aa r X i v : . [ h e p - t h ] F e b Swampland Geometry andthe Gauge Couplings
Sergio Cecotti ∗ SISSA, via Bonomea 265, I-34100 Trieste, ITALY
Abstract
The purpose of this paper is two-fold. First we review in detail the geometric aspectsof the swampland program for supersymmetric 4d effective theories using a new and uni-fying language we dub “domestic geometry”, the generalization of special Kähler geometrywhich does not require the underlying manifold to be Kähler or have a complex structure.All 4d SUGRAs are described by domestic geometry. As special Kähler geometries, do-mestic geometries carry formal brane amplitudes: when the domestic geometry describesthe supersymmetric low-energy limit of a consistent quantum theory of gravity, its formalbrane amplitudes have the right properties to be actual branes. The main datum of thedomestic geometry of a 4d SUGRA is its gauge coupling, seen as a map from a manifoldwhich satisfies the geometric Ooguri-Vafa conjectures to the Siegel variety; to understandthe properties of the quantum-consistent gauge couplings we discuss several novel aspects ofsuch “Ooguri-Vafa” manifolds, including their Liouville properties.Our second goal is to present some novel speculation on the extension of the swamplandprogram to non -supersymmetric effective theories of gravity. The idea is that the domesticgeometric description of the quantum-consistent effective theories extends, possibly withsome qualifications, also to the non-supersymmetric case.February 23, 2021 ∗ e-mail: [email protected] ontents U -duality group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Singularities of moduli spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.1 Singularities from the action of G . . . . . . . . . . . . . . . . . . . . 72.3.2 Example: Type IIB on a 3-CY . . . . . . . . . . . . . . . . . . . . . . 92.3.3 Smoothing surgery . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Behavior at infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4.1 Sign of curvature at infinity . . . . . . . . . . . . . . . . . . . . . . . 132.4.2 Cusps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 P • . . . . . . . . . . . . . . 173.1.2 Liouville property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 τ ( φ ) ab as a map 19 τ ( φ ) ab . . . . . . . . . . . . . . . . . . . . . . . 214.3 DG invariants of the gauge couplings τ ( φ ) ab . . . . . . . . . . . . . . . . . . 224.4 Tension field and harmonic maps . . . . . . . . . . . . . . . . . . . . . . . . 244.4.1 Energy and tension of maps into symmetric spaces . . . . . . . . . . 244.4.2 Cartan gauge couplings . . . . . . . . . . . . . . . . . . . . . . . . . . 25 tt ∗ geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.2.1 HIV brane amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . 325.2.2 Graded tt ∗ geometries & VHS . . . . . . . . . . . . . . . . . . . . . . 345.2.3 Explicit graded HIV brane amplitudes . . . . . . . . . . . . . . . . . 375.2.4 Physical quantities from brane amplitudes . . . . . . . . . . . . . . . 375.2.5 tt ∗ entropy functions & Mumford-Tate groups . . . . . . . . . . . . . 385.3 Domestic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.3.1 Example: 2d SCFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.3.2 Domestic brane amplitudes . . . . . . . . . . . . . . . . . . . . . . . 405.4 Entropy functions in domestic geometry . . . . . . . . . . . . . . . . . . . . 411 Domestic geometry and supergravity 42 L eff . . . . . . . . . . . . . . 507.2.4 Naturalness: Adding gravity . . . . . . . . . . . . . . . . . . . . . . . 517.2.5 A word of caution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537.3 Existence and structure theorem . . . . . . . . . . . . . . . . . . . . . . . . . 547.3.1 The case of OV manifolds . . . . . . . . . . . . . . . . . . . . . . . . 567.3.2 Structure of the gauge coupling µ . . . . . . . . . . . . . . . . . . . . 577.3.3 Swampland conditions for N = 2 SUGRA . . . . . . . . . . . . . . . 587.4 First applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
A SUGRA spaces: tamed maps vs. special holonomy 60B No boundary term in the Bochner argument 62 Introduction
The swampland program [1, 2] (for reviews see [3, 4]) aims to characterize the sparse subsetof effective field theories which arise as low-energy limits of consistent theories of quantumgravity inside the much larger set of formal theories which “look” consistent from a low-energy perspective, but cannot be completed to a fully consistent theory of quantum gravity.A model which cannot be consistently completed is said to belong to the swampland.
The program has taken the form of a rapidly growing list of conjectural necessary con-ditions (the “swampland conjectures” [1–4]) that all effective theories of quantum gravityshould obey. These conditions are motivated by general physical considerations (in particu-lar the thermodynamics of black holes [5]) as well as by lessons drawn from the large supplyof consistent effective theories of gravity which describe the light degrees of freedom in acontrolled vacuum of superstring theory (the string lamppost principle (SLP) [6]). We havefull analytic control on the quantum stability of a candidate string vacuum only when itpreserves some supersymmetry, and we can write a precise effective Lagrangian only whenits couplings are protected by a SUSY non-renormalization theorem: so all reliable examplesat our disposal have extended supersymmetry. The best understood examples are the effective theories of Type IIB compactified to 4don some Calabi-Yau (CY) 3-fold: these examples played a major role in the development ofthe swampland program [2]. The resulting low-energy 4d N = 2 supergravities are described(in the vector sector) by special Kähler geometry [7, 8]. Special Kähler geometry is a verypowerful tool to study the quantum consistency of an N = 2 supergravity; a significant partof the swampland program [9–15, 18, 19] is dedicated to the detailed analysis of the “motivic”special Kähler manifolds which describe the moduli geometry of actual CY manifolds (ormotives). In our opinion, the swampland program has reached a rather satisfactory shape inthis specific N = 2 set-up and “motivic” special Kähler geometry must be seen as a successful model of swampland theory. It would be highly desirable to extend this model theory tobroader contexts.Supersymmetric effective theories are better behaved in many ways, and SUSY seems toplay a crucial role in the consistency of many explicit examples. In some contexts [20] it ishard to grasp how a model can possibly be consistent without been supersymmetric.On the other hand, the real world does not look supersymmetric at low energy, andhence consistent non-supersymmetric effective theories of gravity ought to exist. An impor-tant goal of the swampland program is to say something less vague about the non -SUSYeffective theories of consistent quantum gravity. They are expected to be quite rare andremarkable animals, in a sense even more “magical” than their SUSY counterparts. Morallyspeaking, these theories should enjoy “all the good properties” of SUSY while avoiding itsphenomenological drawbacks as the existence of (unobserved) super-partners. We should notexpect such “magical” theories to share the properties which generically hold for a non-SUSY This limitation in the available explicit examples may a priori lead to some bias in our understandingof quantum gravity. non- generic. From this point of view, the highlynon-generic value of the real-world effective cosmological constant Λ is hardly a surprise.The author’s own prejudice is that there should be a “more general” swampland principle,which under appropriate circumstances reduces to (or implies) SUSY, but which continuesto make sense in some very restricted non-supersymmetric context where most of the “good”facts about SUSY still hold. The reader will find many echoes of the prejudice in this paper.The purpose of this paper is two-fold. First we present a systematic review of the geomet-ric aspects of the swampland program in the SUSY context from a novel unifying perspective.The two key concepts of this approach are: (i) Ooguri-Vafa (OV) manifolds , i.e. the Riemannian spaces with the correct properties(according to [1, 2]) to be the scalars’ spaces of a consistent effective theory; (ii) domestic geometry , with its brane amplitudes and generalized entropy functions.Domestic geometry is the direct generalization of special Kähler geometry which does not require the underlying manifold to carry a complex structure. All 4d supergravities aredescribed by a domestic geometry in the same exact way that the vector-multiplet sectorof 4d N = 2 SUGRA is described by the usual special Kähler geometry. In particular,the vectors’ couplings have an universal expression in terms of domestic brane amplitudes.Domestic geometry looks particularly deep and natural when the underlying manifold is OV.Combining the two ingredients (i) , (ii) one gets arithmetic domestic geometry which is theobvious domestic generalization of the class of “motivic” special Kähler geometries whichdescribe the vector-sector of quantum-consistent 4d N = 2 supergravities. This allowsto phrase the swampland conditions for the general SUSY model in the same suggestivelanguage used in the N = 2 case [15]:A 4d supergravity is described by a domestic geometry from which we compute formal brane amplitudes. If the supergravity is not in the swampland, the formalamplitudes describe actual branes.
OV manifolds have several nice properties. In particular, they are Liouvillic for the sub-harmonic functions. This entails that the rigidity properties of “motivic” special Kählergeometry (the “power of holomorphy” [21]) hold for all OV manifolds even if they do notcarry any complex structure.Our second main purpose is to present some novel speculations about the swamplandconditions in the non-
SUSY case. The space of light scalars M is (conjecturally) an Ooguri-Vafa manifold in all quantum-consistent effective theories of gravity, supersymmetric or not.Arithmetic domestic geometry is naturally defined on all OV manifolds, and its statementsmake perfect sense for non -SUSY effective Lagrangians. One is then led to speculate thatdomestic geometry – which applies to all SUSY cases – may also be relevant to describe vec-tors’ couplings in quantum-consistent non -SUSY 4d effective theories. The speculation may4e stated at different levels of precision. Domestic geometry may be: (a) just qualitativelyvalid, or (b) semi-quantitatively correct, or even (c) exact. Besides its aesthetic geometricappeal, and the evidence from SUSY examples, the speculation rests on some heuristic phys-ical arguments, based on the idea of “naturalness” in the IR description, which suggest thatat least version (b) should hold.The rest of this paper is organized as follows. In section 2 we discuss some generalproperties of 4d effective field theories and some technical aspects of the singularities andasymptotics of the moduli spaces. In section 3 we define the OV manifolds and discusstheir basic properties. In section 4 we describe the gauge couplings seen as a map from themoduli space to the Siegel variety. Section 5 contains the basics of domestic geometry; alarge part of the section is devolved to a detailed review of (generalized) tt ∗ geometry whichis the model which inspires all constructions in domestic geometry. Section 6 describes howdomestic geometry applies to all 4d SUGRAs and may be used to reformulate the swamplandconditions in the SUSY context in a more convenient way. Section 7 describes in more detaildomestic geometry, presenting the math arguments for the existence and uniqueness of theunderlying tamed maps, and rephrasing them as heuristic physical arguments in favour ofour speculation that arithmetic domestic geometry is relevant for the non -SUSY swamplandprogram. Some additional technicality is confined in the appendices. Remark 1.
The recent papers [16, 17] describe a different uniform geometric approach tosupergravity, dubbed bosonic supergravity.
It would be interesting to understand geometricaspects of the swampland conditions from that perspective.
To make the story a bit shorter, in this paper we consider only four-dimensional effectivetheories. Although methods and ideas apply to much wider contexts, we assume a vanishingcosmological constant,
Λ = 0 , and focus on effective Lagrangians valid at parametricallysmall energies. In particular, we make the following two assumptions: A1 all visible IR gauge degrees of freedom (d.o.f.) are in their Coulomb phase. Locallyat generic points in field space we may choose an electro-magnetic duality frame, withrespect to which these d.o.f. are described by Abelian gauge vectors A a ( a = 1 , . . . , h )with field strengths F a = dA a ; A2 the (exactly) massless fields carry no electric or magnetic charge under the A a ’s.The light degrees of freedom then consist of a space-time metric g µν , h vector fields A a , and m massless real scalars φ i , together with spin-1/2 fermions and possibly spin-3/2 gravitini5only in the SUSY case). In the Einstein frame (and the chosen electro-magnetic frame) theeffective Lagrangian takes (locally in field space) the form L eff = −√− g (cid:16) R + 12 G ( φ ) ij ∂ µ φ i ∂ µ φ j − i π τ ( φ ) ab F a + F b + + i π ¯ τ ( φ ) ab F a − F b − + · · · (cid:17) (2.1)where F a ± = (1 − ∓ i ∗ ) F a is the (anti-)self-dual part of the field-strength F a .The swampland program asks for a characterization of the field-dependent couplings inthe Lagrangians L eff which describe low-energy limits of consistent quantum gravities. Inpractice, one looks for necessary conditions they should satisfy, which are usually phrasedas sufficient conditions for the model (2.1) to sink in the swampland.In this paper we limit ourselves to the geometric part of the swampland program, i.e. tothe two-derivative couplings G ( φ ) ij and τ ( φ ) ab .The characterization of the consistent scalars’ kinetic couplings G ( φ ) ij takes advantagefrom their geometrical interpretation [2]. The scalar fields φ i are seen as local coordinateson a connected “manifold” f M , of dimension m , endowed with the Riemannian metric G ij ≡ G ( φ ) ij . The Riemannian metric is smooth at generic points of f M . Remark 2.
In a generic non-SUSY effective field theory we do not expect a non-trivialspace f M of exactly massless scalars since the flat directions of the potential are usuallylifted by quantum effects. This needs not to apply in the present context, since quantum-consistent theories of gravity are highly non-generic. On the other side, the conclusions ofthis paper remain valid even if the scalars parametrizing f M are not exactly massless but onlyhierarchically lighter than the scale above which the low-energy description breaks down. U -duality group A basic datum of the effective field theory is its gauge group . The discrete part of thegauge group is called the U -duality group G . G is a redundancy of the description, so it istautologically an exact symmetry of the full quantum theory not just of its IR sector. Moreprecisely, by G we mean the quotient of the full discrete gauge group of the underlying UVcomplete theory which acts faithfully on the light bosonic fields. In particular, G acts onfluxes of forms of various degrees k π I Σ k F ( k ) (2.2)which are integral by generalized Dirac quantization: the fluxes (2.2) take value in a lattice Λ endowed with a non-degenerated bilinear form Λ ⊗ Λ → Z (the generalized Dirac pairing).The completeness conjecture [5] states that in a consistent theory all fluxes allowed by Diracquantization are realized by some physical state. In 4d the electro-magnetic charges takevalue in a lattice V Z ∼ = Z h with a skew-symmetric principal Dirac pairing. The action of G We use the shorthand F a ± F b ± ≡ ( F a ± ) µν ( F b ± ) µν . Principal means that the Dirac pairing V Z ⊗ V Z → Z yields an identification V Z ≡ V ∨ Z .
6n the scalar fields and the electro-magnetic fluxes defines a group embedding G ֒ → Iso( f M ) × Sp (2 h, Z ) (2.3)The kernel of the map G →
Iso( f M ) is finite; since we always work modulo finite groups, we shall not distinguish G from its image in Iso( f M ) . The map ρ : G → Sp (2 h, Z ) is calledthe monodromy representation , and its image Γ ⊂ Sp (2 h, Z ) the monodromy group. Hencethe U -duality group G is a group extension → G → G → Γ → (2.4)that is, G ∼ = G ⋊ Γ . A version of the π -conjecture [2], which holds true in all knownexamples, states that the U -duality group is isomorphic to a subgroup G ⊂ GL ( k, Z ) (forsome k ) which is generated by k × k unipotent matrices (modulo finite groups). Thus G is asubgroup of the Q -algebraic group GL ( k, Q ) , and in all examples its Q -Zariski closure G Q is semi-simple. Hence, modulo finite groups, G ∼ G × Γ . The moduli space is M ≡ f M / G , (2.5)that is, the space of inequivalent (effective) vacua. The covering space f M may be assumedto be simply-connected with no loss. Then π ( M ) ∼ G . The purpose of this technical sub-section is to argue that we may be cavalier with thesingularities of the moduli space M , and see in which sense we may work as if M was agood ( ≡ complete) Riemannian manifold. The sub-section may be omitted in a first reading. G We write d ( · , · ) : f M × f M → R ≥ for the distance function defined by the metric G ij andreplace f M by its metric completion. Fix a non-trivial unipotent element u ∈ G ; the π -conjecture [2, 3] says that the infimum of the function d u : f M → R ≥ d u ( x ) def = d ( ux, x ) (2.6)is zero. Assuming the conjecture, we consider a sequence of regular points { x i } ⊂ f M reg suchthat d u ( x i ) ց . We have two possibilities: Iso( f M ) is the isometry group of f M with metric G ( φ ) ij . Indeed the kernel is both compact and discrete. The symbol ∼ stands for equivalence up to commensurability [22] i.e. equality up to finite groups. The statement holds in all SUSY consistent theories by a basic fact in domestic geometry, see §.7.3.The reliable examples are all supersymmetric. x i escapes to infinity, i.e. d ( x , x i ) → ∞ . We say that “ u has a fixed point at infinity”.The distance conjecture [2] requires an infinite tower of states to become exponentiallylight as we approach the fixed point at infinity. Their electro-magnetic charges belongto ker ( ρ ( u ) − ;(2) { x i } remains inside some finite-radius ball B ( x , r ) , and hence contains a subsequencewhich converges to a finite-distance point x ∗ ∈ f M which is fixed by u .In the second case the action of G yields an obstruction to a smooth extension of the Rie-mannian metric to x ∗ : indeed its isotropy sub-group G x ∗ ⊂ G ⊂ Iso( f M ) is discrete andcontains the infinite group of isometries u Z , whereas the isotropy sub-group of a regularpoint in Riemannian geometry is always compact. Hence x ∗ is a finite-distance singular-ity where the curvature blows up. In a quantum-consistent effective theory with N ≥ supersymmetry all singularities of the completed covering space f M are of this kind, i.e.fixed-points of non-trivial unipotent elements u ∈ G , while the moduli space M ≡ f M / G has, in addition, orbifold quotient singularities. Thus in the extended SUSY case all finite-and infinite-distance singularities of M are dictated by the action of the U -duality group G – they correspond to vacua where a non-trivial subgroup of G remains unbroken – and noother “accidental” singularity is present. The logic beyond the π and distance conjecturessuggests that this is the case for a general consistent effective gravity theory: all singularpoints are fixed by some non-trivial subgroup of the discrete gauge group G .We give a closed look to the two kinds of finite-distance singularities. Quotient orbifold singularities.
