#  Cobordism, Singularities and the Ricci Flow Conjecture

David Martín Velázquez  Max–Planck–Institut für Physik,
Werner–Heisenberg–Institut,
Föhringer Ring 6, 80805 München, Germany  Davide De Biasio
Max–Planck–Institut für Physik, Werner–Heisenberg–Institut,
Föhringer Ring 6, 80805 München, Germany  Dieter Lüst  Max–Planck–Institut für
Physik, Werner–Heisenberg–Institut,
Föhringer Ring 6, 80805 München, Germany  Arnold Sommerfeld Center for
Theoretical Physics,
Ludwig Maximilians Universität München,
Theresienstrasse 37, 80333 München, Germany

###### Abstract

In the following work, an attempt to conciliate the Ricci flow conjecture and
the Cobordism conjecture, stated as refinements of the Swampland distance
conjecture and of the No global symmetries conjecture respectively, is
presented. This is done by starting from a suitable manifold with trivial
cobordism class, applying surgery techniques to Ricci flow singularities and
trivialising the cobordism class of one of the resulting connected components
via the introduction of appropriate defects. The specific example of
$\Omega^{SO}_{4}$ is studied in detail. A connection between the process of
blowing up a point of a manifold and that of taking the connected sum of such
with $\mathbb{CP}^{n}$ is explored. Hence, the problem of studying the Ricci
flow of a $K3$ whose cobordism class is trivialised by the addition of $16$
copies of $\mathbb{CP}^{2}$ is tackled by applying both the techniques
developed in the previous sections and the classification of singularities in
terms of ADE groups.

###### Contents

  1. 1 Introduction
  2. 2 Brief introduction to bordisms
    1. 2.1 Unoriented bordism
    2. 2.2 Adding more structure to the cobordism group
  3. 3 Brief introduction to the Ricci flow equations
    1. 3.1 The Ricci flow for Einstein manifolds
    2. 3.2 Introducing Ricci solitons
    3. 3.3 The Ricci flow and the swampland
  4. 4 The cobordism class and Ricci flow for $\Omega_{4}^{SO}$
    1. 4.1 Ricci flow and Einstein manifolds in $\Omega_{4}^{SO}$
    2. 4.2 Ricci flow and Ricci solitons
    3. 4.3 Pinching of necks
  5. 5 The main case study: $\Omega^{SO}_{4}$
    1. 5.1 The relevant invariants
    2. 5.2 The case for $\mathbb{CP}^{2}$
    3. 5.3 The case for $K3$
  6. 6 Blowing up points and connected sums of copies of $\mathbb{CP}^{2}$
    1. 6.1 The blow up of a point
    2. 6.2 Blowing up and connected sums
    3. 6.3 Blowups and magnetic monopoles
    4. 6.4 The Burns metric
    5. 6.5 Second form of the Burn’s metric
    6. 6.6 Getting a feel for the flow
    7. 6.7 Addendum: Why the behavior at the singularities may not matter
  7. 7 Distances along the flow
  8. 8 The ADE singularities
    1. 8.1 A lightning fast overview of complex geometry
    2. 8.2 Blow-ups from algebraic geometry
    3. 8.3 The ADE classification of $K3$ singularities
  9. 9 Recovering the ADE singularities from the blow-ups
    1. 9.1 Setting up the stage
    2. 9.2 Introducing the blow-ups
    3. 9.3 Lattice polarizations
    4. 9.4 Restrictions on the allowed ADE singularities from lattice polarizations and the cobordism conjecture.
  10. 10 Other scenarios of interest
  11. 11 Conclusions and Outlook
  12. 12 Appendix: The (co)homology of $\mathbb{CP}^{2}\\#\mathbb{CP}^{2}$

##  1 Introduction

The Swampland Program [Vaf05, Pal19, Bee+21, BCV17] originated as an endeavor
to explore the idea that standard effective field theory techniques might
break down when trying to incorporate gravitational physics into a low energy
quantum field theory. This is expected to hold even at scales which lie way
below Planck’s mass

$M_{P}\equiv\sqrt{\frac{\hbar c}{G}}\ ,$ (1)

where quantum gravity degrees of freedom are supposed to become relevant. A
conspicuous body of evidence for the above-mentioned deviation from the
typical effective field theory intuition, which suggests that quantum gravity
phenomenology could safely be ignored in the low energy regime, has emerged
from inspecting spatio-temporal and thermodynamical properties of black holes,
general quantum gravitational arguments and explicit superstring theory
constructions. Indeed, the belief that quantum gravity effects can strongly
and further constraint the set of apparently consistent quantum field theories
coupled to a dynamical space-time metric is now widely accepted and gets often
expressed in the form of the so-called Swampland conjectures. In order for the
discussion to acquire clarity, it is appropriate to introduce the notion of
the Landscape, being the subset of low energy quantum field theories coupled
to gravity that satisfy the restrictions imposed by Swampland conjectures and
hence admit an ultraviolet completion to quantum gravity. By doing so, one
straightforwardly defines the Swampland as the subset of apparently consistent
low energy quantum field theories coupled to gravity that do not belong to the
Landscape. The aim of the Swampland program is thus to find precise and
unambiguous criteria that allow to classify theories into those belonging to
the Landscape and those that fall into the Swampland. The natural prosecution
of this analysis is to model such a set of theories, partitioned into the
Landscape and the Swampland, as some sort of geometrically characterised
moduli space, locally charted by the vacuum expectation values of some fields
and provided with some notion of distance. The Swampland distance conjecture
was outlined in [OV07] as the claim that large displacements in the moduli
space should force an infinite tower of light states to descend into the
spectrum of the low energy effective theory, lowering the energy cut-off
$\Lambda_{\text{eff}}$ below which it functionally models relevant phenomena.
This idea was pushed into conjecturing that infinite distance limits in the
moduli space should involve significant alterations of the features of a low
energy theory, such as the decompactification of some internal dimensions, the
emergence of asymptotically tensionless and weakly coupled string degrees of
freedom [LLW22] or the restoration of a global symmetry. The latter, in
particular, connects to the long-standing idea that gravity should not allow
for the existence of global symmetries [BD88, BS11, HO18, HO21], which is
strongly backed up by semi-classical black hole physics, holographic arguments
and by the fact that a global worldsheet symmetry in superstring theory always
gets gauged in the target space-time. Both the Swampland distance conjecture
and the No global symmetries conjuecture have been discussed and refined in
many different instances. The Swampland distance conjecture was first
extendend to the case of AdS cosmological constant in [LPV19], showing how the
infinite distance limit in which $\Lambda\to 0$ is accompanied by an infinite
tower of light states. Extending the previously-introduced notion of a moduli
space to regard the space-time metric components themselves as moduli, the
most immediate way of displacing them is by considering paths induced by
geometric flow equations [Pol75, Fri80, Fri85]. Particular attention was
dedicated to Ricci flow [Ham82]

$\frac{\text{d}g_{\mu\nu}}{\text{d}\lambda}=-2R_{\mu\nu}\ ,$ (2)

which precisely reduces to the behaviour discussed in [LPV19] in the case of
AdS, and generalisations thereof. This way, exploiting mathematical methods
which are rooted in string theory via the $\sigma$-model graviton
renormalisation group flow, the Swampland distance conjecture for the metric
was rephrased and specified as the Ricci flow conjecture [KLL20, BL20, LLM21,
DL20]. The reader is strongly suggested to refer to [Per06, CKS04, CZ06, MT07,
KL08, Top06] for comprehensive discussions of the techniques that are usually
employed when studying Ricci flow of differentiable manifolds. On the other
hand, the recent Cobordism conjecture [MV19, OT20, MV21, DH21, Bur+21, ACC22,
Ang+22, Blu+22, Blu+22a], stating that viable quantum gravity backgrounds
should have trivial cobordism class and thus be cobordant to one another, can
arguably be regarded as a refinement of the No global symmetries conjecture.
It must be noted that the Ricci flow conjecture typically deals with locally
defined geometric flow equations and properties, while the Cobordism
conjecture comes as a genuine global statement. In the following work, an
attempt to conciliate the two perspectives is presented. After having
introduced some appropriate mathematical techniques, it is argued that the key
mechanism via which Ricci flow can affect the global and topological
properties of a manifold is for it to encounter a singularity, on which
surgery procedures can be applied. These are expected to preserve cobordism
classes, as they are typically enforced by substituting shrinking throats with
smooth caps and connected sums are cobordant to disjoint unions. From a
physics point of view, it is nevertheless usually pertinent to neglect one of
the two connected components produced by a surgery. It is thus shown that
defects must be introduced in order to trivialise the cobordism class of the
resulting connected manifold and the specific example of $\Omega^{SO}_{4}$ is
studied in detail. A connection between the process of blowing up a point of a
manifold $\mathcal{M}$ and that of taking the connected sum of such manifold
with $\mathbb{CP}^{n}$ is thereafter explored. Hence, the problem of studying
the Ricci flow of a $K3$ whose cobordism class is trivialised by the addition
of $16$ copies of $\mathbb{CP}^{2}$ is tackled both from a geometric flow
perspective and by exploiting the classification of singularities in terms of
ADE groups.

##  2 Brief introduction to bordisms

At its core, the bordism relation is a simple one. We will begin by stating
its simplest definition. Then, we will progressively add more structure,
defining some of the concepts when necessary. Some other introductions are
given in [MV19, ACC22], as well as [Joh74, Chapters 4, 17]. For a more
technical and in-depth treatment of bordisms the reader is redirected to the
classical reference, [Fre12].

###  2.1 Unoriented bordism

We now introduce the most basic bordism relation possible, where we do not
take into account any kind of structure or special property of the bordant
manifolds.

######  Definition 2.1.

Let $M$ and $N$ be two closed (i.e. compact and without boundary)
$n$-dimensional manifolds. An (unoriented) cobordism between them is an
$(n+1)$-dimensional manifold whose boundary is the disjoint union of $M$ and
$N$, which we write as $\partial W=M\sqcup N$.

A more precise statement of the above discussion would involve the
introduction of collar neighbourhoods on the boundary of the
$(n+1)$-dimensional manifold, as well as a pair of maps defined on them such
that, restricted over $\partial W$, they define embeddings of $M$ and $N$ into
$W$. However, such precision would only be needed in formal proofs involving
explicit gluings of the boundaries.

Because $M\sqcup N=N\sqcup M$, it is straightforward to check that cobordisms
define an equivalence relation between manifolds. We denote this relation by
$M\sim N$ without further reference to the type of cobordism which defines it,
and we denote the equivalence class of a manifold $M$ by $[M]$. One may
further endow the set of equivalence classes of closed $n$-manifolds with the
structure of an abelian group by defining the sum of two classes as

$[M]+[N]=[M\sqcup N].$ (3)

We denote this group by $\Omega_{n}$. Here, the zero-element with respect to
the sum is taken to be the empty manifold $\emptyset$. In other words, we say
that, given an $n$-dimensional maniofold, the fact that $[M]=0$ means that
there exists some $(n+1)$-dimensional manifold $W$ such that $\partial W=M$.

Furthermore, we may consider a ”multiplication”, provided by the Cartesian
product of manifolds $[M][N]\coloneqq[M\times N]$. This allows us to build up
$\Omega_{*}\coloneqq\bigoplus_{n=0}^{\infty}\Omega_{n}$, and endow it with the
structure of a graded ring. We call it the unoriented cobordism ring.

$W$
---
$\mathbb{S}^{1}\sqcup\mathbb{S}^{1}$
---
$\mathbb{S}^{1}$
---
Figure 1: Simple example of an unoriented cobordism relation between two
copies of the circle ($\mathbb{S}^{1}\sqcup\mathbb{S}^{1}$) and a single
$\mathbb{S}^{1}$ by a surface of genus 3 with boundary.

A property that any unoriented cobordism ring is that all of its elements have
characteristic 2, and thus it is an algebra over $\mathbb{Z}_{2}$. In other
words, the grading of the ring is given by $\mathbb{Z}_{2}$, and all of its
(nontrivial) cobordism classes are isomorphic to a $\mathbb{Z}_{2}$ factor.
This is easy to see by noting that, if we neglect orientations,
$[M]+[M]=[M\sqcup M]$, and it is simple to construct a manifold whose boundary
is $M\sqcup M$: it is $M\times[0,1]$. Hence, in the language of the group
operation, we have that $2[M]=0$ for any manifold $M$.

###  2.2 Adding more structure to the cobordism group

While the definition we have made in the previous section is enough to
understand the basic idea underlying the bordism relation, it falls flat when
dealing with manifolds which allow for more physically interesting scenarios.
For example, just to consider manifolds which have a definite orientation (and
are, hence, orientable), we need to modify our original definition. The
precise and rigorous way in which this is done is covered in [Fre12], where
tangential structures are introduced and treated in detail. A more condensed
but readable account of this is contained in the introduction of [ACC22].

For our present purposes, it is enough to know that, however the additional
structure is defined on two $n$-dimensional bordant manifolds, it must extend
in a coherent way to the $(n+1)$-dimensional manifold which connects them.
This can be done by means of classifying spaces and bordisms with maps. A
classifying space for a principal $G$-bundle is defined as follows. Given any
Lie group $G$, we can define a space $BG$, together with a fibration given by
its universal cover $EG\to BG$, with the defining property that any $G$-bundle
over $X$ can be obtained as a pullback from $EG$. That is, given any
$G$-bundle $P\to X$, there exists a map $f:X\to BG$ such that $P\simeq
f^{*}EG$. This construction classifies $G$-bundles in the following sense. In
general, given some $G$-bundle $E\to Y$, and two maps $f,g:X\to Y$, then the
two pullback budles $f^{*}E$ and $g^{*}E$ over X are isomorphic if and only if
$f$ and $g$ are homotopic (they belong to the same homotopy class). Therefore,
at the level of the classifying space of a group, we have that all principal
$G$-bundles over some manifold $X$ are classified by the homotopy classes of
maps $X\to BG$. These classes of maps are often denoted by $[X,BG]$.

There is an explicit construction of a classifying space for the structure
groups of frame bundles, given by infinite Grassmannian manifolds. A
Grassmannian manifold $G_{n}(\mathbb{R}^{n+k})$ is a manifold whose points are
$n$-dimensional planes in $\mathbb{R}^{n+k}$. Heuristically, one defines
$BO(n)$ as the direct limit on $k$ of $G_{n}(\mathbb{R}^{n+k})$. In other
words, one considers $k$ planes on higher and higher dimensional Euclidean
spaces. WE highlight this because $BO(n)$ is the relevant classifying space
for the case at hand, as all the structure groups we will consider are at most
reductions of orthogonal ($O(n)$) bundles, which can be pulled back from the
canonical bundle over an infinite Grassmannian.

While this notion is interesting in principle, we would like to have a
practical way in which we could actually classify $G$-bundles over some space.
Namely, we are looking for properties which allow us to tell bundles in
different classes apart. This is where characteristic classes come into play.
Because of their natural behavior under pullblacks, and the fact that they are
invariant under isomorphisms, they are precisely the sort of tools to
differentiate vector (and, in particular, principal) bundles over manifolds.
For example, in the case of bordisms between orientable manifolds, the
relevant characteristic classes are the Pontrjagin and Stiefel-Whitney
classes, which give rise, respectively, to the Pontrjagin and Stiefel-Whitney
numbers.

With classifying spaces, one can define certain structures on manifolds. For
example, given some $n$-dimensional manifold $X$, a map $f:X\to BSO(n)$
induces an $SO$-bundle over $X$, i.e. an orientation. To make this fit with
the bordism relation, we produce the following definition

######  Definition 2.2.

Let $X$ be a topological space. We denote by $(M,f)$ a pair consisting of a
manifold $M$ and a continuous map $f:M\to X$. Now, we say that two pairs
$(M,f)$, $(N,g)$ are bordant if

  * •

There is a cobordism $W$ between $M$ and $N$ in the usual sense, and

  * •

There is a continuous map $F:W\to X$ such that

$F\mid_{M}=f,\hskip 28.45274ptF\mid_{N}=g,$ (4)

where $M$ and $N$ denote the inclusions of these manifolds into $\partial W$.

Let us consider an $SO$-structure as an example again. In the above
definition, this implies taking111Note that we are now taking $BSO$ instead of
$BSO(n)$. This is because we are juggling between two different
dimensionalities. The classifying space $BSO$ is defined as a direct limit on
$n$ of $BSO(n)$ spaces. $X=BSO$. Then, the $SO$-structure on $W$, which is
induced by $F:W\to BSO$ is such that it induces the appropriate orientation on
the bordant manifolds, $M$ and $N$.

Once knowing how extra structure can be treated under the bordism relation in
general, we now come back to the particular case of orientable manifolds,
namely $SO$-structures. The most important result regarding this is that there
are certain topological numbers which fully characterize the oriented bordism
classes.

######  Theorem 2.1.

Two manifolds are (oriented or unoriented) cobordant if and only if their
Stiefel-Whitney and Pontryagin numbers are the same. In particular, a manifold
$M$ is cobordant to $\emptyset$ if and only if all its Stiefel-Whitney and
Pontryagin numbers vanish.

##  3 Brief introduction to the Ricci flow equations

The Ricci flow equations, first introduced by Richard Hamilton in [Ham82] are
a set of PDEs which modify the geometry of a given (differentiable) manifold.
If $(\mathcal{M},g(\lambda))$ is a family of Riemannian manifolds,
parametrized by $\lambda$, we say that they are a solution to the Ricci flow
equations if they satisfy

$\frac{\text{d}g_{\mu\nu}}{\text{d}\lambda}=-2R_{\mu\nu},$ (5)

where $R_{\mu\nu}$ denotes the components of the Ricci tensor with respect to
$g$. Originally, the flow equations were defined as a modification of some
gradient flow equations, so as to make the solutions well-behaved. Indeed, the
Ricci flow can be regarded as a sort of ”heat equation” for manifolds (though
the analogy is not totally rigorous). This is so because it tends to smooth
out certain types of manifolds. In particular, its first use was to show that
manifolds which admitted metrics of positive scalar curvature, also admitted
metrics of constant positive scalar curvature. In recent years, it was
famously used by Perelman to prove Thurston’s geometrization conjecture, which
contained the Poincaré conjecture as a corollary. A great introduction to this
tool, which takes these recent developments, and covers the techniques
developed by Perelman is [Top06]. In this small section, we will not introduce
such technicalities, but rather explore two simple scenarios for the Ricci
flow, and then outline its importance in the context of the swampland program.

###  3.1 The Ricci flow for Einstein manifolds

An Einstein space is a Riemannian manifold $(\mathcal{M},g)$ for which the
Ricci tensor is proportional to the metric, namely $R_{\mu\nu}=\Lambda
g_{\mu\nu}$, with $\Lambda\in\mathbb{R}$. Some examples of this are the well-
known maximally symmetric spaces: de Sitter ($\Lambda>0$), anti de Sitter
($\Lambda<0$), and Euclidean space $(\Lambda=0)$. Clearly, the Einstein
condition greatly simplifies the Ricci flow equations, which are reduced to

$\partial_{s}g_{\mu\nu}(s)=-2\Lambda g_{\mu\nu}$ (6)

The solution to this equation is a rescaling of the metric

$g(s)_{\mu\nu}=(1-2\Lambda s)g(0)_{\mu\nu}$ (7)

This can be parametrized as a Weyl rescaling of the metric

$g(s)_{\mu\nu}=e^{-2\omega(s)}g(0)_{\mu\nu}.$ (8)

Alternatively, given that the solutions to the Ricci flow are given by
rescaling, one is justified in condensing the flow equations by a single ODE
for the cosmological constant. One can check222Though the calculations are
rather cumbersome that under the Ricci flow, the scalar curvature evolves as

$\partial_{s}R=\nabla^{2}R+2R_{\mu\nu}R^{\mu\nu}.$ (9)

For an Einstein space, this reduces to the much simpler equation

$\Lambda^{\prime}(s)=2\Lambda^{2}(s)$ (10)

Whose solution is

$\Lambda(s)=\frac{\Lambda_{0}}{1-2\Lambda_{0}(s-s_{0})},$ (11)

where $\Lambda_{0}=\Lambda(s_{0})$. Clearly, the behavior of the solution
depends on the sign. In particular, we have that for a dS space, the solution
shrinks to a point (the curvature approaches $\infty$) at finite time. On the
other hand, an AdS space flows to flat space in the limit $s\to\infty$. Both
of the perspectives we have seen can of course be related by means of a
rescaling.

###  3.2 Introducing Ricci solitons

In the context of the Ricci flow, there is a further generalization of
Einstein manifolds, which serves as motivation for the definition of new
flows, such as the ones used by Perelman on his landmark works. So-called
Ricci solitons are manifolds that change by a rescaling and a diffeomorphism
under the flow. Let $(\mathcal{M},g(s))$ be a Ricci flow, and define a one-
parameter family of diffeomorphisms $\varphi_{s}:\mathcal{M}\to\mathcal{M}$,
$\forall s\in[0,T)$ such that $\varphi_{0}=\mathds{1}$, as well as a scaling
factor $\sigma(s)$ such that $\sigma(0)=1$. With these definitions, the
condition for a manifold to be a Ricci soliton is phrased as

$g(s)=\sigma(s)\varphi^{*}_{s}(g(0)),$ (12)

where $\varphi^{*}_{s}$ of course denotes the pullback given by the
diffeomorphism. After differentiating the above expression, evaluating it at
$t=0$, and using the explicit form of the Lie derivative defined by
$V=\varphi^{\prime}(s)$, then we have that the PDE defining the flow of a
Ricci soliton is given by

$\frac{\partial g_{\mu\nu}}{\partial
s}=\sigma^{\prime}(0)g_{\mu\nu}(0)+\nabla_{\mu}V_{\nu}+\nabla_{\nu}V_{\mu}.$
(13)

Renaming $\sigma^{\prime}(0)=-2\Lambda$, we obtain that the manifolds which
display this sort of evolution under the Ricci flow must be such that

$-2R_{\mu\nu}=-2\Lambda g_{\mu\nu}+\nabla_{\mu}V_{\nu}+\nabla_{\nu}V_{\mu}.$
(14)

Indeed we can see that any Einstein manifold is a Ricci soliton, with
$V_{\mu}=0$.

Examples of such manifolds are not so easy to come by as with Einstein
manifolds. One such example of great physical signifiacnce is Witten’s cigar,
a two-dimensional geometry given by

$g=\frac{1}{1+x^{2}+y^{2}}\left(dx^{2}+dy^{2}\right)$ (15)

For this metric, the Ricci tensor turns out to be

$R_{\mu\nu}=\frac{2}{1+x^{2}+y^{2}}g_{\mu\nu}.$ (16)

This could in principle look like an Einstein manifold. However, if we define
the radial vector field $Y=-2(x\partial_{x}+y\partial_{y})$, we have that

$\mathcal{L}_{Y}g=\frac{-4}{1+x^{2}+y^{2}}g$ (17)

Thus, the Ricci flow equations are sourced by a pure diffeomorphism, namely,
$\Lambda=0$.

###  3.3 The Ricci flow and the swampland

Recent publications, as [KLL20, BL20, LLM21, DL20], have recast the swampland
distance conjecture in terms of the Ricci flow. To this end, a distance along
the moduli space of metrics was defined, and then used to measure the distance
in moduli space of metrics along the Ricci flow. It was then argued that an
infinite distance along the Ricci flow shuould be accompanied by an infinite
tower of states, whose mass scale is determined by said distance.

Of special importance to this refinement of the distance conjecture are those
backgrounds which are fixed points of the flow, namely those which satisfy

$\frac{\partial g_{\mu\nu}}{\partial s}=0.$ (18)

Clearly, these are given by Ricci-flat manifolds. From the discussion above,
one such example of a manifold flowing to flat space along the flow is $AdS$
space flowing to Euclidean space. From the perspective of the distance along
the space of metrics, this is an incarnation of the $AdS$ distance conjecture.
It states that, as $AdS$ space flows to flat space (namely, $\Lambda\to 0$),
the associated tower of states with mass scale $m$ behaves in Planck units as
$m\sim|\Lambda|^{\alpha}$, with $\alpha$ being a positive order one number.

In [KLL20], the Ricci flow distance conjecture was then refined in terms of
Perelman’s $\mathcal{F}$ and $\mathcal{W}$ entropy functionals (e.g. [Top06,
Chapter 8]). Mathematically, the beauty of these functionals is that they
allow for the consistent recasting of the Ricci flow equations in terms of
gradient flows. Namely, one obtains the equations by means of the variational
principle on certain functionals. Physically, this is useful for adding more
fields to the effective theory, such as the dilaton field. One of most
striking facts about the Perelman’s entropy functionals is that they very
closely resemble certain effective actions which are naturally obtained from
the bosonic string theory. For example, the $\mathcal{F}$-functional has the
form

$\mathcal{F}[g,f]=\int\left(R+\lvert\nabla f\rvert^{2}\right)e^{-f}dV$ (19)

where $f$ is a real scalar function on the background. This is closely related
to the effective action in string theory with a metric and a dilaton field in
the string frame. Indeed, the above $f$ plays the role of the dilaton, and we
have that $f=2\phi$, and thus $g_{s}^{2}=e^{f}$, with $g_{s}$ being the string
coupling constant.

The above observation is important in the context of the infinite distance
conjecture, as it allows us to relate the Ricci flow equations (obtained form
$\mathcal{F}$) with the RG flow equations form the string effective action.
This plays into the fact that the distance conjecture, at its core, studies
the limits where certain theories are strongly and weakly coupled. In this
context, we can also use the $\mathcal{F}$ and $\mathcal{W}$ functionals
themselves as a measure of the distance along the flow. This is basically
because the flow equations are obtained by finding the maximal rate of change
of the functionals, and thus these are monotonic along the flow.

##  4 The cobordism class and Ricci flow for $\Omega_{4}^{SO}$

Our first task is to give solid arguments to justify the behavior of cobordism
classes of manifolds under the Ricci flow. Intuitively, given the smooth
nature of the flow equations, one expects the cobordism class of the initial
manifold to remain unchanged wherever the solution is defined. In other words,
we expect the flow not to change the cobordism class, at least unless we
encounter a singularity. Indeed, given a family of manifolds
$(\mathcal{M},g(s))$ parametrized by the flow time $s$, let $T$ be the maximal
time for which the solution is defined. Then for every $T^{\prime}<T$, we
expect to be able to construct a cobordism between $(\mathcal{M},g(0))$ and
$(\mathcal{M},g(T^{\prime}))$ by following the flow.

In this section we will show that this is the case for Einstein spaces and for
the more general Ricci solitons. For it, we will need a quantity which
classifies the cobordism class of a manifold, and that can be related to the
Ricci Flow in some way. Theorem 2.1 gives us such a quantity: the Pontrjagin
numbers of the frame bundle of the manifold. Indeed, if we were to flow some
manifold, and find that at some value of the flow time one of its Pontrjagin
numbers has changed, then it would follow that it belongs to a different
cobordism class than the original manifold.

Because $p_{1}(\mathcal{R})\in H^{4j}(\mathcal{M},\mathbb{Z})$, the smallest
dimension for which $p_{1}(\mathcal{R})$ does not trivially vanish is $d=4$,
where the first Pontrjagin class is the only one that does not automatically
vanish. In this case, we have that

$p_{1}(\mathcal{R})=-\frac{1}{8\pi^{2}}\text{tr}\mathcal{R}^{2},$ (20)

where $\mathcal{R}$ of course stands for the curvature 2-form of the frame
bundle. Consequently, the trace is defined in terms of the $GL(4,\mathbb{R})$
indices of the curvature, resulting in a top form in $d=4$. There is, however,
one subtlety with the above definition, and it is that $\mathcal{R}$ is
defined in terms of a coordinate frame of the bundle, while the Ricci flow is
more naturally defined in coordinates. In order to make the distinction
explicit, we use latin indices to denote frame indices, and greek indices
otherwise. We can thus write the above definition in a more explicit fashion
as

$p_{1}(\mathcal{R})=-\frac{1}{8\pi^{2}}{\mathcal{R}^{a}}_{b}{\mathcal{R}^{b}}_{a}.$
(21)

The curvature 2-form can be defined in terms of the Riemann tensor as follows

${\mathcal{R}^{a}}_{b}={R^{a}}_{bcd}\hat{\theta}^{c}\wedge\hat{\theta}^{d},$
(22)

where $\\{\hat{\theta}^{a}\\}$ is the orthonormal coframe. We thus have that

$\displaystyle p_{1}(\mathcal{R})$
$\displaystyle={R^{a}}_{bcd}{R^{b}}_{aef}\hat{\theta}^{c}\wedge\hat{\theta}^{d}\wedge\hat{\theta}^{e}\wedge\hat{\theta}^{f}$
$\displaystyle={R^{a}}_{bcd}{R^{b}}_{aef}\varepsilon^{cdef}d\mu,$

where
$d\mu=\hat{\theta}^{c}\wedge\hat{\theta}^{d}\wedge\hat{\theta}^{e}\wedge\hat{\theta}^{f}=\sqrt{g}d^{4}x$.
As we have already said before, the above equation for the first Pontrjagin
class is expressed with respect to an orthonormal frame (i.e. a noncoordinate
basis). Therefore, we must obtain its coordinate form. We can do this by means
of the vielbeins $\\{{e_{\mu}}^{a}\\}\subset GL(4,\mathbb{R})$. There are
quite a few indices to transform, but we see that

${R^{a}}_{bcd}{R^{b}}_{aef}={R^{\alpha}}_{\beta\mu\nu}{R^{\beta}}_{\alpha\rho\sigma}{e_{\alpha}}^{a}{e^{\beta}}_{b}{e_{\beta}}^{b}{e^{\alpha}}_{a}{e^{\mu}}_{c}{e^{\nu}}_{d}{e^{\rho}}_{e}{e^{\sigma}}_{f}$
(23)

Due to the original trace on $\mathcal{R}$, the first four vielbeins turn out
to amount to two traces of $g_{\alpha\beta}$, and hence are constant. Because
we will only be concerned with the variation under the Ricci Flow of $p_{1}$,
we may drop constant prefactors and write

$p_{1}(\mathcal{R})\sim{R^{\alpha}}_{\beta\mu\nu}{R^{\beta}}_{\alpha\rho\sigma}{e^{\mu}}_{c}{e^{\nu}}_{d}{e^{\rho}}_{e}{e^{\sigma}}_{f}\varepsilon^{cdef}\sqrt{g}d^{4}x.$
(24)

###  4.1 Ricci flow and Einstein manifolds in $\Omega_{4}^{SO}$

For an Einstein space, we know that $R_{\mu\nu}=\Lambda g_{\mu\nu}$, and
therefore the Ricci Flow equations are reduced to

$\partial_{s}g_{\mu\nu}(s)=-2\Lambda g_{\mu\nu}$ (25)

The solution to this equation is a rescaling of the metric

$g(s)_{\mu\nu}=(1-2\Lambda s)g(0)_{\mu\nu}$ (26)

This can be parametrized as a Weyl rescaling of the metric

$g(s)_{\mu\nu}=e^{-2\omega(s)}g(0)_{\mu\nu}.$ (27)

In this situation, seeing that $p_{1}(\mathcal{R})$ is invariant is fairly
simple. Under a rescaling, the relevant quantities transform as

$g_{\mu\nu}\mapsto\kappa
g_{\mu\nu}\Rightarrow\begin{cases}{R^{\alpha}}_{\beta\mu\nu}\mapsto{R^{\alpha}}_{\beta\mu\nu}\\\
d\mu\mapsto\kappa^{d/2}d\mu\\\
{e^{\mu}}_{c}\mapsto\kappa^{-1/2}{e^{\mu}}_{c}\end{cases}$ (28)

The second property comes from the appeareance of $\sqrt{g}$ in the measure,
and the third from the fact that $g_{\mu\nu}={e_{\mu}}^{c}e_{\nu c}$. We
conclude that the first Pontrjagin number is invariant under the Ricci flow
for Einstein manifolds (this result is clearly seen to extend to manifolds of
dimension $d=4n$).

###  4.2 Ricci flow and Ricci solitons

We now come to check the invariance of the characteristic numbers of an
orientable 4-dimensional manifold under the Ricci flow. Instinctively, we
expect that they should be, as the only thing that remains is to study their
behavior under a diffeomorphism.

To prove the previous statement, begin by considering some diffeomorphism333We
are using different manifolds for the sake of clarity of the explanation.
$\varphi:\mathcal{M}\to\mathcal{N}$. Not only does such a map define an
isomorphism of the homology and cohomology of both manifolds, but also allows
us to define the pullback bundle $\varphi^{*}T\mathcal{N}$. Due to the
naturality of Stiefel-Whitney and Pontrjagin classes, we have that the
characteristic classes over $\mathcal{M}$ are the pullback of the ones over
$\mathcal{N}$, and thus, $p_{i}(\mathcal{M})=\varphi^{*}p_{i}(\mathcal{N})$,
and analogously with Stiefel-Whitney classes. Furthermore, in the context of
smooth manifolds these are nothing but differential forms. It is a standard
result [Lee13] that under these conditions, and given some generic top form
$\omega$ over $N$,

$\int_{\mathcal{N}}\omega=\pm\int_{\mathcal{M}}\varphi^{*}\omega,$ (29)

where the sign on the right hand side depends on if the diffeomorphism is
orientation preserving or orientation reversing. Now, in our particular case,
for the one-parameter family of diffeomorphisms, $\mathcal{M}=\mathcal{N}$.
Moreover, we know that $\varphi_{0}=\mathds{1}$, and therefore for every
$s\in[0,T)$, $\varphi_{s}:\mathcal{M}\to\mathcal{M}$ is continuously connected
to the identity, making it orientation preserving. Thus, the invariance of the
Pontjagin and Stiefel-Whitney numbers under a flow defined by (14) follows.

###  4.3 Pinching of necks

One of the phenomena which is tightly linked with the Ricci flow is the
appeareance, and the subsequent pinching-off, of necks on a manifold. When a
neck appears, it is very likely that it will shrink faster, compared to other
parts of the manifold. This leads us to the conclusion that, at some point
along the flow, a section of the manifold (located on the neck), will shrink
to a point, thus reaching the end of the flow. However, the flow equations
could still be well defined at all other parts of the manifold. Because of
this, it would be interesting to have a mechanism to allow us to study the
behavior of the manifold under the flow past the pinching of these necks, that
would reproduce the intuitive behavior of a neck pinching off, while at the
same time changing the geometry as little as possible. We will now see that
this procedure is compatible with the bordism relation. First, we will discuss
the notion of a surgery on a manifold, and then we will explain a way to adapt
the notion of convergence in order to allow for such a modification to take
place.

Figure 2: Schematic of the surgery that is to be done when a neckpinch ocurs.

In mathematical terms, a topological neck $\mathcal{N}$ in a manifold
$\mathcal{M}$ is a local diffeomorphism of a cylinder into $\mathcal{M}$:

$\mathcal{N}:\mathbb{S}^{n-1}\times[a,b]\to\mathcal{M}.$ (30)

The treatment of necks in the context of the Ricci flow is a very delicate
one. A very detailed description of it is given in one of Hamilton’s original
papers, [Ham97]. For our present purposes, it is only necessary to know that
the strategy to follow when approaching a singularity in the form of a neck
pinch is to perform a so-called surgery on the manifold. A surgery is a
topological procedure which is based on the observation that the relation

$\partial(\mathcal{M}\times\mathcal{N})=\partial\mathcal{M}\times\mathcal{N}\cup\mathcal{M}\times\partial\mathcal{N}$
(31)

Implies the two following possibilities

$\mathbb{S}^{p}\times\mathbb{S}^{q-1}=\begin{cases}\partial\left(D^{p+1}\times\mathbb{S}^{q-1}\right)\\\
\partial\left(\mathbb{S}^{p}\times D^{q}\right)\end{cases}$

Since the two manifolds on the right hand side have the same boundary, one
could in principle cut one out, and glue the other one along the leftover
boundary. This is precisely a surgery. In mathematical terms, we have the
following

######  Definition 4.1.

Let $\mathcal{M}$ be a manifold of dimension $n=p+q$, and let
$\varphi:\mathbb{S}^{p}\times D^{q}$ be an embedding. Define another manifold
$\mathcal{M}^{\prime}$ as

$\mathcal{M}^{\prime}=\left[\mathcal{M}\setminus\text{Int}(\text{Im}(\varphi))\right]\cup_{\varphi}\left[D^{p+1}\times\mathbb{S}^{q-1}\right].$

We define the surgery to be the triple
$(\mathcal{M},\mathcal{M}^{\prime},\varphi)$444Note that we have not specified
on which manifold we perform the surgery. This is because surgeries define an
equivalence relation, and are in particular reflexive.

In this way, looking at (30), if we perform a surgery, we substitute the
pinched neck by two smooth caps, and then keep flowing the resulting manifold.
In a sense, this procedure realizes the inverse of a disconnected sum, and
thus gives a connected sum decomposition of the original manifold. While this
part is rather simple, in the context of the Ricci flow we are interested not
only on the topology, but also on the geometry of the manifold. Hence, the
problem arises when one tries to extend the metric smoothly from the bases of
the neck to the smooth caps. A detailed account of this is contained in
[Ham97].

What we have defined above is a general surgery, of which the substitution of
a neck for two smooth caps is a particular example of. From the definition of
a neck in (30), we need only choose $p=n-1$ and $q=1$. In fact, the reader
might have noticed that this particular example of a surgery is precisely a
connected sum in reverse. Indeed, one of the topological appeals of the Ricci
flow is that it provides a connected sum decomposition of the manifolds on
which it is defined. This is very convenient for our purposes, as we know that
the process of taking a connected sum defines a bordism
$\mathcal{M}\sqcup\mathcal{N}\sim\mathcal{M}\\#\mathcal{N}$. In general, any
type of surgery in the sense of definition 4.1 defines a bordism relation from
$\mathcal{M}$ to $\mathcal{M}^{\prime}$, by means of taking the so-called
trace of the surgery, defined as follows. In the conditions of definition 4.1,
we define an $n+1$ dimensional manifold $\mathcal{W}$ by first taking
$\mathcal{M}\times I$, where $I$ is the unit interval, and then attaching
$D^{p+1}\times D^{q}$ along one of the ”ends” of $\mathcal{M}\times I$. All
together, we define

$\mathcal{W}=\left[\mathcal{M}\times
I\right]\cup_{\varphi\times\\{1\\}}\left[D^{p+1}\times D^{q}\right].$ (32)

It can be checked that this manifold indeed satisfies
$\partial\mathcal{W}=\mathcal{M}\sqcup\mathcal{M}^{\prime}$, by recalling the
gluing procedure, the definition of $\mathcal{M}^{\prime}$, and noting that
one of the two boundaries of $D^{p+1}\times D^{q}$ is precisely the one that
is glued to $\mathcal{M}^{\prime}$. All in all, what we have learned form this
discussion is that the cobordism class of a manifold is left unchanged by the
Ricci flow, even if necks pinch off, due to the fact that the manifolds before
and after the pinching are bordant.

Finally, there is one more aspect about neck pinching that we should look at,
for the sake of completeness. Since we have just defined a way to extend the
flow past a certain type of singularity, which in particular involves stopping
the flow whenever a neck forms, only to turn it back on after the surgery, we
need to define our notion of convergence. To add to that, this notion of
convergence should be diffeomorphism invariant, so a notion of convergence of
metric as tensors is unsatisfactory. Indeed, we need to specify what we mean
when we say that some manifold $\mathcal{M}$ converges to some other manifold
$\mathcal{M}^{\prime}$ (or has $\mathcal{M}^{\prime}$ as its limit). This is
done by means of the smooth, pointed Cheeger-Gromov convergence of manifolds.
Here, the term pointed means that the manifolds which compose the convergent
sequence have a certain choice of distinguished points. The proper defintion
is as follows

######  Definition 4.2.

A sequence of smooth, complete, and pointed Riemannian manifolds
$(\mathcal{M}_{i},g_{i},p_{i})$, with $p_{i}\in\mathcal{M}_{i}$ is said to
converge smoothly to the smooth, complete, and pointed manifold
$(\mathcal{M},g,p)$ as $i\to\infty$ if there exist

  * •

a sequence of compact sets $\\{\Omega_{i}\\}$, with
$\Omega_{i}\subseteq\mathcal{M}$ exhausting $\mathcal{M}$ with
$p\in\text{Int}(\Omega_{i})$ for each $i$, and

  * •

sequence of smooth maps $\varphi_{i}:\Omega_{i}\to\mathcal{M}_{i}$ which are
diffeomorphic onto their image and satisfy $\varphi_{i}(p)=p_{i}$ for all $i$,

such that $\lim_{i\to\infty}\varphi^{*}_{i}g_{i}\to g$ smoothly in the sense
that for all compact sets $K\subseteq\mathcal{M}$, the tensor
$\varphi^{*}_{i}g_{i}-g$ and its covariant derivatives of all orders converge
uniformly to 0 on $K$

This definition was taken from [Top06]. The interested reader is directed to
this reference for more details on this and almost all basic concepts relative
to the Ricci flow.

##  5 The main case study: $\Omega^{SO}_{4}$

We introduce the Pontrjagin and Stiefel-Whitney numbers as the relevant
topological invariants for the $SO$ structure. For the particular case of the
4-dimensional group, we discuss how the fact that the signatue $\sigma$ of
$C\mathbf{P}^{2}$ is $\sigma(C\mathbb{P}^{2})=1$ can be exploited to kill
cobordism classes by the addition of more and more copies of the complex
projective space. In particular, 16 copies of $C\mathbb{P}^{2}$ may be used to
kill the class of $K3$.

After having killed the class of $K3$, we must give a physical (or otherwise)
interpretation of the complex projective planes. One such interpretation is
given in terms of the blow-up of singularities of the $K3$ surface. These are
classified in terms of the ADE groups. We will explore this in detail in later
sections.

Another interpretation is given by [Leb91], where a metric on the connected
sum of copies of $C\mathbb{P}^{2}$ is provided, together with the
interpretation of the metric of magnetic monopoles in hyperbolic space. Upon
investigation, the given metric is strikingly similar to the Kaluza-Klein
monopole in 4 dimensions (also called the Taub-NUT geometry). Indeed, the
requirements on the connection over the circle bundle present in the monopole
coincide in the two cases.

###  5.1 The relevant invariants

We will now study the particular case of $\Omega_{4}^{SO}\simeq\mathbb{Z}$.
Since the group is nontrivial, we know that the cobordism conjecture forces us
to consider more restricted structures or add defects to the theory. We now
cite two results which will be of use, both of which can be found in [Joh74].
The first is that, in the context of oriented bordism (namely, an
$SO$-structure), the relevant bordism invariants are the Pontrjagin and
Stiefel-Whitney numbers, which we denote for some manifold $M$ by $p_{j}(M)$
and $w_{j}(M)$. Another result is the Hirzebruch signature theorem, which
states that the signature of a manifold can be written in terms of the
Pontrjagin numbers and is thus, by the previous result, a cobordism invariant.

The four dimensional case is special in the above sense, as the Pontrjagin
classes $p_{j}$ are cohomology classes in $H^{4j}(M,\mathbb{Z})$. Therefore
$d=4$ is the first dimension for which we have a class which is a priori
nontrivial555Note that $p_{0}$ is axiomatically set to 1.. Therefore, we will
use the Pontrjagin numbers to label cobordism classes, and thus construct
manifolds such that $p_{1}(M)=0$. We will see that in the examples we
construct, this will also force all Stiefel-Whitney numbers to be 0, thus
implying that their cobordism class is indeed trivial. As we have said in the
introduction, the trivialization of these classes will also mean that for
these examples, the structure group can be further reduced. However, we will
be more interested in the perspective of adding defects.

###  5.2 The case for $\mathbb{CP}^{2}$

It is a known result [Joh74] that $\mathbb{CP}^{2}$ is the generator of
$\Omega_{4}^{SO}$, the oriented cobordism group of dimension 4. Furthermore,
Hirzebruch’s signature theorem states that

$p_{1}(\mathbb{CP}^{2})=3\Leftrightarrow\sigma(\mathbb{CP}^{2})=\frac{p_{1}(\mathbb{CP}^{2})}{3}=1.$
(33)

Since the signature is a cobordism invariant, we assign $[\mathbb{CP}^{2}]=1$
(using additive notation). Constructing a background with trivial cobordism
class is now straightforward: we take two copies of $\mathbb{CP}^{2}$ and
reverse the orientation in one of them. Thus, the background

$\mathcal{M}=\mathbb{CP}^{2}\sqcup\overline{\mathbb{CP}}^{2}$ (34)

bounds. Indeed, we have that, by the properties of Pontrjagin numbers

$\sigma(\mathcal{M})=\sigma(\mathbb{CP}^{2})+\sigma(\overline{\mathbb{CP}}^{2})=1-1=0$
(35)

so we conclude that $[\mathcal{M}]=0$.

As an additional note, we can use the fact that the disjoint union of two
manifolds is cobordant to their connected sum, and then use a long exact
sequence of a pair and a Mayer-Vietoris argument to show that

$H_{p}(\mathbb{CP}^{2}\\#\overline{\mathbb{CP}}^{2})=\left.\begin{cases}\mathbb{Z}\hskip
42.67912pt&p=0,4\\\ \mathbb{Z}\oplus\mathbb{Z}&p=2\\\
0&\text{else}\end{cases}\right\\}=H^{p}(\mathbb{CP}^{2}\\#\overline{\mathbb{CP}}^{2})$
(36)

This tells us that the manifold we have just constructed is not homeomorphic
to $\mathbb{S}^{4}$, even if cobordant to it.

###  5.3 The case for $K3$

One of the manifolds that gives rise to an interesting scenario in our study
of cobordism classes and the Ricci flow is $K3$. In [MV19], the authors have
already studied this manifold from the point of view of its cobordism class.
For our present purposes, an important result that is cited there is that

$\sigma(K3)=-16$ (37)

Again, since the signature of a manifold is a cobordism invariant, this
defines a class in the cobordism group $\Omega_{4}^{SO}$. With
$\mathbb{CP}^{2}$ as a generator, we can write this as

$[K3]=-16[\mathbb{CP}^{2}]$ (38)

where, of course, we use the notation of addition to mean the disjoint union
of 16 projective planes with their orientation reversed. From this, we can
easily construct a manifold which has a trivial cobordism class, given by

$K3\sqcup 16\mathbb{CP}^{2},$ (39)

which by the above argument has vanishing signature and hence bounds a
$5$-manifold.

The interesting part about the above scenario is that, if we turn on the
regular Ricci flow equations, all of the $\mathbb{CP}^{2}$ (being that they
are Einstein spaces with positive sectional curvature) shrink to a point.
However, $K3$ is a Calabi-Yau manifold, and in particular Ricci-flat.
Therefore, it is a fixed point of the flow equations, and remains invariant
under them. Furthermore, because the disjoint union and the connected sum of
manifolds are cobordant, it is irrelevant if we consider the above manifold as
$K3\sqcup\mathbb{CP}^{2}\sqcup\ldots\sqcup\mathbb{CP}^{2}$ or as
$K3\\#\mathbb{CP}^{2}\\#\ldots\\#\mathbb{CP}^{2}$. Therefore, we do not need
to worry about the necks that are formed by the connected sums. It follows
that in this setup, at the end of the flow, what remains is the original $K3$,
and all of the remaining copies of projective planes have shrunk to points.
This loose discussion represents the idea that there is some connection, due
to working under the bordism relation, of trivializing the bordism class at
the topological level, and adding defects to the manifold. The connection will
be given by the blow-ups (minimal resolutions) of the $K3$ singularities.

##  6 Blowing up points and connected sums of copies of $\mathbb{CP}^{2}$

In the context of algebraic geometry, there is a procedure known as ”blowing
up”. The underlying idea is to substitute a certain subspace of some total
space (usually some complex variety) by another, more well-behaved space.
Furthermore, one does it in such a way that the resulting total space looks
like the one we started with away from the blown-up subspace. While this
procedure can be done just for the sake of it, it fully comes into fruition in
the context of the resolution of singularities, where there is some sort of
singular subspace (e.g. a conifold point), which is then substituted by a
smooth space, blended into the total space in such a way that the combination
of the two is smooth throughout. In this section, we will focus on the case
where the singular subspace is just a singular point, we will give the
explicit construction of its blow up, and we will see how it relates to
manifolds containing connected sums of complex projective spaces.

###  6.1 The blow up of a point

We will begin by considering $\mathbb{C}^{2}$. The only generality that is
lost in doing so is that of dimensionality, which will not be much of a
problem, as the construction can be easily generalized to higher dimension.
Since all of the considerations we will make will be local in nature, we can
always reduce the case of an arbitrary complex manifold of dimension $n$ to
that of $\mathbb{C}^{n}$. Now, suppose we wanted to blow up the origin
$\mathcal{O}\in\mathbb{C}^{2}$. This would produce a new space, which we shall
name $Bl_{\mathcal{O}}(\mathbb{C}^{2})$, and define as follows. Consider the
set of all paths in $\mathbb{C}^{2}$ whose endpoint is $\mathcal{O}$, and for
each of them, take their tangent line at the origin. This set of lines through
the origin is precisely the definition of $\mathbb{CP}^{1}$, each line $l$ is
given by a point $[y_{1}:y_{2}]\in\mathbb{CP}^{1}$ (where we have used
homogeneous coordinates) and vice versa. Define
$Bl_{\mathcal{O}}(\mathbb{C}^{2})\subseteq\mathbb{C}^{2}\times\mathbb{CP}^{1}=\\{(x_{1},x_{2},[y_{1}:y_{2}])|(x_{1},x_{2})\in\mathbb{C}^{2},[y_{1}:y_{2}]\in\mathbb{CP}^{1}\\}$
as the zero locus of

$p=x_{1}y_{2}-x_{2}y_{1},$ (40)

namely, all of the points verifying $x_{1}y_{2}=x_{2}y_{1}$. This condition is
equivalent to saying that the point $(x_{1},x_{2})$ is contained in the line
$l\in\mathbb{CP}^{1}$, since the above equations are obtained by imposing
linear dependence of the set of vectors defined by

$\begin{pmatrix}x_{1}&x_{2}\\\ y_{1}&y_{2}\end{pmatrix},$ (41)

by requiring that the rank of the matrix be 1.

