# Algebro-geometrical orientifold and IR dualities

Federico<EMAIL_ADDRESS>Mathematics and Physics
department, Roma Tre, Via della Vasca Navale 84, Rome, Italy  INFN Roma Tre
Section, Physics department, Via della Vasca Navale 84, Rome, Italy

###### Abstract

Orientifold projections are an important ingredient in geometrical engineering
of Quantum Field Theory. However, an orientifold can break down the
superconformal symmetry and no new superconformal fixed points are admitted
(II scenario); nevertheless, in some cases, dubbed I and III scenarios
orientifold, a new IR fixed point is achieved and, for III scenario examples,
some still not fully understood IR duality seems to emerge. Here we give an
algebro-geometrical point of view of III scenario orientifold for toric
varieties and we propose the existence of relevant operators that deform the
starting oriented CFT triggering a flow. We briefly discuss a possible
holographic description of this flow.

###### keywords:

Non-perturbative orientifold , AdS/CFT , Toric geometry , IR dualities.

††journal: Communications in Theoretical Physics

###### Contents

  1. 1 Introduction
  2. 2 Background material: toric geometry and orientifold
    1. 2.1 Toric varieties and toric Calabi-Yau
      1. 2.1.1 Homogeneous coordinates approach to toric varieties
      2. 2.1.2 Moment maps approach and Delzant-like construction
    2. 2.2 Orientifold Projections
      1. 2.2.1 The I scenario orientifold
      2. 2.2.2 The II scenario orientifold
      3. 2.2.3 The III scenario orientifold and the IR duality
  3. 3 Algebro-geometrical orientifold and IR dualities
    1. 3.1 Different number of external points: $\Delta P_{e}=1$
    2. 3.2 Equal number of external points: $\Delta P_{e}=0$
  4. 4 Brief discussion on the holographic dual
    1. 4.1 The toy model for holographic duals of RG flows and beyond
  5. 5 Conclusion

##  1 Introduction

AdS/CFT correspondence is a masterpiece of modern physics: its impact in
linking physics and geometry has opened new doors and laid the foundations for
new research fields. The original Maldacena AdS/CFT correspondence [1] can be
extended to phenomenologically interesting constructions that require $4D$
SUSY gauge theories with $\mathcal{N}=1$ or $\mathcal{N}=2$. The starting
point is to embed our stack of $N$ D$3$-branes into a background space-time of
the form $\mathbb{M}^{3,1}\times C(X_{5})$ where $C(X_{5})$ is a Calabi-Yau
(CY) cone over a Sasaki-Einstein (SE) manifold $X_{5}$. The geometry
properties of the CY define the field theory on the branes but for general
Calabi-Yau cone it is hard to build up the worldvolume gauge theory. We
restrict ourselves to the class of toric CY [2],[3],[4] since this type of
varieties allows an algorithmic construction of the field theories. The first
exemple of this kind of AdS/CFT extensions is due to Klebanov and Witten in
their 1998 pioneering work [5]. The toric condition simplifies a lot the
geometric description of the Calabi-Yau cone, indeed all the information is
stored into the toric diagram, as we will review in the Paragraph 2.1, and the
emerging field theories can be studied using well established brane tiling
techniques [6],[7],[8],[9],[10],[11],[12].

In Superconformal Quantum Field Theories (SQFTs), that emerge from the
geometrical engineering with toric CY cone, we can consider the so-called
orientifold projections [13],[14],[15] that are reviewed in Paragraph 2.2. The
action in the string theory side is to make open oriented string unoriented
and, according to a Chan-Paton-like analysis, real group such as $SO(N)$ and
$Sp(2N)$ are allowed in the emerging SQFTs. In 1995 Polchinski realized [16]
that orientifold projections have a simple and elegant interpretation from the
point of view of the background space-time: they correspond to not dynamical,
mirror-like objects called orientifold planes, defined by $T$-duality as fixed
points of the orientifold projections. On the field theory side, orientifold
projection affects fields and superpotential, and its effect can be understood
using the brane tiling picture [17].

Let us indicate orientifold with $\Omega$, therefore quantities after
orientifold projection will be labelled with the apex $\Omega$. The theory
before orientifold is often referred to as the "parent", "oriented", "pre-
orientifold" or "unorientifolded" theory while the one after orientifold as
"daughter", "unoriented", "post-orientifold" or "orientifolded" theory.
Similar names hold for quantities in the theories.
The presence of orientifold planes modifies the Renormalization Group (RG)
flow, and two different scenarios are mostly investigated in literature. On
the one hand, in the I scenario there is a new fixed point and the $R$-charges
of operators that are not projected out are the same as the charges of the
corresponding parent theory in the large $N$ limit; this give us a central
charge $a^{\Omega}$ that is half the central charge of the parent theory. On
the other hand, in the II scenario the daughter theory does not have a fixed
point.

Despite this, recently [18],[19], a new possibility seems to enter in the
game: the so-called III scenario. In this case it is possible [18],[19] to
construct an orientifold projection such that the central charge $a^{\Omega}$
is less than half of the central charge $a$ of the parent theory; moreover,
the $R$-charges before and after the orientifold are not the same. The
interesting behavior is that the values of the $R$-charges and central charge
coincide, beyond the large $N$ limit, with those of another theory that has
the I scenario orientifold. This seems to suggest an IR duality which is in
the IR regime because the central charge $a$ gives us, due to $c$-theorem in
$4D$, an RG flow ordering of theories; therefore we know that the I and the
III scenario orientifolds are lowering the energy moving the theories into the
IR regime.

Here we present an algebraic-geometrical interpretation of orientifold,
motivated by Greene’s flop transition and the search for a quantum geometry
[20],[21],[22],[23], as morphisms between algebraic varieties: orientifolds
are maps that act both on the states of the theory and on the Calabi-Yau cones
geometry. In Paragraph 2.1 we review toric varieties and toric CY discussing
the construction of manifolds as algebraic subvariety of the complex space
while in Paragraph 2.2 we review orientifold projection and its possible
scenarios. In Sections 3 and 4 we discuss the algebro-geometrical
interpretation of orientifold and the possible way RG flows triggered by the
orientifold itself can be holographically interpreted.

##  2 Background material: toric geometry and orientifold

###  2.1 Toric varieties and toric Calabi-Yau

Toric geometry is a branch of algebraic geometry and a toric variety is, by
definition, an algebraic variety containing an algebraic torus as an open
dense subset, such that the action of the torus on itself extends to the whole
variety. To be more specific, an $n$-dimensional toric variety $\mathcal{M}$
has an algebraic torus action in the sense that the algebraic torus
$\mathbb{T}^{n}=(\mathbb{C}^{*})^{n}$ is a dense open subset and there is an
action $\mathbb{T}^{n}\times\mathcal{M}\rightarrow\mathcal{M}$. The greatest
point in favor of toric geometry is that geometry of a toric variety is fully
determined by combinatorics.
In the case of a toric Calabi-Yau variety the information about the geometry
is summarized in the so-called toric diagram which is a polytope embedded in a
$\mathbb{Z}^{2}$-lattice. Moreover, when we consider a toric Calabi-Yau
threefold cone the action of the algebraic torus enlarges the isometry group
of the variety from $U(1)$, the action induced by the Reeb vector field, to
$U(1)^{3}$. Let us briefly review some possible construction of toric
varieties [2],[3],[4].

####  2.1.1 Homogeneous coordinates approach to toric varieties

The simplest way we can image toric variety is as generalization of weighed
projective space [24]. Let us recall first the definition of the
$m-1$-dimensional weighed projective space

$\mathbb{CP}^{m-1}=\frac{\mathbb{C}^{m}\setminus\\{0\\}}{\mathbb{C}^{*}},$ (1)

where the quotient by $\mathbb{C}^{*}$ is taken into account by the
identification
$(z_{1},...,z_{m})\sim(\lambda^{i_{1}}z_{1},...,\lambda^{i_{m}}z_{m})$ where
$\lambda\in\mathbb{C}^{*}$ and $({i_{1}},...,{i_{m}})$ are the so-called
coordinates weights. An $n$-dimensional toric variety $\mathcal{M}$ is the
generalization where we quotient by more than one $\mathbb{C}^{*}$ action and
the set that we subtract is a subset $U_{\Sigma}$ which contains not only the
origin

$\mathcal{M}=\frac{\mathbb{C}^{m}\setminus\\{U_{\Sigma}\\}}{(\mathbb{C}^{*})^{m-n}\times\Gamma},$
(2)

where $\Gamma$ is an abelian group; this variety has an algebraic torus action
given by $(\mathbb{C}^{*})^{m-m+n}=(\mathbb{C}^{*})^{n}$. Definition 2 emerges
in the context of cones and fans that we are going to discuss.
Let $M$ and $N$ be a dual $n$-dimensional lattices isomorphic to
$\mathbb{Z}^{n}$ and consider the vector spaces $M_{\mathbb{R}}$ and
$N_{\mathbb{R}}$ to be the subspace of $\mathbb{R}^{n}$ spanned, respectively,
by vectors in $M$ and $N$. We define the Strongly Convex Rational Polyhedral
Cone (SCRPC) $\sigma\in N_{\mathbb{R}}\subset\mathbb{R}^{n}$ as the set

$\sigma:=\bigg{\\{}\sum_{i=1}^{m}a_{i}\vec{v}_{i}\bigg{|}a_{i}\in\mathbb{R},a_{i}\geq
0,\vec{v}_{i}\in\mathbb{Z}^{n}\ \forall i\bigg{\\}}$ (3)

for a finite number of vectors $v_{i}$ and satisfying the condition
$(\sigma)\cap(-\sigma)=\\{0\\}$. Let us analyze this definition: we consider
an $n$-dimensional lattice $N\simeq\mathbb{Z}^{n}$, a SCRPC is an $n$ or lower
dimensional cone in $N_{\mathbb{R}}$ with the origin of the lattice as its
apex, bounded by hyperplanes (polyhedral) with its edges spanned by lattice
vectors (rational) and such that it does not contain complete lines (strongly
convex). The dimension of a SCRPC $\sigma$ is the dimension of the smallest
subspace of $\mathbb{R}^{n}$ containing $\sigma$. Two important concepts are
useful in SCRPC theory:

  * 1.

edges: these are the one dimensional faces of $\sigma$, the vectors $\vec{g}$
associated to the edges are the generators of $\sigma$;

  * 2.

facets: these are the codimension one faces.

Figure 1: Examples of SCRPCs. Left: SCRPC in $\mathbb{R}^{3}$, its one
dimensional faces are identified by the vectors
$\vec{g}_{1}=(1,0,0),\vec{g}_{2}=(0,1,0),\vec{g}_{3}=(0,1,1),\vec{g}_{4}=(1,0,1)$
in $\mathbb{Z}^{3}$. Right: SCRPC in $\mathbb{R}^{2}$, its one dimensional
faces are identified by the vectors $\vec{g}_{1}=(0,1),\vec{g}_{2}=(1,-1)$ in
$\mathbb{Z}^{2}$.

A collection $\Sigma$ of SCRPCs in $N_{\mathbb{R}}$ is called fan if each face
of a SCRPC in $\Sigma$ is also a SCRPC in $\Sigma$ and the intersection of two
SCRPCs in $\Sigma$ is a face of each. Examples of SCRPCs and fan are reported
in Figure 1 and Figure 2.

Figure 2: Examples of fan. Left: fan of
$\mathbb{CP}^{1}\times\mathbb{CP}^{1}$, we have four one dimensional SCRPCs
(the vectors) and four two dimensional SCRPCs (the quadrants). Right: fan of
$\mathbb{CP}^{2}$, we have three one dimensional SCRPCs (the vectors) and
three two dimensional ones (the trians).

Let us consider a fan $\Sigma$ and we call $\Sigma_{1d}$ the set of one
dimensional SCRPCs; let $\vec{v}_{i}$ with $i=1,...,m$ be the whole set of
vectors generating the one dimensional SCRPCs in $\Sigma_{1d}$222Note that
obviously $m$ is equal to the number of one dimensional cone and so to the
number of elements in $\Sigma_{1d}$. To each vector $\vec{v}_{i}$ we associate
an homogeneous coordinate $z_{i}\in\mathbb{C}$ and we define the set

$U_{\Sigma}:=\bigcup_{I}\\{(z_{1},...,z_{m})|z_{i}=0\ \forall i\in I\\},$ (4)

where the union is taken over all the sets having $I\subseteq\\{1,...,m\\}$
for which $z_{i}$ with $i\in I$ does not belong to a SCRPC in $\Sigma$.

At this point we need to discuss how the $(\mathbb{C}^{*})^{m-n}\times\Gamma$
acts on $\mathbb{C}^{m}$. First of all, let us clarify the nature of the
abelian group $\Gamma$: this is given by

$\Gamma:=\frac{N}{\tilde{N}},$ (5)

where $\tilde{N}\subseteq N$ is the sublattice generated over $\mathbb{Z}^{n}$
by the vectors $\vec{v}_{i}$. In other words, vectors $\vec{v}_{i}$ not
necessarily generate all $N$, in general they generate only $\tilde{N}$; on
the one hand, if $\tilde{N}=N$ then $\Gamma$ is trivial and it does not play
any role. On the other hand, if $\Gamma$ is no trivial our variety develops
orbifold singularity.
Let us now discuss the algebraic torus $(\mathbb{C}^{*})^{m-n}$ action.
Consider the $n\times m$ matrix build up considering the $m$ vectors
$\vec{v}_{i}$ with $n$ components

$V_{i}^{k}=\begin{pmatrix}v_{1}^{1}&v_{2}^{1}&\dots&v_{m}^{1}\\\
v_{1}^{2}&\ddots&\dots&v_{m}^{2}\\\ \vdots&\vdots&\ddots&\vdots\\\
v_{1}^{n}&v_{2}^{n}&\dots&v_{m}^{n}\end{pmatrix},$ (6)

this induces a map $\phi:\mathbb{C}^{m}\rightarrow\mathbb{C}^{n}$ defined by

$(z_{1},...,z_{m})\mapsto\bigg{(}\prod_{i=1}^{m}z_{i}^{v_{i}^{1}},...,\prod_{i=1}^{m}z_{i}^{v_{i}^{n}}\bigg{)}.$
(7)

Thanks to the rank-nullity theorem333Given $T$ a linear map between two finite
dimensional vector spaces $A$ and $B$ we have that
$dim(Im(T))+dim(Ker(T))=Rank(V)+Null(V)=dim(A)$. In our case we know that
$\vec{v}_{i}$ with $i=1,...,m$ are $m$ linearly independent vectors so
$Rank(V)=dim(Im(T))=m$. the dimension of the Kernel of map 7 must be $m-n$; we
can identify it with $(\mathbb{C}^{*})^{m-n}$. It is now simple to see how
$(\mathbb{C}^{*})^{m-n}$ acts on $\mathbb{C}^{m}$: each $\mathbb{C}^{*}$
action is taken into account by

$(z_{1},...,z_{m})\mapsto(\lambda^{Q_{1}^{a}}z_{1},...,\lambda^{Q_{m}^{a}}z_{m}),$
(8)

with $\lambda\in\mathbb{C}^{*}$. We have $m-n$ actions like 8, where for each
$a=1,...,m-n$ the charge vectors $Q^{a}=(Q_{1}^{a},...,Q_{m}^{a})$ belong to
the Kernel of the map $\phi$ and therefore must satisfy $m-n$ relations

$\sum_{i=1}^{m}V^{k}_{i}Q_{i}^{a}=\vec{0}.$ (9)

Quotient $(\mathbb{C}^{*})^{m-n}$ out means taking into account the
equivalence relations

$(z_{1},...,z_{m})\sim(\lambda^{Q_{1}^{a}}z_{1},...,\lambda^{Q_{m}^{a}}z_{m})$
(10)

for $a=1,...,m-n$.
To summarize, we can define the toric variety as

$\mathcal{M}=\frac{\mathbb{C}^{m}\setminus\\{U_{\Sigma}\\}}{(\mathbb{C}^{*})^{m-n}\times\Gamma};$
(11)

this is a $n$-dimensional variety, with a residual $(\mathbb{C}^{*})^{n}\simeq
U(1)^{n}$ action and the $(\mathbb{C}^{*})^{m-n}$ action is quotient out by
$m-n$ relations 10 with weight that satisfies relations 9. Let us give an
examples.

#### Exemple: $\mathbb{CP}^{2}$

Let us consider the right fan in Figure 2: this is a $\mathbb{Z}^{2}$ lattice;
we have the three vectors
$\vec{v}_{1}=(0,1),\vec{v}_{2}=(1,0),\vec{v}_{3}=(-1,-1)$ that generate one
dimensional cones. We have three homogeneous coordinates
$(z_{1},z_{2},z_{3})\in\mathbb{C}^{3}$ and the set $U_{\Sigma}$ is given by
the origin. Since $m=3$ and $n=2$ we must have one $\mathbb{C}^{*}$ action and
we can find the weight using 9:

$\displaystyle\vec{v}_{1}Q_{1}+\vec{v}_{2}Q_{2}+\vec{v}_{3}Q_{3}=(0,1)Q_{1}+(1,0)Q_{2}+(-1,-1)Q_{3}=$
(12) $\displaystyle=(Q_{2}-Q_{3},Q_{1}-Q_{3})=\vec{0}\Rightarrow
Q_{1}=Q_{2}=Q_{3}=1;$

$\mathbb{C}^{*}$ action is taken into account by the equivalence relation
$(z_{1},z_{2},z_{3})\sim\lambda(z_{1},z_{2},z_{3})$. Finally we note that the
vectors $\vec{v}_{1}$ and $\vec{v}_{2}$ generate the whole lattice
$\mathbb{Z}^{2}$ and so $\Gamma$ is trivial. The toric variety corresponds to

$\mathcal{M}=\frac{\mathbb{C}^{3}\setminus\\{0\\}}{\mathbb{C}^{*}}\equiv\mathbb{CP}^{2},$
(13)

as we expected.

We now give some interesting properties about toric varieties and their fans:

  * 1.

a fan $\Sigma$ is smooth if every SCRPC in $\Sigma$ is smooth, a SCRPC is
smooth if is generated by a subset of a basis of $N\simeq\mathbb{Z}^{n}$;

  * 2.

a fan $\Sigma$ is simplicial if every SCRPC in $\Sigma$ is simplicial, a SCRPC
is simplicial if it is generated by a subset of a basis of $\mathbb{R}^{n}$;

These conditions are important since if a fan is smooth the corresponding
toric variety also is smooth and if a fan is simplicial then the corresponding
toric variety can have at most orbifold singularities. We see immediately that
the two spaces described by the fans in Figure 2 are smooth since every SCRPC
is generated by a subset of a $\mathbb{Z}^{2}$ basis. An example of toric
variety with orbifold singularities is the weighted projective space
$\mathbb{CP}_{2,3,1}$; its fan is given by the Figure 3 below. Orbifold
singularities can be removed by the so-called blow up procedure: roughly
speaking, for a $n$-dimensional toric variety we replace the singular locus by
$\mathbb{CP}^{n-1}$.

Figure 3: Fan of $\mathbb{CP}_{2,3,1}$, we have
$\vec{v}_{1}=(1,0),\vec{v}_{2}=(0,1),\vec{v}_{3}=(-2,-3)$. It is not smooth
but it is simplicial: $\mathbb{CP}_{2,3,1}$ has orbifold singularities.

Let us now specialize to the case of our interest: CY threefolds. First of
all, we have an $U(1)^{3}\simeq(\mathbb{C}^{*})^{3}$ action but there is more:
the condition of trivial canonical bundle implies that all the vectors of the
fan belong to the same hyperplane, so we can project on this hyperplane
obtaining a two dimensional object whose convex hull takes the name of toric
diagram. The CY condition implies

$CY\ condition\Rightarrow\sum_{i=1}^{m}Q_{i}^{a}=0\ \forall a.$ (14)

#### Example: the conifold $\mathcal{C}$

Consider the three dimensional fan given by the four vectors
$\vec{v}_{1}=(1,0,1),\vec{v}_{2}=(0,0,1),\vec{v}_{3}=(0,1,1),\vec{v}_{4}=(1,1,1)$.
We note that these vectors belong to the same hyperplane, so we can project
out the third component: we obtain
$\vec{w}_{1}=(0,0),\vec{w}_{2}=(1,0),\vec{w}_{3}=(0,1),\vec{w}_{4}=(1,1)$.
Hence the conifold is a CY variety and its toric diagram is given below.

