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\begin{document}

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\section*{RR-field tadpole cancellation}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{string_theory}{}\paragraph*{{String theory}}\label{string_theory}

[[!include string theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{ForFractionalDBranes}{In plane and toroidal orientifolds}\dotfill \pageref*{ForFractionalDBranes} \linebreak
\noindent\hyperlink{InTermsOfEquivariantKTheory}{In terms of equivariant K-theory / the representation ring}\dotfill \pageref*{InTermsOfEquivariantKTheory} \linebreak
\noindent\hyperlink{TwistedTadpoleCancellation}{Local/twisted tadpole cancellation}\dotfill \pageref*{TwistedTadpoleCancellation} \linebreak
\noindent\hyperlink{UntwistedTadpoleCancellation}{Global/untwisted tadpole cancellation}\dotfill \pageref*{UntwistedTadpoleCancellation} \linebreak
\noindent\hyperlink{ExamplesForToroidalOrientifolds}{Examples for toroidal orientifolds}\dotfill \pageref*{ExamplesForToroidalOrientifolds} \linebreak
\noindent\hyperlink{ExamplesForFractionalDBranes}{Examples of non-compact singularities}\dotfill \pageref*{ExamplesForFractionalDBranes} \linebreak
\noindent\hyperlink{at_a_orientifold_singularity}{At a $\mathbb{Z}_2$-orientifold singularity}\dotfill \pageref*{at_a_orientifold_singularity} \linebreak
\noindent\hyperlink{at_a_orientifold_singularity_2}{At a $\mathbb{Z}_4$-orientifold singularity}\dotfill \pageref*{at_a_orientifold_singularity_2} \linebreak
\noindent\hyperlink{At2d4Singularity}{At a $2 D_4$-orientifold singularity}\dotfill \pageref*{At2d4Singularity} \linebreak
\noindent\hyperlink{At2D6Singularity}{At a $2 D_6$-orientifold singularity}\dotfill \pageref*{At2D6Singularity} \linebreak
\noindent\hyperlink{At2D8Singularity}{At a $2 D_8$-orientifold singularity}\dotfill \pageref*{At2D8Singularity} \linebreak
\noindent\hyperlink{At2D10Singularity}{At a $2 D_{10}$-orientifold singularity}\dotfill \pageref*{At2D10Singularity} \linebreak
\noindent\hyperlink{At2D12Singularity}{At a $2 D_{12}$-orientifold singularity}\dotfill \pageref*{At2D12Singularity} \linebreak
\noindent\hyperlink{At2D14Singularity}{At a $2 D_{14}$-orientifold singularity}\dotfill \pageref*{At2D14Singularity} \linebreak
\noindent\hyperlink{At2D16Singularity}{At a $2 D_{16}$-orientifold singularity}\dotfill \pageref*{At2D16Singularity} \linebreak
\noindent\hyperlink{At2TSingularity}{At a $2 T$-orientifold singularity}\dotfill \pageref*{At2TSingularity} \linebreak
\noindent\hyperlink{At2OSingularity}{At a $2 O$-orientifold singularity}\dotfill \pageref*{At2OSingularity} \linebreak
\noindent\hyperlink{At2ISingularity}{At a $2 I$-orientifold singularity}\dotfill \pageref*{At2ISingularity} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{examples_and_models}{Examples and Models}\dotfill \pageref*{examples_and_models} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In the presence of [[D-branes]], plain [[type II string theory]] in fact has a [[quantum anomaly]] reflected on the [[worldsheet]] by [[tadpole]] [[Feynman diagrams]] in the [[string perturbation series]] for [[RR-fields]]

$\backslash$begin\{center\} $\backslash$begin\{imagefromfile\} ``file\_name'': ``TadpoleCancellationFactorization.jpg'', ``width'': 600 $\backslash$end\{imagefromfile\} $\backslash$end\{center\}

\begin{quote}%
graphics grabbed from \hyperlink{BlumenhagenLustTheisen13}{Blumenhagen-Lüst-Theisen 13}


\end{quote}
and reflected in [[target spacetime]] by non-trivial total [[RR-field]] [[flux]] on [[compact topological space|compact spaces]]

$\backslash$begin\{center\} $\backslash$begin\{imagefromfile\} ``file\_name'': ``RRTadpoleCancellation.jpg'', ``width'': 600 $\backslash$end\{imagefromfile\} $\backslash$end\{center\}

\begin{quote}%
graphics grabbed from \hyperlink{IbanezUranga12}{Ibanez-Uranga 12}


\end{quote}
This anomaly cancels if the [[D-branes]] are accompanied by a suitable collection of [[O-planes]], hence if one considers [[orientifold]] backgrounds (\hyperlink{Sagnotti88}{Sagnotti 88, pp. 5}, \hyperlink{GimonPolchinski96}{Gimon-Polchinski 96, section 3}). (For space-filling [[O-planes]] this means to consider [[type I string theory]] instead.)

Accordingly, tadpole cancellation via [[orientifolds|orientifolding]] is a key consistency condition in the construction of [[intersecting D-brane models]] for [[string phenomenology]].

Traditionally RR-tadpole cancellation is discussed in [[ordinary cohomology]], the common arguments notwithstanding that [[D-brane charge]] should be in [[K-theory]].

Discussion of tadpole cancellation with [[D-brane charge]] regarded in [[K-theory]] was initated in \hyperlink{Uranga00}{Uranga 00, Section 5}, see also \hyperlink{GarciaUranga05}{Garcia-Uranga 05}, \hyperlink{Marchesano03}{Marchesano 03, Section 4}, \hyperlink{MarchesanoShiu04}{Marchesano-Shiu 04}, \hyperlink{CKMNW05}{CKMNW 05, Section 2.2}, \hyperlink{MaidenShiuStefanski06}{Maiden-Shiu-Stefanski 06, Section 5}, \hyperlink{DFM09}{DFM 09, p. 6-7}.

But the situation remains somewhat inconclusive (see also \hyperlink{Moore14}{Moore 14, p. 21-22}).

$\backslash$linebreak

\hypertarget{ForFractionalDBranes}{}\subsection*{{In plane and toroidal orientifolds}}\label{ForFractionalDBranes}

More details are understood in the special case of plane orbifolds $V^{cpt} \!\sslash\! G$ and [[toroidal orbifold|toroidal orientifolds]] $\mathbb{T}^V \!\sslash\! G$ where [[fractional D-branes]] may be stuck at [[orbifold]]/[[orientifold]] singularities, whose [[D-brane charge]] is supposed to be in the [[equivariant K-theory]] of the point, hence the [[representation ring]] of the given [[isotropy group]].

\hypertarget{InTermsOfEquivariantKTheory}{}\subsubsection*{{In terms of equivariant K-theory / the representation ring}}\label{InTermsOfEquivariantKTheory}

In this case tadpole cancellation conditions are given by [[representation theory|representation theoretic]] [[equations]], constraining the [[character of a linear representation|characters]] of the [[linear representations]] corresponding to the [[fractional D-branes]].

Let $G$ be a [[finite group]]. Let

\begin{displaymath}
[1]
    \subset 
  [H_1]
    \subset 
  [H_2]
    \subset
    \cdots
    \subset
  [G]
\end{displaymath}
be a [[linear extension of a partial order|linear extension]] of its [[partially ordered set|partially ordered]] [[lattice of subgroups|lattice of]] [[conjugacy classes]] of [[subgroups]], with [[subset|sub-]] [[linear order]] of [[cyclic group|cyclic]] [[subgroups]]

\begin{displaymath}
[1]
    \subset 
  \left[
   \left\langle
     g_1
   \right\rangle
  \right]
    \subset 
  \left[
   \left\langle
     g_2
   \right\rangle
  \right]
    \subset
    \cdots
    \subset
  \left[
   \left\langle
     g_{\vert ConjCl(G)\vert}
   \right\rangle
  \right]
  \,.
\end{displaymath}
This way every [[virtual representation]] $[V] \in RU(G) = KU_G(\ast)$ (the [[D-brane charge]] of a [[bound state]] of [[fractional D-branes]]/[[anti-branes]]) has a [[character of a linear representation|character]] which is a list of [[complex numbers]] of the form

\begin{tabular}{l|l|l|l|l|l}
$[H] =$&$\left[\langle e\rangle\right]$&$\left[\langle g_1\rangle\right]$&$\left[\langle g_2\rangle\right]$&$\cdots$&$\left[\langle g_{\vert ConjCl(G)\vert}\rangle\right]$\\
\hline 
$\chi_V =$&$dim(V)$&$tr_V\left( g_1\right)$&$tr_V\left(g_2\right)$&$\cdots$&$tr_V\left(g_{\vert ConjCl(G)\vert}\right)$\\
${{\text{fractional} \atop \text{D-brane/anti-brane}} \atop \text{bound state}}$&${ {\text{mass} =} \atop {{\text{net number} \atop \text{of branes}}}}$&${\text{RR-charge in} \atop {g_1\text{-twisted sector}}}$&${\text{RR-charge in} \atop {g_2\text{-twisted sector}}}$&$\cdots$&$\cdots$\\
\end{tabular}

Here $dim(V) \in \mathbb{Z}$ is the [[mass]], hence the net number of [[fractional D-branes]]/[[anti-branes]] in the [[bound state]], while $tr_V\left(g_k\right)$ is (up to a global [[rational number]]-factor $1/{\vert G \vert}$) supposed to be its [[charge]] as seen by the [[RR-fields]] in the $g_k$-[[twisted sector]].

In fact, since we are dealing with fractional D-branes, both the charge and mass in the above table are in factional units $1/{\vert G\vert}$ of the [[order of a group|order]] of the [[isotropy group]] $G$ (by \href{fractional+D-brane#eq:RRChargeOfFractionalDBraneInGTwistedSector}{this formula}), so that normalized [[mass]] and [[charge]] is

\begin{equation}
M
  \;=\;
  \tfrac{1}{{\vert G\vert}}
  dim(V)
  \,,
  \phantom{AAA}
  Q_V(g)
  \;=\;
  \tfrac{1}{\vert G\vert}
  \chi_V(g) 
  \coloneqq 
    \tfrac{1}{\vert G\vert}
   tr_V\left( g\right)
  \,.
\label{ChargeMass}\end{equation}
\hypertarget{TwistedTadpoleCancellation}{}\paragraph*{{Local/twisted tadpole cancellation}}\label{TwistedTadpoleCancellation}

The \textbf{twisted (local) tadpole cancellation condition} for [[fractional D-branes]] at orbifold singularities is that the RR-charges in all non-trivially twisted sectors vanish:

\begin{equation}
Q_V(g) 
  =
  0
  \phantom{AA}\text{hence equivalently} \phantom{AA}
  \chi_{V}\left(g\right) \;=\; 0
  \,,
  \phantom{AAA}
  g \neq e
\label{VanishingOfCharacterValuesOnNontrivialSubgroups}\end{equation}
\begin{example}
\label{UnitRegularRepresentation}\hypertarget{UnitRegularRepresentation}{}
\textbf{([[regular representation]] solves tadpole cancellation for [[fractional D-branes]])}

For every [[finite group]] $G$, the homogeous tadpole cancellation condition \eqref{VanishingOfCharacterValuesOnNontrivialSubgroups} is satisfied by all multiples $n \cdot k[G/1]$ of the [[regular representation]] $k[G/1]$ (since no non-trivial element $g \in G$ has [[fixed points]] when acting on $G$, and using \href{table+of+marks#MarkHomomorphismIsCharactersOfPermutationRepresentation}{this Prop.}). Hence the [[mass]] and [[charge]] \eqref{ChargeMass} of the [[fractional D-brane]] corresponding to the [[regular representation]] is

\begin{displaymath}
M_{{}_{k[G/1]}}
  \;=\;
  1
  \,,
  \phantom{AA}
  Q_{{}_{k[G/1]}}(g)
  \;=\;
  0
  \,.
\end{displaymath}
These multiples of the [[regular representation]] are regarded as trivial solutions to \eqref{VanishingOfCharacterValuesOnNontrivialSubgroups}.

\end{example}
\begin{prop}
\label{RegularRepSpansSolutionToHomogeneousTadpoleCancellation}\hypertarget{RegularRepSpansSolutionToHomogeneousTadpoleCancellation}{}
In fact, the multiples of the [[regular representation]] (Example \ref{UnitRegularRepresentation}) are the \emph{only} solutions to the local/twisted tadpole cancellation condition \eqref{VanishingOfCharacterValuesOnNontrivialSubgroups} for [[fractional D-branes]].

\end{prop}
\begin{proof}
Consider the truncated [[character morphism]]

\begin{displaymath}
Q \cdot {\vert G \vert}
  \;\colon\;
  Rep_k(G)
  \overset{\chi}{\longrightarrow}
  k^{\left\vert ConjCl(G)  \right\vert}
  \overset{ \text{forget dimension/mass}  }{\longrightarrow}
  k^{\left\vert ConjCl(G)\right\vert -1 }
  \,.
\end{displaymath}
We have to show that the [[kernel]] of this map is the [[free abelian group]] generated by the [[regular representation]]:

\begin{displaymath}
ker\big( 
    Q \cdot {\vert G \vert}
  \big)
  \;\simeq\;
  \mathbb{Z} \cdot k[G/1]
  \,.
\end{displaymath}
Now over a [[ground field]] $k$ of [[characteristic zero]] (such as the [[real numbers]] or [[complex numbers]], in the case at hand) we have (from \href{character+of+a+linear+representation#NormalizedSumOfCharacters}{this Example}) that

\begin{enumerate}%
\item for $\rho \neq \mathbf{1}$ a non-[[trivial representation|trivial]] [[irreducible representation]] we have

\begin{displaymath}
\underset{g \in G \setminus \{e\}}{\sum} Q_{\rho}(g) \cdot {\vert G \vert}
  \;\coloneqq\;
  \underset{g \in G \setminus \{e\}}{\sum} \chi_\rho(g)
  \;=\;
  - dim(\rho)
\end{displaymath}

\item for $\rho = \mathbf{1}$ the [[trivial representation|trivial]] [[irreducible representation]] we have

\begin{displaymath}
\underset{g \in G \setminus \{e\}}{\sum} Q_{\rho}(g) \cdot {\vert G \vert}
  \;\coloneqq\;
  \underset{g \in G \setminus \{e\}}{\sum} \chi_\rho(g)
  \;=\;
  {\left\vert G\right \vert}  - 1
  \;=\;
  - dim(\mathbf{1}) \;mod\; {\vert G\vert}
\end{displaymath}


\end{enumerate}
Since every $V \in R_{k}(G)$ is a $\mathbb{Z}$-[[linear combination]] of these [[irreps]], it follows generally that the \emph{fractional part} of the [[mass]] of a [[fractional D-brane]] is recovered from its [[charges]]:

\begin{displaymath}
dim(V)
   \;mod\;
   {\vert G \vert}
   \;=\;
   -
   \underset{g \in G \setminus \{e\}}{\sum} Q_{V}(g) \cdot {\vert G \vert}
   \;\coloneqq\;
   -
   \underset{g \in G \setminus \{e\}}{\sum} \chi_V(g)
   \,.
\end{displaymath}
But this means that all $V$ in the [[kernel]] of $Q \cdot {\vert G \vert}$ must have

\begin{displaymath}
dim(V) \;=\; 0 \;mod\; {\vert G \vert}
  \,.
\end{displaymath}
This is indeed the case for the multiples $V = n\cdot k[G/1]$ of the [[regular representation]] (Example \ref{UnitRegularRepresentation}). Conversely, the injectivity of the full [[character morphism]] $\chi$ (\href{character+of+a+linear+representation#InCharZeroCharacterMorphismIsInjective}{this Prop.}) says that every $V$ with $dim(V) = n \cdot {\vert G\vert }$ and $Q_V(g) = 0$ must be the $n$th multiple of the regular representation.

\end{proof}
\hypertarget{UntwistedTadpoleCancellation}{}\paragraph*{{Global/untwisted tadpole cancellation}}\label{UntwistedTadpoleCancellation}

On the other hand, at an [[orientifold]] singularity, the [[O-plane]] itself carries such charge -- [[O-plane charge]] (see \href{orientifold+plane#OPlaneCharge}{there}):

\begin{tabular}{l|l|l|l|l|l}
$[H] =$&$\left[\langle e\rangle\right]$&$\left[\langle g_1\rangle\right]$&$\left[\langle g_2\rangle\right]$&$\cdots$&$\left[\langle g_{\vert ConjCl(G)\vert}\rangle\right]$\\
\hline 
$\chi_O =$&$dim(O)$&$tr_O\left( g_1\right)$&$tr_O\left(g_2\right)$&$\cdots$&$tr_O\left(g_{\vert ConjCl(G)\vert}\right)$\\
$\text{O-plane}$&&${\text{O-plane charge in} \atop {g_1\text{-twisted sector}}}$&${\text{O-plane-charge in} \atop {g_2\text{-twisted sector}}}$&$\cdots$&$\cdots$\\
\end{tabular}

(These are $O^-$-plane charges. There may also be $O^+$-plane charges. Alternatively, these are $O^-$-branes with a fractional D-brane stuck on them.)

Now the \textbf{untwisted (global) tadpole cancellation condition} is that (all representations are real and) this [[O-plane charge]] is cancelled against the [[D-brane charge]]:

\begin{equation}
\chi_{V}\left(g_{k \geq 1}\right)  \;=\; 0
  \phantom{AA}
  \text{and}
  \phantom{AA}
  dim(V) = dim(O)
  \,.
\label{InhomogeneousTadpoleCancellation}\end{equation}
By Prop. \ref{RegularRepSpansSolutionToHomogeneousTadpoleCancellation} the only possible solution of this is the $n$th multiple of the [[regular representation]], if $dim(O)$ is $n$ times the dimension of the [[regular representation]]:

\begin{equation}
V = N \cdot k[G/1]
  \,.
\label{VIsMultipleOfRegularRepresentation}\end{equation}
In basic examples the [[O-plane]]-charge

\begin{displaymath}
O = 2^{p-4} n \cdot  \mathbf{1}
\end{displaymath}
is for $n_O$ coincident [[O-planes]] is the corresponding multiple by the [[O-plane charge]] $\mu_{Op} = -2^{8-4}$ (\href{orientifold+plane#eq:OpPlaneCharge}{here}) of the [[trivial representation|trivial]] [[irrep]], whence a solution to the tadpole cancellation exists if $\frac{2^{p-4}}{\vert G\vert } \in \mathbb{N} \subset \mathbb{Q}$ and is then given by

\begin{displaymath}
V \;=\; \frac{2^{p-4}}{\vert G\vert } \cdot k[G/1]
  \,.
\end{displaymath}
Sometimes the condition \eqref{VIsMultipleOfRegularRepresentation} is found with an offset by a trivial representation

\begin{equation}
V = N \cdot k[G/1] + \mathbf{p}_{triv}
  \,.
\label{VIsMultipleOfRegularRepresentation}\end{equation}
This corresponds to single fractional D-branes sitting on top of the O-planes, turning $O^-$-planes into $O^+$-planes.

\hypertarget{ExamplesForToroidalOrientifolds}{}\subsubsection*{{Examples for toroidal orientifolds}}\label{ExamplesForToroidalOrientifolds}

[[!include RR-field tadpole cancellation on toroidal orientifolds -- table]]

$\backslash$linebreak

\begin{quote}%
graphics grabbed from \hyperlink{SatiSchreiber19}{Sati-Schreiber 19}


\end{quote}
See also at \emph{[[equivariant Hopf degree theorem]]}.

\begin{quote}%
graphics grabbed from \hyperlink{SatiSchreiber19}{Sati-Schreiber 19}


\end{quote}
\hypertarget{ExamplesForFractionalDBranes}{}\subsubsection*{{Examples of non-compact singularities}}\label{ExamplesForFractionalDBranes}

We discuss more explicitly the solutions to the local/twisted tadpole cancellation condition \eqref{VanishingOfCharacterValuesOnNontrivialSubgroups} for [[fractional D-branes]] at [[orbifold]] [[singularities]] for [[isotropy group]] one of the [[nonabelian group|non-abelian]] [[finite subgroups of SU(2)]],

\begin{displaymath}
G_{DE}
  \;\subset\;
  SU(2)
\end{displaymath}
hence those in the [[ADE-classification|D- and E-series]], hence the [[binary dihedral groups]] $2 D_{2n}$ and the three exceptional cases: [[2T]], [[2O]] and [[2I]].

For these groups, by \hyperlink{BurtonSatiSchreiber18}{BSS 18, Theorem 4.1} the [[virtual representation|virtual]] [[permutation representations]] span precisely the sub charge lattice of integral (non-irrational) characters/RR-charges in the [[orientifold]] charge lattice of the corresponding [[ADE-singularity]], namely of the [[equivariant KO-theory]]=[[representation ring|real representation ring]]

\begin{displaymath}
KO^0_{G_{DE}}(\ast)
  \;=\;
  RO\left( G_{DE} \right)
  \,.
\end{displaymath}
Since the tadpole cancellation condition \eqref{VanishingOfCharacterValuesOnNontrivialSubgroups} in particular requires the characters/charges to be integral (specifically: zero) the general solution to the tadpole cancellation condition is indeed in this sub-lattice, and so that is where we may and do solve it, below.

In accord with the general Prop. \ref{RegularRepSpansSolutionToHomogeneousTadpoleCancellation} we find that in each case there is precisely a 1-dimensional (i.e. $\simeq \mathbb{Z}$) sublattice of the charge lattice (the [[representation ring]]) which solves the twisted tadpole cancellation condition \eqref{VanishingOfCharacterValuesOnNontrivialSubgroups}, hence a sublattice given by the [[integer]]-multiples $N \cdot V_0$ of one single [[fractional D-brane]] [[bound state]] $V_0 \in KO^0_G(\ast)$. There are then necessarily two of these generators $\pm V_0$. We check below that in all cases the normalized [[mass]] of these is $\pm$ unity, as it must be for the [[regular representation]], by Prop. \ref{RegularRepSpansSolutionToHomogeneousTadpoleCancellation}.

\hypertarget{at_a_orientifold_singularity}{}\paragraph*{{At a $\mathbb{Z}_2$-orientifold singularity}}\label{at_a_orientifold_singularity}

For $G = \mathbb{Z}_2$ the [[cyclic group]] of [[order of a group|order]] ${\vert \mathbb{Z}_2\vert} = 2$, the [[character of a linear representation|characters]]/[[D-brane charges]] of the elementary [[virtual representation|virtual]] [[permutation representations]]/[[fractional D-branes]] are (e.g. \href{https://people.maths.bris.ac.uk/~matyd/GroupNames/1/C2.html}{here})

\begin{tabular}{l|l|l}
$[H] =$&$\left[\langle e\rangle\right]$&$\left[\langle g_1\rangle\right]$\\
\hline 
$\chi_{V_1} =$&$\phantom{-}1$&$\phantom{-}1$\\
$\chi_{V_2} =$&$\phantom{-}1$&$-1$\\
\end{tabular}

One sees immediately that the general solution to the local/twisted tadpole cancellation condition \eqref{VanishingOfCharacterValuesOnNontrivialSubgroups} for $G = \mathbb{Z}_2$ is

\begin{displaymath}
V 
  \;=\;
    n \cdot 
  \Big( 
    1 \cdot V_1 + 1 \cdot V_2 
  \Big)
  \,,
  \phantom{AAA}
  n \in \mathbb{Z}
  \,.
\end{displaymath}
whose minimal [[positive number|positive]] [[mass]] (net brane number) is

\begin{displaymath}
\begin{aligned}
    M_{\mathbb{Z}_2}
    & =
    dim(V)  / {\vert \mathbb{Z}_2 \vert }
    \\
      & =
    \chi_V([\langle e\rangle]) / {\vert \mathbb{Z}_2 \vert}
    \\
      & =
      \big(
      1 \cdot 1
        +
      1 \cdot 1
     \big) / {2}
      & 
      \\
      & 
      =
      2 / 2
      \\
      & = 1
  \end{aligned}
\end{displaymath}
\hypertarget{at_a_orientifold_singularity_2}{}\paragraph*{{At a $\mathbb{Z}_4$-orientifold singularity}}\label{at_a_orientifold_singularity_2}

For $G = \mathbb{Z}_4$ the [[cyclic group of order 4]], the [[character of a linear representation|characters]]/[[D-brane charges]] of the complex [[irreducible representations]]/[[fractional D-branes]] are (e.g. \href{https://people.maths.bris.ac.uk/~matyd/GroupNames/1/C4.html}{here})

\begin{tabular}{l|l|l|l|l}
$[H] =$&$\left[\langle e\rangle\right]$&$\left[\langle g_1\rangle\right]$&$\left[\langle g_2\rangle\right]$&$\left[\langle g_3\rangle\right]$\\
\hline 
$\chi_{V_1} =$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$\\
$\chi_{V_2} =$&$\phantom{-}1$&$\phantom{-}1$&$-1$&$-1$\\
$\chi_{V_3} =$&$\phantom{-}1$&$-1$&$-i$&$\phantom{-}i$\\
$\chi_{V_4} =$&$\phantom{-}1$&$-1$&$\phantom{-}i$&$-i$\\
\end{tabular}

One sees immediately that the general solution to the local/twisted tadpole cancellation condition \eqref{VanishingOfCharacterValuesOnNontrivialSubgroups} for $G = \mathbb{Z}_3$ is

\begin{displaymath}
V 
  \;=\;
    n \cdot 
  \Big( 
    1 \cdot V_1 
    + 
    1 \cdot V_2 
    + 
    1 \cdot V_3 
    + 
    1 \cdot V_4 
  \Big)
  \,,
  \phantom{AAA}
  n \in \mathbb{Z}
  \,.
\end{displaymath}
whose minimal [[positive number|positive]] [[mass]] (net brane number) is

\begin{displaymath}
\begin{aligned}
    M_{\mathbb{Z}_4}
    & =
    dim(V)  / {\vert \mathbb{Z}_4 \vert }
    \\
      & =
    \chi_V([\langle e\rangle]) / {\vert \mathbb{Z}_4 \vert}
    \\
      & =
      \big(
      1 \cdot 1
        +
      1 \cdot 1
        +
      1 \cdot 1
        +
      1 \cdot 1
     \big) / {4}
      & 
      \\
      & 
      =
      4 / 4
      \\
      & = 1
  \end{aligned}
\end{displaymath}
\hypertarget{At2d4Singularity}{}\paragraph*{{At a $2 D_4$-orientifold singularity}}\label{At2d4Singularity}

For $G = 2 D_4 = Q_8$ the [[binary dihedral group]] of [[order of a group|order]] ${\vert 2 D_4\vert}$ (equivalently: the [[quaternion group]]), the [[character of a linear representation|characters]]/[[D-brane charges]] of the elementary [[virtual representation|virtual]] [[permutation representations]]/[[fractional D-branes]] are (\hyperlink{BurtonSatiSchreiber18}{BSS 18, 4.1}):

\begin{tabular}{l|l|l|l|l|l}
$[H] =$&$\left[\langle e\rangle\right]$&$\left[\langle g_1\rangle\right]$&$\left[\langle g_2\rangle\right]$&$\left[\langle g_3\rangle\right]$&$\left[\langle g_4\rangle\right]$\\
\hline 
$\chi_{V_1} =$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$\\
$\chi_{V_2} =$&$\phantom{-}1$&$\phantom{-}1$&$-1$&$\phantom{-}1$&$-1$\\
$\chi_{V_3} =$&$\phantom{-}1$&$\phantom{-}1$&$-1$&$-1$&$\phantom{-}1$\\
$\chi_{V_4} =$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$-1$&$-1$\\
$\chi_{V_5} =$&$\phantom{-}4$&$-4$&$\phantom{-}0$&$\phantom{-}0$&$\phantom{-}0$\\
\end{tabular}

One sees (\href{https://www.wolframalpha.com/input/?i=Nullspace+Transpose%5B%7B%7B1,1,1,1%7D,%7B1,%E2%88%921,1,-1%7D,%7B1,-1,-1,1%7D,%7B1,1,-1,-1%7D,%7B-4,0,0,0%7D%7D%5D}{here}) that the general solution to the local/twisted tadpole cancellation condition \eqref{VanishingOfCharacterValuesOnNontrivialSubgroups} for $G =2 D_4$ is

\begin{displaymath}
V 
  \;=\;
  n
  \Big(
    1 \cdot V_1 
      +
    1 \cdot V_2
      + 
    1 \cdot V_3
      + 
    1 \cdot V_4
  \Big)
  \,,
  \phantom{AAA}
  n \in \mathbb{Z}
\end{displaymath}
whose minimal [[positive number|positive]] [[mass]] (net brane number) is

\begin{displaymath}
\begin{aligned}
    M_{2 D_4}
    & =
    dim(V)/ {\vert 2 D_4\vert}
    \\
    & = 
    \chi_V\left( [\langle e\rangle]\right) / {\vert 2 D_4\vert}
    & =
    \big( 1 + 1 + 1 + 1 + 4 \big) / 8
    \\
    & = 8 / 8
    \\
    & = 1
  \end{aligned}
\end{displaymath}
\hypertarget{At2D6Singularity}{}\paragraph*{{At a $2 D_6$-orientifold singularity}}\label{At2D6Singularity}

