degree. Here the vanishing 3 ⊕ – 49 – reduction in rank ... 3 α V → ) is the generic form of the Weierstrass model for . Constructing the generic Weierstrass model over + 2 7 4 = F z 7 7.22 4 V f on ,( + F F 3 z by blowing up the curve. 3 cannot be decomposed as in ( ]) and a degenerate limit of the theory. f 3 + 11 + 11 B F S 119 S + ( induce non-CY singularities. As an example, consider the special- – 3 − = 5 = 5 + 13 X 117 . More precisely, the heterotic bundle for some polynomial S T T F = 2 2 α = 1) are the coordinates on the elliptic fiber of the , and for more general polynomials it is ∆) increases from the generic values of (4 = 7 is not base-point-free. Thus by the arguments of section 6 Y factor is generically completely broken and will not concern us further. More + 13 = η E η , it is straightforward to verify that this 8 4 S 3 f, g, g E B X,Y,Z = 7 SU(3). For the case at hand this would indicate that by tuning the complex structure bundle. The second bundle, associated to we were inducing a “splitting” of the vector bundle. In terms of the associated vector η 12) on the same curve, and hence the singularity cannot be resolved without going to For this choice of twist, we have an additional confirmation of this heterotic result in In this case, the restriction on enhancement has a clear interpretation in terms of the For the given twist, all additional tunings of the complex structure that might increase For the twist In the dual heterotic theory, this corresponds to a Let us consider here an example of this type for which only one symmetry is possible , 2 4 → 6 Y . Instead, any such decomposition must lead to non-locally free sheaves (i.e., heterotic G , 2 3 bundle with V “small instantons” [ the fact that would lead to the fact, however, that this tuning leads to a badly singular heterotic bundle geometry.corresponds An in enhancement the of heterotic theG theory symmetry to of a of bundles the following reduction of representations in ization of degree of ( (4 a different F-theory base arbitrary complex structure.Higgsed) The from fact the that pointlow-energy of the 4D view interpretation of on thematter both F-theory available to Weierstrass sides get model of a has the vev the duality in same vacuum. — natural therethe gauge is group simply on no Σ charged (Here ( defines the 7-brane locus (i.e.,to the Kodaira-Tate.) section Σ Assymmetry mentioned is in previous sections, if a hence one explicitly, we have the following Weierstrass equation for and all Higgsing/enhancing is forbidden.bundle This with is twist the casethis for base the base with JHEP08(2014)025 (7.26) in the ]. 4 ), which 2 -bundles g 2 S ]. ) G η n NK )) = 1 for the − S + ( ) (7.28) η m O theories over the , 2(6 ... n , 6 1 ] (7.25) F ( 2 + E F 0 ). Although the spec- ( 6 h theories is particularly 0 z , the generic (only) element 6 6 NK H S g 7.24 E + + η in ( 5 [ )) = z · . 3 5 T ] V 6 (7.27) g 6= 0 and the question of whether 2 η [ 5.2 λ ≤ − λ + 2 4 n z K is odd 4 is even ) = 6 and hence (as described in the previous ≤ g ) that for some integer ∨ N − 3 N 21 ( effective F-theory. To accomplish this, we + ( O if if 5.28 D , X,V can be shown to satisfy: X 2 ( ) , applied to these “non-enhanceable” geometries 1 n B 2 1 F, h F ( 4 – 50 – ) bundle described via a spectral cover the chiral ... 0 Y + theories in the dual heterotic/F-theory geometry. N + H + 9 ) + 6 ). Thus, the chiral index is proportional to a simple is not base-point free, it will be impossible to enhance 4 m, m ] in the base with the matter curve [ . This is easy to check since the coefficient E z 4 η ( − nS 4 ) = F f 5.28 6 X,V = = ( ⊗ 2 + 1 λ T h K 3 ] in the 2-fold base. We will be interested in whether or not z 2 theories it is required that − ⊗ 3 , for an SU( f 2 ) and ( 6 K ⊗ E ) ) = is the toric coordinate associated to the divisor + ( η + 3 5.27 V s ( 5.3.2 3 η O X , 2 Ind( symmetries on Σ = B is a perfect square in these cases 6 . 2 ( ] n 4 0 is non-vanishing. Recall from ( E Y s g . Hence if . There the twists H λ n 1 103 , − ) is ∈ F 4 45 = is defined by ( = g , nS 2 2 n λ S F B ( To illustrate the possibilities, we can consider the four generic To understand the significance of this geometry, the case of As an example, let us consider 0 Note that the line bundle cohomology over H 21 satisfies indicates that divisor of all give rise to Weierstrass equation or not the theoryfor has the chiral matter matter curve can [ this be curve reduced is to reducible a and of whether question or of not intersection itbase theory has non-trivial intersection with [ interesting because for SU(3)the bundles constant described as spectral covers we can guarantee that Thus, for SU(3) bundles/ where geometric intersection of the curve [ information about the matter spectrumreturn of to a the 4 formulas for chiral matter given inAs section discussed in section index is [ by providing general restrictions on possible decompositions7.8 like the one given Examples above. withIn non-trivial the chiral matter previousencodes examples otherwise we hard have to seenon obtain that heterotic information CY the about threefolds. F-theory the fourfold moduli In geometry space this frequently of section, vector we bundles use the heterotic theory to obtain new tral cover construction is notin guaranteed general, to the be consistency conditions representativeindicate on for that the if a bundle generic moduli symmetry the space symmetry for specialtion). values of This the provides complex an structure interesting window (compatible into with the a moduli CY space resolu- of all such any smooth SU(3) spectral cover bundle to play the role of JHEP08(2014)025 5 ], , 98 ) = 3 -type = 4 (7.29) (7.31) ⊗ 2 27 n K ⊗ ) η ( ). Moreover, for O 4 ( 0 F ) can be found with H is effective, and it is 3) (7.32) S symmetry and smooth ∈ 1 ) − c 5.27 6 n 3 α n E − )( − n η − = 0. This is where the = (6 α 2 K ) appear as a SU(3) spectral cover 6 = (6 ) can be used to straightforwardly of the form ( = 0 (7.30) g s 0 + 3 S ˆ base, as expected for an SU(3) spectral + Y 7.28 5.32 η L 27 α 2 ˆ X F 4 + f = 6 have chiral index zero, but for elliptic fiber (the above equation gives three ˆ Z + n that the coefficient 3 ˆ and X no. of 3 ) and ( – 51 – 3 ˆ f − X CY . s + 0 F 5.28 = 5.3.2 3 (rather than the generic, non-split 2 27 ˆ 5.3 Z ˆ = 3 and , it can immediately be determined whether or not the + 9 6 Y 5 η E g S n ) s could be computed using Leray spectral sequences [ n the polynomials in ( 27 is effective and base-point-free, 2 − ) defines the “matter curve”, α η ) = no. of of the form = 7.30 V in the fourfold Weierstrass equation above appear as the coefficients 3 , all CY fourfold geometries with generic 4 = (12 ’s and 6 g η CY , g 6.4 27 Ind( ) in ( 4 f have chiral matter. Given the full defining data of the bundle, the exact S ) ) are coordinates on the n ˆ Z − ]. Progress has been made in understanding how chiral matter in F-theory Weierstrass must ˆ Y, 3 ((6 6 we have the heterotic topology: ˆ 123 X, O – CY , ≤ 2 It would be interesting to investigate the question of chiral matter in this context Of course, we derived the necessity of a non-vanishing chiral index for an SU(3) bundle Thus, for this class of bundles ( In the case that n F 120 ( 6= 0. Thus, we are guaranteed that smooth spectral cover bundles exist (stable in the 0 ≤ spectral cover bundles as in section more directly on thegeneral F-theory aspects side in of the howin future, G-flux [ in can particular be by incorporated including G-flux. in 4D Some F-theory models are described described as a spectrala cover, topological but invariant in thespectral the third bundle cover Chern moduli class bundle space. (andfull exists, As hence moduli we a the space can result, Chiral ofdiscussed so use index) hermitian in long section is it bundles, as all as aheterotic of good a duals which satisfy SU(3) probe must the to have base-point-free condition non-vanishing extract and index! the can be As structure described of by well-behaved the multiplicity of the but even at this preliminarydual level, geometries, the given results a of valuetheory of the contains chiral chiral matter. index are intriguing. For these compute From this we see thatthe theory the cases Recall from the argumentsH of section matter is localized in both