$z_{1}$$z_{3}$$z_{2}$$z_{4}$ Figure 4: Toric diagram of the conifold.

We have $m-n=4-3=1$ charge vector given by relation 9:

$\begin{pmatrix}1&0&0&1\\\ 0&0&1&1\\\ 1&1&1&1\\\
\end{pmatrix}\begin{pmatrix}Q_{1}\\\ Q_{2}\\\ Q_{3}\\\ Q_{4}\\\
\end{pmatrix}=\vec{0}\Rightarrow\begin{cases}Q_{1}+Q_{4}&=0\\\
Q_{3}+Q_{4}&=0\\\ \sum_{i=1}^{4}Q_{i}&=0\end{cases}$ (15)

and a possible solution is $Q=(1,-1,1-1)$. As one can note the CY condition is
automatically implemented by the fact that the vectors are coplanar. The
action of the algebraic torus $\mathbb{C}^{*}$ is quotient out by the
equivalence relation

$(z_{1},z_{2},z_{3},z_{4})\sim(\lambda z_{1},\lambda^{-1}z_{2},\lambda
z_{3},\lambda^{-1}z_{4}).$ (16)

Let us consider the $\mathbb{C}^{*}$-invariant polynomials: by 16 we note that

$Z_{1}=z_{1}z_{2},\ Z_{2}=z_{1}z_{4},\ Z_{3}=z_{2}z_{3},\ Z_{4}=z_{3}z_{4}$
(17)

are invariant and this is the minimal basis with which to write all the
$\mathbb{C}^{*}$-invariant polynomials. However, note that these polynomials
are not independent but they must satisfy the relation

$Z_{1}Z_{4}=Z_{2}Z_{3}$ (18)

which is the defining polynomial of the conifold [5].

#### Example: $\frac{\mathbb{C}^{2}}{\mathbb{Z}_{2}}\times\mathbb{C}$

Consider the fan generated by the vectors
$\vec{v}_{1}=(0,0,1),\vec{v}_{2}=(0,1,1),\vec{v}_{3}=(1,0,1),\vec{v}_{4}=(-1,0,1)$;
projecting out the third component we get
$\vec{w}_{1}=(0,0),\vec{w}_{2}=(0,1),\vec{w}_{3}=(1,0),\vec{w}_{4}=(-1,0)$ and
the toric diagram is

$z_{1}$$z_{2}$$z_{3}$$z_{4}$ Figure 5: Toric diagram of
$\frac{\mathbb{C}^{2}}{\mathbb{Z}_{2}}\times\mathbb{C}$.

We have $m-n=4-3=1$ charge vector given by

$\begin{pmatrix}0&0&1&-1\\\ 1&0&0&0\\\ 1&1&1&1\\\
\end{pmatrix}\begin{pmatrix}Q_{1}\\\ Q_{2}\\\ Q_{3}\\\ Q_{4}\\\
\end{pmatrix}=\vec{0}\Rightarrow\begin{cases}Q_{3}-Q_{4}&=0\\\ Q_{1}&=0\\\
\sum_{i=1}^{4}Q_{i}&=0\end{cases}$ (19)

and so $Q=(0,-2,1,1)$. Since we have one vanishing component, $Q_{1}=0$, the
coordinate $z_{1}$ has no role and so we will expect a CY toric manifold of
the form $X\times\mathbb{C}$, where $X$ is unknown for the moment and
$\mathbb{C}$ is the space associated to $z_{1}$. We have the equivalence
relation

$(z_{1},z_{2},z_{3},z_{4})\sim(z_{1},\lambda^{-2}z_{2},\lambda z_{3},\lambda
z_{4});$ (20)

since on $z_{2}$ the algebraic torus action is trivial we not consider it and
so we must find $m-1=4-1=3$ $\mathbb{C}^{*}$-invariant polynomials, for
example

$Z_{1}=z_{1}z_{3}z_{4},\ Z_{2}=z_{1}z^{2}_{3},\ Z_{3}=z_{1}z_{4}^{2},$ (21)

and they satisfy the relation

$Z_{2}Z_{3}=Z_{1}^{2}.$ (22)

Relation 22 is the realization of $\frac{\mathbb{C}^{2}}{\mathbb{Z}_{2}}$ as
subvariety of $\mathbb{C}^{3}$. In the end, the toric CY variety is
$\frac{\mathbb{C}^{2}}{\mathbb{Z}_{2}}\times\mathbb{C}$.

The general algorithm to identify to which manifold a toric diagram belongs is
the following: given the toric diagram we look at the equivalent relations
that quotient out the algebraic torus action $(\mathbb{C}^{*})^{m-n}$ and we
construct a minimal basis of $m$ $(\mathbb{C}^{*})^{m-n}$-invariant
polynomials; the $m-n$ relations that this polynomials must satisfy identifies
the toric variety as subvariety of $\mathbb{C}^{m}$.

####  2.1.2 Moment maps approach and Delzant-like construction

Let us take a step back and consider a different way to define toric
varieties. Let $\mathcal{M}$ a symplectic manifold of real dimension $2n$ with
symplectic form $\omega$. Given the action
$U(1)^{n}\times\mathcal{M}\rightarrow\mathcal{M}$ this is said to be
hamiltonian if its restriction to any $U(1)\subset U(1)^{n}$ is hamiltonian
and any two of them commute. In this context emerge the so-called moment map
$\mu:\mathcal{M}\rightarrow\mathbb{R}^{n}$ whose components are the
hamiltonians of each $U(1)$ action. Since all the $U(1)$ commute, for any
$\vec{r}\in\mathbb{R}^{n}$, its preimage $\mu^{-1}(\vec{r})$ is invariant
under the action of the full $U(1)^{n}$. Moreover, if $\mathcal{M}$ has a
Kähler structure, the existence of the hamiltonian action implies that the
isometry group contains the algebraic torus
$U(1)^{n}\simeq(\mathbb{C}^{*})^{n}$. So a toric variety emerges in a simple
way as a real $2n$ dimensional symplectic manifold $\mathcal{M}$ that has an
hamiltonian action of the algebraic torus on it444There is a little caveat: if
$\mathcal{M}$ is a cone the algebraic torus action must commute with the
homothetic action induced by the Euler vector field..
We are interested in non-compact toric varieties but let us talk a little
about compact ones. If $\mathcal{M}$ is compact, Delzant [25] showed that the
image through the moment map of the variety, $\mu(\mathcal{M})$, is a convex
polytope $\Delta$ called Delzant polytope; however we are interested in CY
cones, i.e. non-compact toric varieties. The generalization to a Deltzan-like
construction is possible [26] but the image under the moment map it is no
longer a polytope but a cone:

$\Theta=\\{\vec{r}\in\mathbb{R}^{n}|\vec{r}\cdot\vec{v}_{i}\leq
0,\vec{v}_{i}\in\mathbb{Z}^{n}\\}$ (23)

and its dual graph is still a fan generated by the normal vectors
$\vec{v}_{i}$. Calabi-Yau condition impose that $\vec{v}_{i}$ are coplanar so
we can project out the common component and get an $n-1$-dimensional object
that encode the geometry: the toric diagram. Since the components of the
vectors $\vec{v}_{i}$ are integer, the toric diagram is the convex hull of a
set of point in a $\mathbb{Z}^{n-1}$-lattice. In Deltzan-like approach, toric
variety is built up using Kähler quotient and the toric variety is a
$U(1)^{n}$ fibration over $\Theta$: one $U(1)\subset U(1)^{n}$ shrinks on the
edges and so acts trivially, moreover, since in a vertex $n$ edges meet each
others the full $U(1)^{n}$ fiber shrinks. Hence the vertexes are the fixed
loci of the full algebraic torus action. It is interesting to note that this
construction is well known by physicists: this is the moduli space of the
Gauged Linear Sigma Model (GLSM), it is a SUSY gauge theory with abelian gauge
group $U(1)^{m-n}$ and $m$ chiral superfields $Z_{1},...,Z_{m}$ with charges
$Q^{1},...,Q^{m-n}$ under the $m-n$ $U(1)$.

In the case of our interest, namely CY threefold, the toric diagram is a two
dimensional object and the CY cone threefold exhibits a three dimensional
algebraic torus action. From the toric diagram we can get information on the
dual QFT using the five brane system and brane tiling constructions; moreover
we can calculate some useful quantities of the dual field theory directly from
the toric diagram, such as the central charge, using the Butti and Zaffaroni
procedure [27] or the symplexic decomposition procedure [28].

###  2.2 Orientifold Projections

Denoting by $0\leq\sigma\leq\pi$ the coordinate describing the open string at
a given time, the two ends $\sigma=0,\pi$ contain, thanks to Chan-Paton
indexes, the gauge group degrees of freedom and the corresponding charged
matter fields. At the endpoints we can apply Dirichlet or Neumann boundary
conditions and we know that under $T$-duality these are interchanged.
Orientifold projections of Type IIB theory are obtained by projecting the Type
IIB spectrum by the involution $\Omega$ ($\Omega^{2}=1$), exchanging the left
and right closed oscillators $\hat{\alpha}_{m}^{\mu}$, $\hat{\beta}_{m}^{\mu}$
and acting on the open strings as phases:

$\displaystyle closed\ strings\Rightarrow\Omega
X^{\mu}(\tau,\sigma)\Omega^{-1}=X^{\mu}(\tau,-\sigma)\Rightarrow\hat{\alpha}_{m}^{\mu}\leftrightarrow\hat{\beta}_{m}^{\mu},$
(24) $\displaystyle open\ strings\Rightarrow\Omega
X^{\mu}(\tau,\sigma)\Omega^{-1}=X^{\mu}(\tau,\pi-\sigma)\Rightarrow\hat{\alpha}_{m}^{\mu}\rightarrow\pm(-1)^{m}\hat{\alpha}_{m}^{\mu}.$

The overall effect is that orientifold projections map oriented strings to
unoriented ones and type I superstring theory is nothing but an orientifold
projection of type IIB superstring theory [29]: $\mathrm{type\
I}=\frac{\mathrm{type\ IIB}}{\Omega}$.

Orientifold introduces, from the worldsheet point of view, new surfaces in the
Polyakov topological expansion. Indeed, due to the presence of unoriented
strings, also non-orientable surfaces such as Klein bottle or Möbius strip are
allowed. From the space-time viewpoint, these correspond to not dynamical
objects, called O-planes, defined by $T$-duality as fixed point of the
orientifold projection [30].

However this picture is not the complete one since, nowadays, there are no
results in actual string theory for what one might expect to be the curved
back-reacted geometry of orientifold. Indeed as reported in [31], for any
solution with O-planes, the presence of such a plane is inferred by comparison
with their flat-space behavior. However, since the orientifolded space-times
have strong curvature and coupling, stringy corrections come into play, and it
is impossible to decide with supergravity alone whether the solutions are
valid. Therefore, it is natural to think of more general behaviors in which
geometric phase transitions due to the presence of orientifold can play a
crucial role. Since this kind of behaviors are highly non-perturbative we
expect something happens in the field theory that goes beyond the large $N$
limit.

This orientifold construction has important consequences on the emergent field
theory build up with the AdS/CFT and toric CY machinery: some degrees of
freedom are projected out and orthogonal and symplectic group are now allowed
together with symmetric and antisymmetric representations of unitary groups.
Rules for construction of these kind of theories are studied for example in
[17],[32]. The crucial point here is that orientifold can give rise to
different scenarios:

  * 1.

in I scenario there is a new superconformal fixed point and the $R$-charges of
the operators that are not projected out by $\Omega$ are the same as the
charges of the corresponding one in the oriented theory and in the large $N$
limit. In the end, the post-orientifold central charge $a^{\Omega}$ turns out
to be half of the pre-orientifold central charge $a$;

  * 2.

II scenario does not admit a new superconformal RG fixed point;

  * 3.

in III scenario there is a new superconformal fixed point but the $R$-charges
of the operators that are not projected out by $\Omega$ are different from the
charges of the corresponding one in the oriented theory and not only at large
$N$. In the end, the post-orientifold central charge $a^{\Omega}$ turns out to
be less than half of the pre-orientifold central charge $a$. However,
something very interesting happens here: this could be (one of) the right
field theory dual to the non-perturbative action of orientifold in the string
theory side.

####  2.2.1 The I scenario orientifold

The I scenario orientifold occurs when the pre and post orientifold
$R$-charges of the theories are the same and the central charge post
orientifold is half of that pre orientifold:

$a^{\Omega}_{i}=a_{i},a^{\Omega}=\frac{a}{2}.$ (25)

The Calabi-Yau cone describing the theory before and after the orientifold is
then the same but with different volume due to the orientifold action.

####  2.2.2 The II scenario orientifold

The II scenario seems to be quite trivial: there is no new superconformal
point and the orientifold breaks conformal symmetry. However, as known in
literature, these situation can give rise to a duality cascade or conformal
symmetry can be restored with flavor branes.

####  2.2.3 The III scenario orientifold and the IR duality

Let us consider two different field theories and call them A and B. Since they
are different theories, they are defined by two different Calabi-Yau cones:
CYA and CYB. Suppose that the theory B has I scenario orientifold while theory
A has III scenario orientifold; namely, the pre and post-orientifold
$R$-charges of theory $\mathrm{A}$ are not the same and the post-orientifold
central charge is less that half of that pre-orientifold [18],[19]:

$a^{\Omega}_{i}\neq a_{i}\ ,\
a_{\mathrm{A}}^{\Omega}<\frac{a_{\mathrm{A}}}{2}.$ (26)

This is our set up. A duality emerges when we compare the post orientifold
quantities of the two theories [18],[19]; we find

$a_{i}^{\Omega}=b_{i}^{\Omega}=b_{i}\ ,\
a_{A}^{\Omega}=a_{B}^{\Omega}=\frac{a_{B}}{2}$ (27)

and also, since the $c$-theorem in $4D$ tells us that $a_{IR}<a_{UV}$,
theories A and B are flown in the IR regime due to the orientifold action and
the duality is an IR one. What we have said in words is summarized in the
following scheme.

$\displaystyle(a_{i},a_{\mathrm{A}})\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left(a^{\Omega}_{i}\neq a_{i},a_{\mathrm{A}}^{\Omega}<\frac{a_{A}}{2}\right)$
(28) $\displaystyle oriented\ model\ \mathrm{A}\xrightarrow{III\ scenario\
orientifold\ projection}unoriented\ model\ \mathrm{A}$ $\displaystyle\ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathrm{IR}\ duals\Bigg{\updownarrow}$
$\displaystyle oriented\ model\ \mathrm{B}\xrightarrow{\ I\ scenario\
orientifold\ projection\ }unoriented\ model\ \mathrm{B}$
$\displaystyle(b_{i},a_{\mathrm{B}})\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left(b^{\Omega}_{i}=b_{i},a_{\mathrm{B}}^{\Omega}=\frac{a_{\mathrm{B}}}{2}\right)$

Some examples of duality due to orientifold in scenario III seem to be
explained and interpreted, from the field theory point of view, in term of
inherited S-duality from the $\mathcal{N}=2$ case [33]. The authors observe
that the duality for the $\mathcal{N}=1$ models discussed in [19] corresponds
to S-duality at different points of the conformal manifold. The crucial
property behind this result is the presence in the spectrum of two-index
tensor fields with R-charge equal to one, uncharged with respect to the other
global symmetries.

##  3 Algebro-geometrical orientifold and IR dualities

Our main goal is to better understand III scenario orientifold and the IR
duality from the point of view of the geometry. The guiding ideas are the flop
transition and the other geometry/topology changing in string theory
compactifications, where the geometry/topology of CY manifold are modified due
to a quantum behavior of the geometry itself [20],[21],[22],[23].
Heuristically, one suspects that geometry/topology might be able to change by
means of the violent curvature fluctuations, such as induced by orientifold,
which would be expected in any quantum theory of gravity. Therefore, the basic
idea is that III scenario orientifold projection, map the algebraic equations
describing the geometry of the CYA into those that describe the geometry of
the CYB. III scenario orientifold modifies the geometry of the CYA cone due to
its intrinsic non-perturbative nature and this geometrical change induces $k$
new degrees of freedom with $R$-charges $\tilde{a}_{i}$ to emerge (we will
call them orientifold-mapping operators or $o\mu$-operators) and they make the
original field theory no longer fulfilling the condition
$\sum_{1}^{d}a_{i}+\sum_{1}^{k}\tilde{a}_{i}=2$; hence old $R$-charges $a_{i}$
and $o\mu$-operators $R$-charges $\tilde{a}_{i}$ must mix together to return a
set of new $R$-charges $a^{\Omega}_{i}=b_{i}$ consistent with the modified
CY${}^{\Omega}_{\mathrm{A}}$ cone geometry, that is now CYB cone, and
satisfying the condition $\sum_{1}^{d}b_{i}=2$. Steps are summarized in the
following scheme.

$\displaystyle(a_{i},\mathrm{CY_{{A}}})\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left(a^{\Omega}_{i}\neq
a_{i},\tilde{a}_{i},\mathrm{CY}^{\Omega}_{\mathrm{A}}=\mathrm{CY_{B}}\right)$
(29) $\displaystyle oriented\ model\ \mathrm{A}\ \ \ \ \xrightarrow{III\
scenario\ orientifold\ projection}\ \ \ unoriented\ model\ \mathrm{A}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathrm{IR}\
duals\Bigg{\downarrow}(a_{i}\ \mathrm{and}\ \tilde{a}_{i}\ R\mathrm{-charges\
mixing})$ $\displaystyle oriented\ model\ \mathrm{B}\ \ \ \ \xrightarrow{\ I\
scenario\ orientifold\ projection\ }\ \ \ unoriented\ model\ \mathrm{B}$
$\displaystyle(b_{i},\mathrm{CY_{{B}}})\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left(b^{\Omega}_{i}=b_{i},\mathrm{CY_{{B}}}\right)$

The procedure is the following: from toric diagrams of theories A and B we
determine the sets of algebraic equations using toric geometry tools exposed
in Section 2.1; then we map these two sets into each other and, finally, we
associate at every homogeneous coordinate $z_{i}$ a trial $R$-charges, adding
the minimal number of new degrees of freedom. The analysis must consider two
different cases:

  1. 1.

$\Delta P_{e}:=P^{(\mathrm{B})}_{e}-P_{e}^{(\mathrm{A})}=0$;

  2. 2.

$\Delta P_{e}:=P^{(\mathrm{B})}_{e}-P_{e}^{(\mathrm{A})}=1$;

$P^{(\bullet)}_{e}$ is the number of extremal point of the toric diagram of
the theory $\bullet$. Since in the oriented theory only the trial $R$-chagres
associated to extremal points matter, in the first case the two oriented
theories have the same number of $R$-charges while in the second case the
number of $R$-charges of the oriented theories are not equal and we expect
that some not extremal points of the toric diagram of theory $\mathrm{A}$
became extremal points after the orientifold action. Cases with $\Delta
P_{e}>1$ are not of interest at the moment since no examples have been found.
In following paragraph we study these two cases with specific examples but
this construction can be repeated for many other pairs and the result is that
it is always possible to construct relevant operators with the new degrees of
freedom inserted by matching the systems containing the polynomial equations
that define the geometries of the pair.