Let us look at (40) more closely. Whenever $(x_{1},x_{2})\neq(0,0)$, then
$(y_{1},y_{2})$ are completely determined up to scaling, which means that they
are actually completely determined in $\mathbb{CP}^{1}$. On the other hand, at
the origin, $y_{1}$ and $y_{2}$ can take any value, and (40) vanishes
tivially, so both coordinates range over all of $\mathbb{CP}^{1}$. These
considerations lead us to define the following projection map

$\begin{split}\pi:\hskip 5.69046pt\mathbb{C}^{2}\times\mathbb{CP}^{1}\hskip
5.69046pt&\to\hskip 8.5359pt\mathbb{C}^{2}\\\
(x_{1},x_{2},[y_{1}:y_{2}])&\mapsto(x_{1},x_{2})\end{split}$ (42)

By the above discussion, this map is an isomorphism away from the origin, but
at the origin itself we have that $\pi^{-1}(0,0)=\mathbb{CP}^{1}$.

###  6.2 Blowing up and connected sums

In this section, we will try to draw a connection between the process of
blowing up a point and that of taking the connected sum of the original
manifold with $\mathbb{CP}^{n}$. This is encapsulated in the following
proposition:

######  Proposition 6.1.

Let $x\in\mathcal{M}$ be a point in a complex $n$-dimensional manifold
$\mathcal{M}$. Then the blow up $Bl_{x}(\mathcal{M})$ is diffeomorphic as an
oriented manifold to $\mathcal{M}\\#\overline{\mathbb{CP}^{n}}$.

###### Proof.

Because the blow up of a point is a local transformation, we can set, without
loss of generality, $\mathcal{M}=D^{n}=\\{z\in\mathbb{C}^{n}|\lVert
z\rVert^{2}<1\\}$, and $x=0$. By the definition introduced in the preceding
section, the blow up is given by

$Bl_{0}(D)=\\{(x,z)\in D\times\mathbb{CP}^{n-1}|x_{i}z_{j}=x_{j}z_{i},\
\forall\ i\neq j\\}$ (43)

Now we have to construct the connected sum. To do so, we must first embed two
disks $D^{n}$ in our original manifolds, quotient out by their images, and
then glue along the remaining boundaries. Because we have set $X=D^{n}$, one
of the embeddings can just be taken as the identity $\text{id}:D^{n}\to X$.
For the remaining one, in order to maintain the overall orientation of the
manifold, we must choose an orientation reversing map
$f:D^{n}\to\mathbb{CP}^{n}$. We thus define it to be $x\mapsto[1:x]$. The fact
that this is indeed an orientation reversing map can be seen by considering
inward pointing rays on each of the coordinates of $D^{n}$. This allows us to
restrict the image rays to $\mathbb{CP}^{1}\subset\mathbb{CP}^{n}$ by setting
the remaining (homogeneous) coordinates to 0. Then, since
$\mathbb{CP}^{1}\simeq\mathbb{S}^{2}$, and $f$ can be seen locally as an
inversion ($z\mapsto 1/z$) at the level of the regular coordinates obtained
from the homogeneous ones, we see that the orientation of the rays is
reversed.

Once we have the embedding maps, we consider the original manifolds and remove
an annulus inside the embedded disk. The fact that it is an annulus and not a
proper disk is topologically irrelevant. For $X$ this is trivial, and for
$\mathbb{CP}^{n}$ this is given by

$\mathbb{CP}^{n}\setminus
f\left(\frac{1}{2}\overline{D}\right)\coloneqq\\{[x_{0}:x]\mid|x_{0}|<2\lVert
x\rVert\\}$ (44)

Note that we are using homogeneous coordinates to describe the manifold, and
hence the particular numerical value of $x_{0}$ is irrelevant, and the removed
disk can be encapsulated in the restriction on the right hand side of (44).
All that remains to show our desired result is to construct the gluing
diffeomorphism (so that we can define $X\\#\mathbb{CP}^{n}$), and a
diffeomorphism from the connected sum to $Bl_{0}(D)$. To do so, we will kill
two birds with one stone, and use the fact that $Bl_{0}(D)$ is isomorphic to
$D$ away from the origin. Hence, the gluing map from $\mathbb{CP}^{n}$ to $D$
will be the same as the one to $Bl_{0}(D)$ (recall that we have chosen $X=D$).
We can give an explicit orientation preserving map which serves this purpose:

$\displaystyle\hat{\xi}:\mathbb{CP}^{n}\setminus
f\left(\frac{1}{2}\overline{D}\right)$ $\displaystyle\to Bl_{0}(D)$
$\displaystyle[x_{0}:x]$ $\displaystyle\mapsto\left(\frac{x_{0}}{2\lVert
x\rVert^{2}}x,[x]\right)$

We will first check that it restricts to the gluing map on the embedded disk,
namely on $f(D)\setminus f(\frac{1}{2}D)$. In the above notation, this means
that we must set $x_{0}=1$, and let $\lVert x\rVert<1$ (since this is the
coordinate on the disk). Both of these conditions restrict the values of $x$
to be on the annulus $\frac{1}{2}<\lVert x\rVert<1$. Therefore, $x$ itself is
defined on the annulus (as opposed to the embedded annulus on
$\mathbb{CP}^{2}$). At the level of $\hat{\xi}$, the image of this set is
easily seen to be given by

$\hat{\xi}\left(f(D)\setminus
f\left(\frac{1}{2}D\right)\right)=\left\\{\left(\frac{x}{2\lVert
x\rVert^{2}},[x]\right)\in Bl_{0}(D)\mid\frac{1}{2}<\lVert x\rVert<1\right\\}$
(45)

Note that the fact that the image is defined on $Bl_{0}(D)$ means that it is
still subject to the set of equations which defined it (c.f. (43)). In
particular the above restriction is seen to be a diffeomorphism of the annulus
$D\setminus\frac{1}{2}D$ to itself, namely $x\mapsto(\frac{x}{2\lVert
x\rVert})$. This map precisely defines the gluing scheme. Finally, due to the
blow-up equations, the remaining components of the image are completely fixed
in $\mathbb{CP}^{n-1}$

To see that this map also defines a diffeomorphism to $Bl_{0}(D)$, we go back
(40). The only possibility for $\hat{\xi}$ to not be differentiable is that
$x=0$. However, this would imply $|x_{0}|<0$, which is impossible by (44).
Furthermore, we can see that the behavior at the singular point $0\in D$,
which was the original purpose of the blow-up, is reproduced by this map,
since $x_{0}=0$ is an allowed value which, by both (44) and the equations that
define the blow-up, lets $x$ range freely over all of $\mathbb{CP}^{n-1}$.
Finally, we can see that $\text{Im}(\hat{\xi})$ is all of $Bl_{0}(D)$ by
taking the norm of the first $n$ components of (45) and again noting that by
(44) $|x_{0}|<2\lVert x\rVert$, thus arriving at

$\frac{|x_{0}|\lVert x\rVert}{2\lVert x\rVert^{2}}<1$ (46)

∎

###  6.3 Blowups and magnetic monopoles

Our end goal is to somehow apply the Ricci flow machinery to the manifold
which is imposed by the cobordism conjecture. In principle, to be able to do
so, we first need to write down an explicit form of the metric for the whole
manifold. This is tackled in [Leb91], and both the result, and the method are
relevant for our purposes, since a connection is drawn between the successive
blowup of points on a manifold and the connected sum of projective planes.
This is further supported by the ADE classification of K3 singularities, and
its interplay with blowups of said singularities. This last point will be left
for a following section, and we will rather focus our attention here on the
geometric side of the construction. We will leave out some details, which can
be read on the original publication.

The starting point for the construction of the metric is the Gibbons-Hawking
ansatz for gravitational multi-instantons, first proposed in [GH78]. This is
generalized to allow for an isometric circle action on the manifold, and
reformulated in geometrical terms, as solutions of a certain set of partial
differential equations. The central result of the paper, out of which all of
the results are derived, is the following:

######  Proposition 6.2.

Let $\xi>0$ and $u$ be smooth real-valued functions on an open set
$U\subseteq\mathbb{R}^{3}$ which satisfy

$\displaystyle u_{xx}+u_{yy}+(e^{u})_{zz}$ $\displaystyle=0$ (47)
$\displaystyle\xi_{xx}+\xi_{yy}+(\xi e^{u})_{zz}$ $\displaystyle=0$ (48)

Suppose further that the de Rham class of the closed 2-form

$\frac{1}{2\pi}\alpha\coloneqq\frac{1}{2\pi}\left(\xi_{x}dy\wedge
dz+\xi_{y}dz\wedge dx+(\xi e^{u})_{z}dx\wedge dy\right)$ (49)

is contained in $H_{dr}^{2}(U,\mathbb{Z})$ (i.e., it is integral). Let $M\to
U$ be a circle bundle such that its first Chern class is given by
$\left[c_{1}(M)\right]_{\mathbb{R}}=\left[\frac{\alpha}{2\pi}\right]$, and let
$\omega$ be a connection 1-form on the bundle whose curvature is $\alpha$. If
$U$ is simply connected, both $M$ and $\omega$ are determined up to gauge
equivlence. Then

$g=e^{u}\xi(dx^{2}+dy^{2})+\xi dz^{2}+\xi^{-1}\omega^{2}$ (50)

is a Kähler metric on $M$ whose scalar curvature vanishes. Conversely, any
scalar-flat Kähler surface with an $\mathbb{S}^{1}$ action can be locally
described by the above result.

Note that the above metric reduces to the Gibbons-Hawking ansatz if $u=0$. The
space that this metric represents is also knwon as the Taub-NUT space. This is
very relevant to our goal for two reasons. The first one is that the Taub-NUT
space describes a KK monopole: a purely geometrical solution to Einstein’s
equations in $d+1$ dimensions, which from the perspective of the $d$
dimensions transverse to the circle bundle describes a magnetic monopole.
Moreover, the Taub-NUT solution allows us to superpose $N$ of these monopoles.
While we will not deal with Taub-NUT spaces, but rather with a space that is
very similar to it, a lot of the properties will be shared between the two.
Thus, it is useful to keep this solution in mind for the following.

The connection that all of this has with blow-ups (and thus with connected
sums of copies of $\mathbb{CP}^{2}$) is given by the so-called Burns metric.
This is a scalar flat Kähler metric on $Bl_{0}(\mathbb{C}^{2})$, the blow-up
of $\mathbb{C}^{2}$ at the origin. There are several coordinate
representations of this metric, and we will come back to this in a later
section. For now, it suffices to know that this metric fits into the
description of proposition 6.2 by setting

$\displaystyle u$ $\displaystyle=\log 2z$ (51) $\displaystyle\xi$
$\displaystyle=\frac{1}{2z}+F_{m}$ (52)

where $m$ is some parameter which appears on the metric, and $F_{m}(x,y,z)$ is
a positive function on $z>0$, and is defined on the complement of
$(x,y,z)=(0,0,m/2)$. In particular, this makes sense when the base space is
$\mathbb{H}^{3}\coloneqq\\{(x,y,z)\in\mathbb{R}^{3}\mid z>0\\}$.

If one defines a new variable $V=2z\xi$, and further defines $q=\sqrt{2z}$, it
turns out that the second PDE from proposition 6.2 can be recast as

$\Delta V=0,$ (53)

where $\Delta$ is the Laplace-Beltrami operator associated to the metric

$h=\frac{dx^{2}+dy^{2}+dq^{2}}{q^{2}}.$ (54)

This is just the usual metric for hyperbolic three-space, which we denote by
$\mathcal{H}^{3}$. In terms of $V$ and $h$, the metric can be written as

$g=q^{2}[Vh+V^{-1}\omega^{2}]$ (55)

Furthermore, we can use the hyperbolic metric distance to rewrite the function
$\xi$ (and thus $V$) that appears in the metric. With the metric (54) for
$\mathbb{H}^{3}$, the hyperbolic distance from a generic point
$(x,y,q)\in\mathbb{H}^{3}$ to $(0,0,q_{0})\in\mathbb{H}^{3}$ can be written as

$\rho=\cosh^{-1}\left(\frac{x^{2}+y^{2}+q^{2}}{2qq_{0}}\right)$ (56)

After a couple of algebraic manipulations, we can recast $V$ as

$V=1+\frac{1}{e^{2\rho}-1}$ (57)

But this last summand is precisely the Green’s function $G$ for the Laplace-
Beltrami operator relative to the hyperbolic volume form, and with
normalization given by $\Delta G=-2\pi\delta$. Consequently, the Burns metric
can also be thought as describing a magnetic monopole in hyperbolic 3-space.

Up to now, we have seen that the metric for a blowup of $\mathbb{C}^{2}$ at
the origin can be recast (or, rather, understood) as a kind of generalized
Gibbons-Hawking metric, which looks eerily similar to a Taub-NUT geometry.
Now, the next step, as is commonplace precisely with the Taub-NUT metric, is
to allow for multiple monopoles for our metric. This can be done as follows:
let $\\{p_{j}=(x_{j},y_{j},z_{j})\\}_{j=1}^{n}\subset\mathbb{H}^{3}$ be a
collection of points on the upper half-space, and let

$G_{j}=\frac{1}{e^{2\rho_{j}}-1}$ (58)

be the Green’s functions associated to each of the points in the sense of the
above paragraph. Define

$V\coloneqq 1+\sum_{i=1}^{n}G_{j}$ (59)

Then, we have that $\xi=\frac{V}{2z}$ is a positive solution for

$\xi_{xx}+\xi_{yy}+(2z\xi)_{zz}=-2\pi\sum_{i=1}^{n}\delta_{p_{i}}.$ (60)

This defines an integral cohomology class as

$\frac{1}{2\pi}\alpha=\frac{1}{2\pi}\left(\xi_{x}dy\wedge dz+\xi_{y}dz\wedge
dx+(\xi e^{u})_{z}dx\wedge dy\right)\in
H^{2}(\mathbb{H}^{3}\setminus\\{p_{i}\\},\mathbb{Z})$ (61)

Finally, let $X\to\mathbb{H}^{3}\setminus\\{p_{i}\\}_{i=1}^{n}$ be the circle
bundle whose first Chern class is $\frac{1}{2\pi}[\alpha]$. In turn, this
means that $X$ has a connection 1-form $\omega$ whose curvature is given (up
to gauge equivalence) by $\alpha$. Then, define a Riemannian metric on $X$ by

$g=q^{2}[Vh+V^{-1}\omega^{2}]$ (62)

By proposition 6.2, this is a Kähler metric with vanishing scalar curvature.
However, it is still not quite the metric that we are looking for. For one, it
is not defined over the ”monopole locations”, namely the
$\\{p_{i}\\}_{i=1}^{n}$. However, it can be smoothly extended to them by
attaching points at the monopole locations. This is done in detail in [Leb91],
and contains very technical details which are not very relevant for our
purposes. The only part of this discussion which we will use is a local
expression for the metric around the monopole locations. This will be
discussed in a later section.

In summary, we are able to construct the Kähler metric (62) which, by the
above discussion, can be interpreted as describing the blow-up of
$\mathbb{C}^{2}$ at $n$ points, which can be furhter viewed as the locations
of magnetic monopoles in hyperbolic 3-space. Since blow-ups and connected sums
with copies of $\mathbb{CP}^{2}$ are diffeomorphic, a consequence of this
construction is that the conformal class of the metric (62) represents a
metric on $n\mathbb{CP}^{2}\coloneqq\mathbb{CP}^{2}\\#\stackrel{{\scriptstyle
n)}}{{\ldots}}\\#\mathbb{CP}^{2}$. In fact, a global representative for the
conformal metric on $n\mathbb{CP}^{2}$ is provided by

$g=\text{sech}^{2}(\rho)[Vh+V^{-1}\omega^{2}]$ (63)

In the following, we will use this very metric, as well as some of the
intermediate steps that led to it, to try to apply the Ricci flow equations to
$n\mathbb{CP}^{2}$. In this way, we hope to use the flow equations to obtain
information about backgrounds which are allowed by the cobordism conjecture.

###  6.4 The Burns metric

Now that we have some geometrical information about the manifold
$n\mathbb{CP}^{2}$, the next step is to set up some kind of flow equations, or
at least to approximate the manifold’s behavior under them. Because of the
above discussion, the first step that we will take in this endeavor is to look
closely at the Burn’s metric. As we have mentioned before, this is a zero
scalar curvature metric on $Bl_{0}(\mathbb{C}^{2})$ (we are using the notation
from the previous sections for the blowup). Because we have seen that blowups
at a point in a manifold are diffeomorphic to connected sums of said manifold
with $\mathbb{CP}^{2}$, we hope to extract some information out of this.

####  6.4.1 First form of the Burns metric

There is more than one way to represent the Burn’s metric. Here we will only
make use of two of them. In the first representation, the Kähler nature of the
metric is manifest, as it is given by the following Kähler potential defined
over $\mathbb{C}^{2}\setminus\\{0\\}$:

$K(z,\overline{z})=\frac{1}{2}\left(\lVert z\rVert^{2}+m\log\lVert
z\rVert^{2}\right)$ (64)

where $m>0$ is a positive real constant, and $\lVert
z\rVert^{2}=z_{1}\overline{z}_{1}+z_{2}\overline{z}_{2}$. Thus, the metric can
be obtained as

$g_{i\overline{j}}=\partial_{i}\overline{\partial}_{\overline{j}}K(z,\overline{z})$
(65)

Going through the computations, this yields

$g_{1\overline{1}}=\frac{1}{2}\left(1+\frac{m\mid z_{2}\mid^{2}}{\left(\lVert
z\rVert^{2}\right)^{2}}\right);\hskip
42.67912ptg_{2\overline{1}}=-\frac{m}{2}\left(\frac{z_{1}}{\left(\lVert
z\rVert^{2}\right)^{2}}\overline{z}_{2}\right)$ (66)

with all other entries of the metric given by the fact that it is Hermitean
(so we need only exchange $1\Leftrightarrow 2$):

$g=\frac{1}{2}\begin{pmatrix}\left(1+\frac{m\mid z_{2}\mid^{2}}{\left(\lVert
z\rVert^{2}\right)^{2}}\right)&-m\left(\frac{z_{2}}{\left(\lVert
z\rVert^{2}\right)^{2}}\overline{z}_{1}\right)\\\
-m\left(\frac{z_{1}}{\left(\lVert
z\rVert^{2}\right)^{2}}\overline{z}_{2}\right)&\left(1+\frac{m\mid
z_{1}\mid^{2}}{\left(\lVert z\rVert^{2}\right)^{2}}\right)\end{pmatrix}$ (67)

For Kähler manifolds, there is a quick way of computing the Ricci tensor,
namely by

$R_{i\overline{j}}=-\partial_{i}\overline{\partial}_{\overline{j}}\log\det g.$
(68)

A quieck computation reveals that

$4\det g=1+\frac{m}{\lVert z\rVert^{2}}.$ (69)

Note that setting $m=0$ is consistent with recovering the usual flat metric on
$\mathbb{C}^{2}$. Since we are taking derivatives of logs, we may also drop
the overall factor of $\frac{1}{4}$, thus arriving at

$R_{i\overline{j}}=-\partial_{i}\overline{\partial}_{\overline{j}}\log\left(1+\frac{m}{\lVert
z\rVert^{2}}\right)$ (70)

The resulting tensor is given by

$R_{2\overline{1}}=\frac{mz_{1}}{\left[\left(\lVert
z\rVert^{2}\right)^{2}+m\lVert z\rVert^{2}\right]^{2}}\left(2\lVert
z\rVert^{2}\overline{z}_{2}+m\overline{z}_{2}\right);\hskip
14.22636ptR_{1\overline{1}}=-\frac{m}{\left(\lVert
z\rVert^{2}\right)^{2}+m\lVert z\rVert^{2}}+\frac{\lvert
z_{1}\rvert^{2}\left(2\lVert z\rVert^{2}+m\right)}{\left[\left(\lVert
z\rVert^{2}\right)^{2}+m\lVert z\rVert^{2}\right]^{2}}$ (71)

and the other components are again obtained by the exchange of indices. As one
can appreciate, the resulting Ricci flow equations become rather complicated,
and far form the case for an Einstein manifolds. It would have been our hope
to be able to get some sort of flow equation for the parameter $m$, but it
seems that our hopes are in vain. Even if we tried to do so out of the Ricci
flow equation for the scalar curvatrue, we are met with the fact that this
metric is, by construction, of zero scalar curvature. Hence, if we were to set
$m=m(s)$ for some flow parameter $s$, our equation would render $m$ constant
along the flow.

Given the explicit formula for the Ricci tensor (68), another idea that one
could have is to try to take the limit where $m\ll\lVert z\rVert$, in order to
get rid of the logarithm. Unfortunately, this has led me nowhere. The only
interesting point in the ”moduli space”, where the Ricci tensor turns out to
vanish, requires $\lVert z\rVert^{2}\to 0$, which is inconsistent with the
limit. On the other hand, we have noted that (even at the level of the Kähler
potential itself) the limit $m\to 0$ corresponds to flat space.

###  6.5 Second form of the Burn’s metric

There is yet another way to represent the Burn’s metric, in which its Kähler
nature is rather implicit. However, the upside of it is that we can obtain a
(real) coordinate representation of it. Hence, we can use a symbolic
computation program, such as Mathematica, to more easily compute all the
quantities that we need. Moreover, this representation is instructive when we
compare it with local representations of the metric (63).

Let $\\{\sigma_{1},\sigma_{2},\sigma_{3}\\}$ be the coframe of $SU(2)$ left-
invariant 1-forms, normalized such that
$\text{d}\sigma_{1}+2\sigma_{2}\wedge\sigma_{3}=0$ (and cyclically). Since
$SU(2)\simeq\mathbb{S}^{3}$, these forms also serve as an orthonormal coframe
on the three-sphere. Then, the Burns metric can be given the following
coordinate representation:

$g_{B}=dr^{2}+r^{2}\left(\sigma_{1}^{2}+\sigma_{2}^{2}+\sigma_{3}^{2}\right)+\sigma_{1}^{2}+\sigma_{2}^{2}$
(72)

As a comment, note that in this representation the usual Euclidean metric is
given by $g=dr^{2}+r^{2}(\sigma_{1}^{2}+\sigma_{2}^{2}+\sigma_{3}^{2})$. As we
have previously discussed, the metric (72) describes the blow-up of
$\mathbb{C}^{2}$ at the origin.

In order to do any kind of calculation with the Burns metric, we need to
introduce an explicit coordinate representation of the left-invariant $SU(2)$
one-forms. We do so in terms of the Euler angles, which we denote by
$\\{\theta,\varphi,\psi\\}$. The one-forms then take the following explicit
form:

$\begin{cases}2\sigma_{1}&=\sin(\psi)d\theta-\cos(\psi)\sin(\theta)d\varphi\\\
2\sigma_{1}&=\cos(\psi)d\theta+\sin(\psi)\sin(\theta)d\varphi\\\
2\sigma_{3}&=d\psi+\cos(\theta)d\varphi\end{cases}$ (73)

With these explicit coordinate representations, we are able to compute both
the Ricci scalar and the Ricci tensor. As should be expected, we have that
$R_{B}=0$ identically. We will come back to this point in a later section

###  6.6 Getting a feel for the flow

Seeing that the treatment of the metric for
$C\mathbb{P}^{2}\\#\stackrel{{\scriptstyle n)}}{{\ldots}}\\#C\mathbb{P}^{2}$
explicitly and analitically is rather complicated, we turn to approximating
the behavior of the manifold under the Ricci flow. To this end, we will use
less information than the required to solve the full flow equations. Namely,
we turn to a result about the metric (63) which is stated in [Leb91]. While
the original metric (62) was constructed as a scalar flat metric, the actual
representative for the metric on $n\mathbb{CP}^{2}$ has a different choice of
conformal factor, and thus the scalar curvature need not vanish. In fact, it
does not, and it turns out to be

$R=\frac{12}{V}.$ (74)

Where $V$ is given by (59). Since $\rho_{j}$ is the distance from a point in
$\mathcal{H}^{3}$ to one of the monopole locations, it only vanishes at these
singular loci. As a consequence of this, at these loci, $V\to\infty$.
Conversely, the scalar curvature vanishes at (and only at) these singular
points.

In order to use this information to study the behavior of the Ricci flow on
this manifold, we need an additional result, coming from the general theory of
the flow equations. In particular, as a consequence of the maximum principle,
we have that

######  Theorem 6.1.

Let $g(t)$ be a Ricci flow on a closed $d$-dimensional manifold $X$, for
$t\in[0,T]$. If $R\geqslant\alpha\in\mathbb{R}$ at time $t=0$, then for all
times $t\in[0,T]$,

$R\geqslant\frac{\alpha}{1-\left(\frac{2\alpha}{d}\right)t}$ (75)

What is important about this result is that, whenever the lower bound is
strictly positive, the scalar curvature blows up at some finite flow time.
Going back to (74), and recalling the explicit formula for $V$, we see that
$R$ is precisely a non-negative function over $n\mathbb{CP}^{2}$. Moreover,
the only points at which $R$ tends to 0 are precisely the singular loci which
define the location of the magnetic monopoles. This is enough information to
know that, at every point but the singularities, there is a lower bound on the
scalar curvature which goes to infinity at finite flow time. Hence, we can
guarantee that the curvature blows up and the manifold shrinks infinitely (but
at finite time) at all points but at the singular loci.

####  6.6.1 The behavior of the flow at the monopole locations

A failure of the above argument is that the scalar curvature (74) vanishes at
the location of the monopoles. To be more precise, $R$ is not technically
defined at the $\\{p_{i}\\}$. However, we have already said that it can be
smoothly extended to the whole manifold by essentially attaching points at the
monopole locations. Then, after a series of approximations of the metric
around said locations, one finds that it can indeed be smoothly extended to
them. Because of the smoothness of this extension, it follows that the Ricci
scalar also extends smoothly, and hence by continuity it must vanish at the
monopole locations, since

$\lim_{p\to p_{i}}\frac{1}{V}=0.$ (76)

This, of course, renders the argument from the previous section invalid, and
hence we cannot use it to justify that the manifold shrinks at every point.
One could, in principle, follow the approximations which lead to the extension
of the metric, and try to see if anything can be done to set up a flow around
the singularities. This is what we will attempt now.

As we have said, the full manifold is obtained by an extension process which,
at the monopole locations in particular, begins by attaching points at the
missing $\\{p_{i}\\}$. This can be done in such a way that the metric that we
had previously built extends smoothly to the the full manifold. To this end,
we begin by introducing exponential polar coordinates on $\mathbb{H}^{3}$ near
one of these points. The metric then becomes

$g=q^{2}[V(d\rho^{2}+\sinh^{2}(\rho)g_{\mathbb{S}^{2}})+V^{-1}\omega^{2}]$
(77)

where $V=\frac{1}{2\rho}+F$ (cf. (52)), for $F$ some smooth function on a
neighbourhood of the origin, and $g_{\mathbb{S}^{2}}$ denotes the standard
metric on the unit 2-sphere. In terms of the $SU(2)$ left-invariant one-forms
that we introduced earlier on, this can be written as
$g_{\mathbb{S}^{2}}=\sigma^{2}_{1}+\sigma_{2}^{2}$. For small values fo the
radial coordinate $\rho$, we can use the polar coordinates on the base to
identify it with $\mathbb{R}^{+}\times\mathbb{S}^{2}$, and the circle bundle
with $\mathbb{R}^{+}\times\mathbb{S}^{3}$. In turn, the bundle map
$\pi:X\to\mathbb{H}^{3}\setminus\\{p_{i}\\}$ is identified with

$\displaystyle p:\mathbb{R}^{+}\times\mathbb{S}^{3}$
$\displaystyle\to\mathbb{R}^{+}\times\mathbb{S}^{2}$ $\displaystyle(r,x)\hskip
5.69046pt$ $\displaystyle\mapsto\left(\frac{r^{2}}{2},\mu(x)\right).$

Here, the map $r\mapsto r^{2}/2$ is chosen as such for consistency with the
previously constructed bundle map,
$pi:X\to\mathbb{H}^{3}\setminus\\{p_{i}\\}$, and
$\mu:\mathbb{S}^{3}\to\mathbb{S}^{2}$ is the usual Hopf fibration. Because of
this identification with unit spheres, it is now useful to fully introduce the
left-invariant $SU(2)$ coframe $\\{\sigma_{1},\sigma_{2},\sigma_{3}\\}$ on
$\mathbb{S}^{3}$, again normalized so that
$\text{d}\sigma_{1}+2\sigma_{2}\wedge\sigma_{3}=0$. Because of this relation,
and the fact that the curvature of the connection form is defined over the
base space, we have the (gauge) freedom of choosing
$\omega=\sigma_{3}+p^{*}\theta$, with $\theta$ some one-form defined on the
base space. With all these choices, and the change of variable $\rho\to
r^{2}/2$, the metric turns out to be

$g=q^{2}\left[(1+r^{2}F)dr^{2}+r^{2}(1+\sinh^{2}\left(r^{2}/2\right)r^{2}F)(\sigma_{1}^{2}+\sigma^{2}_{2})+r^{2}(1+r^{2}F)^{-1}(\sigma_{3}+p^{*}\theta)^{2}\right]$
(78)

Identifying $\mathbb{R}^{+}\times\mathbb{S}^{3}$ with
$\mathbb{R}^{4}\setminus\\{0\\}$, the map $p$ extends to

$\displaystyle p:\mathbb{C}^{2}$ $\displaystyle\to\mathbb{R}^{3}$
$\displaystyle(z_{1},z_{2})$
$\displaystyle\mapsto\left(\frac{|z_{1}|^{2}-|z_{2}|^{2}}{2},\text{Re}(z_{1}\overline{z}_{2}),\text{Im}(z_{1}\overline{z_{2}})\right)$

In particular, this identification allows us to identify the metric at the
origin as precisely the Euclidean metric, expressed as
$g=dr^{2}+r^{2}(\sigma_{1}^{2}+\sigma_{2}^{2}+\sigma_{3}^{2})$. In any case,
the fact that the original metric agrees with the Euclidean one at the origin
in no way allows us to conclude that it is locally flat. As a final remark,
note that one still needs to implement the appropriate conformal factor
$\text{sech}^{2}(\rho)$ in front of the metric for it to be the actual metric
for $n\mathbb{CP}^{2}$.

After this whole discussion, one might be tempted to try to set up the Ricci
flow equations for (78), or an approximation at the origin thereof. This is
rather complicated. Most importantly, while we could explicitly compute the
function $F$, the one-form $\theta$ is arbitrary. This prevents us from giving
an explicit form of the metric, and thus from making explicit computations
with it. Because in the context of [Leb91], the explicit form of $\theta$ is
irrelevant, no further properties are given for it. If one were to drastically
simplify the metric (78), and assume both that $F$ is constant, and that
$p^{*}\theta=0$, then the metric that we would be dealing with would be very
similar to the Euclidean metric, or the Burns metric, with different
prefactors, which have simple dependences on the coordinate $r$. With this
very drastic oversimplification, we obtain that $R=2+8F$, which, by virtue of
$F$ being positive, is itself positive. This tells us that this approximation
is not terribly useful, since it contradicts the argument given around (76),
and thus renders the approximation invalid. We could try to relax the
conditions, and take $p^{*}\theta$ to have constant coefficients in the
$\\{dr,\sigma_{1},\sigma_{2}\\}$ basis for the base space. This indeed allows
us to force $R$ to vanish, depending on the value of the coefficients.
However, we are still approximating $F$ for a constant, when it should be a
function of the coordinates in the base. Hence, the approximation seems too
drastic to be meaningful.

###  6.7 Addendum: Why the behavior at the singularities may not matter

In the previous sections, we have tried to argue that the Ricci flow equations
make the whole $n\mathbb{CP}^{2}$ manifold, endowed with the metric (63)
shrink. While we were able to check that this does indeed happen almost
everywhere, we found some trouble on the set of points $\\{p_{i}\\}\subset
n\mathbb{CP}^{2}$, where the monopoles are located, and the original space was
blown up. In this section, we argue that the behaviour of the manifold at
these discrete points might not be all that important.

What we know for sure is that the Ricci flow makes
$n\mathbb{CP}^{2}\setminus\\{p_{i}\\}$ shrink indefinitely until, at the limit
where the flow ends, it shrinks to a point. Notice that we have used the word
limit here, since turning an $n$-dimensional manifold to a point is a
procedure that does great violence to almost any conceivable topological
property, and thus, cannot be a part of a smooth deformation. However, the
interpretation still stands. If we take the manifold to shrink to a point,
then taking so much care about the exact behavior at the monopole locations
might be pointless, since we are already heavily modifying the manifold.

