Comparing elliptic and toric hypersurface Calabi-Yau threefolds at large Hodge numbers

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Citation Huang, Yu-Chien, and Washington Taylor. “Comparing Elliptic and Toric Hypersurface Calabi-Yau Threefolds at Large Hodge Numbers.” Journal of High Energy Physics, vol. 2019, no. 2, Feb. 2019. © The Authors

As Published https://doi.org/10.1007/JHEP02(2019)087

Publisher Springer Berlin Heidelberg

Version Final published version

Citable link http://hdl.handle.net/1721.1/120533

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Detailed Terms https://creativecommons.org/licenses/by/4.0/ JHEP02(2019)087 Springer August 13, 2018 January 22, 2019 : February 14, 2019 December 29, 2018 : : : Received Revised Accepted Published Published for SISSA by https://doi.org/10.1007/JHEP02(2019)087 240 that are associated with threefolds in the ≥ 1 , 2 h or 1 , 1 [email protected] h , ) with . 1 3 , 2 , h 1805.05907 1 , 1 The Authors. h c Differential and Algebraic Geometry, F-Theory, Superstring Vacua [email protected]

We compare the sets of Calabi-Yau threefolds with large Hodge numbers that , yc Center for Theoretical Physics, Department77 of Massachusetts Physics, Avenue, Massachusetts Cambridge, Institute MA of 02139, Technology, E-mail: U.S.A. Open Access Article funded by SCOAP U(1) factors in the corresponding 6D theory. Keywords: ArXiv ePrint: Kreuzer-Skarke database can bemodels realized for explicitly elliptic by fibrations genericnumber over of or complex somewhat tuned exotic base constructions, Weierstrass/Tate surfaces.models including with elliptic This new fibrations Tate includes over tunings non-toric that a bases, cantunings relatively give of rise gauge small to groups exotic over matter non-toricand in curves, the associated tunings 6D nonabelian with F-theory picture, very gauge large groups, Hodge and number tuned shifts Mordell-Weil sections associated with are constructed using toric hypersurfacefibrations methods using with those Weierstrass can modelcorrespondence be between constructed techniques the as structure motivated of elliptic “tops” byform in F-theory. tunings the toric of polytope There Weierstrass constructionnumber models and is pairs Tate for ( a elliptic close fibrations. We find that all of the Hodge Abstract: Yu-Chien Huang and Washington Taylor Comparing elliptic and toricthreefolds hypersurface at Calabi-Yau large Hodge numbers JHEP02(2019)087 55 25 19 5 53 8 e 36 29 23 18 or 3 7 e 7 and/or having 45 ≤ − 14 m 35 28 35 4 37 ) gauge algebras 16 n ( 12 58 21 so 17 – i – 8 51 48 31 18 11 42 59 -fibered polytopes and corresponding Tate models 1 , 38 3 , 2 P 1 54 curves of non-negative self intersection models 4.3.3 Reflexive polytopes from Tate tunings 4.3.1 NHCs4.3.2 with immediately reflexive polytopes Other NHCs: reflexive polytopes from the dual of the dual 5.2 Main structure5.3 of the algorithm: Special cases: bases bases with lacking a curves of non-Higgsable self-intersection 5.4 Tate-tuned models 4.8 Bases with large Hodge numbers 5.1 Algorithm: global symmetries and Zariski decomposition for Weierstrass 4.4 Multiple tops 4.5 Combining tunings 4.6 Tate tunings4.7 and polytope models Multiplicity of in the KS database 4.1 Reflexive polytopes4.2 from elliptic fibrations Tate tuning without4.3 and singular polytope fibers tops Reflexive polytopes for NHCs and Tate tunings 3.2 Batyrev’s construction3.3 of Calabi-Yau manifolds Fibered from3.4 polytopes reflexive in polytopes the Standard KS3.5 database A method for analyzing fibered polytopes: fiber types and 2D toric bases 2.7 Global symmetry2.8 constraints in F-theory Base surfaces for 6D F-theory models 3.1 Brief review of toric varieties 2.1 Hodge2.2 numbers and the 6D Generic massless and2.3 spectrum tuned Weierstrass models Tate for form2.4 elliptic and Calabi-Yau the manifolds Tate The algorithm Zariski2.5 decomposition Zariski decomposition2.6 of a Tate Matter tuning content from anomaly constraints in F-theory 5 Systematic construction of Tate-tuned models in the KS database 4 Tate tunings and the Kreuzer-Skarke database 3 Elliptic fibrations in the toric reflexive polytope construction Contents 1 Introduction 2 F-theory physics and elliptic Calabi-Yau threefold geometry JHEP02(2019)087 ]. 74 3 83 ] and then 11 78 – 85 8 69 67 ] over each such base, it is 1 13 , , 1 h 12 known mechanism are actually 62 11 F any -fibered polytopes 1 , 3 ] is that a very large fraction of the set 62 , 2 , and 81 19 P – 10 F 14 , 65 9 – 1 – ] has represented the bulk of the known set of F 4 80 61 ], in 2000 Kreuzer and Skarke carried out a complete 2 72 79 Tate tunings and exotic matter n s 2 ], the set of these geometries is still relatively poorly understood. I 1 -fibered polytope tuning 1 , 3 , 79 2 P ] has motivated an alternative method for the systematic construction of 7 – 5 6.2.1 A6.2.2 warmup example The6.2.3 eight remaining missing cases Example: at a large model with a tuned genus one curve in the base 6.1.1 Type 6.1.2 Large6.1.3 Hodge number shifts Tate-tuned models corresponding to non-toric bases 7.1 Summary of7.2 results Possible extensions of this work 6.2 Weierstrass models from non-standard and other exotic constructions 6.1 Fibered polytope models with Tate forms approach, at large Hodge numbersset this of method possibilities. gives One afrom perhaps reasonably this surprising complete work result picture and that of fromof the has other Calabi-Yau recently perspectives threefolds become [ apparent thatelliptically both can fibered, be particularly constructed at by large Hodge numbers. F-theory [ Calabi-Yau threefolds that havesystematically the classifying structure all of bases thatsystematically an considering support elliptic all an possible fibration elliptically Weierstrass (with tunings fiberedpossible [ CY section). in [ principle to By constructare all some elliptically technical fibered issues Calabi-Yau that threefolds. must While still there be resolved for a complete classification from this Following the approach of Batyrevanalysis [ of all reflexivethose polytopes Calabi-Yau (CY) in threefolds four that can dimensions,For be giving many realized as a years hypersurfaces systematic the inCalabi-Yau toric classification resulting varieties threefolds, of [ database particularly [ at large Hodge numbers. More recently, the study of 1 Introduction While Calabi-Yau threefolds have played andays important of role the in string subject theory since [ the early B Elliptic fibrations over the bases C An example with a nonabelian tuning that forces a U(1) factor A Standard 7 Conclusions 6 Polytope analysis for cases missing from the simple tuning construction, JHEP02(2019)087 ], 13 , all the with a 12 1 , 7 1 h we describe 3.4 ] we will describe 20 and at large , where we describe an 1 , 6 2 h and section 5 at least 240 are reproduced by an elliptic , we restrict attention to toric base surfaces 1 , 2 4 h – 2 – , we review some of the basics of F-theory and or 2 1 , 1 h ) fiber type. At large Hodge numbers, for reasons discussed 1 , , is that these simply constructed sets match almost perfectly 3 , 2 5 P , we expect that this should give most or all elliptic fibrations that , we review the Batyrev construction and reflexive polytopes, and the 4.8 3 ], and even over toric bases there are non-toric Weierstrass tunings [ , we find that all these exceptions also correspond to elliptic fibrations though 11 6 , 10 Note that in this paper the focus is on understanding in some detail the connection The main results of the paper are in section The goal of this paper is to carry out a direct comparison of the set of elliptically and identify the set of tuned Weierstrass/Tate models over such bases that naturally Hodge number pairs with either 2 between elliptic fibration geometry and polytopeconstruction geometry of for elliptic these different Calabi-Yau approaches threefolds.a to more In direct a analysis companion ofthere paper is the [ a polytopes toric in fiberat the associated large KS with Hodge database an numbers. that elliptic The also fibration principal for shows class every explicitly of polytope that Tate in tunings the that database we consider in this paper the base that takethe the elliptic base fiber. outside theall The toric upshot class, is and/or that forceCalabi-Yau when Mordell-Weil over sections some these explicitly on more determined exoticsummary base of constructions surface. the are We results included, conclude and in some section related open questions. Calabi-Yau threefolds produced usingnumbers this (18 approach. out There of isare 1,827) a not associated reproduced small with by set toricin our of hypersurface section initial large Calabi-Yaus, scan. Hodge however, By that with examining more these exotic individual structure, cases, such as as described non-flat fibrations resolved through extra blow-ups in compare the results of runningresult, this described algorithm in to section thein Kreuzer-Skarke the database. large Hodge The number initial regimesmodels that constructed we study: by an both appropriate atdatabase, set large of and Tate virtually tunings over all toric the bases Hodge appear in numbers the in KS the database are reproduced by elliptic database with a specific ( further in section arise in the KS database; we find that thisalgorithm is to in systematically fact the run case. through all tuned Tate models over toric bases and we and Tate form Weierstrass models.B In section correspond to a reflexive polytopeway of in the constructing Batyrev from construction. theset point This of of gives elliptic us view a of Calabi-Yau systematic elliptic threefolds fibrations that over are a chosen expected base to a be large seen in the Kreuzer-Skarke through a reflexive polytope.the systematic In construction section ofbase elliptic and Calabi-Yau threefolds the through tuningbase. the of geometry In Weierstrass of or section the structure Tate models of from elliptic the fibrationsthe generic precise in correspondence structure this between over a context. each particular fibration structure In for particular, a reflexive in polytope section fibered Calabi-Yau threefolds thatbased methods can with be thosegeneral constructed that methods arise using through for Weierstrass/Tate reflexive F-theory constructionbases polytope [ of constructions. elliptic While Calabi-Yauwe the focus threefolds here can on include the subset non-toric of constructions that have the potential for a toric description JHEP02(2019)087 . ]. 2 ]). B 34 25 , , ], with 31 24 23 ]. In a few ]. 37 – 27 , 35 , 26 , 33 7 , – 32 5 ]. The construction of many 21 There can also be abelian gauge 1 . 2.3 and in the decompactification limit of M-theory, 2.2 ˜ – 3 – X ]). Therefore the physics data can be extracted by studying 30 – 28 ). At the intersections of seven-branes there are localized matter fields and then specifying an elliptic fibration in Weierstrass form over that 2 2 that we review in sections B X ], and a systematic approach to constructing toric hypersurface Calabi-Yau threefold F-theory compactified on a (possibly singular) elliptically fibered Calabi-Yau threefold (complex dimensions). F-theory models can then be systematically studied by first The short form Weierstrass model is the most general form for an elliptic Calabi-Yau threefold. The 14 1 gives a 6D effective supergravity theory. Such a compactification of F-theory is equivalent 2 cases discussed in thistherefore paper in are principle admit elliptically aCalabi-Yau fibered short threefolds Calabi-Yau form threefolds (lacking Weierstrass that form aCalabi-Yau threefolds always realization. through global have There the a Jacobian section), can construction section alsoThe (described which and be from physics can the genus physics of one be perspective these in fibered cases related [ threefolds we to is find Weierstrass more ittransformation models subtle, useful to of involving the to discrete short elliptic use Weierstrass gauge the form. groups Jacobian [ construction even for cases with a section, giving an explicit determined from the detailedin the form base of (see thethe e.g. singular singularities [ fibers over the by(long means form) codimension-two of of points the Weierstrass models (short form) or the Tate models the base where theeffective theory fibration arise degenerates. from theof The seven-branes the non-abelian and elliptic can gauge fibers be symmetriesclassification along inferred (table of the from the codimension-one the singularity loci 6D that types in are the hypermultiplets base, in according the to 6D the theory; Kodaira the representations of the matter fields can be to M-theory on thewhere resolved in Calabi-Yau theto F-theory zero picture size. theIIB resolved F-theory string components can theory. of alsoIn the be In this elliptic thought this F-theory of fiber description, picture as spacetime are the a filling shrunk type 7-branes nonperturbative IIB sit formulation theory at of the is type codimension-one compactified loci on in the base B choosing a base base. Further background on F-theory can be found inX for example [ 2 F-theory physics andWe elliptic briefly Calabi-Yau summarize threefold here how geometry thefrom massless F-theory spectrum compactification of is awhich six-dimensional related effective is to theory an the elliptically geometric data fibered of Calabi-Yau the threefold internal (CY3) manifold, over a two-dimensional base with certain classes ofTate Tate tunings tunings. from This observed leads polytopein in structures, the some KS and database cases the have to observation ain that the more terms some of identification complex standard structure polytopes of tops. that new doesthe On the not construction other admit of hand, a polytopes new direct structures via description of the tops are correspondence also with found Tate through tunings. polytopes in the KS database throughin combining [ K3 tops and “bottoms”with was accomplished a given baseparticular and application gauge to group models usingOne with of the gauge the language group main of SU(5) results tops as of is this also developed paper studied in is in [ the e.g. systematic [ relationship of such constructions have a complementary description in the language of “tops” [ JHEP02(2019)087 (2.1) (2.2) (2.3) (2.4) is the ]. The 7 V , 6 we review . The gravi- 2.7 , where 2.6 T , we review the 6D 29 can be related to the 2.6 − ˜ X , i , of 1 , = 273 G 2 i h V , with a detailed example worked we describe how this method can − , to the associated 6D supergravity 6 1 and , 2.5 ˜ , H X 1 − , 1 1 Y + 2 h − T ) 2 neutral + ) gives B ( r non-abelian factors H 1 , – 4 – 1 × 2.33 h ) = ) = ˜ ˜ = X X ( ( 1 abelian 1 T , , r 1 2 , we review the Zariski decomposition, which allows us h h 2.4 of the 6D F-theory. = U(1) G G is the total rank of the gauge group, i r i . In section P C + is the number of hypermultiplets that are neutral under the Cartan subal- with a summary of the systematic classification of complex surfaces that can ]. In particular, the Hodge numbers, (1) symmetries is more subtle in that it relates to the global structure of the abelian 38 2.8 u is the number of tensor multiplets, which is determined already by the choice of , r neutral , 7 of the gauge group 2 T H = 2 B r The spectra of 6D theories are constrained by consistency conditions associated with In other words, this counts fields that are neutral matter fields in the 5D M-theory sense but may 2 the absence of anomalies,tational which anomaly we cancellation describe condition ( in further detail intransform section under the unhiggsednon-abelian non-abelian factors factors is of stillnon-abelian the charged groups 6D there under F-theory. can the be Often, Cartan charged matter subalgebra, matter charged but that under is for the neutral certain under representations the of Cartan some subalgebra. where gebra of the 6D effective theory. We also have where base and respondence can be usedtheory to [ relate the(massless) geometry matter of content of the 6D theory: in section support elliptic Calabi-Yau threefolds and can be used2.1 for F-theory compactification. Hodge numbers andBy the going 6D to massless the spectrum 5D Coulomb branch after reduction on a circle, the F-/M-theory cor- form description of an elliptic fibration,be and applied in systematically section in theanomaly cancellation context conditions of and Tate their tunings.and connection the In to Hodge section the numbers of matter the contentthe corresponding of constraints Calabi-Yau a threefold. imposed 6D by In theory global section supported symmetry on groups on curves the intersecting set a of given gauge curve, groups and that we can conclude be the overview of F-theory study of fibration, as opposed tolocally. non-abelian We will symmetries see where casesout with we in abelian can factors appendix just in studyto section determine singular the fibers order of vanishing and consequent gauge group of a Weierstrass or Tate symmetries, which arise from additional rational sections of the elliptic fibration [ JHEP02(2019)087 ) 2.6 does (2.8) (2.6) (2.7) (2.5) . The 1 , 2 2 B h ]) base, and 39 , 9 ). In particular, as we is the total number of hy- 2.4 . moduli are used to implement 1 , 2 . neutral charged h . 4 H H f 6 , + 1) + .) − gz r charged (7) (9) ( 2 T 2 + H so g so so can be described by the Weierstrass model 4 29 , , , , ∆ ) charged B − r − (4) (3) ( (8) H fxz V r, V – 5 – so so su su Algebras + = 3 x H r = ∆ = ∆ 5 1 1 = , , 4 3 2 ≥ 2 1 2 h h r y ) = 272 + , and Rank ∆ ∆ tunings, however, the Hodge numbers do not change (see ˜ X G ( 1 , 2 h is fixed for a given base. Thus, if we start with known Hodge numbers T of course does not change in a rank-preserving enhancement; rank-preserving 1 , for the generic elliptic fibration over a given (e.g. toric [ 1 h 1 , 2 . Rank preserving tunings: tunings of these four classes of gauge algebras do not change h 1 , ) are always non-negative and non-positive respectively for any tuning. In most ). So we have another expression 2 )— h G 2.7 1 and or 1 1 , , 1 1 charged under the Cartan subalgebra (cf. footnote 2.2 Generic andAn tuned elliptic Weierstrass fibration models with a for section elliptic over a Calabi-Yau base manifolds and ( cases, the gauge group increasesthe in rank tuning. and some In oftable the not change either in these tunings, as one can check by considering carefully the matter the second of theseof relations matter simply degrees expresses ofthe the freedom number physical that of expectation are gauge thathand bosons lost the sides lost (“eaten”) number to are in symmetry a associatedin breaking. Higgsing some in special transition Note general cases is that with involve the equal tuned abelian data to non-abelian factors. on gauge the Note right symmetries also that but the also right-hand sides of ( generic models respectively the shifts Such a specialization/tuning amounts physically to undoing a Higgsing transition, and specializations (tunings) of anumber of generic tensors elliptically fiberedh CY3 over a givenspecialize/tune base to a modelthe with Hodge a numbers larger of the gauge tuned group model and can increased be matter simply content, calculated then by adding to those of the group This is more usefuldiscuss for in some further of detail our in purposes the than following equation section, ( we are interested in studying various Table 1. h dimension of the gaugepermultiplets group (separated into neutral and charged matter under the Cartan of the gauge JHEP02(2019)087 ), ∈ . B , so 2 , y (2.9) ] are K 2 . The 4.4 (2.11) (2.12) (2.13) (2.10) 4 B } L ) in the − 2 ], which 44 ∗ together ζ , ( 8 π ( − k O , 29 ⊗ 3 respectively. σ, ζ ∆ , − 6 ) , (2) L ζ 2 in section ∗ ( P j π (NHCs) [ {− 11 , g ) ∈ O ζ and ( . We summarize these x i } 4 f , and are sections of L } 3, and clusters of multiple ∗ 3 = 0 π , and ∆ in an expansion in − , g f, g σ , B, ≤ − { , , , 2 f ··· − → corresponds physically to a maxi- m transversely so that , ] ··· ··· 2 2 + } ≤ B B + ) by the type of the singular fiber. + σ ) {− 2 12 σ σ ζ = 0 , ) ) g ⊕ O ( − } ζ ζ 1 σ 3 ( ( , 3 non-Higgsable clusters { 1 1 requires that − m f g + 27 is characterized by the Kodaira singularity , ⊕ L 3 X 2 } ) + ∆ 2 f ) + ) + – 6 – ζ L ζ ζ {− ( [ ( ( 0 1 monodromies , 0 0 ] can be viewed as weighted projective coordinates 3 2: f g , z 2 ∆ = 0 ∆ = 4 P : { P ) = ) = ) = ∆ y ≤ − : are carried by = , where some factorizability conditions are imposed on the 2 x σ, ζ σ, ζ σ, ζ ∗ P ( ( B : g f ∆( π are sections of, to be more precise, g is the canonical class of the base. More abstractly, we take the ,IV,IV (1) and [ ∗ n P and B , in terms of the first non-vanishing sections ,I ) is required by the Calabi-Yau condition and ∗ f 0 K 3 B ∈ O ,I K n defines a divisor that intersects I − , z ). The physics interpretation is that there are seven-branes on which open ∆ of lowest degrees of vanishing order along ( } 3 ∆ automatically vanish to higher orders along these curves in any Weierstrass 2 O , while L 1 ∗ , = 0 of the discriminant locus 3 ), where f, g, π = , f, g, ζ 2 B } ⊗ L { P K 6 A generic Weierstrass model (i.e. with coefficients at a generic point in the moduli Consider an elliptic Calabi-Yau threefold over a complex two-dimensional base = 0 (3) − (see table ( P σ sections model over the giventhe base. curvature over This themaintain can negative the self-intersection be Calabi-Yau structure curves understood of must thecorresponding geometrically be elliptic minimal as fibration. cancelled gauge an The by groups orders effect 7-branes on of these in vanishing to and NHCs which the are listed in table space) for an elliptically fibered CY3mally over Higgsed a phase. given base In the maximallymatter Higgsed content phase are over many still bases the nontrivial.associated gauge with group The and a minimal given gaugeare base algebras isolated and rational matter curves configuration ofrational self-intersection curves of self-intersection where serve as local coordinates on an open patch of base. conditions in table local expansions σ strings (and junctions)symmetries end can located be atThe determined the gauge (up discriminant algebras to locus,those that of and are types the further resultingterms determined of gauge by monodromy conditions [ vanishes. The type{ of singular fibertype, at which a is generic determined point by along the an orders irreducible of vanishing component of O of the the divisors in the base areone curves. loci The elliptic in fiber the becomes base singular over where the the codimension- discriminant O weighted projective bundle where The Calabi-Yau condition on the total space JHEP02(2019)087 ). ) and 2.13 2.8 )–( 5) 2 ) − g c n 2.11 2 n n (2) (2 n/ 3 0 3 1 (8) condition) b 4 u u f ( su so so (7) or 2 2 27 27 7 8 ) sp (2) or e e so b none none su 6 + ) = ) = e 4) or ) (3) or ζ ζ ) or a ( ( b − n 0 3 su ( + (8) or g g + ) ) n ) ) (but not x ζ ζ su so ( ( ζ ζ (2 ax )( ( ( 3 3 b and and g g so ), which is a perfect square ); , b , b + 2 0 2 1 ) ) ζ ζ in the Weierstrass model ( − + + 2 ( ( ζ ζ u u nonabelian symmetry algebra 0 1 ( ( x x x x 1 3 1 3 ) ) u a a u = 0, locally = 0, locally )( )( f, g − − ζ ζ 0 6 a a ( ( 2 2 is a perfect square foris odd a perfect square for even does not occur in F-theory − − f f 2 1 ) ) ) = ) = ) is a perfect square ) is a perfect square ζ ζ 4 1 2 6 7 8 − x x − ζ ζ ζ ζ + + ( ( 3 1 2 1 n ( ( n ( ( n n u u E E E A A D 3 3 0 2 2 4 none none A ∆ ∆ D monodromy condition f for some since ∆ otherwise g otherwise x = ( for some x = ( for some otherwise since ∆ f for some otherwise g otherwise singularity ) – 7 – 4) 5) c ) 2 − − n 2 7 2 6 4 (3) (2) (8) (7) ( n/ f e n n g 12 b 0 2 3 4 6 8 9 ≥ ≥ so so su su ( su 10 (2 (2 algebra ≥ n n sp ) are coefficients of the expansions in equations ( so so ord (∆) ζ ( k ) 7 ∆ g , 0 2 3 5 6 ) 4 6 8 n ≥ 0 2 3 4 5 1 ζ ( ≥ ≥ ≥ ≥ ≥ n j ord(∆) ord ( , g ) ) ζ ( ) g 3 i f f 0 2 3 4 0 2 2 3 4 1 4 ≥ 0 1 2 3 ord( ≥ ≥ ≥ ≥ ≥ ≥ ≥ ∆: ord ( ) f f, g, 2 2 3 0 2 Kodaira classification of singularities in the elliptic fiber along codimension one loci in Monodromy conditions for certain algebras to satisfy in additional to the desired orders of ≥ ≥ ≥ ∗ ∗ ∗ ord( ∗ ∗ 0 0 n n I I I I II IV II III IV III Type ∗ ∗ ∗ non-min 0 n n I I I IV IV Table 3. vanishing of Table 2. the base in termsthe of discriminant orders locus of ∆. vanishing of the parameters JHEP02(2019)087 (2.14) (2.15) (2.16) (2.17) (2.18) (2.19) (2.20) (2.21) , increasing ] that appeared B 43 ). The general , vanish to orders B 42 , f, g nK 6 − z ( 6 a O 6) loci by blowing up such ) by completing the square , + . , 4 has a simple description (see 2.8 ) 6 6 b xz , b 4 4 2 4 b , we can systematically tune the f, g a 2 a b B 216 + − 2 − 2 3 + 9 z a 4 2 2 6 2 b b x 2 a 2 b , ) 27 a + 4 4 b − + a + 36 3 4 , 3 3 24 b – 8 – 4 3 2 x a 8 b , a , 1 − 2 6 a = − 2 2 − a a ( b to increase the order of vanishing over various curves 3 + 2 8 − ( b g 3 6 1 yz 2 2 + 4 + 4 1 are sections of line bundles a 48 864 a 3 b 2 1 1 2 3 2 a n − − − a a a b and a + ======]. The associated “Tate forms” for the different singularities f 2 4 6 8 g f ∆ = b b b b 44 xyz , 1 a 29 + 2 y , which gives the relations x and a resolution of the singularity can generally be found by first blowing ]; the explicit connection between resolutions giving non-flat fibrations and flat fibrations 3 41 , 6) point in the base, which modifies the geometry of the base ) can be related to the standard Weierstrass form ( 40 , ]) and we have very strong control over the parameters of the Weierstrass model. 13 ) by one. While in many cases the extent to which enhanced gauge groups can be 2.14 , ]. While over some bases there is a great deal of freedom to tune many different gauge 9 and shifting B Starting from the generic model over a given base ( Resolution of non-flat fibers in related cases of tuned Weierstrass models has recently been considered for 13 6). In this paper we also avoid cases with codimension two (4 1 y 3 , , 1 after the initial appearance of this preprint. example in [ over a resolved base through sequences of flops are described explicitly in the papers [ where for anform elliptic ( fibration in the context of F-theory [ match up neatly with thean toric equation construction for that an we elliptic focus curve on in in the this general paper. form We start with e.g. [ 2.3 Tate formThe and Tate the algorithm is Tate a algorithm systematicof procedure an for elliptic determining the fibration, Kodaira and singularity type provides a convenient way to study Kodaira singularities in h tuned in the Weierstrasssuch model as over the any low-energy anomaly given consistencymost base conditions, clearly the can described precise be set in determined of termsthe possible of by case tunings an considerations is of explicit toric description bases, of the the Weierstrass complete coefficients. set In of monomials in by the constraint that(4 there be no codimensionpoints on one the loci base as oversuperconformal part which field of theories; the in resolution the process. geometricnon-flat picture Such fibers such singularities singularities can are be associatedup related with to the 6D (4 Weierstrass model coefficients beyond what is imposed bysome curves the in NHCs, the producing base. additionalin Many or aspects [ of enhanced such gauge tunings groups aregroup on described factors in on a various systematic curves fashion in the Weierstrass model, there are also limitations imposed JHEP02(2019)087 ) k n n n + 1 + 1 + 3 + 3 + 4 + 4 0 1 2 3 3 2 3 4 4 6 6 6 8 8 9 ∆ 10 12 2 2 2 n n n n n n 2 2 2 2 2 2 where ∆ is ) and SO(2 n ∗ ? m I ? ) or n + 1) ? ? ? ∆) can be arranged n m I 6 3) n n n n + 1 + 1 + 1 0 1 2 3 3 1 2 2 3 4 4 5 5 6 , a 2 2 2 2 n n n f, g, (4 2 2 2 3 (2) 5 (4) + 1 (2 n + 2 (2 2 n ) are kept fixed, the vanishing 2 g 4 0 1 1 2 2 1 1 2 2 2 2 2 3 3 3 4 4 n n + 1 + 1 + 1 + 1 + 1 + 1 + 1 a n n n n n n n require additional monodromy conditions. ? , or ∆ are also satisfied automatically by ) and ord( 1 g f 3 , 0 1 1 2 1 1 1 1 1 2 2 2 2 2 3 3 3 n n n n n − + 1 + 1 + 1 a f n n n n – 9 – 2 ) is that by requiring specific vanishing orders of 0 0 0 0 1 0 1 2 0 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 a ) of SU(6) gives alternate exotic matter content; see text ◦ 2.14 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 a ◦ ) , specific desired vanishing orders of ( ) ) ) n + 1) + 2) + 3) + 4) + 1) n n n 4 2 6 7 4 8 n n n n n — — — — F E E G E Sp(1) Sp(1) Sp( Sp( group SU(2) SU(3) SU(2) SU(3) SO(7) SO(8) SU(2 SU(2 SU(2 SO(4 SO(4 SO(4 SO(4 3 3 2 2 s ∗ ns s s ∗ ns ∗ n ss n +1 +1 − − − − ns s 0 1 2 3 ∗ ns s ns s s ns ns ∗ 2 3 2 0 n n n n n n ∗ I I I ∗ 0 I 0 II Tate forms: extends earlier versions of table by including alternative SU(2 ns s ∗ ∗ ∗ ∗ I I I I 2 2 2 2 2 2 II III IV I I type III IV IV I I I I I I IV (2nd version) imposed by some gauge algebras on non-min ’s according to table n n 3 s 2 a I the to implement any of thetable possible gauge algebras.these Moreover, the “Tate monodromy form” conditions models. in required to For vanish example, to for a tunings certain order of while fiber ord( types tunings that canstraints. be In realized particular, purely alternatefor tuning by further ( orders details. of Groups and vanishing tunings without marked additional with monodromy con- An advantage of the general form ( Table 4. JHEP02(2019)087 ] , 2 by a ∆) +1 45 n , 2 a, b ]. The 44 f, g, , m 44 +1 with- ), with +1 ). On any n n n n contain only ’s; this arises i (2 to arrange for 4 a is of order 4 is g su , a m, m 6 2 a a +2) that do not re- are also connected n and of order 2 4 6 (4 f a so , so without loss of generality we 3 ) give a vanishing of ( is a constant — i.e. on curves g 1 = 0 there is a codimension two + a x 1 2.21 ] and later modified in [ 2 a 2. f / 29 )–( 2 + / ; when applied to the polytope toric 1 , so that condition implies the perfect square 3 ) g 2 x 4 ) 2.15 a a 4 +4), like those of This can be satisfied if + and n − b 4 2 2 ]. (2 (4 f a , note that the leading term in the discriminant when 2 ) so 44 b 3 – 10 – ’s to be specified, in contrast to the Weierstrass , in cases where so that ( n + of the cubic 6 + ( n a k / a b 2 (2 a − 2 − ) a b (8), the monodromy condition when − = − b so a ( a, b, + 1, then the necessary monodromy condition is that [ − 3. This can immediately be seen from the fact that at the = 3, in particular, this Tate form gives a tuning with exotic n gives + 2, so can be guaranteed simply by setting certain monomials . For higher n b n n > is a perfect square. This condition is clearly automatically satisfied is a perfect square [ ), which corresponds geometrically to removing certain points from has incorporated some results of the present study into the Tate 6.1.1 vanishes to order 2 — this simply gives an alternate Tate tuning of SU(2 =0 4 k =0 z − z | a | 2.14 are each constrained to be powers of a single monomial 2 6 +2 is of order 2 , but this kind of Tate model for SU(8) and higher would be relevant in a 12). Resolving this singularity generally involves blowing up a point on n /z 1 6 , 2 , a ) , 4 a 6 1 4 3 is one of the roots , / /z a h 2 ) , a ’s prescribed by the Tate algorithm immediately give the desired ord(∆). This 3, and solving for 4 a 6 2 / n ) where it is necessary to carefully tune the coefficients of 2 a a a = 0, each − a 2 2 2 1 a 2.8 = + 4), if a a 4 is actually of order 2 a We have also identified Tate tunings of Note that table n To relate this to the condition stated in table − 6 4 (4 2 4 a a have that ( that condition is satisfied becomes condition. Going thenoting other that way, when the perfect square condition is satisfied we can determine in the Tate coefficients tothe vanish (in toric a context local used coordinate interms the system, in later which can sections become of global thethen in paper). the monodromy On condition the will otherout be hand, automatically specifying if satisfied any the with leading particularof coefficients situation for in these this monomials. paper. We For encounter both kinds naturally in the contexthow of these the two types geometricso of constructions Tate of tunings polytopes. are( relevant We in discuss the briefly constructionsif of this paper. For the base, so thatwith the larger resulting elliptic fibrationcomplete is analysis naturally of thought all of reflexive as polytopes. living on aquire base an extra monodromy condition and only require the vanishing order of of self-intersection other kind of curve,(4, at the 6) codimension singularity two when locilocus where to orders (4 table originally described inmost the significant F-theory new context feature invanishing to [ is order an 2. alternate3-index For Tate antisymmetric form SU(6) matter. foris the An described example algebras in section of a polytope that realizes this tuning sections, tuning a Tate formthe can general be form described ( by simplya removing lattice certain monomials in from in the contrast toric to construction. tuningsconstruction, of we We refer the refer to coefficients to Tate of tunings tunings as of “polytope this tunings”. type as “Tate tunings” makes the Tate form muchrequiring more the convenient order for of constructingform vanishing these ( of singular the fibers bya only vanishing of ∆ tovery higher directly order. to The the Tate forms geometry described of in reflexive table polytopes. As we discuss in the subsequent order of JHEP02(2019)087 0 is of ; in ≥ 6 4.6 A } a i (2.23) (2.24) (2.25) (2.22) C · ], which Y 29 are each a 6 are rational, 2 , a ) has signature 3 + 2), with of negative self- B 1, 1, 3, 3, 6 Z a { i n C B, (4 ( is of order 3, though 2 so 6 , the conditions can be and ∆, it is convenient H j a described in [ g C are not single monomials } , f 6 is a perfect square, satisfied , a =0 4 z | , , a n , 2 2 Q such that the conditions a Y 2 i /z ∈ c ) + − 1, 1, 3, 3, 5 i 6 i i. { g a 0 are rigid. A simple but useful algebraic ∀ C i . Here, however, we are simply interested c , Y, q + 4 Q of a section of the line bundle corresponding 2 3 ) = 2 i i and ∆ along some irreducible divisors in the + C < ∈ a · X B = 0 i C i g . Mathematically, the Zariski decomposition is i q C C + 1 or if the leading terms in K – 11 – e , = i i C q f n q · + . We explore further, for example, in section B . The intersection form on d compatible with the decomposition over the ring of 2 i 2 2. All toric curves on a toric base i Y C X ( = B nK C · − i = c − . The situation is similar when m, m C i C Y A a · at leading order, but cannot be imposed by simply setting B 2 K . We are primarily interested in rational (genus 0) curves, for g − m, m = C · is actually of order 2 6 must be at least of genus C a A C . This is the only gauge algebra with no monodromy-independent Tate tuning is effective and satisfies 3 , the monodromy condition is that ( n Y , m (12) the subtleties in using the Tate tuning 2 ). Curves of negative self-intersection To study the orders of vanishing of so T integers. For example, consider the decomposition The goal is to findare the satisfied. minimal Taking set the of intersection product integer values on both sides with Then the order of vanishing alongto the the curve divisor normally considered over the rationals,in so the smallest integer coefficient of intersection and a residual part where following section. base, aside from looking explicitlyto at consider the the sets more ofdecomposed abstract monomials into “Zariski of (minimal) decomposition”, multiples in of which an irreducible effective effective divisor divisors for any curve which therefore and the intersection product of toric curves has a simple structure that we review in the 2.4 The ZariskiA decomposition central feature of thesection geometry form of on an curves F-theory(1, (divisors) base in surface is thegeometry structure identity, of which the follows inter- from the Riemann-Roch theorem, is that in particular if single monomial proportional to for requires an additional monodromyfact, condition, analogous vs. situations our occur alternative in tuning tuning all gauge algebras with monodromies. the orders of vanishingthe of monodromy each condition ism, m more complicated when except through this kindorder of 2 single monomial condition. Finally, for a single monomial each JHEP02(2019)087 , 4, Y the − . If, ) for = D (2.26) (2.27) = j,n B c on each 2.26 C or K n · ) and self- a in the base C 12 j D j 4 − C 6= − that satisfy the . This gives rise i compatible with j,n . We can then use nK c j,n c − , since ] to analyze the non- 0 8 NHC j,n that intersect at a point Y c 6. We denote by . , + ≥ j 4 C , must therefore also carry at C C,D 3 · so that ∆ vanishes to order 4 , C 2 2) B = 2 ) to be larger than the minimal tuned j,n , D c to increase. As a simple example, − D j, j nK ). 4 ≥ c 2.25 ∀ = 1 was used in [ − that satisfy the inequalities ( , j,n B , n ≡ − B in ( 2.26 0 c ) , so that i K K c ≥ 6 C j,n NHC j,n i c v 12 , and we have a set of curves nK − tuned j,n c c − − B = ji ( = M and – 12 – j,n , and c j ∈ O i (0) j,n B c X n m K ), so the discriminant need not vanish on a 4 − (4) gauge algebra on ≡ − j su j,n 2.22 C v associated with each non-Higgsable cluster, which we list · j . In doing a Tate tuning, we impose the additional condition j,n . The minimal necessary order of vanishing of each } C j c , the vanishing order is at least some specified value that is 6 j C , = C 4 , jj on 3 = 0 from ( , n M are pairwise intersection numbers (non-negative for 2 a D , i · 2 curves (i.e. curves of self-intersection 1 C , we begin with an initial assignment of vanishing orders B − · ), we proceed iteratively, following an algorithm described in appendix A n j ) over a complex surface base ∈ { K C n = 2.26 2.14 ≡ (2) gauge algebra. C · . These are the minimal values can be determined by applying the Zariski decomposition for ji su 5 j B M = 1). The Zariski decomposition of the discriminant locus gives simply C then we have the Zariski decomposition K ]. For each More concretely, to determine the unique minimum set of values The Zariski decomposition of D 8 D · C of [ when we are imposingNHC’s a without given tuning. tuning When we we simply are use computing the the minimal minimal values order from of vanishing from the Zariski higher than the minimumthe imposed Zariski by decomposition the NHCs, tothis determine lower bound the that minimum also values satisfy of the the inequalities ( inequalities ( curve to a set ofin vanishing table orders each value of that over certain curves context of a TateTate tuning. form ( In particular,that assume includes that all we curves havefibration of an is negative given elliptic by self-intersection. fibration theorder in five The of sections vanishing the parameter of space of the elliptic implying that ∆ must alsoleast vanish an to order 2 on 2.5 Zariski decompositionA of particular a application Tate of tuning the Zariski decomposition that we use here extensively is in the consider a pair of ( since however, we tune for exampleon an Higgsable clusters that canapproach to arise analyze models in where we 6Dthe tune minimal theories. a content given gauge specified factor More bywe on would the generally, a choose specific non-Higgsable we by divisor cluster can beyond hand structure.possible to use values; In take this the some may such in values a same turn of situation, force other coefficients where intersection numbers rewritten as the set of inequalities JHEP02(2019)087 = = + } ji (0) ,n c (2.28) (2.29) (2.30) 2 M c , = { }} (1) , so 0 -th step all c } , f and 2 6 − } . , 3 {− 3 , , 2 } '! , 0 ) {− 2 , ) , 4 k 1 ( i , c , {− . 1 jj ∆ , { , which in the toric context } M n 0 =1 '! = a . , l k , = ) on the r.h.s. with 3 3 3 } . j P (0) i ,n − are obtained iteratively this way (0) }} m {− , 1 c c + , 0 ( c } does not necessarily increase the ≤ − > j , j } { c 2 j j ji 0 to be the NHC + 0 5 : m (0) i j { , { } c , } M 2 ) ( m j } k i ··· ( j 0 ji C ) are , m c {− , + P { , m 2 ∆ M } ) { i { − j 0 (2) j , 2.30 j , c m } j P v 1 0 m – 13 – , & − (2+ 1 , + ∆ n j , m { 0 {{− v l 0 , ) to determine the minimal correction that is needed

} (1) j & =    0 c , , } 0 = 1 2.26 6 .

{ dropped for clarity of the notation), , + ∆ . The final solutions , 4 5 (0) j,n j } n , = Max c 0 3 j (0) , , c 1 (1) j 2 c , = Max = {{ ∆ j c = 1 = +1) l } ( j n = 0 for all c n }| ) , and 2 }| f ( j ,n }} , v c (0) 2 2 1 v , c − , as shown in table , ,n {{ } 1 (0) 1 2 { c ). Note that in some cases, the algorithm leads to a runaway behavior when there , where ∆ , 1 } {{ , 1 1 . We continue to repeat this procedure until the corrections in the , 2.26 , Note that a tuning beyond that shown in table As an example, consider the set of curves 3 1 (1) j , c 1 toric bases we are studyingexplicitly here, working an with the essentially various equivalent monomials analysisare in could simply the be sections points carried in outprocedure a by because dual it lattice, is as morehowever, efficient match we and an discuss more explicit in general; toric the the computation next results in of section. each this case. We analysis use should, the Zariski gauge group on any ofpossible the Tate tunings. curves. Both In for particular,fibration the for generic over some gauge a gauge group associated groups giventhrough with there a base the are Tate generic and tuning, multiple elliptic this forphysical properties. means constructions that As with there we will may gauge seedistinct be later, groups polytopes these distinct distinct to that Tate Tate tunings tunings different are can with Calabi-Yau correspond enhanced the threefold through same constructions. Note also that for the { Then the vanishing orders calculated from ( of the gauge algebrado exceeds not that consider desired this by tuning the as tuning, a we viable terminate possibility. {{− the algorithm and At each step this algorithmwhen clearly the increases algorithm the terminatesties orders the ( of solution vanishing is in clearlyis a a no minimal acceptable minimal way, so solution solution without of (4, the 6) inequali- loci. When this occurs, or when one of the factors become zero, ∆ by adding the non-negative correction values The second corrections are obtained∆ similarly, replacing We can then use theto inequalities each ( vanishing order (label decomposition on each isolated curve of self-intersection JHEP02(2019)087 , and (2.32) (2.31) R are normaliza- i λ }} }} over NHCs. 6 , }} , which is built from the 4 , 8 , and the explicit numbers 3 I , 2 , 1, 1, 1, 1, 2 1, 1, 1, 1, 1 1 . { { a 2 i , , . Then, using the notation and i } } }} }} }} }} }} }} }} and the Riemann tensor factorizes for some constant coef- F T,V,H G 8 tr I F i } 1, 1, 1, 1, 2 2 λ , { α i β 4 , b NHC j,n } X c i { α 4 X 1, 2, 2, 2, 4 1, 1, 2, 2, 2 1, 2, 3, 4, 5 1, 2, 3, 3, 5 1, 2, 3, 3, 5 1, 2, 2, 3, 4 1, 2, 2, 3, 4 1, 1, 2, 2, 3 1, 1, 1, 2, 2 X { { + {{ {{ {{ {{ {{ {{ {{ αβ , , 2 } } – 14 – Ω R 1 2 tr 1, 1, 2, 2, 3 α = associated with self-dual and anti self-dual two-forms a {{ 8 1 2 ,T I 1 = R α 1, 1, 1, 1, 2 1, 1, 2, 2, 3 4 ) inner product on the vector space, and X {{ {{ ,T 0) supergravity theories potentially suffer from gravitational, , } } } } } } } } } } The minimal vanishing orders of sections = (1 -8 -7 -6 -5 -4 -3 -12 { { { { { { NHC -3, -2 { ], the conditions for anomaly cancellation are obtained by equating the { N -2, -3, -2 -3, -2, -2 46 { { in the vector space β i Table 5. is a signature (1 , b α a αβ in the gravity and tensor multiplets, µν Here Ω tion constants for theconventions non-abelian of gauge [ group factors where cancelled through a generalized Green-Schwarz termficients if B gauge anomalies, though similar considerationshand, hold the for anomaly abelian gauge information2-forms factors. can characterizing On be the the encoded one non-abelianwhich in field has an coefficients strength 8-form that canof be chiral matter computed fields in in terms different representations. of On the other hand, the anomalies can be 2.6 Matter content from anomaly constraintsSix-dimensional in F-theory gauge, and mixed gauge-gravitational anomalies. We focus here primarily on nonabelian JHEP02(2019)087 , i G (2.41) (2.42) (2.33) (2.34) (2.35) (2.36) (2.37) (2.39) (2.40) (2.38) adjoint g are related are related . i j ,T G 1 . For example, a, b i R ⊗ ]); this simplifies ) two-index anti- i g a, b 13 2 G − +2 , , generically has n ) are irreducible curves in ! , ! b of SU(2), the representation is Z 2 i R j, , ) i Adj 2 2 of the non-abelian factor 2 A 2 (see e.g. [ 6= C / B i R F ) ( R , x 2 for different non-abelian gauge groups n − i R , i i H i R + R j S B fund. λ X b C i R ∈ A · (tr x i R i − a i R )-representation of x R C A T, R C , defined by i Adj X R , B ij RS R,S 5 i 29 X + A , x − K C 4 2

= 1+(

− T, , and i 2 i 2 F F g i Adj ↔ ↔ R,S − λ λ X – 15 – B i 1 6 1 3 B a b fund. fund. and = 273 = 9 = = = 2 tr tr b i i j · a b b R R in the representation V b through the relations b · · · · A B 6 2 i a − i a = b b B = = ) fundamental matter fields, and ( n H 2 4 g : : : : : : 0 = ]. F F − 2 2 2 2 4 4 j in different representations and R R 13 ) ) R R F F 2 2 tr tr R 2 2 i R F ) gauge factor associated with a vector ( F ( F ,C +16(1 R N n ) ,B are group theory coefficients R is the canonical class of N R A B − ,C K R ,B is the number of matter fields in the ( R A ij RS x In principle, the matter content of a 6D theory can be determined by a careful analysis For 6D theories coming from an F-theory compactification, the vectors is the number of matter fields A summary of For each representation the matter content contains one complex scalar field and a corresponding field 5 6 . With this identification, the Dirac inner products between vectors in i i R can be found in appendix B in [ in the conjugatepseudoreal, representation. so that Forhypermultiplet”. the special conjugate representations need like not the be included; such a field is generally referred to as a “half- generic matter content of acontent and low-energy anomaly theory cancellation is simply uniquelya from determined theory the by with values of the an the gaugematter SU( vectors fields, group (8 symmetric matter fields, where the base supporting the singularG fibers associated with the non-abelianto gauge intersection group products factors between divisors in the base. of the codimension two singularities in the geometry. In many situations, however, the to homology classes in the base where, again, x and where coefficients of each term from the two polynomials JHEP02(2019)087 8 0 } 3 E ,C n < 2 ,C 1 . In some C i { a, b ]. This will be useful 47 ], though the low-energy of self-intersection +2 two-index antisymmet- 13 C n 1 curve that does not carry . While the global symmetry ]. We use their results in our 8 − e 47 is the maximal global symmetry (glob) 2 g 0; in the low-energy supergravity theory this or larger in the Kodaira classification cannot 2 > ]. While in most cases these global symmetry g , where – 16 – 0 b 47 · [ b with non-vanishing inner products in the low-energy i 1 are worked out in [ (glob) 2 i G g b . These can be either minimal or enhanced algebras, fundamental fields. For most of the theories we consider } 3 ⊂ , as enumerated in the tables in [ ≤ − n g 2 ) ; i.e., the direct sum of the gauge algebras carried by the 1 , having 8 C g 2 m 0 e N g , ⊕ − 1 b, b 3 g g 1 curve should be a subalgebra of { − can be derived from the low-energy theory by considering the maximum 2 on the curve B 2 g ] states that the maximal global symmetry on a = 0 case of primary interest to us here to a spectrum of 48 g The maximal global symmetry groups realized in 6D F-theory for each possible algebra In general, the anomaly constraints on 6D theories provide a powerful set of consistency sum of the algebrasglobal on symmetry algebra. the curves For instance, intersectingcarrying consider it gauge a should linear algebras chain be ofbut a three they curves subalgebra have of to thealgebra satisfy given maximal theory for the algebras onexplanation a for set this of curves is intersecting not a understood 0 in curve [ terms ofon global a constraints curve from of SCFT’s. algorithm self-intersection to constrainalgebra possible on a gauge curve, algebra the maximal tunings. global symmetry on More the curve explicitly, is given determined, so a the gauge direct rule” [ a nontrivial gauge algebracurves is intersecting the constraints are completely consistent withand F-theory geometry, sufficient they set may not of be constraints; a complete for example a similar constraint appears to hold in F- curves in a base global symmetry of anthat SCFT that supports arises a by given shrinkingconditions gauge simply a match curve factor with thestances expectation the from global anomaly symmetry cancellation, condition in imposes some stronger circum- constraints. For example the “ be associated with vectors is ruled out by theto anomaly conditions a while configuration in the withcorresponding F-theory curves. a picture this In codimension would addition correspond twoon to (4, what these combination 6) types of of point constraints, gauge at another groups the set can of intersection be constraints between tuned the on specific negative self-intersection 2.7 Global symmetryIn constraints in many F-theory cases, thematter content anomaly of the cancellation theory conditions buting curves, also impose corresponding on constraints to what vectors gauge nottheory. groups only may For be example, on combined two on the gauge intersect- factors of conditions that we use in many placesarise in through this tunings; paper to in analyze particular, andspectrum using check various gives the models a anomaly that conditions direct tothe and determine associated simple the elliptic way matter Calabi-Yau in manifolda many that toric cases can hypersurface to be construction. compute matched the to the Hodge Hodge numbers numbers of of ric matter fields and 16+(8 here the matter contentsituations, however, follows more exotic uniquely matter in representationsof can this this arise; later way we in encounter from this some cases the paper, such values as of the three-index antisymmetric representation of SU(6). in the JHEP02(2019)087 , ], In 9 6). , , the 5.1 7 4.7 13 or below arises. − vanish to degrees (4 , there are cases in the that can support elliptic 2 f, g B 6.1.3 12, and the Enriques surface. ≤ ], the Weierstrass model over such 8 m 11 in the set of bases we consider for rule” in our algorithm in section ≤ − , section 8 , E is trivial. 10 , and discussed in more detail in section 4.7 2 − g , ] in classifying non-toric bases, we focus here action. Despite this complication, this blow- 2.2 9 with 0 ∗ − 11 – 17 – , m C F 1 and 10 , the blown up base still has a toric description; in , − 2 10 P F = 11. As discussed in [ ] and section m − and 31 , the resulting surface is a “semi-toric” surface admitting only 11 ] that all such bases can be realized by blowing up a finite 9 F F 49 . We also include the “ 10, and − 4.6 , ], but on surfaces with, for example, multiple curves of self-intersection 9 ] to classify the full set of toric bases 9 − 10 action [ 11, the blow-up of all (4, 6) points in the base gives generally a non-toric base ∗ − , C 10 − More precisely, as described in [ , 7 9 where it was shown thatelliptic there Calabi-Yau threefolds are over 61,539 these toric bases bases give that rise support to ellipticoriginal a CY3’s. base supports range an Generic of elliptic fibration Hodgeat that is some number “non-flat,” points, meaning pairs while that the the fiber elliptic becomes two fibration dimensional over the blown up base is a flat elliptic fibration. that is neither toricup nor and admits resolution a process single the is toric automatically hypersurface handled in construction,these a so types natural that of way (non-flat) curves intoric elliptic bases arise the with fibrations framework naturally curves of over in ofTate/Weierstrass bases self-intersection the constructions. with Kreuzer-Skarke The database. complete Thus, set we of include such bases was enumerated in [ actual base supporting thethe elliptic simplest fibration cases, is such generally as other simple a cases, non-toric such complex as a surface. single − Kreuzer-Skarke database where a toricnon-toric polytope base. corresponds to The anof primary elliptic self-intersection context fibration in over a whichcurves this automatically distinction has is 1, 2, relevantSuch involves or points curves 3 on points the on base the can curve be where blown up for a smooth Calabi-Yau resolution, so that the Calabi-Yau threefolds (toric geometry isWhile described further in progress more has detail beenprimarily in made the on [ following section). torichypersurface base construction surfaces, of aslater Calabi-Yau these threefolds. in are the the Note, paper, primary however, bases particularly that that in as arise e.g. we in section discuss the toric bases. The structureintersection of curves non-Higgsable that clusters can limitsallowed arise bases the by on blowing configurations up any points of given inwhen negative all base, a possible self- ways so disallowed and we truncating configuration theThis can such set of in was as possibilities principle used a construct curve in all of [ self-intersection 2.8 Base surfacesThere for is 6D a F-theory finite models It set was of complex shown base byset surfaces Grassi of that [ points support onThis elliptic the leads Calabi-Yau to threefolds. minimal a bases systematic constructive approach to classifying the set of allowed F-theory neighboring curves are known, making ouris search also over convenient possible sometimes tunings for moreconditions us efficient. to without This determine having the to gaugeis figure algebras described that out in have the section monodromy corresponding monodromy to directly; the the case trick where to doing this for us to constrain the possible tunings on a curve when the gauge symmetries on its JHEP02(2019)087 is a (3.1) N associated } -dimensional i , z n {   = j,n , in some cases the v j z S 6 k =1 Y j corresponding to the one- i ,..., z 1 j, , v j in the coordinate representation z of the toric variety. A fan Σ is a G j v k =1 Y j , fan   (Σ) 7→ may be constructed as the quotient of an Z ]. As we see in section ) G Σ k − 39 X , which are generated by a finite number of vectors in k C – 18 – , . . . , z = 1 z Σ ( , X , each with the apex at the origin, and where n ) R ∗ that do not span a cone of Σ. ⊗ C ( } th component of the ray i l N v → { = strongly convex cones k ) R ∗ N C is the number of rays), by a group (also called rays) of Σ; in :( i k v ( 8 φ is the union of the zero sets of the polynomial sets ], we review some basic features of toric geometry. An can be constructed by defining the k is the kernel of the map we consider Tate tunings over the toric bases and compare to the toric specifies the k C k Σ C 4 51 ) j,l , ∗ X v ⊂ C . 50 11. In these more complicated cases as well, however, the general story is the ( can be described by the homogeneous coordinates n − C ⊂ , Σ (Σ) lattice, satisfying the conditions that X where in Z with the sets of rays G Each face of a cone in ΣThe is intersection also of a two cones cone in in Σ Σ. is a face of each. 10 n In section − More rigorously, these are • • • • , 8 and which contain no line through the origin. 9 N dimensional cones open subset in where rank Then 3.1 Brief reviewFollowing of [ toric varieties toric variety collection of cones hypersurface construction ofmore elliptic detail. Calabi-Yau threefolds, which we3 now describe Elliptic in fibrations in the toric reflexive polytope construction two (4, 6) points on thenon-toric, toric base. base Blowing these with curves up multipleWeierstrass gives model additional rise giving to curves. another, a generically (flat)is elliptic After what fibration these arises over most blow-ups, thebase clearly new there admitting from base. is the the While flat a polytope the ellipticanomaly construction, tuned toric fibration cancellation. base the is structure relevant when of considering the F-theory blown models up and peaks at (11, 491), (251,base 251), needed and for (491, a 11)even [ tuned more Weierstrass exotic model than to− those match arising a from toric blowing hypersurfacesame. up construction The points polytope is on construction curves gives rise of to self-intersection a non-flat elliptic fibration with codimension that fill out the range of known Calabi-Yau Hodge numbers, in a “shield” shape with JHEP02(2019)087 (3.2) (3.3) (3.4) (3.5) . The i v by taking rays to be the vertices associated to all the rays , } +1 i -dimensional) cones to correspond to . v i , n , i = 0 D + face fan i m 1 z = 1 i − = X i i v +1 ≡ { i = D i = – 19 – · D i K · i D v i i D − In this way, a lattice polytope can be associated with D m 9 − of a toric variety is given by the sum of toric divisors , which is the convex hull of a finite set of lattice points (in particular, K − . are given by the sets N i . A fan is smooth if the fan is simplicial and for each top-dimensional cone its generators R D , etc.) 1 is such that i D lattice polytope m ≡ In the context of this paper, toric varieties play two distinct but related roles. On the A fan is simplicial if all its cones are simplicial. A cone is simplicial if its generators are linearly +1 9 k depending upon the refinementadded, there of can the be face many different fan. triangulations of Even the for fan. independent over a given setgenerate the of lattice additional rays of the lattice polytope,the facets and of the the top-dimensionalpolytope polytope. ( as By rays, including and moremultiple smaller thus lattice top-dimensional points subdividing cones, in (“triangulating”) wesuch addition the can as to refine simpliciality facets vertices or the of of smoothness. fana the the to toric impose polytope variety. further into properties In general, a given lattice polytope can lead to many different varieties, in the next section. 3.2 Batyrev’s construction ofGiven a Calabi-Yau manifolds fromthe reflexive vertices polytopes are lattice points), we may define a intersections, up to lattice automorphisms.) one hand, we can usefibered toric geometry Calabi-Yau to threefolds. describe On manyambient the of fourfolds the other in bases which hand, that CY toric threefolds support geometry can elliptically be can embedded be as used hypersurfaces, to as we describe describe where and zero otherwise.by We the will sequence generally denote of the self-intersection data numbers. defining a (The smooth 2D 2D fan toric can base be recovered given the and the self-intersection of each curve is toric divisors arechain. rational curves This with isfan one easily corresponds intersecting seen to another fromalso an forming the easy irreducible 2D a to effective read toric closedD off toric fan linear from divisor. description, the where fan The diagram, each where intersection ray (including of products divisors the are cyclically 2D by setting anti-canonical class Smooth two-dimensional toric varieties are particularly simple. The irreducible effective Toric divisors JHEP02(2019)087 , ∗ is j . of N 1 z ∇ , ) is j 2 = 0. 10 z (3.6) (3.7) (3.8) h p 3.8 , ’s, and ∇ ⊂ =1 j 1 ) indeed , # rays j v 1 containing h Q 3.7 N is given by an ) H: i which are defined ∇ m ∇ ⊂ in the fan associated , j v ∈ ∇} v ∀ , 1 , i ]. A pair of reflexive polytopes in m i reflexive polytopes, 53 , c is reflexive, its dual polytope ∆ = , i ≥ − 4 and each monomial +1 . A lattice polytope i ∇ j R C 4, with the anti-canonical embedding, a u, v ,v describes a toric ambient variety, in which . We can check that equation ( i h ⊗ w : ≤ ∇ h j ∇ z N R i ) is a section in the anti-canonical class. For n X j ≡ – 20 – ⊗ in terms of the homogeneous coordinates Y 3.2 is also a lattice polytope. The dual of a polytope R i = M N m i = # lattice points in ∆ m R = M p lattice, ∈ . We call the pair of reflexive polytopes a mirror pair. Both n ) is the dual lattice. If u ∇ { in ∆ dual polytope Z ] are built out of reflexive polytopes. Given a mirror pair i = = s by the reflexivity of w N, 52 ∗ j ∗ ) z ∗ ∇ ∇ be a rank such that the fan is simplicial so the ambient toric variety will have at = Hom( ∇ N ∗ N , the (possibly refined) face fan of ) define the exceptional divisors in the resolution of the ambient space. The added lattice points is the homogeneous coordinate associated with the ray = M are generic coefficients taking values in 3.8 is defined to be j i ⊂ z c M N . The well-definedness of each M. Kreuzer and H. Skarke have classified all 473,800,776 four-dimensional reflexive We will be interested in particular in the fans from Appropriate subdivisions of the face fan of the toric ambient variety by additional lattice points in the ∇ in 10 M:# lattice points, # vertices (of ∆) N:# lattice points, # vertices (of facets of the polytope giveequation the ( resolved description of thethat embedded do Calabi-Yau, where not extra lie coordinatesCalabi-Yau. in in (Generic the hypersurface CYs interior do of not meet the the facets divisors that also correspond correspond to interior to points of exceptional facets.) divisors in the resolution of the hypersurface will generically avoid these singularities and therefore ispolytopes generically for smooth. the Batyrev Calabi-Yauthe construction KS [ database are described in the format: since we know it issimply the always product an of all interior homogeneous point.which coordinates associated immediately Its to all associated we toric monomial see divisors the by by smoothness equation equation of ( ( thethe Calabi-Yau, as face mentioned fan previously, of theremost exists orbifold a singularities. refinement In the case of to guaranteed by the linearholomorphicity equivalence in relations the among thedefines divisors a associated section of to theWe anti-canonical can class, determine so that the a class CY by hypersurface is looking cut at out any by one of the monomials; we pick the origin where associated lattice point where variety, so that thethe hypersurface anti-canonical itself bundle has is trivial given canonical by class. Explicitly, a section of is also reflexive as ( of them containBatyrev’s the construction origin [ asand the ∆ only interiora lattice Calabi-Yau hypersurface point. is embedded Calabi-Yau using the manifolds anti-canonical in class of the ambient toric as follows. Let the origin is reflexive if∇ its where JHEP02(2019)087 , ]. 2 of ∇ ∇ 56 θ (3.9) – (3.10) (3.11) 54 , (0) = is one that 1 31 [ − ) := number , and mirror N π ˜ 1 θ , 2 ∈ θ/ . h 1 is the complex v } θ − , , 5 ) with a superscript, 5 and ray n -dimensional faces 1 − − -dimensional faces of , l and ∆ are a pair of 4D such that l = 1 ) ) ∗ ∗ h B or ∆, int( ∇ ˜ d θ θ N . fibered polytopes ∇ ) makes the duality between [2] primitive )int( )int( . As 3.9 ˜ θ θ 1 , ∇ → kreuzer/CY/ 2 : ∼ h int( int( ), where . (A by the projection. We can construct π ∆ ∇ ∆ 2 ∇ 2 that are vertices of F N F and ∈ ∈ X X CY ˜ θ θ pt ( 1 lattice that contains a lower-dimensional , denotes the set of 1 p,q h ) + ) + ∆ N − ˜ / θ θ d ∇ h l for all – 21 – F | int( int( related by the dual operation ) of a ray in 1 , i ∆ ∇ 3 3 v ) = − ∇ F F ∇ ( ∇ ∈ ∈ X X π ˜ = θ θ of . The correspondence ( i ˜ CY θ − − ˜ θ or M:165 11 N:18 9 H:9,129 ( ) , the Hodge numbers are given by or p,q [1] ∇ y, pt , which passes through the origin. We are interested in the h h ∇ 2 θ , admits a projection map http://hep.itp.tuwien.ac.at/ Z that are the primitive rays with the property that an integer ∆) := number of lattice points of are faces of / ∇ ˜ = B θ ∈ ∇ is a polytope in the = pts(∆) = pts( ∇ N 2 y 1 1 , , { ∇ 2 1 N as the (fa)N polytope and ∆ as the M(onomial) polytope to remind in h h = ⊂ ) ∇ gives the fan of the ambient toric variety for the CY anti-canonical embed- ∗ arises as the image B 2 θ is itself a reflexive 2D polytope, containing the origin as an interior point. ( i ) ∇ is the quotient of the original lattice v ∇ 2 B )-dimensional faces ( i 2 ), and pts( l ∇ v Z − are faces of ∆, = θ l < n B . Then the hypersurface CY is mirror to the previous one. The Hodge numbers of N Note that conversely, we can start from ∆ and associate it with the polytope that ∇ case where Such a fibered polytope and a set of rays multiple of 3.3 Fibered polytopesFor in the the purpose of KS studying database arise (often in singular) the elliptically KS fibered database,A Calabi-Yau we threefolds will fibered that be polytope interested insubpolytope, 4D reflexive where or ∆ ( of lattice points interiorthe to Hodge number formulaepolytope manifest. and Note not that on the the Hodge detailed numbers refinement depend of the only fan. on the ∆ and (4 For the CY associated with mirror pairs are relateddimension by of the CY;CY in threefolds, particular, where we the willCY only hypersurfaces look have non-trivial at exchanged 4 Hodgereflexive values dimensional numbers polytopes, for reflexive are there polytopes is for a one-to-one correspondence between precisely, or indicate its numericalthe order KS as database it website appearse.g., ( among M:165 11 those N:18 with 9 identical H:9,129 data in defines the fan of the ambientin space, and calculate monomials associated with lattice points ourselves that ding and ∆ determinesit the monomials is of sufficient the tosometimes anti-canonical specify there hypersurface. can polytopes In be distinct with manycase polytopes cases, the we with will information identical either information given give of further in this the the type, vertices in format of which the above, N but polytope to specify the polytope more We will refer to JHEP02(2019)087 . ). ∗ ) 2 with ∇ 2.14 1.) We B taken to Σ = ( ) 2 . As far as n > . Thus, to B 2 → ( i . Such a fan B v B . Such a map ]; this gives an B Σ 2 . 31 :Σ B , with → 2 with π ]. The hypersurface → N -fibered type, with a ∇ B 1 Σ , ∈ 57 3 , , X :Σ 2 w : ]. A primary goal of this P 31 , π ; these models can be con- π 55 , and develop an algorithm 2 [ 23 B 4 2 B is a projection standard with respect to the fiber subpolytope to the maximal torus in 2 of another ray B Σ X are each associated with fibered polytopes nw m = F , the dual ∆ admits a projection to ∆ v 2 – 22 – fan morphism ∇ -fibered polytopes are in the form of equation ( ), of all 16 types of fibered polytopes can be brought 1 , 3 , 2 3.7 . The Calabi-Yau manifold defined by an anti-canonical P defining the base i v ) by the methods described in appendix A in [ ⊂ ∇ , with the result that the anti-canonical hypersurface equations 2.8 2 given in ( ∇ and a given fiber type, there can be a variety of different polytopes 3.4 2 , distinguished by the different twists of the fibration. For a fibered p 1 B P through the Batyrev construction with reflexive polytopes will then be , which is an equivariant map with respect to the torus action on the to be the 2D toric variety given by the 2D fan Σ Σ 2 B X = 0, with with a reflexive subpolytope p ∇ toric morphism One of the findings of this paper is that the bulk of KS models with large Hodge For a given base There are in total 16 2D reflexive polytopes, which give slightly different realizations of In the toric geometry language, a . For example, the Hirzebruch surfaces 2 that point of view directly.dard On polytope the in one section hand, wefrom describe (suitably the refined) structure standard of thisOn type the of other stan- hand,tunings we describe on the the direct effective construction curves of in polytopes the by toric carrying out bases Tate in section with fiber and base polytope numbers appear to have aspecific description form in for the the formnected twist of directly of a to the the fiber Tate form over for the elliptic base fibrations, surface and in fact can be constructed from different examples of the 16 fiber types. corresponding to configurations with differentferent “twists” of embeddings the of fibration, the associated rays ∇ with dif- equations into the Weierstrass form ( equivalent description of the sameThe Calabi-Yau Kreuzer-Skarke as database long of reflexive as polytopes the andconstructions fibration associated contains Calabi-Yau has a hypersurface a wide global range section. of polytopes that include fibered polytopes with many an elliptically fibered Calabi-Yaupaper threefold is over the to base relateto reflexive elliptic polytopes fibrations in of the tuned Weierstrass Kreuzer-Skarke models database as that described havean in elliptic this curve section form when an anti-canonical hypersurface is taken [ that the elliptic fibertriangulation structure with of respect therecognize to polytope an which does elliptic there not Calabi-Yau depend threefold isa in upon reflexive a the the subpolytope fan KS existence morphism hypersurface database, of in it a is only necessary to find the property that formorphism any can cone be in translated Σis to the a a image map is betweentoric contained toric varieties in varieties that a maps cone thethe of authors maximal are Σ torus aware, in ittriangulation is leading not to known a whether compatible in general fan every morphism fibered and polytope toric admits morphism. a Note, however, define the base be the 1D cones;dimensions, the the 2D cones base are ofcone uniquely the structure defined of fibration for the is a base not 2D may variety. uniquely not Note be defined that unique. as in a higher toric variety since the cannot be described as an integer multiple JHEP02(2019)087 2 2 ). y M , 3 1a ; this x (3.12) 1 , 3 , 2 in the P 2 plane projects ∇ 2 to M 2 3) (see figure , . in the Tate form of the , respectively, and to the 3) 6 6 ) transformation to put , = (2 , 4 , z Z 2 , z 4 3 , , , 0 2 , , v , 1 , xz a 1) 3 ambient toric fiber. − 1 , , = (0 , yz 3 , 2 z 2 z P , where the fiber is defined by three 2 . These lattice points correspond to the , v 2 = (0 The dual polytope ∆ 2 1) B y − , , v five sections Weierstrass model, indicated inxyz, the x figure by coefficients of the remaining twoin terms the hypersurface equation. (b) lattice. Projection onto the the lattice points in ∆ intoin seven lattice ∆ points 0 there are generally many distinct polytopes 0) , , 2 0 1 , B − – 23 – z = (0 = ( , v y y x , v v , v x over a toric base 0) v , 1 ∇ − -fibered poly- -fibered polytope, and we compare the two constructions in = 0, we can always perform a SL(2 , 1 1 , , -fibered structure; as we describe in the following section, these 0 z 3 3 1 , , , , v 2 2 3 . For a given base , . The convex hull P + P 2 1 6 that provides a natural correspondence with the Tate form mod- , -fibered polytopes and corresponding Tate models y P 3 = (0 , The reflexive polytope pair for the 1 v 2 , 2 x 3 P ∇ , v 2 + 3 P x lattice. The rays v standard N -fibered polytope 1 for , to systematically classify models of this type that give polytopes and elliptic Figure 1. 3 , , respectively, in the hypersurface 2 2 5 ∇ plays the role of the reflexive sub- P ) is associated with the toric fan giving the weighted projective space in the 1 , ∇ x, y, z 3 , 2.14 2 The toric fan for P is a toric varietyGiven given a by the rays rays satisfying 2 the rays in the fiber into the coordinates correspond to different Tate tunings over the same3.4 base. Standard The fiber polytope els ( in section Calabi-Yau threefolds with large Hodgecorresponding numbers; standard these models arethe remainder all of expected section tothat have have a the standard of polytope topes are associated with thenates homogeneous coordi- equation. (a) JHEP02(2019)087 ) ). 0; B ( 3)) B , v ≥ ) are 2 2 (3.14) , nK ) m − B ) 3.14 ( ( 2 . B ( 2 } , v v ) ) makes this ) then takes 1) ∈ O B ( + , 1 v 1 ]. n 3.7 1 3.13 , a (( 20 m in this coordinate , ) π , in much the same B ( ( 2 1 M , = as one where there is v B from ( ) 2 1) ⊂ K , B p B ( 3) of the fiber, we sometimes 6 0 , v . Note that in fact, these , − fiber is the only one of the B , associated with the ray . An analogous computation of the ambient toric variety. 1 , ). In particular, the condition d ) ( 3 ]. As mentioned above, these , , B 1 + ( 2 ) are associated with the mono- K 0) z 56 P n − , ) in Σ ) 1 − = 0 with 2 ). And the natural correspondence , 3.14 3) (3.13) B ( i, , = p , the condition is 2 base [ , v 2 , 3) in the fiber over which the base is stacked ( 3 ) 4.8 ) have a single overall coefficient, and the , = 0, so the first two points in ( , 1 for each ray 1 1 x m 2 B (see figure B ( ) i, 3 ( 2 i, P 1) ) over the vertex (2 v } B x since the only interior point of a reflexive , ( 2 m 6 , v 1 ≥ − ) v and 3.13 1 , a − B ∇ = ( = 2 ( + i, 4 , and in the variable – 24 – ) y 1 v 1 a , 2 B , d ( m i m y 3 ( ) v = ( ) represents a section of , confirms that the corresponding monomials satisfy B -fibered polytopes play a central role in the analysis 2 , a ) + 2 + 3 ) transformation such that the vectors ( 1 1 a 2 0) , v precisely imposes the condition that ( Z 2 n i , 3 , , v , m a 0 , a -fibered polytope over the base 2 m 1 , ) represent sections of 1 ∇ 1 ) P , a 3 m B , , { ( 2 3.8 2 ( ). The sets of points in ( v and the specific point (2 P , 1 + , 1) 3 , so ( 11 , 1 -fibered polytope, the lattice points of ∆ , 2.14 2 for every ray 2 1 1 respectively; 2 , P , and the degree m B 3 ) is given by ) , − n ∇ 2 we have the condition on the monomial associated with the point 2 , B ( 1 0 P 6 v , , m , y, x, a 1 ≥ − 2 (0 standard 2 , m m 1) ) − B , xy, x 1) that , ( 2 3 , 1 v 1 , , x , 0 + 2 2 , 1 y (0 For a standard m , m { Because the rays of the base are “stacked” in ( ) 1 B 11 ( 1 m refer to constructions ofcorresponding this form to as the “stacking” fiber plays fibrations. a The special “standard role stacking”of we in a have the defined “stacking” analysis here, with ofwhich different generalize this fibers the paper. and/or specific We different “standard describe specified stacking” more used points here, general in in polytopes the the that fiber companion have supporting paper the the [ form stacking, A similar computationv for each of a monomial ( shows that for the pointsfor associated any with compact base this implies that that ∆ is theFor dual example, polytope for of ( in the fan ofway the that base the monomials in ( The elliptically fiberedprecisely CY the hypersurface Tate form equation ( mials monomials in the base associatedmonomials with in the the five other sections five sets of points correspond precisely to twist structure particularly compatible with the reflexiveWe polytope do, Calabi-Yau construction. however, encounter some specific examples of othersystem twists organize in into later the sections. following sets of points: polytopes appear to benumbers. highly This prevalent seems in to16 the occur reflexive Kreuzer for 2D Skarke several polytopes databaseintersection reasons. that at in The is large the possible Hodge basebetween in (see tuned Tate the discussion models presence in and the of section particular curves twist of structure defined very by negative ( self polytope is the origin.specific This class particular of choice fibered of polytopes fiberhypersurfaces. and that twist produce These geometry elliptically standard represents fiberedof a Calabi-Yau threefolds very this as paper,K3 and surface are that a is generalization an of elliptic the fibration well-studied over a 3D reflexive polytope for a a coordinate system after an SL(4 are contained within lattice points are all on the boundary of We can define a JHEP02(2019)087 ) 3.12 -fibered polytope, -fibered type; i.e., ], which forms the 1 1 , , 3 3 , , 59 2 2 , P P ). This corresponds to are simply given by the 58 , ∇ 3.13 1) and are associated with 21 , [ by removing Tate monomials. 