###  3.1 Different number of external points: $\Delta P_{e}=1$

Let us consider the conifold $\mathcal{C}$ and
$\frac{\mathbb{C}^{2}}{\mathbb{Z}_{2}}\times\mathbb{C}$; for toric diagrams
and toric data we refer to Examples 2.1 and 2.1. Moreover, in those examples
determination of the equations that describe the CY cones have already been
done. The conifold has I scenario orientifold while
$\frac{\mathbb{C}^{3}}{\mathbb{Z}_{2}}$ has III scenario orientifold and the
two theories are IR duals. The set of equations for the two theories are

$\displaystyle\frac{\mathbb{C}^{2}}{\mathbb{Z}_{2}}\times\mathbb{C}:\ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathcal{C}:$ (30)
$\displaystyle Y_{2}Y_{3}=Y_{1}^{2};\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ X_{1}X_{4}=X_{2}X_{3};$
$\displaystyle\begin{cases}Y_{1}&=z_{1}z_{3}z_{4};\\\
Y_{2}&=z_{1}z_{3}^{2};\\\ Y_{3}&=z_{1}z_{4}^{2}.\end{cases}\ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \begin{cases}X_{1}&=z_{1}z_{2};\\\ X_{2}&=z_{1}z_{4};\\\
X_{3}&=z_{2}z_{3};\\\ X_{4}&=z_{3}z_{4}.\end{cases}$

Now we have to map the two equations in all possible ways remembering that
after the orientifold action the toric diagram are mapped into each other and
so the coordinate $z_{i}$ are effectively the same. However, if we want to
keep track of the different $R$-charges associated to the homogeneous
coordinate $z_{i}$ of the two different toric diagram, so the $R$-chagres
$b_{i}=b_{i}^{\Omega}$ and $a_{i}$, we must remember that we have to associate
a different set of $R$-charges at those $z_{i}$ belonging to different
equations and so different theories. We have to consider the following
mappings:

$\displaystyle\mathrm{I}:$
$\displaystyle\begin{cases}Y^{2}_{1}&=X_{2}X_{3};\\\ Y_{2}&=X_{1};\\\
Y_{3}&=X_{4};\end{cases}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathrm{II}:\
\begin{cases}Y^{2}_{1}&=X_{2}X_{3};\\\ Y_{2}&=X_{4};\\\
Y_{3}&=X_{1};\end{cases}$ (31) $\displaystyle\mathrm{III}:$
$\displaystyle\begin{cases}Y^{2}_{1}&=X_{1}X_{4};\\\ Y_{2}&=X_{2};\\\
Y_{3}&=X_{3};\end{cases}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathrm{IV}:\
\begin{cases}Y^{2}_{1}&=X_{1}X_{4};\\\ Y_{2}&=X_{3};\\\
Y_{3}&=X_{2};\end{cases}$

By associating the set of oriented $R$-charges
$\\{z_{1},z_{2},z_{3},z_{4}\\}\rightarrow\\{0,a_{2},a_{3},a_{4}\\}$ to the
$z_{i}$ coordinates of $Y_{i}$ polynomials and the set of of oriented
$R$-charges
$\\{z_{1},z_{2},z_{3},z_{4}\\}\rightarrow\\{b_{1},b_{2},b_{3},b_{4}\\}$ to the
$z_{i}$ coordinates of $X_{i}$ polynomials, we get the following relations:

$\displaystyle\mathrm{I}:$
$\displaystyle\begin{cases}2a_{3}+2a_{4}&=b_{1}+b_{2}+b_{3}+b_{4};\\\
2a_{3}&=b_{1}+b_{2};\\\ 2a_{4}&=b_{3}+b_{4};\end{cases}\ \ \ \ \ \ \ \
\mathrm{II}:\ \begin{cases}2a_{3}+2a_{4}&=b_{1}+b_{2}+b_{3}+b_{4};\\\
2a_{3}&=b_{3}+b_{4};\\\ 2a_{4}&=b_{1}+b_{2};\end{cases}$ (32)
$\displaystyle\mathrm{III}:$
$\displaystyle\begin{cases}2a_{3}+2a_{4}&=b_{1}+b_{2}+b_{3}+b_{4};\\\
2a_{3}&=b_{1}+b_{4};\\\ 2a_{4}&=b_{2}+b_{3};\end{cases}\ \ \ \ \ \ \ \
\mathrm{IV}:\ \begin{cases}2a_{3}+2a_{4}&=b_{1}+b_{2}+b_{3}+b_{4};\\\
2a_{3}&=b_{2}+b_{3};\\\ 2a_{4}&=b_{1}+b_{4};\end{cases}$

and they are not all independent. If we make together systems I, II, III and
IV we obtain the following relations:

$\begin{cases}\mathrm{I}=\mathrm{III}\Rightarrow
b_{1}+b_{2}=b_{1}+b_{4}&\Rightarrow b_{2}=b_{4};\\\
\mathrm{I}=\mathrm{IV}\Rightarrow b_{3}+b_{4}=b_{1}+b_{2}&\Rightarrow
b_{1}=b_{3};\end{cases}$ (33)

and we have also the relation $\sum_{i=1}^{4}b_{i}=2$. These conditions are
not sufficient to fix unequivocally the values of the $R$-charges to those of
the conifold. However this is right, indeed using the fact we know the values
at the superconformal fixed point of the two set of $R$-charges, namely
$\\{a_{1},a_{2},a_{3},a_{4}\\}=\\{0,2/3,2/3,2/3\\}$ and
$\\{b_{1},b_{2},b_{3},b_{4}\\}=\\{1/2,1/2,1/2,1/2\\}$, we may note that the
equations 32 do not do the right job. Nevertheless, if we admit the existence
of a new operator (an orientifold-mapping operator) with $R$-charge
$\tilde{a}=\frac{1}{3}$ we note that, for example,

$\mathrm{I}:\begin{cases}2a_{3}+2a_{4}-2\tilde{a}&=b_{1}+b_{2}+b_{3}+b_{4};\\\
2a_{3}-\tilde{a}&=b_{1}+b_{2};\\\
2a_{4}-\tilde{a}&=b_{3}+b_{4};\end{cases}\Rightarrow\begin{cases}2\frac{2}{3}+2\frac{2}{3}-2\frac{1}{3}&=\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2};\\\
2\frac{2}{3}-\frac{1}{3}&=\frac{1}{2}+\frac{1}{2};\\\
2\frac{2}{3}-\frac{1}{3}&=\frac{1}{2}+\frac{1}{2};\end{cases}$ (34)

now we have the right matching.

Finally, let us now consider the fields associated to
$\frac{\mathbb{C}^{2}}{\mathbb{Z}_{2}}\times\mathbb{C}$ and their $R$-charges
[27],[28]:

$\displaystyle\langle w_{1},w_{2}\rangle=2\rightarrow a_{3}$ (35)
$\displaystyle\langle w_{2},w_{3}\rangle=2\rightarrow a_{2}$
$\displaystyle\langle w_{3},w_{1}\rangle=2\rightarrow a_{4},a_{4}+a_{1}$

where

$\langle u,v\rangle:=det\begin{bmatrix}u^{(1)}&u^{(2)}\\\
v^{(1)}&v^{(2)}\end{bmatrix}\;.$

Hence we have 2 fields with $R$-charge $a_{3}$, 2 fields with $R$-charge
$a_{2}$, 1 field with $R$-charge $a_{4}$ and 1 field with $R$-charge
$a_{4}+a_{1}$. Despite in the oriented theory charge $a_{1}$ does not play any
role, here, in the unoriented one, we can identify $a_{1}\equiv\tilde{a}$ and
so there is a field with a new $R$-charge due to orietifold action; we call it
$\Pi$555Its bosonic part will be dubbed puppon while its fermionic one
puchino. This field is the one associated, according to [27] or [28], to the
$R$-charge given by the combination of the new charge entered in the game,
$\tilde{a}$, and the other charges.. It is interesting to note that field
$\Pi$ associated with the $R$-charge $a_{4}+\tilde{a}$ has
$R=a_{4}+\tilde{a}=\frac{2}{3}+\frac{1}{3}=1$. Since superpotential must have
$R$-charges two, we can build up a quadratic deformation with $\Pi^{2}$ and
its scaling dimension is given by $\Delta_{\Pi^{2}}=\frac{3}{2}R[\Pi^{2}]=3$;
therefore this is a relevat deformation. This makes sense given that
orientifold has the effect to make the theory flow into the IR regime; so this
operator turns on and the theory becomes IR dual to another theory that has
the correct set of $R$-charges to match systems 32.

###  3.2 Equal number of external points: $\Delta P_{e}=0$

Consider the two theories $\frac{\mathrm{SPP}}{\mathbb{Z}_{2}}$ and
$L^{(3,3,3)}$; their toric diagrams are drown below

$z_{1}$$z_{2}$$z_{3}$$z_{4}$$z_{5}$$z_{6}$$z_{7}$$z_{8}$

$z_{1}$$z_{2}$$z_{3}$$z_{4}$$z_{5}$$z_{6}$$z_{7}$$z_{8}$

Figure 6: Toric diagrams of $\frac{\mathrm{SPP}}{\mathbb{Z}_{2}}$ (right) and
$L^{(3,3,3)}$ (left).

