§ INTRODUCTION: THE INVERSE PROBLEM

A first fundamental step in any program to understand/construct/classify all 4d $\cn=2$ SCFTs
is to understand/construct/classify their special geometries (a.k.a.
Seiberg-Witten geometries [1, 2, 3, 4]). Such a geometric program has been especially pursued
by Argyres and coworkers [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20][ For a recent review see [21].]; for other work in the same line see e.g. [22, 23, 24, 25, 26].
This paper aims to introduce new powerful techniques and results in this direction.

A special geometry[ See section <ref> for precise definitions.] is a very nice mathematical
gadget, with multiple physical interpretations and deep relations with several other topics, which may be studied from
diverse math perspectives. A special geometry may be seen as the (complexification)
of a classical integrable mechanical system [3, 4] or as a non-perturbative description of
a four-dimensional SUSY QFT [1, 2]: we have two physical intuitions about them – one classical and one quantum – and the interplay between the two yields new unexpected insights.
Special geometry in our SCFT sense looks very similar to the special geometry of
$\cn=2$ supergravity [27, 28, 29, 30, 31, 32, 33]: formally the two are related by a simple “Wick rotation” in the signature of the fiber metric. The supergravity geometry describes the variation of Hodge structure (VHS [34, 35, 36, 37, 38, 39, 40, 41, 42])
of a family of Calabi-Yau 3-folds (3-CY) [29, 30, 31, 32, 33], and all ideas and techniques introduced in the study
of moduli geometries of 3-CY can be equally well applied to the special geometries of $\cn=2$ SCFT.
Strangely enough, it seems that the previous SCFT literature did not leverage on the
deep extensive work done in the Calabi-Yau contest. The 3-CY methods will be instead central in our discussion.
Morally speaking, the program of the geometric classification of all 4d $\cn=2$ SCFT may be
seen as a simpler analogue of classifying all Calabi-Yau 3-folds, a basic problem with fundamental implications for quantum gravity [32].

The main focus of this note is the inverse method in special geometry.
While the paper is somehow written in a review format, essentially all results in this direction are novel,
including the very notion of “inverse problem”. In the rest of this introduction we outline what we mean
by “inverse problem”: a more precise description of the method  is given in section <ref> in the proper
geometric context.

Special geometry.
In this paper a special geometry  is a
holomorphic integrable system[ The integral systems of interest are actually algebraic not just holomorphic.]
associated to the Seiberg-Witten geometry of a 4d $\cn=2$ SCFT
with all its implied structures [3, 4]. Thus a special geometry is a (smooth) complex symplectic $\mathscr{X}$
variety together with a holomorphic fibration $\pi\colon \mathscr{X}\to \mathscr{C}$
over the Coulomb branch $\mathscr{C}$ with Lagrangian fibers
which generically are polarized Abelian varieties.
The Coulomb branch $\mathscr{C}$ is the spectrum of the chiral ring $\mathscr{R}$,
hence an affine complex variety.
Most of our arguments work in the broader context of 4d $\cn=2$
QFT, but we focus on the superconformal case where the geometry carries an additional structure:
a $\C^\times$-action on $\mathscr{X}$ by automorphisms
of all the relevant geometric structures.[ Special geometries with this additional structure will be called $\C^\times$-isoinvariant.]

The $\C^\times$-action reflects
the existence of a conformal $U(1)_R$ symmetry which is spontaneously broken
to a discrete subgroup everywhere in $\mathscr{C}$ but at a single closed point $0$ (the origin).
When the Coulomb branch is smooth at $0$ (as a $\C$-affine variety)
$\mathscr{C}\simeq\C^r$. We shall be concerned mainly, but not exclusively, with this case.

The quantum field theoretic intuition points out that a special geometry $\mathscr{X}\to \mathscr{C}$ comes with a natural  “stratification” in
lower-dimension special geometries [16, 17]. We write the word “stratification”
between quotes since the QFT intuition leads to a fairly new[ Fairly new $\equiv$ the author is not aware of any previous mention of this idea in the math literature.]
notion in symplectic geometry, which is not a stratification in any usual sense, and lays well
outside the horizon of classical Liouvillian physical intuition.
In math terms the “stratification” is a generalization of the Marsden-Weinstein-Meyer symplectic quotient [43, 44] which applies to
singular loci with suitable physical/geometric properties where the usual symplectic quotient becomes meaningless. The “stratification” of $\mathscr{X}$ arises from an ordinary stratification of the Coulomb branch $\mathscr{C}$ [16, 17].

We outline the physical heuristics beyond “stratification”.
At the generic point of $\mathscr{C}$ the IR physics, while not necessarily boring, is relatively simple,
but there are special loci $\mathscr{S}_a\subset\mathscr{C}$ where something “more interesting” happens. At a special point  $u\in\mathscr{C}$
three new phenomena may happen:

(1) additional degrees of freedom become massless;

(2) the unbroken subgroup $R_u\subset U(1)_R$ of R-symmetry enhances;

(3) we have both extra light degrees of freedom and R-symmetry enhancement.

In an interacting theory the special locus
$\cd\subset \mathscr{C}$ where we have additional massless degrees of freedom (called the discriminant)
is a complex analytic subspace of pure codimension 1.
We identify $\cd$ with a simple effective divisor and write $\cd=\sum_a\cd_a$ for its decomposition into irreducible components.
In the SCFT case the $\cd_a$'s are preserved by the $\C^\times$-action.

We focus on a point $u\in\mathscr{C}\setminus\cd$ which is at distance $\epsilon$ from a generic point $u_0\in\cd_a$ of the $a$-th discriminant component.
The BPS states that are massless at $u_0$ are parametrically light at $u$;
since $\cd_a$ has codimension-$1$ in $\mathscr{C}$
these states are charged under a rank-$1$ gauge group. Thus, asymptotically as $u\to u_0$  the light degrees of freedom
are described by
an effective IR $\cn=2$ theory which
looks like a system composed of $(r-1)$ IR-free vector-multiplets and a rank-$1$ model which can be either IR-free or
an interacting SCFT in its own right.
This suggests that the special geometry in an infinitesimal neighborhood of $\cd_a$
should look as a rank-$1$ special geometry describing the light particles
fibered over the local rank-$(r-1)$ geometry which describes the heavy states.
While the actual situation may be a bit more intricate,[ See [26].]
this physical intuition yields a first rough cartoon of the geometry in the vicinity of $u_0$.
Making the idea a little more precise, one defines a rank-$(r-1)$ special geometry
$\mathscr{X}_a\to\cd_a$ over the codimension-$1$ locus $\cd_a\subset \mathscr{C}$.
The local geometry in an infinitesimal neighborhood of $\cd_a$ is fibered over this
rank-$(r-1)$ special geometry.
Iterating the construction, one gets a rank-$(r-s)$ special geometry
$\mathscr{X}_{a_s}\to\mathscr{C}_{a_s}$ over each subvariety in a finite set
$\{\mathscr{C}_{a_s}\colon a_s\in A_s\}$ of
codimension-$s$ subvarieties of $\mathscr{C}$ for all $0\leq s\leq r$.
By the “stratification” of the special geometry $\mathscr{X}\to\mathscr{C}$ we mean the full collection
of special geometries

\begin{equation}
\big\{\mathscr{X}_{a_s}\to\mathscr{C}_{a_s}\colon a_s\in A_s,\ 0\leq s\leq r\big\}.
\end{equation}

The open sets $\mathring{\mathscr{C}}_{a_s}\equiv \mathscr{C}_{a_s}\setminus (\cup_{a_{s-1}\in A_{s-1}}
\mathscr{C}_{a_{s-1}})\cap \mathscr{C}_{a_s}$ form a stratification of the Coulomb branch in the usual sense.
The closures of the codimension-$(s+1)$ strata, $\mathscr{C}_{a_{s+1}}\subset \mathscr{C}$, are the irreducible
components of the discriminants of the special geometries $\mathscr{X}_{a_s}\to\mathscr{C}_{a_s}$ over the (closed) codimension-$s$ strata.
In particular the $\mathscr{C}_{a_1}$'s coincide with the discriminant component $\cd_{a_1}$ of $\mathscr{X}$. The geometry along a codimension-1 stratum $\mathring{\cd_a}$ is determined by the singular fiber $\mathscr{X}_{u_\star} $ at a generic point $u_\star\in \cd_a$. The most important invariant of a divisor $\cd_a$
is the conjugacy class $[\varrho_a]$ of the local monodromy $\varrho_a$ around it; the type of the fiber $\mathscr{X}_{u_\star}$ is determined by $[\varrho_a]$ together with some subtler invariants [45, 46, 47, 48, 49, 50].
The local monodromy classes $[\varrho_a]$ follows a
Kodaira-like $ADE$ classification [51, 52, 53, 54]. Thus to each component $\cd_a$
we can associate the (unique)  rank-1 asymptotic special geometry over the (small) disk $\Delta$
with the same monodromy class.
Roughly speaking, the effective IR field theory which governs the degrees of freedom which are light along $\cd_a$ is described by this rank-1 geometry defined by the local monodromy $\gamma_a$. The detailed geometry/physics is a bit trickier [26] and depends on the particular point $u_\star$, not just on the discriminant component $\cd_a$, but the most important aspects are already captured by $[\gamma_a]$.

The Coulomb branch $\mathscr{C}$ carries a second stratification by unbroken $U(1)_R$ symmetry.
At a generic point in $\mathscr{C}$ the unbroken subgroup is $\Z_k$
with $k\in\{1,2,3,4,6\}$ [25].[ If $k\not\in\{1,2\}$ the model is isotrivial
and the geometry and physics
can be determined by group-theoretical methods [25, 55].]
The codimension-1 loci in $\mathscr{C}$ where the unbroken R-symmetry enhances to a bigger group
form a divisor $E=\sum_f E_f$. The unbroken R-symmetry at the generic point of $E_f$
is $\Z_{\ell_f}$ where $\ell_f$ is a proper multiple of $k$.
On multiple intersections $\cap_jE_{f_j}$ we have further enhancement to
$\Z_{\text{lcm}(\ell_{f_j})}$, etc.
Some $D_a$ may coincide with some $E_f$ (case (3)).

By the stratification data of a $\C^\times$-isoinvariant special geometry we mean

where $\{\cs_a\}_{a\in A}$ is the finite set of ($\C^\times$-invariant) special divisors in $\mathscr{C}$ where something interesting happens,
and $[\varrho_a]$ (resp. $\ell_a$) is the conjugacy class of the local monodromy around $\cs_a$ (resp. the order of the unbroken R-symmetry along $\cs_a$).

Inverse problem.
The very fact that all special geometry has a “stratification” whose non-trivial strata
are special geometries of strictly smaller dimension, suggests an inverse engineering strategy to construct
all special geometries which, in particular, could lead to a complete classification of all 4d $\cn=2$ SCFT.
The idea is that, once we have constructed all $\C^\times$-isoinvariant special
geometries up to rank $r$, we can construct the rank $r+1$ ones
by “gluing together” a finite set of rank-$r$ geometries together with a rank-$(r+1)$ open stratum $\mathring{\mathscr{C}}=\mathscr{C}\setminus \mathscr{S}$
of “boring” generic vacua whose geometry is well understood. In addition to the stratification data (<ref>),
the problem then requires to specify the
generic vacuum which is encoded in two data: the action  of the R-symmetry $U(1)_R$ on $\mathring{\mathscr{C}}$
and the generic Higgs branch. For smooth Coulomb branches the first datum is more conveniently replaced by the $r$-tuple of the Coulomb dimensions $\{\Delta_i\}$.
For simplicity we assume that the generic Higgs branch is trivial,
but this is not an essential assumption for our analysis.

This inverse engineering program is what we call “the inverse problem of special geometry”.
It requires to answer two main questions:

Characterize the sets of the form $\{(\Delta_i)_{i=1}^r,(\cs_a,[\varrho_a],\ell_a)_{a=1}^s\}$
where $\{\Delta_i\}$ are positive rational numbers, $\cs_a\in \C^r$ are homogeneous irreducible divisors, $[\varrho_a]$ Kodaira types, and $\ell_a$ positive integers,
which are the stratification data of indecomposable,[ A geometry is indecomposable if it is not the product of lower-rank geometries, i.e. if the corresponding SCFT
is does not split in two non-interacting sectors. The notion of “indecomposable” should not be confused with the notion of “irreducible”.] rank-$r$, global special geometries.

When $\{(\Delta_i)_{i=1}^r,(\cs_a,[\varrho_a],\ell_a)_{a=1}^s\}$ is an allowed stratification datum,
construct all global special geometries that have this “stratification”.

The answer to Question 1 may be given at different levels.
There are necessary conditions on $\{(\Delta_i)_{i=1}^r,(\cs_a,[\varrho_a],\ell_a)_{a=1}^s\}$ for the existence of
a global special geometry and also sufficient conditions.
Necessary conditions are physical properties which all
4d $\cn=2$ SCFT should enjoy, and hence are statements of direct physical interest. Sufficient conditions produce
examples which may lead to new insights. It would be desirable to have a precise
necessary and sufficient condition. In this paper we shall come “close”
to this goal but some work is still needed to get the complete story.
The trickier part of Question 1  is the classification of the
special divisors $\sum_a\cs_a$ which may arise in a special geometry.
This aspect is the main topic of section <ref>. When the base $\mathscr{C}$
of the (algebraic) Lagrangian fibration is compact, i.e. $\mathbb{P}^r$,
the possible discriminants are well-understood [56], but in our set-up $\mathscr{C}$
is affine, not projective, and a priori the discriminant is much less constrained.

In this long note we shall give essentially complete
answers to the two Questions in rank-2.
For higher rank we give some necessary conditions, discuss general aspects and properties, provide
examples, and pave the way for a detailed study (which should be doable at least
for  low-ranks such as $3$ or $4$).

Our basic tool to answer the two Questions is rigidity of various geometric structures implied by the definition of special geometry.
Several inequivalent notions of rigidity are relevant in the present context: the “weaker one” (Faltings-Peter rigidity [57, 58, 59, 60, 61, 62]) applies
to general (irreducible, non-isotrivial) special geometries, while the stronger Deligne-Simpson rigidity [63, 64, 65] refers to the underlying monodromy representation and yields a typically sufficient condition. In addition we have (non)-rigidity in the special geometric sense,
see section <ref>.

Classification scheme.
The arguments in this paper suggest the
classifying scheme:

\begin{tabular}{|c|p{11cm}|}\hline
I & isotrivial geometries\\\hline
II & quasi-isotrivial geometries not of class I\\\hline
III & reducible indecomposable geometries not of classes I, II\\\hline
IV & non-rigid geometries not of classes I, II, III\\\hline
V & $\mu$-rigid geometries not of classes I-IV\\\hline
VI & others\\\hline
\end{tabular}

Class I can be explored by elementary means for all rank $r$ [55].
Classes II and III are severely
constrained, “few” such geometries exist, and a systematic analysis may be possible
also in higher rank. These geometries look rather “bizarre”, but there is
strong evidence that a small number of them do exist.
In view of the Folk-theorem (. <ref>) class
IV is also essentially charted for all ranks.
For classes V and VI one falls back to the rank by rank strategy via the inverse method.
One tries to construct them by induction on the rank by “gluing together” known lower rank “strata”.
Class VI is the less amenable to analysis and a priori the largest one, but
not a single example of class VI geometry is known to the author, so there is hope
that this hardest class is actually empty or at least very sparse.

Each one of the classes I-V is naturally decomposed into subclasses which are conveniently studied separately.

The paper is organized as follows. Section 2 contains a review of the basic geometric structures with new results and complements perhaps known to the experts but
without a good reference in print. In section 3 we state the inverse problem and explain its relation to the classification program. In section 4 we discuss the families of Abelian varieties which arise from a $\C^\times$-isoinvariant special geometry. In section 5 we discuss the deformation theory of special geometry, the conformal manifold, and the Folk-theorem. To illustrate the main issues here we also  discuss in detail the case of rank 2. In section 6 we consider the rigidity of implied geometric structures such as
the Abelian family and the monodromy representation. Here we state the Deligne-Simpson problem.
In section 7 we discuss how rigidity can be used to understand/construct/classify $\C^\times$-isoinvariant special geometries. In section 8 the program is carried on in rank 2 in detail. Here several tricky points are analyzed such as reducible representations. Finally in section 9 we lay the groundwork for an extension of the program to higher rank, focusing on the problem of understanding the class of allowed discriminant divisors. Some technicality is deferred to the appendices.

§ SPECIAL GEOMETRY: REVIEW WITH COMPLEMENTS

In this section we review special geometry to fix notation and terminology.
We also present some complements which, while
probably known to experts, are not spelled out in the literature (to the best of our knowledge).
Some results are novel, at least in the present generality.

§.§ Defining structures

We first review the geometric structures implied by special geometry.
We start with a rough sketch of the physical foundations of the topic.
We omit all details and gloss over all subtle points since our purpose here
is merely to motivate
the precise  definitions given in the next subsection (. <ref>) which we shall take as the geometric “axioms”
for the rest of the paper.
An Euclidean 4d $\cn=2$ SQFT admits a topological twisting á la
Witten [66, 67] by modifying the couplings of the various fields to the holonomy connection of
the 4-dimensional spacetime manifold $M$. The twisted model is a Topological Field Theory (TFT)
invariant under all oriented diffeomorphisms $M\to M$ (see [68] for an extensive review).
By construction the twisting does not change the functional measure when
$M$ is locally flat.

