Charge Spectra in 6D Supergravity Theories

An infinite swampland of U(1) charge spectra in 6D supergravity theories

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As Published https://doi.org/10.1007/JHEP06(2018)010

Publisher Springer Berlin Heidelberg

Version Final published version

Citable link http://hdl.handle.net/1721.1/116230

Detailed Terms http://creativecommons.org/licenses/by/4.0/ JHEP06(2018)010 2 and no Springer June 1, 2018 April 2, 2018 May 24, 2018 , : : : 1 = q Received Published Accepted charge spectra in 6D Published for SISSA by https://doi.org/10.1007/JHEP06(2018)010 U(1) [email protected] , . 3 1803.04447 The Authors. c F-Theory, Field Theories in Higher Dimensions, Supergravity Models

We analyze the anomaly constraints on 6D supergravity theories with a single , [email protected] Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, U.S.A. E-mail: Open Access Article funded by SCOAP Keywords: ArXiv ePrint: satisfy all known quantumF-theory consistency conditions or but any do otherspectra not known of admit approach charged a matter to in realization stringnonabelian abelian through theories compactification. SU(2) with and those We higher that also rank can gauge compare be symmetries. realized the from breaking abelian U(1) gaugetensor factor. multiplets, anomaly-free For models theoriestheory match compactifications with almost those perfectly. charges models For restricted theories thatcharges, with to can the tensor multiplets F-theory be or constraints with realized larger is are from an less F- infinite well class understood. of We distinct show, massless however, charge that spectra there in the “swampland” of theories that Abstract: Washington Taylor and Andrew P. Turner An infinite swampland of supergravity theories JHEP06(2018)010 34 16 9 2 models 20 , 1 35 34 ) bounds 5 28 N ) models 26 7 27 2 N 8 3 > | 4 – i – 36 q 25 3 | 2 ) models 29 > 3 U(1) models N 22 | ) anomaly cancellation equations 34 ) models 27 q | 28 N N | ≤ q | b ˜ 35 6 21 35 33 2 5 , , 4 12 3 U(1) models 1 23 11 11 | ≤ 2 | ≤ q 1 = | q 3 | q | ≤ 13 15 q | = 0, = 0 = 1 T 6.3.1 UnHiggsable6.3.2 U(1)s Non-unHiggsable U(1)s T T 2.1.1 Anomalies2.1.2 for SU( Anomalies2.1.3 for U(1) models Geometric interpretation in F-theory 7.2 Evenness condition7.3 on 7.4 Bounds from7.5 F-theory Morrison-Park U(1)7.6 bounds and UFD Models SU( with U(1) charges 6.4 Charges 7.1 Anomaly constraints on U(1) and SU( 6.1 Charge 6.2 SU(2) models6.3 with 3-symmetric matter Classification of charge 5.1 Large charge5.2 families A continuous5.3 approximation Two charges 5.4 Fixed number of distinct charges 4.1 U(1) models4.2 SU(2) models4.3 4.4 4.5 F-theory constraints and the swampland for charge 2.4 Generic and non-generic U(1) models 2.1 Anomaly cancellation conditions 2.2 Additional constraints 2.3 Generic and non-generic SU( 7 F-theory and the swampland 6 Classifying models with charges 5 Asymptotics and infinite anomaly-free charge families 3 Relating the U(1) and4 SU( Charge Contents 1 Introduction 2 Background JHEP06(2018)010 9 T < ] of apparently 7 , 6 ]. Thus, it seems that ] further limit the set of 8 4 – 2 37 ]. While there are still a number of ways 5 – 1 – 37 = 0 tensor multiplets. On the other hand, it is known that T 38 ], and additional quantum consistency constraints [ 2 ]. One of the goals of this paper is to inquire whether a similar finiteness bound , 1 8 , 3 While in this paper we focus primarily on quantum consistency constraints from In this paper we investigate the constraints that 6D anomaly cancellation conditions 7.8 The completeness hypothesis 7.7 Constraints on the charge lattice and positivity there is an infinite swampland of U(1) theories inanomalies, six independent dimensions. ofous any places UV in completion theF-theory paper of point additional the of constraints view. theory, that may we In limit also particular, the for consider set theories of at with theories vari- an from abelian the U(1) gauge group and seems that it cannot, atare least infinite without families some of further theoriesconsistent constraints. with from distinct We anomaly find sets cancellation that of andsimplest there U(1)-charged other class matter indeed known of fields constraints, models that even with appear F-theory, restricting the to most the general approach knowngives to only constructing a string vacua finite in number six dimensions, of distinct possible massless spectra [ tensor multiplets) there areand a charged finite matter numberother representations of simple that distinct constraints are such masslessfields consistent as [ spectra with the of proper anomaly gaugecan sign cancellation be fields demonstrated of and for the 6D theories gauge with kinetic abelian gauge term factors. for The all short answer gauge is that it been a productive approachconsistency to conditions developing on our gravity understanding theories of and both the structure the of setplace string on of theory the quantum charged vacua. matterwith content of only theories with nonabelian a single gauge U(1) fields, gauge field. it For theories is known that (at least for theories with theory compactifications through F-theory,supergravity suggesting theory that may arise any from consistent stringin low-energy theory which [ 6D the setthan of the theories set that of satisfyconsistent known all theories string known with vacua, quantum no understanding consistency description the theories in “swampland” is terms [ larger of any known class of string theories has matter in physical theoriesitational, with gauge, supersymmetry. andcontent Furthermore, mixed [ in anomalies six strongly dimensions,possible low-energy constrain grav- supergravity theories. the The gaugefairly theories that closely group satisfy these in and constraints certain match matter regimes with the set of theories that can be realized from string 1 Introduction The largest spacetime dimension inin which any a supersymmetric representation theory ofdimensions can the a have matter gauge natural fields place group to other begin than to the try adjoint to is systematically six. understand the This structure makes of six 8 Conclusions JHEP06(2018)010 = 0), known T , we show that 3 approach to string , where we characterize known = 1 supergravity theories. 2 ) or other gauge group with N N 2 and identify an infinite swampland > | q , we focus on models with abelian matter | ) or U(1) gauge group in terms of a pair of 4 N – 2 – 2, and for corresponding unHiggsed theories with , 1 = q , we include charges 5 2 only. We show that for theories with no tensor multiplets ( , 1 = q ) and U(1) anomaly conditions can be directly related, giving closely parallel N We systematically investigate the structure of 6D supergravity theories with a single Note that different authors use the term “swampland” in slightly different ways. Here, cancellation conditions of the low-energytheory theory constructions, appear though in this aconstructions. stronger stronger form form In for section can standard be F- of violated charge for spectra, more even exoticnumber for F-theory of theories anomaly-consistent without spectra tensor grows multiplets. as We the then natural consider parameter how from the the anomaly the SU( constraints to the chargedwith spectra. charges In section the constraints from anomalytheory cancellation and directly there match is no simpleof swampland UV the except gauge constraints for group from one becomes specific F- relevant. model For theories where with the more global tensor structure multiplets the anomaly U(1) gauge field andnonabelian their and possible abelian mattergeneric spectra. anomaly spectra for conditions We theories begin in withparameters with a 6D appearing some single in in SU( background the section on anomaly cancellation conditions. In section struction is possible;swampland in models, the where latter the situation,understanding adjective we of “possible” refer how reflects to to specificallyof answer such the on the models finite our set well-defined incomplete hererealize of mathematical as a distinct question specific “possible” Weierstrass massless of models spectrum. whether over one an allowed F-theory base surface can Note that with this definition,ery the swampland of is a a new time-dependentswampland class quantum in of consistency theories; scope. condition discov- Note orstruction also a is that possible new while in string for F-theoryto construction (as some 6D can currently spectra string formulated) reduce it compactification, or the can for in be other any other theories argued known it that approach is no not con- known whether an F-theory con- arise through explicit Weierstrass model constructions. to be precise,quantum we consistency define conditions the yetcompactification. cannot swampland be We to focus realized be here in the on any massless set spectra of of theories 6D that obey all matter charged in various representations. Thisnonabelian is theories a useful are perspective much inand particular better in because understood, terms both of F-theory inquestions, realizations. terms understanding In of the addition low-energy allowed totionship constraints structure providing with insight of nonabelian on abelian theories swampland-type of matter may understanding also charges how provide and F-theory new solutions their with insight rela- U(1) into factors the and difficult various problem matter charges can SU(2) gauge groups, standardthan but F-theory similar constructions in structure givequestion to constraints that the that constraints we from are address anomalyfind stronger cancellation. is more whether One generally a particular for giventhrough spectrum many the of Higgs of U(1)-charged mechanism matter the a fields anomaly-free nonabelian can theory be U(1) with realized spectra SU( by breaking we matter restricted to charges JHEP06(2018)010 = q (2.1) (2.2e) (2.2c) (2.2a) (2.2b) (2.2d) ) or U(1) ], in which N 9 ] can be written as . The numbers of massless ) or U(1) gauge group fac- , . 8 R N R 8 d , d R . ! respectively, are then given by x be the number of hypermultiplets ! Adj R H R X A , Adj x C = − and Adj − R . B of dimension V T, A R T ]. We briefly review in this section the local R − C R ,H x 2 R R , 1 + 29 x 1 – 3 – B R − X R V R 2 T, x X

− N 1 6 −

R 1 3 H − X = ], which in the notation of [ , we consider more explicitly theories with charge = 9 = = 2 , 6 Adj b b ) models 0 = 1 a d · · · N 273 = b a a = , we summarize our results on the spectra that inhabit the 7 V ) gauge group factor, and let N 5 matter, in order to further understand the types of models that appear in the , 4 The structure of the anomaly polynomial for 6D supergravity gives the gravitational As this work was being completed, we received a preliminary version of [ , 3 and nonabelian AC conditions [ The number of tensor multiplets is denoted by tors, so we will make use of the2.1.1 anomaly cancellation (AC) conditions Anomalies only for forConsider SU( these a cases. single SU( transforming in the irreducible representation vector multiplets and hypermultiplets, denoted matter content, and describerelated how to theories generic containing models. matter with higher2.1 charges can be Anomaly cancellationFor conditions the purposes of this paper, we are interested in single SU( In six dimensions,strict gauge, constraints gravitational, on and the spectrumto mixed gravity of anomaly in gauge a cancellation fields consistent andanomaly imposes quantum cancellation matter theory fairly conditions content [ for that a cangauge 6D be field. supergravity coupled We theory use with these a conditions single to SU( parameterize the theories with the simplest, generic and which has someconditions overlap we with identify and the use analysis inanalyzed of this in paper this that are paper paper. special — cases of in the particular, more some general2 conditions of the Background “swampland.” In section swampland, and we finish with concluding remarks in section general constraints on 6D supergravity theories from anomaly cancellation are analyzed, constraints increases. In section JHEP06(2018)010 )- ,T (2.3) (2.4a) (2.4b) +5) N + 4) R 2 N 6 C N +4)( is obtained by N 3( R 3( C denotes the trace , 2 ) R 2 +96 F N tr +54 8 3 27 6 ), the group theory coeffi- N R +23 N R + 8 3 − 6 2 N − 2 B 2 , and the dimensions, associated 2 C N +17 N N R ( 2 N + C denotes the associated SO(1 N 4 +4) of SU( y F . N , · ( R 2 tr . x , and q 4 4.3 R q R q 4. For SU(2) and SU(3), no quartic Casimir x B q B 0 , ≥ x +4) ], among other sources. Values for several R = 0 N is as given, and the value of X n> N 3 +3) A 4 2 11 X 1 6 n> R , N F R − – 4 – +3)( + 2 − 6 2 N A 1 3 1 1 0 − 2 R A N 10 ), for , +2)( N N = = tr N 3 N N b b ˜ ˜ ( +2)( ); the notation · · , )-dimensional real vector space that carries a symmet- N b . ˜ a ( 2 T ,T . These group theory coefficients can be computed for 1 F ], of SU( R tr to be the number of hypermultiplets transforming in the R 12 , R +3) q 1) x N A . For each representation 10 +2) − β 1 2 1) = y N N are defined by +2)( +1) − α − 2 6 2 3 2 24 N x N N N 2 R F ( ( +1)( +1)( C αβ N N N R N N ( ( +1)( Dimension tr N N N ( 2 using the values given here. ; we define / N , and are vectors in a (1 + R Z = 0. In this case, the value of R b B B R ∈ ) in a manner described in [ + , irreducible representation of U(1). The anomaly polynomial in this case yields B q N R R and . Values of the group theory coefficients q R C A a Adjoint Note that the anomaly equations really depend only on the local structure of the gauge 2.1.2 Anomalies forConsider U(1) now models a singlecharges U(1) gauge factor.charge- The irreduciblethe representations abelian of AC equations U(1) [ have representations are given in table algebra. Thus, forFor example, the the most part anomaly we conditionsdistinction ignore becomes the are relevant global the is structure same discussed of for in the gauge section SU(2) group, and but SO(3). a case where this where tr denotes thein trace an in arbitrary the representation any fundamental SU( representation and tr Here, ric bilinear form Ωinvariant of product signature Ω (1 cients Table 1 with a variety of representations exists, so taking JHEP06(2018)010 ) b ˜ q ,T (2.9) (2.6) (2.8) (2.7c) (2.7e) (2.5c) (2.7a) (2.5a) (2.7b) (2.7d) (2.5b) through n, γ . Note that in q , we can identify − B 0 because each charge- ) associated with a codi- , N . , ! q > Σ ! Adj · A , Adj K C − + Adj γ . − R , n B − T, A R ]) , Σ, we can then use the Riemann-Roch . R − C = z ], is · ]. For an F-theory compactification on a x K, Σ R Σ R b and a field with charge ˜ · · 14 ) as + 29 · x · + [ B R q 13 , X K K R V + Σ) = R 2.2 K 8 T, x := Σ X

