Type IIB Calabi–Yau orientifold flux background conventions

Use natural units $\mathrm{\hslash }=c=1$ and ${\alpha }^{\mathrm{\prime }}>0$ with dimension length squared. Let $M^{3,1}={\mathbb{R}}^4$ with global coordinates $x^{\mu }$, $\mu =0,\dots ,3$, and metric ${\eta }_{\mu \nu }=\mathrm{diag}(-,+,+,+)$; set $M^{10}=M^{3,1}\times X$. Let $X$ be a compact, oriented, smooth six-manifold without boundary, explicitly identified in each use as the orientifold covering space or as a smooth quotient. If the chosen quotient has singular fixed loci, perform current and period integrals on the smooth cover, or supply an orbifold-current framework before integrating on the quotient. Let $S\subseteq X$ be the union of localized source supports. Classical fields and metrics are assumed smooth on $M^{3,1}\times (X\setminus S)$; no smoothness through $S$ is assumed. Localized equations across $S$ are stated in a current or weak formulation with each term's extension specified. The broad Calabi–Yau convention is the one in the earlier mathematical definition: connected, compact, Kähler, and with holomorphically trivial canonical bundle; exact $SU(3)$ holonomy is not included unless separately assumed. Orient $M^{10}$ by $-{\mathrm{v}\mathrm{o}\mathrm{l}}_4\wedge {\mathrm{v}\mathrm{o}\mathrm{l}}_X$, with ${\mathrm{v}\mathrm{o}\mathrm{l}}_4=d{x}^0\wedge d{x}^1\wedge d{x}^2\wedge d{x}^3$. This choice is part of the GKP-to-R4 dictionary: the Lorentzian Hodge operator and every ten-dimensional form integral use this same orientation. The internal integration orientation is unchanged.

The precise O3/O7 sector datum is a worldsheet-parity quotient by $O=(-1{)}^{F_L}{\mathrm{\Omega }}_p{\sigma }^{\ast }$ on a smooth connected compact Calabi–Yau threefold $X$. Here $I$ is its complex structure, $g(u,v)=J(u,Iv)$ its Kähler metric, ${\mathrm{\Omega }}_p$ the worldsheet-parity operator that reverses worldsheet orientation and exchanges left- and right-moving sectors, $F_L$ the left-moving spacetime fermion number, and ${\sigma }^{\ast }$ pullback by a holomorphic isometric involution $\sigma :X\to X$ satisfying ${\sigma }^2=\mathrm{id}$, ${\sigma }^{\ast }J=J$, and ${\sigma }^{\ast }\mathrm{\Omega }=-\mathrm{\Omega }$. The broad Calabi–Yau and chosen $(J,\mathrm{\Omega })$ data are those of Type IIB Calabi–Yau orientifold data with O3/O7 fixed loci and Compact Calabi–Yau manifold conventions. The orientifold-data definition uses ${\alpha }^{\mathrm{\prime }}=1$ and dimensionless $J$; when restoring units as here, take $J_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}={\alpha }^{\mathrm{\prime }}J$, which leaves the pullback parity unchanged. For smooth complex-valued forms, $d$ is the exterior derivative defined by $d\omega (Y_0,\dots ,Y_r)={\sum }_{j=0}^r(-1{)}^j{Y}_j(\omega (Y_0,\dots ,{\widehat{Y}}_j,\dots ,Y_r))+{\sum }_{0\le i<j\le r}(-1{)}^{i+j}\omega ([Y_i,Y_j],Y_0,\dots ,{\widehat{Y}}_i,\dots ,{\widehat{Y}}_j,\dots ,Y_r)$ for every smooth $r$-form $\omega$ and smooth vector fields $Y_0,\dots ,Y_r$. For $0\le p,q\le 3$, let ${\mathcal{A}}^{p,q}(X)={\mathrm{\Gamma }}^{\mathrm{\infty }}(X,{\mathrm{\Lambda }}^p{T}^{\ast 1,0}X\otimes {\mathrm{\Lambda }}^q{T}^{\ast 0,1}X)$ be the space of smooth complex-valued forms of bidegree $(p,q)$, and let $\bar{\mathrm{\partial }}:{\mathcal{A}}^{p,q}\to {\mathcal{A}}^{p,q+1}$ be the $(0,1)$ part of $d$. Define $H^{p,q}(X)=\frac{\ker (\bar{\mathrm{\partial }}:{\mathcal{A}}^{p,q}\to {\mathcal{A}}^{p,q+1})}{\mathrm{im}(\bar{\mathrm{\partial }}:{\mathcal{A}}^{p,q-1}\to {\mathcal{A}}^{p,q})},{\mathcal{A}}^{p,-1}=0.$ On real $r$-forms, the pointwise inner product induced by $g$ is specified on decomposable covectors by $⟨{\theta }_1\wedge \cdots \wedge {\theta }_r,{\eta }_1\wedge \cdots \wedge {\eta }_r{⟩}_g=\det (g^{-1}({\theta }_i,{\eta }_j))$ and extended bilinearly. Write ${\mathrm{v}\mathrm{o}\mathrm{l}}_g=J\wedge J\wedge J\mathrm{/}3!$ for the oriented Riemannian volume form. Let ${\ast }_g$ be the real Hodge star determined by $u\wedge {\ast }_{g}v=⟨u,v{⟩}_g{\mathrm{v}\mathrm{o}\mathrm{l}}_g$, then extend it complex-linearly. The $L^2$ Hermitian product on equal-degree complex forms is $(\alpha ,\beta {)}_{L^2}={\int }_X\alpha \wedge {\ast }_g\bar{\beta }$. For adjacent degrees, define the formal adjoint $d_g^{\ast }$ by $(d\alpha ,\beta {)}_{L^2}=(\alpha ,d_g^{\ast }\beta {)}_{L^2}$ for all smooth $\alpha ,\beta$; $X$ is compact and boundaryless. Set ${\mathrm{\Delta }}_g=d{d}_g^{\ast }+d_g^{\ast }d$ and ${\mathcal{H}}_g^{p,q}(X)=\{\eta \in {\mathcal{A}}^{p,q}(X):{\mathrm{\Delta }}_g\eta =0\}$. The exact geometric parity result Geometric parity consequences of an O3/O7 involution establishes their ${\sigma }^{\ast }$-eigenspace decompositions and that every fixed component has complex dimension zero or two, giving space-filling O3- and/or O7-plane loci. This string quotient is not merely the smooth geometric quotient $X\mathrm{/}\sigma$: its fixed-locus source charge, tadpole conditions, required branes, and any background flux are separate physical data.

Use ten-dimensional coordinates $x^M$, $M=0,\dots ,9$, four-dimensional coordinates $x^{\mu }$, $\mu =0,\dots ,3$, and internal coordinates $y^m$, $m=1,\dots ,6$. The ten-dimensional Lorentzian metric has signature $(-,+,\dots ,+)$. Write the string-frame metric as $g_{MN}^{(s)}$ and the Einstein metric as $g_{MN}^{(E)}=e^{-\varphi \mathrm{/}2}{g}_{MN}^{(s)}$ on the same spacetime. The dilaton $\varphi$ and axion $C_0$ are smooth dimensionless real scalars on the source-free region and define $\tau =C_0+i{e}^{-\varphi }$; the local, dimensionless string coupling is $g_{\mathrm{s}\mathrm{t}\mathrm{r}}=e^{\varphi }=1\mathrm{/}\mathrm{Im}\tau$ and is constant only in the constant-dilaton specialization.

Let $F_3$ be the raw real RR three-form field strength and $H_3$ the real NS three-form field strength, smooth on $X\setminus S$. On a local potential patch use $F_3=d{C}_2$ and $H_3=d{B}_2$. Define the real tilded form ${\tilde{F}}_3=F_3-C_0{H}_3$ and the complex flux $G_3=F_3-\tau H_3$; they are different forms. For a four-dimensional Poincaré-invariant warped ansatz, take $A:X\setminus S\to \mathbb{R}$ smooth and $\tilde{g}$ a smooth Riemannian metric on $X\setminus S$, and use $d{s}_E^2=e^{2A(y)}{\eta }_{\mu \nu }d{x}^{\mu }d{x}^{\nu }+e^{-2A(y)}{\tilde{g}}_{mn}(y)d{y}^{m}d{y}^n,$ where $A$ is a dimensionless real warp function and $\tilde{g}$ is the chosen internal Riemannian metric with dimensionless components in these length-valued coordinates. On each local potential patch, write $F_5:=d{C}_4$ and define ${\tilde{F}}_5:=F_5-\frac{1}{2}{C}_2\wedge H_3+\frac{1}{2}{B}_2\wedge F_3\mathrm{.}$ This is the GKP convention of §2.1, Eq. (2.2). It equals ${\tilde{G}}_5$ of Type IIB Two Derivative Supergravity Pseudoaction because that supplier sets $G_5=d{C}_4$, $G_3=F_3$, and degree-two forms commute with degree-three forms under the wedge product. The five-form ansatz is ${\tilde{F}}_5=(1+{\star }_{10,E})\text{\hspace{0.17em}⁣}[d\alpha (y)\wedge d{x}^0\wedge d{x}^1\wedge d{x}^2\wedge d{x}^3],$ where $\alpha :X\setminus S\to \mathbb{R}$ is a smooth dimensionless scalar profile and ${\star }_{10,E}$ is defined by $u\wedge {\star }_{10,E}v=⟨u,v{⟩}_{g^{(E)}}{\mathrm{v}\mathrm{o}\mathrm{l}}_{10,E}$ for forms of the same degree, using the stated orientation. For $g^{\mathrm{\prime }}=e^{2\omega }g$ on an $n$-manifold, the Hodge star on $p$-forms scales as ${\ast }_{g^{\mathrm{\prime }}}=e^{(n-2p)\omega }{\ast }_g$. Here $g^{(E)}=e^{2\omega }{g}^{(s)}$ with $\omega =-\varphi \mathrm{/}4$, and $n=10,p=5$, so ${\star }_{10,E}={\star }_{10,s}$ on five-forms. The Einstein-frame self-duality in this ansatz is therefore the same constraint as the separate string-frame postulate Type IIB Five Form Self Duality Constraint, imposed after varying the undoubled pseudoaction. Let $\tilde{\ast }$ be the ordinary wedge-defined Hodge star of the oriented unwarped internal metric $\tilde{g}$. Distinguish from it the transverse component convention used in GKP's flux constraint: on three-forms define ${\mathsf{S}}_6=-\tilde{\ast }$. This explicit dictionary realizes the simultaneous signs of GKP equations (2.23), (2.29), and (2.31); Kachru–Schulz–Trivedi, §2.1 equation (2.9), footnote 3, explicitly records the GKP real-coordinate convention $H_3\wedge {\star }_6{H}_3=-H_{mnp}{H}^{mnp}\mathrm{V}\mathrm{o}\mathrm{l}\mathrm{/}3!$, confirming this negative-wedge transverse convention. The ten-dimensional orientation/current realization above is the explicit translation into the separate arbitrary-oriented R4 wedge formulation. The name ISD in this GKP constraint convention means ${\mathsf{S}}_6{G}_3=i{G}_3$, equivalently $\tilde{\ast }{G}_3=-i{G}_3$. It must not be confused with $\tilde{\ast }{G}_3=i{G}_3$ in the fixed internal orientation. The mathematical star ${\ast }_g$ used above in the geometric and $L^2$ definitions retains its original wedge definition. For the present ten-dimensional orientation the ansatz gives ${\tilde{F}}_5^{\mathrm{i}\mathrm{n}\mathrm{t}}=e^{-8A}\tilde{\ast }d\alpha \mathrm{.}$ Indeed the five-form ${\mathrm{v}\mathrm{o}\mathrm{l}}_4\wedge d\alpha$ has negative squared Lorentzian norm; its dual is $-e^{-8A}\tilde{\ast }d\alpha$ for the product orientation and therefore the displayed positive form for the selected opposite orientation. A space-filling D3 worldvolume is oriented by ${\mathrm{v}\mathrm{o}\mathrm{l}}_4$; its ten-dimensional WZ current is represented internally by minus the positive point current on $X$. This is the current sign needed when comparing the R4 and GKP sourced equations. No pointwise smoothness through $S$ is assumed.

Fix $2{\kappa }_0^2=(2\pi {)}^7{\alpha }^{\mathrm{\prime }4}$ and ${\mu }_3=(2\pi {)}^{-3}{\alpha }^{\mathrm{\prime }-2}$, so $2{\kappa }_0^2{\mu }_3=(2\pi {)}^4{\alpha }^{\mathrm{\prime }2}$ has dimension ${\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}^4$. The dimensionless signed source current pairs with $1$ to give the dimensionless localized charge; that pairing does not change the coefficient's dimension. With length-valued coordinates and the Type IIB field-strength components $[F_3]=[H_3]={\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}^{-1}$ of Supergravity Conventions And Normalizations, the integrated term ${\int }_X{H}_3\wedge F_3$ has dimension ${\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}^4$; division by $(2\pi {)}^4{\alpha }^{\mathrm{\prime }2}$ gives the dimensionless Maxwell flux charge. Here $[{\kappa }_0^2]={\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}^8$ and $[{\mu }_3]={\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}^{-4}$. In the Einstein-frame RR convention ${\kappa }_E={\kappa }_0$; at constant dilaton the string-frame gravitational coupling is $2{\kappa }_{10,s}^2=g_{\mathrm{s}\mathrm{t}\mathrm{r}}^{2}2{\kappa }_0^2$. Every period, Hodge star, localized source current, D3 charge sum, and cover/quotient multiplicity in an application must use one declared $X$ and this same normalization. Localized equations are current-valued and are not applied as smooth equations through their source support.

Geometric parity consequences of an O3/O7 involution

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

Compact Calabi–Yau manifold conventions

Kähler class, canonical bundle, and Ricci form

Supergravity Conventions And Normalizations

Type IIB Five Form Self Duality Constraint

Type IIB Two Derivative Supergravity Pseudoaction

Giddings–Kachru–Polchinski, §2.1 and §4.1

Graña, §2.2 and §4.2

Van Riet–Zoccarato, §§4.1 and 4.4

Kachru, Schulz, and Trivedi, §2.1, equation (2.9), footnote 3

The exact orientifold hypotheses are: $X$ is a smooth connected compact Calabi–Yau threefold with chosen Kähler form $J$ and holomorphic volume form $\mathrm{\Omega }$; $\sigma$ is an order-two holomorphic isometry with ${\sigma }^{\ast }J=J$ and ${\sigma }^{\ast }\mathrm{\Omega }=-\mathrm{\Omega }$; and all localized fixed-locus sources, D-branes, fluxes, and tadpole constraints are declared separately. The local orientifold-data carrier and its full geometric-parity proof were inspected and match this interface. Its generic Calabi–Yau dependency is current-ready, but the orientifold carrier is absent from the active run manifest and has no current Step 1 record; its linked parity lemma also has no active-run readiness record. This definition preserves that interface conditionally and does not copy or replace either supplier.

The current pseudoaction and five-form self-duality carriers are present, but their current Step 1 decisions are escalated. The stated frames, flux names, and units are exact convention inputs; the definition is not a claim that those upstream items have completed authoring readiness or that a solution exists.