M2 Regulated Protected Sector Data

A regulated protected-sector data package for a specified background and charge sector is the following supplied physical input record.

1. For each regulator value $ϵ>0$, specify a pre-Hilbert space ${\mathcal{K}}_{ϵ}^0$ with inner product and norm, its Hilbert completion ${\widehat{\mathcal{K}}}_{ϵ}$, and a Hilbert-orthogonal ghost-number decomposition ${\widehat{\mathcal{K}}}_{ϵ}={\widehat{⨁}}_{g\in \mathbb{Z}}{\widehat{\mathcal{K}}}_{ϵ}^g$. The associated ghost-number operator is the self-adjoint $N_{ϵ}x=gx$ on ${\widehat{\mathcal{K}}}_{ϵ}^g$, with domain $\{x=(x_g):{\sum }_g{g}^2\mathrm{\parallel }{x}_g{\mathrm{\parallel }}^2<\mathrm{\infty }\}$. Supply a densely defined closed differential $d_{ϵ}$ on ${\widehat{\mathcal{K}}}_{ϵ}$ such that $\mathrm{ran}{d}_{ϵ}\subseteq \mathrm{dom}{d}_{ϵ}$, $d_{ϵ}^{2}x=0$ for every $x\in \mathrm{dom}{d}_{ϵ}$, and $\mathrm{dom}{d}_{ϵ}$ is invariant under the ghost-sector projections. Its ghost-number action is required to be $d_{ϵ}(\mathrm{dom}{d}_{ϵ}\cap {\widehat{\mathcal{K}}}_{ϵ}^g)\subseteq {\widehat{\mathcal{K}}}_{ϵ}^{g+1}$. Specify the physical ghost number $g_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ and take the physical Hilbert space to be $${\mathcal{H}}_{ϵ}:=\ker d_{ϵ,g_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}}/\stackrel{‾}{\mathrm{ran}{d}_{ϵ,g_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}-1}},d_{ϵ,g}:=d_{ϵ}{\mathrm{\mid }}_{\mathrm{dom}{d}_{ϵ}\cap {\widehat{\mathcal{K}}}_{ϵ}^g}\mathrm{.}$$ Thus $[x]=[y]$ exactly when $x-y\in \stackrel{‾}{\mathrm{ran}{d}_{ϵ,g_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}-1}}$, and $\mathrm{\parallel }[x]\mathrm{\parallel }={\inf }_{r\in \stackrel{‾}{\mathrm{ran}{d}_{ϵ,g_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}-1}}}\mathrm{\parallel }x+r\mathrm{\parallel }$. Supply a self-adjoint unitary involution ${\mathrm{\Gamma }}_{ϵ}$ on the BRST Hilbert space, preserving each ghost sector and $\mathrm{dom}{d}_{ϵ}$, with $d_{ϵ}{\mathrm{\Gamma }}_{ϵ}=-{\mathrm{\Gamma }}_{ϵ}{d}_{ϵ}$ on that domain. Its induced quotient involution is the selected physical four-dimensional chirality grading. Fix the convention $\eta =(-,+,+,+)$ and ${\gamma }_5=i{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3$ in a positively oriented orthonormal Lorentz frame, with $\{{\gamma }^{\mu },{\gamma }^{\nu }\}=2{\eta }^{\mu \nu }$: the positive eigenspace is right-handed and the negative eigenspace is left-handed. Write ${\mathcal{H}}_{ϵ}^{+}$ and ${\mathcal{H}}_{ϵ}^{-}$ for the $+1$ and $-1$ eigenspaces of the induced quotient grading, and set $n_R:=\dim {\mathcal{H}}_{ϵ}^{+}$ and $n_L:=\dim {\mathcal{H}}_{ϵ}^{-}$ when these are finite. The graded net chirality is then $n_R-n_L$; no finiteness is asserted here.
2. Supply on ${\mathcal{H}}_{ϵ}$ a densely defined closed self-adjoint charge $Q_{ϵ}$ with ${\mathrm{\Gamma }}_{ϵ}{Q}_{ϵ}=-Q_{ϵ}{\mathrm{\Gamma }}_{ϵ}$ on its invariant domain, its normalization and square, and the exact symmetry and commutation relations used in the model. Supply a self-adjoint Hamiltonian $H_{ϵ}$ whose unitary evolution $e^{-it{H}_{ϵ}}$ preserves the physical quotient and grading, preserves $\mathrm{dom}(Q_{ϵ})$, and commutes with $Q_{ϵ}$ there.
3. For regulator removal, supply a common graded Hilbert space $\mathcal{K}$, graded Hilbert spaces ${\mathcal{H}}_0$ and ${\mathcal{H}}_{ϵ}$ as above, and grading-preserving linear isometries $J_{ϵ}:{\mathcal{H}}_{ϵ}\to \mathcal{K}$ for $ϵ\ge 0$. Set ${\mathrm{\Pi }}_{ϵ}=J_{ϵ}{J}_{ϵ}^{\ast }$. Supply an odd self-adjoint $Q_0$ on a specified dense invariant domain in ${\mathcal{H}}_0$. The convergence required here is generalized strong-resolvent convergence: for every $x\in \mathcal{K}$ and $z\in \mathbb{C}\setminus \mathbb{R}$, $$\mathrm{\parallel }{\mathrm{\Pi }}_{ϵ}x-{\mathrm{\Pi }}_{0}x\mathrm{\parallel }⟶0,\mathrm{\parallel }{J}_{ϵ}(Q_{ϵ}-z{)}^{-1}{J}_{ϵ}^{\ast }x-J_0(Q_0-z{)}^{-1}{J}_0^{\ast }x\mathrm{\parallel }⟶0(ϵ\downarrow 0)\mathrm{.}$$ Strong operator convergence means precisely convergence on each fixed vector as in these formulas; stronger estimates must be separately stated. If a protected index survives removal, require each chiral charge to be Fredholm on its graph-norm domain and prove convergence of its kernel-minus-cokernel dimension to the corresponding index at zero. Record the spectral projections ${\mathbf{1}}_{I_{ϵ}}(Q_{ϵ})$ onto every specified Borel energy window $I_{ϵ}\subset \mathbb{R}$ used to track states entering or leaving zero, with their transported ranges under $J_{ϵ}$ and their stated convergence. Strong-resolvent convergence alone asserts no index stability or uniform gap.
4. If a curve reduction is claimed, specify a finite-dimensional smooth parameter manifold $B_{\mathrm{r}\mathrm{e}\mathrm{d}}$, a smooth Hilbert bundle $\mathcal{H}\to B_{\mathrm{r}\mathrm{e}\mathrm{d}}$ with fiber ${\mathcal{H}}_b$ equal to the physical quotient at the regulator value $ϵ(b)>0$ and other declared parameters, and a unitary ambient connection ${\mathrm{\nabla }}^{\mathcal{H}}$ preserving the physical grading. Supply a smooth operator-norm family $b\mapsto P_b$ of orthogonal projections of locally constant finite rank on these fibers, with $P_b{\mathrm{\Gamma }}_b={\mathrm{\Gamma }}_b{P}_b$ for the induced fiber grading ${\mathrm{\Gamma }}_b$ and with each $P_b$ preserving every operator domain used in the reduction. The light bundle is $\mathcal{L}=\mathrm{im}P$ and its Berry connection is, by definition, ${\mathrm{\nabla }}^{\mathrm{B}}s=P{\mathrm{\nabla }}^{\mathcal{H}}s$ for smooth sections $s$ of $\mathcal{L}$; the heavy bundle is $\mathrm{im}(I-P)$. When the only varying parameter is the regulator, write $P_{ϵ}$ for this family. State the closed block operators, their domains, and the bounds used for every block comparison. A mode here means a vector in ${\mathcal{L}}_b=\mathrm{im}{P}_b$. Supply a smooth even orthogonal subprojection $U_b\le P_b$, of locally constant rank, whose range is the designated universal mode bundle. “Universal” is a supplied sector designation, not a consequence of a rank calculation; any physical justification of this designation must be stated separately. Supply a finite index set $A$ and smooth even mutually orthogonal projections $V_{a,b}$ of locally constant rank with $U_b+{\sum }_{a\in A}{V}_{a,b}=P_b$; the additional light constituents are exactly their ranges, including zero ranges if present. These equations are the completeness criterion for this record. All projections preserve the domains used by their stated block restrictions. Record ranks, gradings and those restrictions. A claim that this record includes every physical mode in a specified energy window additionally requires $P_b={\mathbf{1}}_{[-{\delta }_b,{\delta }_b]}(Q_b)$ with supplied ${\delta }_b>0$ in the charge's declared units; it is not inferred from the word light or from finite rank.