= = = Σ − – 5 – − K 1 6 −

n ( b b R a n , a 1 3 · H − X · · · b a = 2( a = ˜ = 9 = = with the divisor class Σ. In this case, the SO(1 b ˜ b -component of the section that generates the Mordell- ˜ 0 = n n b · z K · b 273 = ˜ − . ) vector. We sum over only 1) = Σ K and nonabelian gauge group SU( 1) and − ,T 2.2 g , as elaborated in [ − K b ˜ K g 2( 2( = 1 is not necessarily the fundamental unit charge for the U(1). We q ) have a geometric interpretation [ is the genus of curve Σ, to rewrite ( ] is the divisor class of the is once again an SO(1 with canonical class z g 2.2 b ˜ B When the gauge group is abelian and the only gauge group factor is a single U(1), the with the canonical class where [ Weil group of rational sectionsmust be of both the effective and elliptic even. fibration. In In analogy with an the F-theory nonabelian case, model, we define therefore, ˜ geometric interpretation of Defining the self-intersection offormula, Σ to be where product is identified with the intersection product on divisor classes. Thus, we have From the F-theorytions perspective, ( the quantitiesbase appearing in themension one nonabelian fiber AC singularity overa condi- a divisor in the homology class Σ in hypermultiplet contains a fieldthese with equations, charge + discuss this further in section 2.1.3 Geometric interpretation in F-theory Here, JHEP06(2018)010 ,T to 1 R R . We x b ˜ (2.10a) (2.10b) a, b, , and on ], the global anomaly ˜ n, γ ], for SU(2) and SU(3) 16 19 contains an equal number – must take integer values in n, g, R b · 16 should also be an element of , a directly from the group theory 9 b ˜ Z 2 and b ∈ · b b , . · 2 4 must be an integer for any anomaly-free = b q q q q ], it was shown that the lattice Γ of string n n + x x 4 b · q q X X a , while a half-hypermultiplet has half the field – 6 – ] that ¯ R = = 8 γ . For our purposes, this amounts to allowing n are integers, although the anomaly conditions only 6 3˜ ]. R γ 15 : the number of hypermultiplets transforming in a given q and in is an integer. (For example, the abelian equations are ) groups. Since, as described in [ and , x ) then become N n R , we expect that global anomalies are therefore also satisfied R ] shows that n x 9 R 2.4 ,C R , which are a priori simply vectors in the real vector space , it was shown in [ is always an integer. In [ ,B R a, b R g ,C A R ,B R A We now consider integrality constraints on the parameters In addition to the local anomaly cancellation conditions we have described above, there with inner product Ω,associated must in with fact the be stringsimilar elements charges consideration of of the in gauge string thethe charge and abelian lattice lattice gravitational Γ, sector Γ, instantons, and as suggestsdirectly respectively; hence they that that constrain, are a ˜ e.g., that 3˜ any theory satisfying the ACtheory. conditions. A Thus, similar argument [ coefficients, so that charges coupled to thea two-form fields unimodular of lattice. theaction The theory, suggests with structure Dirac that of product the defined Green-Schwarz by terms Ω, in is the 6D supergravity theory coefficients in all theories we consider here that satisfy thebegin local with anomaly the cancellation nonabelian parameters. conditions. coefficients By simply using the properties of the group theory theories with only fundamental anddamental representation adjoint must matter be fields, congruent theis to automatically number 4 satisfied of and by 0 fields the mod in genericglobal 6, spectra the anomalies respectively. described fun- for in This larger the condition constraints SU( next depend section. only on There the are net no contribution from all matter representations to the group content, all transforming only under be a (non-negative) half-integerirreducible for representations such of representations. SU(2) (Note are that quaternionic.) all even-dimensional are also global anomalies that must cancel. In particular [ representation must be a non-negative integer.representations of The a exception nonabelian is gauge intransforming group. the in case We these can of representations, have quaternionic so-called whichtiplet; “half-hypermultiplets” have a half full the hypermultiplet field transforming content inof of the fields a representation full transforming hypermul- in both framework of F-theory is given in [ 2.2 Additional constraints In addition to thenon-negativity local conditions anomaly on cancellation conditions, we trivially have integrality and A unified geometric description of the abelian and nonabelian anomaly conditions in the The abelian AC conditions ( JHEP06(2018)010 = q (2.11a) (2.11b) ), there lie in this . At the 2.4 b ˜ , ,... are physically 2 3 . q , , x 4 , and b , = 2 ≥ = automatically takes = 1 a 2, and the positivity q b ˜ 0 2 k − , , 1 b, x ,N ,N ˜ 4.3 , for ) gauge theory that contains q = 4 . This can be understood in Adj ) spectra for 6D supergravity as defined above, solving the × some of the representations in x 0 N b ˜ × N g )] n, g = g g n, g . Note that in most of the examples − + 0 kq for larger representations will be given 4.4 , and return to this general question in R theory may naturally be taken to be b, x ˜ × 0 2 C of the abelian AC equations ( + 2(1 k )] 4.3 , x q and genus n 0 g b ˜ = n − b, x ˜ , and 0 b – 7 – ˜ R ) models a, B a, = 1, although there is no theory with an integral N Adj + [ , 2 R = × + 6)(1 , x A 0 g N ), we can determine the other possible spectra for given a + = 3 for which this massless spectrum is possible (non-negative), 1 2.11 + 2( = 1 models in section = 0, where, as discussed in section × , x n T 6 n, g [6 )] ) yields models of the form T / g 0 also arises naturally. 2.7 − = 7 has an absolute scaling relative to the lattice Γ, these models must be b > ˜ b ˜ , γ 3 Γ that gives these values). The string charge lattice also has associated / a, b, + 16(1 ∈ − = 1, in particular in a scenario where there are massive states in the spectrum ]. Note, however, that for some values of n b ˜ ) q . The group coefficients = 19 22 N – n n, g − 20 ) AC conditions ( [(8 Finally, note that for every solution From the generic models ( N they must include other matter representations. R We can now determine allonly anomaly-free fundamentals, models adjoints, forSU(2) and an SU( two-index and antisymmetric SU(3), matteradjoints). which fields have In (in no terms theSU( quartic of case Casimir, the of self-intersection we consider only fundamentals and with that charge. Wecharges give related an by explicit scaling example inthe of this discussion way a of in pair the section of infinite distinct apparent physical swampland2.3 theories for U(1) with models later Generic in and the non-generic paper. SU( charge, but because considered as physically distinct.A distinct questionthe is the U(1) fundamental gauge unit of group. chargedistinct, under the Even fundamental though unit of the charge models2, in or the with may be e.g. we consider here, we take (even) integer values forcondition models that with generic abelian charges is an infinitelevel family of of massless spectra, solutions these solutions may seem equivalent up to a scaling of the overall lattice Γ and with it a positive conepositive supporting cone. BPS strings, The andfrom consistency we the constraints expect supergravity on point that of thisrelevant view, in positive although the cone in discussion this of are paper not this well condition understood is primarily only solved by ˜ these models have negative multiplicity;possible in and such if cases, there are this solutionsx “generic” to type the of anomaly equations matter with is non-negative not multiplicities values of the resulting model willsatisfying have more anomaly uncharged cancellation scalar withF-theory fields the than from same any the values other of types fact mass spectrum that of the codimensiontuning representations two to used special singularities; points here in insmaller come the general, [ moduli from space more the where the exotic most number representations generic of uncharged involve scalar fields is These are in severaltheories. senses For the any most values of generic types of SU( JHEP06(2018)010 ]. 2. 23 , , 1 (2.14) (2.13) (2.12a) (2.15a) (2.12b) (2.15b) 20 [ ), we see that 2.7 ]. ) (2.16) 22 ) for the corresponding , 1 ) and ( 21 ( , ) yields models of the form 2.11 ) 2.10 anomaly equivalent × × 2 2.10 ( , . + 14 × = 6; thus, we can freely make the ) + 15 . 3 n 2 , , representation but a negative number , allowing us to determine an anomaly c γ A C 4 2 4 2 2 2 q ( 2 2 c c 4 2 2 − c c × × + + ˜ n 1 and 1 + + 2 2 4 14 and q 1 1 A C 1 1 − 1 1 – 8 – ←→ ←→ c c c c ) + ) = 1 = = = = 2 1 representation of SU(2); the anomaly coefficients of 2 4 c ( 3 3 . Note that these exchanges need not be realizable by ( × × × ) are simply C A ˜ n R + 6 gives − C 1 ) + 6 1 γ exchange 0 × ( (8 and 14 × R 2. Solving the U(1) AC conditions ( A ) case, we can, in principle, use this result to construct all other 10 , N 1 = 6, so we can freely make the exchange representation, to a consistent model. 3, for example, solving the system 2 , and the resulting spectrum will automatically satisfy the AC conditions. R = , c q = 15 q ) indicates a representation with charge − n = 1 c For example, consider the As in the SU( Representations that are related in this way are termed always exchange this linear combinationrepresentation of representations in ( by a linear combination of those for the fundamental, adjoint, and antisymmetric, so we can this representation can be relatedsystem to those of of equations, the generic matter representations by a linear Using the values fromexchange table gives anomaly-free U(1) models bythe way U(1) analogues of of exchanges:equivalence comparing between ( any higher charge andFor a charge linear combination of fields of charges only charges where ( sifying solutions to thethese anomaly exchanges equations. can actually There be are realized many as instances,2.4 physical however, transitions in [ which Generic andWe non-generic can U(1) similarly models compute all anomaly-free models for a U(1) gauge theory that contains Here we refer to thealent transformation representations between as one an modelany and physical another process; using here anomaly we equiv- simply use these exchanges as an organizational tool for clas- in any SU(2) model satisfyingconditions. the AC Note conditions that tothe we yield exchange, another must but model check note satisfying that also themodel, the that AC such non-negativity in as one some constraint with cases is more such than satisfied an 6 after fields exchange in may the take an inconsistent of fields in the JHEP06(2018)010 , ) = 14 N γ = 0. (3.1a) (3.1b) b ˜ = · ]. Such b ˜ n ]). 26 = 24 , b ˜ 25 · a = 9 where the gauge , since the number of ) theory. T T N . that satisfy K b ˜ − a, . ) and U(1) cases directly. The U(1) , in such a situation, so only a finite 2 N 2 x x N + 4 + 16 1 1 x x – 9 – = = 2 are b b ˜ ˜ ) anomaly cancellation equations · · , 1 b ˜ a N 3 6 − ) gauge group, and we describe a general class of Higgs = q N ]). Since there are strict bounds on the finite set of nonabelian ) to U(1). 24 N 9, there are possible values of ≥ T ) for charges 2.4 ]). In particular, if a nonabelian gauge group and matter spectrum can be realized in We first compare the AC conditions for the SU( In this section, we directly relate the anomaly cancellation conditions and the structure Note that when 28 , If we do notexchanged impose to non-negativity conditions, a then modeltwo any with constraints. anomaly-free only U(1) these model charges, can and be the matter content will satisfy these spectra in theories withtransitions an that SU( break SU( AC conditions ( gauge factor that satisfyhypermultiplets the is anomaly bounded conditions for abovenumber any of for value U(1) any of models given can be unHiggsed to anof anomaly-free matter SU( spectra in theories with a U(1) gauge group to the AC conditions and matter gauge groups and mattermodels that can can be be unHiggsedthere to realized are a in also nonabelian F-theory, U(1) modela only models with good a that an SU(2) limited can F-theory model be set realization.Note in realized of also Note F-theory; in abelian that that F-theory we that encounter there some cannot are examples be only of unHiggsed a this to later finite in number the of paper. massless spectra with a single SU( 27 F-theory and leads in field theorya to Higgsing a process, given abelian then gauge theretheory group must and (though also matter be the spectrum an explicit after F-theorymodel realization realization can of of be the rather the tricky Higgsing resulting [ abelian process in an F-theory Weierstrass 3 Relating the U(1)Since and SU( nonabelian gauge groupsabelian and gauge matter groups are andtheory, it easier matter, is to often both helpful understand to frombe relate and broken the allowed constrain abelian to low-energy structures than reproduce to point abelian nonabelian of models gauge that view groups can and and matter from through F- Higgsing (see, e.g., [ 0. Explicitmodels examples are of generally associated such withrepresentation; Higgsing theories for SU(2) example, have models this that been describes havefactor any only identified F-theory a is model in single supported with adjoint on F-theory a [ curve in the genus one class Σ = to non-negativity). As withway the of nonabelian relating theories, low-energy we theoriescases treat there with these are distinct exchanges corresponding as spectra, physical a though transitions formal we (see, expect e.g., that theThese in discussion correspond many in to [ non-Higgsable U(1) models with no matter content, and ˜ in an anomaly-free U(1) model to yield another anomaly-free U(1) model (again, subject JHEP06(2018)010 . 2 Z − 2 of ∈ (3.5) (3.6) N (3.2a) (3.3a) (3.4a) (3.2b) (3.3b) (3.4b) − N , x . Adj ) follows from , , x , x . N . 2) x ) x 3.2b 2 − x , 2) 2) ) to U(1) that gives N N, x 2) N 3, ( − N − + x − 1 2)( , N 2) N − )( ( N − N > N ( , − k x ) in order to break x k 2 for any 2 ( ( 2 k − 2) N N , ( x , Z k 1) × 24 by comparing the AC equations ) , and then giving VEVs to − 1 2 (8 ) 2) 1 ∈ b − 6 k 1) − N 1) 1) + N − 2 ( N N − 1) + k − − 1) + ( , x N and 1 N 1) + − + ( − ) was found by considering only the AC × b x ˜ N Adj = 0 here; for U(1) 2 − ( 1) + x 1) − 3.4 Adj Adj ) + ( 1) + ( → N x − ( x x − . In such circumstances, we can equate the Adj 2 ) x ]( − x Z 2 × ), appropriately scaled, yielding + 6)( N N N Adj ( – 10 – 2) ∈ 2 x Adj + 6)( N ( 3.2 ) yield × ( − x k − 2 ( ) + 1 + 6)]( N k 2) . To see this, first note that one possible Higgsing of 2.2 ) + 2( 1 kN N ). N 0 N N + ( − ( + ( 3, we take + 6) ( , for + ( + 2 k , + 2 × ≤ 2.2d N 2 × kb N x 2 − x x 2 ( 2 m = 2 1 2 k 1)( 2 x 1) 1) = [ k N kx + 4 ≤ − b ˜ N = = − − k [8 12 = = 1 ) model can be exchanged to a model with only fundamentals, 3 k b b x N 2 2 is 0 or 2 modulo 6, then N N · · 2 (mod 6), for any valid Higgsing from SU( + ) with those of ( ( N ( ( x x , + b a = k 0 3 6 + 1); then we see that under this Higgsing, the fundamental, adjoint, 3.1 b , for 2 ˜ → → → x + 4 ) AC conditions ( b · − ≡ x + 16 ) gauge theory, we can explicitly construct a Higgsing pattern to U(1) 1 N ) 2) . Note that the relationship ( a 1) N 1 k x k k 6 − ) to half of eq. ( N Adj x − , − − − 1 k , ( m 2.2e (8 ( k 6 k 24 ··· m ) to a U(1) using two adjoints is achieved by giving a VEV to a Cartan generator 3. In the case of , = = 1 for some N = , 1 2 b ˜ x x Similarly, any SU( We now assume that By inspection, if For an SU( kb N > = adjoints, and antisymmetrics if wea do model, not the impose SU( the non-negativity conditions. For such for adding eq. ( right-hand sides of ( From these equations, we find We can then determine thefor the relationship two between theories. After the Higgsing, the resulting U(1) model must satisfy diag(1 and antisymmetric representations decompose as with an SU( in one adjoint matter fieldof that breaks the SU( U(1) chargesthe coming remaining from U(1) the factors. second Without adjoint loss of of SU( generality, let the surviving U(1) generator be We thus expect b conditions, without any reference to the explicit symmetry-breaking pattern. ˜ JHEP06(2018)010 . b m 1) (3.7) (4.4) (4.1) (4.2) (4.3) (3.8) − m ( . ) m b · = , in the case of a b ˜ 6 1) 3. For arbitrary − g, n − ), 1)( | ≥ 2 (mod 6) for any Adj − q , | , x 2.14 0 N U(1) with ( 1)( ≡ x . N γ , → ) − 2) 1) 8 2 ) = N − − = 0 for ≤ N ( q . ˜ n m N × x ( T. ) 2 x = 2( ≤ γ m 1)( 2) N γ γ 10 − 2 + 29 2 − 4 − 2 4 N by first Higgsing on an adjoint charge that − 2, i.e., x the number of trivial representations, and N , , x ( n b ˜ n 0 Z + 3(˜ = x 2) 4 1 x | ≤ 2 − q − x ∈ ) becomes | 1) + ( = – 11 – N ) + T γ + . in the case of U(1) models and 1 − 1 2 b 0 , x − 29 ˜ , with 2.2a n ) yields 1)( x 1) 2 − N ( Adj − x ≥ γ, ˜ n − x x × 3.2a ( N + 1) ) N , x 1 N , 274 ( ˜ n 274 = x U(1), and then carrying out the Higgsing described above. − = ( Z N 2, we have U(1) models of the form ( − b ˜ + x × + 2 N ∈ 0 γ = 1) 1) x n | ≤ b ˜ (8 = ( x 3˜ q − ) that | − = 1 in any consistent model are non-negative. x N 1) N 3.5 H 2 R − . Note that, as anticipated above, = ( , x SU( 2 q N N | ≤ ( x − q → ≤ N N | ) x m = N b ˜ ≤ · ) and comparing with eq. ( a 6 ) models with generic matter spectra. 3.6 The anomaly equations and integrality/non-negativity of charges impose the conditions We could instead achieve This analysis gives us a way of relating a broad class of generic U(1) theories to Higgsed N − = 1, so the gravitational condition ( , models will be parameterized by eq. ( noting that Higgsing on an adjoint charge uses up one of the adjoints. Plugging these into and we see from eq. ( V Considering only the charged matter, this yields the condition Restricting ourselves to In this case, we have by considering models withT U(1) charges SU(2) models, where these parametersof are matter constrained fields by the condition that the4.1 numbers U(1) models SU( 4 Charge We now consider the explicit classification of U(1) models and their unHiggsings. We begin breaks SU( By iterating this step, we canfor thus any achieve 2 Higgsings of SU( Thus, this Higgsing gives JHEP06(2018)010 , . 2 1 3 x x (4.7) (4.8) (4.5) (4.6) 16 ≥ (4.9a) (4.10) (4.11) (4.9b) 1 ≤ x 2 x ) becomes 2.2a . ) 2 ( . . × . ) ) g 1) . T, b . ˜ 2 ) · − − × g . Note that we can also derive the a ( g g 8 − + 29 × + 4.5 T , we find + 4(1 , ≤ − ) ) + 2( n x b ˜ 1 1 4.1 × ) + 2 · + 29( 0 b )] ˜ = 2 + 3 ( ( ( g – 12 – n ≤ ) models with only generic matter representations × × × − 12 , which agrees with our expectation from section b ˜ is presumably constrained to be an integer in any x 2 1 b N 2, which is the current case of interest. We have )] · 2 ≥ g n a n , γ → → , ≥ , ˜ 2 = 2 − 1 ); the first bound follows from the condition 4 b ˜ 244 − + 16(1 = 4 276 2.2 3.1 ] in the case of a single abelian gauge factor. Note also that = 3, so the gravitational condition ( n ˜ n 9 = [6 V q + 32(1 n ) can be written as and unbroken (by this, we mean that we give an adjoint hypermultiplet [12 4.4 3 T in the given representation. Thus, x corresponds to 3 ). Specifically, for SU(2) the anomaly-free models are of the form T n ) directly from ( + 3 2.11 4.5 = 4 x n + 2 0 x We can now Higgs these down to U(1) models. We Higgs on an adjoint charge that = leaves the generator a VEV equal to thisof adjoint the charge). generator The resulting U(1) charges are twice the eigenvalues or Matching these with the models from section Note that ˜ for an SU(2) Higgsing. Higgsing uses up one of the adjoints, so we find models of the form As we have seen, theare anomaly-free SU( given by ( This form of theon constraint certain will classes be ofconstraints helpful ( F-theory in models connecting toand in the a second section can related be found stronger by multiplying condition the first equation4.2 by 4 and noting that SU(2) 4 models quantum-consistent low-energy theory. Notethe that constraint the numbered 5 second in ofthe [ these last conditions condition matches in ( though, as discussed in section We can obtain modelsefficients to containing exchange larger theseand representations additional adjoints. by representations using However, for models theU(1) some with models group number only of theory with fundamentals fundamentals H co- charges and adjoints will Higgs down to JHEP06(2018)010 ) 0 ≥ 2.4 = 1, b ˜ (4.13) ). On 2 that (4.12a) (4.12b) V b/ ˜ , 4.11 2 = x b + 3; in order for . 1 − x 7 = + 0 a = 0. In this case, the and , which all otherwise must be positive, and x 2 T b x = 4.5 + . ) 1 H 2 = 24, there are actually two x b ˜ ( ≥ × 2. The abelian AC conditions ( 2 after Higgsing, up to the condition , is an even integer. We see that 6 , , 2 b ˜ = 1 2 − 4 x ). 2 when there is at least one adjoint field. x q b ˜ ]. We discuss this condition further in the , b ˜ 9 2.10 + 4 + 16 1 ), depending on the choice of unit charge; we 1 1 = 1) that do not take the form of ( is an even element of the string charge lattice x x ) + – 13 – b ˜ 1 = 4.13 = = , γ 2 can be unHiggsed to a consistent SU(2) model. to be negative. In this case, it is useful to use the is an integer, there are solutions of the second and q b 2 ˜ ) must be a non-negative integer, which gives us the implies that ( a b ˜ n is an even element of the string charge lattice Γ (i.e., 3 = 2 0 18 × ≥ b ˜ n 2.14 Z are integers such that the number of each representation b ˜ g = 24 from the condition that the numbers of fields of charge ) rather than ( − N ≤ 24 2.4 and b ∈ ˜ b ˜ n ≤ 6) ) (such as ˜ ). By contrast, for the abelian theories we only know that the − 4 and there is always an anomaly-free SU(2) model with b ˜ g ( 4.4 b ˜ Z simply become numbers. In particular, we have 4 b ˜ ∈ and ), and 6 n ). n 4.4 , and condition fixes the sign of b 2 is non-negative. In this set of models, we have 4.12a b ). Recently the condition that , = 0 · a = 0, so the gravitational condition becomes 274 a ) is non-negative (the number of each representation is always integral due to the 4.9 Γ), then ˜ T T Now we consider SU(2) models containing only fundamentals and adjoints, which we We restrict the maximum U(1) charge to be Thus, it is not guaranteed from these considerations that every anomaly-free U(1) We know that both ∈ 4.6 2 1 and anomaly-consistent models satisfy.distinct Note U(1) that models in with the thediscuss spectrum case this ( issue further below. can Higgs to U(1) models with charges Note that requiring from eq. ( and yielding models of the form AC conditions in the form ( give us 4.3 We now specialize to thevectors case of models with nothe tensor kinetic multiplets, terms of theso gauge the fields to have the appropriate sign, that the U(1) theoryfrom has ( sufficient uncharged scalarswas to proven match under the some chargecontext mild 0 of assumptions F-theory contribution in constructions [ and the swampland in sections model with matter ofIn charge particular, at even most if wethird assume conditions that in ˜ ( the other hand, ifb/ we assume that gives the corresponding U(1) model with charges in ( integrality of number of each representation inconditions ( ( ˜ JHEP06(2018)010 2 = 0 ) has = (4.14) (4.15) (4.17) (4.16) T q 4.13 . This relies unbroken, as 3 (which follows 3 is T Z 3 so that there is g = 108. However, (2) gauge algebra. 2 ∈ ≥ su b b , x = 0 2 is the fundamental U(1) . . 1 , . ) x ) ) 2 = 0, so the condition becomes = 1 2 q T ( × ( ( × 2) × × . 3) − = 24 has 2) b − 108 108 b ˜ 2 − b = 3, and ( 1)( b . Furthermore, the gravitational condition b 2 can have two distinct realizations; in one, 2 V Z − ←→ ←→ 1)( , b ) + ∈ . The corresponding value of ( 2 − = 1 – 14 – b , in fact each of the spectra of the form ( b + can be physically distinct from those that satisfy q ( x ) down to U(1) models when x ( = × × k 4.5 = , models with only hypermultiplets of charge n × + 3 55 × 4.15 g ) + 1 ) b b 2.2 1. In the case under consideration here, the model with x − − 2 can be unHiggsed to a model with an 12 in order to have a non-negative integer number of each × + 2 = (12 0 (12 ≤ b 54 x q b 4 b , once again matching the expectation from section 2 ) implies that b = = 0 and the model with n 2 H does not reduce the number of solutions in this case (as in the U(1) = 2 , x = 24, there are some additional subtleties that are useful to discuss. , which all otherwise anomaly-consistent models satisfy. b ˜ T must be an even integer, as discussed previously. = 12, this indicates that in this simple context there is no swampland. b ˜ b ˜ , b x + 29 = 108 1 V + 3 ), we find models of the form x = 24 model with 108 charges = 24 − b ˜ b ˜ is simply a number, we have x ) if we take 2.11 H 2 b 1 is the fundamental U(1) charge, and in the other, 2.4 In general, we can include half-hypermultiplets of any quaternionic (pseudoreal) irre- For the case Thus, we have determined that every anomaly-free U(1) massless spectrum with We can now Higgs the models ( ≥ = 6 has = From ( Since 276 from the integrality of Note that we mustrepresentation. have ducible representation. In this case, however, the requirement that 273 = case above): we have b each of these modelsa can distinct be spectrum: unHiggsed in this case to an anomaly-free SU(2) model with the case Note that, as discussed inare section equivalent at thehypermultiplets level of of charge the anomaly conditions to those with the same number of ing ( on the fact that and matter of charge at most As we discuss in more detailan in explicit section realization in F-theory. Except for some further subtleties that are relevant for at least one adjoint field.before, and We Higgs find on models a of charge the that form leaves the generator Note that we have an exact matching between these models and those we found by solv- This illustrates the point mentioned earlierare that all models multiples with of masslessthe a fields condition of non-unit charges that integer that the GCDthat of the the charges isq 1. There is a further issue, however, which is ˜ JHEP06(2018)010 . 0 0 0 1). is not (4.18) (4.19) , b ˜ lattices, , and Γ 1 0 = (4 1) signature a , ,Γ 1 − = 6 due to the odd. and associated and Γ b ˜ = 1 theory must 2 0 ; this is discussed x b ˜ 1).) Note that for 1), the odd lattice , the only possible , N with , n, γ 1). For brevity, we means that 1 , 0 in the (1 2 1 and values. Note that the x b ˜ b ˜ = (4 vector by Γ are exactly the same; each a, b, 0 0 even and a . For Γ 0 0) and (0 a 314 distinct allowed spectra 0 b ˜ , , 1), up to symmetries. For Γ Γ − , − 1 , = 195 , we have carried out a complete and Γ , 0 0 a , so an odd 1 Γ 2 . , − = (3 x 0 ! 1 , i.e., the lattice Γ Γ ! that are consistent with anomaly cancel- 0 0 a , 1 = 24 unHiggs to two distinct nonabelian 1 0 1 1 0 − b − ˜

= 2) and 1 0 0 , =

a , only the latter of these two models is realized ), respectively, that satisfy the AC conditions, – 15 – − U 2 where we can have = n, γ, n, g values includes a multiplicity for symmetries, i.e., , = (2 1 4.5 , x = 0 0 b ˜ 1 Ω 0 0 x a Ω − = 8 is are realized in string theory with a variety of different T 0 = a − − and Γ ) are both counted. = 9 1 1 a . The two models with , b b ˜ · 2 always gives an even value of b a b ˜ = ( ]. There are two unimodular lattices of signature (1 4 b ˜ 1), which all anomaly-free solutions satisfy. (In fact, all solutions on the , 370 possible distinct combinations of spectra and = 1 U(1) and SU(2) models from the supergravity point of view, we must , so that T an even . For the 3 classes of models there are 195 a b ) and 383 ˜ 0 2 , = 1 , b T 1 b 1) and (0 The sets of possible U(1) spectra on the lattices Γ In this enumeration we have not imposed the positivity condition on For each of the possible classes of models Γ and Γ − , = ( 1 even lattice are inΓ the stricter positive coneeven. generated This by relation (1 does not hold for Γ anomaly-free U(1) model on the even lattice has a corresponding anomaly-free U(1) model number of spectra andb corresponding further in the following section,there but the is basic observation a is(1 choice that for of the positive Γ cone compatible with F-theory, generated by the elements enumeration of the setvalues of of possible U(1)(i.e., models, integer non-negative including values valueswith of of 251 ˜ there are two possibilities: denote the possible combinationsNote of that string charge while lattice bothchoices and Γ of positiverealization cone from (as string we theory of discuss models further of in type Γ the next section), there is no known We denote the correspondingchoice string for charge lattices by Γ with inner product and the even lattice with inner product To classify determine not only the possiblelation values of but also ˜ which ofstring these charge values lattice can be Γ. realized Webe by know unimodular vectors that the [ string charge lattice of a 6D difference in values of models, the first of whichSO(3). has gauge As group we SU(2) discuss andin further the F-theory in second (at section of least which using has known gauge constructions). group 4.4 charge. The latter is still distinct from the model with 108 charges ˜ JHEP06(2018)010 = 1 (4.22) (4.21) T 20). An , . For H 70). Note that ) on the lattice = (13 , b ˜ b ˜ 2 models = 3 on one lattice that = (9 b , by K ˜ b ˜ 1 − 0 2 that can be realized , ), with the inner product by 2 we have the 6D charge Z 3, there are non-Higgsable 0 , ], so a pure U(1) or SU(2) = 1 33), neither of which can be ≥ B, | 31 ( − q 1 = 0 , | , m is odd for both choices of lattice. 1 b ˜ H m 3) and on Γ , , − = (43 , b ˜ 26) and on Γ , = 97 = 142 (4.20) = 236 12. When 2 2 2 = (23 9) or ≤ b , ˜ = (44 , x b , x ˜ – 16 – , x 38), which can be unHiggsed to the SU(2) model , m by , , and the canonical class is m by 1 1 = (29 2. In the cases = 8 = 8 F = 4 1 , P b 1 1 ˜ 1 1 x x , = (10 x is b ˜ , giving 314 spectra, including all 195 of the spectra from B . They all satisfy positivity for both cones described above. 0 0 = 0 a by 0 m lattice by 19). , ]. 1 30 , = (5 is not even in either case. On the other hand, this spectrum can be . For all the remaining models that are not unHiggsable on either lattice 29 , b , b ˜ b ˜ 8 is odd, implying that one component of = 0 the surface 2 is a Hirzebruch surface = 72 T x B . The 6D string lattice is then associated with B , x ), 150 (280) models (corresponding to 105 (140) spectra) are not unHiggsable to an 0 In general, 6D supergravity models come from compactifying F-theory on a complex There are 370 models on Γ Of the 251 (383) anomaly-free U(1) models (including spectrum and = 4 (Γ 1 can be realized onunHiggsed the since Γ realized on the even latticex Γ many U(1) models thatmodel. satisfy the In AC some conditions cases,admits that a an cannot spectrum SU(2) be unHiggsing, can unHiggsed butthe be to cannot spectrum associated an be with SU(2) unHiggsed a on value the of other lattice. For example, (or several) on the odd lattice with the same spectrum, and vice versa. There are, however, arising from intersection of divisordivisors. classes and For the positivethe cone surface being the cone ofnonabelian effective gauge groups everywheretheory in is the only moduli possible space for [ We now consider explicitlyin the F-theory set both of directlyof U(1) an as models SU(2) a model. with U(1)background see This Weierstrass [ model section assumes and some indirectly basic through familiarity Higgsing withsurface F-theory; for more the other choices of lattice and None of these models have F-theory realizations, and we4.5 do not explore them F-theory further constraints here. and the swampland for charge bound is which can be realized onthere the are lattice 18 Γ spectra that cannot be unHiggsed on either lattice for both of these reasons. (50 spectra), An example of such a model is which can be realizedexample on of the a lattice model Γ that cannot be unHiggsed due to the SU(2) gravitational anomaly Γ SU(2) model. Ninety ofspectra) these of spectra the are models notoccur unHiggsable that because on cannot they either be would lattice. unHiggsed violateis on For the independent many SU(2) either of (40 gravitational lattice, anomaly the bound unHiggsing — cannot this condition JHEP06(2018)010 ) is K are 4 2 ] or f 4.25 30 (4.23) (4.24) (4.26) (4.27) (4.28) ), (4.25a) (4.25b) to have f, g ≤ − b 1), so for b − − f, g is generated , K 4 0 . Here, 1). In the case − 2 ( − σ , O . Thus, for a good = 0 of divisor class b σ 1) and (1 = (3 , ≤ is effective. . The construction also b ˜ A K K for the curve to have genus ) on a smooth divisor of at − − − ≤ b B ), , 4.25 K ), lies in the positive cone of the 2 2 is a section of . with 2.8 σ , − 1 , 2 1 is not effective, that is if f ˜ σ g 2 ∆) 4.21 ( . σ b 2 + 2 g , K. ]) 2, the effective cone on Γ K. σ f 8 − σ , z 4 + 1 + = K + [ 2 φf σ 4 = 0 fx ≤ − g ≤ − ). Thus, 1 − 1 b ˜ f K + 12 b m K 3 − is even, and 6 + ≤ – 17 – ≤ − x + 27 2 b − ˜ = 1 we have Γ is used to indicate that 3 ( 3 K φ = K φ 2 f = 2( B O m 1 all vanish and ∆ vanishes identically, which gives a 2 , respectively. 48 is generated by the vectors (0 − b ˜ 1 } y ≤ − 2 864 − 1 g A 1) ] can be explicitly described in the Morrison-Park model , = = ∆ = 4 z ) and (0 g f , K + [ . Furthermore, the condition on ] so that the discriminant vanishes to order 4 1) , and 2 − K K 20 2) and for − f ( ), and so forth. If 4 , , − b , O ] is effective, so 2 1 (1 z f { ≤ − − = (2 2 = b 3), as encountered in the example ( , or as in the nonabelian SU(2) model above the discriminant would K b/ ˜ K and − 4 K then , − = 8 } − 1 ( b 1) O , ≤ − = (23 with (0 , b b ˜ ˜ 0 0) , ] has a Weierstrass model = 1, the effective cone on Γ σ (1 The Morrison-Park direct construction gives, as in ( A 6D F-theory model with SU(2) gauge group realized on a divisor associated with Note that for divisors, the notation { m 1 = [ vanish identically, so we have Furthermore, as the locus that supports an SU(2) after an explicit unHiggsing, and satisfies exactly In this construction, [ requires F-theory model with anleast SU(2) genus realized one, through corresponding the to form the ( existence of at least one adjoint field, we need sections of line bundles a section of does not hold, globally singular and unphysicalcannot F-theory exist model. unless Soone an or SU(2) higher model so of that the an form adjoint ( Higgsing is possible is that which can be arranged eitheran using order the by Tate order tuning tuning of [ a general Weierstrass model [ A fairly general class ofthe F-theory SU(2) form models can be constructed by choosing the vanishing locus ofb a function (really a section of a line bundle) where the discriminant vanishes to quadratic order in of example corresponding 6D supergravity theory.by For lattice Γ JHEP06(2018)010 ] ) ). is 2. 24 σ 4.26 4.25 (4.29) (4.30) = ], if the q is even in 22 b ˜ Note that for 2 ) can take a more 4.25 ) in the nonabelian case ) necessarily holds, as “UFD 4.26 ] show that the upper bound 24 ) is precisely equivalent to the 2. The SU(2) constraint ( 4.26 / ). Such exotic constructions exist even b . ˜ 2) · 4.28 4.28 a − 8 b satisfying ( b 1)( ≤ − K ) and ( 6 b ˜ − = 2 · can fail globally to be a UFD even when 1 and one with fundamental charge 4.26 b b b ˜ ˜ ≤ − – 18 – σ ≤ = 24 model potentially has two distinct realiza- b = ( = b ˜ b ˜ g · must be effective for the discriminant to have the q a = 0 there is a match between anomaly-allowed U(1) 2 2 is not in the ring of global functions on the divisor, so does not hold. Similar considerations motivated by the σ T ), where the constraint ( b , the − φ ) are possible. In particular, as described in [ − are simply numbers, ( 2 and the models that can be constructed from F-theory. is nonzero. For brevity in the subsequent discussion we refer 4.25 b ˜ 4.3 K , ) on 4.28 σ a, 1 12 − 4.26 12, and the corresponding condition for the Morrison-Park U(1) = q ) and ( ). In the absence of additional nonabelian gauge groups, which might ≤ 2. These models can be unHiggsed to non-UFD SU(2) models, so that in fact there b for an SU(2) model is simply the degree of the curve on which the SU(2) > 4.26 4.26 b | ) also is not necessary when the U(1) is realized in a fashion that does not ). Because q ≤ | ), which states = 0, we see that for 4.28 4.5 = 0, 4.28 ). This is a simple example of the “Kodaira constraint,” discussed more generally ) in the abelian case. T T 4.24 4.28 from ( ]. As mentioned in section For Note that F-theory constructions of SU(2) models and U(1) models that do not satisfy We would like to thank Yinan Wang for pointing out that some models of the form constructed in [ 8 b ˜ 2 tions, one with fundamentalThese charge two models unHiggs to models withhave nonabelian no charges gaugeexist groups F-theory SU(2) models and thatin violate SO(3), some the cases constraints where ( the curve supporting the gauge factor is smooth. Since the anomaly cancellationthe conditions already case impose themodels constraint with that charges states that 3 model is ( condition ( must be satisfiedform since ( in [ is tuned. The genus of this curve is to SU(2) models of theSU(2) form models,” ( and modelsany not SU(2) of model this in form F-theory, as the “non-UFD constraint SU(2) models.” on take the Morrison-Park form.F-theory Thus, constructions these stronger described constraints abovemodels. only and hold are In for not fact, the universal thesmooth, standard constraints ring when on the of all genus functions of F-theory on the conditions ( divisor supporting thegeneral SU(2) “non-UFD” is form where singular,the the upper bound leading from termsconstruction ( of in U(1) ( models that can have higher charges [ cause singularities upon unHiggsing ofsufficient the condition U(1), for therefore, the there existencethe is of necessary a a and Morrison-Park precise sufficient U(1) condition necessary for model, and The the which existence condition is of equivalent is an to SU(2) that modeland there of ( the exist form a ( divisor the conditions ( JHEP06(2018)010 ). 3.2 1, or (4.32) (4.33) (4.34) (4.31) ). The = 2 and no 3.1 q . , 1 7.8 = q 0. As in the abelian = 0, which as shown ≥ 6 ) b . ˜ g , a · 3 − a , a 8 2 a torsion; this can be checked by ≤ − 2 +16(1 a . b ˜ Z b , n 8 · · b ˜ a 4 a . ≤ ≤ − 4 b b ˜ ˜ (2) gauge algebra. The group is, however, ) impose the constraints · ≤ − ≤ su a ≤ − b 4.4 After the unHiggsing of the Morrison-Park 2 a · b 2 even if the fundamental charge is 2 0 and the second from the constraint that the − b 3 – 19 – − center of SU(2) in this case. On the other hand, ≤ ≤ , 2 a = g > b , Z Z q · − Z 4 ) is non-negative, 6 a 2 ∈ ∈ − 4.6 b ˜ b · ˜ a 0. This follows directly from the anomaly relations ( also hold for SU(2) theories with at least one adjoint, from ( + 2 , the anomaly cancellation conditions from 6D supergravity are b b ˜ ≥ T · b ˜ b ˜ · b ˜ in this case due to the presence of 2 Z / 0 and ≥ b ˜ · a ] gives the SO(3) gauge group. This is natural, from the absence of massless fields − For larger values of We would like to thank Ling Lin for discussions on this point. 32 3 have analogous inequalities for Thus, while the anomalytheory cancellation conditions conditions are on a specific necessary models, consequence of not these all F- models satisfying the anomaly conditions SU(2) model in F-theory, In both the abelianconstraints are and strictly stronger nonabelian than cases, butWhile closely the the signature parallel Morrison-Park of to the the and inner constraints product UFD from is anomalies. SU(2) indefinite, in F-theory the cases we consider here we must where the first inequalitynumber comes of from fundamental fields in ( case, these constraints are closely related to the conditions for the existence of a UFD Similarly, the anomaly constraints oncharges, an no SU(2) other model gauge with groups, only and fundamental at and least adjoint one adjoint field are additional gauge groups, the AC conditions ( The existence of anHiggsing F-theory a UFD model SU(2) either model with from only the fundamentals and Morrison-Park adjoints construction imposes or the constraints from models are likely butaspects not of certain this candidates in for connection the with swampland. the We “completeness discuss hypothesis”weaker some than in further the section conditions fromto F-theory. have These some two interesting sets parallels. of For conditions U(1) do models seem, with however only charges that transform nontrivially under the there is no clear reasontheory from cannot the consist low-energy only theory of thatthat the charges there massless cannot spectrum be ofwe an the do U(1) SU(2) not model have with analso only F-theory cannot adjoints realization in completely of the these rule massless models, out spectrum. and an suspect While exotic they do non-UFD not type exist, F-theory we realization. So these through the Morrison-Parkmodel, construction. the resulting Weierstrass modelSO(3) has = an SU(2) noting that the Weierstrassin model [ takes a Tate form with respectively. Only the latter of the two models has a natural realization in F-theory JHEP06(2018)010 , 0 a a 0 2, 8 8 = 1, , for , b ˜ 1 T ≤ − ≤ − b ˜ b ˜ = q = 1 models, we have T 40) on the even lattice Γ . 150), which are particularly ) are satisfied. If we assume, , , 7 4.31 = (10 ) = (0 2 b ˜ , x 1 , where there is a similar issue. Thus, automatically even, so that the anomaly 1 x is not even, and models for which b ˜ can be realized in F-theory. For most, but b ˜ a 8 – 20 – 138), that have SU(2) unHiggsings on one of Γ , ≤ − b ˜ ]. 24 on any effective cone from F-theory. Furthermore, the ) = (12 2 ) to be violated makes it clear that these cannot really be a 8 , x 1 4.34 x ≤ − ) is satisfied for some choice of effective cone that is allowed from b ˜ non-negative integers and ) and ( 4.29 b, b ˜ 146) and ( 4.32 must be even, it is unclear whether or not the charge spectra with , b ˜ 2 that violate the condition ) = (4 2 20) and violates the Kodaira bound for any effective cone from F-theory. This , , and that violate the Kodaira bound for any effective cone from F-theory; each of , x 1 1 0, the story becomes more complicated. From our analysis of 1 There are some spectra that arise, such as ( The fact that the anomaly and F-theory constraints precisely agree for x = 0. In this way, we can further understand the proliferation of apparently consistent = 0 theories follows from the fact that the inner product is simply a product of numbers = (5 = In this section,study we the construct growth some of infiniteT the families number of of anomaly-free anomaly-freeU(1) models U(1) charge at models spectra, larger in and charges. the limit of large the associated SU(2) models ismodels therefore are in also the not swampland. realizableany from It way F-theory seems of and likely are proving that in this these the at U(1) swampland, this but time. we do not have 5 Asymptotics and infinite anomaly-free charge families for both lattices, the resultingviolate SU(2) any model known is quantum indefinitely consistency the cannot condition “swampland” be based of realized on theoriesas in the that ( F-theory. low-energy do spectrum In not fact, but or there Γ are eight other U(1) models, such interesting. This spectrumwhich can does be not associated satisfy model can with be unHiggsed tob an SU(2) model thatmodel satisfies can the anomaly also conditions, be but realized has on the odd lattice Γ F-theory. Understanding which, ifand any, can of or the cannot models bemore that unHiggsed complete to violate understanding an the of SU(2) condition likely model non-UFD building are or on in non-Morrison-Park the the F-theory approach swampland of constructions, would [ require a is not satisfied, even thoughhowever, in that both cases the conditionsq ( not all, of thoseKodaira spectra constraint that ( admit an unHiggsing to an anomaly-free SU(2) model, the T in this case, with constraints immediately imply theT Morrison-Park > U(1) and UFDidentified anomaly-free SU(2) U(1) constraints. models for For which the constraints ( low-energy constraints. But itclasses does of suggest models, some and interestingmodels raises structure have a some for special certain questionsupergravity characteristic generic of theory. structure whether We that Morrison-Park return can U(1) to be these and identified questions UFD in in the SU(2) section low-energy satisfy these F-theory conditions. The fact that exotic F-theory constructions can allow JHEP06(2018)010 ) (5.3) (5.1) (5.2) gives r a, b, c, d . 2 Z ) = 3, and 2 9 is finite. It ∈ n , and thus we b ˜ + 2, and define T < a, b, c, d mn , q, r m/ − . The set of ( 2 ≤ c 274, and there are thus 2 r = 1 and arbitrary n m = ≤ + q b q x qr + q with + = 12 a + P 2 b ˜ ) with q Z . ∈ , 5.1 2 ) n = 6 (mod 3), then gcd( d b ˜ + n m, n ( , , 2 , × mn ) n 2 mn , ≡ − ) 2 n − r − 2 – 21 – m − − + m 2 2 ) + 54 into integral parts, mn q c , since the total number of charged hypermultiplet . m m ( c Q Q ( = 2 = = 2 = ( b c ≤ d a ≤ × × q q , each solution to the abelian AC equations trivially gives rise 2 dividing ) + 54 ) + 54 ), we can similarly construct an infinite family of four-charge b 2.2 r d/ 5.1 ( ( × × ) + 54 ) + 54 a ) = 1 otherwise. These numbers have a geometric interpretation, as shown in q is the integral side length of an equilateral triangle such that there exists an c ( ( : × × 1 a, b, c, d Note also that issues related to the fundamental unit charge under the U(1) do not Note that there cannot be an infinite number of anomaly-free models with charges These infinite families of anomaly-free U(1) spectra are particularly interesting, be- In analogy with ( 54 54 fields is bounded by thea finite gravitational anomaly number of condition spectra with affect these infinite families. Foran example, infinite the family family with ( unit charge 1. from F-theory is finite. ThisThe indicates that size there of is this an swampland infinite couldconstraints swampland be that of abelian rule reduced models. either out by some findingcompactifications of further these that quantum models, does consistency or not by come finding from some F-theory. other approach tobelow string any given upper bound primitive integer tuples satisfying this property. cause it is knowngauge that groups the and number matter ofis representations also distinct in known anomaly-consistent supergravity that spectra theories the with for number of nonabelian consistent spectra (abelian or nonabelian) that can arise gcd( figure integral cevian (a lineopposite segment side) connecting of a length vertexgenerated of in the this triangle with way any (after point dividing on by the the GCD, if necessary) is precisely the set of Then, is an anomaly-free model. Note that if models in the following fashion. Choose distinct there are additional nontrivial infinite familiesOne of such U(1) family models is satisfying the given AC by conditions. Before moving to asymptotics,number we of can different directly spectra addresssaw that the at satisfy the question the end of anomaly ofto whether equations section an the is infinite finite total or familyalready infinite. of know models As that we the by number scaling of the distinct charges anomaly-free and U(1) the spectra associated is infinite. However, 5.1 Large charge families JHEP06(2018)010 ) c 5.5 , we (5.6) b ˜ (5.4a) (5.5a) (5.4b) (5.5b) in the range . Such tuples c x = b , with the nonzero + a , . . . , h is automatically an integer = 1 2 i xB = . 2 , for generic values of C 4 i 2 B i , q q b , i ˜ 2 i b ˜ B, C, P P = = 108, so = 18 = 3 2 c 2 4 / i i d/ 2 4 i i = q q at large q q – 22 – ). 2 h h h = 1 =1 =1 C h h i i =1 =1 5.3 X X B i i X X x = . x x ) such that there exists an equilateral triangle of side length a b 2 dividing the side into integral parts, , and ) can be written as b for fixed ˜ d/ 2 2.4 / 6) a, b, c, d = 18 − h ( B B and this value of are integers. We further define the quantity b ˜ C possibly degenerate. and i . An integer tuple ( q B A simple scaling argument suggests that the number of solutions of the equations ( To estimate the number of solutions of these equations for any given value of In our case offor interest, any integer should go roughly as where can consider this as aequations special case of the problem of finding the number of solutions of the where the sum runs overcharges all charged matter representations, The abelian AC conditions ( Figure 1 with an integral cevianproduce of anomaly-free length models of the form ( 5.2 A continuous approximation JHEP06(2018)010 , , . 1 2 2 = N / b ˜ 1)- 2 B i 274. 6) q (5.8) (5.7) − . By − . The i 2 to h h ≤ 2 / ( / P 4) 2) B /h − − 2 h that satisfy mN ( h = ( b i B ˜ 2 . Indeed, this q ≤ B 2 / ). 1 are realized by /B 1 2 b ˜ . / both exceed this 5.5 /h, 2) − B, x h 5.1 ( = 1 that should approach a can have singular peaks B x x x will then go as b ˜ 108, the number of solutions , as long as 108 . 2 2 / ≥ 6) . do not all have a common factor, h xB − 2 i / q m 0), respectively. The possible values ( = 4) b ˜ − 4 r m ( ) should scale as ,..., b ˜ + 0 4 , 5.5 ∼ 0 q, should go as the integrated value, – 23 – b ˜ B , q = should scale as ) and ( 2 approximate m r limit, where the limiting values m N + , as in the continuum approximation it is the volume of an . The number of solutions that satisfy the equality B 2 2 gives a distribution on the set of B q , suggesting large infinite families of solutions, as the total h/ i 2 √ q / , . . . , q B 1 q, q, . . . , q 6) q − h ( b ˜ the number of solutions of ( charges x m clearly scales as 1. This can be seen as follows: the number of charge combinations = 2, and the system of equations . (rounded integer value) are then distributed across the range from h B ≤ 2 ) will scale as x ≤ h, m 2 i The result of these features is that the scaling estimate just given is generally an We may also wish to restrict to cases where the Note, however, that the infinite families identified in section This simple scaling argument suggests that when ≤ 5.4 xB q will go as the derivative of the number of solutions of the inequality, so as i -dimensional ball of radius /h 5.3 Two charges As a simplecharges, toy example to illustrate the issues involved, we consider the case of two small underestimate of the totalof number charge of multiplicities. solutions Toexample understand that in these can further issues arise detail. more at clearly, we favorable consider combinations a simplified analysis. First, the continuous distributionthat on enhance possible values the of relevant number in of affecting the solutions; number of second, solutions number-theoretic for particular considerationsin values can order of to be suppresscharges. the contribution This of only the affects trivial the infinite analysis families by produced an by overall scaling factor, of and is only really relevant for number of five charges, etc. estimate, with the integratedand number the of integrated number models ofcontinuous in models scaling of the approximation the misses three-charge four-charge out family family on scaling scaling several features as as that modify this simplified The number of integrated solutions up to a maximum value So, for example,an there equal should number be of four a charges, logarithmically a divergent linearly number divergent number of of solutions solutions with with an equal of ( number of solutions upchoosing to simpler an charge arbitrary combinations, whereof a charges large arises, this multiple of estimatecopies suggests, each each for of of example, a that smaller the number number of solutions with the charge combinations ( of so for any given This scaling can be understooddimensional geometrically sphere as arising and from quartic the hypersurface intersection defined of the by ( the equations ( h B set of charge combinations smooth distribution in the large 1 P JHEP06(2018)010 ) 1. ) as 5.8 B, x = 2. (5.9) , i.e., (5.12) (5.13) (5.11) (5.10) 2 + 1. n 5.9 -plane, / B /B 2 2 B = , B, xB C B ) when , so the number 2 1 2 contributes to the q √ . B 2 to B, x 2 scales as 1 = / 2 / = 2 2 2 B 2 xB = 1 xB d . Note, however, that the √ q, B faster than the asymptotic = 2 − x 2 − − 2 2 2 = , 4 / C B 3 | r 2 r /B ) b ˜ q ) i 1 0. i = + 0 x x − ) with ( 4 2 Z ( . Thus, the number of solutions at 0 / 2 appears to diverge. From the fact q g f B / B 6) = | 0 0 5.8 B > δ 8 − 1 i 2, and the average number of solutions , + √ (2 x = 1 B X / 4 − x B = 1. Thus, in the continuous approxima- x B/ q x − x ) d − 2 , x < , x > 2 p , x δ x r 1 )) = 0 1 2 2 √ 1 over the entire integration range. Thus, we ), which scales x / q 2 2 − √ ( + ( – 24 – q 1 2 is 2 ≈ 2 f ) corresponding to a given ( − 2 are those where 5.1 (1 / ( q − = 1 / − θ x /B δ ) = B ) q, r x x x B 1) − x = 1 ( 2 p r δ 2+1 ( = 1 θ 1 the number of solutions of the integer equations ( θ / 2 − 2 d x √ 1 / ) in the relevant integration range, we can write ( x 2 √ q x 1 q x x g x 2 )-plane that maps to a unit area in the d Z d ( B d (2 2 f √ and θ ∞ ∞ = Z q, r 0 √ ) for 0 B 2 Z 1 Z 5.8 ≤ − = , this yields . In the 3-charge case, a similar extra factor of 2 ) = x ) = x B r x 2 C x d ( ( d 2 2 √ + 2 , the number of solutions for I 2 I B 2 2 q B − = 2 is expected to be an integer, we can estimate the effect of this divergence B C 2 are the zeroes of ) diverges at the values ). 