For $G = 2 D_6$ the [[binary dihedral group]] of [[order of a group|order]] ${\vert 2 D_6\vert} = 12$, the [[character of a linear representation|characters]]/[[D-brane charges]] of the elementary [[virtual representation|virtual]] [[permutation representations]]/[[fractional D-branes]] are (\hyperlink{BurtonSatiSchreiber18}{BSS 18, 4.2}):

\begin{tabular}{l|l|l|l|l|l|l}
$[H] =$&$\left[\langle e\rangle\right]$&$\left[\langle g_1\rangle\right]$&$\left[\langle g_2\rangle\right]$&$\left[\langle g_3\rangle\right]$&$\left[\langle g_4\rangle\right]$&$\left[\langle g_5\rangle\right]$\\
\hline 
$\chi_{V_1} =$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$\\
$\chi_{V_2} =$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$-1$&$-1$&$\phantom{-}1$\\
$\chi_{V_3} =$&$\phantom{-}2$&$\phantom{-}2$&$-1$&$\phantom{-}0$&$\phantom{-}0$&$-1$\\
$\chi_{V_4} =$&$\phantom{-}2$&$-2$&$\phantom{-}2$&$\phantom{-}0$&$\phantom{-}0$&$-2$\\
$\chi_{V_5} =$&$\phantom{-}4$&$-4$&$-2$&$\phantom{-}0$&$\phantom{-}0$&$\phantom{-}2$\\
\end{tabular}

One finds (\href{https://www.wolframalpha.com/input/?i=Nullspace+Transpose%5B%7B%7B1,1,1,1,1%7D,%7B1,1,%E2%88%921,%E2%88%921,1%7D,%7B2,%E2%88%921,0,0,%E2%88%921%7D,%7B%E2%88%922,2,0,0,%E2%88%922%7D,%7B%E2%88%924,%E2%88%922,0,0,2%7D%7D%5D}{here}) that the general solution to the local/twisted tadpole cancellation condition \eqref{VanishingOfCharacterValuesOnNontrivialSubgroups} for $G =2 D_6$ is

\begin{displaymath}
V 
  \;=\;
  n
  \Big(
    1 \cdot V_1 
      +
    1 \cdot V_2
      + 
    2 \cdot V_3
      + 
    1 \cdot V_4
      + 
    1 \cdot V_5
  \Big)
  \,,
  \phantom{AAA}
  n \in \mathbb{Z}
\end{displaymath}
whose minimal [[positive number|positive]] [[mass]] (net brane number) is

\begin{displaymath}
\begin{aligned}
    M_{2 D_6}
    & =
    dim(V)  / {\vert 2 D_6 \vert }
    \\
      & =
    \chi_V([\langle e\rangle]) / {\vert 2 D_6\vert}
    \\
      & =
      \big(
      1 \cdot 1
        +
      1 \cdot 1
        + 
      2 \cdot 2
        + 
      1 \cdot 2
        + 
      1 \cdot 4
     \big) / {12}
      & 
      \\
      & 
      =
      12 / 12
      \\
      & = 1
  \end{aligned}
\end{displaymath}
\hypertarget{At2D8Singularity}{}\paragraph*{{At a $2 D_8$-orientifold singularity}}\label{At2D8Singularity}

For $G = 2 D_8$ the [[binary dihedral group]] of [[order of a group|order]] ${\vert 2 D_8\vert} = 16$, the [[character of a linear representation|characters]]/[[D-brane charges]] of the elementary [[virtual representation|virtual]] [[permutation representations]]/[[fractional D-branes]] are (\hyperlink{BurtonSatiSchreiber18}{BSS 18, 4.3}):

\begin{tabular}{l|l|l|l|l|l|l|l}
$[H] =$&$\left[\langle e\rangle\right]$&$\left[\langle g_1\rangle\right]$&$\left[\langle g_2\rangle\right]$&$\left[\langle g_3\rangle\right]$&$\left[\langle g_4\rangle\right]$&$\left[\langle g_5\rangle\right]$&$\left[\langle g_6\rangle\right]$\\
\hline 
$\chi_{V_1} =$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$\\
$\chi_{V_2} =$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$-1$&$\phantom{-}1$&$-1$&$-1$\\
$\chi_{V_3} =$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$-1$&$-1$&$\phantom{-}1$&$\phantom{-}1$\\
$\chi_{V_4} =$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$-1$&$-1$&$-1$\\
$\chi_{V_5} =$&$\phantom{-}2$&$\phantom{-}2$&$-2$&$\phantom{-}0$&$\phantom{-}0$&$\phantom{-}0$&$\phantom{-}0$\\
$\chi_{V_6} =$&$\phantom{-}8$&$-8$&$\phantom{-}0$&$\phantom{-}0$&$\phantom{-}0$&$\phantom{-}0$&$\phantom{-}0$\\
\end{tabular}

One finds (\href{https://www.wolframalpha.com/input/?i=Nullspace+Transpose%5B%7B%7B1,1,1,1,1,1%7D,%7B1,1,-1,1,-1,-1%7D,%7B1,1,-1,-1,1,1%7D,%7B1,1,1,-1,-1,-1%7D,%7B2,-2,0,0,0,0%7D,%7B-8,0,0,0,0,0%7D%7D%5D}{here}), that the general solution to the local/twisted tadpole cancellation condition \eqref{VanishingOfCharacterValuesOnNontrivialSubgroups} for $G =2 D_8$ is

\begin{displaymath}
V 
  \;=\;
  n
  \Big(
    1 \cdot V_1 
      +
    1 \cdot V_2
      + 
    1 \cdot V_3
      + 
    1 \cdot V_4
      + 
    2 \cdot V_5
      + 
    1 \cdot V_6
  \Big)
  \,,
  \phantom{AAA}
  n \in \mathbb{Z}
\end{displaymath}
whose minimal [[positive number|positive]] [[mass]] (net brane number) is

\begin{displaymath}
\begin{aligned}
    M_{2 D_8}
     & =
    dim(V)  / {\vert 2 D_8\vert}
    \\
      & =
    \chi_V([\langle e\rangle]) / { \vert 2 D_8\vert }
    \\
      & =
      \big(
      1 \cdot 1 
        +
      1 \cdot 1
        + 
      1 \cdot 1
        + 
      1 \cdot 1
        + 
      2 \cdot 2
        + 
      1 \cdot 8
      \big) / 16
      & 
      \\
      & 
      =
      16 / 16
      \\
      & = 
      1
  \end{aligned}
\end{displaymath}
\hypertarget{At2D10Singularity}{}\paragraph*{{At a $2 D_{10}$-orientifold singularity}}\label{At2D10Singularity}

For $G = 2 D_{10}$ the [[binary dihedral group]] of [[order of a group|order]] ${\vert 2 D_{10}\vert} = 20$, the [[character of a linear representation|characters]]/[[D-brane charges]] of the elementary [[virtual representation|virtual]] [[permutation representations]]/[[fractional D-branes]] are (\hyperlink{BurtonSatiSchreiber18}{BSS 18, 4.4}):

\begin{tabular}{l|l|l|l|l|l|l|l|l}
$[H] =$&$\left[\langle e\rangle\right]$&$\left[\langle g_1\rangle\right]$&$\left[\langle g_2\rangle\right]$&$\left[\langle g_3\rangle\right]$&$\left[\langle g_4\rangle\right]$&$\left[\langle g_5\rangle\right]$&$\left[\langle g_6\rangle\right]$&$\left[\langle g_7\rangle\right]$\\
\hline 
$\chi_{V_1} =$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$\\
$\chi_{V_2} =$&$\phantom{-}1$&$\phantom{-}1$&$-1$&$-1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$\\
$\chi_{V_3} =$&$\phantom{-}2$&$-2$&$\phantom{-}0$&$\phantom{-}0$&$\phantom{-}2$&$\phantom{-}2$&$-2$&$-2$\\
$\chi_{V_4} =$&$\phantom{-}4$&$\phantom{-}4$&$\phantom{-}0$&$\phantom{-}0$&$-1$&$-1$&$-1$&$-1$\\
$\chi_{V_5} =$&$\phantom{-}8$&$-8$&$\phantom{-}0$&$\phantom{-}0$&$-2$&$-2$&$\phantom{-}2$&$\phantom{-}2$\\
\end{tabular}