2 , the set of new lattice points 2 − B , other than those given by ( top 4 2 ) and ( ∇ , m 1 3.12 m ). This corresponds to the generic elliptic 3.13 1) and ( – 25 – ) is known as a ) − B , ( i 1 v , ( 2 1 − , m π 1 (6) tunings associated with exotic matter representations, in m su without non-Higgsable clusters. In other cases, however, there ) a 2 ( i v B the affine root (this is the only inverse image when the fiber is other than those given by ( ) a we describe in more detail the dictionary between Tate tunings and ( i ∇ v 4 ) for the generic elliptic fibration over a given base. In the case of NHCs, over the base, and the fact that the only global holomorphic functions on B O nK ) and the vectors of the form ( , with ) − (1) tunings, novel B ( u ( i ) reflect the singular nature of the resulting Calabi-Yau hypersurface. Up to some 3.12 O D 3.13 In the simplest cases, all the lattice points of the polytope in the original sieveinvolving because they areor tunings accompanied of by generic more modelswe over sophisticated encounter non-toric constructions a bases, which type webecause of had novel they not models considered. are that Moreover, involved did with not tunings arise from on our non-toric systematic curves construction in the base; they turn out considered separately those few exampleswere not in found the by KS our F-theory databasein construction. in the We the KS have found range database that of thathave there interest can Tate are that be forms, a few described but polytopes in werebe terms nonetheless because of not they the found involve standard with suchthe extensive the range tunings initial we that considered. sieve. the There This starting are turns bases also out needed data to are in outside the KS database that we did not identify Our primary approach in this paperhave is counterparts to as systematically construct reflexive Tatefrom polytopes tunings that the in should the F-theory Kreuzer-Skarkedatabase. construction database. and This Thus, gives match we usof the start something KS results like data with a not “sieve” the produced that known by leaves our data behind algorithm. a in set the After of implementing KS special this cases sieve, we have then toric/polytope geometry for specificin gauge the groups base on geometry. particular local curve configurations 3.5 A method for analyzing fibered polytopes: fiber types and 2D toric bases monodromy subtleties that weintroduced discuss together further with inextended section Dynkin diagram ofdivisor the gauge algebrasmooth). of In the section singular fiber over the associated bundles in particular, the monomialsof in gauge ∆ span group theview tunings, appropriate of some the set Calabi-Yau of of geometry,and sections, the these ( while lattice monomials points in in are the case set to zero. From the point of vectors ( fibration over a toric base are lattice points in situations with NHCs or gauge groups tunedThe over curves set in of monomials in ∆ completely span the set of sections of the appropriate line constant functions on thetrivial base. bundle This matches withany the compact base fact are that constants. thesethe This are lattice proves sections points that in for of any ∆over the standard the are base, associated as precisely stated with above. the Tate form of a Weierstrass model the only points of the form ( JHEP02(2019)087 } is z 0 . If v + 1, that , v y (3.15) (3.16) (3.17) max ∇ , v + 3 max with the x shown in v v M { 2 ∇ ∇ w > M · . Let us call the v 2 ∆ . → } , the presence of a fiber . that we are focusing on), 2 1 in ∆ with , 3.3 ∆ = 3. We can then check for 1 , the maximum exponent of w h ∈ 2 , and check to see if 2 . The other fiber types can be ∆ max 2 S , w . , we want to find rays vertex of ∆ 2 M 1 ∇ , → in 3 , w , 2 = 0 0 2 ∀ , 1 ∈ ∇ z P , 1 v . We collect the subset of rays in v, v P 2 max + ∇ y M w, v v that satisfy the necessary linear relations to · ≤ has a fiber S v + 3 – 26 – w ∇ · x v v 2 : . As mentioned in section = 5, and for 2 = max ∇ ∈ ∇ . For example, for max 2 max v ] with built in routines to identify the reflexive subpolytopes whether there exists a vertex ) and the projection ∆ { ∇ M M precisely contains the 7 points in the polytope 60 , 0 = 1 implies that there is a projection from ∆ , . If so, we can then check that the intersection of 3.15 3 ∈ ∇ , S S 2 v, v v P ), ( ⊂ ∇ cannot be a ray in a fiber 2 3.8 v ∇ is large (which is true in the case of large ∇ . If this is the case than 1a . To increase the efficiency of the algorithm in the case that the number of lattice 2 plane spanned by figure checked in a similar fashion. that satisfy In this case we canalso look an at element all of pairs of rays are not ruled out by this condition: be elements of the fiber For example, for each lattice point there is, then points in we begin by focusing onrole only as a the subset points ofsubpolytope in these lattice a points fiber that canmaximum possibly value play of a the inner product for any pair of vectors in the fiber and its dual ∇ We then look for a subset of rays of We assume that we are interested in a fiber described by the 2D reflexive polytope By equations ( We had to explicitly study these specific polytopes with Hodge numbers that we did not 2. 1. all monomials in the variables associated with the rays in the fiber should be software programs like Sage [ of a given polytope. the 16 reflexive fibertoric types base is from a thedetermine fiber fibered the of polytope. singularities the of polytope Ascheck that the we in the elliptic describe question; Hodge fiber later numbers wepolytope of over in then the model. the the define inferred curves tuned Here paper, the model we into we 2D are briefly the determine can consistent summarize if with base, thereby the those a and first of given then the piece reflexive we of 2D this polytope analysis: is the a algorithm fiber of a 4D polytope. There are also elliptic fibrations. immediately identify from Weierstrass/Tate tunings togive hypersurfaces determine that whether are these actually polytopes algorithm elliptically to fibered. analyze We provide reflexive here polytopes. a summary We of can our learn from this analysis whether one of nonetheless to also be described by reflexive polytopes with toric fibers associated with JHEP02(2019)087 , is S 3.4 Note (3.21) (3.18) (3.19) (3.20) 12 . 2 B among all fiber of the fibration. z 2 ] we will describe . For example, for B 2 20 is large, this speedup ) are 5 ∇ } , 4 o . z, x } 3 6 are in one-to-one correspondence ∈ ∇ is 680, while the number in , x , z i 2 -fibered polytope from section , we associate a 2D cone with each 2 4 v ) 1 , , z ∇ ∀ 3 B 2 , ( i , 2 , xz v to be  , x P 2 are the only two independent of the base = 0 ) 3 z 2 2 2 z B 2 ( i, v y B ) is done to restrict to primitive rays in the , xz , x ) , v 4 + 3 ) 2 B is the projection along the fiber subpolytope 1 ( y i, B and , x ( i, – 27 – v y, z 3 π v , v 3 2 )  x fiber. These polytopes can be described using two 1 -fibered polytope (see figure B + 2 ( i, 1 2 , , x v ), we have the maximum exponent in 3 1 , , v 2 1 ) and GCD 1b P ) P . Given the rays , xyz, x , xyz, yz / 2 2 B 2 ) ( i, y B ( i 3.3 { , v , yz v ) 2 1 n B y ( i, are in one-to-one correspondence with the sections of the line { v 2 ) = ( + 1 = 6, and the lattice points in ∆ i ), and the monomials v ( 11, the number of lattice points in B , π max nK M ]. We discuss this point in more detail in later sections. ≡ − + 1 = 4. The first step in the algorithm above is only used to speed up the ) ( 31 B O ( i v max ) = 0). The division by GCD( -fibered polytopes (see figure 2 M In the regions of the Hodge numbers that we study in this paper, we also encounter Once we have determined the fiber, we can then compute the base Similarly, the sections of the 1 , Note that in higher dimensions, the cone structure of the fan is not uniquely determined by the rays. ∇ 3 , ( 12 2 π fibration [ polytopes that have nodifferent standard types of models. One of these other types of model that we encounter is very where ( image, as discussed inpair section of adjacent rays,that the giving base a defined unique this way toric gives structure a flat to toric the fibration, base but not geometry necessarily a flat elliptic We define the set of rays of the fan describing considered, this represents amany cases. speedup While by in ais this useful factor paper when of we considering are hundredsthe larger only systematic or datasets. considering application thousands a of In of few thisHodge the times algorithm examples, numbers. companion to such in paper all a elements speedup [ of the KS database with large and algorithm, but particularly whenis the significant. number of For lattice example,numbers points for H:491 in the polytopeonly associated 9. with Since the the Calabi-Yau second with step Hodge of the algorithm is quadratic in the number of lattice points when the associated rays are such that Note that, following thethe definition lattice of a points standard bundles in ∆ coordinates. sets that are in one-to-oneP correspondence with the latticecoordinates; points in ∆ with the sections and the monomials can be grouped according to the powers of the fiber coordinates into JHEP02(2019)087 ) 2 y 4.1 3.7 as a ]. We 2 , . The 1 , 36 1 , ] to bring P 6.2 23 31 non-standard associated with are taken to vanish. 2 fiber as special cases 6 1 ). In particular, there , a 3 , . Models with this fiber 2 P -fibered polytopes. 2.14 C is identical to that for the 1 1] , , 2 3 0 , , 2 [0 P 1), so that the number of lattice − , . To simplify the set of possibilities, ) can not be constituted by a set of 1 − with two more blowups) rather than ) for the models of this type that we 3.3 3.21 2 , do not all lie in a plane that contains the 2.8 1 polytope, but with a different “twist” to for comparison). This allows us to follow , 1 1 ∇ 5 , P 3 , 2 P – 28 – (1) symmetry that we mentioned earlier [ u models in terms of models with a blow-up of -fibered models described in appendix A of [ 1 , 2 , 3 -fibered polytopes, but has a fiber that is a single blowup , 1 , fiber, which is one of the other 16 reflexive 2D fiber types, 1 2 , 1 3 P , 1 P , 2 3 , fiber, the base rays in P ). The consequence of this is that the hypersurface equations ( 2 given by the usual 1 , P 2 3 , 1] forces a global , 2 3.13 0 . The corresponding fiber subpolytope ∆ , P 6 -fibered polytopes (see figure [0 a 7 1 , 3 , 2 ] and consider Tate tunings over these bases. polytopes, and describe their analysis in more detail in section 9 P of the 2 . To determine the Weierstrass form ( bundle over the base. In other words, while there is a projection of ∆ to the dual 3 ; i.e., the base of the polytope defined in ( Weierstrass/Tate models. 1 . This Bl z x , 1 v 1 3 , -fibered polytopes” (or more precisely, , , fiber except that it has an additional vertex at ( 3 3 2 2 -fibered , , , In principle, to find all the elliptically fibered threefolds in the Kreuzer-Skarke database The other unusual kind of fibration that we encounter in a few models is a fibered 2 1 1 2 1 P , , , P 3 3 P 1 , , 2 2 P we might want to considerto a all 16 variety of of the tunings torichowever, and we fiber focus singularity types on structures mentioned a in that region section dominate. correspond of Hodge A numbers generic where Tate-form we elliptic expectstructed fibration a starting over single from a toric the fiber given “standard type toric stacking” to base procedure can as always we be will con- describe in section constructions through reflexive polytopesto in the the general Kreuzer-Skarkemodels. database construction can The of approach be elliptic we take related with Calabi-Yau is toric to threefolds identify hypersurface through a constructions. specificidentified tuned subclass in In of Weierstrass [ particular, tuned models we that begin match with the set of toric bases of 4 Tate tunings and theWe want to Kreuzer-Skarke understand database how the set of Calabi-Yau threefolds produced by toric hypersurface them into Weierstrass form.non-toric This curves type as of we mentioned novelP earlier. model We gives refer rise totreatment to this of an type non-standard enhancement of over modelsfiber as is closely analogous to the analysis of models with a Bl and/or found and analyze their structure,“ we found that itthe was standard useful to viewthe them method as essentially for analyzing polytope ∆ vector rays all in the form ( for these Calabi-Yau threefoldsis do generically not more take than on one the lattice Tate point form projected ( to the points in ∆ This vanishing of describe an explicit examplecan of be this treated type in of essentially the model same in fashion appendix aspolytope standard with fiber ∆ the of is shown in figure P points in the planeview, of such a the fiber fiber occurs subpolytope when is all the 8 monomials rather in than the 7. coefficient From the Tate point of similar to the standard JHEP02(2019)087 , B (4.1) by Σ -fibered ]. . Such a 1 , 3 B 56 , A 2 By focusing P , in the smooth } fiber and a 2D 13 ) 1) 1 B , -fibered polytope ( i 3 1 − , , v , 2 3 , 0 P 2 , P 0 , (0 , 3) should be in the plane of to be the ambient space of , 0) 2 , 1 , , 1 3 0 , − 2 , , P 0 , base as described in e.g. [ = (0 1 (0 ] of the polytopes. We review polytope z some constraints on the other 15 fiber P v is associated with the trivial bundle over . Since in the Weierstrass or Tate model ∪ { } . 59 z , 4.8 o 3) , fiber type. Tuning the resulting generic Tate B 2.3 58 2 , , and give an example in appendix 1 , , 3 0 , – 29 – , 2 56 -fibered polytope, and showed that there is always , to be the convex hull of the set 4.5 P 1 (0 , , 3 21 ˜ , ∇ 2 1) rays in Σ P ) − ]. . Taking the fan of , B } 20 ( i 0 ) , v B | ( 0 i  fiber type will dominate the set of polytopes. , v -fibered polytope, we start with the 2D 3 { and section 1 1 , (0 , , ) the fiber coordinate 3 , 3 2 , , , 2 2 ) 0) 2.9 P P 2 , B 4.3.3 ( i, 1 , which are related to the monodromy choices of the Tate tunings of − , v . , ) , and we address some subtle issues about the multiple tops of a gauge 1 0 B 4.4 ( , i, are the first and the second components of the 1D ray 4.7 v . From the definition in the previous section, this is a standard ) (0 ; this procedure uses the 4.2 2 { B B we defined a standard n ( i, and = 4.3 , v ) } 3.4 , the simplest 1 z B 4.6 ( i, 5 v , v y To construct a 4D The other 15 fiber types, however, implicitly constrain the Weierstrass model associated That this expectation is correctly borne out is also verified explicitly with a systematic analysis of the , v x 13 v where 2D toric base KS database in the companion paper [ { framework of equation ( the base, the lattice pointthe associated base. with Thus, the we ray define a polytope base, and we construct thestructure polytope over in a the straightforwardwith way base. to the have the set We desired of denotethe fibration elliptic rays the fiber, being toric we fan can associated embed with this in the the base 4D coordinates such that the three rays are a corresponding Tate model. Nowa we are corresponding trying Tate to model, do thealluded we converse wish — to given to earlier, a construct the toric basefor recipe a and an corresponding for elliptically the reflexive fibered polytope. construction threefoldfor is of As the a a natural K3 4D generalization surface standard of that the 3D is reflexive an polytope elliptic fibration over a pair with our systematicsections tuning construction, we will explore4.1 a bit more Reflexive polytopes thisIn from aspect section elliptic in fibrations without singular fibers that when we confinesection the range of Hodgeon numbers this simple to class relatively of largethe constructions, range values, therefore, as of we realize we interest almostwe with do all will a in the single not Hodge class deal numbers in of with Tate-tuned the elliptic matching fibration of models. the Although multiplicity in KS database of a given Hodge tops in section algebra in section the algebra that we have discussed in section with an elliptic fibration.types, We which explain are in related section to the structure of the base. Based on these constraints, we expect model by removing monomials incorresponding the to dual further polytope reflexive thenthis polytopes leads process that to can in a appear set section construction of in can possible the be tunings database; carried wetunings describe out can for be any read off base. from the The tops gauge [ symmetries associated with the and section JHEP02(2019)087 ). is in are 6 w ∇ (4.2) ) B ( j that are } v 1 (isolated ˜ ∇ , , where } and ≥ − 1) ) kw 2 (see table B − -2, -3, -2 ( i − , = v 1 ) , > B 3) can be vertices of 0 ( j , i , v 2 D , (0 is immediately reflexive, ) − , · 2 ) B i ˜ ∇ ( i, 1) B , ( D i + 1 (including the boundary , v 2 v }{ ) i − 1 B , ( i, = 0 v , j 2 lie on the 1D faces of lattice only when the denominators (0 = ( − i -2, -2, -3 v M ∪ { !) 3). We can now check directly that in these 1 , , 2 ), in any case, are 1 , , 0 , so we again have a cancellation and the vertex – 30 – ] , ) k 4.1 ) ) 1 B 2 curves. In this case B ( j ( j, }{ − ] = v ) , v ) B ) will lie on the − of ( j B ( ) i -2, -3 k 1 v , v 4.2 B ) ( i, is reflexive, so it can be used as the reflexive polytope v , which is the 2D face associated with the base; and there B ( i B ˜ 6( v ∇ Det[ other than (0 2). In this case, the Weierstrass/Tate model of the Calabi-Yau θ that contains no non-Higgsable clusters (i.e., no curves with − − B , θ ] B ) ) ) 2 B B ( j ( j, v , v ); i.e., we get lattice points as long as there are no non-convex base ) }{ n − B -3 ( ) i is reflexive without further modification. The vertices of the polytope dual are associated with curves of self-intersection { 2 v B = ( i, ) . The vertices in ( ] are cleared so] = that 1, all so we entries have are a lattice integers. point whenever For a smooth 2D base fan, ˜ ∇ ) ) v B B ( i , i B B +1 θ ( ( 6( j i Det[ v ) are taken to run over all pairs of labels of base rays that correspond to adjacent Non-convexity of NHCs: the rays corresponding to an NHC cannot be a vertices; hence, , v , v ) ) i, j = 1 ( B B ( ( i i j v v We start with the simplest case, in which we have a generic elliptically fibered Calabi- Fan NHC only if ˜ Det[ case rays, which would beseparated skipped. only by We some also number geta primitive a vector, lattice and point Det[ as long as are no interior points in simple cases to the convex hull of the set of vertices ( where ( vertices of Det[ Yau over a toricself-intersection less base than is smooth and therelattice points is associated no to gaugeboundaries curves of group of self-intersection in the the 2D face 6D supergravity theory. In this context, ∇ We now wish to checkBatyrev’s that construction of a Calabi-Yau threefold.and In in some other cases more complicated cases it must be modified to make it reflexive. the vertex contribution from the-2 base curves can will only be come on from curves a of 1D self-intersection face, and also cannotpolytope; be vertices). we refer sometimesconstruction to of this a construction polytope. as the Note “standard that stacking” the approach 4D to rays Table 6. JHEP02(2019)087 . , 2 ) ), } 14 B − ( i 1) 4.2 (4.3) , D 1 , from the 12 , ∇ contains no 6 − ) and another B ( is of the form B , ) 1) 1) (4.4) correspond to the B , nK ( i − 1 , − model gives the only , ∇ D ( 6 polytopes ∆ are ( 2 (0 , − P , M ) correspond precisely to 0) ∈ O , given the respective base , ). The lattice points of ∆ } 1 n 1 (12 . In the previous subsection , a , 3 1) − . The ,  x ( 2.14 1) } 4 1 , , , , 3 1) 1 ) 0) 6 (figure , , , and − 6 2 6 , ] of the enhanced gauge symmetries 2 (0 − − , y (0 , ( , 59 6 , 1) , ) , − n 1) ( -fibered polytope, the lattice points of ∆ , (0 58 1 − { , , 1 , , 3 , , 1 ) immediately provides a reflexive polytope. 1) 6 2 56 , , − P , ( ) still gives a reflexive polytope in certain cases when the 4.1 1, 2, 3, 4, 5, 6, or 7 – 31 – (1 , 21 , , whose vertex sets of the pt +1) 4.1 2 1) , ( 2) , n 1 , , , 2 , − (0 1 (1 =0 , ) , n ) (6( 0) F , B 3) , , ( i , the set ( , 2 1) (1 lv P , { ( 1 ,  , = (2 4.3.1 } 6 7 , 1) − 6 , , − 5 ) , that for a standard , 4 n , 1 s to vanish to some specified order along a particular base divisor 3 , − n 2 3.4 ( , a 1 , models give exactly the three data points with Hodge numbers H:3,243. pt (6(1+ 1) varieties, and correspond to reflexive 2D polytopes, as we have just verified 2 , , , 1 , 1) (0 , , =0 1 n . In coordinate representation, a lattice point in the top of ) 0) , F , 6 B ( i − (1 , D , corresponding to a physics model with no nonabelian gauge group. We now wish { The presence of an NHC in the base or an explicit Tate tuning can force some of the The bases described by toric varieties with no curves of self-intersection less than Simple examples of polytopes realized in this way are the elliptically fibered Calabi-Yau 6 weak Fano As we will discuss in section ∇ − 14 ( where base contain NHCs, butare those not lattice vertices. points corresponding to the curves in the base that carry the NHCs additional lattice points form theover “top” [ coefficients in the This absence of monomialsstandard in stacking. ∆ gives The rise additionalexceptional to lattice divisors a that points corresponding resolve in the enlargement the singularities of fan of the polytope associated fibered geometry. These 2 points corresponding to thewe constant described coefficients generic of elliptic fibrations overprocedure weak Fano bases, immediately where gives the “standard ain stacking” reflexive 4D polytope,to and consider no how additional this storyeither rays changes to are when an needed there NHC is in a the nontrivial base nonabelian or gauge a group Tate tuning due of the monomials in the Tate form. We saw in section that project to eachthe of sets the different of lattice monomialsare points in thus of the divided ∆ coefficients into 5 of groups the corresponding Tate to form the ( 5 sections where there is aTate gauge tuning. group The arising realization either ofrelationship from reflexive between a 4D Tate tunings non-Higgsable polytopes and cluster in “tops” in these in the cases the arises base toric from or4.2 language. a a general Tate tuning and polytope tops rays polytope (up to latticeand automorphism) the with Hodge numbers H:2,272 in theare KS database explicitly. We now want to describe the generalization of this construction to situations non-Higgsable clusters, the set of vertices ( threefolds over the toric bases with the first set{ of vertices respectively being of the dual polyhedra is an integral lattice point. Thus, as long as the base JHEP02(2019)087 - , 17 1 7 8 , , M 3) 3 , 16 , − 2 (4.5) , 15 ). We P , 3 2 14 , − } ( 13 , , ) algebras, 0 17 12 n 4) and table , pt ( } , 1) associated − 11 7 , , , su 0 14 0 14 1 5 10 , , pt pt 9 2 − , } , , , 8 ( 1 , , 0 14 0 12 0 12 7 is the associated 2D , ) and m 6 ) 3) , pt pt pt 5 } , , , n B , − ( ( i 4 , , 0 11 0 11 0 11 0 11 v 3 4 , so ] in the dual space, which , 2 pt pt pt pt , } , , , , 1 − . The tops of the various 1 , ( 22 } 3 0 11 0 10 0 10 0 10 0 10 , , pt 4 0 2 2) pt pt pt pt pt P } , , , , , 2). Recall that the lattice point , pt 0 9 0 9 0 9 0 9 0 9 0 9 − , 3 0 , 1 pt pt pt pt pt pt 2 , } , , , , , , ) 0 8 0 8 0 8 0 8 0 8 0 8 0 8 − , pt 2 , B ( 2 00 ( 1 , pt pt pt pt pt pt pt . , , , , , , , v ) tops. The tops in table 0 6 0 6 0 6 0 6 0 6 0 6 0 6 2) n , pt 4) ( 2 0 pt pt pt pt pt pt pt − = ( , , , , , , , − , 0 5 0 5 0 5 0 5 0 5 0 5 0 5 su , 3 0 2 , pt 4 pt pt pt pt pt pt pt − pt 1 000 , , , , , , , − ( 0 4 0 4 0 4 0 4 0 4 0 4 0 4 ( , , where the number of primes specifies the level , – 32 – 0 pt pt pt pt pt pt pt , pt 1) ) and , , , , , , , 6) 1 00 ··· 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 j n − ], Vincent Bouchard and Harald Skarke generalized − ( , pt pt pt pt pt pt pt pt , 2 , pt , , , , , , , 7 22 so 1 0 0 2 0 2 0 2 0 2 0 2 0 2 0 2 − ) cases with the results of [ or − ( pt pt pt pt pt pt pt pt ( , n , , , , , , , ) { , ( l 0 1 0 1 1 0 0 1 0 1 0 1 0 1 ( j ). All lattice points in these tops are of level one, and correspond 1) 5) so pt pt pt pt pt pt pt = − pt − Top/Affine Dynkin nodes { { { { { { { 4.5 , , 1 } 6 0 4 − − } } } } ( ( } } } , pt ) algebras. The coordinates of the points ) and ( = 0 3 n ( ) transformation. Note that for higher rank 17 4.4 , , pt su Z contains the lattice point 00 2 , , respectively. In [ 16 ]. We have explicitly calculated a few more cases, including the tops , ) gauge algebras to rank 12 in both cases and list the results in tables 8 ∇ n 15 , pt 21 , ( 000 1 3) imposes the conditions on the dual lattice points ( specifies the “level” of the point away from the fiber plane (see figure 14 , , su 0, 1, 3, 4, 7 0, 1, 4, 4, 8 0, 1, 4, 5, 9 0, 1, 5, 5, 10 0, 1, 5, 6, 11 0, 1, 6, 6, 12 0, 1, 6, 7, 13 pt 2 Tate form { { { { { { { N s projecting to the fiber plane are involved. We list the ones we need in table 13 . When we denote a top, the points with fewer than the maximal number of { , : , l ) pt ∈ 2 12 8 , B The tops of , l ( 1 7 8 9 n v 10 11 12 13 11 ) and and table , polytope grows in the fiber subpolytope direction (as opposed to the level direction), n 10 7 ( , There is a simple and precise correspondence between tunings of the Tate form and = ( ∇ 9 , so 8 0 1 pt tops. This correspondence holdsNHC independent or of whether an the explicitfibered Tate tuning. polytope form corresponds Consider to for an pt example a situation where the standard and more and table lattice space), including higher rank were explicitly obtained from reflexiveand polytope we constructed have cross-checked from the successiveagree Tate up tunings, to a GL(2 the in table 3.2 inof [ table the notion of tops (including thoseto which include may all not have fiber a types, completion and to reflexive they polytopes) classified all such “tops in the dual space” (i.e., the parameter primes over each point areindex; omitted and e.g. implied by thegauge point algebras of have most been primes worked withrank out the in no same the greater previous than literature. eight Tops that for arise gauge in algebras of reflexive polytopes can be looked up for example to affine Dynkin nodes. The rank of each algebra is theare number the of 7 the lattice points nodes inray, minus and the one. 2D reflexive fiber subpolytope adopt the shorthand notation Table 7. are given in equations ( JHEP02(2019)087 . ) 16 2) , } B , } ( 15 00 10 00 9 1 , 6+2, , D , , , , , , , pt pt 14 1 } , , } , , , 00 10 0 10 0 10 0 10 00 10 00 10 00 10 0 8 00 9 0 9 00 9 00 9 13 0 , ≥ − pt pt pt pt pt pt pt pt pt pt pt pt , , , , , , , , } , , , , , 12 0 9 00 9 00 9 00 9 00 9 00 9 00 9 00 9 0 8 0 8 0 9 00 6 00 6 , m · 11 pt pt pt pt pt pt pt pt pt pt pt pt pt ) , , , , , , , , , , , , , , } } 0 6 0 6 00 6 00 6 00 6 00 6 00 6 00 6 00 6 00 6 00 6 00 6 00 6 B is listed in each 10 ( , 00 15 00 16 9 pt pt pt pt pt pt pt pt pt pt pt pt pt v s vanish to these , , , , , , , , , , , , , , ) in the table have pt over that ray. We 8 pt pt n 00 4 00 4 00 4 00 4 00 4 00 4 00 4 00 4 00 4 00 4 00 4 00 4 00 4 , } } , , n 7 a ] after an appropriate ⇒ , pt pt pt pt pt pt pt pt pt pt pt pt pt 14 0 00 15 0 14 00 15 ∇ (4 6 , , , , , , , , , , , , , , 1 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 22 5 pt pt pt pt so , } } , , , , 4 pt pt pt pt pt pt pt pt pt pt pt pt pt , , , , , , , , , , , , , , 00 12 00 13 13 00 0 14 00 13 00 13 3 00 2 00 2 00 2 00 2 00 2 00 2 00 2 00 2 00 2 00 2 00 2 00 2 00 2 , ≥ − 2 pt pt pt pt pt pt , pt pt pt pt pt pt pt pt pt pt pt pt pt } , , , , , , } 1 , , , , , , , , , , , , , 00 1 00 1 00 1 00 1 00 1 00 1 00 1 00 1 00 1 00 1 00 1 00 1 00 1 0 11 00 12 0 11 00 12 12 00 00 13 00 12 00 12 1) and pt +3 6 as expected for a section of pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt , , , , , , , , , , , , , − , , , , , , , , m 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 · n ) 00 10 0 11 00 10 00 10 10 00 00 12 0 12 0 12 . Note that just as multiple Tate pt pt pt pt pt pt pt pt pt pt pt pt pt ≥ − B dual to ∆. Thus, there is a precise (4 Affine Dynkin nodes { { { { { { pt { pt { pt { pt pt { { pt { pt { pt ( 9 v m so coefficients over a certain ray in the ∇ · } , , , , , , , , n ) a 00 10 00 10 00 10 00 10 00 10 00 10 00 10 00 10 00 10 ) vanish to orders at least (0 B ( 6 , and indeed if all the pt pt pt pt pt pt pt pt pt v } } , , , , , , , , , . Indeed, this goes both ways: only when ∇ 00 9 00 9 00 9 00 9 00 9 00 9 00 9 00 9 00 9 00 9 00 9 , a 4 ∇ ⇒ pt pt pt pt pt pt pt pt pt pt pt } } , , , , , , , , , , , , a 0 8 0 8 0 8 0 8 0 8 0 8 0 8 0 8 0 8 0 8 0 8 0 8 1 00 16 3 pt pt pt pt pt pt pt pt pt pt pt pt – 33 – 2), associated with the absence of a certain set of pt } , , , , , , , , , , , , vanish to order 2 over the corresponding } } , , a , 00 6 00 6 00 6 00 6 00 6 00 6 00 6 00 6 00 6 00 6 00 6 00 6 00 6 2 ≥ − 1 6 00 15 00 15 00 15 pt pt pt pt pt pt pt pt pt pt pt pt pt , , , , , , , , , , , , , , a , a 1 pt pt pt appear in 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 } , , , 1 , 0 2 + 5 a pt pt pt pt pt pt pt pt pt pt pt pt pt 0 0 14 0 14 0 14 0 14 , , , , , , , , , , , , , ). (Only the highest level point for each , 00 4 00 4 00 4 00 4 00 4 00 4 00 4 4 00 00 4 00 4 00 4 00 4 00 4 pt pt pt pt pt m is present in } , , , , · pt pt pt pt pt pt pt pt pt pt pt pt pt 4.5 , , , , , , , , , , , , , 0 2 00 13 00 13 00 13 00 13 00 13 ) 0 3 0 3 0 3 0 3 0 3 0 3 0 3 3 0 0 3 0 3 0 3 0 3 0 3 B pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt ( } } , , , , , , , , , , , , , , , , , , v 00 2 00 2 00 2 00 2 00 2 00 2 00 2 2 00 00 2 00 2 00 2 00 2 00 2 00 12 00 12 00 12 00 12 00 12 00 12 00 12 must appear in the polytope pt pt pt pt pt pt pt pt pt pt pt pt pt ) and ( pt pt pt pt pt pt pt , , , , , , , , , , , , , ) algebras. The coordinates of the points } , , , , , , , imposes the stronger condition 0 2 00 1 00 1 00 1 00 1 00 1 00 1 00 1 1 00 00 1 00 1 00 1 00 1 00 1 n 0 2 0 11 0 11 0 11 0 11 0 11 0 11 0 11 0 11 that 4.4 ( pt pt pt pt pt pt pt pt pt pt pt pt pt pt 6 Top { { { { { { pt { pt pt { { pt { pt { pt { pt { pt pt so a } } } } } } } } } } } } } 6 7 7 8 8 9 9 10 10 11 11 12 12 , , , , , , , , , , , , , 4 4 4 4 5 5 5 5 6 6 6 6 7 , , , , , , , , , , , , , 3 3 4 4 4 4 5 5 5 5 6 6 6 , , , , , , , , , , , , , 1 1 1 1 1 1 1 The tops of 1 1 1 1 1 1 , , , , , , , , , , , , , ). The point 1 1 1 1 1 1 1 Tate form 1 1 1 1 1 1 { { { { { { { s vanish at least to orders (0 { { { { { { K n 6 a − ( n 13 14 15 16 17 18 19 20 21 22 23 24 25 lattice points in ∆, canorders the then point the point local correspondence between Tatebase, tunings associated of with the lattice points absenttabulate from a ∆, few and examples the of toric top this in correspondence in table with monomials in O corresponding to theA condition similar that calculation showsrespectively that when the ( point the the same top butnumber of different the (numbers nodes of) by one)coordinate affine are transformation. different. Dynkin These nodes tops match as those the found in ranks [ (which differ from the Table 8. are given in equationstop, ( and the lattice points of the lower levels are implied.) JHEP02(2019)087 ) over 6 , a 4 associated , a ∇ 3 : in standard ) , a B 2 ( i v , a 1 a . ns s ∗ ns 2 3 0 4 4.4 type I I I I ) is a set of lattice points stacked H 2 )) is the only point in the top; while G ]. One particular situation in which Sp(2) group SU(2) SU(3) 3.13 23 (in the X-Y plane). The level (the multiple ) 6 1 , 3 , a , ) forming the extended Dynkin diagram of the 2 4 P 2) 3) 3) 4) , a , , , , 11 3 (in the direction 1 2 2 2 (equation ( – 34 – , , , , ) 0 1 , a 1 1 2 2 B 2 , , , , ( i pt v 0 1 1 0 , , a , , , , ) 1 B (0 (0 (1 (0 a ( i D ord( over nearby divisors and the absence of monomials in ∆ leads 0 2 0 3 0 4 1 00 ∇ pt pt pt pt point the affine node. 0 1 pt in the base and the associated tuning of the Tate coefficients ( ) B A 3D visualization of the lattice points that appear in a top over ( v Some examples of the correspondence between additional lattice points in ) where points are located is indicated by the number of primes. When the gauge algebra ) models, a top over a ray in the base This correspondence leads to a natural association of reflexive polytopes with elliptic B 1 ( i , 3 v , 2 can arise from abetween combination extra of vertices in non-Higgsableto clusters local and interactions Tate between tunings. theadjacent sets The rays of in interplay lattice the pointsleads base. in to the consistent We polytope reflexive consider that polytopes more are in affected explicitly both by in the NHC the and following Tate section tuning how cases. this algebra was studied from the pointmultiple of tops view are of tops possible in forTate [ tuning a configurations, fixed gauge which we algebra discuss corresponds further to in monodromy-dependent section fibrations over toric bases that have Tate forms. Over a given base, various gauge groups otherwise there are additionalgauge points algebra with (cf. table tunings can correspond tocorrespond the to same the same gauge gauge algebra, algebra. the The corresponding multiplicity of multiple constructions tops for also a given gauge Figure 2. P over the 7 lattice pointsof of the fiber subpolytope is trivial over the associated divisor Table 9. with a ray the associated divisor. JHEP02(2019)087 , 4, ∇ m , ) is 3 − (4.6) (4.7) and , leads 4.1 6, and = 1 ≤ 4.1 i 4.3.1 in the base 2 ) , B } , m ( 3 6 , so with these 1) } , v ) − 1) , B . ≥ − ( , 2 1 = 3, the polytope , 1 (4 , v ), those cases where a curve. Other than the 0 /m ) , , , m m defined through ( ) B m 12 ( 4) 1 4.2 (0 , v m , ∇ − over the divisor associated 12 are fine as well because > − (3 1) 6 2 − , , , . a 1 ) with 2 m 3) (the self-intersection numbers of − , − 4.6 , } (1 0 4.3.3 = ( { , 4 and 1) , ) − (0 B = , (0 ( 3 , } 1) , v ) , 1 12) 1) , controlled by the i, j − − 6 ( curve, 6 , , { − a 1 lattice (up to lattice automorphism) associated ) are then bounded by , m – 35 – − 6 B ( − M = (0 , − K ( ) 6 1) , B ( − 2 − 1) ( ); the vertices of the 2D convex polygon are , , } , v 1 is the order of vanishing of (0 ] = 3, or 6 also give lattice points; i.e., when the skipped , 0) 12 ) , 6 ∈ O , , B 0) ( 0 j − 6 /m , , , a , v compatible with the standard stacking can also be understood 12 ) (1 = (1 12 6; furthermore, the NHCs { B (78 ) ( − i − , m − B , ) in v = ( 1 2 0 1) v } { , corresponds to the ) 12, so that the standard stacking polytope 1 , m 2 , all vertices of ∆ come from lattice points associated with monomials B , ( − 1 i 2 v 1 , 3 and v , m 6 { , y 6 − 3 . The first and the third constraints intersect at an integral point precisely − − 1 x , ( 4 has , and constructions with tunings in section { m ) should also be sufficient to give the ∆ polytopes of the models with the − ) gives the vertices of the dual polytope ∆, which is a lattice polytope. Indeed, since there are no points in the dual lattice with 12 , ≥ 4.2 12. This intersection point is a vertex of ∆, so ∆ can only be a lattice polytope 12. Note that 6 F 3 ) | | 2 4.2 B 12. In these cases, the standard stacking construction described in section 4.3.2 − ( | 2 m m v . Choosing local toric coordinates for a set of adjacent rays m We can understand the reflexive polytopes formed in this way in terms of the dual Tate As an example, the reflexive polytope model for the generic elliptically fibered CY over mm 6 so that the ray a 3 curve NHC. Using again the local toric coordinates ( − and is the only reflexivewith polytope the in Hodge the pair H:11,491 in the KS database. tunings and tops described in the− previous subsection. For