2+1 x i / 2 ( 2 x / xB 5.7 2 2 B I B = Z C Let us consider the more precise number-theoretic aspects of this question with In accord with the general analysis of the preceding section, the number of solutions estimate ( In fact, the solutionsof to solutions ( with of the equality goes as the derivative of this, or 1 where we have approximated may expect that thewith number an of extra integer factor solutions of ofscaling of ( the infinite family of charges ( that by integrating the divergentNoting integrand that over d the range from in this continuous approximation thusfunction scales as tion, while for genericshould values scale as of with the Heaviside step function. Here, we have assumed where the corresponding to the Jacobian ofthe the transformation number between of these spaces, integerUsing which estimates solutions the relation for ( by the integral This gives the area in the ( In the continuous approximation, we can estimate the number of solutions for fixed JHEP06(2018)010 . x . B 2 1.0 ≤ 2.2 with solu- 68, as . 2 k r B + = 0 ) are rela- 2 0.8 x q , . . . , q q, r 1 compared with q as well as from 2 ) 68 that provides x 2 . 0.6 r of the total + B 2 q √ ( 25 = 0 / 0.4 / ) 4 for the set of all r x + = 17 4 . This is illustrated in figure distinct charges q 0.2 , x B k 2 , but exhibit number-theoretic ir- q = ( √ B , such as the large peak at x (x) √ B 2 = 5 / √ 3.0 2.5 2.0 1.5 1.0 0.5 I ∫dB / q, B = 2 – 25 – r 10 000, where numerical values are averaged over bins x ≤ 1.0 2 r ), and always give sets of order , the overall distribution should be multiplied by this + 2 . This pattern represents a breakdown of the continuous 2 B q /π 0.8 na, nb ≤ 2 r limit, using the fact that the probability that a given pair of ) with ) = ( + B 2 0.6 q, r q 200 (right). Irregularities in the right diagram indicate number-theoretical q, r . These families of solutions arise for every relatively prime pair of / r = 2 0.4 q ), where ( a, b 5 solutions to the equations with the inequality, including a factor of 2 to include . The prediction for the distribution of values of 0.2 20 (left) and 1 / 2 exceeds even the above continuum estimate adjusted to take account for the singu- B/ / 4 We can carry out a similar analysis for the case of three distinct charges, four distinct Related to this, we may wish to restrict our attention to theories where ( (x) 2 p = 1 0.5 2.0 1.5 1.0 5.4 Fixed number of distinct charges From the preceding analyses, wethe can now number consider of the solutions more of general the problem of anomaly estimating equations with charges, and so growthna¨ıve on. coming from singular As peaks in innumber-theoretic the effects. the distribution case on values of of two charges, there can be enhancements to the tively prime, to suppress theNote contribution of that the in trivial the infiniteintegers families large is seen relatively in prime section factor is when 6 considering only relatively prime charges. tions to the inequality approximation. Thus, we expect thatwill a histogram look of fairly values of smoothregularities if when the the bins number are of larger bins than increases 1 beyond larity. Note, however, that thisThere family of is solutions another seems family to∼ of be solutions unrelated with to the singularity. solutions with integers ( patterns that become relevantmentioned for in bins the smaller text. than 1 x Figure 2 numerical values for all ( of size 1 I ∫dB JHEP06(2018)010 , , is B B i 5.3 n ≤ show i as the b ˜ (5.14a) = 0 for (5.14b) b P ˜ q 5.2 x 3, i.e., . In this analysis, 2 / should then scale as 3 | ≤ q B B | arise with slightly larger ). 5.1 5.1 5. , 2 2 in the swampland. We then carry out 4 and B > 1 not . Thus, we would expect based on these 108 | B, 2 q / | = = 2) 2 4 i i − – 26 – the integrated number of solutions will scale as q q k i i ( i b n n ˜ k k =1 =1 k, n i i X X there are a finite number of possible relevant combinations k , we have a similar set of equations to those studied above. This i . The equations that must be solved are then , there are, in fact, an infinite number of anomaly-free spectra for n k . The asymptotic number of solutions at large i 5 , the number of solutions with 4 distinct charges should scale as n 2 / 1 is an integer divisible by 18. Since the total number of charges , . . . , n B = 0. We analyze the types of models that arise, and note mechanisms by 1 b ˜ n T = 18 , and that at specific values associated with singularities like those described above, B 2 4, and / and multiplicities One of our primary tools for identifying a model as not being in the swampland is Summarizing the results of the analysis, the continuum estimates from section 4) − k | ≥ k ( q unHiggsing; if a U(1) modelan can F-theory be compactification, then unHiggsed we tothe can a associated realize nonabelian nonabelian the model F-theory U(1) thatU(1) model. model is model in is realizable In to F-theory by particular, an by SU(2) the Higgsing model, most and so direct we unHiggsing will classify of the a U(1) models we find, in part, by specific U(1) model doesthis not infinite have swampland, therefore, an it F-theory isin realization. illustrative this to section consider In particular we order cases. first| to consider To better this models end, understand with maximumwhich U(1) we charge can identifythe models same that analysis are for clearly models with charges up to As we saw inlow-energy section U(1) models. As theis number finite, of consistent this spectra meansU(1) that models. can that arise As there from discussed is F-theory in an the previous infinite section, swampland however, it of is distinct difficult massless to show spectra that for any number of distinct charges increases,why and singularities certain and infinite number theoretic familiespowers effects explain in like the those growth identified rates in than those section predicted by6 the continuous approximation. Classifying models with charges and the number of solutionswe with have 5 not distinct accounted chargeswhich for should are number-theoretic scale needed effects as to like account those for encountered the in scaling of section that families infinite like families ( of charges are expected, with increasing rates of growth in of suggests that for genericb choices of the number ofanalyses solutions that will the scaleshould total as scale number as of solutions with 3 distinct charges with any 18 where bounded above by 274, for each of the multiplicities the number of solutions for any combination of multiplicities. For each distinct combination multiplicities ˜ JHEP06(2018)010 (6.1) ) from = 0. 2.16 . 3 ) x 3 ( = 0. × 3 T x 3. We allow for half- ) + , 2 2 , which reduces the number 3 ( , x 1 × is a non-negative, even integer. + ). Thus, we have models of the b 2 ˜   3 x = x 2.11 + 6 q 1 − x are all non-negative integers is 260. The ≥ 6 3 − 4 , x b ˜ 2 – 27 – b ˜ , x 1   x , although we will once again find that all models have ) + 1 = 0 can be found via exchanges of the form ( T ( × i 3 and x ). From this, we find that anomaly-free U(1) models with charges 3 U(1) models + 15 4.13 b ˜ ) from the generic matter models ( ], despite it apparently not being unHiggsable to any anomaly-consistent | ≤ − q | 24 3 are of the form 2.13 24 , b ˜ 2 h 2 models ( , 1 | ≤ q Throughout this section we restrict our attention to models with Ultimately, we have not managed to identify any specific U(1) models for which we can There are also cases, however, where we can directly construct a Weierstrass model for | = Now we consider SU(2) modelswhich containing we fundamentals, can adjoints, and Higgshypermultiplets of triple-symmetrics, to both U(1) models with charges The number of suchgravitational models condition for in which this caseof becomes anomaly-free 274 models to 245. Note that these6.2 numbers include models SU(2) with models with 3-symmetric matter The requirement that the numberresentation of must hypermultiplets be transforming a in each non-negative irreducible integer rep- tells us that 6.1 Charge Anomaly-free U(1) models with the q Nonetheless, the analysis here givesconsider us ever an larger idea charges, of andare the may kinds to give of finding some models hint F-theory as that compactifications to may that what arise realize the as them. potential we obstructions models. definitively say, “This anomaly-free modelin cannot the swampland,” be though realized we in knowhigh that F-theory charges, as and infinitely we many therefore continue (in is to fact, classify models cofinitely with many) arbitrarily models must be in the swampland. a U(1) theory [ nonabelian model. Thus,theory while realization having is an sufficient to unHiggsingthat show to that there a a does U(1) nonabelian model notbe model is exist found not with in in such an the F-theory, an F- swampland, and unHiggsing showing is is therefore not not a a complete guarantee criterion that for the identifying model swampland cannot models that cannot be unHiggsedanomaly-free to nonabelian an anomaly-free model SU(2) with model arealized may larger in be gauge F-theory, unHiggsed group. this to similarly If an identifiesWe this the additionally nonabelian model present model as arguments can not that be unHiggsed being may to part help of any identify the nonabelian cases swampland. model. where a model cannot be their ability to be unHiggsed to an anomaly-free SU(2) model. We will also see cases where full hypermultiplets in the fundamental.of Anomaly-free the models form can be ( found via exchanges JHEP06(2018)010 , (6.3) (6.2) . 3 that x . . 2 ) 3 +4 | ≤ q | ( x × × +3 3 and integer non- x x x 2 | ≤ + q ≥ | ) + 2 2 ( × are all non-negative integers and × 3, which includes all models with | ≤ q x | x 2 6 x , − 12 unbroken, which gives models of the form 2) − 3 , x T − 3) – 28 – b . x − 2 3 U(1) models 2 199) that there are U(1) models with b 1)( ( b, b − > | ≤ x b q ( | ) + = 2 1 + 3 x ( × × = 0. x x ), weakening the integrality condition to allow quaternionic irreducible , these solutions match the form coming from the abelian AC conditions , as well as b N 4.3 x ) + 30 = 2 ) + 14 b b b ˜ − − 1 is 223 (we require the number of adjoints to be at least 1 so that we can Higgs on 2. ≥ (12 (12 We can see already (since 245 Each anomaly-free U(1) model can be unHiggsed na¨ıvely to some (potentially incon- The total set of U(1) models we are considering are those that satisfy the AC conditions, b b 4 2 | ≤ q need to look at theon Higgsing an of adjoint these charge SU(2) leaving models the down generator to U(1) models. We again Higgs cannot be obtained by Higgsing an anomaly-free SU(2) model. To explore this further, we by taking As in section | 6.3.1 UnHiggsable U(1)s First, we consider theThese are models subdivided that into two can classes. be unHiggsed to an anomaly-free SU(2) model. sistent) SU(2) model; we saycorresponding that unHiggsed a SU(2) U(1) model satisfiesmodel satisfies satisfies these the the conditions. SU(2) SU(2) gravitational NGIN boundthis Similarly, conditions if bound. we its if corresponding say Note its unHiggsed that that SU(2) we satisfies a consider U(1) all models with number of hypermultiplets transforming underinteger. any We representation collectively must refer be toNon-negativity). a these non-negative conditions Note by NGIN that (Non-Gravitational,ditions we Integrality, for will SU( alsorepresentations use to have the half-hypermultiplets. initialism NGIN to refer to these con- including the gravitational bound.the gravitational In bound classifying these fromnon-gravitational models, the (NG), it other despite is AC theanomaly useful conditions, fact condition. to that which distinguish they we Along include will with the loosely the mixed refer NG gauge-gravitational to conditions, as we include the requirement that the 6.3 Classification of charge We can now dividenegative the spectra anomaly-free into five U(1) classes, models grouped with into two charge broad types, as shown in table which reduces the number of anomaly-freemodels models with to 199. Note that these numbers include x an adjoint). In this case, the gravitational condition becomes 276 The number of such models for which form JHEP06(2018)010 . 3 to be ), but 2 6.1 ,T 1 T ). This U(1) 0 in ( ) that any U(1) < 2.16 X ) 6.3 ; thus, they satisfy b ˜ 3 x − (24 b ˜ 2 model because the result ), and in this case none of the 1) and the adjoint has charges X XX = . None of these models are U(1) − 1 , 24; in fact, the exchangeable U(1) 2 model, but can nevertheless be q x 2.13 (0 ≤ , ≤ ; we can see from ( b ˜ 0) 3 | ≤ 1 , q x | 1 Second, we have models that cannot be > x − ( 3 , x 1) , XX XXX – 29 – First, there are those models that can be reached ) representations. There are 127 such models. First, we have models that cannot be unHiggsed to 1 2 using exchanges of the form given in ( 7 2 ) representations due to the value of 37 72 127 1 | ≤ q | 3 models satisfying the U(1) AC conditions, as well as the integrality and | ≤ q | Type # Exchangeable SU(2) NGIN SU(2) Grav UnHiggsable Non-unHiggsable . Classes of We can obtain at least some of these models by Higgsing from larger gauge groups. exchangeable, and they all satisfy the SU(2) gravitational bound. For example, consider Higgsing SU(3)that down exactly one to of a the singlesuch generators U(1). is that left To the unbroken. do fundamental Take so, the has Cartan we charges generators must (1 Higgs so NGIN non-unHiggsable U(1)s. an SU(2) modelfundamentals because (thus the violating resultant NGIN).These SU(2) There models model are are precisely 37 wouldmodel those such have obtained that by U(1) a have Higgsing models, negative an listed SU(2) number must in of have table 6.3.2 Non-unHiggsable U(1)s Next, we consider the modelsThese that are cannot further be unHiggsedbecause subdivided to we into an have three anomaly-free SU(2)realized the classes. model. in least F-theory. Such understanding models of are how the to most show interesting whether or not they can be unHiggsed to an anomaly-free SU(2)have a model. positive Such number modelsAC, of have ( but cannot bewould have exchanged a to negative number a of maximum ( charge that these models allmodels satisfy the (which SU(2) include Tate two bound precisely non-unHiggsable those models, models as that discussed satisfy in the the SU(2)Non-exchangeable, next Tate section) bound. unHiggsable are U(1)s. reached by U(1) exchanges from an acceptable Exchangeable, unHiggsable U(1)s. from a U(1) modelexchange with corresponds exactly tocorresponding the SU(2) SU(2) exchanges exchange results ( all in U(1) a half-integer models number in of this adjoints, class thus to allowing be unHiggsed to SU(2). There are 72 such models. Note Table 2 non-negativity conditions. Classes arefree grouped SU(2) into model two and types: those that those are unHiggsable not. to an anomaly- JHEP06(2018)010 3 that cannot be unHiggsed = q 3 x 2 x – 30 – 1 b x ˜ 54 0 0 108 52 29 4 99 52 14 10 98 50 65 4 91 50 50 10 90 50 35 16 89 50 20 22 88 50 5 28 87 48 78 12 82 48 63 18 81 48 48 24 80 48 33 30 79 48 18 36 78 48 3 42 77 46 68 28 72 46 53 34 71 46 38 40 70 46 23 46 69 46 8 52 68 44 50 46 62 44 35 52 61 44 20 58 60 44 5 64 59 42 39 60 53 42 24 66 52 42 9 72 51 40 35 70 45 40 20 76 44 40 5 82 43 38 23 82 37 38 8 88 36 36 18 90 30 36 3 96 29 34 20 94 24 34 5 100 23 32 14 100 18 30 0 108 12 = 0 U(1) models with maximum charge T . Anomaly-free Table 3 to an anomaly-free SU(2) model because of the NGIN conditions. JHEP06(2018)010 (6.6) (6.5) (6.4a) (6.4b) . , 3 ) , is less strict than the 3 Finally, we have non- the generators of the two 3 that cannot be unHiggsed 2 ( | ≤ ,T × 1) so that only the generator x q 1 | , − T ) , . 2 ) + 34 Second, we have exchangeable U(1) 3 ) + 80 ( , we can Higgs down to a single U(1) 3 × 2 r x ( × x ( × 2 + 21 U(1) with ] = 16. × x b + 3 ) + 1 × η [ 1 ) + 4 · × 1 0 ] x , as expected from section – 31 – ( ( = 24, can be found in F-theory by a non-UFD a 2 b ) + 24 η b ˜ 1 × × ≥ = 0 U(1) models with 2 4 = 6 b x ˜ ( 0) under these generators. We can then Higgs on a 24 240 12 16 22 254 4 14 b T ˜ , . → → × 5 (0 = 8: 24 ], with [ b 3, 276 × 24 2 , = 1) ∓ q , = 48: 48 1 linearly independent roots (adjoint charges) in the root space and 1 b ˜ − ( , r 2) . 4 , 1 ( , survives. Under this Higgsing, we have . Anomaly-free, exchangeable 1) 2 T We believe that the gravitational bound for SU(2) models that Higgs to U(1) models The second of these models, with In general, for some larger gauge group of rank , + 2 1 gravitational bounds associated with othernine nonabelian U(1) groups, which models would in imply that this the and the previous class cannot be unHiggsed to any nonabelian seven such models, listed in table with maximum charge construction, as described in [ Non-exchangeable, gravityexchangeable U(1) models non-unHiggsable that nevertheless satisfybe U(1)s. the unHiggsed SU(2) NGIN to conditions, an but anomaly-free cannot SU(2) model due to the gravitational bound. There are Exchangeable, gravity non-unHiggsable U(1)s. models that satisfy the SU(2)free NGIN SU(2) conditions, model but due cannotcontain be enough to uncharged unHiggsed the scalars to gravitational to anlisted perform bound. anomaly- in the table unHiggsing. In There other are words, two such these models, models do not which appears in the listto of the 37 U(1) NGIN models conditions. that Note cannot that be unHiggsed to SU(2)by models choosing due some projecting these out to see how the irreducible representations split. to yield the U(1) model ( charge such that SU(3) breaksfactors, down and to then U(1) weT can further Higgs on the charge (1 Table 4 to an anomaly-free SU(2) model because of the gravitational bound. We can then Higgs the anomaly-free SU(3) model JHEP06(2018)010 (6.7) ) model ) adjoint, N N 3 that cannot be unHig- | ≤ q | . 4 − N ) are those that become the genera- N 3 2 more times on charges coming from x 1) uncharged scalars from Higgsing the − 2) = 3 − 2 N − x N ( N 1 Adj – 32 – = 0 U(1) models with x T 1) + ( 2) additional uncharged scalars from the subsequent − b x ˜ − 38 218 4 50 36 213 12 43 34 215 16 37 32 224 16 32 30 225 18 27 28 233 16 23 26 233 16 19 N N 2( 3)( , and then Higgs ≥ 1 . Every adjoint (including the one given the VEV) contributes − − 0 1 2, the number of uncharged scalars must then be at least x N − Adj N ≥ x Adj x . In order to carry out this process, the original model must have at least ) adjoints to break to a single U(1). This is the same Higgsing procedure 3 N ) gauge group, then the “quickest” route (in the sense of producing the fewest . Anomaly-free, non-exchangeable N 1 uncharged scalars, which are precisely Cartan charges. In each of the subsequent Thus, in the entire process, we gain In the first step, the Cartan generators of the SU( We can see that some of these models cannot be unHiggsed to any SU( − first adjoint, and then (2 Higgsings. Because the negative of this charge alsoeaten, exists; and one the of other these becomesadjoints becomes an become the uncharged would-be uncharged scalar. Goldstone scalars. that Both is such charges from any additional which to Higgs away all but one of the remaining U(1)s. tors of the unbroken U(1) N Higgsings, because we give a VEV to a charge coming originally from an SU( the original SU( we saw in section two adjoints. The adjoint toGoldstone which bosons we that give a are VEV eatenare in to uncharged; the we give first must the step have gauge splits at bosons into least mass, the one and would-be other the adjoint remaining in modes order to have adjoint charges on via adjoint Higgsings byan considering SU( the numberuncharged of scalars) uncharged to scalars. acharge If single that we breaks U(1) begin to unbroken U(1) with symmetry group is to Higgs on an adjoint gauge group. We do notbut have we a give completely a rigorous partial and analysis exhaustive here. proof of this assertion, Table 5 gsed to an anomaly-free SU(2) model because of the gravitational condition. JHEP06(2018)010 | ≤ q (6.8) | 4 and X | ≤ q | , , , , ) ) ) ) 3 3 3 3 ( ( X XX 3 only five of these classes ( ( . There are potentially eight × × × × 6.3 | ≤ q | ) + 32 ) + 23 ) + 50 ) + 14 2 2 2 2 , for ( ( ( ( × × ) grows rapidly. Indeed, the gravitational X XX XX XXX × × 6.3 6 – 33 – ) + 16 ) + 4 ) + 4 ) + 16 Exchangeable SU(2) NGIN SU(2) Grav 1 1 1 1 ( ( ( ( 5 5 models satisfying the U(1) AC conditions, as well as the × × × × # | ≤ q ) model, by the argument above. | N 4 7 8 7 11 42 43 88 89 426 22 182 358 7 344 2 model (“U(1) exchangeability”), the SU(2) NGIN conditions, 3 204 81 320 1 157 11 320 5, there exist models in each of the eight possible classes, as shown 5 4 and , 4 = 22 := 254 28 := 233 32 := 224 38 : 218 | ≤ | ≤ | ≤ b b b b ˜ ˜ ˜ ˜ q q q | | | | ≤ q | 4 and Type # = 2 and are not unHiggsable to SU(2) models, and thus cannot be reached by | ≤ 0 . Note that the number of exchangeable models is relatively small. Note also that q x | . Classes of UnHiggsable 6 Non-unHiggsable For anomaly bound for U(1)so enforces a we cutoff expect onsatisfying that the NGIN as allowed will number we of notsection, include hypermultiplets, satisfy and there higher these bounds. are andgravitational still higher Nonetheless, anomaly large charges, as condition. infinite discussed a in families vast the majority of previous models of models that satisfy NGIN and also the actually contain models. in table as the maximum chargeU(1) grows, gravitational the bound number (not of listed models in that table satisfy NGIN but violate the We can carry out5. the same classification We for classifyequivalence U(1) with these models a U(1) withand models charges the by SU(2) gravitational which bound,distinct as of explained classifications. three in section properties As they we satisfy: saw in anomaly section all have adjoint Higgsings of any SU( 6.4 Charges Table 6 integrality and non-negativity conditions.to Classes an are anomaly-free SU(2) grouped model into and two those types: that those are unHiggsable not. The models JHEP06(2018)010 b ˜ ), 4.5 (7.3) (7.4) (7.5) (7.1) (7.2) is even ) model b ˜ N 0, the condition . b 1) T > is likely a necessary − b ˜ N ( N = b ˜ b , ˜ b . b . · · · a ) models a a 8 b . ˜ 4 3 · N b . ˜ a · 2 ≤ − is an even integer. For ] showed that the condition that a ≤ − ≤ − 9 Q b ˜ b ˜ b b 2 18 · · · b ˜ b b ≤ – 34 – ≤ b ˜ ≤ − 2, the anomaly constraints imply the condition ≤ ≤ · b ˜ b ˜ b b b · ˜ · , · · 1 b a ˜ a a 2 − − − b ˜ = , following the same logic as described just following ( is in the “effective” positivity cone of the theory. When the q Q x ≤ − q y means that y ≤ be an even element of the charge lattice is necessary for an F-theory realization, x = 3, this gives the constraint = 0, the anomaly conditions imply that b ˜ These constraints match nicely with the observation that Higgsing an SU( For SU(2) models with only fundamental and adjoint matter, and at least one adjoint The main conclusion is that we have found there is an infinite swampland of apparently Q T follows from some fairlybe mild compactified assumptions on (specifically, any theyin spin assume every that manifold, topological the andcondition class). theory that for can there the Thus, is consistency itare of an presumably seems the allowed inconsistent quantum that gauge and gravity configuration therefore the theory, not and evenness really models of of with interest non-even or in the swampland. on adjoint fields in a natural way gives7.2 a U(1) model Evenness with conditionFor on that but is not imposed by anomalies. Recently, [ For SU(3), the analogous constraint is field, the anomaly constraints imply the condition the anomaly equations imply that any U(1) model must satisfy the constraint For For U(1) models with charges where charges are in the range consistent massless charge spectra for 6Din U(1) F-theory. supergravity However, models there that is cannotwe no be can single realized prove specific definitively U(1) that model there that is we no7.1 can F-theory identify realization. for which Anomaly constraints on U(1) and SU( One of the principalthat questions of appear this consistent paperconsistency from is conditions, the to but ascertain that point are thesome of not scope of realized view of the in of the main F-theory. set results In anomalies this on of and section such models we “swampland” other summarize models. known quantum 7 F-theory and the swampland JHEP06(2018)010 , ) ). a 4 and 7.4 7.5 (7.6) (7.7) a . The ≤ − ) gauge K b 2. N from ( = 0), and 12 b ), and ( , − T and 1 7.4 a 6 ] can violate this = ), ( q 24 ≤ − 7.1 b = 0, terms in the low-energy 2 T F ) bounds ) seem to be sufficient conditions and N 2. 