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The flat complex three-torus O3 orientifold

Physical scope

Geometric illustration of the O3/O7 parity projection on the flat broad-Calabi–Yau three-torus. It is not a strict-SU(3)-holonomy compactification, a sourced ten-dimensional solution, or evidence of tadpole cancellation, flux consistency, or moduli stabilization.

Example

Let $E = C / (Z + i Z)$ , $X = E^{3}$ , and let $σ ([z_{1}, z_{2}, z_{3}]) = [- z_{1}, - z_{2}, - z_{3}]$ . Equip $X$ with $J = \sum_{j = 1}^{3} d x_{j} \land d y_{j}$ and $Ω = d z_{1} \land d z_{2} \land d z_{3}$ . Under the Type IIB O3/O7 quotient convention of def-cmp-type-iib-o3-o7-orientifold-data, this is an O3 projection: $Fix (σ)$ consists of $64$ isolated points, $h_{+}^{1, 1} = 9$ , $h_{-}^{1, 1} = 0$ , $h_{+}^{2, 1} = 0$ , and $h_{-}^{2, 1} = 9$ . Therefore this example has no odd harmonic two-form modes $b^{a}, c^{a}$ in the projected closed-string sector. It illustrates the geometric parity data only; it does not assert a consistent sourced string background or a four-dimensional vacuum.

Example references

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

Flat complex three-torus O3/O7 parity data

Graña, Ksagoni, Ntokos, and Triendl, §6.2

Grimm and Louis, §3.1

Facts & Assumptions

Given: $E = C / (Z + i Z)$ , $X = E^{3}$ , the negation involution $σ$ , its descended flat Kähler form and holomorphic three-form, and the adopted O3/O7 quotient convention.

[F1]

Flat complex three-torus O3/O7 parity data proves that $σ$ has $64$ isolated fixed points and acts on $H^{p, q} (X)$ by $(- 1)^{p + q}$ , with $h^{p, q} = (\binom{3}{p}) (\binom{3}{q})$ .

[F2]

Type IIB Calabi–Yau orientifold data with O3/O7 fixed loci identifies isolated complex fixed loci with O3-plane loci for the stated spacetime-filling Type IIB orientifold convention; source charges and their cancellation are separate inputs.

Proof

technique · substitution into the torus parity calculation and the orientifold fixed-locus convention

Given: The flat torus and involution in the Statement.

1.1F1F2given

The stated $J$ and $Ω$ are the forms from the torus supplier. Its proof verifies that negation is holomorphic and isometric, preserves $J$ , sends $Ω$ to $- Ω$ , and has exactly $4^{3} = 64$ fixed points, all with zero-dimensional fixed components. Applying [F2] classifies those loci as O3-plane loci and gives no O7-plane locus.

2.1F1F2step 1.1algebra∎

Since $p + q$ is even for $(p, q) = (1, 1)$ and odd for $(2, 1)$ , [F1] gives $h_{+}^{1, 1} = (\binom{3}{1}) (\binom{3}{1}) = 9$ , $h_{-}^{1, 1} = 0$ , $h_{+}^{2, 1} = 0$ , and $h_{-}^{2, 1} = (\binom{3}{2}) (\binom{3}{1}) = 9$ . The O3/O7 field projection retains the odd real harmonic two-forms. By [F1], negation acts by $(- 1)^{p + q} = + 1$ on every harmonic bidegree with $p + q = 2$ , including $(2, 0)$ and $(0, 2)$ as well as $(1, 1)$ . Thus the entire odd real harmonic two-form space vanishes, and no internal $B_{2}, C_{2}$ modes, in particular no $b^{a}, c^{a}$ modes, survive. This count is a geometric projection statement, not a tadpole or solution check.

Remarks

The torus has trivial holonomy, so it is an example under the broad Calabi–Yau convention and must not be used to infer the strict- $S U (3)$ -holonomy spectrum. The $64$ fixed points identify geometric O3 loci only; localized charges, D-brane content, flux, tadpole cancellation, and backreaction are not specified.

The flat complex three-torus O3 orientifold

Example

Example references

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources