Special Geometry and the Swampland

In the context of 4d effective gravity theories with 8 supersymmetries, we propose to unify, strenghten, and refine the several swampland conjectures into a single statement: the structural criterion, modelled on the structure theorem in Hodge theory. In its most abstract form the new swampland criterion applies to all 4d $\mathcal{N}=2$ effective theories (having a quantum-consistent UV completion) whether supersymmetry is \emph{local} or rigid: indeed it may be regarded as the more general version of Seiberg-Witten geometry which holds both in the rigid and local cases. As a first application of the new swampland criterion we show that a quantum-consistent $\mathcal{N}=2$ supergravity with a cubic pre-potential is necessarily a truncation of a higher-$\mathcal{N}$ \textsc{sugra}. More precisely: its moduli space is a Shimura variety of `magic' type. In all other cases a quantum-consistent special K\"ahler geometry is either an arithmetic quotient of the complex hyperbolic space $SU(1,m)/U(m)$ or has no \emph{local} Killing vector. Applied to Calabi-Yau 3-folds this result implies (assuming mirror symmetry) the validity of the Oguiso-Sakurai conjecture in Algebraic Geometry: all Calabi-Yau 3-folds $X$ without rational curves have Picard number $\rho=2,3$; in facts they are finite quotients of Abelian varieties. More generally: the K\"ahler moduli of $X$ do not receive quantum corrections if and only if $X$ has infinite fundamental group. In all other cases the K\"ahler moduli have instanton corrections in (essentially) all possible degrees.

1 Introduction and overview: swampland conjectures vs. VHS structure theorem Passing from Quantum Field Theory (QFT) to Quantum Gravity (QG) requires a radical change of paradigm: in the words of Cumrun Vafa [1] it is like going from Differential Geometry (DG) to the much deeper Number Theory. The fundamental principles of quantum physics -such as the concept of "symmetry" -take new (and much subtler) meanings in QG, and the mathematical formulation of the theory requires totally new foundations in which arithmetics is expected to play a central role.
The swampland program [2,3] (see [4,5] for nice reviews) aims to characterize the effective field theories which describe the low-energy limit of consistent quantum gravities inside the much larger class of effective theories which look consistent from the viewpoint of the traditional field-theoretic paradigm. The consistent QG effective theories form a very sparse zero-measure subset of the naively consistent models. An effective field theory which looks consistent but cannot be completed to a consistent QG is said to belong to the swampland [2][3][4][5].
Up to now the swampland program has taken the form of a dozen or so (conjectural) necessary conditions which all consistent effective theories of gravity should obey; we refer to them as the swampland conjectures [2][3][4][5]. Although these conjectures are arguably the deepest principles in physics, indeed in all science, their current formulation looks a bit unsatisfactory: they consist of several disjoint statements, which are neither logically independent nor related by a clear web of mutual implications. Ideally, one would like to unify the several conjectures into a single basic physical principle.
The reason why the swampland conjectures do not look "elegant" is that they are formulated entirely in the language of the old QFT/DG paradigm: they are expressed using words like "distance", "volume", "geodesic", etc., that is, in the strict DG jargon, while one expects that a language appropriate for the gravitational paradigm should emphasize more the arithmetic aspects. It is natural to think that, if we wish to unify and strengthen the several conjectures in one more fundamental swampland principle, we must state it in a suitable 'arithmetic' language.
To build the appropriate formalism in full generality requires a lot of foundational work in which the usual field-theoretic concepts get uplifted to subtler ones endowed with loads of new additional structures most of which arithmetic in nature [6]. The story is long, technically sophisticate, and an enormous quantity of more work is required.
Luckily enough, however, there is one important special case where we can dispense with new foundations: namely the 4d effective theories invariant under 8 supercharges, more specifically, the vector-multiplet sector of an ungauged 4d N = 2 supergravity. At the level of the traditional QFT/DG paradigm such a sector is described by a special Kähler geometry (see e.g. [7]). In this special case the swampland program reduces to the simple question: Question 1. Which special Kähler geometries belong to the swampland? Question 1 may be re-formulated in two well-known alternative languages which already take care of the relevant arithmetic aspects. This will give us two (equivalent) Answers to Question 1.
The second Answer gives a necessary criterion for quantum consistency which makes sense also for low-energy effective theories L eff having rigid N = 2 supersymmetry. In N = 2 QFT one can prove rigorously the validity of this criterion (see § §. 1.3 and 4.6): it turns out to be equivalent to requiring that the low-energy physics is described by a globally well-defined Seiberg-Witten geometry [8,9]. The swampland criterion we propose may be seen as the more general version of Seiberg-Witten geometry which applies both to rigid and local N = 2 susy: in both cases a quantum-consistent effective theory is described by such a higher Seiberg-Witten geometry. In spite of this, the local and rigid cases differ dramatically: in the gravity case the proposed criterion implies the validity of all the standard swampland conjectures [2,3] (see §. 4.7), in particular the distance conjecture which requires infinite towers of light states at infinity, while in the rigid supersymmetry the same argument shows that at most finitely many states may become massless in any limit (cfr. §.1.3).

First alternative language: BPS branes in tt *
One shows 1 [11,12] that the special Kähler geometries are in one-to-one correspondence with the solutions to the tt * equations [12] for the universal deformation of an abstract 2d (2,2) chiral ring R s (s ∈ S) which -in addition to the usual properties 2 -is required to be a family of local-graded C-algebras 3 , that is, The family of rings {R s } s∈S is parametrized by the universal deformation space S (the "moduli" space, a.k.a. the 2d conformal manifold). For the convenience of the reader, the identification of local-graded tt * geometry with special Kähler geometry is reviewed in section 2.1 below (some technical detail being deferred to appendix A).
In the tt * language the swampland program asks for the characterization of the solutions to the tt * PDEs for local-graded chiral rings which "arise from physics" out of the much bigger space of all such solutions which is equivalent to the space of all special Kähler geometries. The physical subset contains the special geometries of (2,2) SCFTs withĉ = 3, and in particular the special geometry of 2d supersymmetric σ-models with target space a Answer 1. A special Kähler geometry which does not belong to swampland satisfies the "straightforward" local-graded generalization of the tt * diophantine conditions of ref. [14]. 4 To understand the nature of the required "straightforward" generalization, let us recall the physical origin of the arithmetic conditions on the physical tt * geometries for semi-simple chiral rings. The tt * PDEs are the consistency conditions of a system of linear differential equations (depending on an additional twistor parameter ζ ∈ P 1 ) [12, [14][15][16] When the tt * geometry is physical, the solutions Ψ(ζ) to the linear problem (1.4) have the physical interpretation of half-BPS brane amplitudes 6 [17]. The BPS branes belong to a linear triangle category [17,18] whose numerical homology and K-theoretical invariants induce a canonical integral structure on the space of solutions to (1.4) which implies the diophantine conditions of [14]. Nowhere in this argument one uses the fact that R s is semisimple: the tt * equations can be written in the linear form (1.4) for arbitrary chiral rings, and -whenever the tt * geometry arises from a physical situation -its brane amplitudes Ψ(ζ) describe objects of a BPS brane category with a finitely generated Grothendieck group from which the tt * geometry inherits an arithmetic structure. The precursor papers [14,17] focused on semi-simple chiral rings for just one reason: in that case, one may identify half-BPS branes with Lefschetz thimbles [17], and then use Picard-Lefschetz theory [14] to relate them in a simple way to the BPS spectrum and wall-crossing phenomena. The relation with the BPS spectrum is lost when R s is local-graded, since there are no non-trivial BPS states in a (2,2) SCFT; nevertheless the half-BPS branes are still there and, when the corresponding special Kähler geometry arises from physics, they are objects in some nice linear triangle 4 We stress that this is still a necessary condition, not a sufficient one, albeit it is expected to "come close" to be sufficient. 5 Eqn. (1.4) is written more explicitly in eqn. (2.24). 6 The twistor parameter ζ labels the susy sub-algebra which leaves the brane invariant.
category, so that their special Kähler geometry inherits interesting arithmetic properties. In facts, the arithmetics of local-graded tt * amplitudes turns out to be much richer, nicer and deeper than the Picard-Lefshetz one for semi-simple chiral rings, see appendix B.
In conclusion, in the special N = 2 case the swampland conditions may be easily guessed from the tt * viewpoint. However there is a second alternative language which makes the story even more straightforward. In the rest of this note we shall mostly adopt this second viewpoint, except in §.4.2 where we use the equivalent tt * language to give an intuitive interpretation of the swampland criterion we are proposing.
Remark. Requiring that the tt * brane amplitudes Ψ(ζ) satisfy the arithmetic conditions is a priori weaker than requiring that they are physical brane amplitudes. The arithmetic conditions essentially see only the K-theoretic aspects of the relevant linear triangle category of branes. The arithmetic conditions are then expected to be mere necessary conditions. It is conceivable that one gets sufficient conditions for the existence of a quantum gravity UV completion by requiring that the tt * branes carry the full-fledged categorical structure and not merely its numerical avatar. Morally speaking, the procedure of UV completion of a low-energy effective theory looks akin to reconstructing an algebraic variety Y out of its derived sheaf category D(Y ).

Second alternative language: "motivic" VHS
In Table 1 (page 7) we compare the problem of describing the set of consistent 4d N = 2 quantum gravities with the mathematical problem of describing all "algebro-geometric objects" of "Calabi-Yau type". Notice that we do not speak of Calabi-Yau manifolds (or varieties) but more generally of "algebro-geometric objects". Indeed experience with moduli of Abelian varieties has shown that, in order to obtain a deep and nice theory, one has to enlarge the class of geometric objects of interest beyond the actual manifolds [19,20]. The appropriate class of "objects" to consider is some sort of "Calabi-Yau motives". 7 Likewise, if one wishes a classification of consistent "Quantum Gravities" with nice functorial properties, one should accept some more general animals than the strict QG theories such as, for instance, consistent truncations of consistent quantum gravities. Both classification problems are somehow open, in the sense that the precise boundaries of the class of "natural" objects to be called "physical" (resp. "geometric") is not fully understood. In particular, we do not have an explicit set of axioms which define in a precise way what a "Quantum Gravity" is supposed to be.
At the naive level of the DG paradigm, on the physical side we know that the effective theory of a 4d N = 2 quantum gravity must be a 4d N = 2 (ungauged) supergravity, whose vector-multiplet sector is described in DG language by a special Kähler manifold S.

Quantum Gravity
Algebraic Geometry classification problem consistent 4d N = 2 "quantum gravities" CY algebro-geometric "objects" up to deformation type  Table 1: The comparison of two classification problems. Symbols in the right margin describe the logical relation between the two boxes in the same row: ≡ means that the two are known to be strictly equivalent, ? ≡ means that the two are conjectured to be equivalent, while means that the two are trivially the same; finally, ⇐ means that the statement in the second column implies the ones in the first column.
Saying that the effective Lagrangian L is N = 2 supersymmetric amounts to saying that its couplings, seen as functions of the scalar fields ("moduli"), satisfy a set of differential relations which are the defining property of special geometry [22,23].
Passing to the second column of Table 1, we know that the DG description of a deformation family of CY "objects" is given by a polarized real variation of Hodge structure (R-VHS) [24][25][26][27][28][29][30]32] of pure weight 3 and Hodge numbers h 3,0 = 1 and h 2,1 = m ≡ dim C S. (1.5) The R-VHS is encoded in the Griffiths period map [24][25][26][27][28][29][30]32] p : where S is the "moduli" space and Γ ⊂ Sp(2m + 2, Z) the monodromy group. The map p is required to satisfy a set of differential relations, known as the Griffiths infinitesimal period relations (IPR) [24][25][26][27][28]30]. One shows that these differential relations are exactly equivalent to the ones defining N = 2 supersymmetry [22,23], so that the problems in the two columns of Table 1 are strictly equivalent at the naive level of the DG paradigm. We write V for the set of abstract weight-3 polarized R-VHS with the Hodge numbers (1.5), equivalently V is the set of consistent-looking N = 2 special Kähler geometries. It turns out that almost all 4d N = 2 supergravities in V cannot arise as low-energy limits of consistent theories of quantum gravity. We write M QG ⊂ V for the extremely minuscule subset which can be physically realized (the complementary set V \ M QG ⊂ V being the vast N = 2 swampland). Likewise, almost all R-VHS in V do not describe actual families of algebro-geometric objects. Again we have a minuscule subset M Mot ⊂ V of "motivic" R-VHS which do arise from algebraic geometry. In particular, a variation of Hodge structure in M Mot should be defined over Q not R, i.e. it should be a (pure, polarized) variation of Hodge structure in Deligne's sense (a VHS for short) [26][27][28][29][30]: this is a first hint of non-trivial arithmetic structures entering in the game. Both M QG and M Mot are God-given, extremely sparse, subsets of the same naive space V, and both are expected to be characterized by subtle arithmetic-like properties.
It is natural to ask whether there is any simple relation between these two special subsets. The meta-conjecture is that they indeed coincide (for a suitable choice of what we are willing to call a "consistent quantum gravity"). This will follow, for instance, if we assume the string lamppost principle (SLP) [33].
Characterizing M QG and M Mot inside V is a fundamental problem in (respectively) Theoretical Physics and Algebraic Geometry. What is known about these two fundamental problems?
First The two columns of Table 1 look very much the same when restricted to their first five rows. The last row describes the second piece of "known" information about the deep problems: a set of necessary conditions an element v ∈ V should satisfy in order to belong to the sub-set M QG or, respectively, M Mot . In this last row the two columns look quite different. In the column of Quantum Gravity the necessary conditions take the form of a dozen or so conjectural statements: they are the usual swampland conjectures (specialized to the case of ungauged N = 2 sugra) [2][3][4][5]. In the Algebraic Geometry column we have a single statement which, moreover, has the logical status of a mathematical theorem: this fundamental result is known under the name of the structure theorem of (global) variations of Hodge structure, see e.g. §. IV of [27], §. II.B of [28], §. III.A of [29], or Theorem 15.3.14 in [30]. We review the theorem in §.4 below following the two nice surveys [27,28].
It is an established fact that all VHS arising from Algebraic Geometry enjoy the properties stated in the structure theorem. In other words, the statement of the theorem yields a proven necessary condition for a VHS to belong to the God-given subset M Mot .
In §.(III.B.7) of ref. [28] the authors ask whether this necessary condition suffices to completely characterize the set M Mot of "motivic" VHS: they answer in the negative, but their general feeling is that the structure theorem comes "close" to the holy Grail of actually determining M Mot , thus "almost" completing the algebro-geometric analogue of the 4d N = 2 swampland program. 8 What is the logical relation between the two boxes in the last row of table 1? It is easy to see that a (global) special geometry which satisfies the VHS structure theorem automatically satisfies all the applicable swampland conjectures. 9 This is hardly a surprise since Ooguri and Vafa [3] used the general properties of Calabi-Yau moduli spaces as motivating examples for their conjectures. However the inverse implication is false, i.e. the structure theorem is actually a stronger constraint than the several swampland conjectures combined.
The comparison of the two columns of Table 1 suggests to unify and refine the several swampland conjectures into the single statement: Answer 2 (The structural swampland conjecture). A (global) special Kähler geometry belongs to the swampland unless its underlying Griffiths period map p (cfr. eqn.(1.6)) satisfies the VHS structure theorem.

