F4 Singular Wrapped M2 Sector Realization Data

Assume full AC, inherited through the even-dimensional spinor and combined Euclidean characteristic-class prerequisites. This definition specifies required physical realization data and makes no existence assertion.

Fix specified F4 model data over a smooth integral complex algebraic threefold $B$, a specified divisor $D\subset B$, a matter curve $C\subset D$, and a selected matter surface ${\mathcal{S}}_R$, with the meanings supplied by \texttt{def-f4-crepant-resolution-and-matter-surface}. The curve $C$ is reduced, irreducible and closed; the geometric generic fibres of the resolved projection over $C$ and $D$ are nonisomorphic after base change along any chosen pair of embeddings of their residue-field algebraic closures into a common algebraically closed field. The resolution $\rho :\widehat{Y}\to Y_W$ is specified proper, birational and crepant, and the resolution chamber is part of the data. Write $\widehat{\pi }:={\pi }_W\circ \rho$, where ${\pi }_W:Y_W\to B$ is the Weierstrass projection. The surface ${\mathcal{S}}_R$ is a specified reduced irreducible closed surface with $\widehat{\pi }({\mathcal{S}}_R)\subseteq C$; its map $q_R:=\widehat{\pi }{\mathrm{\mid }}_{{\mathcal{S}}_R}:{\mathcal{S}}_R\to C$ is proper and flat with nonempty pure one-dimensional geometric fibres and selected geometric generic fibre isomorphic to ${\mathbb{P}}^1$ and belonging to the specified collection $\mathcal{C}$ of resolution-chamber curve components. These are supplied geometric data; this definition asserts no existence or uniqueness of a model, resolution, curve, or surface.

Let $\mathrm{\Delta }\subset C$ be a specified finite set containing every singular point of $C$ and every specified degeneration point. As part of the supplied data, let $\nu :\tilde{C}\to C$ be the normalization, meaning a finite, surjective birational morphism from a normal integral complex curve; require $\nu$ to be an isomorphism over $C^{\circ }:=C\setminus \mathrm{\Delta }$. For each $p\in \mathrm{\Delta }$, the finite set ${\nu }^{-1}(p)$ labels the local analytic branches over $p$. Put ${\tilde{C}}^{\circ }:={\nu }^{-1}(C^{\circ })$, name each connected component $U_{\alpha }$ of ${\tilde{C}}^{\circ }$, and record its image $V_{\alpha }=\nu (U_{\alpha })\subseteq C^{\circ }$; on these sets the normalization map is an isomorphism. This explicitly identifies the smooth branch components used in the comparison and the branch labels over every point of $\mathrm{\Delta }$. No compactness of $C$ or $\tilde{C}$ is implicit unless it is separately supplied.

Write the grading of the supplied regulated Hilbert space as ${\mathcal{H}}_{\epsilon }={\mathcal{H}}_{\epsilon ,+}\oplus {\mathcal{H}}_{\epsilon ,-}$, with grading operator ${\mathrm{\Gamma }}_{\epsilon }$ equal to $+1$ and $-1$ on the two summands. Require ${\mathrm{\Gamma }}_{\epsilon }\mathrm{Dom}(Q_{\epsilon })=\mathrm{Dom}(Q_{\epsilon })$ and $Q_{\epsilon }{\mathrm{\Gamma }}_{\epsilon }=-{\mathrm{\Gamma }}_{\epsilon }{Q}_{\epsilon }$ there. The chiral component is

When a protected index is claimed, require this closed component to be Fredholm on its graph-norm domain: its range is closed, its kernel and algebraic cokernel are finite-dimensional, and

This fixes the sign as even minus odd. The phrase “protected index” below means this Fredholm index only; any regulator limit is the limit of these specified indices under the supplied regulator-removal identifications.

A singular wrapped-M2 sector realization data package is the following actual data, together with the verification maps and estimates listed below. It is not a claim that these objects exist.

1. Regulated protected sector. Use $c=\mathrm{\hslash }=1$ and the oriented M2 charge and Wess–Zumino sign fixed by the classical worldvolume-action supplier def-m2-brane-worldvolume-action. That supplier is a classical action on a smooth closed oriented timelike three-manifold with nondegenerate Lorentzian induced metric; it supplies no quantization or singular-membrane theory. For the actual background and dimensionless regulator label $\epsilon$, supply the ghost-graded completed BRST space, closed differential, its invariant domains, and physical ghost number $g_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ of def-m2-regulated-protected-sector-data. Use its fixed-ghost-number Hilbert quotient $${\mathcal{H}}_{\epsilon }=\ker d_{\epsilon ,g_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}}\mathrm{/}\stackrel{‾}{\mathrm{ran}{d}_{\epsilon ,g_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}-1}},$$ with the quotient equivalence and norm specified there; the denominator is closed in the numerator. The supplied self-adjoint unitary grading preserves each ghost sector and the BRST domain and anticommutes with the differential, so it induces ${\mathcal{H}}_{\epsilon }={\mathcal{H}}_{\epsilon ,+}\oplus {\mathcal{H}}_{\epsilon ,-}$. Adopt exactly the supplier's positively oriented Lorentz-frame convention $\eta =(-,+,+,+)$ and ${\gamma }_5=i{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3$: the positive summand is right-handed and the negative summand left-handed. The index sign fixed above is consequently right-handed minus left-handed protected zero modes when these spaces are finite. Identifying this grading with the physical four-dimensional chirality is supplied realization data, not a consequence of a formal quotient. Supply a densely defined closed quantum charge $Q_{\epsilon }$, its exact Hilbert-adjoint domain, and its actual adjoint. Require $Q_{\epsilon }=Q_{\epsilon }^{\ast }$ on exactly that domain (self-adjointness); require the grading to preserve the domain and anticommute with $Q_{\epsilon }$ (oddness). Supply the normalization and square of this charge as required by item 2 of def-m2-regulated-protected-sector-data, with $\mathrm{Dom}(Q_{\epsilon }^2)=\{u\in \mathrm{Dom}(Q_{\epsilon }):Q_{\epsilon }u\in \mathrm{Dom}(Q_{\epsilon })\}$. Supply a self-adjoint Hamiltonian $H_{\epsilon }:\mathrm{Dom}(H_{\epsilon })\subseteq {\mathcal{H}}_{\epsilon }\to {\mathcal{H}}_{\epsilon }$ and its unitary evolution $U_{\epsilon }(t)=e^{-it{H}_{\epsilon }}$ for every $t\in \mathbb{R}$. Require $U_{\epsilon }(t)$ to preserve the physical quotient and its grading, to map $\mathrm{Dom}(Q_{\epsilon })$ onto itself, and to satisfy $Q_{\epsilon }{U}_{\epsilon }(t)u=U_{\epsilon }(t)Q_{\epsilon }u$ for every $u\in \mathrm{Dom}(Q_{\epsilon })$. Record the preserved symmetry algebra and its exact operator relations, including all domains, energy/central-charge normalizations and units used to relate $Q_{\epsilon }^2$ to the model's energy and central charges; no universal identity $Q_{\epsilon }^2=H_{\epsilon }$ is imposed. Here $H_{\epsilon }$ has units of energy and $t$ of inverse energy in the adopted $\mathrm{\hslash }=1$ convention. For an operator $A$, its graph norm on its domain is $\mathrm{\parallel }u{\mathrm{\parallel }}_A=(\mathrm{\parallel }u{\mathrm{\parallel }}^2+\mathrm{\parallel }Au{\mathrm{\parallel }}^2{)}^{1\mathrm{/}2}$. A Fredholm comparison means a bounded map between the stated graph Hilbert domain and Hilbert target with closed range and finite-dimensional kernel and cokernel. Supply regulator-removal identifications and convergence of the relevant closed operators or orthogonal projections onto specified spectral subspaces, and prove that the protected-index limit exists with all states entering or leaving zero accounted for.
2. Smooth-branch comparison. Over each named component $U_{\alpha }$ of ${\tilde{C}}^{\circ }$, supply the actual smooth wrapped-fibre state bundle, its graded Hilbert fibres and closed operators, and an explicit grading-preserving intertwiner or graph-Fredholm comparison with the regulated sector over $V_{\alpha }$. For this reduction supply the finite-dimensional smooth parameter base $B_{\mathrm{r}\mathrm{e}\mathrm{d}}$, smooth graded Hilbert bundle $\mathcal{H}\to B_{\mathrm{r}\mathrm{e}\mathrm{d}}$ with its quotient fibres, unitary grading-preserving ambient connection ${\mathrm{\nabla }}^{\mathcal{H}}$, and smooth operator-norm family of orthogonal projections $P_b$ of locally constant finite rank required by def-m2-regulated-protected-sector-data. Each $P_b$ commutes with the physical grading and preserves every operator domain used in the reduction. Supply a smooth map $r_{\alpha }:U_{\alpha }\to B_{\mathrm{r}\mathrm{e}\mathrm{d}}$ and a grading-preserving Hermitian identification of the actual light state bundle with $r_{\alpha }^{\ast }\mathrm{im}P$; its Berry connection is exactly $r_{\alpha }^{\ast }(P{\mathrm{\nabla }}^{\mathcal{H}})$, where $(P{\mathrm{\nabla }}^{\mathcal{H}})s=P{\mathrm{\nabla }}^{\mathcal{H}}s$ for a light section. Give the fibre metrics, domains and norm estimates. No connection or projection family is inferred merely from the expected light-state