2 7.7 R , ) is not, however, necessary for an F- a , 1 8 a , 7.7 1 there are anomaly-free SU(2) models that ] all satisfy the condition = 12 ≤ − q 14 b ˜ T > ≤ − ≤ – 35 – a 2 Nb (giving the corresponding anomaly coefficient) must ) gauge groups that are constructed using standard − b , when and the condition ( 2, there is a precise match between the set of models N b ˜ , 4.5 1 = 108, where only the model with fundamental U(1) charge ) in the physically allowed cases, but is stronger than that = 2 2 q 7.1 , x ], it was pointed out that any F-theory model with an SU( , 8 1 = 0 SU(3), the Kodaira bound becomes, respectively, 1 , x = ) also implies a relation between the 2, while all known F-theory models satisfy the anomaly constraints, there q . Other than this case, there is no swampland for 7.6 > | 7.8 q 0 models with tensor multiplets, and for models with massless fields having U(1) = 0, | ). As discussed in section T 2 has a known realization in F-theory. We discuss this exceptional case further below 7.5 The only completely clear explicit constraint that we are aware of, however, on T > that holds for all F-theory models and is stronger than the anomaly constraints is the = This condition impliescondition. ( In fact, the evennessfor of an F-theory modeltheory to model exist. to exist; Thecondition, exotic condition even constructions ( for using models the with methodology only of charges [ F-theory models with U(1)methodology or obey an SU( interesting setF-theory of constraints models closely of parallel the to Morrison-Park ( form [ U(1) models that arisethat from these Higgsing cannot these be SU(2)swampland. realized models, in but F-theory, we though do it seems not likely have that any7.5 they proof are also Morrison-Park in U(1) the bounds and UFD SU( constraint ( theory. For SU(2) which have a similar form to,and but ( are slightly weaker than, theviolate upper this bounds on Kodaira constraint and are therefore in the swampland. There are associated “Kodaira bound.” In [ symmetry associated with a divisor satisfy the bound associated with the Kodaira condition that the discriminant is in the class are in general a varietyinfinite of low-energy classes models of that anomaly-consistent have modelsin no of F-theory. known which F-theory So only realization, in a and general finite F-theory number constraints can are be stronger realized b than the anomaly constraints. in section 7.4 Bounds from F-theory For charges In the limited classcharges are of restricted U(1) to modelsthat satisfy where the there anomaly are conditionsthe and no model those tensor with that multiplets canq be ( realized in F-theory, except for 7.3 JHEP06(2018)010 ]. ). 22 7.5 (7.8) (7.9) 0 this is a T > = 6 in F-theory , which has three | 2 q P | 3 matter were found ]. At this point only a 6, but there is clearly 24 = 3 matter were studied > = | | q ]. q q | | ], and no models with larger 24 , 24 22 ], based on the approach to charge 37 a a , 4 3 2 ≤ − ≤ − – 36 – b b > ], that it may be impossible to construct consistent | = 6 must be realizable in F-theory; for example, one q | | 38 q | = 5.) It is unknown whether any F-theory model can | q | ]. Another approach would be to develop further the non- as the number of distinct spectra for F-theory models is itself 36 q , = 6 matter by tuning an SU(6) on a quartic on | 35 ), and there are some hints [ q | = 4 have been explicitly constructed [ N | = 0 this is equivalent to the anomaly constraint, but for q | T ), the inequality 4.25 ], and a more systematic construction was described in [ 34 ] by systematically studying the set of possible toric hypersurface fiber types. One , In principle, models with up to On the other hand, our understanding of F-theory models with higher U(1) charges is The connection between these Morrison-Park and UFD constraints and the anomaly For any SU(2) model that is realized in F-theory using the standard Tate/UFD con- 33 33 = 2 constructions in recent work [ have been explicitly constructed. The first examples with | q in principle be constructeda with finite U(1) upper matter boundfinite. having on charge It doesby not breaking seem straightforward an to SU( | get a charge greater than UFD approach to describingHiggsing U(1) of models nonabelian with models higher with charges, exotic either matter directly as in orcan get [ through a model with adjoints, and then Higgsingwith on two SU(5) of gives the matter adjoints with to get a U(1). (A similar construction q in [ approach to trying to identifymore new models fully with the larger set U(1)as charges of would that toric be developed to and investigate in complete intersection [ fiber types using methodology such of anomaly-consistent models with any fixed upper bound onquite the limited. charge. Some specialin cases [ of U(1) modelsfew with models charge with realization. We leave this as a provocative question7.6 for further research. Models withWe U(1) have charges identified infinite familiesIn of these anomaly-consistent families, charge the charges spectra become for arbitrarily large; U(1) indeed, theories. there are only a finite number constraints for abelian andquestion nonabelian of whether theories there is isfrom quite some a signature intriguing. UFD in the Itmodels type low-energy from raises construction theory more the of exotic (including natural models models Morrison-Park) that that involve come that non-UFD algebraic would structure distinguish in the these F-theory which is again closely analogousAs to in but stronger the thanconstraints abelian the that are SU(3) case, so anomaly closely it constraint relatedviolate ( is to these the UFD very anomaly constraints constraints. suggestive can Nonetheless, be that models constructed that standard using the F-theory non-UFD constructions methodology of give [ struction ( must hold. For stronger condition. Similarly, for any UFD SU(3) model we have the inequality JHEP06(2018)010 , 3 ], there is ]. This is 40 39 = 24 model with b ˜ = 12 listed in table 6. Further progress on ] that have no massless = 24, where the funda- 3 b ˜ > 41 | 1. Indeed, there are “non- , x , q | 26 , = 2 with q = 108 25 2 = , x q 1. This model appears not to be realized = 0 1 x = q – 37 – ]. No analogue of this argument is known from the 31 3 is in the positive cone, there must be an associated non- ≤ − C · 1 do not appear to have F-theory realizations and seem to be in C = q = 0 and 108 massless fields of charge T ], which are a bit more difficult to assess from the point of view of the low-energy 8 allowed charge spectrum of thefundamental theory. charge In any case,the models swampland; such as a the furtherthe study completeness of hypothesis these seems models likehowever, an and that interesting others models direction that such for do as further not the research. obviously model Note, satisfy with we are aware of,features however, in that a rules theory out thatHiggsable” the still U(1) existence models has of massive that fields a cancharged be of massless fields realized charge in spectrum in the with F-theory low-energysive these [ spectrum. charged fields, Nonetheless, for these example modelsSU(2). associated should This with contain shows mas- massive that adjoint the fields massless in spectrum an unHiggsing does to not always completely determine the this paper. Thus, thiswe is encounter a should hypothesis be or ruledmodel folk out theorem with by in this this hypothesis. context.mental U(1) In Some unit particular, of charge can we the be have modelsin chosen discussed F-theory, to and the be at least naively violates the completeness hypothesis. There is no principle charges of the gaugerelated group to are the realized ideagravity. by that While some there there state are are in variousrelated no physics, arguments the global and for Hilbert the symmetries these statements space have in statementsno been [ based complete a proven proof on in quantum of the black theory AdS/CFT these holes context statements coupled and [ in the to 6D flat-space supergravity context relevant for be addressed for a complete understanding or7.8 clearing of the 6D The supergravity completeness swampland. Another hypothesis constraint on supergravity modelspleteness that hypothesis,” plays which a states role in that this for discussion a is gauge the theory “com- coupled to gravity, all possible curve of self-intersection Higgsable nonabelian gauge grouppoint [ of view of thement low-energy of theory, the though worldvolume it theory of isconstraints. the natural strings Here to arising imagine we in that have the charge a mostly lattice careful left may treat- these provide such questions to the side, but they certainly must partly in [ theory. In particular, F-theorysociated allows with only the certain conepresent positive we of cones have no for effective understanding charged divisorsbe of strings, in constrained, how as- the outside the the positivity F-theory framework coneis of of compactifications fixed F-theory. the space. to Even low-energy assuming theory be that may At one the that positivity comes cone from F-theory, F-theory has the constraint that whenever a the F-theory side is clearly needed to7.7 understand these kinds of Constraints constraints on moreWe the clearly. have charge focused lattice here and onanomaly positivity explicit constraints. questions There about is the a spectrum further and set constraints of related to constraints imposed by F-theory, discussed F-theory models with singularities carrying U(1) matter with JHEP06(2018)010 ) 2 R N 4, or any systematic > | ]. Some progress in this q | 42 ) models, the Kodaira condition N = 1 supergravity models with a single N 3. We do not know what the upper bound on – 38 – > ], which as in the pure U(1) case may help provide | q 1 but for which the GCD of massless fields is 1, are | 36 , ]. A variety of models of this type have been studied 35 = 44 , q 43 2 will generally give a massive state of charge +1. So there is no − . These F-theory constraints are violated by more exotic construc- N 6. It would also be interesting to further explore the set of consistent U(1) provides a universal constraint on models that can arise from F-theory, which a | ≤ q | 12 All of these questions suggest that further exploration of the matter fields in U(1) Finally, more work is needed on the F-theory side. We do not at this time have any Further work is needed in several directions to explore these questions. It is important We have observed some constraints on certain standard F-theory constructions that ≤ − using explicit toric constructions [ useful data for such analyses. models in 6D supergravity andnear in future. F-theory may be a rich arena for further research in the construction of models withU(1) charges charges in F-theory modelsmay may be be, although ascharge discussed spectra in in the the previous presenceconstraints section, to of it an some additional of nonabeliandirection the gauge has F-theory factor, structure been relating studied made anomaly in, in e.g., [ [ the low-energy anomalylow-energy constraints. theory by All identifying inconsistencies theseterms in have constraints theories coefficients where might that the be are higher too derivative understood small in relative toclear the the explicit gauge construction couplings. of F-theory U(1) models with charges to determine whether some newthe consistency set constraint of may consistent belook U(1) identified for models that explicit would to arguments limit for aKodaira the finite constraint, inequalities space expected or of from the F-theory theories. stronger associated constraints It with the on would be certain classes interesting to of models that parallel physical meaning to these F-theoryNb constraints. For SU( is violated by some anomaly-consistentstanding low-energy of spectra. any We analogous do constraintgauge not from group. have F-theory a clear for under- models with only an abelian U(1) UV realization at present. are closely parallel to anomalymodels constraints with for small U(1) modelstions, with but small the charges close and relationship SU( between these formulae suggests that there may be some We have explored the anomaly constraintsU(1) on gauge 6D factor. Wesatisfy have the found anomaly that constraints therethese and are theories. infinite all families other Since ofthat known F-theory distinct quantum there can spectra consistency is that only conditions an give for infinite a swampland finite of set apparently of consistent distinct low-energy models spectra, with this no means not in tension withstates the of charges completeness +3 hypothesis. and reason that For these such spectra a should model, not be combining possible massless for consistent theories8 of quantum gravity. Conclusions which have no fields of charge JHEP06(2018)010 15 B , 04 06 Nucl. , D (1996) 45 6 JHEP , = 1 JHEP Phys. Lett. , , ]. ]. (2009) 050 B 388 DN ]. 6 12 ]. SPIRE SPIRE IN (1994) 1819 [ IN SPIRE ][ supergravity (2012) 093 35 ]. JHEP IN Adv. Theor. Math. Phys. SPIRE , , D ][ Phys. Lett. IN 01 6 ]. , [ SPIRE IN SPIRE JHEP ][ , IN ]. ][ hep-th/0509212 (1982) 324 supergravities J. Math. 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Morrison and W. Taylor, N. Seiberg and W. Taylor, V. Kumar and W. Taylor, M.B. Green, J.H. Schwarz and P.C. West, A. Sagnotti, V. Kumar and W. Taylor, D.R. Morrison and D.S. Park, D.S. Park, E. Witten, D.S. Park and W. Taylor, V. Sadov, J. Erler, V. Kumar, D.S. Park and W. Taylor, 6 [9] [6] [7] [8] [4] [5] [1] [2] [3] [14] [15] [16] [12] [13] [10] [11] Attribution License ( any medium, provided the original author(s) and source are credited. References We would like toDavid thank Morrison, AndrésCollinucci, Paul Jonathan Oehlmann, Heckman,for Nikhil helpful Craig Raghuram, discussions. Lawrie, Roberto The Valandro, Ling authors and Lin, are Yinan supported Wang Open by DOE Access. grant DE-SC00012567. Acknowledgments JHEP06(2018)010 ] ] 11 ]. , Nucl. Nucl. (2016) (2018) 1, 2, JHEP in ]. 04 SPIRE 05 ]. , , IN ]. ][ F-theory on all SPIRE SU(2) JHEP SPIRE ]. JHEP IN , ]. [ arXiv:1408.4808 SPIRE IN , Central Eur. J. 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