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 $\mathbb{Z}_2$-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.

#### O-plane charge for flat orientifolds

The charge of the spacetime-filling $O9$-plane of plain type I string theory (type II string theory on the orientifold $\mathbb{R}^{9,1}\sslash \mathbb{Z}_2$ with trival spacetime $\mathbb{Z}_2$-action) is found by worldsheet-computation to be

(1)$\mu_{O9} \;=\; -32$

This means that RR-field tadpole cancellation here requires the presence of 32 D-branes, hence a space-filling D9-brane with Chan-Paton bundle of rank $32$, 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 O$p$-brane charge for $p \leq n$ follows from T-duality (as above) with respect to KK-compactification on a d-torus $\mathbb{T}^d$ with $\mathbb{Z}_2$-action given by canonical coordinate reflection

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

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

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

$2^d \cdot \mu_{O_{9-d}} \;=\; 1 \cdot \mu_{O_9} \;=\; -32 \;=\; - 2^5$

which means that the $O_{9-d}$-brane charge is

$\mu_{O_{9-d}} \;=\; - 2^{5-d}$

or equivalently

(2)$\mu_{O_p} \;=\; - 2^{ p - 4 }$

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

$Op$-planecharge
$\mu_{Op}$
transverse
d-torus
fixed points
$\left\vert\left( \mathbb{T}^d\right)^{\mathbb{Z}_2}\right\vert$
$O9$$-32$$\mathbb{T}^0$$\phantom{1}1$
$O8$$-16$$\mathbb{T}^1$$\phantom{1}2$
$O7$$-\phantom{1}8$$\mathbb{T}^2$$\phantom{1}4$
$O6$$-\phantom{1}4$$\mathbb{T}^3$$\phantom{1}8$
$O5$$-\phantom{1}2$$\mathbb{T}^4$$16$
$O4$$-\phantom{1}1$$\mathbb{T}^5$$32$

In particular the O4-plane has negative unit charge, so that the total charge of $-32$ here comes entirely from the number $32 = 2^5$ of fixed points of the $\mathbb{Z}_2$-action on $\mathbb{T}^5$.

O-plane charges of different dimension may be present

graphics grabbed from Johnson 97

#### O-plane charge in differential equivariant KR-theory

A much more general formula for this O-plane charge, regarded in differential equivariant KR-theory is supposed to be the topic of Distler-Freed-Moore 09.

### Duality with M-Theory

The possible O-planes in M-theory are $MO1$ ($\leftrightarrow$M-wave), $MO5$ ($\leftrightarrow$M5-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 $MO5$ (Hori 98, Gimon 98, Sec. III, AKY 98, Sec. II B, Hanany-Kol 00, 3.1.1)

graphics grabbed from Gimon 98

and the $O0$ 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 $\mathbb{Z}/2$-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:

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 account:

O-Plane charge in differential equivariant KR-theory:

reviewed/surveyed in

Lift to M-theory:

• Kentaro Hori, Consistency Conditions for Fivebrane in M Theory on $\mathbb{R}^5/\mathbb{Z}_2$ Orbifold, Nucl.Phys.B539:35-78, 1999 (arXiv:hep-th/9805141)

• Eric G. Gimon, On the M-theory Interpretation of Orientifold Planes (arXiv:hep-th/9806226, spire:472499)

• Changhyun Ahn, Hoil Kim, Hyun Seok Yang, $SO(2N)$ $(0,2)$ SCFT and M Theory on $AdS_7 \times \mathbb{R}P^4$, Phys.Rev. D59 (1999) 106002 (arXiv:hep-th/9808182)

• E. Gorbatov, V.S. Kaplunovsky, J. Sonnenschein, Stefan Theisen, S. Yankielowicz, section 3 of On Heterotic Orbifolds, M Theory and Type I’ Brane Engineering, JHEP 0205:015, 2002 (arXiv:hep-th/0108135)

• Amihay Hanany, Barak Kol, section 3 of On Orientifolds, Discrete Torsion, Branes and M Theory, JHEP 0006 (2000) 013 (arXiv:hep-th/0003025)

• Hirotaka Hayashi, Sung-Soo Kim, Kimyeong Lee, Futoshi Yagi, 6d SCFTs, 5d Dualities and Tao Web Diagrams (arXiv:1509.03300)

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