A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. ~ G. H. Hardy, A Mathematician’s Apology (London 1941). Quotations by Hardy. Gap.dcs.st-and.ac.uk. Retrieved on 27 November 2013.
First draft, part one. Nick Huggett argues that target space cannot be space … “that observed, ‘phenomenal’ space is not target space, since a space cannot have both a determinate and indeterminate radius: instead phenomenal space must be a higher-level phenomenon, not fundamental.” I agree and here, I shall give mathematical ‘support’ for that claim. Let me first derive the action of T-dual Dp-branes with the canonical representation being the supergravity solution. Start by T-dualizing in a direction transverse to a Dp-brane lying in the direction and thus the brane lives on a circle of radius that represents the metaplectic compactification space with a dual radius . So, we have an infinite array of identical branes on the line with coordinates , a distance apart, with:

Since the branes are BPS, the supergravity solution in the multi-brane harmonic function form is:

with:

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 is that via Green’s functional analysis, it yields the string coupling of the dual 25-D theory:

which is key to the T-duality transformation properties of propagating background matter fields in 4-dimensional space-time, with the bosonic field configuration corresponding to the string world-sheet, whose variable is . Now, let and be background fields. One T-dualizes in the bosonic direction , which is the direction of the compactifying circle , whose radius is with dual radius , for circle . In the 2-dimensional sigma model, with:

one can use a Lagrange multiplier to derive the action:

since the Lagrange multiplier equation of motion has solution:

given that its Clifford form is:

for any scalar. So, for :

Now, by solving via a metric:

we get the Dp action:

We are almost there. Now, 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:

and now we are in a position to go for a reductio ad absurdum – if target space = space, the sum:

would imply that 4-dimensional space-time cannot allow the graviton to couple with the Higgs field, and that would violate unitarity and unification; and string-sheet quantum fluctuational creation of the graviton would not be possible: hence by the Atiyah-Singer Index theorem, the universe would not have matter, given . Therefore, Nick Huggett is right … “phenomenal space must be a higher-level phenomenon, not fundamental” and not target-space if it is to have gravitonic/Higgs coupling in 4-dimensional space-time describing a universe with ‘matter’. To be polished and continued…

We are not to tell nature what she’s gotta be. … She’s always got better imagination than we have. ~ Sir Douglas Robb Lectures, University of Auckland (1979); lecture 1, Photons: Corpuscles of Light.