Type IIB Calabi–Yau orientifold data with O3/O7 fixed loci

Let $X$ be the smooth connected compact complex threefold and $I$ its complex structure. Let $J$ and $\mathrm{\Omega }$ be its chosen Kähler form and nowhere-zero holomorphic three-form in the broad Calabi–Yau convention; $g(u,v)=J(u,Iv)$ is the associated Kähler metric. Give $X$ its complex orientation. Use natural units $\mathrm{\hslash }=c=1$ and dimensionless internal coordinates with ${\alpha }^{\mathrm{\prime }}=1$, so $J$ and $g$ are dimensionless; when units are restored, $J_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}={\alpha }^{\mathrm{\prime }}J$. A Type IIB O3/O7 orientifold datum in the smooth massless bosonic sector consists of these data and the following projection. Choose a holomorphic isometry $\sigma :X\to X$ with ${\sigma }^2=\mathrm{id}$, ${\sigma }^{\ast }J=J$, and ${\sigma }^{\ast }\mathrm{\Omega }=-\mathrm{\Omega }$. Put $M={\mathbb{R}}^4\times X$, $s={\mathrm{id}}_{{\mathbb{R}}^4}\times \sigma$, and let $\mathcal{V}={\mathrm{\Gamma }}^{\mathrm{\infty }}(M,{\mathrm{Sym}}^2{T}^{\ast }M)\oplus C^{\mathrm{\infty }}(M;\mathbb{R})\oplus {\mathcal{A}}^2(M;\mathbb{R})\oplus C^{\mathrm{\infty }}(M;\mathbb{R})\oplus {\mathcal{A}}^2(M;\mathbb{R})\oplus {\mathcal{A}}^4(M;\mathbb{R})\mathrm{.}$ Here ${\mathcal{A}}^r(M;\mathbb{R})$ is the vector space of smooth real differential $r$-forms; order the components as $(\widehat{g},\varphi ,B_2,C_0,C_2,C_4)$, with $\widehat{g}$ restricted to Lorentzian metrics when interpreted as a spacetime metric. The potentials are local smooth representatives; global flux and gauge patching require separate data. Define real-linear involutions ${\mathrm{\Omega }}_p,P_L:\mathcal{V}\to \mathcal{V}$ by ${\mathrm{\Omega }}_p(\widehat{g},\varphi ,B_2,C_0,C_2,C_4)=(\widehat{g},\varphi ,-B_2,-C_0,C_2,-C_4),P_L(\widehat{g},\varphi ,B_2,C_0,C_2,C_4)=(\widehat{g},\varphi ,B_2,-C_0,-C_2,-C_4)\mathrm{.}$ The notation $P_L=(-1{)}^{F_L}$ denotes this specified bosonic sign action. These maps are the adopted massless-field actions of world-sheet orientation reversal and left-moving spacetime fermion parity, respectively; a full conformal-field-theory state space and its operators are additional data, not constructed here. Pullback $s^{\ast }:\mathcal{V}\to \mathcal{V}$ acts componentwise, and $\mathcal{O}=P_L{\mathrm{\Omega }}_p{s}^{\ast }:\mathcal{V}\to \mathcal{V},\mathrm{\Pi }=\frac{1}{2}(1+\mathcal{O}):\mathcal{V}\to \ker (\mathcal{O}-1)$ is the projection onto the invariant bosonic fields. All three factors commute and square to one, so ${\mathcal{O}}^2=1$ and ${\mathrm{\Pi }}^2=\mathrm{\Pi }$. On purely internal forms $s^{\ast }$ reduces to ${\sigma }^{\ast }$. Use the Hodge-space definitions of the earlier lemma lem-cmp-o3o7-geometric-parity-consequences: if ${\mathcal{A}}^{p,q}(X)$ denotes smooth complex-valued forms of bidegree $(p,q)$, ${\mathcal{A}}^{p,-1}(X)=0$, and $\bar{\mathrm{\partial }}$ is the Dolbeault differential, then $H^{p,q}(X)=\ker (\bar{\mathrm{\partial }}:{\mathcal{A}}^{p,q}\to {\mathcal{A}}^{p,q+1})\mathrm{/}\mathrm{im}(\bar{\mathrm{\partial }}:{\mathcal{A}}^{p,q-1}\to {\mathcal{A}}^{p,q})$; also ${\mathcal{H}}_g^{p,q}(X)=\{\eta \in {\mathcal{A}}^{p,q}(X):{\mathrm{\Delta }}_g\eta =0\}$, where ${\mathrm{\Delta }}_g=d{d}_g^{\ast }+d_g^{\ast }d$ and $d_g^{\ast }$ is the $L_g^2$ formal adjoint of $d$. The internal cohomology $H^{p,q}(X)$ and the spaces of harmonic forms split into the $+1$ and $-1$ eigenspaces of ${\sigma }^{\ast }$; write their dimensions as $h_{+}^{p,q}$ and $h_{-}^{p,q}$. For a basis of $H^{1,1}(X)$ adapted to this splitting, indices $\alpha ,\beta ,\gamma$ denote even classes and $a,b,c$ denote odd classes. The intersection numbers with an odd number of odd indices vanish. Every connected component of $\mathrm{Fix}(\sigma )$ has complex dimension $0$ or $2$; combined with the spacetime-filling Type IIB orientifold convention, these are O3- and O7-plane loci, respectively. This field projection is the bosonic action of the string orientifold prescription, distinct from the geometric quotient $X\mathrm{/}\sigma$: localized source charges, D-branes, background flux, and tadpole constraints are separate physical data. For the massless Type IIB fields, the same O3/O7 projection keeps the metric, dilaton, RR scalar $C_0$, and RR four-form $C_4$ even under ${\sigma }^{\ast }$, while it keeps the NS–NS two-form $B_2$ and RR two-form $C_2$ in the odd eigenspace. Consequently the surviving purely internal harmonic two-form parts of $B_2$ and $C_2$ take values in ${\mathcal{H}}_{g,-}^2(X;\mathbb{R}):=\{\eta \in {\mathcal{A}}^2(X;\mathbb{R}):{\mathrm{\Delta }}_g\eta =0,{\sigma }^{\ast }\eta =-\eta \}$, while the purely internal harmonic four-form part of $C_4$ takes values in ${\mathcal{H}}_{g,+}^4(X;\mathbb{R}):=\{\eta \in {\mathcal{A}}^4(X;\mathbb{R}):{\mathrm{\Delta }}_g\eta =0,{\sigma }^{\ast }\eta =\eta \}$. The odd real harmonic $(1,1)$ sector gives the conventional coefficients $b^a,c^a$, but the broad Calabi–Yau hypotheses do not exclude additional odd real harmonic forms of type $(2,0)+(0,2)$. Keeping only the $(1,1)$ two-form sector is a further truncation unless these additional modes vanish; it does not follow from $\mathrm{\Pi }$. These are the field parities of the adopted orientifold projection, not consequences of the geometric involution alone.

Calabi–Yau threefold data used by string compactifications

Geometric parity consequences of an O3/O7 involution

Grimm and Louis, §3.1

Graña and Triendl, §6.1

${\mathrm{\Omega }}_p$ above is the explicitly specified massless bosonic representation of world-sheet parity, distinct from the holomorphic volume form $\mathrm{\Omega }$. The fixed-locus classification is geometric; the definition leaves all source charges, D-brane content, fluxes, and cancellation conditions to separately declared data.