Clifford-Kähler Algebras and M-Theory Compactification

Continuing from my work on the relation between Clifford algebraic symmetries and M-theory, here I will initiate an analysis of compactification via the derived Kähler-Atiyah bundle associated with Clifford-Kähler manifolds. Recall that whenever 2 or more D-branes coincide, there is a Clifford algebraic symmetry whose generators allow us to derive the total action:

    \[\begin{array}{*{20}{l}}{{S^{Total}} = \frac{1}{{2\pi {\alpha ^\dagger }12}}\int\limits_{{\rm{world - volumes}}} {{d^{26}}} x{\mkern 1mu} d{\mkern 1mu} \Omega {{\left( {{\phi _{Inst}}} \right)}^2}\sqrt {\frac{{ - {g_{\mu \nu }}}}{{ - \gamma }}} {\mkern 1mu} {e^{ - {c_{2n}}/{\Upsilon _\kappa }(\cos \varphi )}} \cdot }\\{\left( {{R_{icci}} - 4{{\left( {{{\not D}^{SuSy}}\left( {{\phi _{Inst}}} \right)} \right)}^2}} \right) + \frac{1}{{12}}H_{3,\mu \nu \lambda }^bH_3^{b,\mu \nu \lambda }/A_\mu ^H + \sum\limits_{D - p - branes} {S_{Dp}^{WV}} }\end{array}\]

and since D-p-branes are metaplectic solitons in closed string-theory, by the von Neumann boundary condition, there is a natural coupling of the super-Higgs field A_\mu ^H to the world-sheet of a string through its boundary:

    \[{S_{open}} = {S_{cld}} + \int\limits_{{\rm{end - points}}} {d\tau } A_\mu ^H{\bar X^\mu }{e^{ - H_3^b}}d{\mkern 1mu} \Omega {\left( {{\phi _{Inst}}} \right)^{\exp {\kern 1pt} ({c_{2n}}/{\Upsilon _\kappa }(\cos \varphi )}}\]

Hence, A_\mu ^H lives on a p+1 dimensional subspace with a {\Upsilon _\kappa }(\cos \varphi ) contribution, yielding the world-volume action:

    \[S_{Dp}^{WV} = {S_{cld}} + \int\limits_{{\rm{end - points}}} {d\tau } A_\mu ^H{\bar X^\mu }d{\mkern 1mu} \Omega {\left( {{\phi _{Inst}}} \right)^2}{e^{ - \left( {H_3^b} \right)/{\Upsilon _\kappa }(\cos \varphi )}} + {e^{{c_{2n}}/{\Upsilon _\kappa }(\cos \varphi )}}/H_3^b\]

and since world-volumes have conformal invariance, by solving the n-loop level equation of motion:

    \[\left\{ {\begin{array}{*{20}{c}}{{R_{\mu \nu }} = - \frac{1}{4}H_{3,\mu \lambda l}^bH_{3,\nu }^{b,\lambda \rho } + 2{{\not D}_\mu }^S{{\left( {{\phi _{Inst}}} \right)}^2}{{\not D}_\nu }^S{{\left( {{\phi _{Inst}}} \right)}^{1/2}}}\\{{{\not D}_\lambda }^S{\phi _{Inst}}H_{3,\lambda \mu \nu }^b - 4{{\not D}_\mu }^S{{\left( {{\phi _{Inst}}} \right)}^2}}\\{4{{\not D}_l}^S{\phi _{Inst}} - 4\left( {{{\not D}_\mu }^S{{\left( {{\phi _{Inst}}} \right)}^2}} \right) = {R_{icci}} + \frac{1}{{12}}{{\left( {H_3^b} \right)}^2}}\end{array}} \right.\]

we get the total world-volume action:

    \[\begin{array}{*{20}{c}}{S_{Dp}^{WV} = {\mkern 1mu} - {T_p}\int {{d^{p + 1}}} x{\mkern 1mu} {e^{ - {\phi _{Inst}}}}{\rm{Tr}}\left( {1 + \frac{1}{4}\left( {{F_{\mu \nu }} + d{\mkern 1mu} \Omega {{\left( {{\phi _{Inst}}} \right)}^2}} \right) + {{\left( {2\pi {\alpha ^\dagger }} \right)}^{ - 1}}{b_{\mu \nu }}} \right) \cdot }\\{\left( {A_\mu ^H + {{\left( {2\pi {\alpha ^\dagger }} \right)}^{ - 1}}{b_{\mu \nu }}} \right) + \frac{1}{2}{{\not D}^S}{\Upsilon _\kappa }(\cos \varphi ) - \frac{1}{4}\sum\limits_{i \times j} {\left[ {\Upsilon _{2\kappa }^i(\cos \varphi ,\Upsilon _{2\kappa }^j(\cos \varphi } \right]} \cdot }\\{\left[ {\Upsilon _{2\kappa }^j(\cos \varphi ),\Upsilon _{2\kappa }^i(\cos \varphi )} \right]}\end{array}\]

and since Clifford algebras are a quantization of target-space exterior algebras, via Gaussian matrix-elimination, we can expand, via Green’s-function, {\not D^{SuSy}}, the supersymmetry group covariant derivative, which I have redundantly simplified to {\not D^S} and by supersymmetry, the total action of M-Theory becomes:

    \[\begin{array}{*{20}{l}}{{S_M} = \frac{1}{{{k^9}}}\int\limits_{world - vol} {{d^{11}}} \sqrt {\frac{{ - {g_{\mu \nu }}}}{{ - \gamma }}} T_p^{10}d\Omega {{\left( {{\phi _{Inst}}} \right)}^{26}}\left( {{R_{icci}} - A_\mu ^H\frac{1}{{48}}G_4^2} \right)}\\{ + \sum\limits_{Dp} {\not D_\mu ^S} {e^{ - H_3^b}}/S_{Dp}^{WV} + \sum\limits_{Dp} {\not D_\mu ^S} {e^{ - H_3^b}}/{S^{Total}}}\end{array}\]

with {T_p} \sim {\alpha ^\dagger }\frac{{p + 1}}{2} the D-p-brane world-volume tension, and:

    \[{F_{\mu \nu }} = {\partial _\mu }A_\mu ^H - {\partial _\nu }\bar A_\mu ^H + \left[ {A_\mu ^H,\Upsilon _{2\kappa }^i(\cos \varphi )} \right]\]

the Yang-Mills field strength, and by a Paton-Chern-Simons factor, we get:

    \[\left[ {A_\mu ^H,A_\nu ^H} \right] = \sum\limits_{k = 1}^N {A_\mu ^{H,ac}} A_\nu ^{H,cb} - A_\nu ^{H,ac}A_\mu ^{H,cb}\]

{\phi _{Inst}} the instanton field, with:

    \[{e^{ - {\phi _{Inst}}{g_{\mu \nu }}}} = {e^{ - 2{\phi _{Inst}}\left( {{g_{\mu \nu }} - 1} \right)}}\]

here: {g_{\mu \nu }} = {e^{{{\left( {{\phi _{Inst}}} \right)}^2}}}.

In N=2 compactifications of 11-dimensional supergravity down to a 3-dimensional (anti)-deSitter space, the internal eight-D-manifold M comes equipped with a Riemannian metric g and a Kähler 1-form f and a 4-form F encoding the 4-form field strength of the 11-dimensional theory. Moreover, the Majorana spinor is a section of the real spin bundle S of M. The background conserves exactly N=2 supersymmetry in 3-dimensions, equivalent to the condition that the real vector-space of solutions to the following differentio-algebraic system of generalized Killing spinor equations:

    \[{D^K}\xi ' = Q\xi ' = 0\]

is two-dimensional, with Q an endomorphism of S:

    \[\begin{array}{l}Q = \frac{1}{2}{{\tilde \gamma }^m}{\partial _m}\Delta - \frac{1}{{288}}{F_{mpqr}}{{\tilde \gamma }^{mpqr}}\\ - \frac{1}{6}{f_{\hat P}}{{\tilde \gamma }^{\hat P}}{{\tilde \gamma }^{\left( 9 \right)}} - \kappa {{\tilde \gamma }^{\left( 9 \right)}}\end{array}\]

{D^K} = {\nabla ^S} the Kähler connection on S, and {\nabla ^S} is the connection induced on S by the Levi-Civita connection of \left( {M,g} \right) where:

    \[A = {\rm{d}}{x^m} \otimes {A_m} \in {\Omega ^1}\left( {M,{\rm{End}}\left( S \right)} \right)\]

 is an {\rm{End}}\left( S \right)-valued 1-form on M, with:

    \[\begin{array}{c}{A_m} = \frac{1}{4}{f_{\hat P}}{{\tilde \gamma }_m}^{\hat P}{{\tilde \gamma }^{\left( 9 \right)}} + \frac{1}{{24}}{F_{mpqr}}{{\tilde \gamma }^{pqr}}\\ + \kappa {{\tilde \gamma }_m}{{\tilde \gamma }^{\left( 9 \right)}} \in \Gamma \left( {{\rm{End}}\left( S \right)} \right)\end{array}\]

with \kappa related to the cosmological constant \Lambda of {\rm{Ad}}{{\rm{S}}_3} as:

    \[\Lambda = - 8{\kappa ^2}\]

The F-motivated centrality of the chirality constraint:

    \[{\tilde \gamma ^{\left( 9 \right)}}\xi ' = + \xi \]

allows the construction of the spin bundle S of a (pseudo)-Riemannian manifold \left( {M,g} \right) with signature \left( {p,q} \right) of dimension d = p + q describable by an S-bundle of modules over the Clifford bundle {\rm{CL}}\left( {{T^ * }M} \right) of the cotangent bundle of {{T^ * }M} endowed with a metric \hat g induced by {g^ * }. The initial problem is that the Clifford bundle is determined by \left( {M,g} \right) only up to isomorphism, so the association of {\rm{CL}}\left( {{T^ * }M} \right) to \left( {M,g} \right) is not functorial. One typically goes around this problem by the invoking the Kähler-Atiyah Clifford-realization bundle of \left( {M,g} \right).

This is the Chevalley-Riesz realization and it yields:

    \[{\rm{Cl}}\left( {{T^ * }M} \right) \cong \wedge {T^ * }M\]

where \wedge {T^ * }M is the exterior algebra of M and the Clifford product of {\rm{CL}}\left( {{T^ * }M} \right) is non-commutative, and the fiberwise multiplication on \wedge {T^ * }M^\circ, transforms \wedge {T^ * }M into the Kähler-Atiyah bundle \left( { \wedge {T^ * }M, \circ } \right).

Hence, the corresponding \mathbb{Z}-grading admits an expansion into a finite sum of homogeneous-degree - 2\kappa binary operations {\Delta _k},\;k = 0,...,d satisfying:

and the parity automorphism \pi is given by:

    \[\pi \overbrace = ^{def.} \otimes _{k = 0}^d{\left( { - 1} \right)^k}{\rm{i}}{{\rm{d}}_{{ \wedge ^k}{T^ * }M}}\]

with:

    \[{\Delta _k}: \wedge {T^ * }M{ \times _M} \wedge {T^ * }M \to {T^ * }M\]

the Kähler-Atiyah generalized products.

Hence, the expansion:

 

is a semiclassical expansion of the geometric product yielding a geometric quantization where the Planck constant is inversely related to the scale of the metric g, thus allowing us to derive:

    \[\begin{array}{c}\omega {\Delta _{k + 1}}\eta = \frac{1}{{k + 1}}{g^{ab}}\left( {{e_a}{,_\parallel }\omega } \right){\Delta _k}\left( {{e_a}{,_\parallel }\eta } \right)\\ = {g_{ab}}\left( {{\iota _{{e^b}}}\eta } \right)\end{array}\]

with \iota the interior product.

Hence, our spin bundle S can be interpreted as a bundle of modules over the Kähler-Atiyah bundle of \left( {M,g} \right), with the module structure defined by a morphism of bundle of algebras:

    \[\tilde \gamma :\left( { \wedge {T^ * }M, \circ } \right) \to \left( {{\rm{End}}\left( S \right), \circ } \right)\]

where \tilde \gamma is fiberwise-irreducible.

With {\left( {{e_m}} \right)_{m = 1...8}} the local TM-frame, {\left( {{e^m}} \right)^{m...8}} the dual {T^ * }M-co-frame, satisfying:

    \[\left\{ {\begin{array}{*{20}{c}}{{e^m}\left( {{e_n}} \right) = \delta _n^m}\\{\hat g\left( {{e^m},{e^n}} \right) = {g^{mn}}}\end{array}} \right.\]

with:

    \[\Omega \left( M \right)\overbrace = ^{{\rm{def}}}\Gamma \left( {M, \wedge {T^ * }M} \right)\]

the space of smooth inhomogeneous globally defined differential forms on M, and a form \omega expands as:

    \[\omega = \sum\limits_{k = 0}^d {{\omega ^{\left( k \right)}}} { = _U}\sum\limits_{k = 0}^d {\frac{1}{{k!}}\omega _{{a_1}...{a_k}}^{\left( k \right)}} {e^{{a_1}...{a_k}}}\]

    \[{\omega ^{\left( k \right)}} \in {\Omega ^k}\left( M \right)\]

with:

    \[{e^{{a_1}...{a_k}}}\overbrace = ^{{\rm{def}}}{e^{{a_1}}} \wedge ... \wedge {e^{{a^k}}}\]

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