nLab
orientifold plane

Contents

Contents

Idea

In type II string theory on orientifolds (Dai-Lin-Polchinski 89), one says O-plane for the fixed point locus of the 2\mathbb{Z}_2-involution (see at real space).

O-planes carry D-brane charges in KR-theory (Witten 98), see (DMR 13) for a mathematical account. They serve RR-field tadpole cancellation and as such play a key role in the construction of intersecting D-brane models for string phenomenology.

Properties

T-Duality with type I string theory

Under T-duality, type I string theory is dual to type II string theory with orientifold planes (reviewed e.g. in Ibanez-Uranga 12, section 5.3.2 - 5.3.4):

O-Plane charge

O-planes carry effective negative RR-charge which may (must) cancel against the actual RR- D-brane charge via RR-field tadpole cancellation.

For flat orientifolds

The charge of the spacetime-filling O9O9-plane of plain type I string theory (type II string theory on the orientifold 9,1 2\mathbb{R}^{9,1}\sslash \mathbb{Z}_2 with trival spacetime 2\mathbb{Z}_2-action) is found by worldsheet-computation to be 32-32 in units of D9-brane charge:

(1)q O9 =32q D9 q_{O9^-} \;=\; -32 \, q_{D9}

(e.g. Blumenhagen-Lüst-Theisen 13 (9.83)).

Beware that there is some convention involved in assigning an absolute value of unit D-brane charge q D9q_{D9}. A common choice in the literature is to mean by “one D-brane” one of the two pre-images on the covering space, in which case its obsolute charge is to be

(2)q Dp=1/2 q_{Dp} \;=\; 1/2

(e.g. BDHKMMS 01, p. 46-47)

This means that RR-field tadpole cancellation here requires the presence of 32 D-branes (or rather: 16 and their 2\mathbb{Z}_2-mirror images), hence a space-filling D9-brane with Chan-Paton bundle of rank 3232, corresponding to a gauge group SO(32). For more on this see at type I string theoryTadpole cancellation and SO(32)-GUT.

From this the Op p^--brane charge for pnp \leq n follows from T-duality (as above) with respect to KK-compactification on a d-torus 𝕋 d\mathbb{T}^d with 2\mathbb{Z}_2-action given by canonical coordinate reflection

2× 10d1,1×𝕋 d 10d1,1×𝕋 d (σ,(x,y)) (x,y). \array{ \mathbb{Z}_2 \times \mathbb{R}^{10-d-1,1} \times \mathbb{T}^d &\longrightarrow& \mathbb{R}^{10-d-1,1} \times \mathbb{T}^d \\ (\sigma, (\vec x, \vec y)) &\mapsto& (\vec x, - \vec y) } \,.

This results in O(9d) O(9-d)^--planes with worldvolume 10d1,1\mathbb{R}^{10-d-1,1}. But since the orbifold 𝕋 d 2\mathbb{T}^d\sslash \mathbb{Z}_2 now has 2 d2^d singularities /fixed points (this Example) there are now 2 d2^d such O(9d) O(9-d)^--planes.

Since the number of D-branes does not change under T-duality, the total O-plane charge should be the same as before

2 dq O(9d)=1q O9=32q D9=2 5q D9 2^d \cdot q_{O(9-d)} \;=\; 1 \cdot q_{O9} \;=\; -32 \cdot q_{D9} \;=\; - 2^5 \cdot q_{D9}

which means that the O(9d) O(9-d)^--plane charge is

q O(9d) =2 5dq D(pd) q_{O(9-d)^-} \;=\; - 2^{5-d} \cdot q_{D(p-d)}

or equivalently

(3)q Op =2 p4q Dp q_{O p^-} \;=\; - 2^{ p - 4 } \cdot q_{D p}

(e.g. Ibáñez-Uranga 12 (5.52), Blumenhagen-Lüst-Theisen 13 (10.212))

In summary, we have the following table of O-plane charges on flat orbifolds:

O-plane
species
charge
q Op /q Dpq_{O p^-}/q_{D p}
transverse
d-torus
fixed points
|(𝕋 d) 2|\left\vert\left( \mathbb{T}^d\right)^{\mathbb{Z}_2}\right\vert
O9 O9^-32-32𝕋 0\mathbb{T}^011\phantom{1}1
O8 O8^-16-16𝕋 1\mathbb{T}^112\phantom{1}2
O7 O7^-18-\phantom{1}8𝕋 2\mathbb{T}^214\phantom{1}4
O6 O6^-14-\phantom{1}4𝕋 3\mathbb{T}^318\phantom{1}8
O5 O5^-12-\phantom{1}2𝕋 4\mathbb{T}^41616
O4 O4^-11-\phantom{1}1𝕋 5\mathbb{T}^53232

In particular the O4-plane has negative unit charge (in units of D4-brane charge q D4q_{D4}), so that the total charge of 32-32 here comes entirely from the number 32=2 532 = 2^5 of fixed points of the 2\mathbb{Z}_2-action on 𝕋 5\mathbb{T}^5.

O-plane charges of different dimension may be present

graphics grabbed from Johnson 97

In the presence of discrete torsion

In the presence of discrete torsion in the B-field and/or the RR-fields, this charge structure of orientifold planes on flat orbifolds gets further modified (Hanany-Kol 00, Sec. 2.1, see Bergman-Gimon-Sugimoto 01, Sec. 1):

graphics grabbed from Bergman-Gimon-Sugimoto 01

(In comparing this last table with the above table, notice that this shows the Op-plane charge in units of q Dq1/2q_{Dq} \coloneqq 1/2 as in (2).)

In differential equivariant KR-theory

A proposal for a formalization of a much more general formula for O-plane charge, regarded in differential equivariant KR-theory is briefly in Distler-Freed-Moore 09, p. 6.

Duality with M-Theory

The possible O-planes in M-theory are MO1MO1 (\leftrightarrowM-wave), MO5MO5 (\leftrightarrowM5-brane) and MO9 (Hanany-Kol 00 around (3.2), HSS 18, Prop. 4.7).

Under the duality between M-theory and type IIA string theory the O8-plane is identified with the MO9 of Horava-Witten theory:

graphics grabbed from GKSTY 02, section 3

while the O4-plane is dual to the MO5MO5 (Hori 98, Gimon 98, Sec. III, AKY 98, Sec. II B, Hanany-Kol 00, 3.1.1)

graphics grabbed from Gimon 98

and the O0O0 to the MO1 (Hanany-Kol 00 3.3)

Fractional branes at O-planes

By the discussion at D-branes ending on NS5 branes, a black D6-brane may end on a black NS5-brane, and in fact a priori each brane NS5-brane has to be the junction of two black D6-branes.

from GKSTY 02

If in addition the black NS5-brane sits at an O8-plane, hence at the orientifold fixed point-locus, then in the ordinary /2\mathbb{Z}/2-quotient it appears as a “half-brane” with only one copy of D6-branes ending on it:

from GKSTY 02

(In Hanany-Zaffaroni 99 this is interpreted in terms of the 't Hooft-Polyakov monopole.)

The lift to M-theory of this situation is an M5-brane intersecting an M9-brane:

from GKSTY 02

Alternatively the O8-plane may intersect the black D6-branes away from the black NS5-brane:

from HKLY 15

In general, some of the NS5 sit away from the O8-plane, while some sit on top of it:

from Hanany-Zaffaroni 98

See also at intersecting D-brane models the section Intersection of D6s with O8s.

References

The term “orientifold” originates around

  • Jin Dai, R.G. Leigh, Joseph Polchinski, p. 12 of New Connections Between String Theories, Mod.Phys.Lett. A4 (1989) 2073-2083 (spire:25758)

Other early accounts include

Textbook accounts:

O-Plane charge in the presence of discrete torsion:

O-Plane charge in differential equivariant KR-theory:

reviewed/surveyed in

Actual construction of twisted differential orthogonal K-theory and its relation to D-brane charge in type I string theory (on orientifolds):

Lift to M-theory:

The Witten-Sakai-Sugimoto model for QCD on O-planes:

See also

Last revised on May 23, 2019 at 08:47:59. See the history of this page for a list of all contributions to it.