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:
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 to the world-sheet of a string through its boundary:
Hence, lives on a p+1 dimensional subspace with a contribution, yielding the world-volume action:
we get the total world-volume action:
and since Clifford algebras are a quantization of target-space exterior algebras, via Gaussian matrix-elimination, we can expand, via Green’s-function, , the supersymmetry group covariant derivative, which I have redundantly simplified to and by supersymmetry, the total action of M-Theory becomes:
with the D-p-brane world-volume tension, and:
the instanton field, with:
In N=2 compactifications of 11-dimensional supergravity down to a 3-dimensional (anti)-deSitter space, the internal eight-D-manifold comes equipped with a Riemannian metric and a Kähler 1-form and a 4-form encoding the 4-form field strength of the 11-dimensional theory. Moreover, the Majorana spinor is a section of the real spin bundle of . 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:
is two-dimensional, with an endomorphism of :
the Kähler connection on , and is the connection induced on by the Levi-Civita connection of where:
is an -valued 1-form on , with:
with related to the cosmological constant of as:
The F-motivated centrality of the chirality constraint:
allows the construction of the spin bundle of a (pseudo)-Riemannian manifold with signature of dimension describable by an -bundle of modules over the Clifford bundle of the cotangent bundle of endowed with a metric induced by . The initial problem is that the Clifford bundle is determined by only up to isomorphism, so the association of to is not functorial. One typically goes around this problem by the invoking the Kähler-Atiyah Clifford-realization bundle of .
This is the Chevalley-Riesz realization and it yields:
where is the exterior algebra of and the Clifford product of is non-commutative, and the fiberwise multiplication on : , transforms into the Kähler-Atiyah bundle .
Hence, the corresponding -grading admits an expansion into a finite sum of homogeneous-degree binary operations satisfying:
and the parity automorphism is given by:
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 , thus allowing us to derive:
with the interior product.
Hence, our spin bundle can be interpreted as a bundle of modules over the Kähler-Atiyah bundle of , with the module structure defined by a morphism of bundle of algebras:
where is fiberwise-irreducible.
With the local -frame, the dual -co-frame, satisfying:
the space of smooth inhomogeneous globally defined differential forms on , and a form expands as: