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, Gimon-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, DFM 09, p. 6-7.

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

In plane and toroidal orientifolds

More details are understood in the special case of plane orbifolds $V^{cpt} \!\sslash\! G$ and 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.

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.

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 $G$ (by this formula), so that normalized mass and charge is

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

The twisted (local) tadpole cancellation condition for fractional D-branes at orbifold singularities 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 local/twisted 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$

(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 untwisted (global) tadpole cancellation condition is that (all representations are real and) this O-plane charge is cancelled against the D-brane charge:

(3)$\chi_{V}\left(g_{k \geq 1}\right) \;=\; 0 \phantom{AA} \text{and} \phantom{AA} dim(V) = dim(O) \,.$

By Prop. 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:

(4)$V = N \cdot k[G/1] \,.$

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 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

$V \;=\; \frac{2^{p-4}}{\vert G\vert } \cdot k[G/1] \,.$

Sometimes the condition (5) is found with an offset by a trivial representation

(5)$V = N \cdot k[G/1] + \mathbf{p}_{triv} \,.$

This corresponds to single fractional D-branes sitting on top of the O-planes, turning $O^-$-planes into $O^+$-planes.

Examples for toroidal orientifolds

RR-field tadpole cancellation conditions for D-branes wrapped on toroidal orientifolds
in terms of their D-brane charge $V \in K_G(\ast) = R_G$ in equivariant K-theory = representation ring
(here $\mathbf{n}_{reg} = k[G/1]$ denotes the regular representation of dimension $n =$ ord(G))

single D-brane species
on toroidal orientifold
local/twisted
global/untwisted
comment
D5-branes
transv. to $\mathbb{T}^{\mathbf{4}_{\mathbb{H}}}\!\sslash\! \mathbb{Z}_{2}$
$V = N \cdot \mathbf{2}_{reg}$
(Buchel-Shiu-Tye 99 (19))
$V = 16 \cdot \mathbf{2}_{reg}$
(Buchel-Shiu-Tye 99 (18))
following
Gimon-Polchinski 96,
Gimon-Johnson 96
D5-branes
transv. to $\mathbb{T}^{\mathbf{4}_{\mathbb{H}}} \!\sslash\! \mathbb{Z}_{4}$
$V = N \cdot \mathbf{4}_{reg}$
(Buchel-Shiu-Tye 99 (19))
$V = 8 \cdot \mathbf{4}_{reg}$
(Buchel-Shiu-Tye 99 (18))
following
Gimon-Polchinski 96,
Gimon-Johnson 96
D4-branes
transv. to $\mathbb{T}^{\mathbf{1}_{\mathrm{triv}} + \mathbf{4}_{\mathbb{H}}} \!\sslash\! \mathbb{Z}_k$
$V = N \cdot \mathbf{k}_{reg}$
(AFIRU 00a, 4.2.1)
D4-branes
transv. to $\mathbb{T}^{\mathbf{1}_{\mathrm{triv}} + \mathbf{4}_{\mathbb{H}}} \!\sslash\! \mathbb{Z}_3$
$V = N \cdot \mathbf{3}_{reg}$
(AFIRU 00b, (7.2))
$V = 4 \cdot \mathbf{3}_{reg} + 4 \cdot \mathbf{1}_{triv}$
(Kataoka-Shimojo 01, (14)-(17)
D8-branes
on $\mathbb{T}^{\mathbf{1}_{\mathrm{triv}} + \mathbf{4}_{\mathbb{H}}} \!\sslash\! \mathbb{Z}_3$
$V = N \cdot \mathbf{3}_{reg}$
$V = 4 \cdot \mathbf{3}_{reg} + 4 \cdot \mathbf{1}_{triv}$
(Honecker 01, 4,
Honecker 02a, (25) ∧ (28) ⇔ (29),
Honecker 02b, (3.19)-(3.27))
equivalent to D4 case by T-duality:
Honecker 01, p. 2,
Honecker 02a, 6,
Honecker 02b, p. 15
review in:
Marchesano 03, Sec. 4
D6-branes
on $\mathbb{T}^6 \!\sslash\! \mathbb{Z}_4$
$V = 8 \cdot \mathbf{4}_{reg}$
(Ishihara-Kataoka-Sato 99, (4.16))
D3-branes
on $\mathbb{T}^{\mathbf{4}_{\mathbb{H}}} \!\sslash\! \mathbb{Z}_k$
$V = N \cdot \mathbf{k}_{reg}$
(Feng-He-Karch-Uranga 01, (25))
D7-branes
on $\mathbb{T}^{\mathbf{4}_{\mathbb{H}}} \!\sslash\! \mathbb{Z}_k$
$V = N \cdot \mathbf{k}_{reg}$
(Feng-He-Karch-Uranga 01, (5), (6))

