Type I Omega Orientifold Projection

Use the flat ten-dimensional perturbative Type I model obtained from the Type IIB RNS theory of RNS Spin Structures and Sectors with the supersymmetric Type IIB GSO choice of RNS GSO Projections. Let $\mathrm{\Omega }$ reverse worldsheet orientation, exchange left- and right-moving closed-string data, and identify states invariant under the combined action. The quotient sums over unoriented closed surfaces; a crosscap is the worldsheet identification used to represent an orientation-reversing cycle.

Fix the standard negative-charge O9-minus Type I convention, where O9-minus denotes the space-filling negative-RR-charge orientifold nine-plane. Take $M={\mathbb{R}}^{1,9}$ with flat background metric $\eta =\mathrm{diag}(-1,1,\dots ,1)$, length-valued coordinates, and natural units $\mathrm{\hslash }=c=1$. The massless field slots are smooth real fields on $M$: a Lorentzian metric $G\in \mathrm{\Gamma }({\mathrm{Sym}}^2{T}^{\ast }M)$ with dimensionless components, a dimensionless scalar dilaton $\mathrm{\Phi }:M\to \mathbb{R}$, an NS two-form potential $B_2\in {\mathrm{\Omega }}^2(M)$, and RR potentials $C_q\in {\mathrm{\Omega }}^q(M)$ for $q\in \{0,2,4\}$. In the adopted string normalization these potential components are dimensionless. This item fixes their massless field slots and $\mathrm{\Omega }$ parities only; it supplies no action, field equations, or flux quantization rule. On the Type IIB massless closed fields, the projection keeps $G_{MN}$, $\mathrm{\Phi }$, and $C_2$, and removes $B_2$, $C_0$, and $C_4$. These are the stated massless-sector parities; no general parity rule for other backgrounds is asserted.

For $N\in {\mathbb{Z}}_{\ge 0}$ space-filling D9-branes (branes whose worldvolume fills the ten-dimensional target), let $\lambda \in {\mathrm{Mat}}_N(\mathbb{C})$ be the dimensionless Chan–Paton matrix labeling open-string endpoint states, and let ${\gamma }_{\mathrm{\Omega }}$ be a nonsingular dimensionless Chan–Paton matrix implementing $\mathrm{\Omega }$ on endpoint labels. On a massless open-string vector state define

$$\mathrm{\Omega }:\lambda ⟼-{\gamma }_{\mathrm{\Omega }}{\lambda }^T{\gamma }_{\mathrm{\Omega }}^{-1}\mathrm{.}$$

For the standard O9-minus choice take ${\gamma }_{\mathrm{\Omega }}^T=+{\gamma }_{\mathrm{\Omega }}$; in a basis with ${\gamma }_{\mathrm{\Omega }}=I$, the $\mathrm{\Omega }$-invariant complex state-label space is $\{\lambda \in {\mathrm{Mat}}_N(\mathbb{C}):\lambda =-{\lambda }^T\}$, the complexification of the gauge algebra. Choose anti-Hermitian generators for the real gauge algebra, so ${\lambda }^{†}=-\lambda$ as well as $\lambda =-{\lambda }^T$. These equations imply $\bar{\lambda }=\lambda$, making the real generators precisely the real skew-symmetric matrices, which form $\mathfrak{s}\mathfrak{o}(N)$. The conventional connected Chan–Paton group label is $SO(N)$. Tadpole cancellation is a separate result.

RNS Spin Structures and Sectors

RNS GSO Projections

Gimon and Polchinski, §§1–2

Polchinski, Chaudhuri, and Johnson, Lecture II §2.6