# Hwang-Oguiso invariants and frozen singularities in Special Geometry

###### Abstract

In Special Geometry there are two inequivalent notions of “Kodaira type” for a
singular fiber: one associated with its local monodromy and one with its
Hwang-Oguiso characteristic cycle. When the two Kodaira types are _not equal_
the geometry is subtler and its deformation space gets smaller (“partially
frozen” singularities). The paper analyzes the physical interpretation of the
Hwang-Oguiso invariant in the context of 4d ${\cal N}=2$ QFT and describes the
surprising phenomena which appear when it does not coincide with the monodromy
type. The Hwang-Oguiso multiple fibers are in one-to-one correspondence with
the partially frozen singularities in M-theory compactified on a local
elliptic K3: a chain of string dualities relates the two geometric set-ups.
Paying attention to a few subtleties, this correspondence explains in purely
geometric terms how the “same” Kodaira elliptic fiber may have different
deformations spaces. The geometric computation of the number of deformations
agrees with the physical expectations. At the end we briefly outline the
implications of the Hwang-Oguiso invariants for the classification program of
4d ${\cal N}=2$ SCFTs.

###### Contents

  1. 1 Introduction: freezing singularities and all that
  2. 2 Preliminaries: “section” vs. “no-section” geometries
  3. 3 “Section” special geometry: Physical heuristics
    1. 3.1 The discriminant locus
    2. 3.2 Physics at a simple stable singularity
      1. 3.2.1 The language of ’t Hooft lines
      2. 3.2.2 The language of effective Lagrangians and partition functions
      3. 3.2.3 HO invariant vs. flavor symmetry
      4. 3.2.4 The HO invariant as a Higgs branch obstruction
      5. 3.2.5 Wild nature of $\mu(u)$
      6. 3.2.6 The fiber freezing mechanism
  4. 4 Interlude: Geometry
    1. 4.1 Review of the rank-1 case
    2. 4.2 Higher rank: simple fibers
      1. 4.2.1 Multiplicities
      2. 4.2.2 Structure of simple fibers
      3. 4.2.3 Structure of simple fibers: intersection viewpoint
      4. 4.2.4 Simple stable fibers: explicit expressions
      5. 4.2.5 Kodaira type vs. monodromy
    3. 4.3 Multiple fibers
      1. 4.3.1 Explicit construction
      2. 4.3.2 Deformations spaces of multiple Lagrangian fibers
  5. 5 Physics of “no-section” special geometries
    1. 5.1 Comparison with M- and F-theory
      1. 5.1.1 (Partially) frozen singularities in M-theory
      2. 5.1.2 Duality with F-Theory
      3. 5.1.3 Effective theory on the D3 probe: unfrozen singularity
      4. 5.1.4 Subtleties with SUSY factorization
      5. 5.1.5 Effective theory on the D3 probe: (partially) frozen singularity
  6. 6 Implications for the ${\cal N}=2$ SCFT classification program
    1. 6.1 What are we supposed to classify?
    2. 6.2 Rank-$1$ _geometric_ classification reloaded
      1. 6.2.1 Dimensions $\Delta=\tfrac{6}{5},\;\tfrac{4}{3},\;\frac{3}{2}$
      2. 6.2.2 Dimension $\Delta=2$
      3. 6.2.3 Dimension $\Delta=3$
      4. 6.2.4 Dimension $\Delta=4$
      5. 6.2.5 Dimension $\Delta=6$
      6. 6.2.6 The last multiple fiber $(E_{8},m=5)$

##  1 Introduction: freezing singularities and all that

There is a number of supersymmetric complex geometries (such as
compactifications of F-theory [1, 2, 3], Seiberg-Witten geometries [4, 5, 6,
7], etc.) where the possible local monodromies around codimension-1
singularities are in 1-to-1 correspondence with Kodaira’s singular elliptic
fibers [8, 9, 10, 11]. One says that the codimension-1 singularity has the
“Kodaira type” of its local monodromy. The local monodromy is expected to
control the physics in the vicinity of the singularity, but there is the
“surprising” phenomenon that some Kodaira types come in several versions with
distinct physics and deformation spaces of different dimensions. The version
with most deformations is called “unfrozen” and the other ones “(partially)
frozen” [1, 2, 3, 12, 13, 14]. The physical distinction between the various
versions is well understood, but the geometrical side still looks a bit
mysterious. Some authors simply think that analytic geometry _per se_ is too a
coarse tool to capture the full complexity of SUSY quantum systems: hence the
same geometry may be associated to several systems with different physics.
While this sounds reasonable, it cannot explain the variability of the
deformation space since one should be able to study the deformations of the
geometry by purely geometric means.

In this paper we discuss _geometrically_ the distinct singularities with the
same local monodromy in the context of (rigid) _special geometry_ i.e. the
integrable systems [15, 16] associated to the Seiberg-Witten geometry of 4d
${\cal N}=2$ SCFTs and some natural generalizations of them (see §. 2 for
definitions). Although this may look a simplified context, it turns out that
these geometries are related by string dualities to frozen singularities in M-
and F-theory.

For clarity we distinguish two “freezing mechanisms” with somehow different
physical interpretations. At a deeper level they share the same mathematical
origin and follow from the same physical idea. We stress that the two
mechanisms may operate simultaneously: in this case the resulting singular
geometry is pretty subtle (yet fully understood).

In the rest of this introduction we give an informal survey of the mathematics
and physics beyond the freezing and sketch the relation with M- and F-theory.

###### The geometric mechanism.

Special geometry is (in particular) a holomorphic fibration over a base of
dimension $r$ whose generic fibers are Abelian varieties of complex dimension
$r$. When $r=1$ this is an elliptic fibration over a curve; the singular
fibers which may arise111 Multiple fibers cannot arise in an elliptic
fibration which is Lagrangian. in codimension-$1$ were classified by Kodaira
[8, 9, 10]222 We review the ellptic story in §. 4.1 for convenience of the
reader. see tables 1 and 2. To understand special geometry in rank $r\geq 2$
one needs to extend Kodaira’s analysis to higher dimension. In doing this one
may follow different strategies. The simplest one is to use the local
monodromy $\varrho_{\alpha}\in SL(2,{\mathbb{Z}})\subset Sp(2r,{\mathbb{Z}})$
as the guiding principle: one declares the singular fiber to be the higher
dimensional analogue of the Kodaira fiber with the same conjugacy class in
$SL(2,{\mathbb{Z}})$. This is the physicists’s way to assign “Kodaira types”
to higher dimensional singular fibers in codimension-$1$.

A more sophisticate strategy is to look for a maximal connected cycle
$\boldsymbol{\Psi}\equiv\sum_{i}m_{i}\Theta_{i}$ of rational curves
$\Theta_{i}$ inside the singular fiber. The intersection matrix
$\Theta_{i}\cdot\Theta_{j}$ is the negative of the Cartan matrix of an
$\tilde{A}\tilde{D}\tilde{E}$ affine Lie algebra333 The zero and the
$A^{\infty}_{\infty}$ matrices are also allowed. while the reduced
multiplicities $m_{i}/\\!\gcd(m_{j})$ are the Coxeter labels of the nodes in
the affine Dynkin graph. Hwang and Oguiso say that the higher dimensional
fiber is _modelled_ on the Kodaira fiber with the same (reduced) cycle
$\boldsymbol{\Psi}/\\!\gcd(m_{i})$, and assign to it the Kodaira type of the
reduced cycle [19, 20, 21].

We end up with two different notions of “Kodaira type” for the singular fiber:
the monodromy one and the Hwang-Oguiso one. When the two agree, everything
works as naively expected: we shall show in §. 4.2 that in this case the
singular fiber is just a product of the Kodaira elliptic fiber with an Abelian
variety of dimension $(r-1)$. When the two “Kodaira types” are _not equal_ the
geometry is subtler. The existence of a second discrete invariant,444 The two
notions of “Kodaira type” of the singular fiber have quite different nature.
While the monodromy class is invariant under continuous deformations (as long
as there is no collision between codimension-1 singularities) the type of the
characteristic cycle, while a discrete invariant, may jump abruptly under a
small deformation of the complex structure. in addition to the conjugacy class
of the local monodromy, explains why two singular fibers with the same
monodromy ($\equiv$ physicists’ Kodaira type) may be geometrically and
physically distinct. The main purpose of this paper is to give a physical
interpretation of the Hwang-Oguiso Kodaira type and discuss the physical
phenomena controlled by it.

###### The Kodaira puzzle.

The reader may be puzzled: the subtlety with two distinct notions of “Kodaira
type” arises when $r\geq 2$. In rank-$1$ the two notions coincide
tautologically. Yet it looks that certain singular elliptic fibers come in
different “unfrozen” vs. “(partially) frozen” versions. How can be possible?

Of course no geometric subtlety escaped Kodaira. The issue is about physics
not geometry. The sentence: “the ${\cal N}=2$ QFT is described by a rank-1
special geometry over a Coulomb branch of dimension 1” is less innocent that
it may sound. First of all the “Coulomb branch” itself is well-defined only up
to an equivalence relation which is _strictly weaker_ than analytic
isomorphism. Second: we typically define our irreducible QFTs by stripping out
would-be “free” subsectors, and most “rank-1 special geometries” arise as
special _sub-_ geometries of higher dimensional ones. Often it is not
legittimate to “extract out” the sub-geometry from the ambient special
geometry because the would-be “free” sectors, while decoupled at the level of
local physics, globally remain coupled to the “rank-1 interacting QFT” by
subtle topological effects (e.g. flux quantization). In this situation only
the deformations of the sub-geometry which are induced by global deformations
of the ambient special geometry make sense. The puzzle is solved by counting
deformations in the proper geometrical set-up which describes the full
physics. See §. 6.2 for details and examples in rank-1.

###### Relation to frozen singularities in M- and F-theory.

The second freezing mechanism is based on multiple Lagrangian fibers: they
were classified by Hwang and Oguiso [20]. This mechanism applies only to
fibers whose monodromy has _unstable_ type. A quick comparison of the table of
multiple Lagrangian fibers [20] with the table of (partially) frozen
singularities in M-theory [2, 12, 13, 14] shows that the two lists are one and
the same. Mathematically this is quite surprising. The two classifications
refer to objects which are very far apart in the math space: the table in [2,
12, 13, 14] arises from the classification of flat bundles on $T^{3}$ which
builds on deep facts from the representation theory of compact Lie groups [22,
23], while the table [20] pertains to analytic symplectic geometry. How it
happens that they are identical?

While the two stories have no obvious connection in mathematics, physically
they are related by a chain of string dualities: one starts from M-theory with
a frozen singularity, go to the F-theory dual geometry, and probe it with a D3
brane to get a ${\cal N}=2$ QFT on its world-volume. There are a number of
subtleties in the physical interpretation of the F-theory geometry from the D3
viewpoint (the ones mentioned above in reference to the Kodaira puzzle). When
everything is taken into account the _total_ special geometry of the D3 world-
volume theory contains a multiple fiber, and its reduction to a rank-1 special
_sub_ -geometry has a (partially) frozen Kodaira fiber because only the
ambient deformations are meaningful, see §. 5.1.

###### Implications for the ${\cal N}=2$ SCFT classification program.

The discussion in this paper has an important impact on the program of
classifying the ${\cal N}=2$ SCFT using special geometry [6, 7, 28, 29, 30,
31, 27, 32]. In facts it radically changes the rules of the game, see §. 6. In
the last part of the paper we give a rough sketch of the modified
classification program in the baby case $r=1$.

###### Organization of the note.

Section 2 is meant for the pedantic reader: it spells out the basic
definitions of what we mean by “section” and “no-section” special geometries.
Section 3 is purely physical, even heuristic, and aims to introduce the Hwang-
Oguiso (HO) invariant for _simple stable_ fibers from first physical
principles. We do this using two alternative languages: line operators
(Definition 1) versus partition functions (Definition 2). We also discuss some
physical consequences of the HO invariant not being trivial. Section 4 is
purely geometric, and aims to sketch the structure of the singular fibers in
codimension-$1$ following [19, 20, 21, 24, 25, 26]. Multiple fibers are
introduced and their local models and deformation spaces constructed. In
section 5 we return to physics and discuss the physical interpretation of the
“no-section” special geometries, construct the duality chains which relate
geometries with multiple fibers to frozen singularities in M- and F-theory. In
section 6 we discuss how the HO invariants _redefine_ the geometric
classification program of ${\cal N}=2$ SCFTs. We apply these ideas to rank-1
with the aim of solving the Kodaira puzzle.

##  2 Preliminaries: “section” vs. “no-section” geometries

The sole purpose of this section is to specify our basic definitions in order
not to confuse the reader with similar sounding but very distinct notions.
Readers who think we are too pedantic may jump ahead.

Informally speaking, by “special geometry” we mean the geometry of the total
space of a holomorphic integrable system, i.e. a holomorphic fibration of
complex manifolds

$\pi\colon\mathscr{X}\to\mathscr{C}$ (2.1)

where $\mathscr{X}$ carries a holomorphic symplectic $(2,0)$-form $\Omega$
while the fibers $\mathscr{X}_{u}$ ($u\in\mathscr{C}$) are Lagrangian
submanifolds: $\Omega|_{\mathscr{X}_{u}}=0$. There are several distinct
variants of this notion, _all_ of them play important roles in the description
of _some_ supersymmetric quantum systems.555 The physical interpretation of
the various notions of special geometry will be discussed elsewhere. To name
just a few alternative definitions:

  * (a)

one may work in the algebraic category or in the complex analytic one;

  * (b)

one may require $\mathscr{C}$ to be (i) compact, or (ii) quasi-projective, or
(iii) affine, or just (iv) a local domain in ${\mathbb{C}}^{r}$;

  * (c)

we may or not insists that the smooth loci of the fibers should be algebraic
_groups,_ in which case we may further require the smooth fibers to be (i)
proper, or (ii) anti-affine, or just (iii) Abelian algebraic groups. When the
group structure is taken as part of the definition, each fiber
$\mathscr{X}_{u}$ contains a preferred smooth point $0_{u}$, namely the
neutral element of the group law, $\pi$ has a canonical zero-section $s\colon
u\mapsto 0_{u}$, and the base $\mathscr{C}$ is identified with a Lagrangian
submanifold of $\mathscr{X}$;

  * (d)

when the generic fibers are proper, we may or may not require them to carry a
polarization, and when we ask for one we may or may not demand it to be
principal;

  * (e)

the special geometry may or may not be scale invariant, meaning that there is
or not a complete holomorphic vector field ${\cal E}$ (the Euler vector) whose
exponential map acts by automorphisms of all structures implied by the
particular notion of special geometry while $\mathscr{L}_{\cal
E}\mspace{2.0mu}\Omega=\Omega$.

The integrable system of the _usual_ Seiberg-Witten geometry [4, 5] of a UV-
complete 4d ${\cal N}=2$ QFT has an affine base $\mathscr{C}$ (the Coulomb
branch) and polarized proper fibers. The fiber is then a polarized complex
torus of dimension $r\equiv\dim_{\mathbb{C}}\mathscr{C}$ whose physical
interpretation is most conveniently described by compactifing the QFT down to
3d on a very large circle. By dualizing the 3d vector to a scalar, at low-
energy we get a 8-SUSY $\sigma$-model with target-space a hyperKähler
manifold666 The hyperKähler metric on $\mathscr{X}$ is the $c$-map of the
special Kähler metric on $\mathscr{C}$ in the sense of [33]. However we shall
not be interested in the metric aspects of the geometry. $\mathscr{X}$ fibered
over the 4d Coulomb branch $\mathscr{C}$ with generic fiber the
electric/magnetic holonomy group along $S^{1}$ polarized by the Dirac pairing.
The generic smooth fiber of $\pi$ is then automatically a compact group, hence
a polarized Abelian variety, and $\pi$ has a zero-section $s$. When the QFT is
a SCFT the special geometry is, in addition, scale invariant.

The first notion of “special geometry” we shall use in this note is:

######  Definition 1.

A “section” special geometry is a holomorphic fibration
$\pi\colon\mathscr{X}\to\mathscr{C}$ with $\mathscr{X}$ holomorphic symplectic
and Lagrangian fibers, whose generic (smooth) fibers are polarized Abelian
varieties, and we have a Lagrangian zero-section
$s\colon\mathscr{C}\to\mathscr{X}$. To keep the discussion simple we also
assume the polarization to be principal and the geometry to be scale
invariant, although these two conditions may be lifted.

The Seiberg-Witten geometry of a typical ${\cal N}=2$ SCFT is a “section”
special geometry in this sense. The considerations in this paper will be local
on the Coulomb branch $\mathscr{C}$, so we do not specify any particular
condition on the base. In practice we work with
$\mathscr{C}\equiv{\mathbb{C}}^{r}$ ($r$ being the _rank_). We stress that the
total space $\mathscr{X}$ is assumed to be smooth.777 This is a very mild
assumption since typically we can blow up the singularities.

The second definition is:

######  Definition 2.

A special geometry is called “non-section” iff all conditions in Definition 1
hold except that there is no section.

######  Remark 1.

It follows from our previous considerations that a “non-section” special
geometry cannot have the physical interpretation of being the (standard sense)
Seiberg-Witten geometry of some ${\cal N}=2$ QFT. Nevertheless “no-section”
geometries _do have_ physical applications/interpretations in supersymmetric
QFTs as we explain in section 5.

######  Remark 2.

The singular fibers are necessarily _simple_ in presence of a section, while
they are allowed to be _multiple_ when there is no section. See §. 4.2.1.

##  3 “Section” special geometry: Physical heuristics

In this heuristic section we provide physical intuition for the HO invariant
in the case of a “section” special geometry. The HO invariant will be
introduced from first physical principles. It turns out (cf. §. 4) that the
only _simple_ singular fibers which may have a non-trivial HO invariant are
the _stable_ ones, so our heuristic discussion will focus on this case. The
heuristic arguments of this section will exactly match the subtle math
theorems of refs.[19, 20, 21]. In retrospect they may be regarded as elegant
proofs of those theorems.

###  3.1 The discriminant locus

When the ${\cal N}=2$ QFT is interacting, there is a locus
$\mathscr{D}\subset\mathscr{C}$, of pure codimension-1 [24], called the
_discriminant,_ which parametrizes the vacua where the physics gets more
interesting. Geometrically $\mathscr{D}$ is the (reduced) hypersurface in
$\mathscr{C}$ where the fibers are singular:

$\mathscr{D}\equiv\\{u\in\mathscr{C}\colon\mathscr{X}_{u}\
\text{singular}\\}\equiv\sum_{\alpha}D_{\alpha}\quad\text{(as an effective
divisor)}.$ (3.1)

We focus on one irreducible component $D_{\alpha}$ of the discriminant, and
fix a _general_ point $u\in D_{\alpha}$. Going around $D_{\alpha}$ the periods
$(a^{i},a^{D}_{j})$ of the Seiberg-Witten differential888 In our set-up the
Seiberg-Witten differential is just the symplectic dual of the Euler vector
$\lambda\equiv\iota_{\cal E}\Omega$. $\lambda$ get rotated by the local
monodromy $\varrho_{\alpha}\in Sp(2r,{\mathbb{Z}})$ with999 A codimension-1
locus with trivial local monodromy is not part of the discriminant, i.e. the
fiber over its generic point is automatically smooth. $\varrho_{\alpha}\neq
1$. The conjugacy class $[\varrho_{\alpha}]\subset Sp(2r,{\mathbb{Z}})$ of
$\varrho_{\alpha}$ is the basic invariant of $D_{\alpha}$.
$[\varrho_{\alpha}]$ is independent of the general point $u\in D_{\alpha}$. We
say that the generic fiber over $D_{\alpha}$ (and $D_{\alpha}$ itself) is
_stable_ if the monodromy $\varrho_{\alpha}$ is unipotent:

$(\varrho_{\alpha}-1)^{2}=0.$ (3.2)

As we approach a vacuum $u\in D_{\alpha}$ some new degrees of freedom become
massless: they describe light BPS particles whose dynamics is governed by an
${\cal N}=2$ effective IR theory which is either a SCFT in its own right or
just IR-free [7]. Counting dimensions, we see that the effective IR theory is
a rank-1 ${\cal N}=2$ QFT (plus IR-free photons which are massless everywhere
in the Coulomb branch). The effective IR theory at the general point $u$ is
largely determined by $[\varrho_{\alpha}]$. In “section” special geometry we
have the following dictionary between algebraic properties of
$\varrho_{\alpha}$ and physical properties of the effective QFT along
$D_{\alpha}$:

local monodromy $\varrho_{\alpha}$ | physical properties | IR effective theory
---|---|---
stable: | $\mathrm{rank}(\varrho_{\alpha}-1)=1$ | non-semisimple | IR free | non UV complete | ?
|  | non-semisimple | IR free | non UV complete |  $SU(2)$ with matter
unstable: | $\mathrm{rank}(\varrho_{\alpha}-1)=2$ |  |  |  |
|  | semisimple | SCFT | UV complete | known list of rank-1 SCFT

Some reader may wonder why we put a question mark in the last box of the first
row. The answer looks pretty obvious: the IR-free, UV non-complete, effective
theory is SQED coupled to a number of massless hypermultiplets of charges
$q_{i}$, such that the $\beta$-function coefficient $\sum_{i}q_{i}^{2}$ is
equal to $\ell$, the greater positive integer such that
$\varrho_{\alpha}=1\bmod\ell$.

The point is that while this statement may be vaguely correct, one should be
careful with its precise meaning. When the general fiber over the discriminant
component $D_{\alpha}$ is _simple_ and _unstable_ everything is as expected:
as we approach the general vacuum $u\in D_{\alpha}$ the IR effective theory
asymptotically factorizes in two decoupled subsectors, namely the ${\cal N}=2$
theory in the last column of the table and $(r-1)$ massless ${\cal N}=2$
vector multiplets. By this we mean that, as we come close to $u$, the
partition function of the effective IR theory gets asymptotically equal to the
product of the partition functions of the two subsectors. The geometric
origin/meaning of this factorization will be explained in §. 4. For _stable_
singular fibers the story is subtler both geometrically and physically as we
are going to explain.

###  3.2 Physics at a simple stable singularity

We describe the physical side of the simple stable fiber story in two ways: in
the language of ’t Hooft line operators and in the language of effective
Lagrangians and partition functions. While the second language may be simpler
and physically more natural,101010 The reader may prefer to jump directly to
§. 3.2.2. it is the first language which is more directly related to the
geometric description of the singular fiber.

We stress that the stable situation is characterized by the condition
$\mathrm{rank}(\varrho_{\alpha}-1)=1$. Since the image of $\varrho_{\alpha}-1$
is the sublattice of electro-magnetic charges of the BPS particles which get
massless as we approach $D_{\alpha}$, in the stable case _all_ light states
carry a charge which is an integral multiple of a fixed charge $q_{0}$,

$\mathrm{im}(\varrho_{\alpha}-1)={\mathbb{Z}}\,q_{0}.$ (3.3)

We call $q_{0}$ the (unit) _electric_ charge by a redefinition of the duality
frame. In particular all light states are mutually local, are coupled to a
single massless photon $A$, and no light monopole or dyon is present. It is
easy to check that the light electrically charged hypermultiplets produce the
correct $\beta$-function. The effective IR theory _“looks like”_ SQED coupled
to hypers in the sense that it has the same spectrum of light BPS _particles._
However this does not imply that the effective IR theory111111 We take as part
of the definition of effective IR theory the truncation to interactions given
by an operator of dimension $\leq 4$. in the general vacuum $u\in D_{\alpha}$
has the light BPS _objects_ of SQED nor that it factorizes in SQED and
decoupled massless vector multiplets. There are other possibilities; for their
geometric origin see §. 4.

####  3.2.1 The language of ’t Hooft lines

While _dynamical_ light BPS monopoles are not present in the spectrum, we may
mimic them in various ways. One way is to switch on a topologically non-
trivial background for the gauge fields: this is the strategy we shall use in
the following subsection. A second possibility is to insert a BPS ’t Hooft
line operator $H_{m}^{\textsc{ir}}(\zeta)$, carrying $m$ units of the
effective IR theory magnetic charge, which we wrap on a circle $S^{1}_{R}$ of
large radius $R$ with the fixed magnetic holonomy (fmh) periodic boundary
condition. The twistor parameter $\zeta\in\mathbb{P}^{1}$ in the symbol
$H_{m}^{\textsc{ir}}(\zeta)$ specifies the set of 4 supercharges (out of 8)
which leave invariant the $\tfrac{1}{2}$-BPS line operator
$H_{m}^{\textsc{ir}}(\zeta)$. The expectation value of this line operator is
(by definition)

$\big{\langle}H_{m}^{\textsc{ir}}(\zeta)\big{\rangle}_{u,S^{1}_{R}}^{\textsc{fmh}}=\mathcal{X}_{m}(\zeta,R,u,a)^{\text{eff}}\Big{|}_{a\to
0},$ (3.4)

where the rhs is the (${\mathbb{C}}^{\times}$-valued) hyperKähler function of
magnetic charge $m$ defined by GMN [34], as computed in the effective IR
theory, while $a$ is the IR electric period which is a well-defined global
complex coordinate in the Coulomb branch _germ_ 121212 Non UV complete ${\cal
N}=2$ models have a Coulomb branch only in the _germ sense._ of the effective
theory. The (germ of the) discriminant at $u\in D_{\alpha}$ is $a=0$.

For the sake of comparison, we recall the situation in ${\cal N}=2$ SQED. For
large $R$ one has

$\lim_{a\to
0}\mathcal{X}_{m}(\zeta,R,a)^{\textsc{sqed}}=\exp[o(R)]\quad\text{for all
}m\in{\mathbb{Z}},\ \zeta\in\mathbb{P}^{1},\ \text{and }R\gg 1,$ (3.5)

in ’t Hooft’s language [35, 36, 37]: in “ordinary” SQED _all_ BPS ’t Hooft
operators are _light_.

Our situation, eq.(3.4) is a bit more subtle. The magnetic charge $m$ of the
low energy theory is _not_ unambiguously defined: the condition that its Dirac
pairing with the light electric charge $q_{0}$ is $1$ determines $m$ only up
to a linear combination of the $2r-1$ charges which are mutually local with
respect to $q_{0}$. In other words, to correctly compare the low-energy
physics along a simple stable discriminant $D_{\alpha}$ with “ordinary” SQED
with the same light BPS spectrum, eq.(3.5), we have to determine which
operator line of the original theory becomes the would-be SQED ’t Hooft line
as we approach $u$. This leads to the following131313 Recall that we are
assuming a principal polarization!:

######  Definition 3.

$D_{\alpha}\subset\mathscr{C}$ an irreducible component of the discriminant
with simple generic fiber and _unipotent_ local monodromy $\varrho_{\alpha}$;
$u\in D_{\alpha}$ a _general_ point. The _HO invariant_

$\mu(u)\in\mathbb{N}\cup\infty$ (3.6)

is the smallest positive integer such that the effective IR theory along
$D_{\alpha}$ admits a twistorial family of $\tfrac{1}{2}$-BPS ’t Hooft lines
of magnetic charge $\mu(u)$, written $H_{\mu(u)}^{\textsc{ir}}(\zeta)$, which
are _light_ in the SUSY vacuum $u\in D_{\alpha}\subset\mathscr{C}$ for all
values of the twistor parameter $\zeta$, that is,

$\log\big{\langle}H_{\mu(u)}^{\textsc{ir}}(\zeta)\big{\rangle}_{u,S^{1}_{R}}^{\textsc{fmh}}\equiv\log\mathcal{X}_{\mu(u)}(\zeta,R,u)^{\text{eff}}=o(R)\qquad\text{as
}R\to\infty.$ (3.7)

If there is no _light_ BPS ’t Hooft line of any magnetic charge we set
$\mu(u)=\infty$. When $\mu(u)=1$ (light lines of all charges) we say that the
HO invariant of the vacuum $u$ is _trivial_.

The magnetic charges of light effective ’t Hooft lines form the sublattice
${\mathbb{Z}}\,\mu(u)\subset{\mathbb{Z}}$.

###### Explicit formulae for rank $r=2$.

Let $(a^{1},a^{2})$ be a set of special coordinates141414 Local coordinates
with the required properties exist under our assumptions, see [21]. in a
neighborhood $U$ of $u$ with $a^{2}=0$ the germ of $D_{\alpha}$ at $u$. We
write $\\{A_{i},B^{j}\\}$ for the basis of
$H_{1}(\mathscr{X}_{u},{\mathbb{Z}})$, symplectic with respect to the
principal polarization, such that ($i,j,k=1,2$)

$a^{i}=\int_{A_{i}}\lambda,\qquad\quad
a^{D}_{j}=\int_{B^{j}}\lambda=\tau_{jk}\,a^{k},$ (3.8)

where the last equality follows from scale invariance. The period matrix of
the Abelian fiber in the symplectic basis $\\{A_{i},B^{j}\\}$ is

$\tau_{ij}=\frac{\partial^{2}{\cal F}}{\partial a^{i}\partial a^{j}},\qquad
i,j=1,2$ (3.9)

where the prepotential ${\cal F}$ has the local form

${\cal F}(a^{1},a^{2})=\widetilde{{\cal F}}(a^{1},a^{2})+\frac{\ell}{4\pi
i}\,(a^{2})^{2}\log a^{2},\qquad\ell\in\mathbb{N}$ (3.10)

with $\widetilde{{\cal F}}(a^{1},a^{2})$ holomorphic in $U$. From Definition
3, the HO invariant $\mu(a^{1})$ is the smallest positive integer such that we
can find two integers $(s,t)\in{\mathbb{Z}}^{2}$ with

$\lim_{a^{2}\to
0}\Big{(}\mathcal{X}_{B^{2}}(\zeta,R,a^{1},a^{2})^{\mu(a^{1})}\mathcal{X}_{A_{1}}(\zeta,R,a^{1},a^{2})^{s}\mathcal{X}_{B^{1}}(\zeta,R,a^{1},a^{2})^{t}\Big{)}=e^{o(R)}\quad\forall\;\zeta.$
(3.11)

The logarithm of the lhs is [34]

$\frac{R}{\zeta}\,{\cal
B}(a^{1},\mu(a^{1}),s,t)\,a^{1}+R\,\zeta\;\overline{{\cal
B}(a^{1},\mu(a^{1}),s,t)}\;\overline{a}^{1}+o(R)$ (3.12)

where

${\cal B}(a^{1},\mu(a^{1}),s,t)\overset{\rm
def}{=}\mu(a^{1})\mspace{2.0mu}\frac{\partial^{2}\widetilde{{\cal
F}}}{\partial a^{2}\partial
a^{1}}\bigg{|}_{a^{2}=0}+s+t\,\frac{\partial^{2}\widetilde{{\cal F}}}{\partial
a^{1}\partial a^{1}}\bigg{|}_{a^{2}=0}.$ (3.13)

The $O(R)$ term in (3.12) must vanish for all $\zeta\in\mathbb{P}^{1}$. Since
the point $u$ is general in the divisor $a^{2}=0$, in particular its first
coordinate $a^{1}\neq 0$. We conclude that in the context of Definition 3
(i.e. for $u$ generic point in a simple stable discriminant component) one
has:

> In the rank-$2$ case $\mu(a^{1})$ is the smallest positive integer such that
>
> ${\cal B}(a^{1},\mu(a^{1}),s,t)=0\quad\text{for some integers $s$, $t$}.$
> (3.14)

Equivalently:151515 Compare with Remark 5.2 of [21].

> In rank-$2$ $\mu(a^{1})$ is the order of the complex number $\exp[2\pi
> i\,\partial_{a^{2}}\partial_{a^{1}}\widetilde{{\cal F}}(a^{1},0)]$ in the
> cyclic subgroup of ${\mathbb{C}}^{\times}$ generated by $\exp[2\pi
> i\,\partial_{a^{1}}^{2}\widetilde{{\cal F}}(a^{1},0)]$.

###### Generalization to rank $r$.

Let $a^{i}$ ($i=1,\dots,r$) be local special coordinates such that the stable
discriminant germ is given by $a^{r}=0$. We set
$u\equiv(a^{1},\cdots,a^{r-1},0)$. Eq.(3.12) gets replaced by

$\begin{split}\lim_{a^{r}\to 0}&\log\mathcal{X}_{\mu(u)}(\zeta,R,u,a^{r})=\\\
&=\frac{R}{\zeta}\,\sum_{i=1}^{r-1}{\cal
B}(u,\mu(u),s_{i},t^{j})_{i}\,a^{i}+R\,\zeta\,\sum_{i=1}^{r-1}\overline{{\cal
B}(u,\mu(u),s_{i},t^{j})}_{i}\,\overline{a}^{i}+o(R)\end{split}$ (3.15)

for the $(r-1)$-component vector $\big{(}{\cal
B}(u,\mu(u),s_{1},t^{j})_{1},\dots,{\cal
B}(u,\mu(u),s_{r-1},t^{j})_{r-1}\big{)}\in{\mathbb{C}}^{r-1}$

${\cal B}(u,\mu(u),r_{i},s^{j})_{i}=\mu(u)\frac{\partial\widetilde{{\cal
F}}}{\partial a^{i}\partial
a^{r}}\bigg{|}_{a^{r}=0}+r_{i}+s^{j}\frac{\partial^{2}\widetilde{{\cal
F}}}{\partial a^{j}\partial
a^{i}}\bigg{|}_{a^{r}=0},\qquad(r_{i},s^{j})\in{\mathbb{Z}}^{2r-2}.$ (3.16)

The statement that $u\in D_{\alpha}$ is generic means that the complex numbers
$a^{i}$, $a^{D}_{i}$ ($i=1,\dots,r-1$) are linear independent over
$\mathbb{Q}$. Then

> $\mu(u)$ is the order of the element $\exp[2\pi
> i\partial_{a^{i}}\partial_{a^{r}}\widetilde{{\cal
> F}}]\in({\mathbb{C}}^{\times})^{r-1}$ in the subgroup
> ${\mathbb{Z}}^{r-1}\subset({\mathbb{C}}^{\times})^{r-1}$ generated by the
> $(r-1)$ elements $\exp[2\pi
> i\partial_{a^{i}}\partial_{a_{j}}\widetilde{{\cal
> F}}]\in({\mathbb{C}}^{\times})^{r-1}$ for $j=1,2,\cdots,r-1$.

Clearly the integer $\mu(u)$ is an important invariant of the fiber at a
general point $u$ on a stable divisor $D_{\alpha}$, related to the invariant
defined by Hwang and Oguiso [19, 20, 21]. The precise relation of $\mu(u)$
with their characteristic cycle $\boldsymbol{\Psi}_{u}$ will be spelled out in
the section 4.

####  3.2.2 The language of effective Lagrangians and partition functions

The low-energy theory at the general vacuum $u$ in a simple stable divisor
$D_{\alpha}$ consists of $r$ massless photons coupled to a number of hypers
which are charged only with respect to one particular linear combination $A$
of the light gauge fields.

At the global level the gauge field $A$ is not just a vector field, it is a
$U(1)$ gauge connection on some line bundle ${\cal L}$ over spacetime. To
fully specify the low-energy field theory, we have to specify the set of gauge
topological sectors over which we sum in the path integral [18]. The
inequivalent SQED theories are labelled by a non-negative integer $k$. In
$k$-SQED we sum over line bundles such that $k\mid c_{1}({\cal L})$. Fixing a
non-zero $k$ in the partition sum is the second way in which we may mimic the
presence of magnetic monopoles besides the introduction of ’t Hooft lines as
we did in the first part of this section. We write $Z_{k}^{\text{eff}}$ for
the partition function of the effective IR theory at $u\in D_{\alpha}$,
quantized in a periodic Euclidean box, where we restrict $A$ (the field which
couples to the light hypers) to be a connection on line bundles ${\cal L}$
with $c_{1}({\cal L})$ a multiple of $k$, and no other topological
restriction, except the ones already implied everywhere in $\mathscr{C}$ by
Dirac quantization of the original SCFT (which enforces the principal
polarization on the fibers). In this language Definition 3 becomes:

######  Definition 4.

The _HO invariant_ $\mu(u)\in\mathbb{N}\cup\infty$ is the smallest positive
integer such that the path integral of the effective IR theory with $k=\mu(u)$
factorizes _asymptotically_

$Z_{k}^{\text{eff}}\leadsto Z_{k}^{\text{SQED}}\times Z^{\text{free}}$ (3.17)

in the path integral of $k$-SQED and a partition function for $(r-1)$ free
vector multiplets.

