Non-Higgsable clusters for 4D F-theory models

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Citation Morrison, David R., and Washington Taylor. “Non-Higgsable Clusters for 4D F-Theory Models.” J. High Energ. Phys. 2015, no. 5 (May 2015).

As Published http://dx.doi.org/10.1007/jhep05(2015)080

Publisher Springer-Verlag

Version Final published version

Citable link http://hdl.handle.net/1721.1/98237

Terms of Use Creative Commons Attribution

Detailed Terms http://creativecommons.org/licenses/by/4.0/ JHEP05(2015)080 Springer May 18, 2015 April 18, 2015 : : January 6, 2015 : Published Accepted Received 10.1007/JHEP05(2015)080 doi: Published for SISSA by b [email protected] , SU(2), that can be realized through 4D non-Higgsable clusters. × . 3 and Washington Taylor a 1412.6112 The Authors. c F-Theory, Gauge Symmetry, Supergravity Models, Superstring Vacua

We analyze non-Higgsable clusters of gauge groups and matter that can arise , [email protected] Center for Theoretical Physics, Department77 of Massachusetts Physics, Avenue, Massachusetts Cambridge, Institute MA of 02139, Technology, E-mail: U.S.A. Departments of Mathematics andSanta Physics, Barbara, University CA of 93106, California, U.S.A. Santa Barbara, b a Open Access Article funded by SCOAP Keywords: ArXiv ePrint: matter, including SU(3) There are also more complicatedparticular, configurations the involving collection more of thanbranchings, two gauge loops, gauge and group factors; long factors in linear with chains. jointly charged matter can exhibit at the level of geometryfeatures in of 4D F-theory F-theory compactifications, models. and givenonabelian Non-Higgsable rise part clusters naturally to seem of structures to the that be standarddark include generic the model matter gauge candidates. group andgroup certain factors, In specific and particular, types only of five there potential distinct are products nine of two distinct nonabelian single gauge nonabelian group factors gauge with Abstract: David R. Morrison Non-Higgsable clusters for 4D F-theory models JHEP05(2015)080 14 29 26 10 19 – 1 – 10 25 13 23 6 8 14 13 4 9 18 6 11 20 1 23 13 16 20 22 7.3 Loops 8.1 Summary and8.2 open questions Classifying Calabi-Yau8.3 fourfolds Physical consequences of non-Higgsable clusters 7.1 Branchings 7.2 Chains 4.5 Clusters with multiple factors 5.1 Possible single5.2 gauge factors Matter 4.1 Constraints on4.2 individual curves Monodromy 4.3 Matter 4.4 Superconformal fixed points 3.1 Derivation of3.2 local conditions Summary of3.3 local conditions Additional constraints 1 Introduction Many supersymmetric stringgroups theory that cannot compactifications betry. contain In broken the by “non-Higgsable” simplest charged cases, gauge matter the non-Higgsable in gauge a group way is that a single preserves simple supersymme- factor such as A The gauge algebra of maximally Higgsed models 8 Conclusions 6 Products of two factors 7 More complicated “quiver diagrams” 5 4D non-Higgsable clusters with single gauge group factors 4 Warm-up: 6D non-Higgsable clusters 3 Local conditions Contents 1 Introduction 2 Review of F-theory basics JHEP05(2015)080 , 8 8 56 E E × 8 E , and 7 E , 6 E , 4 F arises in the 8 E 1 SU(2), with non-Higgsable matter , we comment on some aspects of × 8.3 carrying a half hypermultiplet in the SO(7) 7 × E – 2 – heterotic factors, corresponding on the F-theory 8 E . For example, a non-Higgsable ]; a simple set of examples are given by 6D supergravity . While in the simplest cases there is no charged matter, m 3 12 – F SU(2) and SU(2) 1 F × 2 G ], we performed a systematic analysis of the possible non-Higgsable under which there are no charged matter fields. Such string vacua 4 8 ], as well as decoupled or weakly interacting sectors that have a natural E 5 factors are 5 and 19, corresponding on the F-theory side to a compactification 8 E , there is a non-Higgsable gauge group 7 There are several issues that make the analysis of non-Higgsable clusters, and F-theory In this paper we initiate a systematic analysis of non-Higgsable clusters for 4D F-theory In heterotic constructions that use smooth bundles over smooth Calabi-Yau manifolds The non-Higgsable structure imposed from geometry determines only the gauge algebra, so that these F 1 els. In sixfibered Calabi-Yau dimensions threefold the matches withthe geometric corresponding a complex 6D continuous moduli supergravity structure theory, space moduli so of that space flat there directions is of a in an close correspondence elliptically groups between may in principle be reduced through a quotient by a finite subgroup in some cases. standard model [ possible interpretation as darkthese constructions matter. that may In be section relevant to phenomenology. vacua in general, more complex for four-dimensional models than for six-dimensional mod- models. Non-Higgsable clusters givethe rise moduli to spaces gauge of groupscompactification. Calabi-Yau and fourfolds From our matter over current many understanding at of basesYau generic the that manifolds, space points of can it in elliptically be fibered seems Calabi- usedhave that for such in F-theory structure. fact the Non-Higgsable vast clusters majority can of give F-theory rise compactifications will to the nonabelian part of the negative self-intersection. The complete listmodels of contains, non-Higgsable in gauge addition groups tothe in the two 6D product single F-theory groups group factorsjointly SU(3), charged SO(8), under the adjacent factors in each gauge group. charged matter. In [ structures that can arise inHiggsable 6D clusters” F-theory of compactifications, gauge and group identified allbe factors possible connected broken by “non- by jointly Higgsing. charged mattersecting The that analysis curves cannot was on carried the out (two by looking complex at dimensional) configurations base of surface, inter- with each curve having a representation. This matter cannotconstraints be cannot Higgsed be in satisfied the by low-energy matter theory in since a the single D-term the real non-Higgsable representation. gauge groups containmany only geometries a give rise single to simple non-Higgsable factor. gauge In groups F-theory, with however, multiple factors and jointly theory are placed inside one to of compactification the on there two are also cases wherematter. a For gauge example, group in isin a the non-Higgsable 6D two even heterotic in compactification theon where presence the of numbers charged of instantons have long been known to arisedual in F-theory heterotic string descriptions compactifications, and [ theories in many arising cases from have heterotictions compactifications on on Hirzebruch surfaces K3heterotic and theory dual when F-theory all compactifica- of the 24 instantons needed for tadpole cancellation in the 10D SU(3), SO(8), or JHEP05(2015)080 – 3 6). 21 , -branes into p ] for an intro- 7 = 2) matter. Some , we begin in section N 2 ] for some initial efforts vanish to degrees (4 g 17 , , , we refer only to the gauge f 16 )) seven-brane stacks, off-diagonal ]. While such excitations can be de- N 15 – ]. Though there has been substantial 5 12 geometric matter – 3 – ]; these degrees of freedom are not encompassed 11 and – 8 ] for more recent developments and further references). For )-branes [ k 20 – + 2 18 p geometric gauge group ]). In four dimensions this connection is obscured by the presence of a superpotential After a brief review of some basic aspects of F-theory in section The effects of G-flux and additional degrees of freedom can not only lift flat directions 6 ]), there is still no completely general way of analyzing these effects in an arbitrary 4D As we discuss in thea next problem section, for it consistency isthe of not analysis 4D yet here. F-theory understood models, whether and such singularities we pose include vacuawith with a such loci general in set of formulae that can be used to give a lower bound for the orders group and non-chiral matterThis analysis associated thus with gives only thein a singularities first-order complete picture of F-theory of models. the thethe space Weierstrass further To of incorporation model. determine possibilities of that the G-flux canfuture effects actual exist is work. physical necessary, and Another gauge we complication group leaveof in this and codimension further the matter analysis three analysis to loci of where 4D the F-theory models Weierstrass coefficients is the presence 27 compactification. In this paper,compactification, we in focus particular only on on the thegiven continuous underlying elliptically moduli fibered geometry space Calabi-Yau of fourfold of the parameterizedwe complex F-theory by refer structures a for to Weierstrass model. a the When break the apparent geometric symmetryspectrum to of a the smaller theory, group. andYau can G-flux geometry give also in rise affects the to the F-theoryof chiral matter picture matter describes these although only issues the non-chiral underlyingwork (4D are Calabi- on discussed various aspects in of more G-flux detail in in 4D F-theory [ compactifications (see for example [ string dynamics is even less transparent from the F-theory complex structurein point of the view. moduli space,the but potential can produced also bywhile modify G-flux on the can the spectrum drive other the of hand theory the flux to theory. in a On the point the world-volume of fields one enhanced on hand, symmetry, a set of seven-branes can also a global characterization in terms offormulation Weierstrass models of there the is no full analogousbrane set general global configurations of on open a stringin general degrees this of compact direction, freedom space and associated (seenonperturbative [ with e.g. seven-brane perturbative configurations [ associated with exceptional groups, the open the F-theory context. Evenexcitations for perturbative of (e.g. the SU( world-volumehigher-dimensional adjoint D( scalar fieldsin encode the expansion complex structure of moduli D ofhave the been elliptically studied fibered in Calabi-Yau that used contextscribed for as locally, F-theory “T-branes” unlike and [ for the complex structure degrees of freedom in F-theory which have ple [ that lifts some ofthe the superpotential flat is directions. producedductory by review). Viewing G-flux F-theory on ThereIIB as a are seven-brane dual Calabi-Yau configurations also fourfold to that additional (see a are degrees [ limit as of of yet freedom M-theory, not on well the understood world-volume or of incorporated into the structure of the physical theory and the geometric data of F-theory (see for exam- JHEP05(2015)080 , 2 K g 4 − (2.1) + 27 ]. The 3 37 f – , with , along with n 35 1 B , where the total n 32 , ∆, but also on . The (geometric) , g B n can be described in , B ]. 31 g → f , ] in a simple and direct 3 4 29 , X : and 28 π , f 7 ) over the base ], is listed in table K 6 32 dimensions is defined by a complex , − ( over any given divisor in a complex n 31 g , O 2 g + − ), K and fx 4 f + − ( 3 – 4 – x O = 2 y and the following sections to analyze the local structure 5 . The Weierstrass parameters n ], but more complicated matter representations can also arise. B ] +1)-fold. The data of such an elliptic fibration can be described 38 , n 30 35 , and ∆ vanish to certain orders on a divisor (i.e., a codimension g ]). In some cases, for compactifications to six dimensions or fewer , f 34 , that supports an elliptic fibration with section 33 n are sections of line bundles with a discussion of some of the possible applications of 4D non-Higgsable B g 8 is a Calabi-Yau ( , f X We consider F-theory as a nonperturbative formulation of type IIB string theory. A -fold base locus on the base.in In a simple simple cases, fashion thecodimension from representation two the content locus of enhancement [ the of matter theA is Kodaira complete determined singularity dictionary type between onhas codimension the not two singular yet singularities been and developed, matter though representations a number of recent works have made progress in this of codimension one singularities, followingthe Kodaira resulting [ gauge algebra factorsin (which are M-theory inferred [ from gaugethe symmetry gauge enhancement groupthe depends more not detailed only monodromy(geometric) on matter structure content the of of orders the the theory singularity of is locus determined vanishing by [ of the codimension two singularity one algebraic subspace) thenbe the viewed total as space asingularities of degenerate can the be elliptic limit interpreted fibration oftypes in is than a occur terms singular, smooth in of and the Calabi-Yau perturbativeappearance coincident can manifold; IIB of seven-branes string). a in (but nonabelian In of IIB either gauge picture, more symmetry language the general in physical the the result is supergravity the theory. The classification the anti-canonical class on terms of polynomials of fixedgauge degrees group in of a local theone coordinate corresponding singularity system structure supergravity on of theory thevanishes. is Weierstrass model, determined where When by the the discriminant ∆ codimension = 4 n space by a Weierstrass model [ where Here we summarize a fewof of this the paper. basic More features comprehensive of reviews F-theory can that be aresupersymmetric found central F-theory in to compactification [ the to analysis 10 of 4D clusters usingbranchings, loops, the and general long formulae.in linear section chains We of find connectedclusters. a gauge group rich factors. range of We conclude behavior, including 2 Review of F-theory basics threefold base. Thesemodel formulae and control determine the the local factorsas that a singularity can warm-up structure appear exercise of we innon-Higgsable the clusters use a in Weierstrass a non-Higgsable 6D simplified cluster. theories, version andway. In reproduce of We section the then the results proceed local of in [ divisor section formulae to describe of vanishing of the Weierstrass coefficients JHEP05(2015)080 . of 6) D , f ]; while 48 5) 0 cannot be 2 ) − g c 2 n n > vanish to degrees (2) (2 n/ b 4 g f ( so su 6) codimension two (7) or 6) on a codimension 2 respectively on , , 8 7 sp (2) , when all sections or e e f so ], we do not investigate ≥ none none su 6 D e 4) or 53 γ (3) or ) or – − n ]; the degree of vanishing is su ( 1, (8) or n 49 54 , and/or ∆ are enhanced. In su so (2 ≥ g so , φ f nonabelian symmetry algebra singularities with ∗ n force a gauge group factor according to 2 1 vanish to orders (4 does not occur for susy vacua 4 2 1 − − 8 7 6 γ g n n e e e A A D , , none none A D φ f singularity and type I – 5 – n 7 2 ) vanish to orders 12 9 8 6 4 3 2 0 ≥ ≥ K 10 6 ≥ n n ord(∆) . Note that only certain gauge groups can be forced to − ( 1 6) on a divisor, then there is a “non-minimal singularity” ) O , g 6 5 3 2 0 5 4 3 2 1 0 of ≥ ≥ ≥ ≥ ≥ ord( g ) f 4 4 3 2 2 1 0 3 2 1 0 ]. For the purposes of this paper, the most relevant fact is that, in ≥ ≥ ≥ ≥ ≥ ≥ ≥ 47 ord( – , the precise gauge algebra is fixed by monodromy conditions that can be identified g 39 , , ∗ ∗ ∗ vanish to orders (4 f 0 ∗ 0 ∗ n n II I I I I 37 IV III II IV III g Type . Table of codimension one singularity types for elliptic fibrations and associated non- , ) and all sections non-min f K 4 If A non-Higgsable gauge group factor arises on a given divisor 6) at a codimension three locus (point). At such points, like at (4 − , ( such field theories have been thesuch subject structure of here. some recent In(4 work 4D [ models, the situationloci, is it seems less that clear extra when not massless sufficient states appear at in such the points, theory however, [ to lead directly to a blowup of the point — for this not describe a supersymmetrictwo vacuum). locus If in thecodimension base, two locus then in there the isthat base a the again new total a base space arises non-minimal ofafter with singularity. the a further fibration reduced By blowups. may degree blowing either of Itvanishing singularity, be up is so on resolvable the also a into possible a codimension Calabi-Yau to two directly, locus describe or in the terms structure of associated a with superconformal a field (4 theory [ the Kodaira conditions inappear table in this way.forced In to particular, arise type in I a generic Weierstrass model overthat any cannot base. be resolved to give a total space that is Calabi-Yau (and hence the data does curve in the case ofcarries 4D a compactifications), charge then under there both is of generally the (geometric) gauge matter group that O factors. In such a situation, the orders of vanishing direction [ general, matter arises atgauge codimension group factors, two loci whereparticular, the within when degrees codimension two of divisors onecodimension each vanishing divisors carry of two carrying a locus gauge (a group set factor, of and points they in intersect the along case a of 6D compactifications, or a complex Table 1 abelian symmetry algebras. In casesvanishing of where the algebra isfrom not determined the uniquely form by of the the degrees Weierstrass of model. JHEP05(2015)080 , 2 z B

] for T ) 58 k restrict had the − k 3 ˆ f We do not (6 B have certain 2 − ] to compute g Σ , 56 is the canonical f K 6 Σ . The results of this − K B Σ O , and can be made more precise Γ L (of any dimension). In the whose coefficients g ∈ B Σ | ··· k ]. Thus, we develop here some and g + 55 f 2 z 2 ˆ , constructed from the projectivization f 2 bundle (as is true, for example, when 12). Thus, while it is possible that there + B 1 , ] to classify non-Higgsable clusters on base z P 4 1 ˆ f + – 6 – . The key way that this expansion is used is in 3 0 ˆ f B = in a succinct fashion useful for explicit computations. ⊂ ] for base surfaces (where it was used to analyze the on Σ, and similarly ˆ f 2
57 3.2 B T ) on any given divisor in a generic Weierstrass model. This = ∼ k g − , (4 f . As described in that paper, when there is a good coordinate 2 − B Σ K 4 over − bundle over a complex surface , we derive local conditions that can be used to show that L ) characterizes the “twist” of the line bundle 1 Σ L P 3.1 ( O 1 c must vanish to certain orders) by checking that the corresponding line bundles g = or T ], a general class of 4D F-theory models were considered where the base f This characterization of vanishing conditions for In section For a local or global base geometry with a toric description, it is straightforward to 58 We thank Antonella Grassi for discussions on this point. 2 on Σ have no non-vanishing sections at all. and generalized to anabove description, arbitrary the divisor local in geometry around a the general divisor base Σ is characterized by the normal to sections of where class of the complexdemonstrating surface that Σ certain of thesethat coefficients must vanish upon restriction to Σ (showing In [ structure of a of a line bundle in an open region containingis a toric), section Σ there of is the a series expansion minimal orders of vanishing onanalysis divisors are on summarized a in completely section general base 3.1 Derivation of local conditions the orders of vanishingmethod of is described explicitlyset in of [ all toric basesthreefold that bases. support We elliptically use fibered thisthis Calabi-Yau approach threefolds), paper, for and complementing explicit in the calculations [ general in some methods specific developed examples in in this section. to three-dimensional basesgeneral has local many methods complications forclusters [ for placing 4D constraints F-theory on models. the possible structureuse of the non-Higgsable lattice of monomials dual to the lattice containing the toric fan [ 3 Local conditions Classifying the possible non-Higgsable clustersto that four can dimensions arise is in more F-theory difficultlevel compactifications than of for pure compactifications geometry. to six Thesurfaces dimensions, approach even incorporated used at a in the method [ known as the Zariski decomposition, whose generalization is some problem or inconsistencyalso in plausible models that with such such modelstry codimension represent to three perfectly singularities, acceptable resolve it F-theory this is vacua. issue question may in affect some this of paper, the but structures we we do describe for note 4D some non-Higgsable circumstances clusters. when this we would need additional vanishing to order (8 JHEP05(2015)080 ) z g 2 ˆ f : (3.6) (3.7) (3.8) (3.1) (3.2) (3.4) = D 1 , and in , and in ˆ , and we f a section . . D D B 0 0 k a section of g a section of . along ⊂ 1 → → 1 00 (3.5) . ˆ g f g .

D D (respectively → → 00 (3.3) D D with ˆ N f ) ) then we can write and N N k depend only on the ) ) → → + with D D z with ˆ k + 1 along f k k + 1 along k ) ) . The corresponding g N N z D g z k k 1 D D − − 1 D ˆ K f ) is an alternate way of g = ˆ N N + 3 + 5 − and D to = g = ˆ D D − + (4 + (6 f = + 4 + 6 k f ( g ˆ K K f D D D D D D 4 6 | , K K O − − K K

B 4 6 ( ( ) ) 4 6 . K . D D D D bundle. In general, therefore, if we − − − − ( ( − +1 N N 1 +1 D D k k P z z → O → O + 4 + 6 ) ) +1 vanishes to order 2. For this purpose, we +1 → O → O k D D → O → O k ˆ , we initially have D D ) ) f g g ) and restrict ) ) K K then we can write D − − 4 6 B B = = ˆ or kD kD k kD − − B B K K vanishes to order at least f g – 7 – ( ( f 4 6 − − ). This happens if and only if vanishes to order at least K K vanishes to order at least − f D D 2 , so the conditions on − − 4 6 B B . The corresponding exact sequence is ( ( z g f B O O T − − 2 K K D ( ( g K − 4 6 Γ Γ . 4 − − so that so that to = ˆ = vanishes then we can write → O → O ∈ ∈ ( ( − D vanishes then we can write ( z k z ) ) 3 g → O → O ), and we have used this equivalence in the exact sequence. D D D g | | | O D ) ) D D D +1 /B +1 | g f f k k g Σ → O → O D D N ˆ (or − − f g are the canonical and normal line bundles for 2 2

N − 2 B B ( = = ˆ z − − D D = 2 D K K k k ˆ f ˆ D/B B B 6 4 f g Σ O , and if one of them is zero, then we will be able to write N − − K K + 1) + 1) = N ( ( 4 6 D k k = f ( ( − − a section of vanishes to order at least , when they exist, are sections of the middle term in the exact sequence; ( ( ) and restrict ˆ D 1 k − − g ˆ → O → O g f N containing an effective divisor kD B B 0 0 ). Similarly, if ). or ˆ → O → O K K ), i.e., − B D D 1 6 4 0 0 z and ˆ f B with 2 − − − − g D K k B B 6 z K = ˆ k K K − ˆ f 1 4 6 ( Similarly, if We now see the general pattern: if We can continue, and try to detect if There are exact sequences that help to measure the vanishing of g → O → O − − O = ( ( 0 0 of The restriction vanishes ifthat and case only we if can write ˆ The restriction vanishes ifthat and case only we if can write vanishes to order at least 2 along f exact sequence is To understand these sequences itwriting the is line helpful bundle toNote recall that that we restrict them to (or ˆ use the pair of exact sequences Thanks to these sequences, if O O where have used the adjunction formula, which tells us that local geometry and not onhave a the base global structure as a line bundle, which is JHEP05(2015)080 in a arise (3.9) D , then , then (3.10) (3.14) (3.13) (3.11) (3.12) | ] a k ab , to de- a [ k γ C B g , ˆ K a 6 D | − ] a [ k ˆ , f ! !   ab ab C C b . instead of to order at least b φ γ b ab a ab a in a self-consistent fashion. a D C C 6= D 6= a b b b b X b together with the information X γ ) specify a collection of bundles γ φ D , and we use − a as additional vanishings on the a a − B ab ) 6= 6= 6= ) b a a b b X X a C 3.12 ( in ( D P − − N a N ) ) ) on ) ) can be used to determine the minimal . The terms proportional to b D − a a k b k a ( ( b φ b γ b γ z B vanishes on z ) and ( − D N N − a a , 3.12 ) ) g K b 6= 6= k k . 6 . If the order of vanishing is at least Y b Y b ] φ b 3.11 ] ] a − [ + (4 − − a a + (6 D [ [ on each divisor appear on g and ,( ) – 8 – ) f g a a a ( b a ) and ( g ( D φ = = , + (6 + (4 D K K and , and f ) ) 4 g f 6 we define corresponding families of divisors ] 3.11 a a B ( ( a [ − of must vanish if the line bundle of which it is a section − K B f K K  4 a a  4 6 a γ a D in − | D − − around , ] D a a a O [ k g O = = φ D f

, ) )

a a Γ f ( ( k Γ k ∈ F G ∈ to order at least a , considered as a curve in a ) that vanish to orders instead of b a D D | | ] b D D ] a 3.2 [ a k [ k D ∩ f b g a φ ), ( a D 6= 3.1 b = . P is a local coordinate vanishing on ab ab b vanishes on C − C z f B For the monodromy conditions associated with each divisor, and to identify the com- The properties of the bundles in ( We now go through the same reasoning with analogous exact sequences, starting from More generally, given a set of effective divisors K , for each effective divisor 4 B such situations. 3.2 Summary ofWe local summarize here conditions the constraints derived inthat the will previous section be and useful define some for notation explicit calculations. For a compactification of F-theory on a base the corresponding leading coeffients mustthan vanish (i.e., might the have order been of expected). vanishing will be greater plete (geometric) matter content, wesections need of to ( consider these coefficients more generally as possible orders of vanishing By using the factcorresponds that to a e.g. non-effective divisor on which can be checked for the existence of non-zero sections, and if those sections are absent, where because the vanishing of curves the restricted leading coeffient lies in or where − termine the orders of vanishing of that we can write JHEP05(2015)080 , k b + 1, F ) D − a ( , and (3.16) (3.17) (3.15) 2 around K D 6 f , − , . . . , k 1 1 on different is the curve ) to analyze , D = g ab k , = 0 C 3.14 ˆ f G j , associated with ) a a for ( as sections of the line , and ) D a N a ) and ( a ( ) j . D D k a F | a ] φ a γ 3.13 − [ k on on each divisor through the must vanish to at least order g = = g g g = ˆ , , f + (4 ) f g ) ( . . a g ( k , k ab , ˆ , k 1, then K
a C
) 4 to particular orders guarantees the ex- ) · D − k | k ] − g F a ab [ G ( k , ˆ ( C a f = a f . Similarly, when there is no effective divisor D a D = . k = , . . . , k O ˆ a – 9 – F O D 1 ) ba a , D Γ ( k C Γ ˆ can be computed as an intersection of two divisors f on · ∈ 3 ∈ = 0 ) k a b can similarly be described as a section of ( D a j D , with · | order of vanishing of g D
] N | 2 ] ) a [ for k a k [ k D . f ) ˆ g · a G a ( j ( 1 = D are the divisors associated with the anti-canonical and normal a = G ) D ) D are the orders of vanishing of ) a a a ( k ( ( k O a to vanish to higher orders in some circumstances. f g minimum , though in special cases such as toric bases where there are good γ N g g , , , a ) ,Γ f a φ
( ) , and k a K ˆ F f − D ( can be described as a section of the line bundle over a a D . D O ) a ( considered as a divisor class on . This determines a set of conditions on the vanishing orders of b must vanish to at least order a N ) D D f k This constraint follows from a general fact about intersection theory: if In addition to the constraints described in the preceding sections, we have the geometric As discussed above, when determining monodromy conditions and matter content, it In general the restriction of the leading non-vanishing term in an expansion of In the following sections we use the divisors specified in ( When there is no effective divisor in any of the divisor classes are three divisors, then ∩ − on 3 a As can be inferred fromwhile the that notation, on the the intersection right on is the in left isD carried out within directly in the analysisset of of possible this local paper, divisorF-theory configurations but models. and which associated may non-Higgsable clusters be in useful general inconstraint further analyzing the bundles Γ (6 3.3 Additional constraints We conclude this section by briefly mentioning a further local constraint that is not used Note that for a generalterm base these in equations each are of onlyglobal meaningful coordinates, for the these first expressions non-vanishing are valid for all is useful to consider the leading terms and the restricted leading term in various situations in whichistence the of vanishing non-Higgsable of clusterscorresponding of to different F-theory kinds compactifications in to Calabi-Yau 6D threefolds and and 4D fourfolds respectively. the divisor k divisors in the baseconditions that determine a must bestructure of satisfied the in local geometry. astructure We self-consistent have may not force fashion. ruled out the Note possibility that that further these nonlocal line bundles to D then in any of the divisor classes Here, as above, JHEP05(2015)080 , ) a ), a ab C is a Z (4.1) (4.2) a 3.17 C ’s gives , i.e., as i ), we see a D = deg D | b ab , y, z 0), and there p 1 D , 1 . (When − , a C is not effective (i.e. = ( with ) d = (0 a a v ( D b | v b K , . D − . Permuting the ab ab 1), 3 , Z Z = D 0) and b b 0 | , , 2 φ γ C . On the other hand, the same 0 . Specializing the discussion of from the fact that on projection . In the left-hand side, we have , D K ba Z b b y y is the class of the normal bundle, X X C − = (0 − ) − , then (dropping the superscript ( a − − = (1 = and a ( a , and on the right-hand side we have ) ) v c b 3 y a a C N v ( ( D D − | | , . On the one hand, this can be evaluated = a 1 N N a a ) ) = D C D k k C D ) follows directly from the structure of the · ba − − b C as the intersection of – 10 – · 3.17 with D , considered as a zero-cycle on a b ) · ). b b + (4 + (6 D b ( D C ) ) C N D a a N · ( ( a = K K with C b 6 4 . a D . In this situation, the divisors that support codimension d | − − 2 v C b B = 1, so = = D + ) ) are associated with rays c a a ba v k ( ( is a curve of self-intersection k b C F G ) are identified with the value = · D ab is the canonical class of b , C bd a ) 3.17 yv a C D ( K and = 0, z bd ]. Here we show how these results can be reproduced easily using the constraint is the intersection of yC 4 . − ab ab as the intersection of , namely, the intersection of the divisors Z = C b 3 We begin by noting that if the anti-canonical class In the toric situation, the relation ( · ) D D b ( ab while rational curve, the only thingwhich that matters is about the this intersection zero-cycle number is its degree a nonnegative integer class) for a given curve Here, as before, and codimension two singularitiessignificantly, are since associated with all points. pointsrepresent This all on simplifies divisors a the on curve analysis the a previous represent curve section simply the to as same thesince an case they homology integer of are class, in a now curves base so on of we the dimension can base, 2, we and find denoting the divisors by 4.1 Constraints onIn individual six curves dimensions, we areover concerned a with Calabi-Yau complex threefolds base thatone surface are elliptically singularities fibered of the elliptic fibration associated with gauge group factors are curves, As an illustration ofcharacterize how non-Higgsable the clusters, constraintsA we derived complete begin in classification the as of previousgiven non-Higgsable a in section clusters warm-up [ for can exercise 6D beequations with F-theory used derived compactifications the in to was 6D the previous case. section. the same result since to the plane 4 Warm-up: 6D non-Higgsable clusters fan for a toriccoordinates threefold. where Assuming thatare the 3D cones geometry connecting is thesethat smooth, two rays both and to sides taking the of rays aN ( choice of multiple ways to computewe the consider same the intersection intersection property. product on To apply this totriple derive intersection ( can be evaluatedC on on JHEP05(2015)080 , is k 1, ]. f 4 − K (4.3) (4.4) is the = − a of given contains C = 0 then C . · could not a · a C K k a K C C 4 G K C − − , − 2. In this case, on k = − F ) g on an irreducible a ], in which ) , ( 4 g = 0. We then have , with f C 2 N = 0), which satisfies 0 C − φ G 3. In this case, = 0, so there is no forced − , the gauge group of the = 2 ∗ φ = . Similarly, if 3 curve C = deg ) 0 = 2, and a − g K 0 ( ) must support a Kodaira type IV . If , a . From the data in this table, we F − ( 0 a N ab (assuming 2 f were effective, in which case no . The method of analysis used here, p C K b φ γ . , and IV C C − K ∗ 0 φ 6 − − a N supporting a non-Higgsable cluster are − 6= k k a b C (0) = P , then deg a 4 + 3 6 + 3 unless is a curve of self-intersection = C − − ∈ O a k φ – 11 – C 2 = = g k k 6) singularity at a point; and we can classify non- are easily computed and determine the orders of , F = ˆ G k ). We have then 2 1 , so there is no forced vanishing of G g , where , as tabulated in table 0 P deg deg a g , γ k C − F k must all vanish, so the curve -)effective (actually nef) from which it follows that 1 = 2 g Q k ( , would vanish for all 0 . G g k X a , which we take to be assumed for any given curve , g C k 1 , g f k , this leads us to the conclusion that there cannot be a non-Higgsable over the curve , can be nonvanishing or none can. Since , , deg f as irreducible components, as does g with 0 k g on any curves that intersect φ k , and similarly for f f , C g X f C − f , k k 3+ f = 2 C/ (0) = k 4) singularity. Furthermore, = Systematically applying these methods for any irreducible rational curve Now consider the case of a curve of self-intersection We assume then that all irreducible curves Consider for example the case where , F 2 could be non-vanishing unless the same were true of ∈ O , K 0 k 4.2 Monodromy In the caseslow-energy of theory Kodaira depends upon singularities andescribing additional of the monodromy set types condition; of the IV, cycles Dynkinmapped produced I diagram when to a itself codimension non-trivially one under singularity a is closed resolved path can be in the relevant divisor that goes around group, including effects ofcan monodromy; identify we cases can whereHiggsable ascertain there clusters the is containing a generic multiple (4 mattereach gauge of content; group these we factors. aspects in We the describe following a subsections. few details of self-intersection, the divisors vanishing of can determine many featuresin of complex the structure gauge moduli groupsof space and negative matter for self-intersection. that bases arise that In at contain one generic particular, or points we more can intersecting determine curves the precise minimal gauge decomposed over the rationals, so that− for a self-intersection two factors of however, generalizes more readily toZariski four-dimensional approach. F-theory compactifications than the It follows that (2 the monodromy conditionis so essentially that equivalent to the the associated Zariski gauge decomposition method group used is in SU(3). [ This analysis f rational curves (i.e., equivalent to self-intersection of deg vanishing of be effective and g either all curve of genus cluster on any curve of higher genus; this was shown from a different point of view in [ henceforth on JHEP05(2015)080 g is a = 0. . For 4 ). (In gauge . This g g can be X g 6 e 3 ) ) ) , ) ) ) ) , and are ) 8 8 3 g 7 7 D/B 6 4
f 8 e e e e f e ) ( ( ( ( ( ( su N + ˆ ∗ ∗ so ∗ ∗ ∗ ∗ and ( X x II II ∗ 0 6 respectively. 4 2 IV( I IV IV III III singularity, we + 2 (3 f ˆ f − ⇒ ⇒ ∗ 0 O D ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ + 5, to generically be a K 3 0) 0) 4) 6) 8) 8) 9) 9) 10) 10) 3 − , , , , , , , , , , ,Γ x 2 0 0 2 3 4 4 5 5 5 5
g − (no monodromy) only , , , , , , , , , , ) 4 curve, where ( 8 − D singularity type (0 (0 (2 (2 (3 (3 (3 (3 (4 (4 X (no monodromy) if ˆ O so (2 6 . For ˆ γ γ γ e O 2 γ γ γ γ γ − − − γ in the one-dimensional space of su − − n n n − − − ) for generic choices of γ k − u 1 k n n n / − C G k 2 / − = = 3 = 4 = 6 ) that comes from the restriction of a 5 = 2 x n n n = 2 + 5 = 5 + 7 = 4 + 8 = 6 + = 3 k G )( n n n 2+ 3+ 4+ k 5 G D/B B G G G G 4+ 5+ 5+ Divisors G deg 12. N − G G G deg in the expansion around the divisor. These − . In this case, the cubic x are generic so that maximal Higgsing has been is a perfect square, then there is no nontrivial deg deg deg deg deg factorizes into the product of a quadratic times + 4 g )( g 2 deg deg deg 3 – 12 – ; otherwise it is g , A D D/B 3 g and resulting non-Higgsable singularity types on single f K N k − +ˆ su φ φ φ and 6 x G x φ + φ φ φ 2 − − − , − f ˆ 1 through φ ( k f − φ φ D − n n n − − F − D k n − + k n n K − − F 3 3 curve that does not intersect any other curves where O k (0) so the non-Higgsable gauge group there has an algebra − x − singularity, the gauge algebra is singularity the gauge algebra is = 4 = 8 ]; when considering the generic structure in the moduli space = 8 = 8 = = 2 . When ∗ n n ∗ 0 = 2 + 3 = 3 + 5 = 1 + 7 4 4 = 4 + ∈ O ) must be the square of a section of g k = 2 + 6 37 F F n n n X , 2+ 3+ , k F 2 4 D when g F F , the monodromy is determined by the leading coefficient in f F Divisors 2+ 3+ 3+ 2 F 35 ∗ 7 deg deg F F F deg otherwise. This allows us to immediately read off the − so deg deg deg 4 deg f B deg deg deg , where K
of self-intersection 6 ) 1 2 3 4 5 6 7 8 ], and analyzed and explained further in appendix A. These monodromy C − 11 12 C − − − − − − − − · X ( / 58 ( − are both in one-dimensional spaces of sections Γ B = 0), where ˆ C 10 O 3 O / γ g 9 , ˆ . Table of degrees of divisors − 2 ˆ f For type IV and IV . Similarly, for a type IV 3 algebraically factorized to a productcondition ( is clearly satisfiedThe for remaining the non-Higgsable monodromy cluster conditionhave over a is a gauge that algebra fora a linear term non-Higgsable for type generic I perfect square, and algebras of the non-HiggsableFor cluster a over non-Higgsable curves of typewhen self-intersection I proportional to second and thirdsections powers Γ of some section section of particular, it’s not hard to seeThis that the is space clearly of these the sectionsvanishes case can ( only for be one-dimensional.) a su also be used in analyzing compactifications to 4D in latera sections. type IV codimension onemonodromy singularity, and if the ˆ gaugeperfect algebra square, is every section of as is relevant for non-Higgsablefrom clusters, the the monodromy form condition can ofmonodromy be the conditions read leading off on directly terms monomialssection in for 9 non-Higgsable of [ clustersconditions are are valid described for briefly F-theory in compactifications in any dimension below eight, and will Table 2 rational curves a codimension two singularity.case The are details worked out of in how [ the gauge group is determined in this JHEP05(2015)080 ]. ∈ 4 0 10). 4 or on a g , 6) or − , , 5 3 (1), so 0 , 6) point f su (4 , ∗ ∈ O 6) point in , 3 ˆ f 2 curves, it is − add to (4 g and vanishes, there is a , representation. For φ f 5 g 3 and 56 − ]. − condition does not occur 7 63 = 1 , 3 curves. Any combination 3 so − is not a perfect square; in all F 62 , , 2 2 6 ˆ f − curve where 6) points, clearly there cannot be vanishes on each to order at least . The , ∗ 2 ]. By blowing up the (4 g g = 0 on each, so there is a (4 2 or 2 3 curve and a curve of self-intersection 61 − g , − , , and there is again no matter in this case 2 1 60 P − 7 curve, where 5 curve, there are generically two points where − – 13 – − 3 curves, since vanishes and the singularity type becomes II ]; these are branch points in the moduli space that are − 3 ˆ 48 f 2, which implies vanishes identically and 3 ≥ g = 0. For a ]. Such tensionless string transitions unify the space of all 6D γ γ ∆). Such points correspond to theories where gravity is coupled 49 , 2 curve. Considering configurations with 3 − f, g, when 2 or below are between 2 as discussed below. − C − , = ]). 3
on each − ) we can also determine the set of possible non-Higgsable clusters containing , 59 γ , C with two branch points is again a 2 2 − 1 − ( 39 = P O
12) vanishing of ( Γ 2 curves alone cannot give rise to a non-Higgsable factor since we can have (0) , 6 vanishes. These, however, are the points around which there is monodromy; a double ∈ − O , 4 or below is even worse. So the only intersections that we need to consider for curves 3 curve, there are no points where the Kodaira singularity type is enhanced, since e.g. 4 2 at the intersection. Any− intersection between a of self-intersection of Γ multiple intersecting curvesRestricting of attention negative to self-intersection, basesan reproducing that intersection the between do results two notmore. curves of include where This [ (4 the rules ordersbelow. out of any vanishing Even intersection of between fortwo, two two we curves must intersecting each have of for each self-intersection extra tensor multiplet [ F-theory compactifications into a single connected space4.5 [ Clusters withFrom multiple table factors Analogous to the appearance of(4 matter, at loci onto a a II superconformal field theoryassociated [ with tensionless stringthe transitions base, [ one enters a different branch of the moduli space where the 6D theory has an g cover of (see e.g. [ 4.4 Superconformal fixed points generically there is a point where This corresponds to the appearancethe of other a single half-hypermultiplet non-Higgsable in− gauge the group factors withoutg monodromy, such as Matter can arise in6D F-theory compactifications with constructions the eitheror genus from from of nonlocal local the structure, divisor structurewithout associated on associated monodromy, which the in with presence a codimension of gaugetwo matter two group loci can singularities. is on be supported, identified For thespecific when gauge gauge example there groups group of are divisor codimension this where can be the seen Kodaira for singularity a type is enhanced. A other cases the gaugefor algebra any (of of generic the non-Higgsable models) clustersof is over curves a single curve, but does occur over4.3 a combination Matter done, this occurs only when ˆ JHEP05(2015)080 2 2 2 − − − and , , 3 3 3,
− − − . While , 3 (0) 2 cluster 2 B − O − , 3 Γ − that intersects is nonzero and , ∈ 1, which pushes ] for the
2 4 C 2 − ≥ f (1) g O C 1 curves as enumerated Γ ]. A complete analysis of − 2 curve ∈ we describe the possible , ord 67 − = 2, so cannot factorize completely D 3 f x g φ 5.2 2 C ˆ ). Hundreds of distinct choices f 1 − P and III singularities on the 3 since x algebras and the ∗ 0 ∗ 0 2 g . The only possibilities from the Kodaira = 0, while ⊕ 7 gauge algebra found in [ . In section 2 3 2 so g su su 5.1 ⊕ and – 14 – ]. For example, on a 2 2 4 g 2, which implies ord su = 4, so ≥ γ C , γ C φ 3, and we have then type I , as found in [ . Further analysis of this type can be used to confirm the various ≥ 2 7 has two distinct roots (corresponding to the points of intersection C so su 2 γ 3 combinations giving ˆ f ⊕ − , 7 2 3 curve we have so − − , 2 curves) and is not a perfect square, so ⊕ = 0, so 2 , we must have . In base threefolds, on the other hand, a vast range of surfaces can be realized , where − 2 1 −
D 2 D g P su (2) O 1, so 3 and ] can arise as a codimension one divisor in a threefold base that supports an elliptic Γ ]. The simplest non-Higgsable gauge groups are single nonabelian factors. We describe − 3 curve ≥ 57 4 , ∈ − 2 D 2 ˆ 5.1 Possible single gaugeThe factors set of possible isolated6D, simple with gauge the algebras additional for possibilities 4Dtable models of is that basically are the ruled same out, as for in fact, are those where the order of vanishing of ∆ exceeds that a substantial project for future investigation. the possible single factorappearance (at groups the in level section which of presents geometry) richer of possibilities matter than localized in on 6D. curves in the threefold base, as divisors. Restricting to toricin surfaces [ alone, each of thefibration 61,539 (in toric surfaces the enumerated simplest caseof these by divisor taking geometries a can productall support algebraic with non-Higgsable surfaces clusters that [ can act as divisors supporting non-Higgsable clusters represents ysis of non-Higgsable clusters inthe threefold wide bases range more of complicatedcase possible than of surfaces in twofold that Calabi-Yau bases threefolds, can is base arise as are as discussed classified divisors by in in genus, thecluster and the is previous the threefold a only section, base. curve curves topology In in that the can the support twofold a non-Higgsable We now turn to F-theorycation compactifications on Calabi-Yau to fourfolds four thatthe dimensions, are story which elliptically is fibered involve in compactifi- some over a wayscomplications parallel threefold for to base that four-dimensional of theories, six andseems dimensions, there the to are set be a of substantially number of richer possible additional than non-Higgsable for clusters 6D models. One issue that makes a general anal- combinations of non-Higgsable clusters thatin can [ be connected by 5 4D non-Higgsable clusters with single gauge group factors is not a constant.non-Higgsable This cluster. reproduces The the storycluster, is on similar the the otherf cases. Note thatwith for the two the and the monodromy is − that gives a γ curves respectively, with generic monodromy on the I easy to check that the only nontrivial combinations giving non-Higgsable clusters are the JHEP05(2015)080 . , 2 8 = S P e k (5.5) (5.3) (5.1) (5.2) G , ]; in one H 4) 58 bundle over , 5 1 − P k , so the relevant 12, the associated H 11 in the 6D case. , all divisors in the 1 = (4 − P , k ) become ,..., that is a F 10 6 are the curves associated , n . − can be realized on divisors ˜ , F 8 3.14 F e 9 . 3 . In this case, similar to the = 5 , = 4, = , 2 −
= 18 the gauge algebra is S su ) 3 n 7 P n g e B n = , ). Such geometries were described ) and ( . 6 )) (5.4) S e H H, does not intersect other divisors on

where F F nH 2 ord( , and we have , 3.13 vanish, n n , S 4 − ) ) F ) 6) locus associated with a curve on F + + f g ( 3 k k , f , , S S O , and for − f − − + 2 8 7 e 4)( 6)( S S so (4 (6 3 , 3 ord( − − , 7 . This local condition can be realized explicitly − − − e – 15 – 7 2), corresponding to a type III codimension one = 2 n k k , , = so 12 18 7 . For example, for K e , max n N − , = (3 = (3 2 = = 6 > g k k e k k F , G , F G 4 3 f , gauge algebra. Similarly, for su 4 by different constant complex factors and preserve the section f 2 , ord(∆) 2 g , for any integer su = 0 so that the divisor 8 , which has su 1 γ ]. In this case, the divisors ( so . P and nH , 7 66 = 2 6) points on curves of self-intersection × f – − g , so : 1 ) vanish to degrees (1 φ g = P 64 factors, we can take , , . But such a cancellation cannot occur between generic sections, since we f, g and 17, there is a codimension two (4 = f cannot be realized as a single-factor non-Higgsable cluster on a divisor N 2 1 58 g 0 3 P 3 ≤ F su su must vanish, in accord with the assumption that this is an isolated single-factor , so ( n are linearly equivalent to a multiple of the hyperplane class = g + 27 H , S 3 S ≤ f f 6) While Thus, the only possible nonabelian gauge algebra components that can be realized are We can identify explicit examples in which each of these gauge algebras is realized − formed by projectivization of the line bundle associated with different values of k 2 case, if with the two This gives explicit examples ofother all than the single-factor non-Higgsable gauge group possibilities that does not intersect otherthat divisors realize on other which types of surfaces. Several explicit examples were given in [ (4 singularity supporting an gauge algebras are For 13 analogous to the (4 where we assume which non-Higgsable cluster. ItS is straightforward to read off the Kodaira singularity types on in the context of a globalP model by taking a compactpreviously base in [ the divisor supporting the6D gauge case group where is non-Higgsable thesurface gauge surface groups arise online rational bundles curves can bechosen classified so by that a single integer. In particular, the normal bundle can be in a non-Higgsable cluster. The simplest set of examples corresponds to the case where For this to occur,∆ = we 4 wouldcan need always to multiply haveproperty a while cancellation eliminating between the cancellation the of leading leading terms. terms in determined by JHEP05(2015)080 , , 3 g g K F , N 3 f , (5.7) (5.8) (5.9) (5.6) − D + 2 with a ). , 2). The F K S B − 2 , + 3 − 1 − S = , bundle over the (3 2 as a single-factor 1 − O X , A. P 7 1 + so − non-Higgsable cluster. , H 2 7 − H + so , ), where 1 + G H. X − G , ∼ − (3 . This is not true, however, for 2 H, + E − G X + , 3 F + 1 ∈ O -divisor) without a realization as an F − + − 3 D Q may be an effective divisor, while + g F E + 3 K D ), (0), so we have a type IV singularity with + 2 . We then have a toric surface − non-Higgsable cluster that illustrates this . An explicit computation of the orders of + X − G D E N 7 (2 ∈ O . This geometry can be realized in the context B , 4 + 3 − 2 so E – 16 – = ˆ g − C su , , = ∈ O G,C 2 N D + C g 2 2 T + , f B + 2 ]. − A F , while in integer homology cannot be written as a sum + = 1. Now blow the surface up at these four intersection 58 C is effective then so is 3 K E + , and [ A 4 3 1 − 7 g H is generically nonzero. In this geometry, however, the divisor E X − = − · = 2 so = ∼ f F H K having self-intersections ( 0 N C g − = H + 3 + = , produced from projectivization of a line bundle – F 0 1 B B A · F f + on the divisors in this threefold base using toric methods confirms the D = + 3 A 0 g = A f , f is not a perfect square, and we have an isolated D = 3 · 2 ˆ with an appropriate twist that each have self-intersection zero and intersect cyclically with intersections f B N 0 S bundle over F = 1 + 3 P An explicit example of an isolated It may seem that na¨ıvely it should not be possible to realize , 4D constructions can provide a much richer range of matter associated with isolated B 7 on K · e 6 5.2 Matter While in 6D theis only single gauge factornon-Higgsable that gauge can factors. have associated The non-Higgsable matter primary reason for this difference is that while the vanishes, This local geometry cansurface be realized globallyvanishing by of simply considering a presence of a non-Higgsable We can then write which is clearly effective, so − of effective irreducible divisors with nonnegative integer coefficients. Thus, in this case ˆ Now, we consider embedding this surface into a threefold with normal bundle anti-canonical class is and there are two equivalence relations (from the Stanley-Reisner ideal) phenomenon can beH constructed as follows.H Consider apoints set giving of exceptional toric divisors set divisors of toric divisors and one might thinksurfaces that in if threefold 2 bases, illustratingdecomposition one to of threefolds. the The subtle issuemay aspects is only of that generalizing be the in Zariski integer the linear effective combination cone of (e.g. irreducible an algebraic hypersurfaces. effective no monodromy, and the gaugeof algebra is a non-Higgsable cluster on anyvanish. divisor that In does this not case intersect other we divisors must on have which It follows that JHEP05(2015)080 , ∗ 3 2 1, = B P 3 − that that E ⊂ when (5.10) = 5) III S S 2 − , 3 i . P 2 B E E H · ∼ = in i 7 curve that − 11 D E 1 ) codimension − − 7 D bundle over E e ( = 1 . has the form of a . For each gauge − , by simply taking P ∗
with 3 2 D N ) H a P i 3 , where the (3 to be the line bundle 3 3 E ˆ E f B = 3 D . This local geometry can when + 2 i K 2 E is a del Pezzo surface formed algebra; the matter locus is − H E 3 7 e , it is straightforward to verify within a divisor + 2 23 1 bundle over dP L = dP C 6 E 1 D P , analogous to the bound on surfaces − Like the 6D case of the representation, a divisor 3 + 2 2 B . This example is one of the complete set 3 E carries an H 56 4 N 1 2 (2 H − O that is a 11 12 – 17 – 3 − L 4 B = Γ bound on the number of curves that may support or singularity.
− , so the geometric matter lies on a conic or pair of )
∗ 1 exceeds that of generic points on ) H N E N , and . The anti-canonical class is C 10 j 7 + . If we take the normal bundle of 5) II e − 3 + E = 2 , a priori K E is − 4 = H ; for example, in the example above of , and on a single line in the class for an elliptically fibered Calabi-Yau threefold can nonetheless N ], this does not imply that the resolved Calabi-Yau manifold has a fiber of i + 3 − D ( E H 2 N S B (12 41 , O E − 10 O = 40 , which cannot support an elliptically fibered Calabi-Yau threefold. + − Γ 2 H Γ 18 1 B ∈ F = = at three points, giving three exceptional curves E ∈ vanish. For example, on a divisor carrying a type III ]. 3 ij f 2 N 3 + k ˆ 58 f P L g 23 bundles over del Pezzo and generalized del Pezzo surfaces that were classified , gauge algebra and matter in the k L 1 f 7 P e + 13 = 18, one of the divisors associated with a curve in the base L n As an example of a situation where multiple curves support geometric matter in a non- In general, localized matter appears on a curve In fact, there is no clear + The “Kodaira type” along the matter curve is simply the type associated with the specified order of to be the projectivization of the line bundle 3 3 12 of possible and studied in [ vanishing. As observed in [ that particular type in the codimension two locus. that the gauge algebra on It follows that matter is supportedbe on embedded the in three a disjoint globalB curves threefold base Higgsable cluster, consider the case whereby the divisor blowing up and three lines L associated with the divisor carries an with a normal bundledetermined of by lines in the case the Kodaira singularity typealgebra on type, for genericcoefficients coefficients in theone singularity, Weierstrass matter model, will be this localizedsingularity along occurs is the when vanishing enhanced locus certain to of a (4 cannot act as bases arise as divisors in awith base Hirzebruch surface sufficiently complicated topology. Innumber principle, of mutually a disjoint complex complex surface curves,each each supporting could appearing matter. have in a We the expect anti-canonical verycan that class large there arise and is as a divisors boundthat on in can the complex set act of threefold as surfacehave bases bases types an explicit for statement elliptically about fibered the Calabi-Yau existence threefolds. of such a We bound. do not, Note that however, surfaces theory are labeled bysupported a on single curves, integer and (thesingle there degree), surface can in that be 4D, could many the each topologically in codimension principle distinct two support curve matter matter. classesmatter is within associated a with any given kind of gauge group factor on a complex surface of possible classes of zero-cycles that could support a codimension two matter locus for a 6D JHEP05(2015)080 , , 3 ab − on has C su with g (6.1) 3 (5.11) ⊕ , 2 B 2 f g , 2 component. bundle over su 3 1 ]. , which is itself ⊕ P su 5 − 2 g , is a 3 bundle over dP 2), we cannot have a 3 su carry matter can arise , , 1 B intersect on a curve ⊕ 3 P , and the minimal orders D 3 . b g su
D su ) ⊕ is a , 12 3 3 3 L su and B su (2 a ⊕ O 2 D 3) singularity. Furthermore, as noted , su , (2 , = Γ 2 ∗ 0 2 3
. su ) su su and is a perfect square. So the gauge algebra 12 , corresponding to the nonabelian part of the ⊕ N 3 ⊕ ⊕ L 2 12 3) non-Higgsable cluster; in these cases 7 2 – 18 – su , the same is true for L su + 4 − b so su , ⊕ D 2 K 2 realized through a type IV singularity are described , 6 − a 3 su − ( D ] will be described. These models contain a wide range algebra can only arise when the orders of vanishing are su 6) singularity when combined with an , O , , 1 curve, and hence a rigid divisor within the surface Σ 7 57 2 2 − Γ so . su su 3 . We have ∈ N su ⊕ ⊕ is a 2 from [ g 2 2 contains a ( 2 ⊕ g 2 2 su E 8 itself carries a gauge group but also intersects with multiple other B B so − D , which gives a type IV singularity over the section Σ , 1 vanishes to order 2 on 2 12 E 2 su ; note that the generic monodromy condition is not modified in this case L g 2 − where ]. In one of these examples, for instance, g ⊕ − 5 2 H 8 2 ⊕ B E so ], a systematic analysis of all models where the base 2 = , × + 2 su 67 12 1 1 with normal bundle L P su , and matter is localized on the curve E 2 3 In fact, these are the only possible algebras associated with two-factor gauge products. Examples with a gauge algebra In [ Some simple examples of product groups can be found by taking bases of the form Other examples in which multiple curves within a divisor Similar examples of non-Higgsable gauge groups associated with geometric matter on is at least the sum of that on ⊕ = 4), which would lead to a (4 = su 7 , 3 ab combinations: All other possibilities can be ruled out by a local analysis. gauge group of the standard model of particle physics,one were of described the in toric [ of bases examples of non-Higgsable gauge groups, including examples of five possible two-factor B a non-Higgsable cluster associated withalgebra the lifts of theas corresponding the divisors, available set with of gauge Weierstrass monomials simply increases in the product space. of vanishing that giveproduct containing a anything non-Higgsable larger than gaugeabove a group and type I in factor appendix are(2 A, (1 an This restricts us to theso eight possibilities 6 Products of twoWe factors now consider situationsand where both a pair carry of non-HiggsableC divisors gauge groups. Because the order of vanishing of in situations where divisors that also carry nonabelian gaugeof group factors. this kind We describe in some the explicit examples following sections. Since this means that is one or more curvesof can this be type constructed with for aexplicitly other non-Higgsable in choices [ of gaugeN algebra. Two examples a dP JHEP05(2015)080 . 8 or so 8 ⊕ so = 0 on 2 ⊕ z . If there su 3 3 cannot be a su w 2 z =0 ), it is possible can be realized would not be a w | 2 2 6.1 g =0 w | ⊕ 3 f/w 3 . If such a term exists, su B cannot be type IV, since g/w corresponds to matter in a ∩ of the form A nor A g H 8 , and cannot support a non-Higgsable 3 so . But then 2 w A . Again we have a contradiction, so ⊕ 2 8 z 3 and we have a type III singularity on zw so 3 su w 2 z 6) singularity on to a group factor , – 19 – of the form of the form G g f = 2 type matter in the 4D theory. ). Since we are here only focused on the geometric . We can choose local coordinates so that H ). A N of the form ¯ R g , 6.1 G R intersect and support non-Higgsable algebra factors cannot be a perfect square. So supported on A, B 3 =0 cannot support a non-Higgsable z . Then there must be a leading term in | su 2 B B g/z = 0 on . , with the 8 2 w g so ]. In a quiver diagram, each gauge group factor is represented by a node in a graph, that have nonzero intersection and that support gauge algebra factors The graphs describing non-Higgsable clusters for 4D theories are conveniently described This means that there is no obvious constraint that limits the number of gauge factors Thus, all two-factor products are ruled out in non-Higgsable clusters except for those Now assume that Considering the remaining possibilities in turn, first assume that there exist divisors ⊕ ⊕ 68 , giving a contradiction. It follows that neither and 3 2 3 , then there must be a leading term in in [ and a directed arrow frombifundamental a representation group factor ( aspect of thedirectional gauge arrows, groups corresponding to and matter involved, all matter will be represented by bi- more other factors. Furthermore,other since factors, a with gauge no factor constraintto can other have in long than general chains the be connecting limitprinciple connected one on produce branching to pairings graphs point two from of to ( arbitrary another, complexity or in to the itself, absence whichby could of the in some standard global diagrammatic bound. convention of “quivers”, discussed in the physics literature group factor of its own, along a set ofthat distinct a curves. given non-Higgsablematter gauge in 4D group models. factor Thecan can set of thus be connected contain gauge connected “branchings” group to factors where in through a the single non-Higgsable (geometric) cluster gauge group factor is connected to three or can arise from a non-Higgsableof cluster, matter, in this 4D occurs thehomology, on because structure a surface while is there much all can richer. beand points many are As distinct on non-homologous. curves in a that the A are curvecan case mutually given non-intersecting represent thus divisor the intersect that same with carries many element a other of non-Higgsable divisors, gauge group each factor of which carries a non-Higgsable gauge having the five algebras listed in ( 7 More complicated “quiverWhile diagrams” in 6D, there are only two possible gauge algebras with multiple components that A perfect square, so there is no non-Higgsable gaugesu group containing a connected pair of factors with algebra su in a non-Higgsable cluster. Using the same coordinate systemthere as would above, again the be singularityperfect a on cube. leading term If there in is no term in A, B su A were no such term thanhowever, we then would have a (4 JHEP05(2015)080 , g ∗ 0 , f with (7.2) (7.1) vanish 9 g , 6 for f . Note that − ,...,C , 1 1 4 C . The branched ]. and the sections 2 g i that intersects two 67 . C S i ≥ − i 0) chosen from the bases , cluster) in which a non- 4 , and 2 . 2 2 − m, v g B h , 5) su , 1 , 2 2 − ⊕ , , su 7 gauge algebra, and type IV, I 3 4 , on the various divisors using the 2 so − 1 , , g su ⊕ 1 1 , , 2 − f 0 , , su that satisfy 3 1 ∗ − − , , from the projectivization of the line bundle N 1 0 2 , − , = B carrying a . The monomials in the Weierstrass model are 3 – 20 – 3 − − Z M , ) = (0 9 = a ], and will be described further in [ carrying gauge factors − that gives a non-Higgsable cluster exhibiting branching . We leave a global analysis of these issues for future N 57 8 ) = (2 3 3 9 C B in n B bundle over , i ,..., 6 − 1 1 v C P a , − ’s from [ 4 ( 2 ,..., C 1 as a B n 3 i − B ( n − ], characterized by a sequence of toric divisors (curves) 57 , where i C i bundles over a 1 i singularities on P ∗ 0 P The local rules that we have derived here do not seem to place any significant con- are represented as rays = ± In six dimensions, there isHiggsable only cluster one contains situation (the awith) gauge more than factor one thata other intersects gauge variety (carries of factor. ways, jointly Inin as charged four which discussed dimensions, matter a above. however, non-Higgsable this gauge In can group particular, can happen be in there supported are on many a local divisor configurations to orders 4, 6.points. Many Further examples constructions of exhibiting non-Higgsableset branching clusters of do exhibiting have branching such appear codimension in the three full 7.2 Chains respectively. The orders of vanishingconsidering on the any set divisor of orthat curve available can there monomials. be is In determined a this byand type simply case, I III carrying singularity out onquiver this diagram Σ analysis representing shows thisfor non-Higgsable this cluster particular is construction, depicted there in are figure no codimension three points where using the methods we haveeasier derived to here, in simply practice analyze fortoric the a approach, orders toric which base of like is vanishingΣ this of easily it automated. is generally then To do the this, elements the of divisors the dual lattice We then construct N While the non-Higgsable cluster associated with this base can be worked out in principle computed in [ self-intersections work. Here, we simply givementioned a are few realized examples explicitly. in which the branching7.1 and chain features just Branchings An explicit example ofcan a base be constructed as follows. We begin with a toric base straints on theF-theory complexity constructions of of quivers 4Dthe that number vacua. can of distinct It arisefinite topological seems from and classes likely hence non-Higgsable of that, that in geometriesfrom there as elliptically in compact are fibered for some threefold Calabi-Yau Calabi-Yau actual fourfolds bases threefolds, bounds is on the complexity of possible quivers JHEP05(2015)080 2 ⊕ 4, in su 3 . In − ˜ + S,F with 2 su Σ m − ˜ S 2 ⊕ m ∩ 2 ∩ 3 2 2 6) divisors will be the su , E as described ,..., ⊕ +1 ...,E i 2 K 2 S = 3 ,...,E − su i E + ∩ i Σ = S for , and denote by ∩ bundle, giving further N . We then blow up the 1 m , giving an exceptional 2 = 2 1 − F 0 − i i F − P . On each surface in such m E C m ∩ 2 2 ∩ ˜ E S i 5.2 E , 0 on the adjoining surfaces, where i ,...,E E γ 2 2 times. We then blow up the curves g , ∩ , then on the new curve bundle over E i φ ˜ S . This geometry contains within it a 1 m E  } P ∩ 2 , consider the following construction: we on the chain of divisors

1 3 2 − E © B with normal bundles 2 2 su 6 ? m 3 del Pezzo surfaces, each with normal bundle 2 ⊕ su su 3 3 – 21 – @@R su summands can go up to at least 11. @@ ,..., 2 3 . If we blow up a curve on one such surface, it forces @I @
su 2 ∈ { g ⊕· · ·⊕ (0) i 3 6) singularities over , using the notation of section O , su for +1 Γ are two sections of the original trivial i ⊕ 2 00 S in ascending order, and the curves i ∈ ± 8, considered as a trivial E su 0 − in , ≤ 0 Σ i , g to be a chain of dP that both carry non-Higgsable gauge groups themselves. This opens 0 8, and can be analyzed most easily by toric methods to have a non- 23 3 del Pezzo surfaces dP and the fiber. We blow up on the curve E ∩ m f L 3 00 2 , − ≤ m S B − , with bifundamental (geometrically non-chiral) matter connecting the first m 2 , m m m 0 F g 2 and in S i × ⊕ . This leads to (4 i 0 2 S , where the number of 1 2 S 2 g P E ; we repeat, blowing up on the curve in . The quiver diagram associated with a non-Higgsable cluster having gauge algebra 1 . The surfaces will be connected so that, for example, su ⊕ ∩ ,...,E 1 2 2 E K − ⊕ must also vanish, etc. E to vanish on that surface. This contributes to E su 3 Σ − 0 To realize this explicitly in a 3D toric base As an explicit example, we demonstrate how it is possible inConceptually, a the simple idea class of of the toric examples we describe here is that we can take a set 0 g the new exceptional divisor, etc., for a total of 2 ⊕ ∩ su = , , g 2 2 2 0 0 above; these are the propercurve transforms of the surfaces which are resolved byor blowing curves up when these curves.Higgsable gauge The algebra final geometry has no (4 divisor E E descending order, where Σ exceptional divisors linear chain of 2 f start with the divisors given by theself-intersection lift + of curves in the base that are in the classes of the section with of surfaces N curve a sequence we have f in which relatively long non-Higgsable chains of this kind canmodels arise. to realize non-Higgsable clusters··· with gauge algebras of the form other divisors the possibility of a longSuch linear chains cannot chain be of ruled connected outsome gauge by exploration factors the of in local the analysis a space we non-Higgsable of have cluster. presented possible here. configurations shows Furthermore, that there are global models Figure 1 su component with the other three gauge factors. JHEP05(2015)080 , , i 1 ’s, P 1 D +1), P (7.8) × ∩ , 1 0 ± , P Z × . Blowing singularity i 1 8 = (0 P D e each carry an 1 i ∩ v 5 (7.12) 224 (7.9) (7.10) (7.11) D + i 0), corresponding − − − − , factors, so that the E 0 2 , m m m m m 6) divisors, curves, or 1 2 2 2 2 2 , su − ≤ ≤ ≤ ≤ ≤ i i i i i = ( ≤ ≤ ≤ ≤ ≤ 6 v 1 2 2 2 2 0), ; we do not analyze the details of , , ˜ 1 S − 1) S, , , , , 4, we first blow up the curves , , , 1) 1) + 1 3 = (0 , i, i 0) 1) − 2 5 , i, i, i, v , with matter curves supporting matter in the + 2 + 2 2 0) (7.7) + + + – 22 – = 1 m m 0), su 1)0) (7.3) (7.6) , m, m m m 2 2 i , 0 ⊕ ± 0)0) (7.4) (7.5) 1 − − − − − − , , , , , , , , , , 2 . We then blow up on the curves 0 1 0 − 1 1 2 2 1 1 for , , , , ± su i − − − − − − − = (1 E ⊕ 4 2 = ( = ( = ( = ( = (0 = (0 = (1 = (0 = ( = ( ,X v i i i i i , and then blowing up on the appropriate edges of the toric 2 3 4 5 6 − su , + − v v v v − 1 associated with two distinct points on each of the three 12, the situation becomes more complicated as an 1+ 6+ 6+ 0), ,Y v m m ⊕ v X − − , + ± 2 1 , m m Z , = 4. 5+2 2+6 6 8 − su ,X v v v v , Y + m ± , < m < Y Y ) ) ) ) ) ) ) = (0 + 0 i i 00 ˜ i ± , S S 3 F 6) curves gives us a good F-theory base with no (4 = X E E ( ( ( v E ± , ( ( i ( , (Σ X + D 1), . For 8 Y 3 = 8, we thus have a chain of 13 non-Higgsable gauge group factors, including 11 − , , su − m 0 6) matter curves develops on the surfaces for the case Z , , , 2 + We have identified a number of examples where a loop arises in a non-Higgsable cluster This family of examples illustrates the possibility that long chains of connected gauge This construction is most easily visualized in the toric language, where the toric divisors gauge factor, and they are connected in a cyclic chain as described above. This can all Z = (0 2 2 points. In the final geometry,su the proper transforms ofbe the initial done divisors in thev toric language, starting withto the fan spanned by the rays bifundamental of each adjacent pairand final of factors. gauge This group example factors,with can as divisors be well constructed as as follows:and between the labeling beginning initial with giving new exceptional divisors up four more (4 the previous subsection, andconsiderations. there is no way to rule out suchgeometry. loops In based one on simple purely example,total local there gauge is algebra a is loop consisting of four group factors can arise in a 4D7.3 non-Higgsable cluster geometry. Loops Another complication that can arise inloop a 4D in non-Higgsable cluster the is quiver the presence diagram. of a Locally, closed such a loop looks much like the chains described in The triangulation associated with thefigure cone structure for this fan is shown schematically in copies of with (4 these geometries here. can be defined through the fan the case JHEP05(2015)080 7 . 4 F × 1 8 24 15 25 P 9 23 16 10 17 . 2 1 22 (B) 3 11 21 18 are not shown.) Large blue dots in the linear chain. 3 12 = 1 supergravity theories in four 14 19 20 v su N and . (A) The quiver diagram of the chain, (B) a 6 , the divisors associated with points 9, 10, 11 13 2 v 25 – su v 3 v ⊕ 3 , from an initial fan describing the threefold su – 23 – 6) curves that must be blown up to divisors after the ⊕ , 3 ,... ) 2 su 1 ⊕ E su 3 gauge summand, smaller red dots are divisors supporting an su 3 ⊕ ←→ su 2 3 su surfaces (as can be seen from the structure of solid lines). Similar su 3 ←→ 3 su (A) . Note that before blowing up to 25 v for F-theory compactifications that give ←→ 3 3 B su . A global model with a non-Higgsable geometry giving rise to a linear chain in the quiver ←→ gauge algebra. Open circles represent (4 2 2 su truly present in the low-energypresence supergravity of theory G-flux, of anyyet the specific incorporated corresponding F-theory systematically vacuum. superpotential, into The these and F-theory factors extra may causes degrees affect anon-Higgsable of generic this structures freedom we change conclusion. have not in analyzed qualitative Unless here will behavior, one be however, of generic it features seems of F-theory that vacua. the We have initiated a systematic analysis ofthreefolds geometric non-Higgsable clusters that can arisedimensions. in These structuresat describe the gauge geometric groups level andarise by at matter generic deformation that points of cannot inwork complex the be must moduli structure Higgsed be space moduli, done of and the to corresponding ascertain which Calabi-Yau whether therefore fourfold. these More geometrically non-Higgsable structures are 8 Conclusions 8.1 Summary and open questions are connected del Pezzoconstructions dP are possible with up to (at least) 11 factors of fan in the sequencefan, described with above. vertices A labeled schematic in picture the of order the of triangulation blowups, of is the given in final figure schematic depiction of the triangulationfrom of a the sequence 3D of blowups toric(Note at fan points that describing 7( the points global associatedrepresent geometry, divisors formed with supporting the an toricsu rays blow-ups up to Figure 2 structure of the gauge algebra, JHEP05(2015)080 × 9 13 17 5 . (Note that the 1 P 6) singularities in , × 1 10 14 P 18 6 × 1 P 1 7 3 2 (B) 15 11 8 4 16 12 ]. Unlike in 6D, however, the topological 5 – 24 – 9 5 13 17 2 2 6 ? , from an initial fan describing the threefold su su - - ,... (A)   2 2 6 ? su su . A global model with a non-Higgsable geometry giving rise to a loop in the quiver 6) singularities at points need to be treated in any special way. Such singularities , One issue that we have not addressed here is whether geometries with codimension The set of individual gauge factors that can arise in a non-Higgsable cluster is quite We have developed a set of local equations that govern the possible non-Higgsable for 4D F-theory compactifications. three (4 cannot be blown up withoutdo tuning not additional represent complex additional structure branches in moduli. the These space therefore of fourfolds, unlike (4 we have identified explicit globalinteresting geometries open that question contain is each thethe of extent graph these to structures three which that features. globalfor can An constraints elliptically arise limit fibered the for Calabi-Yau complexity largedistinct threefolds, of non-Higgsable topological there clusters. types is of Indeed, asclear elliptically since yet argument no unlike fibered at proof this Calabi-Yau fourfolds that time the is that number would finite, of bound there the is size no of possible non-Higgsable clusters five possibilities, includingSU(2), that of which the wasstructure nonabelian analyzed of part specifically of the inquiver the gauge [ diagram, standard groups can model apparently inconstraint be SU(3) a that quite prohibits complicated. non-Higgsable branching, From cluster, loops, the or which local long analysis can linear there be chains is of depicted no gauge in factors. a Indeed, toric bases, a straightforwardstructures monomial in analysis any can specific be case. used to determine non-Higgsable limited, and contains only nineof distinct possible products simple of Lie two algebras. gauge Similarly, the factors number that can arise is also quite limited, and consists of only blowups at points 7 points on the left should be identified with the points on the right with thestructures same that labels.) can appear on a combination of divisors in the F-theory base manifold. For Figure 3 structure of the gaugethe algebra. triangulation (A) of The the quiver 3D diagram toric of the fan loop, describing (B) the a global schematic geometry, depiction formed of from a sequence of JHEP05(2015)080 ], the 83 – ] and the 77 73 ] and the complete 75 ] has enabled a systematic anal- 4 ]. F-theory geometry also places 74 , 6 6) singularities, however, which correspond to , – 25 – ]. While both these approaches give a large class ]. It is hoped that the results presented here will 76 , and the Enriques surface. For elliptic Calabi-Yau 2 72 P – 69 12, , ≤ 57 , m m F ] give a simple characterization of the set of possible bases as blowups of 62 ], but it seems likely that as our understanding of the global space of F-theory 66 , 58 The approach of analyzing elliptic fibrations through the geometry of the base provides One physical motivation for attempting a systematic global characterization of the as it reduces theit complexity is of in the principle geometryexpect possible involved, that to and the complementary describe a insights all provided moreare by elliptic complete currently these Calabi-Yau’s different under approach approaches, from active as all this investigation, ofof will point which elliptic provide of Calabi-Yau many view. fourfolds new in insights We the into the near geometry future. a complementary approach tomanifolds the as long-studied hypersurfaces in toricintersection toric (CICY) approach approach varieties to taken pioneered in describing by [ of Calabi-Yau Calabi-Yau Batyrev manifolds, and [ can beclassification used to of study bases elliptic Calabi-Yau’s using (see e.g. non-Higgsable [ clusters is in principle both a simpler approach strong constraints on theenergy compactification 4D geometry that supergravity have theory.out manifestations in Some in [ the initial low- explorationvacua of matures further such constraints insightsemerge. into was the carried constraints produced on low-energy theories will the goal of identifyingsupergravity generic theories. features In or six specificthe dimensions, possible constraints effective F-theory that theories geometry that F-theory can places placesin be various realized. on the constraints 4D Some on of physical theseconsistency theory constraints are conditions understood as on anomaly the constraints, low-energy while theory other [ constraints place additional theory) in higher dimensions is anclusters interesting given open here may problem. provide The someproblem. analysis guidance of in non-Higgsable attempting to systematically address this space of possible elliptic Calabi-Yau fourfolds and associated non-Higgsable clusters is threefolds. For ellipticwork Calabi-Yau of threefolds, Grassi [ thethe minimal Hirzebruch model surfaces programfourfolds, [ where the base is abases complex is threefold, known; no constructing such such simple a characterization of classification minimal using the minimal model approach (Mori ysis of the space ofspace elliptic of Calabi-Yau threefolds, 6D giving F-theory asimilarly models fairly provide [ clear a global useful picturefications tool of to for the four exploring the dimensions.be space Such significantly of analysis more 3D of complicated bases elliptic than for the Calabi-Yau F-theory fourfolds, corresponding compacti- however, analysis will for elliptic Calabi-Yau superconformal field theories coupledalso to represent gravity, some the kind codimension of three more exotic singularities gravitationally may coupled theory. 8.2 Classifying Calabi-Yau fourfolds The classification of non-Higgsable clusters on 2D bases [ codimension two. Like codimension two (4 JHEP05(2015)080 ] that support 70 , 57 SU(2), is one of only five × ] and the potentially even greater 84 ]. On the other hand, the apparently infinite 7 – 26 – . We expect a similar story to hold for fourfolds, 2 ]. And since the Hodge numbers of generic elliptic su 58 ⊕ 2 g , 4 f , 8 e ]. 67 ], non-Higgsable clusters provide a mechanism that may make light matter fields 5 While the possible structures that can arise in non-Higgsable clusters for 4D models Non-Higgsable structures seem to be highly prevalent in F-theory vacua. In six dimen- the nonabelian part of thepossible standard two-summand model Lie gauge algebra group,If structures SU(3) we that can assume arise thatnecessary from components matter, for non-Higgsable and “interesting” clusters. (i.e.,nonabelian at anthropic) part least physics of in the two standard the nonabelian model landscape, arises then gauge as the simply factors, one of are five natural the minimal minimal possibil- non-Higgsable gauge group factors.will Some be initial presented systematic in investigation [ in this direction may be quite complicated, theof set two of gauge groups possible that gauge can groups, appear and in in these clusters, particular is the actually products quite limited. Specifically, elliptic Calabi-Yau threefolds, onlythreefolds 27 with large lack Hodge non-Higgsable numbers have gaugetors many non-Higgsable of groups. clusters, the with numerous gauge Typical fac- algebras though elliptic the types of factorsanalyzed. that In are particular, typical we inlikely expect to 4D that give models for rise fourfolds has to with not the large yet largest Hodge been number numbers, systematically of which distinct are flux vacua, there will typically be many also in [ and gauge symmetries arequiring natural any consequence special of tuning. generic string compactifications, without sions, of the more than 100,000 possible base manifolds studied in [ observations, it seems thatbehavior F-theory of gives a one “generic” ofas class the G-flux of best do nonperturbative pictures string not weby vacua. have substantially non-Higgsable so change Assuming clusters far that the in forphysics issues structure the the of such of F-theory a the geometry, “typical” gauge a F-theory factors suggestive vacuum that picture in are emerges the of forced landscape. the In particular, as emphasized number of topologically distinctnumber type of IIA non-geometric string flux compactifications vacua maythan [ provide those even larger realized families through of vacua provides standard a F-theory compactifications. window on F-theoryavailable the also, from nonperturbative however, other dynamics string of constructions string that vacua require in a a weak way coupling that limit. is not Given these example, F-theory vacua withof a possible smooth F-theory heterotic constructionsCalabi-Yau dual [ fourfolds are are a much smallformed larger from subset than F-theory of those constructions the of isother full estimated threefolds, constructions set to such the be as number much IIB of much flux larger flux vacua than vacua [ those from At the present time, F-theory representsvacuum one of solutions the of most general string approaches6D, theory to F-theory constructing for in which its analytic currentof tools formulation 4D are does available. string not seem While,and vacua, to unlike broader the in in sample any vacua than sense formed those cover the 4D from full vacua F-theory space constructed seem from to many other represent approaches. a For much larger 8.3 Physical consequences of non-Higgsable clusters JHEP05(2015)080 ], G 90 , 70 SU(2) lie × ) gauge group N ]. In the context of 87 sector with a spectrum of that would be restricted to 8 e G or 7 e , ]. 6 5 e , 4 . In fact, these are the only possibilities f 7 , 8 so of the standard model in any non-Higgsable so 2 , , or su 7 – 27 – 2 g so , , 3 2 g su , , 3 2 , with additional matter charged under the SU(2) and su G su structure appears in [ , 2 2 × su su ⊕ 3 SU(2) ], many significant questions must be answered to provide realistic su × 89 3 would rule out the hypothesis that the low-energy spectrum seen in – 85 N > Combining these features, the simplest picture of the “typical” F-theory vacuum that If we assume that all nonabelian gauge groups and matter arising in nature come from sector where nature arises from non-Higgsable geometric structures in generic F-theorywe models. have available atone this or time more would single be or a model multi-factor gauge where the groups visible with light or fields without consist matter, of with only five it would have to behave associated one with of an the internal gaugefor gauge algebras gauge group algebras that cancluster. connect to The the discovery of matterthus charged fit under naturally a with hiddendiscovery gauge of the weakly group predictions interacting in matter of this with, a family for example, would generic an F-theory additional SU( model. On the other hand, in a non-Higgsable cluster, butFor example, with the other standard gauge model factors factorsgauge could connected algebra arise SU(3) in as the part of quiverfactors; a diagram. this non-Higgsable would cluster with correspondconsequence to is a that weakly if interacting dark such matter a sector. dark matter An sector interesting arises from a non-Higgsable cluster then non-Higgsable clusters, this wouldpossible correspond gauge to groups hidden and sectoradditional matter dark supersymmetric content. matter In with theglueballs specific that simplest would cases, interact only this gravitationally couldfor with ordinary dark be matter. matter simply Another can an possibility arise if the nonabelian standard model components SU(3) generic (i.e. non-Higgsable) structures,factors then would the also structure placematter interesting of is a constraints non-Higgsable non-Higgsable on cluster gauge (or dark multiplethat group clusters) are matter. with completely one or disconnected One more from possibility gaugesectors the group for have factors standard model; also dark such been disconnected considered dark in matter the F-theory GUT literature [ 4D F-theory model withgeometrically the non-Higgsable matter SU(2) field in chargedmass a under the through non-Higgsable SU(2) radiative cluster, that corrections if acquiresof after a there the negative SUSY is non-Higgsable is an broken; appropriate further discussion and analysis such a mechanism isunderstand not better yet how well geometric understoodof non-Higgsability in the relates four low-energy to dimensions.Yukawa theory, the couplings and field It would theory is for need also description gauge to a necessary groups be more to and computed realistic matter inmodel model content. any seen further specific in Note structure nature geometry that is such with while broken the as of by proper the the course Higgs the field, SU(2) this of could the in standard principle occur even in a provide an alternative framework for F-theoryGUT phenomenology approach to [ the well-studied F-theory models of particle physicsmodel, from for this example, approach. mustfashion. either The be While abelian the tuned latter by U(1) possibility hand, factor has or in been must the also shown to standard arise be in possible a in non-Higgsable 6D models [ ities that may arise throughout the landscape. While this is certainly suggestive, and may JHEP05(2015)080 × – 28 – SU(2) for two-factor visible gauge group products carrying × as just discussed above. Clearly, supersymmetry breaking, which we 7 so , or 2 g , 3 su , 2 No. PHYS-1066293. Slatyer, and Yinan Wang forwould helpful also discussions like and to the thank refereewere the for Aspen carried helpful Center out. for comments. Physics, We under The where grant research the Contract initial Number of DE-SC00012567. stages W.T. ofthe The is this research National work of supported Science D.R.M. by Foundation is under supported thefor by grant Physics U.S. by PHY-1307513. for Department hospitality We of and thank Energy the partial Aspen support Center by the National Science Foundation Grant Acknowledgments We would like toJonathan thank Heckman, Lara Sam Johnson, Anderson, Shamit Mboyo Kachru, Esole, Shu-Heng Shao, Antonella Eva Grassi, Silverstein, Tracy Jim Halverson, world-volume degrees of freedom ininto F-theory, supersymmetric and to 4D incorporate F-theoryas supersymmetry vacuum our breaking models. understanding improves It theplay seems non-Higgsable an possible, clusters important however, described and that heresupersymmetric perhaps even may vacua predictive continue of to role F-theory. in describing the generic properties of 4D predictions for low-energy non-supersymmetricnon-Higgsable physics. clusters While in theconnection this between analysis the paper of underlying is geometric geometryF-theory and based models low-energy physics as on is in rigorous notgeometry. six as mathematical dimensions, Much direct where in reasoning, more 4D the work the low-energy must physics be precisely done mirrors to the understand the role of G-flux and 7-brane have completely ignored here, wouldwould also need give to rise to be possible incorporated pseudoscalarfrom axion-like in the fields from any supersymmetric the realistic model. lifting of model.universe scalar The is moduli This structure not of too lightfrom different fields a from generic that F-theory this we vacuum. seetypical highly At in F-theory simplified our the vacuum current picture is observed state of still of what understanding, quite this incomplete we picture and might of cannot expect the yet be used to make specific jointly charged matter.additional, There completely could decoupled, also non-Higgsable befrom clusters, the dark or same matter cluster additional and sectors sectors thatSU(2) that corresponding could product be come to as charged well other for as examplesu another under hidden an SU(2) gauge in group an that SU(3) would have to have the algebra possibilities including SU(3) JHEP05(2015)080 . , , )
∗ 0 } g 2 ) β , =0 , we , we z f ∗ 0 0 ∗ + { D/B | ) N 2 αβ + 2 + f/z 2 D α ) and such that ( K = ( 4 − ˆ f − ( D/B = D N ˆ ) f ) O + β Γ ) D D/B 2 + . µ ) K ∈ 3 . α β 2 kN 3 + α . µ − α 3 + ) + ( 1 4 2 + 2 α such that λx x = 2 D λ D − α
)( and g + O ( ) − = 2 β 2 2 =
kK ) ) x µ g 2 g − β D/B and ˆ )( , ˆ (4 − x 2 ( + 2 λ N 2 g . D/B ) , respectively. We let )( α and ˆ D α 2 α − 3 4 N + α . 3 + ˆ D λ 2 O 2 (2 x − α − D + g 2 − ˆ 3 ) fx + 2 K x = D = (all in the local coordinate chart). These are β ) = = ( − 2 – 29 – ˆ + µ ˆ f K f } g − − 3 = = ( 2 ( D/B x =0 α + ˆ ˆ g − f D z ( ( { | O ˆ + ˆ − ˆ kN ∆ = ( D fx ) (7). To ensure that the seven-brane has type I (8). To ensure that the seven-brane has type I 6 O Γ − ˆ so + so fx /z 3 B ∈ Γ ˆ 3, and 6 along ∆ = , then + x ∈ 3 kK ≥ λµ x 2 λ α, β − − ( 2, ), then is not the square of a section of = ˆ ∆ = (∆ D β g µ ≥ O 4 + , which would imply , which would imply describes the location of the brane (in local coordinates), then − α and α α and ˆ ( 1 2 2 2 , which would imply } } λ 2 − − α αβ λ =0 z = = = = 0 6, respectively. { − = | , β β β ) g µ 3 z 3 , { = (i) (ii) ˆ g/z (iii) = to not vanish identically. In all other cases, the gauge algebra is and the gaugealso algebra need in If there are sections such that f to be not identically zero. That is, three things must be avoided: and the gaugealso algebra need is If there are sections and ˆ = 2 We point out two particular ways to solve these constraints (although these solutions The monodromy and gauge algebra are now determined by the behavior of the cubic k • • • D = ( ˆ are not the most general ones possible). polyomial for In this appendix we discuss theIf monodromy conditions for F-theory seven-branesand of type ∆ I have orders g sections of line bundles A The gauge algebra of maximally Higgsed models JHEP05(2015)080 , . α is 3 + + } 0. B ˆ u
can and D D + 6≡ ) B . We (A.2) = 0
K β K
. This ) 0 with ˆ ∆ 4 ) 2 =
β + D/B . ) AX (7). For { − ≡ − g η ( ( ), namely, , N α D/B j so + } D/B D D − 3 N D/B + N for O O N and ˆ D/B N X D = 0 β and the divisor 2 β + +2 N j Γ K ˆ , and u α will be a generic α D B D 2 + 3 β { ∈
j A ). Suppose that +3 ) K − g , K and D β K ( D 4 D 2 = α provided that ξ K D − ˆ − K − α, β f g 6 P ( ( 6 .) O kN α , − D g D for ( − ] into homogeneous linear ( + ( Γ N O O and ˆ D , while ˆ α D αβ D ∈ 2 ˆ ξ, η O f . and ) (A.1) [ ρ O i u so that ,Γ ) = C r kK
f u ) i 2 g B ∈ ) we get the same locus: − r − ˆ g j (8), and let ( . 2 − X − D/B } D ( so and x N N ˆ ( O 3 ) and ˆ f, c η =1 A i 2 Y 1 j + = 0 3 =1 c ξ Γ β are generic, i.e., the model is maximally i Y ( j D ˆ g = are linearly independent in Γ { g + b = K K → → } 2 g =
2 + ) ) P to satisfy the criterion given above. Note that − αβ ˆ g } ( must decompose as a union of + ˆ and – 30 – ˆ f (8) for any choice of such that the constant polynomial ˆ + D } f, kD f AX ˆ 2 = 0 3 fx O so = r + , αβ, β α + ˆ g = 0 ( 2 , + µ 2 B 3 2 α ˆ c g 3 − r { { { x X )). If , kK = 1 D 2 = 0, r ˆ are linearly independent in Γ f , are all one-dimensional and ˆ N − λ is not a square, then the gauge algebra must be
( ( } ) 2 ˆ D B f is not a square. . Here is how we will use the “maximally Higgsed” property: if ), we should obtain the same gauge algebra for general constants O O 3 ˆ g µ D/B ρ 2 , would lead to a linear dependence relation among powers of degree N β, αβ j 2 ˆ 0 and K f, c α :Γ are assumed to be linearly independent, none of + 3 1 { ) with c k ≡ ( D will be a generic element of the image of β 0 since ρ D . After generic scaling ( g ( } K ˆ f 6≡ → 6 D 3 ) and − O ˆ g = 0 µ ( . If ˆ . If the spaces of sections Γ α ˆ D in a model which has been maximally Higgsed. Fixing the base β and that f, be sections such that ∗ 0 O
+ ) are linearly independent in the complex vector space Γ
ˆ ) ∆ = 4 In this case, we get gauge algebra in this case, we can set and there is a corresponding factorization of a section of Γ not identically zero, then thereThere are are constants then constants can be factored into linear factors: α . (If not, then further Higgsing is possible by scaling β { 2 Since Consider first the case that the gauge algebra is We now consider what monodromies can occur in the case of an F-theory seven-brane c D/B at which the seven-brane is located, there are restriction maps , D/B N Solution 2 Solution 1 1 factors and choosing a factor which vanishes upon subtituting vanish identically. Thus, the locus and first remark that this implies2 that is because a linear dependenceconstant relation coefficients among powers of1 degree by factoring the homogeneous polynomial we scale ( c N and (identifying Higgsed, then element of the image of of type I D JHEP05(2015)080 0 , . if is 3 1 + } µ c 2 } 7 (8), and . It D = 0. β 2 4 . We ), we = so } so c 2 K β and 2 4 3 , but 2 − λ λµ c c , so the 0 ) β, αβ u 2 3 . On the − .
2 3 2 α = 0 λ − c ( ) 2 4 c = α and B c − λ D { = must vanish = 4 − { 1). Then the α O c = g D/B 2 2 1 λ 2 3 c c > 4 and c N ( Γ α, β, and c
(respectively, the 3 { ) A } = − and ˆ c + ∈ 2 2 ), so the maximally is a perfect square, 2 2 µ ˆ u 3 D λ = ) c λ 2 ˆ u D/B 4 A 2 3 K λ c c − B N 2
1 − µ . , αβ, β − = 2 ( 2 +3 1 c − 1 c = α D is the union of D c , { ˆ ), the new sections 1 1 f O K } must be 0 and K ˆ g c 6 2 (8) gauge symmetry must take must be linearly dependent and K and ( − (7), and let = 0 = so ( which implies that ˆ 2 4 f, c , we see that ˆ 1. Then not every element of g = 1 with generator ˆ so 2 D β . c 1 2 λ 2 3 { 4 3 c λ c for some constants O c ( c αβ 3 > − =
2 4 = = ˆ u ) and → c µ 4 0 2 3 c B
, we see that ) β c . Thus. if the gauge algebra is c µ ) 2 ˆ g u (8). Thus, D/B = = is not a perfect square. This is only possible β must vanish to order at least 4 (rather than + ˆ 2 4 does not have the form f, 1 so 2 N g c does not have the form g D/B ˆ c β f
– 31 – ; 2
) N + ) α + α that are needed after scaling take the form 4 and ˆ 3 contains only a single non-even monomial in a local be sections such that + c 6= 0, then 0 . Thus, c must in fact be linearly dependent in the complex D/B αβ 2 2 3 2 2 D for any ˆ
D/B β 4 ˆ c ˆ c , so both of them can be written as constant multiples ) f u 0, then the locus N c D = β K K
N 2 3 , algebra, A 0 ) 2 K c ˆ 1 u 6≡ α 7 2 c + 2 and − D/X = ) = ), which permits any use, distribution and reproduction in +3 A (8). λ + ( 1 (respectively dim Γ 2 0 − . It follows that and so ˆ D D/B N D f ( 2 would be a perfect square and the gauge algebra would be D α µ so > must be 1. N K 4 α D α K αβ ˆ O f c 2 3 4 which does not hold for general
6 + 2
O c + ) is the square of an element of Γ ) + − 3 1 − = D ( c 0). But if D (
when either 0 β ) D K 2 D K µ ) = D/B = 0 identically, and 2 D/B = 6≡ 4 ˆ 2 2 α O f , CC-BY 4.0 O 3 N 3 4 ( N − β λ µ − g c 2 D/B ( 3 ( 3 3 c 0 or else This article is distributed under the terms of the Creative Commons + c + D +2 N c D D D O ≡ = O = 0. If in addition αβ 0 + 2 K K . It follows that 2 2 Γ µ (since λ 2 4 c u 6≡ are linearly independent, then + D ∈ − − is not of the form 2 2 ( ( µ K µ λ α ˆ Kµ 4 D D f . Thus, if we scale the coefficients ( ( , 1 } − O O
= c ( ) and D Consider now the case that the gauge algebra is Suppose that dim Γ It follows that any maximally Higgsed model with Suppose that dim Γ The conclusion is that 2 = 0 λ O µ (8). So D/B µ that for a maximallyorder Higgsed 3) along the gauge divisor. Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. which would force theidentically, gauge and algebra we to have be only solution possible 2 when as ˆ above.for generic choices Thus, of wecoordinate find that system, a or non-Higgsable contains multiple independent monomials. Note that this implies { must satisfy If but this is notso true for general the form of Solution 1 above. 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