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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

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

and so, 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:

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

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 hence:

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

with {X^\mu }\left( {\tau ,\sigma } \right) the embedding of the string in target space, {\gamma _{ab}} the worldsheet metric, and {G_{\mu \nu }} the spacetime metric. The renormalization Lie-equation for the sigma-model on the string worldsheet entails that

    \[{\beta _{\mu \nu }} = \frac{{{\rm{d}}{G_{\mu \nu }}}}{{{\rm{dlog}}\Lambda }} = \alpha '{R_{\mu \nu }} + O\left( {{{\alpha '}^2}} \right) = 0\]

So, from

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

the Einstein vacuum field equation is encoded in the quantum structure of the conformal worldsheet theory as characterized by

    \[{\vartheta _\mu }\left( u \right){\vartheta _\nu }\left( v \right) = \lambda _{\mu \nu }^\rho \frac{{{\vartheta _\rho }}}{{{{\left( {u - v} \right)}^\# }}} + ...\]

Therefore, the β-function equation gives a definition of the stress-energy tensor for the worldsheet theory

    \[{\beta ^{\mu \nu }} = {T^{\mu \nu }} = \frac{{\delta \,{\Gamma _{eff}}}}{{\delta {G_{\mu \nu }}}}\]

with

    \[{e^{ - {\Gamma _{eff}}}} = \exp \left( {\int {{d^2}\sigma {G_{\mu \nu }}{T^{\mu \nu }}} } \right)\]

By holographic renormalization, it follows that the worldsheet quantum theory knows all about spacetime physics

as attested by

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

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