5. If the smooth Euclidean fermion/electric package is used, supply the exact mathematical data in the combined Euclidean M2 fermion/C-field descent package. Its interface fixes a finite-dimensional Hausdorff second-countable parameter manifold $B$, a closed Riemannian spin eleven-manifold $Y$, and a proper smooth submersion $\pi :\mathrm{\Sigma }\to B$ with compact closed oriented spin three-dimensional fibers, embedded fiberwise in $Y$. It supplies the rank-eight quotient normal bundle $N=f^{\ast }TY\mathrm{/}df(T^v\mathrm{\Sigma })$ with metric, spin lift, orientation, and connection; here $df{\mathrm{\mid }}_{T^v\mathrm{\Sigma }}$ is the injective bundle map supplied by the fibrewise embedding. It supplies a shifted cocycle $(u,a,t,\omega )$, with $u,a\in Z_{\mathrm{s}\mathrm{m}}^4(Y;\mathbb{Z})$, $t\in C_{\mathrm{s}\mathrm{m}}^3(Y;\mathbb{R})$, $d\omega =0$, and $\delta t=\mathrm{Int}(\omega )-a_{\mathbb{R}}-u_{\mathbb{R}}\mathrm{/}2$. The shift $u$ is fixed, and the declared changes $(a,t,\omega )\mapsto (a+\delta b,t-b_{\mathbb{R}}-\delta s,\omega )$ retain the specified differential representative. It also supplies an actual principal $E_8$ bundle/connection and ordinary differential refinement $x_A$ with its stated curvature and integral class. The fixed shift $u$ remains arbitrary in the general package. Its local normalized total potentials satisfy $d{\mathcal{C}}_i=\omega$, equivalently the unit-coupled potentials $C_{M,i}=2\pi {\mathcal{C}}_i$ satisfy $d{C}_{M,i}=2\pi \omega$, with formal phase $\exp (i\int C_{M,i})$. If the exact Freed--Moore source presentation is used, separately supply its matching refinement: $[u]=\lambda (TY)$, the global normalized correction $c_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}}$, curvature equation $\omega ={\mathrm{\Omega }}_A-\mathrm{\Lambda }\mathrm{/}2+d{c}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}}$, and actual integral/ differential crosswalk $[a]=\alpha -[u]$, including torsion and flat data. None of these source-matching conditions is deduced for an arbitrary general shifted-cocycle package. The source-model Dirac families are $D_f$ on $S({\mathrm{\Sigma }}_b)\otimes S^{+}(N_b)$ and $D_0$ on $S({\mathrm{\Sigma }}_b)$. Their dense Sobolev domains, self-adjoint compact-resolvent properties, smooth parameter dependence, real/quaternionic structures, a vertical metric and horizontal splitting, compatible Clifford connections, spectral-cut charts $a\mathrm{\notin }\mathrm{spec}(D_b^2)$ and their finite-rank bands are supplied hypotheses. The exact source refinement additionally supplies the doubled graded module, volume operator and parallel quaternionic $J$ with $J^2=-1$ commuting with the Clifford and Dirac operators, together with compatible invertible-band Pfaffian transitions, as typed in the supplier. The package includes the Pfaffian and electric eta lines, their transition maps, metrics and geometric connections, and the source electric factor, when that matching refinement is supplied, $C{S}_{{\mathrm{\Sigma }}_b}(A){\tau }_{D_0}^{-1\mathrm{/}4}\exp (2\pi i{\int }_{{\mathrm{\Sigma }}_b}{c}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}})$. The eta convention is the one recorded in the exact supplier. It does not include a tensor pairing of these lines and does not identify a Lorentzian fluxed Hessian or construct the physical Hilbert quotient above.

This is a definition of required inputs, not an assertion that any wrapped membrane realizes them. The classical M2 action remains physical context; classical $\kappa$-symmetry motivates the quantum gauge problem but does not construct its nonlinear quotient.

M2 Combined Fermion Cfield Descent Data