§ INTRODUCTION

The swampland program [1, 2]  (for reviews see [3, 4]) aims to characterize the sparse subset of effective field theories which arise
as low-energy limits of consistent theories of quantum gravity 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 conditions (the “swampland conjectures” [1, 2, 3, 4]) that all effective theories of quantum gravity should
obey. These conditions are motivated by general physical considerations (in particular the thermodynamics of black holes [5]) as well as by lessons drawn from  the large supply of consistent effective theories of gravity which describe the light degrees of freedom in a controlled vacuum of superstring theory (the string lamppost principle (SLP) [6]). We have full analytic control on the quantum stability of a candidate string  vacuum only when it preserves some supersymmetry,
and we can write a precise effective Lagrangian only when its couplings are protected by a SUSY non-renormalization theorem:
so all reliable examples at our disposal have extended supersymmetry.[ This limitation in the available explicit examples may a priori lead to some bias in our understanding of quantum gravity.]

The best understood examples are the effective theories of Type IIB compactified to 4d on some Calabi-Yau (CY) 3-fold: these examples played a major role in the development of the swampland program
[2]. The resulting low-energy 4d $\cn=2$ supergravities are described
(in the vector sector) by special Kähler geometry [7, 8].
Special Kähler geometry is a very powerful tool to study the quantum consistency
of an $\cn=2$ supergravity; a significant part of the swampland program
[9, 10, 11, 12, 13, 14, 15, 20, 19] is dedicated to the detailed analysis of the “motivic”
special Kähler manifolds which describe the moduli geometry of actual CY manifolds (or motives).  In our opinion,  the swampland program has reached a rather satisfactory shape in this specific $\cn=2$ 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 to broader contexts.

Supersymmetric effective theories are better behaved in many ways, and SUSY
seems to play a crucial role in the consistency of many explicit examples. In some contexts  [21] it is hard to grasp how a model can possibly be consistent  without being supersymmetric.

On the other hand, the real world does not look supersymmetric at low energy, and hence consistent non-supersymmetric effective theories of gravity ought to exist. An important goal of the swampland program is to say something less vague about the non-SUSY effective theories of consistent quantum gravity. They are expected to be quite rare and remarkable animals, in a sense
even more “magical” than their SUSY counterparts.
Morally speaking, these theories should enjoy
“all the good properties” of SUSY while avoiding its phenomenological drawbacks as the existence of (unobserved) super-partners. We should not expect such “magical” theories to share the properties which generically hold
for a non-SUSY field theory, because they are extremely non-generic. From this point of view, the highly non-generic value of the real-world effective cosmological constant $\Lambda$ 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 continues to 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 geometric 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 are described by a domestic geometry
in the same exact way that the vector-multiplet sector of 4d $\cn=2$ 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 the obvious
domestic generalization of the class of “motivic” special Kähler geometries which describe the vector-sector of quantum-consistent 4d $\cn=2$ supergravities.
This allows to phrase the swampland conditions for the general SUSY model
in the same suggestive language used in the $\cn=2$ 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 formal amplitudes 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 Kähler geometry (the “power of holomorphy”  [22])
hold for all OV manifolds even if they do not carry any complex structure.

Our second main purpose is to present some novel speculations about the
swampland conditions in the non-SUSY case.
The space of light scalars $\mathscr{M}$ 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 statements make
perfect sense
for non-SUSY effective Lagrangians.
One is then led to speculate that domestic geometry – which applies
to all SUSY cases – may also be relevant to describe vectors' couplings
in quantum-consistent
non-SUSY 4d effective theories. The speculation may be stated
at different levels of precision.
Domestic geometry may be: (a) just qualitatively
valid, or (b) semi-quantitatively correct, or even
(c) exact.
Besides its aesthetic geometric appeal, and the evidence from SUSY examples,
the speculation rests on
some heuristic physical arguments, based on the idea of “naturalness” in the IR description,
which suggest that
at least
version (b) should hold.

Warnings to the reader.
The materials presented in this paper are
of different nature and logical status. There are: (a) mathematically rigorous
geometric constructions/results; (b) physical statements which aim only
to a physicists' level of rigor; (c) widely accepted swampland conjectures (which
are taken as “facts” in most of the recent literature); (d) newly proposed
swampland conjectures which are still open to discussion by the community; (e)
speculative statements which are meant as an invitation to further work
in the hope of re-formulating them as precise swampland conjectures in the near future.
To make the distinction clear
to the reader,
statements related to (a) are labelled Theorem, Lemma, and so.
Most statements in the logical classes (b) and (c) are dubbed Facts.
New conjectures and speculations are called by their proper name.

In the paper we often use the phrase:  or similar ones.  Since there is
an ongoing debate in the community about which examples should be considered
“reliable”, we need to specify the class of examples we have in mind. Our notion of
reliability is the most conservative one, that is, we restrict to examples which are
under full analytic control and are pacifically considered “reliable” by all experts in the field.

Organization of the paper.
The rest of this paper is organized as follows. In section 2 we discuss some general properties of 4d effective field theories and some technical aspects of the singularities and asymptotics of the moduli spaces. In section 3  we define the OV manifolds and discuss their basic properties. In section 4 we describe the gauge couplings seen as a map
from the moduli space to the Siegel variety. Section 5 contains the basics of
domestic geometry; a large part of the section is a detailed review of
(generalized) $tt^*$ geometry which is the model which inspires
all constructions in domestic geometry.  Section 6 describes how
domestic geometry applies to all 4d SUGRAs and may be used to reformulate the swampland conditions in the SUSY context in a more convenient way. Section 7 describes in more detail domestic geometry,
presenting the math arguments for the existence and uniqueness of the underlying tamed maps, and
rephrasing them as heuristic physical arguments in favour of our speculation that
arithmetic domestic geometry is relevant for the non-SUSY swampland program.
Some additional technicality is confined in the appendices.
The recent papers [17, 18] describe a different uniform geometric approach to supergravity, dubbed bosonic supergravity. It would be interesting to understand geometric aspects of the swampland conditions from that perspective.

§ GENERALITIES ON EFFECTIVE THEORIES

§.§ Our set-up

To make the story a bit shorter, in this paper we consider only four-dimensional effective theories. Although methods and ideas apply to much wider contexts, we assume a vanishing cosmological constant, $\Lambda=0$, and focus on effective Lagrangians valid at parametrically small 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. Locally at generic points in field space we may choose
an electro-magnetic duality frame, with respect to which these d.o.f. are  described by Abelian gauge vectors $A^a$ ($a=1,\dots, h$) 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 $g_{\mu\nu}$, $h$ vector fields $A^a$, and $m$ massless real scalars $\phi^i$, together with
spin-1/2 fermions and possibly spin-3/2 gravitini (only in the SUSY case). In the Einstein frame (and the chosen electro-magnetic frame) the effective Lagrangian takes (locally in field space) the form

\begin{equation}\label{lag}
\mathscr{L}_\text{eff}= -\sqrt{-g}\Big(\frac{1}{2}R+\frac{1}{2}G(\phi)_{ij}\,\partial^\mu \phi^i\partial_\mu \phi^j -
\frac{i}{16\pi}\tau(\phi)_{ab}F^a_+F^b_++\frac{i}{16\pi}\bar\tau(\phi)_{ab}F^a_-F^b_-+\cdots\Big)
\end{equation}

where[ We use the shorthand $F^a_\pm F^b_\pm\equiv (F^a_\pm)_{\mu\nu}(F^b_\pm)^{\mu\nu}$.] $F_\pm^a =\tfrac{1}{2}(1-\mp i\,\ast)F^a$ is the (anti-)self-dual part of the field-strength $F^a$.

The swampland program asks for a characterization of the field-dependent couplings
in the Lagrangians $\mathscr{L}_\text{eff}$ which describe low-energy limits  of consistent quantum gravities.
In practice, one looks for necessary conditions they should satisfy, which
are usually phrased as sufficient conditions for the model (<ref>) to sink in the swampland.
In this paper we limit ourselves to the geometric part of the swampland program, i.e. to the two-derivative couplings
$G(\phi)_{ij}$ and $\tau(\phi)_{ab}$.
The characterization of the consistent scalars' kinetic couplings $G(\phi)_{ij}$ takes advantage from their geometrical interpretation [2].
The scalar fields $\phi^i$ are seen as local coordinates on a connected “manifold” $\widetilde{\mathscr{M}}$, of dimension $m$,
endowed with the Riemannian metric $G_{ij}\equiv G(\phi)_{ij}$.
The Riemannian metric is smooth at generic points of $\widetilde{\mathscr{M}}$.

In a generic non-SUSY effective field theory we do not expect
a non-trivial space $\widetilde{\mathscr{M}}$ of exactly massless scalars since the
flat directions of the potential are usually lifted 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 of this paper remain valid even if the scalars parametrizing $\widetilde{\mathscr{M}}$
are not exactly massless but only hierarchically lighter than the scale above which
the low-energy description breaks down.

§.§ The $U$-duality group

A basic datum of the effective field theory is its gauge group.
The discrete part of the gauge group is called the $U$-duality group
$\cg$. $\cg$ is a redundancy of the description, so it is tautologically
an exact symmetry of the full quantum theory not just of its IR sector.
More precisely, by $\cg$ we mean the quotient of the full discrete gauge group of the underlying UV complete theory
which acts faithfully on the light bosonic fields.
In particular, $\cg$ acts on fluxes of forms of various degrees $k$

1/2π∮_Σ_k F^(k)
which are integral by generalized Dirac quantization:
the fluxes (<ref>) take value in a lattice $\Lambda$ endowed with
a non-degenerated bilinear form $\Lambda\otimes \Lambda\to \Z$ (the generalized Dirac pairing).
The completeness conjecture [5] states that in a consistent
theory all fluxes allowed by Dirac quantization are realized by some physical state.
In 4d the electro-magnetic charges take value in a lattice
$V_\Z\cong \Z^{2h}$ with a skew-symmetric principal[ Principal means that the
Dirac pairing $V_\Z\otimes V_\Z\to \Z$
yields an identification $V_\Z\equiv V_\Z^\vee$.] Dirac pairing. The action of $\cg$ on
the scalar fields and the electro-magnetic fluxes
defines a group embedding[ $\mathrm{Iso}(\widetilde{\mathscr{M}})$ is the isometry group of
$\widetilde{\mathscr{M}}$ with metric $G(\phi)_{ij}$.]
↪Iso(ℳ )×Sp(2h,)
The kernel of the map $\cg\to \mathrm{Iso}(\widetilde{\mathscr{M}}\;)$ is finite;[ Indeed the kernel is both compact and discrete.]
since we always work modulo finite groups,[ The symbol $\sim$ stands for equivalence up to commensurability [23] i.e. equality up to finite groups.] we shall not distinguish $\cg$ from its
image in $\mathrm{Iso}(\widetilde{\mathscr{M}}\;)$. The map
\cg\to Sp(2h,\Z)$ is called the monodromy representation, and
its image $\Gamma\subset Sp(2h,\Z)$ the monodromy group.
Hence the $U$-duality group $\cg$ is a group extension

that is,

$\cg \cong \cg^0\rtimes \Gamma$.

A version of the $\pi_1$-conjecture [2],
which holds true in all known examples, states that the $U$-duality group is isomorphic to
a subgroup

$\cg\subset GL(k,\Z)$ (for some $k$)
which is generated by $k\times k$ unipotent matrices (modulo finite groups). Thus $\cg$ is a subgroup of the $\mathbb{Q}$-algebraic
group $GL(k,\mathbb{Q})$, and in all examples[ The statement holds in all SUSY consistent theories
by a basic fact in domestic geometry, see .<ref>. The reliable examples are all supersymmetric.] its $\mathbb{Q}$-Zariski closure $\overline{\cg}^{\,\mathbb{Q}}$ is semi-simple. Hence, modulo finite groups,
$\cg\sim \cg^0\times \Gamma$.

The moduli space is
that is, the space of inequivalent (effective) vacua. The covering space $\widetilde{\mathscr{M}}$
may be assumed to be simply-connected with no loss. Then $\pi_1(\mathscr{M})\sim \cg$.

§.§ Singularities of moduli spaces

The purpose of this technical sub-section is to argue that
we may be cavalier with the singularities
of the moduli space $\mathscr{M}$,
and see in which sense
we may work
as if $\mathscr{M}$ was a good ($\equiv$ complete)
Riemannian manifold. The sub-section may be omitted in a first reading.

§.§.§ Singularities from the action of $\cg$

We write $d(\cdot,\cdot)\colon \widetilde{\mathscr{M}}\times\widetilde{\mathscr{M}}\to \R_{\geq0}$
for the distance function defined by the metric $G_{ij}$ and replace $ \widetilde{\mathscr{M}}$
by its metric completion.
Fix a non-trivial unipotent element $u\in \cg$; the $\pi_1$-conjecture [2, 3] says that
the infimum of the  function

d_uℳ→_≥0    d_u(x)def=d(ux,x)
is zero. Assuming the conjecture, we consider a sequence of regular points $\{x_i\}\subset \widetilde{\mathscr{M}}_\text{reg}$ such that $d_u(x_i)\searrow 0$.
We have two possibilities:

(1) $x_i$ escapes to infinity, i.e. $d(x_1,x_i)\to\infty$. We say that “$u$ has a fixed point at infinity”. The distance conjecture [2] requires an infinite tower of
states to become exponentially light as we approach the fixed point at infinity. Their electro-magnetic charges belong to

(2) $\{x_i\}$ remains inside some finite-radius ball $B(x_1,r)$, and hence contains a subsequence which converges to a finite-distance point $x_\ast \in\widetilde{\mathscr{M}}$ which is fixed by $u$.

In the second case
the action of $\cg$ yields an obstruction to a smooth extension of the Riemannian metric to $x_\ast$:
indeed its isotropy sub-group $\cg_{x_\ast}\subset\cg\subset \mathrm{Iso}(\widetilde{\mathscr{M}})$ is discrete and contains the infinite group of isometries $u^\Z$, whereas the isotropy sub-group of a regular point in Riemannian geometry is always compact. Hence $x_\ast$ is
a finite-distance singularity where the curvature blows up. In a quantum-consistent effective theory  with $\cn\geq2$ supersymmetry
all singularities of the completed covering space $\widetilde{\mathscr{M}}$
are of this kind, i.e. fixed-points of non-trivial unipotent elements $u\in\cg$, while the moduli space $\mathscr{M}\equiv\widetilde{\mathscr{M}}/\cg$ has, in addition, orbifold quotient singularities.
Thus in the extended SUSY case all finite- and infinite-distance singularities of $\mathscr{M}$
are dictated by the action of the $U$-duality group $\cg$ – they correspond to vacua where
a non-trivial subgroup of $\cg$ remains unbroken – and
no other “accidental” singularity is present.
The logic beyond the $\pi_1$ and distance conjectures suggests that this is the case for a general
consistent effective gravity theory:
all singular points are fixed by some non-trivial subgroup of the discrete gauge group $\cg$.

We give a closed look to the two kinds of finite-distance singularities.

Quotient orbifold singularities.
In general the $U$-duality group $\cg$ does not act
freely on the regular locus $\widetilde{\mathscr{M}}_\text{reg}\subset\widetilde{\mathscr{M}}$. The isotropy sub-group $\cg_{\tilde x}$ of a smooth point $\tilde x\in \widetilde{\mathscr{M}}_\text{reg}$ is finite. The image of such a point in the moduli space $\mathscr{M}$ is then a mild orbifold singularity.
Orbifold points correspond physically to vacua where a finite sub-group $\cg_{\tilde x}\subset\cg$ remains unbroken. In some stringy
examples such points correspond to vacua
with a non-Abelian enhancement of the effective continuous gauge symmetry,
i.e. loci where our standing assumptions A1, A2 break down and $\mathscr{L}_\text{eff}$ ceases to be a complete description of the IR physics.

When we have an embedding $\iota\colon \cg\to GL(k,\Z)$ we can “repair” the orbifold singularities in a cheap way.
Fix an integer $n\geq 3$; let $r_n\colon
GL(k,\Z)\to GL(k,\Z/n\Z)$ be the reduction $\mod n$, and write $\iota_n\equiv r_n\circ \iota$.
Consider the exact sequence of groups
1→_n → GL(k,/n)→1.
It follows from the Minkowski theorem that the matrix group $\cg_n$
is a normal subgroup of $\cg$ of finite index which is torsion-free, in facts neat [23].
This implies that $\cg_n$ acts freely on $\widetilde{\mathscr{M}}_\text{reg}$ so that
$\mathscr{M}_n\equiv \widetilde{\mathscr{M}}/\cg_n$ is a finite Galois cover of $\mathscr{M}$,
with Galois group $\iota_n(\cg)\equiv \cg/\cg_n$, free of orbifold singularities.
The finite-quotient singularities may be cured by replacing $\mathscr{M}$ with
$\mathscr{M}_n$ and $\cg$ with $\cg_n$: this is the standard strategy in the math literature
when studying moduli spaces of projective varieties (in particular of Calabi-Yau 3-folds), and we adopt it.
From now on by $\cg$ we always mean a finite-index, neat, normal subgroup of
the actual $U$-duality group. Correspondingly $\mathscr{M}\equiv \widetilde{\mathscr{M}}/\cg$ is free of finite-quotient

Finite-distance curvature singularities.
$\widetilde{\mathscr{M}}_\text{reg}$ is not geodesically complete in general;
that is, $\widetilde{\mathscr{M}}_\text{reg}$ may contain half-geodesics $\ell(t)$ originating from a smooth point $\phi_0\equiv \ell(0)$
which cannot be continued after some finite value of the proper length $t$. Such a finite-length maximal half-geodesic
represents a physical transition – which takes finite time and costs finite energy per unit volume  –
from our initial configuration $\phi_0$ to a physical situation where the IR description provided by  $\mathscr{L}_\text{eff}$
is no longer valid. This finite-time process should be perfectly regular from the viewpoint of the UV complete theory.
What happens is that $\ell(t)$ stops at a vacuum where “new physics” becomes relevant in the infra-red:
some additional degrees of freedom of the fundamental UV theory get massless, and
our IR description breaks down.
From the viewpoint of the UV fundamental theory, these singularities typically correspond to points where different branches of the
space of vacua meet each other (transition points).
Since the process involves a finite energy density,
one expects that there exists a refined effective Lagrangian $\mathscr{L}_\text{new}$ which includes the relevant
“new physics”
and is valid up to  some higher but still finite energy scale. $\mathscr{L}_\text{new}$ allows us to extend the IR description, and hence the physical process described by the half-geodesic $\ell(t)$, beyond
the domain $\widetilde{\mathscr{M}}_\text{reg}$. This means that family of finite-length half-geodesics in $(\widetilde{\mathscr{M}}_\text{reg}, G_{ij})$
ending at a given singular point $x_\ast$ is associated to
a finite number of new states becoming massless; in addition their spins must be $\leq \tfrac{3}{2}$. This is to be contrasted with the case of a fixed point at $\infty$, where
an infinite tower of states get light [2].

