Why Target Space ≠ Space: The Role Of T-Duality In String Theory In Proving The Inequality

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 {X^1},...,{X^p} direction and thus the brane lives on a circle of radius R that represents the metaplectic compactification space with a dual radius R' = \alpha '/R. So, we have an infinite array of identical branes on the line with coordinates {X^{p + 1}}, a distance 2\pi R apart, with:

{X^{p + 1}} \sim {X^{p + 1}} + 2\pi R

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

    \[H_p^{{\rm{array}}} = 1 + \sum\limits_{n = - \infty }^{ + \infty } {\frac{{r_p^{7 - p}}}{{{{\left| {{{\widetilde r}^2} + {{\left( {{X^{p + 1}} - 2\pi nR} \right)}^2}} \right|}^{(7 - p)/2}}}}} \]

with:

    \[{r^2} = {\left( {{X^{p + 1}}} \right)^2} + {\left( {{X^{p + 2}}} \right)^2} + ... + {\left( {{X^{p + 9}}} \right)^2} = {\widetilde r^2} + {\left( {{X^{p + 1}}} \right)^2}\]

I can now derive:

    \[H_p^{{\rm{array}}} \sim 1 + \frac{{r_p^{7 - p}}}{{2\pi R}}\frac{1}{{{{\widetilde r}^{6 - p}}}}\int\limits_{ - \infty }^\infty {\frac{{du}}{{{{\left( {1 + {u^2}} \right)}^{\left( {7 - p} \right)/2}}}}} \]

Hence, the integral is:

    \[\int\limits_{n = - \infty }^{ + \infty } {\frac{{du}}{{{{\left( {1 + {u^2}} \right)}^{\left( {7 - {p^n}} \right)/2}}}}} = \frac{{\sqrt {2\pi n{R^n}} {\mkern 1mu} \Gamma \left[ {\frac{1}{2}\left( {6 - {p^n}} \right)} \right]}}{{\Gamma \left[ {\frac{1}{2}\left( {7 - {p^n}} \right)} \right]}}\]

After checking renormalization, one gets:

    \[H_p^{{\rm{array}}} \sim H_{p + 1}^{{\rm{array}}} = 1 + \frac{{\sqrt {\alpha '} r_{p + 1}^{7 - \,\left( {p + 1} \right)}}}{R}\frac{1}{{{{\widetilde r}^{7 - \,\left( {p + 1} \right)}}}}\]

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}}. Now, let {G_{\mu \nu }} and {B_{\mu \nu }} be background fields. One T-dualizes in the bosonic direction X_{bos}^{25}, which is the direction of the compactifying circle , whose radius is R with dual radius R' = \alpha '/R, for circle X_{bos}^{\dagger ,25}. In the 2-dimensional sigma model, with:

{G_{\mu \nu }}\quad \quad {B_{\mu \nu }}\quad \quad {\Phi _{bos}}

one can use a Lagrange multiplier to derive the action:

    \[\begin{array}{c}{S_\sigma } = \frac{1}{{4\pi \alpha '}} + \int\limits_{{\rm{endpoints}}} {{d^2}} \sigma {g^{1/2}}\left\{ {{g^{ab}}d\,\Omega {{\left( {{\phi _{INST}}} \right)}^2}\left[ {{G_{25,25,{v_a}{v_b}}} + 2{G_{25,\mu }}{v_a}{{\not \partial }_b}{X^\mu } + {G_{\mu \nu }}{{\not \partial }_a}{X^\nu }} \right]} \right\} + \\\frac{1}{{{{\left( {4\pi \alpha '} \right)}^2}}}\int\limits_{{\rm{worldsheets}}} {i{\varepsilon ^{ab}}} \left[ {2{B_{25,\mu }}{v_a}{{\not \partial }_b}{X^\mu } + {B_{\mu \nu }}{{\not \partial }_a}{X^\mu }{{\not \partial }_b}{X^\nu }d\,\Omega {{\left( {{\phi _{INST}}} \right)}^{ - 2}} + 2{{X'}_{25}}\,{{\not \partial }_{a,{v_b}}}} \right] + \\\frac{1}{{{{\left( {4\pi \alpha '} \right)}^3}}}\int\limits_{{\rm{worldvolumes}}} {i{\varepsilon ^{ab}}} \not D_\mu ^{susy}{\phi _{si}}\left( {\exp \left( {{e^{{\Phi _{bos}}}}\frac{{{{\alpha '}^{1/2}}}}{R}} \right)} \right) + \not D_\nu ^{susy}\widetilde {{\phi _{si}}}\left( {\exp \left( {{e^{{{\widetilde \Phi }_{bos}}}}\frac{{{{\alpha '}^{1/2}}}}{R}} \right)} \right)\end{array}\]

since the Lagrange multiplier equation of motion has solution:

    \[{v_b} = {\not \partial _b}\Psi _{scalar}^{{e^{H_{p + 1}^{{\rm{array}}}}}}\]

given that its Clifford form is:

    \[\frac{{{{\not D}^{susy}}L}}{{\not \partial {{X'}^{25}}}} = i{\varepsilon ^{ab}}{\not \partial _a}{v_b} = 0\]

for {\Psi _{scalar}} any scalar. So, for {v_a}:

    \[\frac{{\not D_\mu ^{susy}L}}{{{{\not \partial }_{{v_a}}}}} - {\mkern 1mu} \frac{\partial }{{{{\not \partial }_{{\sigma _b}}}}}\left( {\frac{{\not D_\nu ^{susy}L}}{{\not \partial \left( {{{\not \partial }_{b,{v_a}}}} \right)}}} \right) = 0 = {g^{ab}}\left[ {{G_{25,25,{v_a}}} + {G_{25,\mu }}{X^\mu }} \right] + i{\varepsilon ^{ab}}\left[ {{B_{25,\mu }}\not \partial {X^\mu } + {{\not \partial }_b}{X^{25}}} \right]\]

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

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

we get the Dp action:

    \[S_p^D = \, - {T_p}\int\limits_{{\rm{worldvolumes}}} {{d^{p + 1}}} \xi \frac{{\not D_{\mu \nu }^{susy}L}}{{{{\not \partial }_{{v_a}}}}}{e^{ - {\Phi _{bos}}}}{\rm{de}}{{\rm{t}}^{1/2}}G_{ab}^{\exp \left( {H_{p + 1}^{{\rm{array}}}} \right)}\]

We are almost there. Now, since in 4-dimensional space-time, the mass of a Dp-brane can be derived as:

    \[{T_p}{e^{ - {\Phi _{bos}}}}\prod\limits_{i = 1}^p {\left( {2\pi nR} \right)} \]

by T-dualizing in the

    \[{X^p}\]

direction and factoring the dilaton, the dual is hence:

    \[\begin{array}{c}({T_p}\left( {2\pi \sqrt {\alpha '} } \right){e^{ - {\Phi _{bos}}}}\prod\limits_{i = 1}^{p - 1} {\left( {2\pi {R_p}} \right)} = \\{T_{p - 1}}{e^{ - \Phi _{bos}^{1/2}}}\prod\limits_{i = 1}^{p - 1} {\left( {2\pi {R_{p - 1}}} \right)} \end{array}\]

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

    \[\frac{d}{{{d_{{\sigma _p}}}}}\int\limits_{{\rm{worldvolumes}}}^p {{e^{H_{p + 1}^{{\rm{array}}}}}} + \underbrace {\sum\limits_{{\sigma _p}}^D {{{\left( {S_p^D} \right)}^{ - H_{p + 1}^{{\rm{array}}}}}} }_{{\rm{topologies}}}\]

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 e = m{c^2}. 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.