§.§.§ Chiral ring, Coulomb branch, Coulomb dimensions
The algebra of local operator of the twisted TFT is a subquotient $\mathscr{R}$ of the algebra  $\mathfrak{A}$ of local operators of the parent $\cn=2$ SQFT
called the chiral ring. The correlation functions of operators in $\mathscr{R}$
are independent of the choice of their representatives in $\mathfrak{A}$.
$\mathscr{R}$ is a commutative, associative $\C$-algebra with unit, which
in a sensible SQFT[ A non finitely-generated chiral ring will violate the entropy bound any good QFT should obey.] must be a finitely-generated integral domain.
Its spectrum $\mathscr{C}= \mathsf{Spec}\,\mathscr{R}$ is then an
affine variety over $\C$
called the Coulomb branch. Its complex dimension $r=\dim_\C\mathscr{C}$ is the rank.
The basic observation is that when the Riemannian spacetime $M$ is flat  the correlation functions of operators in $\mathscr{R}$  computed in the SQFT coincide with the ones computed in the corresponding twisted
TFT.[ This equality was the starting point of $tt^*$ [69].]

By replacing $\mathscr{R}$ with its normalization [70], if necessary, we assume
$\mathscr{C}$ to be normal. In particular, the Coulomb branch is regular
(as an affine variety[ We stress that some other relevant geometric structures on $\mathscr{C}$, such as the special Kähler metric, are typically non-regular along a divisor
in the geometries we are interested in. This caveat applies to the question of smoothness in the next paragraph.]) in codimension 1.
We are mainly concerned with the situation where the original $\cn=2$ SQFT was a SCFT.
In this special case the $\C$-algebra $\mathscr{R}$ is graded by the scaling dimension $\Delta$, that is:
ℛ= ⊕_Δ∈1/mℤ_≥0ℛ_Δ,    ℛ_0=,    ℛ_Δ·ℛ_Δ^'⊆ℛ_Δ+Δ^',
for some minimal positive integer $m$.

In this paper[ Unless explicitly stated otherwise !!] by a chiral ring $\mathscr{R}$ of rank $r$
we shall always mean a normal affine $\C$-algebra of dimension $r$ with a grading as in  (<ref>). By its Coulomb branch $\mathscr{C}$ we mean the normal affine $r$-variety $\mathsf{Spec}\,\mathscr{R}$ (or its underlying complex $r$-fold).

By definition the Coulomb branch $\mathscr{C}$ is an affine quasi-cone with an algebraic $\C^\times$-action. The maximal ideal $\mathscr{R}_+\equiv \oplus_{
\Delta>0}\mathscr{R}_\Delta$
defines a closed point $0\in\mathscr{C}$, the tip of the quasi-cone, which physically represents the unique vacuum where the superconformal symmetry is not spontaneously broken. $\{0\}$ is the only closed $\C^\times$-orbit, hence it is contained in the closure of all $\C^\times$-orbits.

Smooth vs. non-smooth Coulomb branches.

The Coulomb branch $\mathscr{C}$ is smooth at $0$
(hence everywhere) iff the chiral ring is regular at the tip i.e.
in which case $\mathscr{R}=\C[u_1,\cdots\!, u_r]$
where $u_i$ has dimension $\Delta_i\in \tfrac{1}{m}\mathbb{Z}_{>0}$.

In the physical applications one usually assumes the affine quasi-cone $\mathscr{C}$ to be smooth  because the typical examples of Coulomb branches which arise in physics are smooth affine varieties. However, as it will be clear in the following, to construct a deep theory
(i.e. to really understand the smooth case)
one needs to allow for non-smooth Coulomb branches as well.
The situation leads to two questions: (1)
Which Coulomb branch singularities are allowed?
What is the physical meaning of a singular Coulomb branch?
A plausible physical mechanism for a singularity is a “discrete gauging”
which leads to Coulomb branches of the form $\C^r/G$ with $G$ a finite group.
In this paper we shall assume the Coulomb branch to be smooth,
except when explicitly stated otherwise.

Coulomb dimensions.

When $\mathscr{C}$ is smooth, hence a copy of $\C^r\simeq\mathsf{Spec}\,\C[u_1,\dots,u_r]$
with $u_i\in\mathscr{R}_{\Delta_i}$, the $r$-tuple of rational numbers
$\{\Delta_1,\dots,\Delta_r\}$ is called the Coulomb dimensions.
We shall deduce various restrictions on the allowed Coulomb dimensions below:
only finitely many $r$-tuples of Coulomb dimensions are allowed for a given $r$ [22].

The projective Coulomb branch $\mathscr{P}$.

In the SCFT case it is natural to replace the ordinary Coulomb branch $\mathscr{C}\equiv\mathsf{Spec}\,\mathscr{R}$ by its projective counterpart

\begin{equation}
\mathscr{P}=\mathsf{Proj}\,\mathscr{R}.
\end{equation}

When the Coulomb branch is smooth, $\mathscr{P}$ is a weighted projective space [71]
𝒫≡ℙ(d_1,⋯,d_r),    d_i=m  Δ_i/{m  Δ_j}.
In particular $\mathscr{P}$ is simply-connected $\pi_1(\mathscr{P})=1$.
$\pi_1(\mathscr{P})$ is at most finite even when $\mathscr{C}$ is not smooth: indeed it is a quotient of the finite group $F$.
In the smooth case $\mathbb{P}(d_1,\cdots\!,d_r)$ is isomorphic to a unique well-formed
weighted projective space $\mathbb{P}(q_1,\cdots,q_r)$ such that $\gcd(q_1,\dots,\widehat{q_j},\dots,q_r)=1$ for all $j$ [71].

§.§.§ The Abelian fibration

The arguments in this subsection are heuristics to motivate the definitions in . <ref> below.

By their very definition in all TFT the RG flow is trivial. Hence all topological correlation functions can be exactly
computed using the low-energy effective theory [72, 73] which, roughly speaking, is a supersymmetric $\sigma$-model
with target space the set of SUSY vacua. We say “roughly speaking” because the vacuum space is typically singular, and one should be careful with the precise definition of the low-energy theory.
For us the main point is that there are several distinct “IR effective theories” and
we may use any one of them to compute TFT quantities. Independence of the
TFT amplitudes from the choice of effective description
implies compatibility conditions between these effective theories
which we realize as additional geometric structures on $\mathscr{C}$.

Consider, for instance, the 4-manifold $M\equiv \R^3\times S^1$ where the circle has
radius $R$. The local TFT observables are equal to their SQFT siblings (since $M$ is flat)
and independent of $R$ by topological invariance.
As $R\to\infty$ they may be computed using the 4d IR effective $\cn=2$ SQFT [72, 73],
while as $R\to0$ the very same local TFT observables may be computed using the 3d IR effective SQFT which now is a 3d $\cn=4$ SUSY theory.[  The physical low-energy effective theory depends in an intricate way from $R$ [74]. Since the topological theory is independent of $R$, we are free to fix $R$ in any convenient way. We use the asymptotic limit $R\to\infty$ where things simplify dramatically.]
We take for granted that the 3d Coulomb branch $\mathscr{X}$ satisfies the same “regularity” conditions as its 4d counterpart $\mathscr{C}$, and hence that $\mathscr{X}$ is a smooth[ Just as for the Coulomb branch, the condition that $\mathscr{X}$ is smooth may be relaxed.]
scheme of finite type over $\C$ (in facts over $\mathscr{C}$). This follows from the physical heuristics sketched below,
as well as from the usual special geometry folklore in the physical literature.[  Of course, if $\mathscr{X}$ is not an algebraic scheme (but still smooth), essentially everything we say will remain true in the analytic category.]
On SUSY grounds [33], the 3d Coulomb branch $\mathscr{X}$
is a holomorphic symplectic manifold of quaternionic dimension $r$ (complex dimension $2r$):
indeed on the real tangent bundle $T_\R\mspace{-1mu}\mathscr{X}$ we have a $\mathbb{H}$-action induced by the
3d $\cn=4$ R-symmetry which endows $\mathscr{X}$ with a torsionless $Sp(r,\C)$-structure.
We write $\Omega$ for its $(2,0)$ holomorphic symplectic form.
The compatibility of the TFT computation from the 4d and 3d viewpoints
requires the algebra isomorphism

which produces the canonical scheme-theoretic fibration

π𝒳→𝖲𝗉𝖾𝖼 Γ(𝒳,_𝒳)≃𝒞
over the Coulomb branch $\mathscr{C}$. The compatibility of the actions of the 3d and 4d R-symmetries
implies that
the fibers of $\pi$ are Lagrangian submanifolds, i.e. $\Omega|_{\mathscr{X}_u}=0$
for all $u\in\mathscr{C}$.
The map $\pi$ has an obvious physical meaning in terms of the RG flow from $R=0$ to $R=\infty$.
We claim that $\mathscr{X}$ is a commutative group-scheme over $\mathscr{C}$.
In particular the fiber $\mathscr{X}_u$ over the generic point $u$ is a commutative algebraic group.
Then $\pi$ has a section $s$ given by the zero of the group, $s(u)=0_u\in\mathscr{X}_u$,
and $\mathscr{C}$
can be identified with the submanifold $s(\mathscr{C})\subset\mathscr{X}$ which should also be Lagrangian by
comparison of R-symmetry actions in 3d and 4d. The claim can be easily checked
in Lagrangian SCFTs and hence in full generality at generic points in $\mathscr{C}$:
this suffices to establish that $\mathscr{X}$ is a (smooth) birational model of a commutative group-scheme over
$\mathscr{C}$. In the usual physical situation where the electromagnetic charges obey
a Dirac quantization, the general fiber is actually a polarized Abelian variety.
These physical requirements lead to the “conventional”
special geometries which are the object of these notes.
More exotic possibilities will be explored elsewhere.

§.§ Basic definitions

We codify the above informal discussion in a definition.

A rank-$r$ special geometry is
a holomorphic fibration $\pi$ with section $s$
<EMAIL_ADDRESS><EMAIL_ADDRESS>  π2mus=Id_𝒞,
over a complex (normal) affine variety $\mathscr{C}$ of dimension $r$, such that:

(a) the total space $\mathscr{X}$ has a holomorphic symplectic structure $\Omega$,
and both the fibers $\mathscr{X}_u$ of $\pi$ and the section $s$ are Lagrangian submanifolds:

the generic fiber $\mathscr{X}_u$ ($u\in\mathscr{C}$) is smooth and a polarized Abelian variety of dimension $r$ with zero element $s(u)$.

The locus $\cd\subset \mathscr{C}$ of points $u$ with non-smooth fiber $\mathscr{X}_u$
is called the discriminant.

Although it is not part of the definition, we shall assume the polarization of the fiber to be
principal for definiteness. Physically this is the most natural situation.

With this definition, a special geometry is (in particular) a birational model of an Abelian scheme over a
(quasi-)projective variety, that is, of an Abelian variety defined over the function field $\C(\mathscr{C})$ (e.g. in rank-1 $\mathscr{X}$ is the Néron model [75]
of an elliptic curve over the field of rational functions on a curve $\overline{\mathscr{C}}\simeq\mathbb{P}^1$).
$\mathscr{X}$ is also an Algebraic Integrable System
where the Hamiltonians in involution are the holomorphic functions on $\mathscr{C}$,
i.e. the chiral ring[ More precisely its Frechét completion.] $\mathscr{R}$,
while the conjugate angle variables are the doubly periodic coordinates along the fibers.
Two weaker notions are often used:

* a non UV-complete special geometry is a fibration $\mathscr{X}\to\mathscr{C}$
which satisfies all the properties of a special geometry except that $\mathscr{C}$
is not quasi-projective. A non UV-complete geometry may or may not have a UV completion, i.e. an analytic extension over some affine base. Non UV-complete special geometries describe low-energy effective
$\cn=2$ QFTs which are not UV-complete, and may or may not have a completion;

* a special Kähler geometry
satisfies all the local conditions predicted by the $\cn=2$ superspace
approach to supersymmetry: existence of a local pre-potential $\cf$, existence of a special Kähler metric on the Coulomb branch, etc., but whose underlying global structures (such as the VHS[ VHS=variation of Hodge structure.]) are not defined over $\mathbb{Q}$, or some other number field $\mathbb{F}$, but over a local field such as $\R$ or $\C$.

We stress that the results in this paper refer only to special geometries in the strict sense.
In particular for us it will be absolutely essential that the Coulomb branch $\mathscr{C}$ is quasi-projective. Most of our statements are just false
when this condition does not hold.

A special geometry is called:

* decomposable if it is the product of two lower-rank special geometries. Decomposable geometries
describe QFTs
which decompose in two non-interacting sub-systems;

* free if the fibration is trivial i.e. $\mathscr{X}=A\times \mathscr{C}$ for a fixed Abelian variety $A$.
Free geometries describe free $\cn=2$ QFT;

* isotrivial if all smooth fibers are isomorphic to a fixed
polarized Abelian variety $A$. All $\cn\geq 3$ QFTs have isotrivial geometries,
but there are also genuine $\cn=2$ interacting SCFT with isotrivial geometries. The structure
of isotrivial geometries is discussed in . <ref> below;

* quasi-isotrivial if all smooth fibers are isogeneous to a product
$A\times Y_u$ where the first factor, $A$, is a fixed polarized Abelian variety while $Y_u$
depends on $u\in\mathscr{C}$;

* unitary if it is complete as a Hamiltonian system, that is,
iff for all Hamiltonian $h\in\mathscr{R}$
the corresponding Hamiltonian vector field $v(h)$
is complete, i.e. iff the solution of the (holomorphic) Hamilton equations
dx^i/dt= v(h)^i
exists for all times $t\in \C$ and initial condition.
The special geometries of unitary QFTs are unitary in this geometric sense. It is amusing that quantum unitarity of
the $\cn=2$ QFT amounts to classical “unitarity” of the underlying Hamiltonian system.

In . <ref> we introduce the notion of reducible special geometry.
While decomposable implies reducible, there exist indecomposable geometries which are not irreducible.
However their structure is very constrained, and their number rather small.

The discriminant $\cd$ is either empty or of pure codimension $1$. When $\cd=\varnothing$ the geometry is free. Otherwise there are finitely-many elements $p_a\in\mathscr{R}$
with $(p_a)$ prime, such that $\cd=\{p_1 p_2\cdots p_s=0\}\subset\mathscr{C}$.

The first assertion is e.g. Proposition 3.1(2) in [45]. The second one follows from the well-known fact that all families of smooth projective varieties parametrized by $\C$ are trivial (cf. . 13.7 of [40]). The third statement is well known.

The above definitions refer to the geometry of a general 4d $\cn=2$ SQFT.
The geometries of $\cn=2$ SCFTs have an extra geometric structure.

A $\C^\times$-isoinvariant special geometry is a special geometry with a complete holomorphic vector field $\ce\in \Gamma(\mathscr{X},T\mspace{-2mu}\mathscr{X})$, called the Euler vector,
such that for all $z\in\C$ the exponential map $\exp(z\,\ce)\colon\mathscr{X}\to\mathscr{X}$ is a complex automorphism preserving all special geometric structures: i.e. $\exp(z\,\ce)$ preserves set-wise the section $s$ and maps fibers into fibers, while[ Here and below $\mathscr{L}_v$ stands for the Lie derivative in the direction of the vector field $v$.]

ℒ_2muΩ=Ω,  that is,  exp(z )^-2mu*1muΩ= e^z Ω.

The Euler vector $\ce$ induces a grading of the chiral ring $\Gamma(\mathscr{X},\co_\mathscr{X})\simeq \mathscr{R}=\oplus_\Delta\mathscr{R}_\Delta$ by
h∈ℛ_Δ  ⇔  ℒ_2mu h=Δ h.
A Hamiltonian $h\in\mathscr{R}_\Delta$ is said to have dimension $\Delta$.
The identity has dimension zero. We shall see momentarily that in a unitary special geometry
the identity $1$ is the only element of $\mathscr{R}$ with dimension zero, while all other Hamiltonians have $\Delta\geq1$.
We shall show below that the $\Delta$'s are rational numbers and that only a short list of rational numbers may appear at a given rank $r$.
Then $\mathscr{R}$ is a finitely generated local graded $\C$-algebra of the expected form
(<ref>). When the Coulomb branch is smooth $\mathscr{R}=\C[u_1,\cdots, u_r]$ and the coordinates have definite dimension
$\mathscr{L}_\ce\, u_i=\Delta_i\, u_i$.
In particular $\exp(z\,\ce)$ induces a complex automorphism of the Coulomb branch,
$\exp(z\mspace{2mu}\boldsymbol{\Delta})\colon \mathscr{C}\to \mathscr{C}$,
such that
πexp(z )=exp(z2muΔ) π,
which for smooth $\mathscr{C}$ is simply $u_i\mapsto e^{z \Delta_i} u_i$.
It follows from Definition <ref> that the isoclass
of the fiber, as a polarized Abelian variety, is invariant along an orbit of
the group of automorphisms $\exp(z\mspace{2mu}\boldsymbol{\Delta})$
for all z∈, u∈𝒞∖  𝒳_exp(z2muΔ)u≃𝒳_u
as polarized Abelian varieties.
The kernel of group homomorphism
$\exp(\cdot \,\ce)\colon\C\to \mathsf{Aut}(\mathscr{X})$ is $2\pi i m\Z$ where $m$ is the integer in eq.(<ref>).
Indeed the induced map $\exp(2\pi i m\mspace{2mu}\boldsymbol{\Delta})$ is the identity on $\mathscr{C}$, so $\exp(2\pi i m\mspace{2mu}\ce)$ fixes the zero element $0_u$ of each Abelian fiber $\mathscr{X}_u$, while it acts as the identity on its Lie algebra
because of the isomorphism $\Omega\colon T_{0_u}\mspace{-2mu}\mathscr{X}_u\simeq T^*_u\mspace{2mu}\mathscr{C}$, so it is the identity in each fiber.
The automorphism group acting effectively on $\mathscr{X}$ is $\C^\times\simeq \C/2\pi i m\Z$.
The prime divisors $p_a$ in the discriminant $\cd$ are irreducible homogeneous elements of $\mathscr{R}$, i.e.
$\mathscr{L}_\ce\, p_a= \Delta(p_a)\, p_a$ for some $\Delta(p_a)\in \tfrac{1}{m}\mathbb{N}$.

In the physical literature it is more common to require the existence of a
Seiberg-Witten (SW) differential $\lambda$ such that $\Omega=d\lambda$. $\lambda$ is simply the image of the vector $\ce$
under the natural isomorphism $\Omega\colon T\mspace{-1mu}\mathscr{X}\to T^*\mspace{-3mu}\mathscr{X}$, and the two data $\lambda$, $\ce$ are equivalent in the $\C^\times$-isoinvariant situation. It is more convenient to work with $\ce$ since it makes manifest that $\lambda$ corresponds to an automorphisms ($\equiv$ a symmetry) of the special geometry which is fully determined by its symplectic structure.

§.§.§ Geometric version of the unitarity bound

(1) In a unitary $\C^\times$-isoinvariant special geometry all homogeneous elements
of $\mathscr{R}$ except the identity, have dimension $\Delta\geq1$. (2) In a unitary, indecomposable, non-free $\C^\times$-isoinvariant geometry the inequality is strict $\Delta>1$.

(1) Let $h$ be a non-constant Hamiltonian of dimension $\Delta$. $dh$ is not zero
and $\mathscr{L}_\ce dh= d\mathscr{L}_\ce h=\Delta\, dh$.
The corresponding Hamiltonian vector $v(h)$ is non-zero and satisfies
Δ  i_v(h)Ω=Δ dh= ℒ_dh=
ℒ_(i_v(h)Ω)=i_ℒ_v(h)Ω+ i_v(h)ℒ_Ω= i_ℒ_v(h)Ω+i_v(h)Ωor $\mathscr{L}_\ce\mspace{2mu} v(h)=(\Delta-1)v(h)$. Therefore
exp(z2mu)_∗2mu v(h)= z^Δ-1 v(h).
If $\Delta<1$ the Hamiltonian vector field
$v(h)$ diverges at the tip of the quasi-cone and so cannot be complete. (2)
follows from the following result whose proof is sketched in appendix <ref>.
The connected component $(\mathscr{X}_0^\text{\rm reg})^0$
of the smooth locus $\mathscr{X}_0^\text{\rm reg}\subset\mathscr{X}$ of the fiber over the origin $0$
has the structure

(𝒳_0^reg)^0= A_0×^r-ℓ,  ℓ≡A_0=ℛ_Δ=1,
where $A_0$ is a polarized Abelian variety,
while the smooth fibers have the form $\mathscr{X}_u=A_0\times\mathscr{Y}_u$ with
$\mathscr{Y}_u$ a polarized Abelian variety of dimension $r-\ell$.

Clearly the geometry is the product of the free geometry $(A_0\times \C^\ell)\to\C^\ell$
with a special geometry $\mathscr{Y}\to \C^{r-\ell}$ with fiber $\mathscr{Y}_u$ and no
dimension 1 coordinate.

Eq.(<ref>) says that the fiber over the origin has additive (as contrasted to multiplicative) degeneration, thus in a non-free $\C^\times$-isoinvariant geometry $\mathscr{X}_0$ is always an unstable degeneration of an Abelian variety.

The structure of the fiber over the origin, eq.(<ref>) is one of the many properties of the special geometry which, taken together, guarantee that the corresponding $\cn=2$ QFT is free of Landau poles. For instance, in rank-1 it rules out singular fibers of Kodaira type $I_n$, which
would correspond to $\cn=2$ SQCD a formal QFT affected by Landau poles.

§.§.§ Implied structures: metric, SW periods, prepotential

Let $L$ be the fibers' polarization seen as a integral $(1,1)$  class which
restricts to a Kähler class in each fiber. On the complement $\mathring{\mathscr{C}}=\mathscr{C}\setminus\cd$ of the discriminant we have the function
Ku↦i∫_𝒳_u L^r-1λ∧λ^*=
where $\{A_i,B^j\}$ is a symplectic basis of 1-cycles on the fibers $\mathscr{X}_u$ defined locally in some $U\subset \mathring{\mathscr{C}}$ with respect to the polarization $L$
and $\lambda\equiv \iota_\ce\Omega$.
The function $K$ is smooth and pluri-subharmonic in $\mathscr{C}\setminus \cd$, so $K$ is the potential of a Kähler metric with Kähler form
ω=i∫_𝒳_u L^r-1∧Ω∧Ω̅.
Except in the free case (when $\cd=\varnothing$) this “special Kähler” metric is not complete since it has singularities along the
discriminant $\cd$ which are at finite distance in the special metric.

We define the SW period $\Pi=(a^D_i,a^j)^t$ to be the local functions in $U$
a^D_i(u)=∫_B^iι_1mu Ω|_𝒳_u,    a^j(u)=∫_A_iι_1mu Ω|_𝒳_u,  u∈U, i,j=1,…,r
so that
K= i  Π^†ΩΠ,  where  Ω=( 0     1_r

-1_r     0).
The periods are locally-defined holomorphic functions on $\mathscr{C}$.
Locally on $\pi^{-1}(\mathring{\mathscr{C}})$ we may find Darboux coordinates such that
$\Omega=da^i\wedge dz_i$ where $dz_i$ are holomorphic differentials on the fiber
normalized as
∫_A_idz_j=δ^i_j,    ∫_B^idz_j=τ_ij.
Then $\iota_\ce\mspace{1mu} \Omega=a^i dz_i$ and
a^D_i=∫_B^iι_1mu Ω|_𝒳_u=τ_ij a^j.
In particular the $A$-periods $(a^1,\dots, a^r)$ are local complex coordinates in $\mathring{\mathscr{C}}$. Clearly
ℒ_ Π=Πso that on local functions on $\mathring{\mathscr{C}}$ we have $\mathscr{\ce}=a^i\partial_{a^i}$ and $\mathscr{L}_\ce\,\tau_{ij}=0$ gives $a^i\partial_{a^i}\tau_{jl}=0$ so that

da^D_i=d(τ_ij a^j)= τ_ij da^j  ⇒  da_i^D∧da^i=0.
The last equation says that, locally in $U\subset\mathring{\mathscr{C}}$ there exists a holomorphic function
$\cf(a^j)$, called the prepotential, such that
ℒ_=2   and  a^D_i=∂/∂a^i,  τ_ij=∂^2/∂a^i∂a^j.
The special coordinates yield an affine structure to $\mathring{\mathscr{C}}$. This is the holomorphic version of the integral affine structure on the base of a Liouville integrable system [76].

§.§ Unbroken R-symmetry and the “characteristic dimension”

In the smooth case it is convenient to see
the $r$-tuple of rational numbers[ We assume the $\Delta_i$ to be ordered in a non-decreasing sequence, i.e. $\Delta_{i+1}\geq \Delta_i$.] $\{\Delta_1,\cdots, \Delta_r\}$
as a point in the projective space over $\mathbb{Q}$ and write
(Δ_1,Δ_2,…,Δ_r)=λ(d_1,d_2,…, d_r)
where $(d_1,d_2,\dots,d_r)$ are the unique positive integers with $\gcd(d_1,\dots\,d_r)=1$
which represent them same point in $\mathbb{P}^{\mspace{1mu}r-1}(\mathbb{Q})$.
We shall refer to $d_i$ as the degree of $u_i$, while $\Delta_i\equiv \lambda d_i$
is its dimension.
$\lambda$ is
a positive rational number which we write in minimal terms as
λ=n/m    with
$\lambda$ is a basic invariant of the $\C^\times$-isoinvariant special geometry.
It is well-defined also for non-smooth Coulomb branches: if $\{\phi_1,\dots,\phi_\ell\}$ is
a set of generators of $\mathscr{R}$, one sets
(Δ(ϕ_1),…,Δ(ϕ_ℓ))=λ (d(ϕ_1),…,d(ϕ_ℓ))
with the
$d(\phi_i)$ integral and coprime.
By construction
the automorphism $\exp(2\pi i\mspace{1mu}\ce/\lambda)$ acts trivially on $\mathscr{R}$,
so it fixes all points of $\mathscr{C}$,
while it multiplies $\Omega$ by a $n$-th root of unity
exp(2πi /λ)^*2muΩ= e^2πi m/nΩ,
which depends only on $m\bmod n$. The root $\zeta\equiv e^{2\pi im/n}$, or equivalently
the pair $(n,m\bmod n)$ with $\gcd(m,n)=1$, was called the
characteristic dimension in [25] and written $\varkappa=n/\langle m\rangle_n$
where  $\langle m \rangle_n$ is the unique positive integer $\leq n$ congruent to $m$ mod $n$.
The physical meanings of $m$ and $n$ are obvious:

(1) the group which acts effectively on the geometry is
$\C^\times=\exp[\C/(z\sim z+2\pi i m)]$;

(2) at the generic point in $\mathscr{C}$
the $\C^\times$ symmetry is spontaneously broken to $\Z_n$;

(3) at any point $u\in\mathscr{C}$ the $U(1)_R$
symmetry is broken to a group which contains $\Z_n$. Hence, a part
for the tip $0$, to a discrete group of the form $\Z_{nk}$.

One shows [25] that $n$ can take only five values

hence we have only 8 possible $\varkappa$

ϰ∈{1, 65, 43, 32, 2, 3, 4, 6}
in one-to-one correspondence with the  conjugacy classes of torsion elements of $SL(2,\Z)$,
or equivalently with the Kodaira elliptic fibers with semisimple monodromy [51, 52]. The possible values of $\varkappa$ are just the allowed dimensions of the Coulomb dimension $\Delta$ in a $r=1$ special geometry: they are in 1-to-1 correspondence with the
semisimple Kodaira fibers, see table <ref>.
The basic result of [25] was that the Abelian fiber over the generic point $u\in\mathscr{C}$
admits complex multiplication by the cyclotomic field $\mathbb{Q}(\zeta)\equiv \mathbb{Q}(e^{2\pi i/\varkappa})$ and hence its isoclass is uniquely determined unless $\mathbb{Q}(\zeta)\equiv\mathbb{Q}$, i.e. unless $\varkappa\in\{1,2\}$.

When $\varkappa\not\in\{1,2\}$ all smooth fibers are isomorphic as polarized Abelian varieties, and the full geometry is isotrivial, hence easily constructed and studied by simple group theory [25]. The statement holds independently of the assumptions that $\mathscr{C}$ is smooth and that the polarization is principal.

Semi-simple Kodaira fibers9pt

Kodaira fiber     $\mathbf{I}_0$     $\mathbf{I}_0^*$     $\mathbf{II}$     $\mathbf{II}^*$     $\mathbf{III}$     $\mathbf{III}^*$     $\mathbf{IV}$     $\mathbf{IV}^*$

$\varkappa$     $1$     $2$     $\tfrac{6}{5}$     $6$     $\tfrac{4}{3}$     $4$     $\tfrac{3}{2}$     $3$

monodromy     $\left[\begin{smallmatrix}1 & 0\\
0 & 1\end{smallmatrix}\right]$     $\left[\begin{smallmatrix}-1 & 0\\
0 & -1\end{smallmatrix}\right]$     $\left[\begin{smallmatrix}1 & 1\\
-1 & 0\end{smallmatrix}\right]$    $\left[\begin{smallmatrix}0 & -1\\
1 & 1\end{smallmatrix}\right]$     $\left[\begin{smallmatrix}0 & 1\\
-1 & 0\end{smallmatrix}\right]$     $\left[\begin{smallmatrix}0 & -1\\
1 & 0\end{smallmatrix}\right]$     $\left[\begin{smallmatrix}0 & 1\\
-1 & -1\end{smallmatrix}\right]$    $\left[\begin{smallmatrix}-1 & -1\\
1 & 0\end{smallmatrix}\right]$

Euler no.     $0$     $6$     $2$     $10$     $3$     $9$     $4$     $8$

Lie algebra     $-$     $\mathfrak{so}(8)$     $-$     $E_8$     $\mathfrak{su}(2)$     $E_7$     $\mathfrak{su}(3)$     $E_6$

For later reference we sketch a proof of these statements. Preliminary we recall
a few well know facts about polarized Abelian varieties.[ For the facts about complex Abelian varieties used here and below see [77] expecially chapter 13.] Let $A$ be a polarized
Abelian variety and $\mathsf{Aut}(A)$ the group of automorphisms $A\to A$
which preserve the polarization. $\mathsf{Aut}(A)$ is a finite group with two
natural representations. First we have the $\mathbb{Q}$-defined rational representation
$\chi$ acting on $H_1(A,\Z)\otimes_\Z\mathbb{Q}$. For principal polarizations $\chi$ is
a group embedding $\chi\colon \mathsf{Aut}(A)\to Sp(2r,\Z)$; for a general polarization see
e.g. chapter 2 of [78].
Then we have the analytic representation $\sigma\colon \mathsf{Aut}(A)\to GL(r,\C)$ which acts on the holomorphic tangent space
at $0$, $T_0A\equiv\mathsf{Lie}(A)$. We have the isomorphism of complex representations
Since $\chi$ preserves the lattice $H_1(A,\Z)\subset H_1(A,\mathbb{Q})$, this implies that for all $\eta\in\mathsf{Aut}(A)$
the characteristic polynomial $\det[z-\chi(\eta)]$ is a product of cyclotomic polynomials
whose roots are the eigenvalues of $\sigma(\eta)$ and their conjugates.
The automorphism
ξdef=exp(2πi 2mu/λ)𝒳→𝒳
generates a $\Z_n$ group of automorphisms of $\mathscr{X}$ which fixes point-wise the Coulomb branch $\mathscr{C}$ and hence acts trivially on its holomorphic cotangent bundle
$\xi$ then restricts to an automorphism $\xi_u\colon\mathscr{X}_u\to\mathscr{X}_u$ of each fiber (smooth or not).

Let $u\in\mathscr{C}$ be a generic point, and $\mathscr{X}_u$ its smooth Abelian fiber. The symplectic structure gives a linear isomorphism
The automorphism $\xi$ acts as the identity
on $T_u^*\mspace{1mu}\mathscr{C}$, hence by eq.(<ref>) as multiplication by $\zeta^{-1}$
on $T_{s(u)}\mathscr{X}_u$, that is,
σ(ξ_u)=ζ^-1·1∈GL(r,)  ⇒  [z-χ(ξ_u)]=((z-ζ^-1)(z-ζ))^r∈[z]
which says that the root of unity $\zeta$ is either $\pm1$ or
an imaginary quadratic integer. This proves eqs.(<ref>),(<ref>).
The final step is
to recall[ See Corollary 13.3.5 in [77].] the fact that when multiplication by $\zeta= i$ or $e^{2\pi i/3}$
is an automorphism of the Abelian variety $A$ we have[ Through out this paper $E_\tau$ stands for the elliptic curve of period $\tau$.]
A=E_ζ×E_ζ×⋯×E_ζ^r factors
and the geometry is (in particular) isotrivial, hence well understood [25, 55].
It remains to study the two cases $\varkappa=1$ and $\varkappa=2$
where the geometry may or may not be isotrivial.

§.§ The index of an irreducible homogeneous divisor

By an irreducible homogeneous divisor in $\mathscr{C}$ we mean an irreducible codimension-1
subvariety preserved by the $\C^\times$-action (the closure of the $\C^\times$-orbits it contains).
An irreducible homogeneous divisor has the form $\{h=0\}\subset\mathscr{C}$ for some irreducible quasi-homogeneous[ In the sequel we simplify `quasi-homogenous' in `homogeneous'.]
Hamiltonian $h\in\mathscr{R}$ of degree $d$, i.e.
ℒ_/λ h= d  h.
E.g. for a smooth geometry $h=h(u_i)$ is a polynomial  in the $u_i$'s of degree $d$, i.e.
h(λ^d_i u_i)=λ^d2mu h(u_i)  ∀  λ∈ and some d∈ℕ.
Consider the Hilbert series $P_h(t)$ of the algebra $\mathscr{R}/(h)$ (graded by degree not dimension!).
E.g. for $\C[u_1,\dots, u_r]/(h)$
The enhancement index $\nu(h)$ of the divisor $h$ is the largest positive integer such that
The physical meaning of $\nu(h)$ is:
ν(h)=[(unbroken subgroup of $\C^\times$ at the

generic point
in the $h=0$ locus) :  (unbroken subgroup of $\C^\times$

at the generic point
in $\mathscr{C}$)]
i.e. $\nu(h)$ measures the enhancement of the unbroken R-symmetry along the divisor $h$.
A divisor $h$ is enhanced (resp. non-enhanced) iff $\nu(h)>1$
(resp. $\nu(h)=1$).

The authors of [17] distinguish the divisors in unknotted vs. knotted ones.
Often, but not always, this coincides with the distinction enhanced vs. non-enhanced.

An irreducible homogeneous divisor $(h)$ is knotted
iff the fundamental group of its complement $\pi_1(\C^r\setminus (h))$ is non-Abelian.

Smooth divisors are automatically unknotted.

For $r=2$ the irreducible homogeneous divisors have one of the two
u_i  for i=1,2  or  u_1^d_2- z u_2^d_1  with z∈^×.
For divisors of the second kind $\nu=1$, while $\nu(u_i)=d_1d_2/d_i$.
The divisors of the second kind are knotted unless $d_1=1$,
while $u_i$ is never knotted.

In a general rank-$r$ smooth geometry, the irreducible divisors of degree $d_i\in \{d_1,\dots, d_r\}$
have $\nu=\gcd\{d_1,\dots,\widehat{d_i},\dots, d_r\}$
while divisors with $d\not\in \{d_1,\dots, d_r\}$ have $\nu=1$.

§.§ Coulomb dimensions I: CM-types along submanifolds

Suppose we have an irreducible reduced total intersection $H=\{h_1=\cdots=h_\ell=0\}$
with an enhancement index $\nu(H)\equiv \nu>1$ which is not contained in the discriminant $\cd$.
The automorphism $\eta\equiv\exp(2\pi i\, \ce/(\nu\lambda))$
fixes all points of $H$, hence its acts as the identity on its cotangent bundle $T^*\mspace{-2mu}H\subset T^*\mspace{1mu}\mathscr{C}|_H$ while it acts by polarized automorphisms on the fibers $\mathscr{X}_u$
for $u\in H$.
We write $N^\ast\mspace{-1mu} H\to H$ for the rank-$\ell$ conormal bundle to $H$; it is spanned
by the differentials $dh_a$ ($a=1,\dots, \ell$). Let $u\in H$ be a generic point; by assumption the fiber $\mathscr{X}_u$ is smooth. One has
T_u^*1mu𝒞= T^*_uH⊕N^∗-1muH_u
$\eta_u$ acts on the first summand as the identity
and on the second as multiplication by the
$\ell\times\ell$ matrix
diag{exp(2πi2mu m1mu Δ(h_k))/(nν))}.
Using the isomorphims $\Omega\colon T_{s(u)}\mathscr{X}_u\to T_u^*\mspace{1mu}\mathscr{C}$
we get the eigenvalues of its analytic representation
$\sigma(\eta_u)$ are

(e^-2πi m/(nν),⋯, e^-2πi m/(nν)^(r-ℓ) times,e^-2πi m(1-Δ(h_1))/(nν),⋯,e^-2πi m(1-Δ(h_ℓ)/(nν))
This $r$-tuple of roots of unity, together with their conjugates, are the roots (with multiplicities)
of a product of cyclotomic polynomials.
Enforcing these conditions on all total intersections of elements of $\mathscr{R}$
yield a large web of constraints on the possible $n$, $m$ and $\nu$
and in turn on the Coulomb dimensions $\Delta_i$.
We stress that nothing depends on the assumption that $\mathscr{C}$ is smooth (except for the $\Delta_i$'s).

We give some examples.

§.§.§ Example: dimension formulae along regular axes [22]

For simplicity from now on we assume $\mathscr{C}$ smooth.
Suppose that the $i$-th coordinate axis $H_i=\{u_j=0\ \text{for }j\neq i\}$
is not contained in the discriminant: we say that $H_i$ is a regular axis. Then $\nu=d_i$ and the list of roots in (<ref>) becomes

Note that a priori once we know this ordered list of roots of 1,
$1/\Delta_i$ is determined only mod 1, but since in a unitary
$0<1/\Delta_i \leq 1$ knowing  the $i$-th root in the list
suffices to determine $\Delta_i$ uniquely. Instead $\Delta_j$ for $j\neq i$
is determined by (<ref>) only up to  addition of an integral multiple of $\Delta_i$.
The ambiguity[ The role of this ambiguity was overlooked in [22].] is however small since only a finite list of dimensions are allowed for any given $r$
(see . <ref>).
Moreover often there are several regular axes $H_j\not\subset \cd$ and
we get one such formula for the dimensions for each one of these axes,
getting an overdetermined system which gets rid of all ambiguities.

When the $i$-th coordinate axis $H_i\not\subset\cd$
the list of $r$ $(n\,d_i)$-th roots of unity in eq.(<ref>)
is a subset of the $2r$ roots of a product of cyclotomic polynomials
with the property that the union of this subset with its conjugate
yields the full set of all $2r$ roots. This entails that
is a root in a cyclotomic field of degree at most $2r$. That is,
the dimension $\Delta_i$ of a regular axis has the form

Δ_i= n_i/m_i
where ϕ(n_i)≤2r and (n_i,m_i)=1  1≤m_i<n_i
so that there are only finitely many allowed dimensions at a given rank $r$ [22].
We shall show in . <ref> that this conclusion remains valid even when
$H_i\subset \cd$; in facts in that case we get the stronger condition $\phi(n_i)\leq 2(r-1)$.

§.§.§ New dimensions and CM-type
The above results lead to the

A rational number $\Delta$ is a new dimension in rank $r$
if it is an allowed Coulomb dimension in rank $r$ but not in rank $(r-1)$. The set of new dimensions in rank-$r$ is

Ξ(r)={n/mϕ(n)= 2r, (n,m)=1, 1≤m<n}    for r≥2

{1,6/5,4/3,3/2,2,3,4,6}     for r=1

see [22] for the cardinality $|\Xi(r)|$ and other interesting properties of the set $\Xi(r)$.

We shall see in . <ref> that an axis $H_i$ with $\Delta_i$ a new dimension in rank $r$ is automatically regular.
In facts, we have more:
If the axis $H_i$ has a new dimension $\Delta_i$, the fiber $\mathscr{X}_u$
at the generic point $u\in H_i$ has complex multiplication by the CM field
$\mathbb{Q}(e^{2\pi i/\Delta_i})$ of degree $2r$ and CM-type[ For the notion of CM-type see e.g. chapter 13 of [77].] given by eq.(<ref>).

It is well-known that the number of isomorphism classes of polarized Abelian varieties
of given CM-type is equal to the class number $h$ of the corresponding
number field $\mathbb{Q}(e^{2\pi i/\Delta_i})$. In particular for ranks $r\leq 9$
all new dimensions correspond to fields with  $h=1$ and
the $r$-tuple of dimensions $\{\Delta_1,\dots,\Delta_r\}$ uniquely determines
the fiber $\mathscr{X}_u$ along a new dimension axis.

§.§.§ Sample results

We list a few simple consequences of the above Facts for rank-2 that we shall use later in the examples.
The geometries are assumed to be smooth.
Here we use that an axis with a new dimension is automatically regular (see . <ref>).

(1) A rank-2 special geometry having one of its dimensions in the list
$\{12,12/5,12/7,12/11\}$ is automatically isotrivial;

(2) if both axes are regular of new dimension then the dimensions are
$\{8,12\}$ or $\{\tfrac{8}{7},\tfrac{10}{7}\}$. A geometry with dimensions $\{8/7,10/7\}$ cannot be isotrivial;

(3) the rank-2 geometries where both axes are regular  must have dimensions
$\{5/4,3/2\}$, $\{8/7,10/7\}$, $\{8,12\}$ or $\{5/3,4\}$.
The first two correspond to the Argyres-Douglas theories of types $A_5$ and $A_4$, respectively,
and the third one is the $G_8$ isotrivial geometry constructed in [25];

(4) a rank-2 geometry with a new dimension of the type $5/k$ or $10/l$ is the Jacobian
fibration of a fibration in generically smooth genus-2 curves.

(1) the dimension formulae yields pairs of dimensions with $\varkappa\not\in\{1,2\}$.
(2)(3) if both axes are regular we can compute $\{\Delta_1,\Delta_2\}$ in two ways; consistency
of the two computations yield the list. Second statement in (2): the fibers over the two axes
have multiplication by $\mathbb{Q}(e^{2\pi i/8})$ and respectively $\mathbb{Q}(e^{2\pi i/10})$,
so are not isomorphic. (4) the fiber over the regular axis with the new dimension has muliplication by
$\mathbb{Q}(e^{2\pi i/5})$ hence is simple and not the product of two elliptic curves (cf. Corollary 13.3.6. of [77]).

§.§ SW differential on enhanced axes

Suppose that the $i$-th axis $H_i$ is regular.
The R-symmetry which is unbroken along $H_i$ is generated by $\xi_i\equiv\exp(2\pi i\,\ce/\Delta_i)$. $\xi_i$ acts by an automorphism of the generic (smooth)
fiber $\mathscr{X}_u$ with analytic representation

\begin{equation}
\sigma(\xi_i)=\mathrm{diag}\big\{\!\exp[2\pi i(\Delta_1-1)/\Delta_i],
\exp[2\pi i(\Delta_2-1)/\Delta_i],\dots,\exp[2\pi i(\Delta_r-1)/\Delta_i]\big\}
\end{equation}

and rational representation $\chi(\xi_i)\in Sp(2r,\Z)$ acting on $H_1(\mathscr{X}_u,\Z)$,
as complex representations.
The polarization gives an isomorphism
𝖫𝗂𝖾(𝒳_u)≃H^0,1(𝒳_u)≃H^0(𝒳_u, Ω^1_𝒳_u)
Let $\lambda\equiv \iota_\ce\Omega$ be the SW differential
and $\lambda_u\equiv \lambda|_{\mathscr{X}_u}$ be its restriction to the fiber.
We have
ξ^*_iλ_u= e^2πi/Δ_i λ_u.
Let $\{e_a\}$ be a symplectic basis of $H_1(\mathscr{X}_u,\Z)$. The SW periods transform as
⟨e_a, λ_u⟩= ξ_i ·⟨e_a,λ_u⟩=
⟨χ(ξ_i)_ab e_b·α, ξ_i^*λ_u⟩=
e^2πi/Δ_i χ(ξ_i)_ab  ⟨e_b,λ_u⟩so, setting

χ(ξ_i)_ab Π(u)_b= e^-2πi/Δ_i Π(u)_a
i.e. the SW periods along $H_i$ are eigenvectors of the rational representation of $\xi_i$
of eigenvalue $\exp(-2\pi i/\Delta_i)$ (up to overall scale the period is independent of $u\in H_i$,
since the axis minus the origin is a $\C^\times$-orbit).

§.§ $\varkappa=1$ vs. $\varkappa=2$ special geometries

The characteristic dimension $\varkappa$ is a basic invariant of a $\C^\times$-isoinvariant special geometry. When $\varkappa$ is not $1$ or $2$ the geometry is isotrivial,
all smooth fibers are the product of $r$ copies of $E_{e^{2\pi i/\varkappa}}$, and the possible
geometries are
essentially known globally.[ We stress that a geometry may be isotrivial even when $\varkappa\in\{1,2\}$.] We remain with two cases: $\varkappa=1$ and $\varkappa=2$.
What is the difference between these two (potentially) non-isotrivial situations?

The automorphism $\xi\equiv\exp(2\pi i\,\ce/\lambda)$ which fixes point-wise the Coulomb branch $\mathscr{C}$ acts on the SW differential $\lambda\equiv \iota_\ce\mspace{1mu}\Omega$ as
ξ^*λ=λ    ϰ=1

-λ   ϰ=2

and in facts
ξ^*|_H^1(𝒳_u,)= (-1)^ϰ+1.
This means that when $\varkappa=2$ the SW periods $(a_i^D,a^j)^t$ are well-defined only
up to overall sign.
In Lagrangian SCFTs with $\varkappa=2$ the sign flip is part of the gauge group (it is the longest word in the gauge Weyl group), so
the two signs are obviously physically equivalent and indistinguishable.
This property extends to special geometries with $\varkappa=2$ which do not describe Lagrangian SCFTs.

We can state the difference between $\varkappa=1$ and $2$ in another (related) way.
When $\varkappa=1$ there is one element $h$ of the Coulomb branch function field[ $\C(\mathscr{C})$ is the quotient field of the integral domain $\mathscr{R}$.] $\C(\mathscr{C})$
of dimension $1$, while there is no such object when $\varkappa=2$.
When $\varkappa=2$ we have instead an element $h^2\in\C(\mathscr{C})$ of dimension $2$.

This difference between the two situations will have fundamental consequences
for the global aspects of a non-isotrivial special geometry.

§.§ Discriminant complement and monodromy representation

As before $\cd\subset\mathscr{C}$ is the discriminant i.e. the divisor
of points with singular fibers; we write
$\sum_i \cd_i$ for its decomposition into irreducible
components. $\cd_i=(p_i)$ for irreducible quasi-homogeneous polynomials $p_i$
of degree $g_i$ and index $\nu_i$.
We write $\mathring{\mathscr{C}}\equiv \mathscr{C}\setminus \cd$
for the complement of the discriminant, i.e. the open domain of “good” points with smooth
Abelian fiber.
We have a holomorphic family of polarized Abelian varieties
parametrized by the complement $\mathring{\mathscr{C}}$.
We stress that $\mathring{\mathscr{C}}$ is quasi-projective over $\C$.
Then we may apply the standard methods and results of
the theory of Variations of Hodge Structure (VHS) over
a quasi-projective base [34, 35, 36, 37, 42, 39, 38, 40, 41].

All fibers of $\mathring{\pi}$ are diffeomorphic. This entails that over $\mathring{\mathscr{C}}$
we have a flat Gauss-Manin local system with fiber $H_1(\mathscr{X}_u,\Z)$ [35, 40].
Going along a non-trivial closed curve $\ell\subset\mathring{\mathscr{C}}$
the fiber of the local system gets back to itself up to a rotation by an element $\gamma(\ell)$
of its polarized automorphism group: for principal polarizations this is the Siegel modular group $Sp(2r,\Z)$ (see [78] for the general case). This yields
a group homomorphism

called the monodromy representation [34, 35, 36, 37, 42, 39, 38, 40, 41, 79],
a crucial invariant of the special geometry
which controls many physical properties of the associated SCFT.
The image $\Gamma\subset Sp(2r,\Z)$ of $\varrho$ is the monodromy group.

§.§.§ Topology of the complement

A basic ingredient of special geometry is then the topology of the complement of the discriminant
$\mathring{\mathscr{C}}$, in particular its fundamental group $\pi_1(\mathring{\mathscr{C}})$.
While the Abelianization of $\pi_1(\mathring{\mathscr{C}})$,
i.e. $H_1(\mathring{\mathscr{C}},\Z)$, is a pretty simple group [80]
H_1(𝒞,)≃^k,  where $k$ is the number of components of $\cd$,
the group $\pi_1(\mathring{\mathscr{C}})$ is typically rather complicated. Roughly speaking, its
derived group
is a measure of the interactions in the corresponding SCFT.
A simple example will illustrate the issue.

Consider the special geometry of 4d $\cn=4$ SYM with $G=SU(3)$
(seen as a special instance of $\cn=2$ theory). The Coulomb dimensions $\{\Delta_1,\Delta_2\}$
are the
degrees of $\mathsf{Weyl}(SU(3))$, $\{2,3\}$, while physically we expect the monodromy group
to be the $SU(3)$ Weyl group $\Gamma=\mathsf{Weyl}(SU(3))\simeq \mathfrak{S}_3$.
Let us see how this comes about topologically. The discriminant has a unique irreducible component
corresponding to the locus in $\mathscr{C}$ where a $SU(2)$ gauge symmetry is restored
and the corresponding non-Abelian d.o.f. get light.
Thus $\cd=(u_1^3-u_2^2)$. One has [81]
π_1(𝒞)≡π_1(^2∖{u_1^3-u_2^2=0})=⟨α, βα^3=β^2⟩which is the Artin braid group $\mathcal{B}_3$ of type $A_2$.
Since the geometry is isotrivial with dimensions $\{2,3\}$, the monodromy representation $\varrho$ factors through the finite quotient $\mathsf{Weyl}(SU(3))\simeq \mathcal{B}_3/\mathcal{P}_3$ producing the physically expected results. ($\mathcal{P}_3$ is the pure braid group in 3 strands). A similar story holds for $\cn=4$ with any gauge group.

We list a few useful facts about the topology of hypersurface complements [80].
When the divisor $\cd$ is normal crossing with $k$ irreducible components the
$\pi_1$ of the complement is an Abelian group,
generated by the $k$ lassos circling the irreducible components. By the Zariski conjecture (proved in [82]) the same holds for all  nodal planar curves.
In the general case
we have an explicit description of $\pi_1(\mathring{\mathscr{C}})$ given by the Zariski-van Kampen theorem
[80, 83, 84]
that we shall review in . <ref>.

§.§.§ Subtleties when $\varkappa=2$

We assume our special geometry not to be isotrivial. At a generic point
$\ast\in\mathscr{C}$ the R-symmetry is totally broken for $\varkappa=1$
while we have
a residual unbroken $\Z_2$ symmetry when $\varkappa=2$.
We define the SW periods $\Pi\equiv (a^D_i,a^j)^t$ in a small neighborhood of $\ast$,
and analytically continue them along a non-trivial loop $\ell\subset\mathring{\mathscr{C}}$
based at $\ast$. One would expect that the analytically continued periods $\Pi^\prime$ satisfy
Π^'= ϱ(ℓ) Π.
However when $\varkappa=2$
the periods $\Pi$ are well defined only up to sign (cf. .<ref>)
only the quotient representation
has a well-defined physical meaning.

§.§.§ Algebraic invariants of $\Gamma$: the Hodge ring

We have a vector bundle $\mathscr{V}$ over the complement $\mathring{\mathscr{C}}$
with fiber
H_1(𝒳_u,)≡H_1(𝒳_u,)⊗_equipped with the flat Gauss-Manin connection. We fix a reference point $u_\star$ and write $V$ for the fiber $\mathscr{V}_{u_\star}$.
$V$ carries the representation $\varrho$ of $\Gamma$ (defined over $\mathbb{Q}$)
while[ The isomorphism is induced by the polarization.] $V\simeq V^\vee$.
This induces representations (over $\mathbb{Q}$) of $\Gamma$ on all tensor spaces
$V^{\otimes k}$.
Since $\mathring{\mathscr{C}}$ is quasi-projective, one has:
[See [79]]
The monodromy group $\Gamma$ is finitely generated and semisimple, i.e. all its finite-dimensional representations
split in direct sums of irreducible ones.

