Journal of Mathematics Research | 2021
Motivic Hypercohomology Solutions in Field Theory and Applications in H-States
Abstract
Triangulated derived categories are considered which establish a commutative scheme (triangle) for determine or compute a hypercohomology of motives for the obtaining of solutions of the field equations. The determination of this hypercohomology arises of the derived category $\\textup{DM}_{\\textup {gm}}(k)$,\xa0 which is of the motivic objects whose image is under $\\textup {Spec}(k)$\xa0 that is to say, an equivalence of the underlying triangulated tensor categories, compatible with respective functors on $\\textup{Sm}_{k}^{\\textup{Op}}$. The geometrical motives will be risked with the moduli stack to holomorphic bundles. Likewise, is analysed the special case where complexes $C=\\mathbb{Q}(q)$,\xa0 are obtained when cohomology groups of the isomorphism $H_{\\acute{e}t}^{p}(X,F_{\\acute{e}t})\\cong (X,F_{Nis})$,\xa0\xa0 can be vanished for\xa0 $p>\\textup{dim}(Y)$.\xa0 We observe also the Beilinson-Soul$\\acute{e}$ vanishing\xa0 conjectures where we have the vanishing $H^{p}(F,\\mathbb{Q}(q))=0, \\ \\ \\textup{if} \\ \\ p\\leq0,$ and $q>0$,\xa0\xa0 which confirms the before established. Then survives a hypercohomology $\\mathbb{H}^{q}(X,\\mathbb{Q})$. Then its objects are in $\\textup{Spec(Sm}_{k})$.\xa0 Likewise, for the complex Riemannian manifold the integrals of this hypercohomology are those whose functors image will be in $\\textup{Spec}_{H}\\textup{SymT(OP}_{L_{G}}(D))$, which is the variety of opers on the formal disk $D$, or neighborhood of all point in a surface $\\Sigma$.\xa0 Likewise, will be proved that $\\mathrm{H}^{\\vee}$,\xa0 has the decomposing in components as hyper-cohomology groups which can be characterized as H- states in Vec$_\\mathbb{C}$, for field equations $d \\textup{da}=0$,\xa0 on the general linear group with $k=\\mathbb{C}$.\xa0 A physics re-interpretation of the superposing, to the dual of the spectrum $\\mathrm{H}^{\\vee}$,\xa0 whose hypercohomology is a quantized version of the cohomology space $H^{q}(Bun_{G},\\mathcal{D}^{s})=\\mathbb{H}^{q}_{G[[z]]}(\\mathrm{G},(\\land^{\\bullet}[\\Sigma^{0}]\\otimes \\mathbb{V}_{critical},\\partial))$ is the corresponding deformed derived category for densities $\\mathrm{h} \\in \\mathrm{H}$, in quantum field theory.