example, consider the case of the and the dual verticespairs, ( arise from thethis polytope pairs has vertices when with the base the toric divisors are the monomials ( 6 when reflexive. The values of from the bounds on thevertices set of from monomials in in B to the factor 6 inray the is numerators skipped of and therays Det[ first are two NHCs coordinates inof ( extra factors ofthe 2 set that ( ariseNHCs from the difference terms in the numerators. Therefore 4.3.1 NHCs withNow immediately consider reflexive polytopes modelssimplest where cases, the where base theand has NHC a contains non-Higgsable adirectly cluster. single to curve a of reflexive We polytope. begin negative self-intersection This with can the be understood from several points of view. Due In this subsection we describe thecorresponding construction to of F-theory reflexive models polytopes with from gaugeWe elliptic groups give from fibrations non-Higgsable the clusters or construction tuning. section of generic models over bases with NHCs in section 4.3 Reflexive polytopes for NHCs and Tate tunings JHEP02(2019)087 ; ◦ ˜ ∇ as ∇ 2); / = 6 ∇ 3 in ) and , n } , where 3 ∇ ⊃ 0 4 , 4.2 that is a 2 that con- ∗ ) to be the , pt ∇ in ( 0 3 ∇ ; if we took ∆ along with M over the NHC } } 0). This plane ) = (2 , pt , ∩ ˜ 1) 1) ∇ 00 2 0 ∗ is not a reflexive , 1). This problem , − f, g, , ˜ 1 , ∇ (3 ˜ , pt ∇ ˜ , 0 2 − construction of ∆; the polytope ∆ (0 ( , . X,Y,H 3) ◦ , , , pt 5) ) = (3 0) 000 1 11 (2 , − , intersects the vertical line 1 lattice that are needed to , i, j 3) of the line between the s in the dual polytope are 1 , pt singularity; in this case the , 2) − n 00 1 ∇ 2 , , a − N , ( 3 curve automatically has the (the convex hull of the standard (1 (0 2 , , pt IV , − { / plane, at ( 0 1 ˜ 12. For example, take the generic = convex hull( 1) ∇ 3 | 2) , } pt ◦ = n − / indeed has a dual ∆ = { 1) from ( , (0 3 } , , ) , 1 = 0 ∇ 3 , 1), we have a picture like figure 0) , 5 i, j , 3). Considering a 3D slice of H − ( / , it is , 1) in the figure), so the reflexive polytope , { { (1 } 2 , { 12 0 3 , (3) is automatically satisfied, so this actually 2 , 3 , 6 su = (0 , pt − − 0 2 ) , – 36 – . Using the same methodology as in the 1 B ( 2 , pt ∇ − v ) = (1 3). The boundary of the polytope in the 3D slice is 00 1 ˜ ∆ is not a lattice polytope then , 2 describes the tops that arise over the divisors support- top. While the top in , pt , 0 1 . This gives us the minimal reflexive polytope 3 4 with additional lattice points to make it into a reflexive f ∗ 11 pt − ) ˜ ∆. If X,Y,H , ◦ { 3) and ( ˜ ∇ , 1 (( , we see that this is a type 2 is is a lattice polytope. − , 0 2 ) 4 . The boundary of the polytope ). Let us denote the convex hull of the set of vertices defined and the ray ∗ 0 ) top as described in earlier literature, which is obtained explicitly B , pt = (∆ 2 ( 4 ∇ B 4 ( 4.2 2 v f ∇ v ∇ 2) in the top in ), (1 , (3) top, as indicated in the first line of table into a reflexive polytope in a minimal fashion by taking the “dual of 3) and ( 1 , , su 4.1 ˜ 2 ∇ 1 , , which is perpendicular to the described by the set of rays − 0 } , , to form a complete , and its dual by 5 ) F ˜ so that ∆ = ∇ B construction we just described above. = 3 ( 4 = (0 lattice that we are looking for; for any base with NHCs, as we have confirmed ◦ v ∇ ) as the set of vertices, we would have 0 2 ,Y ) by N pt 4.1 is identified with For each of the NHC’s, table This “dual of the dual” procedure adds points in the We can turn 4.1 = 2 ) B 5-curve ( X i via the ∆ ing the NHC, the correspondingthe Tate resulting forms, gauge and algebra. the vanishingthe The orders dual minimal of of top the associated dual with is the in ∆ each case the first top listed. In a number of cases there are other just ( there would be a non-latticecan point be vertex seen ( as arising from− the absence of a sufficient setstacking of polytope) lattice over points in the latter is exactly the lattice polytope. complete the tops associated withother the gauge than symmetries those coming of from a themodel single NHCs curve over in with all self-intersection cases the dual”. We begin bypolytope defining defined the by lattice polytope thethen ∆ convex has hull itself of ain the dual the set ofby integral explicit points computation of in each case, the resulting polytope described by ( by ( polytope. We have topolytope supply monodromy condition for the gauge algebra corresponds to an 4.3.2 Other NHCs:The reflexive rest polytopes of from the the NHCs dual have of the the issue dual that there are fractions in the vertices of the dual therefore the 2-plane passing throughpasses the points through (2 the point associated with a standardpoint stacking from aexample base above, with we a see(1, that 1, the 1, orders 2, of 2). vanishing of From table the tains the fiber polytope v { this corresponds in thetwo vertices polytope (1 to the midpoint (0 has vertices from ( JHEP02(2019)087 . , } 11 0) 1) (4.8) (4.9) , − (4.10) 1 , , 1 3 gives us , to vanish , 0 4 3 6 , − ; in table , a . } ( (0 4 , } , ∇ , giving a new 1) 1) , a 1) 1) ∇ − , 3 , , , 1 1 2 for each top. 0 , , , a , − 2 5 6 0 , , , ∇ , in the base, which cor- 0 4 , a 6 , which is also a lattice , 1 (0 − − C , which has extra lattice a ∇ , ( ( (0 ∗ , , , 0) ˆ , 1) 1) 1) ∆) , 1 , − 0 1 , − , , = ( , 1 2 curve 4 4 , 0 , , which is the second coordinate in , − ˆ , 0 ∇ 4 2 , C polytope that results from removing associated with the generic elliptic (0 − − , ( (0 ( , , , M ∇ 3) , 1) 1) 1) , , 2 , this polytope follows from the standard , 1 1 1 , , 1 polytope − , 5 6 , , (2) over the 4.1 − 2 (0 5 N , , , – 37 – su − 2 ( 3) , − (18 , ( , 2 , 1) , 1) 1). This is achieved by setting 1) 1 , m, , , 2 − − 1 . Calling the new 6 , , , − , B , 1 4 6 , 0 − associated with the generic elliptic fibration over a given toric − , 6 ( ( , , − (0 ∇ ( . , 3) , 1) } , , 1) 2 1) 1 1) , , , , , polytope after the reduction is 1 in the coordinate associated with 0 1 5 1 , ˆ . This gives a large class of constructions for Tate tunings that should , } , , , ∆, we get an enlarged 6 2 6 6 M 6 ˆ ∇ (1 , − − . As discussed in section − − { , 1 , 2 ( , , 6 { 6 F 1 − , − . The global models describing generic elliptic fibrations over the Hirzebruch ( (18 0 ( { , { 0 4.4 = Conv { s over the divisor(s) in = 2. This is a reflexive polytope, identified in the Kreuzer-Skarke database as , showing how this construction works in the context of the global polytopes. While n , constructed as above using the standard stacking procedure and the dual of the ∇ a m B 10 As a simple example, consider the polytope ˆ ∆ = Conv The resulting new ishing orders is Now consider a Tate tuning ofresponds the to algebra the 2D toric vectorto (0 orders ∆. The set of the lattice points that have to be removed from ∆ to achieve the required van- with M:335 5 N:11 5 H:3,243. The dual polytope ∆ has vertices have reflexive polytope analogues in the KS database. fibration over stacking procedure and has vertices given by the set of lattice pointsin that the should be removed fromthese ∆ lattice associated points with certainpoints. coefficients In general,reflexive each polytope Tate tuning in ∆ gives a corresponding top in with the reflexive polytope base dual if needed for NHC’s.polytope. We We take can ∆ produce tothe an be NHC’s additional by the gauge performing dual group a polytopeto beyond tuning removing of in the certain the minimum vertices Tate from imposed description the from of polytope the ∆. model, Using which a corresponds Tate tuning from table considering the different tops overwe each also base list and the thus possible classifying new polytopes vertices that4.3.3 may arise in the Reflexive polytope polytopesWe from can Tate understand tunings Tate tunings in the polytope in a similar fashion. Consider starting in section surfaces that incorporate eachtable of the single-curvein NHC’s this are paper also we focusforms described on (corresponding explicitly the to in systematic the structure construction of of ∆), polytopes one through could tuning also of construct Tate general polytopes by higher Tate tunings that give different tops but the same gauge algebra, as discussed further JHEP02(2019)087 3- − (4.11) . The base, (3) top; 9 =3 su m , this might . F 9 5. 1)) to the top } , − in terms of the 1) 1 with monodromy , to order 2 and an } (see in particular − (2) on the isolated 1 polytope such that 6 , 6 8 = (3) top. There also e 0 − a su ) with the additional , , , 2.3 − su M 0 , 4.8 (8) ((0 (0 1,1,1,2,2 = 3 so 0 3 , { — the standard Tate form 2 (3) monodromy condition is pt 6 as well: in these cases, as 0) } , − from ( × su 3 , as described in table 1 , 4 C 2 − ∇ (3). In this case, the geometry , , . Naively from table − 1 0 su , , 4 and 1 (0 , 6, which would naively be tunings for − (2)) , , 1 4.3.1 3-curve are ) with a tuning of { , and the . This multiplicity of tops was also dis- su 3) − } , (3) (indeed, we know from the presence of 2.7 ∇ = 4 2 ∆ , su (3) top: adding 1 m (2) top over , su su f, g, curves in these cases look like those appearing = dim( – 38 – (0 respectively, and the vanishing orders along the for { associated with vanishing of , ) and ( 1 , 4 } s m f 2 3) 4 , 2.6 h , − -polytope , 2 ) is already a reflexive polytope model of the generic 2 IV 3 g ∆ given by augmenting , N , , 3 2 ˆ 3-curve become ∇ , 4.11 − 2 , − , 1 1 in terms of − { ( } , (2)) = 1 , 4 } , 3 3) su 2 , , , 2 2 2), which gives the 2 , , , { (2)” top; however looking explicitly at the Tate form associated to 2 (2) is not possible in this geometry.) In section 0 1 , in equation ( , , 1 su su 1 , (1 ∇ = rank( 1 { − . The tops over the that we found at the end of section { 1 , , and 6 , , } n 1 2 0 a ]. These different tops correspond to distinct Tate tunings of the same gauge h =4 m 23 , pt F 0 1 = Conv . Analogous situations arise for the NHCs pt ), we described two distinct Tate tunings for s ∇ { 3 NHC that 4 . In these cases, however, the gauge algebras are actually As an example of this phenomenon, consider the generic model over the In general, we find that the correspondence described in the last few subsections be- − 4 -curves are IV f , m 2 curve: ∆ 2 discussed above, CY over in the literature for− gauge algebras g additional monodromy condition, andexists the a “top” polytope is model agives with non-standard another polytope the model “usual” on M:347 the 7 Tate N:13 side 6 thisthe model H:5,251, vanishing can which orders be has obtained along thefor by standard the the reduction of the five sections satisfied — hence the gaugethe algebra is indeed table matches the alternate Tate form for This is already acurve reflexive polytope, M:348appear 5 to N:12 be 5 H:5,251, anthe with polytope “ the ∆, top the over vanishing the orders along the model depends upon some monodromycertain condition, cases which by may the be Tate tuning. satisfied automatically in One thing that wesponding have models found in in thedistinct considering KS tops the database that variety is can ofcussed that arise Tate in for in tunings [ the many and gaugealgebra. the algebras In corre- many there cases are these multiple arise in situations where the gauge algebra in the Weierstrass tween Tate tunings and tops immediatelyconstructions. provides reflexive There polytopes are for several mostremainder subtleties Tate tuning of in this this section. construction, which we elaborate in4.4 the Multiple tops lattice point (0 resulting polytope corresponds to theThe example Hodge M:329 7 numbers N:12 from 6cancellation the H:4,238 calculation in polytope in the data equations KS ( database. are− consistent with those from the anomaly This gives the reflexive polytope JHEP02(2019)087 7 ). 15 12 − ≤ 3.11 , they m ∇ form the ≤ ∇ , the lattice . The extra 6 ∇ algebra over e 7 formula ( , e 1 , , there are lattice (8) , corresponding to 1 ) NHCs are given in h ∇ B so ( polytope ∆. Another bases for 0 m D − m M F is enhanced, associated with 6 a may also give the } 5 , 3 , 3 , 2 , 1 . Specifically, in the cases where there are { 0 1 pt , there are also generic polytope models over 3 F – 39 – 8 curve. In this case there is no monodromy issue, − polytopes precisely correspond to the reduction in Tate tops and Tate vanishing orders of 6-curve, giving a further reduced ∇ 6 − e on a , 7 and the resolution of the singular fiber associated with a tuned e (8) in addition to just ∇ so ) B ( v polytope ∆. M ) associated with the “standard stacking” with all possible tops. Proceeding top and corresponding reflexive polytopes. . The details of the corresponding Tate forms for the 7 ]. When the gauge algebra is non-trivial over a divisor e 4.1 . Note in particular, that in addition to those models mentioned above, there is a 10 in addition to the monodromy construction and the standard construction discussed 59 6 , 11 that have the usual F We will not deal systematically with the explicit triangulation of In the analysis in the remainder of this paper we focus on classifying the possible elliptic In addition to multiple tops associated with monodromy conditions in Tate tunings, 58 However, note that the same Tate vanishing orders 6 , 15 4 contribute to the secondThe and second term third corresponds terms, toto components respectively, that the in miss third the Batyrev’s term hypersurface, and arise contributions when the singularitycurves where is there not is also resolved charged by matter. a toric divisor but rather points in the top over no lattice points in thepoints top in lying the in the topDynkin interior that of diagram do the of not 2-dimensional the faces liearise gauge of in in algebra. the the resolution interior Thesepoints of of correspond the lying the to corresponding 3-dimensional in the singularities. faces exceptional the of divisors However, interior when that of there the are 3-dimensional lattice and the 2-dimensional faces of the resolution of the Calabi-Yauship threefold, between but extra rays make in someor comments non-Higgsable here gauge on group. the relation- Manyin of [ the details of this correspondence were worked out fibrations constructions through the setclassifying of different Tate tunings. reflexive One polytopesvertices could by ( also, however, considering imagine allin ways this of fashion augmenting would theeach require possible set a tuned of systematic gauge way group. of identifying the complete set of tops for interesting case of multipletype tops of that Tate tuning/top arises for but in a these second tables Tate tuning isa where second the the possibility degree of of vanishing a of second table third polytope model associatedover with the Hodgeabove. numbers of This third the possibility generic involvesstandard a elliptic construction further along fibration the specialization of the vanishing orders of the We have not attempted aa systematic range analysis of of all possibilitiesHodge possibilities, simply numbers but in and we analyzing have associatedthe encountered the possibilities Tate polytopes that tunings arise, of for wehave the the list the KS the dual Hodge database structures numbers polytopes. of with ofin the generic fixed table To polytopes elliptically give in fibered a the CYs KS sense over database of that F lattice points in the tops ofmonomials these of the there are also other Tate tuning/top combinations that can arise for certain gauge groups. conditions satisfied. Just like the case for JHEP02(2019)087 } 6 0 } 0 7 } ”) (7)”) 0 5 } , pt } 4 0 5 5 0 0 5 , pt } f so 0 5 0 5 , pt (” ”) 00 4 (” , pt , pt , pt 2 , pt , pt 00 4 4 00 00 4 } } } -curve } } g 0 4 0 4 0 4 0 4 0 4 n 0 4 0 4 , pt (2)”) (” , pt , pt , pt 000 3 − , pt , pt , pt , pt , pt 00 3 3 00 00 3 , pt } } , pt } su 0 3 0 3 0 3 00 3 00 3 0 3 0 3 0 3 0 3 0 3 , pt (” , pt , pt , pt , pt , pt , pt , pt , pt 0000 2 , pt , pt , pt , pt 000 2 2 000 000 2 } , pt 00 2 00 2 00 2 00 2 00 2 0 2 0 2 0 2 00 2 0 2 0 2 (affine node) (affine node) (affine node) , pt , pt , pt , pt , pt , pt , pt , pt , pt , pt , pt , pt , pt } } } , pt , pt 0 1 0 1 0 1 0 1 0 1 00 1 00 1 00 1 000 1 00 1 000 1 000 1 000 1 000 1 0000 1 1 0000 0000 1 (6) 1 pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt Top over the { { { { { { { { { { { { { { { { { { ) 8 e 56 1 2 6 6 6 4 (3) (3) (8) (8) f e e e so so su su trivial trivial trivial w/o matter w/o matter enhanced on -3 7 7 enhanced on -4 ): (1) Trivial third term: there is no lattice point (w/ matter e e – 40 – 2 4 (9) enhanced on -4 Gauge symmetry 7 f g NHC -12 curve: e so 3.11 2 1 0 3 3 3 4 4 4 4 5 6 6 6 7 8 8 12 base F F F F F F F F F F F F F F F F F F n formula ( F 1 , 1 h models are divided into two types according to whether there is a 3 3 4 1 3 1 2 1 ∇ ). Mult. in KS 4.4 Polytope models associated with generic elliptic fibrations over the Hirzebruch surfaces , as well as all other models with the same Hodge numbers. Alternate constructions 12 , 8 In summary, ,..., (3,243) (5,251) (7,271) (7,295) (9,321) (10,348) (10,376) (11,491) 1 , Hodge pair 0 term: there are latticegive rise points to the lying additional in Tateinvolved forms the with we have monodromy interior described. conditions, of For there example, two-dimensionalthe are in faces. gauge Tate the forms algebras gauge These of algebras by lower satisfying cases singular degrees, the fiber which additional is achieve monodromy (partially) conditions resolved by automatically. deformation. The Therefore, there are fewer exceptional nonzero third term in the lying in the interiorfrom of tops any (the nodes two-dimensional of face. thenot Dynkin Gauge lie diagram in are algebras interior given can by of the facets), beno lattice which points read additional are in monodromy those off the in condition, top directly the which thatalso literature. do again correspond match The to those Tate exceptional forms in are divisors the those resolving literature. with the singular The fiber. nodes (2) Non-vanishing third by a non-toric deformation, sohappens the exactly Dynkin in diagram those isautomatically gauge satisfied. not algebras fully with visible an from additional the monodromy top. condition This that is Table 10. F include multiple tops, some due totunings monodromy conditions (section in Tate tunings, as well as rank-preserving JHEP02(2019)087 8 8 8 2 2 (2) (7) e e e 6 7 7 8 4 g g (3) (2) (2) (2) (8) f G e e e e so su ⊕ ⊕ so su su su su (9) over ⊕ ⊕ so 10) 10) 10) , , , , 4 5 5 5 f 4) 6) 8) 8) 9) 9) 6) 6) 3) 3), 2), 4), 3), 6), ∆) , , , 10) , , , , , , , , , , , , , , , 2 3 4 4 5 5 3 3 2 (4 (4 (4 2 1 2 2 4 5 , , , , , , , , , , , , , , , f, g, (2 (2 (3 (3 (3 (3 (2 (2 (1 , and (1 (1 (2 (1 (2 ( (4 3 } } } 5 5 5 F , , , , , , , , that do not change the 4 4 4 } } } } } } } } } } } } } } } } } } } } , , , 2 3 3 4 4 4 5 5 6 5 5 6 3 3 2 2 1 2 2 4 3 3 3 , , , , , , , , , , , , , , , tunings of certain gauge ∇ , , , , , , , , 2 2 2 3 2 3 3 3 4 3 4 3 2 2 1 1 1 2 1 2 2 2 2 , , , , , , , , , , , , , , , , , , , , , , , 1 2 1 2 2 2 2 3 2 3 3 3 2 2 1 1 1 2 1 2 1 1 1 , , , , , , , , , , , , , , , , , , , , 1 1 1 2 1 2 2 2 2 2 2 2 1 1 1 { { { 1 1 1 1 2 3 curve in , , , , , , , , , , , , , , , , , , , , 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Tate form 1 1 1 1 1 − { { { { { { { { { { { { { { { { { { { { (6) 1 (6) 1 (6) 1 pt pt pt } } } 00 4 rank-preserving over the 0 5 0 5 0 5 0 5 , , } } 2 } } 0 3 0 4 0 7 0 6 00 4 00 1 00 1 000 1 0 2 0 4 g , pt , pt , pt , pt , pt . pt pt pt pt pt pt pt 00 3 00 4 00 4 00 4 pt pt pt 0000 1 none none none none none { { none, none, none, { { pt B – 41 – pt , pt , pt , pt 000 3 000 3 000 3 Possible vertices , pt , pt , pt . 0000 2 0000 2 0000 2 } } 0 6 } 0 7 } } that there are 0 5 } , pt , pt , pt 0 5 0 5 0 5 , pt , 4.3.2 1 , pt } } } 0 5 (6) 1 (6) 1 (6) 1 } , pt 0 5 0 4 0 4 0 4 , pt , pt 0 4 00 4 , pt } } } } 00 4 00 4 pt pt pt 0 4 0 3 0 3 0 3 0 3 , pt , , , { { { , pt , pt , pt , pt } } 00 4 , pt } } } , pt 0 4 0 3 0 3 0 3 0 2 0 2 , , pt , pt 0 3 0 2 0 2 0 2 , pt , pt , pt , pt , pt 000 3 00 3 00 3 } 00 3 0 2 0 2 0 2 0 2 , pt 0 1 is given in appendix , pt , pt , pt , pt , pt , pt 00 3 , pt , pt , pt , pt 00 3 00 2 00 2 0 2 0 1 0 1 , pt , pt , pt 0 2 pt 2 0 1 0 1 0 1 , pt Tops , pt , pt , pt , pt { F . A detailed example of the polytopes associated with different tunings 00 2 000 2 000 2 pt pt 0 1 00 1 00 1 00 1 0000 2 , pt pt pt pt , pt , pt , pt 4 , pt { { , pt { { { 00 2 000 2 pt pt pt pt 00 1 F 000 1 000 1 00 1 , pt , pt , pt { , pt { { { pt pt pt pt 000 1 over , pt , pt 0000 1 0000 1 { { { { (6) 1 pt 000 1 2 pt pt 0000 1 { pt Tops over NHCs and the corresponding Tate vanishing orders. In each case the first g { { include rank-preserving tunings of pt { pt { { 10 and 4 curve in 3 Finally, recall from table − -9 blown up at 3pts -11 blown up at 1pt -3 -4 -5 -6 -7 -8 -3 -3 -2 su -2, -2, -2, -2, -3, -10 blown up at 2pts -12 NHC associated with further Tate tuningsHodge on ∆ numbers and from additional the topsin generic in table elliptic fibration over athe given base. Theof polytopes listed divisors in the top, inalgebra which counterpart. the “Dynkin diagram” would seem to be the loweralgebras rank that gauge leave the Hodge numbers of an elliptic Calabi-Yau unchanged. These are also Table 11. example is theconstruction top described and in associated section minimal Tate tuning associated with the “dual of the dual” JHEP02(2019)087 , } 2), 1 , − , (4.12) (4.13) 1 − 1), (5 , . We wish , 1 } 6 − , , , } 4 1 , 4), (2 3) − 2 , , , , 2 3 2 , , 1 − 1 , , { 3), (7 2 (0 , , , − } , 5 3) 1 , , 3 − 2 , 2), (5 , , over each curve set to be the 2 , 6 0 , , } 2 − , 1 n , , } | ). We consider Tate tunings that 1 1 − { ( 1) 1), (3 3 . As discussed there, we can perform j,n − , , , c , − } { 2 , 3) (1 4 1’s: 2.5 , 0 { , − , 2 − , 3 = , 0 , 2 (figure , } 1 2 ) − , } − (0 B , 2 , ( i , 3 1) , 1 v , 1 0) { – 42 – − , { {− ( 1 , , − } 3) , 0), (0 , 0 , 2 , 1 , . We then carry out the Tate-Zariski iteration procedure 1 (0 − 6 curve: { − , − , 11 } 1), ( (0 , − , 3) in each of these three cases, we have the vertices from the fiber 1 , 2 − , ∇ 2 , (2, 2, 2, 3) { 1), ( (5 , − 2: (3, 1, 1, 2), (2, 0, 2, 3) , 3) { , − 2 , , 2 3: 4 0), (0 − − , , , , and the discussion in the preceding subsection, we see that there are three different 3 2 6 with three choices of different possible top vertices. (7 { 11 − − − For the polytope Let us consider first as a concrete example the generic model over the base with • • • 1), (1 , and vertices from the tops of the NHCs the vertices from the base, which come from the (3 keep the gauge grouptable the same as inpossible the Tate tunings generic over model, the to determined know by which the of NHCs. thesewhich three From corresponding tunings leads polytopes to exist. a consistent Tate-Zariski decomposition, and not all of theglobal possible model. Tate tunings over a given curvetoric in curves the of base self-intersection are numbers for consistent which with the a toric rays take coordinates of this in generalYau but as threefold we constructed see in later,analysis of this at combined least way tunings the arise through Hodgeour Tate-Zariski from systematic numbers is enumeration a of the of every essential polytope Tate elliptic analysisillustrate tunings in the Calabi- we that issues carry should the that out have KS corresponding can in arise polytopes. database. we To give This a couple of simple examples here, where one but a Zariski decomposition, withinitial the values we initial want values inand of table if the Zariski decomposition withgroups the does desired not vanishing values exist, and thereZariski corresponding will decomposition gauge works not out, be there a is a corresponding corresponding polytope polytope. model. We do In not general, have a if proof the A final important issueglobal that models associated we with must polytopes considerall is whether in combinations given attempting of a generic to Tate modelallowed tunings systematically global over model. a construct that given This base, are dependsthe each on Tate-Zariski the possible decomposition global discussed structure locally in of section can the base be and can combined be into tested by an 4.5 Combining tunings JHEP02(2019)087 . = , , , . , , 11 } } } } 0 5 1 }} 0 }} 0 }} (4.14) . The , , , 0 0 0 } 0 , 0 , 0 , , pt , , , 0 0 0 00 3 0 , 0 , 0 , , , , 0 0 0 pt 1 , 0 , 1 , { , , , 0 0 0 1 , 0 , 1 , -3, -2, -2, -1, -6, 0 { 0 { 0 { { 1,2,2,3,5 . , , , { { { { ∇ , } , } , } } } 4 } 5 4 , , , 0 0 0 , , 3 , 3 3 , , , 0 0 0 , , 2 , 2 2 23, 107, , , , 0 0 0 { , , 2 , 2 2 : , , , 0 0 0 : we start with the initial 1 , , , 1 , 1 1 1 [2] { 0 { 0 { 0 h , , , { { { 6 curve: } , } , } , 6 replaced by } 0 } 0 } 0 − , , , 0 0 0 − , 0 , 0 , 0 , , , 0 0 0 0 , 0 , 0 , , , , 0 0 0 : we start with the initial configura- 1 , 1 , 0 , , , , 0 0 0 [1] 1 , 1 , 0 , { 0 { 0 { 0 , , , { { { } , } , } , } 1 } 1 } 1 , , , 0 0 0 , 1 , 1 , 1 , , , 0 0 0 – 43 – , 1 , 1 , 1 , , , 0 0 0 6 curve. This works as well and gives the generic , 1 , 1 , 1 − , , , 0 0 0 , 1 , 1 , 1 0 { { 0 0 { , , , { { { , corresponding to no additional top vertex from table , } } , , } } 2 } } 2 2 } 4 , , , 3 3 3 , corresponding to the additional top vertices , 1-curve in the base corresponds to a vertex of 2 , , 2 2 , } 3 , , , 2 2 2 over the − 5 , 2 , , 2 2 , , 2 , , , 2 2 2 } 3 , 1 , 1 , 1 , , 2 , , , 1) 1 1 1 2 , , 1 , 1 , 1 , 6. 6. , 1 1 { 1 { 1 1 { . Each 2 − − { , , , , { { { , 2 } , } , , } }} 1 , 3 } 3 } 3 } { on on ) but with the vanishing orders along , , , 2 2 2 (4 2 , 2 , 2 , } } , , , , 1 1 1 5 4 2 , 2 , 2 , , , 0) 4.14 , , , 1 1 1 3 3 , 1 , 1 , 1 , , , 0 , , , 1 1 1 2 2 , The toric fan of the base of a generic model with small 1 , 1 , 1 , , , 1 1 1 1 2 2 , { {{ {{ { {{ { , , 1 (2 1 where each curve{ still has their suitable Tate vanishing orders, which persist as { polytope model M:147 12tion in N:35 ( 13 H:23,107 configuration after iteration becomes where each curve{ still has their suitable Tate vanishing orders, which persist as After the iteration procedure, the configuration becomes This construction leads tothe generic a polytope consistent model Zariskiconfiguration M:148 decomposition, 11 which N:33 gives 11 rise H:23,107 to Tate tuning Minimal tuning We now consider each of the tunings in turn over the 2. 1. Figure 3. -1, -2, -3, -1, -1, -1, -1 JHEP02(2019)087 − , , , , by // } } }} 0 3 }} 4 0 , , 12 0 , 0 2 , 0 − , , 0 , 0 0 , 0 // , , 1 curves in 0 , 1; there are 0 1 , 0 − , , 12 0 − , 0 1 , , 0 − { 0 { 2 (see figure { , , { 8 curves). There , // − } , } } − , 6 6 } }} 0 2 12 , , 0 0 , 4 4 , , 0 − − , , 0 8 , , 2 2 , 0 3 − // , , 0 , , 2 2 , − 0 1 , , 12 0 , , 1 1 , 1 − 0 − 0 { { , { − , , { 2 , , // , } } } 3 are disturbed. Hence, unlike 5 − 3 } 0 0 , 12 , , 0 − − , 3 2 , 0 , , 0 − , , 0 2 1 , − 0 , 0 0 , // , , − − 0 , 2 , , 1 , 0 0 12 , , 2 3 0 − , 1 , 0 , 0 − − − { { 0 1 { , , , { , 2 // − } } , } , 3 } 1 − 0 8 12 , , 0 , , 2 , 1 2 − 0 − , , 0 – 44 – , , − 1 , 1 1 0 // , , , 0 , 1 1 , 1 − 1 00 1 00 0 12 , , , 0 , − 1 , 1 2 pt pt 0 − { 0 { { − , , { , , // } , } } 3 } 3 2 3 , which would correspond to the additional top vertex 12 , , 3 − 8 curve. Analyzing the structure of the polytopes, however, (6) 1 (6) 1 , 00 4 00 4 0 6 } , 2 2 , 2 − 6 − , , 2 . pt pt 1 pt pt pt , , , 2 2 y . This does not give a consistent polytope. The iteration of 2 // − 4 , , 2 } , , , , v , 1 1 1 base where there is a generic polytope model of vanishing order 12 5 2 , , x 1 1) , , , 1 1 v 6, the third Tate tuning and corresponding top realization does not 1 − − 6 2 − { 1 { , (-3 in [-3 -2]): ([-8]): ([-8]): (-3 in [-3 -2]): ([-8]): { , F , − , , { H:416,14 H:416,14 ([-11]): ([-11]): , 1 // 1 0 } , } } , { 7 6 } 3 3 − 2 on 13 153 156 162 13 153 162 11 1 , , , 3 , , B B B B B B B , 2 2 3 } − 1 , , 2 6 , (2 − D D D D D D D 2 , 2 , 1 // , { , , 1 4 , 2 1 , 1 , 1 12 = , , 1 − 2 (c) (c) , (a) (a) (b) (d) (b) Vertex contributions from the baseVertices and from the NHC fiber tops are the same as the first case. A vertex from the 0-curve// in do the not base. contribute to InVertices vertices. particular, from note NHC that tops all and vertices 1 , 1 , , 1 } 1 2 2 {− 7 0 { • • • • • {{ { {{ , , − 1 pt , { exist for this base. where the vanishing ordersthe over case the of NHC the becomes { the initial configuration 14 1 M:29 7 N:575 M:26 6 N:576 Tate tuning , As another illustrative example, consider the polytopes associated with Hodge numbers − we denote the sequence of curves 2. 1. 3. tunings possible over each , 416 7 tuning/top structure. Naively onee might expect four models,we since find: there are two different 12 // in total 163 toric curvesare in only the two base, polytope withmodels curves models of 153 in the and CY the 162 with KS being generic database gauge the group with over H:416,14, the given and base, both with different give detailed polytope Tate H:416,14, which match{ those of the generic elliptic Calabi-Yau threefold over the base JHEP02(2019)087 } } 2 0 5 (6) 1 6 − , pt 3 , pt , 00 4 3 ,-8,-1,- , 1 m > 30 2 , pt − , 00 3 1 , exceptions { 1 , , pt even, starting 1 there are mul- 000 2 h n s are powers of a 6 i ,-2,-2,-3,-1,-5,-1,-3,- e , pt a 1 0000 1 ) gauge algebras can − 20 ). We discuss in this n pt and n { ( (2 3 ,-11, ; in the second model, so 1 so su } − 0 6 ) is ,-2,-2,-3,-1,-5,-1,-3,-2, 1) corresponds to the zero curve, 1 , 2, but all curves with . This fact can be seen in the , pt 156 1 − 0 5 − B ∇ − 10 D , pt ) gauge algebras 00 4 n ) gauge algebras with ( , pt n ([-8]) is not a vertex point. 00 3 ( so polytope so 156 , pt 8-curve ( B 000 2 N greater than − 2 3 4 5 6 D , the monodromy condition for the weaker Tate 8-curves; however there is no model of top - - - - - , pt m ) tunings and the associated reflexive polytopes, 4 – 45 – − 0000 1 n ( over . This matches with the observation that there is pt so { 00 4 156 -12,-1,-2,-2,-3,-1,-5,-1,-3,-2,-2, pt { B 10 D ) is , but the entries in that column are not always vertices, - , for each of the over both 4 162 11 416,14, B over { , for some gauge algebras such as } 0 5 D 1 } , 0 6 1 . Two red dots in the fan diagram correspond to points in the tops: 4.4 , pt h , respectively; these points thus do not correspond to rays of the base fan. 00 4 ∇ , pt 20 0 5 153 - , pt B 00 3 , pt D , we find that both kinds of Tate tunings of the 00 4 increases, since additional rays can expand the convex hull of the base 6 , pt of e 1 , , 000 2 (8)) involving a monodromy condition. (Note that these forms in the table , pt 00 1 1 , where the five curves corresponding to vertices of the base are in boldface, and are 3 00 3 . } h pt so su 0 , pt The toric fan (arrows indicating rays are simplified to points for clarity) of the base of a , pt 156 30 0000 1 000 2 B and - = 8, there are two distinct Tate tunings that realize the algebra, with one (both in pt D { 13 n , pt As for As can be seen from table Note that these models also illustrate another point: a vertex of the base can only come B 0000 1 D pt tuning can be realized automaticallysingle when monomial, the corresponding in leading the termsset polytope in of language certain to lattice a points condition contain that only the a associated single elementarise with in appropriate corresponding multiplicity properties. polytopes in the KS database, corresponding to the usual condition with the case of expand on earlier versions of thethese possibilities.) table appearing As in discussed the in literature, section which did not include all As described in section tiple tops associated withmonodromy distinct condition. Tate tunings, This where alsosubsection one some occurs particular tuning for aspects involves the of anwhich gauge have additional algebras some unique features. in the third columnthough of they table are alwaysfirst lattice model points in in the the second example: 4.6 Tate tunings and polytope models of along from curves with self-intersection number will not necessarily be vertices.increase Though as this generally ispolytope. the case Also, for a small vertex associated with a top can only come from those possibilities listed In the firstwhile model, over the theit top second over is the ( { first no corresponding Zariski decomposition — the vanishing orders can not be 2,-3,-2,-1,-8, denoted by black dots in thewhich fan is diagram. also The a point vertex atof the of top ( Figure 4. generic model with large 2,-2,-1,-12//-12//-12//-12//-12//-12//-12//-12//-12//-12//-12, JHEP02(2019)087 , , } } − } 0 3 2 , and , , {· ⊕ 4 // (1). 