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.
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-planes carry effective negative RR-charge which may (must) cancel against the actual RR- D-brane charge via RR-field tadpole cancellation.
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
(e.g. Blumenhagen-Lüst-Theisen 13 (9.83)).
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 theory – Tadpole 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
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
which means that the $O_{9-d}$-brane charge is
or equivalently
(e.g. Ibáñez-Uranga 12 (5.52), Blumenhagen-Lüst-Theisen 13 (10.212))
In summary, we have the following table of O-plane charges:
$Op$-plane | charge $\mu_{Op}$ | 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
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.
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)
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:
from Hanany-Zaffaroni 98
See also at intersecting D-brane models the section Intersection of D6s with O8s.
The term “orientifold” originates around
Other early accounts include
Clifford Johnson, Anatomy of a Duality, Nucl.Phys. B521 (1998) 71-116 (arXiv:hep-th/9711082)
Amihay Hanany, Alberto Zaffaroni, Branes and Six Dimensional Supersymmetric Theories, Nucl.Phys. B529 (1998) 180-206 (arXiv:hep-th/9712145)
Edward Witten, section 5 of D-branes and K-theory, J. High Energy Phys., 1998(12):019, 1998 (arXiv:hep-th/9810188)
Sunil Mukhi, Nemani V. Suryanarayana, Gravitational Couplings, Orientifolds and M-Planes, JHEP 9909 (1999) 017 (arXiv:hep-th/9907215)
Luis Ibáñez, Angel Uranga, section 5.3.2 of String Theory and Particle Physics – An Introduction to String Phenomenology, Cambridge 2012
Charles Doran, Stefan Mendez-Diez, Jonathan Rosenberg, T-duality For Orientifolds and Twisted KR-theory (arXiv:1306.1779)
Textbook account:
O-Plane charge in differential equivariant KR-theory:
Jacques Distler, Dan Freed, Greg Moore, Orientifold Précis in: Hisham Sati, Urs Schreiber (eds.) Mathematical Foundations of Quantum Field and Perturbative String Theory Proceedings of Symposia in Pure Mathematics, AMS (2011) (arXiv:0906.0795, slides)
Jacques Distler, Dan Freed, Greg Moore, Spin structures and superstrings (arXiv:1007.4581)
reviewed/surveyed in
Daniel Freed, Dirac charge quantiation, K-theory, and orientifolds, talk at a workshop Mathematical methods in general relativity and quantum field theories, Paris, November 2009 (pdf, pdf)
Greg Moore, The RR-charge of an orientifold, Oberwolfach talk 2010 (pdf, pdf, ppt)
Daniel Freed, Lectures on twisted K-theory and orientifolds, lecures at K-Theory and Quantum Fields, ESI 2012 (pdf)
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)
John Huerta, Hisham Sati, Urs Schreiber, Real ADE-equivariant (co)homotopy and Super M-branes (arXiv:1805.05987)
The Witten-Sakai-Sugimoto model for QCD on O-planes:
See also
Last revised on March 2, 2019 at 10:41:40. See the history of this page for a list of all contributions to it.