KK Theory and K Theory for Type II Strings Formalism

Corresponding author: Deep Bhattacharjee, itsdeep@live.com

Mathematical Subject Classification: Primary (19-XX, 83-XX), Secondary (19Kxx, 19K35, 83Exx)

I. Introduction

Any map ^[1] from a domain to a codomain with the mapping parameter $\theta: \zeta \rightarrow \zeta^{\prime}$ can provide a continuous set of functions when $\zeta$ and $\zeta^{\prime}$ is endowed with a metric which when attempt for any representation of a Topological structure considering two sets $\{\zeta\}$ and $\left\{\zeta^{\prime}\right\}$ there norms even a bijection ^[1] between them $\zeta \leftrightarrow \zeta^{\prime}$ which for a defined function $f$ over a value of $f(x)$ there involves a structure of a vector space with concerned operations through a continuous linear transformation, that space for that function carries a Topology best known as Hilbert space. The specified module that carries the $c^{*}-$ algebra ^[2] for that space is defined as $c^{*}-$ Hilbert modules ^[3] through the inner product.

For any group $\wedge$ with a subgroup $\ell$ the representations $\Gamma_{\ell}^{\Lambda}$ makes it easier to construct new representations through the subgroup or the smaller group $\ell$ over certain parameters that when categorize through the constructive modules of Hilbert's $c^{*}$ then this extent the $c^{*}-$ module to $c^{*}-$ algebras through the non—commutative formulations ^[4][5].

Furthermore, any derived pathway to construct the noncommutative geometry provides a framework for the moulder category to represent an equivalence over (left — right) — symmetric rings ^[6][7][8][9] as established afterwards with rings $R$ and $R^{\prime}$; then for the ring — representations, studying the category of those modules; there exists Morita equivalence ^[10] for the isomorphic commutative form or in general norms in the case of non — commutative rings ^[11].

For the constructions of $K K$ — Theory; Morita equivalence is an important tool to $c$ * algebras where for the inequality on the two modules $A$ and $B$; for the moulder form $E$ on $A$ and $B$ for the moulder form $E$ on $A$ and $E \cdot$ on $B$ (as appeared later in the paper) a homotopy invariant bifunctor can make a Morita equivalence for the $K K$ — Theory through $K K(A, B)$ and $K K(B, C)$ for $A, B, C$ as $c$ * algebras; there's for the modular form $E$ having elements $\varepsilon, \epsilon$ the inequality represents the form $<\varepsilon, \epsilon><\epsilon, \varepsilon>\leq \|<$ $\varepsilon, \varepsilon>||<\epsilon, \epsilon>$ where for the $A$ — module; the above relation holds and taking the $B$ — module representing the $c^{*}$ — algebraic pair $K K(A, B)$ and $K K(B, C)$ where one finds the combined form over the composition product representing $K K(A, C)$ and the Morita equivalence to be represented in a specific formulation as to be proved throughout the paper ^[12][13].

Over the compact Hausdorff spaces ^[14] and considering the Fredholm modules of Atiyah-Singer Index Theorem ^[15] for a relatable definition of $A, B, C$ in $c^{*}-$ algebras the Kasparov's product $K K(A, C)$ for $K K(A, B)$ and $K K(B, C)$ will be established over an elliptic differential operator $\varrho_{M_{s}}^{0}$ or $\varrho_{M_{n}}^{0}$ for $s-$ smoothness or $n$ —dim and through extensive analysis of that operator which indeed suffice the Fredholm module making a relatable framework for $K$ — Homology and $K$ — Theory ^{[3][5][6][16]}; The Thom isomorphism is established for the Chern Character Ch over a mapping parameter $\iota$ through a rank — $n$ vector bundle $v_{1(n)}$ with $v_{2}$ having the first related to a unit sphere bundle. This in turn induces the categorical correspondence between a relational establishment over noncommutative geometry and noncommutative topology taking the function $f$ over a bounded structure through linear transformations that bounds the concerned subsets $I$ and $J$ for a mapping parameter $\rho_{\eta}$ in the same Hilbert space $H$.

This will deduce for a much more concrete formalism of the $K$— Theory to $K$— Homology with an extension of $c^{*}$— algebras to reduced $c^{*}-a \lg e$ bras for parent group $(\wedge)$ that defined the $\ell^{2}$ norm of Hilbert space taking into consideration the $K K-$ Theory with Gromov's $a-T-$ menable property for all the necessary formulations concerned before except Morita equivalence that when established through 5 parameters through an assembly mapping parameter $\daleth$ over discrete torsions gives the ultimate relation of $K K$ — Theory in Baum — Connes conjecture taking into account both the Novikov conjecture and Kadison — Kaplansky conjecture for injectivity and surjectivity respectively connecting to noncommutative topology ^[17][18][19].

• Extensions have been made in the operator and Topological aspects in the cohomology class where several classifiers are shown with distinct property to suffice the $\mathrm{Sp}_{\mathrm{c}}$ —Structure and the Atiyah —Hirzebruch spectral sequence for the Type II (II-A and II-B) as concerned on the complex Topology space $T^{*}$ where the Atiyah — Singer Index Theorem taking the Fredholm modules as necessary for K-Theory with Bott — Periodicity is taken and a channelization is made to Grothendieck — Riemann — Roch; for the transition of KK-Theory to Strings; Hodge dual, Gauge symmetry, charge density for the required Lagrangian in RR-fields through D-Brane Potential, De Rham Cohomology, and GSO —Projections are shown. P-form electrodynamics with P-Skeleton are considered for the purpose. NS 3-form and its relation to RR-flux in both D-Brane charge density and supergravity is established. The spectral sequence of Atiyah-Hirzebruch is taken and operator over $E_n^{p, q}$ for n taking the values $2,3, \infty$ over a consideration of several orders of K-Theory as such Topological, Algebraic, and Twisted. étale cohomology and its representation is shown for Algebraic K-Theory and the Kähler (without any specific consideration of compact and Ricci flatness) has been shown in general terms for K-Theory in a Twisted formalism in $E_{i}$ for $i=4=$ $\infty$.

II. Establishing Morita Equivalence With KK-Theory For Hilbert C*-Module

For a Hilbert space $\mathrm{H}$ with a $c^{*}$-module $\mathrm{H}_{c}$ one can define a c*-algebra for the metric $g$ on a Riemann manifold $M$ (having the form $M_{g}$ ) with a vector bundle $V$ there exists a compact neighbourhood being locally variant on a small patch; over an isomorphism of the Hilbert space of that vector bundle $V$ in a continuous way for a commutative $c^{*}$-algebra through the vanishing infinity.

For the modular form of $c^{*}$-algebra the Hilbert module for the non-commutative form is the generalized norm taking the algebra over a topological field $T$ in unital formulation for the unit parameter $i$ as such for every $\epsilon$ in the algebra there exists $\epsilon=i \epsilon=\epsilon i$.

Representing over the induced form for any finite group ${ }^{\wedge}$ with $\ell \subset^{\wedge}$ for the vector bundle $\mathcal{V}$ on the Hilbert space $H$, any construction can be defined over the $k$-elements of the group ${ }^{\wedge}$ over $L$ defined a parameter $\mathcal{P}$ as ^[19],

This gives for each $k$, the induced representation through group $\wedge$ in the same $L_{k}^{+} \in L_{k}$ for $\ell \subset \wedge$ through the vector representation $\mathcal{V}$ of subgroup $\ell$ being $\ell \subset \wedge$ in Hilbert space $H$ parametrized through ^[3][6],

Thus, one gets,

Representing $\mathcal{E}_{k} \in \mathcal{V}$, three non-trivial actions can be noted for the constructions ^[20][21],

1.  $\varepsilon_{k} \in \mathcal{V}$
2.  $L_{k}^{+} \in L_{k} \forall \ell \subset^{\wedge}$
3.  $\ell \subset \wedge$

This takes a pre-Hilbert Hausdorff space to construct c*-algebra satisfying the operations of an inner product through the Hilbert A-module being non-negative and self-adjoint. Taking the inner product of the complex manifold representing $\mathcal{M}^{*}$ through,

Thus, for any sequence of set that is countable over the Topological space $T$ with a proper representation for the previously encountered manifolds $\mathcal{M}^{T}$ taking $k^{\text {th }}$ countable order of infinity,

When merged with the unital form taken before $\epsilon=i \epsilon=\epsilon i$ such that for every unit parameter $i$ there exists $\epsilon$ in the algebra; where for any $c^{*}$-algebra there holds the Banach-algebra for a compact $\mathcal{F}$, that if provided there exists three forms taking $B_{0}(\mathcal{F})$^{[3][19][20][22]},

1. Typical form —For the complex space $\mathcal{M}^{*}$; the locally compact Hausdorff space for vanishing infinity norm gives $B_{0}(\mathcal{F})$ for continuous functions on $\mathcal{M}^{*}$.
2. Unital $-\left\{\begin{array}{c}\text { if is commutative } \\ \text { identity element of having norm } 1 \\ \mathcal{F} \text { in } B_{0}(\mathcal{F}) \text { is compact }\end{array}\right.$
3. For Point [2] to have a congruent transformation, there is Banach algebra $B_{0}(\mathcal{F})$ in $A$-form where the congruent transformation is unital for a closet set $[A]$.

For the compact Hausdorff (here parameterizing $\mathcal{F}_{0}^{+}$) with vector bundles $\mathcal{V}$ for the labeling of $\mathcal{F}_{0}^{+}$ - 0 for positive to extend over Bott Periodicity with + as adjoint through 8-periodic homotopy groups from $\pi_{0}$ to $\pi_{7}$ such that ^[12],

$\pi_{0,1,2,3,4,5,6,7}$ gives 3 — category tables in unitary $U$, orthogonal $\mathcal{O}$, symplectic $S p$,

Thus, for Hausdorff $\mathcal{F}$; the underlying K-Theory $K(\mathcal{F})$ there is ^[12][23];

1. Topological K-Theory $\Rightarrow$ on $\mathcal{M}^{T}$ for $K(\mathcal{F})$
2. Reduced K-Theory $\Rightarrow K_{\text {red }}(\mathcal{F})$ for $S^{n} \exists n>0$ relates the Bott for positive 0 for $\mathcal{F}_{0}^{+}$and adjoint + in Hausdorff $\mathcal{F}$ for $K_{\text {red }}\left(\mathcal{F}_{0}^{+}\right)$in non-commutive form.

Where Point [I] relates the Banach-algebras for the locally compact Hausdorff over a abelian module on any sequence of set countable over Topological space $T$ (as previously mentioned) on c*algebras for bivariant forms suffice the proper framework for the Hilbert $\mathrm{c}^{*}$-module on rings $R$ and $R^{\prime}$ for modular homeomorphisms on $R$ such that the biproduct exists in finitary over a defined functor $\delta$ preserving equivalence and additive properties ^[9][16][23],

For the naturally induced isomorphism for functors $\delta$ and $\delta^{\prime}$ for a finite module ring $R$ for the bimodule $\left(R, R^{\prime}\right)$ suffice the natural isomorphism iff for $X_{\left(R, R^{\prime}\right)}$ and $Y_{\left(R^{\prime}, R\right)}$ there is is ^{[2][3][9][16][23]},

Moreover, if we consider $A, B$ and $C$ as $c^{*}$-algebras then if there is a Hilbert $B$-module that is fully countably generated in the form of $E$, then for that $c^{*}$-subalgebras of $B$ there exists a strong Morita equivalence between $A$ and $B$ provided for the $B$ module there is $\varphi(E) \cong A$ and for $A$ module there is $\varphi(E \cdot) \cong B$ where for the $c^{*}$-algebraic pair $(A, B)$, over a homotopy invariant bifunctor the constructions can be taken for $A, B$ and $C$ in such a way that for the defined abelian group $K K(A, B)$ and combining it with $K K(B, C)$ a strong Morita equivalence can be established in the form ^[2][16][24],

$K K(A, B) \cong K K(A, C) \exists$ Combining the elements of $K K(A, B)$ AND $K K(B, C)$, there exists the product and the non-trivial assumptions that $B$ and $C$ are strongly Morita equivalent.

III. Relating Noncommutative Geometry with Noncommutative Topology

Now, for the linkage of $K$ —Theory to $K-$ Homology and $c^{*}$-algebras for the locally compact Hausdorff spaces, there can be a relatable definition of the $c^{*}$-algebra through noncommutative topology where there exists a detailed constructions to be discussed below ^{[13][17][18][25]}.

The mostly related theorem that suffice this duality with an equivalence between noncommutative geometry and noncommutative topology; just like the formulations of the Kasparov's composition product over $A, B, C$ in c*-algebras giving the result $K K(A, C)$ for $K K(A, B) \times K K(B, C)$ with the associated Morita Equivalence; any abelian group taking a trivial parameterization $\gamma(A)$ or can represent the Atiyah —Singer Index Theorem for the vector bundle $V$ having the elements $v_{1}$ and $v_{2}$ which over the smooth manifold $M_{s}$ with ' $s$ ' representing the smoothness property and the elliptic differential operator for the mapping over smooth sections on $M_{S}$ as,

Where for this elliptic differential operator $\varrho_{M_{s}}^{0}$ implying the Fredholm modules on the Hilbert space $H$ for c*-algebras there is the Chern character $\operatorname{Ch}\left(\varrho_{M_{s}}^{0}\right)$ giving Thom isomorphism with the mapping of vector bundles of rank — $n$ through,

Taking the unit sphere bundle $S\left(v_{1}\right)$ and $v_{2}$ representing the Chern character $C h\left(\varrho_{M_{n}}^{0}\right)$ for $n$ — dimensional compact manifold over the relation through a complex Tangent bundle $T^{\prime}$ as,

Where $T^{\prime}$ represents the complex tangent bundle of Todd class $\operatorname{Td}\left(T^{\prime}\right)$

Where $\xi\left(\varrho_{M_{n}}^{0}\right) \Rightarrow$ isomorphisms on $S\left(T^{\prime}\right)$ in $\operatorname{Td}\left(T^{\prime}\right)$

Which establishes the $K K-$ Theory through Fredholm module $\varrho_{M_{n}}^{0}$ or $\varrho_{M_{s}}^{0}$ for $n$ — dim or $s$ — smoothness where both are considered for the purpose of the constructions of Atiyah-Singer Theorem.

For the relation between noncommutative geometry and noncommutative topology it is now easy to show the $c^{*}$-algebras for the dual category of the Hausdorff spaces over *-isomorphism through the operator theory for a bounded structure over a function $f$ operating through linear transformations through two vector spaces $I$ and $J$ that are bounded through the image of the function $f(\eta)$ for the $\eta$ taking control over the mapping parameter $\rho$ as $\rho_{\eta}$ for $\rho_{\eta}: I \rightarrow J$ where $\rho_{\eta}$ makes the transformations that bounds the subsets of $I$ to subsets of $J$ on the same Hilbert space $H$.