2 2 4 , , − 12 B sp 2 , 1 (9) and , , 1 4 curves. D − 1 0 ⊕ so − , the Tate − , , 5 , ; by requir- 1 1 // on 4 } B { { of the gauge (1) 0 − } , , 12 D , (10) algebra on } sp , 5 } } 4 4 1 , 0 4 − so , 3 , ⊃ − , − 3 , , 0 3 , and , by global symmetry // 3 , 1 , . The Tate tuning 4 2 , 0 2 3 (2) 4 , − 1 , 12 , − B B , 1 , 0 , 1 sp (1), so it is not possible , 4 1 , , 1 ⊕ ·} − D D 1 1) gauge algebra cannot { 0 1 ]; when the gauge algebra − sp { { , {− // − {{ 47 1 , +4) consider the sequence of indeed must carry the gauge (10) } 12 n − on k 0 , 4 so , (2 4 − B (4 (8), which has only monodromy 0 }} − , so ⊕ (7) is possible over a given curve, D 0 so so , // 0 , 1 . Without taking into account the , so 4 , which gives the gauge algebras 0 12 (1) }} , − {− 0 }} or 0 , − sp , 2 1 2 0 , {{ − g ⊕ // , 0 , 0 , +3) and 4 , 12 0 k 0 , (9) algebras, there is really a − (11) is not possible on , 0 ; explicitly analysis of the monomials, however, − (10) (1)’s. Thus, , (4 (1) gauge algebra) on 0 , . There is only one model in the KS database so – 46 – 1 so { 0 so sp so sp // − { . There is not a second polytope with the same ( , ⊕ ·} , , 12 } ⊕ } 11 4 } } 5 4 6 2 , , (9) − − , , 3 , 3 (1) 3 1 , , so 1 1) tuning is allowed, there is not generally a polytope , , 2 2 // sp 3 − 1 , 4-curve. This matches with the observation that the ab- ⊕ , − , are indeed , , 1 1 12 ; by requiring vanishing orders of 0 − ⊕ , M:36 9 N:467 9 H:339,21, which corresponds to tuning of 3 0 , , 6 n , } , } 1 (1) 1 ; the algebra next to two − B 0 0 0 { {− { (2 , , { { , 4 sp (9) D 2 (10) in table 1 , , B so ⊕·} } ⊕ } 23 − so − so 2 , , D 5 , 339,21 , , 2 and 4 (3) { 2 (9) 3 , the other vanishing orders are forced to − 2 , − , 335 sp 5 ) is generally not possible, though the higher Tate tuning generally is. , {· ⊕ 1 so , B 3 { 1 , n B 1 , ⊕ D 0 ⊕ (10) monodromy condition is automatically satisfied. This can also be un- 4-curve is 1 − . − D , (2 , 1) can be realized on one polytope over a given curve, then the monodromy , , − 1 4-curve, the maximal global symmetry algebra is so 1 3 so (1) { − (12) { 13 12 − on − , , n sp so − , } } } (10) along the last (10), for which the maximal global symmetry algebra is 1 (2 6 2 4 ⊕ ⊕ and // , , , so so so 3 1 3 {− , , , 12 (1) (9) s, or not. By the same token, the gauge algebra 12 (9) to appear on 3 2 2 i − , , , (9) on a , since the In the examples just given, on certain curves the For a similar example, for tunings of To illustrate these points weFor give a a first few example, examples. consider a chain of curves sp so a so 4 0 0 1 // , , , ⊕ so B 2 0 1 1 to gauge algebras along the last Hodge numbers correspondingalgebra to the weaker Tatesence realization of multiple data in the KS database for a given tuning is due to the existence of the ically satisfied, andThese one facts has can to bethe use two seen the realizations in higher of with contrasting Tate the the tuning Hodge polytope to pair the models, guarantee generic the for model condition. example,12 of tables arise, and thetioned lower above, Tate when tuning the in with the the KS monodromy database condition with is the same realized. Tate tuning As and men- the monodromy condition automat- { g constraints. Examples of these tunings in the context of global constructions are given in curves { derstood from the perspective ofis global symmetry constraints [ for algebra { monodromy conditions, it{· ⊕ would appear inshows that this while case thatD the enhanced algebras were realizations, can only bewhich realized essentially reduces when the neither appearance of this algebra to the NHC structure of ing Tate vanishing orders vanishing orders on each of the curves become algebra realization of This basically corresponds geometricallyTate to model the with question the ofthe weaker whether vanishing condition the has minimally the tuned appropriate single monomials in that a global Tate-Zariski decomposition is possible. We also note, however, that when the JHEP02(2019)087 , − } 21 // , 12 339 − , , { , , , , } } (10) on the // } } } } ⊕ · ⊕ · so 12 ⊕ · − (9) (10) (9) // so 1,1,2,2,4 1,1,2,2,3 , which has a good so so 1,1,2,3,5 1,1,3,3,5 1,1,2,3,5 1,1,2,3,4 12 ⊕ { { } , , { { { { 6 , , , , − ⊕ ⊕ , } } } } } } (1) 3 }} }} }} }} }} }} ⊕ · // , (1) (1) sp 3 12 (9) by the global symmetry , sp sp (8) 1 ⊕ − so is not allowed as mentioned. , so ⊕ ⊕ 1 // 1,0,1,2,3 1,0,3,2,3 1,0,1,2,3 1,0,1,2,2 1,0,1,0,2 1,0,1,0,0 { . The Tate vanishing orders on the { 0,0,0,0,0 { 0,0,0,0,0 { 0,0,0,0,0 { 0,0,0,0,0 { 0,0,0,0,0 { 0,0,0,0,0 12 (11) , , , , , , ⊕·} } { { { { { { , and the Tate tuning along the } , } , } , } , } , } , 0 (10) (10) − so , } } } } } } ⊕ · ⊕ (9) 4 so so // − ⊕ ·} (12). For example, we can tune an by requiring Tate vanishing orders of so , ⊕ ⊕ ⊕ ⊕ (8) 11 4-curve does not exist in the KS database 1 3 ⊕ so − so − F − (9) 1,1,2,3,4 1,1,3,3,4 1,1,2,3,4 1,1,2,3,4 1,1,2,2,4 1,1,2,2,3 (1) (1) (1) , (1) // 4 { 1,1,2,3,5 { 1,1,2,3,4 { 1,1,2,3,4 { 1,1,2,3,4 { 1,1,2,2,4 { 1,1,2,2,3 so , , , , , , are listed in each of the three blocks. In models sp sp sp { { { { { { − } , } , } , } , } , } , 12 sp 4-curve in the model M:38 8 N:466 8 H:338,22. } , ⊕ } } } } } } ⊕ ⊕ ⊕ ⊕ · ⊕ 1 − ⊕ giving the two different Tate realizations of the (9) there. On the other hand, there are two 21 {− − , – 47 – , (1) (11) and so (9) (9) (9) (8) are indicated. All polytope models in the KS database (9) 4 (12), we need to use ⊕·} sp } 339 so − so so so so so 0 { , 1,0,1,2,1 1,0,3,2,1 1,0,1,2,1 1,0,1,2,1 1,0,1,0,1 1,0,1,0,0 so , , 1 ⊕ ⊕ 1,0,1,2,4 1,0,2,2,3 1,0,1,2,3 1,0,1,2,2 1,0,1,0,2 1,0,1,0,0 (9) 4 } {{ { · ⊕ { {{ · ⊕ {{ { {{ { · ⊕ {{ { {{ { · ⊕ − (10) on the last − , so 22 , (1) 3 , so 1 (10) ⊕ − sp − , giving an (9) on the last so , 338 2 ⊕ 4 (1) { } so − , ⊕ − , 4 } sp , (9) 2 , 1 3 ⊕ 23 − so (1) , , − , , both the weaker and the stronger versions of the tuning of , 2 1 } sp 4 , − 225 1 (10) {·⊕ 22 − , { , ⊕ , , so 1 1 12 4-curve. On the other hand, there is only one model with Hodge pair { ⊕ 338 − − (9) 3-curve of the generic model over {− { − // so (1) , which gives rise to M:328 8 N:18 7 H:8,242 in KS database. Then to get a 12 sp } An example contrasting the absence and the existence of multiple realizations: successive − 5 ⊕ {· ⊕ , 4-curve exist — the weaker version can not correspond to // M:36 9 N:467 9 H:339,21 Tuned symmetries M:37 7 N:467 7 H:338,22 Tuned symmetries M:38 8 N:466 8 H:338,22 M:39 7 N:465 7 H:338,22 Tuned symmetries M:40 7 N:460 7 H:335,23 M:41 5 N:457 5 H:335,23 Generic models NHCs 3 − , 4-curve is (9) 12 3 − , There is a similar story between − so 1 , (11) on the (10). In this case, the weaker tuning satisfies the monodromy condition automatically, // 1 which is expected as so { polytope corresponding to a tuning of algebras last models with H:338,22, M:39{·⊕ 7 N:465 7 andso M:38 8 N:466 8, corresponding to the tuning the weaker version of the— the tuning same of Tate tuning gives same Tate tuning appearingM:38 in 8 the N:466 lower 8 rank H:338,22, gauge algebras: corresponding to there tuning is of already the the case same generic model to gauge 12 last seven curves with the Hodge pairs with Hodge pair middle constraint on the Table 12. Tate tunings of generic CYs over the toric base JHEP02(2019)087 } − or 5 , } 3 // , 3 , 3 12 , 2 1 4 curve. , − , 2 − 1 , { // 1 , , , (8), and this } } 12 1 (7) cannot be 0 }} 2 }} { so , , 0 0 − so 0 , 1 , (12) gauge algebra , , 0 0 will automatically // 0 , 3 , (8) is the minimal so , , 0 0 } ⊕ · 12 0 , 0 , so 4 , , 0 0 , − 1 , 1 , 4, (8) algebra and related 2 (3) { 0 { 0 , , , { { − // so 2 sp } , } , , 2 } 3 } 12 4 will lead to a model with, 1 , , ⊕ 0 6 , 2 , 2 , − − 1 , , 0 3 ⊕ · ⊕ · { 1 , 3 , // 4-curve with a , , 0 3 (12) does not give rise to a Tate-Zariski 1 , 1 , − or (8) 12 , , 0 0 } so 1 , 1 , 6 } (8) involve monodromy constraints. so − , 0 0 { { ⊕ 3 3 , , { { so , , , , } } // 2 3 0 } 1 } , (1) , , , 3 5 12 2 1 ⊕ · ⊕ 1 , 1 , , , sp , , 2 3 − 1 1 on the last 0 , 3 , { , ⊕ (3) , , 2 3 1 } // 2 1 , 1 , { 0 , there are in general many distinct polytopes su , g , 1 1 – 48 – , 11 1 1 , 1 , , 1 2 1 1 − (8) curve. Thus, any Tate tuning of − · ⊕ {{ { · ⊕ {{ { , , h 4 so 4. In the case of the NHC 1 // , − 1 − , 12 h 1 − {− , 3 − , 1 (8) matches with the observation that a tuned − , so 5 are indicated. While the weaker version of the Tate form − } , 1 0 , − 1 , 3 − , − (8) monodromy condition in any Tate tuning over a base with a 4 , over a curve with self-intersection greater than 2 − (7) enhancement. An example of the non-existence of the stronger version of the Tate form: a tuning so Generic model, in KS M:85 6 N:379 6 H:274,58 Tuned model, in KS M:35 7 N:387 7 H:280,22 } , − 1 4 so , , , − 2 2 2 , − , 3 g , 2 − 1 , As we have mentioned, there is a special situation for the , 1 − 1 , (7) curve contains that on a tuned , 1 as discussed inchange the the gauge preceding algebra but subsections. not thedeformations Hodge that There numbers. can are And give in additional also somethat multiplicity. cases A rank-preserving there accounts complete are tunings non-toric analysis for of that tracking these the of KS all multiplicities database distinct exactly possible Tate would tunings for require each gauge a algebra combination complete and a and clear systematic Given a pair ofin Hodge the KS numbers database.course, generic There or tuned aresame elliptic many Hodge fibrations ways numbers. over in distinctconstructions which As bases that such discussed may give coincidentally above, a identical give Hodge however, multiplicityalgebra the through there numbers. may different are Tate Different arise. tunings realizations also may contribute, of many Of often the related closely same to related monodromy gauge tunings { if not 4.7 Multiplicity in the KS database algebra only arises overgauge the algebra, NHC so eithersatisfy vanishing the orders This unique aspect of ruled out through the globalso symmetry group since the global symmetry group on a tuned N:19 8 H:9,233. Thefrom Hodge anomalies. numbers of all these examples are consistent withpolytopes calculations in theThus, KS there database: are no all polytopes realizations where of there is a Tate tuning of the algebra exists in the KSdecomposition with database, the the desired stronger gauge version algebras. Zariski decomposition, and therefore a corresponding reflexive polytope exists, M:318 10 {− Table 13. of a generic model12 over the base (which forces gauge algebras on nearby curves). The Tate vanishing orders on the last six curves JHEP02(2019)087 . 4 } 2 − and , and 1 n , ns 2 1 2-curve A I 4-curves , − 1 − , 1 (11) tunings { so on the (2) } su 2 , only. 1 , III } (10) and 1 0 , Tate vanishing orders for , so 0 1 } , 3 0 − , , { (3) realizations for the latter; 2 4 . In general, the realization of , model. Similarly, type su 2 − , 2 , 4.5 1 g 1 , IV − 1 , { 3 − (2) tuning , 1 su . After iteration, the other two 2 } {− I 4 , , where there is no global Zariski decomposition 2 – 49 – , , and H:338,22 of rank 5 2 13 , 6 curve in section (3) for the former since it is just a specialization of 1 10 − , su (12), and the reflexive polytope model does not exist vanishing, giving the second generic model with all 1 { } so 4 IV , 2 , 2 , 1 , . We start with the minimal , where we can see the changes in three different types of trivial ) etc. Over bases with many curves allowing such tunings this could 1 . We have seen several examples in global models: H:7,271 of rank { n ( 12 tunings in table 14 2.1 realization of so 4 : there is no f , 6 } e 6 10 ) tunings do not give rise to different polytopes. Also for three different , , of the trivial algebra, only the one with the lowest vanishing orders that has 3 n 3 (3) tuning, while there are two different ( 4-curve alone to be , su 3 − su sp , 4 curves, which together do have a corresponding Tate-Zariski decomposition, ,II (9), and 1 3 1 − base, but there is no polytope that corresponds to the type I , +1 so 1 ,I 2 n 0 { ns 2 F I I Another source of multiplicity comes from tunings of rank-preserving type as described Note also that further Tate tunings of a given algebra may not give rise to a new reflex- To systematically analyze multiplicities of different Tate tunings of the same algebra, (8), so is illustrated in table algebra under various tunings. in the end of section 4 H:5,251 in table the type and both of thesetype sets have the rank-preservingtypes tuning a Tate-Zariski decomposition has a reflexive polytope construction. An amusing exercise ive polytope, even ifWe the describe higher briefly vanishing several orders exampleswith still here: H:4,226, have which there a corresponds is valid toof only Zariski the the one type decomposition. polytope inIt the is KS even database more interesting to compare the H:5,233 models discussed in appendix example is given by theof models the in table over the given global base, thoughpattern it with would the seem focus to be on fine the if local we sequence were to analyze the tuning realization does not have an acceptablepolytope, Zariski decomposition as and we there have is seen nowith corresponding for the example generic in gauge the groupany failure given over to combination a must realize be the checkedas by third performing local model a information of global may H:23,107 Tate-Zariski not decomposition, be completely adequate to rule in or out a possible tuning. An the middle are forced to also have curves reaching the second realization.appear This to exhausts the in possibilities. principle betent. So from 8 It what possible can might also combinations happen of that tunings, only only the two lower-order are realization exists, actually while consis- the higher-order we would need to consideralgebras all like combinations of monodromy andgive non-monodromy a tunings large of combinatorial multiplicity.the first For block example, of consider table theall two three polytope models in so there is a corresponding polytope construction. Then we tune the vanishing orders on such a systematic analysis here.focus on Rather, constructing in distinct the possiblethe analysis distinct gauge Hodge in groups numbers the through that remainder can Tatewe arise tunings of discuss for and the in reflexive identifying polytopes paper a in we bit this more way. In detail this some section aspects of the multiplicity question. and systematic analysis of the non-toric deformation possibilities. We have not attempted JHEP02(2019)087 . , . s ? 14 IV with [1] 4 6) points 6) points, , , 12-curves, over , over toric bases − 2.8 KS data ]. To illustrate this structure, M:150 5 N:20 5 H:9,129 M:155 7 N:19 6 H:9,129 M:160 9 N:19 8 H:9,129 11 the generic elliptic fibration , there is no model realized by no corresponding KS data M:165 11 N:18 9 H:9,129 3 41 − I , , 6) points to give } } } } } , 10 40 , − , are not subalgebras of each other, and 31 9 base. Notice that the Tate vanishing orders 4 4 4 (9) (9) 0 6 f f f 1 e − so so F 1,2,2,3,4 1,2,2,3,4 1,1,2,3,4 1,2,2,3,4 1,1,2,3,4 { { { { { – 50 – } } } } } (7) and su 0 0 0 0 0 1 I I I I I 0,0,0,0,0 0,0,0,0,0 0,0,0,0,0 0,0,0,0,0 0,0,0,0,0 { { { { { s. 1 1-curve change in the Tate-Zariski decomposition as the vanishing , } } } } } 2 − h (3) (3) (3) 2 2 0 su g g su su s (3) model in the second row realized by 3 3 I I . In these cases, we see explicitly that the fiber over the (4 su 1,1,1,2,3 1,1,2,2,3 1,1,2,2,3 0,1,1,2,3 0,1,1,2,3 IV { { { { { ]. In some cases, however, the base is still toric after blowing up one or 11 11-curves contains extra irreducible components that may naturally be shift. For example, F − 6) points; in such cases there will be multiple entries in the KS database 1 , 10 , , } } } } } , , 2 we analyze the non-flat elliptic fibration structure of the toric hypersurfaces 10 9 h 10 F B , − , and H:9,129 of different combinations of rank preserving tunings in table Some rank-preserving tunings over the 1 1 0 9 , -1 I I I II II F 9 12 − 1,1,1,1,1 1,1,1,1,1 1,0,1,1,1 0,1,1,1,1 0,0,0,1,1 { { { { { Lastly, multiplicity can come from situations where the elliptic fibration over a toric flat elliptic fibrations of thisin appendix type are analyzed inassociated [ with (flat) toric fibrationssurfaces of the reflexive fiberedin polytopes over the the Hirzebruch cancellation [ more of the (4 associated with these distinctCalabi-Yau bases. threefolds In that general we canor expect be flat that these viewed elliptic all as fibrationsthough represent we non-flat over smooth have the elliptic not checked non-toric fibrations explicitly bases that over this resolved toric is at bases true the in all non-toric cases. (4 Examples of some non- base has (4, 6) pointscontaining that curves must with be self-intersection blown number is up. non-flat As and discussed in thewhich section base there must is be awill flat blown be elliptic up non-toric, fibration. at and the In the (4 general blowups the give base extra resulting tensor from multiplets these contributing blow-ups to anomaly in table Notice that it isto not the always true same thatthe tuning tunings gauge give algebras different with the same rank will lead (but not the case ofwhich gauge involves algebras with realized the by requirements differentpolytope monodromy of in tunings additional the listed conditions), KS in there database table wouldthat even not since if there be the is Tate-Zariski a the configuration corresponding is stable. The example illustrates Table 14. of the trivial algebraorders on of the the two 0-curveswhen get the higher. gauge algebra The tuning last is row gives only an a example further of specialization a of general an observation existing that gauge algebra tuning JHEP02(2019)087 ) K 2 376). fiber. − , ( 1 , 3 O , 2 , ) P K for the generic − ( 1 , O 2 h 12 curve to at least − fiber type polytopes. In part 1 , 3 , 2 P 12 curve in the base. Similar issues arise must vanish over a . Thus, by restricting to Hodge numbers − 1 , 2 curves arise). 2 nK h over any base would seem to give a Calabi- − − 1 , – 51 – 3 , 2 P 8. The base of this type with the largest value that can be achieved for a tuning with any fiber other , the tops over the -9/-10/-11-curves are listed in the second fiber type. Carrying out the analogue of the standard stack- − 1 , 2 2 15 P h 8. This immediately constrains the set of constructions at large fiber, we find that there are 10 dual monomials analogous to the − 2 . These 10 monomials are sections of line bundles , over which the generic elliptic fibration has Hodge numbers (10 P 6 8 5 by the Zariski decomposition. This immediately leads to the presence F 12 curves in the base (though there are notable exceptions to this general . This illustrates the way in which the same smooth Calabi-Yau threefold that are possible; the generic fibration with each of these fiber types gives − n < 2 , . . . , a , 11 , the largest 1 1 ). Any section of a line bundle , 8 is quite restricted; over this base, for example, there are 5 other fiber types a 1 F K P 1 , ] and identify the bases with largest Hodge numbers that have curves of self- when 3 are listed in table 3 9 , − n 2 ( 11 P / O Considering the toric bases, we can simply consider the complete enumeration carried We leave a more detailed analysis of the constraints on different fiber types for future action (including various limits in which 10 / ∗ 9 than including an elliptic Calabi-Yau threefold withthese Hodge or numbers any (11, otherYau 227). threefold fibers Any with other other an fibration than even with smaller value of for the other fiber types. out in [ intersection no smaller than elliptic fibration is Even over ing procedure for a coefficients and degree of a codimension one (4, 6) singularity over any always involve principle, including the other one of the two fibrationswork, of but the briefly H:491:11 outline polytope). theConsider issue for example that the arises for other fiber types besides the types all lead tobase problematic when codimension the base one containsnone (4, curves of 6) of the singularities sufficiently other on negativeintersection 15 self-intersection. some less fibers In divisor than can particular, in beHodge the number, supported since over the any generic base elliptic that fibrations with contains the a largest curve Hodge of numbers self- almost In this work we have confinedthis our is study because to the the standardmodel simplest fiber as type discussed matches previously. withlarge the Hodge Also, Tate numbers. however, structure In this of particular,can the fiber we be can Weierstrass type explicitly used dominates identify constraints for the on the the structure other bases at that 15 fiber types. These constraints are such that the other fiber realized in a differenttable there polytope are in 6 distinct therealizations polytopes of KS at elliptic Hodge database. fibrations numbersC H:14,404, over For different which correspond example, “semi-toric” to bases as toric that illustrated admit in only4.8 a the single Bases with large Hodge numbers pairs for the generic elliptic fibrationF models over the suitably blown upblock Hirzebruch in surfaces table can be realizedsometimes as a a flat non-flat elliptic elliptic fibration fibration over over another one toric base, or with more each toric fibration bases structure as well as associated with divisors in the blown up space. The multiplicity with which the Hodge JHEP02(2019)087 } } } } } } -1, -1, -10, 0, 9 -1, -1, -11, 0, 10 -1, -1, -12, 0, 11 -1, -2, -1, -11, 0, 9 -1, -2, -1, -12, 0, 10 -1, -2, -2, -1, -12, 0, 9 }{ Bases }{ }{ }{ }{ . The possibilities are enumerated in this }{ 11 / 10 / 9 F 0, -9, 0, 9 – 52 – 0, -10, 0, 10 0, -11, 0, 11 { { { -1, -1, -11, -1, -1, 9 -1, -1, -12, -1, -1, 10 { -1, -2, -1, -12, -1, -1, 9 { { 2 curve in the base. For each polytope the base of the fibration is a 2 4 6 − Mult. in KS 11 curve are at non-toric points and the elliptic fibration is non-flat. In these cases (12,462) (13,433) (14,404) A variety of polytope models arise for the Hodge pairs associated with the generic − Hodge pair 10, or − , 9 toric surface given by the curves on the outside of the diagram, with self-intersections as labeled. − the blow-ups are handledmodel automatically corresponding by to the a resolutionthat flat of elliptic arise the fibration by toric over blowingtoric a geometry, up fibrations giving “semi-toric” over one base. a blow-ups resolved or of Therecorresponds the are more to Hirzebruch also of a surfaces. toric limit the bases When with (4, multiple a (4, 6) 6) points points at coincide this toric points, giving polytopes with Table 15. elliptic fibrations over thetable. Hirzebruch The surfaces first graph for each Hodge pair is the generic model, where the (4, 6) singularities on the JHEP02(2019)087 1 , of ]), 3 8), , 9 } 2 1 − P , fibers 2 240 or 1 // h , 7 ≥ 3 , , 1 2 1 , − , 1 P 1 h h // -fibered poly- { , the polytope 8 1 , 3 , − . Therefore, our 8 so it seems that 2 3.5 − P // chains (see e.g. [ 8 4.8 7 − admit extreme tunings E 1 , // 1 8 h − fiber type. // 1 , 8 3 , − 2 P // . 8 − 5.1 1 associated with ] that the set of Hodge pairs // − 8 , 39 2 240 should again restrict us to primarily − − satisfies both inequalities). To compare the two without producing curves of highly negative self- , ≥ – 53 – // } 3 1 8 that can arise for a base with no curves of self- 1 , , 1 − 1 − 1 , h ], we study the fibration structure of the hypersurface , 251 h a general construction of reflexive polytope models of 2 1 , // h 20 − 8 , however, there are a few unusual cases in which bases , 251 1 − 6 { 4.3.2 − // 8 − // 7 8 is 224. The corresponding base has a set of toric curves of self- − − // 8 − , 240, we can expect that the threefolds in the KS database that admit elliptic , there is a close relation between Tate-tuned models and ≥ 4 denotes the sequence 1 , 2 240 (only the Hodge pair // h ≥ All Hodge pairs of the Tate-tuned models from this algorithm fall within those in Of the 30,108 distinct Hodge pairs in the KS database, 1,827 have either Similarly, the largest value of 1 , 2 the logic of this analysis in more detail inthe section KS database.regions However, of there interest are at certainWe which Hodge therefore our have numbers analyzed initial in directly, analysis via the identified the KS no method database matching described in Tate-tuned in model. section the in section topes, which dominate atapproach large is Hodge to construct numbersmodels systematically as all over argued Tate-tuned 2D in models toric section simpler via bases. systematic tunings analysis of As of generic ageneral which Tate preliminary Weierstrass gauge to tunings, group this and tunings then analysis, may we however, be refine we possible the begin based analysis with on to a more Tate tunings. We describe h constructions at large Hodgenumber numbers, of the distinct Weierstrass next modelsmodels step of correspond would tuned to be CYs reflexive to polytope in construct these constructions, however. regions. roughly Nonetheless, this Not as all discussed Weierstrass these generic elliptic fibrations overelliptic toric fibration bases models over with toric NHCs, basesin and have these the we corresponding KS expect reflexive database. that polytope alltuned models We wish Weierstrass generic models to of carry CYs outCalabi-Yau over a hypersurfaces 2D more at toric comprehensive large bases comparison Hodge with by 4D numbers. matching reflexive polytope models of Kreuzer and Skarke have classified all30,108 473,800,776 distinct 4D Hodge reflexive polytope pairs. models,all It which generic was give found elliptically in fibered [ CYsKS database. over toric We gave bases in is section a subset of all the Hodge pairs in the intersection. In a companion paper [ models in the KSat database large more Hodge directly, numbers and and confirm the both existence the of prevalence exceptions of involving5 extreme tunings. Systematic construction of Tate-tuned models in the KS database and a generic elliptic fibrationtuned with over Hodge this base numbers without (224,18). producingconfining a There attention curve is of to nothing self-intersection threefolds that below with canfiber be types. As wethat see have in generic section elliptic fibrationsthat with dramatically increase rather the small value values of of fibrations should be all or almost all described byintersection the below intersection (0 where above JHEP02(2019)087 , 2.7 , there 4.5 , where we collect some 5.4 ]. The more detailed analysis 20 240. As we saw in section ≥ 1 , 2 h . It turns out that all these remaining 6 240 or – 54 – ≥ 1 , 1 h . This can be tested by the Zariski decomposition. More 2 B to determine which combinations are globally compatible. While 2.5 on the curves in the base, applying the variant of the Zariski decomposition we use the results of the Weierstrass tunings to check which Tate tunings 2.3 5.4 models Although ultimately we wish to analyze Tate tunings, we perform an initial analysis We also discuss briefly the limits of Tate tuning in section This gives us agauge set algebras. of Not possible all tunings thesewith constructions, that polytopes. however, we are compatible We expect with begin may Tatein the tunings be section and discussion possible by at focusingare the on possible. level the of Weierstrass tunings the and then efficient. Global symmetriesdependent are gauge also groups that helpful can innot arise apparent on limiting at sequences the the of set intersecting level curves of of in the possible ways Zariski that monodromy- decomposition. are of Weierstrass tunings using global symmetry constraints and the Zariski decomposition. specifically, our goal isfrom to section carry out explicitlydescribed arbitrary in combinations section of thein Tate principle tunings we could simplyglobal iterate over symmetry all constraints possible on gaugea what algebra curve combinations, gauge using of algebras the can negative arise self on intersection the helps curves prune intersecting the tree and make the algorithm more with Hodge numbers inglobal the symmetry regions constraints oncan each be curve tuned put on upperis intersecting bounds curves. an on On issue the the gauge ofmodel other algebras whether hand, over that as local some discussed toric tunings in base on section subsets of curves can be combined into a global We give an algorithmgauge in groups this over section a toconstruct given systematically 2D all construct toric Tate-tuned all base, models starting tunings over with of toric the enhanced bases generic that model. give Our elliptic goal Calabi-Yau is threefolds to results on tunings that arebe compatible realized with by the Tate global tunings.such symmetry cases constraints These give but tunings new can’t may Hodge not be pairs realized outside by the Weierstrass KS models database. and5.1 in Algorithm: global symmetries and Zariski decomposition for Weierstrass through generic or tuned ellipticanalysis fibrations. of This the matches fibration with structurewe the in results carry the through out companion a paper here, direct [ however,and gives the much complex more variety insightsimple of into toric Weierstrass the hypersurface tunings structure constructions. and of these geometries fibrations that are realized through the analysis of thesepolytope cases is models described can in beover section bases described that as are somewhat eitherof more toric the exotic bases two constructions or Weierstrass or at blow-upsanalysis thereof. is the Tate that level tunings This in of the completes Hodge regions the of numbers. comparison interest At all the a Hodge basic pairs level in the the result KS database of are this realized models with the Hodge numbers that were not found in our systematic tuning construction; JHEP02(2019)087 , ] j 8 on e C 61 and = 6, 10 or 8 8 e n e − (glob) j , g 9 = or − 6) points in = 4, 7 , e and first con- n gauge algebra. (glob) j 8 8 g e e on each curve . Additionally, the j , and ∆ along each = 2, the subsequent j g g ) with g j , are constrained by the irreducible toric curves f 3 2.30 , we use C · K of 12 curve supporting an )–( − } = j 12 g j, 2.27 be the set of algebras that satisfy , c 6 has to be a trivial algebra. } can be determined by the Zariski j, algebra cannot be further enhanced; 2 ,α } g 3 , c to obtain the restricted set of algebras 8 of tuned algebras 4 e . g 12 j, { j, } c 1 1 c j { and − { − g (glob) j , without producing a (4, 6) singularity at . For the global symmetries, we used the (glob) j 8 { 8 g g e e ⊂ carrying a gauge algebra = (glob) 2 – 55 – 1 , and g . g m j,α } ] for the maximal global symmetry group 6 g ⊂ 2, respectively. We used the results tabulated in [ . The possible tunings on j, . Let the set 3 4.7 , which is determined by the self-intersection number c 47 ⊕ − 2 2 { C ,α 2 C 3 C = − , g j 6) points in the case where the base contains } on g , 4 on m ⊕ on 3 j, are fixed: 2 g , which is represented by a set of c 8 g e 2 { B ; therefore in such cases when C = (glob) 2 1 and and section g the orders of vanishing ,II − 9; such a curve necessarily carries a non-Higgsable 1 ,α and 3 B 2.8 intersecting each other in a linear chain, we first obtain for the generic = ,I g 1 = 1, the curve we attempt the first tuning on by 0 } ≤ − I C . j m ⊕ } 1 j m g C { ,...,K of negative self-intersection = 3, etc. This choice of the initial configuration is convenient to serve as the (2) for 2 , j j su C satisfying the constraint = = 1 = 12, respectively, or can be directly read off from the non-Higgsable cluster structure and the gauge algebra } We then pass to tunings Now let us consider all possible (Weierstrass) tunings of the generic model. We describe Given a 2D toric base n 2 , j j using the index C j,α 11 curves; in these cases we essentially treat the curve as a (glob) j g C 8 curves of negative self-intersectionalgebras, that of do types not supportg an NHC can carryfor the trivial subgroups gauge of a global{ symmetry group global symmetry group of the constraint results in table 6.1 anda table curve 6.2 in [ starting point of a branchingmoreover, algorithm nothing because can an bea tuned toric next point, to which an gauge we algebras are on assuming does not happen as discussed above. Therefore, the self-intersection We start the procedure bysidering choosing the a possible specific tunings curve one with one a of non-Higgsable the adjacent curves.curve Let by us label the curve with the 5.2 Main structure of theWe algorithm: consider bases first with the simplest a cases, non-Higgsable where there is at least one curve in the toric base of base, and the tunings overbase. the blown We up do base allow would− be non-toric found (4 directly by tuningsunderstanding that over that the polytope hypersurfacesingularities construction will and automatically effectively resolve these blowdiscussions up in the section non-toric points in the base, in accord with the of the curves a procedure to determinein an the allowed base. pattern Notethe that base, in even this after algorithm the we assume tuning; that such there a are point no would toric be (4 blown up to form a different toric { model over the base curve. The setsdecomposition via of the values procedureand described in equations ( JHEP02(2019)087 ] . 2 } } j , 3 [ in 1 3 , j ,β C 4 g 4 g ,α g g . We 3 , { 1 = on g , ] 3 with the k 3[4] after the g [ +1 g j is met. As j } , g ,α = · 3 1 is the gauge 12 , g 8 j, − have changed, e c 3[3] K { { 3 4[3] g stays unchanged can be continued C g ; in such cases we C . First, the usual 3 = j 1 , is again determined 2] C 4 C .) } should be the trivial − g 4 , and 2 K ≤ } 14 is the gauge algebra on 1 j g 6 , 2[ g (glob) 4 j, − { are increased, but without g c denotes a choice of gauge 5[4] (glob) K { 3 ] g g j , [ C j } , we need to consider also the 4 g ⊂ can obtain new values after the 1 j, , c } 2] C j { − . The branch 12 , = K 3 is not allowed on . Note that we must have 3[ i the gauge algebra on curve j> 1] − is determined by the self-intersection of ,α . For each possible algebra we replace (glob) 3[3] ] i − 3 6 and 12, respectively. If all the gauge g [ K , c is satisfied, where now with the desired orders of vanishing cor- j ,α stay unchanged after the iteration, tuning j g , [ 6 g 3 , j g ⊂ 3 } 3 g g to be allowed: in the new configuration after ⊕ . For of the tuned gauge algebra plugged into the (glob) 3[3] C stay unchanged under the iterations. Indeed, 12 i j> = 4 , g ⊇ 3 is satisfied, where (glob) 3[3] ,γ 2[3] ,γ } C } ] , c 1 1 g g – 56 – n j 4 , c 12 [ 12 − − , , (cf. also examples in table , j 6 do not produce the desired gauge algebra . For example, let us start with 3 ⊕ , 4 3 ⊂ K K g 1 branch, and attempt the next tuning } 3 , branch and attempt the next branch of tuning } ≤ g g for ,β , c j> (glob) 4 II j . ,α 4 , c 1 6 c g ,α c 12 3 , , 1 4 } , , g 2[3] { 3 , 3 . In other words, in this case 4 { g to close the tuning pattern. The set of possible tunings 3 4 3 g g 2 g ⊂ 12 , g c , , but the orders of vanishing realizing the gauge algebra , c , and , or , c C } 3 K 4 { 1 . We then perform the Zariski iteration procedure on all } ⊕ 6 , 6 ∈ { , , 4 C , c 3[4] 3 3 ,I 2 k>j c j,α 6 g and 0 , is violated in the new configuration. ≤ { 4[3] , c g C 3 I j 3[3] 4 4 g c ⊕ , g , c 3 { C 4 ∈ { c 1 , , , { ] 3 5[4] (glob) 3 } j c branch with the new configuration: the set of possible tunings [ g 4 g , j { 1 3 , g ⊂ 3 is constrained by ≤ using table is connected back to the first curve j g is constrained by c 1 passes the tests above. We then continue the procedure similarly to tune { 4 ,α 2[3] K − 3 g 1 C , as long as C K , g in the new configuration and 4 3 as we require in the branch that the gauge algebra on C ⊕ in the 4 g C passes the two tests (1) the set of gauge algebras 4 C 1 on 4[3] , C the iterations in the newest updated configuration, and 4 g } g k > j if ,γ 1 3 is not ruled out. If any of the gauge algebras on the curves prior to The procedure continues similarly until the second to the last curve Assume Note that the vanishing orders We attempt tunings one-by-one for each − and the gauge algebra after C K ,α 5 3 3 g the last curve global symmetry constraint on { two conditions have to be satisfied for stays unchanged after performingdesired Zariski degrees iterations of on vanishing configuration, and (2) C by the self-intersection of on the curve we attempt on only if may be different. Wealgebras can on proceed with thealgebra new on configuration to theC next step of tuning terminate the procedure with the further iterations of the tuning.iteration procedure We denote associated by withalgebra in curve a branch, for all in tuning gauge algebras on modifying the gauge algebraalgebra, on but it may be type initial set of iterationsinitial just NHC described. configuration, If we these use increase the beyond larger those vanishing determined orders by the as the starting points in or the vanishing orders the new configuration after theterminate iteration, the tuning procedure with this Note, however, that thethe fact set that of the values gaugeoften algebras it stay is unchanged the case does that not the orders mean of that vanishing on curves near the original orders ofresponding vanishing to curves with the new algebras on the curves priorg to and including JHEP02(2019)087 ; 7 } e 4 , (and 2 , . After 1 already +1 2] { j has to be − C = K in the base; K f, g region as we } 2[ C 7 − 1 e 12 , . , · 1 K g h +1 = j 2 , c g . 6 in the simplest cases, , , ,..., and the gauge algebra algebra, the algorithm 7 is only allowed to be a 8 ·} e e , +1 8 ) pushes up the order of j K 3[3] e K 1] } = g C g , c − 1 4 , n , K g 0 1[ , +1 − j 0 7 in the base. This is because c { 240, we need only consider toric K 240. In our initial scan, we also { g is satisfied. Moreover, there is the 220. As we describe in more detail algebra, so no tuning is allowed on , ≥ ≥ 1 is satisfied, we are basically done to ≤ − . In fact, 1] 8 ≥ ). In such situations, we modify the 1 − e 1 (1) the prior gauge algebras are held , − , ,γ 1 K 2 , m K 2 1 K 1 h ,γ C h − C 2[ 1 h 2.10 with respect to the generic model over a K − − ) tuning ( g is held fixed and is 1 K K n , ( (where some required orders of 1 2 g g , g h su 1] carries an – 57 – ) also change in general, but the new vanishing − f, g 1 , not only do the orders of vanishing on K j C 6) points that must be blown up. And no non-trivial 3[ ,... C , − algebra. Thus, in this case we similarly can make the , where in equation ( +2 K j 1] 7 g g e − C K [ can in some cases correspond to a different gauge algebra. and decrease . On the other hand, as bases associated with generic models (glob) K and to fit with those that correspond to the gauge algebra that ,..., 1 g , 1 } , , etc. so that all possible combinations of gauge group tunings 1] 1 f 1 2 − 12 +1 h ⊂ , h ’s in order similarly to complete the scan through all possible − j K 1 ,γ K C +1 4[ g 1 j g g − , ⊕ , c K 1] 6 , ,γ g − 1 +1 − K j 3[ K 224 always contain at least one curve carrying an g , c g is determined by the self-intersection of the curve , 4 , · > , 1] 8 1 +1 ’s are processed, we proceed iteratively with a nested loop, continuing with the , − e j 1 ,γ c { K h 1 [ { − (glob) K algebra would give additional (4 g K We make some final comments on two technical issues in the tuning procedure. As All tunings increase We check all 8 after the Zariski interations for the tuning . Hence, if the global symmetry constraint on g e K K this can happen forvanishing example of if ∆ a previous moreimposed on significantly the than curve); however,in note that, a this complete can model neverorders happen from of in a vanishing real on ∆ as calculated mentioned above, in tuning thein curve some cases onorders further curves However, because the threethese vanishing Zariski orders iterations do not wereencountered correspond performed a to few independently, any cases algebras sometimes of in the this Kodaira type, classification. for example We where algebras also cannot be further enhancedan without modifying the base —algebra an can enhancement to be tunedconvenient next choice to that an the initial configuration is fixed to be having as described above isalways quite have effective a in simple dealingwork starting with in point tunings the in for same the thei.e., way large iteration. generic for models tunings In with of fact, a generic curve the of models algorithm self-intersection with actually a can curve carrying given base. Thus, tobases classify for all which tuned therestricted models generic to of bases elliptic that fibration havein generic has models the with following section,that this significantly changes misses a few cases where there is a large amount of tuning tuning patterns compatible with theall initial viable possibility for next possible value of are considered. trivial algebra in the simplestC cases as this point in the procedurepattern searching for a tuning pattern. In this case, we obtain a tuning fixed, and (2) the globaladditional symmetry third constraint condition: on (3)satisfied; the i.e., global symmetry constraint onand the curve g the iterations associated with the tuning of JHEP02(2019)087 , 7 e }} 4 , 1 − , so that , } 1 1 − − , 7 and/or , 6 1 − − , , 1 ≤ − 4 , − 1 , m 1 − , 1 {− , . In principle, however, for − 7 or below. In the region 1 , − 6 C 263 , which are the generic gauge compose 9 of these 14 models, , 1 {− 2 C 11 P { and such that there are fewer allowed com- 6 over a curve of minimal self-intersection 2 6 curve ≤ . Note that in this case there cannot be , so no tunings are allowed on any inter- } 7 − m 2 ·} ,C e ≤ g , 1 0 7 , C e 1 F { g minimally. Then we perform the iteration again { – 58 – 1 6 , C gauge algebras. Every base with a generic elliptic +1 j 8 to ensure that all possible tunings are considered. e , c } 4 , or +1 7 j e c algebra , as the global symmetry algebra is always the trivial algebra on 220 has a curve of self-intersection 12 j, 2 7 in the base, where we can begin the iteration process in a simple ≥ C , c 6 1 , j, ], the constraints are weaker and less completely understood. So we do ≤ − 1 . For example, for the generic model , c h have three consecutive curves of non-negative self intersection. Any blow- 13 } m 4 2 j, and an enhancement on m g c , F { 1 ·} g , 6 { e and { 2 240, there are 14 generic models that contain no curve in the base carrying an having curves of non-negative self intersection P algebra; the generic models over bases ≥ From the minimal model point of view we can fairly easily classify the types and A further issue arises for bases that have curves of non-negative self intersection. On Another detail to take care is the tuning of algebras only distinguished by monodromy 8 e 1 , 6-curves without an further enhancement to 2 relevant is rather limited and can be handled efficientlyconfigurations using of more non-negative specific self methods. bases intersection curves that canup arise. of The one minimal of model these bases has either only one such curve or two adjacent such curves, since such curves, there isof no view. global While constraint there onself are the intersection some [ adjacent analogous algebras constraintsnot from in impose the the SCFT global case of point constraintsiterating curves in over of all these non-negative gauge cases. groups, however In in principle practice the this number can of be cases where handled this by issue simply is or and are discussed furtherconfiguration below. that uniquely In determines the theand initial remaining specify configuration, cases, different and initial we there configurations haveto is to perform enumerate no the choice algorithm. of the initial − secting curves. In fact, incases the that Hodge lack number regions non-Higgsable fibration we are having considering, there areh very few binations we may choose to rotatethere the are sequence only of two thealgebra initial curves configurations to on be the any enhancement on fashion as the gauge algebrathere on are the very initial few curve bases ishandled. that fixed. lack Of In such course, the curves; one regions westarting could we describe point simply are here and considering, use looping briefly over a how allefficiency brute these tunings we cases force on would are the algorithm like initial of to curve choosing choose an the arbitrary curves fourth entry 5.3 Special cases: bases lacking curves ofThe self-intersection algorithm described inself-intersection the preceding subsection relies on the presence of a curve of after the modification, and use the resulting configuration to test theconditions. conditions (1) and (2). For thosemonodromy conditions, cases we where retain all therelist the of are possibilities possible allowed distinct by tunings algebras global we symmetries; associated in attach the with an different additional label to the orders of vanishing using a arises by increasing the values JHEP02(2019)087 ], 13 gives 231). , 12 and four F 2 P 272) to (3 , , there is generally 1 240 are also realized , 1 h ≥ 12. Indeed, an explicit , 1 , 6 2 , h = 4 -fibered polytope constructions. n 1 , 3 , (2) on the +12 curve of 2 P su , an exhaustive search is straightforward 12 ≤ m ≤ 0 – 59 – F 318). But it turns out (as we see explicitly from , below the value 240 of interest. For example [ changes the Hodge numbers from (2 and 1 to the fact that monodromy conditions are not really , that have three non-negative curves, most tunings in 2 2 2 h P P 12 5.2 ≤ m ≤ 0 F and is by performing the algorithm separately in both opposite directions 2 1 , P 2 h , run the algorithm in both directions, and stop the tuning procedures 1 C ). (2) on a +1 curve of m su F Although we have only focused on tuning models in the large Hodge number regions, For the cases In general, the way we deal with the cases having one or two non-negative curves for these tunings on toricFurther, curves some will of not the have combinations standard and of gauge global groups constraints that still are cannotfor allowed example be by at the realized Zariski the in analysis endtaken practice of care in section a of global properly model in — the we Zariski alluded decompositions of 5.4 Tate-tuned models The analysis described sopossible gauge far algebra in tunings termsbinations over correspond of each to Weierstrass toric hypersurfaces models base. inthis. reflexive gives Not First, polytopes. a all not There large every of are Weierstrass collection these several tuning reasons gauge of can for algebra be com- realized through a Tate form, so some of one can in principleany classify toric all base allowed tuning withneeded patterns for the of tunings algorithm non-abelian over the described gaugein bases here; algebras these on though cases slightly ascurves different there in are methods only are a few curves in these bases (three curves in the analysis of thein following section) other that ways all byexplicitly these generic cases include or with these tuned inmultiplicity models of our over models analysis other at since toric each Hodge we bases. number are pair. So not we trying do to not reproduce need the to precise fact decrease the Hodgetuning number an There are some exceptions:a Calabi-Yau for with example Hodge tuning numbers an (12 one or two non-negativeat curves most in one the non-negative base.constrained self by intersection For global curve bases symmetries and withthese the and large cases nearby nearby we gauge large group havenon-negative negative is self-intersection used self-interactions; curves. generally simpler in some heuristics of to complete the analysis in the presence of from a good starting pointpatterns” (curve of of maximally tunings, negative and selfchosen intersection) connect curve to them get two appropriately. “half- when the In first other non-negative curve words,conditions is for we the met start curves in of from bothof non-negative a directions. the self half-patterns intersection. We connected do The in combination not this of impose way the gives any two all global sets tuning patterns of a generic model with introduce new curves ofagain non-negative have at self most intersection.they two are curves always Blow-ups adjacent. of of So non-negative the the self possibilities resulting intersection are bases and actuallybases with when quite large limited. there are two blow-ups reduce the self-intersection of curves containing the blow-up point and do not JHEP02(2019)087 for and 4 . We 4 4.5 ) from the Tate tuning 2.7 n s 2 I 240. Only 18 of ≥ ) and ( 1 , . An interesting ques- 2 2.6 lie in the KS database. h 16 5.3 240 or +4) algebras); also, we wished to ≥ n 1 (4 , 1 so h and section 5.2 . The second version of the – 60 – 4 version of ? in table . A tuning pattern gives a genuine Tate-tuned model if it ? algebras, some of the alternate monodromy tunings were not or 2.5 so ◦ ]. In the context of this project, we did a local analysis of the Weier- 13 ) was in fact previously not known and was identified through the analysis ◦ , the polytope constructions do not satisfy the monodromy conditions for the 4.6 -fibered polytopes, given the correspondence that we established in section A question that we do not explore further here, but which is relevant to the more gen- These principles give us a set of gauge group and Tate tunings over each toric base We are interested in constraining to a subset of tuning constructions for which we 1 , 3 , 2 strass tuning patterns that areknown not Tate Tate tuning tuning patterns. swampland Weproblematic and reproduced tunings some also parts in found of the the newtion Tate for obstructions. construction further are research Some listed ismodels examples which in when of of table the these the indicated can sequence be of realized curves through arises good as global part Weierstrass of a toric (or non-toric) base. eral problem of understanding the6D full F-theory set models, of Calabi-Yau is threefolds thebe and extent allowed the to from classification which the of tunings Weierstrassdo are Zariski not possible analysis correspond and that to anomaly look Tatewere constructions. cancellation like analyzed conditions, they Various aspects in but should of [ this “Tate tuning swampland” all of the 1,827the distinct Hodge Hodge pairs pairs indescribed in this above. the In range range the were nextconstructions. not section we found consider by the a analysis of “sieve” these using 18 outlying the polytope Tate constructions P have carried out an explicitof comparison this more of limited these class twoin of sets, the Tate-tuned and KS gauge indeed database. groups thethat all Furthermore, Hodge are correspond the numbers not to Hodge values in pairs thatrealized the appear from by KS the Tate database original tuning. are Weierstrass exactly In analysis those fact, of given the this tuning restricted patterns set that of can tunings not we be reproduce almost section higher rank gauge algebras in these cases. that we believe should have direct correspondents in the KS database through standard the tunings marked with (marked with of the next section. Forpreviously the known (for example, the non- restrict attention to cases whereof the algebra the is Tate guaranteed coefficients. simply by In the order general, of as vanishing we have noted in the examples in section each tuning in atunings tuning described pattern. in We section thenhas perform the the Zariski Zariski decompositionsour decomposition of systematic of Tate analysis the tunings. only Tate multiple the In realizations stronger and/or performing version monodromy this of conditions. analysis, the Tate In we forms particular, used for we in the did algebras not with use any of Weierstrass tuning patterns we got from section expect direct polytope constructions.that satisfies the Hence, Weierstrass Zariski for analysis and eachan global explicit gauge conditions, Tate-type we algebra model attempt to tuning by construct specifying combination Tate orders of vanishing according to table check shows that not all the Hodge pairs calculated via equations ( JHEP02(2019)087 - ) 1 , 3 240 , 17 2 ≥ P -fibered 1 , 1 2 , 3 h , 2 P 240 or (2), ≥ -fibered polytopes in rule” that however can sp 1 1 , , 8 1 3 ⊕ , E h 2 P (10) so , and the other 17 (see table } (4), -fibered polytope) has a Tate form, 261 su . In particular, the CY hypersurface 1 , , 3 , ⊕ 45 2 3.4 { P 1] , (10) 0 , Local geometry -2, -2, -2, -1/0 -2, -2, -1/0 -3, -2, -2 so [0 (2) (2), region, – 61 – some further examples in the KS database that 1 sp su (4), (2), , 2 we analyze the standard (4)) sp ⊕ sp ⊕ h 6.2 . All the polytopes associated with these 18 Hodge su ⊕ ⊕ 6.1 (3) (11) (13) 3.5 (3) (9) sp so so su so ⊕ (2)(or ⊕ · 2 g sp (3), (1), (3), (4), (2) fiber that occurs), but not all of them are the standard ⊕ ⊕ sp su sp su su 1 (10), , ⊕ ⊕ we analyze the polytopes that do not have the standard 3 ⊕ (2) (2) ⊕ ⊕ , so region. In this section we analyze the polytopes in the Kreuzer-Skarke 2 -fibered polytope (or Bl fibered polytope structure (though in some cases it is really the more su su 1 P 1 , ⊕ 6.2 , (3) (13) (11) (9) (7) 3 1 1] 1 , , , 2 8 Tate swamp 2 3 0 h e so so so g su Miscellaneous Tate swamp (some examples) Gauge groups · ⊕ · ⊕ so , , P 2 [0 P Tate tuning swamp: we list all subalgebras allowed by the “ By studying these 18 classes of Calabi-Yau manifolds, we have identified new tuning construction, and other exotic constructions the KS database thatmodels. we have In section not obtainedstructure. in our We systematic also construction includeare of in outside Tate-tuned section the rangeexotic structure of associated focus with of gauge groups this on paper non-toric but curves in that the illustrate base. some further interesting specialized Bl fibered polytopes that weof have a defined in standard section while this isdifferent not “twist”. the case WeHodge for analyze pairs other the separately. In fibration two section different structures types that of use polytopes the arising same from fiber the but 18 a constructions that we had notmodels known utilizing previously; these the constructions that KSWe we database study did provides not the us expect fibration with a structurein global priori of in the the our way original 18 analysis. outstanding describedpairs classes have in by a section analyzing the polytopes in the KS databasemissing 18 that Hodge are pairs is notare in produced in the the large by large ourdatabase Tate associated tuning with algorithm. these 18 One Hodge of number pairs. these 6 Polytope analysis for cases missing from theAs simple discussed tuning above, there are only 18 Hodge pairs in the regions Table 16. not be realized by Tatedo tunings. not violate We global also symmetry giveTate-Zariski constraints some decomposition). but examples that can of not the be tuning realized patterns by we Tate tunings found (i.e. that violate JHEP02(2019)087 - 1 , , 3 } } } , 2 } 57 54 37 9 P , , , , regions 1] , (6) on a 0 1 , . , 245 260 240 261 1 [0 su { { { { }} 3-curve) , , , , , h 6 } } } } − 1 ), or are non- , , 0 2 56 62 35 10 , h , , , , 3-curves, and are ). In the non-flat 12 − 6.1.2 254 260 247 250 − , { { { { , , , , , the CY resolution of that we did not obtain 1 } } } } 6.1.3 1 − , -fibered polytope, and is , 1 84 45 35 10 4.7 1 , 2 , , , , h -tuning”) 3 8 , − e 2 , 258 261 251 244 P 2 (“ { { { { 13)-tuning” on a models, which have a Tate form, , , , , − } } } } } , 1 ≥ , 3 3 13) tunings on 55 51 32 48 , n 2 − , , , , ( , P ≥ , has only one polytope. This polytope 1 so } 259 261 263 258, 60 240 models, which involve tunings on non-toric n − { { { { { ( , 1 , 5 261 3 so , , − 2 220 (described in section , -fibered polytope (or a standard Bl P 45 1 1 , { < 3 − , – 62 – , 2 1 singular fiber that is not just a specialization of , . 3 P 1 n h − s 2 , 6.2 I tunings or 1 region, − 8 1 , e , 2 . The polytopes of the nine missing Hodge pairs at large (1) tuning and 6 h u − , 1 6.1.1 − , 2 , and the non-standard − 6) points). Therefore, at these points the dimension of the fiber , - tuning on non- , 2 1 , 6.1 Tate tunings and exotic matter 3 , {− 2 n - huge tuning , s P 2 1 , I 3 , - global 290 2 1 , , The Hodge number pairs in the KS database at large P 3 , 2 38 P { 1] 1-curve gives models with the three-index antisymmetric representation as opposed , 0 , with the standard fibration structure are either extremely tuned models, with bases The Hodge pair in the large [0 1 ≥ − , fibered polytopes toric curve Non-standard fibered polytopes non-toric base (“ Bl fibered polytopes non-toric base Standard 1 U(1)-charged hypermultiplets. 6.1.1 Type The polytope model M:357a 8 Tate-tuned N:65 model 8 of H:45,261 the is generic a model standard jumps to two givingdivisors with the blow-ups non-flat in the ellipticalternatively base fibration allows as us structure. flat to describe ellipticthe Associating the fibrations resulting cases the Calabi-Yau found over threefolds additional here thealso give blown involved with rise up tuned to Mordell-Weil base. sections, which The are associated resulting with models U(1) factors in and flat elliptic fibration modelselliptic over fibration a cases, toric as base we(4,6) (described have singularities discussed in in at section the terms endambient of toric of the section fiber polytope (as modelsatisfied the produces hypersurface over irreducible equation components the restricted of to (4 the the components is trivially m to the generic fundamental andanalysis two-index in antisymmetry representations. detail in We describeh section this having generic elliptic fibrations with itly from these polytopesconstruction. to learn about the Tate tuningsreveals that a we second missed tuningthe in of known our the Tate initial type tuning. We also find that applying this novel Tate tuning 6.1 Fibered polytopeOf models the with 18 Tate missing forms respectively Hodge in which pairs, there there is a arefibered standard 1 polytope), + 9 which Hodge has a pairs Tate in form. the large Therefore, we analyze the Tate models explic- Table 17. from straightforward Tate-tuned models. However,fibrations all these that can we be reproducedare discuss by studied in some in flat this section elliptic curves section: in the the base, standard are studied in section JHEP02(2019)087 , - ) 1 ◦ n ) D n (2 (6.1) , and ; 1. We (2 } su ) repre- }} over su . 0 , , } ≥ − = 0 20 , }} , , is violated. 6 0 0 } }} } }} 1 , , , the matter , m 4 8 0 0 8 b 3 1 0 0 , , , , , , , D , 2 4 0 0 4 3 0 0 , , , , h , ≡ { , , 3 0 0 3 1 , it seems that the 0 0 { { { { , 1 } , { , , , , , 0 , , } 0 D } } } } { 2 } } , { 2 0 0 0 , 261 0 3 , , } , , , , 6-curve , 1 , , 2 } } 1 0 0 0 , 0 3 , 0 3 − , , , , 3 , , 45 4 , , 1 0 1 0 , 0 3 { , 0 4 { { { { 2 , , 0 , , , , , , , 0 2 , 0 2 } } } } 2-curve, 1 , , 2 , , 6 4 0 { 1 , − 0 2 tuning is equivalent to that of , , 10 10 , { { 1 , , 3 , 2 , , } , ◦ 0 { 1 , , 5 5 5 } } , { { 2 , 2 , , 1 5 , } , 4 { { 4 (6) 3 , , 5 } } , , { { , 2 4 , 1 5 , } } , in the literature. Via this example, we su 3 , , 4 , , 3 } 0 } , 3 3 , 4 4 } , , 0 0 2 , , 2 , , 6 2 , 0 , , 2 2 , 1 3 , , , 0 0 1 , , 2 , , 3 1 , 1 , { 1 1 , 2 2 , { 3 3 { , { { 1 , , )), the orders of vanishing are 3 , , { { } , , { 1 1 , } , , } , are not specializations of the standard } } 4 , { { 1 – 63 – 9 } 8 } } , 4 0 } , , , , , 2 2 n 2.21 , , 1 } } 4 0 5 , 4 , , 2 3 3 , 4 0 { 240. , , 1 1 , , , 3 , , 4 . 3 , 3 , )–( ) tuning, which we have included in the Tate tunings , 3 3 , 4 4 ◦ { 3 { 3 ≥ , , 1 n ) , , 0 , , { { , 2 2 1 + 1 , 2 2 n , } , } , 2 curve. After incorporating these alternative (2 , , 0 , , 2 2 2.15 2 } 0 } { 1 1 , (6) enhanced on the first 2 2 (2 − , , 4 4 , n su , { { , , 0 , h 0 , 0 , 1 } , , 1 { 1 su su 1 , , 2 2 , } 5 } , { { 0 , 0 , 1 − , 1 1 , } , of the Tate form show the orders of vanishing along each curve { 3 { 3 h , 2 , } 5 } , , { { , 2 curve. However, the corresponding Tate tuning is not just a 1 3 } , 1 1 , n } , } , , 3 , 6 , 4 , 2 − } 4 3 } , 3 3 , 4 4 2 curve, as in the example encountered at large , , , 6 6 , , 1 , (6) Tate tuning , a , 1 , 0 , 0 0 , gives rise to the non-generic three-index antisymmetric ( } . , 4 2 2 , − 2 0 (equations ( , , { 3 3 } , 0 , su , 2 , } ∆ , 0 0 , , a { 1 1 , n } 2 2 { 3 { 3 n 3 , { { , 0 , 2 , , { { 2 ∆ } , , 1 { 1 ) tuning is somehow not allowed in this configuration. We checked explic- } , } , , , a 6 } } , { { , indicated by f, g, 6 } } 6 n 0 2 , 3 2 , } , (6) when the tuning is performed on curves of self-intersection , , { , 2 2 3 , , (6) tuning over a } 6 } 4 (2 0 , , 0 0 , a , f, g, 2 3 , 2 3 , n, n, su , , 1 1 tunings, 1 { 3 , , { su , 4 , 1 su 0 , , 0 a , 3 3 , 4 4 ◦ , 2 2 { 1 , , ) , 2 , = 0 { {{ {{ { , 2 2 , 1 2 n { , 0 , } 2 , , 1 1 , 2 2 (2 ∆ (3) on the second { {{ { 0 , , In the case of a As this is the only polytope associated with the Hodge pair 1 1 su su {{ { { f, g, tunings into our algorithm,{ however, we discoveredsentation that of this seconddescribe Tate an realization example of of thisregions explicitly, of in the primary context interest of a global model that lies outside the obtain a polytopethe of these Hodge numberstunings, using the standard tuning methods because content associated with thea standard physics of the exotic which is problematic asThis the confirms global again symmetry thatthe constraint there on Tate-Zariski has the decomposition to for be a a corresponding Tate-tuned polytope pattern that to is exist. consistent under And we cannot or in terms of traditional itly by performing a tuning whereand we substitute perform in the the vanishing Tate-Zariski order decomposition. The vanishing orders after iteration become an specialization of the found the second version oflisted the in table which shows that there is an In terms of The data JHEP02(2019)087 . , . } ). 3 1 2 + + } 6 2 φ , 2 2 b D 6 4 b 1 (6.2) (6.3) 27 207 ) and , × , 3 2 2 − b , 2 . 18 2 f 2 { , 1 f 0 2 0 φ 2-curve { (37 + 62 1 3 φ 2 − + 818194 b 1 . − 12 2 2 3 2) + 656328 = 2 + 5 = 7, and b b f / − 2 2 2 0 1 , b 2 1 φ 1 57. ψ h 1 12 1 4 = (1 − 1-curves, to obtain the − , ; (6.4) = 0, which gives rise to = − 1-curve in polytopes that , ν 0 = 2 to support the three-index = 57 + 46 2 ··· ··· − φ g 3 1 0 ψ g + 1733688 + 3 + φ = . D 972 + 321867 2 . 2 2 − b (6) on 3 and (3) tuned on the σ − }} σ σ f and ( ]. To see this, we fix the complex ) 4 ) 2 0 su 1 2 = su 2 , 2 0 (shared)) = φ b φ b b 30 0 φ ) (6) on a ( 1-curve leads to this exotic matter ( 1 3 2 of , ) 2 6 from the polytope data automatically 2 2 6 2 b − 12) f su g , φ b − × / − 0 15 values to avoid accidental cancellations, 4 , + + 3 φ f, g (1 f 1 6= 0. Although there is no corresponding × base M:335 6 N:11 6 H:3,243. Both models Z ) does not lead to a good global Zariski and an 2 0 σ σ − − ) − ) 1 φ = β 1 2 2 , F 6.2 b 1 b 2 + 1 12 (37+62 = (37 + 62 D ( ( β 20 2 2 1 1 − 6 ]) b b ; we choose to set 2 f – 64 – , − g 1 curve; when we tune the Tate form × 6 0 1 ψ ⇒ × (358621 + 1496554 2 2 2) 6) 30 − 2 φ from anomaly cancellation: ∆ 3 2 / / = 0 over a point on the curve, while the Tate form ψ / b ) + ) + {− 1 1 4 1 , 2 2 ⇒ where the coefficients are in terms of a second local (1 864 , β 2 b b -tuning on the 1-curve ( ( + 1 72) ◦ − h 1 2 = = (1 (6), as described in [ . 0 0 ∼ b / − 2 ] = 1 ψ 6 4 263 f g λ 1 = b 2 0 0 su , (6) g (6) matter in case of the global model studied above (there 2 φ g ≡ φ × 1 φ , ψ su 11 1 2 su ψ = (1 over the base ( ) = ) = σ 0 { ⇒ + . 2 2 6= 0 but φ λ 1 } 1 and 4 = ⇒ f 6 ) and + 15 α f φ 4 0 2 0 , 4 2 1 σ, b σ, b 1 0 2 1 ⇒ 3 b ( ( 3 to some general enough = 0 over a point but φ φ ψ φ φ φ , g f 0 1 3 1 1 1 3 1 3 × 3 48 12 α λβ φ = 0 over the codimension-two point , 3 1 2 f, g − − tuned on the 1 + = GCD[ β , 5 − ◦ ∼ − 2 0 ∼ − ∼ = f in terms of f 2 α { 2 0 1 0 1 1 + 257716 g (6) φ f g f ψ νφ 3 2 1 leads to 1 3 12 su b } − − and 6= 0 and 6 = 0 : = 0 : = 0 : = 0 : = 0 : = 0 : , as opposed to the ordinary 14 , = = f 3 4 5 1 2 3 0 α 3 5 , = (8 + 35) + (6 20 ∆ 770316 ∆ f g ∆ ∆ ∆ ∆ 3 1 , The conclusion that the The polytope model M:280 11 N:28 9 H:18,206 is a Tate-tuned model of , × • • • • • • 1 2 , 2 h 0 / polytope model with ordinary is no polytope in theand KS the database that tuning givesdecomposition), a we Calabi-Yau with can Hodge contrast the numbers describe two tunings of tunings the of generic model over the ∆ representations is not particular toperformed this a specific local global analysis model. on an Followingon isolated the the same curve, steps, we we see{ that Hence, a 3-index antisymmetric matter field.1 Indeed, we have tocorrect use shifts the of representations the 15 Hodge number then we find (following the notation in [ antisymmetric representation of structure moduli of expand coordinate that we choose to be There is an Interestingly, by explicit analysis, we find thatsatisfy the the conditions for the codimension-two singularity on JHEP02(2019)087 4 − , 2. (6.5) 1 − ≥ − . These now with 20 m 1 2 1 , 17 F 4 , 1 − + 2) , − very large to be 1 , , . Explicit analysis m 4 − n } , − }} 6 4 + ( , 0 ). − 1 , − , 6 , − 18 } 12 1 , 4 − − ) with 126 240 turn out to come from + 2) , , { 220, outside the region we n − 4 1 ( , ≥ m = 1 − − < so , , 1 } , − 1 2 1 1 1 , , , h 2 4 1 − − , , h h 0, 2, 2, 4, 6 − 4 2 , Exotic matter { (16 + 3 ∆ (6)-tuning gives the model with ordi- 1 − − , (6)-tuning gives the model with exotic , , 1 , − su 1 3 1 su , 4 h − − 15 , , − ∆ -tuning on curves of self-intersection 4 1 , ◦ { representation) M:197 9 N:16 8 H:8,150. The 1 − − + 2) , , s, which allows (6) − – 65 – 1 4 } , m 20 su 4 3 − − (6) automatically give different representations on , , − 1 (there is only matter in the fundamental represen- } − + ( 4 1 , su , 1 2 − − 6 ) , , − ≥ − 1 4 {− , m (6)-tuning gives the model M:242 12 N:16 9 H:8,179 and (6) and 2 − − m , , su su − 1 4 , 1 − − , , 0, 1, 3, 3, 6 − 1 4 Ordinary matter { (16 + 2 , − − 12 , , 4 1 (from the four “extra” models of this type) is the polytope M:20 6 {− − − 1 , , , , 4-curves, producing huge shifts of the Hodge numbers. While there are 1 1 4 respectively on a 0-curve. The − h 15 − − ◦ , , , 4 1 Representations of (6) 135 2-curves). For example, consider tuning the generic model over − − { su − -tuning gives the model M:236 10 N:16 8 H:8,178. ◦ Tate form Representations (6) We work out one example here in detail; the others have similar structure. The example The two different Tate forms of su Table 18. (6) and as can beTherefore, determined the by enhanced tunings explicitly should computing give the base polytope of the toric fibration. with the largest N:352 7 H:261,9, which is a Tate-tuned model of the generic polytope model not found by Tatecurves tunings, is it a seems common thatboth this feature increasing large and multiplicities tuning there atrealizations, structure are large and on Hodge many also chains other numbers occurring of in examples at cases of Hodge that numbers this also outside in the have the range Tate database, tuned of interest here. standard Tate tuningsconsidered of for generic starting modelsmodels points. that each have contain These aenhanced are on chain listed the of asonly “huge four tunings” specific in models table of this type among the 18 Hodge pairs in the region of interest cancellation with the respective matter representations (see table 6.1.2 Large Hodge number shifts Four of the “extra” Hodge number pairs in the region tation on su nary matter M:204 11matter N:16 (two 9 half-hypermultiplets H:8,152 in whileHodge the the numbers from the polytope data are consistent with the calculation from anomaly exist in the KSthe database: the all curves with self-intersection JHEP02(2019)087 , , , , , , , , , , } } so }} }} }} , 4 is } } , , , }} }} − 12 11 }} 12 , , , (6) (2) (17) (19) }} , , , }} , }} 12 12 2 , , (5) for , 0 , 0 , 0 , , , so sp so sp 3 3 1 sp 10 }} }} }} 10 11 }} }} }} , , , , , 0 , 0 , 0 , , , 4 5 7 1 8 7 4 1 4 1 , { 2 { 2 { 2 3 , , , 5 3 , , , 4) on , , , } , { , { , { 2 0 3 0 3 0 } , } , } , 2 , , , 4 2 , , , {− {− {− {− − } } } 2 { { 0 2 0 2 0 { , , , , (1) , , , 12 9 12 9 11 1 { { { { { { n } } } } (20) rather than -curve , , , , , , , , } } , , , , , } sp 6 5 6 5 6 5 (2 } } } 5 7 } } } 8 , so ·} m , , , , , , (4) (6) , , , 5 4 2 5 7 1 1 , 7 5 7 5 7 5 so (20) (13) , , , 4 5 , , , 5 2 , , , , , , sp sp , , , 6 5 5 5 6 6 , , 0 1 0 1 0 1 so so , , , 3 5 , , , 5 {− 4 curve to be satisfied; , , 1 1 , , , , , , , , , , 7 5 5 5 7 7 {− 4 4 1 1 1 1 1 1 } − , , , , 2 1 , , , 1 , { { , , { { { { } 0 1 1 1 0 0 5) or {− {− cannot be adjacent to of each , , , , , , , , , 1 1 , , , 1 , , 4’s connecting to {− {− (9) 4 1 1 4 1 2 { { 1 1 1 1 1 { 1 − , , } } (2) , , , − { { { { { { } } su so }} n 4 , , , , , 4 , , {− {− {− {− {− {− su 4 2 12 1 1 4 1 ∆ 4 ], , , , , , , , (2 (6) (3) (15) (20) 2 {− {− 13 }} }} , , }} }} }} }} {− {− {− {− {− {− {− so so sp so sp {− , , , , 0 , , , , , , , , {− f, g, , 12 }} }} 12 12 12 12 4 1 4 1 , { ·} , , , , , 0 }} }} }} 9 }} }} }} }} 9 } , , , 3 , , 3 0 3 0 9 3 6 0 9 6 3 2 1 } {− {− {− {− , , , , , 2 , , , 0 , , , , 0 = 126 as expected above). g , , , , 6 2 , , 2 0 2 0 { 3 0 3 0 3 3 0 , } } } } 1 , , , , { { { { 0 , , , , 0 { 3 , {− , a , , , , , } { 2 0 2 2 2 { 2 0 1 , 4 } } } } , 0 , } { { { { { { { } h (3) (6) {− , } , , , , , } 9 9 9 , , 2 (20) (15) , a , , , 5 , } } } 12 9 } } } } 9 12 g sp sp 3 , , , 5 , 5 , 5 , , , 6 3 3 0 3 3 6 . The choice so so ·} , , 5 , , , 6 6 , , , , , 6 6 3 , , 1 1 , – 66 – , a , , , 5 , 5 , 5 1 curve to be satisfied. Hence, the corresponding 5 6 5 5 5 6 5 sp 4 4 1 2 , , 1 , , , 7 7 , , , , , 7 7 − {− {− {− , , , 1 , 1 , 1 5 7 5 5 5 7 5 , a , , , {− {− {− , , 1 , , , 0 0 , , , , , 0 0 1 } , , } } , , , 1 , 1 , 1 { 1 0 1 1 1 0 1 a } } } , , , 1 , , { 1 { , , 1 { 1 , { , , , 1 1 curves are only determined from this analysis up to 1 1 1 1 { { { { 1 1 1 (2) , 4 , 4 , 4 , { { { { { { { (4) (6) (20) (9) (13) − , , , , 1 1 1 1 , , , su m, {− 3 1 4 1 1 4 3 , (19) has to be chosen for two so sp so so sp { , {− {− {− 2 , , , , , , , , {− {− {− {− 4 1 4 4 1 so , , , , }} {− {− {− {− {− {− {− , , , , , }} }} , , }} , {− . , }} }} }} }} 10 4 and {− {− {− {− {− , , , , , , }} }} }} }} 12 }} }} 11 12 }} , ·} − , , , 5 } } } } } 7 8 1 2 11 12 12 7 5 10 4 (17) , }}} , , , , , , , 0 , , , 0 , 0 , 2 (6) algebras for the global symmetry on the 0 , , , 4 0 3 0 1 3 3 3 3 0 3 2 so ·}} , (6) (1) (2) 1 curves have to be , , , , , , 0 , , , 0 , 0 , { , sp 0 (17) (19) , 0 2 0 2 2 { 2 { 2 { 2 0 2 2 0 {− − , sp sp sp , , , , } { { { { { { { { { { { { , , , 0 so so , , , , , , } , } , } , , , 5 , ·} , , 1 1 1 { , } } } } } } } } } } } } , 4 4 , 8 4 5 1 1 8 7 9 12 9 12 4 5 7 11 2 1 } e , , , , , , , , , , , , , , , {− {− {− , 0 3 , , , 5 6 5 5 6 5 6 5 6 5 6 5 6 5 {− {− , , , , , , , , , , , , , , , , {− , , } } } 12 0 , 2 5 7 5 5 7 5 7 5 7 5 7 5 7 5 } } , , } , , , , , , , , , , , , , , 0 8 1 0 1 1 1 0 1 0 1 0 1 0 1 1 0 {− , e (5) (5) (11) (11) (20) , , , , , , , , , , , , , { , , , 0 , 1 1 1 1 1 1 1 1 1 1 1 1 1 1 , so ·} so sp so sp { { { { { { { { { { { { { { , , , , , , 12 0 , , , , , , , , , , , 12 , , , 4 1 4 1 4 1 { 1 2 4 1 4 4 1 4 1 4 1 1 2 4 , (19) between two 0 {− {− {− {− {− {− {{− {− {− { {− {− {− {− {− {− {− {− {− {{− {− {− {− which give the correctaccording to Hodge the number rank of shifts the (in gauge algebras particular, ∆ one can quickly check that so while the lower rank the global symmetry constraint ongauge the algebras are The gauge algebras on monodromies. We can, however, determine themials. algebras without First, explicitly analyzing from mono- so the the anomaly algebras constraint on analyzeddetermined in from global [ symmetry constraints. For example, it has to be of the polytope gives the data JHEP02(2019)087 8, 0) 6) 3), , , }} , over 0 , we 0 2 , gauge , , 2 ≥ − , 97 B 1 1 , where 1 8 ) e − − D 6) points m ) , , , 100 B 2 1 ( 97 D v − − ( , , 1 . − 3) C , , , respectively. As 8 2 in appendix } , . It describes a non- 2 − 6 } , 11 , 1 // F 3) , (7 , and there are in total / , 1 (2) tuned on , 12 } 2 , 6) points on the 10 1 , , 0 1) su 3-curves are also ruled out − F intersects the 9 components , − / − 13 − 0 , 9 p , // { 2 − 0 F , , − 12 12-curve intersects with the four , 0 15 , 2 − and an − and ) of the non-generic toric fiber over − − (0 13) on 12-curve, which carries the ( , } 97 { // , 8 curves and models involving tunings 3 5 − B.2 8-curve, 8-curve. This occurs in the model M:88 D , ≥ 3) − 12 − − 4 , − , , n 2 1 − 3 ( , , − so 2 , // 36 on a , 5 1 − 12 8 − , – 67 – { e tuned on , − 1 42 8 algebra on any curve of self-intersection e − − // , 8 ( e , 3 12 top in equation ( − 0) , − , 8 2 ) restricting to each irreducible component, which corre- e 1 algebras tuned on − // 8-curve to become a − polytope has vertices 8 11 curves. As we discussed already in that context, over (4 , e 3.7 , extended Dynkin diagram, but over four distinct (4 − 2 12 0 − = 0 is trivially satisfied over these four points, and the elliptic ∇ 8 , − e − , 97 1 D // | 1-curves. Now we can calculate the Hodge number shifts of the flat p 3), (0 {− 11 10 and , 3-curves. In the latter case, the “extra” models in the KS database 6) points, we expect that there are four (4 − 2 , − − − , , 6 ) that are the boundary of the 3-dimensional face in a locus comprising 9 , in equation ( // are blown up and the proper transform − in the base. There is an (0 p B.3 12 , i 6) points and the 2D fiber can be understood by an explicit analysis of the 97 , D 2) 13) on {− D intersects also the whole irreducible component corresponding to (( stands for , , 1 p ≥ 6) singularity. Similarly, tunings of , ’s, which form the , 78 , // 1 1 , n 97 P ( − The corresponding flat elliptic fibration model has a non-toric base where the four The (4 We begin with an example of a tuned ) in the top; i.e., , . Analogous to the models over Hirzebruch surfaces D 0 5 so 1 252 97 { pt − fiber over the toric basethese contains four this points. irreducible component, which is two-dimensional, at points on exceptional divisor find in this casein that over equation a ( generic pointnine on the on ( algebra without (4 which the resolved fiber become two-dimensional. hypersurface sponds to a lattice pointD in the where 101 curves the orders of vanishingit are needs enhanced four blowups to for a 8 N:356 8 H:258,60.