Let us explain. The kinetic terms at $u$ mix the gauge field $A$ with the
other $(r-1)$ massless vectors. At the level of local physics, we can always
diagonalize the vector fields’ kinetic terms undoing the mixing. However, to
achieve this diagonalization, we have to redefine the vector fields by taking
linear combinations of them with _real_ coefficients. This operation will
diagonalize the vectors’ kinetic terms, but not (in general) also the
topological $\theta$-terms $\theta^{ab}F_{a}\tilde{F}_{b}/16\pi^{2}$ which are
irrelevant for the local physics but have observable global effects in non-
trivial backgrounds. Thus _as far as the local physics is concerned_ the
factorization (asymptotically) holds. In particular the perturbative
amplitudes factorize. However the field redefinition is illegitimate at the
global level: a linear combination with real coefficients of $U(1)$
connections in general is not a $U(1)$ connection since its fluxes are not
properly quantized. So, in general, the factorization (3.17) does not hold for
any $k$ because the two would-be sub-sectors are still coupled by flux
quantization (and also by the $\theta$-terms in a topologically non-trivial
background). When the coefficients happen to be integral, and the same linear
combination also diagonalizes the $\theta$-terms, the factorization (3.17)
holds in all backgrounds.161616  _A priori_ the statement is valid whenever
the field redefinition matrix is in $GL(r,{\mathbb{Z}})$. In our case the
redefinition matrix may be taken to be unipotent (cf. eq.(3.18)) so it is
automatically in $GL(r,{\mathbb{Z}})$ if it has integral coefficients.

An intermediate situation is when the coefficients are rational. Suppose that
the linear combinations which diagonalize both the kinetic terms and the
$\theta$-terms are (say for $r=2$)

$A_{1}^{\prime}=A_{1}+\frac{a}{b}A_{2},\qquad A_{2}^{\prime}=A_{2}$ (3.18)

with $A_{2}^{\prime}\equiv A_{2}$ the field coupled to the light hypers (which
is automatically a _bona fide_ connection). In eq.(3.18) $a$, $b$ are coprime
integers. Let ${\cal L}$ be the line bundle associated to $A_{2}$ (so the
light hypers are sections of some powers of ${\cal L}$). From eq.(3.18) we see
that $A_{1}^{\prime}$ is a genuine gauge connection iff $b$ divides
$c_{1}({\cal L})$. Therefore we have the factorization (3.17) iff we choose
$k$ in the definition of the theory to be a multiple of $b$. In other words:
$k\equiv b$ is the minimal integer for which we have factorization.

In our set-up $b$ coincides with the HO invariant $\mu(u)$, as it is already
clear from the previous subsection and the fact that we can define only
sectors with magnetic charges (i.e. fluxes) $c_{1}({\cal L})$ which are
multiples of $b$. To see the equality $b\equiv\mu(u)$ more explicitly, let us
write the part of the effective Lagrangian in the Coulomb vacuum $u$ which
contains the vectors’ kinetic and $\theta$-terms

$\begin{split}\frac{1}{16\pi
i}\,&\begin{pmatrix}F^{+}_{1}&F^{+}_{2}\end{pmatrix}\\!\begin{pmatrix}\tau_{11}&\tau_{12}\\\
\tau_{12}&\tau_{22}\end{pmatrix}\\!\begin{pmatrix}F^{+}_{1}\\\
F_{2}^{+}\end{pmatrix}+\text{h.c.}=\\\ &=\frac{1}{16\pi
i}\left[\left(\tau_{22}-\frac{\tau^{2}_{12}}{\tau_{11}}\right)\\!(F_{2}^{+})^{2}+\tau_{11}\left(F_{1}^{+}+\frac{\tau_{12}}{\tau_{11}}F_{2}^{+}\right)^{\\!2}\right]+\text{h.c.},\end{split}$
(3.19)

where $F_{i}=dA_{i}$ are the gauge field-strengths and $F_{i}^{\pm}=(F_{i}\pm
i\tilde{F}_{i})/2$ their (anti)selfdual projections. Comparing with (3.18) we
see that the minimal $b$ is the order of $\tau_{12}/\tau_{11}$ in
${\mathbb{C}}/{\mathbb{Z}}$ or, more precisely, (given that we may change
$\tau_{11}$ by any integer by shifting the corresponding $\theta$-angle by a
multiple of $2\pi$ leaving the partition function unchanged), it is given by
the order of $\exp(2\pi i\mspace{2.0mu}\tau_{12})$ in the multiplicative group
generated by $\exp(2\pi i\mspace{2.0mu}\tau_{11})$. But this is precisely the
definition of the HO invariant $\mu(u)$, eq.(3.14). The extension to arbitrary
rank $r\geq 2$ is straightforward.

####  3.2.3 HO invariant vs. flavor symmetry

In the vicinity of the vacuum $u\in D_{\alpha}$ the effective gauge coupling
$g_{\text{eff}}$ of the Abelian field $A$, coupled to the light BPS states, is

$\frac{\theta_{\text{eff}}}{2\pi}+\frac{4\pi
i}{g_{\text{eff}}^{2}}=\frac{\ell}{2\pi i}\log a+\text{regular}$ (3.20)

and hence $g_{\text{eff}}\to 0$ as we approach $u$ (i.e. as $a\to 0$). At
zero coupling a gauge symmetry reduces to a flavor (global) symmetry. Hence,
_a priori,_ we get a $U(1)$ flavor symmetry for the residual rank-$(r-1)$
${\cal N}=2$ QFT which remains after decoupling the IR-free light sector. (The
residual rank-$(r-1)$ theory is well-defined, at least as an IR effective
theory, see §. 4).

We may ask: does this $U(1)$ flavor symmetry act non-trivially on the
rank-$(r-1)$ residual model? For instance, if the original rank-$r$ theory was
the decoupled product of a rank-$(r-1)$ and a rank-1 QFT, and $D_{\alpha}$ was
the component of the discriminant ${\mathbb{C}}^{r-1}\times\\{0\\}$, the
$U(1)$ would not act on the rank-$(r-1)$ model. However physical intuition
says that the $U(1)$ action on the rank-$(r-1)$ QFT is “typically” non-
trivial. How can we determine whether this is really the case?

A detailed answer requires the special geometric theory of flavor which will
be introduced elsewhere [38]. We sketch the physics of the reasoning. In
${\cal N}=2$ supersymmetry, the mass parameters may be seen as Coulomb branch
coordinates of a zero-coupling SUSY gauge theory with the flavor group as
gauge group. On the other hand, the mass parameters enter the meromorphic
Seiberg-Witten differential as residues at the poles. Hence at $a=\epsilon$,
the Seiberg-Witten differential of the rank-$(r-1)$ QFT has poles with residue
of order $\epsilon$ if and only if the group $U(1)$ acts effectively on the
rank-$(r-1)$ theory. One checks that a pole with residue $O(\epsilon)$ is
present if and only if the HO invariant is non-trivial. The previous example
of the product of a rank-$(r-1)$ and a rank-$1$ theory makes this conclusion
physically intuitive.

####  3.2.4 The HO invariant as a Higgs branch obstruction

In §. 2.7 of [7] Argyres and Martone distinguish two physical situations
around a divisor with $\mathrm{rank}(\varrho_{\alpha}-1)=1$ (a stable
degeneration). While their geometric characterization of the two cases is a
bit vague, we gather that their two different behaviors correspond in our
language to, respectively, trivial and non-trivial HO invariant. Motivated by
some physical considerations, they suggest the following (in our language):

> A non-trivial HO invariant is an obstruction to the existence of a Higgs
> branch for the rank-$1$ effective IR theory governing the new light degrees
> of freedom.

They also present a number of examples supporting the statement.

This is yet another way of saying that the two sectors does not asymptotically
factorize because they remain coupled through flux quantization. The Higgs
phase is characterized by the presence of stable magnetic flux-tubes which do
not spread out [35, 36, 37]. Suppose the gauge field $A$ of the asymptotic
rank-1 theory is in the Higgs phase, so that we have stable magnetic flux-
tubes of unit flux. From eq.(3.18) we see that this is also a stable magnetic
flux tube for the $A^{\prime}_{1}$ gauge field with $\tfrac{a}{b}\bmod 1$
units of flux. So the residual sector, which contains the light gauge field
$A^{\prime}_{1}$, is not in its Coulomb phase, contrary to the assumptions.

####  3.2.5 Wild nature of $\mu(u)$

We have seen that the integer $\mu(u)$ has a direct physical meaning. Indeed,
$\mu(u)$ controls the “quantum phase”171717 In the sense of ’t Hooft’s theory
of phases of 4d quantum gauge models [35, 36, 37]. of the low-energy effective
gauge theory in the vacuum $u\in D_{\alpha}\subset\mathscr{C}$. Despite this,
the invariant behaves quite wildly under infinitesimal deformations of the
reduced prepotential $\widetilde{{\cal F}}$. In (say) rank-2
$\mu(a^{1})<\infty$ requires the three complex numbers

$1,\qquad\tau_{11}(a^{1})\equiv\frac{\partial^{2}\widetilde{{\cal
F}}}{\partial a^{1}\partial
a^{1}}\bigg{|}_{a^{2}=0}\\!,\qquad\tau_{21}(a^{1})\equiv\frac{\partial^{2}\widetilde{{\cal
F}}}{\partial a^{2}\partial a^{1}}\bigg{|}_{a^{2}=0}$ (3.21)

to be linearly dependent over $\mathbb{Q}$, which is an extraordinarily non-
generic case. Only for special and quite sharp values of the coefficients in
the reduced pre-potential $\widetilde{{\cal F}}$ this number-theoretic
condition will be fulfilled. When the condition holds, it will be typically
destroyed by any deformation, however small. Consider, for instance, deforming
$\widetilde{{\cal F}}\leadsto\widetilde{{\cal F}}+t\,(a^{1})^{2}/2$ with
$|t|<\epsilon$. Now we need $1$, $\tau_{11}+t$ and $\tau_{21}$ to be linearly
dependent over $\mathbb{Q}$: this is most certainly false for almost all $t$’s
in any small disk $|t|<\epsilon$ (unless $\tau_{21}\in\mathbb{Q}$).

######  Remark 3.

In rank-$2$ SCFT the HO invariant has the same value at all general points of
a simple stable irreducible component of the discriminant. This follows from
the fact that ${\mathbb{C}}^{\times}$ acts by complex analytic isomorphisms of
the fibers, so it preserves the HO invariant, while $D_{\alpha}$ is the
closure of an open ${\mathbb{C}}^{\times}$-orbit. On the contrary, in higher
rank (or in a non-conformal $r=2$ model) the invariant is expected to behave
quite unruly as we vary $u$ in $D_{\alpha}$: a _generic_ pre-potential will
lead to $\mu(u)=\infty$ almost everywhere in $D_{\alpha}$, except for an
everywhere dense zero-measure subset, see the examples in §. 5 of [21].

######  Remark 4.

The case $\mu(u)=\infty$ has a characterization in terms of algebraic groups.
The connected component of the smooth locus of the fiber over $u$ is an _anti-
affine algebraic group_ if and only if $\mu(u)=\infty$ i.e. if there is no
light magnetic line. The bearing of this fact will be explored elsewhere [38].

####  3.2.6 The fiber freezing mechanism

To preserve the value of $\mu(a^{1})$ the infinitesimal deformation should
have quite a specific form. Usually there will be no deformation with this
“magic” property. Thus, when physical consistency (which coincides with
_geometric_ consistency) requires the HO invariant $\mu$ to have a specific
_finite_ value, the singular fiber over $u$ gets rigidified, or _frozen_ as
people like to say, and very few (or no) deformations survive. This is a
subtle geometric mechanism to freeze exceptional fiber which cannot be
detected by the local monodromy around it.

The case of HO invariant 1 looks different. A stable fiber with $\mu(u)=1$
shares many of the geometric properties of the unstable case (we shall make
this precise in §. 4), and just as we do not expect any freezing for the
_simple_ unstable fibers, we neither expect it for stable fibers of invariant
1. There are several physical arguments which point in this direction. First:
there is the evidence from the known examples: in ref.[7] the term _frozen_
appears to be used interchangeably with their condition which we interpret as
non-trivial HO invariant. Second: from the QFT viewpoint trivial HO invariant
is equivalent to the (asymptotic) factorization of the $k=1$ partition
function, eq.(3.17). All perturbations of the effective QFT which preserve
factorization of the partition function should correspond to deformations of
the geometry which keep the HO invariant equal 1. In particular, this applies
to deformations which act independently on the two sectors. Third: the
presence of a discrete symmetry typically forces the invariant to be 1. For
instance, when the pre-potential $\widetilde{{\cal F}}(u,a)$ is invariant
under $a\leftrightarrow-a$, we have automatically HO invariant 1. All
deformations which preserve the discrete symmetry also preserve $\mu(u)=1$.
Fourth: when a Higgs branch is present (and the electric charge of the
hypermultiplets is 1) all deformations which do not destroy the Higgs branch
should preserve $\mu(u)=1$.

Geometrically a trivial HO invariant means that the fiber is the product of a
rank-$(r-1)$ Abelian variety and stable elliptic singular fibers. The
deformations of the singularity coincide with the deformation of the stable
elliptic fiber, i.e. the deformation space is the one naively associated to
the conjugacy class of the local monodromy.

##  4 Interlude: Geometry

In this section we sketch the geometric structure of the singular fibers in
codimension-1 of the holomorphic Lagrangian fibration

$\pi\colon\mathscr{X}\to\mathscr{C},\qquad\dim_{\mathbb{C}}\mathscr{X}=2\,\dim_{\mathbb{C}}\mathscr{C}=2r,$
(4.1)

following [19, 20, 21, 24, 25, 26]. Our discussion will be rather informal,
aiming to convey the general picture of the geometry. Readers looking for
precise theorems and rigorous proofs are referred to the original math
literature [19, 20, 21, 24, 25, 26]. We assume the polarization to be
principal, although most results stand without this condition.

The singular fibers in codimension-1 are modelled181818 For the precise
meaning of the term “modelled” see Definition 4.1 in [20]. on the elliptic
Kodaira fibers [8, 9, 10, 11] which are the rank-$1$ case. We start by
reviewing that story, and then look for its possible generalizations in rank
$r$.

###  4.1 Review of the rank-1 case

In rank-1 we have a holomorphic fibration $\pi\colon S\to B$ of a smooth
complex surface $S$ over a (non-compact) curve $B$ whose generic (smooth)
fibers are elliptic curves. We assume the fibration to be relatively _mimimal_
i.e. no curve with self-intersection $-1$ is contained in a fiber. The
considerations are local in $B$, and we may take $B\equiv\Delta$, a small disk
centered at a point with an exceptional non-smooth fiber. This is the
classical situation studied by Kodaira in his seminal papers [8, 9, 10] where
he classifies the allowed singular fibers into _types_. One may give several
different mathematical interpretations of Kodaira’s classification: most of
them were already spelled out in the original papers [8, 9, 10]. We briefly
outline some of them for later comparison with the corresponding situations in
higher dimension.

###### Intersection theory viewpoint.

The exceptional fiber $S_{u}$ is a divisor of the form

$S_{u}\equiv\pi^{-1}(u)=\sum_{i}m_{i}\mspace{2.0mu}\Theta_{i}$ (4.2)

where $\Theta_{i}\subset S$ are rational curves (copies of the Riemann sphere
$\mathbb{P}^{1}$), $m_{i}$ positive integers (_multiplicities_), and
$S_{u}^{2}=S_{u}\cdot S_{u^{\prime}}\equiv 0$ ($u^{\prime}\neq u$) since all
fibers are homologous. The divisor $S_{u}$ has connected support. The
multiplicity of the fiber is $m\equiv\gcd(m_{i})$: the fiber is _simple_ when
$m=1$. We focus on the simple fibers. The fiber is necessary simple when there
is a local section $s$; indeed

$1=S_{u}\\!\cdot\\!s\equiv\sum_{i}m_{i}\mspace{2.0mu}(\Theta_{i}\\!\cdot\\!s),$
(4.3)

which means that the zero-section $s$ meets a single component $\Theta_{0}$
with $m_{0}=1$.

The _type_ of the simple fiber $S_{u}$ is determined by the multiplicities
$\\{m_{i}\\}$ and the way the various irreducible components $\Theta_{i}$
cross each other. By Zariski’s lemma [11] the intersection form
$\Theta_{i}\\!\cdot\\!\Theta_{j}\equiv-C_{ij}$ is a negative semi-definite,
symmetric, integral matrix, with diagonal entries191919 Self-crossing number
$-1$ is ruled out by the minimality assumption. $0$ or $-2$ and non-negative
off-diagonal ones. Moreover $C_{ij}$ has a zero-eigenvector with positive
integral entries, $C_{ij}m_{j}=0$. Indeed

$0\equiv\Theta_{i}\\!\cdot\\!S_{u^{\prime}}=\Theta_{i}\\!\cdot\\!S_{u}=\sum_{j}m_{j}\,\Theta_{i}\\!\cdot\\!\Theta_{j}=-C_{ij}\,m_{j},\qquad
u^{\prime}\neq u.$ (4.4)

Then $C_{ij}$ must be the direct sum of Cartan matrices of simply-laced affine
Kac-Moody Lie algebras and zero-matrices. Since $S_{u}$ is connected, $C_{ij}$
is zero iff the fiber is irreducible $S_{u}=\Theta$. When the fiber is
reducible $-\Theta_{i}\\!\cdot\\!\Theta_{j}$ is the Cartan matrix of a simple,
simply-laced, affine Lie algebra of type $\widetilde{A}_{n-1}$,
$\widetilde{D}_{4+n}$, or $\widetilde{E}_{s}$ ($s=6,7,8$). In the irreducible
case the fiber is a singular rational curve of virtual genus 1 (a singular
cubic), and we have two possibilities: either the rational curve $\Theta$ has
a simple node (fiber of Kodaira type $\mathbf{I}_{1}$) or a cusp (Kodaira type
$\mathbf{II}$). The intersection matrices of types $\widetilde{A}_{1}$ (resp.
$\widetilde{A}_{2}$) is realized by two different configurations of
$\mathbb{P}^{1}$’s: either two (resp. three) rational curves tangent (resp.
crossing transversely) at a single point which is Kodaira type $\mathbf{III}$
(resp. $\mathbf{IV}$), or two (resp. three) rational curves crossing
transversely at distinct points which is type $\mathbf{I}_{2}$ (resp. type
$\mathbf{I}_{3}$). All other $\widetilde{A}\widetilde{D}\widetilde{E}$
intersection matrices are realized only by the configuration of rational
curves which is dual to the affine Dynkin graph
$\widetilde{\Gamma}_{\mathfrak{g}}$ of the corresponding Kac-Moody algebra
$\widetilde{\mathfrak{g}}$: a simple root $\alpha_{i}\in\widetilde{\Gamma}$
represents a rational component $\Theta_{i}$, and two distinct components,
$\Theta_{i}$ and $\Theta_{j}$ ($i\neq j$), cross transversely at a number of
distinct points equal to $-C_{ij}$, each curve having intersection number $-2$
with itself. The equation $C_{ij}\mspace{1.0mu}m_{j}=0$ identifies the
multiplicity $m_{i}$ of $\Theta_{i}$ with the Coxeter label of the simple root
$\alpha_{i}$. The dual configurations to the affine Dynkin graphs of types
$\widetilde{A}_{n-1}$, $\widetilde{D}_{4+n}$, $\widetilde{E}_{6}$,
$\widetilde{E}_{7}$, and $\widetilde{E}_{8}$ are called, respectively,
exceptional fibers of Kodaira type $\mathbf{I}_{n}$, $\mathbf{I}^{*}_{n}$,
$\mathbf{IV}^{*}$, $\mathbf{III}^{*}$, and $\mathbf{II}^{*}$. Here $n\geq 0$:
$\mathbf{I}_{0}$ stands for a smooth elliptic curve, and $\mathbf{I}_{1}$ for
the rational curve with a simple node.

###### Monodromy perspective.

Alternatively we can classify singular elliptic fibers using the period map
(called the _functional invariant_ by Kodaira [9]) and the associated
monodromy representation (his _homological invariant_ [9]). The possible
(simple) fibers are naturally in one-to-one correspondence with the conjugacy
classes of local monodromies around a point which are consistent with the
strong monodromy theorem [40, 45] and the nilpotent orbit theorem [46] for
variations of Hodge structure [39, 40, 41, 42, 43, 44, 45, 46].

The allowed local monodromies in rank-1 are the elements $\rho\in
SL(2,{\mathbb{Z}})$ which are quasi-unipotent and satisfy the non-negativity
condition of the nilpotent orbit theorem202020 $\Omega$ is the symplectic form
preserved by $SL(2,{\mathbb{Z}})\equiv Sp(1,{\mathbb{Z}})$.

$(\rho^{m}-1)^{2}=0\quad\text{and}\quad\mathrm{tr}\big{(}\Omega(\rho^{m}-1)\big{)}\geq
0,\quad\text{for some $m\in\mathbb{N}$.}$ (4.5)

The explicit correspondence between Kodaira types and conjugacy classes [8] is
recalled in table 1 for the fibers with semisimple monodromy and in table 2
for the ones with non-semisimple monodromy.

Table 1:  Semisimple Kodaira fibers Kodaira type |  $\mathbf{I}_{0}$ |  $\mathbf{I}_{0}^{*}$ |  $\mathbf{II}$ |  $\mathbf{II}^{*}$ |  $\mathbf{III}$ |  $\mathbf{III}^{*}$ |  $\mathbf{IV}$ |  $\mathbf{IV}^{*}$
---|---|---|---|---|---|---|---|---
$m$ |  $1$ |  $2$ |  $6$ |  $6$ |  $4$ |  $4$ |  $3$ |  $3$
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 $\mathfrak{g}$ |  $-$ |  $\mathfrak{so}(8)$ |  $-$ |  $E_{8}$ |  $\mathfrak{su}(2)$ |  $E_{7}$ |  $\mathfrak{su}(3)$ |  $E_{8}$
Table 2:  Non-semisimple Kodaira fibers. Here $n\geq 1$ Kodaira type |  $\mathbf{I}_{n}$ |  $\mathbf{I}_{n}^{*}$
---|---|---
monodromy |  $\left[\begin{smallmatrix}1&n\\\ 0&1\end{smallmatrix}\right]$ |  $\left[\begin{smallmatrix}-1&-n\\\ 0&-1\end{smallmatrix}\right]$
Euler number |  $b$ |  $b+6$
Lie algebra $\mathfrak{g}$ |  $\mathfrak{su}(n)$ |  $\mathfrak{so}(2n+8)$

###### Group theory viewpoint.

We are assuming the fibration $\pi$ to have a section $s$. The _generic_ fiber
$S_{u^{\prime}}$ is an elliptic curve with a marked zero point,
$s(u^{\prime})\subset S_{u^{\prime}}$, hence (in particular) is an algebraic
Abelian group of complex dimension 1 [47]. A simple singular fiber $S_{u}$ is
not a commutative group, but its subset $S^{\sharp}_{u}\subset S_{u}$ of
smooth points inherits a commutative group structure from the nearby smooth
fibers. Thus $S^{\sharp}_{u}$ is a non-proper algebraic Abelian group of
complex dimension 1, hence the extension by a finite commutative group of
either ${\mathbb{C}}^{\times}$ (multiplicative case) or ${\mathbb{C}}$
(additive case). One can show that the finite group is isomorphic to the
center of the simply-connected Lie group with the Lie algebra associated to
the exceptional fiber (cf. last row of tables 1, 2). The classification of
fibers can be recovered by looking to the blow-ups and blow-downs of
components of such a group-fiber which are needed to make the total space $S$
locally smooth around the singular fiber. See [8, 9, 10] for details.

######  Remark 5.

In complex dimension 1 the multiple fibers are necessary of type
$\mathbf{I}_{n}$ and they cannot appear in Lagrangian fibrations. In higher
rank the story will be radically different.

###  4.2 Higher rank: simple fibers

Our task is to generalize the Kodaira theory to higher rank. We have the
embarrassment of riches: which one of the several interpretations of the
rank-1 classification are we supposed to generalize to higher dimension?

Even if we focus on the intersection viewpoint, we still remain with a two-
fold embarrassment: have we to look for an intersection pattern of the
$r$-dimensional irreducible components of the exceptional fiber, or have we to
look for a maximal connected cycle of rational curves in the fibers? In rank-1
the two objects are tautologically identical, but they are obviously quite
different in higher $r$. We shall see in this subsection that both strategies
are possible, both lead to a classification in terms of Kodaira types,212121
The author of [24, 25] makes the first choice, the authors of [19, 20, 21] the
second one. but the two viewpoints may lead to two _different_ Kodaira types
for a given fiber $\mathscr{X}_{u}$. The “discrepancy” between the two is what
we call the “HO invariant” (when the fiber is simple): it is a measure of how
far from the naive picture the actual geometry is.

To generalize the intersection theoretic approach we need to define the
multiplicities and the intersection numbers of the relevant objects.

####  4.2.1 Multiplicities

In a Lagrangian fibration the inverse image of the irreducible discriminant
divisor $D_{\alpha}\subset\mathscr{C}$ has the form

$\pi^{-1}(D_{\alpha})=\sum_{a}m_{\alpha,a}\,E_{\alpha,a}$ (4.6)

for certain positive integers $m_{\alpha,a}$ and irreducible effective
divisors $E_{\alpha,a}\subset\mathscr{X}$. Let $u\in D_{\alpha}$ be a
_generic_ point, and $\\{F_{u,i}\\}_{i\in I_{u}}$ be the (finite) set of
irreducible components of the singular fiber
$\mathscr{X}_{u}\equiv\pi^{-1}(u)$.

######  Definition 5.

The _multiplicity_ $m_{u,i}$ of the fiber component $F_{u,i}$ is the
multiplicity $m_{u,i}\equiv m_{u,a(i)}$ of the irreducible divisor
$E_{\alpha,a(i)}$ which contains it. We identify the fiber $\mathscr{X}_{u}$
with the cycle

$\mathscr{X}_{u}\leadsto\sum_{i\in I_{u}}m_{u,i}\,F_{u,i}.$ (4.7)

We call $m_{u}\equiv\gcd(m_{u,i})$ the multiplicity of the fiber at $u\in
D_{\alpha}$. The fiber $\mathscr{X}_{u}$ is _simple_ iff $m_{u}=1$, otherwise
it is _multiple._

When our Lagrangian fibration has a section $\mathscr{C}$ (whose image in a
regular fiber is the role of neutral element of the Abelian fiber), for $u$ a
generic point in the discriminant we have

$1=\mathscr{C}\cdot\mathscr{X}_{u}=m_{u}\,\sum_{i}\frac{m_{u,i}}{m_{u}}\;\mathscr{C}\cdot
F_{u,i}$ (4.8)

which implies that $\mathscr{C}$ meets a single component $F_{u,0}$ of
$\mathscr{X}_{u}$ with has $m_{u,0}=1$. In particular in this case $m_{u}=1$
and all fibers are simple.

####  4.2.2 Structure of simple fibers

We focus first on the simple fibers. In this subsection we assume that $\pi$
has a local zero section $s$. We start from the group theory viewpoint which
allow us to get the result quickly and produces explicit expressions.

##### The components $F_{u,i}$: group perspective.

Let $u$ a generic point in the discriminant component $D_{\alpha}$. Just as in
dimension 1, the regular locus $\mathscr{X}_{u}^{\sharp}$ of the fiber
$\mathscr{X}_{u}$ is also an Abelian algebraic group with connected component
$F^{\sharp}_{u,0}$ the regular locus of the zero-component $F_{u,0}$.
$\mathscr{X}_{u}^{\sharp}$ is the union of all smooth points in the components
$F_{u,i}$ with multiplicity $m_{u,i}=1$. Hence $\mathscr{X}_{u}^{\sharp}$ is
the extension of a finite Abelian group by $F_{u,0}^{\sharp}$, and all
components of multiplicity $1$ have isomorphic normalizations:

$m_{u,i}=1\quad\Rightarrow\quad F^{\text{nor}}_{u,i}\simeq
F_{u,0}^{\text{nor}}.$ (4.9)

By the Barsotti-Chevalley theorem [47], $F_{u,0}^{\sharp}$ is a fibration over
an Abelian variety, namely its Albanese variety $A_{u}$, with fiber a
commutative group. Since at most two cycles vanish at the generic point $u$
(in physical language: the degrees of freedom which get light are described by
a rank-$1$ theory), we have $\mathsf{rank}\,H_{1}(A_{u},{\mathbb{Z}})=2r-2$
and $A_{u}$ is an Abelian variety of complex dimension $(r-1)$. The fiber of
$\varpi_{u,0}^{\sharp}\colon F^{\sharp}_{u,0}\to A_{u}$ is then a connected,
non-compact, algebraic Abelian group of dimension 1, hence a copy of either
${\mathbb{C}}$ or ${\mathbb{C}}^{\times}$.

$1\to{\mathbb{C}}^{\times}\to F^{\sharp}_{u,0}\to A_{u}\to
0\qquad\text{or}\qquad 0\to{\mathbb{C}}\to F^{\sharp}_{u,0}\to A_{u}\to 0.$
(4.10)

We call a simple singular fiber of _multiplicative_ (resp. _additive_) type
if the first (resp. second) extension sequence applies. The group extensions
of the Abelian variety $A_{u}$ by ${\mathbb{C}}^{\times}$ (multiplicative) or
respectively by ${\mathbb{C}}$ (additive) are parametrized by the groups [54]

$\mathrm{Ext}^{1}(A_{u},{\mathbb{C}}^{\times})\simeq\mathrm{Pic}(A_{u})^{0}\equiv
A_{u}^{\vee}\quad\text{resp.}\quad\mathrm{Ext}^{1}(A_{u},{\mathbb{C}})\simeq
H^{1}(A_{u},{\cal O}).$ (4.11)

In the multiplicative case, $F^{\sharp}_{u,0}$ is the group extension defined
by a point $w_{u}\in A_{u}^{\vee}$. We write ${\cal L}_{u}\to A_{u}$ for the
corresponding line bundle of vanishing Chern class. $F^{\sharp}_{u,0}$ is then
the complement of the zero and infinity sections $s_{0}(0)$, $s_{\infty}(0)$
in total space of the bundle $\mathbb{P}({\cal L}_{u}\oplus{\cal O})\to A_{u}$
[26]. The normalization $F^{\text{nor}}_{u,0}$ of $F_{u,0}$ is then the ruled
variety $\mathbb{P}({\cal L}_{u}\oplus{\cal O})$ over the base $A_{u}$. For
example, when $r=2$ $A_{u}\equiv A_{u}^{\vee}$ is an elliptic curve of period
$\tau$ and the group extension defined by $w_{u}\in A_{u}$ is explicitly

$F^{\sharp}_{u,0}\Big{|}_{\text{multiplicative}\atop w_{u}\in
A_{u}}=\big{(}{\mathbb{C}}\times{\mathbb{C}}^{\times}\big{)}\Big{/}\Big{\\{}(z,y)\sim(z+m+n\tau_{u},e^{2\pi
inw_{u}}y),\ m,n\in{\mathbb{Z}},\ w_{u}\in A_{u}\Big{\\}}$ (4.12)

We shall see in a moment that the HO invariant is the order of $w_{u}$ and
hence is finite precisely when $w_{u}$ is a torsion point. The distinction
$w_{u}$ torsion vs. non-torsion has an alternative characterization. Recall
that an algebraic group $G$ is _anti-affine_ iff its ring of regular functions
is trivial ${\cal O}(G)\simeq{\mathbb{C}}$.

######  Fact 1.

In the multiplicative case $F^{\sharp}_{u,0}$ is an anti-affine group (in
facts a semi-Abelian algebraic group) iff the point $w_{u}$ is not torsion.

In the additive case the group extension (4.10) is specified by a
$(0,1)$-class $\phi_{u}\in H^{1}(A_{u},{\cal O})\subset
H^{1}(A_{u},{\mathbb{C}})$; explicitly

$F^{\sharp}_{u,0}\Big{|}_{\text{additive}\atop\phi_{u}\in H^{1}(A_{u},{\cal
O})}=({\mathbb{C}}^{r-1}\times{\mathbb{C}})/\Lambda_{u}\quad\text{where}\
\Lambda_{u}\ni\lambda\colon(z,y)\mapsto\big{(}z+\lambda,y+\langle\lambda,\phi_{u}\rangle\big{)}$
(4.13)

where $\langle-,-\rangle\colon H_{1}(A_{u},{\mathbb{Z}})\times
H^{1}(A_{u},{\mathbb{C}})\to{\mathbb{C}}$ is the natural pairing. We can see
its normalization as a ruled variety over $A_{u}$ with a preferred
$\infty$-section $s_{\infty}(0)$.

Translations in the Abelian variety $A_{u}$ are generated by holomorphic
Hamiltonian vectors.222222 In classical mechanics language: the coordinates on
$A_{u}$ are a set of angle variables for the integrable system
$\mathscr{X}\to\mathscr{C}$ and the Hamiltonians $H_{i}$ are their canonically
conjugate momenta. The corresponding Hamiltonian functions
$H_{1},\dots,H_{r-1}\in\Gamma(\mathscr{C},{\cal O})$ generate a
${\mathbb{C}}^{r-1}$ action on the fiber component $F_{u,0}$ whose kernel is
the lattice $\Lambda_{u}$ so that $A_{u}={\mathbb{C}}^{r-1}/\Lambda_{u}$. This
${\mathbb{C}}^{r-1}$ action extends to the full fiber $\mathscr{X}_{u}$, and
preserves each irreducible component $F_{u,i}$. Then the normalization
$F^{\text{nor}}_{u,i}$ of the $i$-th component is a $\mathbb{P}^{1}$-fibration
over the _same_ Albanese variety $A_{u}$ for all $i$ [19]. The
$F^{\text{nor}}_{u,i}$’s are ruled varieties whose lines (the fibers of the
Albanese fibration $\varpi_{u,i}\colon F^{\text{nor}}_{u,i}\to A_{u}$) are the
${\mathbb{C}}$-orbits of the Hamiltonian vector field whose Hamiltonian $h$ is
the irreducible element in the chiral ring
$\mathscr{R}\equiv{\mathbb{C}}[u_{1},\dots,u_{r}]$ such that $D_{\alpha}$ has
equation $h=0$ [19]. Each ruled surface $F^{\text{nor}}_{u,i}\to A_{u}$ has a
finite number $k_{i}\geq 1$ of preferred sections; when $k_{i}\geq 3$ we have
$F^{\text{nor}}_{u,i}\simeq\mathbb{P}^{1}\times A_{u}$ (since the sphere with
3 punctures is rigid) while when $k_{i}=2$ (resp. $1$) it may be a non-trivial
fibration of the multiplicative (resp. additive) kind as shown above.

An exceptional fiber in codimension-1, $\mathscr{X}_{u}$, then has the form

$\mathscr{X}_{u}=\coprod_{i}F_{u,i}^{\text{nor}}\big{/}\text{(identifications)}$
(4.14)

where $F_{u,i}^{\text{nor}}$ are ruled varieties over $A_{u}$ which are glued
together along the preferred sections (with multiplicities $m_{u,i}$). A
fundamental datum of the fiber is the combinatorics of these identifications
encoded in the intersection matrix.

##### The cohomology class $\eta_{u}$ and its physical interpretation.

Anticipating our conclusions, we stress that, while each irreducible component
$F_{u,i}$ is a fibration over a fixed Abelian variety $A_{u}$, the fiber
itself $\mathscr{X}_{u}$ may or may not be globally a fibration over $A_{u}$
depending on the details of the gluings in eq.(4.14). The point is that the
gluing of a section $s_{i}\simeq A_{u}$ in the $i$-th component with a section
$s_{j}\simeq A_{u}$ in the $j$-th component is through a complex automorphism
of $A_{u}$ which may not preserve the zero element, that is, be a
translation232323 Non-trivial automorphisms which fix zero are ruled out for
the following reason: the Lie group of $A_{u}$ is intrinsically identified
with the Hamiltonian vector fields of the $H_{i}$’s and hence the
identifications should act on the sections’ Lie groups as the identity.

$\boldsymbol{z}_{j}=\boldsymbol{z}_{i}+\eta_{ij}\in{\mathbb{C}}^{r-1}/\Lambda_{u}\simeq
A_{u},$ (4.15)

where $\boldsymbol{z}_{i}$ are coordinates in the base $\simeq A_{u}$ of
$F_{u,i}$. Let $\Upsilon$ be the graph dual to the components’ arrangement:
the $i$-th vertex of $\Upsilon$ represents an irreducible component $F_{u,i}$
and an edge of $\Upsilon$ between nodes $i$, $j$ stands for a section along
which the corresponding components are glued. $\eta_{u}\equiv\\{\eta_{ij}\\}$
is a $1$-cocyle on $\Upsilon$ with coefficients in
${\mathbb{C}}^{r-1}/\Lambda_{u}$. Clearly we are free to redefine the
coordinates $\boldsymbol{z}_{i}$ in each component by a translation. Thus only
the class $[\eta]\in H^{1}(\Upsilon,{\mathbb{C}}^{r-1}/\Lambda_{u})$ is
relevant. Clearly

######  Fact 2.

The fiber $\mathscr{X}_{u}$ is a global fibration $\Pi\colon\mathscr{X}_{u}\to
A_{u}$ over the dimension $(r-1)$ Abelian variety $A_{u}$ if and only if the
class $[\eta]\in H^{1}(\Upsilon,{\mathbb{C}}^{r-1}/\Lambda_{u})$ of the gluing
cocycle $\eta$ is trivial. For simple fibers _HO invariant_ is the order $m$
of $[\eta]$ in $H^{1}(\Upsilon,{\mathbb{C}}^{r-1}/\Lambda_{u})$.

