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.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:~ G. H. Hardy, A Mathematician’s Apology (London 1941). Quotations by Hardy. Gap.dcs.st-and.ac.uk. Retrieved on 27 November 2013.

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.

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