( The flat Tate-tuning of the generic elliptic fibration elliptic fibration model over adescribe blown models up that base involve that isof generally non-toric. In thisin section our we region of interestgroup involve is a generated further associated complicationdetailed with in example an which with abelian a that U(1) nontrivial additional factor Mordell-Weil structure in is the relegated F-theory to gauge appendix group; a obtained in Tatemore tunings complicated of generalizations Kodaira offibrations the type. over non-flat structure Thispoints we set the have resolved already of fiber described in tuningsthe for the Calabi-Yau geometry can polytope by resolving model be the is base understood two-dimensional, at these but as points we to can obtain understand a corresponding flat We have not consideredas tuning it an leads toof a a violation (4 ofby the anomaly anomaly cancellation. conditionsthat that appear corresponds Nonetheless, to to there the contain are appearance these polytope tunings, models which in give rise the to KS Hodge database pairs that we have not 6.1.3 Tate-tuned models corresponding to non-toric bases JHEP02(2019)087 = 3- − c (12) H so 12, all 4-curves ≤ } − 0 6 n )-tunings on n ( ≤ } , pt } , M:348 5 N:12 0 5 0 5 0 5 (11) tuning, in so 18, which gives 3 − so F ): , pt , pt } , pt } -fibered form over 3 13)-tunings on 0 4 0 4 00 4 00 4 0 4 1 3-curve , 6) singularities and = F 3 , -fibered polytopes at , − ≥ 1 1 2 , , pt , pt , pt , pt , pt 4-curves that support , 2 0 3 0 3 0 3 0 3 0 3 3 133) + 3, and ∆ P n , for discussion. − that do not also involve h ( 2 (10) to 1 − P , pt , pt , pt , pt , pt , so 4.6 0 2 00 2 00 2 00 2 00 2 1 so h 3-curve. For 7 . Note that the tuning from , pt , pt , pt , pt , pt − 00 1 00 1 00 1 00 1 00 1 19 pt pt pt pt pt = (248 ), so the Hodge numbers for these top over the { { { { { = 4 (each blowup contributes one 3-curve. This also gives a reflexive = 6 and ∆ 1 3-curves to − V 1 T , − 1 ) on the h n 3-curves, there is no bound on ( − ), ∆ 3-curves give rise to (4 so − 13. While the anomaly conditions impose an 3 curves, and also have extra sections associated 2.7 7) + 1, ∆ – 68 – − ≥ − give a model in the KS database as expected, and n 16 ) and ( = (8 (8), 2.6 , as needed. r )-tunings over so } )-tunings on n ( ]. Therefore, in these cases the corresponding flat elliptic n 60 ( polytope model M:342 8 N:15 7 H:6,248 M:339 8 N:16 7 H:7,247 M:332 10 N:17 8 H:8,242 M:328 8 N:18 7 H:8,242 M:318 10 N:19 8 H:9,233 , so 13 13) tunings on so (8) in reflexive polytope models; see section (13) polytope tuning on the } } } } } 258 so ≥ 4 4 5 5 6 { so , , , , , n 13. These are flat elliptic fibration models, where the Hodge numbers can ( 2 3 3 3 3 = , , , , , so } 2 2 2 3 3 = 12 for , , , , , n < 1 1 1 1 1 18 n , , , , , − 1 1 1 1 1 Tate form , { { { { { 6 (11) is a rank-preserving tuning (see table { , and focus in the rest of this section on the issue of Polytope tunings of M:348 5 N:12 5 H:5,251 (generic model over that were missed in our Tate tuning set have a combination of two novel features: so + 7 9 n C 10 11 12 1 } , 13) without suffering from (4,6) points. base. But this is a non-flat elliptic fibration. In fact, we know immediately from 1 ; therefore, by equations ( 78 h 3 3-curve with 2 (13) would be a rank-preserving tuning, and, as for the ≥ , Consider now the We start with a generic polytope model over the Hirzebruch surface As mentioned above, The remaining four Hodge pairs corresponding to standard F − We do not expect tuned n so × (10) to ( 16 252 polytope, M:312 8 N:20the 7 H:10,232, which isthe still Hodge of numbers the that standard to there is some additional subtlety in this tuning. Naively, the Hodge numbers offrom these anomalies. polytope We modelsso list agree these with polytope thecases models Hodge are in numbers identical. table calculated fibration models can beso obtained by resolving the 5 H:5,251, and performthese successive polytope tunings tunings, of except curves in the context ofthe simpler U(1) models issue. with relatively small two-dimensional resolved fibers when upper bound of from anomaly conditions [ large they have apparent with a nontrivial Mordell-Weil rank and correspondingFor U(1) clarity, factors we in delegate the F-theory aappendix physics. complete example of one of the “extra” models of this type to elliptic fibration model viaadditional anomaly tensor cancellation: multiplet), ∆ 10 ∆ { Table 19. the be directly calculated from the anomaly cancellation conditions. JHEP02(2019)087 1 0 5 , = − 4- 1- 3 , pt c − − 2 base 29 P H 3 − ∆ 4-curve 4-curve F 26, over − − − 8) T − n < ) with suitable ≤ , the components for an explicit 29∆ ) on a 19. The two Hodge (22) is tuned on + 1) is the same as 13 C − 3.7 n ’s which form the − = (78 ( 1 so N × 3-curve, the hyper- V 3-curve, over which c = ), 13 P 3-curve is blown up. ) is not apparent in so 6) points. Therefore, − n − , H ( − m 12 ( ∆ polytopes are still = ∆ so × 3-curve gives rise to the 108, which agree with the 1 sp 4 − , ∇ − 2 − − T h . } 29 4-curve are 5 − − 29∆ -fibered polytopes 251 1 8) , , − 5 3 − , 3-curve, and the (26), at which point all monomials in V in the KS database that were missed 2 6) point on the − 6) point on the , P (22) on the so 1 , , 1 } − { so 4 (shared)) = = (66 = ∆ h c × ) is forced to arise on the exceptional 1 , (12), we can also do the calculation as if it were a 1-curve. Therefore, the shifts of the Hodge 232 H 2 m , so 22 − ∆ h ( ) (the Cartan subgroup of SO(2 2)+2) = 12 and ∆ 1-curve. The forced 10 1 × − – 69 – sp (13) tuning on an { 2 − − 1-curve that intersects the proper transform T − / so (13) top in a locus containing − 1 13 ) tunings without producing (4 ), the CY hypersurface equations ( ( 29∆ so − n ( 4 × − 17 so × V (2) on the sp = 1+((11 2) = 5 and ∆ in the . 1-curve replacing the = ∆ r 1-curve in computing the Hodge numbers from the anomaly +12 } 2 − − , − 00 6 20 1 22 h +∆ ). For example, tuning × , pt T 0 6 which agrees with 2.5 (14 , pt 17 = 17: a simple local analysis shows that tuning in table 00 4 − = ∆ = 1 + (6 4) on an intersecting n 3 1 19, r , 29 F , pt 1 − 0 3 − h e − ) and ( (13) is a rank-preserving tuning of 4 + ∆ 1-curve does not exist. But we have to carefully consider this forced gauge so , pt ) = 2.1 00 2 − T n/ 1 d ∆) of the polytope model, which is the non-flat model over the original ( , pt − 8)+10) 00 1 sp = ∆ )). As 4-curve, which forces an 3 curve of − f, g, 13 1 ) tuning, an additional gauge factor N In the flat elliptic fibration model over the resolved base, as we keep increasing the Again, we can calculate the Hodge numbers by considering the corresponding flat , pt , − − ( Note that although the representations of 1 0 1 n are tuned off, and a U(1) global factor comes into play. See appendix ( (13) extended Dynkin diagram, and there is a (4 h 17 (12) tuning, in which case ∆ × 6 pt fibered. The failure to be in thethat Tate are form charged arises underSO(2 from the the Cartan are featureso 5 that there arenumber lattice shifts are the same. 6.2 Weierstrass models fromFor non-standard the remaining eight Hodgeby our pairs Tate with construction large (seehomogeneous table coordinates cannot be in Tate form, although the successive tunings can be calculateda this way up to analysis of one ofbecomes the relevant. models We associated list with thethe the non-flat polytope missing models Hodge of pairs tuning where such a U(1) intersecting an exceptional the numbers are ∆ ((231 Hodge numbers from the polytope. The Hodge numbers of the polytope models from the the ( where the algebra on the exceptional equations ( model M:179 10 N:29 8 H:17,143. The corresponding flat fibration model has a 5 so curve starting at forces in the top. elliptic fibration model overThis the blow-up base produces where an thecurve, exceptional (4 and which can support any ∆ do not. An explicitsurface analysis equation intersects shows with that seven over{ components a associated generic with point theso on seven lattice the points the fiber contains the whole irreducible component associated with the lattice point the absence of other issues these should have the same Hodge numbers, but they clearly JHEP02(2019)087 3 x 1), , non- 1 − , , , , , , , )-tunings on }} }} }} }} }} } } } } } 3 n 3 3 3 3 3 3 3 3 3 ( b b b b b b b b b b 3 0), ( 4 5 4 5 4 5 4 4 3 so c , c c c c c c c c c 0 ): + , + + + + + + + + + 3 1 } . Therefore, the set F , 1 1 1 1 1 1 1 1 1 b b b b b b b b b b 2 }} 3 4 3 4 3 4 3 3 2 c , and the lattice points 0 c c c c c c c c c , } , , , , , , , , , , 0 1), ( 0 8 0 8 0 8 0 8 0 8 0 8 0 8 0 8 8 0 0 14 1) , , , 0 pt pt pt pt pt pt pt pt pt 2 6) point pt , , { { { { { { { { { { (1 , , , , , , , , , − , 0 , , }} }} }} }} } } } } } } } } } }} }} }} }} } { 3 3 3 3 3 3 3 3 3 3 3 3 3 ), and such that the projection 3 3 3 3 3 , 1) , b b b b b b b b b b b b b b b b b b } 4 5 4 5 4 5 4 5 4 5 4 4 3 4 5 4 4 3 − 0 c c c c c c c c c c c c c -fibered polytopes that we have de- c c c c c , , 3.12 1 , 0 + + + + + + + + + + + + + + + + + + 3 (1 , 1), ( , , 1 1 1 1 1 1 1 1 1 1 1 1 1 2 0 1 1 1 1 1 b b b b b b b b b b b b b − , b b b b b P 1) 3 4 3 4 3 4 3 4 3 4 3 3 2 , 3 4 3 3 2 0 , c c c c c c c c c c c c c fibers. We refer to such polytopes as c c c c c 1 6)-point are blown up. { -fibered polytope. Geometrically this fea- 2 , , , , , , , , , , , , , , , , 1 , , , , , 1 , , 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 5 0 } − 3 , 3 , 0 11 0 11 0 11 0 11 0 11 , ( 0 pt pt pt pt pt pt pt pt pt pt pt pt pt 2 2 ( , { 2D component, (4 pt pt pt pt pt P P { 0 { {{ {{ {{ {{ {{ {{ {{ {{ {{ { {{ { {{ { {{ { {{ { , is – 70 – 1 2 , 0 { , -fibered polytope construction in the coordinates given } 1 , 0 3 , , 2 0 , P . Let us first show the forward direction: we already showed 0 are as given in equation ( , } 1 2 1) 26. These are non-flat elliptic fibration models. The last column gives , {{ ∇ 1 , = n < , π ≤ , although they still have in tops). The Hodge numbers can be calculated from the associated flat elliptic 1), ( , 3.4 0 -fibered polytopes. In fact, the feature of having base-dependent terms in pt polytope model M:312 8 N:20 7 H:10,232 M:299 10 N:21 8 H:11,221 M:292 8 N:22 7 H:11,221 M:276 10 N:23 8 H:12,206 M:267 8 N:24 7 H:13,205 M:248 10 N:25 8 H:14,188 M:238 8 N:26 7 H:14,188 M:216 10 N:27 8 H:15,167 M:204 8 N:28 7 H:16,166 M:179 10 N:29 8 H:17,143 M:166 8 N:30 7 H:17,143 M:138 8 N:31 7 H:18,116 M:123 6 N:32 6 H:19,115 , 1 polytopes do not the form of standard , that the standard subpolytope 3 , , 2 1 Polytope tunings of M:348 5 N:12 5 H:5,251 (generic model over ∇ for , n 3 P 3.4 13 14 15 16 17 18 19 20 21 22 23 24 25 , 8 ; i.e., these should be sections of non-trivial line bundles over the base. 2 2 0), ( P 6) points and the corresponding 2D toric fiber components contained in the hypersurface CY y , is equivalent to being a non-standard , 3-curve with 13 1 These 2 − , or y , 3 ( in section matrix to the base is of the vertices of thein dual subpolytope ∆ ∆ are all in one of the forms in the set standard or ture corresponds to the conditionto that each there is of only the a verticeslows: single associated lattice without with point loss in these of ∆ generality, monomials.of we that the choose We projects prove a this coordinate system equivalence as such that fol- the three vertices points in ∆ that give risex to non-trivial base dependence in the coefficients of the monomials fined in section Table 20. the the (4 (see table fibration model over the non-toric base where the (4 JHEP02(2019)087 - 2 ]. , 1 1. 1) , 1 , 3 33 , 1 − − 2 , P ξ P 1 2 , 1) over -fibered 1 , − , ≥ 1). There 3 , , 1 − 2 , 3), we treat , , η P 0 , 1 -fibered poly- , 2 , 1 , , + 1 3 , , 0 fiber, as depicted 2 1) that project to ξ , 1) and ( 2 P 1) that are at least , , − ≤ 1 1), (0 2 , , − 2). The blowup rays , 1 1 , − η , 2 , -fibered polytope, and = (0 P , 1 2 0 , − , -fibered polytope model , , z 1 , terms (assuming that the 1) and ( , 2 0 , v 1 0 , , 1 3 , , 2 P 1 -fibered polytope. − , x P 1 1), , , 2 3 1) and (1 = (0 , y − , , 2 , z -fibered polytope can be put in a 2 P 0 1) and (0 v that satisfy 1 , ] to obtain the Weierstrass model of , , 3) will be included in the polytope. − 3 0 2 , , , 2 31 2 − 1), , ξ, η P in this coordinate system. This proves , ) 3); the conditions over these two points − 0 , 2 B , , ( -fibered polytope, we can obtain similarly i, 2 0 = (0 2 , , , ) , v ∈ ∇ 1 0 , ) y 2 B 1); this linear transformation will also leave 1 , ( 1 i, B 3) ( , v − P i, , , , v v ) – 71 – 2 0) fiber as a twice blown up 1 1 , , , = (0 B 1 ) ( i, 1 , ) — the hypersurface equation of a non-standard 0 2 3 y v B , , ( − i, 2 , , v 5b , and embed the elliptic fiber in this blownup P 1) associated with the 0 , v ). The presence of these two points in ∆ shows that 2 ) , 0) , − 1 ) has coordinates 1 B , , , ( i, 1 1 v 3.12 (1 P − , = (0 , , ξ, η 1) and (0 has the form of a standard ) 0 , 1) x 2 , B , 2 v ( i, -fibered polytope. Note that because even non-standard ∇ 2 fiber resolve the singularities of the tunings. 2 − . We can prove the backward direction as follows: assume there is , , v , − 1 2 2 ) , , 0 = (0 1 1 1 B , , ( i, P x 1 v v P 1) in ∆ − , ; following the procedure in appendix A of [ fiber as a blownup ), non-toric curves can be of higher genus, and this class of global Weierstrass 1 reflect the fact that two of the nine sections of a 1 5a P , ) gives a dual polytope ∆ that contains at most the single points corresponding 3 2 , , ambient fiber with 2 1 -fibered polytope is a specialization of that of a generic , 1) and (1 (0) at the vertices ( The Weierstrass models obtained in this way from non-standard We would like to have a Weierstrass description of the non-standard P 1 1 1 , 3.13 , , P 3 3 O 2 , , 2 2 − topes have the novel featurein that the they base. can Moreover, have unlikephic the gauge to toric groups curves, tuned which over aremodels non-toric always gives curves genus examples zero of curves tunings (isomor- a of gauge check on groups over this higher picture, genuscalculated we curves from can in anomaly verify the that cancellation bases. the match As Hodge with numbers those of of these the Weierstrass polytope models data. of are completely tuned offP (see figure the blowups of the this would not beExplicitly true in coordinates, for example insteadP of if treating the the originalthe elliptic elliptic fiber fibration as hadambient being no fiber embedded section with in [ the in figure the associated Jacobian fibration model ofthat a of the blownup fibered polytopes giveJacobian elliptic fibration Calabi-Yau should have hypersurfaces the same that geometry have as the a original global Calabi-Yau hypersurface; section, the that the presence ofimplies a that single the lattice polytope point of each of thepolytopes forms so ( that wegeometry. can use To this the end, methodology we of treat F-theory the to understand and analyze the on the points in ∆as strong over each as those of imposed thecan by points the be other ray than ( weaker, ( butthese two as points long in the as dualThus, there fiber, for the is each ray ray only ( in the the one base ( point (0 only a single lattice pointis in always ∆ taking a each linearthese of to transformation the forms the that ( points leavesthe (0 the form last of the twoevery fiber coordinates lattice point fixed fixed ( as thatFor ( each moves ray in the base, however, the presence of any such lattice point imposes conditions to base is compact), and boththe origin of as these an points interiorcoordinate must point. be system where present Thus, it for any has standard the( only polytope the ∆ points to (0 contain in ( JHEP02(2019)087 = 1 , 1 3 , , 3 2 . , 3 2 P in the x P or (2) tuning 2 is the dual x, y, z y su -fibered poly- is the dual of 1 231 , 3 , 2 , labeled in terms 112 P 112 . This interpretation allows the y , we analyze the eight remaining 2 x -fibered polytope, in which there are . The monomials in different sections 2 and the red triangle ∆ , 1 The blue triangle ∆ 231 , 6.2.2 , we give a model with an 1 and -fibered polytopes, we contrast the two ∇ 1 P 112 , 4 , we give a simple example that illustrates ambient toric fiber (in blue) and the 3 of the homogeneous coordinates fiber. The equationssurface Calabi-Yau describing for a the non-standard polytope hyper- can be characterizeda as tunings for no nonzero monomials in the sectionsx labeled possibility of having sections in (b) ∇ of are categorized byent projection lattice to points the in differ- ∆ , 2 2 , 6.2.3 P 1 , 1 6.2.1 P – 72 – 231 112 ∇ ∇ (red) 1 , 3 , for stan- 2 P 1a from the KS database that were missing in the Tate-tuned 1 , 1 h (blue). The red triangle -fibered polytopes). The reflexive polytope pairs for the 1 2 , , ambient toric fiber (in red). 3 1 , : M:346 8 N:16 7 H:8,250 and M:345 8 N:17 7 H:8,250. , 2 2 1 } , P P 1 -fibered polytope construction. In section , Comparing the polytopes for 1 We give some examples of Weierstrass models from non-standard 1 , 250 P 3 , , 2 2 8 comes from blowing up the (fiber)twice fan (cf. of indicated rays indard figure (a) and on a non-toric curve of genus one6.2.1 in the base. A warmupAs example an illustration of thepolytopes two in different the types KS of database associated{ with Calabi-Yau threefolds having Hodge numbers the non-toric curve enhancement feature.polytope In data section with large construction. We also givedatabase some at further smaller examples HodgeP of numbers interesting to geometries illustrate from the the unusual KS nature of the non-standard Figure 5. Bl topes in the following subsections. In section JHEP02(2019)087 } 4 1st 2); 2), 0), (1) 2nd are , , , , b ∇ 1 1 ∇ } sp 3 , , 1) and 3) 0 0 , b ⊕ -fibered , , , 2 − 35 4 1 , 2 , b 1), (1 3 , , b 9 3 , , (9) 1 4 2 b b (0 P so − { 3) from { 3), (1 -fibered form. , , 1 , 1 , 3) that we have 2 2 3 , , , -fibered polytope , − and those for the -fibered polytope 3 2 2 22 4 0 ( 1 is a non-standard 1 2 b , b , , , } , P , , and calculate the 3 . 1 6 3 0 , with , , 2 } ) b 2 , 4 1 3) , 1st , P F polytope of either one 1 70 4 P 3) 2 , b 2 ∇ , , 1 2 2 3 2), (0 1) give monomials of the 17 3 , which becomes a vertex 1 (8) , , b , , b 0 4 so 1 − 2 10 2 , , , polytope, which is then the 2nd − b 0 − ) by requiring the vanishing 2 , { 4 , 15 1 1 is a non-standard − 0 F b + 6291082311776640 , 2nd , 2 2 − b ( respectively. 3), (0 1st ∇ (0 528582381600 , 7 , 3 , ··· 1 1 . The base rays are ∇ 2 1) to ∆ 3) b (9) top over the base ray (0 x 3 -fibered polytope, with vertices (in , } , 1) , 2 corresponds to the one lattice point b 1 3), (0 0 2 , , b 3) to be 2 so , , 3 2 , , , − · · · − 2 , 2 2 1 = 0 , (1 , . The ) − 2 3 2nd , P 0 , 4 b , } − 2 0 and , 24 4 0 4 ∇ , b , − 3) b 1 and the condition that ∆ has lattice points , , 0) lacks the preimage (1 3 { , 6 3 , 1 3 (0 , + 217077176379771 − (1 x b 1 values in the complex structure moduli): 2 , , ∇ 4 , − 2 − b 69 4 – 73 – 1 Z 2) , b , , ! 1 that have base dependence): the base rays of 1 , 17 3 2 (0 , b + 1344055426083360 1 b 2 0), ( { 2 0 { , 3 y b , (109370724968448 b 1 , 0) comes from the projection of the 4D ray (1 2 6 2 ) polytope, has an (1 , ) − b or , as a subpolytope, and therefore 4 , 36 4 3 b 16 1 b + 388841808 0 (8) 3) 1 b 9 3 , , x , so b 3 , but the ray (1 23 4 , 2 (0 2 b , 3) to be { + 24 6 3 P 1 , 2nd b , 2 2 b , ∇ . This is a Tate-tuned model over the base (0 1 polytope is a standard (344205633835899813888000 } { 3 3 − (35 . Let us now show explicitly that 0). b 1009274573279509056 + 34622237106205930350000 , 7 3 1) , 4-curve and the 0-curve b polytope is, on the other hand, of a non-standard 2nd − − (0 2 2nd ( , and in particular, − still has , ∇ / 2 3 } + 681083735457852 864) b 1st / 4) ! 1st ∇ 0, 0 48 ··· 274589065851262777907525390625 22258660320 (1 − , / 3 ∇ , − b + 1926547706542277636888364004147200 + 27125536688271 + 588208065199776 + − − 1 (8) KS models (a generic elliptic fibration over − = = 1 with base-dependent coefficients, 3), (0 g . The one lattice point reduction of f so 3 ∆ = 19683 , x (1) top over (1 2 -fibered polytope. The equivalent condition for a non-standard 1st 1), ( Although we do not have a Weierstrass model from a Tate form for this polytope, The first The second , 1 , sp 0 − 3 , , , 2 we instead have a(where Weierstrass form we have for substituted the some hypersurface generic in the Bl P is also satisfied fromto the the ∆ set point of ofset view: 4D of rays let monomials. us Theform associate two base lattice coordinates points ( of an appropriate preimageassociated of with the monomials base in in are the same as thoseremoved; of instead, the basenonetheless ray (1 The data of thisor polytope equivalently by can adding be theof obtained lattice ∆ point by ( removing theenhancement of vertex ∆ (1 polytope and check that it satisfies each of the two equivalent conditions (i.e., the absence coordinate dual of the reduction ofan the ∆ of the orders with respect to the coordinate the standard coordinates) (1 enhanced on the (0 lattice points. This polytope can be obtained by tuning the ∆ JHEP02(2019)087 , 2 11 31 = } b 2 } 54 5 77 18 , so , 28 30 b b z (6.8) (6.7) (6.6) } 0 b , region 53 7 76 11 . , v 77 11 3 0 b b -fibered } y 1 , b , { , fibration 1 as a Bl , 0 , 52 9 75 15 1 } , , v 3 17 29 b b 1 , h 1) , x 1 b 2 1) ∇ 257 3 , = 0) and an 51 11 74 19 v , , P { − 23 28 1 b b 2 { − 3 , b 96 , , P b 50 2 73 23 b 2 1 1 29 27 b b , , b − 5 95 9 13 72 0 b b , (0 35 26 b , 2 , b 49 94 4 term but with base 11 71 b b − (0 1) 6 25 b , , 2 , b 48 93 25 1 y 9 70 b b 1 4-curve ( have a preimage of the 1) 37 24 b − b − 47 8 7 2 92 21 − − , b b b , . Note that this non-toric , 9 31 23 2-curve. 101 1 } , 1 b 46 7 69 91 17 , , 1 − b b b − , 1 25 22 1 100 1 , in both cases. While in this b 45 12 68 3 90 13 , , = 0 − b b b is a non-standard } // h 1 , where ( 4- and the 4-curve at one point. 19 21 99 4 , , } b 44 5 67 15 9 22 − b 12 − b b b ) ( ∇ 1) 250 32 20 B , . We analyze the Weierstrass model , =98 b 43 13 66 2 89 9 − ( = 0 i 1 b b b } 8 v 12 2 , { + 24 b 42 8 65 3 88 23 // 1 32 b b b 2 , 13 19 b − 12 3 b 41 11 64 4 87 14 , b b b − 35 4 33 18 − the vanishing order is enhanced to ( non-toric { b { 40 14 63 5 86 19 , (1) can be trivially transformed into a toric b b b b } 20 17 // 1) b sp 4 17 62 85 10 – 74 – , b b b 12 ≡ { 27 16 = 0 (9) enhancement on the 1 b 39 3 61 7 84 24 shows that , − b b b so 7 34 15 } 100 b ), and find the associated tuned Weierstrass model. 29 38 19 60 6 83 − b // b b b ∆ } , 41 14 b 5 37 17 6 5 82 12 2) (1 non-toric (2) gauge symmetry on the b b b , , ≡ { 7 13 1 − f, g, b D 31 36 16 59 4 81 su , 1) 99 b b b { 36 12 , 0 b 100 // b 26 35 13 58 7 80 , 2 polytope; as in the preceding example we treat 98 b b b D 29 11 − b 12 1 b 21 34 10 57 23 8 , , b b b 3 0 (1) algebra intersects both the − , 22 10 101 , 2 b 16 33 17 56 3 79 b 1), (0 sp 1 b b b P // (0 b − , 6 27 32 7 55 8 78 100 , c b b b 12 b 0 1) 7 , , c + − 0 1 , , = // 3 (0 , 3). − 11 , , 0) 2 3 , − , 1 − ( , non-toric // − { b Over the toric curve which corresponds to an Over the non-toric curve , -fibered polytope (in particular, the fiber coordinates are associated to The generic Weierstrass model over this base has the Hodge numbers As a specific example, we consider the polytope M:65 8 N:357 8 H:261,45. The vertex 0 2 12 , • • , 1 , (1) enhancement on the non-toric 0-curve 1 {− (0 the tunings must beof such that the the non-standard shiftsP are { The resulting computation of There are in totalform 102 ( base rays, and all rays but where both the latticedependence. points Performing in the thepolytope projection second over the line we base contribute find to that a that we did notstructure. obtain We through go Tate through tunings one and example that in have detail; non-standard theset others of have ∆ similar is structure. case, the non-toric curvecurve by supporting a the simpleexamples linear that change we of consider variables, in this the is later not6.2.2 subsections. the case in The the eight moreNow remaining complicated let missing us cases come at back large to the polytopes of the eight Hodge pairs in the large sp curve is a (rational) 0-curvecurve because supporting it the is inThis is the essentially same the class same asan configuration the as anomaly the two calculation second toric model, also 0-curves. so give The the the Hodge numbers same from result, According to this analysis, there is an JHEP02(2019)087 ) 1 − 11 31 6.8 , b (6.9) 2 18 54 5 77 (6.10) 28 30 b b − are the 9-curve b , S. 11 53 7 76 2 − 11 3 b b (3) gauge (see also 97 ∈ 6) point is b b − , 15 52 9 75 . , su I 6) point, we 17 29 b b 6)-points, as } 1 , . b ∀ , 4.7 19 51 11 74 , } 1) − 23 28 b b and , b 96 0 12-curve, and the 23 50 2 73 b , = 0 29 27 b b − 102 b 1 9 5 95 b 13 72 b b ( − 35 26 b Dynkin diagram, just , b 4 49 94 } 8 11 71 1 b b 6 25 b ). The divisor classes of E b − 25 48 93 are constant coefficients, . Note that the non-toric ( non-toric 9 70 b b k = 0 37 24 , b b i ), so the 9 components that 15 b 21 47 8 7 2 92 c { D 2) b b b 97 31 23 , , as can be seen from the first b b 17 46 7 69 91 B.2 1 { b b b ), and there are no intersections , and 25 22 101 9-curve into a 6) point. Over this point the CY 1 b 13 45 12 68 3 90 , ), we know that the (4 } D − b b b 6.8 − in table 19 21 and , b 22 44 5 67 15 9 + 1 } b b b } = 0 6.10 32 20 − b 9 43 13 66 2 89 ( , which corresponds to an 100 b b b 97 404 , } 12 2 b = 0 , D b 3 23 42 8 65 3 88 { 2) b b b , , 13 19 14 + 0 b 102 1 14 41 11 64 4 87 { ) and ( , 9-curve), there is a two-dimensional resolved b , b b b 0 99 33 18 { 2 − { b 19 40 14 63 5 86 6.8 – 75 – D b b b − 20 17 , (a b + 10 4 17 62 85 2 b b b } 27 16 − 98 s that compose the extended b 24 39 3 61 7 84 ( 9-curve such as we discussed in section 1 b b b , D 34 15 P − = 0 b 29 38 19 60 6 83 2) b b b , 41 14 97 1 b 5 37 17 6 5 82 b , b b b 7 13 3 9-curve, there is only one (4 b 31 36 16 59 4 81 − b b b 99 − ≡ { , 36 12 b ). However, as opposed to having three distinct (4 b 3 26 35 13 58 7 80 b b b 98 97 29 11 − b ( b 21 34 10 57 23 8 D B.3 { b b b 22 10 101 b 16 33 17 56 3 79 b = 1 b b b ). The complicated combination of powers in the second term in ( b ): the top is the same as that in equation ( S 6 27 32 7 55 8 78 100 c b b b b 6.8 B are the variables associated with toric divisors 7 9-curve in a locus of c + k − b = I | p 9-curve, there are some differences between the fiber structure over this case, there is only one irreducible component that intersects the CY) Explicitly, are the boundary of thein 3-dimensional the face intersect with theas CY over in a equation generic point ( occur in the isolated intersects the four irreducible components interior to the 3-face (while in the previous Over the curve fiber; due, however,− to the enhancement overand the the non-toric one curve inappendix intersecting an the isolated in the preceding example, thisto non-toric the curve combination is of a curves term 0-curve, and in is ( linearly equivalent arise from the structurebuild those of rays the from toric a fiber rays of and a the minimal sequence model of Hirzebruch base. blow-ups needed to symmetry on the non-toricwhile curve. Incurve this intersects the expression, two toriconly coordinates curves that do notbetween appear the in divisors equation associated ( with the variables in the first and second terms). As the vanishing order is enhanced to Moreover, by comparing equations ( exactly at the intersection of the divisors We now find the associated flat elliptic fibration model, so that we can use F-theory • techniques to compute the Hodgeis number semi-toric, shifts. and We first then identifyblow determine up the the successively resolved three tunings. base, times at which Since thisnon-toric point there 0-curve to is is turn replaced the by the a only chain(similar of one to curves of (4 the self-intersection numbers last graph of the Hodge pair JHEP02(2019)087 × are from 5. . The + 4 (6.12) (6.13) (6.11) 1 − , 97 3 2 ˜ 12-curve D = h × − (2) on the (3) on the 2 non-toric . Note that su ˜ su (12 D × } and 4 − 1 32 , 1 − − and h , , in the middle chain 4 } { 97 = 3 96 ˜ D = D = (3 + 8) . } 1 , 3 1 charge , 2 ,. . . , 2 E h 2 H . Left: before resolution. Right: h . This forces an } . ∆ ∆ ,D , } and the bottom curve is , 45 , 1 3 and 1 , − 1 98 , D E E 1 , 102 { V 3 h 261 ,D − non-toric D − { 1, connecting E ∆ 99 ˜ 2 D { − . = ∆ E − , ,D 1 2 2 6 , − 2 E 100 − 1 h , non-toric = +1 + 2 and ∆ ) are E 2 = ,D 1 D , 2 − 1-curve, − 1 – 76 – ˜ 2.7 101 E = h − 97 D , { . The configuration of the intersecting curves and D 2 : ∆ 9-curve after the three blowups E } = − ) and ( − = +1 and ∆ non-toric 97 1 ˜ 2.6 r ˜ non-toric D D E ˜ 27. D non-toric = − ˜ = ∆ D 1 , ˜ 1 , 1 E 1 ˜ E h { , in the right chain } : ∆ 3 , which is now on the are the exceptional divisors associated with the three blowups. The proper 100 ,E (shared)) = (3) on 3 2 D ˜ 3 E 1-curve. The three curves, su , connecting to , ,E 1 1 2 − × ˜ ⊕ ˜ non-toric E The base of the example with Hodge numbers E 2 , ,E D 1 (2) (2) on − E su 2 su is the proper transform of the Remarkably, the described configuration gives the correct counting of the shifts in • • non-toric 97 2-curve, ˜ D ˜ The total gives thethe desired final Hodge Hodge numbers numberand correspond shifts correctly to reflect the the flat associated elliptic contribution of fibration three over tensor the multiplets non-toric from base, the three Hodge numbers through the anomalythe calculation. tunings through The equations contributions ( to Now we figure out0-curve, the gauge symmetries− on these divisors.the symmetry There enhancements was are an drawn in figure which is a respectively D where transform of the non-toric curve is after resolving (4,6) pointscurves in in the the base. left{ The chain top from curve top is to down are in the order the curves after the blow-up process can be determined in the usual fashion: the Figure 6. JHEP02(2019)087 . , 1 } , by by 1 54 1 h 1 , , , 2 (6.14) 2 h non-toric h and the 260 D { , 100 } , D and }} 62 12 1 0 , , E − 1 − 260 − // { , that we have treated (2)) = 1 and 96 2 17. , 12 } 5. (2)) = 1 and D 11-curve are understood to su } − − ( , − − 45 − r su 51 = 1 = , ( , / = r − // 2 96 , by 10 2 9-curve in the base and one ˜ 261 7 12 261 D × 1 − { , by − { − × 1 / − , 9 h 4 10 1 1 , fibered “extra” cases at large − 1 // has a gauge symmetry that is en- − − − h 1 , , } 12 3 2 , 8-curve . intersecting the 0-curve 2 1 − 57 − − P ˜ 6) singularities involved, which is similar , , E , 3 is the exceptional divisor of the blowup. // − 1 245 , 12 { E 1 non-toric − − , which shifts (2) on . D , 2-curve) = 3 1-curve) = 3 – 77 – 5 // 96 su − − − , which shifts D 12 , 98 1 , where − 1 non-toric 98 D − are all similar to that of 6) point. ˜ E , D , D is resolved into a (2) on (2) on // } 3 − − su su 11 96 56 7-curve ( ( , , 2 D − − − 2-curve 254 , // 1-curve 2-curve { 2 at the (4 − non-toric charged charged − The correspondence between the non-flat and the flat models may − 12 − ; we must, however, be careful to properly include the shared matter 96 D , H H 1 18 {− D − − = , − , and } 79 6.2.1 , 84 (2)) (2)) , su su 6) is blown up. 6) point on the (2) on the (2) on the (2) on the (2) on the non-toric 0-curve 251 ( ( 7-curve , non-toric , { 258 ˜ su V su V su su − { D , } 2. 1. 1. 2. into enhanced gauge symmetries: base structure: The (4 gauge symmetry enhancements one (4 generic model (total 100 toric curves in the base) 55 , We conclude the discussion of these cases by briefly summarizing the details of the We now consider the remaining non-standard The Hodge numbers denoted in a generic model containing a • • • • • 6) point transform to the three divisors resolving the , 18 259 be those of the flat elliptic fibration over the base that has been resolved at the (4,6) points. The corresponding flat elliptic fibration model has model M:82 10 N:351 10the H:254,56, Hodge where number we counting. need to include one extra tensor multiplet in hanced on the non-toric curve, butto there the are example no in (4 contribution to the matter multiplets,ries as a a gauge curve symmetry. intersecting{ the The non-toric models curve with alsoin car- Hodge detail numbers here. be considered as that(4 the four irreducible componentsdivisor of in the the 2-dimensional fiber fiber to over resolve the the forced The model associated with Hodge numbers blow-ups of the base. JHEP02(2019)087 = ∇ (in 1 , 2 } again (6.17) (6.15) (6.16) h 2) ∇ , ∆ , 1 , 1 , , − 0), so none of } , , on the curve in 1) 1 -fibered models , 1 , } 1)) − 1 3 ( 3 2 , , − b , , 2 4 , 2 29 + 28 = 56, which 0 P . 1 = 1 + 1 = 2 b 2) , − 3 3 , 1 , , 0 − 1 b b 0 , 1 { 3 1 , , (8 c 22 , h 1 182 polytope: treating (0 , , c + , 1) − } 1 , , (0 3 2 3 3 + 1) , b , 1 , 2 2 2 3 , , b 1 c 2) b 2 4 P , 1 , = 79 − 8 b , b 1 + 1 , , 1 , , 2 b 4 4 0 2 b 181 { term with base dependence. Then , h 2 1 c − − b ( ( 3 (1 , ) we find the associated tuned Weier- + x are enhanced to { that do not project to (0 } ! 185 1) 3 . The base rays are } c , b } 2) 2 ∇ 2 2 2 , (2) tuning on a non-toric curve of genus one ∆ P b + 1) 1 − , 3 su , − 0 b 180 2 , 3).) = 28 due to the fact that there are half-hyper , 2 1 c – 78 – , f, g, , 1 7-curve, but there are no localized matter fields b 29 1 { 2 (0 + − − , , − ( − 56 2 184 2 3 , ( , 1 2 c b 1) 1 , = 2 ) 1) ) b − + 1 1) , , = , B , 2 2 2 ( 0 (0 2 1 b 179 , where h , , v , 1 c } 2 0 − b ∆ , , h 0) , + 1 , (0 183 3 1 = 0 = 251 + 2 + 1 = 254 and − , c b (1 on the NHC , = 1 1 { 8-curve. 1 0) , c (2) + 1 1 (2 , , − 22 -fibered polytope over su , 1 56 1 h = 1 I , 1 2 − h − 1) 3 , , , 2 (2) 0 2 ≡ { P su , − I -fibered polytope (in particular, the fiber coordinates are associated to 17 = , (0 2 , 1 { 1 − , − 1 5 , = P 1 non-toric the base ∆ multiplets on the NHC − } contribution from the tensor multiplet associated with the one extracompensation blowup of in ∆ contributions from the enhanced gauge symmetries ∆ 2 − D z 3. 1. 2. (( , v { In total we have agree with the Hodge numbers of the polytope model. Hodge numbers y We analyze the Weierstrass model of the non-standard We study in particular a model with an • , v x v strass model. The ordersthe of vanishing base of the preimages are in the form (( { fact, these are the only three lattice points in as the Bl where the first three latticeis points a contribute non-standard to a which come from the projection of the 4D rays in the base: M:223 7 N:10 6 H:3,165. The vertex set of ∆ is large Hodge numbers thatstructure we can have arise focusedseems even on sufficiently interesting in here, and the the novelstructure that context fact of we provide that models of some of this toric details this for non-toric hypersurface type. understanding tuning the Calabi-Yau constructions 6.2.3 Example: aIn model the with final a partthat this tuned section has genus we one the consider curve further anthat in interesting additional has feature the non-standard nonzero that base genus. a While gauge this group phenomenon is does tuned not on occur a in non-toric the curve “extra” models at JHEP02(2019)087 . } 107 (6.18) (6.19) − , = 1 and r ], but only +1 { 13 [ = = ∆ 3 } 1 , 1 . + h 272 2 , = 1 2 × g }−{ 240 that are known to be ⇒ 2 ≥ 165 − , 1 , 3 g 2 { = 9 is 54 h , are charged under the Cartan (see the 1 240 or 2 non-toric I 3 ])) = 0 = 2 D (2) ≥ 3 = 108 + 2 = 110. Then b 2 su 1 , I 1 – 79 – h ] + [ 2 b ∆ = charged 228. These results provide additional evidence that H 107, which agree with ] + [ ≥ -fibered reflexive polytopes, all of which give Calabi-Yau 1 − 1 1 b , , 3 2 ([ , 150 have a genus one fibration, and have an elliptic fibration 2 h − P ≥ ] 1 1 110 = , b 2 − h (3[ 195 or ). Therefore, · = 3 ] is a smooth curve of genus one, which can be calculated by the for- ≥ 1 b 2.1 1 , ]. Since the number of elliptic Calabi-Yau threefolds is finite, this in turn 150 or 3[ 1 = 1 curve of self-intersection h 19 ≥ charged g – component of ∆ is a degree 30 polynomial in the homogeneous coordinates. 1 , ] showing that all polytopes in the KS database giving Calabi-Yau threefolds non-toric H 1 12 1 D h I − ) 20 V 2.22 with whenever virtually all known Calabi-Yau threefoldsfibered, with building large on Hodge ations numbers variety [ are of elliptically other recent work that has led to similar observa- forms. Thus, we have foundduce explicit realizations all of Hodge elliptic numberpossible Calabi-Yau pairs for threefolds Calabi-Yau with that threefolds. pro- Thisper matches [ with the results of a companion pa- this fashion that are associatedrestricted with class an that explicit we Tate/Weierstrass construction considered of in the ourWe initial have explicitly analysis. analyzed the structure ofSkarke the database Calabi-Yau for threefolds the in 18 the Kreuzer- Tate Hodge constructions. number pairs All not of found these in admit our elliptic initial fibrations analysis of from slightly more complicated Almost all Hodge numberelliptically pairs fibered of Calabi-Yau known threefolds CY3’s associated in with the polytopes regime constructed studied in come from (2) on a = ∆ • • • su 1 , 2 a specific class of “standard” threefolds with Hodge numbers that appear in the Kreuzer-Skarke database. In this paper wefolds have with large carried Hodge out numbersover a toric that systematic bases are and realized comparison by thoseBatyrev of tuning that construction. elliptic Tate-form are Weierstrass Calabi-Yau realized Specifically, models three- as wetoric hypersurfaces bases have that in considered have toric a nonabelian varieties gauge class through groups of tuned the over Tate-tuned toric models divisors. over These tunings give h 7 Conclusions 7.1 Summary of results We calculate Hodge numbersof from the anomalytwo conditions: components of the thefootnote matter adjoint in representations representation section where the Note that mula ( In particular, the result for the discriminant ∆ is JHEP02(2019)087 ], ]. ), n 25, 20 22 ( ≤ so n . In [ 2 ≤ P ) tops is that 13. For ), 13 n ( 4 curves to sat- ≤ n ( − so n so ≤ ), 7 n ( su ) cases the resolved geometry also n ( 11 curves, but also more exotic struc- so − , types have the feature that they develop 10 ]. ∗ n − I , (20). 62 9 ] after an appropriate linear transformation; our , – 80 – − so 22 30 and n 8 curve that must be blown up four times, or tunings I − 3 curves that must be blown up to − over a ], we carry out a complementary analysis to that of this paper. 8 13 on e 20 ≥ − , n ) fiber subpolytope. Another interesting feature of the n ( 1 , so 3 , 2 these match the tops found in [ construction gives explicit realizations of theserange tops of in reflexive algebras polytopes listed, forThe which the is tops not associated guaranteed with along from the the fiber construction direction, ofP and [ the projection tothere the can fiber be plane multiple falls distinctmonodromy outside tops conditions the for on certain the gauge associated algebras, Tate corresponding tunings. to including not only blow-ups of ture such as an of isfy anomaly conditions. Ingives some rise of to the athe nontrivial gauge Mordell-Weil group. group associatedSome with polytopes a in U(1) the factorwhich KS in nonabelian database gauge algebras have are elliptic tuned fibrationsWe over worked over non-toric out toric curves the bases in topsas associated the in well base. with as the the gauge tops algebras associated with gauge algebras A novel Tate tuning ofof SU(6) gives a rise form to recently exotic studied 3-index inSome antisymmetric [ matter, polytopes in thealgebras KS with database components correspond like to tuningsPolytopes of in the very KS large databasethat gauge include non-flat resolve elliptic into fibrations flat over toric elliptic bases fibrations over more complicated non-toric bases – – – – – fact finite. In the courseTate/Weierstrass of tunings this needed to analysistopes reproduce we in certain have the CY3’s KS encountered associatedwell database. as some with in This novel poly- the has structures structure led of in fibrations to the that new may insights occur into through the polytopes. Tate algorithm as suggests that the number of distinct topological classes of Calabi-Yau threefolds is in ], in which the database was scanned for elliptic fibrations over the base • 31 derive the elliptic fibrationin [ structure. Thiswe is take essentially that the pointKS approach of database taken view directly. by andHodge Braun The analyze numbers approach most the taken Calabi-Yau fibration in threefolds structure this have of paper, a the however, standard polytopes shows elliptic in that fibration at the structure; large 7.2 Possible extensions ofIn this the companion work paper [ Here we have startedSkarke from database. the Tate tuning One picture can and instead matched start to with data the in the polytopes Kreuzer- in the database and try to JHEP02(2019)087 , ] 23 64 over the curve of 2 g , 3 -fibered polytope tuning su 1 , 3 , 2 P fiber structure for the polytope. 1 , 3 . , 2 2 P F – 81 – ]. We leave such an endeavor for future work. It 20 , this would be a more complicated analysis. In many 4.7 -fibered polytope tuning 1 , 3 , 2 2 in the Hirzebruch surface base ]; each of these represents an opportunity for further work that would give P − 20 ], each leading to a distinct class of Weierstrass tuning types with characteristic Second, we have restricted to large Hodge numbers in part because we have only focused First, we have started from the point of view of tuning Tate models and used the output There are several closely related analyses that could be carried out that we have not 57 , A Standard In this appendix, wewith go an through example the ofself-intersection details polytopes of for a elliptic standard fibrations with tuned Acknowledgments We would like to thank LaraRaghuram, David Anderson, Morrison, Andreas Andrew Braun, Turner, James andauthors Gray, Yinan Sam are Wang Johnson, supported for Nikhil helpful by discussions. DOE grant The DE-SC00012567. other fiber types becomewould also prevalent be [ interesting tomay see give whether further the insights morecomplete into general intersection class other fibers. of Weierstrass fibers tuning considered types in that [ may be possible with There are 16 distinct possible31 toric fibers, analyzed in detailnonabelian in and the abelian F-theory context gauge inall structure, [ tunings and that in correspond principleto we to extend could each the systematically of analysis analyze the of different this fiber paper types. systematically to This smaller would Hodge be numbers, necessary where the reproduced by distinct Tate/Weierstrass tuningsproached of this elliptic systematically. fibrations, This would but be wemight a reveal have natural additional next not novel step ap- structures for this for kind the of elliptic analysis, fibrations and found inon the KS Tate database. models associated with the most generic order choice with no further monodromyFor bases condition with required many for toric a divisors, given thecan gauge number become group of quite combinatorial tuning. large. possibilities of There localout are tunings explicitly also different many equivalent subsets models of thathave toric correspond checked blow-ups to in to carrying some partially resolve cases (4, that 6) the singularities. multiplicity We of Hodge numbers in the KS database is of that analysis to matchtried Hodge to numbers reproduce in all the theFor polytopes KS reasons database. in discussed the in In database, principle, section cases i.e. we there included could are multiplicity have multiple information. localin Tate each tunings case that systematically give taken equivalent gauge only groups, the and lowest we possible have choice for NHCs and the lowest that present novel features. done here or in [ increased understanding of theand set the of landscape Calabi-Yau threefolds, of 6D the F-theory role models. of elliptic fibrations, the “sieve” approach taken here enables us to identify some of the most interesting cases JHEP02(2019)087 ; } } 19 4 2), (3) − su (A.3) (A.4) (A.5) (A.6) (A.7) (A.8) (A.1) (A.2) , 1 s 3 I − . should be x, y, z, b { in the base 2 , . 1), ( 6 ζ , , } = 6.1.3 2 curve in the 3) in the base 4 4 , , , a − b 1) } . } 2 1 ) , , j 6) singularities, , 6 ζ 12 4 1 z , 1) 4 ( b , { 0), (0 to tune an (2) , , B 4 ) − , a 6 , , 9, 25, 49, 81, 169 , 6 4 8 4 0 , ζ 2 , 4 { 6 , b b , v ( expansion. According 0 b (1 a 6 6 ) ) z 4 , { 6 ζ ζ 4 12 − (1) B , 0 ( ( , a + b a in ( 6 , v are 6 8 1 . We calculate the hyper- = , , , , 3 4 a } } 3 4 4 b + i (0 } ) 2 1) 3 a a , ) a 4 + , ζ , a − − B ( 0 2 0) + + ( i , , , 3 11 4 xz , , , 4 v 5 4 7 4 0 1 b 4 1 2 1 { b b ) , , a , a , a ) ) these compose the total set of 335 ζ 2 0 − 1 0 ζ ζ ( , , , , + , 2 ( ( + 3 0 , 0 0 5 7 11 y 2 (0 , , , 2 4 2 4 { , { , z 3 4 6 , a b b 2 ) ) 0 a a a (0 , 1) ζ ζ x , 2 (3) gauge algebra on the , ( ( and 2 + + + a 2 2 1 3) a , , and some second coordinate 3 su , , 1 2 x 6 2 a a ··· ··· ··· 4 s + 3 , b I − 3 2 + + + + + , x – 82 – − 4 4 4 4 4 in the fiber plane and ( b b b b b , = (18 ) ) ) ) ) 1 z , ζ ζ ζ ζ ζ 3 ( ( ( ( ( − , v ( 1) 1 1 1 1 1 , , , , , y yz , , 1 2 3 4 6 3 2 a a a a a , v a 3) x − , v , + 2 6) singularities still lead to reflexive polytopes that can be associated 0 , , ) + ) + ) + ) + ) + , 1 ζ ζ ζ ζ ζ , ( ( ( ( ( (0 xyz 0 0 0 0 0 , , , , , , 1 (0 1 2 3 4 6 , a in the coordinates a a a a a 1) , i 3) + 1 , a ) and take the set of homogeneous coordinates 2 , 2 ) = ) = ) = ) = ) = y 6 is , -fibered polytopes naturally correspond to Tate (tuned) models. In 1 0 3.7 , ζ , ζ , ζ , ζ , ζ , − , 3 4 4 4 4 4 ∇ , , b b b b b 2 6 (1 ( ( ( ( ( , the vanishing orders have to reach P { 1 2 3 4 6 − ( a a a a a 4 { removing the lattice points corresponding to a given tuning gives a different , so all lattice points contributing to corresponding to curves of self-intersection 4 20 . The polytope model for the generic CY is M:335 5 N:11 5 H:3,243, of which the B } 2 F D 1) As an example, consider tuning a type Standard Although in some cases such (4 None of the lattice points corresponding to such further specialization are vertices of ∆, so removing − , 19 20 where the 5 sections have the forms The projection along the(0 fiber gives the rays insurface the equation base ( associated respectively to rays plane to get over with flat elliptic fibrations over blown-up bases, asthose encountered points in does the examples not of affect section the polytope. The numbers of monomialstogether with (lattice the points) two points inlattice associated points the with in sections thefurther divided M according polytope toto ∆. the Tate power table The of the number monomials of in monomials the in each section can be and the set of vertices of ∆ is the polytopes or through F-theory by anomaly cancellation. base set of vertices of principle, as long as the Tateand tunings are on adjacent not curves merely do further notalgebra, lead specialization to of (4 existing tuningsreflexive that polytope, do not associated changeHodge the with numbers gauge the of the resolved new CY resolved polytope of model the can tuned be computed singular either model. directly from The JHEP02(2019)087 2 ). 3) g 1), , , 2 1 2.1 , (A.9) , 1 (A.18) (A.17) (A.10) (A.11) (A.12) (A.13) (A.14) (A.15) (A.16) 1 -fibered 2-curve. − 1 − , , , − 3 , 2 . The other } P tuning model . match exactly 2 } 6)-points in the 3), (0 } g , 233 , , 1) , 2 , 10 5 , 1) 11 = 2, comes from 1 { , 2 − , and the number of , 1 , − } 4 − , 2 , 3 , ; (A.19) 3 { , 2 , } (0 2 , 13 6 − , { = 0) ( 2 − N , , } ( , 11 1 (3) on an isolated , 1 , 1) F for the tuned model becomes , , 2 su 1) 243 0 1 , 1 , { , top. The Hodge numbers are the 2 − 3 . 4 , 2 − , } 1 ( g 3 , , 1) , and 0 − , } − { , 0) , , ( 1 10 , } } , , (0 is a rank-preserving tuning (section 1 F 11-curves produces a flat toric fibration , 3 233 , 1) 1) 1) 2 , , , − ) of a polytope for a standard , , , 2 g 9 1) (3) model , 5 , 0 1 1 , (0 4 F { , , , 1 , 1 su → 10 3 5 4 , − , , , 3) form the 0) , 6 ( − , 2 4 4 , – 83 – } , , 2 − (3) 1 2-curve becomes , which together with the affine node (0 9 − − − , , , 0) 0) } ( ( ( − 1 − su 2 , , , , , , 0 1 − (18 1 , , 1) 1) 1) , , and the increased number , , , , 0 1 2 − , , 0 1 1 ( 1) , , , 1 3 , , 3 5 4 2 − − , , , 0) ( ( 3 5 5 − , , , 0 − − − } ∇ → ∇ 0 , , , , ( ( ( , 1 ↔ { ↔ { } , } , , } 1) , , 0 1 (0 1 , , 1) 1) 1) 1) 1) 0) 1 , 1 3 , , , , , , − (3) model, since a a 0 0 1 1 1 − 1 ( 1) , , , , , , , , , su 4 3 6 5 4 2 3 1 (3) top. The Hodge shifts , , , , , , , 0) , 4 4 6 6 6 2 3 , 6 su 1 − − − − − − − − , (0, -1, 1, 1), (0, -1, 1, 2) ( ( ( ( ( ( ( , 2 { 6 , , which together with (0 3 − } ↔ { ↔ { ↔ { ↔ { ↔ { ↔ { ↔ { 11-curves where the fiber becomes two-dimensional; the singular fiber is resolved ( − 320 = 15, is the number of the lattice points being removed. On the other hand, { ( 0 1 2 0 0 0 1 2) − , , , , , , , , , 6 6 6 2 3 4 4 − 1 a a a a a a a , 10 1 There are two polytopes in the KS database with Hodge numbers − − , , 335 , 0 9 The “standard stacking” construction (section model over a basethat leads surface to containing a hypersurface that− is a non-flat elliptic fibration.into There are an (4 irreducible component of the non-generic toric fiber, which is two-dimensional, as to that of the generic(0 model, there are threesame more as those lattice of points the B Elliptic fibrations over the bases so that the vanishing orderslattice along points in the ∆ (M) decreases by 4. Comparing the fan polytope of the with the calculation from anomalies for tuning the algebra polytope M:316 6gauge N:14 algebra 6 by H:5,233 further removing is from the the polytope arising from an enhancement to a H:5,233 in the KSmodels), database. the difference Comparing in the∆ the two sets number of of lattice datathe points fan (for polytope of the is the generic enlarged monomiallattice and points polytopes tuned ∆ and form exactly the This new polytope is again reflexive, and corresponds to the example M:320 9 N:13 7 After reduction, the vertex set of the new dual polytope ∆ removed. As one can check those are JHEP02(2019)087 ]. ), 1). for 31 1 3.7 − (B.2) (B.3) (B.4) (B.1) , P whose } , , . 2) projects to , , , m of the form } , 0) } 0 restricted to I 3) , 1 , , 1) 0 ∇ p 1)) 2 , , 3 m, , − 1 1 − − , 1 , , − , 0 3 − , 0 , form the other two 0) over three points (0 , − -curve ( { 0 . The set of lattice , , (0 , 3) is the affine node.) , } } 0 (0 , , m , 1) , (0 2 (0 1) , 1) 1 , 3) − , , given by equation ( , , 7 4 , 3) , 1 1 − 1 b , , 1) 2 p 3) ) , , 2 , , 3 − , 3 3 3 , , 0 2 b 2 − 2 , ), (0 3 − , , − 2 c in the , 1 − , m , (0 , − } (0 . + , (0 − , , (0 (0 } , , 2 3 polytope are , 3) , b (0 1 3) = 0 , 3) 2) 1 , , 3) − 3) b , , 2 ∇ 3 3 2 , ( , , 2) 2 1 b , , 2 2 , , , 4 3 285 6 , to the toric curves , 1 1 2 c c 1) − 0 3 , − , } , − − 4 + , − 4 + − , , , 2 3 (0 (1 , − 3 (0 b , b , , (0 (0 , b (0 { 1 3 (0 , , 2 1 , b 6)-points in the base must be blown up, which 2) 3) , (0 b – 84 – , , b , , 1) 2) 0) 2 3) , , 1 2 285 , , , 6) points in the base and flat elliptic fibrations over 284 and 1 1 c and projects to the corresponding base ray (0 , b , c 2 , , 4 (1 1 m, , } } 2 1 + affine Dynkin diagram. These nine components are { b − 4 + 3 − { , − − 1) 8 3 1 b , − , , , curve, the hypersurface CY, b 2 1 = 0 (0 1 E , 0 4 b , (0 (0 , 2 − c . The vertices of the (0 , , m b 2 ( ( -curve is given by the set of lattice points in 3) , 284 { 5 2 , , 1) 1) − − =9 c b m 3) , 2 , , 3) m are some complex structure moduli. This vanishes identically , , 0 1 + , − = 2 5 , , (0 F 2 3 1 , , 2 1 0) is I , b − | 5 , 285 4 6 , − − 2) -curve p c 0 c − , , , − , (0 { , m 1 , 11. 1 , (0 (0 , , − (0 , , 4 − (0 3) -curve), and is otherwise a constant. It is these three points in the toric , , , 10 , 3) 1) − , and , m 3) , , 2 , also intersects the full irreducible component (0 0) 3) is the node where three legs connect, and (0 , , 2 0 , − , p 2 , , 6 284 1 = (0 2 , 3 1 = 9 , − 6 , c − I , 6 − − 4 , -curve, but does not meet the component over the other points: 3), (0 − , , 0 , − (0 , m , ), , c m , , 2 (0 (0 { 3 m , c − ((0 F 6 ((0 { { However, The flat toric fibration of M:560 6 N:26 6 H:14,404 gives a non-flat elliptic fibration − , , a, x, y (0 where over the three points the ray of the { legs. ((0 in the the divisor where the set of components in first line forms the longest leg of the diagram, and the sets Each of these points representstoric an fiber irreducible over component the of theOver 2-dimensional a non-generic generic pointintersects on with only the the irreducibleeach, components which on combine the to boundary form of the the top giving a (0 We associate the basecorresponding coordinates rays inpoints the in base the are top over the blown up bases, whichthis provide appendix Calabi-Yau we threefolds go withbases through the the same details Hodge of numbers. these constructions In for themodel Hirzebruch over surface the toric base points. For a flatin elliptic general fibration, leads the (4 tosome a flops non-toric may base. beNonetheless, necessary this Note before provides that a the inassociated clear with blow-ups the polytopes correspondence can Calabi-Yau leading between to hypersurface be (4 the picture, done non-flat in elliptic the fibrations toric picture [ the hypersurface CY equation restricting to the component is trivially satisfied over these JHEP02(2019)087 , p = 0 =11 (B.5) (B.6) (C.1) m 6 ), and F a -curves 3.7 m − and 10-curve (resp. 12-curve, so we =10 (1) symmetry is − − u m F , which are therefore is non-vanishing, and -fibered over the base 17 . 1 12-curve in the base. I , | } 3 − , p that corresponds to a node 2 2 , 7 4 1 P 0 b -curve over which the resolved P , 1] 3 , 4 b m 0 , ) to vanish (the condition in table − − 2 3 [0 , ), and the tops over the 6 b -curve, } 7 4 ). We know that a 1 3 a occurs in this way in the four miss- b c m ) 37 − 6 3 B.1 , , − -curve to a B.2 b + a 4 3 ). 3 m c − b 240 , − 7 1 { + b 1 , 1 (torsion subgroup). If an elliptically fibered } − b 305 , ), each giving a 4 c 35 4 × c 12-curve and the semi-toric base over which the , – 85 – ( − + 5 2 B.3 − , = 0. Additional sections can be produced through b 2 1 rank 1 247 b z 3 curve, and the further feature of a forced nontrivial Z ]. For instance, an abelian global { 4 = ), and we find − , c − , I ( } 23 | 3 5 2 p B.2 b − 35 , , = 1 9 curve in equation ( I − − | 251 , p { 4 , } ]. The Weierstrass model of an elliptically fibered Calabi-Yau au- {− 7 : M:47 11 N:362 11 H:263,32. This example involves huge tunings, 6) points (resp. one point) in the [ ) tuning on a 32 , , n ( 17 Dynkin diagram. rank so 263 8 { = 11. Over a generic point in the = 10, and E m m -fibered polytope models (see figure polytope of M:47 11 N:362 11 H:263,32 is Bl 1 , 6) point and divisors that resolve the 3 , ∇ , 2 P ].) While this can be simply imposed as a constraint to tune a U(1) factor, this con- 1] , The The rational sections of an elliptic fibration form the Mordell-Weil group, which is a The correspondence between the non-flat and the flat models may be thought of as Similarly, the flat toric fibrations of M:600 6 N:26 6 H:13,433 and M:640 6 N:26 6 0 11-curve) would need two blowups (resp. one blowup) to become a , 36 [0 − on the toric curves.ing The Hodge lack of pairs theBl monomials in tomatically comes with a zeroconstraints section in the toricforced geometry when [ we setin all [ the monomialsdition in can the also section be imposed when we tune a large enough set of nonabelian gauge algebras calculation of the Hodge numbers through the associated flatfinitely elliptic generated fibration group model. ofCalabi-Yau has the a form non-trivial Mordell-Weil rank,abelian the sector F-theory U(1) compactification on it has an C An example withIn a this nonabelian appendix, we tuning workthe that through last the forces part details a of of table U(1)a an blow-up example factor from of an the missingMordell-Weil Hodge group pairs giving in a U(1) factor. After describing the geometry, we do a detailed in the extended encoding the relationship betweenover a the (4 irreducible component of the 2-dimensional fiber in the case of in the case of intersects with the nine components in ( a expect there are two (4 fiber is two-dimensional. Indeed,restrict we it calculate on the each CY component hypersurface in in ( equation ( elliptic fibration model becomes flat and gives a good modelH:12,462 for give F-theory compactification. non-flatrespectively. elliptic Both fibration vertex sets models are givenare over by the equation same the ( as that toric over bases the base that must be blown up to give the JHEP02(2019)087 1 , 3 are , 2 6 P (C.2) a 1] , -fibered 0 , . , 1 , } [0 3 , ·}} . 2 · , P , 2 (24) · , Bl { , 1 , , sp 3 , , ·} 2 1 , (20) P -fibered polytopes. 0 1 { , {− so , 3 , , , } 2 } P 1] , , (12) is a standard (20) . The polytope of interest 0 (68) 2 ambient toric fiber. , } , sp [0 so ∇ 1 so 0 , , , , 3 , , 4 4 160 4 2 , P − (44) {− {− , 1] 28 , , , 1 0 so { } , } : all monomials in the section , − [0 2 , ). Indeed, ∆ 4 3-curve, as it exceeds the upper bound (12) (36) (24) − − becomes a blowup of sp , sp (1) factor in Bl C.1 sp , 2 1 , (b) removed in the tuning,u which leads to a global , 1 1 ∇ − , {− 4 {− , (68) 1 , – 86 – , − } } 3 , , so 2 , 1 P − (92) on the (44) (92) (C.3) , (36) 1 so so 4 so , , − 4 − sp 3 (C.4) , , 1 {− {− , − (19) } , (92) 4 sp so − , (29) sp , (29) 1 The reflexive polytope pair for the Bl sp , {− , the dual fiber subpolytope , 6 } a (38) ). (38) so 7 (1)-tuned models. so Figure 7. u , : the additional ray blown up from 4 2 Now we compute the Hodge numbers from the associated flat elliptic fibration model ∇ {{− (12) associated with anomaly conditions. As the non-abelian tuning uses all of the with tuned gauge symmetries (see figure over the resolved base The non-flat fiber resultsso from the monomials in The generic model overcan this be base obtained has by Hodgeassociated Tate-tuning numbers with a the polytope generic model model, overpolytope, the for base where example ( the M:225 tunings 6 are N:31 6 H:28,160, resolves (a) JHEP02(2019)087 , ). 4- · n , is a − · ( . ) by )) = , · 3 (C.7) (C.5) so , p · 20 , C.1 C.6 · associated 20 24 )), and the , and 2 · × 2 b b 2 1 , 12 2.10 , correspond to the component (away 24 · 24 3 1 , p I 0 × 44 , · = 9 in this example. 0 32 , 2 1 , , , and 2 44 190 p 44 , sum of all terms in ( (92) on the intersecting , · 3 × with respect to one of the two so )) − 1 48 2 300 36 I b · , ) , ( ]. Concretely, as described for ex- 3 2 1 abelian charged p , 48 63 946 ) are C.5 ( H , 2 48 × · )) C.5 2 56 b 1176 , 68 ( , is a polynomial of degree 9 in 2 · p 1-curves. 2 68 associated with the 2-curve and 2 1 p )( − , 1 × 2 , 2278 – 87 – b b 2 , component [ 68 ( b 60 · 1 1 ] , p I 13 72 2628 72 , respectively. The hypermultiplets charged only under 1-curve are shared. All representations on the blown · , 1 ) = − I 1-curve is forced by the 1 2 × 2 , b 2(38 + 58 + 72) = 84, so the gauge symmetries can not be − ( 80 / 741 72 1 28 + 8) = 14739. The representations of the gauge group 4186 discriminant locus factors into I (C.6) , · , on the . The degrees of the polynomials = (sum of all terms in ( , 2 1 ∆ 2 × I 38 38 I b 92 92 · · (4 × × 1711 1 2 92 , − 46 · , 84 , 1 2 703 92 58 · × 58 · 2 factors come from the group theoretic normalization constant of = 14859 ), are: 66 / ], we calculate the discriminant locus of the 1 2 non-abelian charged , m , ( 32 H is a polynomial of degree 76 in 38 58 sp 1 (19) on the exceptional · × p ⊕ 4-curve are also shared: 1 sp ) The final piece needed is the U(1) charged matter. These fields are not charged under 38 30 − non-abelian n · 2(92) = 46 on the exceptional ( V / 1 2 where polynomial of degree 63 in number of nodes and cusps onthe the U(1) are localized at the nodes, and therefore ∆ which the fiber isample Kodaira in [ local coordinates, which wewith choose the to 0-curve; be then the the non-abelian group, andtations. therefore have These not matter yet fields beenfrom are taken the localized into non-abelian at account codimension innumber components) our two of of on compu- the the the U(1) discriminant charged locus matter (equation fields ( corresponds to the number of the nodes, over Hence, ∆ 14830. Note1 that all representationsup of a forcedenhanced non-abelian further on gauge the three group intersecting are shared: where the 1 But some representationsrepresentations are that shared areso between charged each under pair both of of intersecting the curves. two corresponding The group factors, which differ from∆ the total dimensionfactors on of the the individual curves gauge are [ groups in the NHCs in ( curve. Again, the tunedthe non-abelian non-abelian gauge symmetries group force factors a in global equation U(1). ( The dimensions of The JHEP02(2019)087 , . , + ]. ]. Nucl. ]. Nucl. ] (2012) , = 1 + ] , SPIRE D SPIRE 60 6 IN IN SPIRE ]. ][ ][ IN abelian r (2014) 23 ][ . } kreuzer/CY.html 10 SPIRE non-abelian charged Central Eur. J. + ∆ ∼ IN 128 , H − ][ , arXiv:1405.2073 (∆ Fortsch. Phys. arXiv:1207.4792 ]. [ JHEP − , [ , Fibrations from Polyhedra 235 1 T , { F-theory models and tuned 3 hep-th/9602022 2 non-abelian h [ = r K SPIRE D arXiv:1605.08052 29∆ Vacuum Configurations for 6 } IN [ arXiv:1504.07689 − ][ + ∆ ) (2014) 093 [ F-theory models (2013) 937 160 , T 09 D ]. 128, which agrees with the differences arXiv:1404.6300 28 ]. 6 (1996) 403 ]. 324 [ ]. abelian F-theory models − = ∆ V (2016) 581 D 1 , 6 } − { JHEP SPIRE 1 SPIRE (2017) 1063 , 64 +∆ SPIRE h SPIRE IN http://hep.itp.tuwien.ac.at/ B 469 32 IN , An Abundance of IN , [ IN ][ 21 ][ hep-th/0002240 ][ – 88 – (2015) 061 263 [ { 06 Topological Invariants and Fibration Structure of (14830+9) = non-abelian ]. V (1985) 46 − Nucl. Phys. F-theory models and elliptically fibered Calabi-Yau threefolds Classifying bases for Toric bases for , 29 Fortsch. Phys. JHEP ), which permits any use, distribution and reproduction in Enhanced gauge symmetry in Calabi-Yau threefolds with large . ]. , , Non-toric bases for elliptic Calabi-Yau threefolds and D Compactifications of F-theory on Calabi-Yau threefolds. I Compactifications of F-theory on Calabi-Yau threefolds. II − = (∆ Commun. Math. Phys. SPIRE (2002) 1209 Complete classification of reflexive polyhedra in four-dimensions Calabi-Yau data , 1 IN B 258 hep-th/9603161 4 , 2 hep-th/9602114 [ ][ h [ arXiv:1201.1943 SPIRE [ IN ][ (1993) 349 CC-BY 4.0 Adv. Theor. Math. Phys. 69 , This article is distributed under the terms of the Creative Commons (1996) 74 (1996) 437 Nucl. Phys. Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tori , ) = (14739+1) (2012) 1072 Evidence for F-theory ]. ]. 10 B 473 B 476 arXiv:1204.0283 [ SPIRE SPIRE 18)+1 = 235 and ∆ IN IN arXiv:1406.0514 elliptic Calabi-Yau threefolds Duke Math. J. with Interchangeable Parts [ Complete Intersection Calabi-Yau Four-Folds [ [ 1187 over semi-toric base surfaces F-theory vacua Phys. Phys. Phys. Superstrings Adv. Theor. Math. Phys. − Summing up all the pieces, we obtain ∆ D.R. Morrison and C. Vafa, D.R. Morrison and W. Taylor, D.R. Morrison and W. Taylor, M. Kreuzer and H. Skarke, C. Vafa, D.R. Morrison and C. Vafa, V. Batyrev, M. Kreuzer and H. Skarke, P. Candelas, G.T. Horowitz, A. Strominger and E. Witten, P. Candelas, A. Constantin and H. Skarke, J. Gray, A.S. Haupt and A. Lukas, S.B. Johnson and W. Taylor, S.B. Johnson and W. Taylor, G. Martini and W. Taylor, 6 W. Taylor and Y.-N. Wang, abelian charged V [7] [8] [9] [4] [5] [6] [2] [3] [1] [14] [15] [12] [13] [10] [11] any medium, provided the original author(s) and source are credited. References (251 ∆ in Hodge numbers from the polytopes: Open Access. Attribution License ( JHEP02(2019)087 ] s in - and 10 M 11 , ] Nucl. U(1) -, , , E JHEP (2013) 016 , JHEP (1996) 215 ]. (2014) 132 , 01 ]. 08 ]. ]. arXiv:1303.5054 SPIRE ]. B 481 [ IN JHEP SPIRE ]. , (2012) 022 (1997) 146 SPIRE ][ IN JHEP Strings, branes and gravity. IN , SPIRE ][ fields in F-theory ]. 01 SPIRE arXiv:1507.03235 ][ IN Tops with Multiple [ IN , in [ SPIRE ][ B 497 , Boulder, U.S.A., A new construction of Calabi-Yau IN U(1) SPIRE Nucl. Phys. ][ JHEP Elliptic fibrations for , IN (2013) 046005 , ][ ]. (2016) 441 D 88 arXiv:1306.0577 hep-th/0303218 Tools for CICYs in F-theory Multiple Fibrations in Calabi-Yau Geometry Fibrations in CICY Threefolds Nucl. Phys. [ Refined stable pair invariants for [ SPIRE , IN hep-th/0411120 B 906 arXiv:1608.07555 [ ][ ]. ]. arXiv:1809.05160 – 89 – , arXiv:1307.2902 Geometric Engineering in Toric F-theory and GUTs Phys. Rev. [ (2013) 069 , (2003) 205 arXiv:1308.0619 SPIRE SPIRE [ 7 12 IN IN (2016) 105 Nucl. Phys. Sections, multisections and Matter and singularities ][ ][ , F-theory on Genus-One Fibrations (2014) 1 On the prevalence of elliptic and genus one fibrations among Affine Kac-Moody algebras, CHL strings and the classification ]. ]. ]. 10 ]. ]. ]. ]. JHEP Duality between the webs of heterotic and type-II vacua hep-th/9603170 , [ Matter from geometry (2013) 112 SPIRE SPIRE SPIRE B 882 F-theory vacua JHEP SPIRE SPIRE SPIRE SPIRE IN IN IN 11 , IN IN IN IN [ [ ][ ][ ][ ][ ][ U(1) TASI lectures on compactification and duality × (1998) 295 JHEP , arXiv:1608.07554 arXiv:1708.07907 TASI Lectures on Supergravity and String Vacua in Various Dimensions Toric Elliptic Fibrations and F-theory Compactifications [ [ Gauge Factors Nucl. Phys. ]. ]. U(1) , Adv. Theor. Math. Phys. , × B 511 U(1) -strings ] SPIRE SPIRE arXiv:1401.7844 p, q arXiv:1106.3563 arXiv:1110.4883 hep-th/9606086 hep-th/9605200 IN IN [ arXiv:1404.1527 [ [ [ [ Geometric singularities and enhanced gauge[ symmetries Proceedings, Theoretical Advanced Study Institute, TASI’99 May 31–June 25, 1999, pp. 653–719 (1999) [ arXiv:1104.2051 with SU(5) [ F-theory Phys. of tops (2017) 077 toric hypersurface Calabi-Yau threefolds manifolds: Generalized CICYs [ (2016) 004 and String Dualities D.R. Morrison and W. Taylor, M.-X. Huang, A. Klemm and M. Poretschkin, D.R. Morrison and W. Taylor, V. Braun, V. Braun and D.R. Morrison, S.H. Katz and C. Vafa, M. Bershadsky, K.A. Intriligator, S. Kachru, D.R. Morrison, V. Sadov and C. Vafa, D.R. Morrison, W. Taylor, J. Borchmann, C. Mayrhofer, E. Palti and T. Weigand, J. Borchmann, C. Mayrhofer, E. Palti and T. Weigand, SU(5) P. Candelas and A. Font, V. Bouchard and H. Skarke, V. Braun, T.W. Grimm and J. Keitel, L.B. Anderson, X. Gao, J. Gray and S.-J. Lee, Y.-C. Huang and W. Taylor, L.B. Anderson, X. Gao, J. Gray and S.-J. Lee, L.B. Anderson, X. Gao, J. Gray and S.-J. Lee, L.B. Anderson, F. Apruzzi, X. Gao, J. Gray and S.-J. Lee, [33] [34] [30] [31] [32] [28] [29] [26] [27] [24] [25] [21] [22] [23] [19] [20] [17] [18] [16] JHEP02(2019)087 , ]. 3 Z , Nucl. 26 08 , ]. SPIRE , ]. ]. IN . F-theory ][ = 1 JHEP ]. SPIRE , , IN ]. fibrations D SPIRE SPIRE 6 ][ IN 3 DN IN 6 SPIRE ][ K SO(10) ][ IN J. Geom. Phys. SPIRE (1991) 287 , (2015) 017] ][ IN ]. F-theory vacua with ][ Physics of F-theory ]. 06 ]. 290 ]. SPIRE arXiv:1405.3656 The Toric SCFTs: Arbitrary rank [ IN SPIRE d SPIRE Searching for ][ IN Tate’s algorithm and F-theory 6 SPIRE arXiv:1406.5180 IN Mordell-Weil Torsion and the Global . ][ [ IN ][ arXiv:1109.0042 ][ [ ]. effective action of F-theory via M-theory arXiv:1510.08056 Math. Ann. SCFTs from M-/F-theory on Non-Flat Erratum ibid. (2014) 16 [ , ]. 0) d [ arXiv:1502.06953 , 5 arXiv:1112.1082 [ 10 [ (1 SPIRE On the global symmetries of SCFTs via (2014) 156 IN Global Tensor-Matter Transitions in F-Theory SPIRE On the Classification of 6D SCFTs and d (2012) 51 Global aspects of the space of , Annals of Mathematics Study 131, Princeton ][ IN 5 6 12 – 90 – ][ ]. JHEP ]. (2016) 005 ]. (2014) 028 , alg-geom/9310003 Phases of (2015) 736 arXiv:1008.1062 07 (2012) 019 [ 05 arXiv:1804.07386 JHEP SPIRE SPIRE arXiv:1709.06609 [ 05 , SPIRE [ , American Mathematical Society (2003). IN IN [ IN Classifying Anomalies and the Euler characteristic of elliptic Calabi-Yau Six-dimensional ][ JHEP B 898 kreuzer/CY/CYpalp.html ]. ]. ][ JHEP , ∼ PALP: a Package for Analyzing Lattice Polytopes Calabi-Yau four folds and toric fibrations , (1994) 493 [ JHEP hep-th/9610154 (2010) 118 3 , [ SPIRE SPIRE arXiv:1106.3854 (2017) 035 11 [ IN IN [ ][ (2018) 1800037 12 Nucl. Phys. , 66 Mirror Symmetry JHEP Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric (1997) 567 , hep-th/9701175 arXiv:1205.0952 Introduction to Toric Varieties On the Hodge structure of elliptically fibered Calabi-Yau threefolds arXiv:1811.12400 JHEP On minimal models of elliptic threefolds Commun. Num. Theor. Phys. [ [ (2011) 094 , , J. Alg. Geom. , , 08 B 494 arXiv:1312.5746 http://hep.itp.tuwien.ac.at/ Phys. (1998) 272 University Press, Princeton (1993). varieties Generalized ADE Orbifolds [ threefolds supergravities superconformal field theories Fibrations JHEP Landscape Fortsch. Phys. arXiv:1811.10616 on Calabi-Yau threefolds (2012) 032 compactifications without section Structure of Gauge Groups in F-theory gauge symmetry A.C. Avram, M. Kreuzer, M. Mandelberg and H. Skarke, M. Kreuzer and H. Skarke, K. Hori et al., V.V. Batyrev, M. Kreuzer and H. Skarke, J.J. Heckman, D.R. Morrison and C. Vafa, A. Grassi, W. Fulton, V. Kumar, D.R. Morrison and W. Taylor, M. Bertolini, P.R. Merkx and D.R. Morrison, F. Apruzzi, L. Lin and C. Mayrhofer, S. Katz, D.R. Morrison, S. Sch¨afer-Namekiand J. Sully, A. Grassi and D.R. Morrison, M. Dierigl, P.-K. Oehlmann and F. Ruehle, L. Bhardwaj and P. Jefferson, F. Bonetti and T.W. Grimm, W. Taylor, W. Buchm¨uller,M. Dierigl, P.-K. Oehlmann and F. Ruehle, C. Mayrhofer, D.R. Morrison, O. Till and T. Weigand, M. Cvetiˇc,R. Donagi, D. Klevers, H. Piragua and M. Poretschkin, L.B. Anderson, I. Garc´ıa-Etxebarria,T.W. Grimm and J. Keitel, [54] [55] [51] [52] [53] [48] [49] [50] [46] [47] [43] [44] [45] [41] [42] [38] [39] [40] [36] [37] [35] JHEP02(2019)087 10 ]. 03 (2016) ]. 04 ]. SPIRE JHEP IN , [ SPIRE F-Theory on all IN JHEP , ][ ]. (1997) 679 (1981) 1 SPIRE 79 IN (2015) 142 Chaos Solitons Fractals B 505 ][ , 01 hep-th/9704097 [ JHEP Matter in transition Phys. Rept. , , Nucl. Phys. , , version 7.1, The Sage Developers (2016) (1997) 445 arXiv:1208.2695 [ Toric geometry and enhanced gauge symmetry of – 91 – ]. Complete Intersection Fibers in F-theory ]. B 507 ]. (2012) 128 SPIRE SPIRE IN F-Theory and the Mordell-Weil Group of Elliptically-Fibered IN 10 ][ ]. Enhanced gauged symmetry in type-II and F theory ][ SPIRE ]. IN Nucl. Phys. , ][ JHEP SPIRE , SPIRE IN IN ][ ][ hep-th/9806059 arXiv:1411.2615 Group Theory for Unified Model Building String dualities and toric geometry: An Introduction [ [ arXiv:1512.05791 [ hep-th/9704129 arXiv:1408.4808 http://sagemath.org/doc/reference/schemes/sage/schemes/toric/weierstrass.html Calabi-Yau Threefolds (2015) 125 080 compactifications: Dynkin diagrams from polyhedra [ SageMath, the Sage Mathematics[ Software System (1999) 543 Toric Hypersurface Fibrations and its[ Higgs Branches F-theory/heterotic vacua V. Braun, T.W. Grimm and J. Keitel, R. Slansky, L.B. Anderson, J. Gray, N. Raghuram and W. Taylor, D.R. Morrison and D.S. Park, E. Perevalov and H. Skarke, D. Klevers, D.K. Mayorga Pena, P.-K. Oehlmann, H. Piragua and J. Reuter, P. Candelas, E. Perevalov and G. Rajesh, H. Skarke, [64] [61] [62] [63] [59] [60] [57] [58] [56]