One finds (\href{https://www.wolframalpha.com/input/?i=Nullspace+Transpose%5B%7B%7B1,1,1,1,1,1,1%7D,%7B1,-1,-1,1,1,1,1%7D,%7B-2,0,0,2,2,-2,-2%7D,%7B4,0,0,-1,-1,-1,-1%7D,%7B-8,0,0,-2,-2,2,2%7D%7D%5D}{here}) that the general solution to the local/twisted tadpole cancellation condition \eqref{VanishingOfCharacterValuesOnNontrivialSubgroups} for $G =2 D_{10}$ is

\begin{displaymath}
V 
  \;=\;
  n
  \Big(
    1 \cdot V_1 
      +
    1 \cdot V_2
      + 
    1 \cdot V_3
      + 
    2 \cdot V_4
      + 
    1 \cdot V_5
  \Big)
  \,,
  \phantom{AAA}
  n \in \mathbb{Z}
\end{displaymath}
whose minimal [[positive number|positive]] [[mass]] (net brane number) is

\begin{displaymath}
\begin{aligned}
    M_{2 D_{10}}
      & =
    dim(V) / {\vert 2 D_{10}\vert}
    \\
      & =
    \chi_V([\langle e\rangle]) /  {\vert 2 D_{10}\vert}
    \\
      & =
      \big(
      1 \cdot 1
        +
      1 \cdot 1
        + 
      1 \cdot 2
        + 
      2 \cdot 4
        + 
      1 \cdot 8
      \big) / 20
      & 
      \\
      & 
      =
      20 / 20 
      \\
      & = 1
  \end{aligned}
\end{displaymath}
\hypertarget{At2D12Singularity}{}\paragraph*{{At a $2 D_{12}$-orientifold singularity}}\label{At2D12Singularity}

For $G = 2 D_{12}$ the [[binary dihedral group]] of [[order of a group|order]] ${\vert 2 D_{12}\vert} = 24$, the [[character of a linear representation|characters]]/[[D-brane charges]] of the elementary [[virtual representation|virtual]] [[permutation representations]]/[[fractional D-branes]] are (\hyperlink{BurtonSatiSchreiber18}{BSS 18, 4.5}):

\begin{tabular}{l|l|l|l|l|l|l|l|l|l}
$[H] =$&$\left[\langle e\rangle\right]$&$\left[\langle g_1\rangle\right]$&$\left[\langle g_2\rangle\right]$&$\left[\langle g_3\rangle\right]$&$\left[\langle g_4\rangle\right]$&$\left[\langle g_5\rangle\right]$&$\left[\langle g_6\rangle\right]$&$\left[\langle g_7\rangle\right]$&$\left[\langle g_8\rangle\right]$\\
\hline 
$\chi_{V_1} =$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$\\
$\chi_{V_2} =$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$-1$&$-1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$\\
$\chi_{V_3} =$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$-1$&$\phantom{-}1$&$-1$&$\phantom{-}1$&$-1$&$-1$\\
$\chi_{V_4} =$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$-1$&$-1$&$\phantom{-}1$&$-1$&$-1$\\
$\chi_{V_5} =$&$\phantom{-}2$&$\phantom{-}2$&$-1$&$\phantom{-}0$&$\phantom{-}0$&$\phantom{-}2$&$-1$&$-1$&$-1$\\
$\chi_{V_6} =$&$\phantom{-}2$&$\phantom{-}2$&$-1$&$\phantom{-}0$&$\phantom{-}0$&$-2$&$-1$&$\phantom{-}1$&$\phantom{-}1$\\
$\chi_{V_7} =$&$\phantom{-}4$&$-4$&$\phantom{-}4$&$\phantom{-}0$&$\phantom{-}0$&$\phantom{-}0$&$-4$&$\phantom{-}0$&$\phantom{-}0$\\
$\chi_{V_8} =$&$\phantom{-}8$&$-8$&$-4$&$\phantom{-}0$&$\phantom{-}0$&$\phantom{-}0$&$\phantom{-}4$&$\phantom{-}0$&$\phantom{-}0$\\
\end{tabular}

One sees (\href{https://www.wolframalpha.com/input/?i=Nullspace+Transpose%5B%7B%7B1,1,1,1,1,1,1,1%7D,%7B1,1,-1,-1,1,1,1,1%7D,%7B1,1,-1,1,-1,1,-1,-1%7D,%7B1,1,1,-1,-1,1,-1,-1%7D,%7B2,-1,0,0,2,-1,-1,-1%7D,%7B2,-1,0,0,-2,-1,1,1%7D,%7B-4,4,0,0,0,-4,0,0%7D,%7B-8,-4,0,0,0,4,0,0%7D%7D%5D}{here}) that the general solution to the local/twisted tadpole cancellation condition \eqref{VanishingOfCharacterValuesOnNontrivialSubgroups} for $G =2 D_{12}$ is

\begin{displaymath}
V 
  \;=\;
  n
  \Big(
    1 \cdot V_1 
      +
    1 \cdot V_2
      + 
    1 \cdot V_3
      + 
    1 \cdot V_4
      + 
    2 \cdot V_5
      + 
    2 \cdot V_6
      + 
    1 \cdot V_7
      + 
    1 \cdot V_8
  \Big)
  \,,
  \phantom{AAA}
  n \in \mathbb{Z}
\end{displaymath}
whose minimal [[positive number|positive]] [[mass]] (net brane number) is

\begin{displaymath}
\begin{aligned}
    M_{2 D_{12}}
      & =
    dim(V)  / {\vert 2 D_{12}\vert }
    \\
      & =
    \chi_V([\langle e\rangle]) / {\vert 2 D_{12}\vert }
    \\
      & =
      \big( 
      1 \cdot 1 
        +
      1 \cdot 1
        + 
      1 \cdot 1
        + 
      1 \cdot 1
        + 
      2 \cdot 2
        + 
      2 \cdot 2
        + 
      1 \cdot 4
        + 
      1 \cdot 8
      \big) / 24
      & 
      \\
      & 
      =
      24 / 24
      \\
      & = 1
  \end{aligned}
\end{displaymath}
\hypertarget{At2D14Singularity}{}\paragraph*{{At a $2 D_{14}$-orientifold singularity}}\label{At2D14Singularity}

For $G = 2 D_{14}$ the [[binary dihedral group]] of [[order of a group|order]] ${\vert 2 D_{14}\vert} = 28$, the [[character of a linear representation|characters]]/[[D-brane charges]] of the elementary [[virtual representation|virtual]] [[permutation representations]]/[[fractional D-branes]] are (\hyperlink{BurtonSatiSchreiber18}{BSS 18, 4.6}):

\begin{tabular}{l|l|l|l|l|l|l|l|l|l|l}
$[H] =$&$\left[\langle e\rangle\right]$&$\left[\langle g_1\rangle\right]$&$\left[\langle g_2\rangle\right]$&$\left[\langle g_3\rangle\right]$&$\left[\langle g_4\rangle\right]$&$\left[\langle g_5\rangle\right]$&$\left[\langle g_6\rangle\right]$&$\left[\langle g_7\rangle\right]$&$\left[\langle g_8\rangle\right]$&$\left[\langle g_9\rangle\right]$\\
\hline 
$\chi_{V_1} =$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$\\
$\chi_{V_2} =$&$\phantom{-}1$&$\phantom{-}1$&$-1$&$-1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$\\
$\chi_{V_3} =$&$\phantom{-}2$&$-2$&$\phantom{-}0$&$\phantom{-}0$&$\phantom{-}2$&$\phantom{-}2$&$\phantom{-}2$&$-2$&$-2$&$-2$\\
$\chi_{V_4} =$&$\phantom{-}6$&$\phantom{-}6$&$\phantom{-}0$&$\phantom{-}0$&$-1$&$-1$&$-1$&$-1$&$-1$&$-1$\\
$\chi_{V_5} =$&$\phantom{-1}\mathllap{12}$&$\phantom{-1}\mathllap{-12}$&$\phantom{-}0$&$\phantom{-}0$&$-2$&$-2$&$-2$&$\phantom{-}2$&$\phantom{-}2$&$\phantom{-}2$\\
\end{tabular}

One sees by immediate inspection, that the general solution to the local/twisted tadpole cancellation condition \eqref{VanishingOfCharacterValuesOnNontrivialSubgroups} for $G =2 D_{14}$ is

\begin{displaymath}
V 
  \;=\;
  n
  \Big(
    1 \cdot V_1 
      +
    1 \cdot V_2
      + 
    1 \cdot V_3
      + 
    2 \cdot V_4
      + 
    1 \cdot V_5
  \Big)
  \,,
  \phantom{AAA}
  n \in \mathbb{Z}
\end{displaymath}
whose minimal [[positive number|positive]] [[mass]] (net brane number) is

\begin{displaymath}
\begin{aligned}
    M_{2 D_{14}}
      & =
    dim(V)  / {\vert 2 D_{14}\vert }
    \\
      & =
    \chi_V([\langle e\rangle]) / {\vert 2 D_{14}\vert }
    \\
      & =
      \big(
      1 \cdot 1
        +
      1 \cdot 1
        + 
      1 \cdot 2
        + 
      2 \cdot 6
        + 
      1 \cdot 12
      \big) / 28
      \\
      & 
      =
      28 /28
      \\
      & = 1
  \end{aligned}
\end{displaymath}
\hypertarget{At2D16Singularity}{}\paragraph*{{At a $2 D_{16}$-orientifold singularity}}\label{At2D16Singularity}

For $G = 2 D_{16}$ the [[binary dihedral group]] of [[order of a group|order]] ${\vert 2 D_{16}\vert} = 32$, the [[character of a linear representation|characters]]/[[D-brane charges]] of the elementary [[virtual representation|virtual]] [[permutation representations]]/[[fractional D-branes]] are (\hyperlink{BurtonSatiSchreiber18}{BSS 18, 4.7}):

\begin{tabular}{l|l|l|l|l|l|l|l|l|l|l|l}
$[H] =$&$\left[\langle e\rangle\right]$&$\left[\langle g_1\rangle\right]$&$\left[\langle g_2\rangle\right]$&$\left[\langle g_3\rangle\right]$&$\left[\langle g_4\rangle\right]$&$\left[\langle g_5\rangle\right]$&$\left[\langle g_6\rangle\right]$&$\left[\langle g_7\rangle\right]$&$\left[\langle g_8\rangle\right]$&$\left[\langle g_9\rangle\right]$&$\left[\langle g_9\rangle\right]$\\
\hline 
$\chi_{V_1} =$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$\\
$\chi_{V_2} =$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$-1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$-1$&$-1$&$-1$&$-1$\\
$\chi_{V_3} =$&$\phantom{-}1$&$\phantom{-}1$&$-1$&$-1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$\\
$\chi_{V_4} =$&$\phantom{-}1$&$\phantom{-}1$&$-1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$-1$&$-1$&$-1$&$-1$\\
$\chi_{V_5} =$&$\phantom{-}2$&$\phantom{-}2$&$\phantom{-}0$&$\phantom{-}0$&$\phantom{-}2$&$-2$&$-2$&$\phantom{-}0$&$\phantom{-}0$&$\phantom{-}0$&$\phantom{-}0$\\
$\chi_{V_6} =$&$\phantom{-}4$&$\phantom{-}4$&$\phantom{-}0$&$\phantom{-}0$&$-4$&$\phantom{-}0$&$\phantom{-}0$&$\phantom{-}0$&$\phantom{-}0$&$\phantom{-}0$&$\phantom{-}0$\\
$\chi_{V_7} =$&$\phantom{-1}\mathllap{16}$&$\phantom{-1}\mathllap{-16}$&$\phantom{-}0$&$\phantom{-}0$&$\phantom{-}0$&$\phantom{-}0$&$\phantom{-}0$&$\phantom{-}0$&$\phantom{-}0$&$\phantom{-}0$&$\phantom{-}0$\\
\end{tabular}

One sees by immediate inspection, that the general solution to the local/twisted tadpole cancellation condition \eqref{VanishingOfCharacterValuesOnNontrivialSubgroups} for $G =2 D_{16}$ is

\begin{displaymath}
V 
  \;=\;
  n
  \Big(
    1 \cdot V_1 
      +
    1 \cdot V_2
      + 
    1 \cdot V_3
      + 
    1 \cdot V_4
      + 
    2 \cdot V_5
      + 
    2 \cdot V_6
      + 
    1 \cdot V_7
  \Big)
  \,,
  \phantom{AAA}
  n \in \mathbb{Z}
\end{displaymath}
whose minimal [[positive number|positive]] [[mass]] (net brane number) is

\begin{displaymath}
\begin{aligned}
    M_{2 D_{16}}
      & =
    dim(V) / {\vert 2 D_{16}\vert}
    \\
      & =
    \chi_V([\langle e\rangle]) / {\vert 2 D_{16}\vert}
    \\
      & =
      \big(
      1 \cdot 1   
        +
      1 \cdot 1
        + 
      1 \cdot 1
        + 
      1 \cdot 1
        + 
      2 \cdot 2
        + 
      2 \cdot 4
        + 
      1 \cdot 16
      \big) /32
      \\
      & 
      =
      32 / 32
      \\
      & = 1
  \end{aligned}
\end{displaymath}
\hypertarget{At2TSingularity}{}\paragraph*{{At a $2 T$-orientifold singularity}}\label{At2TSingularity}

For $G = 2 T$ the [[binary tetrahedral group]] (whose [[order of a group|order]] is ${\vert 2T \vert} =24$), the [[character of a linear representation|characters]]/[[D-brane charges]] of the elementary [[virtual representation|virtual]] [[permutation representations]]/[[fractional D-branes]] are (\hyperlink{BurtonSatiSchreiber18}{BSS 18, 4.8}):

\begin{tabular}{l|l|l|l|l|l|l|l}
$[H] =$&$\left[\langle e\rangle\right]$&$\left[\langle g_1\rangle\right]$&$\left[\langle g_2\rangle\right]$&$\left[\langle g_3\rangle\right]$&$\left[\langle g_4\rangle\right]$&$\left[\langle g_5\rangle\right]$&$\left[\langle g_6\rangle\right]$\\
\hline 
$\chi_{V_1} =$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$\\
$\chi_{V_2} =$&$\phantom{-}2$&$\phantom{-}2$&$-1$&$-1$&$\phantom{-}2$&$-1$&$-1$\\
$\chi_{V_3} =$&$\phantom{-}3$&$\phantom{-}3$&$\phantom{-}0$&$\phantom{-}0$&$-1$&$\phantom{-}0$&$\phantom{-}0$\\
$\chi_{V_4} =$&$\phantom{-}4$&$-4$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}0$&$-1$&$-1$\\
$\chi_{V_5} =$&$\phantom{-}4$&$-4$&$-2$&$-2$&$\phantom{-}0$&$\phantom{-}2$&$\phantom{-}2$\\
\end{tabular}

One finds (\href{https://www.wolframalpha.com/input/?i=Nullspace+Transpose%5B%7B%7B1,1,1,1,1,1%7D,%7B2,-1,-1,2,-1,-1%7D,%7B3,0,0,-1,0,0%7D,%7B-4,1,1,0,-1,-1%7D,%7B-4,-2,-2,0,2,2%7D%7D%5D}{here}) that the general solution to the local/twisted tadpole cancellation condition \eqref{VanishingOfCharacterValuesOnNontrivialSubgroups} for $G = 2T$ is

\begin{displaymath}
V 
  \;=\;
  n
  \Big(
    1 \cdot V_1
    + 
    1 \cdot V_2
    + 
    3 \cdot V_3
    + 
    2 \cdot V_4
    + 
    1 \cdot V_5 
  \Big)
  \,,
  \phantom{AAA}
  n \in \mathbb{Z}
\end{displaymath}
whose minimal [[positive number|positive]] [[mass]] (net brane number) is

\begin{displaymath}
\begin{aligned}
    M_{2I}
      & =
    dim(V) / {\vert 2 I \vert }
    \\
      & =
    \chi_V([\langle e\rangle]) / {\vert 2 I \vert }
    \\
      & =
      \big(
        1 \cdot 1
        + 
        1 \cdot 2
        + 
        3 \cdot 3
        + 
        2 \cdot 4
        + 
        1 \cdot 4
      \big) / 24
      & 
      \\
      & 
      =
      24 / 24
      \\
      & = 1
  \end{aligned}
\end{displaymath}
\hypertarget{At2OSingularity}{}\paragraph*{{At a $2 O$-orientifold singularity}}\label{At2OSingularity}

For $G = 2 O$ the [[binary octahedral group]] (whose [[order of a group|order]] is ${\vert 2O \vert} = 48$), the [[character of a linear representation|characters]]/[[D-brane charges]] of the elementary [[virtual representation|virtual]] [[permutation representations]]/[[fractional D-branes]] are (\hyperlink{BurtonSatiSchreiber18}{BSS 18, 4.9}):

\begin{tabular}{l|l|l|l|l|l|l|l|l}
$[H] =$&$\left[\langle e\rangle\right]$&$\left[\langle g_1\rangle\right]$&$\left[\langle g_2\rangle\right]$&$\left[\langle g_3\rangle\right]$&$\left[\langle g_4\rangle\right]$&$\left[\langle g_5\rangle\right]$&$\left[\langle g_6\rangle\right]$&$\left[\langle g_7\rangle\right]$\\
\hline 
$\chi_{V_1} =$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$\\
$\chi_{V_2} =$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$-1$&$\phantom{-}1$&$-1$&$-1$\\
$\chi_{V_3} =$&$\phantom{-}2$&$\phantom{-}2$&$-1$&$\phantom{-}2$&$\phantom{-}0$&$-1$&$\phantom{-}0$&$\phantom{-}0$\\
$\chi_{V_4} =$&$\phantom{-}3$&$\phantom{-}3$&$\phantom{-}0$&$-1$&$\phantom{-}1$&$\phantom{-}0$&$-1$&$-1$\\
$\chi_{V_5} =$&$\phantom{-}3$&$\phantom{-}3$&$\phantom{-}0$&$-1$&$-1$&$\phantom{-}0$&$\phantom{-}1$&$\phantom{-}1$\\
$\chi_{V_6} =$&$\phantom{-}8$&$-8$&$\phantom{-}2$&$\phantom{-}0$&$\phantom{-}0$&$-2$&$\phantom{-}0$&$\phantom{-}0$\\
$\chi_{V_7} =$&$\phantom{-}8$&$-8$&$-4$&$\phantom{-}0$&$\phantom{-}0$&$\phantom{-}4$&$\phantom{-}0$&$\phantom{-}0$\\
\end{tabular}

One finds (\href{https://www.wolframalpha.com/input/?i=Nullspace+Transpose%5B%7B%7B1,1,1,1,1,1,1%7D,%7B1,1,1,-1,1,-1,-1%7D,%7B2,-1,2,0,-1,0,0%7D,%7B3,0,-1,1,0,-1,-1%7D,%7B3,0,-1,-1,0,1,1%7D,%7B-8,2,0,0,-2,0,0%7D,%7B-8,-4,0,0,4,0,0%7D%7D%5D}{here}) that the general solution to the local/twisted tadpole cancellation condition \eqref{VanishingOfCharacterValuesOnNontrivialSubgroups} for $G = 2O$ is

\begin{displaymath}
V 
  \;=\;
  n
  \Big(
    1 \cdot V_1
    + 
    1 \cdot V_2
    + 
    2 \cdot V_3
    + 
    3 \cdot V_4
    + 
    3 \cdot V_5 
    + 
    2 \cdot V_6 
    + 
    1 \cdot V_7 
  \Big)
  \,,
  \phantom{AAA}
  n \in \mathbb{Z}
\end{displaymath}
whose minimal [[positive number|positive]] [[mass]] (net brane number) is

\begin{displaymath}
\begin{aligned}
    M_{2O}
      & =
    dim(V) / {\vert 2 O \vert }
    \\
      & =
    \chi_V([\langle e\rangle]) / {\vert 2 O \vert }
    \\
      & =
      \big( 
      1 \cdot 1
      + 
      1 \cdot 1
      + 
      2 \cdot 2
      + 
      3 \cdot 3
      + 
      3 \cdot 3
      + 
      2 \cdot 8
      + 
      1 \cdot 8
      \big) / 48
      & 
      \\
      & 
      =
      48 / 48
      \\
      & = 1
  \end{aligned}
\end{displaymath}
\hypertarget{At2ISingularity}{}\paragraph*{{At a $2 I$-orientifold singularity}}\label{At2ISingularity}

