Riemann Revived II
Recall that the complex number form of S duality has a modular group symmetry. This group appears all over the place in M theory. For example, we looked at the Banach-Tarski paradox in terms of a ternary tiling of hyperbolic space as a Poincare disc. This tiling marks the boundary of the circle with a nice triple pattern of accumulating points.
Alain Connes and Matilde Marcolli say that the Riemann zeta function is related to the problem of mass, which in turn we have seen is related to three stranded braids and M theory triality, of which S duality is a piece. It appears that no part of mathematics is left untouched by gravity.
Update: a new paper by Witten et al on 3D gravity, prominently featuring the modular group, has appeared on the arxiv. See the picture on page 49, and the J invariant on page 52. The paper shows that the partition function cannot be given a conventional Hilbert space interpretation. Holomorphic factorisation is suggested as a possible mechanism for extending the degrees of freedom. For instance, the complexified Einstein equations are considered. They say:
we think another possibility is that the non-perturbative framework of quantum gravity really involves a sum not over ordinary geometries in the usual sense, but over some more abstract structures that can be defined independently for holomorphic and antiholomorphic variables. Only when the two structures coincide can the result be interpreted in terms of a classical geometry.I confess to finding this statement a little ill-phrased, since some more abstract structure presumably does not begin with traditional complex analysis.