Towards the establishment of noncommutative topology as described above in the paper the relation between noncommutative topology with noncommutative geometry over a non-trivial prescriptions of $c$ *—Hilbert modules and Hausdorff space that gets channelized further to establish the $K K-T h e o r y$ and Morita equivalence; a considerable fact is that for the proper extensions of $c *$ —algebras there is a defined category of the operator formalisms in the algebraic notions of $K$ —Theory where it can be shown that for the parent group (taken before) ${ }^{\wedge}$ with the $c^{*}$ —algebra, any reduced category for the completion of $c_{r e d}^{*}\left({ }^{\wedge}\right)$ formalism through a locally compact Topological group (denoting with a trivial notation just for the formulations) as $\wedge^{\prime}$ for a translation invariant norm through bounded functions; this $c_{\text {red }}^{*}\left({ }^{\wedge}\right)$ has an isomorphism for $c^{*}\left({ }^{\wedge}\right)$ where any defined $c^{*}$ —algebra can be expressed taking the $c_{r e d}^{*}\left({ }^{\wedge}\right)$ as a quotient of $c^{*}\left(^{\wedge}\right)$ for the Hilbert space $H$ having the defined norm $H_{l^{2}}$ there exists $5-$ parameters that connects the $K$ —Theory to $K$ —Homology for making $K K-$ Theory which provides a relation to Gromov hyperbolic groups ^[26] along with the groups that defined ^' for a translation invariant norm through bounded functions in $S L_{3}$ (Z)along with other rank-1 Lie Groups and other discrete Lie Groups $S O(n, 1)$ and $S U(n, 1)$ with Gromov's $a-T$ —menable property for the assembly mapping parameter $\daleth$ (which will be extremely useful later in the paper) for isomorphism having the representation of ^[16][18][25],

For $\rho_{\text {free }}(\wedge) \exists \rho$ represents discrete torsion for group $({ }^{\wedge})$

Where the 5 —parameters are the 5 —classifiers viz.,

1. $A$ — module
2.  $B$ — module
3. $\rho_{\text {free }}^{*}-$ Kadison — Kaplansky
4. $c^{*}$— algebra
5. $K K-$ Theory

Where $a-T-$ menable group for Hilbert space $H$ on the (previously taken) $\ell \subset \wedge$ giving three non-trivial connections to conclude this section,

For $\Sigma_{n}^{\ell}$ summing over $n$-elements

For $H^{*} \in H$ where in $\Sigma_{n}^{\ell} H^{*} \rightarrow \infty$

For $c_{r e d}^{*}\left(^{\wedge}\right) \rightarrow$ assembly mapping parameter $\daleth$ over a norm $|N|^{2}$ provides the relation, 

IV. Gelfand Transform and Poincaré Duality For C*-Algebra In $k^{\text {th }}$ Homology Group in $S p_{\mathrm{c}}$

Considering an involution $\iota_{0}$ for the Topological group ${}^{\wedge}$ with the defined Harr measure $\mu$ in a locally compact Hausdorff space $\mathrm{F}$ there exists a commutative spectrum $S^{\sigma}$ where for the unital element $i$ being the element of $S^{\sigma}$ for the Gelfand space $G$ representing,

There exists a commutative form for an algebraic isomorphism $\alpha^{*}$ in two categories of algebras,

1.  $c^{*}$ — algebra for an enveloping $c^{*}$ norm in $\alpha^{*}-$ isomorphism.
2. Banach algebra for the continuous function $f_{c}$

Considering Point [2], one gets the transform of $G$ representing as $G_{c}$ for $c$ —continuous form through $2$ — norms for the group action of group $\wedge$ defined $\ell_{+}\left(^{\wedge}\right)$ and $\ell_{++}\left(^{\wedge}\right)$ where for the spectrum $S^{\sigma}$ in Point [1], gives the modified form of a Fourier Transform as Gelfand Transform for $G_{c}$.

$\exists$ for $\ell_{+}(\mathcal{R})$ in $G_{c}$ and $f_{c} \in \ell_{+}(\mathcal{R})$ any $c^{*}$ —algebra for the Hausdorff space $\mathcal{F}$ over a two-way mapping $\pi: \mathcal{F} \leftrightarrow \mathcal{F}^{\prime}$ where $\mathcal{F}^{\prime}$ is also a Hausdorff space there exists;

Gelfand—Naimark Transform $\Rightarrow c^{*}(\mathcal{F})$ and $c^{*}\left(\mathcal{F}^{\prime}\right)$ in noncommutative $c^{*}$ —algebras the spectrum $S^{\sigma}$ can be defined over $\pi^{\prime}$ for Hausdorff $\mathcal{F}$ in $G_{c}-$ norm in $\alpha^{*}$ — isomorphism as,

Where $c_{+}(\mathcal{F}) \rightarrow c_{+}\left(\alpha_{c^{T}}^{*}\right)$

The two norms in group action for group ${}^{\wedge}$ namely, $\ell_{+}\left({ }^{\wedge}\right)$ and $\ell_{++}\left({ }^{\wedge}\right)$ for a Borel measure $\beta$ for $\ell_{+}\left({ }^{\wedge}\right) \cong$ $\mu \exists \mu$ represents the Harr measure (as considered earlier) through the involution $\iota_{0}$ one gets a generalized notion as,

Noncommutative geometry established over $f_{c} \forall \ell_{++}^{2}\left({ }^{\wedge}\right)$—norm — norm $\exists f_{c} \in \ell^{2}\left({ }^{\wedge}\right)$ there are,

1. Subspace $c_{s u b}^{*}$ for $c^{*}-a \lg e$ bra
2. The generalized norm $\ell_{++}\left({ }^{\wedge}\right)$
3. Banach algebra $B_{0}$ in Banach space $B_{0}(X)$ with a Borel measure $\beta$ in continuous $f_{c}^{*}$ for subgroup $\ell \subset {}^{\wedge}$ in $c^{*}-$ subspace in $c^{*}-a \lg e$ bra suffice the form,

Where $\mu_{\epsilon}$ acts on the Haar measure $\mu$ for group ${ }^{\wedge}$ over the action $\mu_{\epsilon}\left({ }^{\wedge}\right)$.

Now, for the Gelfand-Naimark Transform; a generalized application of the Fourier Transform (rather Gelfand Transform) with its application in noncommutative geometry for the isomorphism over a 'assembly mapping parameter $\daleth^{\prime}$ that we considered earlier, there can be the application for both $c_{r e d}^{*}$ and $c^{*}-a \lg e$ bra for an inde $x_{1}^{0}$ in *-over $c_{r e d}^{*}-a \lg e$ bras the common notion that arises is of,

1. Taking a discrete torsion-free parameter $\psi$ —compact for an equivariant $k$ —homology for the norm of 'right-side accessible form is always difficult than left-side accessible form' —the Baum-Connes conjecture can be extended for a proper action (without considering any classifying space $S_{c}^{c l}$ for $\psi S_{c}^{c l}$; ; the 'assembly mapping parameter' $T$ takes the mapping denoted by $\psi-$ subscript for action $\dot{S}$ given,
   $\dot{S}=\daleth_{\psi-\operatorname{com}} \varphi_{0 o r 1}^{\psi-\operatorname{com}(R)}\left(E \psi_{-\operatorname{com}}^{S\cdot}\right) \rightarrow \varphi_{0 o r 1}\left(c_{r e d}^{*}(\varphi-\operatorname{com})\right)$
2. Taking the same discrete and torsion-free parameter (which has been considered $\varphi$ or $\rho$ throughout the paper for notational significance (without any reduced form); which now acts on the classifying space $\varphi S_{c}^{c l}$ for the complex integers denoting $\Lambda$; the $c^{*}-a \lg e$ bra gets the Gelfand-Naimark Transform in the commutative way that becomes accessible for the Poincare duality to consider upon. However, for the case considered here, any automorphisms acting on $c^{*}$ for $A-\operatorname{module}$ gives the Baum-Connes conjecture in the form of $\daleth$ with $A$ and $\varphi$ as,
   $\daleth_{A, \varphi}: R \varphi \varphi^{\psi(R)}\left(E \varphi_{A}^{S}\right) \rightarrow \varphi_{0 o r 1}\left(A \ltimes_{\gamma} \psi\right)$

For the trivial parameterization of as considered where any parameter-less $A$ for $\delta$ as considered above else the parameter $A$ for the $A-\operatorname{module}$ without $\delta$ being considered otherwise.

Thereby, Poincaré duality can be defined through $K K-$ Theory for complex integer $\Lambda$ on classifying space $\Lambda S_{c}^{c l}$ in $c^{*}-a \lg e$ bra over discrete parameter $\psi$; taking the Thom Isomorphism for Topological $K-$ Theory in the homology theory for a generalized norm defining ^[22][24],

$\operatorname{Spin}^{c}$ —structure $S p_{c}$ on Riemann manifold $\mathcal{M}$ with metric representation $\mathcal{M}_{g}$ for the parameter (mentioned above) as $\Lambda$ structuring,

$\exists q$ Representing morphisms over $\Lambda_{2}$ for the sequence $S$,

Representing the Chern class for $U(1)_{\textit {Chern class }}^{B} \in H^{2}\left(\mathcal{M}_{g} \Lambda\right)$

Thus taking $\mathcal{M}_{n}$ as the $n-\operatorname{dim}$ manifold; Poincaré duality can be expressed in,

For $n^{\circ}$— integers; whereas expressed earlier $S p_{c}^{q}$ being the spin—structure on morphisms for the action on a manifold that is orientable in Topological $K$ — Theory. For isomorphisms on any integers of $n^{\circ}$ in $\mathcal{M}_{n, n^{\circ}}$ any $\textit{mod} -2$ (without any orientation assumption) —Poincare duality holds for,

$H^{n^{\circ}}\left(\mathcal{M}_{n}, \Lambda\right)$ in $(n-k)$-homology group of $n$ in $\left[\mathcal{M}_{M}\right]$ class taking the Thom Isomorphism in $M_{\text {cross-product }}^{\text {hom } o \text { log } y}$ for $H_{\wedge}$ as,

Suffice the form $H_{n^{\circ}-k}(M) \cong H^{n^{\circ}}$ for integers $n^{\circ}$; thereby establishing the Poincare duality.

V. $S p_{c}$ with Type-II Strings In Atiyah-Hirzebruch For Ramond-Ramond Sector

The $K$ — Theory for the operator and Topological aspects in the cohomology class; there exists distinct classifiers for the $D$ — Branes or Dirichlet Branes in the Ramond-Ramond (RR)—Sector of Type II-B Strings sufficing the $3$ — dim integral class property. There is the cohomology class for the transformationtwist giving the $\textit{mod} -2$ torsion quantum corrections considering the Freed-Witten discrepancies as and when considered in the peculiar $K$ — Theory in the reconciled aspects over Atiyah-Hirzebruch spectral sequence.

The non-trivial aspect to discuss in high energy physics for the Topological K-Theory taking the TypeII (II-A and II-B) superstrings is to consider the RR-fields in $P$ — form electrodynamics considering the $10\ -$ dim Supergravity for the potential $\mho^{\circ}$ over $\Omega_{P+1}$ —field defined through the Hodge duals $*_{d}$ in the form $\Omega_{9-P}^{* d}$ there exists $4\ -$ classifiers that will ultimately result the approach of $K\ -$ Theory in the complex Topological space $T^{*}$ on manifold $M$ over a representation $M_{T}^{*}$ relates not only the Atiyah-Singer Index Theorem (for the Fredholm modules, Bott-Periodicity as taken earlier) but also gives the Grothendieck—Riemann—Roch Theorem on bounded complex $\Lambda^{*}$ on sheaves $S_{\prime \prime}$ over a relation $S_{\prime \prime}^{\Lambda^{*}}$ taking the morphism $\sigma_{m}: X \rightarrow Y$ for $\sigma_{m}: A(X) \rightarrow A(Y)$ over the Tangent sheaf $T_{\Lambda^{*}}$ of $\Lambda^{*}$ on $\sigma_{m}$ ! to suffice $\operatorname{ch}\left(\sigma_{m} ! \Lambda^{*}\right)$ gives,

All suffice through the $4-$ classifiers as mentioned above $\mathrm{e}$^[27][28],

1. Hodge dual $*_{d}$
2. Gauge symmetry $g_{P-\text { form }}^{\circ}$
3. Equations of motion $\partial * g^{\circ}=* \mathbf{J}$ for $\mathbf{J}_{\mathrm{P} \text {-vector }}$
4. Charge density $C_{\rho}$ through the Lagrangian for $\zeta_{C_{\rho}}$ in $R R-$ fields for $\varpi_{10-P}$ through the D-Brane potential $(10-P)$ gives the equations of motion $S^{\times}$for $(10-P)$ having a replacement order of $P$ to $(7-P)$ for the previously taken charge density $C_{\rho}$ giving two non-trivial relations ^[29][30],
   1. De Rham Co hom $o \log y$ with $H$-twist for the exterior derivative $\partial$ with charge density $C_{\rho}$ for the parameter $\chi$ gives,
      $\partial \chi_{9-P}+H \times \Omega_{9-P}$
      $\begin{aligned}& =\partial \chi_{P+1} \\& =\partial^{2} \varpi_{7-P} \\& =C_{9-P}\end{aligned}$
   2. The action for Type II (II-B being both $T$ and S-dual to itself) for non-invariant GSO projections in subdomains where for the existence of 32-supercharges in Type II-B $\left(\mathcal{R}^{8,1} \times S^{1}\right)$ the action $S_{\prime \prime}$ of P-form electrodynamics on a manifold $M$ through gauge symmetry can be represented by $g_{P-\text { form }}^{\circ}$ gives,$S_{\prime \prime}=\int_{M}\left[\frac{1}{2} g^{\circ} \chi * g^{\circ}+(-1)^{P} B \chi * J\right]$
      Which gives the nilpotent potential in manifold $M$ over a spacetime coordinates $(\sigma, \tau)$ as,
      $\begin{gathered}\partial \Omega_{P+1}+\chi_{9-P}+\Omega_{(\sigma, \tau)} \\=\partial \Omega_{P+1}(\sigma, \tau) \\=\partial^{2} \varpi_{P(\sigma, \tau)} \\=0_{(\sigma, \tau)}\end{gathered}$

All of these suffice for $S p_{c}$ in the extension of Poincare duality in a generalized norm of orientability of homology theory taking the Thom Isomorphism in complex form of Topological $K-$ Theory relating Atiyah—Singer Index Theorem and Fredholm modules, Bott—Periodicity, Atiyah—Hirzebruch, Grothendieck—Riemann—Roch with KK-Theory ^{[31][32][33][34]}.

Additionally, to discuss furthermore about the Type II Superstrings formalism as associated with supergravity for a homology class there is a relation between the Dirac quantization conditions and RR-fields where in the Lie group structure ^[27][35][36]

The Photon being represented by $U(1)$ the related methodology of the charge quantization and the magnetic monopoles where their independent nature relates the breaking of gauge group from $D(1)$ heavy branes when the distance is infinite for a path $v$ suffice the relation^[35][36],

Considering a cycle $\sigma_{c y}$ in the homogeneous Lie group, the movement can ultimately results in lifting the Lie group that originates over identity structures through,

Where the $2-\operatorname{times}\left(\sigma_{c y}\right)$ where a covering parameter $\mathfrak{J}$ for $S O(2)$ can maintain the Type II superstring actions over the Twisted $K$ — Theory (over Topo log i cal norms). One category of Type II superstrings (Type II-B) which has been extended to $12-\operatorname{dim}$ where in the $\mathrm{t}^{\prime}$ Hooft limit, for Yang-Mills $N=4$, F-Theory being encountered under $\operatorname{SL}(2, \mathbb{Z})$, the D-Brane analogy being extended where there exists some non-trivial aspects being existent over RR-Fields and its relation to the Twisted K-Theory making up these points ^{[27][37][38][39]},

1. GSO — Projections for an eliminated Tachyon and preserved Supersymmetry.
2. Distinct classifiers for Type II into $I I_{I I-B}^{I I-A}$.
3. $S L(2, \mathbb{Z})$ for a $C F T$ for a worldsheet periodicity as concerned for Fermion—projections giving 3 sub—relations,
   1. Invariance over $S L(2, \mathbb{Z})$.
   2. Modular diffeomorphisms as expressed on Torus for Point [3] to get rid of gravitational anomalies.
      1. This in turn establishes the integral for Kalb — Ramond $(K-R)$ field with the relation to the $\mathbf{B}$ field for $\lambda$ as,
         $-\int_{K R} \lambda^{i} \lambda^{j} B_{i j}$

Thus, for the correspondence to $K R N S-N S B$ — field ; a far more concrete relation can be attained for $H-f l u x_{D-B r a n e}^{N S}$ where the $P-$ form for $P-$ skeleton represents a complicated structure later but for the cohomology integral coefficients for a D-Brane absent RR-flux the relation can be stated over ^{[27][37][38][35][36][39]}

Extending Type II for Type II-B the representation when made for a manifold $M$ for the group operators $O g$ in the quotient space $q$ with $q^{\partial-\text { rescalling }}$ Type II-B represents the Orientifold over the operator relation where $\partial$ in $\partial-$ rescalling being taken trivially for the involution parameter, the nonempty operator represents the orientifold for the operator $O g_{p}^{2}$ such that for the operator $P \sim$ there is Type II-B for,

Where through the splitting another structure represents $I I-A$ for the $(1-1)\ -$ form.

The $P\ -$ skeleton as stated above in turn gives the Topological K-Theory over $\mathbf{B}$ for the fibre $f$ in the cohomological space $M$. over a Serre fibration parameter $S_{f .}: M . \rightarrow B$. in the $(p, q)\ -$ norm representing the cohomolgy pair $\left(M_{\cdot(p)}, M_{\cdot(p, q)}\right)$ for $k^{t h}-$ co hom $o \log y$ group through,

For the Atiyah—Hirzebruch taking the space $M$. and the spectral sequence associated with it for the fibres $f$. there exists the $E_{n}$-sheet taking $(p, q)$ — norms for $E_{n}^{p, q}$ for $n$ taking the values $2,3, \infty$; the spectral sequence can be in respect of the differentials $E_{d}$ where there is,

1. Atiyah—Hirzebruch spectral sequence
2. Twisted K—Theory
3. Topological K—Theory
4. Algebraic K—Theory
5. Complex $\delta$
6. $E_{n}$ for different values of $n$ providing;
   1. Serre spectral sequence for $E_{1}$
   2. Topological K-Theory for $E_{2}^{p}$
   3. Twisted K-Theory for $E_{3}$ over the differential $E_{3_{(d)}}$ such that for the $E_{n}^{p, q} ; n$ takes an equality for $E_{2}$ and $E_{3}$.
   4. For [Point $b$ ] in complex parameter $\delta=2 k+1$ denoting complex projective $\mathbb{C P}^{\delta}$ there exists two foundations,
      1. Collapsing for even $2 k$
      2. Non collapsing for odd $2 k+1$
         1. Where Topological K-Theory as associated with Atiyah-Hirzebruch for $2 k+$ 1 over space $M$. ; a nice relation can be expressed in $E_{2}^{p, \delta .}(M$.$) .$
7. For the Kähler where any compact Kahler having Ricci flatness is a Calabi—Yau ^[40][41] for all the threefold being non—trivial in superstring theory, any Kähler (without any consideration of being compact) can give the twisted formalism of K—Theory for $E_{i}$ such that $i \equiv 4=\infty$.
8. Algebraic K—Theory having a relation to the étale cohomology ^[42][43][44] for the scheme $M_{T}^{e}$ where $M_{T}$ is a Topological space; any representation can be done in the local isomorphism such that for the category taking $M_{T}^{e}$ for etale representation et $\left(M_{T}^{e}\right)$ suffice isomorphism for the Topological space^{[45][46][47][48][49][50]} $T\left(\right.$ or $M_{T}$ ) which provides the Atiyah-Hirzebruch spectral sequence ^[27] for $E_{2}$ in $(p, q)$ — norms thereby establishing the Quillen—Lichtenbaum conjecture for $E_{2}^{p, q}$ with the $\dot{e} t a l e$ cohomology $M_{T}^{e}$.