For $G = 2 I$ the [[binary icosahedral group]] (whose [[order of a group|order]] is ${\vert 2I \vert} = 120$), the [[character of a linear representation|characters]]/[[D-brane charges]] of the elementary [[virtual representation|virtual]] [[permutation representations]]/[[fractional D-branes]] are (\hyperlink{BurtonSatiSchreiber18}{BSS 18, 4.10}):

\begin{tabular}{l|l|l|l|l|l|l|l|l|l}
$[H] =$&$\left[\langle e\rangle\right]$&$\left[\langle g_1\rangle\right]$&$\left[\langle g_2\rangle\right]$&$\left[\langle g_3\rangle\right]$&$\left[\langle g_4\rangle\right]$&$\left[\langle g_5\rangle\right]$&$\left[\langle g_6\rangle\right]$&$\left[\langle g_7\rangle\right]$&$\left[\langle g_8\rangle\right]$\\
\hline 
$\chi_{V_1} =$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}1$\\
$\chi_{V_2} =$&$\phantom{-}4$&$\phantom{-}4$&$\phantom{-}1$&$\phantom{-}0$&$-1$&$-1$&$\phantom{-}1$&$-1$&$-1$\\
$\chi_{V_3} =$&$\phantom{-}5$&$\phantom{-}5$&$-1$&$\phantom{-}1$&$\phantom{-}0$&$\phantom{-}0$&$-1$&$\phantom{-}0$&$\phantom{-}0$\\
$\chi_{V_4} =$&$\phantom{-}6$&$\phantom{-}6$&$\phantom{-}0$&$-2$&$\phantom{-}1$&$\phantom{-}1$&$\phantom{-}0$&$\phantom{-}1$&$\phantom{-}1$\\
$\chi_{V_5} =$&$\phantom{-1}\mathllap{12}$&$\phantom{-1}\mathllap{-12}$&$\phantom{-}0$&$\phantom{-}0$&$\phantom{-}2$&$\phantom{-}2$&$\phantom{-}0$&$-2$&$-2$\\
$\chi_{V_6} =$&$\phantom{-}8$&$-8$&$\phantom{-}2$&$\phantom{-}0$&$-2$&$-2$&$-2$&$\phantom{-}2$&$\phantom{-}2$\\
$\chi_{V_7} =$&$\phantom{-}8$&$-8$&$-4$&$\phantom{-}0$&$-2$&$-2$&$\phantom{-}4$&$\phantom{-}2$&$\phantom{-}2$\\
\end{tabular}

One finds (\href{https://www.wolframalpha.com/input/?i=Nullspace+Transpose%5B%7B%7B1,1,1,1,1,1,1,1%7D,%7B4,1,0,-1,-1,1,-1,-1%7D,%7B5,-1,1,0,0,-1,0,0%7D,%7B6,0,-2,1,1,0,1,1%7D,%7B-12,0,0,2,2,0,-2,-2%7D,%7B-8,2,0,-2,-2,-2,2,2%7D,%7B-8,-4,0,-2,-2,4,2,2%7D%7D%5D}{here}) that the general solution to the local/twisted tadpole cancellation condition \eqref{VanishingOfCharacterValuesOnNontrivialSubgroups} for $G = 2I$ is

\begin{displaymath}
V 
  \;=\;
  n
  \Big(
    1 \cdot V_1
    + 
    4 \cdot V_2
    + 
    5 \cdot V_3
    + 
    3 \cdot V_4
    + 
    3 \cdot V_5 
    + 
    2 \cdot V_6 
    + 
    1 \cdot V_7 
  \Big)
  \,,
  \phantom{AAA}
  n \in \mathbb{Z}
\end{displaymath}
whose minimal [[positive number|positive]] [[mass]] (net brane number) is

\begin{displaymath}
\begin{aligned}
    M_{2I}
      & =
    dim(V) / {\vert 2I \vert }
    \\
      & =
    \chi_V([\langle e\rangle]) / {\vert 2I \vert }
    \\
      & =
      \big(
      1 \cdot 1
      + 
      4 \cdot 4
      + 
      5 \cdot 5
      + 
      3 \cdot 6
      + 
      3 \cdot 12
      + 
      2 \cdot 8
      + 
      1 \cdot 8
      \big) / 120
      \\
      & 
      =
      120 / 120
      \\
      & = 1
  \end{aligned}
\end{displaymath}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[C-field tadpole cancellation]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

The issue was first highlighted in

\begin{itemize}%
\item [[Augusto Sagnotti]], \emph{Open strings and their symmetry groups} in G. Mack et. al. (eds.) Cargese ’87, ``Non-perturbative Quantum Field Theory,'' (Pergamon Press, 1988) p. 521 (\href{https://arxiv.org/abs/hep-th/0208020}{arXiv:hep-th/0208020})

\end{itemize}
The argument is recalled in

\begin{itemize}%
\item [[Eric Gimon]], [[Joseph Polchinski]], section 3 of \emph{Consistency Conditions for Orientifolds and D-Manifolds}, Phys.Rev.D54:1667-1676, 1996 (\href{https://arxiv.org/abs/hep-th/9601038}{arXiv:hep-th/9601038})

\end{itemize}
Details are in

\begin{itemize}%
\item [[Edward Witten]], section 9.3 of \emph{Superstring Perturbation Theory Revisited} (\href{https://arxiv.org/abs/1209.5461}{arXiv:1209.5461})

\end{itemize}
Textbook accounts include

\begin{itemize}%
\item [[Luis Ibáñez]], [[Angel Uranga]], section 4.4 of \emph{[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]}, Cambridge 2012


\item [[Ralph Blumenhagen]], [[Dieter Lüst]], [[Stefan Theisen]], section 9.4 of \emph{Basic Concepts of String Theory} Part of the series Theoretical and Mathematical Physics, Springer 2013



\end{itemize}
Quick illustrations include:

\begin{itemize}%
\item Marcus Berg, p. 26-36 in \emph{Introduction to Orientifolds} (\href{https://tp.hotell.kau.se/marcus/physics/talks/orienti_short.pdf}{pdf}, [[BergOrientifolds.pdf:file]])


\item \href{http://www2.kek.jp/theory-center/project/string2higgsflavor/wp-content/uploads/2016/08/part-2.pdf}{slide 18 here}



\end{itemize}
Critical outlook in

\begin{itemize}%
\item [[Gregory Moore]] p. 21-22 of \emph{[[Physical Mathematics and the Future]]}, talk at \href{http://physics.princeton.edu/strings2014/}{Strings 2014} (\href{http://physics.princeton.edu/strings2014/slides/Moore.pdf}{talk slides}, \href{http://www.physics.rutgers.edu/~gmoore/PhysicalMathematicsAndFuture.pdf}{companion text pdf}, [[MooreVisionTalk2014.pdf:file]])

\end{itemize}
The above discussion follows

\begin{itemize}%
\item [[nLab:Hisham Sati]], [[nLab:Urs Schreiber]], \emph{[[schreiber:Equivariant Cohomotopy implies orientifold tadpole cancellation]]} (\href{https://arxiv.org/abs/1909.12277}{arXiv:1909.12277})

\end{itemize}
See also

\begin{itemize}%
\item G. Aldazabal, D. Badagnani, [[Luis Ibáñez]], [[Angel Uranga]], \emph{Tadpole versus anomaly cancellation in $D=4,6$ compact IIB orientifolds}, JHEP 9906:031, 1999 (\href{https://arxiv.org/abs/hep-th/9904071}{arXiv:hep-th/9904071})


\item [[Angel Uranga]], \emph{D-brane probes, RR tadpole cancellation and K-theory charge}, Nucl.Phys.B598:225-246, 2001 (\href{https://arxiv.org/abs/hep-th/0011048}{arXiv:hep-th/0011048})


\item Maria E. Angulo, David Bailin, Huan-Xiong Yang, \emph{Tadpole and Anomaly Cancellation Conditions in D-brane Orbifold Models}, Int.J.Mod.Phys.A18:3637-3694, 2003 (\href{https://arxiv.org/abs/hep-th/0210150}{arXiv:hep-th/0210150})


\item [[Fernando Marchesano]], section 4 of \emph{Intersecting D-brane Models} (\href{https://arxiv.org/abs/hep-th/0307252}{arXiv:hep-th/0307252})


\item [[Fernando Marchesano]], [[Gary Shiu]], \emph{Building MSSM Flux Vacua}, JHEP0411:041, 2004 (\href{https://arxiv.org/abs/hep-th/0409132}{arXiv:hep-th/0409132})


\item C.-M. Chen, G. V. Kraniotis, V. E. Mayes, D. V. Nanopoulos, J. W. Walker, \emph{A K-theory Anomaly Free Supersymmetric Flipped SU(5) Model from Intersecting Branes}, Phys.Lett. B625 (2005) 96-105 (\href{https://arxiv.org/abs/hep-th/0507232}{arXiv:hep-th/0507232})


