How 4-D Physics Emerges from SuperString Theory Based on Exotic 4-D Manifolds

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

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

with its Clifford form

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

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

\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)}

And 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}

By matrix world-volume integral reduction on

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)}

the Polyakov action for the string is given as:

{S^p} - \frac{1}{{4\pi \alpha '}}\int {{d^2}} \sigma \sqrt { - \gamma } {\gamma ^{ab}}{\not \partial _a}{X^\mu }{\not \partial _b}{X^\nu }{G_{\mu \nu }}

How the String Theory Worldsheet 'Knows' All About Spacetime Physics

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:

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}

Thus, 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}}, yielding the two following key relations:

\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

\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}}}