Friday, February 20, 2015

Twistorial questions to ponder

For a non-specialist lacking the technical skills, the work related to twistors (for TGD view and references see
this) is a garden of mysteries and there are a lot of questions to be answered: most of them of course trivial for the specialist. Here a just few of them.

1. How the twistor string approach of Witten and its possible TGD generalization relate to the approach involving residue integration over projective sub-manifolds of Grassmannians G(k,n). Nima et al argue that one can transform Grassmannian representation to twistor string representation for tree amplitudes. The integration over G(k,n) translates to integration over the moduli space of complex curves of degree d= k-1+l, l≥ g is the number of loops. The moduli correspond to complex coefficients of the polynomial of degree d and they form naturally a projective space since an overall scaling of coefficients does not change the surfaces. One can expect also in the general case that moduli space of partonic 2-surfaces is projective sub-manifold of a projective space. What is so nice that loop corrections would correspond to the inclusion of higher degree surfaces.

This connection gives hopes for understanding the integration contours in G(k,n) at deeper level in terms of the moduli spaces of partonic 2-surfaces possibly restricted by conformal gauge conditions.

2. The notion of positive Grassmannian is one of the central notions introduced by Nima et al. The claim is that the sub-spaces of the real Grassmannian G(k,n) contributing to the amplitudes for ++-- signature are such that the determinants of the k× k minors associated with ordered columns of the k× n matrix C representing point of G(k,n) are positive. To be precise, the signs of all minors are positive or negative simultaneously: only the ratios of the determinants defining projective invariants are positive.

At the boundaries of positive regions some of the determinants vanish. What happens that some k-volumes degenerate to a lower-dimensional volume. Boundaries are responsible for the leading singularities of the scattering amplitudes and the integration measure associated with G(k,n) has logarithmic singularity at the boundaries. These boundaries obviously correspond to the boundaries of the moduli space for the partonic 2-surfaces.

This condition has a partial generalization to the complex case: the determinants are non-vanishing. A possible further manner to generalize this condition would be that the determinants have positive real part so that apart from rotation by π/2 they would be in the upper half plane of complex plane - the hyperbolic plane playing key role in the theory of hyperbolic 2-manifolds for which it serves as universal covering space by a finite discrete subgroup of Lorentz group SL(2,C). The upper half-plane has therefore a deep meaning in the theory of Riemann surfaces and might have counterpart at the moduli space of partonic 2-surfaces. The projective space would be based - not on projectivization of Cn but that of Hn, H the upper half plane.

3. Could positivity have some even deeper meaning? Why positivity? In TGD framework the number theoretical universality of amplitudes suggests this. Canonical identification maps ∑ xnpn→ ∑ xnp-n p-adic number to non-negative reals. p-Adicization is possible for angle variables by replacing them by discrete phases, which are roots of unity. For non-angle like variables, which are non-negative by using canonical identification or its variant. The positivity should hold true for all structures involved, the G(2,n) points defined by the twistors characterizing momenta and helicities of particles (actually pairs of orthogonal planes defined by twistors and their conjugates), the moduli space of partonic 2-surfaces, etc...

For a summary about recent TGD view about construction of twistor amplitudes and TGD counterpart of N= 4 SUSY as dynamical and broken supersymmetry with Dirac spinors see the chapter Classical part of the twistor story or the article Classical part of the twistor story.