We expect all finite-distance curvature singularities to correspond to fixed points
$x_\ast$ under a parabolic subgroup of $\cg_\ast\subset \cg$
(cfr. the discussion after eqn.(<ref>)). We have a finite set of vectors $\{q\}\subset V_\Z$
which correspond to the electro-magnetic charges of the finitely-many states which becomes
massless at $x_\ast$. They should be invariant under $\cg_\ast$ (modulo finite groups),
i.e. $\rho(u)\,q=q$ for $u\in \cg_\ast$.
At such a singularity, the metric is continuous
(in appropriate local coordinates) but the curvature invariants blow-up.
If our effective theory has $\cn\geq5$ supersymmetry, there is no matter supermultiplets which may become massless, so
no finite-distance curvature singularities may be present. The same holds for $\cn=3,4$ as a result of a SUSY non-renormalization theorem.[ The curvature singularity is proportional to the contribution of the gauge coupling beta-function from the states (with e.m. charges $\{q\}$) which become massless at $x_\ast$, so it vanishes for $\cn=3,4$ where the only matter supermultiplets are vector-multiplets which yield a zero net contribution to the $\beta$-function.
Geometrically, this non-renormalization theorem corresponds to the fact that, for $\cn\geq3$ SUGRA, $\widetilde{\mathscr{M}}$ is a symmetric space,
whose curvature is parallel, $\nabla_i R_{jklm}=0$, so cannot blow up anywhere.]

§.§.§ Example: Type IIB on a 3-CY

The various kinds of singularities are well illustrated by the vector-scalars' moduli space $\mathscr{M}_v$ of Type IIB compactified on a Calabi-Yau 3-fold $X$. The regular locus of its covering space, $\widetilde{\mathscr{M}}_{v,\,\text{reg}}$, may be identified with the
moduli space of marked and polarized Calabi-Yau's in the smooth deformation class of $X$.
$\widetilde{\mathscr{M}}_{v,\,\text{reg}}$ is equipped with its Weil-Petersson (WP) Kähler metric $G_{i\bar j}$,
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 a conifold transition (more generally, an extremal transition) to a CY with different Hodge numbers [24]. According to a celebrated suggestion by Reid [25], all CY
moduli spaces are expected to be connected through such transitions.  The full Type IIB string theory remains regular
at those transitions, but the low-energy description based on the effective $\cn=2$ supergravity breaks down.
On $\widetilde{\mathscr{M}}_{v,\,\text{reg}}$ there is also another, better behaved, canonical Kähler metric $K_{i\bar j}$, the Hodge one (a.k.a. the $K$-metric [26, 27]),
which has the expression (with $n=\dim_\C\mathscr{M}_v$) [26, 27, 28, 29]

\begin{align}\label{kkaq123}
K_{i\bar j}&= (n+3)G_{i\bar j}+R_{i\bar j}\geq 2 \,G_{i\bar j},\\
R_{i\bar j}&\geq -(n+1)G_{i\bar j},\label{brrricssi}
\end{align}

where $R_{i\bar j}$ is the Ricci curvature of the WP metric $G_{i\bar j}$. It is important to notice that, while the Ricci curvature[ More in detail, each holomorphic sectional curvature of $G_{i\bar j}$ is bounded below by $-2$.]
$R_{i\bar j}$ of the WP metric satisfies the lower bound (<ref>), the Ricci curvature of the $K$-metric
$R^K_{i\bar j}$ satisfies an upper bound [28]

R^K_ij̅≤-1/(√(n)+1)^2+1 K_ij̅.
From eqn.(<ref>) we see that points at infinite distance in the WP metric are also at infinite distance 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 points the metric $G_{i\bar j}$ remains bounded while the Ricci curvature blows up so that the $K$-metric blows up.

The Torelli space is the completion of $\widetilde{\mathscr{M}}_{v,\,\text{reg}}$
with respect to the $K$-metric [30, 31, 32]
ℳ^4muK_v≡(ℳ_v, reg)^K-metricm|
$\widetilde{\mathscr{M}}^{\mspace{4mu}K}_v$ is a smooth space diffeomorphic to $\R^{2m}$
[30, 31, 32]; since $\cg$ is torsion-less,
the space
ℳ^K_v≡ℳ^4muK_v/is a version of a finite cover of the moduli space which is a smooth Kähler manifold, complete for the $K$-metric,
with a contractible universal cover. Its fundamental group $\pi_1(\mathscr{M}^K_v)\cong \cg$ satisfies (a refined version of) the $\pi_1$-conjecture [2]. Unfortunately the nice manifold $\mathscr{M}^K_v\equiv \widetilde{\mathscr{M}}^{\mspace{4mu}K}_v/\cg$
is not complete for the physical WP metric.

$\mathscr{M}^K_v$ is the natural space to parametrize the complex structures on a fixed smooth-class of CY 3-folds: $\mathscr{M}^K_v$ does not talk to moduli spaces of CY's with different topologies which are pushed infinitely away in the $K$-metric. The physical moduli space of Type IIB compactified on $X$ instead
consists of several branches of vacua, with extremal transitions between them;
since the different branches are not
infinitely separated, the physically relevant metric $G_{i\bar j}$ on each branch cannot be complete.

There is another description of these moduli spaces more in the spirit of Algebraic Geometry.
There is a compact projective variety $\bar{\mathscr{M}}_v$ and an effective divisor $D$ such that

By Hironaka theorem, we can choose the pair $(\bar{\mathscr{M}}_v, D)$ so that $\bar{\mathscr{M}}_v$ is smooth and
$D=\sum_i D_i$ is a simple normal crossing divisor.  The points in the support of $D$ are at infinite distance in the $K$-metric.
With respect to the WP metric $D$ splits as $D=D_\infty+D_f$, where $D_f$ (resp. $D_\infty$) is the singular locus at finite (resp. infinite)

§.§.§ Smoothing surgery

Returning to the general case, the presence of finite-distance curvature singularities
in $\widetilde{\mathscr{M}}$ seems to be unavoidable when $m\geq2$, unless the low-energy theory is a $\cn\geq3$ supergravity or some truncation thereof.
In all other cases the moduli space cannot be both
smooth and complete for the kinetic-terms metric $G_{ij}$.

Working with non-complete and/or non-smooth Riemannian spaces is technically inconvenient. We try to improve
the situation by smoothing out the singularities with some local surgery,
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 the analysis of the geometry
of the moduli space $\mathscr{M}$.
However it is suggestive to phrase the surgery as it was an actual modification of
the effective Lagrangian $\mathscr{L}_\text{eff}$.
The modification would be almost “harmless”, since the original Lagrangian
itself gave a poor description of the IR physics
near the singularity in field space,
so the local modification affects physical processes
which were already outside the scope of $\mathscr{L}_\text{eff}$.
In a sense the locally modified Lagrangian is still a “good”
effective Lagrangian, and should satisfy the swampland consistency conditions
as far as they do not involve the region near the
“bad”  points in field space.

The allowed deformations of $\mathscr{L}_\text{eff}$ are quite restricted since
they should preserve all gauge symmetries.
In particular: (1) the deformation of the kinetic-terms metric should leave
the discrete gauge group $\cg$ as an exact symmetry of the problem;
(2) the holonomy and isometry Lie algebras of $G_{ij}$,
$\mathfrak{hol}(\mathscr{M})$ and $\mathfrak{iso}(\mathscr{M})$,
should be preserved.

Under the hypothesis that all singularities are fixed by a subgroup $\cg_\ast$ of the neat group
it suffices to modify the metric on the universal cover $\widetilde{\mathscr{M}}$ in such a way
that $\cg$ is a freely acting group of isometries of the deformed metric.
Since the Ricci curvature is expected to blow up at a fixed point of an unipotent isometry,
a perturbation of the form

G_ij→G^ϵ_ij≡G_ij+ϵ  R_ij,  ϵ>0
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 the metric is negligible for $\epsilon$ very small, so (<ref>) is essentially a local modification of the geometry
around the finite-distance singular loci. $\cg$ is still an exact isometry of the perturbed metric, so we do not spoil the discrete gauge symmetry. More generally, the surgery (<ref>) does not spoil any symmetry the original geometry may have.
Condition (2) is also satisfied.[ Let us check that the holonomy algebra  $\mathfrak{hol}(\mathscr{M})$ is preserved.
We may assume $\widetilde{\mathscr{M}}$ to be irreducible without loss.
If the Riemannian metric $G_{ij}$ is locally symmetric, it is Einstein, and the modification (<ref>) is just
a change of overall normalization of the metric by a factor $1+O(\epsilon)$. Otherwise $\mathfrak{hol}(\mathscr{M})$
is one of the seven Berger holonomies [33]. For generic holonomy $\mathfrak{so}(m)$ there is nothing to show. If $G_{ij}$ is Kähler, so
is $G^\epsilon_{ij}$. For $\mathscr{M}$ a Calabi-Yau, hyperKähler, quaternionic-Kähler, $G_2$- or $Spin(7)$-manifold,
$G_{ij}$ is either Ricci-flat or Einstein so the deformation (<ref>) is either trivial or a slight rescaling of the metric.] We can see the modification (<ref>) as the result of a
backward Ricci-flow [34] of the metric  by the small time $t=-\epsilon/2$, so that, heuristically, it looks like
a RG flow to a slightly larger energy scale, in line with the physical interpretation of the finite-distance singularities
as loci where new physics comes in.
In view of eqn.(<ref>), replacing the WP metric $G_{j\bar k}$ by the Hodge one $K_{j\bar k}$ on
a 3-CY moduli space amounts to the surgery (<ref>)
with $\epsilon=1/(m+1)$; indeed our proposed prescription (<ref>) is modelled
on the standard math treatment of 3-CY complex moduli spaces.

The modification (<ref>) makes sense provided  the Ricci curvature is bounded below, $R_{ij}\geq -K G_{ij}$, so that
$G^\epsilon_{ij}$ is positive-definite for small $\epsilon$. The idea is that the modification replaces
a small region around the singular point $x_\ast$ by a cusp of infinite length but finite volume[ The basic reason of the finiteness of the volume is that we mod out the infinite group $\cg_\ast$ which maps a small neighborhood of $x_\ast$ into itself.]
of order $O(\epsilon^k)$ for some $k>0$. When this happens, the singularity is pushed at infinite distance, and geodesic completeness is restored. Thus the volume conjecture still holds after the surgery, however the new “spurious”
cusps are not associated to towers of light states as the genuine infinite ends of $\mathscr{M}$.

In  this paper we assume that it is always possible to modify the metric locally at the singularities, while preserving $\cg$ and $\mathfrak{hol}(\mathscr{M})$,
by replacing a neighborhood of the finite-distance singularity with a cusp of volume $O(\epsilon^k)$, so that the resulting Riemannian space is complete and smooth.
All our arguments below are meant to apply to the “regularized” moduli manifold so constructed, which we shall denote simply $\mathscr{M}$.
In the known examples the assumption holds true.

After the modification the geometric swampland conjectures still hold if they were satisfied by the original
$\mathscr{L}_\text{eff}$, with the only exception that the distance conjecture does not apply to the spurious infinite ends
introduced by blowing-up finite-distance singularities.

§.§ Behavior at infinity

In some argument below we need some more technical aspect
of the geometry of $\mathscr{M}$ at infinite distance.
In this sub-section we sketch the main issues; it  may be omitted in a quick reading.

§.§.§ Sign of curvature at infinity

In their original paper [2] Ooguri and Vafa conjectured that
the scalars' space $\mathscr{M}$ is non-compact of finite-volume. They also
conjectured that the moduli-space scalar curvature $R$ is negative at infinity.
In ref.[36] Trenner and Wilson constructed an explicit “counter-example”[ See Theorem 3.2 of ref.[36].] to the last statement in the context of
Type IIB on a certain 3-CY with $h^{2,1}=3$. In that example there is a real curve $C$ in moduli space, parametrized by $s\in\R$,
such that, as we approach a “large complex structure limit” along this curve, the WP scalar curvature $R$ behaves as [36]
R= 32/81 s^3+O(s^2)  as s→+∞ along C,
so in this limit $R$ is positive and unbounded. We wrote “counter-example” between quotes
because this example does not contradict the physical picture of [2].
In the language of eqn.(<ref>), the physical intuition for Calabi-Yau moduli spaces goes roughly as follows: as we approach a generic point on the divisor $D_\infty\subset \bar{\mathscr{M}}$ (the infinite-distance locus)
the Ricci curvature of the WP metric becomes negative hence bounded by (<ref>), while as we approach
a generic point on $D_f$ (finite-distance singularities) $R_{ij}$ becomes positive and divergent because of the contributions from
loops of the finitely many additional light particles which can be computed in some “enlarged” effective field theory.
What happens at the special points at infinity $D_\infty \cap D_f$?
The obvious guess is that if we approach the intersection point following a curve $C$ along which
m_f→0,  m_∞→0  with  m_f/m_∞→0,
where $m_f$ (resp. $m_\infty$) is the mass scales of the particles getting light along $D_f$ (resp. $D_\infty$),
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 states along $D_\infty$.
This is what happens in the Trenner-Wilson example; along their curve $C$
$m_f=O(s^{-2})$ and $m_\infty=O(s^{-1})$.
We see that the  non-positivity of $R$ at infinity has the same physical origin as the failure of the WP
metric to be complete [24].
Then after replacing $G_{i\bar j}$ with its regularized version $G^\epsilon_{i\bar j}$, which it
makes $\mathscr{M}$ 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 $R^\epsilon$
to be negative and bounded (for fixed $\epsilon$) everywhere at infinity.
Indeed, along $D_\infty$ the $\epsilon$-modification is inessential while along $D_f$ we have
$G^\epsilon_{i\bar j}\approx \epsilon\, K_{i\bar j}$ so that from eqn.(<ref>) the scalar curvature is
asymptotically negative and large (of order $O(1/\epsilon)$). As a check we have computed (using Mathematica)
the scalar curvature $R^\epsilon$
in the Trenner-Wilson example along the curve $C$:
R^ϵ= -6/ϵ(1+O(ϵ^2))+O(1/s),  as s→+∞ along C.
In other words: the points on the curve $C$ for sufficiently large (but finite) $s$
are outside the region where the Lagrangian $\mathscr{L}_\text{eff}$
yields a reliable IR description of the physics, and the coupling $G_{i\bar j}$
(i.e. the WP metric) needs not to behave in a “physically reasonable” way at these
Fix a regular point $p_0\in\mathscr{M}$; for all points at a sufficiently large
distance from $p_0$ which are not too close to special loci where
some “new physics” appears, $R$ is negative.

