How the String Theory Worldsheet ‘Knows’ All About Spacetime Physics

By George Shiber Sunday, June 26, 2016Friday, October 26, 2018 In Cosmology, Grand Unification Physics, M Theory, Mathematical Physics, Mathematics, Philosophy Of Physics, Physics, Super String Theory, Supersymmetry Dp-Branes, String/M-Theory

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:

By matrix world-volume integral reduction on

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

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

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