To determine the set of equations describing the two varieties we follow the
step highlighted in Section 2.1.
Let us start with $\frac{\mathrm{SPP}}{\mathbb{Z}_{2}}$. We have $m-n=8-3=5$
charge vectors solution of

$\begin{pmatrix}1&0&0&0&1&1&1&1\\\ 3&3&2&1&-1&0&1&2\\\ 1&1&1&1&1&1&1&1\\\
\end{pmatrix}\begin{pmatrix}Q_{1}^{a}\\\ Q_{2}^{a}\\\ Q_{3}^{a}\\\
Q_{4}^{a}\\\ Q_{5}^{a}\\\ Q_{6}^{a}\\\ Q_{7}^{a}\\\ Q_{8}^{a}\\\
\end{pmatrix}=\vec{0}\Rightarrow\begin{cases}Q^{a}_{1}+Q^{a}_{5}+Q^{a}_{6}+Q^{a}_{7}+Q^{a}_{8}&=0\\\
3Q_{1}^{a}+3Q_{2}^{a}+2Q_{3}^{a}+Q_{4}^{a}-Q_{5}^{a}+Q_{7}^{a}+2Q_{8}^{a}&=0\\\
\sum_{i=1}^{8}Q^{a}_{i}&=0\end{cases};$ (36)

where $a=1,...,5$. Solution space can be parametrized as follows:

$\begin{cases}&Q_{1}^{a}=-Q^{a}_{5}-Q^{a}_{6}-Q^{a}_{7}-Q^{a}_{8};\\\
&Q_{2}^{a}=Q^{a}_{4}+4Q^{a}_{5}+3Q^{a}_{6}+2Q^{a}_{7}+Q^{a}_{8};\\\
&Q_{3}^{a}=-2Q_{4}^{a}-4Q_{5}^{a}-3Q_{6}^{a}-2Q_{7}^{a}-Q_{8}^{a};\end{cases}$
(37)

the equivalence relations are given by

$\displaystyle(z_{1},z_{2},z_{3},z_{4},z_{5},z_{6},z_{7},z_{8})\sim$ (38)
$\displaystyle\sim(\lambda^{Q_{1}^{a}}z_{1},\lambda^{Q_{2}^{a}}z_{2},\lambda^{Q_{3}^{a}}z_{3},\lambda^{Q_{4}^{a}}z_{4},\lambda^{Q_{5}^{a}}z_{5},\lambda^{Q_{6}^{a}}z_{6},\lambda^{Q_{7}^{a}}z_{7},\lambda^{Q_{8}^{a}}z_{8}),$

where $Q_{1}^{a},Q_{2}^{a},Q_{3}^{a}$ satisfy 37. The 8 invariant polynomials
can be constructed in a simply way: as first step we consider the homogeneous
coordinates with not independent charges (in this case $z_{1},z_{2},z_{3}$)
and then we multiply them by the right combination of the other $z_{i}$, in
order to compensate the transformation under the equivalence relations; the
second step is to built up the other polynomials taking product or quotient of
polynomial constructed at the first step. This procedure it not only gives us
the invariant polynomials but also the relations between them. Invariant
polynomials and relations are given by

$\begin{cases}&X_{1}=z_{1}z_{5}z_{6}z_{7}z_{8};\\\
&X_{2}=z_{2}z_{4}^{-1}z_{5}^{-4}z_{6}^{-3}z_{7}^{-2}z_{8}^{-1};\\\
&X_{3}=z_{3}z_{4}^{2}z_{5}^{4}z_{6}^{3}z_{7}^{2}z_{8};\\\
&X_{4}=z_{1}z_{2}z_{4}^{-1}z_{5}^{-3}z_{6}^{-2}z_{7}^{-1}=X_{1}X_{2};\\\
&X_{5}=z_{1}z_{3}z_{4}^{2}z_{5}^{5}z_{6}^{4}z_{7}^{3}z_{8}^{2}=X_{1}X_{3};\\\
&X_{6}=z_{2}z_{3}z_{4}=X_{2}X_{3};\\\
&X_{7}=z_{1}^{-1}z_{3}z_{4}^{2}z_{5}^{3}z_{6}^{2}z_{7}=\frac{X_{3}}{X_{1}};\\\
&X_{8}=z_{2}^{-1}z_{3}z_{4}^{3}z_{5}^{8}z_{7}^{4}z_{8}^{2}=\frac{X_{3}}{X_{2}}.\end{cases}$
(39)

For $L^{(3,3,3)}$ we follow the same steps. We have $m-n=8-3=5$ charge vectors
solution of

$\begin{pmatrix}1&0&0&0&0&1&1&1\\\ 3&3&2&1&0&0&1&2\\\ 1&1&1&1&1&1&1&1\\\
\end{pmatrix}\begin{pmatrix}Q_{1}^{a}\\\ Q_{2}^{a}\\\ Q_{3}^{a}\\\
Q_{4}^{a}\\\ Q_{5}^{a}\\\ Q_{6}^{a}\\\ Q_{7}^{a}\\\ Q_{8}^{a}\\\
\end{pmatrix}=\vec{0}\Rightarrow\begin{cases}Q^{a}_{1}+Q^{a}_{6}+Q^{a}_{7}+Q^{a}_{8}&=0\\\
3Q_{1}^{a}+3Q_{2}^{a}+2Q_{3}^{a}+Q_{4}^{a}+Q_{7}^{a}+2Q_{8}^{a}&=0\\\
\sum_{i=1}^{8}Q^{a}_{i}&=0\end{cases};$ (40)

where $a=1,...,5$. Solution space can be parametrized as follows:

$\begin{cases}&Q_{1}^{a}=-Q^{a}_{6}-Q^{a}_{7}-Q^{a}_{8};\\\
&Q_{2}^{a}=Q^{a}_{4}+2Q^{a}_{5}+3Q^{a}_{6}+2Q^{a}_{7}+Q^{a}_{8};\\\
&Q_{3}^{a}=-2Q_{4}^{a}-3Q_{5}^{a}-3Q_{6}^{a}-2Q_{7}^{a}-Q_{8}^{a};\end{cases}$
(41)

the equivalence relations are given by

$\displaystyle(z_{1},z_{2},z_{3},z_{4},z_{5},z_{6},z_{7},z_{8})\sim$ (42)
$\displaystyle\sim(\lambda^{Q_{1}^{a}}z_{1},\lambda^{Q_{2}^{a}}z_{2},\lambda^{Q_{3}^{a}}z_{3},\lambda^{Q_{4}^{a}}z_{4},\lambda^{Q_{5}^{a}}z_{5},\lambda^{Q_{6}^{a}}z_{6},\lambda^{Q_{7}^{a}}z_{7},\lambda^{Q_{8}^{a}}z_{8}),$

where $Q_{1}^{a},Q_{2}^{a},Q_{3}^{a}$ satisfy 41. Invariant polynomials and
relations are given by

$\begin{cases}&Y_{1}=z_{1}z_{6}z_{7}z_{8};\\\
&Y_{2}=z_{2}z_{4}^{-1}z_{5}^{-2}z_{6}^{-3}z_{7}^{-2}z_{8}^{-1};\\\
&Y_{3}=z_{3}z_{4}^{2}z_{5}^{3}z_{6}^{3}z_{7}^{2}z_{8};\\\
&Y_{4}=z_{1}z_{2}z_{4}^{-1}z_{5}^{-2}z_{6}^{-2}z_{7}^{-1}=Y_{1}Y_{2};\\\
&Y_{5}=z_{1}z_{3}z_{4}^{2}z_{5}^{3}z_{6}^{4}z_{7}^{3}z_{8}^{2}=Y_{1}Y_{3};\\\
&Y_{6}=z_{2}z_{3}z_{4}z_{5}=Y_{2}Y_{3};\\\
&Y_{7}=z_{1}^{-1}z_{3}z_{4}^{2}z_{5}^{3}z_{6}^{2}z_{7}=\frac{Y_{3}}{Y_{1}};\\\
&Y_{8}=z_{2}^{-1}z_{3}z_{4}^{3}z_{5}^{5}z_{6}^{6}z_{7}^{4}z_{8}^{2}=\frac{Y_{3}}{Y_{2}}.\end{cases}$
(43)

Let us now proceed with the matching between sets 39 and 43. Since we have
$\Delta P_{e}=0$ we can not consider all the not extremal $z_{i}$, indeed we
expect that no new points enter in the game. We note that if we match the
first three polynomials of the two sets we get automatically that all other
polynomials match, so

$\begin{cases}&X_{1}=Y_{1}\Rightarrow z_{1}z_{5}=z_{1}z_{6}\Rightarrow
a_{3}+a_{4}=b_{1}+b_{6};\\\ &X_{2}=Y_{2}\Rightarrow
z_{2}z_{4}^{-1}z_{5}^{-4}=z_{2}z_{5}^{-2}z_{6}^{-3}\Rightarrow
a_{2}-a_{1}-4a_{4}=b_{2}-2b_{5}-3b_{6};\\\ &X_{3}=Y_{3}\Rightarrow
z_{4}^{2}z_{5}^{4}=z_{5}^{3}z_{6}^{3}\Rightarrow
4a_{4}+2a_{1}=3b_{5}+3b_{6}.\end{cases}$ (44)

If we now consider the set of $R$-charges of
$\frac{\mathrm{SPP}}{\mathbb{Z}_{2}}$
($\\{a_{1},a_{2},a_{3},a_{4}\\}=\\{1-\frac{1}{\sqrt{3}},1-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}\\}$)
and $L^{(3,3,3)}$
($\\{b_{1},b_{2},b_{3},b_{4}\\}=\\{\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\\}$)
we note that system 44 does not match unless we suppose the existence of a new
operator (an orientifold-mapping operator) with $R$-charge
$\tilde{a}=\frac{2-\sqrt{3}}{\sqrt{3}}$. This seems very similar to the
$\Delta P_{e}=1$ example but this is not the case since here we have no
obvious candidate to play the role of $\Pi$. Despite this, we can build
relevant operators that perturb the conformal theory and induce the RG flow to
the dual theory in the IR.

##  4 Brief discussion on the holographic dual

In Section 3 we have shown that, assuming that the orientifold in III scenario
modifies the geometry by transforming the CYA cone into the CYB cone we are
forced to consider relevant operators which induce an RG flow that we know
must to end in a new superconformal fixed point. In holography we are able to
build up gravity solutions that are dual to RG flows.

###  4.1 The toy model for holographic duals of RG flows and beyond

Let us consider a set of scalar fields $\phi_{a}$ in a AdS5 background with
coordinate $(x^{0},x^{1},x^{2},x^{3},z)$666We are going to use Greek indices
for $4$-dimensional quantities and capital Latin indices for $5$-dimensional
ones and action given by

$S=\int
d^{4}xdz\sqrt{-g}\bigg{[}\frac{\mathcal{R}}{4}-\frac{1}{2}G_{ab}\partial_{A}\phi_{a}\partial^{A}\phi_{b}-V(\phi)\bigg{]}.$
(45)

We assume a $4D$ Poincaré invariant metric of the form
$ds=dz^{2}+e^{2Y(z)}dx_{\mu}dx^{\mu}$ that we recognize as Poincaré patch AdS
with radius $R$ after the substitutions $Y(z)=\frac{z}{R}$ and
$e^{\frac{z}{R}}=\frac{R}{z}$. Fields equations read

$\displaystyle
3(\partial_{z}Y)^{2}-\frac{1}{2}G_{ab}\partial_{z}\phi_{a}\partial_{z}\phi_{b}+V=0;$
(46) $\displaystyle
G_{ab}\partial_{z}^{2}\phi_{b}+4\partial_{z}YG_{ab}\partial_{z}\phi_{b}=\frac{\partial
V}{\partial\phi_{a}};$

an obvious solution is given by the negative critical points of the potential
$V_{crit}$ and a set of constant scalars,

$\frac{\partial V}{\partial\phi_{a}}=0,\ \ \ \partial_{z}\phi_{a}=0;$ (47)

then the second equation is trivially satisfied while the first gives us
$(\partial_{z}Y)^{2}=-\frac{V_{crit}}{3}$. Up to coordinate redefinition we
can then write

$Y(z)=\frac{z}{R}\ \ \ with\ \ \frac{1}{R^{2}}=-\frac{V_{crit}}{3}$ (48)

this is nothing but AdS with its radius controlled by the critical point of
the potential. Now we can expand the action around this solution and we can
infer the conformal dimension of a CFT operator $O_{a}$ from the masses of the
quadratic fluctuations $m_{a}$,

$R^{2}m_{a}^{2}=\Delta_{a}(\Delta_{a}-4).$ (49)

At this point we looking for more generic solutions that are asymptotically
AdS and thanks to this last point we can use the holograpic dictionary: in the
asymptotically AdS region the scalar behaves as

$\phi_{a}(z)\simeq
A_{a}z^{4-\Delta_{a}}+B_{a}z^{\Delta_{a}}=A_{a}e^{(\Delta_{a}-4)z}+B_{a}e^{-\Delta_{a}z}$
(50)

in which $A_{a}$ is the source of $O_{a}$ and $B_{a}$ is its VEV. If
$A_{a}\neq 0$ the solution describe a deformation of the CFT by operator
$O_{a}$ while if $A_{a}=0$ and $B_{a}\neq 0$ it describes a different vacuum
of the CFT where the operator $O_{a}$ develops a non vanishing VEV. In both
cases conformal invariance is broken and a RG flow is triggered; the gravity
solution is dual to this RG flow.
Particularly interesting is the case in which the $O_{a}$ operator is relevant
and the RG flow leads to a new fixed point and this is possible if and only if
the potential $V(\phi)$ has more than one critical point (in such a way that
another AdS solution with different radius is possible); the gravity solution
is a kink interpolating between the two critical point AdS solutions. In order
to have an RG flow that starts from a CFTUV we need a relevant operator (very
similar to what happens in IR duality linked to III scenario orientifold) and
to hit a CFTIR we need the operator to become irrelevant there. Looking at the
mass-dimension formula this means that in the relevant operator case the
squared mass of the scalar fluctuation must be negative (a maximum of the
potential $V(\phi)$) while in the irrelevant operator case the squared mass
must be positive (a minimum of the potential $V(\phi)$).
The construction of holographic dual of RG flows could be applied to explain
the IR duality arising from III scenario. We have seen how the duality
requires the presence of relevant operators that deform the initial CFT
associated to $\mathrm{CY_{A}}$ cone into the CFT associated to
$\mathrm{CY_{B}}$ cone. In the spirit of the toy model presented above, this
entire IR duality mechanism could be interpreted as the holographic dual of a
suitable effective supergravity solution that has to interpolate between the
supergravity solution of oriented model $\mathrm{A}$ and the supergravity
solution of unoriented model $\mathrm{B}$. This kind of approach poses
technical and conceptual challenges. First of all the Kaluza-Klein (KK)
reduction on the compact Sasaki-Einstein manifold $X^{5}$ is far more
difficult than for $S^{5}$. Therefore, it is not easy at all to build up
explicit potentials for these supergravity descriptions. Moreover, from the
conceptual point of view, the supergravity description cannot be the full
story since the orientifold is non-perturbative. The supergravity description
has to be uplifted in a full string theory description which must match with
the string theories associated with the two CFTs $\mathrm{A}$ and
$\mathrm{B}$.

##  5 Conclusion

Orinetifold projections give the opportunity of new possibilities to the
creation of field theory models from string background called unoriented
models [13]. The appellation "unoriented" is mainly due to the action of the
orientifold map on the string oscillators: the involution makes oriented
string unoriented one and a Chan-Paton like analysis suggests the possibility
of new gauge group like $SO(N)$ and $Sp(2N)$. However, orientifold behaviour
is inferred by comparison with the flat-space behavior but, probably,
orientifold have an intrinsic non-perturbative nature [31]. This is also
suggested by a recent interesting new feature proposed in [18] and expanded in
[19]: the so-called III scenario. What happens is that a theory after a III
scenario orientifold shares the same $R$-charges, central charge and
superconformal index of another orientifolded theory coming from a I scenario
orientifold. The crucial point is that these physical quantities are the same
not only in the large $N$ limit but for every $N$ [18],[19]: this is a non-
perturbative effect. Since the physical quantities are the same in the IR we
refer to these theories as IR duals and, since in the geometrical engineering
the field theory depends on the geometry of the CY cone, these two
orientifolded theories can be considered to have the same CY cone geometry.
This means that a changing in the geometry of the cone has to happen due to
the orientifold action. The change, due to the orientifold, can be traced back
to the equations that describe the algebraic CY varieties and we require the
matching between these sets of equations. In this sense the orientifold has an
algebro-geometrical interpretation as a morphism between algebraic varieties
given locally by polynomials. This interpretation has as its guiding idea the
flop transition and the other geometrical changing in string theory
compactifications [20],[21],[22],[23]. The properties and a more deep
understanding of this interpretation, such as the answers to the questions:
"Are orientifold regular maps or rational maps? Are they biregular or
birational?" could be fertile ground for future works. We stress that if
orientifold are such kind of maps for the CY cones they open the door to the
use of the theory of category with orientifold as morphisms and field theory
as objects. Therefore, orientifold act both on string states and on the
geometry of the space-time.
From the requirement of the matching between these sets of equations
describing the CY cone of the two theories and using some tools of algebraic
geometry and toric geometry, reported in 2.1, we infer the existence of
relevant operators that deform the initial CFT triggering an RG flow. The flow
could be described using a holographic dual, whose toy model is presented in
4.1, but the prohibitive Kaluza-Klein reduction on the base of a generic CY
cones makes difficult the explicit construction. Moreover, due to the intrisic
non-perturbative nature of orientifold, this supergravity dual is expected to
be only an effective description that must necessarily be uplifted in full
string theory.

#### Acknowledgment

We thank first of all Prof. Fabio Riccioni to have inspired the present work
and Prof. Massimo Bianchi and Salvo Mancani for interesting discussions. We
also thank Davide Morgante for discussions and comments and Matteo Romoli for
its reading of the manuscript and comments. Special thanks to Maria Luisa
Limongi who supported the work.

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# LandCoverNet: A global benchmark land cover classification training dataset

Hamed Alemohammad
Radiant Earth Foundation
<EMAIL_ADDRESS>
Kevin Booth
Radiant Earth Foundation
<EMAIL_ADDRESS>

###### Abstract

Regularly updated and accurate land cover maps are essential for monitoring 14
of the 17 Sustainable Development Goals. Multispectral satellite imagery
provide high-quality and valuable information at global scale that can be used
to develop land cover classification models. However, such a global
application requires a geographically diverse training dataset. Here, we
present LandCoverNet, a global training dataset for land cover classification
based on Sentinel-2 observations at 10m spatial resolution. Land cover class
labels are defined based on annual time-series of Sentinel-2, and verified by
consensus among three human annotators.

##  1 Introduction

Land cover (LC) maps play an important role in many applications such as
precision agriculture, urban mapping, and environmental monitoring among
others [1]. Earth observations provide valuable data at global scale that can
be used to build applications for LC mapping. Recently, with the advancements
in machine learning (ML) techniques, several studies have been carried out to
develop ML-based LC classification models [2, 3, 4, 5]. However, the remaining
challenge is lack of high-quality open-access training data that represents
the geographical diversity of LC classes at global scale [6, 7]. Such a
training dataset can open the value of open satellite imagery (such as Landsat
and Sentinel missions) for regional and global monitoring with enhanced
accuracy.

In response to this need, we present LandCoverNet the first open-access,
global and geographically diverse LC classification training dataset.
LandCoverNet is based on multi-spectral imagery from Sentinel-2 satellites,
and consists of globally representative image chips and labels. Specification
of the dataset, including LC taxonomy, is defined from the outputs of an
expert workshop that we hosted two-years ago.

Generating labels for a large-scale training dataset such as LandCoverNet
requires extensive coordination with annotators, and deploying tools to
facilitate pixel-level labeling. Moreover, human interpretation error is
unavoidable at 10 m spatial resolution. Therefore, a consensus algorithm was
implemented to generate the label for each pixel in the dataset using inputs
from multiple users.

In this paper, we discuss the label generation procedure, the sampling scheme,
and specifications of the data. LandCoverNet v1.0 which covers the whole
African continent is now available for public access, and other parts of the
world will be gradually released.

##  2 Dataset Specifications

###  2.1 Land Cover Taxonomy

LC class taxonomies vary across existing global LC products. While there are
multiple taxonomies for LC classification, they do not take into account the
spatial resolution at which the classification map will be produced. This is
essential since some classes might not be resolvable with the resolution of
the input satellite imagery.

To address this problem, we organized an expert workshop to define the
specification of LandCoverNet dataset as a benchmark for the community. The
participants who were domain experts in LC mapping and ML developed a
hierarchical LC class taxonomy. Based on the hierarchical definitions, they
recommended the taxonomy presented in Table 1 for LandCoverNet dataset.

Class definition in the taxonomy presented in Table 1 is based on annual time-
series of each location. This means in order to assign one of these classes to
a region, one needs to examine the annual time-series of observations from the
satellite, and then assign the appropriate class.