§.§.§ Cusps

We first consider the following simple but typical[ Typical means, in particular, that this is the situation in all known explicit examples.] situation:
$M$ is a complete non-compact Riemannian manifold and
there is a compact subset $K\subset M$ such that
$M\setminus K$ is the disjoint union of finitely many “ends at infinity” of $M$,
the $\alpha$-th end $E_\alpha$ being diffeomorphic to $\R\times Z_\alpha$ for some connected  manifold $Z_\alpha$, while the metric in $E_\alpha$ has the asymptotic form

\begin{equation}\label{rt5167}
ds^2\equiv G_{ij}\,dx^i dx^j\approx dr^2+g(r,u)_{ab}\, du^a du^b,\qquad \text{for }r\gg1,
\end{equation}

where $r$ is the distance from some base point $\ast\in M$, $u^a$ are local coordinates in $Z_\alpha$, and $g(r,u)_{ab}$ is some $r$-dependent metric on $Z_\alpha$.
Finiteness of the volume of $E_\alpha$ requires $\sqrt{\det g(r,u)}$ to decay more rapidly than
$1/r$ for large $r$. In the region where (<ref>) holds we have
R_rr≈-∂_r^2log√(g)-1/4∂_r g^2
If $g(r,u)_{ab}$ goes to zero as
slowly as a negative power of $r$, the rhs is $O(1/r^2)$
and $R_{rr}/g_{rr}$ is not bounded away from zero.
On the other hand if $g(r,u)_{ab}$ goes to zero more rapidly than an exponential,
say as $\approx\! C\, e^{-c\, r^k}$ with $k>1$ the curvature $R_{rr}$ is
$O(r^{2(k-1)})$ and unbounded below for large $r$.
So, if the Ricci tensor is negative and bounded for large $r$, $g(r,u)_{ab}$ should
be a sum of terms with exponential decay

g(r,u)_ab≈∑_i e^-c_i r h^(i)-2mu(u)_ab.
Assuming the asymptotic metric to be enough regular, this leads to bounds for large $r$ of the form

- K_1  G_ij≤R_ij≤-K_2  G_ij    for large $r$ along $E_\alpha$
for some constants $K_1,K_2>0$. We shall call a finite-volume end $E_\alpha$
with the behaviour (<ref>) a “cusp”.
Prototypical examples are the cusps in an arithmetic quotient of
a non-compact symmetric space[ Here $G(\R)$ is a non-compact real Lie group
seen as a concrete group of matrices via a suitable representation of degree $\ell$, $K\subset G(\R)$ is a maximal compact
subgroup, and $G(\Z)\equiv G(\R)\cap GL(\ell,\Z)$ is the arithmetic subgroup consisting of matrices with integral entries.
More generally, we may replace $G(\Z)$ by a commensurable subgroup of $G(\R)$.]

Eqn.(<ref>) is the general form of the moduli space $\mathscr{M}$ when $\mathscr{L}_\text{eff}$
has a large supersymmetry (more than 8 supercharges).
In these cases the nice geometry of the ends of the moduli space
is directly related to the physics of quantum gravity as
described by the  distance conjecture [37]. In these extended susy
examples the $U$-duality group $\cg\equiv G(\Z)$ acts faithfully
(modulo finite groups) on
the electro-magnetic charge lattice $V_\Z$,
and each point at infinity $x_\infty\in\overline{\mathscr{M}}$ is fixed by a parabolic
subgroup $\cg_{x_\infty}\subset G(\Z)$.
States carrying electro-magnetic charges $q\in V_\Z$
invariant under  $\cg_{x_\infty}$ have masses proportional to the length
of the image in the Siegel domain of the shortest loop in $\mathscr{M}$
based at $x$ in the homotopy class of the elements of $\cg_{x_\infty}$;
since the map $\mu$ is a totally geodesic isometric embedding
for $\cn\geq3$ the mass is also proportional to the length of the pre-image loop
in $\mathscr{M}$, which for a good cusp is exponentially small, cfr. eqn.(<ref>).
The same applies (with some subtlety) in the $\cn=2$ case,
using the relation between the kinetic terms of scalars and vectors
implied by SUSY which replaces the totally geodesic embedding condition.
Conjecturally this extends to the general quantum-consistent effective theory: the length of the loop in target space should
be exponentially small to fit with the predictions of the distance conjecture.
If the pre-image loop has a length which vanishes more rapidly than any
exponential, say $O(e^{-c r^k})$, $k>1$,
the derivative would be of order $e^{ c r^k}$,
$\|d\mu\|^2 =O(e^{3 c r^{k}})$ which looks unreasonable.

Condition $\boldsymbol{(\ast)}$.
Although the large $r$ behaviour (<ref>)
is expected for all quantum-consistent effective theories, to be very conservative
in this paper we shall assume a
much weaker condition on the large $r$ behaviour of the geometry
after the smoothing surgery in .<ref>.
First we assume that
the Ricci curvature of $\mathscr{M}$ is still bounded below
$R_{ij}\geq - K G_{ij}$.
Since $\mathscr{M}$ is complete, for all $R>0$
there exists a Lipschitz continuous function $h_R\colon \mathscr{M}\to \R$ such that for some fixed constant $k > 0$ [38]:[ For instance, we can choose $h_R=\varphi(r/R)$ where $\varphi$ is a smooth function on the real line with $0\leq \varphi\leq 1$, $\varphi=1$ for $x\leq 1$ and $\varphi=2$ for $x\geq2$.]

0≤h_R≤1,    h_R= 1    for r≤R

0    for r≥2R,
For typical asymptotic metrics of the form (<ref>),(<ref>)
the Laplacian of $h_R$ is of order $O(R^{-1})$ for large $R$.
We shall require only the much weaker

The Laplacian of $h_R$ is bounded by a constant $C$ independent of $R$

§ OV MANIFOLDS

In this section we assume that there exists a suitable local surgery, along the lines
described in the previous section, such that the singularities of a suitable finite cover of
the moduli space get repaired resulting in a smooth Riemannian manifold $\mathscr{M}$
which still satisfies the Ooguri-Vafa geometric swampland conjectures [2]. This certainly holds in the
known examples of quantum consistent effective theories of gravity.

A smooth manifold which satisfies the Ooguri-Vafa geometric conditions, together with some
mild “regularity” conditions, will be called
an OV manifold. Understanding the geometry of OV manifolds and its physical implications
is one of the themes of this paper.

We propose the following definition of OV manifold:

The point is a zero-dimensional OV space. In positive dimension an OV manifold is
a connected, complete, Riemannian manifold $M$ with a smooth simply-connected cover $\widetilde{M}$
which has a de Rham decomposition of the form
M= F×M_1×⋯×M_s
and a smooth finite cover of the form

M^♭= F×M_1/𝒢_1×⋯×M_s/𝒢_s,    𝒢_k⊂Iso(M_k)
such that:

OV0. the flat factor $F$ is either trivial or the real line $\mathbb{R}$;

OV1. $M_k\equiv \widetilde{M}_k/\mathcal{G}_k$ is a complete, irreducible, non-compact manifold of finite volume;

OV2. $\widetilde{M}_k$ is diffeomorphic to $\mathbb{R}^{m_k}$
and $\mathcal{G}_k\cong\pi_1(M_k)$ is a torsion-less discrete group of isometries of $\widetilde{M}_k$
generated by elements $\{u_i\}$ such that

x∈M_kinf  d(u_i x,x) =0.

OV3. the Ricci curvature $R_{ij}^{(k)}$ of $M_k$ is bounded below by a negative constant

R_ij^(k)≥- K_k g_ij^(k),    K_k>0,
and condition $\boldsymbol{(\ast)}$ (eqn.(<ref>)) is satisfied.

Some comments on the definition are in order:

* OV1 is the volume conjecture [2] and OV2 is a refined version of the $\pi_1$ conjecture. A stronger version of OV2, which holds in all known examples, would be:

OV2$\boldsymbol{\ast}$ $\cg_k$ is
isomorphic to a strongly approximant[ We recall the definition of “strong approximant” subgroup. We see the arithmetic group $G(\Z)$ as a concrete group of integral matrices, i.e.
it comes with a preferred embedding $G(\Z)\subset GL(n,\Z)$ 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 $G(\Z)\xrightarrow{\pi_p} G(\Z/p\Z)$. $\gamma\colon \Gamma\hookrightarrow G(\Z)$ is said to be a strong approximant iff the group homomorphism $\pi_p\circ \gamma\colon \Gamma\to G(\Z/p\Z)$ is surjective for almost all primes $p$.] subgroup $\mathring{\cg}_k$ of an arithmetic group $G(\Z)\subset GL(n,\Z)$. $\mathring{\cg}_k$ is required to be neat, semi-simple, and to have a finite-index subgroup generated by finitely-many unipotent elements $\{u_i\}\subset \mathring{\cg}_k\cong \cg_k$ which satisfy eqn.(<ref>).

* the point and the real line $\mathbb{R}$ are OV manifolds. This is required by math elegance and is consistent since
all the `magic' properties of OV manifolds
are shared by the point and $\mathbb{R}$.  The point and $\mathbb{R}$ do appear as moduli of consistent gravities,
and even as their factor spaces: think of M-theory on $\mathbb{R}^{10-k,1}\times S^k$ for $k=0,1,2$; in particular
for $k=2$ $\mathscr{M}= \mathbb{R}\times  (SL(2,\mathbb{Z})\backslash SL(2,\mathbb{R})/U(1))$.

* OV3 is a milder version of the “regularity” conditions discussed in . <ref> related to the distance conjecture and the expected behaviour at infinity.

The emergence proposal (.5 of [4])
gives a model-independent bridge between the distance conjecture and the
geometry of $\mathscr{M}$ at infinite distance.
One starts with a compactification $\overline{\mathscr{M}}$ of $\mathscr{M}$;
metric is singular along the loci in $\overline{\mathscr{M}}$ where more
degrees of freedom get massless: the singular contributions to the metric come
from loops of light states. When an infinite tower of
d.o.f. get massless the divergence of the metric is so severe that the locus is pushed at infinite distance.
singular part of the metric due to a infinite tower of light states
is universal, and can be read from any example. Thus one gets back the strong
version of OV3 inferred from examples in . <ref>: there exists a $h_R$ as in (<ref>) with $|\Delta h_R|\lessapprox c/R$. We opted for the
much weaker version $|\Delta h_R| < c^\prime$: we
see OV3 as a mere technical condition, not an additional swampland conjecture, and our OV3 is just the weakest assumption
required to prove
“magical” properties of the OV manifolds.

§.§ First consequences of the definition

§.§.§ Holonomy groups and the graded algebra $\cp^\bullet$

A non-compact, complete manifold of non-negative Ricci curvature cannot have finite volume[ This follows from the Calabi-Yau lower bound on the volume [38, 39] see Theorem 7 and Appendix (iii) in ref.[38].], so the Ricci  curvature of the irreducible factor spaces
$\widetilde{M}_k$ somewhere should have some strictly negative eigenvalue.
This rules out a few Riemannian geometries:

\begin{tabular}{cc||c}\hline\hline
allowed $\widetilde{M}_k$  & $\mathfrak{hol}(\widetilde{M}_k)$ & ruled out \\\hline
strictly generic holonomy & $\mathfrak{so}(m_k)$ & Calabi-Yau\\
strictly K\"ahler  & $\mathfrak{u}(m_k/2)$ & hyperK\"ahler\\
strict negative quaternionic-K\"ahler & $\mathfrak{sp}(1)\!\oplus\mathfrak{sp}(m_k/4)$ & positive quaternionic-K\"ahler\\
non-compact symmetric spaces $G/K$ & $\mathfrak{k}$ & $G_2$ and $Spin(7)$ manifolds\\\hline\hline
\end{tabular}

It 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 for supergravity with more than 2 supercharges. We recall that the holonomy algebra
$\mathfrak{hol}(\mathscr{M})$ of the scalars' manifold $\mathscr{M}$ of a supergravity is determined as follows:
from its SUSY algebra and super-multiplet content we read the R-symmetry Lie algebra $\mathfrak{r}$ and its representation $\sigma$ on the scalar fields;  $\mathfrak{hol}(\mathscr{M})$ then has the form [37] (here $m=\dim_\R\mathscr{M}$)

σ(𝔯)⊆𝔥𝔬𝔩(ℳ)⊆σ(𝔯)+𝔷(σ(𝔯))⊂𝔰𝔬(m)    𝔷(σ(𝔯))def≡centralizer of $\mathfrak{r}$ in $\mathfrak{so}(m)$.
In SUGRA the R-symmetry is a gauge symmetry with composite
gauge fields $\partial_\mu \phi^i\, {\Omega(\phi)_i^a}_b$, where ${\Omega(\phi)_i^a}_b$ is the projection of the Levi-Civita connection of $G(\phi)_{ij}$ on the sub-algebra $\sigma(\mathfrak{r})$.

Let $\mathfrak{h}_k\equiv\mathfrak{hol}(\widetilde{M}_k)\subset \mathfrak{so}(m_k)$ be the irreducible holonomy algebra of the $k$-th factor space $\widetilde{M}_k$.
The graded algebra $\cp_k^\bullet$ of parallel forms on $\widetilde{M}_k$ is[ If $V$ is a vector space carrying a representation of the reductive Lie algebra $\mathfrak{h}$, the symbol $V^\mathfrak{h}$
denotes the $\mathfrak{h}$-invariant subspace of $V$, that is, the subspace on which $\mathfrak{h}$ acts trivially.]
_k^∙≡⊕_j=0^m_k _k^j ≅⊕_j=0^m_k(∧^j ^m_k)^𝔥_k
When $\mathfrak{h}_k=\mathfrak{so}(m_k)$, $\cp_k^\bullet$ is spanned by 1 and a volume form $\varepsilon$:
we say that $\cp_k^\bullet$ is trivial.
When $\widetilde{M}_k$ is strictly Kähler, $\cp_k^\bullet=\R[\omega]/\omega^{1+m_k/2}$ with $\omega$ the Kähler form.
When $\widetilde{M}_k$ is strictly quaternionic-Kähler $\cp_k^\bullet=\R[\Omega]/\Omega^{1+m_k/4}$  with $\Omega$
the canonical 4-form.
When $\widetilde{M}_k$ is a symmetric manifold the algebra $\cp_k^\bullet$ is typically larger;  but there are two exceptions: $SO(n,1)/SO(n)$ which has generic holonomy $\mathfrak{so}(n)$, and $SU(m,1)/U(m)$ which has strict Kähler holonomy $\mathfrak{u}(m)$. We stress that the surgery of . <ref> preserves
the algebra $\cp^\bullet$.

The generic OV manifold has no non-trivial parallel form. Let us consider an example in the other extremum, where $\cp^\bullet$ is so rich that fully determines the metric
(up to overall normalization).

Suppose the irreducible OV manifold $\mathscr{M}$ has R-algebra $\mathfrak{r}=\mathfrak{su}(8)$ and R-representation $\sigma$ the 70. Then – up to finite covers – $\mathscr{M}$ is globally isometric to the
locally symmetric space
where $E_{7}(\Z)$ denotes a maximal arithmetic subgroup of $E_{7}(\R)$ (the split real form of $E_7$).[ With respect to some structure of
$E_{7}(\R)$ as an algebraic group defined over $\mathbb{Q}$. The $\mathbb{Q}$-algebraic structure is determined by quantum consistency: in this case $E_{7}$ is identified with the universal Chevalley group of type $E_7$ (a scheme over $\Z$)
and $E_{7}(\Z)$ is the groups of its points valued in $\Z$.] This is the moduli space of Type II
compactified on $T^6$ with the correct $U$-duality group $\cg=E_{7}(\Z)$ (up to commensurability).

§.§.§ Liouville property

Many nice properties of supersymmetric field theories (with or without gravity), such as the non-renormalization/rigidity theorems,
may be traced back to the fact that their moduli spaces $\mathscr{M}$ are “Liouvillic”.
This property holds automatically for OV manifolds. Morally speaking, “most” of the good aspects of SUSY
appear to be given for free once the swampland conjectures are satisfied.

The manifold $M$ is Liouvillic for the class $\mathcal{C}$ of functions
$f\colon M\to\mathbb{R}$ if
f∈𝒞 and |f|< K<∞  ⇒  f=constant

Irreducible OV manifolds $M$ are Liouvillic for the sub-harmonic functions,
i.e.  Δf≥0 and |f|< K<∞  ⇒  f=constant

For $M\equiv \mathbb{R}$ this reduces to the well-known fact that a bounded
convex function $f\colon\R\to \R$ is a constant. For $M$ a complete, non-compact manifold of finite volume, we write $u\equiv f+K$. $u$ is a non-negative function bounded above by $2K$, and for all $p>1$
∫_ℳ u^p  d1mu𝗏𝗈𝗅 ≤(2K)^p·𝖵𝗈𝗅(ℳ)<∞,
and then $u$ is a constant by Theorem 3 in [38].

The statements holds even for reducible OV manifolds as long as they have no flat factor
in eqn.(<ref>). In particular, it holds for all 4d SUGRA satisfying the Ooguri-Vafa
swampland conjectures [2].
The Liouville property seems to be crucial for the swampland story:
this is the case for all supersymmetric consistent effective theories.

Standard Seiberg-Witten theory [43, 44] is grounded on the fact that
the Coulomb branch of an UV complete $\mathcal{N}=2$ QFTs satisfies the much weaker property of being
Liouvillic for the sub-pluriharmonic functions.

§ THE GAUGE COUPLINGS $\TAU(\PHI)_{AB}$ AS A MAP

One wishes to understand which gauge coupling $\tau(\phi)_{ab}$ may appear in a consistent effective
Lagrangian (<ref>) of
quantum gravity. These couplings have a natural geometric description which cries for
an intrinsic characterization of the allowed $\tau(\phi)_{ab}$.

§.§ Generalities

For fixed values of the scalar fields $\phi^i$ (and given duality-frame), the gauge coupling $\tau(\phi)_{ab}$ is an element of Siegel's upper half-space
H_h≡{τ∈𝖬𝖺𝗍_h×h() |  τ=τ^t, Im τ>0}≅Sp(2h,)/U(h).
$\boldsymbol{H}_h$ is a non-compact Kähler symmetric space
on which the group $Sp(2h,\R)$ acts transitively by isometries.
Its maximal arithmetic subgroup $Sp(2h,\Z)\subset Sp(2h,\R)$ is the group of electro-magnetic duality-frame rotations.
As a Riemannian space $\boldsymbol{H}_h$ is Hadamard[ That is, a simply-connected manifold with non-positive sectional curvatures; in facts $\boldsymbol{H}_h$ has even non-positive curvature operators.] [40, 41] so diffeomorphic to $\R^{h(h+1)}$. As a complex manifold $\boldsymbol{H}_h$ is biholomorphic to the (symmetric) bounded domain in $\C^{h(h+1)/2}$ of symmetric $h\times h$ matrices $Z$ with $1-Z\bar Z>0$.

In general the complex gauge couplings $\tau(\phi)_{ab}$ are not well-defined
functions on the scalars' space $\mathscr{M}\equiv \widetilde{\mathscr{M}}/\cg$. Indeed, when we go around a non-contractible loop $\gamma$ in $\mathscr{M}$, we may come back
and find that the duality frame was rotated by a non-trivial element of $Sp(2h,\Z)$. This yields a monodromy representation
The image $\rho(\pi_1(\mathscr{M}))\subset Sp(2h,\Z)$ is the monodromy group $\Gamma$,
cfr. eqn.(<ref>).

To get actual coupling functions, it is convenient to lift the gauge couplings to the smooth
universal cover  $\widetilde{\mathscr{M}}$;
then we may identify them with
the map

μℳ→Sp(2h,)/U(h)≡H_h,    μϕ↦τ(ϕ)_ab,
which lifts the intrinsic gauge coupling map $\mu$ [ Throughout the paper double-headed arrows stand for canonical projections.]

ℳ[rr]^(0.28)μ̃@->>[d]        Sp(2h,)/U(h)@->>[d]

ℳ[rr]_(0.28)μ       Sp(2h,)\Sp(2h,)/U(h)

One also says that the lifted map
$\tilde\mu$ is twisted by the monodromy representation $\rho$, i.e. $\tilde\mu$ satisfies the functional equations

μ̃(g·x)= ρ(g)·μ̃(x),  ∀ g∈⊂Iso(ℳ),  x∈ℳ.

The target space of $\mu$ in (<ref>), $Sp(2h,\Z)\backslash \boldsymbol{H}_h$, is an irreducible, non-compact, locally symmetric space of the special form $G_\Z\backslash G/K$ where $G_\Z\subset G$ is an arithmetic (hence Zariski-dense [23]) discrete subgroup[With respect to the obvious structure of $G\equiv Sp(2h,\R)$ as an algebraic group over $\R$.
In facts, $Sp(2h,\Z)\subset Sp(2h,\R)$ is a maximal arithmetic subgroup [42].
In the present context it is more natural to consider $G$ as the real locus of an algebraic group defined over $\mathbb{Q}$, see [15] and references therein.].
Maps between spaces with these special properties are well studied in mathematics.

From an  algebro-geometric viewpoint the target space $Sp(2h,\Z)\backslash \boldsymbol{H}_h$ is a normal quasi-projective variety [45, 46], called the Siegel variety (or Siegel scheme),
a special instance of Shimura variety [47, 48]. $Sp(2h,\Z)\backslash \boldsymbol{H}_h$ is not smooth since  $Sp(2h,\Z)/\{\pm1\}$ does
not act freely on $\boldsymbol{H}_h$. This can be easily repaired by replacing $Sp(2h,\Z)$ with a neat finite-index subgroup
$\Lambda_h\subset Sp(2h,\Z)$
[23]. Then $\Lambda_h\backslash \boldsymbol{H}_h$ is a smooth quasi-projective finite cover
of $Sp(2h,\Z)\backslash \boldsymbol{H}_h$.  $\Lambda_h\backslash \boldsymbol{H}_h$ is a non-compact manifold of finite volume
whose fundamental group $\Lambda_h$ is generated by unipotent elements: $\Lambda_h\backslash \boldsymbol{H}_h$
is a basic example of irreducible
OV manifold.
$\Lambda_h\backslash \boldsymbol{H}_h$ has a canonical compactification to a normal
projective variety, the Baily-Borel compactification $\overline{\Lambda_h\backslash \boldsymbol{H}_h}^{\mspace{1.5mu}BB}$ [45, 46]. $\overline{\Lambda_h\backslash \boldsymbol{H}_h}^{\mspace{1.5mu}BB}$ is not smooth for $h>1$. However we may blow-up it into a smooth projective variety $\boldsymbol{Y}_{\!h}$ such that the divisor at infinity $\boldsymbol{Y}_{\!h}\setminus (\Lambda_h\backslash \boldsymbol{H}_h)$ is simple normal crossing.
This can be done rather explicitly in terms of a suitable toroidal compactification of $\Lambda_h\backslash \boldsymbol{H}_h$ [49, 50, 51].

Then, replacing $\mathscr{M}$ by a finite cover (if necessary),
we see the gauge coupling $\mu$ as a map between smooth OV manifolds

It is remarkable that both the source and target spaces of $\mu$ are OV manifolds.
In all known examples of quantum-consistent theories (such as compactifications of the superstring on
Calabi-Yau's) both the image $\mu(\mathscr{M})\hookrightarrow \Lambda_h\backslash \boldsymbol{H}_h$ and
the fibers $\mu^{-1}(s)\hookrightarrow \mathscr{M}$ (equipped with the Riemannian structure induced by their respective embedding[ In these examples $\mu$ is an embedding not just an immersion.]) are also  OV spaces.[ The statement holds without exceptions because we defined the point to be an OV manifold.]
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

What properties should have the gauge coupling map $\mu$  in eqn. (<ref>) for the Lagrangian (<ref>)
not to sink in the swampland?

If a simple condition on $\mu$ exists at all, it should be invariant under $Sp(2h,\Z)$ rotations of the electro-magnetic frame.
Since we consider only the very extreme IR limit,  this means that
the property should be invariant for $Sp(2h,\R)$.

§.§ Three different viewpoints on $\tau(\phi)_{ab}$

Since the coupling $\tau(\phi)_{ab}$ is multi-valued in $\mathscr{M}$,[
In the SUSY case (and also in the non-SUSY one under the naturalness condition
we propose in section 7), $\tau(\phi)_{ab}$ must be multi-valued unless it is a numerical constant as in the examples discussed in [52].] its value in a given vacuum $\phi$ is not an intrinsic observable, and we should replace it with some invariantly-defined

Automorphic viewpoint.
In the case of a single photon $h=1$, the modular curve $SL(2,\Z)\backslash \boldsymbol{H}_1$
is biholomorphic to the punctured sphere
jSL(2,)\H_1  →  ℙ^1,
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 $\tau$ (here $q\equiv e^{2\pi i\tau}$)
j(τ)≡(θ_2(q)^8+θ_3(q)^8+θ_4(q)^8)^3/131.5mu824 η(q)^24=
1/q+744+1961.5mu884 q+⋯This is the viewpoint taken in [52]. In principle this strategy may be applied for all $h$ since, by the already mentioned
Baily-Borel theorem [45], $\Lambda_h\backslash \boldsymbol{H}_h$ has
enough automorphic invariant functions to fully characterize its points.

Total space viewpoint.
The space $\Lambda_h\backslash \boldsymbol{H}_h$ (resp. $\boldsymbol{H}_h$) is the moduli space of enhanced[ The Siegel variety $Sp(2h,\Z)\backslash \boldsymbol{H}_h$ is the moduli space
of principally polarized Abelian varieties. Going to the smooth covering Shimura variety $\Lambda_h\backslash \boldsymbol{H}_h$
leads to the moduli of polarized Abelian varieties endowed with some extra structure: e.g. if $\Lambda_h$ is the kernel of
$Sp(2h,\Z)\to Sp(2h,\Z/ m\Z)$ ($m\geq3$) the extra structure is a choice of generators of the group of $m$-torsion points.
An Abelian variety with such extra structure is called an enhanced Abelian variety.]
principally polarized Abelian varieties over $\C$ of dimension $h$ (resp. marked, principally polarized, Abelian varieties).
As for the axion-dilaton $\tau\equiv C_0+ie^{-\Phi}$ in $F$-theory [53],
often it is more convenient to think of the gauge couplings as a fibration $\varpi\colon\mathscr{X}\to\mathscr{M}$
whose fibers $\mathscr{X}_\phi$ are (enhanced) principally polarized Abelian varieties of periods $\tau(\phi)_{ab}$, and then describe the properties of the gauge couplings in terms of the intrinsic geometry of the total space
$\mathscr{X}$ of the fibration.

There is a more physical “total space” construction. One compactifies the 4d effective theory down to 3d on a circle;
each 4d vector field yields two new scalars in 3d and we get two additional scalars from the metric by the KK mechanism.
The resulting 3d scalars' manifold $\mathscr{M}_3$ is fibered over the 4d scalars' manifold $\mathscr{M}$,
and the gauge coupling map $\mu$ is encoded in the fiber's geometry (see .<ref> for details).
So the two 4d couplings $G(\phi)_{ij}$ and $\mu(\phi)_{ab}$ get geometrically unified in the intrinsic Riemannian geometry of
the 3d target manifold $\mathscr{M}_3$ which, according to the swampland conjectures, should
also be an Ooguri-Vafa space with its own discrete gauge group $\cg_3\,\triangleright\,\cg\sim \cg^0\times\Gamma$.
We shall use this “total space” viewpoint when convenient.

Elementary viewpoint.
We mostly adopt a more naive strategy: instead of considering subtle automorphic invariants of the gauge couplings, we shall see (<ref>) as a mere smooth map between Riemannian manifolds, and rely on the invariants of smooth maps which are defined in Differential Geometry (DG) textbooks.

§.§ DG invariants of the gauge couplings $\tau(\phi)_{ab}$

The basic DG invariants of a smooth map $\phi\colon X\to Y$ between Riemann manifolds
(with metrics $G_{ij}$ and $h_{ab}$, respectively) are:

* its energy $E[\phi]$ given by the value of the Dirichlet integral

E[ϕ]= 1/2∫_Xd^nx √(G)  G^ij  h_ab  ∂_iϕ^a ∂_j∂^bϕ^a,
that is, the action of the Euclidean $\sigma$-model with target $Y$ and source space-time $X$;

* its tension field on $X$

T(x)^adef= -1muh(ϕ(x))^ab δE[ϕ]/δϕ^b(x)≡D^i∂_iϕ^a(x)∈C^∞(X,ϕ^*TY).

We replace our original Question in .<ref> with a less ambitious one:
What can we say about the DG invariants $E[\mu]$ and $T[\mu]$
of the gauge coupling map $\mu\colon \mathscr{M}\to \Lambda_h\backslash \boldsymbol{H}_h$ in a quantum-consistent 4d effective theory of gravity?

A partial answer will be given in . <ref>.

Although these DG invariants have a simple definition, in quantum-consistent effective theories of gravity
they seem to involve rather deep number-theoretical issues. For instance, the
allowed energy levels $E[\mu]$ of the gauge couplings $\mu$ in a quantum-consistent effective
theory of gravity belong to a certain discrete subset $\Xi\subset \R_{\geq0}$,
the gauge couplings' energy spectrum, which carries a number-theoretic fragrance.
Computing $\Xi$ is very hard except in some extremely simple class of effective models.

The simplest possible quantum-consistent 4d effective theories are the ones with the properties:

$$\boldsymbol{(\ddagger)}\qquad\text{\begin{minipage}{320pt}\it {\bf(1)} the effective theory has $\cn\geq2$ local supersymmetry, and
{\bf(2)} its $U$-duality group $\cg$ is commensurable to the group $\boldsymbol{G}(\Z)$ of ``$\mspace{2mu}\Z$-valued'' points in a universal
Chevalley group-scheme $\boldsymbol{G}$ without simple factors of type $C_{h}$.\end{minipage}}

The prototypical 4d effective theory satisfying $\boldsymbol{(\ddagger)}$
is obtained by compactifying the 10d Type II superstring on a flat 6-torus $T^6$ (see appendix of [54]): in this case
$\cn=8$ and $\boldsymbol{G}$ has type $E_7$ (cfr. Example <ref>).
Whenever $\boldsymbol{(\ddagger)}$ holds one has (for a standard normalization of the metrics)

Ξ= ·m/2 𝖵𝗈𝗅(K) ∏_ℓ=1^r ζ(d_ℓ)
where $\{d_\ell\}$ are the degrees of the independent Casimir invariants of the
real Lie group $\boldsymbol{G}(\R)$, $K\subset G(\R)$
is a maximal compact subgroup,[ One can show that the absolute ranks of the two Lie groups $G(\R)$ and $K$ are equal.] $\mathsf{Vol}(K)$ its volume
(computed by the Macdonald formula [55]); $\zeta(s)$ is the Riemann $\zeta$-function, $m\equiv\dim G-\dim K$, and $\ci\subset \mathbb{N}$ is the set
of indices of finite-index subgroups of the maximal arithmetic group $\boldsymbol{G}(\Z)$.  If the $U$-duality group is precisely $\cg$, the energy of the coupling constants is given by the rhs
of (<ref>) with $\ci$ replaced by $[\boldsymbol{G}(\Z)\colon \cg]$. Eqn.(<ref>) follows from standard susy arguments together with the Langlands volume formula
for arithmetic quotients [56].
In the case of Type II on $T^6$ all degrees $d_\ell$ are even,
so the energy of the gauge coupling has a closed expression in terms of Bernoulli numbers:
$E(\mu)$ is a know rational number times $\pi^{35}$. For Type II on $T^5$, a part for a power of $\pi$,
the energy has a transcendental factor $\zeta(5)\zeta(9)$.

§.§ Tension field and harmonic maps

To answer the Simpler Question,
we start by recalling some basic facts about energies and tensions of
smooth maps. Building on these facts, in the next section we shall introduce a more detailed and elegant structure, that
we call domestic geometry, modelled on the variations of Hodge structures (VHS) [57, 58, 60, 61, 62, 59] and the more general $tt^*$ geometry [63, 64, 65, 66][26].

Let $(X,G)$ and $(Y,h)$ be two Riemannian manifolds. As already mentioned, the energy $E(\phi)$ of a map $\phi\colon X\to Y$ is the action of the Euclidean $\sigma$-model with target space $Y$ and space-time $X$, see eqn.(<ref>).
The map $\phi$ is harmonic iff it is a solutions of the corresponding equations of motion[ In eqn.(<ref>)
$\gamma^k_{ij}$ and $\Gamma^a_{bc}$ are  the Christoffel symbols for, respectively, the metric $G_{ij}$ and $h_{ab}$.]

\begin{equation}\label{har}
0=T[\phi]^a \overset{\rm def}{=} G^{ij}\mspace{1mu}D_i\partial_j\phi^a \equiv G^{ij}\Big(\partial_i\partial_j \phi^a - \gamma^k_{ij}\,\partial_k\phi^a+
\Gamma^a_{bc}\mspace{1mu}\partial_i\phi^b\partial_j\phi^c\Big),
\end{equation}

that is, if its tension
$T[\phi]\in \phi^*TY$ vanishes. A finite-energy harmonic map is simply an instanton of the $\sigma$-model (<ref>), 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 $\phi$ is totally geodesic iff the full matrix $D_i\partial_j\phi^a$ vanishes and not just its trace as for a general harmonic map.

When the source space $X$ is Kähler,
eqn.(<ref>) reduces to
0=G^ik̅ D_i∂_k̅ϕ^a≡G^ik̅(δ^a_b2mu∂_i+Γ^a_bc2mu∂_iϕ^c)∂_k̅ϕ^b
and all dependence on the source-space Christoffel symbols $\gamma^k_{ij}$ drops out. In this situation the map $\phi$ is said to be pluri-harmonic if the full type-(1,1) tensor
$D_i\partial_{\bar k}\phi^a$ vanishes and not just its trace. This condition depends only on the complex structure of $X$, and is independent of the specific Kähler metric $G_{i\bar k}$.
This means that a pluri-harmonic map is a classical solution of the $\sigma$-model for all
choices of the source metric $G_{i\bar k}$ as long as it is Kähler. In particular $\phi$ remains a solution if we repair the singularities in
the Kähler metric $G_{i\bar k}$ by a local surgery which keeps it Kähler as in . <ref>.
If, in addition, the target space $Y$ is also Kähler
with complex coordinates $z^a$, the pluri-harmonic condition reduces to $D_i\partial_{\bar k}z^a=0$
which is automatically satisfied if the stronger condition
$\partial_{\bar k}z^a=0$ holds, i.e. if the map $z\colon X\to Y$ is holomorphic.

A useful fact is that the composition $\iota\circ \phi\colon X\to Z$
of a harmonic  map $\phi\colon X\to Y$ and a totally geodesic map
$\iota\colon Y\to Z$  is also harmonic [67].

§.§.§ Energy and tension of maps into symmetric spaces

The last observation allows us to describe in a simple way the harmonic maps from a Riemannian manifold $M$
to a symmetric space $G/H$. The map of main interest is the covering gauge coupling $\widetilde{\mu}$ in eqn.(<ref>),
and we write explicit expressions for $G/H= Sp(2h,\R)/U(h)$, the general case then being  obvious.
One considers Cartan's totally geodesic embedding
[68][67][ The target space $Sp(2h,\R)$ is endowed with a
$Sp(2h,\R)\times Sp(2h,\R)$-invariant indefinite pseudo-Riemannian metric, which induces on
the image of $\iota$ a positive-definite Riemannian metric so that $\iota$ is an isometry onto its
image (for a proper normalization of the metrics). See eqn.(<ref>) and footnote  <ref>.]

\begin{equation}\label{wSb}
\begin{aligned}
&\iota\colon Sp(2h,\R)/U(h)\to Sp(2h,\R),\\
&\iota\colon\ce U(h) \mapsto \cs\overset{\rm def}{=} \ce\ce^t \in Sp(2h,\R)\quad\text{with}\quad
\cs^t=\cs,\quad \cs > 0
\end{aligned}
\end{equation}

where $\ce\in Sp(2h,\R)$ (called a vielbein [37])
is any chosen representative in $Sp(2h,\R)$ of the given point in the coset $Sp(2h,\R)/U(h)$. $\cs$ is independent of the choice of $\ce$.
Then the map

≡ι∘μℳ→Sp(2h,),    x↦(x)≡(x)(x)^t
is harmonic iff $\widetilde{\mu}$ is harmonic. The energy $E(\widetilde{\mu})$
of the gauge coupling, written in terms of $\cs$,  becomes
the action of the $Sp(2h,\R)$ principal chiral model[ The statement is a bit formal, since the
action integral in the lhs of eqn.(<ref>) does not correspond to a positive-definite
metric on the non-compact group $Sp(2h,\R)$; all formulae are meant to be analytic continuations from the corresponding compact group $U\mspace{-1mu}Sp(2h)$. However the metric is positive-definite when restricted to the image of the Cartan map $\iota$, i.e. on the space of symmetric, positive-definite, real, symplectic $2h\times 2h$ matrices. As a matter of notation, when $y\in\mathfrak{sp}(2h,\R)$,
$y^\text{o}$ stands for the odd part under the Cartan involution $\theta$, $y^\text{o}\equiv (y-y^\theta)/2$, i.e. for  the symmetric
part of the $2h\times 2h$ matrix $y$. The rhs of the identity (<ref>) is manifestly non-negative.]

1/8∫_ℳd^nx  √(G)  G^ij tr((^-1∂_i )(^-1∂_j ))≡1/2∫_ℳd^nx  √(G)  G^ij tr((^-1∂_i)^o(^-1∂_j)^o)
The tension field of the gauge coupling $\widetilde{\mu}$ is more conveniently written as
T≡d∗(^-1 d),
and $\widetilde{\mu}$ is harmonic iff $\cs$ is a symmetric classical soliton of the chiral model
defined on the space-time
$\mathscr{M}$, that is, iff

\begin{equation}\label{tzero}
d\ast\!\big(\cs^{-1} d\cs)=0.
\end{equation}

Symmetric   means that the solution satisfies the two additional conditions
$\cs^t=\cs$ and $\cs>0$.

§.§.§ Cartan gauge couplings

We call the inverse matrix $\cs^{AB}$ to  $\cs_{AB}\equiv (\ce\ce^t)_{AB}$
the Cartan form of the gauge couplings.
$\iota$ is a global isometry between the Siegel space $\boldsymbol{H}_h$ and the manifold of
symmetric, positive, symplectic $2h\times 2h$ matrices, so $\cs^{AB}$ and $\tau_{ab}$
contain exactly the same information.
With respect to the usual gauge coupling $\tau_{ab}$, its Cartan version $\cs^{AB}$ has the advantage
of transforming linearly
under rotations of the duality frame. All observables, being independent of the frame,
have nicer expressions when written in terms of $\cs^{AB}$.
Writing $\tau_{ab}=X_{ab}+i Y_{ab}$, one has
^AB =(Y^-1     -Y^-1X
-X Y^-1     Y+ XY^-1X)^AB

§ DOMESTIC GEOMETRY

In 4d $\cn=2$ supergravity the couplings of the vector-multiplets are described
by special Kähler geometry, which is equivalent [7, 8] to the geometry of variations of
Hodge structure (VHS) with non-zero Hodge numbers $h^{3,0}=h^{0,3}=1$
and $h^{2,1}=h^{1,2}=m$, where $m$ is the complex dimension of the special Kähler manifold $M$.
VHS itself is a special case of Higgs bundle geometry [69], which we shall refer to as generalized
$tt^*$ geometry.
In all these geometries $M$ is a Kähler manifold.
In the $\cn=2$ SUSY case the geometric swampland problem may be rephrased
as the question of which special Kähler geometries do arise as low-energy limits of consistent theories of quantum gravity. A necessary condition [15] is that the corresponding $tt^*$ geometry enjoys certain
arithmetic properties summarized in the VHS structure theorem [60, 61, 62, 59].

In this section we introduce a geometry – dubbed domestic – modelled on
$tt^*$, which does not require the manifold $M$ to have a complex structure.
Whenever $M$ is Kähler, domestic geometry automatically reduces to (generalized)
$tt^*$ geometry.

We start with a review of $tt^*$ geometry from a viewpoint which makes natural its
domestic generalization. The review is rather detailed, because we need results and formulae which
cannot be found in the physical $tt^*$  literature.
Before going to that, we recall some definitions.

We write $G(\R)$ for a non-compact, connected,
semi-simple, real
Lie group with Lie algebra $\mathfrak{g}$, $K\subset G(\R)$ for a maximal compact subgroup,
and $G(\Z)\subset G(\R)$ for a maximal arithmetic
subgroup. $G(\C)$ and $K(\C)$ stand for the complexification of the Lie groups
$G(\R)$ and $K$, respectively.

§.§ (Arithmetic) tamed maps

$M$ is an oriented Riemannian $m$-fold with a graded algebra $\cp^\bullet$ of parallel forms.
$X$ a Riemannian manifold. A smooth map $\mu\colon M\to X$ is
tamed iff

D∗(dμ∧Ω)=0  for all Ω∈𝒫^∙.
It suffices to require (<ref>) for parallel forms $\Omega$ of degree $\leq m/2$
∗D∗(dμ∧Ω) ≡(-1)^m-1  D∗(dμ∧∗  Ω).

Specializing to $\Omega=1$ we see that tamed $\Rightarrow$ harmonic, i.e. the tension field of a tamed map $\mu$ vanishes, $T[\mu]=0$.
Written in components eqn.(<ref>) requires  $D^j \partial_i \mu\in \mathrm{End}(TM)\otimes \mu^* TX$
to satisfy

Ω_j[i_1⋯i_k-1 D^j∂_i_k]μ=0  for all Ω∈^k,  k=0,1,⋯, m.

We are interested only in tamed maps whose target space $X$ is a locally symmetric
space of non-compact type, $X\equiv \Lambda\backslash G(\R)/K$, the prototypical example being
a smooth finite cover of the Siegel variety
Λ\G()/K e.g.= Λ_h\Sp(2h,)/U(h).

In sections 6 and 7 below we address inter alia the following

What does it mean for
the gauge coupling $\mu$ in eqn.(<ref>)
to be tamed?

We shall see that being tamed is a natural condition for the gauge couplings which has important consequences.
Note that in the physical context, $\cp^\bullet$ is an algebra of invariants for the continuous gauge symmetry,
so $\mu$ is tamed iff the algebra of invariant tensors under the “gauge coupling endomorphisms”
$D^i\partial_j\tau_{ab}$ contains all gauge invariants.

Arithmetic tamed maps.
It is convenient to lift the map $\mu$ to a map between simply-connected covers
which is
twisted by the monodromy representation $\rho\colon \pi_1(M)\to \Lambda$ of the fundamental group

μ∘ξ= ρ(ξ)·μ,    ∀ ξ∈(deck group $\widetilde{M}\to M$),
that is, $\widetilde{\mu}$ is the lift which makes the following diagram to commute

M@->>[d][rr]^μ       G()/K@->>[d]

M[rr]^μ       Λ\G()/K

The image $\Gamma\equiv\mu_\ast(\pi_1(M))\subset\Lambda$ is the monodromy group.
The tamed map $\mu$ is arithmetic iff

(i) $\Lambda$, hence $\Gamma$,
is a neat sub-group of a maximal arithmetic subgroup $G(\Z)$:

(ii)  its energy
is finite, $E[\mu]<\infty$.

An arithmetic tamed map is thus an instanton of the
$\Lambda\backslash G(\R)/K$ $\sigma$-model with some additional properties.

§.§.§ Special cases

For $M$ a Riemannian manifold of dimension $m\geq3$
with generic holonomy algebra $\mathfrak{so}(m)$, tamed is equivalent to
harmonic. There are some special cases:

(a) for $M= \mathbb{R}$:  tamed $\equiv$ harmonic $\equiv$ geodesic;

(b) for $M$ strictly Kähler: tamed $\equiv$ pluri-harmonic, [ A holomorphic map between Kähler manifolds is a special instance of pluri-harmonic map.] i.e. $D\overline{\partial}\mu=0$

(c) for $M$ quaternionic-Kähler (with $m\geq8$): tamed $\equiv$ totally geodesic [ 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 $\cm$.
Let $\mathfrak{ann}(\cp^\bullet)\equiv\{A_{ab}\in \mathrm{End}(T)\colon \Omega_{b[a_1\cdot a_{k-1}}A_{a_k]b}=0\ \text{for all }\Omega\in\cp^\bullet\}$ be the annihilator of $\cp^\bullet$. From eqn.(<ref>) we see that $D_a\partial_b\mu\in \mathfrak{ann}\cap \odot^2T$.
For a strict quaternionic-Kähkler manifold of dimension $\geq8$ one has $\cp^\bullet=\R[\Omega]$ with $\Omega$ the canonical 4-form.
Then  $\mathfrak{ann}(\cp^\bullet)\cap\odot^2 T=0$ see Proposition 1.2 of [70].] i.e. $D_i\partial_j\mu=0$,

(d) for $M$ locally isometric to a symmetric space $G/K$ – not of the form $SO(n,1)/SO(n)$ or $SU(n,1)/U(n)$ –
we typically have
tamed $\equiv$ totally geodesic, see appendix <ref> for examples.

Next we consider the special case (b) in some detail.
This leads to $tt^*$ geometry which is the model geometry which inspires all our constructions.

§.§ Generalized $tt^*$ geometry

In this subsection $M$ is a Kähler space with local holomorphic coordinates $t^i$.

A generalized $tt^*$ geometry (or Higgs bundle) is a tamed map
from $M$ into a locally symmetric space $\Lambda\backslash G(\R)/K$ of non-compact type.
We say that the generalized $tt^*$ geometry is
arithmetic iff the underlying tamed map $\mu$ is arithmetic.

Let us see how this definition leads to the usual $tt^*$ formalism [63]. We recall that the maximal compact subgroup $K\subset G(\R)$ is the fixed locus of a Cartan involution $\theta$  of the semi-simple Lie group $G(\R)$. The Lie algebra
$\mathfrak{g}$ of $G(\R)$ splits into $\theta$-even and $\theta$-odd parts,
$\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$, where $\mathfrak{k}$ is the Lie algebra of $K$
while $\mathfrak{p}$ is a $K$-module.
We fix a faithful real representation $\sigma\colon G(\R)\to SL(V,\R)$
such that $V\cong V^\vee$,
so that $G(\R)$ is seen as a concrete group of $n\times n$ real unimodular matrices $g$ ($n\equiv \dim_\R V$).
We choose conventions so that, in terms of matrices, $\theta$ is the inverse of the transpose, and we shall write  $g^t$ for $(g^{-1})^\theta$.
The Maurier-Cartan form $g^{-1}d g$ is a 1-form on the manifold $G(\R)$ with coefficients
in $\mathfrak{g}\subset \mathfrak{sl}(n)$ which may be decomposed into $\theta$-even and $\theta$-odd parts.
Let $\widetilde{\mu}\colon \widetilde{M}\to G(\R)/K$ be any smooth map
twisted by the appropriate monodromy representation $\rho$, eqn.(<ref>).
We choose a lift $f\colon \widetilde{M}\to G(\R)$ of $\widetilde{\mu}$ and use it to pull-back
the Maurier-Cartan form to a 1-form on $\widetilde{M}$ with coefficients in $\mathfrak{g}$
which we
decompose in $\theta$-even and $\theta$-odd parts as well as in $(p,q)$ form type
f^*-2mu(g^-1 dg)=A+A̅+C+C̅,  where {
A   def= f^*-2mu(g^-1 dg)^even_(1,0),
A̅   def= f^*-2mu(g^-1 dg)^even_(0,1)

C   def=f^*-2mu(g^-1 dg)^odd_(1,0),
C̅   def=f^*-2mu(g^-1 dg)^odd_(0,1)
We write $D\equiv\partial+A$, $\bar D\equiv \bar\partial +\bar A$ and $C\equiv C_i\, dt^i$, $\bar C\equiv  \bar C_{\bar k}\, d{\bar t}^{\mspace{1mu}k}$ with
$C_i, \bar C_{\bar k}\in \mathfrak{p}\subset\mathfrak{sl}(n)$.  By construction $D+\bar D$ is a $K$-connection on $\widetilde{M}$,
is a flat $G(\R)$-connection on $\widetilde{M}$.
Under a change of lift $f\to f^\prime= f \cdot u$ (with $u\colon \widetilde{M}\to K$)
both connections change by the $K$-valued gauge transformation $u$; hence the $K$-gauge invariants are
independent of the chosen lift $f$ of $\widetilde{\mu}$.
If $\widetilde{\mu}$ is twisted by a representation $\rho$ as in eqn.(<ref>),
the forms $A$, $\bar A$, $C$ and $\bar C$ are invariant under the action of the
deck group, so they may be seen as forms on the Kähler base $M\equiv \widetilde{M}/\pi_1(M)$
which are canonically defined (modulo $K$-gauge transformations)
by the original map $\mu\colon M\to \Lambda\backslash G(\R)/K$.

Decomposing the identity $\nabla^2=0$ into even/odd and form type we get
the equalities

D^2+C^2=(DC)=DD̅+D̅ D+CC̅+C̅ C=

= (DC̅)+(D̅ C)= D̅^2+C̅^2=(D̅C̅)=0

which hold for all smooth maps $\widetilde{\mu}$.
Now suppose that
the map $\widetilde{\mu}$ is harmonic with respect to some Kähler metric $G_{k\bar l}$ on $\widetilde{M}$, that is,
it satisfies the equation
D̅^k C_k≡G^l̅ kD̅_l̅  C_k=0,
which implies the equality
D̅^i D̅^k tr(C_i C_k)= tr[(D̅^i C_k)(D̅^k C_i)]+ tr(C_kD̅^i D̅^k C_i)
while the identities (<ref>) yield

\begin{align}
&\mathrm{tr}\Big[(\bar D^i C_k)(\bar D^k C_i)\Big]=  \mathrm{tr}\Big[(D_k \bar C^i)(\bar D^k C_i)\Big]\equiv \|\bar D C\|^2\\
&\mathrm{tr}\Big(C_k\bar D^i \bar D^k C_i\Big)=  \mathrm{tr}\Big(C_k[\bar D^i, \bar D^k] C_i\Big)=-\mathrm{tr}\Big(C_k[\bar C^i, \bar C^k] C_i\Big)=\\ &= \mathrm{tr}\Big([C_i, C_k]\,[\bar C^k, \bar C^i]\Big) \equiv \big\|\mspace{1mu}[C_i, C_k]\mspace{1mu}\big\|^2
\equiv \big\|\mspace{2mu} C^2\mspace{1mu}\big\|^2
\end{aligned}
\end{align}

This shows the
Let $M$ be Kähler and $\mu\colon M\to \Lambda\backslash G(\R)/K$ be a harmonic map.

D̅^i D̅^k tr(C_i C_k)= D̅ C^2+2mu C^21mu^2.

The rhs of (<ref>) is the sum of two non-negative terms:
hence the integral over $M$ of the lhs vanishes if and only if the two terms on the right are both identically zero;
this implies (1) $\bar D C=0$ which (by definition) says that $\mu$ is pluri-harmonic,
and (2) the algebra generated by the coefficient matrices $C_i$ is Abelian. In facts from eqn.(<ref>)
we see that (2) is an automatic consequence of (1).[ For an alternative proof that  (1) $\Rightarrow$  (2) see [66] or the appendix of [15].]
The lhs of (<ref>) is a total derivative, so its integral over $M$ is a surface term: in particular, when $M$
is compact a harmonic map is automatically pluri-harmonic [72].
More generally:
A harmonic map $\mu\colon M\to \Lambda\backslash G(\R)/K$ is pluri-harmonic ($\equiv$ tamed) if and only if

∫_∂M ∗ 1mutr(C_k D̅^k C)≡∫_∂M ∗ 1mutr(C_k D C̅^k) =0,
a condition which depends only on the asymptotic behaviour of $\mu$ at infinity in $M$.

The $tt^*$ equations.
In a generalized $tt^*$ geometry, $\mu$ is pluri-harmonic, and hence $\bar D C=0$. In view of (<ref>), (<ref>) this implies the
$tt^*$ PDEs [63]

D^2=C^2=(DC)=DD̅+D̅ D+CC̅+C̅ C=

= (DC̅)=(D̅ C)= D̅^2=C̅^2=(D̅C̅)=0.

These equations may be summarized in the following statement:
For $M$ strictly Kähler, $\mu\colon M\to \Lambda\backslash G(\R)/K$ is tamed if and only if

(∇^(ζ))^2≡(D+D̅+ζ^-1 C+ζ C̅)^2=0  for all ζ∈ℙ^1.

We write $\mathcal{R}$ for a (commutative)  enveloping algebra $\cu\mathfrak{a}$, where $\mathfrak{a}\subset \mathfrak{gl}(V,\C)$
is a maximal commutative $\C$-subalgebra containing the matrices $C_i$. $\mathcal{R}$ is known as a chiral ring [73].

A $tt^*$ geometry is strict if it has a spectral flow i.e. we can choose $\mathcal{R}$ so that
as $\mathcal{R}$-modules [73]. Since $V\cong V^\vee$, this implies $\mathcal{R}\cong \mathcal{R}^\vee$
as $\mathcal{R}$-modules $\Rightarrow$ in a strict $tt^*$ geometry the chiral ring is a (commutative) Frobenius algebra[ See theorem 1.3 in [74].]. The Frobenius pairing is known as
the topological metric $\eta\colon \mathcal{R}^{\otimes 2}\to \C$ [63, 66].

The vacuum geometry of a 2d (2,2) QFT is described by a strict arithmetic $tt^*$ geometry
[63, 65].
The vacuum bundle[ We recall that the vacuum bundle $\mathscr{V}\to\cm$ is the
holomorphic sub-bundle
of the trivial Hilbert space bundle $\mathscr{V}\times\boldsymbol{H}\to \cm$ whose
fiber $\mathscr{V}_t\subset \boldsymbol{H}$ at
$t\in\cm$ is given by the subspace of zero-energy states for the Hamiltonian $H_t$ with $F$-term couplings $t$;
$\mathscr{V}$ is equipped with the sub-bundle Hermitian metric induced by the Hilbert-space Hermitian product in $\boldsymbol{H}$. The Hermitian bundle $\mathscr{V}$ and its metric
are insensitive to deformations of $D$-terms couplings [63]. ] $\mathscr{V}\to \cm$ over the $F$-term coupling space $\cm$
is holomorphic with structure group $K\subset U(\dim_\C\mathscr{V})$, and  $D+\bar D$ is an unitary connection on the Hermitian vector bundle $\mathscr{V}$ which coincides with the Berry connection in the quantum-mechanical sense [63].
A choice of trivialization identifies the fibers of $\mathscr{V}$ with the
complexification $V_\C$ of the representation space $V$ of the real Lie group $G(\R)$.
Therefore the fibers of $\mathscr{V}$ carry a real structure[ A real structure on a $\C$-space is
an anti-linear map which squares to the identity.] to be identified with the physical PCT operation [63].

The chiral ring $\mathcal{R}$ is more invariantly seen as the fiber $\mathscr{R}_t$ of a sub-bundle $\mathscr{R}\hookrightarrow \mathrm{End}(\mathscr{V})$
consisting of commutative endomorphisms.
For a generalized $tt^*$ the $C_i$'s yield the sub-bundle chain

while for a strict $tt^*$ geometry

T-2muM↪𝒱≅ℛ    strict $tt^*$ geometry

$tt^*$ metric.
To simplify the notation,
we write $\boldsymbol{g}$ for $f^*g$.
Since $\boldsymbol{g}\in G(\R)$, the connection $A+\bar A\equiv (\boldsymbol{g}^{-1}d \boldsymbol{g})^\text{even}$ is
the $K$-connection written in an unitary
trivialization of the Hermitian bundle $\mathscr{V}$;
more precisely, the trivialization is orthogonal because of the
reality structure on $\mathscr{V}$ [63].
Since $\bar D^2=0$ the $K$-connection is also holomorphic,
and it is convenient to work in a holomorphic trivialization where $\bar A\equiv0$.
There is a map $U\colon M\to K(\C)$ such that
$U \bar D U^{-1}=\bar\partial$. The transformation between the orthogonal and the holomorphic trivializations is given by
the complex $K(\C)$-gauge transformation $\boldsymbol{g}\to \boldsymbol{g} U^{-1}$.
The connection $D+\bar D$ is both holomorphic and metric, hence is the unique Chern connection:
one has

A≡h∂h^-1 = (g U^-1)^-1 d (g U^-1)|_θ even    A̅=0,
where $h$ is the fiber metric on $\mathscr{V}$ in the chosen holomorphic trivialization (such that
the topological metric $\eta\equiv \boldsymbol{1}$).
Comparing the two complex gauges
h= UU̅^-1≡UU^†The Hermitian metric $h$ is called the $tt^*$ metric on $\mathscr{V}$ [63].
It satisfies the reality condition $h\bar h=1$ [63].[ In our conventions the
topological metric $\eta=1$ as a consequence of our choice $g^\theta=(g^{-1})^t$.]

For a strict $tt^*$ geometry, in view of eqn.(<ref>), the $tt^*$ metric $h$ is identified with
a Hermitian metric on the fibers of $\mathscr{R}$ which induces the sub-bundle metric on $TM$, i.e. a Hermitian metric on $M$.
It is natural to multiply the $tt^*$ metric on $\mathscr{R}$ by a normalization factor so that the section $\boldsymbol{1}\in\mathscr{R}$
has norm 1.
In 2d (2,2) QFT the normalized $tt^*$ metric on the coupling space $\cm$ plays the role of the Zamolodchikov metric [63].

Hodge metric.
In generalized $tt^*$ there is a second, better behaved metric on $M$,
namely the sub-bundle metric on $TM\hookrightarrow \mathrm{End}(\mathscr{V})$ induced by the
$tt^*$ metric on $\mathrm{End}(\mathscr{V})\cong \mathscr{V}\otimes \mathscr{V}^\vee$.
This metric exists independently of the spectral flow and is always Kähler
[26]. Its Kähler form is
i K_ij̅  dt^i∧dt̅^j̅≡i tr(C∧C̅).
Let $G_{i\bar j}$ be any Kähler metric on $M$. The (1,1) tensor on $M$
T_ij̅= K_ij̅-G_ij̅  G^kl̅ K_kl̅.
is automatically conserved

∇^i T_ij̅= G^ih̅ ∇_h̅ K_i j̅- G^kl̅ ∇_j̅K_kl̅=
Eqn.(<ref>) has a simple explanation. The map $\mu$, being pluri-harmonic, is harmonic – hence a classical solution to the $\Lambda\backslash G(\R)/K$ $\sigma$-model –
for all choices of the “spacetime” Kähler metric $G_{i\bar j}$. $T_{i\bar j}$ is just the energy-momentum tensor evaluated on this on-shell field configuration and hence is conserved.

§.§.§ HIVb  brane amplitudes

The $tt^*$ geometry of a 2d (2,2) QFT computes  important physical quantities.
The basic ones are
the Hori-Iqbal-Vafa half-BPS brane amplitudes (HIVb) $\Psi(\zeta)_a$ [75]
which are sections of the bundle
$\mathscr{V}^\vee\cong \mathscr{V}$ over $\widetilde{M}$ [ Eqn.(<ref>) is written in the conventions common in the $tt^*$ literature [63],
in particular the  bracket $\langle \cdots |\cdots \rangle$ is linear in its first argument rather than anti-linear
as in usual conventions. The index $a$ is a quantum number labelling the different fundamental branes.]

Ψ(ζ)_a[ϕ]= ⟨  ϕ | a brane⟩,    ϕ∈𝒱.
The $\tfrac{1}{2}$-BPS brane amplitudes depend on a twistor parameter $\zeta\in \mathbb{P}^1$ which
specifies the two linear combinations of the supercharges which leave them invariant [75].
The brane amplitudes are solutions to the linear PDEs

(D+D̅+ζ^-1 C+ζ C̅)Ψ(ζ)_a=0
and depend on a choice of $K(\C)$-gauge (i.e. of trivialization of $\mathscr{V}\to \widetilde{M}$);
under a change of gauge
Ψ(ζ)_a→U   Ψ(ζ)_a,    UM→K().
The $tt^*$ PDEs (<ref>) are the integrability conditions of the brane equation (<ref>).

Fundamental solutions.
A fundamental solution to eqn.(<ref>) is a map

such that the columns $\Phi(\zeta; t,\bar t)_a$
($a=1,\dots, n$) of the matrix $\Phi(\zeta)\equiv\Phi(\zeta;t,\bar t)$ yield a basis of linearly independent solutions of (<ref>).
In a given $K(\C)$-gauge, the fundamental solution is unique up to multiplication on the right by a matrix
$L(\zeta)\in \sigma(G(\C))$ which depends only on the twistor parameter $\zeta$
Φ(ζ;t,t̅)→Φ(ζ; t,t̅)  L(ζ).
In an orthogonal trivialization of $\mathscr{V}$, the matrix $L(\zeta)$ may be chosen so that $\Phi(\zeta)$ satisfies the
symmetry and reality conditions

Φ(-ζ)=Φ(ζ)^θ≡(Φ(ζ)^t)^-1,    Φ(ζ)= Φ(1/ζ̅),
and $\Phi(e^{i\theta})\in G(\R)$.
In the SUSY literature it is more common to use the
holomorphic gauge (cfr. eqn.(<ref>))
Φ(ζ)_holo= UΦ(ζ),    Φ(ζ)_holo=h  Φ(1/ζ̅)_holo
which yield the usual formula for the $tt^*$ metric $h$ as a bilinear in the solution of the linear problem
[63, 65, 66]
h= Φ(ζ)_holo1muΦ(-1/ζ̅)_holo^ t.

Integral structure.
The HIVb  fundamental brane amplitudes correspond to a special basis
of solutions to (<ref>), so
Ψ(ζ)_a= (Φ(ζ)L(ζ))_a,
for some $L(\zeta)$. To get the appropriate $L(\zeta)$  note that
the space of physical $\tfrac{1}{2}$-BPS branes
has an integral structure: for (2,2) $\sigma$-models it arises because the physical branes
have support on a sub-manifold of the target space, and hence represent integral
elements of the relevant homology group. More generally, the integral structure arises because of Dirac quantization of the brane charge.
Then the representation space $V$ of $G(\R)$ has the form
V= V_⊗_,
with $V_\Z\subset V$ a lattice preserved by the arithmetic subgroup $G(\Z)\subset G(\R)$.
The brane map $\Psi(\zeta)\colon \widetilde{M}\to \sigma(G(\R))$
is twisted by a monodromy representation (cfr. eqn.(<ref>))

ξ^∗1mu Ψ(ζ)= Ψ(ζ)·ρ_ζ(ξ)^-1,  ∀ ξ∈(deck group $\widetilde{M}\to M$)
which should respect the arithmetic structure, so $\rho_\zeta(\xi)\in G(\Z)$ and hence $\zeta$-independent.
Setting $\zeta=1$ and comparing with eqn.(<ref>), we see that the basic brane amplitudes
are given by an integral basis of solutions to (<ref>) on which the monodromy
action is given by multiplication on the right by $\rho(\xi)^{-1}$, where $\rho$ 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(-1/ζ̅)^†_holo,    where  I≡L(ζ)^-1(L(-1/ζ̅)^-1)^†,
with $I$ the intersection form between dual bases of BPS.[ In the most interesting case,
i.e. graded $tt^*$ geometries, we will present the explicit form of $I$ (see eqn.(<ref>)) checking that $I$ is an element of $G(\Z)$ (as physically expected),
hence $\zeta$-independent.

Brane amplitudes in general $tt^*$ geometry.

The equations of the HIVb  brane amplitudes, eqns.(<ref>),(<ref>), as well as their
integral structure, continue to make sense for all generalized arithmetic
$tt^*$ geometry, whether it has a spectral flow or not.

Given a fundamental brane amplitude $\Psi(\zeta)$
we may construct other ones by changing the representation $\sigma$ of $G(\R)$.
Usually the physical branes are given by the fundamental representation $\sigma_\text{fund}$;
all other representations $\sigma$ are sub-representations of some $(\sigma_\text{fund})^{\otimes s}$
defined by an invariant tensor $t\in \sigma\otimes (\sigma_\text{fund}^\vee)^{\otimes s}$;
so the branes amplitudes associated to an arbitrary representation $\sigma$
may be interpreted as physical multi-brane amplitudes
Ψ(ζ)_I^ A≡Ψ(ζ)_I^ A_1⋯A_s= t_I^i_1⋯i_s  Ψ(ζ)_i_1^ A_1⋯Ψ(ζ)_i_s^ A_s      t_I^i_1⋯i_s G(ℝ)-invariant

§.§.§ Graded $tt^*$ geometries & VHS
When the 2d (2,2) QFT is superconformal,
the $tt^*$ geometry has further structure
induced by the superconformal $U(1)_R$ charge. This additional structure characterizes the variations of Hodge structure
inside the larger class of
$tt^*$ geometries.

An arithmetic generalized $tt^*$ geometry is graded
iff there is a grading element $Q\in i\,\mathfrak{g}\equiv i\,\mathfrak{Lie}(G(\mathbb{R}))$ (the “$U(1)_R$ charge”)
such that

[Q, A]=0,  [Q,C]=-C,   [Q,C̅]=C̅.
A graded $tt^*$ geometry is a variation of Hodge structure (VHS) if, in addition,

e^2πi Q |_V=(-1)^ĉ,    e^iπQ∈G(),    𝖠𝖽(e^πi Q)(g)=g^θ  for g∈G()

ĉdef= 2 max{eigenvalues of Q in V}∈ℕ
When (<ref>),(<ref>) hold, the pair $(V, Q)$ is called a Hodge representation of the Lie group $G(\R)$ of weight $\hat c$. Hodge representations are classified in [62].

We identify elements of $\mathfrak{g}$ and respectively $G(\R)$ with the matrices which represent them in the real representation space $V$;
one has $g^\theta\equiv (g^t)^{-1}$ for $g\in G(\R)$.
Then eqn.(<ref>) implies that $\Omega\equiv e^{i\pi Q}$ is a matrix with integral entries which
g^t  Ω  g =Ω  for all g∈G(),
while $\Omega$ is symmetric (resp. anti-symmetric) for $\hat c$ even (resp. odd).
Hence $V$ is an orthogonal (resp. symplectic) real representation of $G(\R)$. The non-degenerate, integral, bilinear form

\begin{equation}
\Omega(v,w)\equiv v^t \Omega w,\qquad\text{with}\qquad \Omega(v,w)= (-1)^{\hat c}\,\Omega(w,v),
\end{equation}

is called the polarization of the VHS.
For a fixed $g\in G(\R)$, we define the Weil operator
C_g= g^-1 e^iπQ g∈G().
Then $\Omega(C_g v,\bar w)$ is a positive-definite Hermitian form on $V_\C\equiv V\otimes_\R \C$; indeed

Ω(C_g v, w̅)= (-1)^ĉ  w^†ΩC_g v=  (-1)^ĉ  w^†Ωg^-1Ωg v =
(-1)^ĉ  w^†g^t Ω^2 g v = w^†g^†g  v.

We write $\mathfrak{h}$ for the Lie subalgebra of $\mathfrak{g}$ which commute with $Q$, and
$H\subset G(\R)$ for the corresponding Lie subgroup. By eqns.(<ref>),(<ref>) the subgroup
$H$ is compact, and hence contained in
a maximal compact subgroup $K$. $G(\R)/H$ is then a reductive homogeneous space
with a canonical projection into the symmetric space $G(\R)/K$ [76].
Given any $H$-module $W$ we construct canonically a homogeneous bundle $\mathcal{O}(W)\to G(\R)/H$,
with typical fiber $W$, endowed with a unique canonical connection and metric (up to overall normalization)
[76]
(W)= G()×W/{(g,w)∼(g h, h^-1·w)  for h∈H}

$G(\R)/H$ is a homogenous complex manifold,
and the homogeneous vector bundle $\co(W)\to G(\R)/H$ is holomorphic for all $H$-module $W$.

Consider the grading of the complexified Lie algebra[ In a VHS the grading is integral, $r\in\Z$.
In this case the first equation (<ref>) is an (adjoint) Hodge decomposition of $\mathfrak{g}$ [60, 61, 59, 62].]

\begin{align}\label{Pmasqw1}
&\mathfrak{g}\otimes \C= (\mathfrak{h}\otimes \C)\oplus\!\left(\bigoplus_{r\neq0} \mathfrak{g}^{-r,r}\right),&&\text{where }
\mathfrak{g}^{r,-r}\overset{\rm def}{=}\Big\{X\in \mathfrak{g}\otimes \C \colon [Q, X]=r X\Big\},\\
&\big[\mathfrak{h},\mathfrak{g}^{-r,r}\big]\subseteq \mathfrak{g}^{-r,r},
&&\big[\mathfrak{g}^{-r,r},\mathfrak{g}^{-s,s}\big]\subseteq \mathfrak{g}^{-r-s,r+s}.\label{pzzz1mm}
\end{align}

By eqn.(<ref>) each summand $\mathfrak{g}^{-r,r}$ is a $H$-module, so it defines a homogeneous vector bundle $\co(\mathfrak{g}^{-r,r})$.
The complexified tangent bundle of the manifold $G(\R)/H$ is
TG()/H⊗= (⊕_r>0 (𝔤^-r,r))⊕(⊕_r<0 (𝔤^-r,r)).
We define an almost complex structure on $G(\R)/H$ by declaring the first summand to be the complex distribution of (1,0) vectors.
The almost complex structure is integrable since this complex distribution is involutive
by the second equation (<ref>).
For the holomorphic structure of $\co(W)$ see e.g. [76][59].

Infinitesimal period relations.
The sub-bundle $\Theta\equiv \co(\mathfrak{g}^{-1,1})$ of the holomorphic tangent bundle is called the
Griffiths holomorphic horizontal bundle [57, 59]. Let $M$ be a complex manifold
and $TM$ its holomorphic tangent bundle. We say that a map
satisfies the Griffiths infinitesimal period relations (IPR) if [57, 59]

p_*(TM) ⊆Θ.
In particular, such a map $p$ is holomorphic.

Let $V_\C=\oplus_i V_{q_i}$ be the decomposition of
the $G(\R)$-module $V_\C=V\otimes \C$ into eigenspaces of $Q$ of eigenvalue $q_i$. Since $[Q,H]=0$,
each $V_{q_i}$ is a $H$-module and yields a homogeneous bundle $\co(V_{q_i})\to G/H$.