In general the U -duality group G does not act freelyon the regular locus f M reg ⊂ f M . The isotropy sub-group G ˜ x of a smooth point ˜ x ∈ f M reg isfinite. The image of such a point in the moduli space M is then a mild orbifold singularity.Orbifold points correspond physically to vacua where a finite sub-group G ˜ x ⊂ G remainsunbroken. In some stringy examples such points correspond to vacua with a non-Abelianenhancement of the effective continuous gauge symmetry, i.e. loci where our standing as-sumptions A1 , A2 break down and L eff ceases to be a complete description of the IRphysics.When we have an embedding ι : G → GL ( k, Z ) we can “repair” the orbifold singularitiesin a cheap way. Fix an integer n ≥ ; let r n : GL ( k, Z ) → GL ( k, Z /n Z ) be the reduction mod n , and write ι n ≡ r n ◦ ι . Consider the exact sequence of groups → G n → G ι n −→ GL ( k, Z /n Z ) → . (2.7)It follows from the Minkowski theorem that the matrix group G n is a normal subgroup of G of finite index which is torsion-free, in facts neat [22]. This implies that G n acts freely on f M reg so that M n ≡ f M / G n is a finite Galois cover of M , with Galois group ι n ( G ) ≡ G / G n ,free of orbifold singularities. The finite-quotient singularities may be cured by replacing M with M n and G with G n : this is the standard strategy in the math literature when studying8oduli spaces of projective varieties (in particular of Calabi-Yau 3-folds), and we adoptit. From now on by G we always mean a finite-index, neat, normal subgroup of the actual U -duality group. Correspondingly M ≡ f M / G is free of finite-quotient singularities. Finite-distance curvature singularities. f M reg is not geodesically complete in gen-eral; that is, f M reg may contain half-geodesics ℓ ( t ) originating from a smooth point φ ≡ ℓ (0) which cannot be continued after some finite value of the proper length t . Such a finite-lengthmaximal half-geodesic represents a physical transition – which takes finite time and costs finite energy per unit volume – from our initial configuration φ to a physical situation wherethe IR description provided by L eff is no longer valid. This finite-time process should beperfectly regular from the viewpoint of the UV complete theory. What happens is that ℓ ( t ) stops at a vacuum where “new physics” becomes relevant in the infra-red: some additionaldegrees of freedom of the fundamental UV theory get massless, and our IR description breaksdown. From the viewpoint of the UV fundamental theory, these singularities typically cor-respond to points where different branches of the space of vacua meet each other ( transitionpoints ). Since the process involves a finite energy density, one expects that there exists arefined effective Lagrangian L new which includes the relevant “new physics” and is valid upto some higher but still finite energy scale. L new allows us to extend the IR description,and hence the physical process described by the half-geodesic ℓ ( t ) , beyond the domain f M reg .This means that family of finite-length half-geodesics in ( f M reg , G ij ) ending at a given sin-gular point x ∗ is associated to a finite number of new states becoming massless; in additiontheir spins must be ≤ . This is to be contrasted with the case of a fixed point at ∞ , wherean infinite tower of states get light [2].We expect all finite-distance curvature singularities to correspond to fixed points x ∗ undera parabolic subgroup of G ∗ ⊂ G (cfr. the discussion after eqn.(2.6)). We have a finite setof vectors { q } ⊂ V Z which correspond to the electro-magnetic charges of the finitely-manystates which becomes massless at x ∗ . They should be invariant under G ∗ (modulo finitegroups), i.e. ρ ( u ) q = q for u ∈ G ∗ . At such a singularity, the metric is continuous (inappropriate local coordinates) but the curvature invariants blow-up. If our effective theoryhas N ≥ supersymmetry, there is no matter supermultiplets which may become massless,so no finite-distance curvature singularities may be present. The same holds for N = 3 , asa result of a SUSY non-renormalization theorem. The various kinds of singularities are well illustrated by the vector-scalars’ moduli space M v of Type IIB compactified on a Calabi-Yau 3-fold X . The regular locus of its covering The curvature singularity is proportional to the contribution of the gauge coupling beta-function fromthe states (with e.m. charges { q } ) which become massless at x ∗ , so it vanishes for N = 3 , where theonly matter supermultiplets are vector-multiplets which yield a zero net contribution to the β -function.Geometrically, this non-renormalization theorem corresponds to the fact that, for N ≥ SUGRA, f M is asymmetric space, whose curvature is parallel, ∇ i R jklm = 0 , so cannot blow up anywhere. f M v, reg , may be identified with the moduli space of marked and polarized Calabi-Yau’s in the smooth deformation class of X . f M v, reg is equipped with its Weil-Petersson(WP) Kähler metric G i ¯ j , which is the metric appearing in the scalars’ kinetic terms [7, 8].Singularities at finite distance in the WP metric correspond to points where there is aconifold transition (more generally, an extremal transition) to a CY with different Hodgenumbers [23]. According to a celebrated suggestion by Reid [24], all CY moduli spacesare expected to be connected through such transitions. The full Type IIB string theoryremains regular at those transitions, but the low-energy description based on the effective N = 2 supergravity breaks down. On f M v, reg there is also another, better behaved, canonicalKähler metric K i ¯ j , the Hodge one (a.k.a. the K -metric [25, 26]), which has the expression(with n = dim C M v ) [25–28] K i ¯ j = ( n + 3) G i ¯ j + R i ¯ j ≥ G i ¯ j , (2.8) R i ¯ j ≥ − ( n + 1) G i ¯ j , (2.9)where R i ¯ j is the Ricci curvature of the WP metric G i ¯ j . It is important to notice that, whilethe Ricci curvature R i ¯ j of the WP metric satisfies the lower bound (2.9), the Ricci curvatureof the K -metric R Ki ¯ j satisfies an upper bound [27] R Ki ¯ j ≤ − √ n + 1) + 1 K i ¯ j . (2.10)From eqn.(2.8) we see that points at infinite distance in the WP metric are also at infinitedistance in the K -metric. The opposite statement is false: in terms of the K -metric all finite-distance curvature singularities are pushed at infinite distance. Indeed, at conifold pointsthe metric G i ¯ j remains bounded while the Ricci curvature blows up so that the K -metricblows up.The Torelli space is the completion of f M v, reg with respect to the K -metric [29–31] f M Kv ≡ (cid:0) f M v, reg (cid:1) K -metriccompletion . (2.11) f M Kv is a smooth space diffeomorphic to R m [29–31]; since G is torsion-less, the space M Kv ≡ f M Kv / G (2.12)is a version of a finite cover of the moduli space which is a smooth Kähler manifold, completefor the K -metric, with a contractible universal cover. Its fundamental group π ( M Kv ) ∼ = G satisfies (a refined version of) the π -conjecture [2]. Unfortunately the nice manifold M Kv ≡ f M Kv / G is not complete for the physical WP metric. M Kv is the natural space to parametrize the complex structures on a fixed smooth-class of More in detail, each holomorphic sectional curvature of G i ¯ j is bounded below by − .
10Y 3-folds: M Kv does not talk to moduli spaces of CY’s with different topologies which arepushed infinitely away in the K -metric. The physical moduli space of Type IIB compactifiedon X instead consists of several branches of vacua, with extremal transitions between them;since the different branches are not infinitely separated, the physically relevant metric G i ¯ j on each branch cannot be complete.There is another description of these moduli spaces more in the spirit of Algebraic Ge-ometry. There is a compact projective variety ¯ M v and an effective divisor D such that M Kv = ¯ M v \ D. (2.13)By Hironaka theorem, we can choose the pair ( ¯ M v , D ) so that ¯ M v is smooth and D = P i D i is a simple normal crossing divisor. The points in the support of D are at infinite distancein the K -metric. With respect to the WP metric D splits as D = D ∞ + D f , where D f (resp. D ∞ ) is the singular locus at finite (resp. infinite) distance. Returning to the general case, the presence of finite-distance curvature singularities in f M seems to be unavoidable when m ≥ , unless the low-energy theory is a N ≥ supergravityor some truncation thereof. In all other cases the moduli space cannot be both smooth andcomplete for the kinetic-terms metric G ij .Working with non-complete and/or non-smooth Riemannian spaces is technically incon-venient. We try to improve the situation by smoothing out the singularities with some localsurgery, i.e. by modifying the metric G ij in the vicinity of the “bad” points.We stress that the modified metric is meant to be a mere technical trick to simplify theanalysis of the geometry of the moduli space M . However it is suggestive to phrase thesurgery as it was an actual modification of the effective Lagrangian L eff . The modificationwould be almost “harmless”, since the original Lagrangian itself gave a poor description ofthe IR physics near the singularity in field space, so the local modification affects physicalprocesses which were already outside the scope of L eff . In a sense the locally modified La-grangian is still a “good” effective Lagrangian, and should satisfy the swampland consistencyconditions as far as they do not involve the region near the “bad” points in field space.The allowed deformations of L eff are quite restricted since they should preserve all gaugesymmetries. In particular: (1) the deformation of the kinetic-terms metric should leave thediscrete gauge group G as an exact symmetry of the problem; (2) the holonomy and isometryLie algebras of G ij , hol ( M ) and iso ( M ) , should be preserved.Under the hypothesis that all singularities are fixed by a subgroup G ∗ of the neat group G , it suffices to modify the metric on the universal cover f M in such a way that G is a freelyacting group of isometries of the deformed metric. Since the Ricci curvature is expected to11low up at a fixed point of an unipotent isometry, a perturbation of the form G ij → G ǫij ≡ G ij + ǫ R ij , ǫ > (2.14)suggests itself. Except at loci where the curvature blows-up while the metric remains finite– which is exactly the characterization of finite-distance singularities – the correction to themetric is negligible for ǫ very small, so (2.14) is essentially a local modification of the geometryaround the finite-distance singular loci. G is still an exact isometry of the perturbed metric,so we do not spoil the discrete gauge symmetry. More generally, the surgery (2.14) does notspoil any symmetry the original geometry may have. Condition (2) is also satisfied. Wecan see the modification (2.14) as the result of a backward Ricci-flow [33] of the metric bythe small time t = − ǫ/ , so that, heuristically, it looks like a RG flow to a slightly largerenergy scale, in line with the physical interpretation of the finite-distance singularities asloci where new physics comes in. In view of eqn.(2.8), replacing the WP metric G j ¯ k by theHodge one K j ¯ k on a 3-CY moduli space amounts to the surgery (2.14) with ǫ = 1 / ( m + 1) ;indeed our proposed prescription (2.14) is modelled on the standard math treatment of 3-CYcomplex moduli spaces.The modification (2.14) makes sense provided the Ricci curvature is bounded below, R ij ≥ − KG ij , so that G ǫij is positive-definite for small ǫ . The idea is that the modificationreplaces a small region around the singular point x ∗ by a cusp of infinite length but finitevolume of order O ( ǫ k ) for some k > . When this happens, the singularity is pushed atinfinite distance, and geodesic completeness is restored. Thus the volume conjecture stillholds after the surgery, however the new “spurious” cusps are not associated to towers oflight states as the genuine infinite ends of M .In this paper we assume that it is always possible to modify the metric locally at thesingularities, while preserving G and hol ( M ) , by replacing a neighborhood of the finite-distance singularity with a cusp of volume O ( ǫ k ) , so that the resulting Riemannian spaceis complete and smooth. All our arguments below are meant to apply to the “regularized”moduli manifold so constructed, which we shall denote simply M . In the known examplesthe assumption holds true.After the modification the geometric swampland conjectures still hold if they were satis-fied by the original L eff , with the only exception that the distance conjecture does not applyto the spurious infinite ends introduced by blowing-up finite-distance singularities. Let us check that the holonomy algebra hol ( M ) is preserved. We may assume f M to be irreduciblewithout loss. If the Riemannian metric G ij is locally symmetric, it is Einstein, and the modification (2.14)is just a change of overall normalization of the metric by a factor O ( ǫ ) . Otherwise hol ( M ) is one ofthe seven Berger holonomies [32]. For generic holonomy so ( m ) there is nothing to show. If G ij is Kähler,so is G ǫij . For M a Calabi-Yau, hyperKähler, quaternionic-Kähler, G - or Spin (7) -manifold, G ij is eitherRicci-flat or Einstein so the deformation (2.14) is either trivial or a slight rescaling of the metric. The basic reason of the finiteness of the volume is that we mod out the infinite group G ∗ which maps asmall neighborhood of x ∗ into itself. .4 Behavior at infinity In some argument below we need some more technical aspect of the geometry of M atinfinite distance. In this sub-section we sketch the main issues; it may be omitted in a quickreading. In their original paper [2] Ooguri and Vafa conjectured that the scalars’ space M is non-compact of finite-volume. They also conjectured that the moduli-space scalar curvature R is negative at infinity. In ref. [35] Trenner and Wilson constructed an explicit “counter-example” to the last statement in the context of Type IIB on a certain 3-CY with h , = 3 .In that example there is a real curve C in moduli space, parametrized by s ∈ R , such that,as we approach a “large complex structure limit” along this curve, the WP scalar curvature R behaves as [35] R = 3281 s + O ( s ) as s → + ∞ along C, (2.15)so in this limit R is positive and unbounded. We wrote “counter-example” between quotesbecause this example does not contradict the physical picture of [2]. In the language ofeqn.(2.13), the physical intuition for Calabi-Yau moduli spaces goes roughly as follows: aswe approach a generic point on the divisor D ∞ ⊂ ¯ M (the infinite-distance locus) the Riccicurvature of the WP metric becomes negative hence bounded by (2.9), while as we approach a generic point on D f (finite-distance singularities) R ij becomes positive and divergent becauseof the contributions from loops of the finitely many additional light particles which can becomputed in some “enlarged” effective field theory. What happens at the special points atinfinity D ∞ ∩ D f ? The obvious guess is that if we approach the intersection point followinga curve C along which m f → , m ∞ → with m f m ∞ → , (2.16)where m f (resp. m ∞ ) is the mass scales of the particles getting light along D f (resp. D ∞ ),then the divergent positive contribution to R from the finitely many massless particles along D f may win over the bounded negative contribution from the infinite tower of light statesalong D ∞ . This is what happens in the Trenner-Wilson example; along their curve Cm f = O ( s − ) and m ∞ = O ( s − ) . We see that the non-positivity of R at infinity hasthe same physical origin as the failure of the WP metric to be complete [23]. Then afterreplacing G i ¯ j with its regularized version G ǫi ¯ j , which is makes M into a complete manifold,we expect that also the problem with the sign of the scalar curvature at infinity is solved,that is, we expect that its scalar curvature R ǫ to be negative and bounded (for fixed ǫ )everywhere at infinity. Indeed, along D ∞ the ǫ -modification is inessential while along D f wehave G ǫi ¯ j ≈ ǫ K i ¯ j so that from eqn.(2.10) the scalar curvature is asymptotically negative and See
Theorem 3.2 of ref. [35]. O (1 /ǫ ) ). As a check we have computed (using Mathematica ) the scalarcurvature R ǫ in the Trenner-Wilson example along the curve C : R ǫ = − ǫ (cid:0) O ( ǫ ) (cid:1) + O (1 /s ) , as s → + ∞ along C. (2.17)In other words: the points on the curve C for sufficiently large (but finite) s are outsidethe region where the Lagrangian L eff yields a reliable IR description of the physics, and thecoupling G i ¯ j (i.e. the WP metric) needs not to behave in a “physically reasonable” way atthese points. Fix a regular point p ∈ M ; for all points at a sufficiently large distance from p which are not too close to special loci where some “new physics” appears, R is negative. We first consider the following simple but typical situation: M is a complete non-compactRiemannian manifold and there is a compact subset K ⊂ M such that M \ K is the disjointunion of finitely many “ends at infinity” of M , the α -th end E α being diffeomorphic to R × Z α for some connected manifold Z α , while the metric in E α has the asymptotic form ds ≡ G ij dx i dx j ≈ dr + g ( r, u ) ab du a du b , for r ≫ , (2.18)where r is the distance from some base point ∗ ∈ M , u a are local coordinates in Z α ,and g ( r, u ) ab is some r -dependent metric on Z α . Finiteness of the volume of E α requires p det g ( r, u ) to decay more rapidly than /r for large r . In the region where (2.18) holds wehave R rr ≈ − ∂ r log p det g − (cid:13)(cid:13) ∂ r g (cid:13)(cid:13) (2.19)If g ( r, u ) ab goes to zero as slowly as a negative power of r , the rhs is O (1 /r ) and R rr /g rr isnot bounded away from zero. On the other hand if g ( r, u ) ab goes to zero more rapidly thanan exponential, say as ≈ C e − c r k with k > the curvature R rr is O ( r k − ) and unboundedbelow for large r . So, if the Ricci tensor is negative and bounded for large r , g ( r, u ) ab shouldbe a sum of terms with exponential decay g ( r, u ) ab ≈ X i e − c i r h ( i ) ( u ) ab . (2.20)Assuming the asymptotic metric to be enough regular, this leads to bounds for large r ofthe form − K G ij ≤ R ij ≤ − K G ij for large r along E α (2.21)for some constants K , K > . We shall call a finite-volume end E α with the behaviour (2.21)a “cusp”. Prototypical examples are the cusps in an arithmetic quotient of a non-compact Typical means, in particular, that this is the situation in all known explicit examples. G ( Z ) \ G ( R ) /K. (2.22)Eqn.(2.22) is the general form of the moduli space M when L eff has a large supersymmetry(more than 8 supercharges). In these cases the nice geometry of the ends of the modulispace is directly related to the physics of quantum gravity as described by the distanceconjecture [36]. In these extended susy examples the U -duality group G ≡ G ( Z ) actsfaithfully (modulo finite groups) on the electro-magnetic charge lattice V Z , and each pointat infinity x ∞ ∈ M is fixed by a parabolic subgroup G x ∞ ⊂ G ( Z ) . States carrying electro-magnetic charges q ∈ V Z invariant under G x ∞ have masses proportional to the length of theimage in the Siegel domain of the shortest loop in M based at x in the homotopy class of theelements of G x ∞ ; since the map µ is a totally geodesic isometric embedding for N ≥ themass is also proportional to the length of the pre-image loop in M , which for a good cuspis exponentially small, cfr. eqn.(2.20). The same applies (with some subtlety) in the N = 2 case, using the relation between the kinetic terms of scalars and vectors implied by SUSYwhich replaces the totally geodesic embedding condition. Conjecturally this extends to thegeneral quantum-consistent effective theory: the length of the loop in target space should beexponentially small to fit with the predictions of the distance conjecture. If the pre-imageloop has a length which vanishes more rapidly than any exponential, say O ( e − cr k ) , k > ,the derivative would be of order e cr k , k dµ k = O ( e cr k ) which looks unreasonable. Condition ( ∗ ) . Although the large r behaviour (2.21) is expected for all quantum-consistent effective theories, to be very conservative in this paper we shall assume a muchweaker condition on the large r behaviour of the geometry after the smoothing surgery in§.2.3.3. First we assume that the Ricci curvature of M is still bounded below R ij ≥ − KG ij .Since M is complete, for all R > there exists a Lipschitz continuous function h R : M → R such that for some fixed constant k > [37]: ≤ h R ≤ , h R = ( for r ≤ R for r ≥ R, (cid:12)(cid:12) dh R (cid:12)(cid:12) < kR . (2.23)For typical asymptotic metrics of the form (2.18),(2.20) the Laplacian of h R is of order O ( R − ) for large R . We shall require only the much weaker Condition ( ∗ ) . The Laplacian of h R is bounded by a constant C independent of R | ∆ h R | < C. (2.24) Here G ( R ) is a non-compact real Lie group seen as a concrete group of matrices via a suitable rep-resentation of degree ℓ , K ⊂ G ( R ) is a maximal compact subgroup, and G ( Z ) ≡ G ( R ) ∩ GL ( ℓ, Z ) is thearithmetic subgroup consisting of matrices with integral entries. More generally, we may replace G ( Z ) by acommensurable subgroup of G ( R ) . For instance, we can choose h R = ϕ ( r/R ) where ϕ is a smooth function on the real line with ≤ ϕ ≤ , ϕ = 1 for x ≤ and ϕ = 2 for x ≥ . OV manifolds
In this section we assume that there exists a suitable local surgery, along the lines described inthe previous section, such that the singularities of a suitable finite cover of the moduli spaceget repaired resulting in a smooth
Riemannian manifold M which still satisfies the Ooguri-Vafa geometric swampland conjectures [2]. This certainly holds in the known examples ofquantum consistent effective theories of gravity.A smooth manifold which satisfies the Ooguri-Vafa geometric conditions, together withsome mild “regularity” conditions, will be called an OV manifold . Understanding the ge-ometry of OV manifolds and its physical implications is one of the themes of this paper.We propose the following definition of OV manifold:
Definition 1.
The point is a zero-dimensional OV space. In positive dimension an
OVmanifold is a connected, complete, Riemannian manifold M with a smooth simply-connectedcover f M which has a de Rham decomposition of the form f M = F × f M × · · · × f M s (3.1)and a smooth finite cover of the form M ♭ = F × f M / G × · · · × f M s / G s , G k ⊂ Iso( f M k ) (3.2)such that: OV0. the flat factor F is either trivial or the real line R ; OV1. M k ≡ f M k / G k is a complete, irreducible, non-compact manifold of finite volume ; OV2. f M k is diffeomorphic to R m k and G k ∼ = π ( M k ) is a torsion-less discrete group of isome-tries of f M k generated by elements { u i } such that inf x ∈ f M k d ( u i x, x ) = 0 . (3.3) OV3. the Ricci curvature R ( k ) ij of M k is bounded below by a negative constant R ( k ) ij ≥ − K k g ( k ) ij , K k > , (3.4)and condition ( ∗ ) (eqn.(2.24)) is satisfied.Some comments on the definition are in order:• OV1 is the volume conjecture [2] and
OV2 is a refined version of the π conjecture.A stronger version of OV2 , which holds in all known examples, would be:16 V2 ∗ G k is isomorphic to a strongly approximant subgroup ˚ G k of an arithmetic group G ( Z ) ⊂ GL ( n, Z ) . ˚ G k is required to be neat, semi-simple, and to have a finite-index subgroup generated by finitely-many unipotent elements { u i } ⊂ ˚ G k ∼ = G k which satisfy eqn.(3.3).• the point and the real line R are OV manifolds. This is required by math elegance andis consistent since all the ‘magic’ properties of OV manifolds are shared by the pointand R . The point and R do appear as moduli of consistent gravities, and even as theirfactor spaces: think of M-theory on R − k, × S k for k = 0 , , ; in particular for k = 2 M = R × ( SL (2 , Z ) \ SL (2 , R ) /U (1)) .• OV3 is a milder version of the “regularity” conditions discussed in §. 2.4.2 related tothe distance conjecture and the expected behaviour at infinity. P • A non-compact, complete manifold of non-negative Ricci curvature cannot have finite vol-ume , so the Ricci curvature of the irreducible factor spaces f M k somewhere should havesome strictly negative eigenvalue. This rules out a few Riemannian geometries:allowed f M k hol ( f M k ) ruled outstrictly generic holonomy so ( m k ) Calabi-Yaustrictly Kähler u ( m k / hyperKählerstrict negative quaternionic-Kähler sp (1) ⊕ sp ( m k / positive quaternionic-Kählernon-compact symmetric spaces G/K k G and Spin (7) manifoldsIt is remarkable that, a part for “strictly generic” (which, roughly, corresponds to the non-SUSY case), the list of allowed geometries is reminiscent of the list of target spaces forsupergravity with more than 2 supercharges. We recall that the holonomy algebra hol ( M ) of the scalars’ manifold M of a supergravity is determined as follows: from its SUSY algebraand super-multiplet content we read the R-symmetry Lie algebra r and its representation σ We recall the definition of “strong approximant” subgroup. We see the arithmetic group G ( Z ) as aconcrete group of integral matrices, i.e. it comes with a preferred embedding G ( Z ) ⊂ GL ( n, Z ) for some n .For p a prime, we write G ( Z /p Z ) for the finite group of Lie type obtained by reducing the matrices mod p .We have the canonical surjection G ( Z ) π p −→ G ( Z /p Z ) . γ : Γ ֒ → G ( Z ) is said to be a strong approximant iffthe group homomorphism π p ◦ γ : Γ → G ( Z /p Z ) is surjective for almost all primes p . This follows from the Calabi-Yau lower bound on the volume [37, 38] see
Theorem 7 and
Appendix (iii) in ref. [37].
17n the scalar fields; hol ( M ) then has the form [36] (here m = dim R M ) σ ( r ) ⊆ hol ( M ) ⊆ σ ( r ) + z ( σ ( r )) ⊂ so ( m ) z ( σ ( r )) def ≡ centralizer of r in so ( m ) . (3.5)In SUGRA the R-symmetry is a gauge symmetry with composite gauge fields ∂ µ φ i Ω( φ ) ai b ,where Ω( φ ) ai b is the projection of the Levi-Civita connection of G ( φ ) ij on the sub-algebra σ ( r ) .Let h k ≡ hol ( f M k ) ⊂ so ( m k ) be the irreducible holonomy algebra of the k -th factor space f M k . The graded algebra P • k of parallel forms on f M k is P • k ≡ m k M j =0 P jk ∼ = m k M j =0 (cid:0) ∧ j R m k (cid:1) h k (3.6)When h k = so ( m k ) , P • k is spanned by 1 and a volume form ε : we say that P • k is trivial.When f M k is strictly Kähler, P • k = R [ ω ] /ω m k / with ω the Kähler form. When f M k isstrictly quaternionic-Kähler P • k = R [Ω] / Ω m k / with Ω the canonical 4-form. When f M k is a symmetric manifold the algebra P • k is typically larger; but there are two exceptions: SO ( n, /SO ( n ) which has generic holonomy so ( n ) , and SU ( m, /U ( m ) which has strictKähler holonomy u ( m ) . We stress that the surgery of §. 2.3.3 preserves the algebra P • .The generic OV manifold has no non-trivial parallel form. Let us consider an examplein the other extremum, where P • is so rich that fully determines the metric (up to overallnormalization). Example 1.
Suppose the irreducible OV manifold M has R-algebra r = su (8) and R-representation σ the . Then – up to finite covers – M is globally isometric to the locallysymmetric space E ( Z ) (cid:15) E ( R ) (cid:14) SU (8) (3.7)where E ( Z ) denotes a maximal arithmetic subgroup of E ( R ) (the split real form of E ). This is the moduli space of Type II compactified on T with the correct U -duality group G = E ( Z ) (up to commensurability). Many nice properties of supersymmetric field theories (with or without gravity), such as thenon-renormalization/rigidity theorems, may be traced back to the fact that their modulispaces M are “Liouvillic”. This property holds automatically for OV manifolds. Morallyspeaking, “most” of the good aspects of SUSY appear to be given for free once the swamplandconjectures are satisfied. With respect to some structure of E ( R ) as an algebraic group defined over Q . The Q -algebraic structureis determined by quantum consistency: in this case E is identified with the universal Chevalley group oftype E (a scheme over Z ) and E ( Z ) is the groups of its points valued in Z . efinition 2. The manifold M is Liouvillic for the class C of functions f : M → R if f ∈ C and | f | < K < ∞ ⇒ f = constant (3.8) Proposition 1.
Irreducible OV manifolds M are Liouvillic for the sub-harmonic functions,i.e. ∆ f ≥ and | f | < K < ∞ ⇒ f = constant (3.9) Proof.
For M ≡ R this reduces to the well-known fact that a bounded convex function f : R → R is a constant. For M a complete, non-compact manifold of finite volume, wewrite u ≡ f + K . u is a non-negative function bounded above by K , and for all p > Z M u p d vol ≤ (2 K ) p · Vol ( M ) < ∞ , (3.10)and then u is a constant by Theorem 3 in [37].The statements holds even for reducible OV manifolds as long as they have no flat factorin eqn.(3.2). In particular, it holds for all 4d SUGRA satisfying the Ooguri-Vafa swamplandconjectures [2]. The Liouville property seems to be crucial for the swampland story: this isthe case for all supersymmetric consistent effective theories.
Remark 3.
Standard Seiberg-Witten theory [42, 43] is grounded on the fact that theCoulomb branch of an UV complete N = 2 QFTs satisfies the much weaker property ofbeing Liouvillic for the sub- pluri harmonic functions. τ ( φ ) ab as a map One wishes to understand which gauge coupling τ ( φ ) ab may appear in a consistent effectiveLagrangian (2.1) of quantum gravity. These couplings have a natural geometric descriptionwhich cries for an intrinsic characterization of the allowed τ ( φ ) ab . For fixed values of the scalar fields φ i (and given duality-frame), the gauge coupling τ ( φ ) ab is an element of Siegel’s upper half-space H h ≡ n τ ∈ Mat h × h ( C ) (cid:12)(cid:12)(cid:12) τ = τ t , Im τ > o ∼ = Sp (2 h, R ) /U ( h ) . (4.1) H h is a non-compact Kähler symmetric space on which the group Sp (2 h, R ) acts transitivelyby isometries. Its maximal arithmetic subgroup Sp (2 h, Z ) ⊂ Sp (2 h, R ) is the group of19lectro-magnetic duality-frame rotations. As a Riemannian space H h is Hadamard [39, 40]so diffeomorphic to R h ( h +1) . As a complex manifold H h is biholomorphic to the (symmetric)bounded domain in C h ( h +1) / of symmetric h × h matrices Z with − Z ¯ Z > .In general the complex gauge couplings τ ( φ ) ab are not well-defined functions on thescalars’ space M ≡ f M / G . Indeed, when we go around a non-contractible loop γ in M ,we may come back and find that the duality frame was rotated by a non-trivial element of Sp (2 h, Z ) . This yields a monodromy representation ρ : π ( M ) ≡ G → Sp (2 h, Z ) . (4.2)The image ρ ( π ( M )) ⊂ Sp (2 h, Z ) is the monodromy group Γ , cfr. eqn.(2.4).To get actual coupling functions, it is convenient to lift the gauge couplings to the smoothuniversal cover f M ; then we may identify them with the map e µ : f M → Sp (2 h, R ) /U ( h ) ≡ H h , e µ : φ τ ( φ ) ab , (4.3)which lifts the intrinsic gauge coupling map µ f M ˜ µ / / (cid:15) (cid:15) (cid:15) (cid:15) Sp (2 h, R ) /U ( h ) (cid:15) (cid:15) (cid:15) (cid:15) M µ / / Sp (2 h, Z ) \ Sp (2 h, R ) /U ( h ) (4.4)One also says that the lifted map ˜ µ is twisted by the monodromy representation ρ , i.e. ˜ µ satisfies the functional equations ˜ µ ( g · x ) = ρ ( g ) · ˜ µ ( x ) , ∀ g ∈ G ⊂ Iso( f M ) , x ∈ f M . (4.5)The target space of µ in (4.4), Sp (2 h, Z ) \ H h , is an irreducible, non-compact, locallysymmetric space of the special form G Z \ G/K where G Z ⊂ G is an arithmetic (hence Zariski-dense [22]) discrete subgroup . Maps between spaces with these special properties are wellstudied in mathematics.From an algebro-geometric viewpoint the target space Sp (2 h, Z ) \ H h is a normal quasi-projective variety [44, 45], called the Siegel variety (or Siegel scheme), a special instanceof Shimura variety [46, 47]. Sp (2 h, Z ) \ H h is not smooth since Sp (2 h, Z ) / {± } does notact freely on H h . This can be easily repaired by replacing Sp (2 h, Z ) with a neat finite-index subgroup Λ h ⊂ Sp (2 h, Z ) [22]. Then Λ h \ H h is a smooth quasi-projective finite cover That is, a simply-connected manifold with non-positive sectional curvatures; in facts H h has evennon-positive curvature operators. Throughout the paper double-headed arrows stand for canonical projections. With respect to the obvious structure of G ≡ Sp (2 h, R ) as an algebraic group over R . In facts, Sp (2 h, Z ) ⊂ Sp (2 h, R ) is a maximal arithmetic subgroup [41]. In the present context it is more natural toconsider G as the real locus of an algebraic group defined over Q , see [15] and references therein. Sp (2 h, Z ) \ H h . Λ h \ H h is a non-compact manifold of finite volume whose fundamentalgroup Λ h is generated by unipotent elements: Λ h \ H h is a basic example of irreducible OVmanifold. Λ h \ H h has a canonical compactification to a normal projective variety, the Baily-Borel compactification Λ h \ H h BB [44, 45]. Λ h \ H h BB is not smooth for h > . Howeverwe may blow-up it into a smooth projective variety Y h such that the divisor at infinity Y h \ (Λ h \ H h ) is simple normal crossing. This can be done rather explicitly in terms of asuitable toroidal compactification of Λ h \ H h [48–50].Then, replacing M by a finite cover (if necessary), we see the gauge coupling µ as a mapbetween smooth OV manifolds µ : M → Λ h \ H h . (4.6)It is remarkable that both the source and target spaces of µ are OV manifolds. In all knownexamples of quantum-consistent theories (such as compactifications of the superstring onCalabi-Yau’s) both the image µ ( M ) ֒ → Λ h \ H h and the fibers µ − ( s ) ֒ → M (equipped withthe Riemannian structure induced by their respective embedding ) are also OV spaces. This is likely to remain true for all effective theories of quantum-consistent gravity.One goal of this note is to give a preliminary discussion of the following
Question.
What properties should have the gauge coupling map µ in eqn. (4.6) for the La-grangian (2.1) not to sink in the swampland? If a simple condition on µ exists at all, it should be invariant under Sp (2 h, Z ) rotationsof the electro-magnetic frame. Since we consider only the very extreme IR limit, this meansthat the property should be invariant for Sp (2 h, R ) . τ ( φ ) ab Since the coupling τ ( φ ) ab is multi-valued in M , its value in a given vacuum φ is not anintrinsic observable, and we should replace it with some invariantly-defined quantity. Automorphic viewpoint.
In the case of a single photon h = 1 , the modular curve SL (2 , Z ) \ H is biholomorphic to the punctured sphere j : SL (2 , Z ) \ H f → P , (4.7)and one may use as the intrinsically-defined gauge coupling the value of the Hauptmodul j which is independent of the branch of the multi-valued function τ (here q ≡ e πiτ ) j ( τ ) ≡ (cid:0) θ ( q ) + θ ( q ) + θ ( q ) (cid:1) η ( q ) = 1 q + 744 + 196884 q + · · · (4.8) In these examples µ is an embedding not just an immersion. The statement holds without exceptions because we defined the point to be an OV manifold. In the SUSY case (and also in the non-SUSY one under the naturalness condition we propose in section7), τ ( φ ) ab must be multi-valued unless it is a numerical constant as in the examples discussed in [51]. h since, bythe already mentioned Baily-Borel theorem [44], Λ h \ H h has enough automorphic invariantfunctions to fully characterize its points. Total space viewpoint.
The space Λ h \ H h (resp. H h ) is the moduli space of en-hanced principally polarized Abelian varieties over C of dimension h (resp. marked, princi-pally polarized, Abelian varieties). As for the axion-dilaton τ ≡ C + ie − Φ in F -theory [52],often it is more convenient to think of the gauge couplings as a fibration ̟ : X → M whosefibers X φ are (enhanced) principally polarized Abelian varieties of periods τ ( φ ) ab , and thendescribe the properties of the gauge couplings in terms of the intrinsic geometry of the totalspace X of the fibration.There is a more physical “total space” construction. One compactifies the 4d effectivetheory down to 3d on a circle; each 4d vector field yields two new scalars in 3d and we get twoadditional scalars from the metric by the KK mechanism. The resulting 3d scalars’ manifold M is fibered over the 4d scalars’ manifold M , and the gauge coupling map µ is encodedin the fiber’s geometry (see §.7.2.4 for details). So the two 4d couplings G ( φ ) ij and µ ( φ ) ab get geometrically unified in the intrinsic Riemannian geometry of the 3d target manifold M which, according to the swampland conjectures, should also be an Ooguri-Vafa space withits own discrete gauge group G ⊲ G ∼ G × Γ . We shall use this “total space” viewpointwhen convenient. Elementary viewpoint.
We mostly adopt a more naive strategy: instead of consider-ing subtle automorphic invariants of the gauge couplings, we shall see (4.6) as a mere smoothmap between Riemannian manifolds, and rely on the invariants of smooth maps which aredefined in Differential Geometry (DG) textbooks. τ ( φ ) ab The basic DG invariants of a smooth map φ : X → Y between Riemann manifolds (withmetrics G ij and h ab , respectively) are:• its energy E [ φ ] given by the value of the Dirichlet integral E [ φ ] = 12 Z X d n x √ det G G ij h ab ∂ i φ a ∂ j ∂ b φ a , (4.9)that is, the action of the Euclidean σ -model with target Y and source space-time X ; The Siegel variety Sp (2 h, Z ) \ H h is the moduli space of principally polarized Abelian varieties. Goingto the smooth covering Shimura variety Λ h \ H h leads to the moduli of polarized Abelian varieties endowedwith some extra structure: e.g. if Λ h is the kernel of Sp (2 h, Z ) → Sp (2 h, Z /m Z ) ( m ≥ ) the extra structureis a choice of generators of the group of m -torsion points. An Abelian variety with such extra structure iscalled an enhanced Abelian variety.
22 its tension field on XT ( x ) a def = − h ( φ ( x )) ab δE [ φ ] δφ b ( x ) ≡ D i ∂ i φ a ( x ) ∈ C ∞ ( X, φ ∗ T Y ) . (4.10)We replace our original Question in §.4.1 with a less ambitious one:
Simpler Question.
What can we say about the DG invariants E [ µ ] and T [ µ ] of the gaugecoupling map µ : M → Λ h \ H h in a quantum-consistent 4d effective theory of gravity? A partial answer will be given in §. 6.3.Although these DG invariants have a simple definition, in quantum-consistent effectivetheories of gravity they seem to involve rather deep number-theoretical issues. For instance,the allowed energy levels E [ µ ] of the gauge couplings µ in a quantum-consistent effectivetheory of gravity belong to a certain discrete subset Ξ ⊂ R ≥ , the gauge couplings’ energyspectrum, which carries a number-theoretic fragrance. Computing Ξ is very hard except insome extremely simple class of effective models. Extremely simple situation.
The simplest possible quantum-consistent 4d effective the-ories are the ones with the properties: ( ‡ ) (1) the effective theory has N ≥ local supersymmetry, and (2) its U -duality group G is commensurable to the group G ( Z ) of “ Z -valued” points in a universal Chevalley group-scheme G without simple factors of type C h . The prototypical 4d effective theory satisfying ( ‡ ) is obtained by compactifying the 10dType II superstring on a flat 6-torus T (see appendix of [53]): in this case N = 8 and G has type E (cfr. Example 1 ). Whenever ( ‡ ) holds one has (for a standard normalizationof the metrics) Ξ =
I · m Vol ( K ) r Y ℓ =1 ζ ( d ℓ ) (4.11)where { d ℓ } are the degrees of the independent Casimir invariants of the real Lie group G ( R ) , K ⊂ G ( R ) is a maximal compact subgroup, Vol ( K ) its volume (computed by theMacdonald formula [54]); ζ ( s ) is the Riemann ζ -function, m ≡ dim G − dim K , and I ⊂ N is the set of indices of finite-index subgroups of the maximal arithmetic group G ( Z ) . If the U -duality group is precisely G , the energy of the coupling constants is given by the rhs of (4.11) with I replaced by [ G ( Z ) : G ] . Eqn.(4.11) follows from standard susy argumentstogether with the Langlands volume formula for arithmetic quotients [55]. In the case ofType II on T all degrees d ℓ are even, so the energy of the gauge coupling has a closedexpression in terms of Bernoulli numbers: E ( µ ) is a know rational number times π . ForType II on T , a part for a power of π , the energy has a transcendental factor ζ (5) ζ (9) . One can show that the absolute ranks of the two Lie groups G ( R ) and K are equal. .4 Tension field and harmonic maps To answer the
Simpler Question , we start by recalling some basic facts about energiesand tensions of smooth maps. Building on these facts, in the next section we shall introducea more detailed and elegant structure, that we call domestic geometry, modelled on the variations of Hodge structures (VHS) [56–61] and the more general tt ∗ geometry [62–65][25].Let ( X, G ) and ( Y, h ) be two Riemannian manifolds. As already mentioned, the energy E ( φ ) of a map φ : X → Y is the action of the Euclidean σ -model with target space Y andspace-time X , see eqn.(4.9). The map φ is harmonic iff it is a solutions of the correspondingequations of motion T [ φ ] a def = G ij D i ∂ j φ a ≡ G ij (cid:16) ∂ i ∂ j φ a − γ kij ∂ k φ a + Γ abc ∂ i φ b ∂ j φ c (cid:17) , (4.12)that is, if its tension T [ φ ] ∈ φ ∗ T Y vanishes. A finite-energy harmonic map is simply an instanton of the σ -model (4.9), i.e. a classical Euclidean-signature solution of finite action.When X has dimension 1, a harmonic map is just a geodesic on Y of constant velocity.More generally, a map φ is totally geodesic iff the full matrix D i ∂ j φ a vanishes and not justits trace as for a general harmonic map.When the source space X is Kähler, eqn.(4.12) reduces to G i ¯ k D i ∂ ¯ k φ a ≡ G i ¯ k (cid:16) δ ab ∂ i + Γ abc ∂ i φ c (cid:17) ∂ ¯ k φ b (39)and all dependence on the source-space Christoffel symbols γ kij drops out. In this situationthe map φ is said to be pluri-harmonic if the full type-(1,1) tensor D i ∂ ¯ k φ a vanishes andnot just its trace. This condition depends only on the complex structure of X , and isindependent of the specific Kähler metric G i ¯ k . This means that a pluri-harmonic map is aclassical solution of the σ -model for all choices of the source metric G i ¯ k as long as it is Kähler.In particular φ remains a solution if we repair the singularities in the Kähler metric G i ¯ k bya local surgery which keeps it Kähler as in §. 2.3.3. If, in addition, the target space Y is alsoKähler with complex coordinates z a , the pluri-harmonic condition reduces to D i ∂ ¯ k z a = 0 which is automatically satisfied if the stronger condition ∂ ¯ k z a = 0 holds, i.e. if the map z : X → Y is holomorphic .A useful fact is that the composition ι ◦ φ : X → Z of a harmonic map φ : X → Y and atotally geodesic map ι : Y → Z is also harmonic [66]. The last observation allows us to describe in a simple way the harmonic maps from a Rieman-nian manifold M to a symmetric space G/H . The map of main interest is the covering gaugecoupling e µ in eqn.(4.3), and we write explicit expressions for G/H = Sp (2 h, R ) /U ( h ) , the In eqn.(4.12) γ kij and Γ abc are the Christoffel symbols for, respectively, the metric G ij and h ab . ι : Sp (2 h, R ) /U ( h ) → Sp (2 h, R ) ,ι : E U ( h )
7→ S def = E E t ∈ Sp (2 h, R ) with S t = S , S > (4.13)where E ∈ Sp (2 h, R ) (called a vielbein [36]) is any chosen representative in Sp (2 h, R ) of thegiven point in the coset Sp (2 h, R ) /U ( h ) . S is independent of the choice of E . Then the map S ≡ ι ◦ e µ : f M → Sp (2 h, R ) , S : x
7→ S ( x ) ≡ E ( x ) E ( x ) t (4.14)is harmonic iff e µ is harmonic. The energy E ( e µ ) of the gauge coupling, written in terms of S , becomes the action of the Sp (2 h, R ) principal chiral model Z M d n x √ G G ij tr (cid:16) ( S − ∂ i S )( S − ∂ j S ) (cid:17) ≡ Z M d n x √ G G ij tr (cid:16) ( E − ∂ i E ) o ( E − ∂ j E ) o (cid:17) (4.15)The tension field of the gauge coupling e µ is more conveniently written as T ≡ d ∗ (cid:0) S − d S ) , (4.16)and e µ is harmonic iff S is a symmetric classical soliton of the chiral model defined on thespace-time M , that is, iff d ∗ (cid:0) S − d S ) = 0 . (4.17) Symmetric means that the solution satisfies the two additional conditions S t = S and S > . We call the inverse matrix S AB to S AB ≡ ( E E t ) AB the Cartan form of the gauge couplings. ι is a global isometry between the Siegel space H h and the manifold of symmetric, positive,symplectic h × h matrices, so S AB and τ ab contain exactly the same information. Withrespect to the usual gauge coupling τ ab , its Cartan version S AB has the advantage of trans-forming linearly under rotations of the duality frame. All observables, being independent ofthe frame, have nicer expressions when written in terms of S AB . Writing τ ab = X ab + iY ab , The target space Sp (2 h, R ) is endowed with a Sp (2 h, R ) × Sp (2 h, R ) -invariant indefinite pseudo-Riemannian metric, which induces on the image of ι a positive-definite Riemannian metric so that ι isan isometry onto its image (for a proper normalization of the metrics). See eqn.(4.15) and footnote 29. The statement is a bit formal, since the action integral in the lhs of eqn.(4.15) does not correspondsto a positive-definite metric on the non-compact group Sp (2 h, R ) ; all formulae are meant to be analyticcontinuations from the corresponding compact group USp (2 h ) . However the metric is positive-definite whenrestricted to the image of the Cartan map ι , i.e. on the space of symmetric, positive-definite, real, symplectic h × h matrices. As a matter of notation, when y ∈ sp (2 h, R ) , y o stands for the odd part under the Cartaninvolution θ , y o ≡ ( y − y θ ) / , i.e. for the symmetric part of the h × h matrix y . The rhs of the identity(4.15) is manifestly non-negative. S AB = Y − − Y − X − XY − Y + XY − X AB (4.18) In 4d N = 2 supergravity the couplings of the vector-multiplets are described by specialKähler geometry, which is equivalent [7, 8] to the geometry of variations of Hodge structure(VHS) with non-zero Hodge numbers h , = h , = 1 and h , = h , = m , where m isthe complex dimension of the special Kähler manifold M . VHS itself is a special case ofHiggs bundle geometry [68], which we shall refer to as generalized tt ∗ geometry. In all thesegeometries M is a Kähler manifold. In the N = 2 SUSY case the geometric swamplandproblem may be rephrased as the question of which special Kähler geometries do arise aslow-energy limits of consistent theories of quantum gravity. A necessary condition [15] isthat the corresponding tt ∗ geometry enjoys certain arithmetic properties summarized in theVHS structure theorem [58–61].In this section we introduce a geometry – dubbed domestic – modelled on tt ∗ , which doesnot require the manifold M to have a complex structure. Whenever M is Kähler, domesticgeometry automatically reduces to (generalized) tt ∗ geometry.We start with a review of tt ∗ geometry from a viewpoint which makes natural its domesticgeneralization. The review is rather detailed, because we need results and formulae whichcannot be found in the physical tt ∗ literature. Before going to that, we recall some definitions. Notation.
We write G ( R ) for a non-compact, connected, semi-simple, real Lie group withLie algebra g , K ⊂ G ( R ) for a maximal compact subgroup, and G ( Z ) ⊂ G ( R ) for a maximalarithmetic subgroup. G ( C ) and K ( C ) stand for the complexification of the Lie groups G ( R ) and K , respectively. M is an oriented Riemannian m -fold with a graded algebra P • of parallel forms. Definition 3. X a Riemannian manifold. A smooth map µ : M → X is tamed iff D ∗ ( dµ ∧ Ω) = 0 for all Ω ∈ P • . (5.1)It suffices to require (5.1) for parallel forms Ω of degree ≤ m/ since ∗ D ∗ ( dµ ∧ Ω) ≡ ( − m − D ∗ ( dµ ∧ ∗ Ω) . (5.2)26pecializing to Ω = 1 we see that tamed ⇒ harmonic , i.e. the tension field of a tamed map µ vanishes, T [ µ ] = 0 . Written in components eqn.(5.1) requires D j ∂ i µ ∈ End(
T M ) ⊗ µ ∗ T X to satisfy Ω j [ i ··· i k − D j ∂ i k ] µ = 0 for all Ω ∈ P k , k = 0 , , · · · , m. (5.3)We are interested only in tamed maps whose target space X is a locally symmetric spaceof non-compact type, X ≡ Λ \ G ( R ) /K , the prototypical example being a smooth finite coverof the Siegel variety Λ \ G ( R ) /K e . g . = Λ h \ Sp (2 h, R ) /U ( h ) . (5.4)In sections 6 and 7 below we address inter alia the following Question.
What it means for the gauge coupling µ in eqn. (4.6) to be tamed?We shall see that being tamed is a natural condition for the gauge couplings which hasimportant consequences. Note that in the physical context, P • is an algebra of invariantsfor the continuous gauge symmetry, so µ is tamed iff the algebra of invariant tensors underthe “gauge coupling endomorphisms” D i ∂ j τ ab contains all gauge invariants. Arithmetic tamed maps.
It is convenient to lift the map µ to a map between simply-connected covers e µ : f M → G ( R ) /K (5.5)which is twisted by the monodromy representation ρ : π ( M ) → Λ of the fundamental group e µ ◦ ξ = ρ ( ξ ) · e µ, ∀ ξ ∈ (deck group f M → M ), (5.6)that is, e µ is the lift which makes the following diagram to commute f M (cid:15) (cid:15) (cid:15) (cid:15) e µ / / G ( R ) /K (cid:15) (cid:15) (cid:15) (cid:15) M µ / / Λ \ G ( R ) /K (5.7)The image Γ ≡ µ ∗ ( π ( M )) ⊂ Λ is the monodromy group . The tamed map µ is arithmetic iff(i) Λ , hence Γ , is a neat sub-group of a maximal arithmetic subgroup G ( Z ) : Γ ⊂ Λ ⊂ G ( Z ) ⊂ G ( R ); (5.8)(ii) its energy is finite, E [ µ ] < ∞ .An arithmetic tamed map is thus an instanton of the Λ \ G ( R ) /K σ -model with someadditional properties. 27 .1.1 Special cases For M a Riemannian manifold of dimension m ≥ with generic holonomy algebra so ( m ) , tamed is equivalent to harmonic . There are some special cases:(a) for M = R : tamed ≡ harmonic ≡ geodesic; (b) for M strictly Kähler: tamed ≡ pluri-harmonic, i.e. D∂µ = 0 (c) for M quaternionic-Kähler (with m ≥ ): tamed ≡ totally geodesic i.e. D i ∂ j µ = 0 ,(d) for M locally isometric to a symmetric space G/K – not of the form SO ( n, /SO ( n ) or SU ( n, /U ( n ) – we typically have tamed ≡ totally geodesic, see appendix A forexamples.Next we consider the special case (b) in some detail. This leads to tt ∗ geometry which isthe model geometry which inspires all our constructions. tt ∗ geometry In this subsection M is a Kähler space with local holomorphic coordinates t i . Definition 4. A generalized tt ∗ geometry (or Higgs bundle) is a tamed map from M into alocally symmetric space Λ \ G ( R ) /K of non-compact type. We say that the generalized tt ∗ geometry is arithmetic iff the underlying tamed map µ is arithmetic.Let us see how this definition leads to the usual tt ∗ formalism [62]. We recall that themaximal compact subgroup K ⊂ G ( R ) is the fixed locus of a Cartan involution θ of thesemi-simple Lie group G ( R ) . The Lie algebra g of G ( R ) splits into θ -even and θ -odd parts, g = k ⊕ p , where k is the Lie algebra of K while p is a K -module. We fix a faithful realrepresentation σ : G ( R ) → SL ( V, R ) such that V ∼ = V ∨ , so that G ( R ) is seen as a concretegroup of n × n real unimodular matrices g ( n ≡ dim R V ). We choose conventions so that,in terms of matrices, θ is the inverse of the transpose, and we shall write g t for ( g − ) θ . TheMaurier-Cartan form g − dg is a 1-form on the manifold G ( R ) with coefficients in g ⊂ sl ( n ) which may be decomposed into θ -even and θ -odd parts.Let e µ : f M → G ( R ) /K be any smooth map twisted by the appropriate monodromy rep-resentation ρ , eqn.(5.6). We choose a lift f : f M → G ( R ) of e µ and use it to pull-back the A holomorphic map between Kähler manifolds is a special instance of pluri-harmonic map. Let us sketch a proof. We write a , b for the “flat” indices of an orthonormal frame in the tangent space T at base point in M . Let ann ( P • ) ≡ { A ab ∈ End( T ) : Ω b [ a · a k − A a k ] b = 0 for all Ω ∈ P • } be the annihilator of P • . From eqn.(5.3) we see that D a ∂ b µ ∈ ann ∩⊙ T . For a strict quaternionic-Kähkler manifold of dimension ≥ one has P • = R [Ω] with Ω the canonical 4-form. Then ann ( P • ) ∩ ⊙ T = 0 see Proposition 1.2 of [69]. f M with coefficients in g which we decompose in θ -evenand θ -odd parts as well as in ( p, q ) form type f ∗ ( g − dg ) = A + ¯ A + C + ¯ C, where A def = f ∗ ( g − dg ) even (1 , , ¯ A def = f ∗ ( g − dg ) even (0 , C def = f ∗ ( g − dg ) odd (1 , , ¯ C def = f ∗ ( g − dg ) odd (0 , (5.9)We write D ≡ ∂ + A , ¯ D ≡ ¯ ∂ + ¯ A and C ≡ C i dt i , ¯ C ≡ ¯ C ¯ k d ¯ t k with C i , ¯ C ¯ k ∈ p ⊂ sl ( n ) . Byconstruction D + ¯ D is a K -connection on f M , while ∇ ≡ D + ¯ D + C + ¯ C ≡ f ∗ ( d + g − dg ) (5.10)is a flat G ( R ) -connection on f M . Under a change of lift f → f ′ = f · u (with u : f M → K ) bothconnections change by the K -valued gauge transformation u ; hence the K -gauge invariantsare independent of the chosen lift f of e µ . If e µ is twisted by a representation ρ as in eqn.(5.6),the forms A , ¯ A , C and ¯ C are invariant under the action of the deck group, so they may beseen as forms on the Kähler base M ≡ f M /π ( M ) which are canonically defined (modulo K -gauge transformations) by the original map µ : M → Λ \ G ( R ) /K .Decomposing the identity ∇ = 0 into even/odd and form type we get the equalities D + C = ( DC ) = D ¯ D + ¯ DD + C ¯ C + ¯ CC == ( D ¯ C ) + ( ¯ DC ) = ¯ D + ¯ C = ( ¯ D ¯ C ) = 0 (5.11)which hold for all smooth maps e µ . Now suppose that the map e µ is harmonic with respectto some Kähler metric G k ¯ l on f M , that is, it satisfies the equation ¯ D k C k ≡ G ¯ lk ¯ D ¯ l C k = 0 , (5.12)which implies the equality ¯ D i ¯ D k tr( C i C k ) = tr h ( ¯ D i C k )( ¯ D k C i ) i + tr (cid:16) C k ¯ D i ¯ D k C i (cid:17) (5.13)while the identities (5.11) yield tr h ( ¯ D i C k )( ¯ D k C i ) i = tr h ( D k ¯ C i )( ¯ D k C i ) i ≡ k ¯ DC k (5.14) tr (cid:16) C k ¯ D i ¯ D k C i (cid:17) = tr (cid:16) C k [ ¯ D i , ¯ D k ] C i (cid:17) = − tr (cid:16) C k [ ¯ C i , ¯ C k ] C i (cid:17) == tr (cid:16) [ C i , C k ] [ ¯ C k , ¯ C i ] (cid:17) ≡ (cid:13)(cid:13) [ C i , C k ] (cid:13)(cid:13) ≡ (cid:13)(cid:13) C (cid:13)(cid:13) (5.15)This shows the Lemma 1 (Sampson’s Bochner-formula [70,71]) . Let M be Kähler and µ : M → Λ \ G ( R ) /K e a harmonic map. Then ¯ D i ¯ D k tr (cid:16) C i C k (cid:17) = (cid:13)(cid:13) ¯ DC (cid:13)(cid:13) + (cid:13)(cid:13) C (cid:13)(cid:13) . (5.16)The rhs of (5.16) is the sum of two non-negative terms: hence the integral over M ofthe lhs vanishes if and only if the two terms on the right are both identically zero; thisimplies (1) ¯ DC = 0 which (by definition) says that µ is pluri-harmonic, and (2) the algebragenerated by the coefficient matrices C i is Abelian. In facts from eqn.(5.16) we see that (2) is an automatic consequence of (1) . The lhs of (5.16) is a total derivative, so its integralover M is a surface term: in particular, when M is compact a harmonic map is automaticallypluri-harmonic [71]. More generally: A harmonic map µ : M → Λ \ G ( R ) /K is pluri-harmonic ( ≡ tamed) if and only if Z ∂M ∗ tr (cid:0) C k ¯ D k C (cid:1) ≡ Z ∂M ∗ tr (cid:0) C k D ¯ C k (cid:1) = 0 , (5.17)a condition which depends only on the asymptotic behaviour of µ at infinity in M . The tt ∗ equations. In a generalized tt ∗ geometry, µ is pluri-harmonic, and hence ¯ DC = 0 . In view of (5.11), (5.16) this implies the tt ∗ PDEs [62] D = C = ( DC ) = D ¯ D + ¯ DD + C ¯ C + ¯ CC == ( D ¯ C ) = ( ¯ DC ) = ¯ D = ¯ C = ( ¯ D ¯ C ) = 0 . (5.18)These equations may be summarized in the following statement: Proposition 2.
For M strictly Kähler, µ : M → Λ \ G ( R ) /K is tamed if and only if (cid:0) ∇ ( ζ ) (cid:1) ≡ (cid:0) D + ¯ D + ζ − C + ζ ¯ C (cid:1) = 0 for all ζ ∈ P . (5.19)We write R for a (commutative) enveloping algebra U a , where a ⊂ gl ( V, C ) is a maximalcommutative C -subalgebra containing the matrices C i . R is known as a chiral ring [72]. Definition 5. A tt ∗ geometry is strict if it has a spectral flow i.e. we can choose R so that V ∼ = R as R -modules [72]. Since V ∼ = V ∨ , this implies R ∼ = R ∨ as R -modules ⇒ in a strict tt ∗ geometry the chiral ring is a (commutative) Frobenius algebra . The Frobenius pairingis known as the topological metric η : R ⊗ → C [62, 65].The vacuum geometry of a 2d (2,2) QFT is described by a strict arithmetic tt ∗ geometry[62, 64]. The vacuum bundle V → M over the F -term coupling space M is holomorphicwith structure group K ⊂ U (dim C V ) , and D + ¯ D is an unitary connection on the Hermitian For an alternative proof that (1) ⇒ (2) see [65] or the appendix of [15]. See theorem 1.3 in [73]. We recall that the vacuum bundle V → M is the holomorphic sub-bundle of the trivial Hilbert space V which coincides with the Berry connection in the quantum-mechanicalsense [62]. A choice of trivialization identifies the fibers of V with the complexification V C of the representation space V of the real Lie group G ( R ) . Therefore the fibers of V carry areal structure to be identified with the physical PCT operation [62].The chiral ring R is more invariantly seen as the fiber R t of a sub-bundle R ֒ → End( V ) consisting of commutative endomorphisms. For a generalized tt ∗ the C i ’s yield the sub-bundlechain TM ֒ → R ֒ → End( V ) , (5.20)while for a strict tt ∗ geometry TM ֒ → V ∼ = R strict tt ∗ geometry (5.21) tt ∗ metric. To simplify the notation, we write g for f ∗ g . Since g ∈ G ( R ) , the connection A + ¯ A ≡ ( g − d g ) even is the K -connection written in an unitary trivialization of the Hermitianbundle V ; more precisely, the trivialization is orthogonal because of the reality structure on V [62]. Since ¯ D = 0 the K -connection is also holomorphic, and it is convenient to workin a holomorphic trivialization where ¯ A ≡ . There is a map U : M → K ( C ) such that U ¯ DU − = ¯ ∂ . The transformation between the orthogonal and the holomorphic trivializationsis given by the complex K ( C ) -gauge transformation g → g U − . The connection D + ¯ D isboth holomorphic and metric, hence is the unique Chern connection: one has A ≡ h∂h − = ( g U − ) − d ( g U − ) (cid:12)(cid:12)(cid:12) θ even ¯ A = 0 , (5.22)where h is the fiber metric on V in the chosen holomorphic trivialization (such that thetopological metric η ≡ ). Comparing the two complex gauges h = U ¯ U − ≡ U U † (5.23)The Hermitian metric h is called the tt ∗ metric on V [62]. It satisfies the reality condition h ¯ h = 1 [62]. For a strict tt ∗ geometry, in view of eqn.(5.21), the tt ∗ metric h is identified with aHermitian metric on the fibers of R which induces the sub-bundle metric on T M , i.e. aHermitian metric on M . It is natural to multiply the tt ∗ metric on R by a normalizationfactor so that the section ∈ R has norm 1. In 2d (2,2) QFT the normalized tt ∗ metric onthe coupling space M plays the role of the Zamolodchikov metric [62]. bundle V × H → M whose fiber V t ⊂ H at t ∈ M is given by the subspace of zero-energy states forthe Hamiltonian H t with F -term couplings t ; V is equipped with the sub-bundle Hermitian metric inducedby the Hilbert-space Hermitian product in H . The Hermitian bundle V and its metric are insensitive todeformations of D -terms couplings [62]. A real structure on a C -space is an anti-linear map which squares to the identity. In our conventions the topological metric η = 1 as a consequence of our choice g θ = ( g − ) t . odge metric. In generalized tt ∗ there is a second, better behaved metric on M ,namely the sub-bundle metric on T M ֒ → End( V ) induced by the tt ∗ metric on End( V ) ∼ = V ⊗ V ∨ . This metric exists independently of the spectral flow and is always Kähler [25]. ItsKähler form is i K i ¯ j dt i ∧ d ¯ t ¯ j ≡ i tr( C ∧ ¯ C ) . (5.24)Let G i ¯ j be any Kähler metric on M . The (1,1) tensor on MT i ¯ j = K i ¯ j − G i ¯ j G k ¯ l K k ¯ l . (5.25)is automatically conserved ∇ i T i ¯ j = G i ¯ h ∇ ¯ h K i ¯ j − G k ¯ l ∇ ¯ j K k ¯ l = G i ¯ h (cid:16) ∇ ¯ h K i ¯ j − ∇ ¯ j K i ¯ h (cid:17) ≡ . (5.26)Eqn.(5.26) has a simple explanation. The map µ , being pluri-harmonic, is harmonic – hence aclassical solution to the Λ \ G ( R ) /K σ -model – for all choices of the “spacetime” Kähler metric G i ¯ j . T i ¯ j is just the energy-momentum tensor evaluated on this on-shell field configurationand hence is conserved. The tt ∗ geometry of a 2d (2,2) QFT computes important physical quantities. The basic onesare the Hori-Iqbal-Vafa (HIS) half-BPS brane amplitudes Ψ( ζ ) a [74] which are sections ofthe bundle V ∨ ∼ = V over f M Ψ( ζ ) a [ φ ] = (cid:10) φ (cid:12)(cid:12) a brane (cid:11) , φ ∈ V . (5.27)The -BPS brane amplitudes depend on a twistor parameter ζ ∈ P which specifies thetwo linear combinations of the supercharges which leave them invariant [74]. The braneamplitudes are solutions to the linear PDEs (cid:0) D + ¯ D + ζ − C + ζ ¯ C (cid:1) Ψ( ζ ) a = 0 (5.28)and depend on a choice of K ( C ) -gauge (i.e. of trivialization of V → f M ); under a change ofgauge Ψ( ζ ) a → U Ψ( ζ ) a , U : f M → K ( C ) . (5.29)The tt ∗ PDEs (5.19) are the integrability conditions of the brane equation (5.28). Eqn.(5.27) is written in the conventions common in the tt ∗ literature [62], in particular the bracket h· · · | · · · i is linear in its first argument rather than anti-linear as in usual conventions. The index a is aquantum number labelling the different fundamental branes. undamental solutions. A fundamental solution to eqn.(5.28) is a map Φ( ζ ) : f M → σ ( G ( C )) ⊂ SL ( n, C ) (5.30)such that the columns Φ( ζ ; t, ¯ t ) a ( a = 1 , . . . , n ) of the matrix Φ( ζ ) ≡ Φ( ζ ; t, ¯ t ) yield a basisof linearly independent solutions of (5.28). In a given K ( C ) -gauge, the fundamental solutionis unique up to multiplication on the right by a matrix L ( ζ ) ∈ σ ( G ( C )) which depends onlyon the twistor parameter ζ Φ( ζ ; t, ¯ t ) → Φ( ζ ; t, ¯ t ) L ( ζ ) . (5.31)In an orthogonal trivialization of V , the matrix L ( ζ ) may be chosen so that Φ( ζ ) satisfiesthe symmetry and reality conditions Φ( − ζ ) = Φ( ζ ) θ ≡ (Φ( ζ ) t ) − , Φ( ζ ) = Φ(1 / ¯ ζ ) , (5.32)and Φ( e iθ ) ∈ G ( R ) . In the SUSY literature it is more common to use the holomorphic gauge(cfr. eqn.(5.22)) Φ( ζ ) holo = U Φ( ζ ) , Φ( ζ ) holo = h Φ(1 / ¯ ζ ) holo (5.33)which yield the usual formula for the tt ∗ metric h as a bilinear in the solution of the linearproblem (5.28) [62, 64, 65] h = Φ( ζ ) holo Φ( − / ¯ ζ ) t holo . (5.34) Integral structure.
The HIV fundamental brane amplitudes correspond to a specialbasis of solutions to (5.28), so Ψ( ζ ) a = (cid:0) Φ( ζ ) L ( ζ ) (cid:1) a , (5.35)for some L ( ζ ) . To get the appropriate L ( ζ ) note that the space of physical -BPS branes hasan integral structure: for (2,2) σ -models it arises because the physical branes have supporton a sub-manifold of the target space, and hence represent integral elements of the relevanthomology group. More generally, the integral structure arises because of Dirac quantizationof the brane charge. Then the representation space V of G ( R ) has the form V = V Z ⊗ Z R , (5.36)with V Z ⊂ V a lattice preserved by the arithmetic subgroup G ( Z ) ⊂ G ( R ) . The brane map Ψ( ζ ) : f M → σ ( G ( R )) is twisted by a monodromy representation (cfr. eqn.(5.6)) ξ ∗ Ψ( ζ ) = Ψ( ζ ) · ρ ζ ( ξ ) − , ∀ ξ ∈ (deck group f M → M ) (5.37)which should respect the arithmetic structure, so ρ ζ ( ξ ) ∈ G ( Z ) and hence ζ -independent.Setting ζ = 1 and comparing with eqn.(5.6), we see that the basic brane amplitudes aregiven by an integral basis of solutions to (5.28) on which the monodromy action is given33y multiplication on the right by ρ ( ξ ) − , where ρ is the monodromy representation of the tt ∗ geometry. Then the tt ∗ metric h is given by a Hermitian form in the brane amplitudes(written in a holomorphic K ( C ) -gauge) h = Ψ( ζ ) holo I Ψ( − / ¯ ζ ) † holo , where I ≡ L ( ζ ) − ( L ( − / ¯ ζ ) − ) † , (5.38)with I the intersection form between dual bases of BPS. Brane amplitudes in general tt ∗ geometry. The equations of the HIV brane am-plitudes, eqns.(5.28),(5.32), as well as their integral structure, continue to make sense for allgeneralized arithmetic tt ∗ geometry, whether it has a spectral flow or not.Given a fundamental brane amplitude Ψ( ζ ) we may construct other ones by changingthe representation σ of G ( R ) . Usually the physical branes are given by the fundamentalrepresentation σ fund ; all other representations σ are sub-representations of some ( σ fund ) ⊗ s defined by an invariant tensor t ∈ σ ⊗ ( σ ∨ fund ) ⊗ s ; so the branes amplitudes associated to anarbitrary representation σ may be interpreted as physical multi- brane amplitudes Ψ ( ζ ) AI ≡ Ψ ( ζ ) A ··· A s I = t i ··· i s I Ψ( ζ ) A i · · · Ψ( ζ ) A s i s t i ··· is I G ( R ) -invarianttensor (5.39) tt ∗ geometries & VHS When the 2d (2,2) QFT is superconformal, the tt ∗ geometry has further structure inducedby the superconformal U (1) R charge. This additional structure characterizes the variationsof Hodge structure inside the larger class of tt ∗ geometries. Definition 6.
An arithmetic generalized tt ∗ geometry is graded iff there is a grading element Q ∈ i g ≡ i Lie ( G ( R )) (the “ U (1) R charge” ) such that [ Q, A ] = 0 , [ Q, C ] = − C, [ Q, ¯ C ] = ¯ C. (5.40)A graded tt ∗ geometry is a variation of Hodge structure (VHS) if, in addition, e πiQ (cid:12)(cid:12) V = ( − ˆ c , e iπQ ∈ G ( Z ) , Ad ( e πiQ )( g ) = g θ for g ∈ G ( R ) (5.41)where ˆ c def = 2 max (cid:8) eigenvalues of Q in V (cid:9) ∈ N (5.42)When (5.41),(5.42) hold, the pair ( V, Q ) is called a Hodge representation of the Lie group G ( R ) of weight ˆ c . Hodge representations are classified in [61].We identify elements of g and respectively G ( R ) with the matrices which represent themin the real representation space V ; one has g θ ≡ ( g t ) − for g ∈ G ( R ) . Then eqn.(5.41) In the most interesting case, i.e. graded tt ∗ geometries, we will present the explicit form of I (seeeqn.(5.58)) checking that I is an element of G ( Z ) (as physically expected), hence ζ -independent. Ω ≡ e iπQ is a matrix with integral entries which satisfies g t Ω g = Ω for all g ∈ G ( R ) , (5.43)while Ω is symmetric (resp. anti-symmetric) for ˆ c even (resp. odd). Hence V is an orthogonal(resp. symplectic) real representation of G ( R ) . The non-degenerate, integral, bilinear form Ω( v, w ) ≡ v t Ω w, with Ω( v, w ) = ( − ˆ c Ω( w, v ) , (5.44)is called the polarization of the VHS. For a fixed g ∈ G ( R ) , we define the Weil operator C g = g − e iπQ g ∈ G ( R ) . (5.45)Then Ω( C g v, ¯ w ) is a positive-definite Hermitian form on V C ≡ V ⊗ R C ; indeed Ω( C g v, ¯ w ) = ( − ˆ c w † Ω C g v = ( − ˆ c w † Ω g − Ω gv = ( − ˆ c w † g t Ω gv = w † g † g v. (5.46)We write h for the Lie subalgebra of g which commute with Q , and H ⊂ G ( R ) for thecorresponding Lie subgroup. By eqns.(5.41),(5.46) the subgroup H is compact, and hencecontained in a maximal compact subgroup K . G ( R ) /H is then a reductive homogeneousspace with a canonical projection into the symmetric space G ( R ) /K [75]. Given any H -module W we construct canonically a homogeneous bundle O ( W ) → G ( R ) /H , with typicalfiber W , endowed with a unique canonical connection and metric (up to overall normaliza-tion) [75] O ( W ) = G ( R ) × W .(cid:8) ( g, w ) ∼ ( gh, h − · w ) for h ∈ H (cid:9) (5.47) Lemma 2. G ( R ) /H is a homogenous complex manifold, and the homogeneous vector bundle O ( W ) → G ( R ) /H is holomorphic for all H -module W .Proof. Consider the grading of the complexified Lie algebra g ⊗ C = ( h ⊗ C ) ⊕ M r =0 g − r,r ! , where g r, − r def = n X ∈ g ⊗ C : [ Q, X ] = rX o , (5.48) (cid:2) h , g − r,r (cid:3) ⊆ g − r,r , (cid:2) g − r,r , g − s,s (cid:3) ⊆ g − r − s,r + s . (5.49)By eqn.(5.49) each summand g − r,r is a H -module, so it defines a homogeneous vector bundle O ( g − r,r ) . The complexified tangent bundle of the manifold G ( R ) /H is T G ( R ) /H ⊗ C = M r> O ( g − r,r ) ! ⊕ M r< O ( g − r,r ) ! . (5.50) In a VHS the grading is integral, r ∈ Z . In this case the first equation (5.48) is an (adjoint) Hodgedecomposition of g [58–61].
35e define an almost complex structure on G ( R ) /H by declaring the first summand to bethe complex distribution of (1,0) vectors. The almost complex structure is integrable sincethis complex distribution is involutive by the second equation (5.49). For the holomorphicstructure of O ( W ) see e.g. [75][58]. Infinitesimal period relations.
The sub-bundle Θ ≡ O ( g − , ) of the holomorphictangent bundle is called the Griffiths holomorphic horizontal bundle [56, 58]. Let M be acomplex manifold and T M its holomorphic tangent bundle. We say that a map p : M → G ( R ) /H (5.51)satisfies the Griffiths infinitesimal period relations (IPR) if [56, 58] p ∗ ( T M ) ⊆ Θ . (5.52)In particular, such a map p is holomorphic.Let V C = ⊕ i V q i be the decomposition of the G ( R ) -module V C = V ⊗ C into eigenspacesof Q of eigenvalue q i . Since [ Q, H ] = 0 , each V q i is a H -module and yields a homogeneousbundle O ( V q i ) → G/H . Lemma 3.
Let ˜ µ : f M → G ( R ) /K be a lift of the tamed map µ of a graded tt ∗ geometry.Then we have the factorization f M ˜ µ ( ( ˜ p / / G ( R ) /H / / / / G ( R ) /K (5.53) ˜ p , which is called the (Griffiths) period map of the graded tt ∗ geometry, satisfies the IPR (5.52) . The bundles V q i ≡ ˜ p ∗ O ( V q i ) → f M are called Hodge bundles .Proof.
In view of eqn.(5.48), eqns.(5.40) are equivalent to the IPR (5.52).Usually one identifies the graded tt ∗ geometry with the period map p : M → Γ \ G ( R ) /H (5.54)which lifts to e p on the universal cover f M . By definition, a period map satisfies the IPR. Applications to 2d QFT.
The vacuum geometry of (2,2) 2d SCFT over the chiralconformal manifold M is a graded strict tt ∗ geometry. If, in addition, the U (1) R charges ofthe chiral primaries [72] are integral, the SCFT vacuum geometry is a VHS of CY type, For a VHS being of CY type is equivalent to being strict as a tt ∗ geometry, i.e. that there is a spectralflow isomorphism. h ˆ c, = h , ˆ c = 1 . The vacuum bundle V → M then has an orthogonaldecomposition (preserved by parallel transport with the Berry connection) V = ˆ c/ M q = − ˆ c/ V q , where Q (cid:12)(cid:12) V q = q ∈ N − ˆ c . (5.55)In this set-up ˆ c is one-third the Virasoro central charge [62]. The spectral-flow isomorphismis graded by the U (1) R -charge, R q ∼ = V q − ˆ c/ , with R = ˆ c M q =0 R q , (5.56)and implies the “local Torelli” property T M ∼ = V − ˆ c/ . Comparing with eqn.(5.28) we get an explicit formula for the twistorial multi-brane ampli-tudes of a graded tt ∗ geometry: in the orthogonal trivialization they are Z -linear combina-tions of the columns of the matrix Ψ ( ζ ) AI = (cid:0) ζ − Q g − (cid:1) AI monodromy groupacts on the right (5.57)with g IA the matrix elements of g ∈ G ( R ) in the Hodge representation V . Comparing witheqns.(5.30)-(5.38), we see that L ( ζ ) = ζ − Q and I ≡ L ( ζ ) − (cid:0) L ( − / ¯ ζ ) − (cid:1) † = e − πiQ VHS ≡ ( − ˆ c Ω ∈ G ( Z ) , (5.58)where the last equality holds in the VHS case. We see that I is the natural intersection form. We can use the brane amplitudes to compute several physical quantities, that is, K -gaugeinvariant expressions which are independent of the choice of ζ ∈ P . These physical quan-tities are well-defined for all generalized tt ∗ geometries, graded or non-graded, strict or not.Examples are • Kähler form of Hodge metric: i K i ¯ j dt i ∧ d ¯ t j = i tr (cid:0) C ∧ ¯ C ) (5.59) • Cartan gauge coupling: S AB = (cid:0) Ψ( ζ ) t Ψ( ζ ) (cid:1) AB (5.60) • Hodge bilinears: S AB = (cid:0) Ψ ( ζ ) t Ψ ( ζ ) (cid:1) AB (5.61)We think of the Hodge bilinears as “higher versions” of the gauge coupling.37 .2.5 tt ∗ entropy functions & Mumford-Tate groups Suppose our tamed (covering) map ˜ µ : f M → Sp (2 h, Z ) \ Sp (2 h, R ) /U ( h ) (5.62)is actually the gauge coupling of a 4d effective gravity theory. The classical entropy function(in Sen’s sense [76–78]) of an extremal black-hole with electro-magnetic charge q ∈ V Z ∼ = Z h ,(well-defined on the universal cover f M ) is E q = π q A S AB q B ≡ π k q k h , (5.63)that is, (up to a factor π ) the Hodge norm-squared k q k h of the charge vector.For a general G -representation σ , contained in ⊗ s V , we define the “generalized entropyfunctions” of the generalized tt ∗ geometry to be S q : f M → R + , x q A S ( x ) AB q B , q ∈ ⊗ s V Z , (5.64) S q reduces to E q /π when σ is the fundamental representation (i.e. s = 1 ). Proposition 3.
In (generalized) tt ∗ geometry the generalized entropy functions S q ( t ) are sub-pluriharmonic (in particular sub-harmonic ) for all q ∈ ⊗ s V Z , i.e. the matrix ∂ t i ∂ ¯ t j S q issemi-definite positive.Proof. Set L = Ψ ( − q , so that DL = CL , ¯ DL = ¯ CL , L ∗ = L , and S q = L t L ≡ L † L . Then ¯ ∂S q = 2 L t ¯ DL = 2 L t ¯ CL (5.65)and ∂ ¯ ∂S q = 2( DL ) t ¯ DL + 2 L t D ( ¯ CL ) = 2( DL ) t ¯ DL − L t ¯ CCL == 2 L † ( C i C † j + C † j C i ) L dt i ∧ d ¯ t j (5.66)The actual value of the classical entropy for an extremal black hole of charge q is givenby the value π S q at a critical point ∂S q = 0 (if it exists!!). As an example, we consider thespecial case of a VHS of CY type with ˆ c = 3 , which describes the gauge coupling e µ of aneffective 4d theory with N = 2 [7, 8]. A critical point t ∈ f M of S q ( t ) corresponds to a L ≡ Ψ( − t ) q which is an eigenvector of Q with eigenvalue / . This observation is calledthe attractor mechanism [80–85]. Proposition 3 implies that the critical point is actuallya minimum for the entropy function. The attractor mechanism illustrates the point thatbeing sub-harmonic is a very natural property for a physical entropy function, being strictlyrelated to the convexity of thermodynamical potentials. By “classical” we mean the classical entropy function as computed by the truncation of the effectiveLagrangian to two-derivatives; the classical entropy becomes exact asymptotically for large charged | q | → ∞ . he Mumford-Tate group. If the multi-charge q is Γ -invariant, the generalized en-tropy function is well-defined on M not just on f M . Suppose that we have an arithmeticgraded tt ∗ geometry (say, a VHS) whose base M is quasi-projective, i.e. M = M \ D ∞ for M a smooth (compact) projective variety and D ∞ a simple normal crossing divisor; then M is Liouvillic for the sub-pluriharmonic functions. Under the above assumptions, one checks(by a careful asymptotic analysis [79]) that the generalized entropy of a Γ -invariant charge q is bounded along the divisors at infinity in the projective closure M of M . Then, beingsub-pluriharmonic, S q ( t ) must be a constant in M . We conclude that for a Γ -invariant multi-charge q ∈ ⊗ s V Z the multi-brane amplitude Ψ ( ζ ) · q is H -gauge equivalent to a constant;this can be expressed as Proposition 4.
Let Hg be the ring of all Γ -invariant multi-charges in ⊕ s ( ⊗ s V Z ) and let M ( R ) ⊂ G ( R ) be the subgroup which fixes all elements q ∈ Hg and H M ≡ H ∩ M a maximalcompact subgroup. Then the map p in eqn. (5.54) factorizes as M µ p ) ) m / / Γ \ M ( R ) /H M / / / / Λ \ G ( R ) /H / / / / Λ \ G ( R ) /K (5.67)Eqn.(5.67) is almost the structure theorem of VHS [58–61], except that the theorem yieldsmore details on the group M ( R ) ; we shall discuss these results in the more general domesticcontext below. In the VHS literature the elements of the Q -algebra Hg ⊗ Q are called Hodgetensors and the Q -algebraic group M ( Q ) is called the Mumford-Tate group [58–61]. (Arithmetic) domestic geometry is defined by the very same
Definition 4 of tt ∗ geometry,except that now we forget that the source Riemannian space M was assumed to be Kähler.An arithmetic domestic geometry on the Riemannian manifold M is specified by a tamed map M → Λ \ G ( R ) /K of finite energy. The geometric structures implied by domestic geometrydepend crucially on the algebra P • of parallel forms on M (which we assume to be irreduciblewith no loss). In particular, a domestic geometry on M is a generalized tt ∗ geometry if andonly if P • contains a subalgebra R [ ω ] /ω m +1 with ω a parallel 2-form.Domestic geometry is more general than tt ∗ geometry as the following example shows. The (universal covering of the) conformal manifold f M of a 2d (2,2) SCFT splits in a productof spaces associated to the two non-conjugate chiral rings f M = f M chiral × f M twisted chiral , (5.68)39nd the Berry geometry on each irreducible factor space is a domestic geometry of the u (1) graded tt ∗ kind (§. 5.2.2).The Berry geometry on the moduli of a 2d (4,4) SCFT is still a domestic geometry, but not a tt ∗ geometry. Indeed the domestic geometry is sp (1) ⊕ sp (1) graded rather than u (1) -graded. The moduli space M is quaternionic-Kähler, and hence the underlying tamedmap µ is totally geodesic (cfr. §. 5.1.1): in (real) local coordinates x i ∇ i C j = 0 , where C j dx j = f ∗ ( g − dg ) odd . (5.69)Eqn.(5.69) implies that f M is a non-compact symmetric space with holonomy algebra of theform sp (1) ⊕ sp (1) ⊕ j ⊂ so (4 k ) , and hence f M = SO (4 , k ) / [ SO (4) × SO ( k )] . (5.70)For a (much longer) proof not using domestic geometry, see [86]. Grading.
As the 2d example illustrates, in physical applications the domestic geometryis graded by the effective R-symmetry Lie algebra σ ( r ) , cfr. eqn.(3.5), that is, we have adecomposition g ⊗ C = M α ∈ Irr g α , g adj = σ ( r ) , g triv = j (5.71)where the sum is over the irreducible representations of σ ( r ) . Eqn.(5.71) generalizes theadjoint Hodge decomposition (5.48) of VHS theory to a possibly non-Abelian σ ( r ) . We may repeat much of the tt ∗ story in this more general setting. However now, in general,we cannot distinguish differential forms by type, so we decompose f ∗ ( g − dg ) in just twopieces A = ( f ∗ g − dg ) even , Φ = ( f ∗ g − dg ) odd , (5.72)and we do not have a twistorial P -family of flat connections but only two of them ∇ ( ± ) = d + A ± Φ . (5.73)Consequently, we have only two “HIV brane amplitudes” Ψ ± , which satisfy the equations ∇ ( ± ) Ψ ± = 0 . (5.74) Of course, a (4,4) SCFT is in particular a (2,2) SCFT; however the space M of marginal deforma-tions which preserve (4,4) SUSY is a non-complex submanifold of the complex manifold M of marginaldeformations which preserve only (2,2) SUSY.
40s a consequence of Dirac quantization of charge, in our applications the group G ( R ) pre-serves some bilinear pairing V Z ⊗ V Z → Z given by an integral matrix Ω ∈ G ( Z ) (symmetricor antisymmetric) with ΩΩ t = 1 . In this case Ψ − = Ψ θ + Ω , Ψ ∗± = Ψ ± . (5.75)Again we may consider higher domestic brane amplitudes Ψ ± associated to higher represen-tations of the Lie group G ( R ) which may be written as multi-linear products of basic braneamplitudes.The physical quantities of tt ∗ are still well-defined (we assume (5.75)): • the Riemannian metric: ds = K ij dx i dx j ≡ tr(Φ i Φ j ) dx i dx j (5.76) • Hodge bilinears: S AB = (cid:0) Ψ t ± Ψ ± (cid:1) AB (5.77) • generalized entropy functions: S q = q A S AB q B (5.78)and the symmetric tensor T ij = K ij − G ij G kl K kl (5.79)is still conserved ( G ij is the metric on M ). When the tamed map e µ : f M → Sp (2 h, R ) /U ( h ) which defines the domestic geometry is thegauge coupling of some effective theory of gravity, Sen’s classical entropy function for anextremal black hole of charge q (if it exists!) is given by πS q , where S q is the domesticentropy function for the fundamental representation. The same argument as in §.5.2.4 showsthat the generalized entropy functions are sub-tamed , in particular,• sub-harmonic for M of generic holonomy: ∆ S q ≥ • sub-pluriharmonic for M Kähler: ∂ i ¯ ∂ ¯ j S q ≥ • convex for M quaternionic-Kähler and most symmetric spaces: ∇ i ∂ j S q ≥ . Structure of µ . If M is Liouvillic for the sub-tamed functions and the generalizedentropy function S q of a Γ -invariant multi-charge q is bounded (a property natural for µ offinite energy), one concludes that the structure theorem (5.67) holds in the domestic case aswell: M µ ) ) m / / Γ \ M ( R ) /K M / / / / Λ \ G ( R ) /K , K M = M ( R ) ∩ K. (5.80)We shall show in §. 7.3.2 below that this conclusion is valid in the situations of physicalinterest using a different and more direct approach. Here we give a rough sketch of an41rgument along the lines of §. 5.2.5. We have already seen in §. 3.1.2 that all OV manifolds M are Liouvillic for the sub-harmonic functions hence a fortiori for the sub-tamed ones. Itremains to show that S q is bounded for q Γ -invariant. We have to check the behaviour of S q at infinity in f M . Under our assumptions in §. 2.4.2, a point at infinity x ∞ is fixed by someinfinite subgroup P ⊂ Γ ⊂ Sp (2 h, Z ) and hence is mapped by a continuous extension of e µ in a point at infinity y ∞ ∈ Sp (2 h, R ) /U ( h ) in the compactification of Sp (2 h, R ) /U ( h ) [45]which is also fixed by P , i.e. such that P y ∞ = y ∞ . Since the entropy function S q = e µ ∗ k q k h (5.81)is the pull-back of the Hodge norm-square of q computed on Sp (2 h, R ) /U ( h ) , it suffices tocheck that, whenever q is fixed by P , k q k h is bounded in a neighborhood of the point atinfinity y ∞ fixed by P . The last statement is a purely group theoretical fact.The property of being sub-tamed is rather natural for a physical entropy function, asillustrated by the attractor mechanism in the N = 2 case. Consider a 4d supergravity with any number N of light gravitini, matter content, and cou-plings. Its scalars’ universal covering manifold f M = f M (1) × f M (2) × · · · × f M ( s ) (6.1)is a product of non-compact spaces in one-to-one correspondence with the types of super-multiplets present in the model. For instance, in 4d N = 2 SUGRA the scalars’ manifoldis the space of hypermultiplet scalars times the space of vector-multiplet scalars. The map e µ , which describes the coupling of the scalars to the vectors, splits into a set of maps { e µ ( i ) } which describe the coupling of vectors to the scalars in supermultiplets of the i -th type.The geometry of each factor space f M ( i ) depends on the corresponding supermultiplet. Prima facie these geometries look quite different one from the other: in some cases f M ( i ) isKähler (possibly with additional structures), in other situations f M ( i ) does not even admita natural complex structure. A general feature is that each space f M ( i ) carries a non-trivialalgebra P • ( i ) of parallel forms determined by the representation of R-symmetry on the scalarsas described in §. 3.1.1.Domestic geometry unifies all these seemingly different geometries in a single one. Thetraditional supergravity theory, as well as the geometric swampland conjectures, may besummarized in the following statement: As always, we work modulo finite groups and finite covers. act 1. All 4d SUGRA models are defined by a domestic geometry: the Γ -twisted (covering) gauge coupling map ˜ µ : f M → Sp (2 h, R ) /U ( h ) is always tamed. More precisely, for each factor manifold f M ( i ) in eqn. (6.1) there is a Γ ( i ) -twisted, P • ( i ) -tamed map e µ ( i ) : f M ( i ) → G ( i ) /K ( i ) , Γ ( i ) ⊂ G ( i ) (6.2) with target a symmetric space G ( i ) /K ( i ) and ˜ µ factorizes as in the commutative diagram f M ˜ µ + + f M (1) × · · · × f M ( s ) ˜ µ (1) ×···× ˜ µ ( s ) / / G (1) /K (1) × · · · × G ( s ) /K ( s ) (cid:31) (cid:127) ι / / Sp (2 h, R ) /U ( h ) where ι is the totally geodesic embedding induced by a subgroup embedding G (1) × · · · × G ( s ) ι ֒ → Sp (2 h, R ) (6.3) and Γ ∼ Q i Γ ( i ) modulo finite groups; most couplings in the Lagrangian L of an (ungauged) 4d SUGRA are given by universalexpressions in terms of brane amplitudes of the domestic geometries e µ ( i ) . For N ≥ and the vector sector of N = 2 ALL couplings are so expressed; the domestic geometry rigidity theorems/structure theorems reproduce and extend theusual SUSY non-renormalization theorems; if the domestic geometry defined by a map e µ ( i ) in (6.2) is non-arithmetic the SUGRAfalls in the swampland.The statement holds, with the appropriate modifications, in other spacetime dimensions. As we have seen in §. 5.1.1, there are several special cases of domestic geometry dependingon the particular algebra P • ; when the domestic geometry is a VHS, we may talk of its Hodgenumbers and its Mumford-Tate (MT) group. In table 1 we list the domestic geometries whicharise in 4d SUGRA with the special properties of each one. The above
Fact allows for a more physical interpretation of the Ooguri-Vafa geometricswampland conditions. For
N ≥ the couplings are expressed in terms of brane amplitudes,and the swampland conjectures just say that these brane amplitudes have the properties weexpect on physical grounds for actual extended objects.The brane viewpoint makes the swampland story a lot less mysterious. Tautologically,a theory is consistent iff it leads to physically sound predictions for all observables. In, say, N = 2 supergravity the brane amplitudes are important physical quantities: if they don’t43 & supermultiplet namespace C ? • kind of domestic geometry • algebra P • if special • for VHS h p,q = 0 • MT group if special notes N = 1 chiral X graded ˆ c = 1 non -strict tt ∗ { h , }N = 2 vector SKG X graded ˆ c = 3 strict tt ∗ { h , = 1 , h , }N = 2 hyper QK NO domestic tamed by P • = C [1 , Ω] ( Ω canonical 4-form) N = 3 vector X graded ˆ c = 3 non-strict tt ∗ tamedby P • = [ ∧ ∗ ( C ⊗ C k )] u (3) ⊕ su ( k ) { h , = 3 , h , = k } Γ Q ( R ) = SU (3 , k ) LU, SH N = 4 gravity H X graded ˆ c = 1 strict tt ∗ { h , = 1 } LU, SH N = 4 vector NO domestic tamed by P • = [ ∧ ∗ ( R ⊗ R k ) so (6) ⊕ so ( k ) LU N = 5 gravity X graded ˆ c = 1 strict tt ∗ tamedby P • = [ ∧ ∗ ( ∧ C ) ∨ ] u (5) { h , = 10 } Γ Q ( R ) = SU (5 , LU, SH N = 6 gravity ] SKG X graded ˆ c = 3 strict tt ∗ tamed by P • = [ ∧ ∗ ( ∧ C )] u (6) { h , = 15 , h , = 1 } Γ Q ( R ) = SO (6 , H ) LU, SH N = 8 gravity NO domestic tamed by P • = [ ∧ ∗ ( ∧ C )] su (8) LU Table 1:
4d SUGRAs as domestic geometries. In second column SKG stands for special Kählergeometry, QK for quaternionic-Kähler, ] SKG for twisted special Kähler (with e iπQ → − e iπQ ), and H for the upper half-plane. In third column X means that the manifold has a natural complexstructure. Forth column specifies the class of tamed geometries and its taming algebra P • . In fifthcolumn we specify the data for domestic geometries which are in fact VHS. In the last column LUmeans that the geometry is locally unique (so f M ( i ) is a symmetric space) and SH that, under theassumption that the domestic geometry is arithmetic M ( i ) ≡ Γ ( i ) \ ˜ M ( i ) is a Shimura variety . behave in the correct way, the theory is doomed to be inconsistent. This is what (typically)happens when the swampland conjectures of [2] are not obeyed. From table 1 we see that in all 4d SUGRAs the gauge coupling maps µ ( i ) : Γ ( i ) \ f M ( i ) → Γ ( i ) \ G ( i ) /K ( i ) (6.4)are harmonic, hence solutions to the σ -model with target space Γ ( i ) \ G ( i ) /K ( i ) and action S [ µ ( i ) ] = 12 Z Γ ( i ) \ f M ( i ) d n ( i ) x q det g ( i ) g kl ( i ) h ( µ ) ab ∂ k µ a ( i ) ∂ l µ b ( i ) (6.5)where g ( i ) kl is the kinetic-term metric on the i -th factor space f M ( i ) .If our SUGRA model is not in the swampland, the solutions µ ( i ) have finite energy (action) S [ µ ( i ) ] < ∞ . Thus a partial answer to the Simpler Question on page 23 is that, in theSUSY case, the tension field of the gauge couplings should vanish while their energy mustbe finite.Although the above statements are fully correct, they looks a bit unsatisfactory. We aretreating the couplings g ( i ) and µ ( i ) asymmetrically – the first one as a background metricon f M ( i ) and the second one as a classical dynamical field – while the two couplings are on44he same footing in the swampland story, in facts geometrically unified in the 3d scalars’manifold M (cfr. the “total space” viewpoint in §.4.2). Then it is natural to treat also themoduli metrics g ( i ) as classical dynamical fields.A naive proposal will be to replace the i -th σ -model action by its minimal coupling togravity (allowing for a cosmological constant), that is, to consider the following theory livingon the moduli space M ( i ) S [ g ( i ) , µ ( i ) ] = Z Γ i \ f M i d n i ϕ q det g ( i ) − κ i ) R ( i ) + Λ ( i ) + 12 g kl ( i ) h ( µ ) ab ∂ k µ a ( i ) ∂ l µ b ( i ) ! (6.6)However the M “total space” viewpoint of §.4.2 suggests that additional KK fields mustlive on M , so the proposal (6.6) looks a bit naive, and we should not expect it to workin full generality. If we are lucky, S [ g ( i ) , µ ( i ) ] may at best be a consistent truncation of themoduli-space gravity theory (if it exists!). Claim.
In 4d
N ≥ SUGRA all couplings g ( i ) , µ ( i ) are solutions to the classical equationsof motion following from the action S [ g ( i ) , µ ( i ) ] for appropriate constants κ ( i ) , Λ ( i ) . If theSUGRA arises as the low-energy description of a consistent quantum theory of gravity, thesolutions have finite action, i.e. are gravitational instantons on M ( i ) ≡ Γ ( i ) \ f M ( i ) . Indeed, the equation of motion for the moduli-scalars are satisfied since the µ ( i ) areharmonic. One has only to check that the Einstein equations R ( i ) kl − g ( i ) kl R ( i ) + κ i ) Λ ( i ) g ( i ) kl = κ i ) T ( i ) kl (6.7)hold for some constants κ i ) , Λ ( i ) . Equivalently, it is enough to show that the three symmetrictensors R ( i ) kl − g ( i ) kl R ( i ) , g ( i ) kl , and T ( i ) kl are not linearly independent. For N ≥ allfactor spaces M ( i ) are locally symmetric, hence Einstein R ( i ) kl = − λ ( i ) g ( i ) kl , while the gaugecoupling e µ ( i ) : f M ( i ) → G ( i ) /K ( i ) is an isometry (up to overall normalization), so that thethree tensors are equal up to an overall constant and Claim holds. For N = 2 one hastwo factor spaces f M hyper and f M vector . e µ hyper is the contant map, so T hyper kl ≡ , while f M hyper , being negative quaternionic-Kähler, is Einstein, so eqn.(6.7) holds. The tricky caseis f M vector , which is a special Kähler manifold. That the Einstein equations (6.7) hold in thiscase was shown in [28]. Finally the last statement follows from the fact that evaluated onthe appropriate classical solution the action density is proportional to the volume form [28],so that finite action is equivalent to finite volume of M ( i ) , which is one of the swamplandconjectures.The Claim may be regarded as a general geometric rigidity theorem, alias a generalSUSY non-renormalization theorem. E.g. the N = 2 case yields the two non-renormalizationtheorems of N = 2 SUGRA. More precisely, we may choose G ( i ) as small as possible and then e µ ( i ) is an isometry; if we choose G ( i ) non-minimal, e µ ( i ) is just a totally geodesic embedding. = 1 SUGRA.
In the N = 1 case we have weaker non-renormalization theorems, sowe cannot expect that the story is as simple as for N ≥ . We have a non-renormalizationtheorem for F -term couplings, so we expect the gauge coupling µ to be still a tamed map; thisis of course correct, since the gauge coupling is holomorphic and a fortiori pluri-harmonic.The moduli metric, however, is not expected to be a solution of the Einstein equationsfollowing from a simple action of the form (6.6) since we don’t have the corresponding non-renormalization theorem. One may speculate about more complicated “dynamical” equationsfor the N = 1 moduli metric g ( φ ) ij with additional degrees of freedom propagating on thescalars’ moduli space M . Some proposal will be discussed elsewhere. In the supersymmetric context the geometric swampland conditions may be convenientlyrephrased as the requirement that the underlying domestic geometry is arithmetic, i.e. asthe statement that quantum-consistent effective models have formal brane amplitudes withthe right properties to correspond to actual physical branes. Since domestic geometry andits brane amplitudes Ψ ± make sense on all Riemannian manifolds, one is naturally led toask whether the statement is true for all quantum-consistent effective theories of gravity,supersymmetric or not.We have no quantitative control on the quantum-consistency of non -supersymmetriceffective theories, so the question is really a matter for speculation. There are, however,several reasons to believe that domestic geometry is somehow on the right track even in the non -SUSY case:• evidence from the SUSY examples;• the elegant and deep connection between domestic geometry and the Ooguri-Vafa ge-ometric swampland conjectures which apply in general, not just in the SUSY context;• domestic geometry implies physically desirable properties of the entropy functions;• physical “naturalness” considerations, see the discussion in §§. 7.2.3 and 7.2.4.
OV manifolds are the natural arena for tamed maps. Indeed
Tamed property. M a OV manifold and G ( R ) /K a symmetric space of non-compact type.All maps µ : M → Λ \ G ( R ) /K of finite energy (action) may be continuously deformed intoa unique tamed map ˚ µ which is the map of minimum energy in its homotopy class. Inparticular, all harmonic maps µ : M → Λ \ G ( R ) /K which have finite energy (action) areautomatically tamed.
46e defer the argument to the end of §. 7.3.1.We assume as our working hypothesis that, in a quantum-consistent theory of gravity, thelow-energy gauge coupling µ : M → Sp (2 h, Z ) \ Sp (2 h, R ) /U ( h ) has finite energy (action).This holds in all known (supersymmetric) examples constructed from the superstring, wherethis condition is strictly related to various swampland considerations [15], and looks quitereasonable in general. Under this hypothesis, the actual gauge coupling µ is a continuousdeformation of a well-defined tamed map ˚ µ , so domestic geometry is at least “qualitativelycorrect”. The “correction” µ − ˚ µ vanishes in the SUSY case, and we shall speculate that itshould be small (or even zero) in general.We stress that, for M an OV manifold, the tamed map ˚ µ is uniquely determined bythe action on the scalars of the continuous and discrete (bosonic) gauge symmetries. Underour working assumption, the action of the gauge symmetries on M is then restricted bythe condition that a finite-energy tamed map ˚ µ does exist. This yields a severe conditionon π ( M ) which may be regarded as a stringent refinement of the swampland π -conjectureof [2]. When the source is an OV manifold and the target is an arithmetic quotient of a symmetricspace without compact factors, the conditions of being arithmetic tamed and being harmonic are equivalent for maps of finite energy by the
Tamed property . Then, to construct anarithmetic domestic geometry on an OV space M , it suffices to impose that the relevantmap µ : M → Λ \ G ( R ) /K is harmonic of finite energy. This last condition has a simpleinterpretation which we now review. Let M , N be Riemannian manifolds. In order to construct a harmonic map φ : M → N in a given homotopy class (equivalently, a covering harmonic map ˜ φ : f M → e N twisted by agiven monodromy representation ρ of π ( M ) ) one may think of starting with an arbitrarysmooth map φ in that class which has finite energy E ( φ ) < ∞ (if it exists !), and thencontinuously deform it to decrease its energy until we reach a minimum value. A convenientway of implementing this variational strategy, pionereed by Eells and Sampson [87], is the tension flow. One considers a family of maps φ t : M → N , parametrized by t ∈ R , whichsatisfies the differential equation dφ t dt = D ∗ dφ t ≡ T ( φ t ) ≡ − grad E ( φ t ) (7.1)with initial value the finite energy map φ . If the solution to the PDE (7.1) exists and itslimit as t → + ∞ is smooth, φ ∞ is a finite-energy harmonic map in the homotopy class of grad stands for the gradient in the Banach manifold of smooth maps from M to N . Cfr. eqn.(4.10). . The existence problem for harmonic maps is then reduced to showing that the solutionto the gradient flow (7.1) exists and its limit is regular.For a family of ρ -twisted maps e µ t : f M → Sp (2 h, R ) /U ( h ) the flow equation takes theelegant form (cfr. eqn.(4.17)) S − t d S t dt = D i ( S − t ∂ i S t ) (7.2)where S t = ( S t ) t > is the composition of e µ t with the Cartan diffeomorphism (4.13) ι : Sp (2 h, R ) /U ( h ) ≡ H h ∼ → Sp (2 h, R ) ∩ P (2 h, R ) . (7.3)The derivative D i in (7.2) is covariant only with respect to the Levi-Civita connection onthe source space f M . From the form eqn.(7.2), it is obvious that along the flow one has S t ∈ Sp (2 h, R ) and ( S t ) t = S t > for all t .The tension flow has many analogies with the well known Ricci flow on a manifold M (fora review see [33]) which we may roughly see as the RG flow of the 2d σ -model with target M . The analogy is not accidental; indeed the tension flow is a special instance of Ricci flowas we are going to show. Before addressing the question of existence of harmonic maps, let us explain the relation ofthe gradient flow (7.1) with the Ricci flow in a context where the map µ : M → Λ \ H h isthe gauge coupling of a 4d field theory. For simplicity we first consider an effective modelwith only scalars and Abelian vectors (no gravity or fermions) L eff = − F π G ( φ ) ij ∂ µ φ i ∂ µ φ j − i π τ ( φ ) ab F a + F b + + i π ¯ τ ( φ ) ab F a − F b − (7.4)which we interpret as a field theory with an explicit UV cut-off Λ eff at the energy scalewhere the IR description breaks down. The scalar fields φ i are seen as adimensional localcoordinates on M , and F π is the overall mass scale of their kinetic terms. Except in §. 7.2.3we set F π = 1 .We compactify this 4d model to 3d on a circle of radius R . Each 4d vector yields tworeal scalars in 3d: one from the internal component A a and one from the dual to A aµ . The h scalars arising from the 4d vectors are periodic since they correspond to the electric andmagnetic U (1) h holonomies along the circle; we parametrize the h holonomies as exp(2 πiy A ) ( h = 1 , . . . , h ).The resulting 3d effective theory is the σ -model with scalars’ manifold the total space ofthe fibration X → M mentioned in §. 4.2: the fiber X φ is a h -dimensional Abelian variety P (2 h, R ) stands for the space of positive-definite h × h real symmetric matrices. A different application of the Ricci flow to the swampland program has been discussed in [34]. C ) with periods τ ( φ ) ab and fixed principal polarization (which we identify with itsKähler class). The total space X is equipped with the metric ds = R G ( φ ) ij dφ i dφ j + 1 R S ( φ ) AB dy A dy B + (cid:0) exponentially small as R → ∞ (cid:1) (7.5)where S AB ≡ ( E E t ) AB is the inverse of the Cartan coupling S AB . The exponentially smallcorrections are due to 4d massive particles, carrying electro-magnetic charges, whose world-line wrap the circle; such corrections are well studied in the context of 3d compactificationsof 4d N = 2 QFT [89]. We shall take R large and ignore the exponential corrections.Compactifying further down to two dimensions, we get a 2d σ -model with target spacemetric proportional to (7.5) whose RG flow is given (in the one-loop approximation) by theRicci flow. The flow preserves the structure of the metric (7.5) so it decomposes into a pairof equations of the form R ddt G ( φ ) ij = − R ( φ ) ij , S AC ddt S ( φ ) CB = − R ( φ ) AB (7.6)The Ricci curvature in the fiber directions is easily computed to be R AB = − D i (cid:0) S − ∂ i S (cid:1) AB . (7.7)Comparing with eqn.(7.2), we see that the Ricci flow restricted to the fibers has the sameform as the original tension flow. However, in general, the two flows in (7.6) are coupledtogether because the covariant derivative D i depends on the evolving metric G ij on the base.When the base M is Ricci-flat or Einstein, the two flows are identical.Note that R AA = 0 , so that the Ricci flow preserves the volume of the Abelian fibers. Infacts, the flow preserves the fiber’s Kähler form ( ≡ polarization) while changing its complexstructure. To see this, we introduce the orthonormal co-frame e m ≡ E Am dy A E ∈ Sp (2 h, R ) (7.8)where E is the vielbein in (4.13). The isotropy group U ( h ) acts on the “flat” index m inthe representation h ⊕ ¯ h , defining on the fiber a torsion-less flat U ( h ) -structure hence anintegrable complex structure and a closed Kähler form which is given in “flat” indices by theconstant symplectic matrix Ω mn . Then the fiber Kähler form is ω fiber def = Ω mn e m ∧ e n = ( E Ω E t ) AB dy A ∧ dy B ≡ Ω AB dy A ∧ dy B (7.9)which is preserved by the Ricci flow of the fiber metric S − ∂ t S = D i ( S − ∂ i S ) . (7.10)49 o/o/o/o/o/o/o/o/o/o/o • P◆❊✸✤ ☛ ② ♣ ♥ ♥ ♣ ② ☛✤✸❊◆P /o/o/o/o/o/o/o/o/o/o/o /o/o/o/o/o/o/o • ❴❴❴❴❴❴ G(cid:7) E(cid:5) C(cid:3) @(cid:0) <| 8x 3s /o +k &f "b (cid:30)^ (cid:27)[ (cid:25)Y (cid:23)W • /o/o/o/o/o/o/o Figure 1: One-loop corrections to the photon propagator arising from the scalar-vectorcouplings in the vectors’ kinetic terms. Wavy lines are photons, dashed ones scalars.The Ricci flow preserves the form (7.5) of the 3d target metric with S AB a positive symmetricmatrix in Sp (2 h, R ) . Thus the Ricci flow for the metric (7.5) evolves the complex moduliof the Abelian fibers of X → M but not their Kähler moduli. At a weakly-coupled fixed-point of the 2d RG flow the tension D i ( S − ∂ i S ) vanishes and hence the fixed-point map e µ is harmonic. L eff In terms of the original 4d effective theory (7.4) the vanishing of the tension T ( µ ) meansthat the two (quadratically divergent) one-loop Feymann graphs in figure 1 cancel. Indeedthe leading correction to gauge couplings is proportional to the tension of the gauge coupling δτ ab ∝ Λ eff F π T ( µ ) ab ≡ Λ eff F π D i ∂ i τ ab . (7.11)In order for the 4d Lagrangian (7.4) to be meaningful as a weakly-coupled effective descriptionof the low-energy dynamics, the correction (7.11) should be rather small. If F π . Λ eff , thetension should be approximately zero T ( µ ) ab ≈ . Thus the vanishing of the tension may beseen as a “naturalness” requirement for the weakly-coupled 4d effective theory L eff . This isequivalent to the statement that the gauge coupling µ is (approximately) harmonic, i.e. thatits energy is near the minimum value consistent with the monodromy representation ρ .The interpretation of T ( µ ) ab = 0 as a naturalness condition on L eff is reflected in thefact that the gauge coupling tension vanishes for all
4d supersymmetric theories – whetherthey are rigid SUSY QFTs or supergravities. This is one of the many ways in which SUSYimproves naturalness. At weak coupling, naturalness of the gauge couplings does not require τ ( φ ) ab to satisfy all the detailed constraints of supersymmetry: the much weaker conditionthat the gauge coupling map µ is harmonic suffices. When M is an OV manifold and µ hasfinite energy, the weaker condition implies that the gauge coupling µ is tamed and arithmetic;whenever the holonomy of the OV manifold is not generic, we either get an arithmetic tt ∗ There is a third one-loop graph (the tadpole graph) proportional to the Christoffel symbols γ ijk . Thetadpole graph may be set to zero by using normal coordinates in the perturbative expansion. Assuming our working hypothesis that the energy of µ is finite. In other words, when M is an OV manifold with a sufficiently large algebra P • , the weakernaturalness condition on µ is equivalent to the detailed constraints from supersymmetry.Let µ : M → Λ \ Sp (2 h, R ) /U ( h ) be the gauge coupling of an effective theory L eff , with M an OV space. Under our working hypothesis that µ has finite energy, the homotopy class [ µ ] contains a unique tensionless map ˚ µ , which is automatically tamed. The naturalnessargument above suggests that the torsion T ( µ ) , and hence the correction µ − ˚ µ , is small,that is, that domestic geometry is (at least) approximatively correct. We add gravity to the model (7.4), and return to the original Lagrangian (2.1) coupled togravity, and again compactify the model on S . In a gravity theory this is just a topolog-ical sector of the 4d theory, and the 3d effective low-energy physics ought to be quantum-consistent if the 4d theory is. In presence of gravity the 3d effective theory has two additionallight scalars: ρ corresponding to g and the dual z of the KK vector g µ . The 3d scalars’manifold M is now a fibration over M × R ρ with fiber a copy of the locally homogeneousspace H ( Z ) \ H ( R ) where H ( R ) is the real Heisenberg Lie group H ( R ) = { ( z, y A ) ∈ R h +1 } ( z, y A ) · ( w, u A ) = ( z + w + Ω AB y A u B , y A + u A ) . (7.12)Ignoring quantum corrections (which are supressed as ρ → ∞ ), the Einstein frame 3d scalars’metric takes the form ds = ds base + ds fiber == (cid:18) G ij dx i dx j + 12 ρ dρ (cid:19) + 12 ρ (cid:18) ρ S AB dy A dy B + (cid:0) dz + Ω AB y A dy B (cid:1) (cid:19) , (7.13)with G ij , S AB and Ω AB as before. The Killing vector ∂ z defines on each fiber a transverselyholomorphic foliation (THF) endowed with a THF-compatible transversely Kähler metric:the normal bundle to the leaves of the transversely holomorphic foliation carries a complexstructure as well as the transverse Hermitian metric ρ − S AB dy A dy B which is Kähler whenrestricted to a fiber with Kähler form ρ − Ω AB dy A ∧ dy B (cfr. eqn.(7.9)). The complexstructure of the normal bundle to the THF foliation on each fiber is again specified by theuni-modular matrix S AB .In view of the Heisenberg symmetry and the scaling invariance ( ρ, z, y A ) → ( λ ρ, λ z, λy A ) , That is, a totally geodesic embedding into the locally symmetric space Λ \ G ( R ) /K . Probably the best reference for transversely holomorphic foliations (THF) is the physics paper [90]which discuss them in the context of supersymmetry on curved 3d manifolds. They discuss the 3d case; theextension to an arbitrary odd number of dimensions is straightforward. R AB dy A dy B + R zz dz = 1 ρ A AB dy A dy B + 1 ρ B (cid:0) dz + Ω AB y A dy B (cid:1) (7.14)where A AB and B are functions only of the coordinates x i of M . In particular the Ricci flowpreserves the THF of each fiber, while evolves its transverse complex structure and Kählerform ddt (cid:18) ρ S AB (cid:19) = − ρ A AB (7.15)where A = − S (cid:0) D i ( S − ∂ i S ) − c (cid:1) (7.16)for some constant c which depends only on h and m . Writing S = f S ′ , with det S ′ = 1 ,we get for the uni-modular matrix S ′ the same flow equation as before, eqn.(7.10), so theevolution of the transverse complex structure of the fiber-wise THF is again given by the4d tension flow. The new aspect is that now the transverse Kähler form gets rescaled bythe factor e ct . This is due to the fact that gravity modifies the Ricci flow equation so thatthe appropriate 3d “fixed point condition” requires M to be Einstein rather than Ricci-flat;the easiest way to see this is to compare 3d N ≥ rigid SUSY field theories, which haveRicci-flat target spaces, with the N ≥ supergravities which have Einstein target spaceswith R ij = − λg ij , where λ > is a universal constant which depends only on the fieldcontent. In “natural” effective theories the additional term proportional to S AB /ρ in the rhs of eqn.(7.15) should be cancelled by diagrams of the form in figure 1 with the scalar internallines replaced by graviton propagators whose contribution is ∝ S AB /ρ . Thus the absence oftorsion of µ can again be interpreted as a naturalness condition on the 4d effective gravitytheory. All weakly-coupled infrared fixed points then have tensionless gauge couplings µ .In presence of fermions, the quadratic divergence of the photon propagators, in additionto the diagrams in figure 1, gets contributions from one-loop of fermions /o/o/o ◦ (cid:30) (cid:30) ◦ ^ ^ /o/o/o (7.17)where ◦ stands for Pauli coupling. In the susy case these new contributions would notchange our conclusions because of the relation of Pauli couplings to the gauge couplingsimplied by supersymmetry: the net effect is just a further contribution to the constant c in(7.16), that is, the Pauli couplings will not modify the flow of the fiber’s transverse complexstructure. It seems plausible that this conclusion remains valid in all consistent effectivetheories, possibly with more complicate flows for the fiber’s transverse Kähler moduli, butwithout affecting the flow of the fiber’s transverse complex moduli. If this is the case, the“natural” value of the tension of the gauge coupling is still zero (or very small). We expectthe fiber’s flow to have this property for the following reason: according to the definition of52V manifold, π ( M ) is “big” and – unless µ is the constant map – its image Γ in the Siegelmodular group should also be “big”. Since Γ is a gauge symmetry, the Pauli couplingsshould be exactly invariant under it. In the extreme IR this implies invariance under thecontinuous group Sp (2 h, R ) . Defining (as it is usual) the fermi fields to be invariant under Γ , the electro-magnetic field strengths F ≡ ( G a , F a ) can enter the Pauli interactions onlythrough the Γ -invariant combination E − F : these couplings are automatically Sp (2 h, R ) -invariant and hence cannot spoil the tension flow of the uni-modular matrix S ′ ab which isa flow in the group Sp (2 h, R ) . In particular, when µ is harmonic, hence tamed, the Paulicoupling must be a domestic brane amplitude, as expected.More generally, the tension field in the rhs of eqn.(7.2) is the only Sp (2 h, R ) -covarianttensor we can write to second order in the scalar field derivatives which flows the transversecomplex structure of the Heisenberg fiber of M ; all possible covariant corrections (to thisorder) will affect the fiber’s transverse Kähler moduli but not its transverse complex moduli.Therefore the condition that µ is harmonic seems to be a rather plausible “naturalness”requirement, at least under certain circumstances, especially when the scalars’ manifold M is an OV space. If the couplings flow in the extreme IR to a “weak-coupling” regime (meaninga regime in which the geometric interpretation of the couplings is sound), at the fixed pointthe gauge coupling µ should be tensionless, i.e. µ harmonic. As already stressed, we knowno reliable example of consistent effective theory where the tension of µ is not zero. In the heuristics above we neglected the issues related to finite-distance singularities inscalars’ space. At such points finitely many charged states become light, and we have addi-tional effective one-loop contributions from diagrams of the form (7.17) where the internallines correspond to the new light charged particles and the vertices ◦ are minimal gaugecouplings. In terms of the dependence on the effective cut-off Λ eff , these contributions aresuppressed by a factor Λ − eff log Λ eff , but there is no reason to expect that – as functions ofthe background value of the scalars – they are proportional to the tension of the gaugecouplings µ . This suggests that near the finite-distance singular points in M the effectivetension (Λ eff /F π ) T ( µ ) ab , or more precisely its invariant norm (Λ eff /F π ) k T ( µ ) ab k , is smallbut non-zero. Morally speaking, we expect the gauge couplings τ ( φ ) ab to satisfy some kindof “wave-equation with sources” on M roughly of the form T ( µ ) ab ≡ D i ∂ i τ ( φ ) ab = X s f s ( φ ; φ s ) (7.18) This exceptional case does happen in quantum-consistent theories: consider the 4d N = 2 SUGRAwhich describes Type IIB on a rigid
Calabi-Yau 3-fold. Assuming µ not to be the constant map (which is a special case of harmonic map), the technicalstatement is that Γ is Zariski dense in some non-compact Lie subgroup G ( R ) ⊂ Sp (2 h, R ) : see §.7.3. Thismeans that all algebraic invariants for the group Γ are invariants for the full group G ( R ) , which genericallyis the full group Sp (2 h, R ) . The dual field strengths G a are defined as G a = ∗ ∂ L /∂F a . φ s ∈ M are the finite-distance singular points at which finitely-many new degrees offreedom become massless, and the f s ( φ ; φ s ) ’s are functions (or distributions) with supportin some small region centered at φ s which capture the local physics at these special pointsin moduli space. The most classical existence theorem for harmonic maps is due to Eells and Sampson [87];their strategy was to show existence and regularity of the tension flow (7.1): let M and N be compact Riemannian manifolds and N have non-positive sectional curvatures. Then everysmooth map f : M → N is homotopic to a harmonic map. The harmonic map is essentiallyunique in its homotopy class and it is the map of minimal energy in its class. For the gaugecoupling map µ : M → Λ \ Sp (2 h, R ) /U ( h ) (7.19)the condition on the sectional curvatures of the target space is satisfied, but both the sourceand the target manifolds are non-compact. However they are expected to behave “almostas they were compact” because they have finite volume and enjoy other good properties.Thanks to these special properties, we may invoke other, more powerful, existence theorems.Our target space is locally symmetric; in this situation one has a stronger result [88]:let G be a real Lie group, K ⊂ G a maximal compact subgroup, and Λ ⊂ G any discretesubgroup. Assume the Riemann manifold M is compact. A harmonic map f : M → Λ \ G/K exists if and only if the monodromy group ρ ( π ( M )) ≡ Γ ⊆ Λ has reductive Zariski closure Γ R in G . When the harmonic map exists, it is essentially unique. In our case, eqn.(7.19), the target space has the required form with G = Sp (2 h, R ) , K = U ( h ) . (7.20)Suppose the real Lie group G ′ ≡ Γ R ⊂ G is semi-simple and non-compact: in the appli-cations we have in mind these properties follow from the swampland conditions on Γ . Thenthe image of a ρ -twisted harmonic map ˜ φ : f M → G/K has dimension ≥ and hence thecorresponding harmonic map φ : M → Γ \ G/K (7.21)is uniquely determined by ρ . K ′ ≡ K ∩ G ′ is a maximal compact subgroup of G ′ , and wehave the totally geodesic embedding ι : Γ \ G ′ /K ′ ֒ → Γ \ G/K, (7.22) We see the Lie group as an algebraic group over R through its adjoint representation. In facts it is unique except if it is a constant map or its image is a geodesic; in the last case the differentharmonic maps correspond to different linear parametrizations of the same geodesic. More correctly: the real Lie group underlying the R -algebraic group Γ R [91]. G ′ ֒ → G . Since Γ is neat, ι induces isomorphisms of allhomotopy groups π n , so by the Whitehead theorem [92] there exists a deformation retract r : Γ \ G/K → Γ \ G ′ /K ′ , ι ◦ r = Id ′ . (7.23)Hence we have a map φ ′ ≡ r ◦ φ : M → Γ \ G ′ /K ′ such that ι ◦ φ ′ is in the same homotopyclass as φ . Applying the theorem of ref. [88], we conclude that there is a ρ -twisted harmonicmap ˜ φ ′ : f M → G ′ /K ′ (7.24)which we may think of as a ρ -twisted harmonic map f M → G/K whose image is fullycontained in the submanifold G ′ /K ′ ֒ → G/K . By uniqueness, the corresponding harmonicmap should be the same one as the harmonic map φ . We have proven the Structure theorem. M compact. Let G ′ ≡ ρ ( π ( M )) R ⊂ G be the R -Zariski closure ofthe monodromy group and K ′ ≡ K ∩ G ′ . The harmonic map φ : M → Λ \ G/K (resp. thecovering ρ -twisted harmonic map ˜ φ ) factors through φ : M φ ′ −−→ Γ \ G ′ /K ′ ֒ → Λ \ G/K, ˜ φ : f M ˜ φ ′ −−→ G ′ /K ′ ֒ → G/K (7.25)Were not for the assumption that M is compact, the above theorem would be essentiallythe same statement as the structure theorem for the period map p in Hodge theory [58–61]which, in general, factorizes as in the commutative diagram M / / µ p ( ( φ ′ & & ▼▼▼▼▼▼▼▼▼▼▼▼ Γ \ G ′ / [ H ∩ G ′ ] (cid:15) (cid:15) (cid:15) (cid:15) (cid:31) (cid:127) / / Γ \ G/H (cid:15) (cid:15) (cid:15) (cid:15) Γ \ G ′ / [ K ∩ G ′ ] (cid:31) (cid:127) / / Γ \ G/K (7.26)where for a Hodge structure of odd weight k and Hodge numbers { h p,q } [56, 58–60] H = Y p + q = kp We may drop the assumption that M is compact and replace it by two conditions: (i) thesource space is an OV manifold M , and (ii) there exists a ρ -twisted smooth map ˜ φ which,when seen as a map φ : M → Γ \ G/K , has finite energy E ( φ ) < ∞ . Under this lasthypothesis, it makes perfect sense to talk of continuous deformations of φ which decreaseits energy, so the variational strategy is still meaningful: we may think of deforming contin-uously the map until we reach the absolute minimum value of the energy in the homotopyclass defined by the monodromy representation ρ . The tension-flow is an efficient way of im-plementing the deformation in the direction of steepest descent of the energy, cfr. eqn.(7.1).Then Theorem (Corlette [69]) . Suppose ρ : π ( M ) → G is a homomorphism with Zariski denseimage and there exists a ρ -twisted map ˜ µ from f M to G/K with finite energy. Then there isa ρ -twisted harmonic map with finite energy from f M to G/K . Roughly speaking, finite energy corresponds to finite volume of the image µ ( M ) : thus ifthe scalars’ manifold M satisfies the (slightly stronger version of) the standard swamplandconjectures, any gauge-coupling µ : M → Λ \ Sp (2 h, Z ) /U ( g ) is homotopic to a harmonicone, namely the fixed point µ harm of the tension flow with initial condition µ . The heuristicphysical arguments of §.7.2 suggests that, in the extreme IR limit of a consistent quantum56ravity, the physical coupling µ phys actually coincides with µ harm at least approximately andaway from finite-distance singularities. We stress that this statement is literally true in allknown examples of reliable quantum-consistent effective theories of gravity.Moreover, if we can show that the surface term in the integration over M of the lhs of (7.29) vanishes, we conclude that the harmonic map µ is actually tamed. Thus to showthe Tamed property for OV manifolds stated at the beginning of §. 7.1, we have only tojustify the dropping of the boundary term in the integration by parts of the lhs in eqn.(7.29)under our assumptions that M is OV and µ has finite energy. We defer this technicality toappendix B. µ The case of a finite-energy harmonic map µ whose source is an OV space is similar to thecase where the source space is compact, since µ is “trivial at infinity”. In particular, theessential uniqueness of the harmonic map in its homotopy class is typically still true. Thenwe can apply the argument in eqns.(7.21)–(7.25) to deduce the Structure theorem for thegauge coupling µ which is the statement that it factorizes as in the commutative diagram M φ ′ / / µ ' ' Γ \ Γ R / [Γ R ∩ U ( h )] (cid:31) (cid:127) ι / / Γ \ Sp (2 h, R ) /U ( h ) (7.30)If µ is harmonic, the real Lie group (or, rather, algebraic group over R ) Γ R ⊂ Sp (2 h, R ) must be reductive. Hence, modulo finite groups, it has the form Γ R ∼ = A × G × · · · × G s (7.31)with A Abelian and G ℓ simple. Correspondingly (up to commensurability) Γ ∼ = Γ A × Γ × · · · × Γ s (7.32)with Γ A ⊂ A and Γ ℓ ⊂ G ℓ . Since we are assuming Γ to be neat and generated by unipotents, Γ A must be trivial. Then the real Lie group Γ R ⊂ Sp (2 h, R ) is either trivial or semi-simple.In the fist case the gauge couplings µ are field-independent numerical constants. An exampleof this situation is given by the compactification of Type IIB on a rigid Calabi-Yau [51].If Γ is not trivial, the harmonic map φ ′ decomposes into a s -tuple of partial maps φ ℓ : M → Γ ℓ \ G ℓ / [ G ℓ ∩ U ( h )] , ℓ = 1 , · · · , s. (7.33)We stress that all spaces through which the gauge coupling µ factorizes, i.e. Γ \ Γ R / [ K ∩ Γ R ] and the Γ ℓ \ G ℓ / [ G ℓ ∩ K ] are OV manifolds. 57he “structure theorem” (7.30) is identical in form to the structure theorem for Griffithsperiod maps in modern Hodge theory [58–61], which is satisfied by the low-energy effectivetheories of Type II compactified on a geometric family of Calabi-Yau, whose couplings aredetermined by the Griffiths period map [7,8], and which is the main condition discriminatingthe quantum consistent N = 2 supergravities from the ones belonging to the swampland [15].In particular, whenever the N = 2 SUGRA satisfies the structure theorem automaticallysatisfies all the relevant swampland conjectures, see [15].It is remarkable that the very same structure statement holds in full generality – evenwhen the moduli space M has general holonomy so ( m ) and no natural complex structure –by virtue of the properties of the OV manifolds, provided we add to the list of the swamplandconjectures the statement that the IR gauge coupling µ is harmonic of finite-energy.We take the above state of affairs as evidence that our working hypothesis is somehowon the right track. N = 2 SUGRA This paragraph is a comment about reference [15]. As we know, on SUSY grounds, the vector-multiplet couplings of a 4d N = 2 SUGRA are dictated by special Kähler geometry which isequivalent to a variation of Hodge structure of weight 3 with h , = 1 . In [15] it was observedthat a deep problem in math is to determine which VHS arise from geometry, i.e. describesan actual family of Calabi-Yau manifolds. A condition mathematicians have proven to benecessary is that the period map p of the VHS satisfies the structure theorem [58–61]. It wasproposed in [15] that this same condition is also a swampland criterion for 4d N = 2 effectivetheories. Then the question was the logic relation between this new criterion and the Ooguri-Vafa geometric swampland conjectures [2]. The fact that the validity of the structure theoremof [61] implies the swampland conjecture is easy to see [15]. It was initially believed that thestructure theorem was a stronger requirement than the Ooguri-Vafa ones: the naive feelingwas that the structure theorem is a stringent condition with lots of Number Theoreticaland Algebro-Geometric aspects whereas the OV statements looked like simple qualitativeproperties of the relevant geometries. However, now we see that the OV properties arestrong enough (when supplemented by the conditions following from N = 2 supersymmetryand some mild regularity assumption) to actually imply the structure theorem of [61], sothat the two set of conditions are essentially equivalent. In the context of 4d N = 2 supergravity, the Hodge-theoretic structure theorem – whichholds only in a tiny subset of the space of all formal N = 2 sugra which includes all the The Griffiths period map satisfies in addition the IPR, so we have the more detailed factorization of µ in the diagram (7.26). N =2 situation arise from the interplay between two pieces of information: the structure theoremand special Kähler geometry. In the non-SUSY case, where M has generic holonomy, thesecond ingredient is lacking, and we are able to extract from the structure theorem muchweaker physical consequences – which however have the merit of being (conjecturally) truein full generality. The N = 2 case. Let us briefly recall the situation in the N = 2 context. In thiscase the gauge coupling µ , seen as a fibration over its image B , µ : M → B ≡ µ ( M ) ⊂ Γ \ Sp (2 h, R ) /U ( h ) (7.34)is essentially trivial in the sense that M = M hyper × M vector , and the Griffiths’ infinitesimalperiod relations [56, 58–60], together with the Torelli theorem [29–31], say that the periodmap p is a Griffiths-horizontal, holomorphic embedding of the universal cover of M vector intothe Griffiths period domain D h (cfr. eqn.(7.27)) p : f M vector → D h def = Sp (2 h, R ) (cid:14) [ U (1) × U ( h − . (7.35)Composing with the canonical projection D h ։ H h , we see that the non-holomorphicsmooth map f M vector → e B ≡ e µ ( f M ) is a local isomorphism by horizontality of ˜ p . TheKähler form on f M vector is pull back ˜ p ∗ F of the curvature 2-form F of the Hodge line bundle L → D h (i.e. the homogeneous bundle over D h defined by the fundamental character of the U (1) factor in H = U (1) × U ( h − ). Then in the N = 2 case there is a simple relationbetween the gauge couplings τ ( φ ) ab and the scalar metric G ( φ ) ij expressed by the modulispace Einstein equation (6.7). In particular, all isometries of f M vector ֒ → D h is the restrictionof an Sp (2 h, R ) symmetry of the ambient space D h . The general case. In the case of a general quantum-consistent effective theory – withNO supersymmetry – we do not expect a simple relation between the couplings τ ( φ ) ab and G ( φ ) ij . However, to the extend that µ is harmonic, quantum consistency still implies subtlerelations between the two couplings. In particular the scalars’ metric G ( φ ) ij is constrainedby the condition that the gauge coupling µ is harmonic for G ( φ ) ij . This severely restrictsthe allowed scalar metric. E.g., when M is a complex OV manifold and µ is pluri-harmonic,it requires G ( φ ) ij to be Kähler. The scalars metric satisfies also other strong constraints: (i) the infinite group G acts by isometries on G ( φ ) ij , (ii) the volume is finite, and (iii) Ricci Our summary below is rather rough and the statements are not meant to be technically precise; see [15]for a more precise discussion. susy , fora given gauge coupling µ there is not that much freedom in the choice of the scalars’ metric G ( φ ) ij if we wish to avoid ending in the swampland.Unfortunately, for a non-SUSY theory the relation between µ and the consistent scalar’smetric G ( φ ) ij is rather implicit. For this reason, in absence of susy it is hard to rephrasethe structure theorem for µ in terms of geometric proprieties of G ( φ ) ij .The structure theorem refers to properties of B ≡ µ ( M ) , seen as a submanifold of theSiegel variety Sp (2 h, Z ) \ H h , rather than directly to the intrinsic geometry of M . In the N = 2 case B ∼ = M vector so this is not a limitation, but in general the two spaces are quitedifferent. Anyhow the N = 2 statements of [15] hold for general domestic geometries whenreferred to B . In particular we have the dycothomy: (a) either e B ⊂ H h is a totally geodesic submanifold, hence symmetric, and the fibers ˜ µ − ( b ) ⊂ f M are minimal submanifolds;(b) or no continuous symmetry of the ambient space H h leaves e B fixed (as a set).Possibility (a) corresponds to very special effective theories which look like consistent trun-cations of some N ≥ supegravity. The second possibility is the generic case. In the N = 2 case this implies completeness of instanton corrections (which is expected on physicalgrounds [93]), and this implication is likely to extend to more general situations. A SUGRA spaces: tamed maps vs. special holonomy We want to show that if X is a symmetric space relevant for 4d SUGRA not of the form SO ( m, /SO ( m ) or SU ( m, /U ( m ) all tamed maps f : X → Y are totally geodesic.We consider the symmetric Riemannian manifolds of type III, i.e. of the form G/K where G is some real Lie algebra and K its maximal compact subgroup. By general theoryits holonomy Lie algebra is k ≡ Lie ( K ) . The space SO ( m, /SO ( m ) has dimension m andstrictly generic holonomy algebra so ( m ) , so has no non-trivial parallel forms, and hence inthis case tamed ≡ harmonic . The complex hyperbolic space SU ( m, /U ( m ) has complexdimension m and holonomy Lie algebra su ( m ) , so it is a strict Kähler manifold and hencefor the complex hyperbolic spaces tamed ≡ pluri-harmonic . We shall call SO ( m, /SO ( m ) and SU ( m, /U ( m ) the strict cases. For all other type III symmetric spaces the holonomyalgebra hol ( G/K ) is neither generic nor strict Kähler, so for these spaces tamed is strictlystronger than being harmonic or pluri-harmonic. We consider the cosets G/K relevant for4d SUGRA.We write T ∼ = T ∨ for the (irreducible) holonomy representation of the symmetric space G/K . The corresponding Lie algebras decomposes as g = k ⊕ T and the holonomy represen-tation on T is induced by the adjoint action of g on itself. G/K has a non-trivial algebra P • 60f parallel forms Ω ( s ) ∈ ∧ k s T . We consider their annihilator algebra a def = n a ij ∈ ⊗ T ∼ = End( T ) : a [ i j Ω ( s ) ji ··· i ks ] = 0 ∀ Ω ( s ) ∈ P • o (A.1)A map f is tamed iff D i ∂ j f is contained in a ∩ ⊙ T ; when this space is zero and f is tamedwe must have D i ∂ j f = 0 , that is, a ∩ ⊙ T = 0 = ⇒ all tamed maps are totally geodesic. (A.2) a ⊂ ⊗ T , is a real Lie subalgebra of sl ( T ) , contains k and is a k -invariant subspace; henceit has the form a = k ⊕ b with b ⊂ ( ⊗ T ) traceless . The algorithm goes through the followingsteps. (1) we show that a ∩∧ T = k while all irreducible k -representations in ( ⊙ T ) traceless areself-dual, so we infer that a is a reductive Lie algebra with maximal compact subalgebra k .Writing A , K for the corresponding group, A/K is a, possibly trivial or reducible, symmetricspace. (2) One checks in the Cartan table of symmetric space which groups A , K are allowedand reads from them the candidate b . (3) Finally one checks that the candidate b 6⊂ ⊙ T ,getting a paradox. (4) We conclude that a ∩ ⊙ T = 0 and apply (A.2).We run the algorithm one space at the time. • N = 8 sugra . The scalars’ space is E /SU (8) . K = SU (8) and T is the i.e. T = ∧ F ( F stands for the fundamental of SU (8) ). ∧ T contains a singlet, i.e. on thesymmetric space E /SU (8) (A.3)we have a non-trivial parallel 6-form and so ( T ) a . On the other hand, as su (8) -modules so ( T ) = ∧ ( ∧ F ) = su (8) ⊕ (A.4)so su (8) is the maximal compact subalgebra of a . On the other hand ⊙ T = ⊙ ( ∧ F ) = ⊕ (A.5)and both representations are self-dual. Hence a is semi-simple with maximal compact sub-algebra su (8) and A/K is a non-compact symmetric space. Since E /SU (8) is the onlynon-trivial non-compact symmetric space of holonomy SU (8) , we must have either a ∩ ⊙ T equal zero or T . But T 6⊂ ⊙ T and the second possibility is ruled out. • N = 6 sugra . The scalars’ space SO ∗ (12) /U (6) is Kähler; T = ( ∧ F ⊕ ∧ F ) R ( F isthe fundamental of U (6) ). Then ∧ T = ( ∧ F ) ⊗ ( ∧ ¯ F ) ⊕ (cid:0) ∧ ( ∧ F ) ⊕ ∧ ( ∧ ¯ F ) (cid:1) R = ⊕ su (6) ⊕ ⊕ ( ⊕ ) R (A.6)where is the Kähler form. ∧ ( ∧ F ) ⊗∧ ( ∧ ¯ F ) contains 2 singlets, so that we have a parallel613,3) form different from the cube of the Kähler form and u (15) a . On the other and u (15) = ( ∧ F ) ⊗ ( ∧ ¯ F ) = u (6) ⊕ (A.7)so the maximal compact subalgebra of a is u (6) . Now we have 3 non-trivial symmetricspaces to consider namely SU (6 , /U (6) , Sp (12 , R ) /U (6) and SO ∗ (12) /U (6) with would-be b , ( F ⊕ ¯ F ) R , ( ⊙ F ⊕ ⊙ ¯ F ) R and ( ∧ F ⊕ ∧ ¯ F ) R , respectively. Since ⊙ T contains only the U (1) characters , , − we get a contradiction in all cases. We conclude that b = 0 , so a ∩ ⊙ T = 0 . • N = 5 sugra has G/K = SU (5 , /U (5) which is a strict case. • N = 4 sugra . The (universal cover of) the scalars’ space is reducible SU (1 , /U (1) × SO (6 , k ) / [ SO (6) × SO ( k )] , (A.8)the first factor is strict, as it is the second one when k = 1 . We focus on the second factorand assume k ≥ then T = V ⊗ V k , where V k is the vector of SO ( k ) . We have a parallel6-form and dually a parallel k − form, hence so (6 k ) a . Since ∧ T = so (6) ⊗ ( ⊕ ( ⊙ V k ) traceless ) ⊕ ( ⊕ ) ⊗ so ( k ) (A.9)we have that so (6) ⊕ so ( k ) is the maximal compact subalgebra of a . We have two possiblenon-trivial symmetric spaces SL (6 , R ) /SO (6) × SL ( k, R ) /SO ( k ) and SO (6 , k ) / [ SO (6) × SO ( k )] (A.10)with would-be b ⊂ ( ⊙ V ) traceless ⊕ ( ⊙ V k ) traceless and b ⊂ V ⊗ V k , respectively. The firstone obviously does not preserve the parallel forms, and the second one is not containedin ⊙ T . This rules out also SL (6 , R ) /SO (6) and SL ( k, R ) /SO ( k ) and one remains with a = so (6) ⊕ so ( k ) . • N = 3 sugra . SU (3 , k ) / [ SU (3) × U ( k )] again is Kähler and for k = 1 strict. T =( F ⊗ F k ⊕ ¯ F ⊗ ¯ F k ) R . We have parallel (3 , and ( k, k ) forms, the maximal compact subalgebrais su (3) ⊕ u ( k ) ; going through the various symmetric spaces, one concludes that b = 0 . B No boundary term in the Bochner argument As explained at the end of §.7.3, we have to show that if M is a OV manifold and f : M → Λ \ G/K is a finite-energy harmonic map to a locally symmetric space of non-compact type,the surface term Z M d (cid:16) g ij ∗ ( df i ∧ Ω) ∧ D ∗ ( df j ∧ ∗ Ω) (cid:17) (B.1)62anishes (cfr. eqn.(7.29)). We proceed by adapting the argument in [69]. As in §.2.4.2, forall R > we write h R : M → R for a Lipschitz continuous function such that for some fixedconstant k > [37]: ≤ h R ≤ , h R = ( for r ≤ R for r ≥ R, (cid:12)(cid:12) dh R (cid:12)(cid:12) < kR , (B.2)and assume Condition ( ∗ ) i.e. eqn.(2.24) | ∆ h R | < C. (B.3)We then proceed as in reference [69]: (cid:12)(cid:12)(cid:12)(cid:12)Z M h R ∆ | df | d vol (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z M (∆ h R ) | df | d vol (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z M | ∆ h R | | df | d vol ≤ C E ( f ) (B.4)and taking R → ∞ (cid:12)(cid:12)(cid:12)(cid:12)Z M ∆ | df | d vol (cid:12)(cid:12)(cid:12)(cid:12) ≤ C E ( f ) . (B.5)Now let f : M → Λ \ G/K be harmonic of finite energy, E ( f ) < ∞ . Since the target spacehas non-positive sectional curvatures, and the Ricci tensor of M is bounded below (cfr.eqn.(3.4)), R ij ≥ − K g ij , (B.6)the Bochner formula of Eells and Sampson for harmonic maps [87] 12 ∆ | df | = |∇ df | + R ij h ab ∂ i f a ∂ j f b − R habcd g ik g jl ∂ i f a ∂ j f b ∂ k f c ∂ l f d (B.7)gives a bound of the form | D i ∂ j f | ≤ ∆ | df | + K | df | ⇒ Z M | D i ∂ j f | d vol ≤ (cid:0) C + K (cid:1) E ( f ) . (B.8)This bound implies that both ( df ∧ Ω ( s ) ) and D ∗ ( df ∧ ∗ Ω ( s ) ) have finite L norms. 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