Relation with Seiberg-Witten theory in N = 2 QFT
The structural swampland criterion holds in rigid N = 2 supersymmetry as well. Here we sketch the story without entering in technical details (see §. 4.6 for more). The Lagrangian of a formal N = 2 low-energy effective QFT has the form where X a (a = 1, 2, . . . , h) are restricted N = 2 chiral superfields containing the field strengths of the h Abelian gauge fields and F is the holomorphic pre-potential. Let M be the Coulomb branch where the scalars 10 X a take value and M its smooth simply-connected cover, so that M = G\ M for some discrete group of isometries G. We have a commutative 8 Translated in the physical language, the basic source of non-sufficiency is that the structure theorem does not say which points τ in the upper half-plane may be realized as periods of harmonic (3,0)-forms on rigid 3-CY. Roughly speaking, the problem is that in this case the moduli space S is a single point, and the global aspects of S (central to the swampland program [3]) do not restrict the period map p. However there are good reasons to believe that in the physical context we may strengthen the statement and get rid of this inadequacy: one compactifies the 4d N = 2 theory with constant graviphoton coupling τ = θ/2π + 4πi/g 2 on a circle S 1 . In the resulting 3d N = 4 theory S is promoted to a quaternionic-Kähler manifold Q τ of real dimension 4 [34] whose geometry depends on τ . Applying the 3d N = 4 version of the swampland conditions to the 3d moduli space Q τ should yield some useful constraint on the allowed τ 's. 9 For some conjecture this actually holds after some minor technical refinement of the original statement. 10 We use the same symbol to denote the superfield and its scalar first component.
satisfies all the axioms of a weight-1 local VHS with Hodge numbers h 1,0 = h 0,1 = h. Again we may pose a swampland question: describe the sub-set of formal VHS p which define low-energy effective Lagrangians having a UV completion which is a fully consistent QFT. A necessary condition follows directly from QFT first principles. The operator algebra of a 4d N = 2 QFT contains a susy protected subfactor, the chiral ring R 4d , which is a finitely-generated, commutative, unital C-algebra. Then a 4d N = 2 QFT comes equipped with an affine variety, namely the spectrum Spec R 4d of its chiral ring. The (global) smooth Coulomb branch M is the complement in Spec R 4d of the discriminant divisor D (the locus where additional degrees of freedom become massless). Write Spec R 4d = M \ Y ∞ for M projective and Y ∞ an effective divisor 12 . Hence Then the group Γ ⊂ Sp(2h, Z) satisfies the properties of the monodromy group for a VHS over a quasi-projective base of the form (1.10), and then the period map p satisfies the VHS structure theorem. (For succinct surveys of the underlying mathematical facts, see e.g. § §.5-7 of [36] or §. IV of [27]). Then we have the following Fact 1. A 4d N = 2 low-energy effective field theory which has a quantum-consistent UV completion is described by a period map p : M → Γ\Sp(2h, R)/U(h) which satisfies the VHS structure theorem.
When the above necessary condition is satisfied, we may realize p as the period map of an algebraic family of Abelian varieties parametrized by M. Thus we may rephrase Fact 1 by saying that a quantum-consistent 4d N = 2 effective theory necessarily arises from a Seiberg-Witten geometry [8,9]. Experience in low dimension [37][38][39][40][41][42] suggests that this 11 Notation: throughout this paper double-headed arrows / / / / stand for canonical projections while hook-tail arrows / / represent canonical inclusions. 12 For explicit examples of these algebraic-geometric gymnastics in 4d N = 2 QFT see [35].
condition is almost sufficient. Roughly, the idea is that the UV completion of a low-energy effective theory, satisfying the structural condition, is obtained by using its Seiberg-Witten geometry to engineer the full N = 2 QFT in string theory [43].
Quantum-consistent N = 2 gravity versus N = 2 QFT. Although the structural swampland criterions for N = 2 QFT and N = 2 sugra are in principle identical, they lead to very different physical consequences. In the sugra case the structural swampland condition implies the validity of the Ooguri-Vafa swampland conjectures [3] while nothing of that sort applies in QFT. The most striking differences in the implications of the structural swampland condition in the two physical contexts are: A) the volume conjecture: in a quantum-consistent N = 2 gravity theory the volume of the scalars' space is finite, while this is not true in quantum-consistent N = 2 QFT, as the example of free-field theory shows; B) the distance conjecture: in quantum-consistent N = 2 gravity an infinite tower of states becomes light at infinity in scalars' space, whereas nothing of that sort may happen in a UV-complete QFT which has finitely-many effective degrees of freedom in all regimes.
These dramatic differences arise from three simple elements: 1. In N = 2 QFT the VHS has weight 1 (non-zero Hodge numbers h 1,0 = h 0,1 ), whereas in N = 2 sugra the VHS has weight 3 (non-zero Hodge numbers h 3,0 = h 0,3 = 1 and h 2,1 = h 1,2 ). Let γ i be the monodromy around a prime component Y i of the snc divisor Y = j Y j . By the hard monodromy theorem [30] γ i satisfies 13 As we approach the support of a divisor Y i with k i = 3 an infinite tower of states becomes massless. This cannot happen in QFT in view of (1.11). When we approach a divisor with k i = 1, in the gravity case either an infinite tower or just finitely many states may become massless, depending on the details of the degenerating mixed Hodge structure along Y i (see §. 4.7 and references therein). The same analysis shows that in QFT at most finitely many degrees of freedom may become massless along any Y i ; in showing this eqn.(1.12) below plays a crucial role.
2. In sugra the covering period map p is a local immersion (the local Torelli theorem), while in QFT p just factorizes through the immersion S → Sp(2h, Z)/U(h) of the where R ij is the Ricci tensor of G ij . If we insert back the Newton constant, the first term (m + 3)G ij will carry a relative factor M −2 p (M p being the Planck mass) so that the two formulae agree in the limit M p → ∞.
From eqn.(1.12) we see that being at infinite distance in the physical metric G ij has quite different meanings from the viewpoint of the intrinsic Hodge metric geometry in QFT versus SUGRA. In particular, the volume of the space S ≡ Γ\ S, as computed with the Hodge metric K ij , is finite in both cases, but this implies that the volume of the scalars' manifold M, as computed with the physical metric G ij , is also finite only in SUGRA (see §. 4.7 for details).
Notice that in QFT we may locally identify M and S around points where the Ricci tensor is non-singular i.e. det R ij = 0.
Despite the different physical implications of the structural swampland condition in QFT and SUGRA, its general consequences hold equally in both cases. For instance, the dichotomy introduced in the next subsection holds for a quantum-consistent Torelli manifold S in both cases. Assume there is one point in the smooth, simply-connected, irreducible cover M of the Coulomb branch where det R ij = 0; then either the rigid special Kähler manifold M is symmetric or has no Killing vector. The first possibility is ruled out for a non-free N = 2 QFT, since a symmetric rigid special Kähler manifold is necessarily flat corresponding to a free QFT (see appendix C).

A survey of the first applications
From the structural conjecture one can easily extract some novel property that all consistent N = 2 effective theories should satisfy (in addition to the usual swampland conditions).
To state the first one, we need some generality about "symmetry" in the QG paradigm. For more precise definitions and statements, see §.2.2 below.
"Symmetry" in QG. In QG there are no symmetries in the strict sense of the term [44]. However we can still consider the local Killing vectors of the scalars' space S (with respect to the metric in their kinetic terms). These local Killing vectors are never globally defined on S (so they do not generate symmetries) but they do constrain the couplings in the effective Lagrangian L and in particular the scalars' metric. For instance, if we compactify Type II on a six-torus we get a 4d N = 8 effective theory whose scalars' manifold S N =8 has 133 local Killing vectors forming the Lie algebra e 7 (7) , and this fact determines L almost uniquely [45].
In the N = 2 context, the scalars' space has the form S = Γ\ S for a simply-connected special Kähler manifold S and a discrete group of isometries Γ ⊂ Iso( S). The (naive) symmetry group Sym( S) of the covering special geometry S is a closed sub-group (1.13) The elements of its Lie algebra sym( S) ⊆ iso( S) are our local Killing vectors on S. As we shall see, when the special geometry S does not belong to the swampland the group Sym( S) and its algebra sym( S) -if non-trivial -carry interesting arithmetic structures.
The dichotomy. A first consequence of the structural conjecture is the following Fact 2 (Dichotomy). Assume Answer 2. Let S = Γ\ S be a special Kähler geometry which does not belong to the swampland, where S is its simply-connected cover and Γ its monodromy group (a.k.a. duality group). Then: S has a non-zero local Killing vector ⇒ S is locally symmetric. More precisely: • either S is Hermitian symmetric, its isometry group Iso( S) is one of the Lie groups in Table 2, Sym( S) = Iso( S), and Γ ⊂ Iso( S) is an arithmetic subgroup; • or sym( S) = 0, that is, the symmetry group Sym( S) of the covering special geometry S is discrete.
In other words: sym( S) = 0 if and only if S is a Shimura variety of rank-1 or 'magic' type. 15 Remark. Shimura varieties are the geometries with the richer and most interesting arithmetics [19,20,47,49]: indeed they are the very paradise of arithmetics. Their simplest (and 15 By a Shimura variety [19,20,47,49] we mean an arithmetic quotient of a non-compact Hermitian symmetric manifold, i.e. Γ\G/K where G is a non-compact semi-simple real Lie group, K ⊂ G a maximal compact subgroup of the form U (1) × H, and Γ ⊂ G an arithmetic subgroup (in the strict algebraic-group sense). We say that a Shimura variety Γ\G/K is of 'magical' type if G is a 'magical' Lie group, i.e. one of the groups in the right hand side of Table 2. rank-1 (quadratic) 'magic' (cubic) Table 2: Isometry groups G of symmetric special Kähler manifolds [46]: S ≡ G/K with K ⊂ G a maximal compact subgroup. One has k = m − 1 ≥ 1. The pre-potential F is quadratic and respectively cubic. 'Magic' SL(2, R) has rank-1 as a symmetric space, but it has "rank-3" as a special geometry [32] as the rest of 'magic' special geometries.
most classical) examples are the (non-compact) modular curves [48]. The fact that Shimura varieties arise so naturally provides further evidence for the relevance of arithmetics in QG. Note that last two groups in Table 2 do not correspond to 'classical' Shimura varieties, that is, they are not moduli spaces of Abelian motives.
There are plenty of non-symmetric simply-connected special Kähler geometries S with a non-trivial Killing vector, sym( S) = 0. They and all their quotients belong to the swampland. In particular, Fact 2 is bad news for the author of ref. [50]: all quotients of the several infinite series of homogeneous special Kähler geometries constructed there, however beautiful and elegant, belong to the swampland! (This negative result is already obvious at the naive DG level: see §. 3.2 below).

Fact 2 has important
Corollaries. The first one is in facts a confirmation in the special N = 2 case of a more general and profound prediction by the authors of ref. [51]: Corollary 1 (Completeness of instanton corrections). Suppose we have a pre-potential with an asymptotic expansion at ∞ of the form (1.14) which does not belong to the swampland. Then (i) either c λ = 0 for essentially all possible λ; (ii) or c λ = 0 for all λ, the "Euler characteristic" χ = 0, A IJ ∈ Z, and d ijk z i z j z k is the determinant form of a rank-3 real Jordan algebra (in particular, d ijk ∈ Z).
Remark. The (possibly reducible) rank-3 real Jordan algebras are in 1-to-1 correspondence with the cubic 'magic' groups in the right-hand side of Table 2.
In eqn.(1.14) we labelled the various terms according to their origin in the special case where the pre-potential F describes Type IIA compactified on a Calabi-Yau 3-fold X. In this particular application, the vector-multiplet scalars z i ≡ X i /X 0 parametrize the quantum Kähler moduli of X: the first term is the world-sheet classical (≡ large volume) answer, the second one the perturbative loop corrections, and the last term the world-sheet instantons [52,53]. In this specific geometric set-up the coefficients d ijk are integers (being intersection indices in cohomology [54]) and χ is the Euler characteristic of X.
Roughly speaking Corollary 1 says that we must have all possible world-sheet instanton corrections unless our N = 2 sugra is a consistent truncation of a N > 2 supergravity. This fact was predicted in [51] as an extension of the swampland conjectures. We make the statement more precise in the next three Corollaries. Corollary 2. Suppose the pre-potential (1.14) describes the quantum Kähler moduli of a 2d (2,2) superconformal σ-model (withĉ = 3). Then the "instanton corrections" in eqn. (1.14) vanish if and only if the absence of quantum corrections in the world-sheet theory is implied by the non-renormalization theorem of a higher (p, q) > (2, 2) 2d supersymmetry. In this case, also the loop corrections must vanish. 16 Corollary 3. Let S = Γ\ S be a special Kähler manifold, consistent with our swampland structural criterion, which is described locally by a strictly cubic pre-potential F cub . Then at least one of the following possibility applies: • S is a geodesic submanifold of the scalar's space of some consistent N = 4 supergravity.
That is, the N = 2 sugra is a consistent truncation of N = 4 to configurations invariant under a certain discrete symmetry group; • S is a geodesic submanifold of the scalar's space of some consistent N = 8 supergravity. That is, the N = 2 sugra is a consistent truncation of N = 8 to configurations invariant under a certain discrete symmetry group; • S is a Shimura variety which is an arithmetic quotient of Cartan's domain Using results from [19,20] we get a characterization of the QG cubic scalar manifolds S with well-known moduli spaces of nice algebro-geometric "objects": Corollary 4. S as in Corollary 3 and dim C S = 15, 27. Then S is the moduli space of some family of Abelian motives. Typically, they may be realized as the special geometries of the untwisted sector moduli of Type II/heterotic compactifications on tori [34].
Note that, up to finite covers, there are just five finite-volume, cubic, special Kähler manifolds S which are locally symmetric and locally irreducible. They have complex dimensions 1, 6, 9, 15, and 27.
(1.15) So if S does not belong to the swampland and is locally irreducible with dim C S not in the list (1.15), we conclude that either instanton corrections are present or there is no asymptotic limit in the space S where F gets cubic, i.e. the special geometry is "orphan".

Applications to Algebraic Geometry
Fact 2 has important implications in Algebraic Geometry. For instance: Corollary 5. Let X be a Calabi-Yau 3-fold which admits a mirror X ∨ . Suppose that X does not contain rational curves. Then X is a finite quotient of an Abelian variety; in particular X has Picard number ρ = 2 or 3.
The assumption of the existence of a mirror may be replaced by some much weaker technical requirement. Modulo this aspect, Corollary 2 answers in the positive the Question posed by Oguiso and Sakurai in ref. [55]. We stress that under the stated assumption, the result is mathematically fully rigorous, since it is a direct consequence of the VHS structure theorem applied to the universal deformation space of X ∨ .

Organization of the paper
The rest of this paper is organized as follows. In section 2 we review special Kähler geometry from a viewpoint convenient for studying its global and arithmetic aspects, emphasizing its relations with tt * geometry and Griffiths' theory of variations of Hodge structures. For later convenience we collect here some facts about "symmetries" in special geometry. In section 3 we present (for comparative purposes) two purely differential-geometric results which go in the same directions as our main conclusions, but are significantly weaker. In subsection 3.3 we collect some facts about the geometry of symmetric special Kähler geometries for later use. In section 4 we introduce our structural swampland criterion first at an intuitive level in the language of tt * and then systematically using the mathematical framework of VHS. Then we discuss how this statement implies, in suitable senses, the validity of the original Ooguri-Vafa swampland conjectures [3]. In section 5 we show how the criterion may -in principle -be used to compute the pre-potential F of a N = 2 supergravity which does not belong to the swampland. In section 6 we prove the main result of the present paper: the dichotomy for quantum-consistent special geometries. Section 7 describes the behaviour of a quantum-consistent special geometry when we approach at infinity a MUM (maximal unipotent monodromy) point, and presents applications to Type IIA compactifications on Calabi-Yau 3-folds. In section 8 we discuss the Oguiso-Sakurai question and argue that the answer is positive. In section 9 we briefly comment on the case of Picard number 1. In m , a = 1, 2, 3. The nodes are decorated by a color (white vs. black) and a shape (circle vs. lozenge). Forgetting all decorations we recover the Dynkin graph of C m+1 which specifies the underlying complex Lie group G m (C). Forgetting the shape of the nodes, one gets the Vogan diagram [60] of the real Lie algebra g m of the group G m (R). Forgetting the color, one gets the diagrams [61] of the complex parabolic subgroups P appendix A we review the equivalence between the tt * PDEs and the condition that the "Weil map" w is pluri-harmonic. In appendix B we show that the tt * brane amplitudes of superconformal 2d (2,2) models are characterized by special arithmetic properties not shared by their massive counterparts.

Review of special Kähler geometry
In this section we review the basic facts of special Kähler geometry mainly to fix language and notation. By convention, "special Kähler manifold" stands for "integral special Kähler manifold", that is, one whose global structure is consistent with Dirac quantization of electromagnetic fluxes and charges. In particular the underlying variations of Hodge structures are genuine VHS and not mere R-VHS.
Notations and conventions. Through this paper, S will denote a (integral) special Kähler manifold, S its smooth, simply-connected cover, and m its complex dimension. We write G m (K) for the group Sp(2m + 2, K) where K is one of the rings C, R, Q, Z. We see G m (K) as the group of K-valued points in the universal Chevalley group-scheme 17 of type C m+1 . G m (C) and G m (R) have (in particular) the structure of (respectively) a complex and a real Lie group, and G m (Z) ⊂ G m (R) is a maximal arithmetic subgroup with respect to the natural structure of algebraic group over Q provided by 18 G m (Q). We stress that all these fancy algebraic-geometric structures on Sp(2m + 2, R) have an intrinsic physical meaning, being unambiguously implied by Dirac quantization of charge/fluxes. 19

Basic geometric structures
We are concerned with three geometries which eventually will turn out to be equivalent: (1) tt * geometry for local-graded chiral rings as in eqns.
We begin by introducing their respective Klein models.    The diagrams of these three geometries are represented in Figure 1. Diagram (1) yields the Siegel upper half-space, i.e. the space of gauge couplings τ ab (symmetric matrices with positive-definite imaginary part); (2) is obtained by making a lozenge the node associated to the Dirac representation (2m + 2) of the electro-magnetic charges, while one gets (3) from (2) by eliminating all lozenges shared by (1) and (2). From the figure one sees that the G m (C)-homogeneous spacesĎ are rational, projective (hence compact), complex manifolds 20 whose homogeneous (holomorphic) bundles are in one-to-one correspondence with the possible assignments of an integer n i ∈ Z to each node of the corresponding graph in Figure 1 with the restriction that n i ≥ 0 for the circle nodes.Ď     19 See, for instance, the wonderful computations in the appendix of [59] for the particular case of Type II compactified to 4d on a six-torus. 20 For these basic facts, see e.g. chapter 2 of [62].
holomorphic structure from the corresponding bundles over the projective varietyĎ where ̟ 3 is holomorphic while ̟ 1 is just smooth. From the diagrams in Figure 1, we see that the correspondence D m implied by the diagram (2.2) ("supersymmetry") is nothing else than the Penrose correspondence [61]. In particular the Griffiths domain D (II) The holomorphic contact structure. As already anticipated, the domain D is the subgroup fixing a line ℓ in the 2m + 2 fundamental representation of G m (C) ≡ Sp(2m + 2, C). The presentation (2.3) induces on P C 2m+1 a canonical structure of holomorphic contact manifold; in homogeneous coordinates z a (a = 0, 1, . . . , 2m + 1) this contact structure is generated by the holomorphic one-form with coefficients in O(2) where ω = S ⊗ 1 m+1 is the standard symplectic matrix. Written in the homogeneous coordinates z a , the open domain D The one-dimensional representation of the isotropy group on the line ℓ defines a tautological holomorphic line bundle whose fibers are the lines t · z ∈ C 2m+2 . L is clearly Sp(2m + 2, R)-homogeneous; it is also equipped with the canonical, invariant, positive-definite Hermitian form −i Q(z,z).
We recall that a holomorphic Legendre submanifold L in the complex contact manifold (D (3) m , λ) of dimension 2m + 1 is a holomorphic immersion z : L → D (3) m such that z * λ = 0 and dim C L = m.
(III) Griffiths infinitesimal period relations (IPR). The weight lattice of C m+1 is Ze a with inner product (e a , e b ) = δ ab . (2.7) The simple roots (ordered as in Figure 1) are α a = e a − e a+1 for 1 ≤ a ≤ m and α m+1 = 2 e m+1 . Φ evaluated on the simple roots of graph (2) returns zero if the root is a white circle, it returns 2 if the node is black, and −4 if it is a white lozenge. Φ induces a Z-grading on the complexified Lie algebra g C m of G m (C). There is a unique Q ∈ g C m such that 23 Borrowing the tt * terminology, we shall refer to the Lie algebra element Q as the (superconformal) U(1) R charge. Then The adjoint representation induces on each direct summand of g C m the structure of a H (2.14) Griffiths' horizontal holomorphic bundle [25,30] is the homogenous bundle O(g −1,1 m ). Let S be a complex manifold. We say that a map p : S → Γ\D (2) m satisfies Griffiths' IPR iff where T S is the holomorphic tangent bundle to S. In particular p must be holomorphic.
(IV) tt * equations and branes. S a complex manifold, G a semi-simple real Lie group without compact factors K ⊂ G a maximal compact subgroup, and Γ ⊂ G a discrete subgroup. A map is said to be a solution to the tt * PDEs with monodromy group Γ ⊂ G if it is pluri-harmonic (with respect to the symmetric metric), i.e.
see appendix A. The tt * equations describe an isomonodromic problem [67] and their solutions are essentially determined by the monodromy group Γ. We specialize to the case relevant for special geometry, i.e. G ≡ G m (R) Let Ψ : S → G m (R) be any lift of w, and let ω ∈ Λ 1 (G m (R))⊗ g m be the Maurier-Cartan form of G m (R). The usual tt * (Berry) connection A and the tt * (anti)chiral ring valued 1-forms C,C [12] are It is well known that A, C, C solve the tt * equations iff w is pluri-harmonic [15,66]; for the benefit of the reader we review the argument in appendix A. Note that two lifts of w yield gauge-equivalent A, C,C describing the same tt * geometry. We say that w is a U(1) R -preserving solution to tt * if, in addition, Clearly this condition may be satisfied if and only if the chiral rings {R s } s∈S are localgraded (Q being their grading operator). If we have a U(1) R -preserving solution w, we may construct a S 1 -family of lifts by acting with e iθQ , θ ∈ [0, 2π). The S 1 -family is then extended to a full P 1 twistorial family of complexified solutions by analytic continuation 24 Extending from the equator |ζ| = 1 to the full twistor sphere, Ψ(ζ) becomes valued in the complex group G m (C) instead of the real group G m (R) as dictated by unitarity. Ψ(ζ) then satisfies a twistorial reality condition The maps Ψ(ζ) : S → Sp(2m + 2, C) are called (covering) tt * brane amplitudes. They satisfy the linear PDEs (the tt * Lax equations) [12,16] Caveat. The brane amplitudes here are written in the real gauge not in the more traditional holomorphic gauge [12]. In particular, on the twistor equator |ζ| = 1 they are real.
Let S = Γ\ S for a discrete group Γ of isometries acting freely (so π 1 (S) ∼ = Γ). There is a monodromy representation ̺ : Γ → G m (Z) such that the covering brane amplitudes Ψ(ζ) : S → G(C) satisfy the Γ-equivariance property which just reflects the fact that the brane amplitudes Ψ(ζ) : S → ̺(Γ)\G(R) are well defined on the physical space S.
Remark. We see that the specifying a U(1) R grading in the sense of tt * , i.e. Q, is identically to the math procedure of constructing a Hodge representation in Hodge theory (by the integral form Φ) [27][28][29]. In other words, the equator of the tt * twistor sphere gets identified with the Deligne circle of Hodge theory at the reference point in the period domain.
(V) Definition of special geometry. S a complex manifold of dimension m, and Γ ⊂ G m (Z) a subgroup to be called the monodromy (or U-duality) group. Modulo commensurability (i.e. up to finite covers) we may (and do) assume Γ torsion-free (and even neat [64,65]).

Consider the commutative diagram
where, as before, the double-headed arrows ̟a / / / / stand for the Penrose canonical projections (2.2). The arrow w describes tt * geometry, the arrow p is Griffiths' period map describing the variation of Hodge structure (VHS), and the arrow z encodes the supergravity geometry.
If one (hence all) of the conditions is satisfied, we say that the diagram (2.26) is a special (Kähler) geometry.
In this statement "essentially equivalent" means that if we are given any one of the three arrows p, w or z, which satisfies the relevant condition in the above list, one may complete, in an essentially unique way, the commutative diagram by arrows satisfying the stated condition. For instance, p is canonically the 1 st prolongation of z [68,69]. Then z * is everywhere of maximal rank, and the pull-back of the curvature of the Chern connection of the tautological bundle L (2.6), equipped with the Hermitian form (2.5), is positive and hence it defines a Kähler metric on S (the so-called special Kähler metric). If we are given p, we construct the other two arrows, w and z, by compositing it with the two canonical projections; so the most convenient description of a special geometry is through its period map p. Then Question 1 is more conveniently stated in the form Question 2. Which period maps p, satisfying the IPR, do arise from quantum-consistent N = 2 theories of gravity?
(VI) The local pre-potential F . Let S be the simply-connected cover of S. p and z may be lifted to their respective covering maps for the projection on the last (m + 1) homogeneous coordinates z a of D m . Following tradition [70], we shall denote the homogeneous coordinates of P m as X I (I = 0, 1, . . . , m). A generic Legendre submanifold z( S) intersects transversally the generic fiber of π; hence the differential of the map π z : S → P m is generically of maximal rank. The usual coordinates on P m then may serve as local holomorphic coordinates in (small enough) local charts U ⊂ S. We may write the local Legendre manifold z(U) ⊂ L in terms of the homogeneous coordinates on D for some holomorphic Hamilton-Jacobi function F (X I ), homogeneous of degree 2, which is called the pre-potential in the N = 2 supergravity context. A priori the map π z : S → P m may be multi-to-one, and then each local branch of this multi-cover has its own local prepotential. In other words: the prepotential F may be multivalued (even after replacing S by its universal cover S). Since the swampland conditions refer to the global geometry of the scalars' manifold S, we cannot ignore the issue of the several branches of F . This item will be discussed in paragraph (IX). As a preparation, we recall that the Hamilton-Jacobi (2.30) (VII) WP and Hodge metrics. The diagram (2.26) induces on S a countable family of distinct Kähler metrics; the first two are relevant for our present purposes. The first one, G ij , is called the Weil-Petersson (WP) metric (a.k.a. the normalized tt * metric [12]); its Kähler form is the pull-back via z of the curvature of the positive line bundle (2.6). This is the special Kähler metric which appears in the N = 2 sugra kinetic terms [22,23]. The second one, K ij , called in the math literature the Hodge metric [24,25,30,71], was introduced and studied from the tt * viewpoint in [67,72]. Its Kähler form ω H is the pull-back via p of the positive curvature of the Griffiths' canonical line bundle L can over the period domain D (2) m [24]. In terms of the matrix one-forms defined in (2.19) one has [67] For a special Kähler geometry of complex dimension m, the relation between the two Kähler metrics is [67,71] The two metrics G ij and K ij have the same isometry group. The metric K ij is better behaved than G ij in several senses. E.g. the curvatures of K ij have much better nonpositivity properties: for a general special geometry the Ricci curvature of K ij is negative and away from zero by a known constant [71], and its curvature tensor is non-positive in the Griffiths sense [24,25,30] (as well as in the Nakano sense [6]).
(VIII) Torelli properties. In the particular case where our special geometry describes an actual family of Calabi-Yau 3-folds, it is convenient to identify the simply-connected cover S of S with the completion, with respect to the Hodge metric K ij , of the Torelli space T ′ which is the moduli of polarized and marked CY's, see refs. [73][74][75]. S should not be confused with the Teichmüller space T i.e. the universal cover of the uncompleted space T ′ . S is biholomorphic to a domain of holomorphy in C m [74]. We stress that S is metrically complete with respect to the Hodge metric but not necessarily with respect to the Weil-Petersson one. The implications of this fact for physics will be discussed in §. 4.7 (2).
The analysis of [73][74][75] extends from this geometric situation to general N = 2 supergravities with quantized electro-magnetic fluxes. Then the "global Torelli" theorem of refs. [73][74][75] asserts that the lifted period map p in eqn.(2.27) is injective. 25 Again, this holds in any naively-consistent N = 2 sugra including the ones in the swampland. More precisely, we take injectivity of p as part of our definition of an integral special geometry (by convention, all our geometries are integral). Most of our arguments are independent of this property: they rest on the CY strong local Torelli theorem (see e.g. Theorem 16.9 in ref. [78]) which clearly holds for all formal special Kähler geometry. 26 Henceforth we shall identify S with its isomorphic image p( S). Then S = Γ S, where the monodromy group Γ acts on S ≡ p( S) in the obvious way. 25 We stress that the strong version of global Torelli is false for Calabi-Yau 3-folds, as shown by the Aspinwall-Morrison quintic counterexample [76,77]. 26 For these facts expressed in the "old" supergravity language see [70].
(IX) Branching of F . Let us consider the pull-back of the local holomorphic function F to S (again written F ). When defined as in (VIII), S is biholomorphic to a domain of holomorphy in C m ; if the holomorphic function F has no singularity in S, it has a uni-valued global analytic extension to all S. F may become singular only at a locus B ⊂ S where the Legendre sub-manifold L ceases to be transverse to the fibers of π. On the locus B the holomorphic function det(∂ I ∂ J F ) has a pole; hence B has complex codimension at least 1 in S. Indeed, B ⊂ S is just the branch locus of the holomorphic covering map where distinct branches of the cover coalesce together. Around such a locus F should become multivalued to represent the several local branches of the Legendre cover L → P m . Since B has codimension ≥ 1, the several branches of F in S \ B all arise from the (multi-valued) analytic continuation of a single holomorphic function.

Special Kähler symmetries
The special Kähler space of a general N = 2 supergravity has the form with S a smooth, Hodge-metric complete, simply-connected, special Kähler manifold. It is convenient to work with the covering N = 2 sugra, whose scalar fields take value in the covering special Kähler geometry S, and think of Γ as a discrete symmetry of the covering sugra which we must gauge in order to get the actual low-energy quantum gravity with moduli space S. In other words: two field configurations in the covering theory which differ by the action of Γ are regarded as the same physical configuration in QG. We write Iso( S) for the Lie group of holomorphic 27 isometries of the special Kähler metric on S, and Sym( S) ⊆ Iso( S) for the naive 28 symmetry group of the covering N = 2 sugra. For all naive symmetry ξ ∈ Sym( S) we have where the naive symmetry ξ acts on the electro-magnetic field strengths by the real duality rotation S ξ . The map ξ → S ξ yields a Lie group homomorphism whose kernel is trivial by "global Torelli", so that the naive symmetry group Sym( S) gets identified with a closed subgroup of G m (R), that is: all naive symmetries of S ⊂ D (2) m arise from automorphisms of the ambient space The Lie algebra sym( S) of Sym( S) is then seen as a sub-algebra of g R m ≡ sp(2m + 2, R). In facts, the injectivity of already follows from strong local Torelli ([78] Theorem 16.9).
The actual symmetry group of the covering special Kähler geometry S is the subgroup of the naive one which is consistent with the quantization of the electro-magnetic fluxes, i.e. (2.39) The discrete gauge group satisfies Γ ⊂ Sym( S) Z . The normalizer N (Γ) of Γ in Sym( S) Z is an "emergent" symmetry of the QG, and the quotient is the "emergent" global symmetry of the effective theory. Here "emergent" means that N (Γ)/Γ is a symmetry of the low-energy effective theory truncated at the 2-derivative level, which may (and should, according to the swampland conjectures) be explicitly broken by higher derivative couplings. 29 It is natural and convenient to consider the larger group of rational symmetries Rational symmetries are not symmetries in the strict sense of the world, since they do not preserve the Dirac symplectic lattice Λ of electro-magnetic charges. However, given a rational and hence ξ sets a correspondence between the subsector of the theory whose electro-magnetic charges are in the sublattice Λ ξ and the subsector with electro-magnetic charges in the sublattice ξ Λ ξ (having the same index in Λ as Λ ξ ). The correspondence Λ ξ ↔ ξ Λ ξ leaves invariant the classical physics, so it is a kind of sector-wise symmetry. More importantly, its existence has observable consequences. For instance (assuming ξ is a rational symmetry of the full theory and not just of its 2-derivative truncation !!) it implies that the entropy of an extremal black holes with charge v ∈ Λ ξ is equal to the entropy of the corresponding extremal black hole with charge ξv ∈ ξ Λ ξ .
We see G m (Q) ≡ Sp(2m + 2, Q) as a connected (linear) algebraic group defined over the field Q [79]. The subgroup then contains a maximal connected Q-algebraic subgroup This allows to introduce the notion of the Q-Lie algebra of "infinitesimal symmetries" of the special Kähler geometry S, namely the Q-Lie algebra q( S) of the algebraic group Q( S) [79] We shall see that in a quantum-consistent special geometry the last inclusion is an equality.

Warm-up: Two weak results in the DG paradigm
Before addressing the issues of main interest, we state two 'elementary' results which are weaker versions of Fact 2 and Corollary 1, respectively. The merit of these statements is that they may be proven remaining inside the naive DG paradigm by assuming the swampland conjectures in the original Ooguri-Vafa form [3]. In facts, it suffices to impose that the moduli space S is non-compact of finite volume (the other conditions then hold automatically in this special context).
The material in this section will be only marginally used in the rest of the paper, and can be omitted in a first reading. However, comparison of the swampland structural criterion with some well-known phenomena in Differential Geometry may help to understand the nature and effects of the QG 'arithmetic' paradigm (see Comments on the proof below).
The first result is weaker than Fact 2 because it makes additional geometric assumptions on the special Kähler manifold S, so that it applies only to a small subclass of special geometries. The second result holds in full generality, but it is less precise than Corollary 1 in a significant way.
Since both results will be subsumed by the stronger Fact 2, in this section we shall be rather sketchy with the proofs, and be cavalier with several fine points.

A restricted class of special Kähler geometries
As before, Iso( S) stands for the Lie group of holomorphic isometries of the special Kähler manifold S, and iso( S) for its Lie algebra generated by the holomorphic Killing vectors. All The rank of an abstract group Γ [85] Let Γ be an abstract group. For σ ∈ Γ, we write Z Γ (σ) for its centralizer in Γ. We write A i (Γ) for the set of the elements σ ∈ Γ such that Z Γ (σ) contains a free Abelian subgroup of rank ≤ i as a subgroup of finite index. Then and set rank Γ := sup r(Γ * ) : Γ * ⊆ Γ is a finite index subgroup . Properties: • Commensurable groups have the same rank; • if Γ = Γ 1 × · · · × Γs is a product, then rank Γ = j rank Γ j ; • free groups have rank 1; • if G Q is a Q-algebraic group of Q-rank r and Γ ⊂ G Q is an arithmetic subgroup [64], then rank Γ = r.
Killing vectors of S belong to iso( S).
Fact 3. Let S = Γ\ S be a special Kähler manifold which is non-compact with Vol(S) < +∞.
Here S is its smooth simply-connected cover. 30 Assume, in addition, that the Riemannian sectional curvatures S are non-positive. Then we have the dicothomy: • if S has a non-trivial Killing vector (i.e. iso( S) = 0) then S is Hermitian symmetric: Table 2) and Γ ⊂ G is an arithmetic subgroup; • otherwise Iso( S) is a discrete group of rank 1, and Γ ⊂ Iso( S) is a finite-index subgroup.
The definition of the rank of an abstract group Γ is recalled in the box on page 29.
Remark. Fact 3 is Fact 2 except that we restrict to the very small sub-class of nonpositively curved special Kähler manifolds. In this restricted class of manifolds the conclusion is stronger since we have the extra information that in the non-symmetric case the monodromy group Γ has rank 1.
Similar results hold under other special assumptions on S: see e.g. Theorem 1.7 in [81].
By definition [80], under the present special assumption, the simply-connected Riemannian manifold S is a Hadamard manifold without Euclidean factors. 31 Then Fact 3 is a 30 Recall that -replacing S by a finite cover, if necessary -we are assuming Γ to be neat [64,65], hence torsion-free. Then S is the universal cover and π(S) = Γ. 31 The non obvious part of this statement is that S is metrically complete. By construction, S is complete for the Hodge metric; from eqn.(2.32) we see that, when the sectional curvatures of G ij are non-positive, completeness with respect to the Hodge metric implies completeness with respect to the WP metric. direct consequence of the rigidity theorems in Differential Geometry for finite-volume quotients of Hadamard manifolds, see refs. [82][83][84][85]. When S is locally symmetric these theorems reduce to the Mostow rigidity theorem [64,86] while the properties of Γ follow from the Margulis super-rigidity theorem [64,87].
We recall some definitions and the simplest rigidity statement [83].
is said to be reducible iff the manifold M = H/Γ has a finite cover which is reducible as a Riemannian manifold.
Theorem (P. Eberlein [83]). Let H be a Hadamard manifold without Euclidean factors 32 and let Γ be an irreducible lattice in H. Then either: (1) Iso(H) is discrete, Γ has finite index in Iso(H) and H is irreducible; (2) H is isometric to a symmetric space of non-compact type.
Remark. In order to study the interplay between the isometry group and the global properties of a special Kähler geometry, it is convenient to replace its usual WP Kähler metric G ij with the Hodge one K ij , eqn.(2.32). Since the two metrics G ij and K ij have the same isometry group, the statement and proof of Fact 3 holds equally well for both metrics. However as mentioned in §. 2.1(VII) K ij has better chances of having non-positive curvatures. Unfortunately having a non-positive curvature in Griffiths sense is much weaker than the condition required in the above Theorem, i.e. non-positive Riemannian sectional curvatures, so that, while consideration of K ij may seem to enlarge the class of special geometries to which Fact 3 applies, it still consists of a very small portion of all special Kähler geometries. 33 Proof of Fact 3. Let S be the smooth simply-connected cover of our special Kähler manifold S having (by hypothesis) non-positive Riemannian sectional curvatures. We need to show the following claim: the existence of a discrete subgroup Γ ⊂ Iso( S) such that Vol(Γ\ S) < ∞ implies that either iso( S) = 0 or S is symmetric. The simply-connected reducible special Kähler manifolds were classified in [88] (see also [32]): they are all symmetric spaces, corresponding to the non-simple Lie groups in Table 2. Therefore we may assume without loss that S is an irreducible Hadamard manifold without Euclidean factors. The claim then follows from the Theorem quoted above.
Comments on the proof. The basic ingredient of the argument is a rigidity theorem in Differential Geometry. Already in ref. [3] it was observed that the swampland conjectures are easy to prove for negatively-curved moduli spaces. Unfortunately, in general the special Kähler manifolds are not quotients of Hadamard spaces, and no useful general rigidity theorem for them is available while remaining in the DG paradigm. Going to the "new" 32 Proposition 4.4 of [83] has no assumption on the Hadamard manifold H but requires Γ not to contain Clifford translations. Our assumption of no Euclidean factor implies the last condition, see Theorem 2.1 of [82]. 33 See Remark at the end of this subsection.
paradigm, we may replace in the above argument the DG rigidity theorem by the subtler rigidity theorems of Hodge theory [24,25,30] whose most convenient and powerful formulation is the VHS structure theorem. This fancier rigidity theorem has an arithmetic flavor and holds (conjecturally) for all special Kähler geometries 34 which do not belong to the swampland, independently of the sign of their curvature, while having essentially the same implications as the usual rigidity theorems of Differential Geometry and Lie group theory.
We close this subsection mentioning some other results for the non-positively curved case. We recall that the rank, rank(M) of a complete Riemannian manifold is defined as [85] rank(M) = min where PJ v is the vector space of parallel Jacobi fields along the geodesic with initial velocity vector v. If M is locally symmetric this reproduces the standard rank (i.e. the R-rank of its isometry group).
Theorem (Burns-Spatzier [84]). Let S be a complete connected Riemannian manifold of finite volume and non-positive sectional curvature without Euclidean factors. Then S has a finite cover which splits as a Riemannian product of rank 1 spaces and a locally symmetric space.
For a manifold S as in the Theorem we may define the rank of its fundamental group π 1 (S) in purely abstract group-theoretical fashion, see the box on page 29 for the detailed definition. Under the present hypothesis on S: Proposition 1 (Ballmann-Eberlein [85]). S a complete Riemannian manifold with nonpositive sectional curvatures and Vol(S) < ∞. Then rank π 1 (S) = rank(S). (3.2) In particular, rank(S) is a homotopy invariant.
Corollary 6. Let S be a special Kähler manifold with Vol(S) < ∞ and non-positive Riemannian curvatures. Then either S is locally symmetric or rank π 1 (S) = 1.
Remark. Inverting the logic, if a finite-volume special Kähler manifold S is not locally symmetric, while the monodromy group Γ has rank ≥ 2, we conclude that its special Kähler metric G ij , as well as its Hodge metric K ij , should have at some point a positive sectional curvature. This remark applies, say, to the dimension-101 moduli space of quintic hypersurfaces in P 4 where rank Γ = 102 [89]. This observation implies that non-symmetric special Kähler manifolds with everywhere non-positive Riemannian sectional curvatures -if they exist at all -are quite rare.