Let $\tilde\mu\colon \widetilde{M}\to G(\R)/K$ be a lift of the tamed map $\mu$ of a
graded $tt^*$ geometry. Then we have the factorization-12pt

M@/^2pc/[rrr]^μ̃[rr]^p̃        G()/H @->>[r]    G()/K
$\tilde p$, which is called the (Griffiths) period map of the graded $tt^*$ geometry,
satisfies the IPR (<ref>).
The bundles $\mathscr{V}_{q_i}\equiv \tilde p^* \co(V_{q_i})\to \widetilde{M}$ are called Hodge bundles.

In view of eqn.(<ref>), eqns.(<ref>) are equivalent to the IPR (<ref>).

Usually one identifies the graded $tt^*$ geometry with the period map

p M→Γ\G()/H
which lifts to $\widetilde{p}$ on the universal cover $\widetilde{M}$. By definition, a period map
satisfies the IPR.

Applications to 2d QFT.

The vacuum geometry of  (2,2) 2d SCFT over the chiral conformal manifold $M$
is a graded strict $tt^*$ geometry. If, in addition, the
$U(1)_R$ charges of the chiral primaries [73] 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 spectral flow isomorphism.] i.e. with Hodge number $h^{\hat c,0}=h^{0,\hat c}=1$.
The vacuum bundle
$\mathscr{V}\to M$  then has an orthogonal decomposition (preserved by parallel transport with the Berry connection)

𝒱=⊕_q=-ĉ/2^ĉ/2𝒱_q,  where  Q|_𝒱_q= q∈ℕ-ĉ/2.
In this set-up $\hat c$ is one-third the Virasoro central charge [63]. The
spectral-flow isomorphism is graded by the $U(1)_R$-charge,
ℛ_q≅𝒱_q-ĉ/2,  with  ℛ=⊕_q=0^ĉℛ_q,
and implies
the “local Torelli” property
$TM\cong \mathscr{V}_{1-\hat c/2}$.

§.§.§ Explicit graded HIVb  brane amplitudes
Comparing with eqn.(<ref>) we get an explicit formula for the twistorial  multi-brane amplitudes of a graded $tt^*$ geometry: in the orthogonal trivialization they are $\Z$-linear
combinations of the columns of the matrix

Ψ(ζ)_I^ A= (ζ^-Q g^-1)_I^ A    monodromy group
acts on the rightm
with $\boldsymbol{g}_{\boldsymbol{A}}^{\ \, \boldsymbol{I}}$ the matrix elements of $\boldsymbol{g}\in G(\mathbb{R})$ in the Hodge representation $V$.
Comparing with eqns.(<ref>)-(<ref>), we see that $L(\zeta)=\zeta^{-Q}$ and

I≡L(ζ)^-1 (L(-1/ζ̅)^-1)^†= e^-πi QVHS≡ (-1)^ĉ Ω∈G(),
where the last equality holds in the VHS case. We see that $I$ is the natural intersection form.

§.§.§ Physical quantities from brane amplitudes

We can use the brane amplitudes to compute several physical quantities, that is,
$K$-gauge invariant expressions
which are independent of the choice of $\zeta\in\mathbb{P}^1$.
These physical quantities are well-defined for all generalized $tt^*$
geometries, graded or non-graded, strict or not.
Examples are

\begin{align}
&\bullet\ \text{K\"ahler form of Hodge metric:} && i\, K_{i\bar j}\, dt^i\wedge d\bar t^j=i\, \mathrm{tr}\big(C\wedge\bar C)\\
&\bullet\ \text{Cartan gauge coupling:} &&\mathcal{S}^{AB}=\big(\Psi(\zeta)^t\Psi(\zeta)\big)^{AB}\\
&\bullet\ \text{Hodge bilinears:} && \boldsymbol{\mathcal{S}}^{\boldsymbol{AB}}=\big(\boldsymbol{\Psi}(\zeta)^t\boldsymbol{\Psi}(\zeta)\big)^{\boldsymbol{AB}}
\end{align}

We think of the Hodge bilinears as “higher versions” of the gauge coupling.

§.§.§ $tt^*$ entropy functions & Mumford-Tate groups

Suppose our tamed (covering)
is actually the gauge coupling of a 4d effective gravity theory. The classical[ By “classical” we mean the classical entropy function as computed by the truncation of the effective Lagrangian to two-derivatives; the classical entropy becomes exact asymptotically for large charged $|q|\to\infty$.] entropy function (in Sen's sense [77, 78, 79])
of an extremal black-hole with electro-magnetic charge $q\in V_\Z\cong\Z^{2h}$,
(well-defined on the universal cover $\widetilde{\mathscr{M}}$) is
E_q=π  q_A ^AB q_B ≡π q_h^2,
that is, (up to a factor $\pi$) the Hodge norm-squared  $\|q\|_h^2$
of the charge vector.

For a general $G$-representation $\sigma$, contained
in $\otimes^s V$, we define the “generalized entropy functions”
of the generalized $tt^*$ geometry to be
S_qℳ→ℝ_+,    x↦q_A  𝒮(x)^AB q_B,    q∈⊗^s V_,
$S_{\boldsymbol{q}}$ reduces to $E_q/\pi$ when $\sigma$ is the fundamental representation
(i.e. $s=1$).

In (generalized) $tt^*$ geometry the generalized entropy functions
$S_{\boldsymbol{q}}(t)$ are
sub-pluriharmonic (in particular sub-harmonic) for all $\boldsymbol{q}\in \otimes^s V_\Z$, i.e. the matrix $\partial_{t^i}\partial_{\bar t_j} S_{\boldsymbol{q}}$ is semi-definite positive.

Set $L=\boldsymbol{\Psi}(-1)\boldsymbol{q}$, so that
$DL=CL$, $\bar DL=\bar CL$, $L^*= L$, and
$S_{\boldsymbol{q}}=L^t L\equiv L^\dagger L$. Then
∂̅S_q = 2  L^t D̅ L= 2  L^t C̅ L

∂∂̅S_q   = 2 (DL)^t D̅L+
2 L^t D(C̅ L)= 2 (DL)^t D̅L- 2  L^t C̅ C L =

=2  L^†(C_i C^†_j +C^†_j C_i)L  dt^i∧dt̅^j

The actual value of the classical entropy for an extremal black hole of charge $q$
is given by the value $\pi\, S_q$ at a critical point $\partial S_q=0$ (if it exists!!).
As an example, we consider the special case of a VHS of CY type with $\hat c=3$, which describes the
gauge coupling $\widetilde{\mu}$ of an effective 4d theory with $\cn=2$
[7, 8]. A critical point $t_0\in\widetilde{\mathscr{M}}$ of $S_q(t)$
corresponds to a $L\equiv\Psi(-1;t_0)q$ which is an eigenvector of $Q^2$
with eigenvalue $9/4$. This observation is called the attractor mechanism
[81, 82, 83, 84, 85, 86]. Proposition <ref> implies that the critical point is actually a minimum for the entropy function.
The attractor mechanism illustrates the point that being sub-harmonic is a very natural property for a physical entropy function, being strictly related to the convexity of
thermodynamical potentials.

The Mumford-Tate group.

If the multi-charge $\boldsymbol{q}$ is $\Gamma$-invariant, the generalized entropy function is well-defined on $\mathscr{M}$
not just on $\widetilde{\mathscr{M}}$. Suppose that we have an arithmetic graded $tt^*$ geometry (say, a VHS) whose base
$\mathscr{M}$ is quasi-projective, i.e. $\mathscr{M}=\overline{\mathscr{M}}\setminus D_\infty$
for $\overline{\mathscr{M}}$ a smooth (compact) projective variety and
$D_\infty$ a simple normal crossing divisor; then $\mathscr{M}$ is Liouvillic for the sub-pluriharmonic functions.
Under the above assumptions, one checks (by a careful asymptotic analysis [80])
that the generalized entropy of a $\Gamma$-invariant charge $\boldsymbol{q}$ is bounded  along the divisors at infinity in the projective closure $\overline{\mathscr{M}}$ of $\mathscr{M}$. Then, being sub-pluriharmonic, $S_{\boldsymbol{q}}(t)$ must be a constant in $\mathscr{M}$. We conclude that  for
a $\Gamma$-invariant multi-charge $\boldsymbol{q}\in\otimes^s V_\Z$
the multi-brane amplitude
$\boldsymbol{\Psi}(\zeta)\cdot\boldsymbol{q}$ is $H$-gauge equivalent
to a constant; this can be expressed as

Let $\mathsf{Hg}$ be the ring of all $\Gamma$-invariant
multi-charges in $\oplus_s (\otimes^s V_\Z)$ and let $M(\R)\subset G(\R)$ be the subgroup
which fixes all elements $\boldsymbol{q}\in \mathsf{Hg}$
and $H_M\equiv H\cap M$ a maximal compact subgroup. Then
the map $p$ in eqn.(<ref>)
factorizes as

ℳ@/_2.2pc/[rrrr]^μ@/^2pc/[rrr]^p[rr]^(0.4)m        Γ\M()/H_M@->>[r]    Λ\G()/H@->>[r]    Λ\G()/K

Eqn.(<ref>) is almost the structure theorem of VHS [60, 61, 62, 59], except that the theorem yields more details on the group $M(\R)$; we shall discuss these results in  the more general domestic context below. In the VHS literature the elements of the $\mathbb{Q}$-algebra $\mathsf{Hg}\otimes\mathbb{Q}$ are called Hodge tensors and the $\mathbb{Q}$-algebraic group $M(\mathbb{Q})$ is called the Mumford-Tate group [60, 61, 62, 59].

§.§ Domestic geometry

(Arithmetic) domestic geometry is defined by  the very same Definition <ref> 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
$M\to \Lambda\backslash G(\R)/K$ of finite energy. The geometric structures implied
by domestic geometry  depend crucially on the algebra $\mathcal{P}^\bullet$ of parallel forms
on $M$ (which we assume to be irreducible with no loss).
In particular, a domestic geometry on $M$ is a generalized $tt^*$ geometry
if and only if $\cp^\bullet$ contains a subalgebra $\R[\omega]/\omega^{m+1}$ with $\omega$ a parallel 2-form.

Domestic geometry is more general than $tt^*$ geometry as the following example shows.

§.§.§ Example: 2d SCFT

The (universal covering of the) conformal manifold $\widetilde{\cm}$ of a 2d (2,2) SCFT splits in a product of spaces associated to the two non-conjugate chiral rings
=_chiral×_twisted chiral,
and the
Berry geometry on each irreducible factor space is a domestic geometry
of the $\mathfrak{u}(1)$ graded $tt^*$ kind (. <ref>).

The Berry geometry on the moduli of a 2d (4,4)
SCFT is still a domestic geometry, but not a $tt^*$ geometry.[ Of course, a (4,4) SCFT is in particular
a (2,2) SCFT; however the space $M$ of marginal deformations which preserve (4,4) SUSY is a non-complex
submanifold of the complex manifold $\cm$ of marginal deformations which preserve only (2,2) SUSY. ]
Indeed the domestic geometry is $\mathfrak{sp}(1)\oplus\mathfrak{sp}(1)$ graded
rather than $\mathfrak{u}(1)$-graded.
The moduli space $M$ is quaternionic-Kähler, and hence the underlying tamed
map $\mu$ is totally geodesic (cfr. . <ref>): in (real) local coordinates $x^i$

∇_i C_j=0,  where   C_j  dx^j= f^*(g^-1dg)^odd.
Eqn.(<ref>) implies that $\widetilde{M}$ is a non-compact symmetric space with holonomy algebra of the form
$\mathfrak{sp}(1)\oplus\mathfrak{sp}(1)\oplus \mathfrak{j}\subset \mathfrak{so}(4k)$, and hence
For a (much longer) proof not using domestic geometry, see [87].

As the 2d example illustrates, in physical applications the domestic geometry is graded by the effective R-symmetry Lie algebra $\sigma(\mathfrak{r})$,
cfr. eqn.(<ref>), that is, we have a decomposition

𝔤⊗= ⊕_α∈Irr 𝔤_α,    𝔤_adj=σ(𝔯),    𝔤_triv=𝔧
where the sum is over the irreducible representations of $\sigma(\mathfrak{r})$.
Eqn.(<ref>) generalizes the adjoint Hodge decomposition (<ref>) of VHS theory to a possibly non-Abelian

§.§.§ Domestic brane amplitudes
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 $f^*(g^{-1}dg)$ in just two pieces
A= (f^* g^-1d g)^even,    Φ=(f^* g^-1dg)^odd,
and we do not have a twistorial $\mathbb{P}^1$-family of flat connections but
only two of them
∇^(±) = d+A ±Φ.
Consequently, we have only two “HIVb  brane amplitudes” $\Psi_\pm$,
which satisfy the equations
As a consequence of Dirac quantization of charge, in our applications the group $G(\R)$ preserves some bilinear pairing $V_\Z\otimes V_\Z\to \Z$
given by an integral matrix $\Omega\in G(\Z)$ (symmetric or antisymmetric) with $\Omega\Omega^t=1$.
In this case

Ψ_-=Ψ_+^θ Ω,    Ψ_±^*=Ψ_±.
Again we may consider higher domestic brane amplitudes $\boldsymbol{\Psi}_\pm$ associated to higher representations of the Lie group $G(\R)$ which may be written as multi-linear products of basic brane amplitudes.

The physical quantities of $tt^*$ are  still well-defined (we assume (<ref>)):

\begin{align}
&\bullet\ \text{the Riemannian metric:} && ds^2= K_{ij}\,dx^i dx^j\equiv\mathrm{tr}(\Phi_i\Phi_j)dx^i dx^j\\
&\bullet\ \text{Hodge bilinears:} && \boldsymbol{\mathcal{S}}^{\boldsymbol{AB}}=\big(\boldsymbol{\Psi}_\pm^t\boldsymbol{\Psi}_\pm\big)^{\boldsymbol{AB}}\\
&\bullet\ \text{generalized entropy functions:} && S_{\boldsymbol{q}}=\boldsymbol{q_A}\,\boldsymbol{\mathcal{S}^{AB}}\boldsymbol{q_B}
\end{align}

and the symmetric tensor
T_ij= K_ij-1/2G_ij  G^kl K_kl
is still conserved ($G_{ij}$ is the metric on $M$).

§.§ Entropy functions in domestic geometry

When the tamed map $\widetilde{\mu}\colon \widetilde{M}\to Sp(2h,\R)/U(h)$ which defines the domestic geometry
is the gauge coupling of some effective theory of gravity,
Sen's classical entropy function for an extremal black hole of charge $q$ (if it exists!)
is given by $\pi S_q$, where $S_q$ is the domestic entropy function for the fundamental representation.
The same argument as in .<ref>
shows that the generalized entropy functions are sub-tamed,
in particular,

* sub-harmonic for $M$ of generic holonomy: $\Delta S_{\boldsymbol{q}}\geq0$

* sub-pluriharmonic for $M$ Kähler:
$\partial_i\bar\partial_{\bar j} S_{\boldsymbol{q}}\geq0$

* convex for $M$ quaternionic-Kähler and most symmetric spaces:
$\nabla_i\partial_j S_{\boldsymbol{q}}\geq0$.

Structure of $\mu$.

If $M$ is Liouvillic for the sub-tamed functions
and the generalized entropy function $S_{\boldsymbol{q}}$ of a $\Gamma$-invariant
multi-charge $\boldsymbol{q}$ is bounded (a property natural for $\mu$ of finite energy), one concludes that the structure theorem (<ref>)
holds in the domestic case as well:-14pt

M@/^2pc/[rrr]^μ[rr]_(0.4)m        Γ\M()/K_M@->>[r]    Λ\G()/K,    K_M=M()∩K.
We shall show in . <ref> below that this conclusion is valid in the situations of physical interest using a different and more direct approach.
Here we give a rough sketch of an argument along the lines of . <ref>. We have already seen in . <ref> that all OV manifolds $M$ are Liouvillic for the
sub-harmonic functions hence a fortiori for the sub-tamed ones. It remains to show that $S_{\boldsymbol{q}}$ is bounded for $\boldsymbol{q}$ $\Gamma$-invariant.
We have to check the behaviour of $S_{\boldsymbol{q}}$ at infinity in $\widetilde{M}$.
Under our assumptions in . <ref>, a point at infinity $x_\infty$ is fixed by some
infinite subgroup $P\subset\Gamma\subset Sp(2h,\Z)$ and hence
is mapped by a continuous extension of
$\widetilde{\mu}$ in a point at infinity $y_\infty\in\overline{Sp(2h,\R)/U(h)}$ in the compactification  of $Sp(2h,\R)/U(h)$
[46] which is also fixed by $P$, i.e. such that $Py_\infty=y_\infty$. Since the entropy function
is the pull-back
of the Hodge norm-square of $\boldsymbol{q}$
computed on $Sp(2h,\R)/U(h)$, it suffices to check that, whenever $\boldsymbol{q}$ is fixed by $P$, $\|\boldsymbol{q}\|_h^2$ is bounded in a neighborhood of the point at infinity $y_\infty$ 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, as
illustrated by the attractor mechanism in the $\cn=2$ case.

§ DOMESTIC GEOMETRY AND SUPERGRAVITY

§.§ Supergravity in 4d

Consider a 4d supergravity with any number $\cn$ of light gravitini, matter content,
and couplings. Its scalars' universal covering manifold[ As always, we work modulo finite groups and finite covers.]

is a product of non-compact spaces in one-to-one correspondence with
the types of supermultiplets present in the model. For instance, in 4d $\cn=2$ SUGRA the scalars'
manifold is the space of hypermultiplet scalars
times the space of vector-multiplet scalars. The map $\widetilde{\mu}$,
which describes the coupling of the scalars to the vectors, splits
into a set of maps $\{\widetilde{\mu}_{(i)}\}$ which describe the coupling of vectors to
the scalars in supermultiplets of the $i$-th type.

The geometry of each factor space $\widetilde{M}_{(i)}$
depends on the corresponding supermultiplet.
Prima facie
these geometries look quite different one from the other:
in some cases $\widetilde{M}_{(i)}$ is Kähler (possibly with additional structures), in other situations
$\widetilde{M}_{(i)}$ does not even admit a natural complex structure. A general feature is that each
space $\widetilde{M}_{(i)}$ carries a non-trivial algebra $\cp_{(i)}^\bullet$
of parallel forms determined by the representation of R-symmetry on
the scalars as described in . <ref>.

Domestic geometry unifies all these seemingly different geometries in a single one.
The traditional supergravity theory, as well as the geometric swampland
conjectures, may be summarized in the following statement:

All 4d SUGRA models are defined by a domestic geometry:

1. the $\Gamma$-twisted (covering) gauge coupling map $\tilde\mu\colon \widetilde{\mathscr{M}}\to Sp(2h,\R)/U(h)$
is always tamed.

2. More precisely, for each factor manifold $\widetilde{M}_{(i)}$ in eqn.(<ref>)
there is a $\Gamma_{\!(i)}$-twisted, $\cp_{(i)}^\bullet$-tamed map

μ_(i)M_(i)→G_(i)/K_(i),    Γ_(i)⊂G_(i)
with target a symmetric space $G_{(i)}/K_{(i)}$ and $\tilde\mu$ factorizes as in the commutative diagram-26pt

<EMAIL_ADDRESS>\ar@{=}[r]& \widetilde{M}_{(1)}\times\cdots\times \widetilde{M}_{(s)}\ar[rr]^(0.42){\;\tilde \mu_{(1)}\times
\cdots\times \tilde \mu_{(s)}\;} && G_{(1)}/K_{(1)}\times \cdots\times G_{(s)}/K_{(s)}\ \ar@{^{(}->}[r]^(0.62){\iota} & Sp(2h,\R)/U(h)}

where $\iota$ is the totally geodesic embedding induced by a subgroup embedding
G_(1)×⋯×G_(s)ι↪ Sp(2h,)
$\Gamma\sim \prod_i\Gamma_{\!(i)}$ modulo finite groups;

3. most couplings in the Lagrangian $\mathscr{L}$ of an (ungauged) 4d SUGRA are given by
universal expressions in terms of brane amplitudes of the domestic geometries $\widetilde{\mu}_{(i)}$. For $\cn\geq3$ and the vector sector
of $\cn=2$ ALL couplings are so expressed;

4. the domestic geometry rigidity theorems/structure theorems reproduce and extend the usual SUSY non-renormalization theorems;

5. if the domestic geometry defined by a map $\widetilde{\mu}_{(i)}$ in (<ref>)
is non-arithmetic the SUGRA falls in the swampland.

The statement holds, with the appropriate modifications, in other spacetime dimensions.

As we have seen in . <ref>, there are several special cases of domestic geometry depending
on the particular algebra $\cp^\bullet$; when the domestic geometry is a VHS,
we may talk of its Hodge numbers and its Mumford-Tate (MT) group. In table <ref> we list
the domestic geometries which arise in 4d SUGRA with the special properties of each one.

\begin{tabular}{l|p{0.6cm}|c|p{4cm}|p{3.5cm}|c}\hline\hline
$\mathcal{N}$ \& supermultiplet & name

space &$\C?$ & $\bullet$ kind of domestic geometry

$\bullet$ algebra $\mathcal{P}^\bullet$ if special & $\bullet$ for VHS $h^{p,q}\neq0$

$\bullet$ MT group if special & notes\\\hline\hline
$\mathcal{N}=1$ chiral & & $\checkmark$ & graded $\hat c=1$
\textbf{non}-strict $tt^*$ & $\{h^{1,0}\}$ &\\\hline
$\mathcal{N}=2$ vector & SKG & $\checkmark$ & graded $\hat c=3$
strict $tt^*$ & $\{h^{3,0}=1,h^{2,1}\}$\\\hline
$\mathcal{N}=2$ hyper & QK & NO & \textbf{domestic} tamed by

$\mathcal{P}^\bullet=\mathbb{C}[1,\Omega]$ ($\Omega$ canonical 4-form) &\\\hline
$\mathcal{N}=3$ vector &&$\checkmark$ & graded $\hat c=3$
\text{non}-strict $tt^*$

by $\mathcal{P}^\bullet=[\wedge^*(\mathbb{C}^{3}\otimes \mathbb{C}^k)]^{\mathfrak{u}(3)\oplus\mathfrak{s\mspace{-2mu}u}(k)}$ & $\{h^{3,0}=3,h^{2,1}=k\}$

& LU, SH\\\hline
$\mathcal{N}=4$ gravity & $\boldsymbol{H}_1$ &$\checkmark$ & graded $\hat c=1$
strict $tt^*$ & $\{h^{1,0}=1\}$ & LU, SH\\\hline
$\mathcal{N}=4$ vector && NO & \textbf{domestic} tamed by

$\mathcal{P}^\bullet=[\wedge^* (\mathbb{R}^{6}\otimes \mathbb{R}^{k})^{\mathfrak{s\mspace{-2mu}o}(6)\oplus\mathfrak{s\mspace{-2mu}o}(k)}$ & & LU\\\hline
$\mathcal{N}=5$ gravity && $\checkmark$ &
graded $\hat c=1$
strict $tt^*$ tamed

by $\mathcal{P}^\bullet=[\wedge^* (\wedge^4\mathbb{C}^5)^\vee]^{\mathfrak{u}(5)}$
& $\{h^{1,0}=10\}$

& LU, SH \\\hline
$\mathcal{N}=6$ gravity & $\widetilde{\text{SKG}}$ & $\checkmark$ & graded $\hat c=3$
strict $tt^*$
by $\mathcal{P}^\bullet=[\wedge^*(\wedge^4\mathbb{C}^6)]^{\mathfrak{u}(6)}$ & $\{h^{3,0}=15, h^{2,1}=1\}$

$\Gamma^{\mathbb{Q}}(\mathbb{R})= SO(6,\mathbb{H})$ & LU, SH\\\hline
$\mathcal{N}=8$ gravity & & NO & \textbf{domestic} tamed by

$\mathcal{P}^\bullet=[\wedge^*(\wedge^4_+\mathbb{C}^8)]^{\mathfrak{s\mspace{-2mu}u}(8)}$ & & LU\\\hline\hline
\end{tabular}

4d SUGRAs as domestic geometries. In second column SKG stands for special Kähler geometry, QK for quaternionic-Kähler,
$\widetilde{\text{SKG}}$ for twisted special Kähler (with $e^{i\pi Q}\to -e^{i\pi Q}$), and $\boldsymbol{H}_1$ for the upper half-plane.
In third column $\checkmark$ means that the manifold has a natural complex structure. Forth column specifies
the class of tamed geometries and its taming algebra $\cp^\bullet$. In fifth column we specify the data for domestic geometries which
are in fact VHS. In the last column LU means that the geometry is locally unique (so $\widetilde{M}_{(i)}$ is a symmetric space)
and SH that, under the assumption that the domestic geometry is arithmetic
$M_{(i)}\equiv \Gamma_{(i)}\backslash \tilde M_{(i)}$ is a Shimura variety.

For the benefit of the reader we recall on which grounds
each statement in Fact <ref> rests. Items 1. and 3.  are a rephrasing of well-known
facts about 4d SUGRA (see e.g. the textbook [37]); they are perfectly rigorous in the sense of classical Lagrangian field theory. Item 2 follows from 1. and 3. together with
the original Ooguri-Vafa swampland conjectures (which we take as “facts” for the purpose of the present paper). In other words: it follows (rigorously) from the above standard facts in supergravity together with the assumption that the scalars' manifold is of the OV class. More in detail: for $\cn\geq3$ SUGRA item 2. is shown in .4.9 of [37] and for
$\cn=2$ SUGRA in refs.[15, 29]. In $\cn=1$ SUGRA the gauge coupling $\tau_{ij}$
is a holomorphic map between the scalars' Kähler manifold and
the Siegel variety $Sp(2h,\Z)\backslash Sp(2h,\R)/U(h)$. Then, by definition, $\tau_{ij}$ is the
formal period map
a weight-1 VHS; since item 2 is just the structure theorem for period  maps in VHS,
we need only to check that the formal period map $\tau_{ij}$
satisfies the conditions of the structure theorem.
The VHS structure theorem is a consequence of the theorem of the fixed part in VHS: hence
it is enough to verify that this last theorem holds for $\tau_{ij}$; going through
the details of Schmid's proof [80] of the fixed part theorem, we see that its
statement holds if the scalars' Kähler space is Liouvillic for the sub-pluriharmonic functions.
This property holds for Kählerian OV manifolds, and item 2 follows for $\cn=1$. Item 4. is a consequence of item 3:
non-renormalization of a Lagrangian coupling means that it cannot
be deformed continuously (in a consistent way); traditionally the SUSY non-renormalization
properties are attributed to “the power of holomorphy” [22]. However holomorphy is just the
cheaper way of implementing rigidity of the HIVb  which determines the coupling. It is obvious
that rigidity of the branes suffices to rule out corrections to the associated couplings, while rigidity applies to
a larger class of situations than holomorphy. Conversely, if the coupling is undeformable,
the associated brane is rigid. Item 5. summarizes the previous discussion of arithmetic
domestic geometry and its relation to Dirac quantization of charge.

§.§ Swampland vs. brane amplitudes

The above Fact allows for a more physical interpretation of the
Ooguri-Vafa geometric swampland conditions. For $\cn\geq2$
the couplings are expressed in terms of brane amplitudes, and
the swampland conjectures just say that these brane amplitudes have the properties we expect 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, $\cn=2$ supergravity
the brane amplitudes are important physical quantities: if they don't 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.

§.§ Supergravity as moduli-space gravitational instantons

From table <ref> we see that in all 4d SUGRAs the gauge coupling maps
are harmonic, hence solutions to the $\sigma$-model with target space
$\Gamma_{\!(i)}\backslash G_{(i)}/K_{(i)}
$ and action
S[μ_(i)]= 1/2∫_Γ_(i)\M_(i) d^1.5mun_(i)x √(g_(i))  g_(i)^kl h(μ)_ab ∂_k μ^a_(i) ∂_l μ^b_(i)
where $g_{(i)\,kl}$ is the kinetic-term metric on the $i$-th
factor space $\widetilde{M}_{(i)}$.

If our SUGRA model is not in the swampland, the solutions $\mu_{(i)}$
have finite energy (action) $S[\mu_{(i)}]<\infty$.
Thus a partial answer to
the Simpler Question on page que???  is that, in the SUSY case, the tension field of the gauge couplings should vanish while their energy must be finite.

Although the above statements are fully correct, they looks a bit unsatisfactory.
We are treating the couplings $g_{(i)}$ and $\mu_{(i)}$
asymmetrically – the first one as a background metric on $\widetilde{M}_{(i)}$ and the second one as a classical dynamical field – while
the two couplings are on the same footing in the swampland story, in facts
geometrically unified in the 3d scalars' manifold $\mathscr{M}_3$
(cfr. the “total space” viewpoint in .<ref>).
it is natural to treat also the
moduli metrics $g_{(i)}$ as classical dynamical fields.

A naive proposal will be to replace the $i$-th $\sigma$-model action by its
minimal coupling to gravity (allowing for a cosmological constant),
that is, to consider the following theory living on the moduli space $M_{(i)}$

S[g_(i),μ_(i)]= ∫_Γ_i\M_i d^1.5mun_iφ √(g_(i))(-1/2κ_(i)^2 R_(i)+Λ_(i)+1/2 g_(i)^kl h(μ)_ab ∂_k μ_(i)^a ∂_l μ_(i)^b)
However the $\mathscr{M}_3$ “total space” viewpoint of .<ref>
suggests that additional KK fields must live on $\mathscr{M}$,
so the proposal (<ref>) looks a bit naive, and we should not expect it to
work in full generality.
If we are lucky, $S[g_{(i)},\mu_{(i)}]$ may at best be a consistent truncation of the moduli-space gravity theory
(if it exists!).

In 4d $\cn\geq2$ SUGRA all couplings $g_{(i)}$, $\mu_{(i)}$ are solutions to the classical equations of motion
following from the action $S[g_{(i)}, \mu_{(i)}]$ for appropriate constants $\kappa_{(i)}$, $\Lambda_{(i)}$. If the SUGRA arises as the low-energy description of
a consistent quantum theory of gravity, the solutions have finite action, i.e. are gravitational instantons
on $M_{(i)}\equiv \Gamma_{(i)}\backslash\widetilde{M}_{(i)}$.

Indeed, the equation of motion for the moduli-scalars are satisfied since the $\mu_{(i)}$ are harmonic.
One has only to check that the Einstein equations

R_(i) kl-12g_(i)kl R_(i)+κ_(i)^2 Λ_(i)  g_(i)kl=κ_(i)^2 T_(i)kl
hold for some constants $\kappa^2_{(i)}$, $\Lambda_{(i)}$.
Equivalently, it is enough to show that the three symmetric tensors $R_{(i)kl}-\tfrac{1}{2}g_{(i)kl}R_{(i)}$, $g_{(i)kl}$, and $T_{(i)kl}$
are not linearly independent. For $\mathcal{N}\geq 3$ all factor spaces $M_{(i)}$ are locally symmetric,
hence Einstein $R_{(i)kl}=-\lambda_{(i)}\, g_{(i)kl}$, while the gauge coupling $\widetilde{\mu}_{(i)}\colon \widetilde{M}_{(i)}\to G_{(i)}/K_{(i)}$ is an isometry[ More precisely, we may choose $G_{(i)}$ as small as possible and then $\widetilde{\mu}_{(i)}$ is an isometry; if we choose $G_{(i)}$ non-minimal, $\widetilde{\mu}_{(i)}$ is just a totally geodesic
embedding.] (up to overall normalization), so that the three tensors are equal up to an overall constant
and Claim holds. For $\cn=2$ one has two factor spaces $\widetilde{M}_\text{hyper}$
and $\widetilde{M}_\text{vector}$. $\widetilde{\mu}_\text{hyper}$ is the contant map, so $T_{\text{hyper}\,kl}\equiv0$, while
$\widetilde{M}_\text{hyper}$, being negative quaternionic-Kähler, is Einstein, so eqn.(<ref>) holds. The tricky case is
$\widetilde{M}_\text{vector}$, which is a special Kähler manifold. That the Einstein equations (<ref>)
hold in this case was shown in [29].
Finally the last statement follows from the fact that evaluated on the appropriate classical solution the action density is proportional to the volume form [29], so that finite action is equivalent to finite volume of $M_{(i)}$,
which is one of the swampland conjectures.

The Claim may be regarded as a general geometric rigidity theorem, alias a general SUSY non-renormalization theorem.
E.g. the $\mathcal{N}=2$ case yields the two non-renormalization theorems of $\mathcal{N}=2$ SUGRA.

$\cn=1$ SUGRA.

In the  $\cn=1$ case we have
weaker non-renormalization theorems, so we cannot expect that the
story is as simple as for $\cn\geq2$. We have a non-renormalization theorem
for $F$-term couplings, so we expect
the gauge coupling $\mu$ to  be still a tamed map; this is 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 equations following from
a simple action of the form (<ref>) since we don't have the corresponding non-renormalization
One may speculate about more complicated “dynamical” equations for the $\cn=1$ moduli metric $g(\phi)_{ij}$ with additional degrees of freedom propagating on the scalars' moduli space $\mathscr{M}$.
Some proposal will be discussed elsewhere.

§ DOMESTIC GEOMETRY AND THE SWAMPLAND

In the supersymmetric context
the geometric swampland conditions may be conveniently
rephrased as the requirement that the underlying domestic geometry
is arithmetic, i.e. as the statement that quantum-consistent
effective models have formal brane amplitudes with the
right properties to correspond to actual physical branes.
Since domestic geometry and its brane amplitudes $\Psi_\pm$
make sense on all Riemannian manifolds,  one is naturally
led to ask 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-supersymmetric effective 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 geometric 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 . <ref> and <ref>.

§.§ Domestic geometry vs. OV manifolds

OV manifolds are the natural arena for tamed maps. Indeed

$\mathscr{M}$ a OV manifold and $G(\R)/K$ a symmetric space of non-compact type.
All maps $\mu\colon \mathscr{M}\to \Lambda\backslash G(\R)/K$ of finite energy (action)
may be continuously deformed into a unique tamed map $\mathring{\mu}$ which is the map of minimum energy in its homotopy class.
In particular,
all harmonic maps
$\mu\colon\mathscr{M}\to \Lambda\backslash G(\R)/K$ which have finite energy
are automatically tamed.

We defer the argument to the end of . <ref>.

We assume as our working hypothesis that, in a quantum-consistent theory of gravity, the low-energy