rank. Equality of expected ranks alone is not a comparison map. The exact adopted smooth F4 index context is the postulate \texttt{post-f4-m2-matter-curve-zero-mode-dictionary}: it assumes a smooth compact resolved flat sectioned elliptic Calabi--Yau fourfold, with holomorphically trivial canonical bundle under the broad convention used by that postulate, a selected smooth compact matter surface $S_R^{\mathrm{s}\mathrm{m}}$ embedded by $i:S_R^{\mathrm{s}\mathrm{m}}\hookrightarrow {\widehat{Y}}_4$, and a proper smooth submersion $q:S_R^{\mathrm{s}\mathrm{m}}\to C_R$ with closed oriented real two-dimensional fibres equal to the chosen rational curve, over a smooth compact matter curve $C_R$, algebraic gauge-background flux, the oriented fibre-integrated real class $\alpha =q_{\ast }(i^{\ast }[g_4])\in H_{\mathrm{d}\mathrm{R}}^2(C_R;\mathbb{R})$, a supplied integral lift $c\in H^2(C_R;\mathbb{Z})$ of $\alpha$, a supplied holomorphic line bundle ${\mathcal{L}}_R$ with $c_1({\mathcal{L}}_R)=c$, and a supplied spin line ${\mathcal{S}}_R^{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}}$ with $({\mathcal{S}}_R^{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}}{)}^{\otimes 2}\cong K_{C_R}$. The source postulate denotes the smooth matter surface and spin line by $S_R$ and ${\mathcal{S}}_R$, respectively, and its gauge line bundle by ${\mathcal{L}}_R$; the superscripts here distinguish the smooth surface and spin line from the singular-model surface ${\mathcal{S}}_R$ fixed above. Under those adopted inputs it states $$\mathrm{deg}({\mathcal{L}}_R)={\int }_{S_R^{\mathrm{s}\mathrm{m}}}{g}_4,{\chi }_R=h^0(C_R,{\mathcal{L}}_R\otimes {\mathcal{S}}_R^{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}})-h^1(C_R,{\mathcal{L}}_R\otimes {\mathcal{S}}_R^{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}})\mathrm{.}$$ Here $h^q(C_R,\mathcal{E}):={\dim }_{\mathbb{C}}{H}^q(C_R,\mathcal{E})$ when finite. The separate conditional flux convention in \texttt{post-f4-mtheory-g4-shifted-quantization} uses the normalized real cohomology class $g_4=[G_{4,\mathrm{C}\mathrm{J}\mathrm{S}}]\mathrm{/}((2\pi {)}^3{\mathrm{\ell }}_p^3)$ after adopting $T_{M2}=(2\pi {)}^{-2}{\mathrm{\ell }}_p^{-3}$, where ${\mathrm{\ell }}_p$ is the Planck length in the adopted $c=\mathrm{\hslash }=1$ units and $T_{M2}$ has dimension ${\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}^{-3}$, and identifying the unit-M2-coupled CJS curvature with the curvature in the Witten shift. Under its further hypotheses of a smooth compact connected resolved Calabi--Yau fourfold ${\widehat{Y}}_4$ with holomorphically trivial canonical bundle, a smooth second-countable spin eleven-manifold $Y_{11}$ and internal inclusion $i:{\widehat{Y}}_4\hookrightarrow Y_{11}$ with $i^{\ast }T{Y}_{11}\cong T{\widehat{Y}}_4\oplus {\epsilon }^3$, that postulate states only $$g_4+\frac{1}{2}{\rho }_{{\widehat{Y}}_4}(c_2(T{\widehat{Y}}_4))\in \mathrm{im}{\rho }_{{\widehat{Y}}_4},{\rho }_{{\widehat{Y}}_4}:H^4({\widehat{Y}}_4;\mathbb{Z})\to H_{\mathrm{d}\mathrm{R}}^4({\widehat{Y}}_4;\mathbb{R})\mathrm{.}$$ It neither chooses the integral lift $c$ and holomorphic line bundle above nor retains torsion or differential C-field data; it has no singular-space conclusion. These two postulates identify smooth comparison conventions only; neither supplies any singular state, transfer map or index proof. This zero-mode postulate supplies only that smooth compact line-bundle/spin-line index dictionary; it does not supply the branch Hilbert spaces or comparison maps required here. It explicitly excludes singular matter curves, general non-algebraic integral flux and microscopic M2 quantization. In particular, it cannot be applied across $\mathrm{\Delta }$ or used as a derivation of singular branch or vertex states. No index on the punctured open branch is inferred from the smooth compact postulate. These adopted postulates identify target conventions only; the package must still supply the actual singular state spaces, operators, comparison maps and twist identification as data.
3. Light branch and vertex data. At every point of $\mathrm{\Delta }$, supply the state spaces of every light branch, constituent membrane, and vertex degree of freedom, with their physical chiral gradings and actual interaction/transfer operators. If a high-sector gap closes, include the corresponding states in the light sector before applying an inverse to the high block.
4. Closed singular domain and limit. Give the closed regulated domain, the actual limiting operator, the convergence mode that permits the claimed Fredholm/index comparison, and the computed local index defects. Any finite branch/vertex reduction must have graph-norm-bounded transfer. A value prescribed only by point evaluation on smooth tests is not a closed-domain construction.
5. Combined fermion/C-field twist. Supply the mathematical data of def-m2-combined-fermion-cfield-descent-data and prove its actual identification with the protected-state twist, retaining Berry factors, spin weights, spectral-cut conventions, electric-line data, and flat differential-cocycle information. The Euclidean data definition supplies no such physical identification or pairing by naming it.

