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Ordinary periods, differential refinement, and RR K-theory are distinct layers

Physical scope

Conceptual distinction among free flux periods, differential-cohomology data, and the source-motivated Type II K-theory charge model. The Type IIB GKP period postulate and RR/K-theory identification are source-scoped physical inputs, not consequences of the massive IIA quantization postulate or of mathematical definitions.

Statement

Here $F_{3}$ and $H_{3}$ are the Type IIB Ramond–Ramond and Neveu–Schwarz three-form fluxes. In the classical GKP Type IIB model, adopt the source's normalization $(2 π α^{'})^{- 1} \int_{Γ_{3}} F_{3} \in 2 π Z$ and $(2 π α^{'})^{- 1} \int_{Γ_{3}} H_{3} \in 2 π Z$ on free integral three-cycles; equivalently, each period divided by $(2 π)^{2} α^{'}$ is an integer. This statement is Type IIB-specific and does not follow from the massive Type IIA free-period postulate Massive IIA Romans-mass and NS three-form flux quantization. Smooth differential forms determine real cohomology classes and cannot detect integral torsion. An ordinary differential character adds curvature, an integral characteristic class, and holonomy; twisted differential $K$ -theory, obtained from the twist definition with $E = K U$ (complex K-theory), adds the specified twist data. These constructions are distinct mathematical objects. Identifying Type II RR charges with $K$ -theory is a source-motivated physical model, not a theorem of the mathematical definitions or a consequence of the free-period postulate.

Facts & Assumptions

Given: A smooth manifold $M$ , the stated source-scoped GKP Type IIB input, and the current mathematical suppliers.

[A1]

The source-scoped Type IIB input GKP free-period quantization for Type IIB three-form flux records Giddings–Kachru–Polchinski, hep-th/0105097v2, equation (2.25) and §3.1, equations (3.4)–(3.6): the classical Type IIB fluxes satisfy $(2 π α^{'})^{- 1} \int F_{3} \in 2 π Z$ and $(2 π α^{'})^{- 1} \int H_{3} \in 2 π Z$ on free integral three-cycles. This is an adopted Type IIB model input in that paper's normalization, not the separate Type IIA postulate Massive IIA Romans-mass and NS three-form flux quantization, which assumes constant $F_{0}$ , smooth closed internal $H_{3}$ , no NS5 magnetic current, and free periods in the Graña normalization.

[F1]

Cheeger–Simons differential character: a degree- $k$ differential character has a unique closed integral-period curvature $R (h)$ , an integral characteristic class $I (h)$ , and cycle holonomy $h$ .

[F2]

Exact sequences for ordinary differential cohomology: curvature-zero characters form $H^{k - 1} (M; R / Z)$ , and characters with a fixed characteristic class can still differ by differential-form data.

[F3]

Differential twist and twisted differential K-theory: a differential twist includes an invertible topological $E$ -module, a differential-form module, and a real comparison equivalence; a gerbe presentation alone does not supply that comparison.

[A2]

Minasian–Moore, hep-th/9710230v3, equation (1.1): the paper proposes a K-theoretic RR charge expression involving a K-theory pushforward and $\hat{A}$ -class. It is evidence for that physical proposal, not proof of a universal charge classification.

[F4]

Spin and Spin-c obstruction and torsor classification: for an oriented bundle of positive rank over the stated smooth-manifold base, a Spin-c lift exists exactly when $W_{3} = β_{2} w_{2} = 0$ , under full AC; this criterion is proved by the Spin and determinant-reduction suppliers.

[F5]

The de Rham theorem: the de Rham comparison identifies smooth closed forms modulo exact forms with real singular cohomology, by integration.

Proof

1.1F1F2F5givenalgebra

If $c \in H^{k} (M; Z)$ is torsion, choose $N > 0$ with $N c = 0$ . Its image $c_{R} \in H^{k} (M; R)$ satisfies $N c_{R} = 0$ . Since a real vector space has no nonzero element annihilated by a positive integer, $c_{R} = 0$ . By the de Rham comparison in The de Rham theorem, smooth closed forms represent real cohomology, so their periods cannot distinguish $c$ from zero. The exact sequence [F2] also places flat characters in the kernel of curvature, showing that curvature forms alone omit holonomy information.

1.2A2F1F3given

The definition [F3] makes the topological twist, differential-form module, and comparison equivalence part of the mathematical input; its homotopy-pullback construction then defines twisted differential cohomology. It supplies no identification with physical RR fields. The K-theory expression in [A2] is therefore a distinct source-motivated physical proposal, not an inference from [F1]–[F3].

2.1A1F4given∎

The GKP integer-period statement [A1] is scoped to its classical Type IIB model, whereas Massive IIA Romans-mass and NS three-form flux quantization is an adopted massive Type IIA input; neither proves a universal ordinary-cohomology classification of RR flux or charge. If the trivial- $B$ Spin-c criterion is invoked for a positive-rank normal bundle in the Freed–Witten setup on the stated smooth base, [F4] identifies $W_{3} = 0$ with existence of a Spin-c lift under full AC. For a zero-rank normal bundle, the rank conventions give $W_{3} = 0$ , and ${Spin}^{c} (0) = ({\pm 1} \times U (1)) / ⟨ (- 1, - 1) ⟩ ≅ U (1)$ has its trivial principal lift over the trivial $S O (0)$ frame bundle; the negative-square zero-dimensional Clifford norm gives $Spin (0) = {\pm 1}$ . These definitions and rank conventions are supplied by Given Spin and Spin-c structures and their spinor bundles, Normalized Pin products and Clifford norm and Stiefel–Whitney classes from the projective-bundle relation. This mathematical identification does not prove the physical anomaly or a universal RR charge classification.

Source references

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

Massive IIA Romans-mass and NS three-form flux quantization

Cheeger–Simons differential character

Differential twist and twisted differential K-theory

Exact sequences for ordinary differential cohomology

Spin and Spin-c obstruction and torsor classification

Ordinary periods, differential refinement, and RR K-theory are distinct layers

Statement

Facts & Assumptions

Proof

Source references

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Sources