In particular when $\Upsilon$ is simply connected $\mathscr{X}_{u}$ is a
global fibration. We shall see in a moment that when $\mathscr{X}_{u}\to
A_{u}$ is a global fibration the fiber is one of the Kodaira simple fibers.

##### Fiber dychotomy.

_A priori_ we may have 3 different situations:

(a)

the singular fiber $\mathscr{X}_{u}$ is not a global fibration over the
Albanese variety $A_{u}$;

(b)

the singular fiber $\mathscr{X}_{u}$ is a _non-trivial_ global fibration over
$A_{u}$;

(c)

the singular fiber $\mathscr{X}_{u}$ is a product $F\times A_{u}$ with $F$ a
Kodaira fiber and $A_{u}$ Abelian.

At first sight one would think that (b) is the “typical situation”; however we
shall see that (b) is just _not possible:_ the fiber is either not a (global)
fibration or a trivial fibration. The possibility (c) occurs precisely when
the HO invariant is trivial, otherwise we have (a). We shall refer to this
property as _dichotomy_. Preliminarily, we present some simple examples.

######  Example/Claim.

The simplest instance is an _irreducible multiplicative fiber_ (the higher
dimensional analogue of Kodaira type $\mathbf{I}_{1}$). In this case we
identify the zero and infinity sections $s_{0}\sim s_{\infty}$ of the
irreducible multiplicative fiber. E.g. when $r=2$ the fiber is the quotient
of ${\mathbb{C}}\times\mathbb{P}^{1}$ under the identifications (for
$z\in{\mathbb{C}}$, $(y_{1}:y_{2})\in\mathbb{P}^{1}$)

$\displaystyle(z\mid y_{1}:y_{2})\sim(z+m+n\tau_{u}\mid e^{2\pi
inw_{u}}y_{1}:y_{2})\qquad\forall\,m,n\in{\mathbb{Z}}$ (4.16)
$\displaystyle(z\mid 0:y_{2})\sim(z+\eta_{u}\mid y_{1}:0)$

for a pair of points $w_{u}$, $\eta_{u}$ in the elliptic curve $A_{u}$.
However the two points $w_{u}$ and $\eta_{u}$ are not independent: they are
related by the constraints that the fiber is Lagrangian and the polarization
principal. We Claim that under these conditions $\eta_{u}\equiv w_{u}$.
Therefore in the irreducible multiplicative case if the fiber is a global
fibration over $A_{u}$, it is automatically a trivial fibration (trivial HO
invariant).

######  Example/Claim.

Next example in order of complexity is a simple _irreducible additive fiber_
(the higher dimensional analogue of Kodaira type $\mathbf{II}$). In this case
$\Upsilon$ is acyclic, $\eta_{u}\equiv 0$, and we remain with just the point
$w_{u}$. We Claim that the principal Lagrangian fibration condition implies
$w_{u}=0$ i.e. the fiber is a product of a type $\mathbf{II}$ Kodaira fiber
with $A_{u}$.

More generally

######  Claim 1.

Suppose that the local monodromy $\varrho_{\alpha}$ around the divisor
$D_{\alpha}$ is of the unstable type, $\mathrm{rank}(\varrho_{\alpha}-1)=2$,
and the fiber over the generic point $u\in D_{\alpha}$ is simple; then the
fiber is a product of an unstable Kodaira fiber (types $\mathbf{I}_{n}^{*}$,
$\mathbf{II}$, $\mathbf{II}^{*}$, $\mathbf{III}$, $\mathbf{III}^{*}$,
$\mathbf{IV}$, $\mathbf{IV}^{*}$) and an Abelian variety $A_{u}$ of dimension
$(r-1)$.

##### Physical intuition beyond dichotomy.

What is the physical distinction between situations (a), (b), and (c)? Case
(c) is obvious: the quantum system factorizes asymptotically as we approach
the generic point in the discriminant in a rank-$(r-1)$ system described by
the special geometry ${\cal A}\to D_{\alpha}$ with fiber $A_{u}$ and an IR
effective rank-1 quantum system whose Seiberg-Witten geometry is the
germ242424 The rank-1 system may be non UV complete (this happens in the non-
semisimple case); in this case the special geometry exists only in the germ
sense. of the elliptic fibration ${\cal E}\to\Delta$ with central Kodaira
fiber $F$. Situation (a) is also clear: although the special geometry ${\cal
A}\to D_{\alpha}$ and the rank-$(r-1)$ system are still well-defined _per se_
, the two systems do not fully decouple (asymptotically) because they remain
entangled through the flux quantization as described in §. 3. To interprete
physically situation (b) would be quite challenging: we have no physical
observable to define and/or detect this would-be third “quantum phase”.
Physical intuition says “just forget it, it cannot happen” (under our
assumptions including principal polarization). This is precisely the statement
of _dichotomy_.

####  4.2.3 Structure of simple fibers: intersection viewpoint

To generalize the Kodaira story from the intersection-theory side we may try
to do two different things:

  * 1.

define the local intersection numbers $F_{u,i}\\#F_{u,j}$ of the irreducible
components along a section (their multiplicities $m_{u,i}$ being as in
Definition 5);

  * 2.

construct a maximal connected cycle $\sum_{a}m_{a}C_{i(a)}(z_{a})$ of
$\mathbb{P}^{1}$ fibers $C_{i(a)}(z_{a})\subset F_{u,i(a)}$ over points
$z_{a}\in A_{u}$, give appropriate definitions of their multiplicities $m_{a}$
and intersection numbers $C_{i(a)}\\!\cdot\\!C_{j(b)}$.

As already stated, in rank-$1$ the two procedures coincide.

######  Definition 6.

The intersection $F_{u,i}\cap F_{u,j}\subset F_{u,i}$ defines a divisor
$D_{j}$ in $F_{u,i}$ whose support is a section of $\varpi_{u,i}\colon
F_{u,i}\to A_{u}$. We set

$F_{u,i}\\#F_{u,j}\overset{\rm def}{=}C_{i}(z_{i})\cdot
D_{j}=C_{j}(z_{j})\cdot D_{i}$ (4.17)

where $C_{i}(z_{i})$ is the fiber of the ruled variety
$F_{u,i}^{\text{nor}}\to A_{u}$ over $z_{i}\in A_{u}$ ($F_{u,i}\\#F_{u,j}$ is
independent of the chosen point $z_{i}$ since ${\mathbb{C}}^{r-1}$ acts
transitively on $A_{u}$). By convention we also set $F_{u,i}\\#F_{u,i}=-2$
when the fiber is reducible and 0 otherwise.

##### Characteristic cycle.

We consider two fibers $C_{i}(z_{i})$ and $C_{j}(z_{j})$ of
$\varpi_{u,i}\colon F_{u,i}\to A_{u}$ and (respectively) $\varpi_{u,j}\colon
F_{u,j}\to A_{u}$, which cross each other at a point in the divisor

$D_{i}\subset F_{u,i}\cap F_{u,j}\simeq A_{u}.$ (4.18)

By eq.(4.15) in order to cross they must be of the form $C_{i}(z_{i})$ and
$C_{j}(z_{i}+\eta_{ij})$. We define the (local) intersection number of the two
rational curves,

$C_{i}(z_{i})\\#C_{j}(z_{i}+\eta_{ij})\overset{\rm def}{=}F_{u,i}\\#F_{u,j}.$
(4.19)

The multiplicity $m_{i,z_{i}}$ of the rational curve $C_{i}(z_{i})$ is defined
to be the multiplicity $m_{u,i}$ of the component $F_{u,i}$ which contains it.
A maximal connected cycle of $\mathbb{P}^{1}$ fibers

$\boldsymbol{\Psi}=\sum_{v\in V_{u}}m_{u,i(v)}C_{i(v)}(z_{v})$ (4.20)

is called the _characteristic cycle_ of the exceptional fiber [19, 20, 21]. We
stress that the two index sets in eqs.(4.7), (4.20), $I_{u}$ and $V_{u}$, need
_not_ to be equal.

The arguments used by Kodaira for the elliptic fibers apply both to the fiber
cycle $\mathscr{X}_{u}=\sum_{i\in I_{u}}m_{u,i}F_{u,i}$ and to the
characteristic cycle $\boldsymbol{\Psi}$. In particular, both intersection
matrices

$C_{i,j}\equiv F_{u,i}\\#F_{u,j},\qquad K_{v,w}\equiv
C_{i(v)}(z_{v})\\#C_{i(w)}(z_{w}),$ (4.21)

should be either zero or the negative of the Cartan matrix of a simple,
simply-laced affine Lie algebra of type $\widetilde{A}_{n-1}$,
$\widetilde{D}_{4+n}$, $\widetilde{E}_{6}$, $\widetilde{E}_{7}$ or
$\widetilde{E}_{8}$, while for the Cartan matrices of $\widetilde{A}_{1}$ and
$\widetilde{A}_{2}$ we have two different realizations akin to rank-1 case:
for $\widetilde{A}_{1}$ the two components may either cross transversely at
two distinct sections, or tangentially at a single section; for
$\widetilde{A}_{2}$ the three components may meet pairwise transversely at
three distinct points or all at the same point. The conclusion is that both
the fiber cycle $\sum_{i}m_{u,i}F_{u,i}$ and the characteristic cycle
$\boldsymbol{\Psi}$ are “modelled” on a Kodaira fiber, _except that_
$\boldsymbol{\Psi}$, contrary to the fiber cycle, may have _infinitely many_
components so that the type $\mathbf{I}_{\infty}$ is also allowed for the
characteristic cycle.252525 A more careful analysis [20] shows that Type
$\mathbf{I}^{*}_{\infty}$ is ruled out. Cycles modelled on a Kodaira fiber
have (by definition) the same number of components, intersection form, and
multiplicities as their model Kodaira fiber.

######  Remark 6.

The characteristic cycle type is well-defined, i.e. independent of the
various choices. Indeed, ${\mathbb{C}}^{r-1}$ acts transitively on the set of
maximal connected cycles of $\mathbb{P}^{1}$ fibers.

######  Remark 7.

One extends the allowed Kodaira type of the characteristic cycle to type
$\mathbf{I}_{0}$ (relevant only when the fiber is multiple). In this case
$\boldsymbol{\Psi}$ is a smooth elliptic curve.

However, in general, the Kodaira type of the characteristic cycle
$\boldsymbol{\Psi}$ is _different_ from the Kodaira type of the fiber cycle
$\mathscr{X}_{u}$.

######  Claim 2.

The Kodaira types of a _simple_ fiber and of its characteristic cycles
coincide if and only if the class $[\eta]\in H^{1}(\Upsilon,A_{u})$ is
trivial.

In particular when the simple fiber is reducible, and the fiber cycle is not
one of the afore mentioned special types, $\Upsilon$ is the dual of the affine
Dynkin graph associated to the fiber cycle, which is acyclic, except for
$\widetilde{A}_{b-1}$ i.e. type $\mathbf{I}_{b}$. Therefore for simple fibers
$[\eta]$ may be non-zero only if the fiber cycle has type $\mathbf{I}_{b}$
with $b\geq 1$, i.e. for _stable_ singular fibers. It is easy to check that
the statement extends to the special cases.

####  4.2.4 Simple stable fibers: explicit expressions

Recall that the polarization of the special geometry is assumed to be
principal. In a sufficiently small neighborhood of a generic point $u$ in a
_stable_ discriminant $D_{\alpha}$ we can find special holomorphic coordinates
$a^{i}$ ($i=1,\cdots,r$) with local discriminant $a^{r}=0$ and local
prepotential of the form

${\cal F}(a^{i})=\widetilde{{\cal F}}(a^{i})+\frac{\ell}{4\pi
i}(a^{r})^{2}\log a^{r},\qquad\ell\in\mathbb{N}.$ (4.22)

While this is physically obvious (it corresponds to some hypers with electric
charges $q_{i}$ getting massless, while $\ell=\sum_{i}q_{i}^{2}$), the
mathematically oriented reader may wish to have a look to [21] for a math
proof. We write

$\tau_{ij}\overset{\rm def}{=}\frac{\partial^{2}\widetilde{{\cal F}}}{\partial
a^{i}\partial a^{j}}\bigg{|}_{u},\qquad\theta_{i}\overset{\rm
def}{=}\frac{\partial^{2}\widetilde{{\cal F}}}{\partial a^{i}\partial
a^{r}}\bigg{|}_{u}\quad i,j=1,\dots,r-1.$ (4.23)

$\tau_{ij}$ is the period matrix of the Abelian variety $A_{u}$ in the chosen
duality frame (marking in the math jargon). In other words, writing
$\Lambda_{u}$ for the lattice

$\Lambda_{u}\equiv\Big{\\{}m_{i}+\tau_{ij}\mspace{1.0mu}n^{j}\colon
m_{i},n^{j}\in{\mathbb{Z}}^{r-1}\Big{\\}}\subset{\mathbb{C}}^{r-1},$ (4.24)

we have $A_{u}={\mathbb{C}}^{r-1}/\Lambda_{u}$. In the simple stable case all
components are multiplicative of multiplicity 1, and as an algebraic
group262626 Here we use that the integer $\ell$ in eqn.(4.22) is the number of
components in the fiber just as in Kodaira’s rank-1 case. This can be inferred
by starting from the particular case $\theta=0$ where the statement is obvious
(since the fiber is the product of $A_{u}$ and a Kodaira fiber of type
$\mathbf{I}_{\ell}$) and then deform away from that point making $\theta$ non-
zero. Clearly the deformation would not change the monodromy or the number of
components. Readers not satisfied with this argument are referred to [21].

$F_{u}^{\sharp}=F_{u,0}^{\sharp}\rtimes{\mathbb{Z}}_{\ell}.$ (4.25)

In particular the normalized components $F_{u,i}$ are all isomorphic copies of
a ruled variety $\mathbb{P}({\cal L}\oplus{\cal
O})\to{\mathbb{C}}^{r-1}/\Lambda_{u}$. We write
$z_{i}\in{\mathbb{C}}^{r-1}/\Lambda_{u}$ for the coordinate in the $A_{u}$
basis of the $i$-th copy. We think $\theta\equiv(\theta_{i})\bmod\Lambda_{u}$
as a point in $A_{u}$.

Let $\boldsymbol{Z}$ be the _infinite_ cycle of copies $\\{\varpi_{u,i}\colon
F_{u,i}\to A_{u}\\}_{i\in{\mathbb{Z}}}$ of the ruled variety with preferred
zero- and $\infty$-sections, $s_{0}(i),s_{\infty}(i)\colon A_{u}\to F_{u,i}$
obtained by gluing the $0$-section of $F_{u,i+1}$ to the $\infty$-section of
$F_{u,i}$.

$\boldsymbol{Z}=\coprod_{i\in{\mathbb{Z}}}F_{u,i}\Big{/}\big{(}s_{0}(i+1)\sim
s_{\infty}(i)\big{)}.$ (4.26)

Since the dual graph of $\boldsymbol{Z}$, $A^{\infty}_{\infty}$, is acyclic we
may assume with no loss that the gluing is via the identity map $A_{u}\to
A_{u}$, i.e. $z_{i+1}=z_{i}$. On $\boldsymbol{Z}$ we have a maximal connected
cycle of countably-many rational curves

$\boldsymbol{\widetilde{\Psi}}=\sum_{i\in{\mathbb{Z}}}C_{i}(0),$ (4.27)

and all other maximal cycles are obtained by acting with ${\mathbb{C}}^{r-1}$
on it.

The fiber $\mathscr{X}_{u}$ with pre-potential (4.22) is the quotient of
$\boldsymbol{Z}$ under the identification [21]

$(F_{u,i},z_{i})\simeq(F_{u,i+\ell},z_{i}+\theta)$ (4.28)

In other words: we have $\ell$ distinct components $F_{u,i}$ cyclically
identified as in figure 2 of [26] (see also figure 1 of [24]), and the fiber
cycle has Kodaira type $\mathbf{I}_{\ell}$. However when we go around the
components’ cycle following a fiber (i.e. an orbit of the Hamiltonian flow
generated by $a^{r}$) we do not come back where we started from, but rather to
a point of $s_{0}(0)\simeq A_{u}$ translated by $\theta$. In particular, the
components of the maximal 1-cycle (4.27) do not get identified mod $\ell$,
unless $\theta=0\bmod\Lambda_{u}$. Indeed

$C_{i+\ell}(0)=C_{i}(\theta)\ \big{(}\neq C_{i}(0)\ \text{for }\theta\neq
0\big{)}.$ (4.29)

If $\theta$ is a $m$-torsion point, i.e. $m$ is the minimal positive integer
such that $m\mspace{1.0mu}\theta=0\bmod\Lambda_{u}$, the rational curve will
return to itself after going around $m$-times the components’ cycle

$C_{i+m\ell}=C_{i}(m\theta)\equiv C_{i}(0).$ (4.30)

In this case the characteristic cycle is

$\boldsymbol{\Psi}=\sum_{s=0}^{m-1}\sum_{i=0}^{\ell-1}C_{i}(\ell\theta)$
(4.31)

with $m\ell$ components and Kodaira type $\mathbf{I}_{m\ell}$. If $\theta\in
A_{u}$ is not a torsion point, the characteristic cycle will wrap around the
fiber cycle indefinitely, and the Kodaira type of the characteristic cycle
will be $\mathbf{I}_{\infty}$. For a visualization of the fiber geometry we
refer the reader to the beautiful figures in ref.[26].

##### The line bundle ${\cal L}\to A_{u}$.

It remains to show our claim that in the irreducible multiplicative case
$w_{u}\equiv\eta_{u}$, or for the more general stable case that the line
bundle ${\cal L}$ such that $F_{u,i}\simeq\mathbb{P}({\cal L}\oplus{\cal O})$
coincides with the point $\theta\in A_{u}^{\vee}\simeq A_{u}$. Notice that for
$a^{r}$ small but non-zero the fiber is a smooth Abelian variety of dimension
$r$ with period matrix $\partial^{2}{\cal F}/\partial a^{I}\partial a^{J}$. If
$(z_{i},y)$ are coordinates in the covering ${\mathbb{C}}^{r}$ we have (in
particular!) the identifications ($i,j=1,\dots,r-1$)

$(z_{i},y)\sim(z_{i}+m_{i}+\tau_{ij}n^{j},y+m+\theta_{i}n^{i})\quad
m_{i},m,n^{i}\in{\mathbb{Z}}$ (4.32)

or, in the multiplicative notation,

$(z_{i},w)\equiv(z_{i},e^{2\pi iy})\sim(z_{i}+m_{i}+\tau_{ij}n^{j},e^{2\pi
i\theta_{i}n_{i}}w),$ (4.33)

which is (the generalization of) eq.(4.12).

##### The HO invariant.

In the simple stable case we define the HO invariant to be the number of times
the characteristic cycle wraps around the fiber cycle. In the case of simple
unstable fibers the graph $\Upsilon$ is acyclic, and the Kodaira types of the
characteristic cycle and fiber cycle are equal and belong to the list

$\mathbf{I}^{*}_{b},\ \mathbf{II},\ \mathbf{III},\ \mathbf{IV},\
\mathbf{II}^{*},\ \mathbf{III}^{*},\ \mathbf{IV}^{*}.$ (4.34)

We define the HO invariant for _simple fibers_ to be the ratio

$\mu(u)=\frac{\text{(number of components of the fundamental
cycle)}}{\text{(number of irreducible components of the
fiber)}}\in\mathbb{N}\cup\infty$ (4.35)

The HO invariant is trivial precisely when the Kodaira type of the
characteristic cycle agrees with the one of the fiber cycle. In this case the
singular fiber is a product $K\times A_{u}$ with $K$ the Kodaira fiber of
their common type and $A_{u}$ a polarized Abelian variety of dimension
$(r-1)$.

####  4.2.5 Kodaira type vs. monodromy

As in rank-1, we can describe the situation in terms of the local monodromy
$\varrho_{\alpha}$ around $D_{\alpha}$. The $2r-2$ homology classes
corresponding to $H_{1}(A_{u},{\mathbb{Z}})$ are invariant under the local
monodromy, which then lays in a $SL(2,{\mathbb{Z}})$ subgroup of
$Sp(2r,{\mathbb{Z}})$, and can be identified with a Kodaira monodromy (tables
1, 2). In particular

$(\varrho_{\alpha}^{m}-1)^{2}=0\quad\text{where
}m=1,2,3,4,6\quad\text{and}\quad m=3,4,6\ \Rightarrow\
\varrho_{\alpha}^{m}=1.$ (4.36)

######  Claim 3.

Suppose the fiber over the generic point $u\in D_{\alpha}$ is simple. The
conjugacy class of the local monodromy $\varrho_{\alpha}$ around $D_{\alpha}$
is given by the Kodaira monodromy associated to the Kodaira type of the fiber
cycle of the fiber $\mathscr{X}_{u}$ at a generic point $u\in D_{\alpha}$.

We stress that it is the fiber cycle and not the characteristic cycle which
controls the local monodromy. Indeed the local monodromy is independent
(modulo conjugacy) of the point $u$, whereas the characteristic cycle may
depend on $u$ in a very wild way, cf. §. 3.2.5.

We show Claim 3. Let
$\Delta=\\{(h_{1},\dots,h_{r})\colon|h_{i}|<\epsilon\\}\subset\mathscr{C}$ be
a polydisk centered at $u\in D_{\alpha}$ with $h_{r}=0$ the local equation of
$D_{\alpha}$ in $\Delta$ and $\mathscr{X}|\to\Delta$ the restricted Lagrangian
fibration. Pulling back the fibration through the branched cover

$f\colon\widetilde{\Delta}\to\Delta,\qquad f\colon h_{r}\mapsto h_{r}^{m}$
(4.37)

we get a new fibration $\widetilde{\mathscr{X}}|\to\widetilde{\Delta}$ whose
fibers over
$\widetilde{\Delta}^{\sharp}\equiv\widetilde{\Delta}\setminus(\widetilde{h}_{r})$
($\widetilde{h}_{r}\equiv f^{*}h_{r}$) are smooth principally polarized
Abelian varieties. The family of smooth Abelian varieties over the complement
of the discriminant

$\widetilde{\mathscr{X}}|_{\widetilde{\Delta}\setminus(\widetilde{h}_{r})}\to\widetilde{\Delta}\setminus(\widetilde{h}_{r}),$
(4.38)

has now a strictly unipotent monodromy $\widetilde{\varrho}_{\alpha}$ around
the discriminant $(\widetilde{h}_{\alpha})$:

$(\widetilde{\varrho}_{\alpha}-1)^{2}=0.$ (4.39)

In other words (as it is well known [42, 43, 44, 45]) via the covering trick
(4.37) we can always reduce ourselves to the stable situation: see [20] for
the application of the trick in the present context.

Let us consider first the case where original monodromy $\varrho_{\alpha}$ was
_semisimple_ , i.e. $\varrho_{\alpha}^{m}=1$ with $m\in\\{2,3,4,6\\}$. The
pulled back monodromy is now trivial $\widetilde{\varrho}_{\alpha}=1$, and we
may extend the family of Abelian varieties (4.38) along the discriminant
$h_{r}=0$ to get a fibration $\widetilde{\mathscr{X}}\to\widetilde{\Delta}$
(see e.g. Theorem $\boldsymbol{13\\!\cdot\\!7\\!\cdot\\!5}$ of [45]). The
central fiber $\widetilde{\mathscr{X}}_{u}$ of the family is now a principally
polarized, $r$-dimensional, Abelian variety with a polarized automorphism
$\xi$ of order $m$ whose analytic representation272727 Let $\xi\colon A\to A$
an automorphism of the Abelian variety of dimension $r$ (over ${\mathbb{C}}$).
Its _analytic representation_ [48] is the induced complex representation on
the tangent space at the origin $T_{0}A\simeq{\mathbb{C}}^{r}$ which is the
Lie algebra $\mathsf{Lie}(A)$ of the complex Lie group $A$. $\rho_{a}(\xi)$
has $(r-1)$ eigenvalues $1$ while its last eigenvalue is a primitive $m$-th
root of unity. By (say) Theorem 13.2.8 of [48], there exist two polarized
Abelian subvarieties of $\widetilde{\mathscr{X}}_{u}$, $A_{u}^{\vee}$ and
$E_{u}$, respectively of dimension $(r-1)$ and $1$ and an isogeny

$A_{u}^{\vee}\times
E_{u}\to\widetilde{\mathscr{X}}_{u},\qquad\text{with}\quad\xi\in\mathsf{Aut}(E_{u}).$
(4.40)

Since $\widetilde{\mathscr{X}}_{u}$ is principally polarized, we may identify
$A_{u}^{\vee}$ with the dual of the Albanese variety $A_{u}$ i.e. the Picard
variety of $\mathscr{X}_{u}$. $A_{u}^{\vee}\not\simeq A_{u}$ in general, i.e.
$A_{u}$ needs not to be _principally_ polarized. In particular, when
$m=3,\;4,\;6$ the complex structure of the elliptic curve $E_{u}$ is fixed to
$\tau=e^{2\pi i/3}$, $i$, and $e^{2\pi i/3}$ respectively. When $m=2$, $E_{u}$
can be any elliptic curve. When $m\in\\{3,4,6\\}$ the field
$\mathbb{Q}(e^{2\pi i/m})$ has degree $2$, and we have two choices for the CM-
type of $E_{u}$ [48]: as discussed in [49] they correspond to the two possible
Kodaira types with a semisimple monodromy of the given order $m$ namely

  * •

$\mathbf{IV}$, $\mathbf{IV}^{\ast}$ for $m=3$;

  * •

$\mathbf{III}$, $\mathbf{III}^{\ast}$ for $m=4$;

  * •

$\mathbf{II}$, $\mathbf{II}^{\ast}$ for $m=6$.

Taking the quotient of $\widetilde{\mathscr{X}}\to\widetilde{\Delta}$ by
${\mathbb{Z}}_{m}$ we return to a fibration over the original polydisk
$\Delta$, $\mathscr{X}^{\flat}\to\Delta$ whose restriction to
$\Delta\setminus(h_{\alpha})$ is isomorphic to the restriction of our original
fibration $\mathscr{X}\to\Delta$. The central fiber of $\mathscr{X}^{\flat}$
has the form $mF$ where the reduced irreducible variety $F$ is a ruled variety
over $A_{u}$. However $\mathscr{X}^{\flat}$ is not smooth: to get a smooth
total space we need to perform an appropriate sequence of blow-up and blow-
down of various components inside the central fiber. The precise sequence of
these operations for each semisimple Kodaira type can be read in §. 4 of [20]
(or §.V̇.10 of [11]). In particular we conclude that in the semisimple case
the order of the local monodromy agrees with the Kodaira type of the
exceptional fiber which is a product of a Kodaira elliptic singular fiber $K$
times $A_{u}$.

The covering trick (with $m=2$) reduces a fiber of type $\mathbf{I}^{*}_{n}$
to a fiber of type $\mathbf{I}_{2n}$ (again after suitable blow-downs/ups of
components of the central fiber). This shows that Claim 3 is true iff it holds
for the stable case. The stable case was studied in great detail in §. 4 of
[21] (assuming a principal polarization). There it is shown that a stable
fiber with $\ell$ components has a local monodromy conjugate over
$Sp(2r,{\mathbb{Z}})$ to

$\left(\begin{array}[]{c|c}\;\begin{smallmatrix}1&\ell\\\
0&1\end{smallmatrix}&0\\\ \hline\cr 0&\boldsymbol{1}_{r-1}\end{array}\right).$
(4.41)

This completes the argument.

###  4.3 Multiple fibers

Next we consider _multiple_ Lagrangian fibers following [20]. Multiple fibers
may be present only in a “no-section” special geometry. As discussed in §. 2
the “no-section” geometries have not the physical interpretation of standard
Seiberg-Witten geometries: their physical meaning will be discussed in section
5.

In a Lagrangian fibration of a holomorphic symplectic manifold only a handful
of multiplicities $m\geq 2$ are allowed, and only for specific types of the
characteristic cycle $\boldsymbol{\Psi}$. Theorem 1.1 of [20] gives the
following table of allowed multiple fibers:

multiplicity $m$ | characteristic cycle type
---|---
$6$ |  $\mathbf{I}_{0}$ ($E_{\zeta_{3}}$) |  |  |  |
$5$ | $\mathbf{II}$ |  |  |  |
$4$ |  $\mathbf{I}_{0}$ ($E_{\zeta_{4}}$) | $\mathbf{IV}$ |  |  |
$3$ |  $\mathbf{I}_{0}$ ($E_{\zeta_{3}}$) | $\mathbf{III}$ | $\mathbf{I}_{0}^{*}$ |  |
$2$ |  $\mathbf{I}_{2n}$ ($n\geq 0$) | $\mathbf{I}_{\infty}$ | $\mathbf{I}_{0}^{*}$ | $\mathbf{IV}$ | $\mathbf{IV}^{*}$
(4.42)

The notation $\mathbf{I}_{0}$ $(E_{\zeta_{n}})$ means that the
$\mathbf{I}_{0}$ type characteristic cycle is required to be an elliptic curve
whose period $\tau$ is a primitive $n$-th root $\zeta_{n}$ of 1.282828
Throughout this paper $\zeta_{n}$ will always denote a primitive $n$-th root
of unity.

##### Local monodromy around a multiple fiber.

As it will be clear from the explicit construction below, in the semisimple
case the local monodromy $\varrho_{\alpha}$ satisfies (for $m\geq 2$) the
following equation

$[\varrho_{\alpha}^{m}]=\begin{smallmatrix}\textbf{conjugacy class of the
local monodromy}\\\ \textbf{ around the Kodaira fiber on which the}\\\
\textbf{characteristic cycle is modelled}\end{smallmatrix}$ (4.43)

that is, the local monodromy is the one of the Kodaira fiber associated to the
Lie algebra $\mathfrak{g}$, see the second column of table 3. In the non-
semisimple case $m=2$ $\boldsymbol{\Psi}=\mathbf{I}_{2n}$ we have
$\mathfrak{g}=D_{4+\ell}$ where $2\ell\mid 2n$ is the number of components of
the fiber. We notice that monodromies of types $\mathbf{II}$, $\mathbf{III}$
and $\mathbf{IV}$ (whose dual graph is not an affine Dynkin diagram) cannot
appear.

$(m,\boldsymbol{\Psi}\ \text{type})$ | monodromy | equation of ${\cal E}$ |  ${\mathbb{Z}}_{m}$ automorphism $\sigma$
---|---|---|---
$(6,E_{\zeta_{3}})$ | $E_{8}$ | $y^{2}=x^{3}+1$ | $(x,y,t)\mapsto(\zeta_{6}^{2}x,\zeta_{6}^{3}y,\zeta_{6}t)$
$(5,\mathbf{II})$ | $E_{8}$ | $y^{2}=x^{3}+t$ | $(x,y,t)\mapsto(\zeta_{5}^{2}x,\zeta_{5}^{3}y,\zeta_{5}t)$
$(4,E_{\zeta_{4}})$ | $E_{7}$ | $y^{2}=x^{3}+x$ | $(x,y,t)\mapsto(-x,\zeta_{4}^{3}y,\zeta_{4}t)$
$(4,\mathbf{IV})$ | $E_{8}$ | $y^{2}=x^{3}+t^{2}$ | $(x,y,t)\mapsto(-x,\zeta_{4}^{3}y,\zeta_{4}t)$
$(3,E_{\zeta_{3}})$ | $E_{6}$ | $y^{2}=x^{3}+1$ | $(x,y,t)\mapsto(\zeta_{3}^{2}x,y,\zeta_{3}t)$
$(3,\mathbf{III})$ | $E_{7}$ | $y^{2}=x^{3}+tx$ | $(x,y,t)\mapsto(\zeta_{3}^{2}x,y,\zeta_{3}t)$
$(3,\mathbf{I}_{0}^{*})$ | $E_{8}$ | $y^{2}=x^{3}+t^{3}$ | $(x,y,t)\mapsto(\zeta_{3}^{2}x,y,\zeta_{3}t)$
$(2,\mathbf{I}_{0})$ | $D_{4}$ | $y^{2}=x^{3}+ax+b$ | $(x,y,t)\mapsto(x,-y,-t)$
$(2,\mathbf{I}_{0}^{*})$ | $E_{7}$ | $y^{2}=x^{3}+t^{2}x$ | $(x,y,t)\mapsto(x,-y,-t)$
$(2,\mathbf{IV})$ | $E_{6}$ | $y^{2}=x^{3}+t^{2}$ | $(x,y,t)\mapsto(x,-y,-t)$
$(2,\mathbf{IV}^{*})$ | $E_{8}$ | $y^{2}=x^{3}+t^{4}$ | $(x,y,t)\mapsto(x,-y,-t)$
Table 3:  Data for the construction of multiple Lagrangian fibers. The
$2^{\text{nd}}$ column specifies the local monodromy in terms of the
corresponding Lie algebras $\mathfrak{g}$ as listed in tables 1, 2.

####  4.3.1 Explicit construction

Ref.[20] gives an explicit construction in rank 2 of all allowed multiple
fibers in table (4.42). For sake of comparison with physics we review their
construction. We first assume that the characteristic cycle is not of type
$\mathbf{I}_{2n}$ with $n>0$.

For each allowed pair $(m,\boldsymbol{\Psi}\ \text{type})$ in the first column
of table 3, we write ${\cal E}\to{\mathbb{C}}$ for the elliptic fibration
whose central fiber ${\cal E}_{0}$ has the same Kodaira type as
$\boldsymbol{\Psi}$: see third column of table 3 for the explicit Weierstrass
model of each fibration ($t$ is the coordinate on the base ${\mathbb{C}}$).
The rank-1 special geometry ${\cal E}$ has a natural symplectic form

$\omega=\frac{dx\wedge dt}{y}.$ (4.44)

The fibration ${\cal E}\to{\mathbb{C}}$ has a ${\mathbb{Z}}_{m}$
automorphism292929 For the interpretation of this automorphism in the 4d/2d
correspondence, and its relation to the quantum monodromy of the 4d ${\cal
N}=2$ QFT, see [50, 51]. $\sigma$ acting on the base coordinate as
$t\mapsto\zeta_{m}\,t$ and preserving $\omega$. Let $E$ be a _fixed_ elliptic
curve (with holomorphic differential $dz$) and $p_{m}\in E$ an $m$-torsion
point. Consider the fibration

$\mathscr{Y}\equiv{\cal E}\times
E\times{\mathbb{C}}\to{\mathbb{C}}\times{\mathbb{C}},\quad\big{(}(x,y,t),z,s\big{)}\mapsto(t,s),$
(4.45)

with symplectic form

$\Omega=\omega+dz\wedge ds.$ (4.46)

Take the quotient of $\mathscr{Y}$ with respect to the ${\mathbb{Z}}_{m}$
group generated by the order-$m$ symplectic automorphism

$\Sigma\colon\big{(}(x,y,t),z,s\big{)}\mapsto\big{(}(\zeta_{m}^{2}x,\zeta_{m}^{3}y,\zeta_{m}t),z+p_{m},s\big{)}.$
(4.47)

The resulting Lagrangian fibration

$\mathscr{X}\equiv\mathscr{Y}/{\mathbb{Z}}_{m}\longrightarrow{\mathbb{C}}/{\mathbb{Z}}_{m}\times{\mathbb{C}}\simeq{\mathbb{C}}^{2}$
(4.48)

has a central fiber of multiplicity $m$ and type $\boldsymbol{\Psi}$ [20]. The
Albanese variety of an irreducible component of the singular fiber is
$E/{\mathbb{Z}}_{m}$. The above construction has obvious generalizations to
higher dimensions.

##### Multiple fibers with $\mathbf{I}_{2n}$ characteristic cycle.

In the $\mathbf{I}_{2n>0}$ case we have the extra feature of the possibility
of a non-trivial HO invariant in the sense of our discussion in §. 4.3.1. A
part for that, the construction for $\mathbf{I}_{2n}$ is pretty similar to the
one for the other cases; the explicit geometry is constructed in Example 6.3
of [20]. Note that we may also have $n=\infty$.

######  Remark 8.

Fibers of multiplicity 2 and characteristic cycle of type $\mathbf{I}_{2n}$
($n>0$) have an even number of components (i.e. their fiber cycle is of type
$\mathbf{I}_{2n^{\prime}}$ with $n^{\prime}\mid n$).

######  Remark 9.

We stress that we have double fibers of characteristic cycle type
$\mathbf{I}_{2n}$ with any HO invariant $\mu$ (as defined in eq.(4.35)).

####  4.3.2 Deformations spaces of multiple Lagrangian fibers

“(Partially) frozen” singularities have a deformation space ${\cal M}$ of
smaller dimension than naively expected

$\dim_{\mathbb{C}}{\cal M}<\mathrm{rank}\,\mathfrak{g}$ (4.49)

where $\mathfrak{g}$ is the Lie algebra associated to the conjugacy class of
the local monodromy. In order to show that our geometrical description is on
the right track, we need to construct _geometrically_ the deformation space
and check that it has precisely the dimension predicted by physics for the
(partially) frozen singularity of each given kind. To match the physical
expectation the dimension should be equal to the rank of the corresponding
flavor group as described in the following two physical sections.

