Genus-one torsors and discrete gauge data

Assume Full AC. Let $B_3$ be a connected nonsingular compact algebraic threefold over $\mathbb{C}$, let $Y_4$ be a smooth compact projective Calabi–Yau fourfold with holomorphically trivial canonical bundle, and let $\pi :Y_4\to B_3$ be a proper, flat, surjective genus-one fibration in the stated genus-one-fibration convention; no section is required. Choose a dense Zariski-open subset $B_3^{\circ }\subseteq B_3$ such that the restricted morphism ${\pi }^{\circ }:Y_4^{\circ }:={\pi }^{-1}(B_3^{\circ })\to B_3^{\circ }$ is smooth. Assume that a smooth commutative group scheme $J^{\circ }\to B_3^{\circ }$ of relative dimension one, with identity section, is specified. The restriction $Y_4^{\circ }\to B_3^{\circ }$ is a right $J^{\circ }$-torsor only when a right action is supplied and it is étale-locally equivariantly isomorphic to $J^{\circ }$, in the stated étale-torsor convention. A degree-$k$ multisection means a subvariety of $Y_4$ whose map to $B_3$ is generically finite of degree $k>0$.

Separately adopt the source-specific geometric, long-wavelength F-theory regime for this no-section model, in which ${\alpha }^{\mathrm{\prime }}$ has dimensions of length squared and string-loop and ${\alpha }^{\mathrm{\prime }}$ corrections are controlled or explicitly retained. The elliptic fiber is auxiliary geometric data, not two additional ordinary spacetime dimensions. For the source-selected flat genus-one models without a section, adopt the source's physical dictionary that the genus-one fibration and its Jacobian have common axiodilaton and $SL(2,\mathbb{Z})$ monodromy data. Assume separately the source-specific M/F limit: the genus-one fiber area tends to zero while the dual circle decompactifies, with the source's scaling hypotheses. If the model's localized states and its torsor or multisection data are checked, these extra sectors may be interpreted as a discrete gauge sector. The discrete group and the presence of the requisite states are model-specific outputs; absence of a section alone does not determine them.

This is an adopted physical dictionary, conditional on the stated geometry and limit. It does not assert a mathematical construction of the Jacobian for every Calabi–Yau fourfold, an equivalence of all M-theory and F-theory models, or a global compact vacuum.

Flat genus-one fibration

Torsors under a specified relative group scheme on the small étale site

The Axiom of Choice

M-theory/F-theory elliptic-fiber limit

Type-IIB axiodilaton and elliptic-fiber dictionary

Braun and Morrison, §1

Braun and Morrison, §§4.3–4.4

Weigand, §§8.1–8.2

The source's Tate–Shafarevich group is not identified here with the locally defined ${\mathrm{Sha}}_{\mathrm{e}\mathrm{t}}(J^{\circ }\mathrm{/}{B}_3^{\circ })$ of Torsors under a specified relative group scheme on the small étale site. That definition requires $J^{\circ }$ as data, defines torsors as sheaves on the small étale site, and separates representability and extension across $B_3\setminus B_3^{\circ }$; the source's $H^1(B,A)$ description has its own smoothness, irreducible-fiber, and regular-section hypotheses. No comparison theorem between these interfaces is assumed. Full AC is used only for the mathematical class group, as recorded in the axiom-strength field.

The current Type-IIB axiodilaton and elliptic-fiber dictionary carrier adopts the dictionary for an elliptically fibered Calabi–Yau fourfold with a regular section and a Weierstrass patch; it does not by itself supply the same statement for a genus-one fibration without section. The common axiodilaton and monodromy statement for the source-selected no-section models is separately adopted from the cited source above. Likewise, the M/F input for this no-section case is the source's separate genus-one limit assumption. The current M-theory/F-theory elliptic-fiber limit carrier adopts the limit only for a fixed smooth compact resolved flat elliptic Calabi–Yau fourfold with a section. Its current Step 1 readiness record is missing, and it does not supply the no-section extension. The cited Braun–Morrison paper says its Calabi–Yau fourfold Jacobian assertion was not mathematically proved; that limit and Jacobian extension remain conditional here.

The action in the diagram is supplied torsor data; the square only records that it is an action over the base and is not a proof of the physical interpretation.