Is it really possible to formulate all geometric statements as statements of algebraic geometry?

The TGD view of the geometric Langlands correspondence states that there is a correspondence between the algebraic, essentially linguistic view of physics and the geometric view of physics relying on vision. This leads to a kind of language game. The highly non-trivial challenge is to find whether the geometric picture can be formulated using the language of algebraic geometry involving generalized complex variables of which one is hypercomplex and real.

First of all, one must find out whether the known algebraically universal extremals appearing for practically any conceivable action, deduced by geometric and symmetry arguments, have a simple algebraic description as the roots (P,Q)=(0,0) where P and Q are analytic functions of generalized complex coordinates of H=M⁴× CP₂. This is not at all obvious. One should carefully check whether CP₂ type extremals, cosmic strings and monopole flux tubes, and massless extremals allow this kind of formulation.

Inequalities are part of geometric description and involve in an essential manner the notion of distance. The representation of topological boundaries gives rise to inequalities. In TGD a long standing question is whether one should allow boundaries and whether the boundary conditions guaranteeing conservation laws indeed allow space-time boundaries. For instance, could one eliminate CP₂ type extremals defining wormhole contacts glued to the Minkowskian background and leaving partonic orbits as boundaries (see this).

1. The problem is that well-ordering required by inequalities characterizes only real numbers: the notion of inequality is not algebraically universal. Inequalities have no natural place in pure algebraic geometry involving complex numbers or p-adic numbers. In TGD, the natural variables are generalized complex coordinates and inequalities cannot be represented for the complex numbers using only complex analytic functions.

   In TGD, the light-like hypercomplex coordinate u is however an exception. u is real and inequalities make sense for it. For instance, the segment u₁≤u≤u₂ can be defined in the semialgebraic context and the simplest situation corresponds to a position dependent time interval x-u₁≤ t ≤ x+u₂ or propagating pulse. The real part Re(w) of the complex coordinate w of the space-time surface defining the analog of the real axis in complex analysis would be a second coordinate of this kind and could be assigned to the partonic 2-surface.

2. Also in the p-adic topology well-ordering is absent and inequalities would be represented in terms of norm but this is not a notion of algebraic geometry. Only the discrete subsets of p-adic numbers defined by powers of p are well-ordered and inequalities can be defined for them. The hierarchy of discretizations as cognitive representations defined by extensions of rationals could however allow to overcome this problem by reducing them to inequalities.

The notion of semi-algebraic geometry makes it possible to represent these observations formally.

1. In semi-algebraic geometry inequalities are allowed in the real case but do not make sense for complex and p-adic numbers. In TGD, semialgebraic geometry would make sense for the regions of space-time surface for which the generalized complex coordinates of H or space-time surface are real.

   All inequalities should be formulated for the real sub-manifolds, which for ordinary complex 4-manifolds are 2-D. This is the case now. String world sheets parameterized by light-like coordinates u and v, would be naturally 2-D surfaces of this kind but the coordinate v does not appear as the argument of the functions (P,Q). Only the inequalities relating to u seem to make sense.

2. Hamilton-Jacobi structure (see this) means a slicing of M⁴ by pairs of strings world sheets and partonic 2-surfaces and would allow to generalize this representation to the interior of the space-time surface. Could the inequalities related to the geometry of preferred extremals implied by holography=holomorphy correspondence reduce to this kind of inequalities? The two real coordinates u and x= Re(w) could have interpretation as local choices of light-like direction and polarization direction and inequalities in this sense would be consistent with the notion of semialgebraic geometry.

   An interesting question is whether symplectic structure, which is basic element of the WCW geometry and can be seen as a companion of the generalized complex structure, could correspond to the decomposition of the complex space-time coordinate as w= P+iQ and hypercomplex coordinate as (u,v) such that (P,Q) and (u,v) define canonically conjugate coordinate pairs is consistent with the Hamilton-Jacobi structure. Note that the two real coordinates u and x= Re(w) could have interpretation as local choices of light-like direction and polarization direction and inequalities in this sense would be consistent with the notion of semialgebraic geometry.

Could one get rid of inequalities altogether by a suitable choices of the real coordinate variants (u,x)? There is indeed a well-known trick allowing to get rid of an inequalities representable in the form t≥ 0 by a change of the coordinate variable as a replacement t → T= t². Only the points with t≥ 0 are allowed by mere reality conditions. This trick might work to inequalities involving u and x.

See the article About Langlands correspondence in the TGD framework or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.