Another argument that can support this claim is the following. Recall that, in
order to make sense of the idea of a manifold converging along the flow to
some other manifold, we introduced in section 4.3 the notion of smooth,
pointed convergence. In this context, we could pick precisely the set
$\\{p_{i}\\}$ of points along the flow. Indeed, since the curvature increases
much faster away from the monopole locations than around them, we expect necks
to form, and to then pinch off. In this sense, the Ricci flow would perform a
disconnected sum decomposition of $n\mathbb{CP}^{2}$, in such a way that it
would leave a point $p_{i}$ in each of the remaining components. Then, under
the notion of pointed convergence, we conclude that the manifold would
converge to a set of points, which are precisely the monopole locations.

##  7 Distances along the flow

The Swampland distance conjecture [OV07, Pal19] substantially asserts that
large distances in the effective field theory moduli space should be
accompanied by asymptotically massless towers of states, when quantum gravity
is taken into account. This intuition was further extended to the case of
Ricci flow in [KLL20], in which metric components themselves were regarded as
moduli of the theory. Exploiting the technical discussion presented [GM91],
the distance along a family of metrics $g\left(t\right)$ on a Riemannian
manifold $\mathcal{M}$, where $t$ is a real flow parameter, can be computed as

$\Delta\left(t_{1},t_{2}\right)=c\int_{t_{1}}^{t_{2}}\left(\frac{1}{\mathcal{V}_{\mathcal{M}}}\int_{\mathcal{M}}\sqrt{g}g^{\mu\nu}g^{\alpha\beta}\frac{\text{d}g_{\mu\alpha}}{\text{d}t}\frac{\text{d}g_{\nu\beta}}{\text{d}t}\right)^{\frac{1}{2}}\text{d}t\
,$ (79)

where $c$ is a real constant and $\mathcal{V}_{\mathcal{M}}$ is the volume of
$\mathcal{M}$. When the family is taken to follow Ricci flow, the above
reduces to:

$\Delta\left(t_{1},t_{2}\right)=2c\int_{t_{1}}^{t_{2}}\left(\frac{1}{\mathcal{V}_{\mathcal{M}}}\int_{\mathcal{M}}\sqrt{g}R_{\mu\nu}R^{\mu\nu}\right)^{\frac{1}{2}}\text{d}t\
.$ (80)

For a $D$-dimensional Einstein manifold, we get:

$\Delta\left(t_{1},t_{2}\right)\sim\log{\frac{R\left(t_{1}\right)}{R\left(t_{2}\right)}}\
.$ (81)

Therefore, it is clear that both flow singularities and flat spacetime fixed
points lie at infinite distance and should be accompanied by a tower of
asymptotically massless states. For a more general manifold, we can decompose
the Ricci scalar as

$R_{\mu\nu}=\frac{R}{D}g_{\mu\nu}+T_{\mu\nu}\ ,$ (82)

where $T_{\mu\nu}$ is its traceless part. Since

$R_{\mu\nu}R^{\mu\nu}=\frac{R^{2}}{D}+T_{\mu\nu}T^{\mu\nu}$ (83)

we obtain

$\Delta\left(t_{1},t_{2}\right)=2c\int_{t_{1}}^{t_{2}}\biggl{(}D^{-1}\braket{R^{2}}+\braket{T_{\mu\nu}T^{\mu\nu}}\biggr{)}^{\frac{1}{2}}\text{d}t\
,$ (84)

in which $\braket{A}$ referes to the average value of $A$ on $\mathcal{M}$.
Let’s assume that the flow encounters a singularity at $t=t_{s}$, so that
$R\to\infty$ for $t\to t_{s}$. In the broad set of situations in which it
implies $\braket{R^{2}}\to\infty$, we can expect that flow singularities sit
at an infinite distance in the moduli space. This is clearly the case for a
positive-curvature manifold shrinking to a point, as a $\mathbb{CP}^{2}$ term
in a disjoint union. It would be interesting to assess this in a more general
setting and investigate what singularity resolutions imply from the
perspective of the distance along the flow. The fact that they connect
cobordant manifolds gives hope that one could somehow assign a meaningful
moduli space length to the deformation connecting the manifold which is about
to pinch and the resolved one, even though a topology change is involved. We
expect that rephrasing (80) in a more general form, in which the deformation
parametrised by $t$ can generally connect cobordant manifolds, could shed some
light on the issue at hand. The $t$-integral, in that sense, would have to be
regarded as an integral over the extra coordinate of the manifold of which the
initial and the resolved one are boundaries. In particular, one would like to
assess whether singularity resolutions can avoid distance divergencies. This
would be consistent with the general expectation that quantum gravity
processes can naturally induce topology changes, implying that space-time
manifolds with different topological properties should belong to the same
moduli space. Further discussions will be required, but such an investigation
goes beyond the scope of the current work.

##  8 The ADE singularities

One of the most striking features of $K3$ surfaces, which is also very
relevant for physics applications, is that all of their singularities are
classified in terms of the ADE Lie groups. However, when one resolves these
singularities by blowing them up, all the information regarding the ADE
classification is lost. From the connected sum perspective, the manifold
$K3\\#\mathbb{CP}^{2}$, for example, contains no information about the
singularity which was replaced by a smooth manifold. In this section, we will
try to elucidate how to recover the ADE information from the blow-up of the
singularities. The idea is that the cobordism conjecture states that $K3$ is a
valid background only when it is accompanied by a series of blow-ups of its
singularities. This is sometimes solved by adding a series of defects on the
$K3$, sourcing fluxes. Then, it could be that we can reconstruct, or find some
restrictions on, the gauge groups associated to said fluxes.

###  8.1 A lightning fast overview of complex geometry

In this section we review/introduce some concepts in algebraic and complex
geometry which are needed to shape the discussion. They are not, however,
necessary to understand the general idea. We begin with the concept of a
holomorphic line bundle.

####  8.1.1 Holomorphic line bundles and the Picard group.

In general terms, a line bundle is simply a vector bundle or rank $1$, so the
fibers at any point are lines. Note the difference on the nature of the bundle
when its dimension is complex rather than real. We now give the definition of
a holomorphic vector bundle

######  Definition 8.1.

Let $X$ be a complex manifold. A holomorphic vector bundle of rank $r$ on $X$
is a complex manifold $E$ together with a holomorphic projection map $\pi:E\to
X$, and such that each fibre $E(x)\coloneqq\pi^{-1}(x)$ is an $r$-dimensional
complex vector space, which further satisfies the following condition: There
exists an open covering $X=\bigcup_{i}U_{i}$ and biholomorphic666A
biholomorphic function is a function $f:U\to\mathbb{C}^{n}$ which is both
holomorphic and injective. maps
$\psi_{i}:\pi^{-1}(U_{i})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}U_{i}\times\mathbb{C}^{r}$
commuting with the projections from the bundle to $U_{i}$ such that the
induced map
$\pi^{-1}(x)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathbb{C}^{r}$ is
complex linear.

The above definition os basically that of a regular vector bundle, but we
further require that the so-called local trivializations (the $\psi_{i}$
above) are holomorphic. Now, we can just define a holomorphic line bundle as a
holomorphic vector bundle of rank $r=1$. One last comment on this definition
is that the induced transition functions on the vector bundle

$\psi_{ij}(x)\coloneqq\left(\psi_{i}\circ\psi_{j}^{-1}\right)(x,\
):\mathbb{C}^{r}\to\mathbb{C}^{r}$ (85)

are complex linear for all $x\in U_{i}\cap U_{j}$. In the same way as for
differentiable real or complex bundles, a holomorphic rank $r$ vector bundle
is determined by the set of holomorphic cocycles $\\{U_{i},\psi_{ij}:U_{i}\to
GL(r,\mathbb{C})\\}$. This will be important in the following, when we
introduce the Picard and the Néron-Severi (NS) groups.

We now turn to a very specific (and unique) type of line bundle. Over an
$n$-dimensional complex projective space $\mathbb{CP}^{n}$, there is (up to
isomorphism) only one holomorphic line bundle. It is called the tautological
line bundle, and denoted $\mathcal{O}(1)$. Its dual is defined as follows:

######  Proposition 8.1.

The set $\mathcal{O}(-1)\subseteq\mathbb{CP}^{n}\times\mathbb{C}^{n+1}$,
consisting of all pairs $(l,z)\in\mathbb{CP}^{n}\times\mathbb{C}^{n+1}$ such
that $z\in l$ forms in a natural way a holomorphic line bundle over
$\mathbb{CP}^{n}$.

The proof of this proposition is not terribly complicated, and hence we will
skip it. The basic idea is to just take an element in $\mathbb{CP}^{n}$, which
is a complex line, and define the fibre over that point as the line itself.
Rigorously, it can be shown that the fibre $\mathcal{O}(-1)\to\mathbb{CP}^{n}$
over $l\in\mathbb{CP}^{n}$ is naturally isomorphic to
$l\subset\mathbb{C}^{n+1}$. The above construction gives rise to the following
definition

######  Definition 8.2.

The line bundle $\mathcal{O}(1)$ is defined as the dual of $\mathcal{O}(-1)$,
namely $\mathcal{O}(-1)^{*}$. This can be extended. For $k\in\mathbb{Z}^{+}$,
let $\mathcal{O}(k)$ be the line bundle
$\mathbb{O}(1)\otimes\stackrel{{\scriptstyle
k)}}{{\ldots}}\otimes\mathcal{O}(1)$. One defines $\mathcal{O}(-k)$ in an
analogous way. It is standard notation to write $E(k)\coloneqq
E\otimes\mathcal{O}(k)$ for any vector bundle $E\to\mathbb{CP}^{n}$.

Finally, if one defines
$\mathcal{O}(0)\coloneqq\mathbb{CP}^{n}\times\mathbb{C}$ (i.e. the trivial
line bundle), the tensor product and the dual endow the set of all isomorphism
classes of holomorphic line bundles over a complex manifold $X$ with the
structure of an Abelian group. Being that this groups is infinite cyclic, it
must be isomorphic to $\mathbb{Z}$. This is the Picard group of $X$,
$\text{Pic}(X)$.

There is a consequence of the above definition which will help us build up to
our end goal. It is the fact that $\text{Pic}(X)\simeq
H^{1}(X,\mathcal{O}_{X}^{*})$. Here, $\mathcal{O}_{X}^{*}$ denotes the
subsheaf of nonvanishing holomorphic functions on $X$. The precise definition
of a sheaf is not necessary here. To understand the general argument it is
enough to thing about it in terms of homology with coefficients which are
nonvanishing holomorphic functions. The proof for the previous statement,
which we skip here, relies on two facts. The first one is that line bundles
are described in terms of cocycles, given by the transition functions. These
are in fact not any type of cocycles, but fall under the classification of
Čech cocycles. The association is clear with the definition of the Čech
cohomology groups. We will skip this definition for reasons that will be clear
in a moment. Because of this definition in terms of cocycles, one has that
$\text{Pic}(X)\simeq\check{H}^{1}(X,\mathcal{O}_{X}^{*})$, where $\check{H}$
denotes Čech cohomology. The second fact that this proof relies on, is that
for $p=1$, the homomorphism $\check{H}^{p}(X,\mathcal{F})\to
H^{p}(X,\mathcal{F})$ is an isomorphism. Therefore, in the end, we have that
$\text{Pic}(X)\simeq H^{1}(X,\mathcal{O}_{X}^{*})$.

####  8.1.2 The Néron-Severi group.

We will now introduce one of the last concepts which are needed to start
discussing the blowup of singularities on $K3$ surfaces. It builds up from the
previous section in the sense that it is a construction which is defined in
terms of the Picard group of a complex manifold. This is the so-called Néron-
Severi group.

We beign by the fact that there is a short exact sequence of sheaves, called
the exponential sequence (for reasons that will be clear in a moment). It is
the following

$0\to\mathbb{Z}\to\mathcal{O}_{X}\to\mathcal{O}_{X}^{*}\to 0.$ (86)

Again, $\mathcal{O}_{X}$ denotes the sheaf of of holomorphic functions on $X$,
and $\mathcal{O}_{X}^{*}$ the subsheaf of nonvanishing holomorphic functions
on $X$. We may check for exactness as follows. The first map,
$\mathbb{Z}\to\mathcal{O}_{X}$ is an inclusion, so it is trivially injective.
The map in the middle is the one that gives the name to the sequence. It is
given by

$\displaystyle\exp:\mathcal{O}_{X}$ $\displaystyle\to\mathcal{O}_{X}^{*}$
$\displaystyle f$ $\displaystyle\mapsto e^{2\pi if}.$

Thus, it is clear that
$\text{im}(\mathbb{Z}\to\mathcal{O}_{X})=\ker(\mathcal{O}_{X}\to\mathcal{O}_{X}^{*})$.
Finally, surjectivity of $\mathcal{O}_{X}\to\mathcal{O}_{X}^{*}$ is granted by
the (local) existence of the complex logarithm.

The exponential sequence gives rise in the usual sense to a long exact
sequence in cohomology. We look in particular at

$\ldots\to H^{1}(X,\mathbb{Z})\to H^{1}(X,\mathcal{O}_{X})\to
H^{1}(X,\mathcal{O}^{*}_{X})\to H^{2}(X,\mathbb{Z})\to\ldots$ (87)

Inserting the Picard group explicitly, we have

$\ldots\to H^{1}(X,\mathbb{Z})\to H^{1}(X,\mathcal{O}_{X})\to\text{Pic}(x)\to
H^{2}(X,\mathbb{Z})\to\ldots$ (88)

The last map above is known, it is precisely the first Chern class
$c_{1}:\text{Pic}(X)\to H^{2}(X,\mathbb{Z})$. A map which is related to this
will give us the definition of the Néron-Severi group.

The canonical boundary map in the long exact sequence above, arising from the
exponential map, $\text{Pic}(X)\to H^{2}(X,\mathbb{Z})$ can be composed with
the map $H^{2}(X,\mathbb{Z})\to H^{2}(X,\mathbb{C})$, which is simply induced
by the inclusion $\mathbb{Z}\hookrightarrow\mathbb{C}$. On the other hand, a
standard result of Hodge theory is that, for compact Kähler manifolds, one has
the following decomposition

$H^{n}(X,\mathbb{C})=\bigoplus_{p+q=n}H^{p,q}(X)$ (89)

In particular, for $n=2$,

$H^{2}(X,\mathbb{C})=H^{2,0}(X)\oplus H^{1,1}(X)\oplus H^{0,2}(X)$ (90)

Before going further, note that due to complex conjugation on
$H^{n}(X,\mathbb{C})$, we have that $\overline{H^{p,q}}(X)=H^{q,p}(X)$. This,
combined with the fact that $\mathcal{O}_{X}$ is the sheaf of holomorphic
functions on the manifold, yields $H^{2}(X,\mathcal{O}_{X})=H^{0,2}(X)$.

With this information, we know that the long exact sequence given by the
exponential sequence (86) shows that the composition

$\text{Pic}(X)\to H^{2}(X,\mathbb{Z})\to H^{2}(X,\mathbb{C})\to
H^{2}(X,\mathcal{O}_{X})\left(=H^{0,2}(X)\right)$ (91)

is trivial, precisely because it fits into an exact sequence. Moreover, we
have that

$\text{Im}\left(\text{Pic}(X)\to
H^{2}(X,\mathbb{C})\right)\subseteq\text{Im}\left(H^{2}(X,\mathbb{Z})\to
H^{2}(X,\mathbb{C})\right)$

We also have that $\text{Im}\left(\text{Pic}(X)\to
H^{2}(X,\mathbb{C})\right)\subseteq\ker(H^{2}(X,\mathbb{C})\to H^{0,2}(X))$,
where the last map is the usual projection given by the Hodge decomposition.
Consider now $H^{2}(X,\mathbb{R})\subseteq H^{2}(X,\mathbb{C})$. It is
invariant under complex conjugation, and further contains
$\text{Im}(H^{2}(X,\mathbb{Z})\to H^{2}(X,\mathbb{C}))$, which must therefore
also be invariant under complex conjugation, and hence fall into $H^{1,1}(X)$
in the Hodge decomposition. As a conclusion of all this, one finds that the
image $\text{Pic}(X)\to H^{2}(X,\mathbb{C})$ is contained in the following
homology group

$H^{1,1}(X,\mathbb{Z})\coloneqq\left(\text{Im}(H^{2}(X,\mathbb{Z})\to
H^{2}(X,\mathbb{C}))\right)\cap H^{1,1}(X).$ (92)

The above definition is sometimes abbreviated as
$H^{1,1}(X,\mathbb{Z})\coloneqq H^{2}(X,\mathbb{Z})\cap H^{1,1}(X)$. It is a
standard result (Lefschetz theorem on $(1,1)$ classes), that for $X$ a compact
Kähler manifold, the map $\text{Pic}(X)\to H^{1,1}(X,\mathbb{Z})$ is
surjective.

To conclude this section, we look at a slightly different map from the Picard
group of a complex manifold. The image of the map $\text{Pic}(X)\to
H^{2}(X,\mathbb{R})$ is called the Néron-Severi group of the manifold $X$,
denoted $NS(X)$. It spans a finite dimensional real vector space $NS(X)\subset
H^{2}(X,\mathbb{R})\cap H^{1,1}(X)$. Because of the surjectivity of
$\text{Pic}(X)\to H^{1,1}(X,\mathbb{Z})$, it turns out that the natural
inclusion $NS(X)\subseteq H^{1,1}(X,\mathbb{Z})$ is actually an equality.
Finally, the rank of $NS(X)$ is denoted by $\rho(X)$ and called the Picard
number.

In the following, we will not work with general complex manifolds, but rather
with very specific and well behaved ones. In particular, we will deal with
Calabi-Yau manifolds, for which we have that $H^{1}(X,\mathcal{O}_{X})=0$.
This can in fact be read off from the Hodge diamond of any Calabi-Yau
$n$-fold. Because of the exact sequence (88), this means that
$c_{1}:\text{Pic}(X)\to H^{2}(X,Z)$ is injective. This in turn implies that

$\text{NS}(X)=\text{Pic}(X)$

Hence, once we introduce blow-ups from the perspective of algebraic geometry
and line bundles, it will follow that said blow-ups are described by the
Néron-Severi group of the manifold.

###  8.2 Blow-ups from algebraic geometry

In order to make a connection with the rest of the work, we will now introduce
the blowups of a $K3$ manifold in a perspective more akin to algebraic
geometry, and using the notions which we have introduced in the previous
chapter. Before going any further, we have to introduce a bit of nomenclature.

The first definition is that of a complex analytic hypersurface

######  Definition 8.3.

An analytic hypersurface of an analytic variety $X$ is an analytic subvariety
$Y\subset X$ of codimension one, namely $\dim(Y)=\dim(X)-1$.

Above, an analytic variety is a sort of generalization of complex manifolds,
which allows for singularities on the space. More concretely, a complex
analytic variety is locally given as the zero locus of a certain (finite) set
of holomorphic functions. In other words, it is the union of a certain set of
components, which are zero loci of holomorphic functions. In the same fashion,
a hypersurface inherits this property from its ambient variety. Hence, every
hypersurface $Y$ is the union $\cup_{i}Y_{i}$ of its so-called irreducible
components. If the ambient manifold $X$ is compact, this union is finite, but
such in general we only have that the union is locally finite777A locally
finite collection of subsets of a topological space (in this case
$\\{Y_{i}\\}$ as subsets of $X$) is one in which every point
$y\in\cup_{i}Y_{i}$ has a neighbourhood which intersects at most finitely many
elements of the collection . This very property is what allows us to
(pointwise) define a formal linear combination of hypersurfaces as follows

######  Definition 8.4.

Given $\\{Y_{i}\\}_{i\in I}$ a collection of irreducible hypersurfaces of $X$,
a divisor $D$ on $X$ is a formal linear combination

$D=\sum_{i\in I}a_{i}[Y_{i}]$ (93)

with $a_{i}\in\mathbb{Z}$.

Note that local finiteness implies that for any $x\in X$ there exists an open
neighbourhood of $x$ on which $a_{i}\neq 0$ and $Y_{i}\cap U\neq\emptyset$ for
only finitely many $i\in I$. In other words, the sum can be defined because it
is pointwise finite.

Given the previous sections, one can see the importance of introducing
hypersurfaces and divisors. We would like to understand complex manifolds with
tools as simple as possible. One such tool is the idea of line bundles, which
as we have seen can be studied by means of their cohomology. Now, if we
consider sections of said line bundles, it turns out that hypersurfaces are
always given as some global holomorphic zero section of a line bundle. This
makes sense, since the fibers of line bundles are by definition one
dimensional, and so their complement has codimension one. In fact, we state
without proof that there exists a natural group homomorphism between the group
of divisors of a complex manifodl $X$, $\text{Div}(X)$, and $\text{Pic}(X)$.

####  8.2.1 Blow-ups and line bundles

Previously, we have seen how to construct the blow-up of $\mathbb{C}^{2}$ at
the origin, by considering tangent lines at the origin. Now, we realize that
in order to do so, we had to define the blow-up of our original space in a
very similar way to how we later on defined holomorphic line bundles, in the
sense that in both cases we had subsets of
$\mathbb{CP}^{n}\times\mathbb{C}^{n+1}$. However, in one case we had a blow-up
of $\mathbb{C}^{n}$ at some point, and in the other a line bundle over
$\mathbb{CP}^{n}$. We will now see how both are related. Again, blow-ups can
be performed along complex subvarieties, but for the purposes of this work, we
will only consider blow-ups along points.

For a line bundle $\mathcal{O}(-1)\to\mathbb{CP}^{n}$ we can write down the
following diagram of maps

${\mathbb{CP}^{n}}$${\mathcal{O}(-1)}$${\mathbb{CP}^{n}\times\mathbb{C}^{n+1}}$${\mathbb{CP}^{n}}$${\\{0\\}}$${\mathbb{C}^{n+1}}$${\mathbb{C}^{n+1}}$$\scriptstyle{\sigma}$

The line at the top comprises the usual line bundle map, where the fibre over
a $l\in\mathbb{CP}^{n}$ is precisely the line itself. However, instead of
working with $\mathbb{CP}^{n}$ as the base, we can just as well use
$\sigma:\mathcal{O}(-1)\to\mathbb{C}^{n+1}$, while keeping the bundle
construction from before exactly the same. To make eveerythign match, we
define the fibres of $\sigma$ as follows. For $z\neq 0$, $\sigma^{-1}$ is the
unique line $l_{z}\in\mathbb{CP}^{n}$ that passes through
$z\in\mathbb{C}^{n+1}$. For $z=0$, this preimage must be
$\sigma^{-1}(0)=\mathbb{CP}^{n}$, since all lines contain the origin. As a
matter of fact, this preimage is the zero section of
$\mathcal{O}(1)\to\mathbb{CP}^{n}$, as it represents $l\mapsto(l,0)$, where we
are using the notation from our introduction of line bundles. This allows us
to define a blow-up as follows:

######  Definition 8.5.

The blow-up $\sigma:Bl_{0}(\mathbb{C}^{n+1})\to\mathbb{C}^{n+1}$ of
$\mathbb{C}^{n+1}$ at the origin is the holomorphic line bundle
$\mathcal{O}(-1)\to\mathbb{CP}^{n}$ together with the natural projection
$\sigma:\mathcal{O}(-1)\to\mathbb{C}^{n+1}$

For the sake of completeness, we will now give the general definition of a
blow-up along an arbitrary submanifold.

######  Definition 8.6.

Let $Y$ be a complex submanifold of $X$. Then there exists a complex manifold
$\hat{X}=Bl_{Y}(X)$, the blow-up of $X$ along $Y$, together with a holomorphic
map $\sigma:\hat{X}\to X$ such that
$\sigma:\hat{X}\setminus[\sigma^{-1}(Y)]\simeq X\setminus Y$, and
$\sigma:\sigma^{-1}(Y)\to Y$ is isomorphic to
$\mathbb{P}(\mathcal{N}_{Y\setminus X})\to Y$.

Above, $\mathcal{N}_{X}$ denotes the normal bundle of $X$. The two conditions
above may be interpreted as follows. Away from the submanifold along which the
blow-up is performed, the procedure does nothing. On the contrary, this
submanifold is replaced by (a bundle map from) the projectivization of the
normal bundle of that submanifold. If we again consider the particular case of
$Y=\\{x_{0}\\}$, and use the fact that
$\mathbb{P}(\mathbb{C}^{n})=\mathbb{CP}^{n-1}$, we recover the definition from
above.

######  Definition 8.7.

The hypersurface $\sigma^{-1}=\mathbb{P}(\mathcal{N}_{Y\setminus X})\subseteq
Bl_{Y}(X)$ is called the exceptional divisor of the blow-up
$\sigma:Bl_{Y}(X)\to X$

In the particular case that we are dealing with, in which $Y$ is just a point
in an $n$ dimensional manifold $x\in X$, then the blow-up replaces $x$ by
$\mathbb{CP}^{n-1}$. The exceptional divisor $\sigma^{-1}(x)$ is usually
denoted $E$.

###  8.3 The ADE classification of $K3$ singularities

In this short section, we briefly review how singularities are classified on a
$K3$ surface. In contrast with other sections, we will not go into much
detail, as a lot of these concepts are known from $F$-theory. A standard
reference for this is [Wei18].

In the context of $F$-theory, one considers spacetime compactifications of
type IIB string theory of the following form

$\mathcal{M}^{1,9}=\mathbb{R}^{1,9-2n}\times B_{n}$ (94)

with $B_{n}$ a manifold of (complex) dimension $n$. One further considers
7-branes which fill the $\mathbb{R}^{1,9-2n}$ factor, and that wrap an
internal holomorphic cycle $\Sigma_{n-1}\subset B_{n}$ of codimension one. The
7-branes imply the appeareance of the axion $C_{0}$ in the theory which,
together with the dilaton, combine to give a complex field, the axio-dilaton

$\tau=C_{0}+ie^{-\varphi}.$ (95)

Because of the nontrivial variation of the axio-dilaton around the
$D7$-branes, and the fact that $R_{ab}=\nabla_{a}\nabla_{b}\varphi\neq 0$, we
know that $B_{n}$ cannot be Calabi-Yau. However, it turns out that this
variation of $\tau$ can be used to define an elliptic fibration over $B_{n}$.
An elliptic fibration is one in which each fibre has the structure of a torus
with a marked base point (an elliptic curve)

${\pi:\mathbb{E}_{\tau}}$${Y_{n+1}}$${B_{n}}$

Here, the fact that each fiber $\pi^{-1}(b)$ for $b\in B_{n}$ has a marked
point is represented in the requirement that there exists a section
$\sigma:B_{n}\to Y_{n+1}$. The image of the section at each point in the base
is the marking of the fiber. It turns out, by calculations regarding the first
Chern classes of both $Y_{n+1}$ and $B_{n}$, that $c_{1}(Y_{n+1})=0$, and
hence $Y_{n+1}$ is Calabi-Yau. In the case of $n=1$, we have that $Y_{2}=K3$,
and $B_{1}=\mathbb{CP}^{1}$. Note that the $K3$ here is generic. For the
remainder of the work, we will stay in this context.

We will now see how singularities may occur on an elliptic fibration, and how
they are classified. We will try to be as simple as possible here. Most often,
elliptic fibrations are described by a so-called Weierstrass model. The
construction is as follows. First, one considers a special type of projective
space of dimension 3, namely $\mathbb{P}_{231}$, defined as

$\mathbb{P}_{231}\coloneqq\frac{\mathbb{C}^{3}\setminus\\{0\\}}{\sim},\ \ \ \
\ \ \ \text{where }(x,y,z)\sim(\lambda^{2}x,\lambda^{3}y,\lambda z)\ \
\text{for }\lambda\in\mathbb{C}^{*}$ (96)

Then, the elliptic curve is defined pointwise as the zero-locus on
$\mathbb{P}_{231}$ of

$P=y^{2}-(x^{3}+fxz^{4}+gz^{6})$ (97)

Namely, the elliptic curve is defined as the hypersurface $P=0$. Above, $f$
and $g$ are technically understood as sections of a line bundle over the base
$B_{n}$, namely

$\displaystyle f$ $\displaystyle\in\Gamma(B_{n},\mathcal{L}^{4})$
$\displaystyle g$ $\displaystyle\in\Gamma(B_{n},\mathcal{L}^{6})$

Where powers of a line bundle are understood as coming form tensor products
with itself, as in the definition of $\mathcal{O}(k)$. This technicality
arises because both $f$ and $g$ are related to the structure constant $\tau$
of the marked torus which defines the elliptic curve. They thus have certain
transformation properties under $SL(2,\mathbb{Z})$, the isometry group of the
torus, and this is reflected in their definition as sections. In more
practical terms, $f$ and $g$ can be thought of as coefficients at each point
in the base, which define a particular elliptic curve at that point through
(97). We would also like to stress the fact that this strange looking
polynomial is highly non-unique, and depending on certain mathematical details
(such as the characteristic of the field over which the curve is defined), it
can be brought to simpler forms. However, we would like to stick to the
notation used in [Wei18], as it is a very popular reference.

With the Weierstrass model in mind, we will now see what it means for a
singularity to occur, and then we will discuss the sense in which they are
classified. We say that a hypersurface $P=0$ is singular if its gradient also
vanishes. That is, if we both have that $P=0$ and $dP=0$. Let us see what this
means for the Weirstrass model. First, we check that the singularity cannot
happen at $z=0$, since then $P=0$ would reduce to $y^{2}=x^{3}$, whose
gradient is $2y=3x^{2}$. Therefore, the only possible singularity would be at
$x=y=z=0$, which is a point that is not on $\mathbb{P}_{231}$. By the
rescaling properties of this projective space, we can thus restrict ourselves
to the case where $z=1$, where the Weierstrass equation $P=0$ can be written
as

$y^{2}=x^{3}+fx+g.$ (98)

The left hand side is just a degree 3 polynomial, which in these conditions
can be decomposed as

$y^{2}=K\prod_{i=1}^{3}(x-x_{i}),$ (99)

where $\\{x_{i}\\}$ are the roots. A simple calculation of the gradient as
above reveals that a singularity occurs if and only if a root has multiplicity
greater than 1. On the other hand, since we are dealing with a degree 3
polynomial, this only happens when its discriminant $\Delta$ vanishes. For
this polynomial, the explicit form of the determinant is

$\Delta=4f^{3}+27g^{2}.$ (100)

There is a bit more to the singularities of the Weierstrass model. In the
general case, only $\Delta$ vanishes, and $f$ and $g$ take whichever nonzero
values needed for this to happen. However, it can happen that the zero of the
discriminant is also a zero of both $f$ and $g$ as functions of the base
coordinate. In this case, the singularity is enhanced, and the Weierstrass
model becomes itself singular (whereas before, even if $\Delta=0$ at some
points, the whole elliptic fibration $Y_{n+1}$ is still smooth).

####  8.3.1 The ADE classification

In principle, an elliptically fibered $K3$ may present many types of
singularities, depending on how $\Delta$, $f$, and $g$ vanish. Fortunately,
there is a way of classifying these singularities, first introduced in two
landmark works by Kodaira [Kod63] and Néron [Nér64]. We will now briefly
discuss this classification, in order to make our latter arguments clearer.

First, we should mention that the classification in question is over smooth,
minimal surfaces. The reason for smoothness, is that any singular surface can
be made smooth by means of so-called birational maps, of which blow-ups (as in
definition 8.6) are a particular example. This means that we can ”smooth out
surfaces in a non-intrusive way”. Thus, once smooth surfaces are classified,
the non-smooth ones are too. The second condition is minimality. This is
somewhat related to the last condition. We say that a smooth surface is
minimal whenever it is not birationally equivalent (there is a birational map
between them) to another smooth surface. In practical terms, it means that a
minimal smooth surface is one which cannot be obtained by blowing up smooth
points of some other smooth surface.

Once this technicality is understood, we proceed with the classification of
$K3$ singularities. Let $X$ denote some elliptically fibered $K3$ surface,
with a certain number of singular points $\\{p_{1},\ldots,p_{n}\\}$, which we
take to be distinct. In order to obtain the resolution of $X$, which we denote
by $\overline{X}$, we need to perform $m$ blow-ups, where $n\leqslant m$888It
might be that a certain singular point has a higher order singularity and
needs to be blown up more than once. This, as we have said before, produces a
set of $m$ total exceptional divisors $\\{E_{1},\ldots,E_{m}\\}$. Given the
dimensionality, these $E_{i}$ are nothing but rational curves, isomorphic to
$\mathbb{CP}^{1}$. These are sometimes referred to as degenerate fibers, as
over the hypersurface defined by $\\{\Delta=0\\}$, they are degenerate.

