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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 VK G(*)=R GV \in K_G(\ast) = R_G in equivariant K-theory = representation ring
(here n reg=k[G/1]\mathbf{n}_{reg} = k[G/1] denotes the regular representation of dimension n=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 𝕋 4 2\mathbb{T}^{\mathbf{4}_{\mathbb{H}}}\!\sslash\! \mathbb{Z}_{2}
V=N2 regV = N \cdot \mathbf{2}_{reg}
(Buchel-Shiu-Tye 99 (19))
V=162 regV = 16 \cdot \mathbf{2}_{reg}
(Buchel-Shiu-Tye 99 (18))
following
Gimon-Polchinski 96,
Gimon-Johnson 96
D5-branes
transv. to 𝕋 4 4\mathbb{T}^{\mathbf{4}_{\mathbb{H}}} \!\sslash\! \mathbb{Z}_{4}
V=N4 regV = N \cdot \mathbf{4}_{reg}
(Buchel-Shiu-Tye 99 (19))
V=84 regV = 8 \cdot \mathbf{4}_{reg}
(Buchel-Shiu-Tye 99 (18))
following
Gimon-Polchinski 96,
Gimon-Johnson 96
D4-branes
transv. to 𝕋 1 triv+4 k\mathbb{T}^{\mathbf{1}_{\mathrm{triv}} + \mathbf{4}_{\mathbb{H}}} \!\sslash\! \mathbb{Z}_k
V=Nk regV = N \cdot \mathbf{k}_{reg}
(AFIRU 00a, 4.2.1)
D4-branes
transv. to 𝕋 1 triv+4 3\mathbb{T}^{\mathbf{1}_{\mathrm{triv}} + \mathbf{4}_{\mathbb{H}}} \!\sslash\! \mathbb{Z}_3
V=N3 regV = N \cdot \mathbf{3}_{reg}
(AFIRU 00b, (7.2))
V=43 reg+41 trivV = 4 \cdot \mathbf{3}_{reg} + 4 \cdot \mathbf{1}_{triv}
(Kataoka-Shimojo 01, (14)-(17)
D8-branes
on 𝕋 1 triv+4 3\mathbb{T}^{\mathbf{1}_{\mathrm{triv}} + \mathbf{4}_{\mathbb{H}}} \!\sslash\! \mathbb{Z}_3
V=N3 regV = N \cdot \mathbf{3}_{reg}
V=43 reg+41 trivV = 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 𝕋 6 4\mathbb{T}^6 \!\sslash\! \mathbb{Z}_4
V=84 regV = 8 \cdot \mathbf{4}_{reg}
(Ishihara-Kataoka-Sato 99, (4.16))
D3-branes
on 𝕋 4 k\mathbb{T}^{\mathbf{4}_{\mathbb{H}}} \!\sslash\! \mathbb{Z}_k
V=Nk regV = N \cdot \mathbf{k}_{reg}
(Feng-He-Karch-Uranga 01, (25))
D7-branes
on 𝕋 4 k\mathbb{T}^{\mathbf{4}_{\mathbb{H}}} \!\sslash\! \mathbb{Z}_k
V=Nk regV = N \cdot \mathbf{k}_{reg}
(Feng-He-Karch-Uranga 01, (5), (6))

Last revised on September 9, 2019 at 12:31:11. See the history of this page for a list of all contributions to it.