Motivic Hypercohomology Solutions in Field Theory and Applications in H-States

Triangulated derived categories are considered which establish a commutative scheme (triangle) for determine or compute hypercohomology of motives the obtaining solutions field equations. The determination this arises category $\textup{DM}_{\textup {gm}}(k)$,  is motivic objects whose image under $\textup {Spec}(k)$  that to say, an equivalence underlying triangulated tensor categories, compatible with respective functors on $\textup{Sm}_{k}^{\textup{Op}}$. geometrical will be risked moduli stack holomorphic bundles. Likewise, analysed special case where complexes $C=\mathbb{Q}(q)$,  obtained when cohomology groups isomorphism $H_{\acute{e}t}^{p}(X,F_{\acute{e}t})\cong (X,F_{Nis})$,   can vanished for  $p>\textup{dim}(Y)$.  We observe also Beilinson-Soul$\acute{e}$ vanishing  conjectures we have vanishing $H^{p}(F,\mathbb{Q}(q))=0, \ \textup{if} p\leq0,$ and $q>0$,   confirms before established. Then survives $\mathbb{H}^{q}(X,\mathbb{Q})$. its in $\textup{Spec(Sm}_{k})$.  complex Riemannian manifold integrals those $\textup{Spec}_{H}\textup{SymT(OP}_{L_{G}}(D))$, variety opers formal disk $D$, neighborhood all point surface $\Sigma$.  proved $\mathrm{H}^{\vee}$,  has decomposing components as hyper-cohomology characterized H- states Vec$_\mathbb{C}$, equations $d \textup{da}=0$,  general linear group $k=\mathbb{C}$.  A physics re-interpretation superposing, dual spectrum quantized version space $H^{q}(Bun_{G},\mathcal{D}^{s})=\mathbb{H}^{q}_{G[[z]]}(\mathrm{G},(\land^{\bullet}[\Sigma^{0}]\otimes \mathbb{V}_{critical},\partial))$ corresponding deformed densities $\mathrm{h} \in \mathrm{H}$, quantum theory.