nLab RR-field tadpole cancellation on toroidal orientifolds -- table

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 tadpole cancellation condition	global/untwisted tadpole cancellation condition	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))