Table 1: LandCoverNet class taxonomy Level 1 | Level 2 | Level 3
---|---|---
Bare max veg/yr $\leq 10\%$ | Snow/Ice | Snow/Ice
Water | Water
Bare Ground | Artificial
Natural
Vegetation max veg/yr $\geq$ 10% | Woody | Woody
Non-Woody | Cultivated
(Semi) Natural

###  2.2 Sentinel-2 Tile Sampling

Sentinel-2 data are provided as $100\times 100$ $km^{2}$ ortho-images in
UTM/WGS84 projection. Each of the $100\times 100$ $km^{2}$ regions is called a
tile. To build a geographically diverse training dataset, we designed a
sampling scheme and selected 300 tiles of Sentinel-2 for training data
generation.

The number of samples in each continent is set to be proportional to the area
of the continent. To capture diversity of land surface in each continent, we
used an existing global LC map at 250m resolution based on MODIS instrument
[8] to define LC classes within each tile of Sentinel-2. We chose to use MODIS
LC maps as features for sampling our regions as this is the only way to ensure
diversity in our samples. Practically, MODIS LC data is the best estimate of
the distribution of LC classes at global scale, and we use it to sample a set
of tiles and generate an accurate training dataset that can be used for future
LC mapping.

For each tile of Sentinel-2 (approximately 18000 across all land areas), we
calculated a set of features as following: latitude and longitude of the tile
centroid, and percentage of each of the 17 IGBP LC classes in the tile based
on MODIS product. This results in 19 features for each tile. To ensure tiles
do not have a dominant LC class, any tile with more than 60% water or 90%
barren land was excluded from the sampling. Finally, we used the empirical CDF
of the features, and uniformly sampled the tiles in each continent. The
resulting data contains 66 tiles in Africa, 92 tiles in Asia, 20 tiles in
Australia and Oceania, 28 tiles in Europe, 54 tiles in North America and 40
tiles in South America.

###  2.3 Chip Selection

As part of the expert workshop, and a follow up user survey, the chip size for
this training dataset was set to $256\times 256$. For the first set of these
labels, and based on computational resources we decided to generate 30 chips
within each of the 300 selected tiles of Sentinel-2. This results in 9000
chips at global scale, or equivalently $\sim$589 million pixels.

##  3 Methodology

Generating human verified labels for time-series imagery is a challenging
task. In order to facilitate the process, we provided our team of annotators
with a "guess label" based on a model prediction. For this, we trained a
Random Forest model for each tile of Sentinel-2 in our dataset, and overlaid
that in the annotation dashboard. Details of this model training is explained
in Appendix A. In this section, we explain the human validation of labels and
calculation of the consensus score.

###  3.1 Human Validation

To generate a high-quality training dataset, we validated labels of each pixel
with the help of human annotators. For this, we worked with a team of
annotators who had experience labeling satellite imagery, and provided them
with training necessary to work on this dataset, and our labeling dashboard.
We also performed a labeling campaign with them on a test dataset, to ensure
they are familiar with the taxonomy before starting on the main dataset. The
new element for this labeling task was the time-series dimension.

Every user was asked to look through all the 24 scenes of Sentinel-2 for each
chip and then check the model prediction. If the predictions are good, leave
them unchanged, otherwise add the correct label. They had to submit the labels
even if no changes were made to ensure the label was marked as validated.

To facilitate the labeling, each $256\times 256$ chips was split to 64 tasks
of $32\times 32$ pixels. In each task, annotators would see the Sentinel-2 RGB
image, false color NIR visualization, NDVI and NDWI as well SWIR bands. They
had the option to zoom in and out of the task bounding box and examine the
surrounding area to help them with the labeling. Figures A.1 \- A.3 show
screenshots of the labeling dashboard.

We implemented another measure to minimize chances of error in human
validation of our data [9]. Each chip was sent to three individuals in the
annotation team and they were asked to validate the labels. Then we used a
consensus technique (Section 3.2) to generate the final label for each pixel.

###  3.2 Consensus Label

Sentinel-2 imagery has a 10m spatial resolution in the RGB and Near Infrared
(NIR) bands. While distinguishing between the 7 classed outlined in Table 1 is
possible at this resolution, human error is inevitable. As discussed by Elmes
et al. [9], such errors need to be taken into account during the training data
generation process, and quantified in the final product.

To mitigate these error, we took two steps in our label validation phase.
First, about 540 tasks were also labeled by an expert in our team to be used
as reference for assessing each annotators’ work. Second, each task was
independently labeled by three annotators.

To derive a consensus score and label for each pixel using the three labels,
we first calculated a score for each annotator using the expert labeled tasks.
Annotator’s score for each task is defined as the percentage of pixels that
they correctly labeled in reference to the expert labels. Using the 540 expert
labeled tasks, we calculated an overall accuracy score for each annotator.

Finally, using a Bayesian averaging approach we calculate the consensus
probability for each of the 7 classes. Then the class with highest probability
is selected as the label and the probability is reported as the consensus
score [10, 11]. Eq. 1 presents the consensus score for a pixel:

$P(\theta_{j})=\sum_{w_{i}=1}^{n}P(w_{i})\times P(\theta_{j}|w_{i})$ (1)

where: $\theta_{j}\in[1,\dots 7]$ is the true LC class, $n$ is number of
annotators labeling a pixel, and $P(w_{i})$ is the accuracy score of annotator
$w_{i}$.

##  4 Dataset Properties

LandCoverNet v1.0 which covers all the image chips for Africa is published on
Radiant MLHub for public access. This version includes 1980 chips of
$256\times 256$ pixels across Africa. The labels in the dataset cover all the
classes other than permanent snow/ice which is not common in Africa. Figure 1
shows the distribution of pixels within different classes. Majority of the
pixels fall within vegetated class, including about 15% cultivated pixels.
Natural bare ground also has a significant portion with about 17% of the data.
Figure 3 shows four sample chips overlaid on Sentinel-2 imagery from the
dataset.

Overall, the consensus scores are very high. Figure 2 shows the histogram of
consensus scores across all pixels. 60% of the data has a score of 100%
meaning all annotators agreed on the class, and the rest have relatively high
score, with about 34% of the data having a consensus score of more than 60%
and less than 100%. Users who access the data get a per pixel consensus score
that can be used during model training to increase the model confidence in
high score pixels and vice-versa.

Figure 1: Distribution of different classes in the LandCoverNet v1.0 Figure 2:
Distribution of consensus score in the LandCoverNet v1.0 Figure 3: Four sample
chips of LandCoverNet labeled data (color shaded regions) overlaid on one of
their Sentinel-2 scene. (blue: water, dark green: cultivated vegetation, light
green: (semi) natural vegetation, grey: artificial bare ground, brown: woody
vegetation)

##  5 Dataset Access

LandCoverNet is distributed under a Creative Commons Attribution 4.0
International (CC BY 4.0) license. Interested users can access the data on
Radiant MLHub111www.mlhub.earth through an API. The dataset has a description
page accessible on www.landcover.net which includes a Jupyter Notebook
tutorial for downloading the data.

## Acknowledgements

This dataset was generated with a grant awarded to Radiant Earth Foundation
from Schmidt Futures. We would like to acknowledge in kind technology support
from our collaborators at Sinergise.

## References

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##  Appendix A Generating Guess Labels for Human Validation

###  A.1 Model Development

The goal of this LC classification model was to provide a baseline for
annotators to facilitate their labeling. Therefore, we chose a simple Random
Forest architecture for this and trained a pixel-wise classification model
that used time-series of observation from Sentinel-2 and predicted the LC
class.

####  A.1.1 Input Data

For each tile, we queried the Sentinel-2 catalog in calendar year 2018, and
selected 24 scenes for modeling. Scenes were selected in a two step approach:
1) one scene was selected in each calendar month with the minimum cloud cover
percentage, and 2) 12 more scenes were selected throughout the year sorted by
their cloud cover percentage. The logic for this selection was to ensure there
is enough data distributed throughout the year to reconstruct the time-series
for each pixel.

We used 10 bands from Sentinel-2 observations as predictors. Red, Green, Blue
and NIR bands that are at 10m spatial resolution and Red-edge bands 1-4, and
SWIR bands 1-2 that are at 20m spatial resolution. We used a nearest neighbor
interpolation to map 20m bands to 10m for each scene.

Sentinel-2 L2A product has a scene classification layer (SCL) that provides a
quality flag for each pixel, and a four class label (water, vegetation, snow,
not-vegetated). We used this layer as a filter to clean the time-series data
before inputting to the model. Any pixel that was classified as cloudy, cloud
shadow or saturated/defective was removed from the time-series.

Finally, all the valid observations of each pixel were used to interpolate the
missing or filtered-out ones, and reconstruct a time-series of 24 observations
with 10 bands at each time for all pixels within the scene. We used a linear
interpolation with a periodicity of 1 year to reconstruct the time-series.
These 240 features were used as input in the model training.

####  A.1.2 Model Training

For training a LC classification model, we needed yet another target LC
classification product. Therefore, we decided to use the best available
product. Noting that any LC map has its own uncertainties, and our goal was to
generate the best guess for our annotators. We selected GlobeLand30 (GLC)
product which was produced for the year 2010 using Landsat 8 data at 30m
spatial resolution [12].

We used the method presented by [13] to map GLC classes to LandCoverNet
taxonomy. To remove any discrepancy between GLC product from 2010 and our
imagery from 2018, we used a majority voting on the SCL labels from Sentinel-2
L2A and defined a set of annual 4-class filter for each pixel: water,
vegetation, snow, not-vegetated. Comparing these classes to the LC class from
GLC for each pixel, we dropped any pixel that had a mis-match between the two
labels, and didn’t used them in the training. The remaining pixels were split
to train (75%) and test (25%) sets using stratified sampling.

The training (and test) sets naturally have class imbalance as the spatial
coverage of a tile is limited to $100$ $km^{2}$. Therefore, we used a Balanced
Random Forest model, and chose 2 as the number of trees. These parameters were
selected after several hyperparameter tuning on multiple tiles to maximize the
model performance.

Finally, for each tile, the trained model was used to generate a LC label at
10m resolution that was then passed to annotators for validation.

Figure A.1: Labeling dashboard showing Sentinel-2 scene on the left side and
the labels on the right. Figure A.2: Same as Figure A.1 with the temporal
window overlaid on Sentinel-2 scene allowing users to select different times
of the year. Figure A.3: Same as Figure A.2 with the class selection tool open
to show how users select different classes.