Each of the curves that are produced in the blow-up can not only intersect
itself, but it can also intersect one or more of the other curves. When we use
the term intersect, we mean in the usual way in which cycles intersect in
homology theory. This is captured in the intersection product, which assigns
to any two curves $C_{i}$ and $C_{j}$ an integer, $C_{i}\cdot
C_{j}\in\mathbb{Z}$. We will come back to the intersection structure of a $K3$
in a later section. The punchline, and the basis for the classification of ADE
singularities, is the profoundly shocking fact that the intersection structure
of these rational curves exactly mimics one of the extended Dynkin diagrams of
an ADE Lie algebra. Recall that the Dynkin diagram associated to a Lie algebra
is constructed from its simple roots $\\{\alpha_{i}\\}$, with
$i=1,\ldots,\text{rk}(\mathfrak{g})$, by a pairing
$d_{ij}=\langle\alpha_{i},\alpha_{j}\rangle\langle\alpha_{j},\alpha_{i}\rangle$.
Because of technical details, which are discussed in [Wei18], this leads us to
identify $E_{i}\sim(-\alpha_{i})$, for $i=1,\ldots,\text{rk}(\mathfrak{g})$.

The above relation can be thought of in a very pictoric and intuitive way. If
we use the fact that $\mathbb{CP}^{1}\simeq\mathbb{S}^{2}$, the Riemann
sphere, a blow-up (or successive blow-ups) replaces singular points with
Riemann spheres. The intersection of rational curves from above translates
into the intersection of these Riemann spheres at a certain (finite) number of
points. Because of the identification of these spheres with the (negative of)
the simple roots of some ADE algebra, the spheres arrange themselves literally
like in a Dynkin diagram.

There are certainly some subtleties about this classification, and some
singularities whose rational curves do not straightforwardly reproduce an
extended Dynkin diagram, but can nevertheless still be classified by an ADE
group. For our purposes, and the extension of this work, it is enough to have
an intuitive idea of how this classification arises, and the mathematics that
are relevant to it. We redirect the interested reader to [MP89, SS09, Wei18],
the latter one being the standard refence for $F$-theory in physics. We also
note that a full table with the classification of the ADE singulatrities of an
elliptically fibered $K3$ can be found on [Wei18, Table 4.1].

##  9 Recovering the ADE singularities from the blow-ups

Now that we have introduced the necessary mathematics to properly discuss the
topic, and we have a general idea of how singularities occur and are
classified on an elliptically fibered $K3$, it is time to go backwards. In the
end, our goal is to relate all this to the cobordism conjecture, and try to
draw some restrictions on the allowed backgrounds for our theories of quantum
gravity. The problem here is that cobordisms are, in a sense, ”only sensitive
to the topology of a manifold”, and not even completely at that. By this we
mean that the bordism class of a manifold can be affected by such things as
characteristic numbers (the Pontrjagin and Stiefel-Whitney numbers),
tangential structures, such as in [ACC22], etc. However, the main appeal of
bordisms is that they may relate two manifolds whose topologies differ
completely. One explicit example of this is given in appendix 12, where we
check that, while $\mathbb{CP}^{2}\\#\overline{\mathbb{CP}}^{2}$ and
$\mathbb{CP}^{2}\sqcup\overline{\mathbb{CP}}^{2}$ are bordant, their homology
groups differ.

Throughout the last sections, we have seen that line bundles could be
understood through the Picard group of a manifold, and then, through its
Néron-Severi group. In particular, this leads us to the conclusion that blow-
ups, defined in terms of holomorphic line bundles, may be described in the
same terms. Since the ADE classification of $K3$ singularities is based on how
the exceptional divisors coming from the blow-ups of the singularities
intersect, it follows that some aspects of the ADE classification of a $K3$
have to do with the cohomology structure of the manifold. This is really a
problem, as these are liable to change under bordisms. In particular, a
bordism relation (or the bordism class of a manifold) tells us nothing about
its homology structute. It follows that, in order to fully specify the ADE
structure, or the exact allowed gauge groups for backgrounds which are allowed
by the cobordism conjecture, one needs to add some extra information to what
is given by the cobordism conjecture. In particular, from the discussion in
the last sections, we would need to specify the way in which the exceptional
divisors coming from the blow-ups of the singularities intersect. In this
section, we will see what kind of restrictions on the allowed gauge groups are
obtainable from this perspective.

###  9.1 Setting up the stage

In previous introductory sections, we have seen how blow-ups can be understood
in terms of holomorphic line bundles, and how these can be in turn understood
from the Néron-Severi group of the manifold $X$ which has been blown up.
Moreover, we have stated that
$\text{NS}(X)=H^{1,1}(X,\mathbb{Z})=H^{2}(X,\mathbb{Z})\cap H^{1,1}(X)$. We
can draw two conclusions from here: the first is that $\text{NS}(X)$ will
inherit a lattice structure from $H^{2}(X,\mathbb{Z})$, and the second is that
$\rho(X)=\text{rk}(\text{NS}(X))\leqslant 20$. The second conclusion comes
from knowing that the dimensionality of the cohomology groups of a $K3$
surface is completely fixed, and encapsulated in the Hodge diamond

$\begin{matrix}&&1&&\\\ &0&&0&\\\ 1&&20&&1\\\ &0&&0&\\\ &&1&&\\\ \end{matrix}$
(101)

In particular, we have that
$\text{rk}(H^{2}(X,\mathbb{Z}))=h^{2,0}+h^{1,1}+h^{0,2}=22$, whereas
$\text{rk}(H^{1,1}(X))\coloneqq h^{1,1}=20$. The reason why $h^{1,1}$ only
provides an upper bound on $\rho(X)$ is that one is allowed to consider
complex structure deformations on a $K3$. These change the complex structure
of the manifold, which in turn modifies the decomposition (90). Thus, elements
of $\text{NS}(X)$ could in principle pick up $(2,0)$ or $(0,2)$ components,
and hence not belong to $\text{NS}(X)$ anymore. Thus, the best that we can say
about the rank of the Néron-Severi group is that $\rho(X)\leqslant 20$.

Now we come to discussing lattice structures, and how $\text{NS}(X)$ inherits
one. This is a consequence of the fact that it is a subgroup of
$H^{2}(X,\mathbb{Z})$, which can itself be given the structure of an even,
self-dual lattice. In general, we recall that a lattice is essentially a
module999For present purposes, one could substitute ”module” by ”vector space”
over $\mathbb{Z}$ equipped with a non-degenerate symmetric bilinear form

$\langle\cdot,\cdot\rangle:L\times L\to\mathbb{Z}.$ (102)

We now proceed to explain how this comes about. Because of Poincaré duality,
and the dimension of the manifolds we are working with, this can be seen
either form the point of view of homology or cohomology. From the above
discussion, we know that $H^{2}(X,\mathbb{Z})\simeq\mathbb{Z}^{22}$ as a
group, so as a module it will have finite rank. Because we are working with
the so-called middle cohomology101010Namely, we have a manifold of dimension
$d=4k$, and the (co)homology groups taken into account are $H^{2k}$., the cup
product of forms (which in the case of de Rham cohomology is the wedge
product) endows $H^{2}(X,\mathbb{Z})$ with a symmetric, bilinear form

$\begin{split}\langle\cdot,\cdot\rangle:H^{2}(X,\mathbb{Z})\times
H^{2}(X,\mathbb{Z})&\to H^{4}(X,\mathbb{Z})\simeq\mathbb{Z}\\\
(\alpha,\beta)&\mapsto\langle\alpha,\beta\rangle=\int_{X}\alpha\wedge\beta\end{split}$
(103)

where $H^{4}(X,\mathbb{Z})\simeq\mathbb{Z}$ follows form Poincaré duality. The
above defined pairing is usually referred to as the intersection pairing. In
the dual homology picture, this pairing is precisely the one given by the
intersection number, where the pairing between two cycles $a,b\in
H_{2}(X,\mathbb{Z})$ is given by

$a\cdot b=\\#(a\cap b),$

where the symbol $\\#$ denotes the cardinality of the set. With any of these
two dual pairings, we can endow $H^{2}(X\mathbb{Z})$ with the structure of a
lattice, which we shall denote by $\Lambda_{K3}$. We have now come full
circle. The starting point of this work is precisely that the Hirzebruch
signature theorem states that the signature of this lattice is
$\sigma(K3)=-16$. Since its rank is 22, the signature of the lattice must
decompose as $(3,19)$.

The remaining properties of $\Lambda_{K3}$ can be seen as follows. From
Poincaré duality, we know that for any basis $\\{e_{i}\\}$ of 2-forms, we can
find a dual basis $\\{e^{*}_{j}\\}$ such that

$e_{i}\cdot e^{*}_{j}=\delta_{ij}$

Again, by duality, we were able to define a basis for $H^{2}(X,\mathbb{Z})$
from $\\{e_{i}\\}$. Thus, $\Lambda_{K3}$ is self-dual. Lastly, $\Lambda_{K3}$
is even, meaning that for any $\alpha\in H^{2}(X,\mathbb{Z})$, we have that

$\langle\alpha,\alpha\rangle\in 2\mathbb{Z}.$ (104)

In the dual picture, for $a\in H_{2}(X,\mathbb{Z})$, we have that

$a^{2}\coloneqq a\cdot a\in 2\mathbb{Z}.$ (105)

Rather surprisingly, this comes from the fact that $K3$ is spin, and thus
$c_{1}(K3)^{2}=c_{2}(K3)=0$ (mod 2). This is just a restatement of the fact
that the first and second Stiefel-Whitney classes $w_{1}$ and $W_{2}$ vanish.
The proof of this is complicated, and relies on Wu’s formula.

The requirement that $\Lambda_{K3}$ has to be even and self-dual severely
restrict the possibilities for the lattice. Combining this with the signature
and rank of the lattice, we know from intersection theory that the only
(unique up to isometries with respect to the inner product defined above)
choice is

$\Lambda_{K3}=(-E_{8})^{\oplus 2}\oplus H^{\oplus 3}$ (106)

Above, $E_{8}$ denotes the root lattice associated to the Lie algebra of the
same name, and $H$ represents the unique rank 2 hyperbolic lattice. With this
explicit form of the $K3$ lattice, its intersection form, which governs the
inner product defined above, is box-diagonal. Two of these boxes are $8\times
8$ matrices given by minus the Cartan matrix associated to $E_{8}$ (hence the
minus sign in (106)), and the three remaining boxes each contain the
intersection form associated to $H$, which is the following $2\times 2$
matrix111111Note that we are denoting both the lattice and the intersection
form by the same name:

$H=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}$ (107)

After having briefly described $\Lambda_{K3}$, the lattice defined by
$H^{2}(X,\mathbb{Z})$, we move on to seeing how $\text{NS}(X)\subset
H^{2}(X,\mathbb{Z})$ inherits a lattice structure. As we have stated before,
the possibility of deforming the lattice structure of the surface means that
we can only give an upper bound on the rank of this lattice, namely
$\rho(X)\leqslant 20$. Because of the natural inclusion, the symmetric
bilinear pairing on $\text{NS}(X)$ is exactly the same as the one defined on
$H^{2}(X,\mathbb{Z})$ by the intersection product, but just restricted to
$\text{NS}(X)$. The only thing left that we can determine about the Néron-
Severi lattice is the signature of its intersection form. The answer to this
is given by the Hodge index theorem [HB05, Corollary 3.3.16], which we repeat
here

######  Proposition 9.1.

Let $X$ be a compact Kähler surface. Then the intersection pairing (103) has
index $(2h^{2,0}+1,h^{1,1}-1)$. Restricted to $H^{1,1}$ it is of index
$(1,h^{1,1}-1)$

We will not provide a proof, but just note in passing that the offset of $1$
in both indices has its roots in the appeareance of the Kähler class
$[\omega]\in H^{1,1}(X)$. As a consequence of this, given that
$\text{NS}(X)=H^{2}(X,\mathbb{Z})\cap H^{1,1}(X)$, we have that the signature
of the Néron-Severi lattice is $(1,\rho-1)$

###  9.2 Introducing the blow-ups

The only information that the cobordism conjecture gives us, working in
$\Omega_{4}^{SO}$, is that for a generic $K3$ to be a valid background, it
must be accompanied by a certain number of copies of $\mathbb{CP}^{2}$, which
we interpret (through passing from disjoint unions to connected sums under the
bordism relation) as coming from a number of blow-ups performed on ADE
singularities on the original $K3$. We note in passing that a connected sum
$\mathbb{CP}^{2}\\#\stackrel{{\scriptstyle n)}}{{\ldots}}\\#\mathbb{CP}^{2}$
is interpreted as coming from resolution of $m$ points $p_{1},\ldots,p_{m}$ in
the manifold, for $m\leqslant n$. While we know exactly how many blow-ups are
needed, we will keep the number arbitrary for the time being. Now, we need to
see how these blow-ups might relate to the ADE structure of the original $K3$.
In the following, we will name the original $K3$ (with singularities) $X$, and
the resolved $K3$ will be called $\hat{X}$.

As we have seen before, blow-ups are understood in terms of line bundles. For
each blow-up, we produce an exceptional divisor $E_{i}$ on the blown-up
manifold $\hat{X}$, as in definition 8.7. Each of these exceptional divisors
is given by a $\mathbb{CP}^{1}$, and so they are rational curves. Thus, if we
blow up $n$ points, we will end up with $n$ exceptional divisors
$E_{1},\ldots,E_{n}$, which should fit in the Néron-Severi lattice of the
blow-up surface, $NS(\hat{X})$. This, in particular, forces them to satisfy
(105). The fact that they do is guaranteed by the so-called adjunction
formula. This formula deals with the canonical bundle of a manifold and that
of a hypersurface which is embedded in said manifold. In particular, for a
curve $C$ of genus $g$ embedded on a surface $Y$ with canonical
divisor121212The canonical divisor is the divisor associated to the canonical
(line) bundle of a manifold. $K$, it states that

$C\cdot\left(C+K\right)=2g-2=-\chi(C)$ (108)

However, since we are dealing with a Calabi-Yau, the canonical bundle is by
definition trivial, so its canonical divisor is $K=0$, and thus we have that
the self-intersection of a curve is given by its genus $g$. Since
$\mathbb{CP}^{1}\simeq\mathbb{S}^{2}$, $g=0$, and hence

$E_{i}\cdot E_{i}=-2$ (109)

so all of these exceptional divisors fit nicely into
$\text{NS}(\hat{X})\subset\Lambda_{K3}$.

In summary, we have seen that all of the exceptional divisors are algebraic
curves of self-intersection $-2$, which fit into the Néron-Severi latice of
$\hat{X}$. Therefore, they themselves span an even sublattice
$M\coloneqq\text{Span}_{\mathbb{Z}}\\{E_{1},\ldots,E_{n}\\}\subset\text{NS}(X)$.
The ADE classification that we introduced in section 8.3.1, which had to do
with how these exceptional divisors intersect, is now reflected in the fact
that $M$ must be isometric to a direct sum of lattices of the same ADE type as
the original singularities (i.e. given by the Cartan matrices of said
algebras). Since isometry type of a lattice is determined by its intersection
form, this is the same as requiring the singular fibers to intersect in a way
that reproduces the extended Dynkin diagram of the ADE algebras. All in all,
the lattice $M$ spanned by the exceptional divisors induces a decomposition of
the Néron-Severi lattice as $\text{NS}(X)=M\oplus M^{\perp}$, whose rank is
$n\leqslant\text{rk}(M^{\perp})\leqslant 20$.

There is however a problem with this line of thought, and it is the fact that
we have no information about the ADE type of the singularities. Said in
another way, we do not know how the exceptional divisors intersect, or the
isometry type of $M$. The cobordism conjecture says nothing about this, but
only about the number of exceptional divisors that are present. Thus, it
follows that we would need to add information by hand in order to distinguish
the type of ADE algebras to which the singularities correspond. The way of
doing this might be through lattice polarizations131313D.M. would like to
thank Michael Schultz for discussion on these notions.

###  9.3 Lattice polarizations

The answer to endowing the sub-lattice $M$ with the information necessary to
recover the ADE singularities might come from lattice polarizations. The idea
underlying this is to specify the way in which the Néron-Severi lattice embeds
into $\Lambda_{K3}$. In turn, this specifies the intersection structure of the
exceptional divisors, and hence the ADE singularities.

######  Definition 9.1.

Let $L$ be an even, non degenerate lattice of signature $(1,t)$. An
$L$-polarized $K3$ surface is a $K3$ surface $Y$ together with a primitive
lattice embedding $i:L\hookrightarrow\text{NS}(X)$ such that $i(L)$ contains a
pseudo-ample class.

######  Definition 9.2.

A sublattice $L\subseteq M$ is said to be primitive if the quotient $M/L$ is a
torsion-free Abelian group. A lattice embedding $i:L\hookrightarrow M$ is said
to be primitive if $i(L)$ is a primitive sublattice of $M$. Presently, this
implies that the short exact sequence of Abelian groups $0\to L\to M\to M/L\to
0$ always splits, and hence there is an isomorphism of Abelian groups

$M\simeq L\oplus(M/L)$ (110)

As we have seen in the previous section, the exceptional divisors span an even
sublattice $M\subset\text{NS}(X)$. This then leads to a decomposition of the