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 $-32$ in units of D9-brane charge:
(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 $q_{D9}$. 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
(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 $\mathbb{Z}_2$-mirror images), 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 $O(9-d)^-$-plane 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 on flat orbifolds:
O-plane species | charge $q_{O p^-}/q_{D p}$ | 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 (in units of D4-brane charge $q_{D4}$), 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
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 $q_{Dq} \coloneqq 1/2$ as in (2).)
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.
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)
Jan de Boer, Robbert Dijkgraaf, Kentaro Hori, Arjan Keurentjes, John Morgan, David Morrison, Savdeep Sethi, section 3 of Triples, Fluxes, and Strings, Adv.Theor.Math.Phys. 4 (2002) 995-1186 (arXiv:hep-th/0103170)
Textbook accounts:
Luis Ibáñez, Angel Uranga, section 10 of String Theory and Particle Physics – An Introduction to String Phenomenology, Cambridge 2012
Ralph Blumenhagen, Dieter Lüst, Stefan Theisen, Section 9.4 and 10.6 of Basic Concepts of String Theory Part of the series Theoretical and Mathematical Physics, Springer 2013
O-Plane charge in the presence of discrete torsion:
Oren Bergman, Eric Gimon, Shigeki Sugimoto, Orientifolds, RR Torsion, and K-theory, JHEP 0105:047, 2001 (arXiv:hep-th/0103183)
O-Plane charge in differential equivariant KR-theory:
Charles Doran, Stefan Mendez-Diez, Jonathan Rosenberg, T-duality For Orientifolds and Twisted KR-theory (arXiv:1306.1779)
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)
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:
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, 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 May 23, 2019 at 08:47:59. See the history of this page for a list of all contributions to it.