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 ${\Psi _{scalar}}$ any scalar. Hence, for ${v_a}$, we have:

Now, by solving via a $Dp \times Dp$ metric:

$E_{\mu \nu }^m = {G_{\mu \nu }} + {B_{\mu \nu }}$

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:

By matrix world-volume integral reduction on

the Polyakov action for the string is given as:

This is how the worldsheet quantum theory knows all about spacetime physics. Setting the stage first. Since branes are 'generalizations', and are BPS, the supergravity solution in the multi-brane harmonic function form is:

with:

Thus, I can now derive:

Hence, the integral is:

After checking renormalization, one gets:

which is the correct harmonic function for a D(p+1)-brane. The relevance of $H_{p + 1}^{{\rm{array}}}$ is that via Green's functional analysis, it yields the string coupling of the dual 25-D theory:

${e^{{\Phi _{bos}}}} = {e^{\Phi _{bos}^{{e^{{\phi _{si}}}}}}}\frac{{{{\alpha '}^{1/2}}}}{{2\pi nR}}$

which is key to the T-duality transformation properties of propagating background matter fields in 4-dimensional space-time, with ${\Phi _{bos}}$ the bosonic field configuration corresponding to the string world-sheet, whose variable is ${\phi _{si}}$, yielding the two following key relations:

and