For definiteness we take $r=2$. The Hwang-Oguiso construction (§. 4.3.1) of
the multiple fiber started from a covering family of elliptic curves with
Weierstrass equation of the form

$y^{2}=P_{0}(x,t)\quad\text{where}\quad
P_{0}(\zeta_{m}^{2}\mspace{2.0mu}x,\zeta_{m}\mspace{2.0mu}t)=\zeta_{m}^{6}\mspace{2.0mu}P_{0}(x,t),$
(4.50)

which are listed in table 3 for each $\boldsymbol{\Psi}$’s Kodaira type and
multiplicity $m$. A deformation of the geometry is a deformation of the
polynomial

$P_{0}(x,t)\leadsto P(x,t)$ (4.51)

subjected to the following rules:

  * (i)

the deformation does not change the geometry at infinity, i.e. the
deformation consists in adding to $P_{0}(x,t)$ monomials $x^{a}t^{b}$ with
$(a,b)\in{\mathbb{Z}}^{2}\subset{\mathbb{R}}^{2}$ contained in the closure of
the Newton polytope $N$ of $P_{0}(x,t)$, the convex hull
$N\subset{\mathbb{R}}^{2}$ of the points
$\\{(0,0),(a_{1},b_{1}),\dots,(a_{s},b_{s})\\}$ where $(a_{i},b_{i})$ are the
degrees of the monomials in $P_{0}(x,t)$;

  * (ii)

the deformation must preserve the symmetry $\sigma$

$P(\zeta_{m}^{2}\mspace{2.0mu}x,\zeta_{m}\mspace{2.0mu}t)=\zeta_{m}^{6}\mspace{2.0mu}P(x,t).$
(4.52)

  * (iii)

two polynomials $P(x,t)$, $P^{\prime}(x,t)$ which satisfy (i) and (ii) are
considered to be equivalent if they are related by a linear redefinition of
the variables $P(x,t)^{\prime}=P(x+Q(t),t)$ for some polynomial $Q(t)$.

For instance, for the pair $(2,\mathbf{IV}^{*})$ the Newton polytope in
${\mathbb{R}}^{2}$ is

(4.53)

(black/gray dots stand for ${\mathbb{Z}}^{2}\subset{\mathbb{R}}^{2}$). The
monomials satisfying (i) are $1$, $x$, $x^{2}$, $x^{3}$, $t$, $tx$, $tx^{2}$,
$t^{2}$, $t^{2}x$, $t^{3}$, and $t^{4}$, while the monomials satisfying (ii)
are the ones containing $t$ to an even power (black nodes in (4.53)). We
remain with

$P(x,t)=x^{3}+b_{1}x^{2}+b_{2}x+b_{3}+b_{4}t^{2}x+b_{5}t^{2}+t^{4},$ (4.54)

while (iii) the redefinition $x\leadsto x-b_{1}/3$ sets the coefficient of
$x^{2}$ to zero, leaving with a deformation space ${\cal M}$ of dimension $4$.
In table 4 we list the polynomials describing the deformations of each
multiple fiber: the $c_{i}$’s are the effective deformation parameters.

$(m,\boldsymbol{\Psi}\ \text{type})$ | equation of ${\cal E}$ | deformed equation | $\dim_{\mathbb{C}}{\cal M}$
---|---|---|---
$(6,E_{\zeta_{3}})$ | $y^{2}=x^{3}+1$ | $y^{2}=x^{3}+1$ | $0$
$(5,\mathbf{II})$ | $y^{2}=x^{3}+t$ | $y^{2}=x^{3}+t$ | $0$
$(4,E_{\zeta_{4}})$ | $y^{2}=x^{3}+x$ | $y^{2}=x^{3}+x$ | $0$
$(4,\mathbf{IV})$ | $y^{2}=x^{3}+t^{2}$ | $y^{2}=x^{3}+c_{1}x+t^{2}$ | $1$
$(3,E_{\zeta_{3}})$ | $y^{2}=x^{3}+1$ | $y^{2}=x^{3}+1$ | $0$
$(3,\mathbf{III})$ | $y^{2}=x^{3}+tx$ | $y^{2}=x^{3}+tx+c_{1}$ | $1$
$(3,\mathbf{I}_{0}^{*})$ | $y^{2}=x^{3}+t^{3}$ | $y^{2}=x^{3}+a_{1}xt+t^{3}+a_{2}$ | $2$
$(2,\mathbf{I}_{0})$ | $y^{2}=x^{3}+Ax+B$ | $y^{2}=x^{3}+Ax+B$ | $0$
$(2,\mathbf{I}_{0}^{*})$ | $y^{2}=x^{3}+t^{2}x$ | $y^{2}=x^{3}+c_{1}x^{2}+c_{2}x+c_{3}+t^{2}x$ | $3$
$(2,\mathbf{IV})$ | $y^{2}=x^{3}+t^{2}$ | $y^{2}=x^{3}+c_{1}x+c_{2}+t^{2}$ | $2$
$(2,\mathbf{IV}^{*})$ | $y^{2}=x^{3}+t^{4}$ | $y^{2}=x^{3}+(c_{1}+c_{2}t^{2})x+(c_{3}+c_{4}t^{2})+t^{4}$ | $4$
Table 4:  Deformations of the covering Weierstrass equation

##  5 Physics of “no-section” special geometries

We now consider the physical interpretation of special geometries which are
Lagrangian fibrations $\mathscr{X}\to\mathscr{C}$ _without_ a global section.
This opens up the possibility of multiple singular fibers. The allowed
multiple fibers are listed in table (4.42) following ref.[20].

As explained in §. 2 these special geometries cannot have the interpretation
of standard Seiberg-Witten geometries on “usual sense” Coulomb branches. Their
physical interpretation is rather subtle. The easiest way to see that these
more general geometries _do describe_ actual supersymmetric quantum system is
to relate them to M-/F-theory constructions. We start from this perspective.

###  5.1 Comparison with M- and F-theory

Looking at the table (4.42) of multiple Lagrangian fibers in rank $r\geq 2$
one cannot not notice an intriguing feature: the Lagrangian multiple fibers
are in one-to-one correspondence303030 For the precise sense in which the
correspondence is 1-to-1 see Remark 10. with the (partially) frozen
singularities in M-theory (in complex codimension 2) [1, 2, 3, 12, 13]! In
turn these frozen singularities are in one-to-one correspondence with the
“non-obvious” components of the moduli spaces ${\cal N}_{G}$ of flat
$G$-connections on a 3-torus $T^{3}$ [22, 23], that is, components different
from the connected component of the trivial connection

${\cal
N}_{G}^{0}=(\boldsymbol{T}_{G}\times\boldsymbol{T}_{G}\times\boldsymbol{T}_{G})\big{/}\mathsf{Weyl}(G)$
(5.1)

(here $\boldsymbol{T}_{G}\subset G$ is the maximal torus and
$\mathsf{Lie}(G)=\mathfrak{g}$). For a given $G$ we have [22]

${\cal N}_{G}=\mspace{-20.0mu}\coprod_{m\in\mathbf{C}(\mathfrak{g})\atop
n\in\mathsf{Gal}(\mathbb{Q}(\zeta_{m})/\mathbb{Q})}\mspace{-20.0mu}{\cal
N}_{G}^{(m,n)},\qquad{\cal N}_{G}^{0}\equiv{\cal N}_{G}^{(1,1)}$ (5.2)

where $\mathbf{C}(\mathfrak{g})$ is the _set_ (no repetitions!) of dual
Coxeter labels313131 For simply-laced algebras the dual Coxeter labels
coincide with the Coxeter labels. of the affine Dynkin graph of
$\widetilde{\mathfrak{g}}$ and
$n\in({\mathbb{Z}}/m{\mathbb{Z}})^{\times}\simeq\mathsf{Gal}(\mathbb{Q}(\zeta_{m})/\mathbb{Q})$
is a choice of a primitive $m$-root of unity. The connections on the component
${\cal N}_{\mathfrak{g}}^{m,n}$ are characterized by the value of the Chern-
Simons invariant323232 $h^{\vee}$ is the dual Coxeter number of
$\mathfrak{g}$.

$\int_{T^{3}}\omega^{(3)}\equiv\frac{1}{16\pi^{2}h^{\vee}}\int_{T^{3}}\mathrm{tr}\\!\left(A\,dA+\frac{2}{3}\,A^{3}\right)=\frac{n}{m}\bmod
1$ (5.3)

######  Remark 10.

In the statement of Theorem 1.1 of [20] there is no mention of the choice of
primitive $m$-th root. However from the proofs and explicit constructions in
[20] one sees that one needs to choose a primitive root: see e.g. their
eq.(4.47). Different choices lead to distinct multiple fibers with the same
characteristic cycle type.

####  5.1.1 (Partially) frozen singularities in M-theory

Partially frozen singularities in M-theory have been discussed in [12, 13, 2,
14].

An unfrozen singularity of the form ${\mathbb{C}}^{2}/\Gamma_{\mathfrak{g}}$
(where $\Gamma_{\mathfrak{g}}\subset SU(2)$ is the finite subgroup
corresponding to the $ADE$ Lie algebra $\mathfrak{g}$) yields at the singular
locus a 7d 16-SUSY super-Yang-Mills theory with gauge algebra $\mathfrak{g}$.
In addition to its unfrozen version, the singularity has (partially) frozen
variants, still preserving 16 supercharges, characterized by a fractional
3-form flux emerging from the singularity

$\int_{S^{3}/\Gamma_{\mathfrak{g}}}C^{(3)}=\frac{n}{m}\bmod 1.$ (5.4)

As in eq.(5.3) $m$ is a dual Coxeter label on a node of the Dynkin graph of
$\mathfrak{g}$ and $n\in({\mathbb{Z}}/m{\mathbb{Z}})^{\times}$. $m$ is the
_order_ of the singularity. The unfrozen singularity has order $1$. The flux
is fractional iff $m\geq 2$, i.e. if $m$ is the Coxter label of a non-
extension node. Only Lie algebras of type $DE$ lead to frozen singularities,
since all nodes in a $\widetilde{A}_{k}$ Dynkin graph are extension.

##### The associated Lie algebra $\mathfrak{f}^{m}_{g}$.

The 7-brane of a partially frozen singularities has a gauge symmetry Lie
algebra $\mathfrak{f}^{m}_{\mathfrak{g}}$ which is smaller than
$\mathfrak{g}\equiv\mathfrak{f}_{\mathfrak{g}}^{1}$. The dual Coxeter labels
of its affine extension $\widetilde{\mathfrak{f}}^{m}_{\mathfrak{g}}$ satisfy
the following rule [13]: let
$\\{m_{i_{k}}\\}_{k}\subset\mathbf{C}(\mathfrak{g})$ be the list of dual
Coxeter labels of $\widetilde{\mathfrak{g}}$ which are divisible by the
multiplicity $m$; the dual Coxeter labels of
$\widetilde{\mathfrak{f}}^{m}_{g}$ are the integers $\\{m_{i_{k}}/m\\}_{k}$.
This rule uniquely determines $\mathfrak{f}^{m}_{\mathfrak{g}}$ except when
$m=2$ and $\mathfrak{g}=E_{6}$ (resp. $\mathfrak{D}_{k+4}$) where the dual
Coxeter labels of $\mathfrak{f}^{2}_{\mathfrak{g}}$ are $(1,1,1)$ (resp.
$1,\dots,1$) consistent with both $A_{2}$ and $C_{2}$ (resp. $A_{n}$ and
$C_{n}$). The Lie algebras $\mathfrak{f}^{m}_{\mathfrak{g}}$ for $m\geq 2$ are
listed in table 5 (cf. Table 1 in [2]). The deformation space of a partially
frozen singularity ($m\geq 2$) satisfies

$\dim_{\mathbb{C}}\\!\left(\begin{smallmatrix}\textbf{partially frozen}\\\
\textbf{singularity}\end{smallmatrix}\right)=\mathrm{rank}\,\mathfrak{f}^{m}_{\mathfrak{g}}<\mathrm{rank}\,\mathfrak{g}=\dim_{\mathbb{C}}\\!\left(\begin{smallmatrix}\textbf{unfrozen}\\\
\textbf{singularity}\end{smallmatrix}\right).$ (5.5)

To each pair $(\mathfrak{g},m)$ ($m\geq 2$) we associate the Kodaira fiber
type $\mathbf{K}_{\mathfrak{g}}$ dual to its extended Dynkin graph
$\widetilde{\Gamma}_{\mathfrak{g}}$, whose local monodromy
$\varrho_{\mathfrak{g}}$ can be read in tables 1, 2, and also the Kodaira
fiber $\widetilde{\mathbf{K}}_{\mathfrak{g}}$ obtained from it by the _base
change_ $z\mapsto z^{m}$ [52] i.e. the Kodaira fiber with local monodromy
$\varrho_{\mathfrak{g}}^{m}$ (cf. table 2 of [2]). The resulting Kodaira
types are listed in table 5.

pair | $(D_{k+4},2)$ | $(E_{6},2)$ | $(E_{6},3)$ | $(E_{7},2)$ | $(E_{7},3)$ | $(E_{7},4)$ | $(E_{8},2)$ | $(E_{8},3)$ | $(E_{8},4)$ | $(E_{8},5)$ | $(E_{8},6)$
---|---|---|---|---|---|---|---|---|---|---|---
$\mathbf{K}_{\mathfrak{g}}$ | $\mathbf{I}_{k}^{*}$ | $\mathbf{IV}^{*}$ | $\mathbf{IV}^{*}$ | $\mathbf{III}^{*}$ | $\mathbf{III}^{*}$ | $\mathbf{III}^{*}$ | $\mathbf{II}^{*}$ | $\mathbf{II}^{*}$ | $\mathbf{II}^{*}$ | $\mathbf{II}^{*}$ | $\mathbf{II}^{*}$
$\widetilde{\mathbf{K}}_{\mathfrak{g}}$ | $\mathbf{I}_{2k}$ | $\mathbf{IV}$ | $\mathbf{I}_{0}$ | $\mathbf{I}_{0}^{*}$ | $\mathbf{III}$ | $\mathbf{I}_{0}$ | $\mathbf{IV}^{*}$ | $\mathbf{I}_{0}^{*}$ | $\mathbf{IV}$ | $\mathbf{II}$ | $\mathbf{I}_{0}$
$\mathfrak{f}^{m}_{\mathfrak{g}}$ | $C_{k}$ | $A_{2}$ | $0$ | $B_{3}$ | $A_{1}$ | $0$ | $F_{4}$ | $G_{2}$ | $A_{1}$ | $0$ | $0$
Table 5:  In the first row we list the pairs $(\mathfrak{g},m)$ where
$\mathfrak{g}$ is a simply-laced Lie algebra and $m$ a Coxeter label of a non-
extension node of its Dynkin graph. In the second row we list the Kodaira
fiber type $\mathbf{K}_{\mathfrak{g}}$ associated to $\mathfrak{g}$. The third
row gives the Kodaira fibers $\widetilde{\mathbf{K}}_{\mathfrak{g}}$ obtained
from the ones in the second row by the base change $z\mapsto z^{m}$. The forth
row yields the gauge Lie algebra $\mathfrak{f}^{m}_{\mathfrak{g}}$ for the
(partially) frozen M-theory singularity associated to the pair
$(\mathfrak{g},m)$.

Comparing tables 5 and (4.42), and bearing in mind that the explicit
construction of the multiple fibers in §. 4.3 depends on a choice of a
primitive $m$-root of unity $\zeta_{m}$, we get:

######  Fact 3.

There is a one-to-one correspondence between Lagrangian multiple fibers in
rank $r\geq 2$ (with trivial HO invariant in the $\mathbf{I}_{2n}$ case) and
(partially) frozen singularities in M-theory. The correspondence is as in the
following table:

Lagrangian singular fibers | M-theory singularities
---|---
fiber multiplicity $m$ | order $m$ of the singularity
monodromy $\varrho_{\alpha}$ | monodromy $\varrho_{\mathfrak{g}}$
characteristic cycle type | Kodaira type $\widetilde{\mathbf{K}}_{\mathfrak{g}}$
choice of primitive root $\zeta_{m}$ | $n\in({\mathbb{Z}}/m{\mathbb{Z}})^{\times}\simeq\mathsf{Gal}(\mathbb{Q}(e^{2\pi i/m})/\mathbb{Q})$
(5.6)

######  Remark 11.

The correspondence (5.6) extends to _simple_ fibers (with trivial HO
invariant), $m=1$, of types different from $\mathbf{II}$, $\mathbf{III}$,
$\mathbf{IV}$. That these three types should be excluded looks physically
natural since it reflects the QFT _“fiber selection rule”_ stated in the Fact
of §. 3.3.1 in [27] which is the geometric counterpart to the _“no dangerous
irrelevant operator”_ property (a.k.a. _“safely irrelevant conjecture”_)
argued on physical grounds in [6, 28, 29, 30, 31]. This observation may be
seen as a geometric “proof” of the physical conjecture.

In the correspondence

$\left[\begin{smallmatrix}\textbf{\emph{simple} fibers
$\neq\mathbf{II},\,\mathbf{III},\,\mathbf{IV}$ in Special\phantom{m}}\\\
\textbf{Geometry (with trivial HO
invariant)}\end{smallmatrix}\right]\longleftrightarrow\left[\begin{smallmatrix}\textbf{\emph{unfrozen}
singularities}\\\ \textbf{in M-theory\phantom{mmmmn}}\end{smallmatrix}\right]$
(5.7)

the Lie algebra $\mathfrak{g}$ which is the gauge Lie algebra in M-theory
becomes the flavor Lie algebra in the ${\cal N}=2$ SCFT side. We shall see
below using string dualities that this rule extends to the _multiple_ fiber as
well, which then carry a flavor Lie algebra $\mathfrak{f}^{m}_{\mathfrak{g}}$
equal to the gauge algebra of the corresponding partially frozen M-theory
singularity. This also entails that the singularity is (partially) frozen on
both sides of the correspondence, since the allowed deformations are related,
respectively, to the flavor and gauge Lie algebras via eq.(5.5). Comparing
tables 3, 4 and 5 we confirm the expectation

$\dim_{\mathbb{C}}{\cal
M}\equiv\mathrm{rank}\,\mathfrak{f}^{m}_{\mathfrak{g}},$ (5.8)

that is,

######  Fact 4.

The deformation space ${\cal M}$ of a multiple Lagrangian fiber (cf. §. 4.3.2)
coincides with the deformation space of the M-theory singularity with the same
pair $(\mathfrak{g},m)$; in particular both have dimension
$\mathrm{rank}\,\mathfrak{f}^{m}_{\mathfrak{g}}$.

######  Remark 12.

A fiber with monodromy type $\mathbf{I}_{b}^{*}$ may be both double and have a
covering fiber with fiber cycle of type $\mathbf{I}_{2b}$ but characteristic
cycle of type $\mathbf{I}_{2bm}$ for any $m\in\mathbb{N}\cup\infty$. In other
words the “freezing” mechanisms described in §. 3 and in the present section
can operate _simultaneously_ : see Example 6.3 in [20]. This lead to a much
large zoo of partially frozen singular Lagrangian fibers all with the
monodromy type $\mathbf{I}^{*}_{b}$. This opens the possibility of new
phenomena which should be duly explored.

####  5.1.2 Duality with F-Theory

The identifications in Facts 3, 4 are no coincidence. They follow from the
duality between M- and F-theory. We consider M-theory on the geometry

${\cal K}\times S^{1}\times{\mathbb{R}}^{2}\times({\mathbb{R}}^{2,1}\times
S^{1}),$ (5.9)

where ${\cal K}\to\mathbb{P}^{1}$ is an elliptic K3 and the radius of the
second $S^{1}$ is parametrically large. ${\cal K}\to\mathbb{P}^{1}$ is a
Lagrangian fibration. We suppose that the fiber ${\cal K}_{p}$ over
$p\in\mathbb{P}^{1}$ contains a ${\mathbb{C}}^{2}/\Gamma_{\mathfrak{g}}$
singularity. We consider a small disk $\Delta\subset\mathbb{P}^{1}$ centered
at $p$, with complex coordinate $w$, and focus on the local Lagrangian
fibration over it, ${\cal K}|_{\Delta}\to\Delta$. The central fiber has
Kodaira type $\mathbf{K}_{\mathfrak{g}}$, and the local monodromy around the
origin in $\Delta$ of the elliptic fibration is $\varrho_{\mathfrak{g}}$. The
local geometry is then

${\cal K}|_{\Delta}\times
S^{1}\times{\mathbb{R}}^{2}\times({\mathbb{R}}^{2,1}\times S^{1}).$ (5.10)

We also assume that the singularity is (partially) frozen, i.e. we have a
fractional 3-form flux as in equation (5.4).

At a generic point $0\neq w\in\Delta$ we can reduce to type IIA along one
$S^{1}$ in the elliptic fiber ${\cal K}_{w}\simeq(S^{1})^{2}$, and then take
the $T$-dual of the other $S^{1}$ in the fiber. In absence of 3-form flux we
get the type IIB geometry [2, 14]

$\Delta\times S^{1}\times
S^{1}\times{\mathbb{R}}^{2}\times({\mathbb{R}}^{2,1}\times
S^{1})\simeq\Delta\times E\times{\mathbb{C}}\times({\mathbb{R}}^{2,1}\times
S^{1})$ (5.11)

where $E\simeq(S^{1})^{2}$ is a fixed elliptic curve with complex coordinate
$z$. Since the axio-dilaton $\tau$ is not constant, this situation is more
appropriately described by a F-theory geometry: its 12-dimensional
elliptically-fibered space is

${\cal K}|_{\Delta}\times E\times{\mathbb{C}}\times({\mathbb{R}}^{2,1}\times
S^{1}),$ (5.12)

and we have a 7-brane in the 10d base at $w=0$ which is wrapped on
$E\times{\mathbb{C}}\times({\mathbb{R}}^{2,1}\times S^{1})$.

Now we switch on a non-trivial fractional 3-form flux (5.4) in this geometry.
The net effect [2] is that now the monodromy action on the F-theory elliptic
fiber as $w\to e^{2\pi i}w$ is accompanied by a shift by
$2\pi\breve{n}R^{\prime}/m$ of the $T$-dualized circle (the first $S^{1}$ in
the lhs of (5.11) of radius $R^{\prime}$) where $\breve{n}$ is the inverse of
$n$ in $({\mathbb{Z}}/m{\mathbb{Z}})^{\times}$

$\breve{n}\,n=1\bmod m.$ (5.13)

In other words, the monodromy acts on the complex coordinate of elliptic curve
$E$ by a shift $z\mapsto z+p_{m}$ where $p_{m}\in E$ is a certain
primitive333333 I.e. $m$ is the smallest positive integer so that
$mp_{m}=0\in E$. $m$-torsion point.

We probe this F-theory configuration with a D3 brane wrapped on the last
factor space $({\mathbb{R}}^{2,1}\times S^{1})$; its image in the other factor
spaces is a point.

To study this geometry it is convenient to introduce a covering disk with
coordinate $t$

$f\colon\widetilde{\Delta}\to\Delta,\qquad t\mapsto w=t^{m}.$ (5.14)

The pulled-back elliptic fibration $\widetilde{{\cal
K}}|_{\Delta}\to\widetilde{\Delta}$ has now monodromy
$\varrho_{\mathfrak{g}}^{m}$ and Weierstrass equation

$y^{2}=P_{0}(x,t)\quad\text{with}\quad
P_{0}(\zeta_{m}^{2}x,\zeta_{m})=\zeta_{m}^{6}\,P_{0}(x,t).$ (5.15)

Before introducing the D3 probe, our set-up is a 16-susy F-theory
configuration over the 12d elliptic fibration

$(\widetilde{{\cal K}}|\times
E)/{\mathbb{Z}}_{m}\times{\mathbb{C}}\times({\mathbb{R}}^{2,1}\times
S^{1})\to(\widetilde{\Delta}\times
E)/{\mathbb{Z}}_{m}\times{\mathbb{C}}\times({\mathbb{R}}^{2,1}\times S^{1}),$
(5.16)

where ${\mathbb{Z}}_{m}$ acts as [2]

${\mathbb{Z}}_{m}\ni k\colon\widetilde{{\cal K}}|\times
E\to\widetilde{\Delta}\times E,\qquad
k\colon(x,y,t,z)\mapsto(\zeta_{m}^{2k}x,\zeta_{m}^{3k}y,\zeta_{m}^{k}t,z+kp_{m}).$
(5.17)

with $p_{m}\in E$ a $m$-torsion point: the group
$({\mathbb{Z}}/m{\mathbb{Z}})^{\times}$ (i.e. the $n$ in eq.(5.4)) acts by
$p_{m}\mapsto\breve{n}\,p_{m}$. This statement is easily checked by taking the
$T$-dual of both circles in the elliptic fiber in eq.(5.10) and considering
the extra $S^{1}$ factor in that equation as the M-theory circle, see §. 2 of
[2].

The 12-dimensional total space of the F-theory geometry is

$\boldsymbol{Y}=(\widetilde{{\cal K}}|\times
E)/{\mathbb{Z}}_{m}\times{\mathbb{C}}\times({\mathbb{R}}^{2,1}\times
S^{1})\equiv\boldsymbol{X}\times({\mathbb{R}}^{2,1}\times S^{1}).$ (5.18)

We note that $\boldsymbol{X}$ is a complex manifold of dimension 4 which
carries a nowhere vanishing holomorphic 2-form

$\Omega=\omega+dz\wedge ds$ (5.19)

where $\omega$ is the symplectic form on the local 2-CY $\widetilde{{\cal
K}}|$ and $z$ (resp. $s$) is the coordinate on $E$ (resp. ${\mathbb{C}}$).
We conclude that the fibration
$\boldsymbol{\pi}\colon\boldsymbol{X}\to\Delta\times{\mathbb{C}}$ is a rank-2
(germ) special geometry in its own right. By comparison with the Hwang-Oguiso
construction in §. 4.3.1 we see that its central fiber is a multiple fiber of
multiplicity $m$ with characteristic cycle of Kodaira type
$\widetilde{\mathbf{K}}_{\mathfrak{g}}$.

####  5.1.3 Effective theory on the D3 probe: unfrozen singularity

On the world-volume of the D3 probe there is a 4d 8-susy QFT whose flavor
group contains the gauge group of the SYM theory living on the 7-brane. This
4d QFT is compactified down to 3d on a _large_ circle $S^{1}$. The circle
compactification allows us to describe the IR physics of the probe in both the
languages of 4d ${\cal N}=2$ and 3d ${\cal N}=4$ supersymmetry.

We consider first the case of an unfrozen singularity ($m=1$). The light 3-3
string states yield the d.o.f. of a 4d ${\cal N}=4$ vector-multiplet, that
is, a 4d ${\cal N}=2$ vector-multiplet plus a neutral ${\cal N}=2$
hypermultiplet: from the viewpoint of 3d ${\cal N}=4$ susy they correspond
respectively to a twisted-hypermultiplet and a hypermultiplet which combine
into a supermultiplet of 3d ${\cal N}=8$ with R-symmetry algebra
$\mathfrak{spin}(8)$ which from the 3d ${\cal N}=4$ viewpoint yields a
$\mathsf{Spin}(4)$ R-symmetry and a $Sp(1)\times Sp(1)\simeq\mathsf{Spin}(4)$
flavor group. The v.e.v. of the scalars in the two 4d supermultiplets
describe the position of the probe: the vector-multiplet in the directions
orthogonal to the 7-brane (whose support is $D_{\alpha}$) and the
hypermultiplet in the parallel ones. It is well known (see [55, 56]) that the
3d twisted-hypermultiplet scalars describe the position of the D3 brane in the
full elliptic hyperKähler manifold ${\cal K}$. The 3-7 and 7-3 strings produce
interacting d.o.f. coupled to the vector multiplet living on the probe. This
interacting sector has a flavor Lie algebra $\mathfrak{g}$ equal to the gauge
algebra of the theory living on the 7-brane. Thus the IR effective theory on
the D3 probe is an interacting rank-1 4d ${\cal N}=2$ with flavor algebra
$\mathfrak{g}$ plus an IR-decoupled free hypermultiplet described by a 8-susy
$\sigma$-model with target the factor space in (5.16) (for $m=1$) which is
parallel to the 7-brane and normal to the probe, i.e. the flat hyperKähler
manifold ${\cal H}\equiv E\times{\mathbb{C}}$ equipped with its natural
holomorphic symplectic form $ds\wedge dz$. This sector has its own flavor
algebra $\mathfrak{sp}(1)$.

In the 3d description the IR theory is a ${\cal N}=4$ $\sigma$-model with
target space the product two hyperKähler manifolds each of quaternionic
dimension 1

${\cal K}\times{\cal H}$ (5.20)

The factor ${\cal H}$ is the target space of the hypermultiplet scalars (the
same manifold as in the 4d description), while the elliptic 2-CY ${\cal K}$ is
the target space for the twisted-hypermultiplets. In the standard conventions
of 3d ${\cal N}=4$ susy [57] the orientations of the two factor spaces are
chosen so that the Riemann tensor of ${\cal H}$ is selfdual while the Riemann
tensor of ${\cal K}$ is anti-selfdual. Geometrically it may be more natural to
redefine the orientations so that the product space is a single hyperKähler of
double dimension with the holomorphic symplectic form (5.19).

Before turning to the $m>1$ case we digress on fine points about
supersymmetry.

####  5.1.4 Subtleties with SUSY factorization

It is usually stated, and used to deduce countless susy non-renormalization
theorems,343434 See e.g. [17] for basic examples. that when a susy model
contains fields transforming in different supermultiplets, the scalars take
value in a Riemannian product

$M_{1}\times M_{2}\times\cdots\times M_{\ell},$ (5.21)

where $M_{i}$ is the target space of the scalars belonging to supermultiplets
of the $i$-th type. This statement applies to chiral versus twisted-chiral in
2d $(2,2)$ susy [17], or to hypers vs. twisted-hypers in 3d ${\cal N}=4$ susy
[33], or vector-multiplets vs. hypermultiplets in 4d ${\cal N}=2$ susy [33],
etc. However the traditional argument technically proves only a _weaker_
statement [57]:

######  Pedantic Version.

If the scalars’ target space $M$ of a susy field theory is a _complete,_
_simply-connected,_ Riemannian manifold, then $M$ is a metric product as in
eq.(5.21).

There are two clauses in the statement. The target space of a susy effective
theory with $\leq 8$ supercharges is usually non-complete: typically there are
singularities at finite-distance where some additional d.o.f. get light. A
more sophisticated analysis shows that _when we have 8 supercharges_ we can
get rid of the completeness hypothesis:353535 In particular the validity of
the non-renormalization theorems is not spoiled by the subtleties discussed in
this subsection. the statement remains true (with qualifications) in presence
of the mild singularities expected on physical grounds363636 At least when the
IR effective theory _do admit_ a UV completion.. The assumption that the
scalar space is _simply-connected_ is instead essential for the factorization
property to hold, as the (counter)examples below will show.

We can always reduce ourselves to the simply-connected case by replacing the
scalar manifold $M$ by a smooth simply-connected cover373737 A smooth simply-
connected cover is a finite cover of the universal cover which smooths out the
orbifold singularities of the topological universal cover. $\widetilde{M}$ so
that $M\equiv\widetilde{M}/\Gamma$. Thus all susy model is obtained from one
where the scalars take value in the product manifold $\widetilde{M}$ by
gauging a discrete symmetry $\Gamma$ which acts on both factors
simultaneously. In other words: the factorization holds at the level of the
_local physics,_ but it may be spoiled by global topological effects which
couple, say, the hypermultiplets and the twisted-hypermultiplets of a 3d
${\cal N}=4$ $\sigma$-model.

The gauging of the discrete group $\Gamma$ preserves all supersymmetries iff
it acts on each factor space by automorphisms of the relevant geometric
structures. For instance, when the covering space is the product of two
hyperKähler manifolds ${\cal K}\times{\cal H}\equiv\widetilde{M}$ as in
eq.(5.20), $\Gamma$ should act by simultaneous symplectic holomorphic
isometries on both ${\cal K}$ and ${\cal H}$. Isometries of this kind exist,
so we can indeed have 3d ${\cal N}=4$ $\sigma$-models with globally non-
factorized scalar target spaces. When the 3d ${\cal N}=4$ $\sigma$-model arise
as the low-energy limit of a 4d ${\cal N}=2$ QFT compactified on $S^{1}$ one
says that ${\cal K}$ is the “3d Coulomb branch” and ${\cal H}$ the “Higgs
branch”. Both branches are well-defined only up to a cover. There are strong
reasons to believe that in a UV-complete theory the Coulomb (resp. Higgs)
branch is well-defined up to _finite_ covers [62].

Now suppose that the factor ${\cal H}$ is smooth and flat. Before the discrete
gauging, the hypermultiplets form a free sector which decouples from the
rest383838 In the IR, hence in the full theory when it is a SCFT.. After the
gauging of $\Gamma$, the hypermultiplets still look like a free decoupled
sector at the level of the _local physics_ , but _globally_ they are still
coupled to the interacting sector by a topological feature. This is another
version of the mechanism discussed in §.3.2.2:

######  Physical Mechanism.

The local physics contains a free decoupled subsector which however in the
full theory is still coupled by subtle global topological effects.

The difference between the present case and the one in §.3.2.2 is that there
the would-be free sector was a set of $(r-1)$ vector-multiplets whereas now
are hypermultiplets. The physical principle in action is nevertheless the
same, and mathematically they both arises from the fact that the Kodaira type
of the Hwang-Oguiso characteristic cycle $\boldsymbol{\Psi}$ does not match
the type of the local monodromy.

####  5.1.5 Effective theory on the D3 probe: (partially) frozen singularity

We return to our D3 brane probing the F-theory geometry (5.18) which is dual
to a frozen M-theory singularity. We consider its 3d ${\cal N}=4$ IR effective
theory. From the geometry it is obvious that this effective theory is obtained
by gauging a $\pi_{1}(\boldsymbol{X})\equiv{\mathbb{Z}}_{m}$ symmetry of a
system containing a twisted-hypermultiplet taking value in ${\cal K}$ and a
hypermultiplet valued in ${\cal H}$. Since ${\cal H}$ is locally flat, the
hypermultiplet is _locally_ free yet globally coupled to the twisted
hypermultiplet by the ${\mathbb{Z}}_{m}$ discrete gauging. A peculiarity of
this 3d ${\cal N}=4$ model is that also the hyperKähler factor ${\cal H}$ is
elliptically fibered with Lagrangian fibers and so is a (free) special
geometry in its own right.

Since both factor spaces of the universal cover $\widetilde{\boldsymbol{X}}$
are elliptic fibered, we may consider the projection of $\boldsymbol{X}$ on a
2-dimensional complex base
$\widetilde{\Delta}/{\mathbb{Z}}_{m}\times{\mathbb{C}}$ of coordinates
$w=t^{m}\\!,$ $s$

$\boldsymbol{\varpi}\colon\boldsymbol{X}\equiv(\widetilde{{\cal K}}|\times
E)/{\mathbb{Z}}_{m}\times{\mathbb{C}}\longrightarrow\widetilde{\Delta}/{\mathbb{Z}}_{m}\times{\mathbb{C}}\simeq\Delta\times{\mathbb{C}}\subset{\mathbb{C}}^{2},$
(5.22)

which is a Lagrangian fibration whose generic fibers are Abelian varieties of
dimension 2, that is, $\boldsymbol{\varpi}$ is a rank-$2$ special geometry. In
fact $\boldsymbol{\varpi}$ is a “no-section” special geometry (in the sense of
§.2) with a central fiber of multiplicity $m$ equal to the order of the
original M-theory (partially) frozen singularity. This conclusion is not in
contradiction with the physical fact that the Seiberg-Witten special geometry
along the Coulomb branch must be of the “section” kind (cf. §. 2) since the
QFT on the probe _has rank 1_ and the two-dimensional base of the Lagrangian
fibration (5.22) has _not_ the physical interpretation of being its Coulomb
branch. Moreover the statement in §. 2 that the Seiberg-Witten geometry has a
zero section holds in the same sense in which the notion of “Coulomb branch”
is defined, that is, only modulo finite covers.

Comparing with the Hwang-Oguiso construction in §. 4.3, we see that the rank-2
special geometry (5.22) along the discriminant divisor $t=0$ has precisely a
singular fiber of multiplicity $m$ with characteristic cycle
$\boldsymbol{\Psi}$ of type $\widetilde{\mathbf{K}}_{\mathfrak{g}}$. This
explains the correspondence in Fact 3: duality with $F$-theory in presence of
a D3 probe transforms a (partially) frozen M-theory singularity into a
multiple fiber of the “no-section” rank-2 special geometry which describes the
topologically-coupled 3d twisted-hyper/hyper system living on the D3 probe
wrapped on $S^{1}$.