Contents

Contents

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

graphics grabbed from Blumenhagen-Lüst-Theisen 13

and reflected in target spacetime by non-trivial total RR-field flux on compact spaces

graphics grabbed from Ibanez-Uranga 12

This anomaly cancels if the D-branes are accompanied by a suitable collection of O-planes, hence if one considers orientifold backgrounds (Sagnotti 88, pp. 5, Simon-Polchinski 96, section 3). (For space-filling O-planes this means to consider type I string theory instead.)

Accordingly, tadpole cancellation via 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 Uranga 00, Section 5, see also Garcia-Uranga 05, Marchesano 03, Section 4, Marchesano-Shiu 04, CKMNW 05, Section 2.2, Maiden-Shiu-Stefanski 06, Section 5.

But the situation seems to remain somewhat inconclusive (see also Moore 14, p. 21-22).

For fractional D-branes at orbifold singularities

More details are understood in the special case of fractional D-branes 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.

In terms of equivariant K-theory / the representation ring

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

Detailed review of this is in Marchesano 03, Section 4, based on ABIU 99, Honecker 02.

Let $G$ be a finite group. Let

$[1] \subset [H_1] \subset [H_2] \subset \cdots \subset [G]$
$[1] \subset \left[ \left\langle g_1 \right\rangle \right] \subset \left[ \left\langle g_2 \right\rangle \right] \subset \cdots \subset \left[ \left\langle g_{\vert ConjCl(G)\vert} \right\rangle \right] \,.$

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 which is a list of complex numbers of the form

$[H] =$$\left[\langle e\rangle\right]$$\left[\langle g_1\rangle\right]$$\left[\langle g_2\rangle\right]$$\cdots$$\left[\langle g_{\vert ConjCl(G)\vert}\rangle\right]$
$\chi_V =$$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$

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 the isotropy group $G_{DE}$ (by this formula), so that that normalized mass and charge is

(1)$M \;=\; \tfrac{1}{{\vert G\vert}} dim(V) \,, \phantom{AAA} Q_V(g) \;=\; \tfrac{1}{\vert G\vert} \chi_V(g) \coloneqq \tfrac{1}{\vert G\vert} tr_V\left( g_1\right) \,.$

Now in terms of this, the tadpole cancellation condition for fractional D-branes is that the RR-charges in all non-trivially twisted sectors vanish:

(2)$Q_V(g) = 0 \phantom{AA}\text{hence equivalently} \phantom{AA} \chi_{V}\left(g\right) \;=\; 0 \,, \phantom{AAA} g \neq e$
Example

(regular representation solves tadpole cancellation for fractional D-branes)

For every finite group $G$, the homogeous tadpole cancellation condition (2) 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 this Prop.). Hence the mass and charge (1) of the fractional D-brane corresponding to the regular representation is

$M_{{}_{k[G/1]}} \;=\; 1 \,, \phantom{AA} Q_{{}_{k[G/1]}}(g) \;=\; 0 \,.$

These multiples of the regular representation are regarded as trivial solutions to (2).

Proposition

In fact, the multiples of the regular representation (Example ) are the only solutions to the homogeneous tadpole cancellation condition (2) for fractional D-branes.

Proof

Consider the truncated character morphism

$Q \cdot {\vert G \vert} \;\colon\; Rep_k(G) \overset{\chi}{\longrightarrow} k^{\left\vert ConjCl(G) \right\vert} \overset{ \text{forget dimension/mass} }{\longrightarrow} k^{\left\vert ConjCl(G)\right\vert -1 } \,.$

We have to show that the kernel of this map is the free abelian group generated by the regular representation:

$ker\big( Q \cdot {\vert G \vert} \big) \;\simeq\; \mathbb{Z} \cdot k[G/1] \,.$

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 this Example) that

1. for $\rho \neq \mathbf{1}$ a non-trivial irreducible representation we have

$\underset{g \in G \setminus \{e\}}{\sum} Q_{\rho}(g) \cdot {\vert G \vert} \;\coloneqq\; \underset{g \in G \setminus \{e\}}{\sum} \chi_\rho(g) \;=\; - dim(\rho)$
2. for $\rho = \mathbf{1}$ the trivial irreducible representation we have

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

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

$dim(V) \;mod\; {\vert G \vert} \;=\; - \underset{g \in G \setminus \{e\}}{\sum} Q_{V}(g) \cdot {\vert G \vert} \;\coloneqq\; - \underset{g \in G \setminus \{e\}}{\sum} \chi_V(g) \,.$

But this means that all $V$ in the kernel of $Q \cdot {\vert G \vert}$ must have

$dim(V) \;=\; 0 \;mod\; {\vert G \vert} \,.$

This is indeed the case for the multiples $V = n\cdot k[G/1]$ of the regular representation (Example ). Conversely, the injectivity of the full character morphism $\chi$ (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.

On the other hand, at an orientifold singularity, the O-plane itself carries such charge – O-plane charge (see there):

$[H] =$$\left[\langle e\rangle\right]$$\left[\langle g_1\rangle\right]$$\left[\langle g_2\rangle\right]$$\cdots$$\left[\langle g_{\vert ConjCl(G)\vert}\rangle\right]$
$\chi_O =$$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$

Now the tadpole cancellation condition is that (all representations are real and) this O-plane charge is cancelled against the D-brane charge, an affine version of the previous condition:

(3)$\chi_{V}\left(g_{k \geq 1}\right) - \chi_{O}\left( g_{k \geq 1} \right) \;=\; 0 \,.$

(Marchesano 03 (4.15), following Honecker 02 (28))

The general solution to (3) is of course the sum $O$ with any solution to the homogeneous equation (2). By Prop. the latter are exactly the multiples $n \cdot k[G/1]$ of the regular representation, so that the general solution to the tadpole cancellation condition (3) for fractional D-branes is

$V \;=\; O + n \cdot k[G/1] \,.$

In basic examples the O-plane-charge

$O = 2^{p-4} n \cdot \mathbf{1}$

is for $n_O$ coincident O-planes is the corresponding multiple by the O-plane charge $\mu_{Op} = -2^{8-4}$ (here) of the trivial irrep, whence the general solution to the tadpole cancellation condition then is the affine version of (2), being $+ 2^{p-4} \cdot \mathbf{1}$ plus, by Prop. , any numbers of the regular representation:

$V \;=\; 2^{p-4} n_O \cdot \mathbf{1} \;+\; n \cdot k[G/1] \,.$

Examples

We discuss more explicitly the solutions to the homogeneous tadpole cancellation condition (2) for fractional D-branes at orbifold singularities for isotropy group one of the non-abelian finite subgroups of SU(2),

$G_{DE} \;\subset\; SU(2)$

hence those in the 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 BSS 18, Theorem 4.1 the 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=real representation ring

$KO^0_{G_{DE}}(\ast) \;=\; RO\left( G_{DE} \right) \,.$

Since the tadpole cancellation condition (2) 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. 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 homogenous tadpole cancellation condition (2), 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. .

At a $\mathbb{Z}_2$-orientifold singularity

For $G = \mathbb{Z}_2$ the cyclic group of order ${\vert \mathbb{Z}_2\vert} = 2$, the characters/D-brane charges of the elementary virtual permutation representations/fractional D-branes are (e.g. here)

$[H] =$$\left[\langle e\rangle\right]$$\left[\langle g_1\rangle\right]$
$\chi_{V_1} =$$\phantom{-}1$$\phantom{-}1$
$\chi_{V_2} =$$\phantom{-}1$$-1$

One sees immediately that the general solution to the (homogenous part of the) tadpole cancellation condition (2) for $G = \mathbb{Z}_2$ is

$V \;=\; n \cdot \Big( 1 \cdot V_1 + 1 \cdot V_2 \Big) \,, \phantom{AAA} n \in \mathbb{Z} \,.$

whose minimal positive mass (net brane number) is

\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}

At a $\mathbb{Z}_3$-orientifold singularity

For $G = \mathbb{Z}_3$ the cyclic group of order ${\vert \mathbb{Z}_3\vert} = 3$, the characters/D-brane charges of the complex irreducible representations/fractional D-branes are (e.g. here)

$[H] =$$\left[\langle e\rangle\right]$$\left[\langle g_1\rangle\right]$$\left[\langle g_1\rangle\right]$
$\chi_{V_1} =$$\phantom{-}1$$\phantom{-}1$$\phantom{-}1$
$\chi_{V_2} =$$\phantom{-}1$$\exp\left( \tfrac{1}{3} 2 \pi i \right)$$\exp\left( \tfrac{2}{3} 2 \pi i \right)$
$\chi_{V_3} =$$\phantom{-}1$$\exp\left( \tfrac{2}{3} 2 \pi i \right)$$\exp\left( \tfrac{1}{3} 2 \pi i \right)$

One sees immediately that the general solution to the (homogenous part of the) tadpole cancellation condition (2) for $G = \mathbb{Z}_3$ is

$V \;=\; n \cdot \Big( 1 \cdot V_1 + 1 \cdot V_2 + 1 \cdot V_3 \Big) \,, \phantom{AAA} n \in \mathbb{Z} \,.$

whose minimal positive mass (net brane number) is

\begin{aligned} M_{\mathbb{Z}_3} & = dim(V) / {\vert \mathbb{Z}_3 \vert } \\ & = \chi_V([\langle e\rangle]) / {\vert \mathbb{Z}_3 \vert} \\ & = \big( 1 \cdot 1 + 1 \cdot 1 + 1 \cdot 1 \big) / {3} & \\ & = 3 / 3 \\ & = 1 \end{aligned}

This pattern immediately continues for all cyclic groups $\mathbb{Z}_n$.

At a $2 D_4$-orientifold singularity

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

$[H] =$$\left[\langle e\rangle\right]$$\left[\langle g_1\rangle\right]$$\left[\langle g_2\rangle\right]$$\left[\langle g_3\rangle\right]$$\left[\langle g_4\rangle\right]$
$\chi_{V_1} =$$\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$

One sees (here) that the general solution to the (homogenous part of the) tadpole cancellation condition (2) for $G =2 D_4$ is

$V \;=\; n \Big( 1 \cdot V_1 + 1 \cdot V_2 + 1 \cdot V_3 + 1 \cdot V_4 \Big) \,, \phantom{AAA} n \in \mathbb{Z}$

whose minimal positive mass (net brane number) is

\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}

At a $2 D_6$-orientifold singularity

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

$[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]$
$\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$

One finds (here) that the general solution to the (homogenous part of the) tadpole cancellation condition (2) for $G =2 D_6$ is

$V \;=\; n \Big( 1 \cdot V_1 + 1 \cdot V_2 + 2 \cdot V_3 + 1 \cdot V_4 + 1 \cdot V_5 \Big) \,, \phantom{AAA} n \in \mathbb{Z}$

whose minimal positive mass (net brane number) is

\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}

At a $2 D_8$-orientifold singularity

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

$[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]$$\left[\langle g_6\rangle\right]$
$\chi_{V_1} =$$\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$

One finds (here), that the general solution to the (homogenous part of the) tadpole cancellation condition (2) for $G =2 D_8$ is

$V \;=\; n \Big( 1 \cdot V_1 + 1 \cdot V_2 + 1 \cdot V_3 + 1 \cdot V_4 + 2 \cdot V_5 + 1 \cdot V_6 \Big) \,, \phantom{AAA} n \in \mathbb{Z}$

whose minimal positive mass (net brane number) is

\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}

At a $2 D_{10}$-orientifold singularity

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

$[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]$
$\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$

One finds (here) that the general solution to the (homogenous part of the) tadpole cancellation condition (2) for $G =2 D_{10}$ is

$V \;=\; n \Big( 1 \cdot V_1 + 1 \cdot V_2 + 1 \cdot V_3 + 2 \cdot V_4 + 1 \cdot V_5 \Big) \,, \phantom{AAA} n \in \mathbb{Z}$

whose minimal positive mass (net brane number) is

\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}

At a $2 D_{12}$-orientifold singularity

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

$[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]$$\left[\langle g_8\rangle\right]$
$\chi_{V_1} =$$\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$

One sees (here) that the general solution to the (homogenous part of the) tadpole cancellation condition (2) for $G =2 D_{12}$ is

$V \;=\; n \Big( 1 \cdot V_1 + 1 \cdot V_2 + 1 \cdot V_3 + 1 \cdot V_4 + 2 \cdot V_5 + 2 \cdot V_6 + 1 \cdot V_7 + 1 \cdot V_8 \Big) \,, \phantom{AAA} n \in \mathbb{Z}$

whose minimal positive mass (net brane number) is

\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}

At a $2 D_{14}$-orientifold singularity

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

$[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]$
$\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$

One sees by immediate inspection, that the general solution to the (homogenous part of the) tadpole cancellation condition (2) for $G =2 D_{14}$ is

$V \;=\; n \Big( 1 \cdot V_1 + 1 \cdot V_2 + 1 \cdot V_3 + 2 \cdot V_4 + 1 \cdot V_5 \Big) \,, \phantom{AAA} n \in \mathbb{Z}$

whose minimal positive mass (net brane number) is

\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}

At a $2 D_{16}$-orientifold singularity

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

$[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]$$\left[\langle g_9\rangle\right]$
$\chi_{V_1} =$$\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$

One sees by immediate inspection, that the general solution to the (homogenous part of the) tadpole cancellation condition (2) for $G =2 D_{16}$ is

$V \;=\; n \Big( 1 \cdot V_1 + 1 \cdot V_2 + 1 \cdot V_3 + 1 \cdot V_4 + 2 \cdot V_5 + 2 \cdot V_6 + 1 \cdot V_7 \Big) \,, \phantom{AAA} n \in \mathbb{Z}$

whose minimal positive mass (net brane number) is

\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}

At a $2 T$-orientifold singularity

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

$[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]$$\left[\langle g_6\rangle\right]$
$\chi_{V_1} =$$\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$

One finds (here) that the general solution to the (homogenous part of the) tadpole cancellation condition (2) for $G = 2T$ is

$V \;=\; n \Big( 1 \cdot V_1 + 1 \cdot V_2 + 3 \cdot V_3 + 2 \cdot V_4 + 1 \cdot V_5 \Big) \,, \phantom{AAA} n \in \mathbb{Z}$

whose minimal positive mass (net brane number) is

\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}

At a $2 O$-orientifold singularity

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

$[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]$
$\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$

One finds (here) that the general solution to the (homogenous part of the) tadpole cancellation condition (2) for $G = 2O$ is

$V \;=\; n \Big( 1 \cdot V_1 + 1 \cdot V_2 + 2 \cdot V_3 + 3 \cdot V_4 + 3 \cdot V_5 + 2 \cdot V_6 + 1 \cdot V_7 \Big) \,, \phantom{AAA} n \in \mathbb{Z}$

whose minimal positive mass (net brane number) is

\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}

At a $2 I$-orientifold singularity

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

$[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]$$\left[\langle g_8\rangle\right]$
$\chi_{V_1} =$$\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$

One finds (here) that the general solution to the (homogenous part of the) tadpole cancellation condition (2) for $G = 2I$ is

$V \;=\; n \Big( 1 \cdot V_1 + 4 \cdot V_2 + 5 \cdot V_3 + 3 \cdot V_4 + 3 \cdot V_5 + 2 \cdot V_6 + 1 \cdot V_7 \Big) \,, \phantom{AAA} n \in \mathbb{Z}$

whose minimal positive mass (net brane number) is

\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}

References

The issue was first highlighted in

• Augusto Sagnotti, Open strings and their symmetry groups in G. Mack et. al. (eds.) Cargese ’87, “Non-perturbative Quantum Field Theory,” (Pergamon Press, 1988) p. 521 (arXiv:hep-th/0208020)

The argument is recalled in

Details are in

Textbook accounts include

Quick illustrations include:

Critical outlook in

• G. Aldazabal, D. Badagnani, Luis Ibáñez, Angel Uranga, Tadpole versus anomaly cancellation in $D=4,6$ compact IIB orientifolds, JHEP 9906:031, 1999 (arXiv:hep-th/9904071)

• Angel Uranga, D-brane probes, RR tadpole cancellation and K-theory charge, Nucl.Phys.B598:225-246, 2001 (arXiv:hep-th/0011048)

• Gabriele Honecker, 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 (arXiv:hep-th/0201037)

• Maria E. Angulo, David Bailin, Huan-Xiong Yang, Tadpole and Anomaly Cancellation Conditions in D-brane Orbifold Models, Int.J.Mod.Phys.A18:3637-3694, 2003 (arXiv:hep-th/0210150)

• Fernando Marchesano, section 4 of Intersecting D-brane Models (arXiv:hep-th/0307252)

• Fernando Marchesano, Gary Shiu, Building MSSM Flux Vacua, JHEP0411:041, 2004 (arXiv:hep-th/0409132)

• C.-M. Chen, G. V. Kraniotis, V. E. Mayes, D. V. Nanopoulos, J. W. Walker, A K-theory Anomaly Free Supersymmetric Flipped SU(5) Model from Intersecting Branes, Phys.Lett. B625 (2005) 96-105 (arXiv:hep-th/0507232)

• Inaki Garcia-Etxebarria, Angel Uranga, From F/M-theory to K-theory and back, JHEP 0602:008, 2006 (arXiv:hep-th/0510073)

• John Maiden, Gary Shiu, Bogdan Stefanski, D-brane Spectrum and K-theory Constraints of D=4, N=1 Orientifolds, JHEP0604:052,2006 (arXiv:hep-th/0602038)

• Tetsuji Kimura, Mitsuhisa Ohta, Kei-Jiro Takahashi, Type IIA orientifolds and orbifolds on non-factorizable tori, Nucl.Phys.B798:89-123, 2008 (arXiv:0712.2281)

For the topological string:

• Johannes Walcher, Evidence for Tadpole Cancellation in the Topological String (arXiv:0712.2775)

The character tables for virtual permutation representations above are taken from

Last revised on March 19, 2019 at 01:07:17. See the history of this page for a list of all contributions to it.