M-Theory and how the Quantum Worldsheet Knows 4-D Spacetime Physics

Here, I will demonstrate how M-theory implies that the string worldsheet quantum theory ‘knows’ all about four-dimensional physics. Start with the Dp-action:

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

4-dimensionality entails that the mass of a Dp-brane can be derived as:

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

and by T-dualizing in the


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:

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

Flux Compactifications of String Theory and Kähler/Calabi–Yau manifolds

In this post, I will analyse several deep issues pertaining to flux compactifications of string theory, and draw foundational conclusions in relation to Kähler and Calabi–Yau manifolds. I already discussed the connection with Hodge theory and Gromov-Witten invariants of Calabi-Yau 3-Folds. The key thing to realize is that fluxes present in 10/11-D string theory naturally stabilize the moduli of a compactification with a downward reduction to 4-dimensional Minkowskian manifolds with broken supersymmetry minimally affecting the quantum vacuum energy as well as a ‘natural’ non-fine-tuned solution to the hierarchy problem: that is their importance for phenomenology, not just theory. Start with the ten-dimensional type IIB supersymmetry transformations:


    \[\delta \lambda = \frac{1}{\kappa }{\Gamma ^M}{P_M}{B^{\left( {10} \right) * }}{\varepsilon ^ * } + \frac{1}{{24}}{\Gamma ^{MNP}}{G_{MNP}}\varepsilon \]

with the Gromov-Witten genus-one amplitude given by:

    \[{A^{g = 1}} = {\hat R^4}\int_{{{\hat F}_{\left( 1 \right)}}} {\frac{{{d^2}\tau }}{{\tau _2^2}}} \int_{\hat {\rm T}} {\prod\limits_{i = 1}^3 {\frac{{{d^2}{\nu ^{\left( i \right)}}}}{{{\tau _2}}}} } \,{e^{{D_{\left( 1 \right)}}}}\]

with the flux U-duality integral:

    \[\int {{d^{\left( {10} \right)}}} x\sqrt { - {g^{\left( {10} \right)}}} \varepsilon _{\left( {1,0} \right)}^{\left( {10d} \right)}{\hat \not D^4}{\hat R^4}\]


    \[\varepsilon _{\left( {1,0} \right)}^{\left( {10d} \right)}{\hat \not D^4}{\hat R^4} = \upsilon _{10}^{ - 4/8}{E_{4/2}}\left( \Omega \right) + \frac{{2{\pi ^2}}}{3}\upsilon _{10}^{ - 6/8}\]