VI. Discussions

Representing a finite group of two elements for a specified vector bundle acting on the Hilbert space for $A$-module being non—negative and self-adjoint constructions are made over a complex manifold such that taking that topological space through a pre—Hibert Hausdorff order there exists the $c^{*}$-algebra and Banach-algebra where the 3-points emphasized here a 8-periodic homotopy groups can be established through Bott Periodicity for Unitary, Orthogonal and symplectic category over a compact Hausdorff suffice K-Theory in the Topological and reduced form over two functors in (left—right) ring representation makes the well-defined Morita Equivalence taking the naturally induced isomorphism for those two functors in bimodular forms.

The linkage of $K-$ Theory to $K-$ Homology and $c^{*}$ — algebras have been established taking the same Morita equivalence for Kasparov's composition product where through the elliptic differential operator representing the Fredholm modules it has been established by Atiyah-Singer Index Theorem and Thom Isomorphism for the associated Chern character to establish the structures taking over the operator theory for the linear transformations with $n$ — dim vectors through a specified mapping parameter over $n$ — dim spaces for the complex tangent bundle channelizing the way to represent the dual category of the Hausdoff spaces in * — isomorphism that bounds the two subsets taken in this paper giving the relation between noncommutative geometry and noncommutative topology where the noncommutative topology is sufficed over a connectivity channelling from $K$ —Theory to $K$ — Homology considering operator $K-T h e o r y$ taking Baum — Connes conjecture through the 5 — classifiers for a concrete relation to $K K-T h e o r y$.

Considering the two categories of algebra $c^{*}$ and the Banach where for the defined group action for the associated 2-norms as defined for Banach; there's two spectrums satisfying $c^{*}$ taking the same group actions where a modified form of Fourier transform, i.e., a Gelfand Transform can be established and given related to the same Hausdorff space for a Gelfand—Neimark transform over $a^{*}$-isommorphism considered. The properties of noncommutative geometry can be perceived through a generalized notion occupying Borel measures, involutions and Harr measures given the second norms taking group action $\left({ }^{\wedge}\right)$ as $\ell_{++}\left({ }^{\wedge}\right)$. This also provides the subspace of $c^{*}$-algebra as $c_{s u b}^{*}$ over the previously mentioned transforms and associated parameters with the necessary mapping operator in index $x_{1}^{0}$ there is a $c_{r e d}^{*}$-algebras in the $k-h o m o l o g y$ for a right side assemble parameter taking the same Baum-cones conjecture for a classifying space $S_{c}^{c \ell}$. This torsion-free parameter in the A-module with Baum-Connes conjecture and the Hilbert A-module over the complex parameter $\delta$ gives the Poincare duality through the KK — Theory over another complex integer $\wedge$ where Thom isomorphisms have been considered for a Topological K-Theory and the $S p_{c}$-structure over the associated Chern class in the $n$-dim manifold.

Different forms of K—Theory as Twisted, Topological, Algebraic is considered taking $E_{n}^{p, q}$ and $E_{n}^{p}$ — norms for defined value of $n=1,2,3,4=\infty$ where the last value is expressed in terms of Kahler manifold (which if is compact with vanishing Ricci curvature can give the Calabi-Yau and iff this CY is of threefold then a nontrivial expression of string theory is defined for various values of supersymmetry). The K-Theory in the cohomology class with the topological aspects gives the Ramond-Ramond sector sufficing 3-dim integral class property for Type-II(B) strings. For the potentials of supergravity and the classifiers that are concerned through various representation-forms give the Atiyah-Singer Index Theorem with Fredholm modules and the Grothendieck-Riemann-Roch Theorem through the equations presented in the paper.

De Rahm cohomology with $\mathrm{H}$—Twist is taken with a charge density and GSO-Projections for concerned Type-II action (on II-A and II-B) in P-form electrodynamics. Along with RR-fields the NS-NS B-field in the same P-form over P-skeleton where the NS 3-form is shown for the RR-flux on D-Brane RR-flux on supergravity. For the extension of Type-II(A) being considered through splitting and (1-1)-forms where the extended notion of Serre fibration is shown and the Atiyah—Hirzebruch spectral sequence is established for $E_{n}$ — sheets with $n=1,2,3,4=\infty$ giving distinct categories of $\mathrm{K}$-Theories.

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The authors have no conflict of interest related to this paper.