The subtle point about the previous statement is that we must work with representations defined
over $\mathbb{Q}$ instead of the algebraically closed field $\C$.
However there are some simple situations.

Let $V=V_0\oplus V_1$ where $V_0$ is the largest $\C$-subspace where $\Gamma$ acts trivially. Then:

(1) The chiral ring contains precisely $\dim V_0/2$
elements with $\Delta=1$.

(2) The special geometry is a product with one factor a free geometry of rank $\dim V_0/2$.

Let $\{\gamma_I\}\equiv \{B^i,A_j\}$ ($I=1,\dots,2r$, $i,j=1,\dots,r$)
be multi-valued sections of the local system with section $H_1(\mathscr{X}_u,\Z)$
which form a symplectic basis in each fiber. The SW periods
Π_I(u)def=∮_γ_I ι_1mu Ω|_𝒳_u,     Π_I(u)≡a^D_i(u)

-4pt are a set of $2r$ multi-valued holomorphic sections of
$\mathscr{V}^\vee\simeq\mathscr{V}\to \mathring{\mathscr{C}}$ which span
the fibers and satisfy
ℒ_  Π_I= Π_I.
Only half the periods are functionally independent:
$da_i^D(u)\wedge da_i(u)=0$ (cf. eq.(<ref>)) and
where $\tau_{ij}(u)$ is the (multivalued) period matrix of $\mathscr{X}_u$ in the given basis $\{B^j,A_i\}$,
Let ${P_I}^J$ be the projector $V\to V_0$ written in the basis $\{\gamma_I\}$.
By construction  ${P_I}^J a_J$ are now global holomorphic functions on
$\mathring{\mathscr{C}}$. It is easy to check that they are regular along
the discriminant,[ By Hartogs theorem it suffice to check regularity in codimension 1.] and hence are defined and regular everywhere on $\mathscr{C}$. By definition, these holomorphic global functions belong to $\mathscr{R}$ and have $\Delta=1$. It is easy to see that exactly half of them are functionally independent.

We can generalize the story. Suppose that the symmetric $k$-th power $V^{\odot k}$
contains a $\Gamma$-invariant tensor $T^{I_1\cdots I_k}$ (automatically defined over $\mathbb{Q}$); the function
(if not zero) is an element of $\mathscr{R}$ with $\Delta=k$. The invariants of $\Gamma$ form a
subalgebra of the chiral ring $\mathscr{R}$ called the Hodge ring $\mathscr{H}$ [38, 39],
a fundamental invariant of special geometry. For instance, for a $\cn=4$ special geometry
$\mathscr{H}=\mathscr{R}$. More generally, for an isotrivial geometry $\mathscr{R}$
is a finitely-generated $\mathscr{H}$-module. The ring $\mathscr{H}$ is non-trivial for
all quasi-isotrivial geometries as well as for the ones having a non-generic Mumford-Tate group.

More generally, if we have an invariant tensor $T^{I_1\dots I_k;J_1\dots J_l}\in V^{\odot k}\otimes V^{\wedge l}$ we can form the global $l$-form of scaling dimension $k+l$
T^I_1…I_k;J_1…J_l Π_I_1⋯Π_I_k dΠ_J_1∧⋯∧dΠ_J_l
When $T^{J_1J_2}\equiv\Omega^{J_1J_2}$ is the polarization, the corresponding invariant 2-form is trivial, $\Omega^{IJ} d\Pi_I\wedge d\Pi_J=0$, in facts we have the stronger condition $\Omega^{IJ}\Pi_I\wedge d\Pi_J=0$.

§.§.§  The reducible case: subtleties

We consider now the case of a monodromy representation $V$ which decomposes over $\mathbb{Q}$ as
$V_1\oplus V_2$ where the summands are non-trivial and non-isomorphic.

We consider two different scenarios. In the fist one the image
of $\Gamma$ in $GL(V_s,\mathbb{Q})$ is infinite for $s=1,2$.
Working in a sufficiently small domain $U\subset\mathring{\mathscr{C}}$,
we  define the SW periods and use half of them as special coordinates in $U$.
Taking $\mathbb{Q}$-linear combinations, we may split the periods in
two sets, $\Pi^{(1)}_I$ and $\Pi^{(2)}_J$ taking value respectively in $V_1$ and $V_2$.
When we go around a non-trivial path in $\mathring{\mathscr{C}}$
and return back to $U$, each $\mathbb{Q}$-period gets transformed
into linear combinations of $\mathbb{Q}$-periods from the same set. The two sets never mix,
and the splitting of the local special coordinates in two sets $a_{(1)}^i$ and $a_{(2)}^j$
is globally defined.
The cotangent bundle of $\mathscr{C}$ splits into the subbundle spanned by the
differentials $da_{(1)}^i$ and the one spanned by the $da_{(2)}^j$'s.
up to isogeny, the fibers $\mathscr{X}_u$
split into a product $A^{(1)}_{u}\times A^{(2)}_{u}$ of Abelian varieties with $H_1(A_{u}^{(s)},\mathbb{Q})\simeq V_s$. Going to a finite cover we may assume the fiber is a product.
We have

The decomposition (<ref>) defines two complementary integrable distributions
whose leaves have local equations in $U$ of the form
a_(s)^i=const.  s=1,2,
so that (restricting $U$ if necessary) locally
the Coulomb branch can be written as $U= U_{(1)}\times U_{(2)}$
with $a_{(s)}^i$ local coordinates on the factor $U_{(s)}$ ($s=1,2$).
The local pre-potential is a sum
where the period matrix of $A_{u}^{(s)}$ is given by $\partial^2 \cf_{(s)}(a_{(s)})\,\partial a^i_{(s)}\partial a^j_{(s)}$,
and the analytic continuation of $\cf(a_{(1)}^i,a_{(2)}^j)$ (as a multivalued
holomorphic function in $\mathring{\mathscr{C}}$) is the sum of the analytic continuations of the
two multivalued functions in the rhs.

Thus locally in $U$ we have a product of two special geometries
with Coulomb branches  $\mathscr{C}^{(s)}$ and fibers $A^{(s)}$ ($s=1,2$).
This local product structure is preserved by the monodromy so it is a kind of “global”

At first sight it may seem that we have got a decomposable geometry. Is this really so?
Not quite. A decomposable geometry has some peculiar properties which are not implied by the above discussion of reducible monodromy.

The $r=2$ situation.

For definiteness we discuss the case $r=2$ which already presents all essential phenomena. All decomposable $r=2$ geometries are isotrivial
and their discriminant is contained in the union of the two axes $H_1\cup H_2$ (in particular, it is  normal crossing). Neither properties
follow from reducibility of the monodromy representation. Indeed the geometry cannot be isotrivial
and the discriminant cannot be normal crossing
when $\Gamma$ is infinite (as we are assuming).
The point is that while the generic fiber is indeed
the product of two elliptic curves $E_{\tau_{(1)}}\times E_{\tau_{(2)}}$, the periods of these curves need not to be constant: we  have just $\tau_{(s)}=\partial^2\cf_{(s)}(a_{(s)})/\partial^2 a_{(s)}$ for $s=1,2$. The only conclusion that we may draw is that the monodromy representation factors through
while (by assumption) its image is infinite in both factors, in fact the $\mathbb{Q}$-Zariski closure of the image in each factor is the full  $SL(2,\mathbb{Q})$ modulo finite groups.
In particular there are no invariant symmetric elements in the tensor algebra $T^{\bullet,\bullet}V_s$ of each
summand $V_s$,
so while the splitting of the periods in the two sets $\Pi^{(1)}$ and $\Pi^{(2)}$ is
globally defined, we cannot use each set of special coordinates separately
to define global functions on $\mathscr{C}$
i.e. elements in $\mathscr{R}$ which may play the role of coordinates $u_i$ in the factor Coulomb branches of a decomposable geometry. Therefore we expect that the Coulomb coordinates $u_1,u_2$
are non-trivial functions of both $a_{(1)}$ and $a_{(2)}$. However, in order for the functions $u_i=u(a_{(1)},a_{(2)})$, initially defined locally in $U$, to have a univalued analytic continuation everywhere in
$\mathring{\mathscr{C}}$, the monodromy transformations of the two summands should be correlated,
and given the Zariski density of $\Gamma$ the only possibility which comes to mind is a further factorization
π_1(𝒞)→SL(2,ℚ) SL(2,ℚ)×SL(2,ℚ) ↪Sp(4,ℚ).
If this was the case, we would have an element of $\mathscr{R}$ with $\Delta=2$
given by $\epsilon^{ij} \Pi^{(1)}_i \Pi^{(2)}_j$.

In the second scenario (for $r=2$) the image of $\Gamma$ in the first $SL(2,\mathbb{Q})$ factor
is infinite but is finite in the second one. (If both images are finite the geometry is isotrivial, see . <ref>).
When pulled back to a finite cover $\mathscr{C}^\sharp$ of $\mathring{\mathscr{C}}$
the family takes the form
E×𝒴→𝒞^♯with $E$ a constant elliptic curve and $\mathscr{Y}\to  \mathscr{C}^\sharp$ a non-isotrivial
family of elliptic curves. This is what we call a quasi-isotrivial situation. If such
a special geometry exists, it cannot be decomposable. However it looks pretty hard to cook up
such a quasi-isotrivial family in a way which is consistent with all the other required structures
to produce a regular $\C^\times$-isoinvariant special geometry. A quasi-isotrivial geometry is a highly over-constrained
problem which may have a solution only under very specific conditions: peculiar Coulomb dimensions,
specific form of the discriminant, etc. A priori no one will bet a cent on their existence.
However, wait for . <ref>, <ref>, <ref> and see.

This discussion motivates the following

An indecomposable geometry is reducible iff it is not isotrivial
and its underlying monodromy representation is reducible.

Later we shall show that

An indecomposable but reducible geometry is a very rare, bizzarre and ugly animal.
They exist only for some special “troublesome” dimension $r$-tuples $\{\Delta_1,\dots,\Delta_r\}$.
In rank $2$ they are either quasi-isotrivial or have a dimension $2$ Coulomb coordinate.

§.§ Singular fibers along the discriminant

Non-semisimple Kodaira fibers. Here $n\geq1$9pt

Kodaira fiber     $\mathbf{I}_n$     $\mathbf{I}_n^*$

monodromy     $\left[\begin{smallmatrix}1 & n\\
0 & 1\end{smallmatrix}\right]$     $\left[\begin{smallmatrix}-1 & -n\\
0 & -1\end{smallmatrix}\right]$

Euler number     $n$     $n+6$

Lie algebra $\mathfrak{g}$     $\mathfrak{su}(n)$     $\mathfrak{so}(2n+8)$

In this technical subsection we quickly sketch the structure of singular fibers as
described in the math literature [45, 46, 47, 48, 49, 50]
and reviewed in [26].
We stress that our assumption that there is a section $s\colon\mathscr{C}\to\mathscr{X}$ implies that only simple singular fibers may be present, so the phenomena associated with more general kinds of fibers are not present in this restricted setting (for “special geometries” with multiple fibers see [26]).

In rank $1$ the classification of the singular elliptic fibers is given by Kodaira [51, 52, 53, 54];
the local monodromies of
simple exceptional elliptic fibers
with semisimple monodromy are listed in table <ref>[ In the following we omit type $I_0$ which is non-singular.] and the ones with non-semisimple
monodromy in table <ref>.
In rank-$r$ the possible conjugacy classes of the local monodromy around a discriminant component $\cd_a$ are again classified in Kodaira types but new phenomena appear.
The local monodromy is not sufficient to determine the singular fiber in general [45, 46, 47, 48, 49, 50].

§.§.§ Generalities

We stress that the following analysis applies under genericity
assumptions. For the highly non-generic situations in Fact <ref>
one needs to introduce the appropriate modifications.

We consider a generic point $u$ in the $i$-th discriminant component.
The fiber $\mathscr{X}_u$ is a union of components $F_{u,a}$; the
normalization of each component $F_{u,a}^\text{nor}$ is a fibration over an Abelian variety $A_u$ of dimension
$(r-1)$ (the Albanese variety of the fiber)
which is the same for all components. For simple fibers
(i.e. fibers of multiplicity 1) the fiber of $F_{u,a}^\text{nor}\to A_u$ is
a copy of $\mathbb{P}^1$.
However it is not always true
that $\mathscr{X}_u\to A_u$ is globally a fibration: two components
$F_{u,a}$, $F_{u,b}$
are glued together by identifying a section $\simeq A_u$ of the first component with a section
$\simeq A_u$
of the second one, but the identification does not necessarily carry the zero to the zero.
When the fiber is not simply-connected this may result in an obstruction to the fiber being
a global fibration over $A_u$.
One may classify the fibers type in terms of the characteristic cycle:
i.e. a maximal connected sequence of fibers.
The possible (simple) characteristic cycles are classified by the Kodaira types
plus an extra type $I_\infty$ [45, 46, 47].
If the type is semi-simple ($I_0^*$, $II$, $II^*$, $III$, $III^*$, $IV$, and $IV^*$)
we have a fibration over $A_u$ with fiber the Kodaira singularity.
In these cases by some blow-up/down and a finite cover we can
get a fibration with no singular fiber (as we see from the period map) [46, 47].
These same operations allow to reduce the case $I^*_b$ to the case
$I_{2b}$.  The case $I_b$ (including $b=\infty$) is the most subtle one,
but (at least when the
polarization is principal) it is described in great detail in [47]. The rough
picture, is as follows: each irreducible component $F_{u,a}$
is a fibration over $A_u$ with fiber a $\mathbb{P}^1$.
Two components $F_{u,a}$, $F_{u,b}$ cross along finitely many sections.
In facts their intersection matrix $F_{u,a}\# F_{u,b}$ (suitably defined)
is equal to the
intersection matrix $\Theta_a\cdot \Theta_b$ of the components a Kodaira fiber $K$.
If $K$ has type $I_b$, the type of the characteristic cycle
can be $I_{m b}$ where $m$ is a positive integer or
$\infty$ (in this case the characteristic cycle warps $m$ times around the fiber).
When the fiber is simply-connected, $K$ coincides with the type of the characteristic

We stress that, while the type $K$ is locally constant as we move the general point $u$
in the discriminant locus, the type of the characteristic cycle may change abruptly:
as shown in the examples of [47], points whose
fibers are of type $I_\infty$ or type $I_k$ ($k$ finite) may be both dense in the discriminant.[ See Proposition 5.3 of [47].] The physical
meaning of the characteristic cycle was discussed in [26]; we refer the interested reader to that paper.

The type $K$ controls the local monodromy $\varrho_a$ around the $a$-th divisor component.
It is clear from the above discussion that $\varrho_a$ is conjugate in $Sp(2r,\mathbb{Q})$
to the direct sum of the Kodaira monodromy of type $K$, $\varrho_K\in SL(2,\Z)$, and the
$(2r-2)\times (2r-2)$ identity matrix. In particular, its characteristic polynomial
has the form
[z-ϱ_a]=(z-1)^2(r-1) P_K(z)
where $P_K(z)$ is the characteristic polynomial of $\varrho_K$. Notice that the conjugacy class
in $Sp(2r,\mathbb{Q})$ distinguishes types $II$, $III$, $IV$ from the ones with  the
same characteristic polynomial $II^*$, $III^*$, $IV^*$ (respectively). A priori we would have an ambiguity
for types $I^*_b$ and $I_b$. However this is precisely the case where the detailed analysis of [46, 47]
applies, and the relation of the monodromy class to the fiber geometry is established:

[See [47]] Assume principal polarization.
The monodromy $\varrho_a$ around a semi-stable divisor $\cd_a$
(characteristic cycle of type $I_b$) is conjugate over in $Sp(2r,\Z)$ to

1     ℓ

0     1

where $\ell$ is the number of components of the fiber. As a consequence if $\varrho_a$ is a local monodromy around
a divisor of type $I^*_b$, $\varrho_a^2$ is conjugate over $Sp(2r,\Z)$ to the matrix (<ref>)
with $\ell=2b$.

The geometry also depends on the characteristic cycle, see [45, 46, 47]
or [26] for details.

§.§.§ More details on the stable reduction

This subsection may be skipped in a first reading.
Having constructed a putative special geometry, we have to check that it is a genuine one, that is,
analytically regular. One important step is to check its behavior along the discriminant which is the locus where things may go wrong. This is done by comparing the near discriminant geometry with the model local geometry which describes the singular fiber type at the generic point in the discriminant. For the present purposes
the important invariant of the singular fiber is its local monodromy class. As always we assume
principal polarization. The following local models are borrowed from refs.[45, 46, 47].

A. Transformation of the fiber type.

Just as in the Kodaira situation (rank-1), locally in
a neighborhood of $y\in\cd_i$
we can always reduce to the semi-stable
case by a sequence of blow up/down of components of the fiber
together with
a (branched) cyclic base change, that is, by making $\tilde z\to \tilde z^{\mspace{1mu}m}=z$, where $z$
is the local coordinate in $\mathscr{C}$ such that $\cd_i$ is given by $z=0$ and
$m$ is the order of the semi-simple part of the monodromy around $\cd_i$:
model fiber     $m$     semi-simple reduction
$I^*_n$, $n\geq1$     $2$     $I_{2n}$

$II$, $II^*$     $6$     $E(e^{2\pi i/3})$

$III$, $III^*$     $4$     $E(i)$

$IV$, $IV^*$     $3$     $E(e^{2\pi i/3})$

-4.5pt where $E(\tau)$ stands for the elliptic curve of period $\tau$,
and fiber type $E(\tau)$ means that the fiber over the point $\tilde y$
covering $y$ has the form
for an Abelian variety $A$ of dimension $r-1$.

B. Transformation of the symplectic form.

$y$ is a generic point in a  component $\cd_i$ of the discriminant, and
$U\ni y$ a sufficiently small neighborhood. Let $\tilde U\to U$ be the branched cover
of the (local) semi-stable reduction. We write $V=\pi^{-1}(U)$ and $\tilde V=\tilde\pi^{-1}(\tilde U)$,
where tilded symbols refer to the semi-stable reduction.
We can find local coordinates $(x_i,y^i)$ in $V$ (resp. $(\tilde x_i, \tilde y^i)$ in $\tilde V$)
such that

* $x_i$, $\tilde x_i$ are coordinates in the bases $U$ and $\tilde U$;

* the local equation of $\cd_i$ is $x_r=\tilde x_r=0$.

There are three cases [46, 47]:

(I) Type $I_k^*$, $k\geq1$.
In this case
x_r=x̃_r^2,  y^r=ỹ^r/x̃_r,    x_i=x̃_i  y^i=ỹ^i  for  i=1,…,r-1
∑_i dx_i∧dy^i =∑_k=1^r-1 dx̃_i∧dỹ^i+2  dx̃_r∧dỹ^r

(II) Types $I_0^*$, $II^*$, $III^*$ and $IV^*$. One has

x_r=x̃_r^m,  y^r=ỹ^r/x̃_r^m-1,    x_i=x̃_i  y^i=ỹ^i  for  i=1,…,r-1
∑_i dx_i∧dy^i =∑_k=1^r-1 dx̃_i∧dỹ^i+m  dx̃_r∧dỹ^r
where $m=2,6,4,3$, respectively.

(III) Types $II$, $III$ and $IV$. One has
x_r=x̃_r^m,  y^r=ỹ^r/x̃_r,    x_i=x̃_i  y^i=ỹ^i  for  i=1,…,r-1
∑_i dx_i∧dy^i =∑_k=1^r-1 dx̃_i∧dỹ^i+m  x̃^m-2
where $m=6,4,3$, respectively.

The monodromy in the cover geometry as $\tilde x_r\to e^{2\pi i}\tilde x_r$ is equal to the monodromy
in the original space as $x_r\to e^{2\pi i m}x_r$ i.e. to the $m$-th power of the original monodromy
which is trivial for all semisimple types and unipotent of type $I_{2n}$ in the $I^*_n$ case.

C. Local pre-potential in the semi-stable case.

We consider a (sufficiently small) neighborhood $U$ of a generic point $x$ on a semi-stable divisor $\cd_i$.
Locally we can always reduce to this case.

[[47]] In $U$ we can find special coordinates $(a^1,\dots, a^r)$ such that

* the local equation of $\cd_i$ is $a^r=0$;

* the prepotential has the form

(a^1,⋯, a^r)=(a^1,⋯, a^r)+ℓ/4πi (a^r)^2 loga^r
where $\tilde\cf(a^1,\cdots, a^r)$ is holomorphic in $U$ and $\ell$ is a positive integer;

* the period matrix is
τ_ij=∂^2/∂a_i ∂a_j.

* the local monodromy around the semi-stable discriminant with pre-potential
(<ref>) is unipotent of type $I_\ell$.

Indeed under the transformation $a_r\to e^{2\pi i}a_r$ the periods will transform as
a^i→a^i,    a^D_j≡∂/∂a^j→a^D_j  j≠r,    a^D_r ≡∂/∂a^r→a^D_r+ℓa^r.

§.§ Period map and structure theorem

We saw above that the restriction of the fibration $\pi$ to the complement $\mathring{\mathscr{C}}$ is in particular a family $\mathring{\mathscr{X}}$ of polarized Abelian varieties parametrized by the quasi-projective space $\mathring{\mathscr{C}}$.
The moduli space parametrizing isomorphism classes of
principally polarized Abelian varieties of dimension $r$
is the Siegel variety

\begin{equation}
\mathsf{S}_r\equiv Sp(2r,\Z)\backslash Sp(2r,\R)/U(r).
\end{equation}

We then have a holomorphic period map $\check{p}$
which sends a point $u\in  \mathring{\mathscr{C}}$ into the
isoclass $[\mathscr{X}_u]\in \mathsf{S}_r$ of its fiber.  The geometry is isotrivial iff $\check{p}$ is the constant map.

We stress that
the family $\mathring{\mathscr{X}}$ of Abelian varieties over $\mathring{\mathscr{C}}$
is not fully determined by the period map.[ This is well-known in the case
of families of elliptic curves over a curve [51, 52, 53, 85]. There are several families of elliptic curves over a curve
with the same period map: they are related by the so-called
quadratic transform  [85, 86].] Since $\{\pm1\}\subset Sp(2r,\Z)$ acts trivially on the
symmetric space $Sp(2r,\R)/U(r)$,
the period map $\check{p}$ only determines the quotient representation
while the family of Abelian varieties over $\mathring{\mathscr{C}}$ depends on the actual monodromy  i.e. on the specific lift $\varrho$ of $\check{\varrho}$

π_1(𝒞)[rr]^ϱ@/_1.5pc/[rrd]_(0.35)ϱ̌        Sp(2r,)[d]^can

which is a finer invariant of the family $\mathring{\mathscr{X}}$ than $\check{\varrho}$.
There may be at most
finitely many special geometries with the same period map [87]. When $\varkappa=2$
only $\check{\varrho}$ is a a priori well-defined (cf. . <ref>), and one has to be careful with for the proper
lift $\varrho$ which yields the correct family of Abelian varieties arising from  the particular special geometry.

§.§.§ The Abelian family over $\mathring{\mathscr{P}}$

The holomorphic map $\exp(z\,\ce)$
acts by automorphisms, hence
𝒳_exp(z )u≃𝒳_u  for all z∈/2πi m ,
i.e. the fibers over all points in a $\C^\times$-orbit are isomorphic (as polarized Abelian varieties).
Then the map $\check{p}$ factors through a holomorphic period map

p𝒞/^×≃𝖯𝗋𝗈𝗃 ℛ∖Y →𝖲_r
where $Y\subset \mathsf{Proj}\,\mathscr{R}\equiv\mathscr{P}$ is the hypersurface
When $\mathscr{C}$ is smooth
𝖯𝗋𝗈𝗃 ℛ=ℙ(d_1,⋯, d_r)≃ℙ(q_1,…,q_r)  (well-formed),
and $Y$ is the weighted projective hypersurface with quasi-cone the reduced discriminant $\cd$.

However there is a subtlety. $\mathbb{P}(d_1,\cdots, d_r)$ is not
projectively[ The underlying complex space may be nevertheless smooth (this happens when the associated well-formed weighted projective space is $\mathbb{P}^{r-1}\equiv\mathbb{P}(1,\dots,1)$ [71]). See [44] for a discussion. ] smooth on points which corresponds to $\C^\times$-orbits with enhanced R-symmetry,
since the quotient of $U(1)_R$ which acts effectively on $\mathscr{C}$
does not act freely there. We are mainly interested
in singularities in codimension one:
over an enhanced divisor of index $\nu$
we have $\Z_\nu$ quotient singularities.

It is convenient to replace the divisor $Y$ by the special locus divisor $\mathscr{S}\subset \mathscr{P}$
whose irreducible components $\cs_i$ are either projective images $\pi(\cd_i)$ of the discriminant
components or divisors of $\mathscr{P}$ which are the projective images
of R-enhanced irreducible divisors.
Note that a special divisor $\cs_i$ can belong to both the discriminant and the R-enhancement locus.

We consider the projective monodromy representation
μπ_1(𝒫∖𝒮)→ Sp(2r,)     ϰ=1

PSp(2r,)    ϰ=2

which we call the $\mu$-monodromy to distinguish it from the $\varrho$-monodromy representation of
$\pi_1(\mathscr{C}\setminus\cd)$. The two monodromies are related and determine each other.

For $\varkappa=1$ (resp. $\varkappa=2$) the group $\mu(\pi_1(\mathscr{P}\setminus\mathscr{S}))$ is equal to the Coulomb branch monodromy group $\Gamma$
(resp. the image $\check{\Gamma}$ of $\Gamma\subset \mathit{Sp}(2r,\Z)$
in the quotient group $\mathit{Sp}(2r,\Z)/\{\pm1\}$).

Let $\ast\in \mathscr{C}\setminus \pi^{-1}(\mathscr{S})$ be a generic
point in the Coulomb branch. A closed path $\ell\in\mathring{\mathscr{C}}\equiv\mathscr{C}\setminus\cd$,
based at $\ast$, may be continuously deformed to lay in $\mathscr{C}\setminus \pi^{-1}(\mathscr{S})$.
Then $\ell^\sharp\equiv\pi(\ell)$ is a closed path in $\mathscr{P}\setminus\mathscr{S}$ based at $\pi(\ast)$.
On the other hand if $\gamma(t)$ is a closed path in  $\mathscr{P}\setminus\mathscr{S}$ based at $\pi(\ast)$
we can lift it into a path $\tilde\gamma(t)$ starting at $\ast$ and ending at some point $\tilde\gamma(1)$
in the same $\C^\times$-orbit as $\ast$. Hence

∗=exp(a )γ̃(1),    for some $a\in\C/2\pi i m\Z$
and we can define a closed loop $\gamma^\flat$ in  $\mathscr{C}\setminus \pi^{-1}(\mathscr{S})$
based at $\ast$
γ^♭(t)= γ̃(2t)     0≤t≤1/2

exp[(2t-1)a ]γ̃(1)     1/2≤t≤1

Note that the homotopy class of $\gamma^\flat$ is independent of the lift.
When $\varkappa=1$ the two maps
π_1(𝒞∖π^-1(𝒮)) @<0.5ex>[rr]^(0.55)♯       @<0.5ex>[ll]^(0.45)♭π_1(𝒫∖𝒮)
are clearly inverse of each other and $\mu(\ell)=\varrho(\ell^\flat)$.
However when $\varkappa=2$
the base point $\ast$ is fixed by an automorphism of the form  $\exp(2\pi i m\,\ce/2)$ ($m$ odd)
and $a$ in eq.(<ref>) is ambiguous by the shift $a\mapsto a+i\pi m$.
Hence $\mu(\ell)=\pm\varrho(\ell^\flat)$.

We return to Example <ref>: $\cn=4$ with $G=SU(3)$.
As complex manifolds, one has $\mathbb{P}(2,3)\simeq \mathbb{P}^1$ via the map
$(u_1,u_2)\mapsto (u_1^3 :u_2^2)$ [71]. The only component of the discriminant is mapped to the point $(1:1)\in\mathbb{P}^1$. The sphere less one point is simply-connected, so all monodromy groups are trivial, while we expect it to be the non-Abelian (but solvable)
group $\mathsf{Weyl}(SU(3))$. The local monodromy $\mu_1$ around the discriminant point  $1\equiv (1:1)$
is physically expected to generate $\mathsf{Weyl}(SU(2))\simeq \Z_2$,
while the full monodromy group is $\mathsf{Weyl}(SU(3))$ (cf. Example <ref>).
Taking into account the monodromy around the two R-enhanced divisors
we get the correct answer:  the local $\mu$-monodromy around $0$ is of order $\nu_0=2$
and the local monodromy around $\infty$ is of order $\nu_\infty=3$;
hence the $\mu$-monodromy group has the presentation
which is the standard Coxeter presentation of $\mathsf{Weyl}(SU(3))$.

In conclusion: the family of polarized Abelian $r$-varieties $\mathring{\mathscr{X}}\to\mathring{\mathscr{C}}$
defines a family of polarized Abelian $r$-varieties
which is our main object of study in this paper.
A basic property of $\Gamma$ is that it contains finite-index normal subgroups
which are neat (so, in particular, torsionless) [79]. We write $\Upsilon\triangleleft \Gamma$ for such a subgroup and $\mathsf{G}=\Gamma/\Upsilon$ for the quotient finite group. By general theory we have
a finite holomorphic cover $f\colon\mathscr{Q}\to\mathscr{P}$, branched over $\mathscr{S}$, with deck group $\mathsf{G}$ such that the pulled back family
has neat monodromy group
$\Upsilon$. For many purposes it is technically  more convenient to consider the $\mathsf{G}$-twisted period map
which contains the same information but is simpler since we got rid of the finite groups.

§.§.§ Local monodromies at special divisors

The special divisor $\cs_i$ may be of three kinds:

(1) projective discriminant non-enhanced: the local monodromies
satisfy $[\mu_i]=[\rho_i]$;

(2) R-enhanced non-discriminant. In this case the period map $p$
may be extended holomorphically on the divisor, as it is evident from eq.(<ref>).
Let $(u,z)$ be local coordinates along a regular enhanced divisor with local homogeneous
equation $z=0$. If its index is $\nu_i$, we have that the points $p_k\equiv(u, e^{2\pi i k/\nu_i}\epsilon)$
are all in the same $\C^\times$ orbit, indeed $p_{k+1}=\exp[2\pi i \,\ce/(\Delta(z)\nu_i)]p_k$.
A curve in $\mathring{\mathscr{C}}$ from $p_k$ to $p_{k+1}$ will project to a closed loop $\ell$ in $\mathring{\mathscr{P}}$. Going around $\ell$ for $\nu_i$ times we return to the original point $p_0\equiv p_{\nu_i}$ in $\mathring{\mathscr{C}}$; since the locus $z=0$ is not in the determinant, there is no $\varrho$-monodromy.
Then the $\mu$-monodromy along the loop $\ell\subset\mathring{\mathscr{P}}$ satisfies
μ_i^ν_i=1∈𝑆𝑝(2r,)     ϰ=1

𝑃𝑆𝑝(2r,)     ϰ=2.

One can try to be more precise. By continuity as $\epsilon\to0$, $\mu_i$ is (up to conjugacy)
the same as the rational representation of the automorphism $\exp[2\pi i \,\ce/(\Delta(z)\nu_i)]$
of the generic fiber over the R-enhanced discriminant. Hence $\mu_i$ is a $\nu_i$-root of $1\in PSp(2r,\Z)$ which lifts in $Sp(2r,\Z)$ to a matrix with eigenvalues

(3) R-enhanced and discriminant. The same argument as before yields
μ_i^ν_i=ϱ_i∈𝑆𝑝(2r,)     ϰ=1

𝑃𝑆𝑝(2r,)     ϰ=2.

To find the proper root of $\varrho_i$ (more precisely: to determine the minimal polynomial of $\mu_i$) one compares with the case in which the geometry is locally the product of a special geometry with
fiber an Abelian variety of dimension $(r-1)$ – the Albanese variety (cf. . <ref>) – on which $\exp[2\pi i\, \ce/(\Delta(z)\,\nu_i)]$
acts by polarized automorphisms times an elliptic fibration over a small disk with a
Kodaira central fiber of the appropriate type.
Up to conjugacy in $Sp(2r,\mathbb{Q})$
(and overall sign for $\varkappa=2$) $\mu_i$ is the block diagonal
matrix having a $(2r-2)\times (2r-2)$ block of eigenvalues
and a $2\times 2$ block which is the monodromy obtained from the Kodaira monodromy of the fiber
type in $\mathscr{X}$ by the
base change $z\mapsto z^{\nu_i}=w$ (see e.g. table 5.2 of [86]).

§.§.§ The structure theorem

The structure theorem of VHS gives the period map in terms of the neat monodromy group
$\Upsilon$ [37, 38, 39, 40]. By definition $\Upsilon$ is a discrete subgroup of the algebraic
group $Sp(2r,\mathbb{Q})$ defined over $\mathbb{Q}$. The $\mathbb{Q}$-Zariski closure $\overline{\Upsilon}^{\,\mathbb{Q}}$ of $\Upsilon$ is the smallest $\mathbb{Q}$-algebraic
group containing $\Upsilon$. $\Upsilon$ and $\overline{\Upsilon}^\mathbb{Q}$ are
algebraically indistinguishable, in the sense that no algebraic invariant can discriminate between them. $\overline{\Upsilon}^\mathbb{Q}$ is also called the algebraic monodromy group.
Let $\overline{\Upsilon}^\mathbb{Q}(\R)\subset Sp(2r,\R)$ be the real Lie group
of $\R$-valued points of the $\mathbb{Q}$-algebraic group  $\overline{\Upsilon}^\mathbb{Q}$:
it is a semi-simple Lie group with finitely many connected components.[ To show the last statement one uses the fact that the Coulomb branch is - by definition - quasi-projective as a complex manifold hence Liouvillic for the plurisubharmonic functions.]
We write $H$ for a real Lie group such that $\overline{\Upsilon}^\mathbb{Q}(\R)\times H$ is a maximal subgroup of $Sp(2r,\R)$ and
write $K$, $L$ for maximal compact subgroups of the real Lie groups $\overline{\Gamma}^\mathbb{Q}(\R)$ and $H$, respectively. The structure theorem says that the
period map factorizes through
ℬ→(Υ\Γ^ ℚ()/K)×H/L →Sp(2r,)\Sp(2r,)/U(r)≡𝒮_r,
and is the constant map in the factor $H/L$.
In special geometry

(1) if $\Gamma$ is finite $p$ is the constant map;

(2) if $\Gamma$ is Abelian $p$ is the constant map;

(3) if $\Gamma=\Gamma_1\times \Gamma_2$
the period map decomposes into two distinct period maps
$p_1$ and $p_2$
factoring, respectively, through $\Gamma_1\backslash \overline{\Gamma_1}^\mathbb{Q}(\R)/K_1$ and $\Gamma_2\backslash \overline{\Gamma_2}^\mathbb{Q}(\R)/K_2$.

A special geometry where $p$ is the constant map is called isotrivial.
All its smooth fibers are isomorphic as polarized Abelian varieties.
In the situation (3) if one of the two maps is constant we say that the special geometry
is quasi-isotrivial.

The  $\R$-algebraic monodromy group $M\equiv\overline{\Upsilon}^{\,\mathbb{Q}}(\R)$ of a non-isotrivial special geometry is isogeneous to
where $M_i$ are simple, non-compact real Lie subgroup of
$Sp(2r,\R)$ of odd Mumford-Tate type and $\boldsymbol{2r}=\oplus_i V_i$
for a real symplectic representation $V_i$ of $M_i$.

For the list of allowed pairs $(M_i,V_i)$ see e.g.  the table in [88] or [38].
For a given rank $r$ there are finitely many allowed groups $M$, and we may classify
the special geometries group by group. The simpler $\R$-algebraic monodromy $M$
the simpler the corresponding classification program.
The case $M=\{1\}$ is specially easy.

This has a consequence already noted in [17]:

(1) If the discriminant $\cf$ is a simple normal crossing divisor, the geometry is a product of
$r$ rank-1 geometries; (2) let $\mathscr{R}=\C[u_1,\cdots,u_r, v_1,\cdots, v_s]$
and $\cd=(f_1)+(f_2)$ where $f_1\in \C[u_1,\cdots,u_r]$, $f_2\in \C[v_1,\cdots,v_s]$.
Then the special geometry is the product of two special geometries with
chiral rings  $\mathscr{R}_1\in \C[u_1,\cdots,u_r]$, $\mathscr{R}_2\in \C[v_1,\cdots,v_s]$.

The discriminant of an indecomposable rank-2 geometry
contains at least:

(1) one knotted component;

(2) or two
unknotted components crossing non-transversely;

(3) or three unknotted components with
pairwise transverse crossings at the origin.

The geometries of the SCFT numbered
$\#20$, $\#21$, $\#60$ in the tables of [89] have a minimal discriminant of type (1);
those numbered $\#1$, $\#7$, $\#11$, $\#14$, $\#16$, $\#18$, $\#51$-$\#53$, $\#61$-$\#65$, $\#68$
have a minimal discriminant of type (2); the ones numbered
$\#24$, $\#27$, $\#30$ a minimal discriminant of type (3). All other SCFT
in the tables of [89] have non-minimal discriminants with more components.
Model $\#32$ is not indecomposable.

The case of higher $r$ will be sketched in section <ref>.

§.§.§ $\Gamma$ finite: isotrivial special geometries

isotrivial special geometries have several (equivalent)
For a $\C^\times$-isoinvariant special geometry
the following are equivalent:

(1) $\Gamma$ is finite or, equivalently, $M=\{1\}$;

(2) all smooth fibers of the special geometry $\mathscr{X}\to\mathscr{C}$
are isomorphic as polarized Abelian varieties to a fixed variety $A$;

(3) there is a finite cover $f\colon\check{\mathscr{C}}\to \mathscr{C}$, branched only over the discriminant, such that the pulled-back fibration is trivial, i.e. $f^*\mathscr{X}= A\times \check{\mathscr{C}}$ for a fixed polarized Abelian variety $A$. In particular the special geometry
$\pi\colon\mathscr{X}\to\mathscr{C}$ is a global quotient  [37, 38, 39, 40]

A×𝒞̌@->>[d][rr]^p_2        𝒞̌@->>[d]

𝒳≡(A×𝒞̌)/Γ[rr]^π       𝒞=𝒞̌/Γ

(4) all local monodromies around the discriminant components $\cd_i$ have finite order.

$(1)\Leftrightarrow(2)\Leftrightarrow(3)$ is very well-know in VHS
and valid for all holomorphic
fibration in polarized Abelian varieties over a quasi-projective base. We summarize the arguments:
(1) $\Leftrightarrow$ (2) is the structure theorem together with the Godemen criterion [90].
(1) $+$ (2) $\Rightarrow$ (3) follows from the obvious Galois cover of the complement
$\check{\mathring{\mathscr{C}}}= \widetilde{\mathring{\mathscr{C}}}/\mathrm{ker}\,\varrho$
with finite deck group $\Gamma$, together with Riemann's existence theorem [91]
which extends it to a finite branched cover $\check{\mathscr{C}}\to\mathscr{C}$.
(3) $\Rightarrow$ (1) follows from the fact that $\Gamma$ is a subgroup of the finite group of
polarized automorphisms of $A$.

The tricky (and important for non-perturbative physical considerations) part of the statement is the equivalence of (4) with one (hence all) of the first 3 conditions. It is the only point where one uses that  $\mathscr{X}\to\mathscr{C}$ is a $\C^\times$-isoinvariant special geometry and not merely a  fibration in polarized Abelian varieties.  (1) $\Rightarrow$ (4) is trivial, and one needs to prove
(4) $\Rightarrow$ (1). There is a finite cover $\mathscr{B}\to\mathscr{C}$, branched only on the discriminant,
such that the pull-back of the Abelian family to $\mathscr{B}$ has trivial local monodromies.
The pull-back of the special Kähler metric extends to a regular metric over
the inverse image of the discriminant components of types $I_0^*$, $II^*$, $III^*$, $IV^*$.
If we have only local monodromies of these types, the pulled back special Kähler metric is conic and regular,
hence it is the flat metric, which implies that the period $\tau_{ij}$ is constant. Otherwise
we have a covering $\mathscr{B}\to\mathscr{A}$, branched only on discriminant components of types $II$, $II$ and $IV$, such that the singular special Kähler metric on $\mathscr{B}$ is the pull back of a regular one in $\mathscr{A}$ which then must be flat. In all cases the period matrix $\tau_{ij}$ is locally constant.

Many $\cn=2$ SCFT have isotrivial geometries:
(i) all $\cn\geq3$ models
(regarded as special $\cn=2$ SCFT); (ii)
all models with $\varkappa\not\in\{1,2\}$; (iii)
all rank-1 models. In principle
the isotrivial special geometries can be constructed and classified
in all ranks $r$ [55]. More details will be given elsewhere.

§.§ “Stratification” of the special geometry

The Coulomb branch $\mathscr{C}$ is naturally stratified [16, 17]. The closure of a codimension-$s$
stratum is a connected component of a multiple intersection of
discriminant components $\cd_{1_1}\cap\cdots\cap \cd_{i_s}$ of codimension $s$.
To this stratification there corresponds a “stratification” of the special geometry
in the sense that the special geometry $\mathscr{X}\to \mathscr{C}$
induces a Lagrangian fibration
$\mathscr{X}_{a_s}\to\mathscr{C}_{a_s}$ over each codimension-$s$ stratum $\mathscr{C}_{a_s}$ whose generic fibers are polarized Abelian varieties
of dimension $\dim \mathscr{C}_{a_s}$.

Let $\cd_{a_1}$ be an irreducible component of the discriminant and $u\in \cd_{a_1}$ a general point.
To $u$ we may associate a polarized Abelian variety $A_u$,
namely the Albanese variety of the normalization of a connected
component of the fiber $\mathscr{X}_u$. $A_u$ is well-defined
since it does not depend on the chosen component [45, 46, 50]. We can glue the various $A_u$'s
along $\cd_{a_1}$ to get a fibration $\mathscr{X}_{a_1}\to \cd_{a_1}\equiv\mathscr{C}_{a_1}$ in polarized Abelian varieties which we claim is a special geometry
in its own right. The geometry $\mathscr{X}_{a_1}\to \cd_{a_1}\equiv\mathscr{C}_{a_1}$ may be seen
as a symplectic reduction of $\mathscr{X}\to \mathscr{C}$
in a sense which is more general than the usual Marsden-Weinstein-Meyer (MWM) one [43, 44]
which applies when the usual regularity assumptions of the MWM theorem do not hold, but the Lagrangian fibration is holomorphic.

§.§.§ A generalization of Marsden-Weinstein-Meyer symplectic reduction

We recall how it works the usual MWM symplectic reduction of a phase space $X$ with symplectic form $\Omega$:
first one considers a level subset $\iota\colon h^{-1}(c)\hookrightarrow X$
of a Hamiltonian function $h\colon X\to \C$, and then one constructs the space $Y$
which parametrizes the orbits
of the flow generated by the Hamiltonian vector $v(h)$ dual to $dh$. In general $Y$ may be quite weird.
In the lucky case when the orbit space $Y$ happens to be a nice manifold, it inherits a symplectic structure $\omega$ from the one of $X$
such that

ι^*Ω=π^*ωwhere $\pi\colon h^{-1}(c)\to Y$ is the canonical projection sending a point to the orbit it belongs. In this regular case $Y$ is a phase space in its own right, called the symplectic reduction of $X$ along the hypersurface $h=c$. The typical application is when $h$ is the momentum map
of a $U(1)$ action on $X$.
Unfortunately, most often $Y$ is
not a nice manifold, indeed it can be a very ugly space, and the symplectic reduction does not make sense unless the necessary regularity conditions are met.

To get the special geometric “stratification” one may try to apply the MWM strategy to the zero level set of the prime homogeneous Hamiltonian $p_a\in\mathscr{R}$ such that $\cd_a=(p_a)$. However now the level subset
$p_a^{-1}(0)\subset \mathscr{X}$ is non-smooth by definition. The divisor $p_a^{-1}(0)$
contains several non-reduced irreducible components intersecting each other in a Kodaira pattern.
The zero-section $s$ is contained in one component of multiplicity 1
of  $p_a^{-1}(0)$.
We focus on the normalization of this zero component,
and consider the space parametrizing its orbits under the flow generated by the vector $v(p_a)$.
Away from special points in $\cd_a$, this orbit space $\mathscr{X}_a$ is smooth. In facts
the normalization of the zero component of the fiber over
the generic point $u\in \cd_a$ is a fibration over the Albanese polarized Abelian variety $A_u$
and the vector $v(p_a)$ is tangent to the fiber [45],
so we may identify $A_u$ with the fiber at $u\in\cd_a$
of the orbit space $\mathscr{X}_a\to \cd_a$.
In other words $\mathscr{X}_a\to \cd_a$ is the fibration with generic fiber $A_u$.
Note that if we start with the
normalization of any other irreducible component we
would get the same orbit space $\mathscr{X}_a$, which then is an `intrinsic' property of the divisor $\cd_a$.
It is easy to check tha $\mathscr{X}_a$ carries a natural symplectic form such that (<ref>) holds.[ As mentioned before, in the stable case (monodromy type $I_m$) there is some additional subtlety related to the type of the characteristic cycle. This does not affect the conclusion, just requires some more care in the precise definitions of the various entities.]
$\mathscr{X}_a$ may (in facts, should) have singularities
over a codimension-$1$ locus in $\cd_a$. If these singularities admit a crepant resolution,
the resolved $X_a$ is a bona fide special geometry of rank $(r-1)$ and
we can consider recursively its own “stratification”. We assume that this can be done.

§.§ “Stratification” vs. Coulomb dimensions

“Stratification” gives new information on the allowed Coulomb dimensions.
The formulae of [22] reviewed in . <ref> assumed the existence of
a regular coordinate axis $H_i$ not contained in the
discriminant. While this is often the case, there are many geometries where
no such “good” axis is present, and we need a more general analysis.

We may assume without loss that $\Delta_j>1$ for all $j$.
Fix an axis $H_i$ and let $\ell$ be the smallest integer such that $H_i$ is contained
in a codimension-$\ell$ (open) stratum; clearly $1\leq \ell\leq r$.
Let $u\in H_i$ be a generic point.
Let $\mathscr{X}_{u,0}$ be the unique component  of the fiber which contains the smooth point $s(u)$.
The normalization of $\mathscr{X}_{u,0}$ is
a fibration over an Albanese variety $A_{u}$ of dimension $\ell$.
$T_{s(u)} \mathscr{X}_{u,a}$ is isomorphic to $T_u^*\mathscr{C}$.
The automorphism $\xi_u=\exp(2\pi i\, \ce/\Delta_i)$ fixes $u$ and acts
linearly on
$T_{s(u)}\mathscr{X}_{u,a}$ with eigenvalues

exp(2πi (Δ_j-1)/Δ_i),  j=1,2,⋯, r
Since $\xi_u$ is an automorphism of the fiber, it preserves its Albanese variety $A_u$
and acts on it as a polarized automorphism for some polarization.
Hence $\ell$ out of the $r$ eigenvalues (<ref>) are the eigenvalues of
the analytic representation $\sigma$ of a polarized automorphism of the Abelian variety
$A_u$ of dimension $\ell$.
Moreover $\xi_u$ acts trivially on $\C\, du_i|_u \subset T_u\mathscr{C}$
which corresponds to a vector field tangent to the Albanese variety. Hence
$\exp(-2\pi i/\Delta_i)$ is one of the eigenvalues of $\sigma$.
Writing $\Delta_i=m/n$ with $\gcd(m,n)=1$ and $n<m$, we get
$\phi(m)\leq 2\ell$.

Let $H_i$ be an axis of dimension $\Delta_i$ which is new in rank $\ell$. The axis $H_i$ is not contained
in a stratum of $\mathscr{C}$ of dimension $<\ell$.
In particular, if $\Delta_i$ is a new dimension
$H_i$ does not belong to the discriminant.
If $\Delta_i$ is new in rank $r$, the full $r$-tuple $\{\Delta_1,\dots,\Delta_r\}$
is given by the formulae of . <ref>. If $\Delta_i$ is new in rank $\ell$,
at least a subset of $\ell$ dimensions are given by those formulae
with $r$ replaced by $\ell$.

In particular, when the rank is 2 and $\Delta_i$ is new  rank-2
the corresponding axis $H_i$ does not belong to the discriminant, and the dimension formulae of
. <ref>
can be used giving a small set of possibilities for the second dimension.

§ CLASSIFICATION: THE INVERSE PROBLEM

One possible strategy to construct/classify all $\cn=2$ SCFTs is to
construct first all possible $\C^\times$-isoinvariant
special geometries and then work out the (finitely many)
SCFT associated with each one of them. This program was
initiated and carried on to a certain extend by Argyres and coworkers
(see [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17] and references therein). The first step in such a program requires
to construct all $\C^\times$-isoinvariant special geometries,
or at least to chart their “geography/zoology”, by following either
one of two natural strategies:

(A) rank after rank;

(B) or general subclass after general subclass
in increasing order of “complexity”.

Subprogram (A) is completed in rank 1 [7, 8, 9, 23, 24] and in good shape in rank 2 [18, 19, 20],
while subprogram (B) was initiated in [25] with the introduction of a first
numerical invariant, $\kappa$, which partially characterize
the complexity  subclasses. The subclasses with $\kappa\not\in\{1,2\}$ can be worked out explicitly for arbitrary $r$ [55] (at least when $\mathscr{C}$ is smooth). The Lagrangian SCFT are also classified for arbitrary $r$ [95], but not all their global special
geometries are explicitly known. In our view the best course is to follow a “`mixed” strategy where we describe
the simplest subclasses for arbitrary ranks and then discuss the residual “hard” geometries
rank by rank. We propose the following coarse classification of
the indecomposable, non-free, unitary geometries:

\begin{tabular}{|c|p{11cm}|}\hline
I & isotrivial geometries\\\hline
II & quasi-isotrivial geometries not of class I\\\hline
III & reducible indecomposable geometries not of classes I, II\\\hline
IV & non-rigid geometries not of classes I, II, III\\\hline
V & $\mu$-rigid geometries not of classes I-IV\\\hline
VI & others\\\hline
\end{tabular}

Classes I and IV are essentially charted for all ranks. Classes II and III are very
constrained, and a systematic analysis may be doable. In these cases one aims to produce strong necessary conditions which rule out such geometries except under “exceptional” circumstances such as peculiar values of the dimensions $\{\Delta_1,\dots,\Delta_r\}$ and of the local monodromies determining their “zoology”.
For classes V and VI one falls back to the rank by rank strategy.
Class VI is the less amenable to analysis and a priori the largest one, but

[Supported by some experimental evidence] Class VI is hard but empty.

Each class is further divided into subclasses.

§.§ The inverse problem

For the last two classes of geometries at the moment there is no better plan
than studying one rank at the time.
In this direction a
natural idea, already put forward in [16, 17], is to exploit the
“stratification” of the special geometry in lower rank ones,
and try to construct recursively
all possible geometries in rank $r$
by “gluing” together local rank-$(r-1)$ special geometries around codimension-$1$ loci $\cd_a$ and a “boring” rank-$r$ open stratum of generic vacua in the complement
$\C^r\setminus\cup_a\cd_a$ in “all consistent ways”.
This strategy leads to what we call the Inverse problem which now we  state in concrete terms:

Given the following data:

(i) the Coulomb dimensions $\{\Delta_1,\dots,\Delta_r\}$,

(ii) the discriminant $\cd$,

(iii) the conjugacy class of the local monodromy at each irreducible component $\cd_a$  of $\cd$,
which we label by the corresponding Kodaira type $C_a$;

construct the indecomposable special geometries of rank $r$ with these data
or conclude that no such geometry exists.

We can restrict the set of allowed data. First of all we may consider only $r$-tuple $\{\Delta_1,\dots,\Delta_r\}\equiv\lambda\{d_1,\dots,d_r\}$ with $\Delta_i$'s in the finite set $\cup_{s=1}^r \Xi(s)$ of allowed dimensions
in rank $r$,
since otherwise the answer is negative for obvious reasons.
We may also assume that $1/\lambda$
is integral ($\varkappa=1$) or half-integral ($\varkappa=2$) since otherwise the geometry is automatically isotrivial and hence can be constructed/classified by group-theoretical means.
If some of the $\Delta_i$ is equal $2$ the geometry is non-rigid (see . <ref>). A folk-theorem states that a non-rigid
special geometry describes a Lagrangian SCFT which is more conveniently studied by different (physical) means.
There are many other restrictions in the $r$-tuple from the
dimension formulae and their generalizations.
We can limit ourselves to admissible $r$-tuples $\{\Delta_1,\dots,\Delta_r\}$
with $\Delta_i\neq 1,2$ and $2\lambda\in\mathbb{N}$ without essential loss.
We know that when all Kodaira types $C_a$ are semi-simple the geometry is isotrivial,
therefore we may assume that at least one monodromy class $C_a$ is $I_n$ or $I_n^*$ with $n>0$.
This, in particular, requires the discriminant $\cd$ not to be normal crossing; more generally
we have the condition
that $\pi_1(\mathscr{C}\setminus \cd)$ has an infinite quotient $\Gamma$ which is semisimple
with a symplectic action on $\mathbb{Q}^{2r}$. When focusing on the “harder” classes IV, VI we may assume the action to be irreducible.

In physical terms the Inverse problem asks to find a common UV completion to the several
low energy effective theories describing locally the degrees of freedom which are light
along each discriminant component, if this (geometric) global UV completion exists.

There are at most finitely many geometries for a given datum $(\{\Delta_i\},\cd,\{C_a\})$.

The general expectation is that the special geometry associated to an inverse datum, when it exists,
is actually unique.[ In this statement is important that $C_a$ is defined as a Kodaira type, i.e. as a conjugacy class in $SL(2,\Z)$; if $C_a$ is seen as a conjugacy class in a larger group, then multiple geometries do exist.] The idea is that the existence of a global special
geometry is already a very overdetermined problem, which can be solved positively  only for a tiny set
of data $(\{\Delta_i\},\cd,\{C_a\})$. It looks extremely unlikely that we have more than one solution for
the same datum. The finiteness in the Fact follows from the fact that there are finitely many
fibrations in Abelian $r$-varieties over $\mathscr{C}\setminus\cd$ for fixed $\cd$.
For almost all choices of $\cd$, the monodromy of all these finitely-many fibrations would not
be compatible with any Kodaira-type/dimension data, so that
only a “short list” of discriminants are allowed in a given rank $r$ and for a given
discriminant $\cd$ only a “small set” of Kodaira-type/dimension datum is consistent.
We shall discuss in . <ref> the conditions on $\cd$. They turn out to be very restrictive.

For each consistent set of “stratification” data there exist only finitely many Abelian fibrations,
but this is just one of the many geometric structures required for a special geometry.
To be a special geometry the total space of an Abelian family over $\mathscr{C}\setminus\cd$
should have a symplectic structure which extends over the “bad” fibers
such that all fibers are Lagrangian. This is quite a severe restriction to impose on the few surviving
fibrations, and for most consistent-looking data no one will pass the test. The existence
of more than one geometry for the same datum looks totally
The inverse problem may be studied with different aims.
In decreasing order of ambition:

(1) to construct all $\C^\times$-isoinvariant geometries recursively in the rank $r$;

(2) to discover interesting properties which hold for all such geometries;

(3) to get necessary conditions for the existence of a geometry;

(4) to construct
examples with prescribed/desired properties.

§.§ Taking “stratification” seriously

At first sight the weak aspect of the Inverse problem strategy is that we need
to specify
a priori a discriminant $\cd\subset\mathscr{C}$ (or, equivalently, a special divisor
$\mathscr{S}\subset\mathscr{P}$). While the datum of a dimension $r$-tuple $\{\Delta_1,\dots,\Delta_r\}$
is to be chosen from a known finite set, and the Kodaira classes $C_i$ from a known discrete set which becomes
finite when we consider conjugacy classes in $Sp(2r,\R)$ instead of $Sp(2r,\Z)$,
the discriminant $\cd=(P)$ a priori is just a square-free homogeneous element $P\in \mathscr{R}$ which may have (and indeed has) several components, with arbitrarily bad singularities, intersecting each other is all ugly non-transverse ways. At face value, describing the
set of meaningful initial data for the problem looks harder than
finding needles in a haystack.

Luckily it is not so. Consider (the normalization of) an irreducible component $\cd_a$ of the discriminant.
If we take “stratification” seriously we conclude that it is the Coulomb branch
of a $\C^\times$-isoinvariant special geometry on its own right. Thus $\cd_a$
cannot be an arbitrary hypersurface in $\mathscr{C}$, it should satisfy the conditions
of special geometry. $\cd_a$ cannot be, say, a quasi-cone over a smooth Calabi-Yau.
The affine variety $\cd_a$ needs not to be smooth, so its normalization may not be just a copy of $\C^{r-1}$
(although this is rather common in examples), but at the very least
it should be a quasi-cone over a Fano variety or (according to physical intuition)
satisfy the even more restrictive condition of admitting
a polynomial parametrization of the form $u_i=u_i(v_j)$.

Going on, consider the divisor
(or rather the respective normalizations). The pair $(\cd_a,\cd_a^\prime)$
should be in the list of allowed pairs $(\textsf{Coulomb branch},\textsf{Discriminant})$ in rank $(r-1)$,
a condition that we call hereditarity.
This sets a strong condition on $\cd$.

Then there are two other sets of constraints on $\cd$.
The first one comes from topology: the fundamental group of the complement
$\pi_1(\mathscr{C}\setminus\cd)$ should have the right properties
(not to be finite, or solvable, or a product, etc.). Thus $\cd$
cannot be normal crossing, nor $\cd_1+\cd_2$ with all components of $\cd_1$ crossing transversely the ones of $\cd_2$, etc.

However the most strong condition on $\cd$ comes from the hero of the present paper:
rigidity which will be the central topic of the rest of the note.

We introduce a coarse equivalence relation between discriminants
or rather special divisors (“commensurability”).
$\mathscr{S}\sim \mathscr{S}^\prime$ iff their complements $\mathscr{P}\setminus\mathscr{S}$
and $\mathscr{P}\setminus\mathscr{S}^\prime$ admit a common finite unbranched cover,
so that their fundamental groups have an isomorphic finite-index normal subgroup.

[Well supported] In rank-$r$ there are only finitely many equivalence
classes of  (deformations classes of) allowed special divisor.

The validity of Hope is a tautological consequence of the (physically expected) fact that there are only finitely many special geometries at a given rank $r$. But what we really hope for is that the allowed divisor classes
can be listed a priori without working out the classification of the geometries themselves.
In the rest of the paper the hope will be illustrated by several examples in various situations.

§ ABELIAN FAMILIES ARISING FROM A SPECIAL GEOMETRY

We adopt a viewpoint (and language) which is more familiar in the context of
Calabi-Yau moduli spaces and mirror symmetry. As stated in the introduction, the two
problems are formally similar.
Suppose we are given an $\C^\times$-isoinvariant Abelian family $\mathring{\mathscr{X}}\to \mathring{\mathscr{C}}$,
or equivalently a family

with $\mathscr{S}$ the special divisor in $\mathscr{P}$.
The Abelian family $\mathscr{A}$ defines a monodromy representation
$\mu\colon \pi_1(\mathring{\mathscr{P}})\to Sp(2r,\Z)$ and
also a flat Gauss-Manin (GM) differential equation of rank $2r$
of the form

\begin{equation}\label{iuytqz}
\nabla\mspace{1mu}\Phi \equiv d\mspace{1mu}\Phi-\Phi A=0
\end{equation}

where the connection $A$ is locally a $2r\times 2r$ matrix
of holomorphic differentials and $\Phi$ is the fundamental solution
(a $2r\times 2r$ matrix normalized to be the identity at a reference point $\ast\in\mathring{\mathscr{P}}$). The differential equation has only regular singularities along $\mathscr{S}$[96]. When $\dim\mathscr{P}=1$ it reduces to a classical Picard-Fuchs equation, and we use the terms Gauss-Manin connection and Picard-Fuchs system interchangeably. For simplicity we consider the smooth case
where $\mathscr{P}=\mathbb{P}(q_1,\dots,q_r)$ and we work in the affine patch
$z_r\neq0$. In this patch ($\simeq \C^{r-1}$) the connection can be written as
∇=d+∑_i A_i(u)/p_i(u)
where the  $p_i(u)$ are the equations of the irreducible components of $\mathscr{S}$ and the $A_i(u)$ are $2r\times 2r$ matrices of holomorphic differentials. The connection should be flat
∇^2= dA+A^2=0,    Adef=∑_i A_i(u)/p_i(u)

The Riemann-Hilbert correspondence states
that there is a one-to-one equivalence between the GM PDEs and
the monodromy representation up to the obvious equivalences. In our case the monodromy takes values in $Sp(2r,\Z)$.
Since we are mainly interested in irreducible non-isotrivial geometries,
we may assume that the monodromy group $\Gamma$ acts irreducibly and contains
non-trivial unipotent elements.[ If $\Gamma$ does not contain non-trivial unipotent elements it is automatically finite.] Since $\pi_1(\mathscr{P})$ is trivial,
the monodromy is generated by the images of loops encircling the
irreducible components $\cs_i$ of the special divisor $\mathscr{S}=\sum_i \cs_i$.
The monodromy group acts on the solution as

\begin{equation}
\mu([\ell])\cdot \Phi= \Phi\bigg|_{\text{\begin{tiny}$\begin{smallmatrix}\text{analytically}\\ \text{continued\phantom{m}}\\ \text{along $\ell$\phantom{mml}}\end{smallmatrix}$\end{tiny}}}
\end{equation}

where $\ell$ is a loop based at a reference point $\ast\in\mathring{\mathscr{P}}$
which represents a given class in $\pi_1(\mathring{\mathscr{P}},\ast)$.
The Zariski closure over $\C$ of the group $\Gamma\subset Sp(2r,\Z)$
is the differential Galois group of the PDE (<ref>).
We use the symbol $\Pi$ for a column $2r$-vector given by a linear combination of
the columns of $\Phi$ (with complex coefficients). The symplectic $2r\times 2r$ matrix will be written $\Omega$ (so $\gamma\in\Gamma$ $\Rightarrow$ $\gamma^t\Omega\gamma=\Omega$). In a basis where  the monodromy elements are integral symplectic matrices,
the entries of $\Pi$ are the periods of some holomorphic differential $\eta$
with respect to a symplectic basis $\{A^i,B_j\}$ of 1-cycles on the fibers

-4pt The period vector $\Pi$ is multivalued on $\mathring{\mathscr{P}}$. We see $\Pi$ as
a map defined on the universal cover $\widetilde{\mathring{\mathscr{P}}}$
of $\mathring{\mathscr{P}}$
equivariant with respect to the deck group $\pi_1(\mathring{\mathscr{P}})$
σ^∗Π= μ(σ)Π,    σ∈π_1(𝒫).

The family (<ref>) of Abelian varieties arises from a $\C^\times$-isoinvariant special geometry, iff its GM equation has a preferred period
$\Pi$ such that
its image in $\mathbb{P}^{2r-1}$ is a Legendre submanifold of the canonical contact structure of
$\mathbb{P}^{2r-1}$, that is,

Π^t  Ω  dΠ=0
while the $(1,1)$-form $i\, d\Pi^\dagger\,\Omega\,d\Pi$ is positive definite.

(Adapted from [41]) Write $\Pi=(a^D_i,a^j)^t$ ($i,j=1,\dots,r$). For a Legendre submanifold in generic position,
locally there exists a holomorphic function $\cf(a^j)$, homogeneous of degree $2$,
such that
a_i^D= ∂(a)/∂a^i.
On the other hand $a^i=\int_{A_i}\eta$ implies $\eta=\sum_i a^i \omega_i$
where $\{\omega_i\}$ are holomorphic differential normalized as $\int_{A_i}\omega_j={\delta^i}_j$.
Then the period matrix of the Abelian variety is
τ_ij=∫_B^iω_j =∂/∂a^j∫_B^iη=∂/∂a^j a_i^D= ∂^2(a)/∂a^i ∂a^j.
In particular $\mathrm{Im}\,\tau_{ij}>0$ by the second Riemann relation for the Abelian fibers.
A posteriori the differential $\eta$ defined by the Legendre condition is identified (when it
exists) with the Seiberg-Witten differential $\lambda$.
In other words, the Legendre-manifold condition is equivalent to the existence of a Seiberg-Witten differential on the pull-back of the Abelian family from $\mathring{\mathscr{P}}$ to $\mathring{\mathscr{C}}$.
When such differential exists, it pulls back to a  Lagrangian fibration $\mathscr{X}$
over $\mathscr{C}$ by taking as local pre-potential $\cf$ the one defining (locally) the Legendre submanifold.
The Legendre condition is also called the “integrability” condition.

Comparison with the Calabi-Yau case.

A family of Calabi-Yau 3-folds with $h^{2,1}=r-1$ is also described by a GM connection with a period
$\Pi$ satisfying the Legendre condition [41]. The difference is that in the CY case the
matrix $\mathrm{Im}\,\tau_{ij}$ has signature $(r-1,1)$ instead of being positive-definite.
A part for this crucial sign condition, all equations look formally identical, so we can borrow methods developed in the context of Calabi-Yau periods and mirror symmetry.

§.§ The Legendre condition

A priori the Legendre condition may look a severe constraint on the family of
Abelian varieties. However often it is not hard to satisfy. Consider the 1-form $\Pi^t \,\Omega\, d\Pi$
for a generic period. If it extends over the special divisor $\mathscr{S}$, it is a global
holomorphic 1-form on $\mathscr{P}$ hence zero since the latter has $h^{1,0}=0$.
So the question boils down to the existence of a period $\Pi$ which is “sufficiently regular”
along the special divisors $\cs_a$. Since the local monodromy around $\cs_a$ corresponds (by construction) to the IR physics of the degrees of freedom which purportedly becomes light there,
“sufficiently regular” means that the period $\Pi$ approaches along $\cs_a$ the ones
the local physics of the light degrees of freedom, i.e. that $\Pi$ agrees near $\cs_a$
with one particular local period, the “good” one. The “good” local period $\Pi$ exists
because we choose the exponents of the monodromy (encoded in the datum $C_a$
of the inverse problem) precisely to guarantee that the local  geometry follows the appropriate model (cf. . <ref>).

Seen in this way, the Legendre condition is a requirement on the connection formula for the PDE:
the “good” local period near $\cs_a$, when analytically continued to a neighborhood of $\cs_b$, should agree with the “good” local solution there for all $b$.
In the language of the inverse problem:
the several local geometries along the strata should “glue well” together.
Alternatively one has to show that
the holomorphic one-form $\Pi^t\Omega d\Pi$ vanishes in a neighborhood of $\cs_a$
and not just asymptotically when we approach it. This point of view
clarifies that the special period $\Pi$ (when it exists) is unique.

In the classical case of rank-2
we may be very precise. Here we use the analogy with
mirror symmetry for one-modulus Calabi-Yau's [97, 98].

§.§ The classical case $r=2$

In the case $r=2$ the normalization of $\mathscr{P}$ is just
the Riemann sphere $\mathbb{P}^1$ for all pairs of Coulomb dimensions $\{\Delta_1,\Delta_2\}$,
while $\mathring{\mathscr{P}}\simeq \mathbb{P}^1\setminus\{z_1,\dots,z_k\}$
where $\{z_i\}$ are $k$ distinct points on the sphere.
The Gauss-Manin PDE reduces to a Fuchsian ODE. Being primarily interested in irreducible geometries,
we assume the monodromy to be irreducible. Then, by a “choice of gauge”
the ODE can always
be written in the form [99, 100]

\begin{equation}\label{kuyqwert888}
\frac{d}{d z}\Pi=\Pi A,\qquad
A\equiv\sum_i \frac{A_\ell}{z-z_\ell}
\end{equation}

for constant $4\times 4$ matrices $A_\ell$ satisfying the non-resonant condition.[ That is: if two eigenvalues of $A_\ell$ are equal mod 1, then they must be equal.]
The conjugacy class in $GL(4,\C)$ of the local monodromy $\mu_\ell$ at $z_\ell$ is

\begin{equation}\label{uuuu76cz}
\big[\mu_\ell\big]=\big[\exp(2\pi i A_\ell)\big].
\end{equation}

The equation (<ref>) can be recast
in the form of the 4-th order Fuchsian linear equation for a scalar function $y$:

\begin{equation}\label{ODe4}
L y=0,\quad\text{where }L=\frac{d^4}{dz^4}+a_3(z)\,\frac{d^3}{dz^3}+
a_2(z)\frac{d^2}{dz^2}+a_1(z) \frac{d}{dz}+ a_0(z)
\end{equation}

whose coefficients are rational functions on $\mathbb{P}^1$
determined by the points $z_i$ and the monodromy representation of
the original equation.
In [98] (see also [97]) it is shown that the coefficients $a_i(z)$
of a 4-th order ODE (<ref>)
with differential Galois group
contained in $Sp(4,\C)$ satisfy the differential relation[ The inverse statement holds only modulo finite groups (equivalently modulo finite covers).]

\begin{equation}\label{thecondition}
4 a_2 a_3 -8 a_1- a_3^3+8 a^\prime_2-6 a_3a_3^\prime-4 a^{\prime\prime}_3=0.
\end{equation}

Let $y_1$, $y_2$, $y_3$, $y_4$ be four linearly independent solutions to
the ODE (<ref>). Out of these 4 functions we may produce 6 independent
$2\times 2$ Wronskians

\begin{equation}
W_{ab}\equiv\left|\begin{matrix}y_a & y_b\\ y_a^\prime & y^\prime_b\end{matrix}\right|,\quad 1\leq a<b\leq 4.
\end{equation}

In a general 4-th order Fuchsian ODE the Wronskians satisfy a linear ODE of order 6.