graphics grabbed from Sati-Schreiber 19

See also at equivariant Hopf degree theorem.

graphics grabbed from Sati-Schreiber 19

Examples of non-compact singularities

We discuss more explicitly the solutions to the local/twisted 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 twisted 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 local/twisted 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 $\mathbb{Z}_4$-orientifold singularity

For $G = \mathbb{Z}_4$ the cyclic group of order 4, 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]$$\left[\langle g_2\rangle\right]$$\left[\langle g_3\rangle\right]$
$\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$

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

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

whose minimal positive mass (net brane number) is

\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}

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 local/twisted 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 local/twisted 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 local/twisted 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 local/twisted 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 local/twisted 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 local/twisted 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 local/twisted 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 local/twisted 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 local/twisted 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 local/twisted 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

General

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

The above discussion follows and character tables for virtual permutation representations above are taken from

In view of consistency of flux compactifications:

• Philip Betzler, Erik Plauschinn, Type IIB flux vacua and tadpole cancellation (arXiv:1905.08823)

For the topological string:

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

Examples and Models

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

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

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. (p. 4)

• Gabriele Honecker, Non-supersymmetric Orientifolds with D-branes at Angles, Fortsch.Phys. 50 (2002) 896-902 (arXiv:hep-th/0112174)

• 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)

• Gabriele Honecker, Non-supersymmetric orientifolds and chiral fermions from intersecting D6- and D8-branes, thesis 2002 (pdf)

The Witten-Sakai-Sugimoto model on D4-D8-brane bound states for QCD with orthogonal gauge groups on O-planes:

• Toshiya Imoto, Tadakatsu Sakai, Shigeki Sugimoto, $O(N)$ and $USp(N)$ QCD from String Theory, Prog.Theor.Phys.122:1433-1453, 2010 (arXiv:0907.2968)

• Hee-Cheol Kim, Sung-Soo Kim, Kimyeong Lee, 5-dim Superconformal Index with Enhanced $E_n$ Global Symmetry, JHEP 1210 (2012) 142 (arXiv:1206.6781)

Specifically D5 brane models T-dual to D6/D8 models:

Specifically for D6-branes:

• S. Ishihara, H. Kataoka, Hikaru Sato, $D=4$, $N=1$, Type IIA Orientifolds, Phys. Rev. D60 (1999) 126005 (arXiv:hep-th/9908017)

• Mirjam Cvetic, Paul Langacker, Tianjun Li, Tao Liu, D6-brane Splitting on Type IIA Orientifolds, Nucl. Phys. B709:241-266, 2005 (arXiv:hep-th/0407178)

Specifically for D3-branes/D7-branes:

Various:

• Dieter Lüst, S. Reffert, E. Scheidegger, S. Stieberger, Resolved Toroidal Orbifolds and their Orientifolds, Adv.Theor.Math.Phys.12:67-183, 2008 (arXiv:hep-th/0609014)

Last revised on February 7, 2021 at 10:52:37. See the history of this page for a list of all contributions to it.