JHEP09(2020)147 or local Springer April 30, 2020 = 2 effective : August 25, 2020 : N Killing vector. September 22, 2020 : local Received do not receive quantum Accepted Published X ) or has no 3; in facts they are finite quotients m , U( Published for SISSA by https://doi.org/10.1007/JHEP09(2020)147 / = 2 ) ρ , m modelled on the structure theorem in Hodge theory. has infinite fundamental group. In all other cases the K¨ahler X . 3 . More precisely: its moduli space is a Shimura variety of ‘magic’ type. 2004.06929 structural criterion, The Authors. = 2 supergravity with a cubic pre-potential is necessarily a truncation of c Differential and Algebraic Geometry, Effective Field Theories, Supergravity

In the context of 4d effective gravity theories with 8 supersymmetries, we , sugra N [email protected] N : indeed it may be regarded as the more general version of Seiberg-Witten geometry Applied to Calabi-Yau 3-folds this result implies (assuming mirror symmetry) the As a first application of the new swampland criterion we show that a quantum- without rational curves have Picard number SISSA, via Bonomea 265, I-34100 Trieste, Italy E-mail: Open Access Article funded by SCOAP Models ArXiv ePrint: of Abelian varieties.corrections More if and generally: onlymoduli the if have K¨ahlermoduli instanton of corrections in (essentially) all possible degrees. Keywords: In all other casesquotient of a the quantum-consistent complex special hyperbolic K¨ahlergeometry space is SU(1 eithervalidity an of arithmetic the Oguiso-SakuraiX conjecture in Algebraic Geometry: all Calabi-Yau 3-folds rigid which holds both in the rigid and local cases. consistent a higher- Abstract: propose to unify, strenghten,statement: and the refine theIn several its swampland most conjectures abstract intotheories form (having a the a single quantum-consistent new UV swampland completion) criterion whether applies supersymmetry is to all 4d Sergio Cecotti Special geometry and the swampland JHEP09(2020)147 2 1 53 7 41 24 48 36 ’s 39 F 25 32 4 ∗ 61 = 2 QFT 28 tt 54 59 62 N 60 31 40 14 – i – 10 30 47 63 22 15 13 40 31 mirror symmetry

51 solitons and brane rigidity ∗ tt 51 7.3 Applications to Type IIA compactifications 8.1 The mathematical argument 6.1 Proof6.2 of dicothomy Existence of quantum-consistent symmetric geometries 7.1 Necessity of7.2 non-perturbative corrections Non-empty instanton-charge sectors 4.8 More on “no free parameter” 4.2 Intuitive view: 4.3 The structure4.4 theorem of algebro-geometric The VHS case4.5 of “motivic” special The K¨ahlergeometries structural4.6 swampland criterion A direct4.7 physical proof? Recovering the Ooguri-Vafa swampland statements 3.2 The quantum3.3 corrections cannot be Symmetric trivial rank-3 tube domains 4.1 The rough physical idea 2.1 Basic geometric2.2 structures Special K¨ahlersymmetries 3.1 A restricted class of special K¨ahlergeometries 1.2 Second alternative1.3 language: “motivic” Relation VHS with1.4 Seiberg-Witten theory in A survey1.5 of the first Organization applications of the paper theorem 1.1 First alternative language: BPS branes in 8 Answering the Oguiso-Sakurai question 7 Swampland criterion 5 Functional equations for quantum-consistent 6 Dicothomy 4 The swampland structural criterion 3 Warm-up: two weak results in the DG paradigm 2 Review of special K¨ahlergeometry Contents 1 Introduction and overview: swampland conjectures vs. VHS structure JHEP09(2020)147 ]. The story is 6 consistent from the viewpoint look ] it is like going from Differential Ge- 1 74 72 70 ] for nice reviews) aims to characterize the ]. Although these conjectures are arguably 5 5 branes , – – 1 – basic physical principle. 4 PDEs 2 ∗ [ ∗ tt tt 69 single ] (see [ 3 , 2 solution to ⇒ is expected to play a central role. which all consistent effective theories of gravity should obey; we refer superconformal swampland conjectures ]. 5 – 2 consistent but cannot be completed to a consistent QG is said to belong to the [ arithmetics looks To build the appropriate formalism in full generality requires a lot of foundational The reason why the swampland conjectures do not look “elegant” is that they are Up to now the swampland program has taken the form of a dozen or so (conjectural) The swampland program [ theorem a suitable ‘arithmetic’ language. work in which the usualloads field-theoretic of concepts new get uplifted additionallong, to structures technically subtler sophisticate, most ones and endowed of an with which enormous quantity arithmetic of in more nature work is [ required. formulated entirely in the language ofwords the old like QFT/DG “distance”, paradigm: “volume”, theyone are “geodesic”, expects expressed etc., using that that amore is, language the appropriate in arithmetic for the aspects. the strict Itthe gravitational DG is several paradigm natural jargon, conjectures should to in while emphasize think one that, more if fundamental we swampland wish principle, to we unify must and state strengthen it in to them asthe the deepest principles inbit physics, unsatisfactory: indeed they in consistindependent all of nor science, several related their disjoint by currentunify statements, a formulation the which clear several looks are web conjectures of neither a into mutual logically a implications. Ideally, one would like to very sparse zero-measure subset ofwhich the naively consistent models.swampland An effective field theory necessary conditions in which effective field theories which describeinside the the low-energy much limit larger of classof consistent of the quantum effective gravities traditional theories field-theoretic which paradigm. The consistent QG effective theories form a Passing from Quantum Field Theorychange (QFT) of to paradigm: Quantum in Gravityometry the (QG) (DG) words requires to of a the Cumrun radical much Vafa deeperphysics [ Number — Theory. such The as fundamentalin the principles of QG, concept quantum and of the “symmetry” mathematical — take formulation new of (and the much theory subtler) requires meanings totally new foundations 1 Introduction and overview: swampland conjectures vs. VHS structure A Pluri-harmonic B Arithmetics of C Symmetric rigid special K¨ahlermanifolds 9 The case of Picard number 1 JHEP09(2020)147 ): ]. 10 (the (1.1) (1.2) (1.3) 4.6 S ]. . 13 2 and S 1.3 — is required to 2 = 2 supergravity. At = 2 supersymmetry. In N N S ∈ . C ∗ rigid tt ∼ → , 3 s ), in particular the distance conjecture , s 0 q 3 s ]. The swampland criterion we propose R + : q s 9 4.7 having , s 8 ⊕ R tr eff 2 s ⊆ R must be a family of finite-dimensional, commutative, L 0 S q s – 2 – ∈ ] for the universal deformation of an abstract 2d ⊕ R s , } that is, 1 s ·R 1 12 s · 3 q s {R ] (see section geometry with special K¨ahlergeometry is reviewed in C R ⊕ R 3 ∗ , 0 s = 2 tt R 0 s R is parametrized by the universal deformation space = -algebras, equations [ s ]). In this special case the swampland program reduces to the S C ∗ 7 ∈ R : in both cases a quantum-consistent effective theory is described ) which — in addition to the usual properties s tt S } s towers of light states at infinity, while in the rigid supersymmetry ∈ susy s {R ( -algebras with unity which satisfies the axioms of a Frobenius manifold [ ] that the special K¨ahlergeometries are in one-to-one correspondence C s = 2 R 12 , infinite local-graded N Which special K¨ahlergeometries belong to the swampland? 11 [ denotes the Frobenius trace map, i.e. the TFT one-point function on the sphere 1 argument shows that at most finitely many states may become massless in any s Question 1 may be re-formulated in two well-known alternative languages whichThe al- second answer gives a necessary criterion for quantum consistency which makes Luckily enough, however, there is one important special case where we can dispense same In the math literatureBy this the statement usual is sometimes properties known we as mean the that (generalized)Here Simpson tr theorem [ = 2 QFT one can prove rigorously the validity of this criterion (see sections 1 2 3 the identification of local-graded section 2.1 below (some technical detail being deferred to appendix A). associative Frobenius The family of rings “moduli” space, a.k.a. the 2d conformal manifold). For the convenience of the reader, with the solutions to the (2,2) chiral ring be a family of the limit (cfr. section 1.3). 1.1 First alternativeOne language: shows BPS branes in rigid and local by such a higher Seiberg-Wittendramatically: geometry. In in spite the ofstandard swampland this, gravity conjectures the [ case local the andwhich rigid requires proposed cases criterion differ implies the validity of all the sense also for low-energyN effective theories it turns out toglobally well-defined be Seiberg-Witten equivalent geometry to [ may requiring be seen that as the the more low-energy general physics version is of Seiberg-Witten described geometry by which applies a both to Question 1. ready take careanswers of to the question 1. relevant arithmetic aspects. This will give us two (equivalent) with new foundations:more namely specifically, the the 4d vector-multipletthe sector effective of level theories an of invariant ungauged underK¨ahlergeometry the (see 4d 8 e.g. traditional supercharges, [ QFT/DGsimple paradigm question: such a sector is described by a special JHEP09(2020)147 ] 4 5 ]. ] 17 ] to , = 3, 14 (1.4) 16 c – 14 14 sugra ). This is semi- 14 s , 1.3 ) have the R = 2 12 )–( 1.4 )[ ]. In principle, geometries are N 1 1.1 ∗ P 14 tt ∈ ) which implies the geometries for semi- ζ ] studies this problem -theoretical invariants 1.4 ∗ , and then specializing 14 tt K S ∈ ] to arbitrary, that is not -models with target space s ) for arbitrary chiral rings, σ } 14 s 1.4 {R ]. The BPS branes belong to a belong to swampland satisfies the . 17 [ diophantine conditions of ref. [ 6 ∗ ) to the linear problem ( not tt ζ ]. ) = 0 ζ geometries. Ref. [ 11 [ ∗ Ψ(  tt X ) ζ ( ). – 3 – ∇ ], and then use Picard-Lefschetz theory [ 2.24 + 17 sub-algebra which leaves the brane invariant. ) is semi-simple instead of local-graded (for a 2d (2,2) physical ζ ( s ∇ R susy  ] whose numerical homology and generalization of the PDEs are the consistency conditions of a system of linear ]. Nowhere in this argument one uses the fact that 18 ∗ , 14 tt = 2 swampland program becomes a special instance of the more 17 labels the geometry arises from a physical situation — its brane amplitudes such solutions which is equivalent to the space of all special K¨ahler ζ , which coincide with the special geometry of the ∗ N tt all X geometry inherits an arithmetic structure. The precursor papers [ local-graded ∗ PDEs for local-graded chiral rings which “arise from physics” out of the tt equations can be written in the linear form ( language the swampland program asks for the characterization of the solu- ∗ geometry is physical, the solutions Ψ( ∗ ∗ tt A special K¨ahlergeometry which does ∗ tt tt tt ) is written more explicitly in eq. ( 1.4 To understand the nature of the required “straightforward” generalization, let us recall So rephrased, the In the ) describe objects of a BPS brane category with a finitely generated Grothendieck group We stress that this is still aEq. necessary ( condition, notThe a twistor parameter sufficient one, albeit it is expected to “come ζ 4 5 6 BPS branes withrelate Lefschetz thimbles them [ in a simple way toclose” the to be BPS sufficient. spectrum and wall-crossing phenomena. The simple: the and — whenever the Ψ( from which the focused on semi-simple chiral rings for just one reason: in that case, one may identify half- When the physical interpretation of half-BPSlinear brane triangle amplitudes category [ induce a canonical integraldiophantine structure conditions on of [ the space of solutions to ( simple chiral rings.differential The equations (depending on an additional twistor parameter Answer 1. “straightforward” the physical origin of the arithmetic conditions on the physical one should be ableby to get a the “straightforward” desired generalizationnecessarily arithmetic of formulation semi-simple, the of universal analysis the families swampland ofthe conditions general of ref. answer chiral [ to rings theleads particular case to of a local-graded first rings answer of to the question form 1: ( obtained by compactification of Type IIB on general problem of characterizing the in the opposite situation inQFT which this means gappedcharacterized versus by diophantine conformal). equations, that The is, physical by arithmetic semi-simple conditions [ tions to the much bigger space of geometries. The physical subset containsand the in special geometries particular of the (2,2)a special SCFTs Calabi-Yau with geometry ˆ 3-fold of 2d supersymmetric JHEP09(2020)147 ]. Y = 2 20 Like- , N 7 19 language to ∗ necessary con- tt mere amplitudes turns out to ∗ tt = 2 (ungauged) supergravity, Calabi-Yau operators “as describing N ) satisfy the arithmetic conditions ζ abstract is local-graded, since there are no non- branes carry the full-fledged categorical conditions for the existence of a quantum s ∗ R tt ” even when they are not the Picard-Fuchs operators 1 where we use the equivalent P -theoretic aspects of the relevant linear triangle – 4 – K 4.2 sufficient = 2 case the swampland conditions may be easily ). Y ( in the sense that the precise boundaries of the class of N brane amplitudes Ψ( D ∗ tt open, ] Van Straten proposes to consider viewpoint. However there is a second alternative language which 21 ∗ = 2 quantum gravity must be a 4d tt ) we compare the problem of describing the set of consistent 4d 5 N a rank four Calabi-Yau motive over weaker than requiring that they are physical brane amplitudes. The arith- Requiring that the (page 1 At the naive level of the DG paradigm, on the physical side we know that the effective In conclusion, in the special For instance, in ref. [ 7 a priori theory of a 4d something like of an actual one-parameter family of 3-CY. as, for instance, consistenttion truncations problems are of somehow consistent“natural” quantum objects gravities. to be Bothparticular, called classifica- we “physical” do (resp. not“Quantum have “geometric”) Gravity” an is is explicit not supposed set fully to of understood. be. axioms which In define in a precise way what a varieties) but more generally ofuli “algebro-geometric of objects”. Abelian Indeed varieties experience hasto with shown enlarge mod- that, the in class orderThe of to appropriate obtain geometric class a of objects “objects” deep ofwise, to and interest consider if nice is beyond one theory, some one the wishes sort has properties, actual a of “Calabi-Yau one manifolds classification motives”. should of [ accept consistent some “Quantum more Gravities” general with animals nice than functorial the strict QG theories such 1.2 Second alternative language:In table “motivic” VHS quantum gravities with the mathematicaljects” problem of of describing “Calabi-Yau all type”. “algebro-geometric ob- Notice that we do not speak of Calabi-Yau manifolds (or ditions. It is conceivable thatgravity one UV gets completion bystructure requiring and that not merely the its numericalpletion avatar. of Morally a speaking, low-energy the effective procedureout theory of of UV looks its com- akin derived to sheaf reconstructing category an algebraic variety Remark. is metic conditions essentially see onlycategory the of branes. The arithmetic conditions are then expected to be see appendix B. guessed from the makes the story even morethis straightforward. second In viewpoint, the except restgive in of an this section intuitive interpretation note of we the shall swampland mostly criterion adopt we are proposing. trivial BPS states inwhen a the (2,2) corresponding SCFT; special neverthelesssome K¨ahlergeometry nice the arises linear half-BPS triangle from category, branes so physics,arithmetic are that they properties. their still are special In there K¨ahlergeometry inherits objects facts, interesting and, be the in much arithmetics richer, of nicer local-graded and deeper than the Picard-Lefshetz one for semi-simple chiral rings, relation with the BPS spectrum is lost when JHEP09(2020)147 . ), V V S ⊂ ⊂ 1.5 ? (1.6) ≡ X ≡ ⇐ QG Mot M M \ = 1 means that the 0 , V 3 do not describe X h for the extremely V V theorem ]. ⊂ “motives”? 23 , , cannot arise as low-energy )] QG 22 ish ] -VHS in m ], so that the problems in the V . (1.5) M R S U( 32 CY- , 23 C the structure , which VHS arise from up to deformation type × Algebraic Geometry 30 means that the two are known to be 22 – ) the monodromy group. The map families of actual CY 3-folds dim ≡ Z CY algebro-geometric “objects” (VHS) of weight 3 with -VHS with the Hodge numbers ( 24 real variations of Hodge structure , ≡ R [U(1) = 2 special K¨ahlergeometries.  m + 2 ) R N = m ≡ , = 2 supersymmetric amounts to saying that 1 , we know that the DG description of a defor- , 2 N – 5 – + 2 1 h Sp(2 is m sugra = 2 = 2 supergravities in ⊂ L N N Sp(2 means that the statement in the second column implies the = 2 supersymmetry [  ]. One shows that these differential relations are exactly = 1 and Γ ⇐ 0 N , 30 3 conjectures , → be physically realized (the complementary set h the swampland S 28 = 2 ungauged – : Quantum Gravity p “quantum gravities” consistent 4d can N 24 Type II compactifications are strictly equivalent at the naive level of the DG paradigm. We 4d special K¨ahlergeometries 1 which special geometry cancompleted be to a consistent QG? means that the two are conjectured to be equivalent, while ] of pure weight 3 and Hodge numbers = 2 swampland). Likewise, almost all ? ≡ 32 , N is the set of consistent-looking -VHS which do arise from algebraic geometry. In particular, a variation 30 R V – 24 is the “moduli” space and Γ for the set of abstract weight-3 polarized . The comparison of two classification problems. Symbols in the right margin describe the S -VHS is encoded in the Griffiths period map [ V R It turns out that almost all 4d Passing to the second column of table classification problem DG description naive paradigm: what is known examples new paradigm: deep question necessary conditions -VHS) [ is required to satisfy a set of differential relations, known as the Griffiths infinitesimal R limits of consistent theoriesminuscule of subset quantum which gravity.being We the write vast actual families of algebro-geometric objects.of Again “motivic” we have a minuscule subset period relations (IPR) [ equivalent to the ones defining two columns of table write equivalently where p mation family of CY( “objects” is given by a polarized real variation of Hodge structure The whose vector-multiplet sector is describedSaying in that DG the language effective by Lagrangian aits special couplings, K¨ahlermanifold seen asrelations functions which of are the the scalar defining fields property (“moduli”), of special satisfy geometry a [ set of differential Table 1 logical relation between thestrictly two equivalent, boxes in thetwo same are row: trivially theones same; in finally, the first column. JHEP09(2020)147 . 2 of . τ QG ], or πi/g Q : this Mot M ]: this 29 suffices + 4 M = 30 π – 2 (specialized 26 θ/ Mot theorem . However there = p M τ ). The information s is a single point, and the of (global) variations , yields an element of X S S conjectures ∈ s ], section III.A of [ } = 4 version of the swampland s , i.e. it should be a (pure, 28 N X R below following the two nice { 4 should satisfy in order to belong not V Q ∈ v structure theorem ]) do not restrict the period map ]. 3 is a fundamental problem in (respectively) 33 ], section II.B of [ . Applying the 3d is promoted to a quaternionic-K¨ahlermanifold ]. In the Algebraic Geometry column we have V τ . In this last row the two columns look quite 5 S – 6 – 27 – of “motivic” VHS: they answer in the negative, 2 Mot an element )[ M Mot inside (by compactifying Type IIB on 8 M = 2 theory with constant graviphoton coupling Mot QG look very much the same when restricted to their first = 4 theory ] the authors ask whether this necessary condition sugra N 1 M M N should be defined over 28 , thus “almost” completing the algebro-geometric analogue of = 2 are God-given, extremely sparse, subsets of the same naive space and Mot N in the upper half-plane may be realized as periods of harmonic (3,0)-forms on Mot M τ ]. We review the theorem in section Mot QG M or, respectively, 30 M M necessary conditions ] whose geometry depends on QG (central to the swampland program [ necessary condition for a VHS to belong to the God-given subset and 34 M ]. S QG . In the resulting 3d 28 , 1 = 2 swampland program. M S 27 proven statement which, moreover, has the logical status of a mathematical . In the column of Quantum Gravity the necessary conditions take the form of a N as well as an element of 3-CY. Roughly speaking, the problem is that in this case the moduli space It is an established fact that all VHS arisingIn section from III.B.7 Algebraic of ref. Geometry [ enjoy the First of all we have a large supply of explicit examples in the form of moduli spaces of The two columns of table Characterizing Both It is natural to ask whether there is any simple relation between these two special Translated in the physical language, the basic source of non-sufficiency is that the structure theorem 8 Mot , and both are expected to be characterized by subtle arithmetic-like properties. single does not say which points rigid global aspects of are good reasons to believeinadequacy: that in one the compactifies physical the contexton 4d we a may strengthen circle the statementreal and dimension get rid 4 of [ this to completely characterize thebut set their general feelingactually is determining that the structurethe theorem 4d comes “close” to the holy Grail of surveys [ properties stated in theyields structure a theorem. In other words, the statement of the theorem to the case of ungauged a fundamental result is known under theof name Hodge of structure, the seetheorem e.g. 15.3.14 section in IV [ of [ five rows. The last rowproblems: describes a the set second of pieceto of the “known” sub-set information aboutdifferent the deep dozen or so conjectural statements: they are the usual swampland known Calabi-Yau 3-folds. AM deformation family of 3-CYs, we get from such explicit examples tautologically corroborates the idea that subsets. The meta-conjecturewe is are that willing they to indeed callassume a coincide the “consistent (for string quantum lamppost a gravity”). principle suitable This (SLP) choice will [ of follow, for what Theoretical instance, Physics if and we Algebraic Geometry.tal problems? What is known about these two fundamen- polarized) variation of Hodgeis structure a in first Deligne’s hint sense of (a non-trivial VHS arithmetic for structures short) enteringV in [ the game. of Hodge structure in JHEP09(2020)147 )) be (1.8) 1.6 false, M e S ? \ 1 ) Z ’s. τ (cfr. eq. ( ) ) = Γ h, This is hardly a for more). The p f 9 M M ( ( e p p Sp(2 4.6 ) (1.9) ≡ ≡ ⊂ . h e S S Γ U( . We have a commutative     4.2 / G ) its smooth simply-connected R f M h, stand for canonical projections while / / = 2 QFT where = 2 supersymmetry as well. Here Sp(2 A (global) special K¨ahlergeometry N . ∈ N ) suggests to unify and refine the several ) c 1 ) h X h = 2 chiral superfields containing the field ( is the holomorphic pre-potential. Let U( F / U( F b take value and ) N / ∂ – 7 – ) a R a   ) + hypermultiplets + h.c. (1.7) R ∂ X a h, h, X 10 7→ ( 4 * ) Sp(2 h \ Sp(2 θ F / ] used the general properties of Calabi-Yau moduli spaces 2 Γ should yield some useful constraint on the allowed / 3 ,X d τ Q Z ··· , ) are restricted = 2 low-energy effective QFT has the form = 1 for some discrete group of isometries X N L f M :( e represent canonical inclusions. p e   , . . . , h p e p p / G\ 2   Abelian gauge fields and , e S S = / / formal h  = 1 M throughout this paper double-headed arrows (The structural swampland conjecture) a ( a 11 X Answer 2 and answer 1 are mutually consistent, see section The comparison of the two columns of table It is easy to see that a (global) special geometry which satisfies the VHS structure What is the logical relation between the two boxes in the last row of table   For some conjecture this actuallyWe holds use after the some same minorNotation: technical symbol refinement to of denote the the original superfield statement. and its scalar first component. f M M 9 10 11 conditions to the 3d moduli space hook-tail arrows whose covering map where strengths of the the Coulomb branch where the scalars cover, so that diagram The structural swampland criterionwe holds sketch the in story rigid Lagrangian without of entering a in technical details (see section satisfies the VHS structure theorem. 1.3 Relation with Seiberg-Witten theory in conjectures combined. swampland conjectures into the single statement: Answer 2 belongs to the swampland unless its underlying Griffiths period map theorem automatically satisfies all thesurprise applicable swampland since conjectures. Ooguri and Vafaas [ motivating examples fori.e. their the conjectures. structure theorem However is the actually inverse a implication stronger is constraint than the several swampland JHEP09(2020)147 1 M (1.10) for . Again ∞ h (the locus = 2 QFT, Y = , which is a \ satisfies the ]. D which define d N 1 , 4 p 35 0 identical, they e p M R h ] while nothing of = = 3 d ] suggests that this 0 4 , as the period map of ) which satisfies the 1 h 42 R p h – Although the structural U( 37 / = 2 QFT comes equipped ) = 2 QFT see [ in principle ]. R N N . 43 h, case the structural swampland . Thus we may rephrase fact are ∞ = 2 gravity an infinite tower of = 2 gravity theory the volume of M Y Sp(2 N + N \ of its chiral ring. The (global) smooth = 2 QFT. Γ sugra D sugra d of the discriminant divisor 4 = N d → 4 R = 2 ), and then the period map R M = 2 effective theory necessarily arises from a : N ]). Then we have the following p 1.10 – 8 – N Hence -algebra. Then a 4d 27 Y,Y C 12 protected subfactor, the chiral ring \ M = = 2 QFT in string theory [ susy in quantum-consistent ]. Experience in low dimension [ N M ) satisfies the properties of the monodromy group for a VHS in a quantum-consistent = 2 QFT and 9 , Z = 2 gravity versus 8 N h, N ] or section IV of [ Sp(2 36 is the complement in Spec an effective divisor. ⊂ = 2 low-energy effective field theory which has a quantum-consistent UV M ∞ Y N regimes. = 2 QFT contains a distance conjecture: volume conjecture: A 4d all N states becomes light athappen infinity in in a scalars’ UV-complete space, QFTin which whereas has nothing finitely-many of effective that degrees of sort freedom may the scalars’ space is finite,as while the this example is of not free-field true theory in shows; quantum-consistent the the When the above necessary condition is satisfied, we may realize A necessary condition follows directly from QFT first principles. The operator algebra For explicit examples of these algebraic-geometric gymnastics in 4d B) A) 12 swampland condition in the two physical contexts are: Quantum-consistent swampland criterions for lead to very different physicalcondition consequences. implies the In validity of the the Ooguri-Vafathat swampland sort conjectures applies [ in QFT. The most striking differences in the implications of the structural by saying that aSeiberg-Witten quantum-consistent geometry 4d [ condition is almost sufficient. Roughly,effective the theory, satisfying idea the is structural that condition,geometry the is UV to obtained completion engineer by of the using a full its low-energy Seiberg-Witten Fact 1. completion is described byVHS a structure theorem. period map an algebraic family of Abelian varieties parametrized by Then the group Γ over a quasi-projective baseVHS of structure the theorem. forme.g. (For ( succinct sections 5–7 surveys of of [ the underlying mathematical facts, see with an affine variety, namely theCoulomb spectrum branch Spec where additional degrees ofprojective freedom and become massless). Write Spec we may pose alow-energy swampland effective Lagrangians question: having a describe UV the completion sub-set which is of aof formal fully a VHS consistent 4d QFT. finitely-generated, commutative, unital satisfies all the axioms of a weight-1 local VHS with Hodge numbers JHEP09(2020)147 has but = 1 snc j ¯ e 3 i which (1.12) (1.11) S , 0 j ¯ G ) of the i h ≡ h G ), whereas of the = 1 f , i U( M may be locally 0 / 0 Y , ) h 3 induces a nice f M 13 Z h = j ¯ i h, 0 , and K 1 . The (possibly sin- e h S ). When we approach f Sp(2 M satisfies , as computed with the i e → S γ 1.11 1 QFT 3 SUGRA \ ] e by the Torelli property, and ( S Γ 0 so that 30 e S ≤ being the Planck mass) so that ≡ = 3 an infinite tower of states 6≡ i (for details see section 2.1(VII)) j ≡ ¯ p QFT SUGRA i k is a point). i S k 14 R M e ≤ f S M ( j ¯ is i 2 but not necessarily on the (covering) R − f p M with e . S + i M j ¯ Y i and references therein). The same analysis G → ∞ ) below plays a crucial role. p 4.7 is a local immersion (the local Torelli theorem), – 9 – = 0 with 1 + 3) M case we have . If we insert back the Newton constant, the first e p j ¯ j ¯ 1.12 i i +1 m i R ( G k
( i γ = sugra j ¯ i (see section log i be the monodromy around a prime component K Y i γ . In the . By the hard monodromy theorem [ into the Siegel upper half-space. (In many examples the VHS has weight 3 (non-zero Hodge numbers 6= 0 f j M just factorizes through the immersion = 1, in the gravity case either an infinite tower or just finitely many will carry a relative factor i e Y S k i j ¯ e p ). Let
j i k i 2 ) we see that being at infinite distance in the physical metric , γ G ; in showing this eq. ( 1 P i h sugra is the Ricci tensor of Y the covering period map = log 1.12 j ¯ = + 3) i Y 1 , R m = 2 2 = 2 QFT the VHS has weight 1 (non-zero Hodge numbers h is a genuine metric K¨ahler on the covering scalars’ space (to be defined in section 2.1(VII) below) and the special K¨ahlermetric N sugra N j j ¯ ¯ i i where term ( the two formulae agree in the limit gular in the rigid case) pulled back metric on K enters in the scalars’positive-definite kinetic metric on terms. theCoulomb Torelli Indeed, branch space the HodgeK metric while in QFT “Torelli” space this does not hold, e.g., in the free theory where a divisor with states may becomeHodge massless, structure depending along onshows the that details in of QFTalong at the any most degenerating finitely mixed many degrees of freedom may become massless As we approach thebecomes support massless. of This a cannot happen divisor in QFT in view of ( in and divisor In In consequence of 2. we have a very different relation between the VHS Hodge metric In From eq. ( Here we are glossing over some technicality discussedBy in abuse the of main notation text we use below. the In same particular, symbol we for have the metric and its pull-back, partly because in the 2. 3. 1. 13 14 versus SUGRA. In particular, the volume of theassumed (without space loss) that the monodromy group Γrigid is case neat. we are mainlyidentified. interested to special geometries with det quite different meanings from the viewpoint of the intrinsic Hodge metric geometry in QFT These dramatic differences arise from three simple elements: JHEP09(2020)147 ]. ⇒ has 44 . As (1.13) S almost =8 N S L ). The (naive) (with respect to e Killing vector S S ( globally defined on Iso local for a simply-connected S. ⊂ e S C never Killing vectors on \ the couplings in the effective is a closed sub-group = Γ around points where the Ricci be a special K¨ahlergeometry below. local = dim e S S 6= 0; then either the rigid special e S e S m j ¯ 2.2 \ i , is also finite only in SUGRA (see j ¯ R i , and this fact determines and has a non-zero G is its simply-connected cover and Γ its = Γ 7(7) point in the smooth, simply-connected, S f e e M S ) are our do constrain S e S ( one ) where symmetries in the strict sense of the term [ does not belong to the swampland the group iso R , – 10 – S no ⊆ ) + 2 e Killing vectors of the scalars’ space S ( m = 8 effective theory whose scalars’ manifold 6= 0. sym local j ¯ Sp(2 i N ) — if non-trivial — carry interesting arithmetic structures. R ⊂ e S and a discrete group of isometries Γ in both cases, but this implies that the volume of the scalars’ ) ( e S e In QG there are ( S sym ) of the covering special geometry Assume answer 2. Let e S finite . ( Sym of the Coulomb branch where det A first consequence of the structural conjecture is the following is symmetric or has no Killing vector. The first possibility is ruled out generate symmetries) but they , is f = 2 QFT, since a symmetric rigid special K¨ahlermanifold is necessarily Sym f M j ¯ M i and in particular the scalars’ metric. For instance, if we compactify Type N K not = 2 context, the scalars’ space has the form ]. in both cases. Assume there is L , as computed with the physical metric N for details). e 45 S M (Dichotomy) 4.7 ) and its algebra e S In the To state the first one, we need some generality about “symmetry” in the QG paradigm. Notice that in QFT we may locally identify Despite the different physical implications of the structural swampland condition in ( = 2 effective theories should satisfy (in addition to the usual swampland conditions). is locally symmetric. More precisely: (so they do The dichotomy. Fact 2 which does not belongmonodromy to the group swampland, (a.k.a. where dualityS group). Then: The elements of itswe Lie shall algebra see, whenSym the special geometry uniquely [ special K¨ahlermanifold symmetry group the metric in their kineticS terms). These local KillingLagrangian vectors are II on a six-torus133 local we Killing get vectors a forming 4d the Lie algebra N For more precise definitions and statements, see section “Symmetry” in QG. However we can still consider the flat corresponding to a free QFT (see appendix1.4 C). A survey ofFrom the the structural conjecture first one can applications easily extract some novel property that all consistent QFT and SUGRA,the its dichotomy general introduced consequences inmanifold hold the equally next in subsectionirreducible holds both cover for cases. aK¨ahlermanifold quantum-consistent For Torelli instance, for a non-free manifold section tensor is non-singular i.e. det Hodge metric JHEP09(2020)147 G 15 . ⊂ with (1.14) e K S a maximal ) ]: G R with 15) 51 , , − ⊂ 7( Q is quadratic and K E ]. The fact that ∈ G/K F 2 { 48 ) ≡ [ IJ 0 ]: all quotients of the e S X ( ,A subgroup; 50 ]: ) Sp(6 χ ijk ) is one of the Lie groups 46 R 3 d e , S ) (12) ( ∗ π (3) is a ‘magical’ Lie group, i.e. one of their quotients belong to the Iso i ζ G 2(2 quotient of a non-compact Hermitian subgroup (in the strict algebraic-group }| if Suppose we have a pre-potential ) of the covering special geometry loop all where arithmetic + e S . modular curves ( J 1. The pre-potential X I , ‘magic’ (cubic) ) SL(2 Sym do not correspond to ‘classical’ Shimura ) { ≥ X ) is an 0 arithmetic , k 2 arithmetic 1 e S IJ ( − an A ~ X/X ‘magical’ type · Iso SO(2 3) SO 1 2 is a Shimura variety of rank-1 or ‘magic’ type λ G ~ z m , – 11 – × πi ⊂ ⊂ S + 2 = is of ) e { U(3 k R ( k }| 6= 0. They and , 3 below). X of the form . j ) is a non-compact semi-simple real Lie group, ] we mean an Li G/K 2 0 e S \ , and Γ X λ ~ G 3.2 ∞ ( SL(2 i ) has rank-1 as a symmetric space, but it has “rank-3” as a c X 49 H is bad news for the author of ref. [ ), and Γ , }| 6 R ρ + X e instanton corrections S × , 2 Z ( classical sym 47 ∈ , ijk X where z λ ~ if and only if of symmetric special K¨ahlermanifolds [ d Iso 20 , + − z G 19 6= 0 ) = G/K [ \ ) e ) = S ]: indeed they are the very paradise of arithmetics. Their simplest i e ( ) S ( ] as the rest of ‘magic’ special geometries. 49 ,X , , m 0 Sym 32 sym , ) = 0, that is, the symmetry group X is Hermitian symmetric, its isometry group 47 (Completeness of instanton corrections) , ( e S 2 SU(1 e ( S F Shimura varieties are the geometries with the richer and most interesting arith- 20 has important corollaries. The first one is in facts a confirmation in the special discrete. , rank-1 (quadratic) sym 2 Shimura variety is 19 . Isometry groups e S either in table or Fact There are plenty of non-symmetric simply-connected special K¨ahlergeometries By a • • = 2 case of a more general and profound prediction by the authors of ref. [ 15 symmetric manifold, i.e. Γ compact subgroup of thesense). form U(1) We say thatthe a groups Shimura in variety Γ the right hand side of table N Corollary 1 with an asymptotic expansion at a non-trivial Killingswampland. vector, In particular, fact several infinite series of homogeneous specialbeautiful K¨ahlergeometries constructed and there, elegant, however belongat to the the naive swampland! DG level: (This see negative section result is already obvious (and most classical) examplesShimura are varieties the arise (non-compact) sometics naturally in QG. provides Note further thatvarieties, last evidence that two for groups is, in the they table are relevance not of moduli arith- spaces of Abelian motives. In other words: Remark. metics [ special geometry [ Table 2 a maximal compact subgroup.respectively cubic. One has ‘Magic’ SL(2 JHEP09(2020)147 is the . Then k z j cub z F ). i . z Z large volume) X = 4 supergrav- = 8 supergrav- 2 supergravity. ∈ ijk parametrize the ≡ d 0 > N N ijk are integers (being d N /X = 4 to configurations = 8 to configurations i 2) 2d supersymmetry. ijk , and , d X Z N N (2 pre-potential ≡ ∈ > i . z ) IJ 2 A p, q = 0, χ strictly cubic 16 ) describes the quantum K¨ahlermoduli of ; is the Euler characteristic of λ = 3). Then the “instanton corrections” in ~ χ depends on the electro-magnetic duality frame. What c 1.14 – 12 – IJ A is a consistent truncation of a ]) and is a consistent truncation of is a consistent truncation of 54 duality frame. describes Type IIA compactified on a Calabi-Yau 3-fold sugra ] as an extension of the swampland conjectures. We make the absence of quantum corrections in the world-sheet theory ] we get a characterization of the QG cubic scalar manifolds F sugra sugra : the first term is the world-sheet classical ( -model (with ˆ be a special K¨ahlermanifold, consistent with our swampland 51 some σ X 20 e , S = 2 = 2 = 2 \ ]. , the “Euler characteristic” 19 6 N λ ~ N N E = Γ × S vanishes in ]. In this specific geometric set-up the coefficients if and only if ) we labelled the various terms according to their origin in the special IJ Suppose the pre-potential ( 6= 0 for essentially all possible Let 53 A [U(1) , λ ~ / of the following possibility applies: 1.14 c 52 = 0 for all The (possibly reducible) rank-3 real Jordan algebras are in 1-to-1 correspondence 25) λ ~ − is a Shimura variety which is an arithmetic quotient of Cartan’s domain c ) vanish is a geodesic submanifold of the scalar’s space of some consistent is a geodesic submanifold of the scalar’s space of some consistent 7( E S ity. That is, the invariant under a certain discrete symmetry group; S S ity. That is, the invariant under a certain discrete symmetry group; determinant form of a rank-3 real Jordan algebra (in particular, or either 1.14 Using results from [ Roughly speaking corollary 1 says that we must have all possible world-sheet instanton In eq. ( The statement requires specifications since • • • with well-known moduli spaces of nice algebro-geometric “objects”: (i) . In this particular application, the vector-multiplet scalars 16 (ii) S we mean is that at least one eq. ( is implied by the non-renormalization theoremIn of this a case, higher ( also the loop correctionsCorollary must 3. vanish. structural criterion, which is described locally by a This fact was predictedthe in statement [ more precise in the nextCorollary three 2. corollaries. a 2d (2,2) superconformal quantum K¨ahlermoduli of answer, the second one the perturbativeinstantons loop [ corrections, and theintersection last term indices the in world-sheet cohomology [ corrections unless our with the cubic ‘magic’ groups in the right-hand side of table case where the pre-potential X Remark. which does not belong to the swampland. Then JHEP09(2020)147 = 2 first (1.15) not in N S C of a F special K¨ahler . Suppose that ]. ∨ 34 X symmetric is the moduli space of some ]. In section 5 we show how S 3 . ∨ X structural swampland criterion . 27. Then has important implications in Algebraic , 2 and 27 is a finite quotient of an Abelian variety; in , 6= 15 Fact 15 X S and then systematically using the mathematical – 13 – , gets cubic, i.e. the special geometry is “orphan”. C 9 ∗ ]. We stress that under the stated assumption, the , F tt 6 55 , = 2 or 3. 1 ρ and dim where 3 S be a Calabi-Yau 3-fold which admits a mirror geometry and Griffiths’ theory of variations of Hodge structures. For X ∗ tt as in corollary Let S which are locally symmetric and locally irreducible. They have complex has Picard number ), we conclude that either instanton corrections are present or there is no S X 1.15 does not belong to the swampland and is locally irreducible with dim S The assumption of the existence of a mirror may be replaced by some much weaker Note that, up to finite covers, there are just five finite-volume, cubic, special K¨ahler does not contain rational curves. Then of the present paper:7 describes the the dichotomy behaviour forinfinity of a quantum-consistent a MUM special quantum-consistent (maximal special geometries. unipotent geometry monodromy)IIA point, when Section compactifications and we on presents approach applications Calabi-Yau at question to 3-folds. Type and In argue section that 8 the we answer discuss is the positive. Oguiso-Sakurai In section 9 we briefly comment on the geometries for later use. Inat section an 4 intuitive level we in introduceframework the our of language of VHS. Thenvalidity of we the discuss original how Ooguri-Vafathe this swampland criterion conjectures statement may [ implies, —supergravity in in which does suitable principle not — belong senses, to be the the swampland. used In to section compute 6 we the prove the pre-potential main result its relations with later convenience we collectsection here 3 some we facts present aboutwhich (for “symmetries” go comparative in in purposes) special twoIn the geometry. purely subsection same In differential-geometric 3.3 directions results we as collect our some main facts about conclusions, the but geometry are of significantly weaker. theorem applied to the universal deformation space of 1.5 Organization ofThe the rest paper of this paper isfrom organized as a follows. viewpoint In section convenient 2 for we review studying special K¨ahlergeometry its global and arithmetic aspects, emphasizing technical requirement. Modulo this aspect,posed corollary by 2 Oguiso answers and in Sakurai theresult in is positive mathematically ref. the fully question [ rigorous, since it is a direct consequence of the VHS structure Applications to Algebraic Geometry. Geometry. For instance: Corollary 5. X particular So if the list ( asymptotic limit in the space family of Abelian motives. Typically,untwisted they sector may moduli be of realized Type as II/heterotic the special compactifications geometries on of tori the [ manifolds dimensions Corollary 4. JHEP09(2020)147 is ∼ = ]. ) ) and 58 a a ( ( m m Q PDEs L H special 3. The of type ∗ , . We see 2 tt 17 , Z 19 , 2 the domain Q , = 1 , integral a R = 1 , , ). We stress that ) a C a Q ( m ( ] of the real Lie algebra m /H  )  60 G ] of the complex parabolic R ( 61 18 its complex dimension. We m ) have an intrinsic physical ks ks ks G R ] for the particular case of Type II ◦ ◦ • ◦ m , 59 ≡ will denote a (integral) special ) + 2 a S ( m is compact so for ◦ ◦ ◦ m ) D a is one of the rings ( m provided by H K ]; a readable introduction is chapter VII of [ Q 57 ··· ··· , ··· which specifies the underlying complex Lie group 2d (2,2) models are characterized by special -VHS. 56 +1 R is pluri-harmonic. In appendix B we show that – 14 – ) is a maximal arithmetic subgroup with respect ) where m R w C K ◦ ◦ ( ◦ , m G + 2 Through this paper, ⊂ ◦ ◦ ◦ m ) Z ( superconformal m ♦ ◦ ♦ ) have (in particular) the structure of (respectively) a complex G -valued points in the universal Chevalley group-scheme R K ( VHS and not mere ), while forgetting nothing one gets those of the reductive subgroup m ) where # is the total number of lozenges (black and white) and ). Forgetting the color, one gets the diagrams [ (3) (1) (2) C G R ( its smooth, simply-connected cover, and R ( ( m e S m m G G G -valued points. genuine ⊂ Q ) and ⊂ ) a ) for the group Sp(2 C ) ( m ( a K ( m P m ( . The diagrams of the three Klein geometries L m brane amplitudes of G × ) as the group of G ∗ . ). Forgetting the shape of the nodes, one gets the Vogan diagram [ has a homogeneous metric structure. K # tt C The standard sources forWith Chevalley some groups abuse, are [ in this paper we doSee, not for instance, distinguish the between wonderful an computations algebraic in group the defined appendix over of [ ( ) of the group ( +1 a 17 18 19 ( m m m m m meaning, being unambiguously implied by Dirac quantization of charge/fluxes. the group of its compactified to 4d on a six-torus. G C and a real Lie group,to and the natural structureall of these algebraic fancy group algebraic-geometric over structures on Sp(2 structures are Notations and conventions. K¨ahlermanifold, write 2 Review of special K¨ahlergeometry In this sectionguage we and review notation. the By basicK¨ahlermanifold”, convention, that “special is, facts K¨ahlermanifold” one stands of whose for globalof special “ structure electro-magnetic K¨ahlergeometry is consistent fluxes mainly with and Dirac to charges. quantization fix In lan- particular the underlying variations of Hodge and the condition thatthe the “Weil map” arithmetic properties not shared by their massive counterparts. U(1) the real Lie groupsubgraph whose over Vogan the diagram circle isD nodes obtained is by totally deleting white, all hence lozenges. For (1) and (2)case the of Picard number 1. In appendix A we review the equivalence between the nodes are decorated bydecorations we a recover color the Dynkin (white graphG vs. of black) andg a shape (circle vs.subgroups lozenge). Forgetting all Figure 1 JHEP09(2020)147 ) is R ) ( a ( m m D G likewise ) a ( associated to D + 1; = 1; ) Besides a 0 ( , m m 3 D h = . Diagram (1) yields 0 , for polarized weight-3 1 1 ); h . By construction with the restriction that ) (symmetric matrices with 1.3 a 1 whose homogeneous (holo- ; ( m , namely the period domain ab 3 (2.1) )–( m D τ , 20 2 = holomorphic contact domain. )-homogeneous, complex Klein , of 1.1 are in one-to-one correspondence 1 R , ( 2 ) = 1 a h m ( m G D ) for the bundle over ) = 1, → a = 2 supergravity. ( 0 ) , a , a ) 3 ( a N m h ( ( m compact dual Griffiths period domain V O ): gets its canonical complex structure by restrict- /P – 15 – ]. ) ) Siegel upper half-space a 62 C of the electro-magnetic charges, while one gets (3) ( m ( )] which we call the D m R ( )-homogeneous bundles over the domain G 1 )], i.e. the domains R + 2) − ( = is called the def m ]; we write m m ) ) m a G a U( 63 G ( m ( m [ (2 ˇ ˇ × × D ) D + 1), i.e. the a ( m ; the domain m ) a U( [U(1) [U(1) ( m / / / nodes. ˇ ) ) ) D , and the ) R R R a ( ( ( ( m ˇ m m m D to each node of the corresponding graph in figure circle G G G -modules ) Z a ( = = = m def def def ∈ )-homogeneous spaces geometry for local-graded chiral rings as in eqs. ( H i C ]. The homogeneous vector bundles (2) (3) (1) m m m n ( ∗ 63 m D Hodge structures with Hodge numbers D D for polarized weight-1 Hodge structures with Hodge number tt 0 for the [ . special K¨ahlergeometry in the sense of variations of (effective) polarized weight-3 Hodge structures with The diagrams of these three geometries are represented in figure ) G ) For these basic facts, see e.g. chapter 2 of [ a ≥ a ( ( 20 m (2) (3) (1) (3) (1) (2) i ˇ get their holomorphic structure fromD the corresponding bundles over thewith projective the variety m morphic) bundles are ininteger one-to-one correspondence withn the possible assignmentsan of open domain an in ing the one of the are rational, projective (hence compact), complex manifolds the Siegel upper half-space, i.e.positive-definite the imaginary space part); of (2) gauge isto couplings obtained the by Dirac making representation afrom lozenge (2) the by node eliminating associated all lozenges shared by (1) and (2). From the figure one sees that itself, we introducegeometries the (called following henceforth the three reductive, We begin by introducing their respective Klein models. (I) Three Klein geometries and their Penrose correspondence. We are concerned with three geometries which eventually will turn out to be equivalent: 2.1 Basic geometric structures JHEP09(2020)147 a 21 +1 (3) z m = 0 m D (2.3) (2.4) (2.5) (2.6) (2.2) 2 λ C ∗ z ). ) , we see P R ¯ z ) 1 , R , m, z ( ) and is non +2 m Sp(2 m i Q × − U( fundamental rep- such that Sp(2 / ) hence compact, ) U(1) defines a tautological m R (3) & ) (“supersymmetry”) m & ` ≡ + 2 ) induces on D U( m, 2.2 / )-homogeneous; it is also . We give a more explicit (3) m m 2.3 → R 2 (3) m D , / L + 1) L : , . + 2 z × m = Sp(2 3 0 in the complex contact manifold , +1 $ b 1 m m L > z in the − 2 b d (1) m C ` ¯ z a ]. In particular the Griffiths domain = SU( a D P z = U(1) z 61 m is the locus where ab = , ) ) As already anticipated, the domain C (3) ab m ω R , m (3) +1 m P H (3) m i ω ≡ m U( +2 D is 2 ) × m /P 1 C z ) is clearly Sp(2 – 16 – ≡ − P $ C ). The presentation ( , d implied by the diagram ( U(1) ) L Sp(2 ( L → z C ¯ z ⊂ . m ( , , ≡ )-homogeneous, holomorphic, contact manifold. This (3) m G Q z +2 is just smooth. From the diagrams in figure R (3) ( m D / ( m (2) + 2 m ≡ 1 = 2 D def m Penrose correspondence i Q D is always a complex submanifold of the Griffiths period domain. C $ m λ G (3) m 1 − o ˇ ∈ $ D z (1) m . The fiber of · Sp(2 + 1 is a holomorphic immersion t D 1 (3) m m ≡ $ is the standard symplectic matrix. Written in the homogeneous D ) is the Siegel upper half-space ) is the subgroup fixing a line o × +1 C 3 ) C ( m ( (1) R m $ , m 1 +1) this contact structure is generated by the holomorphic one-form with . m (2) D y G +1) +2 y m ⊗ , the open domain m G O 2 a m m S z = U( ⊂ is holomorphic while L Sp(2 = 3 ,..., C (3) ) of dimension 2 m 1 ≡ ω $ , P , λ embeds in We recall that a holomorphic Legendre submanifold Note that we have canonical fibrations between the domains (which are the restrictions (1) m By general theory the fiber of (3) = 0 m D (2) 21 m a D equipped with the canonical, invariant, positive-definite Hermitian form ( and dim The one-dimensional representation of the isotropyholomorphic group on line the bundle line whose fibers are the lines where coordinates resentation of a canonical structure( of holomorphic contact manifold;coefficients in in homogeneous coordinates is obvious from the formdescription. of the First, isotropy group since D while the one of compact. (II) The holomorphic contactcarries a structure. canonical structure of is nothing else than the Penrose correspondence [ of the corresponding fibrations between their compact duals) where that the correspondence JHEP09(2020)147 - is (2) m , for (2.8) (2.9) +1 m H (2.11) (2.12) (2.13) (2.14) (2.15) (2.10) ). Let g m (super- 1 , C 1 ≡ − m 23 g +1 -grading on ( m Z o O as the C . stands for the root C Q kX α ⊗ g . such that ) ) . (2.7) ] = +1 C m ab (2) m must be holomorphic. δ g m the structure of a 2.10 ,X e H p ∈ ( C m Q g ) = . satisfies Griffiths’ IPR iff b Q = 2 :[ )-homogeneous holomorphic Lie α . The weight lattice of g , e R a ≡ ( (2) a m +1 C m ]. In eq. ( ∈ n e g m m D . C α 60 \ α G ∈ +1  =2 ⊗ Γ X a X , m is a compact (resp. non-compact) root. ) . X k,k 1 , (2) m → + α − m n 1 (2) and h m . In particular g 1 − m S for D n S = g def = : ] is the homogenous bundle =1 ( 3 m p 3 k 0 k , α → defines a 0 m − 30 ⊕ ≤ ) , g X k, m 22  ⊆ O ) a 7→ − – 17 – 25 k,k k,k O ) α ). There is a unique a − m ≤ − m ) are e C g = g Φ( ( TS ( a 1 4 if it is a white lozenge. Φ induces a ( 1 2 with inner product ( m n ∗ O (2) m − p G for 1 +1 D =1 , g ] = a a X m of T , α e l +1 k,k Z − a m C − m k ,X e g Φ: g − +1 =1 Q 3 l, − [ a M m − + 3 a . Then k m = e M = k g ) is 0 mod 4 (resp. 2 mod 4) when = W = ⊆ α a ] (where it is written Ψ). The basic property of the linear form Φ defining a Hodge Λ  l C m terminology, we shall refer to the Lie algebra element α charge g − 28 be the integral linear form ∗ , l, m R . g tt α 27 Z , k − → U(1) is the holomorphic tangent bundle to k, m g W  TS Φ is the linear form which defines the relevant HodgeFor structure these statements on see the section Lie X.3 algebra in the second edition of [ be a complex manifold. We say that a map 22 23 where more details see [ representation is that Φ( space of the root Griffiths’ horizontal holomorphic bundleS [ E.g. the holomorphic (1,0) tangent bundle is The adjoint representation induces onmodule, each direct hence summand of eachbundle direct over the summand Griffiths domain Borrowing the conformal) Φ evaluated on thereturns simple 2 roots if of the graphthe node (2) complexified is Lie returns black, algebra zero and if the root is a white circle, it Let Φ: Λ The simple roots (ordered as in figure (III) Griffiths infinitesimal period relations (IPR). JHEP09(2020)147 24 is a , we ]; for (2.20) (2.21) (2.16) (2.17) (2.18) (2.19) (2.23) (2.22) w ) then w ζ 66 , are local- a discrete 15 ] and their S G ∈ 67 s -family is then } ⊂ 1 s S 1) , {R (0

if it is pluri-harmonic m be the Maurier-Cartan s (anti)chiral ring valued G ) becomes valued in the | ). The ζ ∗ m ω ⊂ . π ∗ . g -preserving solution tt 2 a semi-simple real Lie group Γ , R Ψ ⊗ k,k is pluri-harmonic [ [0 G + 1) − m . )) geometry. We say that = 1 g R w ∈ ∗ C. P m ( C tt θ m odd ∈ U( , and the solutions by analytic continuation M ) as dictated by unitarity. Ψ( ] = k G / . Q ( ) R ∗ C A 1 iθ ( ) , = , R e ζ Λ ( m 0) . Q , m [ m G/K G ∈ s , ζ (1 \ 1

G = 0 ω Γ \ − m ) equations iff s
Γ | ) = Ψ( R ∗ Q → ∗ ω , ( – 18 – ≡ ζ complexified ∗ tt C, m S /ζ D∂w k,k Ψ − Ψ a complex manifold, : G (1) m − m g w S = D Ψ(1 , and let ≡ ] = = \ def describing the same a maximal compact subgroup, and Γ w Γ G ) even ¯ ,C if, in addition, ζ PDEs with monodromy group C M G (Berry) connection = 1 to the full twistor sphere, Ψ( k solve the → ∗ , Q ∗ | [ ∗ Ψ( ζ tt ⊂ tt = C S | C ,C tt , : , m equations describe an isomonodromic problem [ m k K k | A w C ∗ ω ) is needed to convert right vs. left actions in order to match the conflicting , ∗ tt A twistorial family of ) instead of the real group Ψ 2.22 1 -family of lifts by acting with C ) be any lift of = ( 1 P R S m ] are ( A . The and Hodge theory. G m 12 ). The usual ∗ [ A G tt R a solution to the ¯ ( being their grading operator). If we have a U(1) C m → , equations and branes. Q G C e S ∗ : -preserving solution to tt R Ψ The inverse in eq. ( yield gauge-equivalent 24 satisfies a twistorial reality condition conventions in extended to a full Extending from the equator complex group Clearly this condition maygraded be ( satisfied ifmay and construct only a if the chiral rings It is well knownthe that benefit of thew reader we reviewU(1) the argument in appendix A. Note that two lifts of where Let form of 1-forms see appendix solutions are essentially determined byrelevant the for monodromy special group geometry, Γ. i.e. We specialize to the case without compact factors subgroup. A map is said to be (with respect to the symmetric metric), i.e. (IV) JHEP09(2020)147 They , and (2.24) (2.25) (2.26) m Γ). There ) are well R ∼ = ( , is identically ) G \ Q S ( 1 . = 1 they are real. (Γ) π | % , i.e. ζ ∗ | encodes the supergrav- ) group. Modulo com- tt → ) = 0 brane amplitudes. is Griffiths’ period map z ζ ∗ p S , (1) Ψ( m tt twistor sphere gets identified Γ
): gauge not in the more tradi- D ∗ ζ \ ∈ C -duality tt Γ ζ ξ & / U / real ∀ + ] (or C stand for the Penrose canonical pro- 1 1 ] may be summarized into: 16 / , $ − / , ζ 69 1 a 12 – − $ + ) ) such that the covering brane amplitudes a complex manifold of dimension ζ Z 66 grading in the sense of A ( , ; 3 S ∗ (2) (3) m m R w $ m geometry, the arrow +   – 19 – ) Ψ( tt 23 D D monodromy G d ∗ ξ , \ \ ( ) are called (covering) tt Γ Γ % / . C → 22 , , = def = ) 1 ζ 15 :Γ + 2 p , − % ) Lax equations) [ . Ψ( ζ m 12 z ∗
– S ) describes ]. In particular, on the twistor equator tt ζ Ψ( ( 10 ∗ 12 w Sp(2 ξ ∇ S → + ): ]. In other words, the equator of the ) e S ζ ( 29 The following three conditions are essentially equivalent for the com- 2.26 – ): ∇ -preserving solution to for a discrete group Γ of isometries acting freely (so ζ 27 R e ) satisfy the Γ-equivariance property S \ C ( ). The arrow ]). G We see that the specifying a U(1) ) a subgroup to be called the = Γ The brane amplitudes here are written in the 2.2 Z 65 → S ( is a U(1) , . satisfies Griffiths’ IPR; e m S p w 64 G Work by many people [ Consider the commutative diagram Let The maps Ψ( ): ζ ⊂ (1) (2) Fact/Definition. mutative diagram ( where, as before,jections the ( double-headed arrows describing the variation of Hodgeity structure (VHS), geometry. and the arrow mensurability (i.e. up toneat finite [ covers) we may (and do) assume Γ torsion-free (and even integral form Φ) [ with the Deligne circle of Hodge theory(V) at the Definition reference of pointΓ in special the period geometry. domain. which just reflectsdefined the on fact the physical that space the braneRemark. amplitudes Ψ( to the math procedure of constructing a Hodge representation in Hodge theory (by the is a monodromy representation Ψ( satisfy the linear PDEs (the Caveat tional holomorphic gauge [ JHEP09(2020)147 z is m P ∗ ). A z and (2.27) (2.29) → special p e S . : ; hence the e S z , . . . , m ) (2.28) π π ) is a 1 , ]. Then +1 , m, m . Following tra- 2 2.26 ), is positive and = 0 69 , z (3) m I ··· : 2.5 ( , D 68 [ 1 the map I , ··· of z X : , by compositing it with a z = 0 z as +2 m in terms of the homogeneous m z and P A priori e : L w e L, ,I,J ⊂ +1 ) ) ⊂ J m ), homogeneous of degree 2, which is U z I o X ( (   ) (2) (3) m m ( e z X prolongation of I U ( D D 7→ / / ( F st e F z I ) π X +1 be the simply-connected cover of 1 2 ∈ e p m is more conveniently stated in the form e 2 ) S e z – 20 – ≡ J z 1 : ) X J Let (the so-called special K¨ahlermetric). :( ··· X is generically of maximal rank. The usual coordinates e ( S S : , satisfying the IPR, do arise from quantum-consistent . ] (3) m p F 1 ) intersects transversally the generic fiber of m F 68 = 2 supergravity context. + 1) homogeneous coordinates z D e ), equipped with the Hermitian form ( P S : ( , m ∈ e N z ) 0 2.6 → ) J ( z is canonically the 1 J e S X L :( p : ( . Then question ,X e z p F in the I I π a holomorphic Legendre submanifold. We write F X ( , we construct the other two arrows, in the form [ ∂ p n (3) m , π = D (3) m def m , which satisfies the relevant condition in the above list, one may complete, P ) = I D ⊂ z Which period maps F U ) ( → e or S e z ( pre-potential (3) e z w m ], we shall denote the homogeneous coordinates of geometry. , D is a Legendre submanifold. then may serve as local holomorphic coordinates in (small enough) local charts ≡ p 70 : . We may write the local Legendre manifold z e L e π m S If we are given P = 2 theories of gravity? ⊂ (3) (VI) The local pre-potential may be lifted to their respective covering maps the two canonical projections;through so its the period most convenient map description of aQuestion special 2. geometry is N in an essentially uniquecondition. way, the For instance, commutativeeverywhere diagram of by maximal rank, arrowsof and satisfying the the tautological the pull-back bundle stated ofhence the it curvature defines of a the K¨ahlermetric on Chern connection If one (hence all)(K¨ahler) of the conditions is satisfied, we sayIn that this the statement diagram “essentially ( equivalent”arrows means that if we are given any one of the three for some holomorphic Hamilton-Jacobi function called the differential of the map on U coordinates on for the projection on thedition [ last ( generic Legendre submanifold with JHEP09(2020)147 e ]. ). S S ]); ] 23 12 2.6 67 , ], was (2.31) (2.30) (2.32) should Torelli 22 over the is better e 71 S , j ¯ is negative is the pull- i . This item ]. geometry of can j metric [ ¯ 30 i . K F , H L ∗ 75 J ω K ) one has [ , of the – tt j ¯ X 25 is i , ) 73 J global a countable family K 2.19 F 24 K ]. We stress that S kinetic terms [ D 74 (even after replacing [ have much better non- KI ) on ) rotation but only under j ¯ m ]). i B K Z 6 ( C , , K I sugra Q X , + 2 1 2 R = 2 . ]. Its K¨ahlerform m ) induces on + j ¯ = i multivalued N J . , the relation between the two K¨ahler 72 R K ,
X 2.26 Sp(2 m ): ) + C 67 )( j ¯ ∈ K i ∧ K ( KJ ( G i.e. the universal cover of the uncompleted ! m B C may be m L ], and its curvature tensor is non-positive in G IL J T G KI F tr 71 B ⊂ + 3) D C i 2 – 21 – ( ) I m K of the curvature of the positive line bundle ( K = I JK F , z The diagram ( A C H = ( viewpoint in [ + under a ω

j special geometry the Ricci curvature of ¯ ∗ + 1 J i via tt F ≡ K m ) have the same isometry group. The metric γ In the particular case where our special geometry describes KJ j ¯ ] (as well as in the Nakano sense [ i C in value general I K 30 , K ). Since the swampland conditions refer to the A ]. In terms of the matrix one-forms defined in ( , we cannot ignore the issue of the several branches of e ( 25 S and , I S 24 , called in the math literature the Hodge metric [ [ j ¯ F j ¯ i 24 in several senses. E.g. the curvatures of i 1 2 G (2) j ¯ K m ] i + D G with the completion, with respect to the Hodge metric 71 is biholomorphic to a domain of holomorphy in , of the positive curvature of the Griffiths’ canonical line bundle is not invariant S e 67 S (2). γ which is the moduli of polarized and marked CY’s, see refs. [ p −−→F F . of , is called the Weil-Petersson (WP) metric (a.k.a. the normalized 0 0 j ¯ 4.7 i e the action (in value) of F T T S via G not be confused withspace the Teichm¨ullerspace is metrically complete withto respect the to Weil-Petersson the one. Hodgesection metric The but implications not of necessarily this with fact respect for physics will be discussed in the Griffiths sense [ (VIII) Torelli properties. an actual family ofcover Calabi-Yau 3-folds, itspace is convenient to identify the simply-connected The two metrics behaved than positivity properties: for a and away from zero by a known constant [ For a special of K¨ahlergeometry complex dimension metrics is [ This is the special K¨ahlermetric whichThe appears second in one, the introduced and studied from the back period domain (VII) WP and Hodgeof metrics. distinct K¨ahlermetrics; theone, first two areits relevant for K¨ahlerform is our present the purposes. pull-back The first its block-diagonal subgroup GL( potential. In other words:by the its prepotential universal cover the scalars’ manifold will be discussed in paragraphfunction (IX). As a preparation, we recall that the Hamilton-Jacobi may be multi-to-one, and then each local branch of this multi-cover has its own local pre- JHEP09(2020)147 . e S m P ⊂ , and (2.34) (2.33) e should S → B , where = 2 su- e e L S F N Again, this = Γ . On the locus , it has a uni- 25 S e π S special geometry 26 ). Then all arise from the (multi- e integral S ( B e p ) is injective. has complex codimension at \ ]. e S B 70 2.27 is biholomorphic to a domain of in e S for Calabi-Yau 3-folds, as shown by the F , whose scalar fields take value in the has no singularity in m physical configuration in QG. We write in eq. ( false P F Torelli theorem (see e.g. theorem 16.9 in e p e S → \ including the ones in the swampland. More sugra may become singular only at a locus e same S local special K¨ahlergeometry. : ]. = Γ F e – 22 – p isometries of the special K¨ahlermetric on ) in the obvious way. = 2 holomorphic function. = 2 supergravity has the form . S 3 77 e , S e ) has a pole; hence S 27 ( sugra N N 76 e p F π$ J ceases to be transverse to the fibers of formal with its isomorphic image ≡ , and think of Γ as a discrete symmetry of the covering ∂ single = 2 I e e as part of our definition of an L S e e S S ∂ e p N is just the branch locus of the holomorphic covering map 1, the several branches of ] extends from this geometric situation to general Let us consider the pull-back of the local holomorphic function e S ≥ ). When defined as in (VIII), 75 . – ⊂ F F 73 B ; if the holomorphic function m . In other words: two field configurations in the covering theory which C S . Indeed, ] asserts that the lifted period map e S 75 has codimension – (again written which we must gauge in order to get the actual low-energy quantum gravity with a smooth, Hodge-metric complete, simply-connected, special K¨ahlermanifold. It is ]) which clearly holds for all B e 73 S e S ) for the Lie group of holomorphic 78 Henceforth we shall identify The analysis of [ e S We stress that the strong versionFor of these global facts expressed TorelliOne in is the can “old” show supergravity that language see the [ connected component of the isometry group of a special K¨ahlermanifold is to ( the holomorphic function det( 25 26 27 Iso Aspinwall-Morrison quintic counterexample [ made of holomorphic isometries, i.e. all Killing vectors are holomorphic. convenient to work with thecovering special covering K¨ahlergeometry sugra moduli space differ by the action of Γ are regarded as the The special space K¨ahler of a general with become multivalued to represent theSince several local branches of thevalued) Legendre analytic cover continuation of a 2.2 Special K¨ahlersymmetries least 1 in where distinct branches of the cover coalesce together. Around such a locus F holomorphy in valued global analytic extensionwhere to the all Legendre sub-manifold B this property: they restref. on [ the CY strong the monodromy group Γ acts on (IX) Branching of pergravities with quantized electro-magneticrefs. fluxes. [ Then the “globalholds Torelli” in theorem any of naively-consistent precisely, we take injectivity of (by convention, all our geometries are integral). Most of our arguments are independent of JHEP09(2020)147 is Z ) ). In arise e ) gets (2.39) (2.41) (2.35) (2.37) S R ( e duality S , Dirac). (2) ( m . For all D + 2 Sym real ´ala Sym ⊂ m e S , (2 is the subgroup sugra ) e symmetries sp S R ( ≡ = 2 m G R m (Γ) of Γ in N g ∈ N ξ rational ) (2.36) . )] R , ] is such an example with an “emer- m ,S . . ) ) ) 76 e S + 2 U( Z Q ) (2.38) ( ( ( R × m m m , Iso G G all naive symmetries of ⊆ ∩ ∩ + 2 Sp(2 ) ) ) [U(1) e S e e / m . The normalizer Γ (2.40) S ≡ S ( ) ( ( Z / ) (2 ) R ] theorem 16.9). e R , S sp ( (Γ) ( Sym Sym Sym 78 – 23 – m ), that is: are regarded as one-form fields instead of Abelian connec- N ∈ + 2 → R G = = I def def ( ) Sym m A e S Z m Q → ( ) ⊂ ) G ) e e S S e S, ξ e ( ( S Sp(2 symmetry group of the covering ) is then seen as a sub-algebra of ( sym e ∈ S ≡ ( 28 s yields a Lie group homomorphism Sym Sym Sym ξ acts on the electro-magnetic field strengths by the (2) m : 29 S ) we have Sym ξ D κ e S naive ) ( 7→ s ( ) of ξ e p e S · Sym ( ξ ∈ S ξ sym ) for the symmetry group we mean the would-be symmetry group of the supergravity theory with e S ) = ( s · . The map Iso ξ ξ naive ( S e p ⊆ symmetry at the 2-derivative level which is explicitly broken by higher derivative operators. ) Γ is a symmetry of the low-energy effective theory truncated at the 2-derivative level, 5 / e Z S It is natural and convenient to consider the larger group of The actual symmetry group of the covering special K¨ahlergeometry ( By the Compactification of Type IIB on the Aspinwall-Morrison 3-CY [ (Γ) 28 29 not preserve the Dirac symplectic lattice Λ ofthe electro-magnetic same Lagrangian charges. but whosetions However, vector (the fields given difference being that in the secondgent” case the electro-magnetic fluxes are quantized Rational symmetries are not symmetries in the strict sense of the world, since they do is the “emergent” globalN symmetry of thewhich effective may theory. (and Here should, “emergent” accordinghigher derivative to means couplings. the that swampland conjectures) be explicitly broken by The discrete gauge group satisfiesan Γ “emergent” symmetry of the QG, and the quotient already follows from strong local Torelli ([ of the naive one which is consistent with the quantization of the electro-magnetic fluxes, i.e. from automorphisms of the ambient space The Lie algebra facts, the injectivity of whose kernel is trivial byidentified with “global a Torelli”, closed so subgroup that of the naive symmetry group naive symmetry where the naive symmetry rotation Sym JHEP09(2020)147 ] Λ 1 ↔ 79 is a ξ ⊂ )[ (2.45) (2.42) (2.44) ξ e ξ S ( Q of special subclass . ) e S ( is equal to the entropy of ξ , in this section we shall be sym Λ 2 . ⊆ ξ ). The correspondence Λ ∈ ]. In facts, it suffices to impose ξ Λ 3 R v ) of the algebraic group ξ Q e S ( ∈ ⊗ ) (2.43) q . ) Q Q e ξv S ( ) ( e q S m , ( there is a finite-index sublattice Λ G Λ Z ⊂ ⊂ ) Sym e ξ S Q -Lie algebra of “infinitesimal symmetries” of the , respectively. The merit of these statements ( ) – 24 – and the subsector with electro-magnetic charges ⊂ Λ Q -Lie algebra e S 1 ξ ) ξ ) and ( because it makes additional geometric assumptions Q e S Q Sym -algebraic subgroup ( 2 , \ Q Q ) as a connected (linear) algebraic group defined over Sym , so that it applies only to a small Q Q + 2 ) , S e S m ( + 2 (2 sp m Sym , namely the and corollary is non-compact of finite volume (the other conditions then hold e ⊂ S ∈ 2 ) S (having the same index in Λ as Λ Sp(2 e ξ S ξ ( ≡ q Λ ) ξ Q ( ]. The subgroup m sets a correspondence between the subsector of the theory whose electro- 79 G ξ [ Q leaves invariant the classical physics, so it is a kind of sector-wise symmetry. More Since both results will be subsumed by the stronger fact The material in this section will be only marginally used in the rest of theThe paper, first result is weaker than fact We see ξ Λ on the special K¨ahlermanifold geometries. The second result holdsin in a full significant generality, way. but it is less precise thanrather corollary sketchy with the proofs, and be cavalier with several fine points. and can be omittedcriterion in with a some first well-known reading.stand phenomena However, the in comparison Differential nature of Geometryproof” the and may swampland below). effects help structural of to under- the QG ‘arithmetic’ paradigm (see “Comments on the weaker versions of fact is that theyswampland may conjectures be in proven the original remainingthat Ooguri-Vafa the inside form moduli [ the space naiveautomatically in DG this paradigm special by context). assuming the We shall see that in a quantum-consistent special geometry the last3 inclusion is an equality. Warm-up: two weakBefore results addressing in the the issues DG of paradigm main interest, we state two ‘elementary’ results which are This allows to introduce the notionspecial of K¨ahlergeometry the the field then contains a maximal connected ξ importantly, its existencerational has symmetry observable of the consequences. fullthat theory and the For not entropy just instance of of anthe (assuming its extremal corresponding 2-derivative truncation black extremal !!) holes black it hole with implies with charge charge such that and hence magnetic charges are in thein sublattice the Λ sublattice a rational symmetry JHEP09(2020)147 ]. < 3 ) 81 non- S ], ( ], hence 64 65 Vol is complete , e S 64  Then fact are non-positive, (Γ) i 31 j ¯ i A subgroup; · G j . γ

j ) is a finite-index sub- [ e S is Hermitian symmetric: ( ; e j S Iso : see e.g. theorem 1.7 in [ arithmetic e S ⊂ is an arithmetic subgroup [ rank Γ j Q is an Assume, in addition, that the Rie- G P ) for its centralizer in Γ. We write ) contains a free Abelian subgroup G 6= 0) then σ σ ⊂ ( ( Γ such that Γ = 30 ) Γ ⊂ Γ e Then we have the dicothomy: S ∈ Z Z ( j γ and Γ iso r is metrically complete. By construction, Γ is a finite index subgroup ) = Γ. e ) and Γ S ]. S ⊆ – 25 – ( 2 ∗ π 85 -rank [ ). Γ, we write Γ such that e Q S Γ ( ):Γ ∈ ∈ ∗ σ σ iso (Γ r are non-positive.  ) we see that, when the sectional curvatures of of an abstract group Γ is recalled in the box below. S is a product, then rank Γ = 2.32 except that we restrict to the very small sub-class of s Γ by a finite cover, if necessary — we are assuming Γ to be neat [ rank 2 a group in table . S belong to r : there exist finitely many be a special K¨ahlermanifold which is non-compact with ) for its Lie algebra generated by the holomorphic Killing vectors. G i e is a Hadamard manifold without Euclidean factors. S ], under the present special assumption, the simply-connected Rie- e ) is a discrete group of rank 1, and Γ S e S  -algebraic group of ( e e S \ S 80 × · · · × Q rank Γ := sup ( is the universal cover and is fact 1 iso special K¨ahlermanifolds. In this restricted class of manifolds the conclu- e (with Iso ) stands for the Lie group of holomorphic isometries of the special K¨ahler S = Γ 3 is a as a subgroup of finite index. Then e S is its smooth simply-connected cover. i Q ( S G e ≤ S , and Fact Iso (Γ) := min G/K Commensurable groups have the same rank; if Γ = Γ free groups have rank 1; if then rank Γ = has a non-trivial Killing vector (i.e. e r S Let e • • • • = S (Γ) for the set of the elements i e group. if S otherwise Properties: and set A of rank The rank of anLet abstract Γ group be an abstract group. For The definition of the Similar results hold under otherBy special assumptions definition on [ . Here Recall that — replacing The non obvious part of this statement is that • • 30 31 ∞ mannian sectional curvatures As before, manifold All Killing vectors of Fact 3. + 3.1 A restricted class of special K¨ahlergeometries mannian manifold torsion-free. Then for the Hodge metric;completeness from with eq. respect ( to the Hodge metric implies completeness with respect to the WP metric. positively curved sion is stronger sincemonodromy we group have Γ the has rank extra 1. information that in the non-symmetric case the Remark. JHEP09(2020)147 < 32 in ) e S \ Γ in a (Γ have the reducible j ¯ i . A lattice Vol K lattice ∞ cover which is < and j ¯ rigidity theorem ]. A i Γ) Going to the “new” finite G 83 H/ ) such that ( e but requires Γ not to contain S is locally symmetric these Vol H ]): they are all symmetric . Therefore we may assume Iso( Γ has a S is irreducible; 32 2 ⊂ H/ H ] while the properties of Γ follow = holds equally well for both metrics. 86 , 3 M ) and ]. When 64 ) such that H 85 may seem to enlarge the class of special ]. H in Differential Geometry for finite-volume ] (see also [ has better chances of having non-positive – j ¯ i j ¯ 88 87 82 i , Iso( ). Since the two metrics K K 64 is symmetric. The simply-connected – 26 – ⊂ 2.32 ] it was observed that the swampland conjectures e S . Then either: 3 Hadamard manifold without Euclidean factors. The H (VII) be a Hadamard manifold without Euclidean factors iff the manifold , eq. ( H j ¯ 2.1 i K while remaining in the DG paradigm. rigidity theorems Let quotients of Hadamard spaces, and no useful general rigidity The basic ingredient of the argument is a ) = 0 or . e S applies, it still consists of a very small portion of all special ]) ( irreducible reducible not 3 83 is a subgroup Γ iso be the smooth simply-connected cover of our special K¨ahlermanifold ] has no assumption on the Hadamard manifold H e is an S 83 33 e S Let . 3 ) is discrete, Γ has finite index in Iso( (P. Eberlein [ In order to study the interplay between the isometry group and the global with the Hodge one ) is said to be H j ¯ i H is isometric to a symmetric space of non-compact type. G H Iso( ]. Iso( We recall some definitions and the simplest rigidity statement [ Proposition 4.4 of [ See remark at the end of this subsection. implies that either 82 having (by hypothesis) non-positive Riemannian sectional curvatures. We need to show ⊂ 32 33 (2) (1) ∞ special K¨ahlermanifolds werespaces, classified corresponding in to [ thewithout non-simple loss Lie that groups inclaim table then follows from the theorem quoted above. geometries to which fact K¨ahlergeometries. Proof of fact S the following claim: the existence of a discrete subgroup Γ same isometry group, the statementHowever and as proof of mentioned fact incurvatures. section Unfortunately having a non-positive curvaturethan in Griffiths the sense condition is much required weaker curvatures, in so the that, above theorem, while i.e. consideration non-positive of Riemannian sectional Remark. properties of a specialmetric K¨ahlergeometry, it is convenient to replace its usual WP K¨ahler Γ reducible as a Riemannian manifold. Theorem and let Γ be an irreducible lattice in quotients of Hadamard manifolds,theorems see reduce to refs. the [ from Mostow the rigidity Margulis theorem super-rigidity [ theorem [ Hadamard manifold is a direct consequence of the theorem for them isparadigm, available we may replace in the above argument theClifford DG translations. rigidity theorem Ourof by assumption [ the of subtler no Euclidean factor implies the last condition, see theorem 2.1 Comments on the proof. Differential Geometry. Already in ref.are [ easy to provecial for K¨ahlermanifolds negatively-curved are moduli spaces. Unfortunately, in general the spe- JHEP09(2020)147 ] S 85 = 2 (3.2) (3.1) . This N p,q -rank of h R is not locally belong to the S for the detailed and non-positive ) = 1. not S 25 ( ∞ 1 π < ) S ( which do 34 Vol 2, we conclude that its special v . ≥ ) PJ S , should have at some point a positive j ¯ i dim K =1 2 a complete Riemannian manifold with non- : | . Then ] whose most convenient and powerful formu- v S | ) = rank( S ∞ ]. This observation implies that non-symmetric S – 27 – 30 be a complete connected Riemannian manifold . ( , 89 < ) = min 1 ]) is locally symmetric or rank S ) π 25 85 M S , S ) of a complete Riemannian manifold is defined as [ This fancier rigidity theorem has an arithmetic flavor ( ( 24 Let M rank ( Vol special K¨ahlergeometries . rank ]) all rank 84 be a special K¨ahlermanifold with as in the theorem we may define the rank of its fundamental group ) is a homotopy invariant. where rank Γ = 102 [ S S S 4 , as well as its Hodge metric structure theorem. j ¯ P and non-positive sectional curvature without Euclidean factors. Then (Ballmann-Eberlein [ i is locally symmetric this reproduces the standard rank (i.e. the Let G ]. 6 cover which splits as a Riemannian product of rank 1 spaces and a locally M (Burns-Spatzier [ Inverting the logic, if a finite-volume special manifold K¨ahler is the vector space of parallel Jacobi fields along the geodesic with initial velocity v . If v finite PJ ) in purely abstract group-theoretical fashion, see the box on page For a manifold We close this subsection mentioning some other results for the non-positively curved In facts, more generally for all “motivic” VHS for arbitrary values of the Hodge numbers S finite volume ( 34 1 special K¨ahlermanifolds with everywhere non-positive Riemannianif sectional they curvatures — exist at all — are quite rare. fact allows to extendedsupergravities the [ swampland structural criterion to effective theories more general than Remark. symmetric, while theK¨ahlermetric monodromy group Γsectional has curvature. rank This remark applies,hypersurfaces say, to in the dimension-101 moduli space of quintic In particular, rank( Corollary 6. Riemannian curvatures. Then either definition. Under the present hypothesis on Proposition 1 positive sectional curvatures and has a symmetric space. π vector its isometry group). Theorem of case. We recall that the rank, where lation is the VHS and holds (conjecturally) for swampland, independently of the signimplications of as their the curvature, while usual having rigidity essentially theorems the of same Differential Geometry and Lie group theory. rigidity theorems of Hodge theory [ JHEP09(2020)147 , . , S Y X ∞ tube (3.6) (3.4) (3.5) (3.3) G/K < ) I ( ). Granted Vol ) are regular, 3.5 e S that is, a pair if and only if . )-invariant measure is locally symmetric, e S ∞ lattice S < is compact, a discrete subgroup with )) ) is a special case of a I , e , S G ) m 3.3 I C ( ⊂ dµ Iso( \ Y ⊂ has finite volume . Z Vol ). Since (Γ o k e S e × S
 V \ X ) and Γ
j and Γ 0 ) Vol e ) must admit a ∈ S e e X S S G X Y ≡ i z Iso( 3! X S described by ( \ Iso( Iso( ). ]. \ e : Im ijk Γ S ) the compact isotropy group of a chosen base 2 91 d Γ e ≡ S m – 28 – admits a lattice if and only if it is unimodular. − C quantum correction, unless G G Vol ) such that Iso( = ∈ Vol of these quantum corrections: it does not even rule e S z ⊂ ) = n cub some ) such that I ) = e S Iso( e F S \ S on the Lie group Iso( ( ⊂ ) = ) has the form nature (Γ ), Iso( V V e Vol ( ( S , with Vol Vol ⊂ T T A Lie group /I . Iso( ) e ]) S says that, in a Type IIA compactification on a Calabi-Yau 3-fold ⊂ m 4 (hence an arithmetic quotient of a Hermitian symmetric space 90 is an invariant measure on should receive C is homogeneous (hence one of the special K¨ahlermanifolds constructed S not locally homogeneous behaves in a similar way provided we can show = Iso( e S S Vol . We recall a well known fact: e S ∞ (See e.g. [ a special K¨ahlermanifold with strictly cubic pre-potential A tube domain < is the appropriate “Fadeev-Popov” induced measure on the generic-orbit space e ) S 35 a group in the right column of table . G may not belong to the swampland only if there exists a Iso( dµ \ says nothing about the Γ) where ]). Then G , S and a discrete subgroup Γ The case of As anticipated, we shall be very cavalier with the argument. Consider first the case In particular fact (Γ 4 Also known as Siegel domain of the first kind [ 50 35 Vol The covering cubic special K¨ahlermanifold domain ( Vol Lemma 1 Tube domains in where We shall dispense with theit, technicalities involved we in again the justification have of that eq. the ( Lie group and Vol that, after possibly theso excision that of we a may zero-measure define subset, a the nice orbits generic-orbit of space Iso( in [ point. Then, for all Γ for some invariant measure since otherwise the purely cubic classicalfact pre-potential belongs to the swampland.out However, that perturbative corrections suffice. In this sensewhere it the is cover much weaker than corollary 1. There exists a subgroupis Γ locally symmetric with the K¨ahlermoduli One has the following Fact 4. 3.2 The quantum corrections cannot be trivial JHEP09(2020)147 i ∂ z pre- (3.7) (3.9) ) cor- (3.14) (3.10) (3.11) (3.12) (3.13) )) and m ] V R V ( 92 ( [ T ( ∂ Aut — a classical ∂ (large-volume) ⊂ Aut 0 X ` X, > ≡ R ] = G , the Abelian subgroup classical ∂, X [ X . with equality if and only if v, ⇔ )); its elements are holomor- +1 1)-forms on ` , + V , m . . be a connected Lie group such ( is also graded by ) )) aut )) e aut T S ≤ V Az ∩ ( G ( G ( , V orders in world-sheet perturbation ∈ g ( T +1 7→ of ( +1 T Aut Aut = ( g are polynomials in the coordinates )) (3.8) z g all ) arises as the aut i ⊂ aut V ) +1 ( V ): ) Aut z ( g ⊆ )) of T ( e S ,X ( T ( ⊆ 1 f V + dim v, A g , ( +1 0 G Iso T Aut m g ⊕ ( – 29 – )) is graded by the adjoint action of ⊆ aut − 0 ⊂ ⊂ ∈ g V aut ⊕ m = )( ( m m we have at least the symmetry -field by harmonic (1 ), while dim ⊕ 0 T V ∂ R ( 1 ( B g V V , ∂ n R ( n R − aut tr n g 0 0 aut ]. The Lie algebra Aut R > ⊕ > aut = 1 R 92 R o = g − ≡ to be unimodular, is that the trace of the adjoint action of together with the following 1 m be the Euler vector field, which generates the overall scaling whose coefficients 0 − 1 G R i ) is the group of linear automorphisms of the ambient aut g i z z R ∂ aut i ∂ i a strict, convex, open cone. Let z ) ∈ m, )) = z vanishes. Since m ( ≡ ∂ V . ( f g R , GL( ∂ V 1 T a strict, convex, open cone. Its group of holomorphic automorphisms, ( ⊂ − . The Lie algebra ⊂ 0 m ) aut V > aut ) corresponds to the geometric automorphism of the classical K¨ahlercone. R V R V ( ]. Let ≡ ( ), we have is either semi-simple or non-unimodular. In the first case )), contains the group ⊂ 92 Consider the (real) Lie algebra V [ m V Aut follows from lemma ( G ( Aut V R T m T ) is a Hermitian symmetric space. ) is a symmetric domain [ 4 Note that for all convex cone When the cubic special K¨ahlermanifold ( C V V ≡ ( ( e A necessary condition for on its Lie algebra with T with Proof. phic vector fields of symmetry that Then T S Fact Lemma 2. continuous symmetry which remainstheory unbroken (while to instantonwhile corrections typically break it down toresponding a in discrete Type subgroup) IIAsymmetry — to may be overall broken rescalings already of by the the loop K¨ahlerform. corrections. This Then, classical for all rescaling cubic tube domain limit of the K¨ahlermoduli of Type IIA compactified on a 3-CY describes the axionic shifts of the Aut where serving the cone with JHEP09(2020)147 ) = e S (3.15) (3.16) )), the ) should V ( e S T ( ). . Aut F      symmetric cubic G ( F 3 ). The determinant ∈ ) = Her real e S ij ( : 3.16 . For the spaces with ))-invariant metric, that ) are precisely the sym- , z 3 3 Iso applied to the Lie group ) is a symmetric domain V ). Their isometry groups x V R ( ( V 2 ( 3 matrices over the normed . As a T ∈ T 3.3 ( T F i × 1). In the remaining four cases , . . . , k, ) = 2 x Aut − , 3 matrix ( ( , x d ×    , k ), see table ), lemma . For later reference let us describe = 1 3 13 23 F x z z 2 R 3.9 . Hence 2 ∗ 12 23 m x z z is the real dimension of = 1 (a.k.a. the Yukawa coupling). For the cu- ∗ ∗ 12 13 m x , i, j z z j R +1    x – 30 – ) of Hermitian 3 i F      x 6 9 15 27 RCHO ( aut → 3 ij 1), it is given by = ) = 3(1 + dim def m A F ≥ F ) Her 0 m -algebras. R ( F 3 k x is symmetric. R 3 denotes the connected component of the group ( 3 ◦ e

S )( and dim Her G ) to be biholomorphic to a symmetric domain and Iso( : , k ] — all of which have cubic pre-potentials — belong to -linear complex conjugation in R Her does not belong to the swampland for some Γ, Iso( ) = 3 -space V d is proportional to the unique +1 I . Then, in view of eq. ( R ( 50 R e ∼ = S x e T )) — the symmetric one — and since 1 S ( \ dim ) aut d SO(2 F V ≡ ( , the connected component of the holomorphic automorphism group R ≡ ) there is a unique metric (up to overall normalization) which is × ◦ ≡ = Γ e T S is 6! times the determinant of the 3 ) U(1) the cubic form is given by . The normed V ( )) m ( / S R ) V +1 R , T ( g Aut R If 3(1+dim T , → is the usual ( . R ∗ m 4 Table 3 , where z R Aut is an integral quadratic form of signature (1 F ) requires 3 e )). On S = 7→ ij

)) = SL(2 )) are listed in the right-hand side of table ). V : z A G V V ( d ( ( V Iso( T is identified with the In particular, all quotients of the homogeneous non-symmetric special K¨ahlerge- ( 36 T T ( Through the paper the symbol ( ( T ≡ 36 -algebra m and of this happens iff Aut invariant under special K¨ahlermetric on is, the special K¨ahlermanifold Proof of fact be unimodular by lemma G where form is given by the usual formula, but we need to pay attention to the order in which we R R Iso where As already anticipated themetric symmetric special rank-3 K¨ahlermanifold tube withIso domains a cubic pre-potentialthe ( symmetric cubicbic form space SL(2 ometries constructed inthe swampland. [ 3.3 Symmetric rank-3 tube domains JHEP09(2020)147 is an (3.17) ) = 0: . E X — but in ( ∗ 12 0 z -equivalent , is a group- ) 2 R L h ∗ 23 G z with the same is endowed with 13 Z we choose in the Σ where ; z d G / G ) Z d K ) + ( G ∈ ∗ × 13 Lagrangian z E cubic form is 23 ( z . ( . The generic integral cubic ≡ G 12 naive real X z X + ∗ 12 z 12 z 3 x − which keeps only the underlying Lie-theoretic ∗ 23 – 31 – z whose underlying Lie of QG “symmetries”. Typically 23 z Z 1 G → x → G − m ∗ 13 Z z An effective field theory belongs to the swampland unless: 3 is a topological invariant of is neither commutative nor associative in general. The correct 13

z F Z 2 carries additional arithmetic structures, that is, : d . For instance, consider the quantum-consistent special geometry to one of these eight quantum-consistent forms, belongs to the x Z d ] G d − . Their difference would look physically irrelevant according to the R d 3 93 x In the applications to Quantum Gravity we are actually concerned with 2 is [ a K3 surface and Σ a freely acting symmetry such that x . 1 . R Z x K d cubic form ) = jk caveat ). Clearly, there are many inequivalent integral cubic forms , z i the gauge group object in some subtler category a canonical forgetful functor structures of the “usual” description of symmetry in field theory. we can have infinite discrete gauge groups; x ( all couplings are consistent with the gauge symmetry its gauge group belongs to the appropriate subtler category, 3.17 d integral (i) (a) (b) 6! (ii) Rough physical principle. One would guess the existence of a fundamental QG principle of the rough form: by the gauge symmetries ofthat the in theory. QG One expects the this gauge principle symmetries to enjoy extend two to novel QG, features: except 4 The swampland structural criterion 4.1 The rough physicalA basic idea physical principle in QFT states that the possible interactions are severely restricted The integral cubic form form, equivalent over swampland. This quantumcriteria, inconsistency but will the notintegral structural be forms one detected is by the strong usual enough swampland to discriminate between the different to the swampland dependingreal equivalence on class the of particularobtained integral from cubic Type form IIAelliptic compactified curve, on thethere manifold are precisely eight diffeomorphism classes of such complex manifolds, see section 8. to ( underlying real form field-theoretic paradigm — sincethe they Quantum all Gravity define the settingbecause same different of integral flux forms quantization. define Indeed, physically a distinct cubic theories special geometry may or may not belong expression over Crucial an perform the products since JHEP09(2020)147 with (4.1) (4.2) is time will be Y not just ]. -model is R X G σ 94 37 , where R . To make the story × . On this space-time i S x X , b is K¨ahler,there is a special class y ¯ k the structures carried by X ∂ moduli space a , y all i and coordinates is “big enough”. ∂ = 0 G γ = 0 (4.3) ab a . Since g (i.e. on the special K¨ahlergeometry) arising a y to be K¨ahler: in our application sugra ¯ k y i Y ¯ k “almost” determines the theory. p ¯ k ∂ i X G – 32 – ∂ γ γ maps — in which the full matrix vanishes → i D √ ¯ k D i X of the . As a consequence, (b) is a much stronger constraint geometry which offers a different physical intuition of X γ : Z e ∗ S -duality group Γ should satisfy in all motivic VHS; y Lie tt U = m . The theorem consists of two parts: We consider a space-time of the form . The (classical) static solitonic particles of this model are the solitons and brane rigidity -model with target space some Riemannian manifold a ∗ pluri-harmonic y σ tt ]. In the special case of an effective theory which is an ungauged 4d 6 -model. means compatible with respect to σ and coordinates is a Riemannian manifold with metric g rough physical principle from Γ. consistent X strong constraints on the period map a list of properties that the Indeed the VHS structure theorem, when interpreted as a statement about the 4d We believe that a suitable technical version of this rough statement is the proper Indeed it uniquely determine the effective theory iff = 2 supergravities having an algebro-geometric origin, has precisely the form of the = 2 supergravity this physical principle takes the form of the VHS structure theorem. 37 (a) (b) whose value at the extremumnothing is else the than mass aof of harmonic such the map solitons soliton. — Thus called a soliton of the which extremize the functional a bit shorter, weidentified with assume the from universal thewe cover start introduce ametric classical (time-independent) solutions to the equations the next subsection. Given itsthat purpose, it is this just subsection a isso rephrasing sketchy it and of can rough; the be but technical proof we made stress of fully the rigorousAn structure by auxiliary theorem adding in a Hodge little theory, and bit of math pedantry. Although we mostly use the VHSfrom language, the it (equivalent) viewpoint is of worthwhile tothe briefly same discuss deep the facts. basic idea Thisphysical subsection arguments is to written having abstract in mathematics. mind readers which All prefer others simple are invited to jump directly to Item (b) follows from (a)as together the with ultimate the extension VHS of rigidity Seiberg’s theorems which principle4.2 may of be the regarded “power of Intuitive holomorphy” view: [ N above the underlying usual consistency in than the corresponding fact in QFT, and swampland condition [ N Here JHEP09(2020)147 = on j ¯ (4.7) (4.8) (4.4) (4.5) (4.6) i γ DC for the p ⊕ algebra. In k d . Under this y l ¯ G = ∂ c g of y . ¯ k ¯ k j ∂ V γ b j is locally isometric to geometry is the original 2d . l ¯ ∂ i ∗ = 0 a γ commutative Y tt ) are exactly the solutions -terms of the dual model. y ). From that diagram we = 0  i C ] D through the Maurier-Cartan l ¯ ∂ 4.3 C ) satisfy Simpson’s Bochner- ∧ p C 2.17 , ∧ C ¯ k ∼ = abcd 4.1 ( = 2 supergravity is nothing else C ) is a special soliton in the above R C X. = geometry defines a dual 2d (2,2) system, 1 ][ , lj ¯ + Y choices of the K¨ahlermetric ∗ j TY 0 generate a C N γ tt 2.26 T , C i ¯ ki ∧ ,C p → i | C γ ∧ ⊕ , see eq. ( C C ω [ ∗ X − C ]: the X  for all y = : 0
, 16 + b + 2 y 1 -model solitons ( = C y σ T j + tr m ∂ – 33 – C D 2 -model with the extra feature of preserving the DC 2 ¯ k ∼ = is actually pluri-harmonic, while the second term . k of the original theory, whose σ + = is identified with D y + Y C of non-compact type. We write = ], that the first condition implies the second one, and X C
C
DC D ⊗ a in the diagram ( b V k ) are the two summands in the type decomposition D TY y 66 l ¯ y D i , ∗ X j ¯ x w ∗ ∂ y = G/K l ∂ ¯ the special ] ). We stress that the special solitons have the property d = 15 ) = T l ¯ a j ⊗ D 2 12 y C 4.1 i ) and C D DC i X ∂ ab R ∗ ≡ arrow G/K g C , = = ab T lj ¯ ∗ g C 2 = tr( γ ∈ tt lj ¯ j are point-wise positive semi-definite; the first one vanishes iff PDEs [ + 2 D ¯ ki ) becomes γ Y DC ∗ D γ dy i ¯ ki m tt 4.4 geometries are invariant under deformations of the γ ] = 0, that is, iff the matrices The = rhs D (resp. j ∗ l ¯ i tt D ,C ] i ¯ dx k i C . Then is the Riemann tensor on = Sp(2 D 95 we show, following [ C , G PDEs. G A ≡ In this setting, all harmonic maps 10 ∗ abcd are seen as matrices acting on some representation space on tt C R l -solitons. ¯ 38 ω . We now specialize to the case in which the target space C Q This is related to the deep fact uncovered in ref. [ , ; in the physical applications one says that these special configurations are protected by ∗ 38 i see that a specialthan a K¨ahlergeometry special in soliton the of sense our of auxiliary with field space the coupling-constant(2,2) space model. Both In other words, when to the tt sense for vanishes iff [ appendix then the full set of Both terms in the 0, i.e. if and only if the harmonic map where C identification, eq. ( corresponding Lie algebra decomposition, and identify form where an irreducible symmetric space of being solutions ofX the equations ofsusy motion formula [ not just its trace as in eq. ( JHEP09(2020)147 ) ) is i 4.4 4.1 ). (4.9) C is not PDEs , with (4.10) whose c 4.8 ∗ -solitons G tt , Q -solitons. G/K of eq. ( → ∗ Q  ) 1 2 → tt ∗ , PDEs ( ) lhs tt X  e X ∗ ( , X ) 1 : tt ( π e 1 y ··· 4.1 : π ) is equivalent to the , % generated by the 1 2 ∼ = the solutions to eq. (
 2.21 R maps , by a discrete subgroup of X 3 2 all , the only stable finite-mass ∓ n → equations, and hence may be  R . If Γ is not torsion, the target ), eq. ( G/K e ∗ X twisted ) are equivalent to the ≡ tt 2.11  4.3 G/K ) (IV). Uniqueness of the soliton with \ Γ + 2 2.1 -preserving special soliton is nothing else → m Q 2 X : deck group of -solitons (and viceversa), so asking which special y – 34 – ). A Q ∈ ∗ ξ -preserving special solutions to ( tt 2.21 (weights ), see section has the correct spectrum Q ζ ∈ Q λ V , which satisfy the equivariance condition In physics we are not interested in e ): X λ geometry of a family of superconformal 2d (2,2) models whose ˆ ) for all charge x ∗ Φ( ), is to quotient the target space ( tt R e y 1 2 ), it represents a solution to the , i.e. the one with the smallest possible mass. If this soliton happens · 4.8 )  is compact, the integral of the total derivative in the acting on ). To ensure that the VHS has the correct Hodge numbers we must ξ 4.3 ). We call such ( = X Q % to ensure that the unital commutative algebra G/K -solitons. \ 2.15 2.10 is no longer contractible, so now the solitons may be stabilized by their brane amplitude Ψ( Γ Q Q ) = is a discrete subgroup, and we consider the ∗ x ∗ ), ( → · tt G When = 2 special geometries are , which is a contractible space diffeomorphic to tt ξ descends to a well-defined map that the U(1) ( 2.9 G/K X ⊂ e e y y N \ , and (2) they are stable against small deformations. Only these solitons describe : 39 charge y ∞ G/K spectrum of R Intuitively, we have at most one stable soliton of finite mass per homotopy class of Stability of solitons requires the existence of some non-trivial topological charge: the All Otherwise we get the = 39 . We are led to the following set-up: we have a homomorphism maps to be special, eq.lifted ( to a necessarily 3. so that space Γ non-trivial homotopy class. of special geometry ( G image Γ domain the universal cover soliton of smallest massY within a givensolution topological is the sector vacuum, is sinceof stable. the there field is configuration For no down to instance,while obstruction zero. to preserving when The continuosly the only decreasing crucial way the to fact energy get that some non-trivial the stable equations soliton, ( Physical but only in solitonsm which < satisfy two additionalphysical states conditions: in the (1) spectrum they of have the a theory. finite mass geometries belong to theare swampland unphysical. is equivalent to asking whichRemark. formal vanishes, and all solitons are automatically special, i.e. solutions of the see eqs. ( a local-graded chiral ring,than a cfr. VHS eq. in ( Griffiths’Griffiths’ sense: IPR indeed, ( in viewrequire of eq. ( U(1) JHEP09(2020)147 ] M 96 (4.13) (4.11) M/K \ is a real M ), must have R ( ? = 0 (4.12) R . Without loss we i ), then there is a not to be torsion, Γ M dS 4.2 G ) as a consequence of
≡ b of finite volume (e.g. it y ⊂ j M/K M 2.26 X ∂ -Zariski closure is the full a R . y PDEs provided i ∂ ∗ = 0 tt ab g  whose  lj ¯ j M γ C M ¯ ki i ) with target -algebraic group, we conclude that its γ ⊂ R . In the actual story of the swampland structural ,C l ¯ M/K R in the diagram ( 4.10 Q \ D  may be continuously retracted to Γ Γ w ) and has finite mass (  and deforming it continuously to decrease its ∂X tr . Again this is a manifestation of the ‘arithmetic’ nature → Z – 35 – ) as an 1 2 Q 4.10 any G/K = R X y − \ , : x -preserving solitons are automatically special since = m commutative, provided we may justify that when we the boundary term any + 2
d Q i j y
C j C m X is simple. Under these assumptions on Γ one shows ([ i solitons requires the subgroup Γ satisfies ( C i C ∗ guarantees that in this process our finite-mass configuration M C M = 2 supersymmetry nor definable while remaining within the Sp(2 tt tr , identified with the real Lie group tr ) over M ≡ N j G 4.4 M/K D G i ⊆ D → R γ Γ e X √ ]), so that the stable solitons are naturally expected to take value in this is the union of a compact set and finitely many cusps). This condition is : generated by the 40 X is K¨ahler,the soliton will be automatically special (i.e. pluri-harmonic), and M Z R X any ∩ e X y and a discrete monodromy subgroup Γ K [ language. M ≡ When We are led to the following situation: we have a non-compact semi-simple real Lie Stability of non-trivial Here the Zariski closure is taken over the ground field = 2 supersymmetry. Indeed, all M , and we consider the twisted maps as in eq. ( , it describes a special geometry K¨ahler enjoying the extra property of “stability” which 40 K criterion one should take theof Zariski QG. closure over vanishes. This is typically trueholds for when finite-mass solitons inautomatically spaces satisfied by theN special soliton the algebra integrate the equality ( unique finite-mass soliton inwith the the same given homotopy generic class.mass finite-mass until map One we reach gets itsmasses this minimal allowed soliton possible in by value. the starting Itsdensity given mass condition saturates topological of the class, Γ lowerdoes just in bound not like on escape the it to was infinity a in BPS field space state. nor develops The singularities. Zariski- theorem 2.1) that if there is any smooth map whose lift semi-simple Lie group without compact factors. group M may reduce to the case that hence infinite. Seeing Zariski closure positive dimension. The target( space Γ smaller space. The special ones yield solutions of the math rigidity theorems wemanifestations use of this in basic physical this idea.Q paper If, in (including addition, the the stable VHSis special neither structure soliton preserves required theorem) by are 4d sugra given topological charge then implies rigidity of the finite-mass stable brane amplitude. All JHEP09(2020)147 i ]: S is Y i 36 Y 6= 1 of and a (4.14) special S -algebraic -algebraic . All other do describe Q Q = 2 w N is Zariski dense theories. This is ). Γ ) divisor. Since Z ) in the -algebraic subgroup , Z Q ( snc + 2 , susy m m G + 1) is minimal in the following ⊂ i m around each component Y , is then non-trivial unipotent. i i i U( Y γ / by Γ. This is the statement of P ) is the smallest R 41 = , Q geometry, these solutions A quantum-consistent Γ Y ∗ + 2 Γ (seen as matrices) have no eigenvalue . The physical meaning of the auxiliary tt e S ∈ m equations arise in this way for some dense \ γ ∗ tt Sp(2 = Γ -Zariski closure of Γ – 36 – \ -soliton version of the conjecture we are proposing: S Γ Q which contains the image of a lift of Q is a simple normal-crossing ( PDEs are an isomonodromic problem, and the solutions are ≡ ) ∗ ∗ R ) cannot be extended along any prime component tt Y In particular the monodromy group . By definition, , tt (1) m When the VHS has a geometric origin, the Q if all elements 1.6 D . Γ \ informal + 2 ]) Γ . The Hironaka desingularization theorem implies that the neat m Y 36 → are all non-trivial unipotent elements of Sp(2 of the VHS with a finite cover, we may (and do) assume Γ neat. physical , \ i in eq. ( S S Y Sp(2 . 28 S : p , ⊂ w = 27 ) is called around each prime divisor at infinity, ]. n S M i 30 (hence torsion-free) finite-index subgroup. We work modulo finite groups, γ GL( 42 ⊂ semi-simple ), that we denote as (See e.g. [ Q so that is ) may be chosen so that ( ) which contains Γ. Q m a stable one of finite-mass. Y Q Γ G ( S,Y ; this is equivalent to saying that the monodromy m The most important property of Γ is that it is semi-simple. To formulate this condition The monodromy group Γ of an algebro-geometric VHS enjoys special properties [ The statement sounds very physical in its wording, but it refers to an auxiliary field This leads to the following All finite-mass stable solutions to the From a different point of view:A the group Γ Y G (the “moduli” space) is quasi-projective, that is, there is a projective variety around the prime divisors 41 42 i γ group determined by the monodromy Γ they preserve. norm 1. In particular, inelements are a unipotent. neat group In there particular, are after no passing non-trivial to finite the order finite elements, cover and as all in quasi-unipotent the main text, the monodromies precisely, we introduce the notion ofgroup the of Theorem Γ is finitely generatedcontains a (in neat facts, inso, by our replacing particular the base context,The even monodromy finitely presented), and pair ( is assumed to besense: complete the for period theof map Hodge metric, non-semi-simple [ 4.3 The structureA theorem basic of fact algebro-geometric about VHS S a VHS which hasdivisor an algebro-geometric origin is that its complex base in a Lie subgroup special K¨ahlergeometries belong to the swampland. theory which lives onfield the theory moduli is space unclear to me. Swampland structural criteriongeometry (informal). is described by a that is, discrete group Γrigidity: and they it are ishardly analogue uniquely a to determined surprise: thefinite-tension in rigidity stable the BPS of usual branes. BPS applications configurations of in JHEP09(2020)147 – (in 29 (4.18) (4.15) (4.17) (4.19) (4.20) (4.21) (4.16) ) have odd Q Γ 45 preserving the G of the VHS, i.e. this is , ), contain a compact with a finite-cover if Q m M ]; detailed treatments [ R Γ ( S is a normal subgroup of = 28 44 , ≡ is a non-compact, simple, A by the monodromy group 1 Q A -valued points, for clarity. , i 27 2 P Γ Q D × h M = 2 supergravity the possible factors entering in /H, ⊂ k ≡ ) ) and P i ` R M N P , . R ( = 1, M ) ) ≡ ( is i G -algebraic group. By definition of 0 ` -algebraic group Q , m ⊂ , 3 Q M Q H ≡ i M × · · · h U( , G. D + 2 +1 m × -algebraic groups ` ⊆ -algebraic) group , but only the ones whose underlying real m Q = Q Q M 1 M Γ , × · · · × with Γ have the same rational tensor invariants so, = 2 with 2 Sp(2 ) is itself compact). Therefore, not all simple ≡ h = U(1) Q – 37 – R , M -algebraic group is an almost-product of simple ≡ R ( ` Γ Q Γ (2) × A m m × = 1, 1 H G 0 P D, , M 3 \ R,R ≡ ]. In our application to = (in particular, the first h ) of the real-valued points of Γ ≈ × 60 R G H × · · · × ( Q → i M -algebraic group with the group of its P 2 -algebraic groups. Hence (replacing Γ of [ Q Γ S M Q = of : × p 1 A M of the Mumford-Tate ( ], a semi-simple M . We have the inclusion of simple ; in our case, 79 and ) is a semi-simple (resp. a torus) 43 Γ = Γ D ] for the list of Lie groups satisfying this more stringent requirement. i (2) m p,q A figure 6.2 is the isotropy group of a reference Hodge decomposition with the given h ) has a Vogan diagram with trivial automorphism, that is, the ones in M D 29 R , ’s are ( ≡ H i i 27 and ) are further restricted by the condition that the weight of the VHS is ) is the group of the real points of the M M (resp. means equality up to finite groups (i.e. modulo isogeny). Since we are working modulo finite i /H R i ) ( ≈ 4.18 M R G M ( G For a general (that is, not necessarily of CY-type) pure polarized VHS, the period map All factors Here For the theory of the Mumford-Tate groups, see e.g.: nice shortIn surveys the [ equation we identify a ]. has target space the quotient of the Griffiths period domain 43 44 45 -algebraic groups may appear as factors of -algebraic groups groups, we shall not distinguish between an almost-simple31 group and its simple quotient. Hodge numbers and where polarization; in the 3-CY case with Hodge numbers In eq. ( groups fact 3), see [ p the property that the Liemaximal groups torus of (so their the realQ points, Abelian group Lie group figure 6.1 where Zariski closure, theroughly two speaking, groups they Γ are “algebraically and indistinguishable”. which entails that the group real Lie group. Onethe can derived show that, group modulo finite groups, Q where the necessary) we may assume By general theory [ JHEP09(2020)147 ] ∗ tt 28 , ] over global (4.25) (4.23) (4.24) (4.26) (4.27) (4.28) (4.29) 27 63 , ) we may . C i 32 , m 4.21 S 30 46 ∈ ) through the adjoint factors. Of course ? . R ] 1) in the associated ` ( i take the form [ , ] (4.22) H 1 M $ H ∩ − -module of ) i )). ∩ , C i R ) H (  m i i R ( 2.15 D i H M \ [  / Γ M ) [ ) ( / ) (4.30) ]. To each factor in ( R / 1 R ( ) , , ( i 1 89 1 i . R , ) − i ( 1 M k,k i M  (cfr. eq. ( R − at a reference point m  − D. ( i ( g g M i +1 S R Γ G k ` O ∩ → , `  M Y k D =1 = i Y i C i D of M ` R ∩ associated to the =1 ` =1 × i m M i Y g p ≡ ≡ i D =
H ] ] – 38 – ⊆ In a VHS of algebro-geometric origin the ≡ P = def × C /H H H ) . = ) D 1 def P Q P , is the component of type ( \ ∩ ]) ∩ 1 i R ) be the homogeneous vector bundle [ D ⊗ D ( 1 1 Γ TS ) ) − i , , i H  ( . For instance, equality holds — modulo finite groups 30 g 1 1 / R R ∗ m – Γ ( ( G − i − M $ g P 27 R m [ [ = → q ( / / ) ) O S ) acts on the complexified Lie algebra R M R : S R ( ( factor. ( , and i $ P R C R M )) factorizes as in the commutative diagram Q D is necessarily non-constant in each one of the = = def def ⊂ ⊗ i ) P R $ i 4.18 H D D M ( (cfr. ( Lie p (Structure theorem [ ] for details). Then the infinitesimal period relations for = is the holomorphic tangent bundle to C i 30 – constant in the m TS q 27 The main result in the theory is the We focus on the non-trivial factor and consider the map The Lie subgroup should still obey the Griffiths infinitesimal period relations (i.e. should describe a 46 Theorem period map We have the obvious embedding — for all universal familiesassociate of a complete sub-domain intersections of [ the period domain For “generic” VHS one has where representation. As explained in the text, this action preserves the Hodge decomposition of adjoint Hodge structure on the Lie algebra (see [ geometry, cfr. section 2). Let the reductive homogeneous space where where Note that the map $ with JHEP09(2020)147 . i 3. , M 2 = 1, , 0 (4.31) (4.32) 1 , 3 , h = 0 ). Hence if , ` S ( is locally sym- ): ··· $ S , 2 4.15 ]. , -Zariski closure 98 Q = 1 . ) ) being locally symmet- , eq. ( Z S , ( P $ ) and hence it an arithmetic ]. ) of its is an isomorphism of complex + 2 Z , k 97 Z ( and its image [ i ( )] m $ i ] of the rational algebraic group ) (4.33) k S M Z M simply-connected reducible special ( 65 thin , Sp(2 m SO( all G 64 ∩ of factors in × i ∩ is onto, the Griffiths infinitesimal period ]. All families of complete intersection ` M Q i 32 $ Γ of algebro-geometric origin has ≡ ) [SO(2) ≡ S / Z ) – 39 – ) ( Z m ( , k i ). (2) If G ] we may identify is of finite index in M ]. However, there are geometric situations with thin ∩ 75 i = 0 (in this case the corresponding algebro-geometric – SO(2 89 ⊂ 4.31 i ] which says that i S 73 × M Γ 88 C is locally isometric to the Hermitian symmetric space = def S ) rigid 3-CY). U(1) / Z = 1. ) ( i ` R is locally symmetric, in which case ) is an arithmetic subgroup [ , M . = 2. Z P P ( k , e.g. a i D D \ SL(2 \ M Γ rigid ) become tautological, and this is equivalent to ∼ = = 3 iff (1) A special K¨ahlergeometry 3 if and only if S ` , 4.30 ), we can say a bit more on the number is of infinite index in the arithmetic subgroup = 0 if and only if dim = 2 (1) By “global Torelli” [ i m is onto iff Γ ]. with in all other cases object is ` ` In this case the monodromy group is said to be the monodromy (sub)group Γ group on its own right; Γ 1 the special K¨ahlergeometry is locally isometric to a product; we then apply the = Simple situations have arithmetic monodromies: in particular, when We write 27 and one has $ • • • • • 1 i , 2 K¨ahlergeometries have the form ( relations ( ric [ manifolds, Proof. ` > Ferrara-van Proeyen theorem [ (2) In the case that our VHSh has the Hodge numbers of a special (i.e. K¨ahlergeometry Lemma 3. 4.4 The case of “motivic” special K¨ahlergeometries metric the monodromyCY is have always arithmetic arithmetic monodromymonodromy: [ [ the simplest such example is the mirror of the quintic [ By construction M by Dirac quantization of charge. We have two possibilities: JHEP09(2020)147 is ]). d and uv 4 is an 100 (4.34) R d M modulo 4 ˙ R α ¯ Z, A ˙ β ¯ ˙ Q α  . When d AB  algebra and locality = 2 supersymmetry. = = 2 case the several ). } ˙ N β ). N susy B belongs to the swampland 4.15 conformal central charges ¯ Q finite, (4.35) S is a simple normal crossing , 4.25 rigid ˙ α ≡ Y A ) finite ¯ Q is also finitely-generated. Indeed . This allows to define the chiral uv for some projective variety { c d ] , carrying a linear representation of 4 H ∞ − 99 Y R H Z, \ uv a operators commuting with αβ M  4(2 47 AB ≤ = 2 SCFT (see discussion in section 5 of [  ) and inherits finite-generation from it. – 40 – d = N uv d 4 uv 4 } -algebra with unit. The second crucial ingredient is R B β R C A special K¨ahlergeometry -anticommutators. From the is graded by the conformal dimension ,Q ¯ . Q is the effective divisor with support on the locus where d A α uv 4 Q D R { Y , hence has the form , = 2 QFT the chiral ring C µ P satisfies the VHS structure theorem ( N ˙ = 2 case we propose to replace in the β µ p α #(generators of of quantum operators acting on γ ), where ]). N . It is convenient to work with the smooth locus in the Coulomb branch, D B is a commutative A , and we may assume without loss that 27 Y is a deformation of A ∞ d algebra Y δ 4 d Y = 2 theory is described by a weight-1 VHS over a smooth quasi-projective + 4 \ = R R (Structural criterion) ∞ N } M Y , whose chiral ring susy B β ( as the subfactor consisting of scalar = 2 quantum gravity setting subjected to two assumptions: uv ≡ \ c ,Q d ˙ 4 = 2 α N polynomial ring we have the unitarity bound [ M M A R From the above we see that we have a proof of the structural swampland criterion also Then, by the Hilbert basis theorem, the spectrum of a quantum-consistent In particular the discrete gauge group Γ should satisfy eq. ( and N ¯ The restriction to scalar operators is for convenience. ≡ Q { free 47 uv group Γ to bee.g. neat. section Then IV the of [ structure theorem holds (forin a the sketch of the argument, see M some extra degree of freedomcomplete becomes 4d massless. Then thebasis low-energy description of adivisor. UV By going to a finite cover, if necessary, we may also assume the monodromy and one reduces to this caseBy in RG all flow, known affine variety defined over effective divisor in this case wea have a well-defined UVa fixed-point, with and of an algebra ring the ones which may bewe written learn as that that in a UV complete Let us focus onof the a crucial well-defined ingredients quantum of (complex)the the Hilbert proof space in QFT: the first one is the existence 4.6 A directAs physical sketched proof? in section 1.3a the necessary structural condition) swampland for criterion low-energy may effective be theories actually with proven (as In the (ungauged) swampland conjectures by the following single one: Criterion 1 unless its period map 4.5 The structural swampland criterion JHEP09(2020)147 ]. S 64 sugra to be (4.38) (4.36) (4.37) R ] while ). This . Then e S j ¯ p is also a i 102 S K ): compare to the is zero-dimensional Y ] are implied by the S 3 . Their K¨ahlerforms are re- This is the most tricky state- which is the one entering the p j ¯ ), the WP metric is tied to the i ) by “global Torelli”, G S ]. ( 3 2.26 $ ∼ = (3) (3) (2) m m m D D D S \ \ Γ Γ 3 $ of the curvature of the canonical homogeneous ], whereas the Legendre map → → −−−→ – 41 – S S S 30 (2) , m : : p z D and 24 by construction. However the OV conjecture refers to is related to the fact that Γ should contain non-trivial L = 2 supersymmetry. In this case the chiral ring (which is locally identified with its image under j ¯ i S N S K by adding to it the points at finite distance in the WP metric The first OV conjecture states that either over the respective target domains. L complete for the Weil-Petersson metric. Indeed it is known that ) a ( m ]. In particular, a conifold point is at finite WP distance [ we found convenient to define the covering special geometry D not is → 2.1 102 , and the Hodge metric to the period map ) reduces to a point, and since ) , S z a S ( ( L is finitely generated. 101 $ sugra is non-compact. unitary representation of is well defined; R the asymptotically-Minkowskian quantum states form a Hilbert space carrying a In general S 2) 1) is geodesically complete for may be extended continuously towords, the we conifold may point complete aseven we if shall we see cannot extend momentarily.peculiar there In situation other is related to the fact that the fiber of the projection cannot be extended to such a point [ line bundles the moduli spaces ofmetric, Calabi-Yau see 3-folds [ is typicallybeing non-complete at infinite for Hodge the distance. Weil-Petersson The period map geodesic completeness with respectscalars’ to kinetic the terms. WPabout Each metric special one geometry: of theseLegendre with two map reference metrics to is diagramstrictions associated to ( to the respective its images own viewpoint Godement non-compactness criterion for finite-volume quotients of(2) symmetric The spaces [ scalars’ manifoldment. is In geodesically section complete. the completion of aS certain “moduli” space with respect to the Hodge metric (1) or it is non-compact.is Since compact a VHSpoint. with a The compact non-compactness base of unipotent has elements a (i.e. the constant monodromies period around map, the if prime divisors in 4.7 Recovering theWe Ooguri-Vafa swampland now statements show thatabove the criterion Ooguri-Vafa i.e. (OV) by swamplandnumbers). the conjectures Most VHS [ arguments structure may theorem already (specialized be found to in the [ appropriate Hodge From the semi-classic viewpoint these two assumptions look reasonable. JHEP09(2020)147 ) (3) m m D finite U( states (4.41) (4.42) (4.39) ]. For matter / ⊂ infinite ) 9 , R e L 8 coupling of ⊂ m, )) t ( some e γ , including the one ( finitely many 3 IJ , with positive-definite τ $ ij τ → ∞ ). We conclude that such z makes sense only at the linearized the situation is akin the ones 2.32 develops a singularity at 48 ¯ a -linear combination of the electric ]. Thus a point at finite distance ; near the singular point the dual C | ) may be extended continuously in da d 22 a (3) [ m | | D vector couplings 0 i.e. Im a D | Z which approaches a conifold point may j ¯ i → × a, a all log Λ) + regular (4.40) R a (2) symmetric matrix m (1) m a/ ∝ + D gravitational sector as D m j 2 ¯ i ⊂ – 42 – × log( ds vs. , be the period associated to the charge of the light ) a e −→ S CG m specifies which describes a a ( is a copy of Siegel’s upper half-space Sp(2 πi q ⊂ (2) m O 2 (1) 3 (3) m m central charge ) D t $ D )) + D = ( e γ z D πiz a 1 matter sector susy 4 Im = and the graviphoton couplings remain finite, but ¯ z = exp(2 = 2 effective Lagrangian, i.e. in Seiberg-Witten theory [ z a Z R = 0 where the hypermultiplet becomes massless would be at 6= 0. Clearly the periods ( N a q 0, while the K¨ahlermetric (where → z has the form a D a central charge Since the problem arises purely in the matter sector, a suitable constant) for instance in the Hodge metric ( We stress that the separation 48 C finite-distance singular points correspond to locibecome in light, scalars’ space spoiling where ourplets low-energy only. effective description in terms of light vectorlevel, multi- i.e. in the first order expansion around a fixed scalar field configuration. so that the point distance in any K¨ahlermetric of the form ( distance. The Ricci tensor hascoordinate the form of the standard Poincar´emetric in the upper-plane hypermultiplet so that itsperiod mass is proportional to for some integer the limit appearing in rigid comparison sake, let usscalars’ metric recall arises what at points happensbecomes in massless, in the making Coulomb the branch incomplete rigid wherefree the some photon case. low-energy super-multiplets charged effective hypermultiplet only. description The Let in singularities terms of of IR- the and magnetic charges isin the the WP metricsusy but infinitely away inthe the matter Hodge gauge one vectors is blows up a (or field vanishes, configuration depending where on the the chosen duality frame). whose physical interpretation ispoint easily in the understood fiber, from seenimaginary as the part, a Penrose is symmetric correspondence. nothingvector fields, else whereas A the that point the in of the matrix graviphoton. of The complexified point couplings in of the stretch to infinite lengthremains in at the finite distance fiber in direction the while base. its We projection may be a little more precise: theis an Penrose embedding, map while the fiber of is non-compact, so, a geodesic JHEP09(2020)147 ) ˇ D Q the ). , . i → (4.46) (4.47) (4.49) (4.43) (4.44) (4.45) H + 2 4.41 πiτ m 2 as an open e m ˇ . The non- : ∆ is a compact D (2 L F ˇ = 0, in which ⊂ ) = D sp i j τ D ( ]. κ q 30 ) and , , = i
i i S N q τ 28 , ∩ Im ^ π 27 U , 2 be a small open set (in the with exists. The extended metric − ∼ = = exp( e 25 S , . z s i , } − the punctured unit disk, 24 U ) γ ⊂ O [

m ’s. For instance, let us approach m e } p i is U 6= 0 1 D ∆ , z ) + N ≡ } a = 0, so that < × =0 (3) U m s j | s , z ··· q e p z i is described — up to corrections which z | at fixed values of the other coordinates , D = 0 | q ( F i Y +1 ··· 3 z < )-homogeneous space such that → s F { 2 C · σ $ 0 z , S , z ` j 1 → ∞ ! s , − H ←−− : i D, z + exponentially small (4.48) N j e + 2 p C , q – 43 – N z s 2 : 3 i ) → m − ∈ τ j ` $ m ··· z z S { j , s { ≡ =1 ∆ κ 1 ∩ i X q e z = ( × ^

U 4(Im From this expression one gets the asymptotic behaviour ) = Sp(2 s ∗ ]. More concretely, let : ) C 7→ = max ∗ ( U = ) 49 j e p G j 105 m (∆ κ of the Griffiths period domain z , takes the form j ) = exp ¯ z , z ∼ = a ) that the map Y , i.e. we take Im G 103 z the universal covering map j S ; ··· i Y -tuple of non-zero, commuting, nilpotent elements of σ , τ ∩ 4.46 ] sot that we may extend continuously the covering Legendre manifold s ( U +1 U compact and Y simple normal crossing, the asymptotic behaviour of ] s e p 101 S , z compact dual s 103 , τ = 0 [ ’s are a i with j is the q N ··· ] , Y ˇ 1 D τ \ . Set 101 a :( S -th prime divisor Let us discuss in more detail the behavior of the special geometry at infinity. Since The original OV conjecture thus holds with the specification that “scalars’ manifold” j , z σ Here j = there. 49 6= i e extends to L projective complex manifold and a domain. so completeness follows from comparisoncase with we the see Poincar´emetric, unless from eq. ( of the metric atthe infinity in terms ofq the action of the Then [ where the such that the monodromyis around a the regular divisor holomorphic map. The local lift of the period map, takes the form [ where ∆ is theupper open half-plane, unit and disk, ∆ the special geometry asare we approach exponentially a small point in in nilpotent orbit the theorem geodesic [ distanceanalytic topology) — where by the multi-variable version of the should be understood totrivial mean part the of WP the statement completion isis of that not the a smooth continuous Legendre extension since manifold of its Ricci curvature should blowS up at the conifold points, see ( JHEP09(2020)147 ] ]. ), 21 S 21 and ( = 1 (4.50) (4.51) (4.52) (4.53) $ 1 S 4) [ , , 2 2 5 1 h x , 1 , This is con- 1 ) = , 3 (1 sx ]. P + There are 2 ) be a loop encircling 107 is below by a positive , sx 1 50 . + − 1 ). For more details and 106 comp 1 ) σ L 0 x 5 ( σ ). Note that near a conifold )( 1 4.48 √ 2 π 2 x is non-positive. ∈ 4.48 + 3 + x ] shows that the volume of 1 } , S σ x 5 24 ∞} )) but this very fact implies that the + √ , Γ (because Γ is assumed torsion-free) 2 4 1 − , ∼ = sx 0 4.41 3 4 points ) , + S \{ 1 2 3 ]. Ref. [ ≥ ( , 1 1 ] states: n 24 sx P π 3 ∞ – 44 – )( , \{ ∼ = 3 1 1 , , is finite when the VHS has a compactifiable base sx 0 P j ¯ i ( + = comp. K 2 groups, \ is finite [ L x S ], i.e. a certain crepant resolution of the one-parameter 1 punctures being non-semisimple). There are plenty of such S P + 4 n 108 ≡ x The above asymptotic expressions, together with compactness )( S . The length of a path in the class 4 Conjecture 4 of [ distance in the WP metric. ] applies to this metric as well, since the two metrics are simply x ∈ 24 ∞} s + . , 1 3 1 x π , finite 0 + degree 8 hypersurface in the weighted projective space 2 x ∈ { ]. The OV conjecture refers to the volume computed with the WP metric ]. It follows from the asymptotic formula ( + x 3 1 24 [ x ( are MUM points, whereas the other three punctures are conifolds points, see [ singular 4 Y x \ 3 In the scalars’ manifoldwithin there a is given no homotopy non-trivial class. 1-cycle with minimum length has finite volume. x ∞} S 2 , , show that the volume of Here we adopt a conservative attitude: we take as “scalars’ space” the nicer S I thank Cumrun Vafa for a discussion on the issue. x . The argument of [ 1 S 1 j = ¯ , 50 i 0 x constant, so the wording of the conjecture needs someslightly minor modify modification. the statementmology of class”. the conjecture Since, by as replacing abstract “homotopy class” with “ho- Alternatively we maymanifold take as “scalars’ manifold” the metricallyin which completed the three Legendre conifoldthe punctures puncture are filled in. Let where { examples. For instance,number the 254 double in octic Meyer’sfamily corresponding of list to [ the eight plane arrangement The precise wording ofCalabi-Yau 3-folds this whose conjecture complex moduli requires space some has refinement. the form (the monodromies around the special geometries consistent with the structure theorem(5) — Properties see of [ computed with the HodgeS metric G related in the asymptoticprecise results regime on as the a WP consequence volumes of of eq. CY moduli ( space — which apply in general to all jecture 3 of [ point the curvature issingular indeed point positive is (cfr. at eq. ( (4) of (3) Asymptotically at infinity the curvature of JHEP09(2020)147 j ¯ i S ]. G 3 ⊂ )— finite S (4.54) (4.55) are at ( Y matrix 1 Y π ⊂ are at i ] whenever Γ j Y Y 109 ). Then -Zariski closure, we . From the discus- Q 4.25 belong to the swampland sugra infrared divergences worse
e j S as in ( \ Y p (Γ) ], for nice and detailed analyses Z particles becoming massless along does not in the algebraic sense. 32 , Sym Γ — has the same tensor invariants as N / . Modulo commensurability, the group P, +2 Z Vol (Γ) ) ), the points on the divisor m ≡ but not necessarily in the WP metric Z  e 2 at least S may be chosen simply-connected. I am not ( j ¯ Q Q i . S Sym Z S 4.41 ) K ∞ N e (Γ): Γ finitely many Sym S – 45 – of the scalars’ manifold with a sub-locus Z ( < = S Sym ]. It is known that these results imply the distance N Note that from the structure theorem Sym  When the special geometry satisfies our structural glob to be simple (otherwise Γ is automatically arithmetic (Γ): Γ] 51 G . In other terms, Q 105 Z All global symmetries should act non-trivially on the ) = , Γ P e Γ be the subgroup generated by all unipotent elements of S Sym \ / ≡ N 103 tower of states get light. (Γ u Q Vol -symmetry is a gauge symmetry in Γ is an infinite normal subgroup. Taking the R ≡ u Q u infinite , one has [ generated by nilpotent elements. In other words: the Ooguri-Vafa Γ 0, Γ ∞ u we see that the global symmetry group is is a special geometry with period map < ]. ) S e S > S ; we may assume 2.2 114 \ Q C -orbit theorems [ (Γ) is the normalizer of Γ in – . In order to have a more severe singularity of the metric, strong enough to 2 Z (Γ ). Then Γ j is automatically trivial. sl Y 3 ⊆ Vol Sym 110 , Q u N rhs Γ 102 We stress that, in the present set-up, the distance conjecture is almost tautological. Let us be general. Let Γ Indeed one has 51 in the and, since bosonic fields sincesion the in section where the locus make the WP distancethan infinite, those produced the by theory any finitepossibility should number is of have that particles at an becoming massless. Clearly, the(7) only No global symmetries. — we have awhere natural compactification the special geometryinfinite gets distance singular. in the All(cfr. nicer points paragraph Hodge in 2). metric the As discussed primeWP around divisors distance eq. if ( the singularity arises from orbit and conjecture. For asee rough [ sketch of the argument see [ Our viewpoint is that — whenever our special geometry a discrete group Γ statement above holds (after refinement) for We feel that this algebraic version is(6) the most The natural formulation distanceswampland of criterion, conjecture conjecture. 4 its of asymptotic [ behaviour at infinity is described by the nilpotent- Γ: when dim have by lemma Corollary 7. seen as a concrete matrix group Γ acting on group, Γ is generated byis unipotent an elements. arithmetic This groupsuch is of as automatically all rank true complete at [ intersections),aware least of or 2 an when example (as where it neither happens conditions in apply. most “elementary” examples the conservative version of the conjecture is equivalent to saying that, as a concrete JHEP09(2020)147 ], all ] (so 116 -space ] refer . Then , which 117 Q q V 120 – 118 , 5 , 4 = 2 Pauli couplings N 0) projection of and these span , } v Γ { charged state exists. That charged = 1 modulo finite groups. These conjectures [ one is a finite-index sublattice of the electromagnetic symmetry group Γ. glob G occ ). The square of the susy central charge The group Γ acts irreducibly on the q , q gauge – 46 – C ). In the present context — low-energy effective ( This conjecture states that there must exists states φ ( Q V = 2 supergravities — the scalar potential is identically zero This conjecture states that the couplings in the effective La- N 6= 0, we have states of charges , and the actual v Z ) gives a first illustration of this state of affairs. Saying that the e S and hence follows from our previous discussion. ( Λ. Equivalently we only show 4 52 ≡ ] ungauged Sym is purely cubic, is equivalent to saying that the occ , e S 115 , F , where Λ is the symplectic lattice of electric-magnetic charges. Thus, if are either the v.e.v. of light fields (so they are non-trivial functions on the 5 asserts that in a consistent theory such numerical Pauli/Yukawa couplings Q = 1 heterotic Yukawa couplings) are field-independent numerical parameters. Z is its Hodge squared-norm 4 L ), or they are frozen to some very specific “magic” numerical value. In the UV ⊗ N S q Since in this note we work modulo commensurability (and so use rational VHS rather than integral VHS) = Λ 52 theories which are and all these conjectures are trivially satisfied. we may only conclude that thelattice occupied Λ charge and lattice Λ not Λ charge (which is the mass foris a smaller BPS by state) the is Schwarz the inequality. Hodge norm(11) of dS the (3 conjecture,to AdS properties conjecture, of the etc. scalar potential (10) the weak gravity conjecture. for which the electromagneticmeans repulsion that is the squared-mass larger shouldboth than be measured the less than gravitational in the attraction. intrinsic square normalizations. of This the electromagnetic The charge invariant square of the electromagnetic Then fact should be the cubic determinantwhose form classification of is a provided (possiblythe by reducible) adjective the rank-3 “magical” Freudenthal-Rozenfeld-Tits real magic is Jordanbit square technically algebra more [ accurate on for these these aspects couplings). in the We next shall subsection. elaborate a and so are givenalso by subjected the to isolatedaway quite critical from strong the points “magical” consistency ofquantum numerical requirements. some level. values high makes Then energy the Fact pre-potential any effective potential, theory small which inconsistent is perturbation at(or the the (9) No free parameter. grangian moduli completed QG these “magic” numerical values become the v.e.v.’s of heavy fields [ Since we are working modulopossible finite charges groups, are this physicallystates statement realized is exist provided equivalent follows, to say, sayingcompleness from that of all electro-magnetic the spectrum validitysymmetries is of [ a the formal distance consequence of conjecture. the absence Alternatively of the global symmetries of (8) Completeness of chargeV spectrum. there is a statethe of charge physically realized electro-magnetic charges make a sublattice of finite-index in Λ. so there is no algebraic invariant which may detect a difference between the group of JHEP09(2020)147 λ ), is P s ), form given 4.17 (4.59) (4.60) (4.56) (4.57) (4.58) in the l S . 4.59 . ⊗ • S ) , Q • ∨ infinitesimal ∈ , eq. ( V Hg ( s M . ⊗ 3 , k 3 ]). Restricting to ) ⊗ 3 , 20 ) V 5 s 54 n T is ≡ ( has pure type ( s ] k,l λ ⊂ T 30 2 . Fix a point – 2.16 in [ ⊗ 53
27 3 . . [ , 0 s in some suitable normalization. ,  p,q ) H ) l sugra p,q s k,l s − H ⊗ invariant under Γ is defined over T k n appearing in the structure theorem: (
• C = 2 ( , ) n = • ) 1 3 2 q Q l P , integers T M ) which fixes the elements of , 2 + s N − Proposition ⊗ ) p k l Q • , , ,H M = =( • Hodge tensors − 0 ] that the Mumford-Tate group q , (see 3 k s + C – 47 – + 2 Hg 29 ( p specifies a particular Hodge decomposition as in H – Q 3 2 ⊗ ∈ m =  27 (2) V m λ C D ] End( ≡ Sp(2 3 ⊗ ) = ]. 29 ∈ C . The Pauli/Yukawa coupling at

≡ – . V k,l ) 0 68 V p, q , s l, T ( 27 m ( ∈ + 25 p , this entails that a polarized Hodge structure over the fixed k G s

T S F , is a numerical parameter independent of the scalar v.e.v. ; such couplings will correspond to elements of some higher tensor ∼ = ∈ . One shows [ k -space , see [ λ F S s X pure type k,l Q ∂ T j X ∂ i X ∂ is polarized which have , specified by a Hodge decomposition (say, of pure weight implies strong restrictions on the group V ) of the fixed V k,l λ T 4.56 In particular, when the dimension of the Hodge tensors of the appropriate type and In view of the VHS rigidity theorems, saying that the Pauli/Yukawa coupling, or any Let us specialize to a VHS underlying an an element of the complexified space of Since The Pauli/Yukawa is the cubic formHere which we is use the the fact only that invariant the of linear space the of corresponding tensors 53 54 55 -space 55 They cannot betensor arbitrary integers, however, since the presence of a non-trivialvariation of Hodge Hodge structure at the generic point of is precisely the subgroup of symmetry is at mostcouplings 1 — (as if it field-independent happens — for should the be “magical” Pauli/Yukawa couplings) such higher tensor coupling equivalent to saying that itis is fixed by the monodromy Γ. Since space eq. ( More generally, we may consider otherof couplings higher defined derivatives as of suitable invariant combinations scalars’ manifold; its image induces functorially Hodge structures onby all the its Hodge tensor decomposition spaces [ Let us revisit the example inexploiting (9) the of numeric fact (i.e. field-independent) thata Pauli/Yukawa couplings the semi-simple polarized Tannakian category VHS’sthe over over fiber a over connected a complexQ point manifold 4.8 More on “no free parameter” JHEP09(2020)147 . 1 (2) m ] of D (5.1) (5.2) (5.3) field- (4.61) 29 , → ). While 27 e S I : X e p ( generic if there is no F ]. ]. With some abuse of 121 = 0, leading to a contra- 30 σ ,   (3) (2) m m is an embedding and hence S ’s D D C 2 25 ( / & $ , F P P P 24 ). In particular, a P P R S has positive dimension (i.e. we are P satisfies our swampland criterion ι ( e P ι e C P e P σ (2) S S m P \ P D P Γ dim \ P Z  ) = P ) belongs to the swampland Γ P ≡ e ≥ e S S P ( ( P F S P → ) which have Hodge representations [ Iso e e p S /H /H ) Sym \ Q e z ) – 48 – , ). In principle one may translate the swampland Γ R R ( ⊂ trivial, and then dim ( ≡ P +2 Γ / P P 4.25 S m C : cannot be too small since p only specifies (locally) a covering period map dim P $ Sp(2 to belong to the swampland (i.e. to be quantum-consistent) F ≡ the pre-potential not m  G F is. On the other hand, when would imply e S λ P in the following commutative diagram e is locally symmetric and S S case), the group ) into a set of functional equations for the analytic function rigid 4.25 giving a pre-potential Let us summarize our previous discussions of the properties of a quantum-consistent The situation looks very much in line with the arguments of [ The fact that field-independent couplings require the existence of non-trivial Hodge -algebraic subgroups of are conceptually relevant. covering geometry satisfies the VHS structure theoremcriterion ( ( these equations do not look promising as an effective tool for explicit computations, they such that the quotient special K¨ahlergeometry Said differently, in order there must be a monodromy group Γ such that the induced quotient period map A priori, which satisfies the Griffiths infinitesimallanguage, period we relations shall [ saysubgroup that the appropriate kind. This yieldsin back terms our of classification determinant of forms the for allowed rank-3 numerical Jordan Yukawas algebras. 5 Functional equations for quantum-consistent independent coupling diction in presence of light vector-multiplets. tensors leads to a classification ofQ their possible “magical” values in terms of the finite list of there are, the smaller not in the with equality iff should leave all Hodge tensor invariants, so the more numerical field-independent couplings JHEP09(2020)147 . . % 56 (5.7) (5.5) (5.6) (5.4) = 27, = % m is locally S and . Since the P , and hence the 25) − symmetry group /H is identified with 25) , so that ) 7( − , the lozenge stands e S R 7( E 7 56 ( (3) E E P D naive ) of the ambient space is completely contained ) = R R e , S is the ( P The image of the composite % (3) . P + 2 .  , ] (3) m Q 6 /H m and ) D E / / R ( ), so that × 25) ◦ 3 P R − , and that its , $ , ) 7( ≡ corresponds to the black node, which is an (2) m R ) E ] the special K¨ahlermanifold node. ( ) we learn that % [U(1) + 2 D R ◦ J  ). (   P % (2) m m defined over 5.3 ) = m 25) × D is a Hermitian symmetric space. By general ⊆ 25) / 5.6 is a fundamental representation associated to − R , faithful, real, symplectic, Hodge representa- G is − ) / ( 7( % e – 49 – 7( is the real Lie group whose diagram is obtained 25) S % . Since ) the Sp(2 P ◦ ◦ E ( − % , E R ) which makes the following diagram to commute: [U(1) ι e J % 7( ⊆ ( P / R (3) E = ) ) ( P ), Sym ) = D we know that the universal cover P R R P R ( ( 25) ( R ) ◦ is a maximal compact subgroup, so irreducible ( − P P /H P R   2.2 7( ) P ( (3) = E R 25) P ( − D . From diagram ( (3) ◦ P P 7( e S E D ⊂ ) is compact, -algebraic group For instance, when 25) Q . − ) 5.6 7( )-homogeneous complex manifold 2 (3) E R is induced by an ( H ι e of the real Lie group P % orbit of the subgroup ] (figure is a . The diagram of the Klein geometries for the example 29 ι e – 3 ) is the subgroup of the automorphism group Sp(2 e 27 $ which fixes (set-wise) S single From the analysis in section ( (2) m a complex submanifold ofSym the Griffiths domain D in a isotropy group in ( theory we conclude thatisometric when to the Cartan symmetric domain ( Example we get the holomorphic domain which in this particular example coincides with the MT domain which may be easilyWhen constructed (as it with looks thea to help node be of in the the the generalby Vogan case) weight-3 deleting diagram Hodge from of the representation Vogan graph of By definition of the map tion [ for the highest weightextension of the node, representation special geometry, is locally symmetric. The inclusion Figure 2 The white/black color specifies the particular real form of the Lie group JHEP09(2020)147 U ): ) in i (5.8) (5.9) K z (5.12) (5.11) (5.10) F ( X analytic ( F I ) relate F + 2) matrix m 5.11 global , m. (2 should be seen as × ··· e L , shows. . . may be multi-valued. 2 ) J ) acting as a group of 2 , + 2) R F Q X that is, the values of all , , m = 1 IJ . This should also be seen ) which fixes the Legendre ) refer to the B + 2 + 2 F is not known. Typically we , ) (3) + P , i m m to the full Z F ) for all elements of a finitely- 0
5.11 , D (or rather the function F K /X 56 Sp(2 Sp(2 5.11 i + 2 X F . X ⊆ ⊂ J m P F = P ), homogeneous of degree 2, be the global J /H i J I ) = Sp(2 Γ the functional equations ( A R X physically invisible, Q ( ( ∈ ⊂ = Γ P F ) is, tautologically, a group of automorphisms γ Γ
, z e – 50 – S ) L = ∈ ( def 0 X ! , L (2) ) P /X Sym i Z K D should be , X D AB CD ( F

+ + 2 F (in facts of its subspace 2 L = ) m . F 0 γ e (3) L m X where the analytic expression of KL D Sp(2 into itself may be used rather straightforwardly to constrain the should be invariant under Γ C e L + 1) functional equations for the holomorphic functions ⊂ U I ) = ( ) whose generating function is the homogeneous pre-potential (3) i m m Γ e F S D ( to all of e ,X z . Then the functional equations associated to elements of the subgroup is not a contact manifold in general, as the example in figure 0 ⊂ F m = X e Let the holomorphic function L (3) P ( C e L D F ⊂ satisfies the system of functional equations ( U )) has a local analytic expression as a series which converges only in some small Γ which maps F 5.9 We conclude The existence of the global analytic continuation of In most applications, the putative pre-potential ⊂ Note that 56 U Then generated group Γ such that observables are independent of theas choice a of fine determination point of in our swampland criterion. Criterion 2. analytic continuation of a local pre-potential which does not belong to the swampland. establish whether its has the right branching properties. a very subtle part ofevidence our that swampland the criterion. correct Asthe statement far global as is holomorphic its that function uniqueness the goes, difference there between is some two determinations of eq. ( domain Λ terms in the series,to but far for away most regions elements do of not even know if a global analytic continuation exists, and when it exists, it is hard to The meaning of these equations isIn a facts bit this subtle is because the thecontinuation generic function of case. The functional equations ( we get a system of ( In particular, automorphisms of the ambient contact domain. Then, for each (2 ent space From the diagram weof also the see contact that manifold submanifold and all naive symmetries arise from automorphisms of the smaller homogeneous ambi- JHEP09(2020)147 ) e ), S R is 3 ( 2 (6.1) (6.2) P ). Let R ( ). If the -algebraic P . However, Q Q I where , ) given by (at F Γ) which lead Z namely, its un- -Zariski closure , + 2 /K . Criterion ) of the covering ⊂ ) Q e F S m R +2 ( ( m P . ]. Then we have one Sp(2 Sym R ). . ]) and are all symmetric semi-simple 89 p ) is purely discrete, or [ F 2 e ≡ S 88 smooth simply-connected). ( 5 , ) per generator of the Siegel ): this happens, say, for the ) = P Q e S e Iso 32 Z S Γ ( , -algebraic group (cfr. lemma for the ⊆ 5.11 Q iso ) + 2 P e S ( m ) = (but more general). e S 3 ( Sym D Sp(2 \ is a Shimura variety. As already mentioned, Γ sym ≡ +3 functional equations for the S → m ). We write – 51 – swampland structural criterion: S Γ (or even infinite subgroups Λ : p ∈ 4.25 γ ) = 0 or e S ( (dicothomy), that is, satisfies our sym 2 for the real Lie algebra of the group of its real points S to be irreducible of positive dimension, since simply-connected . Then e S ] yields an economic set of generators for Sp(2 R e S be a special K¨ahlermanifold (with Q e is isometric to the Hermitian symmetric space either ⊗ 104 S \ p e S Γ ≡ ) or one of the ‘magic’ isometry groups in table ≡ R p , m be the real Lie algebra of the naive symmetry group S ). The simplest possibility is Γ Z R , p and Let of the monodromy group Γ is an almost-simple Q ⊆ + 2 We may assume . In this second case the moduli space Γ ) P is similar in spirit to the DG statement fact e 2 S m That is, either the naive symmetry group of Since Γ is infinite this looks like a huge set of equations. However not all of them are ( 5 ≡ Q Proof. reducible special K¨ahlermanifolds have been classifiedspaces, (see so [ the statement isΓ trivially true in the reducible case. Then the is either SU(1 is a Hermitian symmetrictable space and hence,fact being special K¨ahler,is one of the spaces in special K¨ahlergeometry In the second case derlying Griffiths period map satisfies the VHS structuregroup theorem ( sym In this section we prove fact Fact 5. Assume, in addition, that gravity the gauge group (i.e. Γ) determines the Lagrangian (i.e. 6 Dicothomy 6.1 Proof of dicothomy independent system of functionalmodular equations group. of Ref. the [ formmost) ( 3 explicit matrices. Wetypically get there a set exist of other 3 to elements simpler relations whicha yield concrete elementary (if but unpractical) useful realization constraints of on the physical idea that in a consistent quantum independent: it suffices toof impose Γ. the ones corresponding Formonodromy is to instance, arithmetic, the consider Γ finitely-many is theSp(2 generators a generic finite-index quotient situation ofuniversal where the family maximal arithmetic of subgroup complete intersections of two cubics in JHEP09(2020)147 . e S R p (6.5) (6.6) (6.3) (6.4) (6.7) (6.8) ), i.e. a R p ( ]. Then we ), that is, a ). Therefore -Lie algebra. R 32 p Q -mod , hence by the ( ) is simple. 5.4 ) )[ Q R R closed inclusions. ( ( m Γ ι of vectors fixed by e P -mod ). It is easy to see P , ι R U( . ≡ be its [Γ] / T 5.4 Γ R ) p P is an ad Γ-invariant real ⊂ ∈ , , see eq. ( , m R Γ γ R % R p T , Q ⊆ ⊗ (2) SU(1 m s (2)   2 m D ⊗ ≡ D \ ) ( / Γ . ∨ ) for all 5 m , / e ) R S V C ( Z ι ι e Q ( . H ⊗ ⊗ R P -algebraic group  as in the diagram ( p sym Γ V be simple, and let ) Q (  ∩ % ∈ ∨ ⊆ Q P ) is also an element of End p ⊂ P e 1 ) Γ S . R ( e − S e /H p p ⊗ R ). Suppose that R p ( ) ≡ – 52 – -space defines an element of End /H   p p Q R ) R R ( R ( P γxγ Sym ⊗ → R = , p P sym ( ) = ( P s s ⊂ ∨ \ as in the commutative diagram: P Γ / R ), the real vector subspace p → , Γ Γ e p / T ⇒ V s ⊗ p e $ $ = 0 or ) -space of the VHS, namely the representation space of the GL( ) is preserved by the adjoint action of Γ e S s ). Then e   = ( S Q ( def e S S ⊂ ( R ( -spaces R are also fixed by the T R P sym is non-zero and simple. The structure theorem yields factorizations Γ sym -algebraic group ) ) of the Lie groups induces the inclusion of the corresponding (real) ∈ p ∨ and of its lift Q p x 5.7 , that is, p . Since Γ ⊗ Q for the Lie algebra of % p to be absolutely irreducible without loss. R is irreducible, but not absolutely irreducible, the special K¨ahlermanifold % Q via Let the ) is an ad Γ-invariant real Lie subalgebra of the simple real Lie algebra % e ) ⊗ S ( is the underlying p irreducible, faithful, symplectic Hodge representation -Lie algebra V Q The inclusion of = V then follows from the following lemma. sym R 5 ) intertwiner, hence zero or an isomorphism because the Lie group The inclusion ( End( p R ( ⊂ is induced by the irreducible representation Thus Fact On the other hand, so that the subalgebra ι that when is symmetric, in factsmay a assume complex hyperbolic space Lie algebras where the vertical double-headed arrows are canonical projections, and Γ is defined over All tensors in ( group of its realP points where rational, p Lemma 4. Set Lie subalgebra. Then either Proof. Γ-invariant tensor which belongs to the of the period map and its e JHEP09(2020)147 ] 1. of F are  122 α (6.9) 1 (6.10) (6.11) (6.12) 6= ] there F has no η S 122 and 0 . . In [ F C = 1 1 , . 2 1)] -dimensional represen- − is described in ref. [ , h m C + 2) The Calabi-Yau manifolds as multiplication by . = 61 0 There are known examples of or a product of a K3 surface m , SO( are divided in two classes: (1) 1 3 , 1 (2 A × H 270 h , )) is irreducible but not absolutely . R ( J sugra 264 P X and [SO(2) I , / Σ, where Σ acts non-freely, when we froze / X 1 = 2 1) , 6 2 261 T IJ − h . Seven explicit examples of this situation are N , · A 1 – 53 – ]. 1 2 , m 72 X 11 34 = 1 arises from the classification of Picard-Fuchs , ] of numbers = − 1 , 34 2 and F SO(2 is locally symmetric; and (2) models whose 108 h , 0 × S = 84 X 13 1 , , 1 4 U(1) / ) , h ], have cubic pre-potentials corresponding to arithmetic quotients R 6 , 55 ]: they corresponds to the resolved double octics arising from the ≤ 1 , there are only two linear independent periods so that SL(2 2 123 h C ≤ 1 centralizes Γ, so that its representation becomes reducible over α Killing vector at all. We wish to show that the first class is not empty. We distinguish two cases, namely we have the consistent truncationa of higher discrete supergravities group. to theType subsector Such II invariant truncated on under a models Kummerthe arise, CY twisted 3-fold for sector of degrees instance, the of form this in freedom way the to are compactifications zero. listed in of The the sub-sector tables special of geometries ref. arising [ in with an elliptic curve [ of the reducible symmetric special geometry These examples are discussed in detail in the next two sections. In addition to these ones, Meyer arrangements of eight lines [ Quantum-consistent ‘magic’which cubic are finite pre-potentials. quotients of either an Abelian variety operators for Calabi-Yau manifolds withis one-dimensional quadratic moduli spaces. precisely when It4, is the clear so Picard-Fuchs that operator that has over complex order linear 2 combinations of insteaddescribed of in the generic ref. [ Clearly are 6 explicit examples of such CY 3-folds with Hodge numbers Another source of examples with By the structure theorem, thistation happens of if Γ and only (orirreducible. if equivalently the of A the mechanism real(see, which Lie in guarantees group particular, reducibility his over theoremthe universal 2.5). deformation The space idea of the is CY that which there acts is on a global automorphism Quantum-consistent quadratic special geometries. geometric families of Calabi-Yau 3-foldspre-potential whose period map is given by a purely quadratic By dicothomy, the quantum-consistentmodels 4d whose scalars’ manifold local quadratic versus cubic pre-potentials. We consider only geometries of positive dimension. 6.2 Existence of quantum-consistent symmetric geometries JHEP09(2020)147 ] ]. snc (7.6) (7.2) (7.3) (7.4) (7.5) (7.1) 124 , a 125 52 Y of mirror ) through e ∨ = 0 for all S ( 4 X e . z ) ) , i ≡ Q N X , i is non-trivial. We e L Q λ i Γ such that + 2 Y , , see Deligne [ i P

⊂ F m of x m, 0 J } = Sp(2 S w > , . . . , m. = 0 ) for some nilpotent element i C ⊆ i . N Im x S ) Γ = 1 {

Q dim ∈ ( ⊂ C

P , i ) ∈ = exp( ⊂ Z = 0 , w i a compact complex space and } o ∈ γ  ∈ H ) contains the finite-index subgroup m i m i Z x S ]. Here we focus on just one particularly = n ( = 0 Q = m -algebraic subgroup and local coordinates : along a certain direction, we may end up with H m ∈ 127 is G Q , w x i S ) , i.e. by integral symplectic rotations of its ho- -th component , for S ) – 54 – i ··· n i i i ∩ m q Y γ ⊂ (3) ) m ) ··· = ∗ \ 2 D i Q U :( 2 πiw ( x Y S x (∆ where 1 J i ( x i mirror symmetry = ∼ = = 0. ≡ by the strong monodromy theorem. The point N , = 1 i def \{ ). ) S S 6≡ m x q commute between themselves, and (

R J Z i Λ  U ∩ 3 ) acts on the covering Legendre manifold = exp(2 i ( H ) ∈ Z i i J Γ around the N ,X U i = , = x X I N λ h i ∈ S F S λ i ]. The special situation we have in mind is as follows: our special 7→ + 2 ∩ i γ ∩ i exp m w n U P ^ 105 U ). The Q ) = , Sp(2 Q ( ⊂ ]. As we approach infinity in + 2 J m 103 (2 sp The universal cover of Consider the commutative, unipotent, The validity of the VHS structure theorem implies, in particular, that the asymptotic ∈ = 2 effective gravity theory described by that special geometry. For a deeper perspective i with cover map The group Λ automorphisms of the ambientmogeneous space coordinates ( Its group of integral points N choices of the coefficients is a MUM point iff ( divisor, and there is a small open set while the monodromy may assume with no loss that Γ is neat, so that nilpotent orbits of severalmixed different kinds: Hodge they structures are whichsimple classified possibility, may in namely arise terms what of [ monodromy) happens the point when degenerating [ we approachK¨ahlermanifold a has MUM the (maximal form unipotent N on the asymptotic structure of a quantum-consistent pre-potential behavior of the periodtheorem map [ at infinity is described by the (multi-variable) nilpotent-orbit In this section we study inK¨ahlergeometry slight which more detail satisfies the asymptotic ourreproduces behavior swampland at most infinity criterion, of of a andwithout the special see any properties need how we to it assume usuallyCalabi-Yau 3-folds. automatically that associate the The relevant special with properties geometry mirror just arises follow symmetry from from quantum a [ consistency pair of the 4d 7 Swampland criterion JHEP09(2020)147 ) J (7.9) (7.7) (7.8) ,X ) (see (7.10) (7.11) (7.12) ’s. j i F z I ( w X F ∂ integral. In is non-zero. ) , : this reflects i i { ( ~w IJ j · ’s and ijk -th monodromy S δ i i d z , . . . , m, πi~n + 2 j e and = 1 w ~n i }| j λ J a functional equation of . The f 0 → i \ X ) with respect to elements Hodge theory. The Fourier Fourier series j Q I ∈ X w X z ~n ) 5.11 , i, j i i as of the MUM point into them- ( + IJ ) with , . . . , m for all w S j e the cubic form i U 2 1 7.9 w 3 j ) = 1 f . are rational numbers which need to ⇒ i arithmetic -tant. πi i (3) ) + M = at the MUM point. The requirement → ∞ m c I ζ . ’s takes the form j 0 i i 2(2 Z ): Λ] X ijk F γ ( )-rotations on the full vector ( d w X X Z ∈ + ( and Z F i ` , = J . Then, for a suitable choice of symplectic = k z def I ij k ) has been written with a particular normaliza- f b i j – 55 – ) = = [ X c N + 2 24 z , 0 j . This condition restricts the function ijk f ) 7.10 d X i + m N ijk is the positive ( IJ i i d j j } S det N z 0 ) satisfies for all of ( i f I , so MUM point with z ≥ + X i ij M ( 2 j n b are still arbitrary complex numbers. rhs F : 57 , the action of the ~n 0 + ,X I λ 0 m k X X Z i z X Γ maps small neighborhoods j j ( ∈ = 1 and there is no need to distinguish between z f and i ) )) F ⊂ f z e m + 0 j 2.30 ijk 3! X X d ) stands for a generic holomorphic function, defined in the neighborhood acts on the covering local coordinates , . . . , n − 1 7→ 7→ n )) requires that C acts by an integral Sp(2 7.10 ( j 0 ) where the coefficients { i i ) = ∈ X X i γ is an integral matrix such that w = i z 5.11 i ) may be seen as a strengthened version of the nilpotent-orbit theorem. At this ( i ( γ j j Q )) to the general form : F f f i γ 7.10 5.9 Now we appeal (a bit heuristically) to some more The subgroup Λ ≡ Here j 57 z Eq. ( stage the coefficients series in eq. ( The constant coefficient in the tion for later convenience. From the nilpotent orbit theorem we see that for a certain non-zero matrix ( satisfy strict integrality conditionsparticular, in order to satisfy ( eq. ( the form (cfr. eq. ( for some symmetric integral matrix In most examples det selves. Then we mayof use directly Λ the to functional equationsthat constrain ( each the asymptotic(cfr. structure eq. of ( where element in an integralhomogeneous symplectic coordinates action on the The MUM point corresponds to the limit Im JHEP09(2020)147 , ] ). j i z δ ( 126 L (7.14) (7.13) + i w → i w ]. . However this way ). Q 105 ) may be defined to [ ~w ∈ ~w · ~n e L · } ~n ~n ~w . ( . · ( 3 ) associated to the Abelian Q L , ) L ) ∈ ] we learn that such function πi~n ~n ~w 2 7.9 } πikw · e πi k ) 2 126 { ~n e ~w ( (2 , not to its local expression in a · M¨obiusfunction). (7.15) 1 L ~n F ≥ ( ~n ≡ k X L N ) has an interesting monodromy [ µ { 0 ( w \ , namely ) = ( Q m L ]) of maximal weight 3, so that the natural ∈ X πiw should have interesting arithmetic properties ~n H ) may be rewritten as a series in the around the MUM point, its global analytic 2 ~n e . In particular the swampland philosophy sug- 125 3 e ( N ) prominent in the swampland program. From L → ~n ~n/d d 3 – 56 – 7.10 w λ λ ~w ’s — ( · ⊂ Li ) ~n L 3 d λ ) ( πi~n U 2 µ 1 e πi } the VHS degenerates to a mixed Hodge structure (more 58 ~n ~n not the (2 { λ ~n ) which canonically represents the Tate Hodge structure 0 X = gcd \ def N w | → ∞ Q ( ) d univalued. Indeed ∈ X i 3 L w ~n ) ( w = 2 supergravities with a MUM point. πi L not N . Thus, to procede with the swampland program, we are forced to complex) coefficients e L = (2 ~n ’s — contrary to the N ~n Γ. Such general function was written as a sum over elements of the standard N a priori ⊂ stands for a suitable compactification of the covering Legendre manifold at a MUM point are the ) is the trilogarithm Let us see how these arithmetic conditions arise. While Which basis is the “intrinsic”In one the in MUM the limit Im present context? e L (possibly after restricting it), and periodic under the integral shifts w F 58 (3), i.e. the natural function is the one which satisfies the appropriate Hodge-theoretic ( the global analytic continuationparticular of region the of pre-potential understand the physical implications of the multivaluedness of be univalued in thecontinuation neighborhood is certain which is the basic reasonthis why is this the special very functionthe property describes which the discussion makes relevant in Hodge structure: the previous section, we know that the swampland condition refers to The structural swampland criterion suggestsof that the intrinsic coefficients ingests the that expansion the in quantum-consistent for the new ( Formally, all Fourier seriesTherefore of we the shall form rewrite ( L precisely to a mixed Hodge-Tatecandidate structure is [ theQ function functional equations. From, say, theorem 7.1 in the review [ which yields the general solution tosubgroup the Λ functional equations ( basis for the periodic holomorphic functionsof in writing the solution isby not going intrinsic, to and a the more expression natural may basis be of made such more functions illuminating U JHEP09(2020)147 j ) i f 6= 0 7.10 (7.19) (7.17) (7.18) (7.16) ~n so it is j ¯ i 2. For, say, G )-rotation of , physi- / 2 Z ) { ) , ~z ~w · · ~n + 2 are (non-negative) to zero and setting ( should be integral ( ~n ]. ) the “instanton cor- ~n perturbative) contri- ~n L m ~n , while N 52 N }| ~n [ ≡ should be } Z is the triple intersection l 7.18 N ~ω ) n 59 0 ∈ l · Z \ j i f Q ~n ω k ∈ ∈ X 7.14 n ( z ~n 2 “instanton corrections” ∧ Z k ) i for type IIA compactified on a ijk i 2 2 f + c d { of X { e ∈ F i ) R i ) 3 of class n ) 1 z ( , = 2 ]. For simply-connected 3-CY one gets rhs ~n }| X i πi ≡ (3) h . Indeed the constant term in eq. ( c to be integers. “loop” N ζ i . Z ) where − ~n 128 , 2(2 ~n z w 2 1 N , N of the form + 1 ∈ (0) 7.18 i h e , which follows from our swampland condition, L supergravity, the ambiguity in the global z F = 1 ( – 57 – 2( e 2 2 i (integer) f c 24 2 and ≡ 1 2 = 2 -theoretic quantization of electro-magnetic charges, 3 + e/ c j , K ) + N ) differ by an integral multiple of ( i ) drops out of the special K¨ahlermetric z X I ]. We dubbed the last term in ( c i ~w Z z the same integrality condition which holds for the , · X 52 ( [ ij = ij 7.18 ~n b b ( =0 F e , X ~n L + N ijk k ). . Then criterion i d z of harmonic (1,1) forms, j j Z f in eq. ( } z i i ∈ that is, no physical observable should depend on the particular has exactly the form ( i z ω c 2 { X (mod 24). e/ ijk 3! d Z − ∈ i ω 2. Applying to ) = i ∧ z e/ ( 2 In a quantum-consistent 4d definition of eachcally term invisible, in thedetermination sum we in choose. the c F ≡ R The coefficient The large-volume expansion of the pre-potential The same token leads to the condition Two determinations of Consistency of the swampland scenario we are proposing leads to the following More precisely, one has the slightly weaker condition that gcd =0 = = 1, this reflects to an indeterminacy of 59 ~n i is the inverse of the matrix ambiguous at the differential-geometric level.is However, crucial in the to full reproducesee VHS the e.g. its the value correct (mod detailed analysis 24) c in section 4.1.2 of [ rections” since this term hasfor this the physical same interpretation reason inbution. we this particular denoted In class the the of constant models;integers Type term because IIA as they the large count “loop” CY the ( rational volume curves set-up, on the coefficients Calabi-Yau 3-fold of an integral basis is the Euler characteristic of with integral coefficients so it may be absorbedN in the sum byterms extending then the yields summation index implies that around a MUM point with the electro-magnetic frame. This requires the may be simply written as f and this is invisible precisely when it can be compensated by a Sp(2 principle: JHEP09(2020)147 ), 7.8 of a (7.23) (7.21) (7.22) (7.20) 5 provided . m which is perfectly Z , ) F ⊆ Z , ) makes sense. NE of non-empty instanton- + 2 . This operation formally i ⊂ may contribute to the local 7.18 z m ) which, in view of eq. ( NE    Z ~n Q , and this requires analytic con- Sp(2 → − ⊂ , , the Mori strict convex cone of +2 0 ∈ o z some m 24 m There is one counter-intuitive point Z 1) /X = 1 MUM point at infinity) − i contains 6= 0 in − ⊂ 3 globally, f , X ~n e 3! S z F N e L M ≡ ··· of : i C , ) + and in all determinations of the multivalued z 1 z m in which the sum ( ( e ∞ L − Z – 58 – , L U is outside the domain of validity of the represen- ∈ +1 ~n , ) = 1 z n ) that is i − − , ( = def X . We may however, use the functional equation of the → −∞ L − U F i ··· , z , 0 1 NZ with a non-zero coefficient X − ( ~w of mirror Calabi-Yau 3-folds are valid in full generality for any , curves — whose volume is positive — may possibly contribute in · ∨ → = 2 supergravity with Γ = Sp(2 πi~n ) 2 X i should contain instantons contributing to all asymptotic expansion the Fourier expansionasymptotic of region at e N , : ,X X diag(+1 0 NE effective : m = 2 effective supergravity (with a outside the small domain X of non-empty instanton sectors. Z U ) F ∈ N ) for the function z ( ~n NE ) in = 1. In particular, Γ contains the element L 4d    7.18 f are the ones with instanton charge in 7.18 In view of this situation, it looks natural to define the set Of course, the region Im However the general story is subtler: in order to apply theTo swampland make criterion, our we point sharp, let us consider the example discussed in section We see that essentially all the general predictions that we may infer from the existence F . In particular tribute in all ofF the asymptotic regions of to formally flip the sign ofto the negative. topological The charge price for we some paycompatible primitive is with instanton a the from redefinition positive of required the integrality polynomial properties part of of its coefficients. charge sectors to be the global one, containing the instanton charges which virtually con- tation ( function implies which acts as ( “invertes the sign” of the instanton charges. should look at thetinuation covering of Legendre manifold quantum consistent In the geometric set-up of Typeis IIA obvious: on a only 3-CY, this restrictionthe large-volume to limit. positive instanton-charges Thus, naively, theto only instanton-charge sectors which do contribute effective curves. to be stressed. Atsum first ( sight only positive instanton-charges of an actual pair abstract it satisfies the structuralgravity swampland implies criterion. mirror Morally symmetry speaking, as a quantum special consistency case. of The set JHEP09(2020)147 . ). to 2 the = 0 = 0 ~n ~n 4.54 (7.29) (7.27) (7.28) (7.24) (7.25) (7.26) ≡ ( N N ) of sym- V ). e S ( ) vanishes as , ) with 3.17 Z Sym 7.18 ∈ ), and set 7.18 ⊂ i Z to the determinant ( ) is essentially the full J ) and ( R R ( ), that is, when , n i J NE 3.15 is a Hermitian symmetric X 0 7.18 . e S X i = 0 n , . . . , m m 2 e + R , , ] is that . ) ) ∈ U 51 ( Z equivalent over i = 1 υ ( ⇒ ) symmetric we may choose the frame i Z V /J ( , y )) d NZ T 0 are not uniquely defined since they depend Z Υ ) i ( X ∈ c e S [ 6= 0 = i J υ ⇒ F → F ( its covering space , ( y – 59 – ~n Γ should be zero ij . For plays for the electro-magnetic ones. Therefore, =1 ∞ b + ) of a rank-3 symmetric convex cone [ d e N sym i (3) m V ( ⊃ X ) = ⊂ glob D def i unless T G R 6= 0 implies that the constant term in ( j Υ X → NE 0 I i ~n ∈ δ = 0 for all X i + -fold cover instantons contributing to the local expansion in X ~n d 0 ∂ is uniquely determined by the corresponding Jordan algebra, N 0 X we conclude that a pre-potential of the form ( F i . The physical expectation [ I X δ 5 U , hence on the instanton-charges; morally speaking it plays the same ( i ~z = 0 for all n , commutative, unipotent, real Lie subgroup ~n + should contain m I N I )) be the normalizer in Γ of the unipotent subgroup X Z ) is a homogenous cubic form over NE ( i correspond to → z → F J ( ( U I = 0 also, but the coefficients I Γ . To establish this fact is our next task. F i F X c 6= 0, the real shifts m N ( Z d NZ ~n In particular Let 6= 0) belongs to the swampland ~n Naively, on the choice of the duality frame. Indeed the change of the duality frame in the Jordan algebra whose explicit expression is givenwell, in that eqs. is, ( the “Euler characteristic” positive cone of aIn rank-3 this Euclidean exceptional Jordan case modulo algebra), the trivial see ambiguity due thefor to right-hand possible the side different homogeneous choices of coordinates ofso table of integral-symplectic that frames 6! are non-zero. From fact ( manifold, hence the tube domain for all form a dimension- metries of the special geometry. In particular, its Killing vectors the asymptotic region lattice 7.1 Necessity ofWe non-perturbative note corrections that when there are no “instanton corrections” in eq. ( the very least where Note that Υ has theΥ same acts formal linearly on structure the asrole the for would-be the global instanton-charges symmetry that group ( JHEP09(2020)147 . ] it 3 51 m Z (7.30) (7.31) (7.33) ) since Z , have some + 2 in section X m 4 corrections, the = 0 mod 24. The , ) Γ, because, in this = 1 for simplicity). i Z c f that the perturbative , ⊂ Υ (7.32) is the full lattice ) + 2 Z ≡ ( = 0 mod 24. ) i correspond to a non-empty J m NE . Z ) quantum correction, pertur- implies ( m ⊂ k c necessarily have non zero left- Z Z , , Sp(2 /J } t m ∈ non-perturbative )) γ ∈ -Zariski closure of Γ is semisimple, Z + 2 { some ~n Z . However physically we expect [
Q ( 1 ⊆ m m J − ( Z ) t Γ is not symmetric, the instanton-charges ). NE ) given by the block-diagonal embedding S A Sp(2 e the weaker condition S ( Z ) down to Λ ⊂ , , ⊂ → N R 1 , ) ( ) , U ( J + 2 Z of contributing instanton-charge vectors is big A ξ – 60 – , is not symmetric, the quantum corrections should m 1 U m, at most e S . d NZ NZ m Ξ Sp(2 GL( ∈ Z Diag ∈ [ ξ ∩ ⊂ )) and Ξ ~n ) 7→ Z =1 ∞ [ ( d Z J = Γ A def (while keeping fixed all other coefficients), so that, in absence ) of some of its generators ( ). Indeed, the structure of these equations strongly suggests m, i Γ 3 Ξ n ) in view of the fact that the ) 5.10 7.18 Z ⊆ N 5.11 + 24 m, Ξ i c 6= 0, and the corresponding functional equation cannot be solved by any ) GL( IJ → i C c 7.25 Let us compare this result with the differential-geometric fact belong to the swampland — should receive Killing vector is allowed to survive quantization in a consistent theory. This no not One may make the above heuristics a little more explicit (we take That infinitely many instanton-charges should be realized is already clear from the In facts, dicothomy yields more: if -torsion, the actual congruence seems to be something like then by eq. ( so to show that thesuffices set to of show non-empty that instanton-charge Ξ sectors is “big enough”. This clearly holds when Γ = Sp(2 and let Clearly that instantons of all charges will appear. Consider the subgroup GL( functional equations ( and hence the matrices ( bottom block, finite sum of the form ( 7.2 Non-empty instanton-charge sectors In the previous section weof have the non-empty seen sectors that should when generatea the stronger lattice condition, that is,topological that sector all (in instanton-charges some asymptotic region of does bative or non-perturbative.perturbative Here ones we being not see enoughsence that of to non-perturbative we satisfy correction need in the some swamplandloop exceptional criterion. cases corrections Moreover, vanish the as ab- well. We shall elaborate more on this below. case, condition requires that theenough to set generate the full lattice Remark. There we shown that a non-symmetric asymptotically-cubic special geometry — which of instanton corrections oneactual can assert condition seemsk to be even weaker: ifbreak the the cohomology continuous groups unipotent symmetry of has the effect JHEP09(2020)147 ]. ]. . m 131 135 Z (7.34) (7.36) . Since element (see next m ) “behave Z Z , (ii) was computed or + 2 ) a monodromy Z primitive m (i) (iii) n, is a complex 3-torus, 1 is the full lattice 1)] (7.37) / X GL( − NE ⊂ ρ (i) contains a Γ , SO( . U G ) × 61 Finite mono) NZ ] / ι . σ σ ι 129 ) modulo a finite group. 7.1 [SO(2) and Z ) / i strong approximation of monodromy groups [ Z ) is onto. (7.35) ) = 0 mod 24 is a Shimura variety). ) 1) , is a finite free quotient of m, Z the residue map Z X a Zariski-connected, simply-connected, semi- S − ( X p /p . If the large-volume limit of the pre-potential 2 + 2 O super- 60 P m, ) is arithmetic reduces to the previous one. We ∨ Z – 61 – , ρ ι ( Z m X , ] getting a purely cubic polynomial, in agreement P (iii) , c GL( ), by section / prime SO(2 Z p + 2 is an arithmetic quotient of a Hermitian symmetric 132 Sp(2 )-orbit consists of all primitive elements of / π ), with × n, −−−→ Z ) = 0 m = 0 if and only if its Ricci flat K¨ahlermetric is flat [ S Q Γ X 2 n, ( c that there should exist an electro-magnetic frame with this n, SL( 3 U(1) Sp(2 O c 5 Γ Ξ / i / / ) ⊂ almost all GL( ≡  ] R ) , ⊂ Z 130 because in many respects thin subgroups of Sp(2 ) we see that Ξ is still GL( n, 1 1 Q SL(2 ( has no instanton correction, then NE σι P a 3-CY with a mirror , while its SL( X m X ker Z ≡ We check the corollary in the two classes of known examples in which the pre- (Strong approximation for monodromy [ -group. Then for Q is an elliptic curve times a K3, or The situation when Γ We recall that a 3-CY has The case of a thin monodromy, is less clear, although the same argument will typically contains all multiples of its primitive elements, in this case X There exists a more preciseSee result, also namely theorem the A in [ 60 61 of the lattice NE have the commutative diagram (first row exact, in this case Ξ = GL( property. Indeed, in thesebe cases locally it symmetric is to fairly obvious that the Weil-Petersson metric should potential does not receives instanton(ii) corrections, that is, when section for more details).(in The some pre-potential convenient for frame) allwith in Calabi-Yau the of ref. prediction type [ from fact and the quantummanifold K¨ahlerspace of the “magic” type (in particular, Example. We apply the previous analysis to TypeCorollary IIA 8. compactifications: for Type IIA on Zarinski-dense in simple 7.3 Applications to Type IIA compactifications lead to a large set as they were the full group”.Property For instance: group consistent with the VHS structure theorem, that is, Γ is finitely-generated and and since Ξ JHEP09(2020)147 ρ ]. K- H is a = 0 (8.2) (8.1) 2 134 X , c , each of ⊂ 55 A ). Their C in i of dimension 7.28 ]. An A-type C ]. A 3-CY of has genus zero by X 55 ) is a power 1 Σ is called of 133 [ P / ) 7.37 K K × ) is finite or it is infinite. E X ( 1 , π . ) = 0. Indeed, a CY has X ( 62 2 ) = 0 4 or 3. (8.3) c ) of the Ricci-flat metric is exactly X , ] so 5 X O , ( is A-type 7 and a K3 surface , X, 135 hol , ( X E 1 ) = 0 in agreement with eq. ( is a compact K¨ahlermanifold = 11 ) = 0 and = 2 or 3. This observation is related to our . 134 X , 1 ⇔ ( sense. C , X ρ – 62 – the structure sheaf. All 3-CY (in this sense) are X 2 ( χ 55 , but this contradicts the fact that an Abelian variety does h χ X ) is the disjoint union of connected curves ,H A , while one of the form ( C = O X of an A-type 3-CY is either 2 or 3 [ ( ] has strict 1 1 O , 1 . But any connected unbranched cover of , − 1 ) = 0 1 2 π X h ∼ = P h 134 has a finite unbranched cover which is either an Abelian X , A-type ( be the unbranched cover, and suppose (by absurd) that ≡ ∼ = X 2 ≡ 55 c ∞ K ρ C X 1 , if and only if 1 = ≡ h | H ) Σ ≡ X A/ ( the holonomy Lie algebra ρ is a rational curve in is unbranched, 1 is a 3-CY in the i → π π | ∞ C ]. In both cases Σ is a finite group of automorphisms acting freely. A X : < π | 134 ]. A K-type 3-CY A Calabi-Yau 3-fold (3-CY) ) , Σ is said to be of is the canonical sheaf and X or the product of an elliptic curve ], and ( A/ contains no rational curve for obvious reasons. 1 134 132 X , , A π | K 133 X 55 55 2 being the Picard number) and all such locally symmetric special geometry have a [ There are eight deformation types of K-type 3-CY again explicitly constructed in There are six deformation types of A-type 3-CY explicitly constructed in refs. [ Given this definition, there are two possibilities: either A 3-CY with In facts, let ≥ (3) [ 62 ρ rational curve. Since which is an unbranchedGauss-Bonet. cover Hence of not contain any rational curve. 3-CY refs. [ possible Picard numbers are [ if and only if its Ricci-flat metric is flat [ The Picard number variety the form type Their Ricci-flat K¨ahlermetric is flat, so When su 3 with where smooth projective algebraic varieties over 8 Answering the Oguiso-Sakurai question In the rest ofAlgebraic Geometry. this note we adopt theDefinition. notion of “Calabi-Yau 3-fold” which is natural in purely cubic pre-potential inof a the suitable upper frame. half-plane next Note topic, that the the Oguiso-Sakurai spacemore question. ( precise. In the next section we shall be mathematically ( JHEP09(2020)147 ∨ X X (8.4) (8.5) (8.6) assumed = 2 or 3? should be -model to ∨ ρ σ ∨ X X ), so the corre- has a mirror susy is it true that the 62 X ) for Type IIA on . ) 7.18 corrections X has a mirror ( , ) 2 X ) is an affine (resp. projective) X H no perturbative loop ( 2 W ⊂ H ⇒ o = vanish because of too many fermionic has no rational curve if and only if it dim = 0 F ) = 0. X 3 ) (resp. ≡ X D ( W

( χ ) C susy X – 63 – ( , ρ 2 = 1 will be settled in the next section below using ) in the positive assuming that (same argument as in footnote That is, (in physical language): We recall the ρ H 3 X implies . ( E 2 ∈ 3 Q D higher 2d PH n 2) world-sheet supersymmetry? non-renormalization , ⊂ and uses the (4,4) non-renormalization theorems to show the (2 ) = W K > ) is infinite? W ( ) × X 2. The case of ⇐⇒ ( C defined over 1 E ≥ p, q π ( ρ ] applied to the universal deformation of the mirror Calabi-Yau Is it true that the “instanton corrections” in eq. ( ] Oguiso and Sakurai raise the following higher 30 Is it true that all 3-CY without rational curves have Picard number 55 – 27 indeed the only thing we use in the argument below is that the period map of the corrections no instanton The positive answer to question In this section we answer question In ref. [ A stronger statement would be that a 3-CY whose affine cone is Since the intersection numbers arealgebraic integers, hypersurface 8.1 The mathematicalWe argument consider the projective cubic hypersurface so that the absencethe of perturbative non-perturbative ones, world-sheet as correction weparticular, also found absence entails in of the the rational absence previous curves section of implies by the dichotomy argument. In to exist. From adropped: physical perspective, theVHS hypothesis which that describes the quantumrequired K¨ahlermoduli by satisfies our the swampland VHS criterion. structure theorem, as and Picard number less rigorous physical arguments.rigorous (under the The stated assumptions); results indeedtheorem they of [ follow directly the from present the VHS section structure are mathematically Question 3. vanish if and only if “instanton corrections” vanish if and onlytheorem if of they a are forbidden by the non-renormalization Question. is A-type. An even stronger result will be a positive answer to the following families parametrized by thesponding instanton curve corrections to the pre-potential zero-modes; this fact maythe be finite-cover target stated moreabsence intrinsically: of one corrections. lifts the 2d A K-type 3-CY contains several rational curves, but all of them appear in one-parameter JHEP09(2020)147 ]. 137 ) are Q ( W ; ) are dense Q Q ( W 6= 1. ) = 1, since in this X ( is absolutely reducible 2 H W . dim the rational points X cubic hypersurfaces. ≡ ⇒ ρ that is, if 63 specific has finite fundamental group and the 2) such that its A-model TFT has no , possibly counted with multiplicity. This X ≥ Q , to answer the Oguiso-Sakurai question we ; If ρ 2 Q . 2) without rational curves or, more generally, ]) – 64 – ≥ . contains a rational linear space of dimension 3; 137 ρ ∞ , W = The rational points are dense in the real locus: ]. | 136 ) 19. . this claim gives ]) 137 X ( 2 5, and 1 ρ > . Then we remain with three possibilities: 3 [ 137 π ). | 5 (Meyer’s theorem); satisfies the condition that its rational points Q R ( ρ > ρ > W ρ > W ), then there exists a rational curve on R ( a 3-CY (with mirror and (P.M.H. Wilson [ W which may arise from the triple intersection of divisors in a 3-CY without X 3 D a 3-CY (with mirror and (P.M.H. Wilson [ is irreducible and is irreducible, Note that the proposition says nothing when contains a hyperplane it should be rational, is an irreducible cubic defined over contains a hyperplane and an irreducible quadric, both defined over consists of 3 hyperplanes defined over X is not defined. W W W W cannot happen for W W W If If In case (b) when Together with proposition Below we show the following: The swampland structural criterion sets severe constraints on the possible integral The rational points are trivially dense along the irreducible components which are If For this claim and the list of properties in items (a), (b), (c) see comment after lemma 4.2 in ref. [ 63 (c) (2) (3) (1) (a) (b) Corollary 9. instanton correction. Then From this corollary all other claims follow provided the Picard number Claim. such that its A-modeldense TFT in has the no real instanton locus corrections cubic forms rational curves. Then, inneed view to of study proposition the rational points of only these very hyperplanes. Using generalzero, facts one about shows: quadratic and cubic integralLemma forms 5 representing it is already reducible over projective hypersurface in its real locus Remark. case Proposition 2 JHEP09(2020)147 - 5 is R ≥ α (8.8) (8.9) ρ C of the (8.10) α ), where C Z ( α . 3 the special ) , respectively, M , . Z O ) in the ambient , ⊂ is the quotient of Q = 2 α α , + 2 S , open, strict, convex ρ , and m + 2 H , m 0 (8.11) associated to the various has Lorentzian signature C , Sp(2 > α ij R V k ∩ Sp(2 d x α = j symmetric ⊂ is the set of “forward time-like” x M F i α x ⊂ m for irreducible cubics = 4, since for α ijk R ) for a quad. d α V ρ , V ) ( R T 25) ( α − (1). Part (3) of the lemma implies the claim 7( M 5 19. by the arithmetic subgroup Γ – 65 – ) with Γ ,E ) of real points is is the cone of positive-definite elements in the > α Q R , ( for quadrics or, respectively, cubics. The open cone -rank -Zariski closure of Γ /K α ) implies that, in the absence of instanton corrections, (12) Shimura variety of appropriate type. (8.7) R cub. ) ρ ∗ ∨ + 2 . Each irreducible Shimura variety Q = 27 V M α R = Q 0 respectively ( X ρ m SO α α > 1 or , M M j 3) − Sp(2 x , i ,S ρ . x ) α ⊂ α S = do not get any quantum correction, and that their special K¨ahler ij 1) for quadrics U(3 -rank Q C α 8.6 d , α Q Γ − m X ) for hyperplanes ) Y stands for the integers. The integer quadratic form 3 Hermitian matrices with entries in = ≡ R R -algebraic group As we saw sections 6 and 7, the structure theorem of VHS (applied to , ρ = , Q , α × α α is an hyperplane. From now on we shall consider only quadric and cubic Q Γ ijk S ), eq. ( M d α ’s are defined over where SL(2 SO(2 Sp(6 . The classical cone K¨ahler — which coincides with the quantum one under C α W ( ) of 3 m C (after replacing it with a finite cover, if necessary) is the product of irreducible rank Γ is (biholomorphic to) a tube domain F C R 3.3 ( simple α S 3 ⊂ , resp. 1) so, in the quadric case, the convex cone ] the /K ij α Her ) − d V R From sections 3, 4, 6 and 7 we know that each irreducible Hermitian symmetric space is the 137 ( is a connected component of the open domain in α α , m -algebraic group. The Lie group α vectors, whereas in thespace cubic case see section the present hypothesis —irreducible is components the product of the convex cones V with (1 it is settled by Meyer’sfor theorem, the see last lemma cubic case which has M cone The density of theobvious when rational points incomponents. the real For locus quadricsgeometry of becomes the reducible the only (so irreducible open we are component case back is to the case of hyperplanes) while when where, as before, Q By [ a Hermitian symmetric space M the complex moduli of thethe mirror K¨ahlermoduli of geometry Shimura varieties in one-to-one correspondenceaffine with cone the irreducible components Proof of claim. JHEP09(2020)147 ) ⊂ Q ) , ) of F W (8.13) (8.14) (8.15) (8.18) (8.16) (8.17) ( ( +2 3 C m -algebraic Her Q obtained by . ) ) is the full light- R Q -algebraic structure sugra ( . Q

m, quad. = 1 lmn C GL( d ) is the simple N of the hypersurface Q ∈ =

ijk , α -algebraic group Sp(2 ], and its d kl n ) is the isometry group of the m, 1) C k d Q R g ( , g − . 117 = , ) α ) (8.12) SL( 1 quadrics 2 cubics, m ) a l j resp. Q L j Q F x ( ∩ g , , m · , , g α l ij ) α α i is nothing else than the automorphism k d m, i g Q R L + 2 ) we may find a finite set of real-valued ) = α g M ( α SO(1 SL(3 R . × G ij ijk m W ( = GL( G d d ( α × × X 0 with = 0. Thus m α ∩ C L a has a simple physical interpretation: its real × × G L [ > α Sp(2 – 66 – ): ): R R of α = ) M ∈ m m L α α ) = ) -rank α = ) = ) = C G R 8.11 R , g ( SL( SL( R R α Q ( ( 1 α symmetry group of the 5d , G = ∈ ∈ t C ) of the component ) j j α cub. i i R 1 quad. ( L g g of real points of − G are seen as subgroups of the ). α   G g ◦ ( naive C ) . Then , 8.9 = = R m . rhs ( -rank R α Q 26) cub. ) is identified with the space of Hermitian matrices in G quad. ⊂ − L -algebraic group R diag(1 ) such that the union of the (analytic) closure of their orbits under L ( 6( α Q R V E ( -algebraic group 7→ α cub. ) form the Q g ≡ C C R ) ( α O be the , The is the multiplicative group and L } ∈ α a α m G x 2. Q { G ) is the full real locus ≤ For each irreducible component Let The equation of the associated irreducible component is obtained by replacing in eq. ( R ( α m points G a fact already implied by ( Remark. points compactifying M-theory on theuniversal cover Calabi-Yau of the 5d vector-multipletis scalars’ induced space by [ the 5d flux quantization. In particular, the 5d swampland condition yields where SL(3 The connected group group of the cone Modulo finite groups, one has where group which leaves invariants the rational tensors where both groups in the through the block-diagonal embedding R cone, whereas rank JHEP09(2020)147 ). ) is R ) to 65 ( R α (8.20) (8.21) (8.19) . n, G ) to be Z 8.18 lying in SL( x 8.18 in ( ⊂ = 6), where ) in the real 64 } ) rather than . ρ a Q L 6 in ( Q x )( , Q { } a ∈ W x ( ) ) is dense in { 6 be a connected subgroup C if and only if Q ( L Q α is non-compact of finite = 0 contains a non-zero , ) G j ) is dense in the real locus x Q , . . . , x ]: let i -group SL(3 ( 1 quad x 3 x 64 -algebraic group if and only if 8.19 Q ( ij Q /K d ) Her of ( ). Indeed, if this was the case, the ) is at least one by the Godement . R ) ⊂ ( R R , ( , is defined over ) Q ] ( ) lhs 6 ). Q L α just a single non-zero point Q F ( quad. ( C quad. Γ we invoke the following fact: } 3 n, a cub. ⊂ α x ) is the plain , . . . , x SL( C a L Her { 1 quad. Q x ∩ ( ∈ x · ∈ [ L ) – 67 – Z 0) 0) Q cub. = , def ( , ∈ -rank of Γ L 1 α 1 Q Q  would be given by the plain determinant formula L G     -equivalence, the expression needs not to be valid over , , 5 6 3 ) together with the fact that Q a x x x Q ), this is a direct consequence of, say, proposition (5.3.4) [ 4 2 6 ) be a semi-simple and connected (closed) Lie subgroup. → x x x 8.18 R 8.17 ) . 1 4 5 ] we know that all connected, semi-simple subgroup ). Then apply proposition (5.1.8) of [ -algebraic group R Q x x x R = diag(1 m, ( = diag(1 Q L 3 139    n,  -Zariski closed; then the group  , and we would be done. . Density of the rational points is obvious for the multiplicative is the Lie group of real points of a x R SL( x α Her Q R will follow if we can show that the rational structure of the alge- L : ⊂ = det L ]. d is arithmetic, the } ) must hold up to k × ), it would follow if R a 64 x R x L j m , is the “obvious” one: for instance, for the first cubic case ( { 8.20 x G i quad. ) will follow if we can show that we may choose the points α ) is dense in x Let R L = Q )( . ijk α . For the simple d L m, ) which is almost W G m -Zariski closed in SL( ( 1 ) = SL(3 R 6! G R C R ]. In other words: since the quotient Γ ). This follows from eq. ( SL( n, -valued. ( More generally, in all instances — quadrics as well as cubics — the existence of the For quadrics it is enough to show that the light-cone Putting everything together, the density of the rational points Suppose for the moment that it is possible to choose the points R The equality ( From theorem (4.6.3) of [ ∩ 64 Q ( 65 64 -valued. In this case α cub. R almost of SL( is dense in and the “obvious” points would be defined over L some fancier rational structure onrational the cubic real form group SL(3 criterion; for a Lorentz groupin this is the equivalent light-cone to [ the existence of a non-zero rationalrational point points braic group be rational point. In view ofof eq. [ ( volume, and Γ Lemma 6. Some finite cover of L locus We claim that the union ofC the rational orbits inIndeed, the group the light-cone, while in thedifferent cubic signature, case say we may take a pair ofQ rank 2 Hermitian matrices of Indeed, in the quadric case we may take as JHEP09(2020)147 ) ) ). )) 19 R Q r R , , , − > (8.27) (8.23) (8.25) (8.26) we get √ ] I, IV, ( for rank R Q ( ∨ 140 = 27 ) 3 3 ρ ]. There are R ( Her 64 ∈ ⊕ 3 0) R , ) and we may apply , 1 ) Q r (  ) of Hermitian matrices , − F W √ D ), the same argument leads ( ( ) is the rational structure on 3 C Q , ( Q ] 3 , Her F = diag(1 F )) (8.22) 140 Her r D  ) (8.24) = 6 instance is settled. is not covered by Shimura, since the − F ∈ x ∈ ρ D √ ) = SL(3 . O 3+3 dim -space x ( ,    . Alternatively the rank 2 matrices R F 0 0 Q Q r ( Q ∀ = , 3 = has, respectively, Shimura type [ ≈ F corresponds to Picard number cub. ]). Since we know that the underlying real ) R F r x . Tensoring this representation with L F O 6 Q 64 D , − 4 D ) = SL(3 – 68 – Q ⊗ Q ( x √ = ). A more direct argument which does not use 3 Q F ( ∈ whose precise value is not relevant for our present ) = SL(3 F + , with a unique (canonical) positive-definite Rosati D 8.9 has the property [ ? Q 2 r Her Q ( for rank 2 and the reducible one x cub. x ? . The case 3 x ⊂ L 3 + . Again the points Q cub. R Q x 2 = 9), where L ↔ r x V

ρ x − 1 0 0 0 -algebraic groups whose real locus is the Lie group SL(3 form a dimension 3 linear space in x √ -structure is the correct one for this first cubic example, follows Q 4 − Q Q 2 if it exists , such that (section 3.3) we conclude that SL(3 x ? ∈ 3 x , the division algebra    R ) H independently of the value of 2 4 as well as the positive rational cone ↔

F , x Q x 3 D , and , x 2 C , is a division algebra over , x 1 R F x ) selected by quantum gravity. So the cubic ) is as follows: all rational algebraic groups with underlying real Lie group SL(3 D = -rank equal 2 as required by ( R , 8.9 Q The canonical involution On the basis of these two examples, let us consider the general cubic case. We have In the second cubic case ( That the “obvious” F so that it makeswith sense entries to in talk about the vector and III over the totallyoctonions real do field not formcase; an associative anyhow algebra. we do Theand not argument this need formally case extends it, is also since already to settled this by lemma 5(3). where (anti)involution For where ( lemma 5(2). are defined over us to for some square-free positivepurposes, integer although it should be seen5d as an quantum important gravity physical invariant — of the corresponding have a natural rational representationthe on irreducible representation 1 (see the proofrepresentation is of proposition (6.6.1)SL(3 in [ from classification of the We recall the relevantinfinitely facts, many see non-isomorphic e.g. such rational thehas groups, proof but of only proposition theeq. “obvious” (18.6.4) ( one in SL(3 [ JHEP09(2020)147 ), F ) = D (9.4) (9.1) (9.2) (9.3) ( X (8.28) (8.30) 3 ( χ = 1 has Her ρ ∈ ] since 0) , whose isomor- , . F 1 72 t ) D  ? , , a 11. The existence of , 1)] 7 ) are rational numbers. without rational curves. , = ( def F 5 − † D X , ρ ( -algebra 4 3 , Q ] — a 3-CY with )). (8.29) , a SO( , ) Her , = 3 ¯ z F ) as 136 × z (defined by the Vinberg prescrip- N ∈ F ) are then identified with the ele- 8.26 D ρ G , D Q ∈ m ( m 3 3 )( 10 , (by ( ¯ z . W SL(3 z = [SO(2) ( Her / ¯ z G ∈ z = 1 1) 2 3 cub. ,N 1 R , Q 3 C , a − − 1 z ) + h -space F → – 69 – = ¯ z , ρ 3! N is the upper half-plane with the Poincar´emetric z Q D ) ≡ ¯ z ( F z G − e 3 S ρ D R 4 ( SO(2 has no rational curve. Then, by the previous result 3 Her ≡ ) = × z X ¯ z ∈ ( z Her F K U(1) 2. This set is clearly non-empty, e.g. diag(1 / det: , m ) acts on the ) † F ≤ R D , , 3 we have 3-CY without rational curves i.e. the A-type ones, and in ama is non-commutative/non-associative) , SL(2 F 7→ D . ) of rank = 2 m F ρ ∞ D ( = 3 | For ) ) when Her ) the entries along the main diagonal of any X ( 1 3.17 π 8.26 normalized to that | We assume (by absurd) that ¯ z z -algebraic group SL(3 0 in absence of instanton corrections. that is, 3 times the Poincar´emetric in the standard normalization. The Hodge metric is and it does not receive the contact-term quantum correction described in [ so that the coveringG special geometry In this section we(infinitely argue many) rational that curves. —the as To discussion attach everybody is this meant expects residual to [ be case, just we heuristic. use physical ideas and and 9 The case ofThe Picard argument number of the 1 previous section cannot be applied to Calabi-Yau 3-folds with moduli, namely the K-typethese whose special Picard cases numbers is are geometry is consistent an with arithmetic our quotient findings. of In all these cases the special K¨ahler The rational points ofments the of cubic affine cone and we are done. Remark. addition there are 3-CY with rational curves but no quantum corrections to the K¨ahler The cubic form is justtion the ( determinant of the matrix By ( Again, our arguments are independentphism of class the precise is division anQ important datum of the consistent quantum gravity (if it exists). The i.e. the would-be rational of K¨ahlercone the putative Calabi-Yau JHEP09(2020)147 ∗ tt (9.5) (9.6) (9.7) (9.8) (9.9) (9.10) in the for the ). 66 Z ). Indeed, g G , Z , ⊗ ) G ( 1 for its summands Λ p ), one finds that the ) ∈ generate SL(2 is a maximal compact dg 1 9.4 1 dg − is the full SL(2 G 1 − 10 12 S g − T − ⊂ g ,( , and k = K  ) T 2 of the Lie algebra of , | .  dg ) 1 ] and eq. ( Γ 1 z / F 67 ( − ) 72 z while f g ∂ Z | is a discrete subgroup. , N . 10 . πi 1 . We write ) ) G ∈ 2 20 p 0 mod 24 z ) which contains ) Z ). Then, following the argument around T Z  ⊂ , τ , and ( ) , 6≡ ( PDEs 9.6 . Then the global symmetry group is G η (Im TS →∞ .Γ ∗ T  1 z Γ = SL(2 SL(2 SL(2 p − tt / – 70 – = 10 ) = S Im log ⊕ ≡ ( z 2 ( N/ k T (Γ) c − = lim Γ ) f Z ≡ ∧ 2 = , ) without zero or poles in the upper half-plane; the c 1 g ω z ) = − ( ∧ ¯ z S SL(2 X f ω Z z, N ( X 1 Z solution to F 1 is a K¨ahlermanifold which in the standard application of 12 1 we get ⇒ is a semi-simple real Lie group and − ]; all arguments and quantities will be independent of the chosen is the orthogonal decomposition − X = G 12 p 3 . is ) containing a power of ) Z ⊕ 1 plus the complementary space X , TS ], we get k F K 72 = g 1 and ( − ] we know = 72 2 S The period map factors through the upper half-plane, so that Γ must be a finite-index We use the fact that a normalWith subgroup respect to the Killing form. 66 67 Maurier-Cartan form on thewith group respect manifold to the decomposition from geometry to 2d (2,2) QFTsthe plays chiral the primaries role [ ofK¨ahlermetric on the space of couplingsubgroup. constants associated to Lie algebra of conditions for Quantum Gravity.I. I Zadeh. have benefit of discussions with E. Palti,A S. Katz and Pluri-harmonic In this appendix: Acknowledgments I am very grateful to Cumrun Vafa for sharing his profound insights on the consistency which looks as a contradiction since for some holomorphicexpression function should be modulareq. (8) invariant of by ref. ( [ From [ Using the equations of one-loopgenus holomorphic one anomaly index [ subgroup of SL(2 and if we assumeconclude that that it is trivial (as required by the usual swampland arguments) we JHEP09(2020)147 for (A.8) (A.6) (A.7) (A.1) (A.2) (A.3) (A.4) V . Define f ) C . ∧ f ¯ C be a lift of = 0. Conversely, all + a ¯ is pluri-harmonic. The . G C , f , i f = 0 ∧ 1) 1) ∂ → , , = 0 ¯ k C (0 C (0 e | and the form type: ¯ | ( X D , . C k p ∧ ) ) : p ∧ ¯ ∧ C. φ C dg dg ⊕ = 0 = 0 = 0 = 0 ¯ C i, j, k, l. A. C ∧ 1 1 k = ¯ ¯ ) − − C C C − + + g g and = ¯ ¯ C C ( ( D DC ∧ ∧ ¯ C ∂ ∗ ∗ C g ) if we may prove the two equalities ∧ ∧ ¯ ¯ X = 0 (A.5) φ φ + C C = ∧ = ∧ 2 ) C C ¯ -gauge transformation, so define the same C = = ) + + ( G/K A.4 ¯ C D C \ 2 ¯ = ¯ K ¯ D DC = 0 yields A dg C C − Γ ¯ + ] ∧ 1 D ) = 0 for all C ∧ − ¯ , , 12 C  C → g = ]: † – 71 – D DC C 0) 0) ] , DC , l + ∧ + A, C X (1 + = (1 66 | ¯ + | d : , C C ,C ∧ k p + ( ( ) ) C k f ∗ equations ( D ∧ 15 ∂ differ by a C ¯ φ ∧ C DD D dg dg [ , this yields D ∗ 2 C 1 1 , ¯ = j + C ] = 0 and φ + tt = − − j = 2 e dt equations [ g g D X ¯ = ( = 2 ( ( D D ∗ ,C ∧ ∗ ∗ ¯ i DC and D D C DC i φ φ tt ) C 1 [ dt = be the universal cover of = = = φ ]  j 2 ¯ e DD A C X -covariant Dolbeault differentials D ,C DC i + K = 0. The first one is just the condition that C ¯ [ D solve the 1 2 C is pluri-harmonic, i.e. in local coordinates D ¯ C arise in this way. ∧ ≡ f , ∗ C C C Let tt 0 = ( , Two local lifts ∧ ¯ D C , Consider the Maurier-Cartan identity D = 0 and We wish to show the following [ geometry which depends only on the underlying pluri-harmonic map , and consider the ∗ ¯ Writing These equations reduceDC to the consistency of the condition and split it according to Lie algebra decomposition tt Proof. if and only if solutions to Remark. Then whose coefficients are seenG as square matrices acting on some representation space Lemma 7. be a smooth map.the Let definite-type one-forms on JHEP09(2020)147 . ), is ∗ ⊂ ]. ) R ) ζ V R (B.1) 17 n, ) ], the ( , (A.13) (A.10) (A.11) (A.12) so that G 12 -algebra 14 A.7 s along the C ) = Ψ( R ζ ) along the ∗ ). Then the ζ /ζ . . By ( pluri-harmonic. A.2 1 ij e f − λ ) , ) ζ , i.e. the )), X R n Ψ( d ) and Ψ(1 → , ) C ) takes values in SL( ) ζ SO( ζ e n ). X n, ∩ ( n ) SO( R + Ψ( ( / SO( d ) Deck / G , ) ) = 0 (A.9) R = ∈ ) lays in a proper subgroup ζ Γ is 1 (such a normalized basis always = R ] is a normal matrix it must vanish, ¯ C 1 / ξ n, j ) X ζ ij − Ψ( n, K η ) R
,C + ζ . From the reality condition [ i SL( ¯ C 1 ]). More precisely, for a fixed is pure gauge n, ). Since the branes satisfy the equivariant branes C SL( ζ C P Ψ( \ 1 R 15 π acts by multiplication by e
∗ X , we get the identifications ( + Γ (with − ) −−→ SL( ∈ 1 n, ζ ζ ij tt = 0 since the trace of a commutator vanishes. ) C − C Γ for all → ζ 1 – 72 – → + R V Ψ( /K − | ∈ ) d ] ¯ X S ζ D n, j ξ R : = SL( ( brane amplitudes of a 2d superconformal (2,2) are + ): + − f ,C G ζ i ∗ = Ψ(1) G SL( ¯ d , the image Ψ(1)( D D , γ i C tt g ξ Ψ( → = C 1 + ] are normal and may all be simultaneously diagonalized. γ characterization of the branes of a (2,2) SCFT inside the j − ] are zero; since [ ¯ with X C D j equations as the integrability conditions of the linear system ζ : e ,C f i e Ψ(1) ∗ f ,C , then [ + i C −−−−−→ 2d (2,2) QFT, the brane amplitude Ψ( tt [ k ) may be chosen to take value in SL( C ¯ C ζ e C X 1 superconformal ≡ , − of [ k ij ζ arithmetic C = 1 and normalize the basis elements of the chiral ring C Ψ(1) = Ψ(1) ij ∗ + generic | λ ξ = 1 and taking ζ ¯ A | branes of (2,2) QFTs, we consider the brane amplitudes Ψ( ζ ∗ + ) is a map tt )-connection A ζ = 0. Under suitable algebraic conditions on the chiral ring C + all C is the Witten index (see section 3 of [ n, d ) and we may write ∧ n be a simultaneous eigenspace where R C Conversely, write the To describe the V n, descends to a pluri-harmonic map e f composition yields a pluri-harmonic map condition is flat, so its pull-back on the universal cover Specializing to for all valuesfundamental of solution the Ψ( spectralThe parameter SL( Let left invariant by all So all eigenvalues and Hence the matrices the determinant of the topologicalexists). 2-point function Then, in a where equator, Ψ( much richer and moreelegant than beautiful the than ones for thosethimbles a whose of gapped monodromy (2,2) a is model governed generic where by the (2,2) the BPS classical QFT, branes Picard-Lefschetz are inspace theory simple of particular Lefschetz [ more twistor equator SL( B Arithmetics of The arithmetic properties of the Remark. generated by 1 and the matrices JHEP09(2020)147 (the (B.5) (B.7) (B.8) (B.4) N . Then are local ∈ g is a space S c ∈ S ∈ s } s its Lie algebra. Q t {R ]. ]. Moreover, if 28 is given by )) while 60 g q X. ), and grading of the Lie algebra ⊂ 2.21 of [ ] = k Q R θ , ), ( ) (B.2) ,X R Q = 1  2.10 n, | ] we know that o ζ Γ (B.3) | / , 60 ⇔ SL( ) 6= 0 figure 6.2 \ q,q R q ) q yields a U(1) − − n n, − g q, q, Q and g ) g , yields the corresponding solution to the (see eqs. ( ) a maximal compact subgroup. (B.6) ). This reduction of the “brane group” has SO( SL( R even ∈ n R R M \ ( A Q q ) → n, ) of the universal lift q n, G n – 73 – ) = R ( ⊂ R SL( C SL( SO( G = max def . Since the compact part n, ,X ⊗ → g ( ) is of “Mumford-Tate type” i.e. it contains a compact figure 6.1 → k q,q e ) ∩ S R max ( − S R 0 q : SL( , g ( ): G 0 ˆ c ζ g G − K,K can ˆ c = a maximal torus containing exp( e π Ψ( ⊂ M q charge operator t of chiral rings has special physical properties it may happen that T = R S being straightforward). ∈ C c s } ) = rank ⊗ s R g ( {R ) is a compact one-parameter subgroup since it preserves the positive- G is compact. R ( k G ] = 0, that is, The real Lie group metric. Let ⊂ t rank ) is a real Lie groups in ) must be in the list of odd-weight groups on page 24 of [ , -independent pluri-harmonic map ∗ g ⊂ R R ζ ) of the form ( Q ( tt ∩ ) R From section X.3 of the second edition of [ G G 0 ( , Q 0 G θ We focus on a family of superconformal (2,2) models, i.e. the rings If the family g PDEs. odd of ∗ ⊂ Clearly [ t is Proof. exp( definite Hence Lemma 8. maximal torus, that is, and graded by theof U(1) exactly marginal deformations.extension to fractional For ˆ simplicity ofg notation we also assume ˆ is a proper realimportant Lie implications subgroup for the quantum theory. which, through thett algorithm in appendix the “brane group”, that is, the image we get a with Γ the monodromy group. Composing with the canonical projection JHEP09(2020)147 ) 1.12 (B.9) 3) Nucl. (B.10) Indeed . Then , j ¯ ≥ i n G is the Hodge ]. j ¯ i G SPIRE IN 2d (2,2) model, as a [ . 2] higher arithmetics. n/ [ /K = 0 the symmetric manifold is ≡ is flat. ) ) λ R ( n M & G \ . If ]. superconformal hep-th/0509212 j ¯ Γ i / , / have the above property (for λ G of a SPIRE ) we have not = IN w 6= rank SO( j ¯ i ) ][ groups which have discrete series automorphic ] while the Ricci curvature of /H 2.21 R R w – 74 – ) ) g R 30 n, ) does , ( only , factorizes through a quotient of a Mumford-Tate ∩ ). R G 1 24 0 S , , \ 1 n, 0 Γ − g / g ( ), which permits any use, distribution and reproduction in rank SL( p is Einstein ≡ hep-th/0605264 ⊆ O On the geometry of the string landscape and the swampland M = exp( S [ 1 ) pluriharmonic map ]. In view of eq. ( H − TS ∗ is constant, and the QFT is free. Otherwise we see from eq. ( n coincides, up to a factor, with the special K¨ahlermetric ( tt ∗ CC-BY 4.0 p j ¯ 142 p i , K (2007) 21 This article is distributed under the terms of the Creative Commons , where 141 The be a symmetric rigid special K¨ahlermanifold. Without loss we may assume The String landscape and the swampland 766 /H ) A symmetric rigid special K¨ahlermanifold M R ( G ] of the Shimura varieties, the “paradise” of arithmetics. Let Phys. B 31 C. Vafa, personal communication. C. Vafa, H. Ooguri and C. Vafa, The (arithmetic) quotients of Mumford-Tate domains are the natural generaliza- The above result shows that the conformal brane amplitudes are very different from The generic “brane group” SL( to be irreducible. Then [1] [2] [3] holomorphic with flat, the period map the Hodge metric the Ricci curvatures of the twoof metrics the are equal Hodge and metric non-singular.metric is But and non-positive the hence [ Ricci non-negative. curvature We got a contradiction. C Symmetric rigid specialLemma K¨ahlermanifolds 9. Proof. M p tion [ Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. References domain arithmetic: in particular theyrepresentations are [ the Corollary 10. function of the conformal manifold the well-known one forthe the real massive Lie 2d groups models. of the They “Mumford-Tate” have type are characterized by their interesting JHEP09(2020)147 ]. , SPIRE (2014) (1996) = 2 IN ]. [ 05 N (1991) 755 (1994) 19 (1991) 359 Commun. 1620 Commun. Commun. , , , SPIRE 355 IN 426 JHEP 367 . , (1994) 447, also (1970) 125. ][ (2019) 1900037 (1992) 5. 55 38 67 (1999) 31 75 hep-th/0005247 , (1990) 163. 203 dimensions , Clay Mathematics ]. 4 Lect. Notes Math. Nucl. Phys. B . Nucl. Phys. B Nucl. Phys. B ]. , , 133 , , ]. ]. , Princeton University Press, Publ. IHES hep-th/9408099 and , [ SPIRE 3 SPIRE Fortsch. Phys. supersymmetric theories IN SPIRE SPIRE , IN ][ IN IN ][ = 2 ][ ][ Publ. Math. IHES , N Proc. Symp. Pure Math. (1994) 484 . , https://www.jmilne.org/math , geometry in – 75 – arXiv:1704.00164 The string landscape, the swampland, and the missing Commun. Math. Phys. ∗ 431 ]. , , tt topological field theories Commun. Math. Phys. ]. hep-th/9407087 , ]. Landau-Ginzburg families SPIRE D-branes and mirror symmetry D ]. IN hep-th/9206037 hep-th/9211097 − arXiv:1711.00864 [ [ [ [ ]. = 2 Electric-magnetic duality, monopole condensation, and Monopoles, duality and chiral symmetry breaking in 2 SPIRE SPIRE IN N Topological antitopological fusion On classification of SPIRE IN Nucl. Phys. B ][ supersymmetric Yang-Mills theory IN Dirichlet branes and mirror symmetry ][ , SPIRE ][ (1994) 485] [ IN supergravity, type IIB superstrings and algebraic geometry = 2 (1990) 517 (1993) 539 (1993) 569 ][ Calabi-Yau operators N Special geometry 430 Periods of integrals on algebraic manifolds. III. Some global = 2 Geometry of Geometry and integrability of topological-antitopological fusion 131 152 158 Higgs bundles and local systems Topics in transcendental algebraic geometry Special K¨ahlermanifolds Shimura varieties and moduli Shimura varieties and motives https://www.jmilne.org/math N Geometry of The SISSA lectures on geometry and arithmetics of effective Lagrangians The swampland: introduction and review ]. ]. PoS(TASI2017)015 , arXiv:1312.1008 hep-th/9407018 [ [ SPIRE SPIRE IN IN hep-th/9712042 Erratum ibid. arXiv:1903.06239 differential-geometric properties of the period mapping Princeton, U.S.A (1984). available at Math. Phys. Monographs volume 4, American Mathematical Society, U.S.A (2009). Math. Phys. Math. Phys. 055 [ 120 supersymmetric QCD [ [ confinement in [ corner [ lecture notes in preparation. A. Strominger, P.A. Griffiths, P. Griffiths, J.S. Milne, D. Van Straten, S. Cecotti, K. Hori, A. Iqbal and C.P.S. Vafa, Aspinwall et al., J.S. Milne, B. Dubrovin, S. Cecotti, D. Gaiotto and C. Vafa, S. Cecotti and C. Vafa, B. Dubrovin, S. Cecotti and C. Vafa, C. Simpson, S. Cecotti, N. Seiberg and E. Witten, D.S. Freed, N. Seiberg and E. Witten, E. Palti, S. Cecotti, T.D. Brennan, F. Carta and C. Vafa, [9] [7] [8] [5] [6] [4] [23] [24] [25] [20] [21] [22] [17] [18] [19] [15] [16] [12] [13] [14] [10] [11] JHEP09(2020)147 1 02 = 2 = 2 ]. N N 3 Acta rank , = 2 = 2 JHEP SPIRE , = 2 N N , second IN (2018) 003 N ][ , Cambrigde ]. ]. 02 . (2018) 001 hep-th/0504070 02 SPIRE , SPIRE IN JHEP ]. IN , ][ [ (2018) 057 SYM theory and the Classification of JHEP 08 , , Lecture Notes in Boll. Unione Mat. Ital. SPIRE hep-th/9604034 ]. , = 4 Selfdual strings and [ IN . [ N matkerr/MTD.pdf JHEP superconformal field theories with ∼ SPIRE (1989) 2475 , IN 4 = 2 ][ Geometric constraints on the space of Expanding the landscape of (1996) 746 N Geometric constraints on the space of Geometric constraints on the space of arXiv:1912.06144 [ 477 Period mappings and period domains ]. Geometry of Type II superstrings and the moduli of hep-th/0510226 – 76 – Monodromy of variations of Hodge structure , Four-dimensional Mumford-Tate domains Hodge theory, complex geometry, and representation Mumford-Tate groups and domains: their geometry and SPIRE , IN ][ (2020) 026003 Classification of arXiv:1602.02764 ]. ]. ]. Int. J. Mod. Phys. A [ Nucl. Phys. B , , 102 Special arithmetic of flavor SPIRE SPIRE SPIRE https://www.math.wustl.edu/ IN IN IN ][ ][ ][ (2016) 088 (2003) 183. 05 75 Mumford-Tate groups Phys. Rev. D arXiv:1601.00011 Travaux de Griffiths, Boubaki S´eminaire Exp. 376 Supersymmetric field theories. Geometric structures and dualities , [ , Annals of Mathematics Studies, Princeton University Press, Princeton U.S.A. ]. JHEP SCFTs. Part II. Construction of special K¨ahlergeometries and RG flows , , American Mathematical Society, U.S.A. (2013). = 2 SPIRE arXiv:1609.04404 arXiv:1505.04814 arXiv:1803.00531 IN SCFTs. Part III. Enhanced[ Coulomb branches and central charges supersymmetric field theory N (2018) 002 SCFTs two-dimensional Coulomb branches. II. SCFTs. Part I. Physical[ constraints on relevant deformations [ Appl. Math. superconformal field theories with two-dimensional[ Coulomb branches swampland superconformal field theories theory University Press, Cambridge U.K. (2015). arithmetic (2012). edition, Cambridge studies in advancedPress, mathematics Cambridge volume U.K. 168, (2017). Cambrigde University Mathematics volume 180, Springer, Germany (1970). https://publications.ias.edu/sites/default/files/Trieste.pdf (2010) 281, available at P. Argyres, M. Lotito, Y. L¨uand M. Martone, A. Klemm, W. Lerche, P. Mayr, C. Vafa and N.P. Warner, P.C. Argyres, M. Lotito, Y. L¨uand M. Martone, P.C. Argyres, M. Lotito, Y. L¨uand M. Martone, P.C. Argyres and J.R. Wittig, P. Argyres, M. Lotito, Y. L¨uand M. Martone, C.A.M. Peters and J.H.M. Steenbrink, P.C. Argyres, M. Crescimanno, A.D. Shapere and J.R. Wittig, H.-C. Kim, H.-C. Tarazi and C. Vafa, S. Cecotti, S. Ferrara and L. Girardello, M. Caorsi and S. Cecotti, M. Green, P. Griffiths and M. Kerr, S. Cecotti, M. Green, P. Griffiths and M. Kerr, J. Carlson, S. and M¨uller-Stach C. Peters, P. Griffiths, M. Green, P. Griffiths and M. Kerr, P. Deligne, [42] [43] [40] [41] [38] [39] [36] [37] [33] [34] [35] [31] [32] [29] [30] [27] [28] [26] JHEP09(2020)147 , 213 (2001) JHEP 5 . , , S.G. Frontiers in Vector (1960/61) Phys. Rev. D ]. , . ICTP Summer = 2 13 . N AMS/IP Stud. Adv. , Graduate Texts in (1979) 141 Phys. Lett. B [ SPIRE , supergravity and IN ]. [ 159 Asian J. Math. = 2 , N , Springer, Germany (2005). SPIRE (1991) 21 ]. IN . A pair of Calabi-Yau manifolds as ][ Semin. Bourbaki , lectures notes at matkerr/SV.pdf (1985) 445 359 , ∼ 2 SPIRE IN Several complex variables IV Nucl. Phys. B [ algebras in , , Second Edition, Birkh¨auser,Switzerland , in T , lectures given at the conference https://www.jmilne.org/math ]. , , Yale University, mimeographed notes (reprinted Nucl. Phys. B (1989) 23 hep-th/9910181 A topological formula for the K¨ahlerpotential of , – 77 – [ SPIRE supergravity , March 9–21, Les Houches, France (2003), 124 IN M theory, topological strings and spinning black holes Supersymmetric protection and the swampland Classification of K¨ahlerManifolds in ][ The Penrose transform. Its interactions with Class. Quant. Grav. , SO(8) A first course in modular forms ]. (1999) 1445 Calabi-Yau threefolds of quotient type The Symmetries and strings in field theory and gravity https://pdfs.semanticscholar.org 3 https://www.math.wustl.edu/ SPIRE , Oxford Mathematical Monographs, Oxford University Press, Oxford ]. IN strings and its implications for the moduli problem Introduction to Lie algebras and representation theory . Homogeneous complex manifolds arXiv:2003.10452 ]. = 2 . Lie groups beyond an introduction [ Certains sch´emasde groupes semi-simples Lectures on Chevalley groups SPIRE Commun. Math. Phys. IN , , available at N Introduction to Shimura varieties Homogeneous K¨ahlermanifolds and [ , An introduction to arithmetic groups Shimura varieties: a Hodge-theoretic perspective (1998) 31] [ = 1 9 (2020) 168 (2011) 084019 DN math/9909175 representation theory U.K. (1989). Gindikin and G.M. KhenkinSpringer, eds., Germany Encyclopaedia (1990). of Mathematical Sciences volume 10, number theory, physics andmath/0403390 geometry (2002). by the AMS), available at Mathematics 9, Springer, Germany (1972). (1988) 443 [ 219. Math. Adv. Theor. Math. Phys. 4 superstrings 06 an exactly soluble superconformal theory course 2010 83 Multiplet Supergravity Couplings R.J. Baston and M.G. Eastwood, D.N. Akhiezer, A.W. Knapp, R. Steinberg, J.E. Humphreys, C. Soul´e, K. Oguiso and J. Sakurai, C. Chevalley, S.H. Katz, A. Klemm and C. Vafa, S. Cecotti, S. Ferrara and L. Girardello, E. Palti, C. Vafa and T. Weigand, P. Candelas, X.C. De La Ossa, P.S. Green and L. Parkes, F. Diamond and J. Shurman, M. Kerr, S. Cecotti, E. Cremmer and B. Julia, E. Cremmer and A. Van Proeyen, J.S. Milne, T. Banks and N. Seiberg, [61] [62] [60] [57] [58] [59] [55] [56] [53] [54] [51] [52] [48] [49] [50] [45] [46] [47] [44] JHEP09(2020)147 , ]. , , ]. , 123 Sophia = 2 , SPIRE , N IN . ][ (1996) 655] theories with (2003) 621. Commun. Math. 1 ]. , 2) , arXiv:0909.2201 Acta Math. 132 , = (2 SPIRE IN N . [ , Lectures in Mathematics, hep-th/9406032 Calabi-Yau Manifolds and (2009) 087 [ Proc. AMS [ arXiv:1512.08384 5 , , , in (1984) 89 Holomorphic anomalies in topological Arithmetic and geometry. Papers ]. Sigma math/0106063 245 supersymmetric theories , (1994) 79 , , in AMS/IP Stud. Adv. Math. [ SPIRE = 2 , University Lecture Series volume 73, American 334 IN N ][ – 78 – Variations of Hodge structure considered as an , Birk¨auser,Switzerland (1983). Chiral rings do not suffice: (1993) 279 Nucl. Phys. B Potentials and symmetries of general gauged , (1980) 73. Global Torelli theorem for projective manifolds of Calabi-Yau 405 Locally homogeneous complex manifolds Phys. Lett. B 24 ]. Some observations on the infinitesimal period relations for Isometry groups of simply connected manifolds of non-positive , Ising model and . hep-th/9209085 [ SPIRE From local Torelli to global Torelli Hodge metric completion of the moduli space of Calabi-Yau manifolds Geometry of complex domains with finite-volume quotients IN , M. Gross et al. eds., Universitext, Springer, Germany (2003). . . (2002) 81. ][ (1982) 41. 45 . Introduction to arithmetic groups Nucl. Phys. B On an example of Aspinwall and Morrison , Isometry groups of simply connected manifolds of nonpositive curvature II 149 Algebraic groups. The theory of group schemes of finite type over a field Harmonic maps into symmetric spaces and integrable system theory (1993) 139 Illinois J. Math. Calabi-Yau manifolds and mirror symmetry Introduction to arithmetic groups , Nonpositive curvature: geometric and analytic aspects On the Hodge metric of the universal deformation space of Calabi-Yau threefolds 157 arXiv:1205.4207 , hep-th/9302103 arXiv:1801.00459 curvature Acta Math. Cambridge University Press, Cambridge U.K. (2017). ETH Birk¨auser,Switzerland (1997). Z¨urich, nonzero fundamental group Related Geometries type arXiv:1305.0231 math/0505582 field theories [ dedicated to I.R. Shafarevich volume 2 exterior differential system: old and new results supergravity: Yang-Mills models Kokyu. Math. Phys. regular threefolds with trivial canonical bundle (1969) 253. Mathematical Society, U.S.A. (2019). S.S. Chen and P. Eberlein, P. Eberlein, J.S. Milne, J. Jost, K. Liu and Y. Wu, P.S. Aspinwall and D.R. Morrison, B. Szendr¨oi, M. Gross, X. Chen, K. Liu and Y. Shen, K. Liu and Y. Shen, K. Liu and Y. Shen, Z. Lu, M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, J. Carlson, M. Green and P. Griffiths, B. de Wit and A. Van Proeyen, S. Cecotti and C. Vafa, R. Bryant and P. Griffiths, D.W. Morris, A. Borel, Y. Ohnita, P. Griffiths and W. Schmid, [82] [83] [79] [80] [81] [76] [77] [78] [73] [74] [75] [71] [72] [69] [70] [67] [68] [64] [65] [66] [63] JHEP09(2020)147 J. , , (1992) JHEP , Inv. Publ. , , , (1992) 165. 166 , 135 . , H. Grauert ed., ]. Ann. Math. , ]. SPIRE IN , Modern Surveys in superconformal field theories in Lin. Algebra Appl. Transactions of the Moscow SUSY field theories What is. . . a thin group? , ][ , Princeton University Press, SPIRE D , in = 2 4 Classification and canonical IN special K¨ahlerproduct manifolds arXiv:1210.0523 , Springer, Germany (1972). N ][ , Lectures Notes in Mathematics volume = 2 Transactions of the Moscow Mathematical N Sp(4), ]. ]. , in theories with disconnected gauge groups (1997) 157. 4 arXiv:0804.1957 , Americal Mathematical Society, U.S.A. (1965). – 79 – SPIRE [ = 2 SPIRE IN IN N [ ][ Central charges of d arXiv:1903.00596 A theorem on [ Fundamental groups of manifold of nonpositive curvature (1989) 173. Harmonic mappings of K¨ahlermanifolds to locally symmetric Complex analysis and algebraic geometry (2008) 109 Manifolds of nonpositive curvature and their buildings 69 ]. Thin monodromy in , Americal Mathematical Society, U.S.A. (1965). Swampland distance conjecture for one-parameter Calabi-Yau 09 , in (1989) L243 Math. Res. Lett. , (2019) 086 6 Discrete subgroups of Lie groups SPIRE . IN 08 [ JHEP (1987) 35. , (1987) 1. Discrete subgroups of semisimple Lie groups arXiv:1611.08602 Strong rigidity of locally symmetric spaces The theory of convex homogeneous cones [ Le groupes de monodromie des familles universales d’hypersurfaces et On automorphisms of siegel domains 65 On the incompleteness of the Weil-Petersson metric along degenerations of 25 Archimedian superrigidity and hyperbolic geometry Variation of Hodge structure: the singularities of the period mapping The power of holomorphy: exact results in JHEP (1973) 211. , Publ. Math. IHES A simple presentation of the Siegel modular groups 22 , (2017) 145 threefolds Math. 185. 03 Calabi-Yau manifolds arXiv:1812.02330 four dimensions hep-th/9408013 spaces Society for the Year 1963 286, Springer, Germany (1972). Mathematical Society for the Year 1963 d’intersection completes Lectures Notes in Mathematics volume 1195, Springer, Germany (1986). realization of complex homogeneous domains Mathematics volume 17, Springer, Germany (1991). Class. Quant. Grav. Math. IHES Diff. Geom. Princeton U.S.A. (1973). W. Schmid, N. Lu, P.C. Argyres and M. Martone, 4 C.-L. Wang, A. Joshi and A. Klemm, A. Kontorovich, D.D. Long, A. Lubotzky and A.W. Reid, C. Brav and H. Thomas, A.D. Shapere and Y. Tachikawa, N. Seiberg, J.A. Carlson and D. Toledo, K. Corlette, S. Murakani, E.B. Vinberg, M.S. Raghunathan, S.G. Gindikin, I.I. Pjateckii-Sapiro and E.B. Vinberg, G.A. Margulis, S. Ferrara and A. Van Proeyen, A. Beauville, W. Ballmann and P. Eberlein, G.D. Mostow, K. Burns and R. Spatzier, [97] [98] [99] [94] [95] [96] [92] [93] [90] [91] [87] [88] [89] [85] [86] [84] [103] [104] [100] [101] [102] JHEP09(2020)147 , ] , ]. Ast´erisque , (2019) 134867 Phys. Lett. B , SPIRE math/0510021 IN 797 [ ][ ]. arXiv:1909.10355 , arXiv:1808.03483 , ]. SPIRE IN ]. (2006) 297 ][ SPIRE Phys. Lett. B 261 IN , ]. SPIRE ][ ]. On the cosmological implications of the ]. IN De Sitter space and the swampland Classifying Calabi-Yau threefolds using arXiv:1806.09718 ][ ]. [ SPIRE IN SPIRE The swampland distance conjecture for K¨ahler Exceptional supergravity theories and the SPIRE [ IN [ IN SPIRE Infinite distances in field space and massless ][ IN , Fields Institute Monographs volume 22, Infinite distances and the axion weak gravity – 80 – ][ arXiv:1802.08264 [ (2018) 271 ]. (1983) 72 ]. Infinite distance networks in field space and charge orbits arXiv:1905.00901 784 [ , Clay Mathematics Monographs volume 1, American Commun. Math. Phys. AdS and the swampland . 133 arXiv:1812.07548 , ]. Picard-Fuchs operators for octic arrangements I. The case of (1994) 193. [ ]. SPIRE . Cobordism Classes and the Swampland Fine tuning, sequestering, and the swampland (2018) 143 IN [ 166 Theta-problem and the string swampland 08 SPIRE arXiv:1910.02963 SPIRE On the systems of generators of arithmetic subgroups of higher rank arXiv:1811.02571 , IN (2020) 020 [ IN arXiv:1905.06342 [ ][ On the Weil-Petersson volume and the first Chern class of the moduli Weil-Petersson geometry on moduli space of polarized Calabi-Yau Phys. Lett. B [ (2019) 075 03 , JHEP alg-geom/9304007 Phys. Lett. B , , 08 Mirror symmetry Compactifications of moduli spaces inspired by mirror symmetry arXiv:0902.4592 Maximal automorphisms of Calabi-Yau manifolds versus maximally unipotent , JHEP math/0510020 Modular Calabi-Yau threefolds (2019) 016 , , arXiv:1709.09752 ]. ]. ]. JHEP , Pacific J. Math. 03 , , (2019) 135004 (1993) 243 [ SPIRE SPIRE SPIRE arXiv:1906.05225 IN IN IN monodromy orphans Mathematical Society, U.S.A (2003). [ 798 MAGIC square arXiv:1806.08362 string swampland [ [ moduli conjecture infinite distance limits towers of states JHEP [ American Mathematical Society, U.S.A. (2005). groups 218 manifolds space of Calabi-Yau manifolds S. Cynk and D. Van Straten, K. Hori et al., D. L¨ust,E. Palti and C. Vafa, J.J. Heckman and C. Vafa, J.C. Rohde, G. Obied, H. Ooguri, L. Spodyneiko and C. Vafa, P. Agrawal, G. Obied, P.J. Steinhardt and C. Vafa, J. McNamara and C. Vafa, S. Cecotti and C. Vafa, M. G. G¨unaydin, Sierra and P.K. Townsend, T.W. Grimm and D. Van De Heisteeg, T.W. Grimm, F. Ruehle and D. van de Heisteeg, T.W. Grimm, E. Palti and I. Valenzuela, T.W. Grimm, C. Li and E. Palti, P. Corvilain, T.W. Grimm and I. Valenzuela, C. Meyer, T.N. Venkataramana, Z. Lu and X. Sun, Z. Lu and X. Sun, D. Morrison, [123] [124] [120] [121] [122] [118] [119] [115] [116] [117] [113] [114] [110] [111] [112] [108] [109] [106] [107] [105] JHEP09(2020)147 , , 123 1 ]. , Essays (2012) J. Diff. (1986) 139. , in 22 SPIRE , IN 98 ]. ][ Ann. Math. , (1997) 83. Thin groups and SPIRE 61 IN . (1994) 3. , in ][ Inv. Math. , 55 , Princeton University Press, Geom. Funct. Anal. ANS/IP Studies Adv. Math. (1961) 787. , arXiv:1010.5792 , [ 1 math/0310402 , Congruence properties of Zarinski-dense arXiv:1511.08778 (2011) 111 [ Proc. Symp. Pure Math. Fivebrane instantons, topological wave functions , 03 (1984) 514. Degeneration of Hodge structures – 81 – Proc. Symp. Pure Math. 48 , Soviet Math. Dokl. JHEP , (2016) 157 Calabi-Yau threefolds of type K (II): mirror symmetry Calabi-Yau threefolds of type K (I): Classification , 10 Expansion in perfect groups A geometric construction of the discrete series for semisimple ]. Discrete series (1979) 189. , E. Breuillard and H. Oh et al., MSRI Publications volume 61 ]. . 54 SPIRE , S.-T. Yau ed., International Press, U.S.A. (1992). IN [ Calabi-Yau manifolds with large Picard number K¨ahlerclasses on Calabi-Yau threefolds — An informal survey (1963) 149. Superstrong approximation for monodromy groups Homogeneous cones Differential geometry of complex vector bundles Ratner’s theorems on unipotent flows aie´s¨heinedn la classe primi`ere deVari´et´esK¨ahleriennesdont Chern est nulle On analytic families of polarized Abelian varieties and automorphic functions 78 Classical polylogarithms Inv. Math. Local behavior of hodge structures at infinity Proc. London Math. Soc. (1983) 755. , , arXiv:1210.3757 18 arXiv:1108.4900 Ann. Math. Lie groups Princeton U.S.A. (1987). on mirror manifolds Geom. arXiv:1409.7601 superstrong approximation (2013) [ Commun. Num. Theor. Phys. and hypermultiplet moduli spaces subgroups 1832, (1997) 683. (1986) 457. V. Bolton and W. Schmid, D.W. Morris, G. Shimura, M. Atiyah and W. Schmid, P.M.H. Wilson, P.M.H. Wilson, E.B. Vinberg, A. Beauville, K. Hashimoto and A. Kanazawa, S. Kobayashi, J.S. Ellenberg, K. Hashimoto and A. Kanazawa, C. Matthews, L. Vaserstein and B. Weisfeiler, A. Golsefidy and P.P. Varj´u, R.M. Hain, E. Cattani, A. Kaplan and W. Schmid, S. Alexandrov, D. Persson and B. Pioline, P. Deligne, [142] [139] [140] [141] [136] [137] [138] [133] [134] [135] [131] [132] [129] [130] [126] [127] [128] [125]