Riemann's Brane
By now we've all heard about the relation between the Riemann zeta function and Hermitian operators associated to matrix models. With CFT/AdS in the air now, it is not surprising to find this paper by McGuigan, which discusses brane partition functions.
Somehow, according to McGuigan, on the gravity side we are supposed to end up with modular functions like those appearing in the already notorious Witten paper on 2+1D gravity. In fact, the so-called cosmological constant (just think extra time coordinates) appears as the variable $z$ in a function whose zeroes must lie on the real axis, namely
$\Theta (z) = \zeta (iz + \frac{1}{2}) \Gamma (\frac{z}{2} + \frac{1}{4}) \pi^{- \frac{1}{4} - \frac{iz}{2}} (- \frac{z^2}{2} - \frac{1}{8})$
Who would have thought such stuff could get published on the arxiv?
Somehow, according to McGuigan, on the gravity side we are supposed to end up with modular functions like those appearing in the already notorious Witten paper on 2+1D gravity. In fact, the so-called cosmological constant (just think extra time coordinates) appears as the variable $z$ in a function whose zeroes must lie on the real axis, namely
$\Theta (z) = \zeta (iz + \frac{1}{2}) \Gamma (\frac{z}{2} + \frac{1}{4}) \pi^{- \frac{1}{4} - \frac{iz}{2}} (- \frac{z^2}{2} - \frac{1}{8})$
Who would have thought such stuff could get published on the arxiv?
posted by Kea | 11:13 AM
2 Comments:
A comment about branes. I do not regard cosmology with negative lambda as a solution to the accelerated expansion in TGD context: in pure GRT context it is certainly the only imaginable solution.
Just for curiosity I however wrote a little comment about the imbedding of De Sitter cosmology (accelerated expansion) as 4-surface.
The imbedding is possible as a vacuum extremal but the 1-D CP_2 projection fills densely at least 2-D submanifold of CP_2. Torus is standard example with phi_1=kphi_2, k irrational. For k rational only finite portion of De Sitter is imbedded which would make sense in zero energy ontology.
Could one regard space-time surfaces filling densely D>4-dimensional sub-manifolds of H as analogs of branes? Could D-branes in M-theory be seen as strings filling densely higher-D sub-manifold?
Second observation: TGD does not allow imbedding of Anti De Sitter cosmology whereas M-theory predicts these (with wrong sign of lambda).
Thanks, Matti. I agree that a negative lambda is just as unphysical as a positive lambda in a reasonable version of M theory.
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