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:

with the embedding of the **string in target space**, the **worldsheet metric**, and the **spacetime metric**.

Now, the variation, through all Einstein-indices, of the action with respect to is then:

with

and with **equation of motion** (EOM):

after taking the **determinant**, the following holds:

and inserting into the Polyakov action yields:

with the **Nambu-Goto action** for the string:

hence, the functional derivative of the Polyakov Lagrangian with respect to the derivative of the dynamical field is

giving us the **Euler-Lagrange equations**:

## Thus,

the strings obey a wave equation

as can be seen by the **Laplacian equation**:

It follows then that the renormalization Lie-equation for the sigma-model on the string worldsheet entails

So, from

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

with satisfying:

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

with

By holographic renormalization, it follows that the worldsheet quantum theory knows all about 4-D spacetime physics

as attested by

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