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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:

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

and by T-dualizing in the

direction and factoring the dilaton, the dual is hence:

By matrix world-volume integral reduction on

the Polyakov action for the string is hence:

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:

and:

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

with the flux U-duality integral:

and: