• Among the many truly remarkable properties of M-theory, that it is a unified theory of all interactions, including quantum gravity, and gives a completely well-defined analytic S-matrix satisfying all the axioms for a physically acceptable theory entailing Lorentz invariance, macro-causality and unitarity is perhaps the deepest, and to boot, the only quantum gravity paradigm that has that essential feature. Here, I will discuss some key aspects of nonlocality and space-time uncertainty in string theory. Let us start with an action smoothly interpolating between the area preserving Schild action and the fully reparametrization invariant Nambu–Goto action:

where $\Phi \left( \sigma \right)$ is an auxiliary world-sheet field, ${\gamma _{mn}} \equiv {\eta _{\mu \nu }}{\partial _m}{X^\mu }{\partial _n}{X^\nu }$ the induced metric on the string Euclidean world-sheet ${x^\mu } = {X^\mu }\left( \sigma \right)$, and ${\mu _0} \equiv 1/2\pi \alpha '$ is the string tension. Combining, we get the Nambu-Goto-Schild action:

And to make the Nambu-Goto-Schild action quadratic in space-time coordinates, we use the Virasoro constraint and an auxiliary field that transforms as a world-sheet scalar and as an anti-symmetric tensor with respect to the space-time indices:

to yield:

Before proceeding, let us get some clarity.

• 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 $A_\mu ^H$ to the world-sheet of a string through its boundary:

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

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