the heterotic/F-theory geometries. where ( points on the elliptic fibercover). for As each usual, point onin the the λ appropriate adiabatic region in K¨ahlermoduli space). inside the heterotic 3 For these SU(3) bundles, straightforward to verify that simple line bundles section) the fiber type is split to JHEP08(2014)025 ) 1 1 5 m g V P × 7.28 1 and P ], but a 4 g = symmetry ]): , we would 131 2 8 3 B E 108 factor). For an )). However, this 1 2 P B ( 3-fibered 4-folds) has 0 (7.34) 1 ] which explore Chern no smooth bundles c K carried an ) could yield important → 2 − 134 , 1). For generic choices of ⊕ = 12 . Over the entire elliptically 7.25 , 1 1) + and the chiral index, indepen- 133 (1 , P η η (1 that Σ × alone is not enough to determine 1 4 ⊗O is a simple twist of the poly-stable 1 Y P η P m 1 → O T V ] (7.33) 2 2 satisfies , ⊕ = 2 m 0) 1 V , NK V (1 + η ) exist between – 52 – [ are the hyperplanes in each · ⊕ O ] 2 η 5.32 [ ⊕ ] and has been used more recently in F-theory model S, F 1) , 125 is trivial and = 0. According to the arguments of section , -flux) could be linked (at least in , we can likewise define the stable, slope-zero pull-back (0 1 2 G ) and ( − V B ) and η 124 → O bF 5.28 → 1 ] and more directly from the M-theory description [ + 3 ); i.e., X 130 → V 2 with simple singularities). We hope to explore this in future work. – aS : with vanishing slope; i.e., ( 0 B , consider the poly-stable rank 2 vector bundle defined as a kernel (of 4 ) and ( 0 ( 2 Y π O 1 F B c 124 ]. , 5.21 = 0 (i.e., that , we discussed how G-flux can break the gauge group associated with a 20 ) = = 6 − 132 η , ) via the following short exact sequence (i.e., a “monad” bundle [ T a, b ). To see this, consider the following smooth heterotic geometry. 6.5 ( 20 1 m V O ( with 2 and its links to intersection structure such as that in ( c 3 On the base It would be interesting to study more generally whether simple correlations such as 4 X Y (where appropriately block-diagonal choice oftangent the bundle map of this bundle is slope-stable forfibered all threefold, of the K¨ahlercone of even the map with twist naturally have determined from the F-theory(i.e., geometry the fiber degenerationon is type II*) andis hence too that quick there since were this argument ignores the fact that 7.9 Generic G-fluxIn that breaks section gauge symmetry purely geometric construction, through theheterotic structure bundle. of As the an second example Chern of class this of mechanism, the consider dual the case of base well known that some 4-foldsG-flux. cannot be In good these F-theory cases vacuable their without with second including trivial Chern non-trivial G-flux class in (orof the more presence generally of Wu class) quantizationinformation conditions. is along incompati- A these study lines of (forand the similar Wu investigations topology classes see of [ building [ those between ( dent of the existencecould of explicitly give a indications of heterotic such dual. correlations through its In topology. fact, For example, it it is is possible that the F-theory 4-fold in the F-theory geometryat corresponds points exactly in to the base thesingular — simultaneous that loci vanishing is, in of the the chiral3 4-fold index singularities geometry. in these (and The cases associated observation is thatbeen counted observed chiral by for co-dimension matter some 3 and time co-dimension [ construction [ more complete and directlyshows computable that formulation the is points desirable. determined by An the inspection intersection of ( models can be determined in the presence of G-flux based on aspects of the spectral cover JHEP08(2014)025 )) of V ( factor a c ( 8 ω E M = 0 indicates η ]. 80 , 42 )). Taking into account , which arise from higher from the pull-back of the 20 S 5.27 -type Weierstrass equation, = 0 (7.35) 8 . − E 0 η F and ( theory. In the other ⇒ in 7 itself, focusing on purely geometric 5.3 ] it is clear that more general bundles E ) on Calabi-Yau threefolds and there are 2 ) (as required by anomaly-cancellation, 4 2 ω S, F ω Y 1 ω 136 , 1 ω M ) non-trivial solely ω . At present, very few techniques are known 1 ( (4 3 135 ∗ V ∗ X ( π ) is an example of a smooth, everywhere stable π 2 – 53 – (see section c in the heterotic effective theory. Thus, in con- = 0 [ In the standard heterotic/F-theory dictionary, the S 7.34 7 ) = σ ζ ) + 88 E ( 22 0 ∗ F π , the holomorphic tangent bundle to an elliptically fibered threefold, over ( → 1 S c 8 ) = 0 and L ) = 1 5.3.3 E 1 ) V V ( ζ ( ( ) and we would be tempted to conclude here that the spectral 1 2 ) = 12 ∗ c c 2 π V 5.24 ( 2 c 1)-forms dual to the divisors with = 0 described only the Fourier-Mukai transform of the trivial rank 2 , 1 )) breaks all the symmetry. At first pass it would seem that the bun- η ]. V A.7 ] from ( 137 , ) = σ 1 . However, this is forgetting half of the data of the Fourier-Mukai transform: ) cannot be naively be described by a smooth spectral cover and that as a ) with 2 V 110 ( are the (1 ⊕ ∗ )[ i 7.34 ) and ( O π ] = 2[ S 5.25 ω S -form on the base, 5.2 Unlike in six dimensions, four-dimensional heterotic/F-theory duality provides new As this example illustrates, such possibilities must be taken into account if one hopes The pull-back bundle defined by ( , is frequently found to have a degenerate spectral cover description of this type (i.e., reducible or non- As mentioned in section section are possible after FM transform. 2) 3 , 22 1 many open questions which are ofapplications interest to to both string physics and phenomenology mathematics. (for These include example theTX large scalereduced scans for “Standard arising in heterotic theoriesthis and section we links provide between a bundle brief topology summary of and these structureand results. group. non-trivial insight In semi-stable into sheaves with the fixed topology structurefor on determining of the dimension the and structure heterotic of moduli space 8 Consequences for heteroticMany bundles of theof new the results moduli in space this of paper bundles (more are precisely, conclusions/constraints the regarding moduli properties space of semi-stable sheaves) to fully determine the propertiesdual. of heterotic vector bundle For modulistructure now, space and from properties, we its and consider F-theory leavefor an only future investigation work. of the the intriguing data possibilities of of G-flux the possibility of rankrank sheaves 1 (in sheaves this on caseV rank the 2) non-reduced on scheme data of these rank 1non-reduced or sheaves (whether reducible ordinary case line as bundles above) is or mapped higher into rank G-flux sheaves in [ the dle in ( result the Heterotic/F-theory dictionarythat is [ unclear.cover For ( spectral covers bundle in particular the rank 1 sheaf SU(2) bundle whichtradiction breaks to the conventionalit indication is of clear thegeneric that bundle F-theory this with issee an ( everywhere well-defined (2 where bundle JHEP08(2014)025 − η ) bundle. We N must be effective )) to conditions for 1 V ( Nc c determining part of the . We briefly summarize − η ω η M ], has focused on bundles with ) structure groups, independent of 19 N of the second Chern class need not be η , the parameter 5.3 ) and Sp( (or smaller) corresponding to heterotic theories – 54 – N 6 must be effective, and ,E η 7 ]) as well as more mathematical questions such as the SU(3), independent of the stable degeneration limit or E , 17 – . and section η 15 )) and the computation of higher rank Donaldson-Thomas must be effective in the stable degeneration limit, and V 5.1 ( η c ( ω ) structure groups, constructed using spectral covers in the sta- M N is necessary for SU( η 3 for gauge groups , ) must be effective for a spectral cover construction of an SU( 2 = 2 ) and Sp( B ( N N 1 a broad class of Calabi-Yauin threefolds. explicitly These constructing results or couldintriguing be characterizing to mathematically such utilize useful these bundles. conditions in Infor formulating bundles topological particular, constructed consistency it through conditions the wouldnot cameral be cover as construction explicitly (which described are at as present those for the spectral cover construction). heterotic/F-theory dual models andbase-point identified free a condition large on the rangesatisfied components