I last derived spacetime physics from the string worldsheet, here I shall derive 4-D physics on exotic manifolds in the context of super-string theory. Before proceeding, recall the key steps: since the solution to the Lagrange multiplier equation of motion is
with its Clifford form
for any scalar. Hence, for , we have:
Now, by solving via a metric:
we get the Dp action:
And since in 4-dimensional space-time, the mass of a Dp-brane can be derived as:
by T-dualizing in the
direction and factoring the dilaton, the dual is hence:
the Polyakov action for the string is given as:
with the embedding of the string in target space, the worldsheet metric, and the spacetime metric. The renormalization Lie-equation for the sigma-model on the string worldsheet entails that
the Einstein vacuum field equation is hence encoded in the quantum structure of the conformal worldsheet theory as characterized by
Therefore, the β-function equation gives a definition of the stress-energy tensor for the worldsheet theory
By holographic renormalization, it follows that the worldsheet quantum theory knows all about spacetime physics
as attested by
The key to this post is the avoidance of the standard methods of deriving 4-D physics via compactification, flux stabilization, Dp-brane-theory or first-order AdS/CFT correspondence/holography and will attempt a purely geometric method based on exotic smooth 4-D topological manifolds.
To appreciate the uniqueness of 4-D manifolds, and why M-theory is essential for 4-D physics, absorb the following facts, from Alexandru Scorpan
and so, “there goes the neighborhood”:
This is where the magic of superstring theory comes in. When formulated on backgrounds containing 4-dimensional exotic rather than standard , topological surgical tensor-network methods allow 4-D exotic to orbifoldize with worldsheets of strings. So we re-capture the fact that in string theory, the notion of space-time as a smooth manifold is no longer valid due to vacuum fluctuations of the worldsheet, which is ‘complex’, where 4-D physics emerges as a classical-limit with a geometry (M, g, B), and such exotic relative to the background, are classified by the integral classes of -geometric objects representing the third cohomologies of the associated complex line bundles: hence the existence of Abelian gerbes, and the presence of B-field such that H is the curvature of such a gerbe means that the correct, semi-classical, geometry for string theory is the one based on Abelian gerbes supplementing Riemannian geometry and the dilaton is constant and vanishes, and the β-function forces the background be non-flat unless is zero.
Here’s a deep theorem proven by Torsten Asselmeyer-Maluga and Jerzy Król:
For any family of exotic ‘s in the radial family whose members are embedded in standard , there exists an associated family of 3-spheres in the boundaries of the Akbulut-corks for these exotic ’s and each such exotic 4-fold from that family is determined by the codimension-1 foliation of the corresponding 3-sphere, with non-vanishing Godbillon–Vey class in . The radius in the family, , and value of being related as . Yet, the classification of D-branes in string backgrounds is characterized by K-theory of the background.
Here is where M-theory comes in. Without loss of generality, we can take the bosonic -WZW model. Now, as , D-branes in G-manifolds are determined by wrapping the conjugacy classes of , and so there are k + 1 …