\item Inaki Garcia-Etxebarria, [[Angel Uranga]], \emph{From F/M-theory to K-theory and back}, JHEP 0602:008, 2006 (\href{https://arxiv.org/abs/hep-th/0510073}{arXiv:hep-th/0510073})


\item John Maiden, [[Gary Shiu]], [[Bogdan Stefanski]], \emph{D-brane Spectrum and K-theory Constraints of D=4, N=1 Orientifolds}, JHEP0604:052,2006 (\href{https://arxiv.org/abs/hep-th/0602038}{arXiv:hep-th/0602038})


\item Tetsuji Kimura, Mitsuhisa Ohta, Kei-Jiro Takahashi, \emph{Type IIA orientifolds and orbifolds on non-factorizable tori}, Nucl.Phys.B798:89-123, 2008 (\href{https://arxiv.org/abs/0712.2281}{arXiv:0712.2281})


\item [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], \emph{Orientifold Pr\'e{}cis} in: [[Hisham Sati]], [[Urs Schreiber]] (eds.) \emph{[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]}, Proceedings of Symposia in Pure Mathematics, AMS (2011) (\href{http://arxiv.org/abs/0906.0795}{arXiv:0906.0795}, \href{http://www.ma.utexas.edu/users/dafr/bilbao.pdf}{slides})



\end{itemize}
In view of consistency of [[flux compactifications]]:

\begin{itemize}%
\item Philip Betzler, Erik Plauschinn, \emph{Type IIB flux vacua and tadpole cancellation} (\href{https://arxiv.org/abs/1905.08823}{arXiv:1905.08823})

\end{itemize}
For the [[topological string]]:

\begin{itemize}%
\item Johannes Walcher, \emph{Evidence for Tadpole Cancellation in the Topological String} (\href{https://arxiv.org/abs/0712.2775}{arXiv:0712.2775})

\end{itemize}
The character tables for [[virtual representation|virtual]] [[permutation representations]] above are taken from

\begin{itemize}%
\item [[Simon Burton]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:The image of the Burnside ring in the Representation ring|The image of the Burnside ring in the Representation ring -- for binary Platonic groups]]} (\href{https://arxiv.org/abs/1812.09679}{arXiv:1812.09679}, \href{https://arxiv.org/src/1812.09679v1/anc}{Python code})

\end{itemize}
\hypertarget{examples_and_models}{}\subsubsection*{{Examples and Models}}\label{examples_and_models}

Specifically [[K3]] [[orientifolds]] ($\mathbb{T}^4/G_{ADE}$) in [[type IIB string theory]], hence for [[D9-branes]] and [[D5-branes]]:

\begin{itemize}%
\item [[Eric Gimon]], [[Joseph Polchinski]], Section 3.2 of: \emph{Consistency Conditions for Orientifolds and D-Manifolds}, Phys. Rev. D54: 1667-1676, 1996 (\href{https://arxiv.org/abs/hep-th/9601038}{arXiv:hep-th/9601038})


\item [[Eric Gimon]], [[Clifford Johnson]], \emph{K3 Orientifolds}, Nucl. Phys. B477: 715-745, 1996 (\href{https://arxiv.org/abs/hep-th/9604129}{arXiv:hep-th/9604129})


\item Alex Buchel, [[Gary Shiu]], S.-H. Henry Tye, \emph{Anomaly Cancelations in Orientifolds with Quantized B Flux}, Nucl.Phys. B569 (2000) 329-361 (\href{https://arxiv.org/abs/hep-th/9907203}{arXiv:hep-th/9907203})


\item P. Anastasopoulos, A. B. Hammou, \emph{A Classification of Toroidal Orientifold Models}, Nucl. Phys. B729:49-78, 2005 (\href{https://arxiv.org/abs/hep-th/0503044}{arXiv:hep-th/0503044})



\end{itemize}
Specifically [[K3]] [[orientifolds]] ($\mathbb{T}^4/G_{ADE}$) in [[type IIA string theory]], hence for [[D8-branes]] and [[D4-branes]]:

\begin{itemize}%
\item J. Park, [[Angel Uranga]], \emph{A Note on Superconformal N=2 theories and Orientifolds}, Nucl. Phys. B542:139-156, 1999 (\href{https://arxiv.org/abs/hep-th/9808161}{arXiv:hep-th/9808161})


\item G. Aldazabal, S. Franco, [[Luis Ibanez]], R. Rabadan, [[Angel Uranga]], \emph{D=4 Chiral String Compactifications from Intersecting Branes}, J. Math. Phys. 42:3103-3126, 2001 (\href{https://arxiv.org/abs/hep-th/0011073}{arXiv:hep-th/0011073})


\item G. Aldazabal, S. Franco, [[Luis Ibanez]], R. Rabadan, [[Angel Uranga]], \emph{Intersecting Brane Worlds}, JHEP 0102:047, 2001 (\href{https://arxiv.org/abs/hep-ph/0011132}{arXiv:hep-ph/0011132})


\item H. Kataoka, M. Shimojo, \emph{$SU(3) \times SU(2) \times U(1)$ Chiral Models from Intersecting D4-/D5-branes}, Progress of Theoretical Physics, Volume 107, Issue 6, June 2002, Pages 1291–1296 (\href{https://arxiv.org/abs/hep-th/0112247}{arXiv:hep-th/0112247}, \href{https://doi.org/10.1143/PTP.107.1291}{doi:10.1143/PTP.107.1291})



\end{itemize}
\begin{quote}%
The $\mathbb{Z}_N$ action with even $N$ contains an order 2 element $[ ...]$ Then there will be D8-branes in the type IIA D4-brane theory. Since the concept of intersecting D-branesinvolves use of the same dimensional D-branes, we restrict ourselves to the case that the order $N$ of $\mathbb{Z}_N$ is odd. (\href{https://arxiv.org/pdf/hep-th/0112247.pdf#page=4}{p. 4})


\end{quote}
\begin{itemize}%
\item [[Gabriele Honecker]], \emph{Non-supersymmetric Orientifolds with D-branes at Angles}, Fortsch.Phys. 50 (2002) 896-902 (\href{https://arxiv.org/abs/hep-th/0112174}{arXiv:hep-th/0112174})


\item [[Gabriele Honecker]], \emph{Intersecting brane world models from D8-branes on $(T^2 \times T^4/\mathbb{Z}_3)/\Omega\mathcal{R}_1$ type IIA orientifolds}, JHEP 0201 (2002) 025 (\href{https://arxiv.org/abs/hep-th/0201037}{arXiv:hep-th/0201037})


\item [[Gabriele Honecker]], \emph{Non-supersymmetric orientifolds and chiral fermions from intersecting D6- and D8-branes}, thesis 2002 ([[HoneckerIntersectingDBraneModels02.pdf:file]])



\end{itemize}
The [[Witten-Sakai-Sugimoto model]] on [[D4-D8-brane bound states]] for [[QCD]] with [[orthogonal group|orthogonal]] [[gauge groups]] on O-planes:

\begin{itemize}%
\item Toshiya Imoto, [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], \emph{$O(N)$ and $USp(N)$ QCD from String Theory}, Prog.Theor.Phys.122:1433-1453, 2010 (\href{https://arxiv.org/abs/0907.2968}{arXiv:0907.2968})


\item Hee-Cheol Kim, Sung-Soo Kim, Kimyeong Lee, \emph{5-dim Superconformal Index with Enhanced $E_n$ Global Symmetry}, JHEP 1210 (2012) 142 (\href{https://arxiv.org/abs/1206.6781}{arXiv:1206.6781})



\end{itemize}
Specifically D5 brane models [[T-duality|T-dual]] to D6/D8 models:

\begin{itemize}%
\item [[Angel Uranga]], \emph{A New Orientifold of $\mathbb{C}^2/\mathbb{Z}_N$ and Six-dimensional RG Fixed Points}, Nucl. Phys. B577:73-87, 2000 (\href{https://arxiv.org/abs/hep-th/9910155}{arXiv:hep-th/9910155})


\item Bo Feng, [[Yang-Hui He]], [[Andreas Karch]], [[Angel Uranga]], \emph{Orientifold dual for stuck NS5 branes}, JHEP 0106:065, 2001 (\href{https://arxiv.org/abs/hep-th/0103177}{arXiv:hep-th/0103177})



\end{itemize}
Specifically for [[D6-branes]]:

\begin{itemize}%
\item S. Ishihara, H. Kataoka, Hikaru Sato, \emph{$D=4$, $N=1$, Type IIA Orientifolds}, Phys. Rev. D60 (1999) 126005 (\href{https://arxiv.org/abs/hep-th/9908017}{arXiv:hep-th/9908017})


\item [[Mirjam Cvetic]], Paul Langacker, Tianjun Li, Tao Liu, \emph{D6-brane Splitting on Type IIA Orientifolds}, Nucl. Phys. B709:241-266, 2005 (\href{https://arxiv.org/abs/hep-th/0407178}{arXiv:hep-th/0407178})



\end{itemize}
Specifically for [[D3-branes]]/[[D7-branes]]:

\begin{itemize}%
\item \hyperlink{FengHeKarchUranga01}{Feng-He-Karch-Uranga 01}

\end{itemize}
Various:

\begin{itemize}%
\item [[Dieter Lüst]], S. Reffert, E. Scheidegger, S. Stieberger, \emph{Resolved Toroidal Orbifolds and their Orientifolds}, Adv.Theor.Math.Phys.12:67-183, 2008 (\href{https://arxiv.org/abs/hep-th/0609014}{arXiv:hep-th/0609014})

\end{itemize}
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[[!redirects RR-field tadpole]] [[!redirects RR-field tadpoles]]

[[!redirects RR-tadpole]] [[!redirects RR-tadpoles]]

[[!redirects RR-field tadpole anomaly]]



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