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Type IIB three-form periods and D3 tadpole on the T6/Z2 cover

Physical scope

A finite GKP-cover charge-accounting example using the adopted O3-minus and free-period conventions. It checks two explicit flux periods and the integrated Maxwell tadpole; it does not establish a global supergravity solution, all-cycle quantization, supersymmetry, stability, or a universal O3 charge law.

Example

Let $X = (R / Z)^{6}$ be the smooth covering six-torus with dimensionless coordinates $x^{i}$ of period one, a constant length scale $R > 0$ , flat metric $d s_{6}^{2} = R^{2} \sum_{i = 1}^{6} (d x^{i})^{2}$ , positive orientation $d x^{1} \land \dots \land d x^{6}$ , complex coordinates $z^{a} = x^{2 a - 1} + i x^{2 a}$ , Kähler form $J = R^{2} \sum_{a = 1}^{3} d x^{2 a - 1} \land d x^{2 a}$ , and holomorphic volume form $Ω = d z^{1} \land d z^{2} \land d z^{3}$ . Let $σ (x^{1}, \dots, x^{6}) = (- x^{1}, \dots, - x^{6})$ modulo one and choose the ordinary O3-minus convention for the quotient $X / ⟨ σ ⟩$ , while doing current and flux integrals on the smooth cover $X$ . Exclude exotic O3 planes, choose the source sector with no localized flux on any O3 plane as required in Schulz §2.1, and take no mobile D3-branes, D7-branes, or other induced localized D3 sources. The absence of localized O3 flux is an explicit model choice, separate from absence of induced D3 charge; no localized discrete flux datum is inferred from a smooth differential form. In this cover convention, the 64 fixed-point terms each have source weight $- 1 / 2$ . Set $F_{3} = 2 (2 π)^{2} α^{'} d x^{1} \land d x^{2} \land d x^{3}, H_{3} = - 16 (2 π)^{2} α^{'} d x^{4} \land d x^{5} \land d x^{6} .$ For the oriented coordinate cycles $Σ_{123} = T_{123}^{3}$ and $Σ_{456} = T_{456}^{3}$ , the normalized cover periods are $2$ and $- 16$ , respectively; both are even, as required here when exotic O3 planes are excluded. The degree-six cover source current is $ρ_{3}^{l o c} = - \frac{1}{2} \sum_{p \in Fix (σ)} δ_{p}$ . On the covering space, the dimensionless Maxwell flux charge is $Q_{f l u x} = 32$ and the localized charge is $Q_{3}^{l o c} = - 32$ , so the integrated D3 tadpole balances. This is a worked period/current ledger conditional on the cited physical inputs; it is not a construction of a solution to the local field equations and does not claim that these two cycle checks establish quantization on the full free lattice.

Example references

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

De Rham currents of compact oriented source cycles

Type IIB Calabi–Yau orientifold flux background conventions

GKP free-period quantization for Type IIB three-form flux

Compact Type IIB D3 tadpole from the sourced five-form Bianchi identity

Giddings–Kachru–Polchinski, §2.1 and §4.1

Graña, §§2.2 and 4.2

Van Riet–Zoccarato, §4.4

Kachru, Schulz, Tripathy, and Trivedi, §2.1

Schulz, §2.1

Facts & Assumptions

Given: The smooth covering torus, its coordinates and orientation, the involution $σ$ , the two flux forms and two cycles in the Example section, the adopted GKP free-period input, the GKP O3-minus source convention, and the conditional integrated tadpole theorem.

[F1]

Type IIB Calabi–Yau orientifold flux background conventions: $α^{'}$ has dimension length squared; the internal integration space, source current, and flux periods use the same declared smooth cover.

[F2]

Type IIB Calabi–Yau orientifold flux background conventions: the chosen smooth cover, orientation, and O3/O7 sector are fixed there; its Definition records the worldsheet projection, involution, and pullback parities, while its Remarks records that the local orientifold carrier is outside this run manifest and lacks current Step 1 readiness. This example checks its concrete involution directly and treats the O3 charge as a separately cited conditional input.

[F3]

GKP free-period quantization for Type IIB three-form flux: on the selected smooth cover, each free integral three-cycle has the adopted normalized $F_{3}$ and $H_{3}$ periods in $Z$ ; here only the two displayed cycles are checked.

[F4]

De Rham currents of compact oriented source cycles: for each point $p$ on a compact oriented six-manifold, $⟨ δ_{p}, f ⟩ = f (p)$ and the finite source sum pairs with $1$ as the sum of its coefficients.

[F5]

Compact Type IIB D3 tadpole from the sourced five-form Bianchi identity: under the adopted current Bianchi identity and matching cover/source normalizations, $Q_{f l u x} + Q_{3}^{l o c} = 0$ .

[A1]

Adopt the no-localized-O3-flux sector of Schulz §2.1, after Eq. (2.3), as in the specialized tadpole theorem. The quotient O3-minus charge input is $- 1 / 4$ per plane in GKP's convention. On the $T^{6}$ cover, use the source density $Q_{O 3} = - 1 / 2$ per fixed point and count 64 fixed points; for the no-exotic-O3 sector, Kachru et al. §2.1 footnote 2 require even normalized cover periods. These are source and lattice inputs, not deductions from the mathematical fixed-point count.

Verification

Given: The data in the Example section, with $α^{'} > 0$ , and the separately adopted O3-minus charge and period conventions in [A1] and [F3].

1.1F1F2givenalgebra

The map $σ$ squares to the identity and preserves the flat metric. Since $z^{a} \mapsto - z^{a}$ , it is holomorphic; its pullback fixes each summand of $J$ and multiplies the three factors of $Ω$ by $(- 1)^{3} = - 1$ . Thus $σ^{*} J = J$ and $σ^{*} Ω = - Ω$ , as required by the orientifold interface recorded under [F2].

1.2F2givenalgebra

A point of $R / Z$ is fixed by $x \mapsto - x$ exactly when $2 x = 0$ modulo one, so each of the six coordinates has two fixed values, $0$ and $1 / 2$ . The fixed set therefore has $2^{6} = 64$ points and no positive-dimensional component; under the selected O3 projection these are isolated O3 loci.

1.3A1F1givenalgebra

Each $d x^{i}$ is closed, and $σ^{*} d x^{i} = - d x^{i}$ , so both displayed degree-three fluxes are closed and odd under $σ$ . They have the required O3 parity. The two constant forms extend smoothly over every fixed point, so their displayed bulk representatives introduce no delta-supported flux term or NS5/D5 magnetic current. The no-localized-O3-flux sector is separately chosen in [A1], not deduced from the de Rham forms; hence this example meets the specialized theorem's new source premise without changing either flux or either period.

1.4A1F3givenalgebra

On $Σ_{123}$ , the coordinate integral of $d x^{1} \land d x^{2} \land d x^{3}$ is one and that of $d x^{4} \land d x^{5} \land d x^{6}$ is zero; on $Σ_{456}$ the corresponding integrals are zero and one. Hence the normalized displayed cover periods are $2$ for $F_{3}$ on $Σ_{123}$ and $- 16$ for $H_{3}$ on $Σ_{456}$ , while the cross-periods vanish. Both integers are even and satisfy the adopted free-period lattice condition [A1, F3]; no all-cycle claim is used.

2.1givenalgebrastep 1.4

Since moving a three-form past a three-form contributes $(- 1)^{9} = - 1$ , $d x^{456} \land d x^{123} = - d x^{123456}$ . The minus sign in $H_{3}$ and this permutation give $\int_{X} H_{3} \land F_{3} = 32 (2 π)^{4} α^{' 2},$ and therefore $Q_{f l u x} = 32$ by the definition in the Statement.

2.2A1F4step 1.2algebra

By [A1] and step 1.2, each of the 64 fixed points contributes $- 1 / 2$ to the cover source density, so $⟨ ρ_{3}^{l o c}, 1 ⟩ = 64 (- 1 / 2) = - 32$ . This uses the mathematical current pairing [F4] and the physical cover-source normalization [A1] separately.

3.1F5step 2.1step 2.2givenalgebra∎

Apply [F5] to these matched cover and source normalizations: $Q_{f l u x} + Q_{3}^{l o c} = 32 - 32 = 0$ . This verifies the integrated charge cancellation in this chosen data set only; it does not imply that the current equation admits a solution obeying the remaining local field equations.

Remarks

The flux integrals and delta-current sum are both over the covering torus and use the cover source density. A quotient calculation would need its own quotient-normalized periods and localized source multiplicities. The only period checks here are on $Σ_{123}$ and $Σ_{456}$ ; the example does not claim those cycles exhaust $H_{3} (X; Z) / t o r s i o n$ . The charge identity is inherited from the conditional A-page theorem. No supersymmetry, stability, warp factor, five-form profile, local sourced solution, or global string compactification is asserted.

Type IIB three-form periods and D3 tadpole on the T6/Z2 cover

Example

Example references

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources