Kea said:
Yes, and for the sake of other readers, let me point out again that the j invariant is also associated to a theta function triplet, related to the E8 function. However, unlike with Witten's current use of the j invariant in the context of a non-physical cosmological constant in 2+1d (AdS/CFT), it arises here in a spatial 3d setting because (a) spatial directions are associated with the triplet and (b) \Lambda is completely replaced by a cosmic time parameter.
The twistor dimension (Riemann surface) moduli triple is also interesting as a genus (0,1,2) triple, because genus plays the role of time steps, instead of the usual classical directions.
There has been of late some interest in the idea that constants of physics change, or are variable. João Magueijo has suggested various schemes in which the speed of light could vary. The variability of constants is a bit odd. For instance if \alpha~=~e^2/\hbar c were determined by different values of e,~\hbar,~c, but where \alpha~\simeq~1/137 physics would be absolutely indistinguishable from what we know. So the most proper constants of nature are dimensionless ones, and if we were to track any variation in constants, dimensionless ones would be the choice of what should be detected.
The speed of light is just a conversion factor between space and time: Light travels such a distance in a certain amount of time in a fixed proportion we call c. Much the same is the case with the Planck scale which is that
<br />
\frac{G\hbar}{c^3}~=~L_p^2, <br />
where the Planck length is a conversion factor with units of cm that converts from the Dirac unit of action hbar, to c^3/G that has units of erg-s/cm^2. This is "action per unit area," which is the amount of action associated with an area associated with a black hole horizon area. This leads into Beckenstein bounds and in part what I related yesterday about a homomorphism between gravity and QM and how quantum mechanics and its unit of action hbar involves what might be called a "quantum horizon," which is a limit to the detectability of physics, HUP and so forth.
With the Planck unit if we were to half the speed of light we would find that the Planck length is increased by \sqrt{8}. Our clocks by the Planck unit of time T_p~=~\sqrt{G\hbar/c^5} would tick away at a rate \sqrt{32} times slower, which by c~=~L_p/T_p~=~1/2, would mean that we would observe nothing at all observationally changed by any rescaling of the speed of light! Interestingly if you consider electromagnetism according to Planck units (Planck units of charge, impedance and so forth) you again would observe absolutely nothing at all if you vary the speed of light. The observational consequences for changing the speed of light are absolutely unobservable. One might think that because hbar has not been changed that the fine structure constant should change. But one must realize that the h or hbar was first deduced from the Bohr radius
<br />
a_0~=~\frac{4\pi\hbar}{me^2}~=~\frac{m_p}{m_e\alpha}L_p, <br />
which would appear to change by \sqrt{8} It is assumed the masses of the proton and electron shift with the rescaled Planck mass equally. But there is a hitch here, for we and our experimental set up also rescale by \sqrt{8}, which negates any observable scale change in an atom due to the change in c. In effect the experimenters in a c~\rightarrow~c/2 world would find an \hbar' and the Bohr radius so that nothing at all changes! In effect their physics published results would be indistinguishable from our own.
The speed of light is a spatial measure associated with projective rays, and these can be rescaled arbitrarily. The Planck unit of action or Dirac's unit \hbar~=~h/2\pi also has what might be called projective properties as well, though physics has not explored this terribly much, where hbar rescales (or the quantum horizon as a projective system) according to how one might change c.
This equivalency with respect to projective varieties leads us in some ways to twistor constructions. Twistor geometry is motivated by the fact lightcones are projective geometries, or the projective Lorentz group PSL(n,~C). Twistor space, or twistor theory, applies to four dimensional Minkowksi spacetime, but where the projective structure pertains to the conformal group spin(4,~2)~\simeq~SU(2,~2). The conformal space is six dimensional R^{4,2}, and the blow up of a point in this space is PR^5 (signature information suppressed). The twistor space is constructed from this projective null space, which is the holomorphic twistor space. The projective twistor space contains the null space, with there being PT^\pm massless \pm helicity states of massless particles. The null projective space is a subspace of the projective twistor space, which has 5 real dimensions, where four of these are complex or components of two complex dimensions. The PT^\pm have the same dimension, but have a twist or helicity state.
This leads to some interesting prospects with spin systems. I argue on
https://www.physicsforums.com/showthread.php?t=115826&page=3
that a spin-net for gravity exihibts a quantum phase transition. This is related to how spin fields, such as those associated with twistors, exist on a Fermi surface. This surface also has some topological features. A space of evolution, which can be a spin-net in the LQG sense, or a D-brane in the string theory (I am not partisan to either theory camp) For the space of evolution, which defines a world volume V~=~\Sigma\times M^1, where V is the evolute of the surface \Sigma, which has a target map to the spacetime or super-spacetime M^n. The compactified winding of a D-brane on this world volume is given by a unitary group U(n), where n is the winding number or coincidence number of these branes. These winding numbers define the brane charges on the voume, which define charges in K-theory groups on the manifold M^n, which are closely related to the cohomology H^p(M^n,~R). Within twistor theory for n~=~4 this is the sheaf cohomology, where these charges are the \pm helicity states or frequences for the PT^\pm subspaces of twistor geometry.
The topology of this spin space is physically similar to a Fermi surface, which is the standard system in condensed matter physics. Volvik (gr-qc/0005091) has written on how the vacuum state of quantum gravity, and what determines the cosmological constant \Lambda. What I want to show is that the K-theory index is identified with the homotopy of the group structure of the Fermi surface K(X~\subset~M^n)~\sim~\pi_k({\cal G}), for the space X of k + 1 dimensions.