of in models the dual inheterotic F-theory which side model. the is In an all exceptionalthe these group conditions on models or the SO(8). the second structure Chern Thus,for class group F-theory vector that bundles on allows are with us the necessary, exceptional and to and apparently identify sufficient, SO(8) structure group to be constructed over condition on the the method of bundle construction. Moreover, our study hastor shown bundle that do these not constraints on seem the to topology be of universal. the vec- We have considered a broader class of Previous work aimed at describingcontext vector of bundles heterotic/F-theory over Calabi-YauSU( duality, threefolds in such the asble [ degeneration limit. Herein we a have considered construction-independent consistency way conditions and on topology demonstrated that the base-point-freeness for with structure bundles SU(2) method of bundle construction. second Chern class of the heteroticin bundles must different obey contexts. several effectiveness constraints Nc have found that thesewith constraints a are more smooth general. heterotic dual, For any F-theory construction As discussed in section • • Base-point-freeness and bundles with exceptional structure group. here the main new results: Effectiveness conditions on possible existence of newvanishing/triviality mathematical of rules for linkinginvariants. topology ( Forprovides elliptically a rich fibered setstep Calabi-Yau of in new threefolds, using computational these heterotic/F-theory tools tools and duality to we determine view the this full work structure as of a preliminary Model” bundles undertaken in [ JHEP08(2014)025 2 B duals; F- 2. These 8 is blown up − E 2 P × 8 E models without requiring 8 ) remains unchanged), and E 3 × X ( 8 1 ]. , E 1 h 146 – there is a moduli space of Weierstrass 3 143 B , 56 , – 55 – 22 , theory. This extends to four dimensions results on 8 21 23 E ], we can identify topologically which F-theory models × 9 8 E SO(32) dual heterotic pairs that were previously understood / 8 E × 8 E jumps discontinuously (though 2 ] for similar “Noether-Lefschetz” type-problems and “jumping” in complex struc- B 142 – 138 , 103 the geometry of in higher-dimensional contexts [ those F-theory bases withtheory/heterotic SO(32) duality heterotic may duals, illuminate thereheterotic the are perturbative connection limits also and between thebetween these the resulting two moduli relationship distinct spaces on of Calabi-Yautures SO(8) threefolds associated structure with group the bundles and other bundle struc- geometric moduli space.models Over that each provides a base nonperturbative completionuli of the space perturbative heterotic of mod- bundlesthreefold. (sheaves) over Furthermore, the the dual distinct heteroticsitions bases elliptically that are fibered connected correspond Calabi-Yau by to tensionless small string tran- instanton transitions on the heterotic side. For a stable degeneration limit. Weare have dual explicitly to identified those SO(32) F-theoryCalabi-Yau heterotic threefold models string base, that theory and over showngauge a that group general in of smooth all SO(8), elliptically such which cases fibered cannot the be genericOn broken model further the has by F-theory a Higgsing. side, all the models we have considered are connected in a smooth By using topological termsheterotic F-theory in duality the [ 4Dare dual effective to supergravity heterotic action models for to SO(32) characterize as well as the K¨ahlercone of the Calabi-Yauin threefold decreases the correspondingly. K¨ahlercone Thisthat impacts change can the arise, properties andstructure of groups also the in seems many moduli to cases. restrict space the of existing stable bundles bundles to have exceptional space of the genericdel elliptically Pezzo. fibered This Calabi-Yau limit threefoldexplicit can over be the construction controlled corresponding of precisely thearise and in vector may provide these bundles an cases. withis avenue for exceptional More tuned the structure precisely, to in groups producedivisors these that the in limits generalized as del the complex Pezzo structure surfaces, of the Mori cone of effective base-point free condition isgeneralized violated del is Pezzo thatsurfaces surfaces they are that limits all of contain involve usualare elliptic curves del brought fibrations Pezzo together of surfaces in over self-intersection where specificover the ways. points generalized In where principle, del the Pezzo Calabi-Yau threefolds surfaces formed should simply be special limits in the moduli One interesting feature of the heterotic models with F-theory duals where the See [ • • • 23 ture/bundle moduli space. SO(32) heterotic/F-theory duality and the connectivity of string moduli space. JHEP08(2014)025 ) ). N N (see (8.1) = SU( 2) and no , H . When this non-base-point ) than has been 3 no bundles with V ( hermitian; in these 2) or (1 c , )) — the two integer H . , for some choices of 7 V in table and ⊂ ( E . Phrased differently, in 3 i 8 η 6.5 ) there are generic SU(3) H or H E ζ, c 6 ); this indicates a new layer in E correspond to structure groups ) there exist 3.2 4.2.1 η there is an upper bound on the over the elliptically fibered CY with η G η H in ( cannot be consistently enhanced ... -flux breaks the apparent symmetry ... 2 G ζ . In these cases, the bundle structure 4 H Y × . These conditions (given in the absence which 1 η ⊕ O ⊕ G H In addition to the non-Higgsable symmetries 2 ) gauge symmetries are possible (this would ) vanishing to degree (2 , but also with ( N H → . – 56 – ⊕ V f, g cannot be Higgsed in the 4-dimensional effective . In these cases, the bundle structure group can 1 H H 8 G E with for any bundle in the moduli space. As explained in , the commutant of → V . η or H 3 , with ( H V 7 ], the presence of generic, non-Higgsable symmetries for 4 indicate that the base-point-free condition described in cannot be consistently violated in the case of E Y thus cannot be enhanced to , 38 , 6 G 6.4 6.4 E 37 , 4 . F , H with a generic symmetry 2 , it is clear that there is no way to reduce the structure group to SU( 4 η G Y geometries indicates that for a given and section 4 , if a generic symmetry Y )) and its constraints. As described in section 5.3 6 V ( c -flux) are listed in table ( = SO(8), ). In the dual heterotic geometry, these geometric observations provide constraints ω indicated by the Weierstrass equation of G 6 never be consistently decomposed cases the gauge group In a similaror spirit, SU(2) for symmetries many on possible examples enhancements (see to section higher SU( (i.e., for a hermitianfor bundle, regardless any of technology the appliedfree method choice to of of a construction). bundle As associatedWe find a with that result, all a the consistent non-base-pointH free examples of The results ofsection section of structure linking notvalues only specifying the remainingcorrelation bundle would involve topology a finer notso level studied far of in explored structure linking in this the work. literature. We Such hope a to explore these issues in further work. M bundle/4-fold topology it may beG that generic group may beoccurs, bigger it than must indicated involve by non-trivial the values bounds for on structure group larger than order to definethreefold, a there bundle is with aof structure minimum group “size” for We further observe that these rules are at present only a first step in determining As first explored insingular [ size of the structuresection group theory, this implies that for the given topology ( • • • • with a smaller (reduced) structure group. Lower bounds on structuredescribed group, above and theirmany heterotic examples consequences, of in thesection F-theory geometry we haveon seen when a given vector bundle can be consistently decomposed into a reducible sum: Upper bounds on JHEP08(2014)025 , , i 1 ]. ’s η P V F 2 we 72 B 2 -flux. V/ ] that 2 B G 60 B bundles) F ⊂ itself [ 1 F ⊕ is the only P M 4 = F compatible with V ’s that violate the T 3 in some cases. B . For each base H ’s for some specific 5 3 , ). As in the case of the B 4 N (constructed as 3 is that more than half of the B 4 giving the generic elliptic fibration 3 X the moduli space of semi-stable sheaves cannot decompose as η V ]. may be strengthened by the presence of – 57 – ’s that have smooth heterotic duals. Each choice ’s over that base (the number of possible “twists” ’s), and the number of these 147 , and the group quotient structure of 3 3 , H 3 B B V B 137 , 30 constrain the structure of possible sub-sheaves have been explored in previous work; in particular, the H that support such models are the generalized del Pezzo (gdP) 4 2 B described in the first four lines of the table, and the 18 toric Fano ), and explicitly analyzing the monomial and singularity structure m F , it could act to reduce the generic gauge symmetry arising purely from 4.10 ) non-vanishing and ∆ vanishing to degree , where an example of a topology is given for which H and )–( are base-point free. The resulting enumeration of 2 f, g -flux cannot enhance the gauge group in a way that violates these lower P 4.8 2 G 7.7 bundle giving acceptable . The results of the toric analysis are listed in tables , η 1 1 4 tabulated in table P η ; in these cases for the given value of , with a particular choice in the part of the bundle topology characterized by 3 2 ). The analysis was performed by considering all possible twists B corresponds to a specific Calabi-Yau threefold geometry, which could lead to even stronger lower bounds on Harder-Narasimhan filtration of Finally, these “lower” bounds on While bounds on section consistent structure group. The lower boundsand on determine aetc. bound The below determination which of such substructure has important consequences for the non-base-point free examples above,exceptional this structure constrains groups the can ways be in decomposed whichWe bundles into hermitian with find factors. examples forH which there appearcan to contain bundles be with both exactly one upper allowed and structure lower group only. bounds See on for example require ( B One interesting observation from the data in table 3 3.2 • • • B in the bases bundles over varieties were also described in [ possible toric F-theory geometries violate the base-point free condition on at least one whether matches that described in the previousin section section using the more generalhave constraints indicated described the number ofT distinct base-point free condition for one or both gauge group factors. A subset of the F-theory over in ( the bounds ( of the resulting Weierstrass modelhave a in (4, the 6) toric singularity description. on a For each divisor model or that curve, we does determine not the gauge group content and section. The toric bases surfaces, a subset of 16can of act the complete as set basestoric of for 61,539 gdP toric bases elliptically surfaces we fibered enumeratedof find in Calabi-Yau 4962 [ manifolds distinct with section. Over the 16 9 Enumeration of heterotic/F-theoryWe dual have systematically pairs analyzed withthat all toric have toric smooth bases F-theory heterotic duals bases on Calabi-Yau threefolds that are elliptically fibered with JHEP08(2014)025 3 B bundles 1 P 0 0 0 0 0 0 0 0 0 0 0 32 45 29 217 173 496 # cod 3 for each base i G becomes smaller and 0 0 0 0 0 0 0 0 42 25 69 59 26 140 150 185 696 H that violate the base-point NB (2) 3 B 0 0 0 0 0 8 14 31 18 100 396 213 755 219 149 119 2022 NB (1) ’s 3 , tuning Weierstrass monomials can lead B 3 14 82 24 62 83 36 109 472 173 776 729 312 406 351 214 1119 4962 B # 1 , – 58 – . The final column is the number of models that have 1 1 2 2 2 3 3 4 4 4 4 5 5 6 6 6 7 ± h ) ) ) ) ) ) 2 0 1 2 2 3 vanish to degrees 4, 6. P F F F is characterized by the sequence of self-intersections of toric dP dP f, g 2 . Note that this gauge algebra represents the minimal (most B 2 ⊕G 1 G . The base 2 B 2 we tabulate the number of models in the full set that have each of the possible B 5 compatible with a smooth heterotic dual are generalized del Pezzo’s, we expect . Table of all smooth F-theory bases with smooth heterotic duals that are 2 base (1, 1, 1) ( (0, 0, 0, 0) ( (1, 0, -1, 0) ( (2, 0, -2, 0) ( (0, 0, -1, -1, -1) ( (1, -1, -1, -2, 0) (-1, -1, -1, -1, -1, -1) ( (0, -1, -1, -2, -1, -1) (0, 0, -2, -1, -2, -1) (1, 0, -2, -2, -1, -2) (-1, -1, -2, -1, -2, -1, -1) (0, -1, -1, -2, -2, -1, -2) (-1, -1, -2, -1, -2, -2, -1, -2) (-1, -2, -1, -2, -1, -2, -1, -2) (0, -2, -1, -2, -2, -2, -1, -2) (-1, -2, -2, -1, -2, -2, -1, -2, -2) total B In table correspondingly larger. The minimal gauge algebra summands 2 curves, and for such bases a very high fraction of models violate the base-point free distinct gauge algebras generic) gauge algebra for eachto base. enhanced For gauge each groupsspecial through loci “unHiggsing,” in correspondingG bundle on the moduli heterotic space side where to the structure group condition on at least onebases side. Since the vast majoritythat of the the fraction hundreds of of all possiblestructure non-toric models group with is smooth quite heterotic high. duals that have exceptional or SO(8) of the gauge factors,generic so bundles that with moreviolate exceptional than the or half base-point SO(8) of the− free structure corresponding condition group. heterotic are models The those have only with models generalized that del can Pezzo bases having Table 4 over toric bases divisors. NB (Non-Basefree point condition free) on indicates one (1) thea or toric number both codimension of (2) 3 sides bases locus Σ where JHEP08(2014)025 , 2 6 7 ].) , an su 3 8 so [ with , c . In 2 3 1 B su ⊕ 7 , with 2 3 so cg ⊕ 2 = = 0 on the divisor su is a perfect square, 3 ∆ vanish to degrees 2 contains intersecting f z 8 g 0 e 2 B f, g, 7 4 0 e when 4, the gauge algebra is 8 , a variety of distinct models with 6 2 0 0 0 e 3 so , B to arise as a generic gauge algebra 4 7 2 0 f 32 13 so 8 in the toric picture using table ∆ are 2 3 9 0 2 0 so . Similarly, when ) is only guaranteed to be a perfect square m f, g f, g, 2 2 do not uniquely determine the gauge algebra v 2 0 . In principle, the gauge algebra could be u, v g 34 12 54 14 n ( when the leading part of m 2 ) a perfect square, where 2 2 – 59 – 6 u g ) is not a perfect square, but this does not occur v f, g 3 e n 2 7 1 6 0 0 0 u, v 2 su ( u, v 2 ( bu g 2 2, which support the gauge algebra 2 f . The algebra becomes = 2 0 − 2 11 47 62 14 74 24 su arising in generic models for the 4962 F-theory bases , 3 g 3 2 , g − ) with , m 3 considered here. (Note, however, that for more general bases 3 · 2 ⊕ G z 15 v 712 121 499 589 276 184 890 3 ( 1 = 0 and − are coordinates on the divisor. When there is no restriction other 1245 n G B 3 O 3 g · 3 2 2 8 4 6 7 8 au + u, v f × e e e g 2 so su su = z ) 2 f u, v we see that the great majority of models (85%) have some gauge group ( 2 5 g . When the vanishing degrees of = and smooth heterotic duals. Note that for many bases . Gauge algebras g 3.1 2 B In table automatically imposed from themoduli geometry, which fields. cannot be Furthermore, removedsubgroups by most of Higgsing SU(5), of charged or theas contain models SU(2) discussed or in either SU(3) the have factors previous gauge that sections. factors cannot Note be that that enhanced are the to not gauge SU(5) group described here is purely which can occur when for any of thethat 3D do bases not have smooth heteroticcomponent. duals we This do occurs, expect for example,curves in of 6D self-intersection models when the base 3, 4, 8, thewhich gauge again algebra is factor only possible isthe when only it story is is a slightly single moregeneric even subtle, monomial. but gauge again For algebra vanishing easy degrees to factoroverall 2 analyze is constant in terms (complex) of coefficient, thesingle monomials. which monomial is The only possible when each contains only a section unless locus in question and than the vanishing of certainif monomials, it contains only a single even monomial are determined fromplaces the where dual the monomials degrees oftype, in vanishing the of gauge algebrawhich is can fixed be by the read monodromy off around from the the codimension structure one of divisor, the monomials following the discussion in Table 5 toric enhanced gauge groups candistinct be elliptically fibered realized fourfolds when in moduli the are F-theory picture. tuned to specific loci, corresponding to JHEP08(2014)025 , ; 2 8 e F B 7.8 = 2 for 6D T 2 B . One of the 6 and the base 4 bases for which , 2 B with twist 1 F ]; as discussed in section bundle over the 10th base . Note that, as mentioned in , in some situations the gauge with self-intersection -1. This 4 89 1 2 D P ]. 6.5 B vanish to degrees 4 vanish to degrees (4, 5). We have ∩ bundle over 61 3 , 1 bundles over toric P D f, g f, g locus where the degree of vanishing of 60 1 , ∩ P 8 8 e − in the last column of table that can act as the base of an elliptically 11 curves; along such curves there is an 2 0 3 − B B . We leave further investigation of the models − – 60 – Σ and is the divisor in ∩ , ’s that are 5 3 S 10 D B living in the − , on 3 9 B − f, g arise that are not included in this tabulation because there , where 8 5 e D , in some models that are otherwise well-behaved there are codi- = . The simplest example is the 3 4.1 T B there can also be codimension three (4, 6) singularities arising at non-toric , characterized by divisors in the base with self-intersections (1, 0, -2, -2, -1, -2) vanish to degrees (4, 6) at the point Σ must be blown up to have a base 4 4.1.6 3 g B Another feature that can arise is a codimension two singularity on a curve that does not Several unusual features arise in many of the 4962 models we have constructed. As Note that there are many toric reaches 4, 6. These are a special class of examples of situations where a curve in the where this occurs for each base surface and 3 the presence of abelian U(1)the factors. threefold This bases occurs inthis roughly half base (2495 gives of no thebut 4962 codimension has total) one a of singularity (2, associated 3)with these vanishing with of features nonabelian to gauge future groups, work. lie on any divisorindicate carrying matter a charged gauge under group. thein While nonabelian this in gauge situation general groups this codimension ofare interpretation the two simply singularities corresponding is cusps divisors, not in possible. the discriminant It locus is with possible no that physical meaning, these or singularities they may herald model has no codimensionf one singularities associated withsection nonabelian gauge groups, but points, such as genericallynot occurs attempted on to curves classify where the models with such singularities here. with this feature (orB bug) in thesimplest examples toric of set; a threefold wein with have table this tabulated property is theand the number a of twist threefolds divisor mension three singularities ofherald a order sickness of (4, the associated 6).such 4D codimension supergravity three It theories singularities [ is mayabelian not also gauge symmetries. be known associated In whether sometoric with these cases, point chiral given a matter, singularities by codimension G-flux, the three or intersection (4, of 6) three singularity toric arises divisors. at There a are a total of 496 models F-theory models that contain gauge algebra summand and 3,must 2, be or blown 1 points up where for a smooth F-theory model [ mentioned in section f, g base fibered Calabi-Yau fourfold. After this blow-up,heterotic the dual F-theory model and no is longeralso not has a generically included smooth in non-toric. this These analysis. cases In are this closely situation the analogous blown to up base base surfaces is group may be modifiedof when this G-flux effect is to taken future into work. account. We leave further investigation gauge algebra summands are codimension two curves in that determined by the geometry. As discussed in section JHEP08(2014)025 and 2 B = 1 string N and SO(32) theories on 8 E × 8 E . We have shown that the number 2 ] in which the duality has been most B 19 is toric provide an extensive dataset of dual – 61 – 2 B . Within this class of smooth dual geometries, there bundle on the F-theory side and related components = 1 4D supergravity theories that can be realized in 2 1 B P N is toric. ], we used topological structure, in the form of axion-curvature 2 9 B bundle over a base 1 bundle with section over the same base P 1 P The roughly 5000 models where The results described in this work represent a small step towards a systematic charac- By focusing on the underlying geometrical and topological structure of the theories, we that is a 3 smooth dual geometries where theB F-theory model is described inare terms many of questions a that threefold could base be explored further. heterotic/F-theory constructions that may be useful in a variety of contexts. To aid further In this work, followingsquared [ terms in theometries. 4D This supergravity gives theory, an tothat association identify is between independent dual F-theory of heterotic the constructions andthoroughly stable and degeneration F-theory studied; heterotic limit ge- the bundles [ approach taken here has enabled a systematic classification of all a further systematic expansion ofas our tools understanding for expanding of the heterotic/F-theoryto range duality, string of as applicability well compactification. of bothfuture We heterotic directions. conclude and with F-theory approaches a brief summary of10.1 some of these Detailed possible physics of smooth heterotic/F-theory dual pairs terization of the broaderstring class theory. of There are clearlyther. many directions Many more in which detailed this aspectscan work of be could the explored be physics further expanded of using fur- the the large tools class described of models here. described This here work also provides a basis for of the second Cherna class detailed of characterization thegeneral of structure bundles the groups on circumstances should thesmooth under exist heterotic Calabi-Yau which both threefolds. side. for slope-stable heterotic implications bundles The for These with structure the constraints interplay of give between chiral chiral matter matter and on G-flux the on heterotic the side F-theory side. has have developed tools and identifiedlimits features of or these bundle models thatfrom constructions do the not on F-theory depend the side on aexistence specific heterotic simple of side set a of of constraints Calabi-Yauterms the that compactification of duality. are the geometry; necessary “twist” these and We defining sufficient constraints have the for are identified the expressed in and in F-theory through compactificationis on itself an elliptic a fibration overof a topologically base threefold distinct that vacua inmodels this where class the is base finite, and we have explicitly enumerated all In this paper wevacua have that given admit a both global a heterotic characterizationvacua description we of have and a considered a are broad dual described F-theory classa in description. of smooth the The heterotic 4D Calabi-Yau class theory threefold of through thatcarries compactification is on a elliptically smooth fibered vector with bundle section (and over in a some base cases, 5-branes wrapping the elliptic fiber), 10 Conclusions and open questions JHEP08(2014)025 2 B ] for 158 – is available (which by 9 152 , 7.6 34 . Much work has 7 bundle over that E 1 P , or 6 E ; other non-toric generalized del This file contains a listing for each s http//ctp.lns.mit.edu/wati/data/het-F- 0 3 describing a worked out in section B 24 T 4 ] or at ’s, there are many branches of the moduli 3 148 B – 62 – 5000 toric ] for a review of some of this work, and [ ∼ 151 and associated minimal gauge groups described in section – 3 B 149 6) codimension one or two singularities, as well as the generic gauge , ], is that a large fraction of the elliptically fibered Calabi-Yau manifolds generated by for codimension one singularities over that base (SO(8) in of the complete set of allowed twists 2 61 , 2 60 B ⊕ G . 1 G Although we have focused in this paper on general aspects of heterotic and F-theory For the models considered and enumerated here, many more detailed questions remain The list of 4962 toric bases 24 been done in constructing phenomenologically orientedunification F-theory structure models (see based [ on an SU(5) online in an ancillarydual-bases.m mathematica format text file at [ groups and 4D physics requireswe a have better studied incorporation heresmallest, of are and the among may effects provide those the of in mostterms G-flux, promising of which the candidates potentially the phenomenologically models for relevant minimal realistic gauge models groups,theory geometric of we geometries gauge have physics. contain found groups that In geometric many are SU(2) F- to and/or SU(5), SU(3) factors but that that cannot can be for enhanced example be enhanced to SO(10), be relevant for more phenomenologicaltematic “model study building”. of F-theory One models general bothdimensions in lesson 6D [ from and the in 4D, sys- illustratedthat particularly clearly can in be six usedWhile to a compactify clear F-theory understanding give of rise the to connection large between “non-Higgsable” geometrically non-Higgsable gauge gauge groups. effects of G-flux onF-theory the moduli F-theory and side; produce thishave chiral mechanism developed matter. will here We in will hope general help that lift in the many elucidating explicit geometric these correspondence issues we constructions further. that are independent of specific models, some lessons have emerged that may moduli spaces remain to beconditions investigated. under We have which identified the from thestructure dual F-theory heterotic group; side model specific in shouldknown many admit for such of a models, bundle finding these such with constructionsrelated cases represents exceptional to another there this class work. of is open And, problems no as mentioned explicit throughout the mathematical text, construction we have not incorporated the to be addressed. For eachspace of in the which thea generic tuning of gauge Weierstrass groupspace moduli is in on enhanced the the by F-theory heterotic “unHiggsing”, picture side. and corresponding special to loci Many in general bundle aspects moduli of the branching structure of these more generally for anysmooth of elliptically the fibered several hundred Calabi-Yauwould generalized threefold. be del One the Pezzo natural bases explicitnon-toric) extension that construction bases, of support and along the a classification the workitself lines of here already of this the gives broader example risePezzo class dP surfaces of to are (generically roughly expected to 7000 similarly additional generate large numbers of additional examples). ification geometries in a fileof that the can bases be downloaded. that does not have (4 algebras those cases that have an SO(32) heterotic dual). The methods of this paper can be applied development in this direction, we have provided the details of this set of F-theory compact- JHEP08(2014)025 – ) can 159 3 , X 82 on ) can clearly i V 7.34 for a discussion). ). For example, of N 7.9 = 0 of ( η , there appear to be many 9 or of the bundles 4 Y ]. However, the results of this work 80 , 42 , 41 , – 63 – 19 ], including bases related to non-toric generalized 61 symmetry (on at least one patch), 897 of these fail to ) for SU(2) bundles constructed as generic, irreducible 7 E 3 branes. We hope to explore these topics in future work. 5.29 D ) or b) the bundle moduli space contains no irreducible spectral ] for example, for more exotic possibilities). It is also possible that this 5.3.3 103 As mentioned above, we have not considered here the effects of G-flux in modifying At present, the explicit heterotic/F-theory dual “dictionary” has been most fully de- ] has focused on the role of global U(1) factors in F-theory models. In most cases such the potential in the low energyimpact theory. not only A on better the understanding problemries of of but this moduli could potential also stabilization would lead in have to heterotic/F-theoryfor novel effective dynamical example, theo- effects the in the obstruction 4-dimensionalduality of theories — to tensionless including, non-commutative string/small-instanton transitions and possible the underlying Calabi-Yau geometry ofdimensional F-theory. heterotic/F-theory One duality of is the the most factsymmetry interesting (i.e., that aspects deformations deformations of that of the change 4- complex thebe structure gauge obstructed. of These obstructions can appear both through D- and F-term contributions to parity condition may be a moreassociated general F-theory constraint models, and may for indicate examplea some consistent that problem choice with may of the indicate G-flux. aexplicit In conflict moduli any with case, mapping the there existence of remainsnew of the something insights to heterotic/F-theory into be dual both understood pair theories. in which the could yield important satisfy the parity conditionspectral ( covers. This indicatesbe constructible that as if ordinary good SU(2)described heterotic spectral in covers duals section (i.e., exist they theycovers possess at must one all either (see of [ the a) limitations not not be described asOther an specific ordinary, examples smooth arise spectralconsidering in the cover the set (see class of section of 4962good toric models F-theory dual we geometries pairs for have described which analyzed in wewith here. cannot section standard directly tools, For construct the instance, even heteroticthe when dual the bundles 947 heterotic 4-folds structure with bundle a is generic SU( termined in a cornerspectral of (more moduli generally space cameral) in coversshows which [ that the many consistent, heterotic perturbative bundles heteroticapplication can theories of be these cannot constructions. be described described For via example, by the a bundle naive with U(1) factors arise only atgiven very F-theory special base; tuned it loci wascertain in recently special the found, F-theory Weierstrass however, moduli that basesdel space U(1) in Pezzo over factors 6D any bases can [ considered beheterotic/F-theory here. generic models over Further provides investigation another of interesting U(1) direction factors for in further the work. class of dual generic gauge group structuresmodel that building we approaches, have perhaps exploredwe in here have the focused might context play on ofnot into generic GUT investigated more geometric groups the general structure other tuning in than174 of the SU(5). abelian models As gauge studied here, group we factors. have also Much recent work [ some specific global GUT models). It would be interesting to study more broadly how the JHEP08(2014)025 can ]. It M 82 , ]), which 81 82 bundles) and situations 1 P bundle with section. It is also possible 1 P ). If is not known how F-theory duals to such – 64 – fibration without a section or indeed even more B fibrations (rather than 1 1 is itself a P P 3 B is a 3 B bundle that also has a single section. Since we expect that ]. 1 P 175 , 66 is a 3 B built as more general 3 B Another example of a situation where heterotic/F-theory duality is not well understood More generally, there are classes of geometries that are slightly more general than One clear question is the extent to which heterotic/F-theory duality can be systemati- 4-folds on the F-theory side. This is another interesting avenue for further investigation. comes from the facthave that multiple for components Calabi-Yau (see threefolds,situations appendix can the be moduli understood. spaceerotic A of moduli natural space bundles hypothesis there is would exist that topologically for identical, each non-diffeomorphic component Calabi-Yau of the het- should give a correspondingof construction this on correspondence the have F-theorywhere not side, the been F-theory the worked threefold out. details base to of In consider the this situations physics paper where general we geometries considered [ only cases natural heterotic duals. F-theoryrecently been models incorporated with into a theis multi-section less global but clear, moduli no however, space how global toYau of section threefold construct Weierstrass have an with models F-theory multiple [ dual sectionscan for or be a a realized heterotic multi-section. as model special While onthe limits a in Jacobian in Calabi- principle fibration the such associated Weierstrass threefolds moduli with space threefolds of having elliptic a fibrations multi-section, (using as in [ where either the heterotic or F-theoryMordell-Weil group) elliptic fibrations or have no multiple section sectionselliptically (higher at fibered rank all. but In have particular, morecompletely for clear. than Calabi-Yau one geometries F-theory section, models that with or are are multiple a understood sections multi-section, simply (higher the as rank models story Mordell-Weil group) with is additional not U(1) yet factors, which generally should have ple geometric framework of F-theoryquestions has that the are potential rather to subtle clarify some in of this the context mathematical onthose the heterotic considered side. here include bases which heterotic/F-theory duality is not understood. These in- sheaves. While many examples of thisstructure have such been as studied in enhanced the gauge literature,tensionless groups the string and appearance transitions of additional geometric isWeierstrass moduli more models arising transparent through on geometrically the fromuseful F-theory the context side. point in which of The to view framework systematically of extend developed the here duality may in provide these a directions. The sim- to extend this duality toto a introduce much more broader class complicated of mathematical vacua, objects possibly in at the the theory expensecally on of described one needing when or the both heteroticsingular, sides. Calabi-Yau leading geometry mathematically and/or to bundle a structure description becomes in terms of more singular objects such as In this work weboth have focused sides on have the ellipticallyF-theory simplest fibered base class Calabi-Yau of geometries heterotic/F-theorythe with dualities, heterotic a where and single F-theory section, constructionsdescribing the are and same simply the physical different theory mathematical in approaches distinct to limits, we expect that it should be possible 10.2 Expanding heterotic/F-theory duality JHEP08(2014)025 2 ), B is in base α 3 → D = 1 4D X 3 ( ∗ X N π is given by : -forms dual ). If 3 } = π 3 1 X α , X 1 D { ] for a more complete 30 , (dual to a single point), to be section (and not a 2 19 0 B . For such smooth, minimal D 2 B on ζ = 0 (A.1) γ D that span the Picard group of ∩ β 25 – 65 – D ∩ as an algebraic sub-manifold within α ) + 1. We will denote the basis of 2 2 D B B ( 1 . , ) is an ample divisor on } 1 2 α h B ( , ω 1 0 , ) = 1 ω 3 { X ( 1 and divisors pulled back from the base of the form , , . . . h 1 0 h D = 1 α , base α D By virtue of this simple fibration structure, the triple intersection numbers of these Finally, on the heterotic side there are compactifications on Calabi-Yau threefolds that We focus here on a minimal form of elliptically fibered threefold in which all exceptional curves in the 25 Moreover, from themultisection) very it definition is of guaranteed what that it for means any for two-form fiber have been blown down. to the divisors above as divisors exhibit a universal behavior. First, since the base is a 2-fold it is clear that treatment). We focus onwith a smooth single elliptically section fibered (whichWeierstrass defines Calabi-Yau form, threefolds, a minimal setthe of zero divisors section, where elliptic fibrations, A Properties of elliptically fiberedIn Calabi-Yau this three- appendix and weand fourfolds topology briefly of review elliptically a fibered Calabi-Yau collection manifolds of (see [ useful results regarding the geometry the DOE under contractNational #DE-FC02-94ER40818, Science and Foundation under was Grant alsoSimons No. supported Center PHY-1066293. in for We part Geometry would(WT), by like and and the to the Physics thank Center (LA for the of the and this Theoretical WT), work. Physics at the MIT Aspen (LA) for Center hospitality for during part Physics Acknowledgments We would like toJonathan Heckman, thank Samuel Ron Johnson,Daniel Donagi, Denis Park, Klevers, Antonella and Gabriella Grassi, Lucia Martini, Swanson James David for Morrison, Gray, helpful Thomas discussions. Grimm, This research was supported by that incorporates all degrees ofheterotic/F-theory freedom. duality In may an provide optimistic somethe scenario, insight theory further into and development a a of more broadersupersymmetric complete and string formulation more theory of unified vacua. characterization of the full space of are not elliptically fibered.from Such geometries elliptically can fibered be Calabi-Yau reachedunderlying threefolds, by geometric so nonperturbative transitions moduli should space inknown of principle mechanism F-theory be by compactifications. connected which to F-theory Atan the present, can incomplete there include physical is such theory, no vacua. however; there F-theory is is no at direct present action still principle for the theory JHEP08(2014)025 (A.6) (A.7) (A.9) (A.2) (A.3) (A.4) (A.5) ) (A.8) 2 B ( , the volume 1  , α β 1 t can be written ω h α α 3 t t − 0 X t + 0 β . αβ ω 2 0 0 K 2 = 10 d t ) it is possible to write B ) 2 . With these results, the αβ = α αβ B + 3 A.3 m ( ω ω m 1 α α = 0 β t c = 2 K ) α K 0 α t 00 αβγ C ( d d ) = K ω α forms from ) + 11 2 2 αβ 00 forms in ( ∧ ζ } B 0 B d ( 2 ) = m ( } ω B 1 , + 2 2 2 c 2 ω 0 c ∧ , = B + 3 − 2 ω ( ∧ 3 β + 2 1 3 { }{ K ) c ∧ = A 0 1 0 D X 2 , t ω = η ω ( 1 B ∩ K 3 0 { ∧ Z – 66 – X α ω 000 ) , on ) = Z 2 d of D ∧ V V ,  B = satisfies 0 ∩ ( π ( 2 1 ω 1 0 2 3! β c ) c 2 B D K = ABC B forms, α 3 ( d ω 1 } , K X 2 1 ∧ ) = 12 . These facts are enough to derive the following important , αβ h αβ 3 2 ω m { m base β ∧ TX = = ( D ω 2 c ∩ αβ , are given by X ) = 2 + 000 0 } Z 2 d d α B 1 3! base α ( , ω 2 0 D are pullbacks through c ω 2 ζ ) = { = 3 B Z X = is the canonical class of the base, αβ and A = 1 (that is, the zero section intersects each elliptic fiber precisely once). It follows m ) = η ω K takes the form Vol( 2 ] ζ 3 Finally, using the redundancy relation on The fibration structure guarantees that the second Chern class of With these intersection numbers and a form chosen K¨ahler B 19 ∩ ( X 0 χ B A brief explorationA of novel rigid feature of bundles four-dimensional compactificationsthe of possibility heterotic of string multiple theory/F-theory components is In in the the case dual (vector ofhave bundle/fourfold) heterotic/F-theory not moduli duality, yet spaces. such been multiple studied components in to detail. the Indeed, moduli thus space far in the literature the correspondence where the second Chern class of any bundle, as [ where in addition the topology of of where where triple intersection numbers of from this fact that where cohomological identity on D JHEP08(2014)025 . 0) A . On (B.3) → 3 ∨ X L → 2 V ) will be rigid if factor to → 2 B.3 P L ) (B.1) → 2 H ) = 0 in the K¨ahlercone, − L ( 1 ]. Geometrically the spaces µ H ( 99 [ ) = 0 (B.2) 2 O 3 ⊗ ∨ X ) + , of two projective spaces, consider the 2 ,L 2 3 H P satisfying ∨ X + × ( 3 L 1 1 2 X h + P H – 67 – L − ( ) = ). This reducible bundle in ( 2 O ⊗ 5.6 = ,L ∨ 3 L X may be non-trivially “glued” back into an indecomposable ( 1 + ∨ h L ) constitute the space of non-trivial extensions (for example ) = 0 2 2 = ⊗ L, L ⊗ ) 2 ∨ ∨ V ,L L 3 ( X ( X, 1 ( ]. 1 H are the restrictions of the hyperplanes of each ambient h is rigid, since the dimension of the space of bundle-valued singlets is given by hypersurface in the product, 19 2 2 ), ) parameterizes the space of non-trivial extensions 0 } V 2 2 3 ) = is a holomorphic line bundle on ,H , ⊗ ⊗ 2 1 3 has no infinitesimal deformations, that is, it is rigid. ⊗ { L ,L ,L H ∨ 3 3 L In this appendix, we make a tentative exploration of whether it is possible to obtain Although the example above is interesting from the point of view of vector bundle For elliptically fibered Calabi-Yau threefolds, it is straightforward to come by examples For the moduli space of stable sheaves on Calabi-Yau threefolds a natural case of X X X,L ( ( ( 1 1 + 1 where as required for supersymmetryh by ( isolated components toabove, the over the moduli class of space Calabi-Yauthreefolds threefolds of with considered SU(2) here a — bundles, that singleOnce is, such again, section, elliptically we fibered as obeying can the the search for topological one bundles identities described of listed the form in appendix moduli spaces, it is nottheory clear pairs what constructed the in impact thisit of work. has such no Although examples section the will and(if threefold be any) cannot above in is be is the unknown. elliptically written heterotic-F- fibered, in Weierstrass form. At present its F-theory dual H H which parameterize how SU(2) bundle. Since theL space of such extensions vanishes in these cases, the split bundle where this space, for generic values of the complex structure of degree following poly-stable SU(2) bundle not they reside inbundle the is same rigid (and component the of otherconsists has a local of global deformation its moduli), bundle own it modulirigid is distinct bundles space. clear is component that one the However, to of rigid if the bundle bundle simplest one moduli probes for space. multiple components of Asof bundle such a moduli rigid spaces. result, vector bundles. a search For instance, for over the Calabi-Yau threefold defined as a single uli space of Calabi-Yaucomponents fourfolds [ has only been studied ininterest the is case given when of the single,In local general, moduli connected space given in two fact bundles contains with an the isolated, rigid same component. topology, it is difficult to decide whether or between vector bundle moduli spaces in heterotic theories and the complex structure mod- JHEP08(2014)025 . 2 B and (B.7) (B.9) (B.4) X → satisfies 3 2 on X . B : V T . To see this, π 1 ) can be locally E )) is one dimen- V on ( U ∗ ( 2 1 ]. According to this π ⊗ − m L 98 (pushed forward under f R 0 (B.11) V 0 (B.10) 1 and the spectral sequence → , )) then, we need only consider 2 → ). Moreover, the structure of , and any bundles U α B 2 α 2 = 0 = )) (B.6) D ⊗ B B D σ ( )) (B.5) | V ) = ∗ α m ) )) (B.8) ( b σ l,m V ∞ π → ∗ V α σ ( ( ( α π 2 · E ∗ 3 ∗ D b B π ( m π σ m X ∗ ( + q m : + , we have a natural bi-grading such that → O| 1 π l ,R 2 R X α 3 ) E = π ⊗ O aσ ,R b B p ( X 2 U 3 → O = ( )) + ) = B 1 X ) → ( -th direct image sheaf, − l ) = ∞ U – 68 – σ aσ O 3 ∗ ( → O aσ ( f m E H 3 ( ( 3 π ) X = X X m O σ : = ⊗ X,V L O ( − H ( = π p ( , the V ∗ 3 l,m 1 1 ( → O L π H ∗ X P i E 3 )). This implies that under push-forwards we have the π R U → q X α , since the spectral sequence terminates at on ) built from divisors pulled back from the base R D ) is non-vanishing only for → O α ( +1 -th direct image sheaf of the bundle U r ∗ 0 V D → O m − ( π )) = α ∗ 0 α , α b ) yields r,q b π ( 3 σ ( D + ( m 3 ( O X p 3 ∗ r X R X π E O on α ) is the O b V → V ( ). We need not concern ourselves with the iteration of the sequence via the + ∗ )) can straightforwardly be determined by considering the Koszul sequence: π ) = π p,q r α aσ aσ m E ( ( D : R 3 3 α , X X b r ( d B O O 2 ( ( To determine the cohomology of To see this, we must consider the line bundle cohomology of As we will argue below, however, for the geometry in consideration in this work, such B ∗ ∗ on π π O i i ∗ Twisting this by U R R behaves under the push-forwardbundles functor. of the The form firstπ useful useful observationso-called is “projection that formula”. line For a fibration sional. As a result, terminates at To analyze the cohomology, we further need to observe how a line bundle of the form note that on anyrepresented open by set the pre-sheaf For elliptic fibrations, however, the fiber (locally isomorphic to and and the fibration maps Here we are aidedspectral by sequence the for formalism thefor of fibration any Leray bundle Spectral sequences [ where examples appear to be rareconsistent and with we the have obtained fact nofourfold that explicit (with examples. in a This the resultmoduli single dual is space) F-theory connected fully for geometry (and each we choice non-trivial) find of component vector a bundle to single topology/twisting its Calabi-Yau parameter, complex structure JHEP08(2014)025 on ) = F (B.14) X,L ( 1 H . In order to ) means that 3 . . . (B.17) X B.20 )) = 0)) = 0 (B.15) (B.16) )) = 0 (B.18) ]. With these results σ σ aσ 98 (2 (3 ( 1 (B.19) 3 3 3 0 (B.12) , X X X O O O 0 (B.13) → ( ( ( ) can we have = 0 ∗ ∗ ∗ 2 → π π π B 1 1 ) = 0) = 0 (B.21) (B.22) 1 , i B.8 = 2 2 L L σ R R R | K ∗ ∗ K ) π π ∨ in ( σ 1 1 ) → ⊕ · ) = 0 (B.20) F 2 2 ,R ,R ∗ )) B a, b 2 2 π K σ i 3 ( O ( X,L B B 3 ⊗ ( ( ( 2 R 3 0 1 X K must have entirely vanishing cohomology on H H H O → O )) = ( ⊕ L ∗ σ ) = ( ) – 69 – ) ∗ 2 2 ( 2 π σ 2 3 ) = ) = ) = π ) is the “dualyzing sheaf” [ ⊗ ⊗ 0 2 2 B ( | 1 3 X ∨ 2 L L K 3 R K K ∗ ∗ X R X O as the zero-section this is simply K π π ⊕ ( ω ( X,L ⊕ ⊕ 0 0 → ] ∗ ) it is clear that we require ∗ ( 2 σ 2 2 π 0 2 ⊗ π B 99 0 and → O ,R ,R B K K ∨ H B.5 O 2 2 1 in a similar manner. It is straightforward to demonstrate R 3 ( L F ⊕ ⊕ B B a X ) = ∗ ( ( ( 2 2 ∨ ∗ π 2 2 1 → O gives the short exact sequence 0 B B a > ) = 0 and π K i H H 0 ) = 0 (by Serre duality). This, coupled with ( 2 R O O ( → O − ) and ( ∨ ∗ B 1 0 0 can be found by using Grothendieck duality: for any sheaf π X,L R . . . ( B.4 )) for ⊗ 1 )) = )) = )) = Sym X,L 3 σ σ ( aσ a < H 1 X aσ ( (2 (3 ( 3 3 3 3 H K X ) = 0, and hence [ X X X = L O O O O ( ( 2 ) and the definition of ( ( ) = ( ∗ µ ∗ ∗ ∗ B π | π π ) = 0? By ( i π 3 0 0 0 A.3 ∨ R X the base of a CY threefold, the above sequence splits and we have determined the X,L R R R ( ω 2 2 B X,L H ( , the push-forward functors obey the following relation: . However, as we will see below, this does not occur for simple threefolds of the type we 1 3 2 for the direct image sheaves B are considering here. H in order to satisfy of interest, Thus, to construct acohomology rigid and SU(2) further bundle ask, we for must what use values the of results above for line bundle where in hand, we canbuild now reducible, in rigid principle SU(2) calculate bundles like all those line above bundle we must cohomology note on that for the line bundles Similar results for X that For direct image sheaf: The calculation outlined above cansheaves be iterated inductively to find the higher direct image and pushing forward to But by ( JHEP08(2014)025 ) 5.1 B.3 (B.24) (B.26) ) where α D α 0. Finally, it b ≥ + a σ ( O ... 2 curves, thus without ) (B.23) , )) α − 2 α D B D ] that are similarly rigid in α α b . Putting this together with , volume 1 and 2, Cambridge ⊂ b 68 ( ) 2 ( 2 2 A B B B , systematic searches have found B.8 ) (B.25) D 2 ⊂ 2 · this enforces that B ⊗ O K ⊗ O D D ∨ ) , it is useful to consider the index of 2 − a i L as in ( ∀ ⊗ 2 K A ⊕ ( ( L χ L · + 1) + 0. A = ...K Superstring theory 6= 0. Although we have not rigorously ruled a ) + ) = 0 1 2 α ⊕ V ) L – 70 – ∗ 2 L a > D ∗ π α ) = ( K 0 π b L 0 ( ⊕ ∗ 2 ,R ). For π ) = 1 + 2 B 2 ,R 0 V B 2 A O B ( ( R ( + 1) for some curve ( , no reducible SU(2) bundles of the form shown in ( 2 O B ( i ), which permits any use, distribution and reproduction in = 0, the line bundles can be verified to have non-vanishing χ c ( χ a χ 1 , volume 1 and 2, Cambridge Univ. Pr., Cambridge U.K. (1998). ( A a H = ( H − ) = L ∗ L = ∗ π 0 π D 0 · R R CC-BY 4.0 ( D and using the fact that for any divisor, String theory χ 2 curve. Although there do exist curves of this type (for example the This article is distributed under the terms of the Creative Commons 2 curves in our set of base manifolds α − − D ), it can be verified on a case-by-case basis that here the necessary condition α 2 b F is a = α in , some models with exotic matter were identified in [ D D using the Riemann-Roch theorem. With S 2 α Univ. Pr., Cambridge U.K. (1989). P b L M.B. Green, J.H. Schwarz and E. Witten, J. Polchinski, For the geometries considered in this work, we have at most To see that it is rare for ∗ = = π [1] [2] 0 2 References the sense that they have no moduliOpen that preserve Access. the gaugeAttribution group License and ( matter content. any medium, provided the original author(s) and source are credited. exist as rigid componentsgeneral in the abstract moduli proof space. ofheterotic/F-theory It this duality, would result. it be would If nice,structions. be however, rigid Some to bundles interesting possibly have can related to a F-theory beB investigate more models found may the exist; within dual for the 6D F-theory compactifications context con- over of D divisor is not in factout sufficient all and possible no examples with entirelyometries vanishing cohomology. outlined Thus, in we appendix expect that for the simple ge- places a positivity condition on can be noted that forcohomology. the Thus, case here we will consider loss of generality we can restrict ourselves to line bundles of the form we have If we demand that themology, index we vanishes require as aother necessary geometric condition constraints for in entirely the vanishing problem, coho- we see that the Bogomolov bound of section Letting R we have that for each term in the sum, the index is additive. 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