M2 Curve Comparison Realization Data

Given a regulated protected-sector package, curve-comparison realization data consist of the following actual supplied objects for the same background and charge family.

1. A specified family $\pi :\mathcal{C}\to \mathcal{B}$, where $\mathcal{B}$ is a finite-dimensional Hausdorff second-countable smooth manifold without boundary and $\pi$ is a proper smooth submersion. Every fiber ${\mathcal{C}}_b={\pi }^{-1}(b)$ is a compact closed oriented Riemann surface; its complex structure on $T^v\mathcal{C}:=\ker (d\pi )$ varies smoothly with the point of $\mathcal{C}$, and its complex orientation is the stated orientation. Define the relative canonical line by $K_{\mathcal{C}\mathrm{/}\mathcal{B}}:={\mathrm{\Lambda }}^{1,0}(T^v\mathcal{C}{)}^{\ast }$. For a bundle $F\to \mathcal{C}$, write $F_b:=F{\mathrm{\mid }}_{{\mathcal{C}}_b}$.
2. A finite-rank ${\mathbb{Z}}_2$-graded Hermitian bundle $E=E^{+}\oplus E^{-}$ over $\mathcal{C}$, with orthogonal grading, ranks $r_{\pm }(p):={\dim }_{\mathbb{C}}{E}_p^{\pm }$, and a specified smooth map $r:\mathcal{C}\to B_{\mathrm{r}\mathrm{e}\mathrm{d}}$ to the projection-family base of the M2 Regulated Protected Sector Data package. Require an actual grading-preserving Hermitian identification $E\cong r^{\ast }\mathrm{im}P$; its connection is the pullback of the Berry connection $P{\mathrm{\nabla }}^{\mathcal{H}}$ defined there. The characteristic fields in this record are explicitly $c_{\pm }=[\frac{i}{2\pi }\mathrm{tr}{F}_{\pm }]\in H_{\mathrm{d}\mathrm{R}}^2(\mathcal{C};\mathbb{R})$, where $F_{\pm }$ is the curvature of that pulled-back connection on $E^{\pm }$. Here $H_{\mathrm{d}\mathrm{R}}^2$ is the quotient of closed smooth real two-forms by exact smooth real two-forms. In a unitary frame with connection matrix $A_{\pm }$, curvature means $F_{\pm }=d{A}_{\pm }+A_{\pm }\wedge A_{\pm }$. Record these ranks and classes from the actual identification with $r^{\ast }\mathrm{im}P$, together with the physical derivation of that identification; they are not fitted to a desired index. Supply a fiberwise holomorphic vector-bundle structure on $E$: locally over $\mathcal{B}$, provide frames smooth in the base parameter and holomorphic along each fiber, with transition matrices smooth in the base and holomorphic along fibers. In these frames ${\bar{\mathrm{\partial }}}_{E,b}$ acts componentwise by ${\mathrm{\partial }}_{\bar{z}}$, satisfies the Leibniz rule, preserves $E_b^{\pm }$, and varies smoothly with $b$. Also supply $L$ and the spin line $T$ as smooth line bundles on $\mathcal{C}$ with fiberwise-holomorphic transition functions varying smoothly with $b$, and a fiberwise holomorphic isomorphism $T^{\otimes 2}\cong K_{\mathcal{C}\mathrm{/}\mathcal{B}}$. At the level of this mathematical target, $L$ is an arbitrary supplied holomorphic line bundle. Calling it the physical anomaly-corrected twist requires a separate model-specific construction and identification with the actual protected-state twist, including its transition maps and connection; no such identification or correction formula is inferred from the general Euclidean descent package. This physical construction remains part of the held realization obligation, rather than a property of every supplied $L$.
3. On every fiber set $W_b:=E_b\otimes L_b\otimes T_b$. Supply a smoothly varying Riemannian metrics $g_b$ compatible with the fiber complex structures and smoothly varying Hermitian metrics on $E_b,L_b,T_b$; their tensor product gives $h_{W_b}$. For smooth forms define the first-variable-linear pairing $$⟨u,v{⟩}_{L^2}:={\int }_{{\mathcal{C}}_b}⟨u,v{⟩}_{g_b,h_{W_b}}d{V}_{g_b}\mathrm{.}$$ and let $L^2({\mathcal{C}}_b,{\mathrm{\Omega }}^{0,\ast }(W_b))$ be its Hilbert completion. In fiberwise holomorphic frames define the smooth Dolbeault operator componentwise by ${\mathrm{\partial }}_{\bar{z}}$, extend it as the maximal weak $L^2$ operator, and define its Hilbert adjoint with respect to this pairing. Fix a finite holomorphic chart/frame cover $(U_j,e_j)$ and subordinate smooth partition $({\chi }_j)$; the domain $H^1$ is the completion with $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1}^2:={\sum }_j\mathrm{\parallel }({\chi }_{j}u{)}_{e_j}{\mathrm{\parallel }}_{W^{1,2}(U_j)}^2$, using the Euclidean vector norm on local coefficient vectors. The total Dolbeault-plus-adjoint operator is $D_b:={\bar{\mathrm{\partial }}}_{W_b}+{\bar{\mathrm{\partial }}}_{W_b}^{\ast }$ on this domain. Its total grading is coefficient-bundle parity plus form degree modulo $2$, so $D_b$ is odd.
4. An actual grading-preserving Hilbert identification of the physical protected sector with the stated target space on the declared subspace, including its domain map, together with the norm-continuous Fredholm comparison operators and their exact domains. State which operators are endpoints of any comparison path; a path is data and by itself is not asserted here to prove an index identity.
5. Every relative zero-mode sector, its physical grading, the actual kernels and cokernels used to calculate its graded index, and the contribution included in the comparison. Rank-one or zero-relative- index conditions may be used only when independently demonstrated.

The definition requires an actual physical comparison and its inputs; it does not assert that the comparison exists, that its index equals the curve index, or that a vacuum has rank one.

M2 Regulated Protected Sector Data