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:
with:
Thus, I can now derive:
Hence, the integral is:
After checking renormalization, one gets:
which is the correct harmonic function for a D(p+1)-brane. The relevance of is that via Green’s functional analysis, it yields the string coupling of the dual 25-D theory:
which is key to the T-duality transformation properties of propagating background matter fields in 4-dimensional space-time, with the bosonic field configuration corresponding to the string world-sheet, whose variable is , yielding the two following key relations:
and
and so, one can use a Lagrange multiplier to derive the action:
since the Lagrange multiplier equation of motion has solution:
given that its Clifford form is:
for any scalar. So, for :
Now, by solving via a metric:
we get the Dp action:
Since in 4-dimensional space-time, the mass of a Dp-brane can be derived as:
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. The renormalization Lie-equation for the sigma-model on the string worldsheet entails that
So, from
the Einstein vacuum field equation is encoded in the quantum structure of the conformal worldsheet theory as characterized by
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 spacetime physics
as attested by
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A Critical Analysis of Loop Quantum Gravity: Part 1, the Holst Action
Thursday, October 6, 2016[…] really) and LQG, let us contrast the actions. Realize, and this is key, that, unlike in LQG, as I showed, the worldsheet quantum theory of string-theory 'knows' all about 4-D spacetime physics […]