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 $H_{p + 1}^{{\rm{array}}}$ 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:

and

We saw how the omega-simplified Wheeler-DeWitt equation:

with the corresponding scalar product

which is conserved in $\Omega$-time by virtue of:

allowed us to derive what looks like a Hilbert space inner-product, until one looks at the right-hand-side of:

there can be no selection of an intrinsic time functional hence, the non-existence of a suitable Killing-vector follows: this is the 'Hilbert space problem' for time. Here, I shall try and recover a notion of time via a Dirac-Schrödinger analysis.

Given, as I showed, that the Eulerian fluid action, after the ADM splitting, is

with

the entropy $S$, given the Mandelstam-Tamm formulation of a time-energy uncertainty relation, entails the undefinability of inner products of functional differential operators for this form of the Wheeler-DeWitt equation:

and at face value, that is a no-go theorem for the 'existence' of time outside of string-theory's AdS/CFT duality, but not to be addressed here. Let me
ignore the operator-ordering problem to reduce the WdW-equation to

and try and define an inner product via the Klein-Gordon interpretation of quantum gravitational geometrodynamics and proceed from there.