The quantum corrections cannot be trivial
One has the following Fact 4. S a special Kähler manifold with strictly cubic pre-potential There exists a subgroup Γ ⊂ Iso( S) such that S ≡ Γ\ S has finite volume if and only if S is locally symmetric (hence an arithmetic quotient of a Hermitian symmetric space G/K with G a group in the right column of Table 2).
In particular Fact 4 says that, in a Type IIA compactification on a Calabi-Yau 3-fold X, the Kähler moduli S should receive some quantum correction, unless S is locally symmetric, since otherwise the purely cubic classical pre-potential belongs to the swampland. However, Fact 4 says nothing about the nature of these quantum corrections: it does not even rule out that perturbative corrections suffice. In this sense it is much weaker than Corollary 1.
As anticipated, we shall be very cavalier with the argument. Consider first the case where the cover S is homogeneous (hence one of the special Kähler manifolds constructed in [50]). Then S = Iso( S)/I, with I ⊂ Iso( S) the compact isotropy group of a chosen base point. Then, for all Γ ⊂ Iso( S), for some invariant measure Vol on the Lie group Iso( S). Since I is compact, Vol(I) < ∞, and S may not belong to the swampland only if there exists a Iso( S)-invariant measure Vol and a discrete subgroup Γ ⊂ Iso( S) such that Vol(Γ\Iso( S)) < ∞. The case of S not locally homogeneous behaves in a similar way provided we can show that, after possibly the excision of a zero-measure subset, the orbits of Iso( S) are regular, so that we may define a nice generic-orbit space Y and where dµ is the appropriate "Fadeev-Popov" induced measure on the generic-orbit space Y . We shall dispense with the technicalities involved in the justification of eqn. (3.5). Granted it, we again have that the Lie group G ≡ Iso( S) must admit a lattice that is, a pair (Vol, Γ) where Vol is an invariant measure on G and Γ ⊂ G a discrete subgroup with Vol(Γ\G) < ∞.
We recall a well known fact: Lemma 1 (See e.g. [90]). A Lie group G admits a lattice if and only if it is unimodular.

Tube domains in C m
The covering cubic special Kähler manifold S described by (3.3) is a special case of a tube domain 35 . A tube domain T (V ) has the form with V ⊂ R m a strict, convex, open cone. Its group of holomorphic automorphisms, Aut(T (V )), contains the group where Aut(V ) ⊂ GL(m, R) is the group of linear automorphisms of the ambient R m preserving the cone V . When the cubic special Kähler manifold T (V ) arises as the classical (large-volume) limit of the Kähler moduli of Type IIA compactified on a 3-CY X, the Abelian subgroup describes the axionic shifts of the B-field by harmonic (1, 1)-forms on X -a classical continuous symmetry which remains unbroken to all orders in world-sheet perturbation theory (while instanton corrections typically break it down to a discrete subgroup) -while Aut(V ) corresponds to the geometric automorphism of the classical Kähler cone.
Note that for all convex cone V we have at least the symmetry R >0 ⊂ Aut(V ) corresponding in Type IIA to overall rescalings of the Kähler form. This classical rescaling symmetry may be broken already by the loop corrections. Then, for all cubic tube domain S ≡ T (V ), Fact 4 follows from Lemma 1 together with the following Then G is either semi-simple or non-unimodular. In the first case G ≡ Aut(T (V )) and T (V ) is a Hermitian symmetric space.
Proof. Consider the (real) Lie algebra aut(T (V )) of Aut(T (V )); its elements are holomorphic vector fields f (z) i ∂ z i whose coefficients f (z) i are polynomials in the coordinates z i of C m [92]. Let ∂ ≡ z i ∂ z i be the Euler vector field, which generates the overall scaling symmetry R >0 . 35 Also known as Siegel domain of the first kind [91].
The Lie algebra aut(T (V )) is graded by the adjoint action of ∂ [92] aut is a symmetric domain [92]. The Lie algebra g of G is also graded by ∂ A necessary condition for G to be unimodular, is that the trace of the adjoint action of ∂ on its Lie algebra g vanishes. Since In particular, all quotients of the homogeneous non-symmetric special Kähler geometries constructed in [50] -all of which have cubic pre-potentials -belong to the swampland.

Symmetric rank-3 tube domains
As already anticipated the symmetric rank-3 tube domains T (V ) are precisely the symmetric special Kähler manifold with a cubic pre-potential (3.3). Their isometry groups Iso(T (V )) are listed in the right-hand side of Table 2. For later reference let us describe the symmetric cubic form d : ⊙ 3 R m → R (a.k.a. the Yukawa coupling). For the cubic space SL(2, R)/U(1) the cubic form is given by d(x) = x 3 . For the spaces with Iso(T (V )) = SL(2, R) × SO(2, k) (k ≥ 1), it is given by   where A ij is an integral quadratic form of signature (1, k −1). In the remaining four cases R m is identified with the R-space Table 3: where z → z * is the usual R-linear complex conjugation in F. As a real symmetric cubic form d : ⊙ 3 R m → R is 6! times the determinant of the 3 × 3 matrix (3.16). The determinant is given by the usual formula, but we need to pay attention to the order in which we perform the products since F is neither commutative nor associative in general. The correct expression over R is [93] Crucial caveat. In the applications to Quantum Gravity we are actually concerned with an integral cubic form d Z : ⊙ 3 Z m → Z whose underlying real cubic form is R-equivalent to (3.17). Clearly, there are many inequivalent integral cubic forms d Z with the same underlying real form d. Their difference would look physically irrelevant according to the field-theoretic paradigm -since they all define the same naive Lagrangian L -but in the Quantum Gravity setting different integral forms define physically distinct theories because of flux quantization. Indeed, a cubic special geometry may or may not belong to the swampland depending on the particular integral cubic form d Z we choose in the real equivalence class of d. For instance, consider the quantum-consistent special geometry obtained from Type IIA compactified on the manifold X ≡ (E × K)/Σ where E is an elliptic curve, K a K3 surface and Σ a freely acting symmetry such that h 2,0 (X) = 0: there are precisely eight diffeomorphism classes of such complex manifolds, see section 8. The integral cubic form d Z is a topological invariant of X. The generic integral cubic form, equivalent over R to one of these eight quantum-consistent forms, belongs to the swampland. This quantum inconsistency will not be detected by the usual swampland criteria, but the structural one is strong enough to discriminate between the different integral forms d Z .

The rough physical idea
A basic physical principle in QFT states that the possible interactions are severely restricted by the gauge symmetries of the theory. One expects this principle to extend to QG, except that in QG the gauge symmetries enjoy two novel features: (i) we can have infinite discrete gauge groups; (ii) the gauge group G carries additional arithmetic structures, that is, G is a group-object in some subtler category G of QG "symmetries". Typically G is endowed with a canonical forgetful functor G → Lie which keeps only the underlying Lie-theoretic structures of the "usual" description of symmetry in field theory.
One would guess the existence of a fundamental QG principle of the rough form: Rough physical principle. An effective field theory belongs to the swampland unless: (a) its gauge group belongs to the appropriate subtler category, G ∈ G; (b) all couplings are consistent with the gauge symmetry G.
Here consistent means compatible with respect to all the structures carried by G not just the underlying usual consistency in Lie. As a consequence, (b) is a much stronger constraint than the corresponding fact in QFT, and G "almost" determines the theory. 37 We believe that a suitable technical version of this rough statement is the proper swampland condition [6]. In the special case of an effective theory which is an ungauged 4d N = 2 supergravity this physical principle takes the form of the VHS structure theorem.
Indeed the VHS structure theorem, when interpreted as a statement about the 4d N = 2 supergravities having an algebro-geometric origin, has precisely the form of the above rough physical principle. The theorem consists of two parts: (a) a list of properties that the U-duality group Γ should satisfy in all motivic VHS; (b) strong constraints on the period map p (i.e. on the special Kähler geometry) arising from Γ.
Item (b) follows from (a) together with the VHS rigidity theorems which may be regarded as the ultimate extension of Seiberg's principle of the "power of holomorphy" [94].

Intuitive view: tt * solitons and brane rigidity
Although we mostly use the VHS language, it is worthwhile to briefly discuss the basic idea from the (equivalent) viewpoint of tt * geometry which offers a different physical intuition of the same deep facts. This subsection is written having in mind readers which prefer simple physical arguments to abstract mathematics. All others are invited to jump directly to the next subsection. Given its purpose, this subsection is sketchy and rough; but we stress that it is just a rephrasing of the technical proof of the structure theorem in Hodge theory, so it can be made fully rigorous by adding a little bit of math pedantry.
An auxiliary σ-model. We consider a space-time of the form X × R, where R is time and X is a Riemannian manifold with metric γ and coordinates x i . To make the story a bit shorter, we assume from the start X to be Kähler: in our application X will be identified with the universal cover S of the sugra moduli space S. On this space-time we introduce a classical σ-model with target space some Riemannian manifold Y with metric g and coordinates y a . The (classical) static solitonic particles of this model are the (timeindependent) solutions to the equations whose value at the extremum is the mass of the soliton. Thus a soliton of the σ-model is nothing else than a harmonic map y : X → Y . Since X is Kähler, there is a special class of such solitons -called pluri-harmonic maps -in which the full matrix vanishes not just its trace as in eqn.(4.1). We stress that the special solitons have the property of being solutions of the equations of motion for all choices of the Kähler metric γ ij on X; in the physical applications one says that these special configurations are protected by susy. 38 In this setting, all harmonic maps y : X → Y (4.1) satisfy Simpson's Bochner-formula [10,95] DkDl γk i γl j g ab ∂ i y a ∂ j y where R abcd is the Riemann tensor on Y .
We now specialize to the case in which the target space Y is locally isometric to an irreducible symmetric space G/K of non-compact type. We write g = k⊕p for the corresponding Lie algebra decomposition, and identify T Y ∼ = p through the Maurier-Cartan form ω on G. Then dy ∈ T * X ⊗ y * T Y is identified with where C ≡ C i dx i (resp. C ≡ Cl dxl) are the two summands in the type decomposition C i , Cl are seen as matrices acting on some representation space V of G. Under this identification, eqn.(4.4) becomes Both terms in the rhs are point-wise positive semi-definite; the first one vanishes iff DC = 0, i.e. if and only if the harmonic map y is actually pluri-harmonic, while the second term vanishes iff [C i , C j ] = 0, that is, iff the matrices C i generate a commutative algebra. In appendix A we show, following [15,66], that the first condition implies the second one, and then the full set of tt * PDEs [12] tt * Q-solitons. The tt * arrow w in the diagram (2.26) is a special soliton in the above sense for G = Sp(2m + 2, R) and V = 2m + 2, see eqn. (2.17). From that diagram we see that a special Kähler geometry in the sense of N = 2 supergravity is nothing else than a special soliton of our auxiliary σ-model with the extra feature of preserving the U(1) R charge Q to ensure that the unital commutative algebra R generated by the C i is a local-graded chiral ring, cfr. eqn. (2.21). A Q-preserving special soliton is nothing else than a VHS in Griffiths' sense: indeed, in view of eqn.(2.11), eqn.(2.21) is equivalent to the Griffiths' IPR (2.15). To ensure that the VHS has the correct Hodge numbers we must require 39 that the U(1) R charge Q has the correct spectrum spectrum of Q acting on V = see eqns.(2.9),(2.10). We call such Q-preserving special solutions to (4.1) tt * Q-solitons. All N = 2 special geometries are tt * Q-solitons (and viceversa), so asking which special geometries belong to the swampland is equivalent to asking which formal tt * Q-solitons are unphysical.
Remark. When X is compact, the integral of the total derivative in the lhs of eqn. (4.4) vanishes, and all solitons are automatically special, i.e. solutions of the tt * PDEs (4.8).
Physical tt * Q-solitons. In physics we are not interested in all the solutions to eqn.(4.1) but only in solitons which satisfy two additional conditions: (1) they have a finite mass m < ∞, and (2) they are stable against small deformations. Only these solitons describe physical states in the spectrum of the theory.
Stability of solitons requires the existence of some non-trivial topological charge: the soliton of smallest mass within a given topological sector is stable. For instance, when Y = G/K, which is a contractible space diffeomorphic to R n , the only stable finite-mass solution is the vacuum, since there is no obstruction to continuosly decreasing the energy of the field configuration down to zero. The only way to get some non-trivial stable soliton, while preserving the crucial fact that the equations (4.3) are equivalent to the tt * PDEs of special geometry (4.8), is to quotient the target space G/K by a discrete subgroup of G. We are led to the following set-up: we have a homomorphism ̺ : π 1 (X) → G whose image Γ ⊂ G is a discrete subgroup, and we consider the twisted maps y : X → G/K, with domain the universal cover X, which satisfy the equivariance condition for all ξ ∈ deck group of X → X ∼ = π 1 (X), (4.10) so that y descends to a well-defined map y : X → Γ\G/K. If Γ is not torsion, the target space Γ\G/K is no longer contractible, so now the solitons may be stabilized by their non-trivial homotopy class. Intuitively, we have at most one stable soliton of finite mass per homotopy class of maps y : X → Γ\G/K, i.e. the one with the smallest possible mass. If this soliton happens to be special, eqn.(4.3), it represents a solution to the tt * equations, and hence may be lifted to a tt * brane amplitude Ψ(ζ), see §. 2.1(IV). Uniqueness of the soliton with given topological charge then implies rigidity of the finite-mass stable brane amplitude. All math rigidity theorems we use in this paper (including the VHS structure theorem) are manifestations of this basic physical idea. If, in addition, the stable special soliton preserves Q, it describes a special Kähler geometry enjoying the extra property of "stability" which is neither required by 4d N = 2 supersymmetry nor definable while remaining within the sugra language.
Stability of non-trivial tt * solitons requires the subgroup Γ ⊂ G not to be torsion, hence infinite. Seeing G ≡ Sp(2m + 2, R) as an R-algebraic group, we conclude that its Zariski closure 40 Γ R ⊆ G, identified with the real Lie group M ≡ Γ R (R), must have positive dimension.
The target space Γ\G/K may be continuously retracted to Γ\M/K M (K M ≡ [K ∩ M]), so that the stable solitons are naturally expected to take value in this smaller space. The special ones yield solutions of the tt * PDEs provided M is a real semi-simple Lie group without compact factors.
We are led to the following situation: we have a non-compact semi-simple real Lie group M and a discrete monodromy subgroup Γ ⊂ M whose R-Zariski closure is the full M, and we consider the twisted maps as in eqn. whose lift y any : X → M/K M satisfies (4.10) and has finite mass (4.2), then there is a unique finite-mass soliton in the same homotopy class. One gets this soliton by starting with the given generic finite-mass map y any and deforming it continuously to decrease its mass until we reach its minimal possible value. Its mass saturates the lower bound on the masses allowed in the given topological class, just like it was a BPS state. The Zariski-density condition of Γ in M guarantees that in this process our finite-mass configuration does not escape to infinity in field space nor develops singularities. When X is Kähler, the soliton will be automatically special (i.e. pluri-harmonic), and the algebra R generated by the C i commutative, provided we may justify that when we integrate the equality (4.4) over X the boundary term vanishes. This is typically true for finite-mass solitons in spaces X of finite volume (e.g. it holds when X is the union of a compact set and finitely many cusps). This condition is automatically satisfied by the special soliton w in the diagram (2.26) as a consequence of N = 2 supersymmetry. Indeed, all Q-preserving solitons are automatically special since All finite-mass stable solutions to the tt * equations arise in this way for some dense 40 Here the Zariski closure is taken over the ground field R. In the actual story of the swampland structural criterion one should take the Zariski closure over Q. Again this is a manifestation of the 'arithmetic' nature of QG. discrete group Γ and they are uniquely determined 41 by Γ. This is the statement of rigidity: it is analogue to the rigidity of BPS configurations in susy theories. This is hardly a surprise: in the usual applications of tt * geometry, these solutions do describe finite-tension stable BPS branes.
This leads to the following informal version of the conjecture we are proposing: Swampland structural criterion (Informal). A quantum-consistent N = 2 special geometry is described by a physical tt * Q-soliton that is, a stable one of finite-mass. In particular the monodromy group Γ is Zariski dense in a Lie subgroup M ⊂ Sp(2m + 2, R) which contains the image of a lift of w. All other special Kähler geometries belong to the swampland.
The statement sounds very physical in its wording, but it refers to an auxiliary field theory which lives on the moduli space S = Γ\ S. The physical meaning of the auxiliary field theory is unclear to me.

The structure theorem of algebro-geometric VHS
A basic fact about a VHS which has an algebro-geometric origin is that its complex base S (the "moduli" space) is quasi-projective, that is, there is a projective variety S and a divisor Y so that S = S \ Y . The Hironaka desingularization theorem implies that the pair (S, Y ) may be chosen so that Y is a simple normal-crossing (snc) divisor. Since S is assumed to be complete for the Hodge metric, Y = i Y i is minimal in the following sense: the period map p in eqn.(1.6) cannot be extended along any prime component Y i of Y ; this is equivalent to saying that the monodromy γ i around each component Y i is non-semi-simple [30]. The monodromy group Γ of an algebro-geometric VHS enjoys special properties [36]: Γ is finitely generated (in facts, in our particular context, even finitely presented), and contains a neat 42 (hence torsion-free) finite-index subgroup. We work modulo finite groups, so, by replacing the base S of the VHS with a finite cover, we may (and do) assume Γ neat. The monodromy γ i around each prime divisor at infinity, Y i , is then non-trivial unipotent.
The most important property of Γ is that it is semi-simple. To formulate this condition precisely, we introduce the notion of the Q-Zariski closure of Γ ⊂ G m (Z) in the Q-algebraic group G m (Q), that we denote as Γ Q . By definition, Γ Q is the smallest Q-algebraic subgroup of G m (Q) which contains Γ. 41 From a different point of view: the tt * PDEs are an isomonodromic problem, and the solutions are determined by the monodromy Γ they preserve. 42 A group Γ ⊂ GL(n) is called neat if all elements γ ∈ Γ (seen as matrices) have no eigenvalue = 1 of norm 1. In particular, in a neat group there are no non-trivial finite order elements, and all quasi-unipotent elements are unipotent. In particular, after passing to the finite cover as in the main text, the monodromies γ i around the prime divisors Y i are all non-trivial unipotent elements of Sp(2m + 2, Z).
By general theory [79], a semi-simple Q-algebraic group is an almost-product of simple Qalgebraic groups 43 where the M i 's are simple Q-algebraic groups. Hence (replacing S with a finite-cover if necessary) we may assume where M i (resp. A) is a semi-simple (resp. a torus) Q-algebraic group. By definition of Zariski closure, the two groups Γ and Γ Q have the same rational tensor invariants so, roughly speaking, they are "algebraically indistinguishable". All factors M i and A of M (in particular, the first ℓ factors entering in Γ Q ) have the property that the Lie groups of their real points, M i (R) and A(R), contain a compact maximal torus (so the Abelian group A(R) is itself compact). Therefore, not all simple Qalgebraic groups may appear as factors of Γ Q , but only the ones whose underlying real Lie group M i (R) has a Vogan diagram with trivial automorphism, that is, the ones in Figure  6.1 and Figure 6.2 of [60]. In our application to N = 2 supergravity the possible groups M i are further restricted by the condition that the weight of the VHS is odd (in fact 3), see [27,29] for the list of Lie groups satisfying this more stringent requirement. For a general (that is, not necessarily of CY-type) pure polarized VHS, the period map p has target space the quotient of the Griffiths period domain D by the monodromy group where G(R) is the group of the real points of the Q-algebraic group G preserving the polarization; in the 3-CY case with Hodge numbers h 3,0 = 1, h 2,1 = m this is 45 Here ≈ means equality up to finite groups (i.e. modulo isogeny). Since we are working modulo finite groups, we shall not distinguish between an almost-simple group and its simple quotient. 44 For the theory of the Mumford-Tate groups, see e.g.: nice short surveys: [27,28]. detailed treatments: [29,30,143]. 45 In the equation we identify a Q-algebraic group with the group of its Q-valued points, for clarity.

In eqn.(4.18) H is the isotropy group of a reference Hodge decomposition with the given
Hodge numbers h p,q ; in our case, h 3,0 = 1, h 2,1 = m, H is 20) and G(R)/H ≡ D (2) m . We have the inclusion of Q-algebraic groups For "generic" VHS one has M = G. For instance, equality holds -modulo finite groups -for all universal families of complete intersections [89]. To each factor in (4.21) we may associate a sub-domain of the period domain D We have the obvious embedding D P × D R ֒→ D. with q constant in the D R factor.
We focus on the non-trivial factor and consider the map where m C i = Lie(M i ) ⊗ Q C, and g −1,1 is the component of type (−1, 1) in the associated adjoint Hodge structure on the Lie algebra g of G at a reference point ⋆ ∈ S (see [27][28][29][30] for details). Then the infinitesimal period relations for ̟ take the form [27,28] where T S is the holomorphic tangent bundle to S (cfr. eqn.(2.15)).

The case of "motivic" special Kähler geometries
In the case that our VHS has the Hodge numbers of a special Kähler geometry (i.e. h 3,0 = 1, h 2,1 = m), we can say a bit more on the number ℓ of factors in P , eqn.(4.15): Lemma 3. (1) A special Kähler geometry S of algebro-geometric origin has ℓ = 0, 1, 2, 3.
• ℓ = 0 if and only if dim C S = 0 (in this case the corresponding algebro-geometric object is rigid, e.g. a rigid 3-CY). • in all other cases ℓ = 1.
(2) ̟ is onto iff Γ\D P is locally symmetric, in which case ̟ is an isomorphism of complex manifolds, S ∼ = Γ\D P .
(2) If ̟ is onto, the Griffiths infinitesimal period relations (4.30) become tautological, and this is equivalent to ̟(S) being locally symmetric [27]. 46 The Lie subgroup H i ⊂ M i (R) acts on the complexified Lie algebra m C i of M i (R) through the adjoint representation. As explained in the text, this action preserves the Hodge decomposition of m C i .

We write
By construction M i (Z) is an arithmetic subgroup [64,65] of the rational algebraic group M i and one has by Dirac quantization of charge. We have two possibilities: • the monodromy (sub)group Γ i is of finite index in M i (Z) and hence it an arithmetic group on its own right; • Γ i is of infinite index in the arithmetic subgroup M i (Z) of its Q-Zariski closure M i . In this case the monodromy group is said to be thin [97].
Simple situations have arithmetic monodromies: in particular, when S is locally symmetric the monodromy is always arithmetic [32]. All families of complete intersection CY have arithmetic monodromy [89]. However, there are geometric situations with thin monodromy: the simplest such example is the mirror of the quintic [98].

The structural swampland criterion
In the (ungauged) N = 2 case we propose to replace in the N = 2 case the several swampland conjectures by the following single one: Criterion 1 (Structural criterion). A special Kähler geometry S belongs to the swampland unless its period map p satisfies the VHS structure theorem (4.25).
In particular the discrete gauge group Γ should satisfy eqn.(4.15).

A direct physical proof?
As sketched in §. 1.3 the structural swampland criterion may be actually proven (as a necessary condition) for low-energy effective theories with rigid N = 2 supersymmetry. Let us focus on the crucial ingredients of the proof in QFT: the first one is the existence of a well-defined quantum (complex) Hilbert space H, carrying a linear representation of the N = 2 susy algebra and of an algebra A of quantum operators acting on H. This allows to define the chiral ring R 4d as the subfactor consisting of scalar 47 operators commuting withQ Aα modulo the ones which may be written asQ-anticommutators. From the susy algebra and locality we 47 The restriction to scalar operators is for convenience.
learn that R 4d is a commutative C-algebra with unit. The second crucial ingredient is that in a UV complete N = 2 QFT the chiral ring R 4d is also finitely-generated. Indeed in this case we have a well-defined UV fixed-point, with finite conformal central charges a uv and c uv , whose chiral ring R uv 4d is graded by the conformal dimension d. When R uv 4d is a free polynomial ring we have the unitarity bound [99] #(generators of R uv 4d ) ≤ 4(2a uv − c uv ) ≡ finite, (4.35) and one reduces to this case in all known N = 2 SCFT (see discussion in §. 5 of [100]). By RG flow, R 4d is a deformation of R uv 4d and inherits finite-generation from it. Then, by the Hilbert basis theorem, the spectrum of a quantum-consistent R 4d is an affine variety defined over C, hence has the form M \ Y ∞ for some projective variety M and effective divisor Y ∞ . It is convenient to work with the smooth locus in the Coulomb branch, where Y D is the effective divisor with support on the locus where some extra degree of freedom becomes massless. Then the low-energy description of a UV complete 4d N = 2 theory is described by a weight-1 VHS over a smooth quasi-projective basis M ≡ M \ Y , and we may assume without loss that Y is a simple normal crossing divisor. By going to a finite cover, if necessary, we may also assume the monodromy group Γ to be neat. Then the structure theorem holds (for a sketch of the argument, see e.g. §. IV of [27]).
From the above we see that we have a proof of the structural swampland criterion also in the N = 2 quantum gravity setting subjected to two assumptions: 1) the asymptotically-Minkowskian quantum states form a Hilbert space carrying a unitary representation of N = 2 supersymmetry. In this case the chiral ring R sugra is well defined; 2) R sugra is finitely generated.
From the semi-classic viewpoint these two assumptions look reasonable.

Recovering the Ooguri-Vafa swampland statements
We now show that the Ooguri-Vafa (OV) swampland conjectures [3] are implied by the above Criterion i.e. by the VHS structure theorem (specialized to the appropriate Hodge numbers). Most arguments may already be found in [3].
(1) S is non-compact. The first OV conjecture states that either S is zero-dimensional or it is non-compact. Since a VHS with a compact base has a constant period map, if S is compact ̟(S) reduces to a point, and since S ∼ = ̟(S) by "global Torelli", S is also a point. The non-compactness of S is related to the fact that Γ should contain non-trivial unipotent elements (i.e. the monodromies around the prime divisors in Y ): compare to the Godement non-compactness criterion for finite-volume quotients of symmetric spaces [64].
(2) The scalars' manifold is geodesically complete. This is the most tricky statement. In §.2.1 we found convenient to define the covering special geometry S to be the completion of a certain "moduli" space with respect to the Hodge metric K ij . Then S is geodesically complete for K ij by construction. However the OV conjecture refers to geodesic completeness with respect to the WP metric G ij which is the one entering the scalars' kinetic terms. Each one of these two metrics is associated to its own viewpoint about special geometry: with reference to diagram (2.26), the WP metric is tied to the Legendre map z, and the Hodge metric to the period map p. Their Kähler forms are restrictions to the respective images L and S of the curvature of the canonical homogeneous line bundles L (a) → D (a) m over the respective target domains.
In general S is not complete for the Weil-Petersson metric. Indeed it is known that the moduli spaces of Calabi-Yau 3-folds is typically non-complete for the Weil-Petersson metric, see [101,102]. In particular, a conifold point is at finite WP distance [102] while being at infinite Hodge distance. The period map cannot be extended to such a point [24,30], whereas the Legendre map may be extended continuously to the conifold point as we shall see momentarily. In other words, we may complete L by adding to it the points at finite distance in the WP metric even if we cannot extend there S (which is locally identified with its image under p). This peculiar situation is related to the fact that the fiber of the projection is non-compact, so, a geodesic γ(t) ⊂ S ⊂ D (2) m which approaches a conifold point may stretch to infinite length in the fiber direction while its projection ̟ 3 ( γ(t)) ⊂ L ⊂ D (3) m remains at finite distance in the base. We may be a little more precise: the Penrose map is an embedding, while the fiber of ̟ 3 is a copy of Siegel's upper half-space Sp(2m, R)/U(m) whose physical interpretation is easily understood from the Penrose correspondence. A point in the fiber, seen as a symmetric m × m symmetric matrix τ ij with positive-definite imaginary part, is nothing else that the matrix of complexified couplings of the matter vector fields, whereas the point in D m specifies which C-linear combination of the electric and magnetic charges is the susy central charge Z [22]. Thus a point at finite distance in the WP metric but infinitely away in the Hodge one is a field configuration where the susy central charge Z and the graviphoton couplings remain finite, but some coupling of the matter gauge vectors blows up (or vanishes, depending on the chosen duality frame).
Since the problem arises purely in the matter sector, 48 the situation is akin the ones appearing in rigid N = 2 effective Lagrangian, i.e. in Seiberg-Witten theory [8,9]. For comparison sake, let us recall what happens in the rigid case. The singularities of the scalars' metric arises at points in the Coulomb branch where some charged hypermultiplet becomes massless, making incomplete the low-energy effective description in terms of IRfree photon super-multiplets only. Let a be the period associated to the charge of the light hypermultiplet so that its mass is proportional to |a|; near the singular point the dual period a D has the form a D = q 2πi a log(a/Λ) + regular (4.40) for some integer q = 0. Clearly the periods (a, a D ) may be extended continuously in the limit a → 0, while the Kähler metric ds 2 ∝ log|a| da dā develops a singularity at finite distance. The Ricci tensor has the form of the standard Poincaré metric in the upper-plane coordinate z (where a = exp(2πiz)) so that the point a = 0 where the hypermultiplet becomes massless would be at infinite distance in any Kähler metric of the form (C a suitable constant) for instance in the Hodge metric (2.32). We conclude that such finite-distance singular points correspond to loci in scalars' space where finitely many states become light, spoiling our low-energy effective description in terms of light vector multiplets only. The original OV conjecture thus holds with the specification that "scalars' manifold" should be understood to mean the WP completion of the Legendre manifold L. The nontrivial part of the statement is that a continuous extension of z exists. The extended metric is not smooth since its Ricci curvature should blow up at the conifold points, see (4.41).
Let us discuss in more detail the behavior of the special geometry at infinity. Since S = S \ Y with S compact and Y simple normal crossing, the asymptotic behaviour of the special geometry as we approach a point in Y is described -up to corrections which are exponentially small in the geodesic distance -by the multi-variable version of the nilpotent orbit theorem [103,105]. More concretely, let U ⊂ S be a small open set (in the analytic topology) where Y takes the form z 1 z 2 · · · z s = 0, so that where ∆ is the open unit disk, ∆ * = {z ∈ C, 0 < |z| < 1} the punctured unit disk, H the upper half-plane, and σ the universal covering map σ : (τ 1 , · · · , τ s , z s+1 , · · · , z m ) → (q 1 , · · · , q s , z s+1 , · · · , z m ), with q i = q(τ i ) = e 2πiτ i . (4.44) The local lift of the period map, takes the form [103] p (4.46) where the N i 's are a s-tuple of non-zero, commuting, nilpotent elements of sp(2m + 2, Q) such that the monodromy around the divisor {z i = 0} is γ i = exp(N i ) and F : ∆ m →Ď is a regular holomorphic map. 49 From this expression one gets the asymptotic behaviour of the metric at infinity in terms of the action of the N i 's. For instance, let us approach the j-th prime divisor Y j , i.e. we take Im z j → ∞ at fixed values of the other coordinates q i =j , z a . Set Then [101] Gz j z j = κ j 4(Im z j ) 2 + exponentially small (4.48) so completeness follows from comparison with the Poincaré metric, unless κ j = 0, in which case we see from eqn.(4.46) that the map extends to q j = 0 [101] sot that we may extend continuously the covering Legendre manifold L there.
(3) Asymptotically at infinity the curvature of S is non-positive. This is Conjecture 3 of [3]. It follows from the asymptotic formula (4.48). Note that near a conifold point the curvature is indeed positive (cfr. eqn.(4.41)) but this very fact implies that the singular 49 HereĎ is the compact dual of the Griffiths period domain D [24,25,27,28,30].Ď is a compact projective complex manifold and a G(C) = Sp(2m + 2, C)-homogeneous space such that D ⊂Ď as an open domain. point is at finite distance in the WP metric.
(4) S has finite volume. The above asymptotic expressions, together with compactness of S, show that the volume of S is finite [24]. Ref. [24] shows that the volume of ̟(S), computed with the Hodge metric K ij , is finite when the VHS has a compactifiable base S = S \ Y [24]. The OV conjecture refers to the volume computed with the WP metric G ij . The argument of [24] applies to this metric as well, since the two metrics are simply related in the asymptotic regime as a consequence of eqn. (4.48). For more details and precise results on the WP volumes of CY moduli space -which apply in general to all special geometries consistent with the structure theorem -see [106,107]. In the scalars' manifold there is no non-trivial 1-cycle with minimum length within a given homotopy class.
The precise wording of this conjecture requires some refinement. 50 There are h 2,1 = 1 Calabi-Yau 3-folds whose complex moduli space has the form . (4.52) {0, 1, ∞} are MUM points, whereas the other three punctures are conifolds points, see [21]. Alternatively we may take as "scalars' manifold" the metrically completed Legendre manifold in which the three conifold punctures are filled in. Let σ x ∈ π 1 (L comp ) be a loop encircling the puncture x ∈ {0, 1, ∞}. The length of a path in the class σ 0 σ −1 1 is below by a positive constant, so the wording of the conjecture needs some minor modification.
Here we adopt a conservative attitude: we take as "scalars' space" the nicer S and slightly modify the statement of the conjecture by replacing "homotopy class" with "homology class". Since, as abstract groups, π 1 (S) ∼ = Γ (because Γ is assumed torsion-free) the conservative version of the conjecture is equivalent to saying that, as a concrete matrix group, Γ is generated by unipotent elements. This is automatically true [109] whenever Γ is an arithmetic group of rank at least 2 (as it happens in most "elementary" examples such as all complete intersections), or when S may be chosen simply-connected. I am not aware of an example where neither conditions apply.
Let us be general. Let Γ u ⊳ Γ be the subgroup generated by all unipotent elements of Γ: when dim C S > 0, Γ u is an infinite normal subgroup. Taking the Q-Zariski closure, we have Γ Q u ⊆ Γ Q ; we may assume Γ Q to be simple (otherwise Γ is automatically arithmetic by In other terms, Corollary 7. S is a special geometry with period map p as in (4.25). Then π 1 (S) -seen as a concrete matrix group Γ acting on Q 2m+2 -has the same tensor invariants as a discrete group Γ u generated by nilpotent elements. In other words: the Ooguri-Vafa statement above holds (after refinement) for S at least in the algebraic sense.
We feel that this algebraic version is the most natural formulation of Conjecture 4 of [3].
(6) The distance conjecture. When the special geometry satisfies our structural swampland criterion, its asymptotic behaviour at infinity is described by the nilpotent-orbit and sl 2 -orbit theorems [103,105]. It is known that these results imply the distance conjecture. For a rough sketch of the argument see [32], for nice and detailed analyses see [102,[110][111][112][113][114]. We stress that, in the present set-up, the distance conjecture is almost tautological. Our viewpoint is that -whenever our special geometry does not belong to the swampland -we have a natural compactification S of the scalars' manifold with a sub-locus Y ⊂ S where the special geometry gets singular. All points in the prime divisors Y i ⊂ Y are at infinite distance in the nicer Hodge metric K ij but not necessarily in the WP metric G ij (cfr. §.(2)). As discussed around eqn. (4.41), the points on the divisor Y j are at finite WP distance if the singularity arises from finitely many particles becoming massless along the locus Y j . In order to have a more severe singularity of the metric, strong enough to make the WP distance infinite, the theory should have at Y j infrared divergences worse than those produced by any finite number of particles becoming massless. Clearly, the only possibility is that an infinite tower of states get light.  (4.54) where N Sym Z (Γ) is the normalizer of Γ in Sym( S) Z . Modulo commensurability, the group in the rhs is automatically trivial. 51 Note that from the structure theorem so there is no algebraic invariant which may detect a difference between the group of all symmetries of S, Sym( S) Z , and the actual gauge symmetry group Γ.
(8) Completeness of charge spectrum. The group Γ acts irreducibly on the Q-space V = Λ⊗ Z Q, where Λ is the symplectic lattice of electric-magnetic charges. Thus, if there is a state of charge v = 0, we have states of charges {Γv} and these span V . Then the physically realized electro-magnetic charges make a sublattice of finite-index in Λ. Since we are working modulo finite groups, this statement is equivalent to saying that all possible charges are physically realized provided one charged state exists. That charged states exist follows, say, from the validity of the distance conjecture. Alternatively the compleness of electro-magnetic spectrum is a formal consequence of the absence of global symmetries [115][5] 52 and hence follows from our previous discussion.
(9) No free parameter. This conjecture states that the couplings in the effective Lagrangian L are either the v.e.v. of light fields (so they are non-trivial functions on the moduli S), or they are frozen to some very specific "magic" numerical value. In the UV completed QG these "magic" numerical values become the v.e.v.'s of heavy fields [116], and so are given by the isolated critical points of some high energy potential, which is also subjected to quite strong consistency requirements. Then any small perturbation away from the "magical" numerical values makes the effective theory inconsistent at the quantum level. Fact 4 gives a first illustration of this state of affairs. Saying that the pre-potential F is purely cubic, is equivalent to saying that the N = 2 Pauli couplings (or the N = 1 heterotic Yukawa couplings) are field-independent numerical parameters. Then Fact 4 asserts that in a consistent theory such numerical Pauli/Yukawa couplings should be the cubic determinant form of a (possibly reducible) rank-3 real Jordan algebra whose classification is provided by the Freudenthal-Rozenfeld-Tits magic square [117] (so the adjective "magical" is technically accurate for these couplings). We shall elaborate a bit more on these aspects in the next subsection. 51 Indeed one has Vol(Γ\ S) = N Sym Z (Γ) : Γ Vol N Sym Z (Γ)\ S and, since Vol(Γ\ S) < ∞, one has [N Sym Z (Γ) : Γ] < ∞. 52 Since in this note we work modulo commensurability (and so use rational VHS rather than integral VHS) we may only conclude that the occupied charge lattice Λ occ is a finite-index sublattice of the electromagnetic lattice Λ and not Λ occ ≡ Λ. Equivalently we only show G glob = 1 modulo finite groups.
(10) the weak gravity conjecture. This conjecture states that there must exists states for which the electromagnetic repulsion is larger than the gravitational attraction. This means that the squared-mass should be less than the square of the electromagnetic charge both measured in intrinsic normalizations. The invariant square of the electromagnetic charge q is its Hodge squared-norm Q (Cq, q). The square of the susy central charge (which is the mass for a BPS state) is the Hodge norm of the (3, 0) projection of q, which is smaller by the Schwarz inequality.

More on "no free parameter"
Let us revisit the example in (9) of numeric (i.e. field-independent) Pauli/Yukawa couplings exploiting the fact that the polarized VHS's over a connected complex manifold S form a semi-simple Tannakian category over Q (see Proposition 2.16 in [20]). Restricting to the fiber over a point s ∈ S, this entails that a polarized Hodge structure over the fixed Q-space V , specified by a Hodge decomposition (say, of pure weight n) induces functorially Hodge structures on all its tensor spaces 53 T k,l ≡ V ⊗k ⊗ (V ∨ ) ⊗l given by the Hodge decomposition [27][28][29] T k,l ⊗ C = p+q=(k−l)n (T k,l s ) p,q .
More generally, we may consider other couplings defined as suitable invariant combinations of higher derivatives of F ; such couplings will correspond to elements of some higher tensor 53 Since V is polarized T k,l ∼ = T k+l,0 . 54 The Pauli/Yukawa is the cubic form which is the only invariant of the corresponding infinitesimal variation of Hodge structure, see [25,68]. space T k,l which have pure type In view of the VHS rigidity theorems, saying that the Pauli/Yukawa coupling, or any higher tensor coupling λ, is a numerical parameter independent of the scalar v.e.v. s is equivalent to saying that it is fixed by the monodromy Γ. Since λ has pure type (4.59), λ is 55 an element of the complexified space of Hodge tensors [27][28][29][30] λ ∈ Hg •,• ⊗ Q C (4.60) at the generic point of S. One shows [27][28][29] that the Mumford-Tate group M , eqn. (4.17), is precisely the subgroup of G m ≡ Sp(2m + 2, Q) which fixes the elements of Hg •,• . In particular, when the dimension of the Hodge tensors of the appropriate type and symmetry is at most 1 (as it happens for the "magical" Pauli/Yukawa couplings) such couplings -if field-independent -should be integers in some suitable normalization. They cannot be arbitrary integers, however, since the presence of a non-trivial Hodge tensor λ implies strong restrictions on the group P appearing in the structure theorem: P should leave all Hodge tensor invariants, so the more numerical field-independent couplings there are, the smaller P is. On the other hand, when S has positive dimension (i.e. we are not in the rigid case), the group P cannot be too small since ̟ is an embedding and hence with equality iff S is locally symmetric and Iso( S) = P (R). In particular, a generic fieldindependent coupling λ would imply P trivial, and then dim C S = 0, leading to a contradiction in presence of light vector-multiplets. The fact that field-independent couplings require the existence of non-trivial Hodge tensors leads to a classification of their possible "magical" values in terms of the finite list of Q-algebraic subgroups of G m ≡ Sp(2m + 2, Q) which have Hodge representations [27,29] of the appropriate kind. This yields back our classification of the allowed numerical Yukawas in terms of determinant forms for rank-3 Jordan algebras.
The situation looks very much in line with the arguments of [121].

Functional equations for quantum-consistent F's
A priori, giving a pre-potential F only specifies (locally) a covering period map p : S → D (2) m which satisfies the Griffiths infinitesimal period relations [24,25,30]. With some abuse of language, we shall say that the pre-potential F belongs to the swampland if there is no 55 Here we use the fact that the linear space of tensors T •,• invariant under Γ is defined over Q. such that the quotient special Kähler geometry S ≡ Γ\ S satisfies our swampland Criterion 1. Said differently, in order F not to belong to the swampland (i.e. to be quantum-consistent) there must be a monodromy group Γ such that the induced quotient period map satisfies the VHS structure theorem (4.25). In principle one may translate the swampland criterion (4.25) into a set of functional equations for the analytic function F (X I ). While these equations do not look promising as an effective tool for explicit computations, they are conceptually relevant.
Let us summarize our previous discussions of the properties of a quantum-consistent covering geometry S in the following commutative diagram The inclusion ι is induced by an irreducible, faithful, real, symplectic, Hodge representation [27][28][29] ̺ of the real Lie group P (R) which makes the following diagram to commute: By definition of the Q-algebraic group P, ̺ is defined over Q. The image of the composite map ̟ 3 ι is a P (R)-homogeneous complex manifold which may be easily constructed with the help of the weight-3 Hodge representation ̺. When (as it looks to be the general case) ̺ is a fundamental representation associated to a node in the Vogan diagram of P (R), J is the real Lie group whose diagram is obtained by deleting from the Vogan graph of P (R) the ̺ node.
Example (Figure 2). For instance, when P (R) = E 7(−25) and ̺ is the 56, so that m = 27, The white/black color specifies the particular real form of the Lie group E 7 , the lozenge stands for the highest weight of the representation ̺. Since ̺ corresponds to the black node, which is an extension node, H 25) , and hence the special geometry, is locally symmetric.
we get the holomorphic domain which in this particular example coincides with the MT domain P (R)/H P . Since the isotropy group in (5.6) is compact, D E 7(−25) is a Hermitian symmetric space. By general theory we conclude that when P (R) = E 7(−25) the special Kähler manifold S is locally isometric to the Cartan symmetric domain (5.6).
From the analysis in §. 2.2 we know that the universal cover S is identified with a complex submanifold of the Griffiths domain D (2) m , and that its naive symmetry group Sym( S) is the subgroup of the automorphism group Sp(2m + 2, R) of the ambient space D (2) m which fixes (set-wise) S. From diagram (5.3) we learn that S is completely contained in a single orbit of the subgroup P (R) ⊆ Sp(2m + 2, R), so that and all naive symmetries arise from automorphisms of the smaller homogeneous ambient space D From the diagram we also see that Sym( S) is, tautologically, a group of automorphisms of the contact manifold D P ) which fixes the Legendre submanifold L = z( S) whose generating function is the homogeneous pre-potential F In particular, L ⊂ D m should be invariant under Γ ⊂ Sp(2m + 2, R) acting as a group of 56 Note that D P is not a contact manifold in general, as the Example in figure 2 shows.
automorphisms of the ambient contact domain. Then, for each (2m + 2) × (2m + 2) matrix we get a system of (m + 1) functional equations for the holomorphic functions F I (X K ): The meaning of these equations is a bit subtle because the function F may be multi-valued. In facts this is the generic case. The functional equations (5.11) refer to the global analytic continuation of F to all of L.
In most applications, the putative pre-potential F (or rather the function F (z i ) in eqn.(5.9)) has a local analytic expression as a series which converges only in some small domain U ⊂ C m . Then the functional equations associated to elements of the subgroup Λ U ⊂ Γ which maps U into itself may be used rather straightforwardly to constrain the terms in the series, but for most elements γ ∈ Γ the functional equations (5.11) relate U to far away regions of L where the analytic expression of F is not known. Typically we do not even know if a global analytic continuation exists, and when it exists, it is hard to establish whether its has the right branching properties.
The existence of the global analytic continuation of F to the full L should be seen as a very subtle part of our swampland criterion. As far as its uniqueness goes, there is some evidence that the correct statement is that the difference between two determinations of the global holomorphic function F should be physically invisible, that is, the values of all observables are independent of the choice of determination of F . This should also be seen as a fine point in our swampland criterion.
We conclude Criterion 2. Let the holomorphic function F (X J ), homogeneous of degree 2, be the global analytic continuation of a local pre-potential which does not belong to the swampland. Then F satisfies the system of functional equations (5.11) for all elements of a finitely-generated group Γ such that Since Γ is infinite this looks like a huge set of equations. However not all of them are independent: it suffices to impose the ones corresponding to the finitely-many generators of Γ. For instance, consider the generic situation where Γ Q ≡ Sp(2m+2, Q). If the monodromy is arithmetic, Γ is a finite-index quotient of the maximal arithmetic subgroup Sp(2m + 2, Z). The simplest possibility is Γ ≡ Sp(2m + 2, Z): this happens, say, for the universal family of complete intersections of two cubics in P 5 [89]. Then we have one independent system of functional equations of the form (5.11) per generator of the Siegel modular group. Ref. [104] yields an economic set of generators for Sp(2m + 2, Z) given by (at most) 3 explicit matrices.
We get a set of 3m + 3 functional equations for the F I . However, typically there exist other elements γ ∈ Γ (or even infinite subgroups Λ ⊂ Γ) which lead to simpler relations which yield elementary but useful constraints on F . Criterion 2 is a concrete (if unpractical) realization of the physical idea that in a consistent quantum gravity the gauge group (i.e. Γ) determines the Lagrangian (i.e. F ).

Proof of dicothomy
In this section we prove Fact 2 (dicothomy), that is, Fact 5. Let S ≡ Γ\ S be a special Kähler manifold (with S smooth simply-connected). Assume, in addition, that S satisfies our swampland structural criterion: namely, its underlying Griffiths period map p : S → Γ\D (6.1) satisfies the VHS structure theorem (4.25). We write P for the semi-simple Q-algebraic group Γ Q and p R ≡ p ⊗ Q R for the real Lie algebra of the group of its real points P (R). Let In the second case S is isometric to the Hermitian symmetric space P (R)/K where P (R) is either SU(1, m) or one of the 'magic' isometry groups in Table 2.
That is, either the naive symmetry group of Sym( S) ⊆ Iso( S) is purely discrete, or S is a Hermitian symmetric space and hence, being special Kähler, is one of the spaces in table 2. In this second case the moduli space S is a Shimura variety. As already mentioned, Fact 5 is similar in spirit to the DG statement Fact 3 (but more general).
Proof. We may assume S to be irreducible of positive dimension, since simply-connected reducible special Kähler manifolds have been classified (see [88] [32]) and are all symmetric spaces, so the statement is trivially true in the reducible case. Then the Q-Zariski closure Γ Q ≡ P of the monodromy group Γ is an almost-simple Q-algebraic group (cfr. Lemma 3), and its Q-Lie algebra p is non-zero and simple. The structure theorem yields factorizations of the period map p and of its lift p as in the commutative diagram: where the vertical double-headed arrows are canonical projections, and ι, ι closed inclusions. ι is induced by the irreducible representation ̺ as in the diagram (5.4). It is easy to see that when ̺ is irreducible, but not absolutely irreducible, the special Kähler manifold S is symmetric, in facts a complex hyperbolic space HC m ≡ SU(1, m)/U(m) [32]. Then we may assume ̺ to be absolutely irreducible without loss.
The inclusion (5.7) of the Lie groups induces the inclusion of the corresponding (real) Lie algebras On the other hand, Γ ⊂ Sym( S) ∩ P (Z), (6.5) so that the subalgebra sym( S) is preserved by the adjoint action of Γ x ∈ sym( S) ⇒ γxγ −1 ∈ sym( S) for all γ ∈ Γ. (6.6) Thus sym( S) is an ad Γ-invariant real Lie subalgebra of the simple real Lie algebra p R . Fact 5 then follows from the following Lemma.
Lemma 4. Let the Q-algebraic group P ≡ Γ Q be simple, and let p be its Q-Lie algebra. Set p R = p ⊗ Q R for the Lie algebra of P (R). Suppose that s ⊆ p R is an ad Γ-invariant real Lie subalgebra. Then either s = 0 or s = p R .
Proof. The inclusion of R-spaces s ֒→ p R defines an element of End R[Γ]-mod (p R ), that is, a Γ-invariant tensor which belongs to the R-space where V is the underlying Q-space of the VHS, namely the representation space of the rational, irreducible, faithful, symplectic Hodge representation ̺, see eqn. (5.4). Therefore p ⊂ End(V ) via ̺. Since Γ ⊂ GL(V ), the real vector subspace T Γ R ⊂ T R of vectors fixed by Γ is defined over Q, that is, All tensors in (p ⊗ p ∨ ) Γ are also fixed by the Q-algebraic group P ≡ Γ Q , hence by the group of its real points P (R). Then s ֒→ p R is also an element of End P (R)-mod (p R ), i.e. a P (R) intertwiner, hence zero or an isomorphism because the Lie group P (R) is simple.

Existence of quantum-consistent symmetric geometries
By dicothomy, the quantum-consistent 4d N = 2 sugra are divided in two classes: (1) models whose scalars' manifold S is locally symmetric; and (2) models whose S has no local Killing vector at all. We wish to show that the first class is not empty. We distinguish two cases, namely quadratic versus cubic pre-potentials. We consider only geometries of positive dimension.
Quantum-consistent quadratic special geometries. There are known examples of geometric families of Calabi-Yau 3-folds whose period map is given by a purely quadratic pre-potential By the structure theorem, this happens if and only if the (2m + 2)-dimensional representation of Γ (or equivalently of the real Lie group P (R)) is irreducible but not absolutely irreducible. A mechanism which guarantees reducibility over C is described in ref. [122] (see, in particular, his Theorem 2.5). The idea is that there is a global automorphism α of the universal deformation space of the CY which acts on H 3,0 as multiplication by η = ±1. Clearly α centralizes Γ, so that its representation becomes reducible over C. In [122] there Quantum-consistent 'magic' cubic pre-potentials. The Calabi-Yau manifolds which are finite quotients of either an Abelian variety A or a product of a K3 surface with an elliptic curve [55], have cubic pre-potentials corresponding to arithmetic quotients of the reducible symmetric special geometry These examples are discussed in detail in the next two sections. In addition to these ones, we have the consistent truncation of higher supergravities to the subsector invariant under a discrete group. Such truncated models arise, for instance, in the compactifications of Type II on a Kummer CY 3-fold of the form T 6 /Σ, where Σ acts non-freely, when we froze the twisted sector degrees of freedom to zero. The sub-sector special geometries arising in this way are listed in the tables of ref. [34].

Swampland criterion mirror symmetry
In this section we study in slight more detail the asymptotic behavior at infinity of a special Kähler geometry which satisfies our swampland criterion, and see how it automatically reproduces most of the properties we usually associate with mirror symmetry [52,124] without any need to assume that the special geometry arises from a pair X, X ∨ of mirror Calabi-Yau 3-folds. The relevant properties just follow from quantum consistency of the 4d N = 2 effective gravity theory described by that special geometry. For a deeper perspective on the asymptotic structure of a quantum-consistent pre-potential F , see Deligne [125].
The validity of the VHS structure theorem implies, in particular, that the asymptotic behavior of the period map at infinity is described by the (multi-variable) nilpotent-orbit theorem [103]. As we approach infinity in S along a certain direction, we may end up with nilpotent orbits of several different kinds: they are classified in terms of the degenerating mixed Hodge structures which may arise [127]. Here we focus on just one particularly simple possibility, namely what happens when we approach a MUM (maximal unipotent monodromy) point [105]. The special situation we have in mind is as follows: our special Kähler manifold has the form S = S \ Y , for S a compact complex space and Y a snc divisor, and there is a small open set U ⊂ S and local coordinates x i such that while the monodromy γ i ∈ Γ around the i-th component {x i = 0} of Y is non-trivial. We may assume with no loss that Γ is neat, so that γ i = exp(N i ) for some nilpotent element N i ∈ sp(2m+2, Q). The N i commute between themselves, and ( i λ i N i ) 4 = 0 for all choices of the coefficients λ i ∈ R by the strong monodromy theorem. The point The universal cover of U ∩ S ∼ = (∆ * ) m is with cover map The MUM point corresponds to the limit Im w i → ∞ for all i. The i-th monodromy element γ i ∈ C acts on the covering local coordinates w j as w j → w j + δ j i : this reflects in an integral symplectic action on the X I . Then, for a suitable choice of symplectic homogeneous coordinates X I , the action of the γ i 's takes the form In most examples det f = 1 and there is no need to distinguish between z i 's and w i 's. The subgroup Λ ⊂ Γ maps small neighborhoods U of the MUM point into themselves. Then we may use directly the functional equations (5.11) with respect to elements of Λ to constrain the asymptotic structure of F at the MUM point. The requirement that each γ i acts by an integral Sp(2m + 2, Z)-rotations on the full vector (∂ X I F , X J ) (cfr. eqn.(5.11)) requires that F (X I ) satisfies for all i = 1, . . . , m a functional equation of the form (cfr. eqn.(2.30)) for some symmetric integral matrix S (i) IJ . This condition restricts the function F (z j ) (see eqn. (5.9)) to the general form 57 λ n e 2πi n· w , (7.10) (z j ≡ f j i w i ) where the coefficients d ijk , b ij and c i are rational numbers which need to satisfy strict integrality conditions in order to satisfy (7.9) with f j i and S (i) IJ integral. In particular, The constant coefficient in the rhs of (7.10) has been written with a particular normalization for later convenience. From the nilpotent orbit theorem we see that for a certain non-zero matrix M, so MUM point ⇒ the cubic form d ijk is non-zero. Eqn.(7.10) may be seen as a strengthened version of the nilpotent-orbit theorem. At this stage the coefficients e and λ n are still arbitrary complex numbers. Now we appeal (a bit heuristically) to some more arithmetic Hodge theory. The Fourier series in eqn.(7.10) stands for a generic holomorphic function, defined in the neighborhood U (possibly after restricting it), and periodic under the integral shifts w i → w i +δ i j , which yields the general solution to the functional equations (7.9) associated to the Abelian subgroup Λ ⊂ Γ. Such general function was written as a sum over elements of the standard basis for the periodic holomorphic functions in H m , namely {e 2πi n· w } n∈Q . However this way of writing the solution is not intrinsic, and the expression may be made more illuminating by going to a more natural basis of such functions {L( n · w)} n∈Q .
Which basis is the "intrinsic" one in the present context? In the MUM limit Im w i → ∞ the VHS degenerates to a mixed Hodge structure (more precisely to a mixed Hodge-Tate structure [125]) of maximal weight 3, so that the natural candidate is the function L(w) which canonically represents the Tate Hodge structure Q(3), i.e. the natural function is the one which satisfies the appropriate Hodge-theoretic functional equations. From, say, Theorem 7.1 in the review [126] we learn that such function L(w) is the trilogarithm Formally, all Fourier series of the form (7.10) may be rewritten as a series in the L(z). 57 Here Q = {(n 1 , . . . , n m ) ∈ Z m : n i ≥ 0} is the positive m-tant.
Therefore we shall rewrite n∈Q\0 λ n e 2πi n· w → n∈Q\0 N n L( n · w), (7.14) for the new (a priori complex) coefficients N n The structural swampland criterion suggests that the intrinsic coefficients in the expansion of F at a MUM point are the N n not the λ n . In particular the swampland philosophy suggests that the N n 's -contrary to the λ n 'sshould have interesting arithmetic properties in quantum-consistent N = 2 supergravities with a MUM point.
Let us see how these arithmetic conditions arise. While L( n · w) may be defined to be univalued in the neighborhood 58 U ⊂ L around the MUM point, its global analytic continuation is certain not univalued. Indeed L(w) has an interesting monodromy [126] which is the basic reason why this special function describes the relevant Hodge structure: this is the very property which makes L(w) prominent in the swampland program. From the discussion in the previous section, we know that the swampland condition refers to the global analytic continuation of the pre-potential F , not to its local expression in a particular region of L. Thus, to procede with the swampland program, we are forced to understand the physical implications of the multivaluedness of L( n · w).
Consistency of the swampland scenario we are proposing leads to the following principle: In a quantum-consistent 4d N = 2 supergravity, the ambiguity in the global definition of each term in the sum in the rhs of (7.14) should be physically invisible, that is, no physical observable should depend on the particular determination we choose.
Two determinations of L( n· w) differ by an integral multiple of ( n· w) 2 /2. For, say, f = 1, this reflects to an indeterminacy of F of the form and this is invisible precisely when it can be compensated by a Sp(2m + 2, Z)-rotation of the electro-magnetic frame. This requires the N n to be integers. 59 The same token leads to the condition e ∈ 2Z. Indeed the constant term in eqn.(7.10) may be simply written as e 2 L(0), (7.17) 58 L stands for a suitable compactification of the covering Legendre manifold L [105]. 59 More precisely, one has the slightly weaker condition that gcd{f i k n k f j l n l } N n should be integral (f i j is the inverse of the matrix f j i ).
so it may be absorbed in the sum by extending the summation index n to zero and setting N n=0 ≡ e/2. Applying to N n=0 the same integrality condition which holds for the n = 0 terms then yields e/2 ∈ Z. Then Criterion 2, which follows from our swampland condition, implies that around a MUM point with f = 1 (w i ≡ z i ) with integral coefficients d ijk , b ij , c i , e/2 and N n .
The large-volume expansion of the pre-potential F for type IIA compactified on a Calabi-Yau 3-fold X has exactly the form (7.18) where d ijk ∈ Z is the triple intersection of an integral basis {ω i } of harmonic (1,1) is the Euler characteristic of X [52]. We dubbed the last term in (7.18) the "instanton corrections" since this term has this physical interpretation in this particular class of models; for the same reason we denoted the constant term as the "loop" (≡ perturbative) contribution.
In the Type IIA large CY volume set-up, the coefficients N n are (non-negative) integers because they count the rational curves on X of class n · ω [52]. The coefficient c i in eqn. (7.18) drops out of the special Kähler metric G ij so it is ambiguous at the differential-geometric level. However, in the full VHS its value (mod 24) is crucial to reproduce the correct K-theoretic quantization of electro-magnetic charges, see e.g. the detailed analysis in §. 4.1.2 of [128]. For simply-connected 3-CY one gets c i = c 2 ∧ ω i ∈ Z (mod 24).
We see that essentially all the general predictions that we may infer from the existence of an actual pair X, X ∨ of mirror Calabi-Yau 3-folds are valid in full generality for any abstract 4d N = 2 effective supergravity (with a f = 1 MUM point at infinity) provided it satisfies the structural swampland criterion. Morally speaking, quantum consistency of gravity implies mirror symmetry as a special case.
The set NE of non-empty instanton sectors. There is one counter-intuitive point to be stressed. At first sight only positive instanton-charges n may contribute to the local sum (7.18) in U: In the geometric set-up of Type IIA on a 3-CY, this restriction to positive instanton-charges is obvious: only effective curves -whose volume is positive -may possibly contribute in the large-volume limit. Thus, naively, the only instanton-charge sectors which do contribute to F are the ones with instanton charge in C M ⊂ Z m , the Mori strict convex cone of effective curves. However the general story is subtler: in order to apply the swampland criterion, we should look at the covering Legendre manifold L globally, and this requires analytic continuation of F outside the small domain U in which the sum (7.18) makes sense.
To make our point sharp, let us consider the example discussed in §.5 of a quantum consistent N = 2 supergravity with Γ = Sp(2m + 2, Z) which, in view of eqn.(7.8), implies f = 1. In particular, Γ contains the element diag(+1, −1, · · · , −1, +1, −1, · · · , −1) ∈ Sp(2m + 2, Z), (7.21) which acts as This operation formally "invertes the sign" of the instanton charges. Of course, the region Im z i → −∞ is outside the domain of validity of the representation (7.18) for the function F . We may however, use the functional equation of the function L(z) 22) to formally flip the sign of the topological charge for some primitive instanton from positive to negative. The price we pay is a redefinition of the polynomial part of F which is perfectly compatible with the required integrality properties of its coefficients. In view of this situation, it looks natural to define the set NE of non-empty instantoncharge sectors to be the global one, containing the instanton charges which virtually contribute in all of the asymptotic regions of L and in all determinations of the multivalued F . In particular NE should contain instantons contributing to all asymptotic expansion    n ∈ Z m : the Fourier expansion of F in some asymptotic region at ∞ of S contains e 2πi n· w with a non-zero coefficient Note that Υ has the same formal structure as the would-be global symmetry group (4.54). Υ acts linearly on the z, hence on the instanton-charges; morally speaking it plays the same role for the instanton-charges that G glob plays for the electro-magnetic ones. Therefore, to the very least NE should contain where d NZ U correspond to d-fold cover instantons contributing to the local expansion in the asymptotic region U. The physical expectation [51] is that NE is essentially the full lattice Z m . To establish this fact is our next task.

Necessity of non-perturbative corrections
We note that when there are no "instanton corrections" in eqn. (7.18), that is, when N n = 0 for all n = 0, the real shifts form a dimension-m, commutative, unipotent, real Lie subgroup J(R) ⊂ Sym( S) of symmetries of the special geometry. In particular, its Killing vectors are non-zero. From Fact 5 we conclude that a pre-potential of the form (7.18) with N n = 0 ( n = 0) belongs to the swampland unless its covering space S is a Hermitian symmetric manifold, hence the tube domain T (V ) of a rank-3 symmetric convex cone V (≡ the positive cone of a rank-3 Euclidean Jordan algebra), see the right-hand side of Table 2. In this exceptional case F is uniquely determined by the corresponding Jordan algebra, modulo the trivial ambiguity due to possible different choices of integral-symplectic frames for the homogeneous coordinates of D m . For T (V ) symmetric we may choose the frame so that 6! F (z i ) is a homogenous cubic form over Z equivalent over R to the determinant in the Jordan algebra whose explicit expression is given in eqns. (3.15) and (3.17).
In particular N n = 0 for all n = 0 implies that the constant term in (7.18) vanishes as well, that is, the "Euler characteristic" e should be zero N n = 0 for all n = 0 =⇒ e = 0. (7.28) Naively, c i = 0 also, but the coefficients b ij , c i are not uniquely defined since they depend on the choice of the duality frame. Indeed the change of the duality frame has the effect c i → c i + 24 n i (while keeping fixed all other coefficients), so that, in absence of instanton corrections one can assert at most the weaker condition c i = 0 mod 24. The actual condition seems to be even weaker: if the cohomology groups of X have some k-torsion, the actual congruence seems to be something like k c i = 0 mod 24.
In facts, dicothomy yields more: if S is not symmetric, the quantum corrections should break the continuous unipotent symmetry J(R) down to Λ ⊂ J(Z) ⊂ Γ, because, in this case, no Killing vector is allowed to survive quantization in a consistent theory. This condition requires that the set n ∈ NZ U of contributing instanton-charge vectors is big enough to generate the full lattice Z m .
Remark. Let us compare this result with the differential-geometric Fact 4 in section 3. There we shown that a non-symmetric asymptotically-cubic special geometry -which does not belong to the swampland -should receive some quantum correction, perturbative or non-perturbative. Here we see that we need non-perturbative corrections, the perturbative ones being not enough to satisfy the swampland criterion. Moreover, the absence of non-perturbative correction in some exceptional cases implies that the perturbative loop corrections vanish as well. We shall elaborate more on this below.

Non-empty instanton-charge sectors
In the previous section we have seen that when S is not symmetric, the instanton-charges of the non-empty sectors should generate the lattice Z m . However physically we expect [51] a stronger condition, that is, that all instanton-charges n ∈ Z m correspond to a non-empty topological sector (in some asymptotic region of S).
That infinitely many instanton-charges should be realized is already clear from the functional equations (5.11) in view of the fact that the Q-Zariski closure of Γ is semisimple, and hence the matrices (5.10) of some of its generators {γ t } necessarily have non zero leftbottom block, C IJ = 0, and the corresponding functional equation cannot be solved by any finite sum of the form (7.18). Indeed, the structure of these equations strongly suggests that instantons of all charges will appear.
One may make the above heuristics a little more explicit (we take f = 1 for simplicity). Consider the subgroup GL(m, Z) ⊂ Sp(2m + 2, Z) given by the block-diagonal embedding GL(m, Z) ∋ A → Diag 1, A, 1, (A t ) −1 ∈ Sp(2m + 2, Z), (7.30) and let Ξ def = Γ ∩ GL(m, Z) ⊂ Sp(2m + 2, Z).  33) so to show that the set of non-empty instanton-charge sectors NE is the full lattice Z m it suffices to show that Ξ is "big enough". This clearly holds when Γ = Sp(2m + 2, Z) since in this case Ξ = GL(n, Z) ≡ ±SL(n, Z), by §. 7.1. NZ U contains a primitive element of the lattice Z m , while its SL(n, Z)-orbit consists of all primitive elements of Z m . Since NE contains all multiples of its primitive elements, in this case NE is the full lattice Z m . The situation when Γ ⊂ Sp(2m + 2, Z) is arithmetic reduces to the previous one. We have the commutative diagram (first row exact, i and ι mono) and since Ξ ≡ ker σι we see that Ξ is still GL(m, Z) modulo a finite group. The case of a thin monodromy, is less clear, although the same argument will typically lead to a large set NE because in many respects thin subgroups of Sp(2m + 2, Z) "behave as they were the full group". For instance: 60 Property (Strong approximation for monodromy [129] 61 ). Γ ⊂ GL(n, Z) a monodromy group consistent with the VHS structure theorem, that is, Γ is finitely-generated and Zarinskidense in P (Q) ⊂ GL(n, Q), with P a Zariski-connected, simply-connected, semi-simple Qgroup. Then for almost all prime p the residue map

Applications to Type IIA compactifications
We apply the previous analysis to Type IIA compactifications: Corollary 8. X a 3-CY with a mirror X ∨ . If the large-volume limit of the pre-potential for Type IIA on X has no instanton correction, then c 3 (X) = 0, c 2 (X) = 0 mod 24, (7.36) and the quantum Kähler space S is an arithmetic quotient of a Hermitian symmetric manifold of the "magic" type (in particular, S is a Shimura variety).
We recall that a 3-CY has c 2 = 0 if and only if its Ricci flat Kähler metric is flat [135].
Example. We check the Corollary in the two classes of known examples in which the prepotential does not receives instanton corrections, that is, when (i) X is a complex 3-torus, (ii) X is an elliptic curve times a K3, or (iii) X is a finite free quotient of (i) or (ii) (see next section for more details). The pre-potential for all Calabi-Yau of type (iii) was computed (in some convenient frame) in ref. [132] getting a purely cubic polynomial, in agreement with the prediction from Fact 5 that there should exist an electro-magnetic frame with this property. 60 There exists a more precise result, namely the super-strong approximation of monodromy groups [131]. 61 See also Theorem A in [130] Indeed, in these cases it is fairly obvious that the Weil-Petersson metric should be locally symmetric to SL(2, R)/U(1) × SO(2, ρ − 1)/[SO(2) × SO(ρ − 1)] (7.37) (ρ ≥ 2 being the Picard number) and all such locally symmetric special geometry have a purely cubic pre-potential in a suitable frame. Note that the space (7.37) is a power H ρ of the upper half-plane H if and only if ρ = 2 or 3. This observation is related to our next topic, the the Oguiso-Sakurai question. In the next section we shall be mathematically more precise.

Answering the Oguiso-Sakurai question
In the rest of this note we adopt the notion of "Calabi-Yau 3-fold" which is natural in Algebraic Geometry.
where K X is the canonical sheaf and O X the structure sheaf. All 3-CY (in this sense) are smooth projective algebraic varieties over C.
Given this definition, there are two possibilities: either π 1 (X) is finite or it is infinite. When |π 1 (X)| < ∞ the holonomy Lie algebra hol(X) of the Ricci-flat metric is exactly su(3) [133], and X is a 3-CY in the strict sense.
A 3-CY with |π 1 (X)| = ∞ has a finite unbranched cover which is either an Abelian variety A or the product of an elliptic curve E and a K3 surface K [133]. A 3-CY of the form A/Σ is said to be of A-type, while one of the form (E × K)/Σ is called of K-type [55,132,134]. In both cases Σ is a finite group of automorphisms acting freely.
There are six deformation types of A-type 3-CY explicitly constructed in refs. [55,134]. Their Ricci-flat Kähler metric is flat, so χ(X) = 0 and c 2 (X) = 0. Indeed, a CY has c 2 = 0 if and only if its Ricci-flat metric is flat [55,134,135] so The Picard number ρ ≡ h 1,1 ≡ h 2,1 of an A-type 3-CY is either 2 or 3 [55]. An A-type 3-CY X contains no rational curve for obvious reasons. 62 There are eight deformation types of K-type 3-CY again explicitly constructed in refs. [55,134]. A K-type 3-CY X has χ(X) = 0 in agreement with eqn. (7.28). Their possible Picard numbers are [55,134] ρ ≡ h 1,1 = h 2,1 = 11, 7, 5, 4 or 3. (8.3) A K-type 3-CY contains several rational curves, but all of them appear in one-parameter families parametrized by the curve E (same argument as in footnote 62), so the corresponding instanton corrections to the pre-potential F vanish because of too many fermionic zeromodes; this fact may be stated more intrinsically: one lifts the 2d susy σ-model to the finite-cover target E × K and uses the (4,4) non-renormalization theorems to show the absence of corrections.
In ref. [55] Oguiso and Sakurai raise the following Question. Is it true that all 3-CY without rational curves have Picard number ρ = 2 or 3?
A stronger statement would be that a 3-CY X has no rational curve if and only if it is A-type. An even stronger result will be a positive answer to the following Question 3. Is it true that the "instanton corrections" in eqn. (7.18) for Type IIA on X vanish if and only if π 1 (X) is infinite? That is, (in physical language): is it true that the "instanton corrections" vanish if and only if they are forbidden by the non-renormalization theorem of a higher (p, q) > (2, 2) world-sheet supersymmetry?
In this section we answer Question 3 in the positive assuming that X has a mirror X ∨ and Picard number ρ ≥ 2. The case of ρ = 1 will be settled in the next section below using less rigorous physical arguments. The results of the present section are mathematically rigorous (under the stated assumptions); indeed they follow directly from the VHS structure theorem [27][28][29][30] applied to the universal deformation of the mirror Calabi-Yau X ∨ assumed to exist. From a physical perspective, the hypothesis that X has a mirror X ∨ should be dropped: indeed the only thing we use in the argument below is that the period map of the VHS which describes the quantum Kähler moduli satisfies the VHS structure theorem, as required by our swampland criterion.
The positive answer to Question 3 implies no instanton corrections ⇐⇒ higher 2d susy non-renormalization =⇒ no perturbative loop corrections (8.4) so that the absence of non-perturbative world-sheet correction also entails the absence of the perturbative ones, as we found in the previous section by the dichotomy argument. In particular, absence of rational curves implies χ(X) = 0.

The mathematical argument
We consider the projective cubic hypersurface W ⊂ P H 2 (X), ρ ≡ dim H 2 (X), (8.5) whose affine cone is Since the intersection numbers are integers, C(W ) (resp. W ) is an affine (resp. projective) algebraic hypersurface defined over Q. We recall the Proposition 2 (P.M.H. Wilson [136,137]). If X has finite fundamental group and the projective hypersurface W satisfies the condition that its rational points W (Q) are dense in its real locus W (R), then there exists a rational curve on X.
Remark. Note that the Proposition says nothing when ρ ≡ dim H 2 (X) = 1, since in this case W is not defined.
If W contains a hyperplane it should be rational 63 , that is, if W is absolutely reducible it is already reducible over Q. Then we remain with three possibilities: (a) W is an irreducible cubic defined over Q; (b) W contains a hyperplane and an irreducible quadric, both defined over Q; (c) W consists of 3 hyperplanes defined over Q, possibly counted with multiplicity. This cannot happen for ρ > 3 [137].
The rational points are trivially dense along the irreducible components which are hyperplanes. Using general facts about quadratic and cubic integral forms representing zero, one shows: Lemma 5 (P.M.H. Wilson [137]). The rational points are dense in the real locus: (1) In case (b) when ρ > 5 (Meyer's theorem); (2) If W is irreducible, ρ > 5, and W contains a rational linear space of dimension 3; (3) If W is irreducible and ρ > 19.
The swampland structural criterion sets severe constraints on the possible integral cubic forms D 3 which may arise from the triple intersection of divisors in a 3-CY without rational curves. Then, in view of Proposition 2, to answer the Oguiso-Sakurai question we need to study the rational points of only these very specific cubic hypersurfaces.
Below we show the following: 63 For this claim and the list of properties in items (a),(b),(c) see comment after Lemma 4.2 in ref. [137].
Claim. X a 3-CY (with mirror and ρ ≥ 2) without rational curves or, more generally, such that its A-model TFT has no instanton corrections ⇒ the rational points W (Q) are dense in the real locus W (R).
Together with Proposition 2 this Claim gives Corollary 9. X a 3-CY (with mirror and ρ ≥ 2) such that its A-model TFT has no instanton correction. Then |π 1 (X)| = ∞.
From this Corollary all other claims follow provided the Picard number = 1.
Proof of Claim. As we saw sections 6 and 7, the structure theorem of VHS (applied to the complex moduli of the mirror X ∨ ) implies that, in the absence of instanton corrections, the Kähler moduli of X do not get any quantum correction, and that their special Kähler geometry S (after replacing it with a finite cover, if necessary) is the product of irreducible Shimura varieties in one-to-one correspondence with the irreducible components C α of the affine cone C(W ), eqn. By [137] the C α 's are defined over Q. Each irreducible Shimura variety S α is the quotient of a Hermitian symmetric space with d ij , resp. d ijk integers. The integer quadratic form d ij has Lorentzian signature (1, m−1) so, in the quadric case, the convex cone V quad. is the set of "forward time-like" vectors, whereas in the cubic case V cub. is the cone of positive-definite elements in the R-space Let G α be the Q-algebraic group where both groups in the rhs are seen as subgroups of the Q-algebraic group Sp(2m + 2, Q) through the block-diagonal embedding Modulo finite groups, one has where G m is the multiplicative group and L α = M α ∩ SL(m, Q) is the simple Q-algebraic group which leaves invariants the rational tensors d ij resp. d ijk The connected group G α (R) • of real points of G α is nothing else than the automorphism group of the cone V α ⊂ R m . Then where SL(3, O) ≡ E 6(−26) .
Remark. The Q-algebraic group α L α has a simple physical interpretation: its real points α L α (R) form the naive symmetry group of the 5d N = 1 sugra obtained by compactifying M-theory on the Calabi-Yau X. α L α (R) is the isometry group of the universal cover of the 5d vector-multiplet scalars' space [117], and its Q-algebraic structure is induced by the 5d flux quantization. In particular, the 5d swampland condition yields Q-rank L α = R-rank L α (R) = 1 quadrics 2 cubics, (8.17) a fact already implied by (8.9).
For each irreducible component C α of C(W ) we may find a finite set of real-valued points {x a } ∈ C α (R) such that the union of the (analytic) closure of their orbits under G α (R) is the full real locus C α (R) of the component Indeed, in the quadric case we may take as {x a } just a single non-zero point x lying in the light-cone, while in the cubic case we may take a pair of rank 2 Hermitian matrices of different signature, say x ± = diag(1, ±1, 0) ∈ Her 3 (F). Suppose for the moment that it is possible to choose the points {x a } in (8.18) to be Q-valued. In this case a G α (Q)· x a ⊂ C α (Q). (8.19) We claim that the union of the rational orbits in the lhs of (8.19) is dense in the real locus C α (R). This follows from eqn. (8.18) together with the fact that G α (Q) is dense in G α (R). Indeed, G α = G m × L α . Density of the rational points is obvious for the multiplicative group G m . For the simple Q-algebraic group L α we invoke the following fact 64 Lemma 6. Let L R ⊂ SL(m, R) be a semi-simple and connected (closed) Lie subgroup. Some finite cover of L R is the Lie group of real points of a Q-algebraic group if and only if L R ∩ SL(m, Q) is dense in L R .
Putting everything together, the density of the rational points C(W )(Q) in the real locus C(W )(R) will follow if we can show that we may choose the points {x a } in (8.18) to be Q-valued.
For quadrics it is enough to show that the light-cone d ij x i x j = 0 contains a non-zero rational point. In view of eqn. (8.17), this is a direct consequence of, say, Proposition 64 From Theorem (4.6.3) of [139] we know that all connected, semi-simple subgroup L ⊂ SL(n, R) is almost R-Zariski closed in SL(n, R). Then apply Proposition   [64]. In other words: since the quotient Γ quad. Γ quad. (R)/K quad is non-compact of finite volume, and Γ quad. is arithmetic, the Q-rank of Γ quad. (R) is at least one by the Godement criterion; for a Lorentz group this is equivalent to the existence of a non-zero rational point in the light-cone [64].
More generally, in all instances -quadrics as well as cubics -the existence of the rational points {x a } will follow if we can show that the rational structure of the algebraic group L α is the "obvious" one: for instance, for the first cubic case (ρ = 6), where L cub. (R) = SL(3, R), it would follow if L cub. (Q) is the plain Q-group SL(3, Q) rather than some fancier rational structure on the real group SL(3, R). Indeed, if this was the case, the rational cubic form d : Her 3 (Q) → Q would be given by the plain determinant formula 65 would be defined over Q, and we would be done. That the "obvious" Q-structure is the correct one for this first cubic example, follows from classification of the Q-algebraic groups whose real locus is the Lie group SL(3, R). We recall the relevant facts, see e.g. the proof of Proposition (18.6.4) in [64]. There are infinitely many non-isomorphic such rational groups, but only the "obvious" one SL(3, Q) has Q-rank equal 2 as required by (8.9). A more direct argument which does not use eqn.(8.9) is as follows: all rational algebraic groups with underlying real Lie group SL(3, R) have a natural rational representation on Q 6 . Tensoring this representation with R we get the irreducible representation ⊙ 2 R 3 for rank 2 and the reducible one R 3 ⊕ (R 3 ) ∨ for rank 1 (see the proof of Proposition (6.6.1) in [64]). Since we know that the underlying real representation is ⊙ 2 R 3 ( §. 3.3) we conclude that SL(3, Q) is the rational structure on SL(3, R) selected by quantum gravity. So the cubic ρ = 6 instance is settled.
In the second cubic case (ρ = 9), where L cub. (R) = SL(3, C), the same argument leads us to L cub. (Q) = SL(3, Q( √ −r)) (8. 22) for some square-free positive integer r whose precise value is not relevant for our present purposes, although it should be seen as an important physical invariant of the corresponding 5d quantum gravityif it exists. Again the points x ± = diag(1, ±1, 0) ∈ Her 3 (Q( √ −r)) are defined over Q independently of the value of r. Alternatively the rank 2 matrices       x 1 x 2 + √ −r x 3 0 (8.23) where (x 1 , x 2 , x 3 , x 4 ) ∈ Q 4 form a dimension 3 linear space in W (Q) and we may apply where D F is a division algebra over Q, with a unique (canonical) positive-definite Rosati (anti)involution x ↔ x ⋆ , such that D F ⊗ Q R = F. (8.25) For F = R, C, and H, the division algebra D F has, respectively, Shimura type [140] I, IV, and III over the totally real field Q. The case F = O is not covered by Shimura, since the octonions do not form an associative algebra. The argument formally extends also to this case; anyhow we do not need it, since F = O corresponds to Picard number ρ = 27 > 19 and this case is already settled by Lemma 5(3). The canonical involution x ↔ x ⋆ has the property [140] x + x ⋆ ∈ Q, ∀ x ∈ D F (8.26) so that it makes sense to talk about the vector Q-space Her 3 (D F ) of Hermitian matrices with entries in D F as well as the positive rational cone V Q ⊂ Her 3 (D F ) ≈ Q 3+3 dim F (8.27) i.e. the would-be rational Kähler cone of the putative Calabi-Yau X without rational curves. By (8.26) the entries along the main diagonal of any m ∈ Her 3 (D F ) are rational numbers. Again, our arguments are independent of the precise division Q-algebra D F , whose isomorphism class is an important datum of the consistent quantum gravity (if it exists). The Q-algebraic group SL(3, D F ) acts on the Q-space Her 3 (D F ) as m → a m a † , m ∈ Her 3 (D F ), a ∈ SL(3, D F ), a † def = (a ⋆ ) t . (8.28) The cubic form is just the determinant of the matrix m (defined by the Vinberg prescription (3.17) when D F is non-commutative/non-associative) det : Her 3 (D F ) → Q (by (8.26)). (8.29) The rational points of the cubic affine cone C cub. (W )(Q) are then identified with the elements of Her 3 (D F ) of rank ≤ 2. This set is clearly non-empty, e.g. diag(1, ±1, 0) ∈ Her 3 (D F ), and we are done.
9 The case of Picard number 1 The argument of the previous section cannot be applied to Calabi-Yau 3-folds with ρ ≡ h 1,1 = 1. In this section we argue that -as everybody expects [136] -a 3-CY with ρ = 1 has (infinitely many) rational curves. To attach this residual case, we use physical ideas and the discussion is meant to be just heuristic.
We assume (by absurd) that X has no rational curve. Then, by the previous result so that the covering special geometry S is the upper half-plane with the Poincaré metric G zz normalized to that that is, 3 times the Poincaré metric in the standard normalization. The Hodge metric is and it does not receive the contact-term quantum correction described in [72] since χ(X) = 0 in absence of instanton corrections.
The period map factors through the upper half-plane, so that Γ must be a finite-index subgroup of SL(2, Z) containing a power of T . Then the global symmetry group is 66 N SL(2,Z) (Γ)/Γ = SL(2, Z)/Γ, (9.5) and if we assume that it is trivial (as required by the usual swampland arguments) we conclude that Γ ≡ SL(2, Z). (9.6) Using the equations of one-loop holomorphic anomaly [72] and eqn.(9.4), one finds that the genus one index F 1 is F 1 (z,z) = − log (Im z) 10 |f (z)| 2 , (9.7) for some holomorphic function f (z) without zero or poles in the upper half-plane; the expression should be modular invariant by (9.6). Then, following the argument around eqn. (8) of ref. [72], we get f (z) = η(τ ) 20 . (9.8) From [72] we know which looks as a contradiction since X ω ∧ c 2 = 10 ≡ 0 mod 24. (9.10) K plus the complementary space p. We write g −1 dg ∈ Λ 1 (G) ⊗ g for the Maurier-Cartan form on the group manifold G, and (g −1 dg) k , (g −1 dg) p for its summands with respect to the decomposition g = k ⊕ p. Γ ⊂ G is a discrete subgroup.
We wish to show the following [15,66]: be a smooth map. Let X be the universal cover of X and φ : X → G be a lift of f . Define the definite-type one-forms on X A = φ * (g −1 dg) k | (1,0) ,Ā = φ * (g −1 dg) k | (0,1) , if and only if f is pluri-harmonic, i.e. in local coordinatesDk∂ i f a = 0. Conversely, all solutions to tt * arise in this way.
Remark. Two local lifts φ 1 and φ 2 differ by a K-gauge transformation, so define the same tt * geometry which depends only on the underlying pluri-harmonic map f .
Remark. Under suitable algebraic conditions on the chiral ring R, i.e. the C-algebra generated by 1 and the matrices C i , the image Ψ(1)(X) lays in a proper subgroup G(R) ⊂ SL(n, R) and we may write f : X → G(R)/K (with K = G(R) ∩ SO(n)), f pluri-harmonic.

B Arithmetics of superconformal tt * branes
The arithmetic properties of the tt * brane amplitudes of a 2d superconformal (2,2) are much richer and more beautiful than those of a generic (2,2) QFT, in particular more elegant than the ones for a gapped (2,2) model where the BPS branes are simple Lefschetz thimbles whose monodromy is governed by the classical Picard-Lefschetz theory [14,17].
To describe the arithmetic characterization of the branes of a (2,2) SCFT inside the space of all tt * branes of (2,2) QFTs, we consider the brane amplitudes Ψ(ζ) along the twistor equator |ζ| = 1 and normalize the basis elements of the chiral ring R s so that the determinant of the topological 2-point function η ij is 1 (such a normalized basis always exists). Then, in a generic 2d (2,2) QFT, the brane amplitude Ψ(ζ) takes values in SL(n, R), where n is the Witten index (see §. 3 of [15]). More precisely, for a fixed ζ along the equator, is a proper real Lie subgroup G(R) SL(n, R). This reduction of the "brane group" has important implications for the quantum theory.
We focus on a family of superconformal (2,2) models, i.e. the rings {R s } s∈S are local and graded by the U(1) R charge operator Q (see eqns.(2.10),(2.21)) while S is a space of exactly marginal deformations. For simplicity of notation we also assumeĉ ∈ N (the extension to fractionalĉ being straightforward). Q yields a U(1) R grading of the Lie algebra g of G(R) of the form g ⊗ C =ĉ q=−ĉ g −q,q , X ∈ g q,−q ⇔ Q, X] = q X. is odd G(R) must be in the list of odd-weight groups on page 24 of [28].
Proof. From §. X.3 of the second edition of [60] we know that Q ∈ g. Then exp(θQ) ⊂ G(R) is a compact one-parameter subgroup since it preserves the positive-definite tt * metric. Let T a maximal torus containing exp(θQ), and t its Lie algebra. Clearly [Q, t] = 0, that is, t ⊂ g 0,0 ∩ g. Since the compact part k ⊂ g is given by The generic "brane group" SL(n, R) does not have the above property (for n ≥ 3) n − 1 ≡ rank SL(n, R) = rank SO(n) ≡ [n/2]. (B.9) The above result shows that the conformal brane amplitudes are very different from the well-known one for the massive 2d models. They have higher arithmetics. Indeed the real Lie groups of the "Mumford-Tate" type are characterized by their interesting arithmetic: in particular they are the only groups which have discrete series automorphic representations [141,142]. In view of eqn.(2.21) we have The (arithmetic) quotients of Mumford-Tate domains are the natural generalization [143] of the Shimura varieties, the "paradise" of arithmetics.
C Symmetric rigid special Kähler manifolds Lemma 9. A symmetric rigid special Kähler manifold M is flat.
Proof. Let M be a symmetric rigid special Kähler manifold. Without loss we may assume