Number theoretic aspects of holography = holomorphy  principle

In the sequel, some number theoretic aspects of the holography = holomorphy (H-H) principle are considered.

Holography = holomorphy principle

First some background is needed.

1. The holography = holomorphy (H-H) (see this, this, and this) principle implies that space-time surfaces are determined as roots of the pairs of holomorphic functions f=(f₁,f₂): H=M⁴× CP₂ → C². The function pairs f=(f_º1,f₂) define an algebra with respect to element-wise sum and product of functions f_i. If f₁ or f₂ is not varied, one obtains a function field.
2. The maps g=(g₁,g₂): C²→ C² define symmetries via the composition f→ g_º f and give rise to hierarchies of space-time surfaces with exponentially increasing algebraic complexity in the case of polynomials.

In particular, one obtains function field analogs of p-adic number fields and of adeles with powers of the p-adic prime p replaced with powers of polynomials of prime degree having coefficient polynomials with degree smaller than p. If the polynomial primeness is realized in the sense that the Galois group is not affected in deformations, there are also prime polynomials with non-prime degree having no decomposition to the functional composite of polynomials of lower degree.

The holomorphic maps g: C²→ C² define infinite hierarchies of space-time surfaces. The interpretation is in terms of complexity hierarchies and cognitive hierarchies. In the general case the complexity increases exponentially.

Space-time surfaces are analogs of Bohr orbits for 3-surfaces replacing point-like particles and provide representations for the elements of function algebras and function fields. Space-time surfaces can also be interpreted as theorems: the slight failure of the classical determinism located at the singularities makes possible several analogs of theorems represented as Bohr orbits such that the 3-D holographic data at the boundary of CD provides the fixed premises of the theorem. The classical laws of physics would provide a physical representation for the axioms of mathematics.

The dynamics implied by the H-H principle is universal if the classical action is general coordinate invariant and constructible in terms of the induced geometry of the space-time surface. By holomorphy = holography (H-H) principle (see this, this, and this) the solutions of field equations are holographic minimal surfaces with 3-D singularities at which the minimal surface property and holography fail. At singularities the boundary conditions stating the conservation of classical isometry charges are satisfied and without additional symmetries the singularities, in particular their positions, depend on the classical action.

This suggests that classical action serves in the role of effective action so that the parameters appearing in it can vary. In particular, they could depend on the extension E of rationals appearing in the coefficients of the pairs holomorphic functions f=(f₁,f₂): H=M⁴× CP₂ → C².

WCW would define what might be called Platonia and by adding WCW spinor fields one obtains quantum Platonia and Boolean logic. By adding state function reductions, one obtains conscious Platonia, which remembers and learns more and more about itself. Number theoretic evolution as an increase of number theoretic complexity is unavoidable. How do CP₂ type extremals emerge?

The proposed view seems to give only the Minkowskian regions of the space-time surface: also CP₂ type extremals with Euclidean signature are needed. The space-time surface is not affected if f₁ and f₂ are scaled by a non-vanishing analytic function. What happens if f₁ and f₂ are scaled with the same function which vanishes at some points? A good guess is that the allowance of this kind of scalings leads to the emergence of the CP₂ type regions with Euclidean induced metric and the geometry of CP₂.

1. The scaling invariance implies that in regions z₃≠ 0, the natural interpretation of C² is as a projective space CP₂ obtained from C³ by identifying the points (z₁,z₂,z₃) of C³ differing by complex scaling and one can take z₃=1 are identified so that for instance (ξ₁,ξ₂)=(z₁/z₃,z₂/z₃) serve as coordinates. The regions with z₂= 0 and (z₁,z₂)≠ (0,0) correspond (z₁,z₂)≠ (∞,∞) defining the homologically trivial geodesic sphere of CP₂ at infinity. By using (z₁/z₂,z₃/z₂) or (z₂/z₁,z₃/z₁) as coordinates one obtains the three coordinate patches of CP₂.
2. The points (z₁,z₂,z₃)=(0,0,0) however remain still problematic and they are indeed of fundamental importance since the condition (z₁=0,z₂=0) defines the space-time surface. At these points the ratios z_i/z_j are ill-defined and a blow-up takes place so that these kinds of points must be replaced with CP₂. Is this the mechanism for how CP₂ type extremals as Euclidian regions of the space-time surface emerges as a blow-up? I have indeed proposed the blow-up mechanism earlier.
3. The problem is that CP₂ type extremals have as an M⁴ projection a light-like curve of M⁴, which only in a special case is reduced to the standard embedding of CP₂ to H. This problem can be solved by replacing the notion of H-J structure (see this) with its twisted variant (see this). Twisting means that the canonical embedding of CP₂ is replaced with a twisted embedding for which the geometry of CP₂ induced from H is not affected but a geodesic line of CP₂ is stretched to a light-like curve as a coordinate line for the Hamilton-Jacobi coordinates such that the dual of the light-coordinate is constant along it.
4. The twisting as tilting of M⁴ to the direction of the geodesic circle S¹⊂ CP₂ has also an interpretation as warping of H-J structure. The induced metric of tilted M⁴ is modified but remains flat. A reduction c→ c_#this). Warping is indeed possible only for space-time surfaces, not for abstract 4-D Riemann spaces.
5. Warping also explains the reduction of light-velocity in electrodynamics for non-vacuum systems (see this). The notions of di-electric constant and refraction therefore reduce to the geometry of space-time surfaces. Warping is also a universal critical phenomenon allowing large numbers of almost vacuum extremals X⁴ with energy density determined by the volume term in the classical action proportional to the cosmological constant Λ, which has an extremely small value in long length scales. Therefore it could therefore be behind very many quantum critical phenomena. This view corresponds to the original view based on the huage vacuum degeneracy of Kähler action interpreted as 4-D spin glass degeneracy. The introduction of small cosmological constant associated with thge volume action leaves only the approximate vacuum degeneracy appearing as warping.

The outcome would be that by allowing scaling factors of (f₁,f₂), which vanish at a discrete set of points and twisted H-J structure, one obtains also CP₂ type extremals as solutions of field equations.

Could classical action have interpretation as a number theoretic invariant?

The generalized Langlands duality (see thisand this) and the universality of the classical dynamics suggest that the exponent of the classical action has interpretation as an effective action and can be regarded as a number theoretic invariant. This invariant would assign to the space-time surface a number, which is real, algebraic number or integer depending on how strong assumptions are made about the roots of (f₁-c₁,f₂-c₂)=(0,0) .

The guess motivated by the H-H hypothesis is that this number is an analog of a discriminant D for a polynomial of a single variable expressible as a product of the non-vanishing root differences. In the recent case, the roots of (f₁-c₁,f₂-c₂)=(0,0) are 4-D regions of space-time surfaces. One should be able to pick points of the space-time surface as ordinary roots in order to define D using the standard formula.

There are several kinds of sub-manifolds of the space-time surface.

1. The 3-D singularities X³ are physically in a preferred role since they define loci of non-determinism as memory seats. They also define generalized vertices.
2. The 3-D light-like partonic orbits Y³ are interfaces of regions of the space-time surface with Minkowskian and Euclidean signature. The induced metric is effectively 2-D at them.
3. 2-D string worlds Z² sheets appear as 2-D self-intersections of space-time surfaces X⁴ obtained at the limit of the intersections of X⁴ when a small deformation X⁴ becomes trivial. The intersections Y¹=Z²∩ Y³ of string world sheets and partonic 3-surfaces are identified as fermion lines.
4. 3-D light-like partonic orbits Y³ and singular surfaces X³ intersect along 2-D partonic 2-surfaces Y². The light-curves Y¹ defining the fermion lineas intersect these partonic 2-surfaces Y² at a discrete set of points for which the conditions (f₁,f₂)=(0,0) is true. By 2-dimensionality of Y² the roots are complex numbers. If f_i is polynomial, one can assign to the roots discriminant D_i. As a matter of fact, this is true even when f_i is not a polynomial.
5. One can assign to all intersections Y²_i,j of partonic orbits Y³_i and singular 3-surfaces X³_j discriminant D defined as the product D(i,j)=D₁D₂(Y²_i,j) of the discriminants for f₁ and f₂ at the partonic 2-surface Y²_ij=X³_iY³_i. The roots as points of the partonic surface with coordinate z would be loci for fermions and the idea that fermions code for the classical dynamics is highly attractive.

If the complex coordinate z for Y² is unique, the roots have general coordinate invariant meaning. Whether this kind of generalized complex coordinate can be identified, is not clear. A weaker condition would be that only the discriminant is a general coordinate invariant, say conformal invariant. The Hamilton-Jacobi structure poses strong conditions on the coordinates of M⁴.

The proposal has been that a power of the product D=∏_i,jD_i,j of the discriminants D_i,j be identified as the exponent of the classical action for X⁴. This would give strong constraints to the form of the action, in particular the parameters appearing in it so that the interpretation as an effective action would be appropriate. If one requires that the discriminants are rational numbers or in an extension E of rationals, the conditions are really strong: this kind of condition is not however necessary. The discriminant hypothesis would solve the theory in the sense that the calculations of the exponent of classical action defining the vacuum functional could be reduced to number theory.

How uniquely the value of D characterizes the space-time surface as a root of (f₁,f₂)? D depends on the choice of H-J structure but this conforms with the fact that also the space-time surface depends on this choice.

Could the value of D and the fermionic loci as complex roots fix the space-time surface for a given Hamilton-Jacobi structure and the choice of CD? This would mean a strong form of holography and might be unrealistic. For instance, symmetry related space-time surfaces would have the same value of action and D. Also quantum criticality implies a degeneracy so that there could be a large number of space-time surfaces with the same action exponential. The fermionic loci at the singularities would allow to distinguish between symmetry-related space-time surfaces.

One the other hand, if f_i are polynomials, there is a finite number N= N₁+N₂ of coefficients for both of them: fixing the values of f=(f₁,f₂) at N=N₁+N₂ points associated with the fermionic loci at the singularities could fix the coefficients.

The Gödelian dream would be that D is analogous to Gödel number fixing space-time surface and therefore the statement represented by the space-time surface. The maps g: C²→ C² as symmetries of H-H princple

The maps f=(f₁,f₂): H→ C²) define what might be called functional integers or rationals, depending on whether f_i are polynomials or rational functions. The coefficient field can be assumed to be an extension E of rationals. The functional numbers are fundamental and cognition made possible by the existence of symmetries g:C²→ C² and the slight failure of determinism should make possible the representation of these functional numbers in terms of space-time surfaces satisfying (f₁,f₂)=(c₁,c₂) . How to achieve this?

1. The value pairs (f₁,f₂)= (c₁,c₂) define a representation of the functional number f as space-time surfaces identified as roots of (f₁-c₁,f₂-c₂). A representation of the functional number as a many-particle system would be in question. Only a finite subset of values (c₁,c₂) can be realized in this way in practice.
2. If one is interested in representing only a subset {(c₁^k),c₂^k))} of the spectrum of f one can consider a map g: H→ C defined as F= ∏_k [(f₁-c₁^k))² +(f₂-c₂^k))²] as a representation for the surfaces. Now the polynomials are reducible. c_i should belong to an extension of E. This kind of representation is however rather tricky.
3. For E-rational functions, the roots of (f₁,f₂)=(0,0) (and (f₁-c₁,f₂-c₂)=0) can be solved by first solving from f₂=0 the the one of the 3 complex coordinates of H, say z₁ as an algebraic function z₁(z₂,z₃,u) of the remaining two complex coordinate and one hypercomplex coordinate. After that one has equation f₁(z₁(z₂,z₃,u)), z₂,z₃,u)=0 and for instance z₂ can be solved as algebraic function of z₂(z₃,u). Note that the second light-like coordinate v dual to u is a dummy variable so that one obtains a 4-D algebraic surface.
4. For an E-rational map g= (g₁,g₂): C²→ C² g_º (f₁,f₂)=(0,0) selects a set or parameters (c₁^k),c₂^k)) as the roots of g in turn defining surfaces (f₁,f₂)= (c₁^k),c₂^k)). A number theoretically motivated hypothesis is that the allowed values of (c₁^k),c₂^k)) belong to the extension E of rationals so that one obtains hierarchies of discretizations of WCW.

Some general comments about the role of the maps g are in order.

1. This picture allows to understand why the hierarchies of the compositions of maps g applied to a space-time surface defined by (f₁,f₂)=(0,0) which is prime in the sense that it does not allow a representation as a functional composite f= g_º h is so important. One can say that prime pairs f represent the substrate and the maps g represent the cognition for which the goal is to understand the substrate represented by f.
2. The degree d of rational function R=P₁/P₂, defined as the difference d(P₁)-d(P₂) of the polynomials appearing in it, is analogous to polynomial degree and is multiplicative in the functional composition. Therefore rational functions with prime degree are special. One could define the generalization of functional p-adic numbers. Functional p-adic would be a sum of powers of rational function with prime degree multiplied by coefficients, which are rational functions of a lower degree.
3. Galois groups are an important part of the TGD view of cognition. One can also ask whether one can assign a Galois group to the roots of g identifiable as space-time surfaces. Google LLM informs that the notion of Galois group is extremely general and this makes sense. The Galois group would act as a flow permuting the roots. In the same way, one could assign a Galois group to the roots of (f₁,f₂)=0 having identification as space-time regions. This interpretation was proposed in (see this and this) as an intuitive notion.