The supplied interface of def-m2-regulated-protected-sector-data specifies a Hilbert-completed BRST quotient, its grading and domains, a closed odd self-adjoint preserved charge and compatible evolution, and a controlled regulator limit if an index is claimed. A curve reduction also supplies the actual light projection, state bundle, Berry connection, closed blocks and comparison estimates. This is an input specification, not a physical quantization theorem.

The mathematical interface of def-m2-combined-fermion-cfield-descent-data specifies the closed smooth Euclidean spin family, actual normal-spinor data, and a source-matched shifted differential C-field. Its general shifted-cocycle component retains an arbitrary fixed integral shift. When a Freed--Moore source-presentation refinement is used, its spin shift, ordinary E8 refinement, normalized global correction form, curvature equation and integral/differential label crosswalk must additionally be supplied. These are source-matching conditions, not restrictions inferred for every general package. Integral and flat data and their actual compatible identification are retained. Full AC is inherited from the even-dimensional spinor and characteristic-class inputs. Self-adjointness, compact resolvent, smooth family dependence and geometric line data remain supplied mathematical hypotheses.

Neither interface supplies the physical realization, a Pfaffian/electric line pairing, or a comparison of different operator sign lines. The mathematical family-pairing theorem is not a logical prerequisite of this required-data definition: a requested pairing or physical twist identification is something the actual realization must provide, not a theorem presumed by naming the package.

Crepant resolution and matter surface

M2 Brane Worldvolume Action

M2 Regulated Protected Sector Data

M2 Combined Fermion Cfield Descent Data

M2 matter-curve zero modes for smooth algebraic flux backgrounds

Shifted M-theory G4 quantization on a resolved Calabi–Yau fourfold

This is a required-data definition, not an adopted postulate or a physical existence theorem. It does not extend the classical smooth-embedding action by fiat, assign physical meaning to an arbitrary finite matrix or nodal sheaf, or prove the original matter-surface index relation. Current supplier readiness is a separate native record; it is not inferred from carrier presence or encoded as a permanent absence/ready claim in this definition. The actual physical singular sector and twist identification remain required inputs even if the local input contracts receive current ready records.