Could birational maps g: C²→ C² play a special role?

The roots of g: C²→ C² define the corresponding spectrum (f₁,f₂)=(c₁,c₂). One can argue that since the maps g relate to cognition, the maps g: C²→ C² for which roots can be solved analytically are cognitively very special and might be those, which appear first in the number theoretic evolution.

1. For birational maps g=(g₁,g₂): C²→ C² also the inverses g^-1 are rational maps. This allows us to solve the roots g analytically as (c₁^k), c₂^k))= g^-1(0,0).
2. The birational maps g: C²→ C² are known as Cremona transformations (see this) and have a very rich structure. Their functional composition is possible but they do not form a group unlike in the case of CP₁ (Möbius transformations). Note that Möbius transformations as linear transformations act as symmetries of C².

Cremona transformations are generated by projective transformations of CP₂ and the so-called standard quadratic transformation (z₁,z₂)→ (1/z₁,1/z₂), which is singular at the origin. Projective transformations correspond to the group SL(3,C) containing the isometry group SU(3) of CP₂ as a subgroup. The implication of fundamental importance for TGD is that Cremona transforms a spectrum generating algebra rather than acting as a mere symmetry group.

The so called polynomial automorphisms are birational maps, which are everywhere well-defined isomorphisms of C² to itself. The Jung-van der Kulk theorem states that every polynomial automorphism is a composition of affine maps and elementary maps of form (z₁,z₂)→ z₁,z₂)→ z₁,z₂+P(z₁), where P(z₁) is a composition of affine maps and polynomial. These transformations could be regarded as symmetries.

Every Cremona transformation can be factored into a sequence of point blow-ups followed by a finite sequence of point blow-downs. Geometrically this means blowing up a set of points where the map is undefined and then contracting the curves down to points to create the image surface.

At the blow-up points the Cremona transformation gives an indeterminate expression like 0/0 and the value of the expression depends on the direction from which the point is approached. In the case of Cremona transformation a single point as image or pre-image is substituted by an entire projective in CP₁.

A familiar analog is the Coulomb force of the point charge which has ill-defined direction at the charge. In the case of Coulomb force, the point is replaced by a sphere. By the argument already given, the blow-up means that C² is replaced by CP₂ by adding the sphere CP₁ at infinity.

A concrete example is provided by the standard Cremona transformation (z₁,z₂)→ (z₂/z₁z₂,z₂/z₁z₂). Origin is the problem point. The approach to origin along the line z₂= mz₁ gives in projective coordinates the point (1/m,1) which gives to CP₁ instead of a single point. Blow-up as the addition of CP₁ to infinity replacing C² with CP₂ is the solution of the problem.

For g this would produce a sphere as an image point. There is no problem with the inverse image of (0,0) since it does not belong to CP₁ at infinity so that g^-1(0,0) is well-defined.

For g^-1(0,0), (c₁,c₂) would be ill-defined. For projective coordinates there would be an entire CP₁ of points (c₁,c₂) defining space-time surfaces. One can solve the problem by concluding that the coordinates for CP₂ become ill-defined in this situation and one must use another coordinate patch: 3 patches are enough. In a new coordinate patch g^-1(0,0) is unique and obtains a single space-time surface.

One could also argue that the inverse image is a union of the space-time surfaces labelled by the points of CP₁ and therefore 6-dimensional rather than 4-dimensional. Space-time surfaces can be regarded as intersections of 6-D f₁=0 and f₂=0 surfaces having a twistor space interpretat and the singularity could mean that the intersecting 6-surfaces are identical in this case In the case of Coulomb force, point is replaced by a sphere. The M-theory analog would be 4-dimensional 3-brane.

The degree of the rational map g: C²→ C² defines the analog for the degree of a polynomial. For the iterations of many families of maps this degree behaves like d_n = 3d_n-1 – 2d_n-2. At the limit n→ ∞ the asymptotic behavior d_n+1-d_n → Δ is consistent with this condition and means that complexity as the increase of the degree increases but not exponentially as in the general case. If genuine symmetries were in question, the complexity would not increase. The behavior of d_n can also be periodic and even chaotic. See the article M⁸-H duality viz. Hubble law, gravitational Planck constant viz. Allais effect and warping, and CP₂ type extremals from holography = holomorphy principle 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.

Source: https://matpitka.blogspot.com/2026/05/number-theoretic-aspects-of-holography.html