Motives and twistors in TGD

Motivic cohomology has turned out to pop up in the calculations of the twistorial amplitudes using Grassmannian approach (see this and this). The amplitudes reduce to multiple residue integrals over smooth projective sub-varieties of projective spaces. Therefore they represent the simplest kind of algebraic geometry for which cohomology theory exists.

Also in Grothendieck's vision about motivic cohomology projective spaces are fundamental as spaces to which more general spaces can be mapped in the construction of the cohomology groups (factorization). In the previous posting I gave an abstract of a chapter about motives and TGD explaining a proposal for a non-commutative variant of homology theory based on a hierarchy of Galois groups assigned with the zero locus of polynomial and its restrictions to lover dimension planes obtained by putting variables appearing in it to zero one by one: the basic idea is simple but I would have never discovered it without infinite primes.

The basic problem is to define boundary homomorphism for the hierarchy of Galois groups G_k satisfying the non-abelian generalization of δ²=0 stating that the image under δ² belongs to the commutator subgroup of G_k-2 and therefore is mapped to zero in abelianization, which means division by commutator sub-group.

1. The proposal is also that the roots can be represented as points of 2-D surface (partonic 2-surface) and that Galois groups can be lifted to braid groups acting on a braid of braids of .... to which infinite primes can be mapped. Infinite primes at n:th level of hierarchy describe a states of n times quantized arithmetic SUSY for which the many particles states of the previous level take the role of elementary particles.

2. The basic idea is very physical: the braiding for a braid of braids induces braiding of sub-braids and this is represented as a homomorphism of the Galois group lifted to braid group of the braid to the corresponding groups of sub-braids. This nothing but a representation of symmetries and braiding as a isotopic flow gives excellent hopes about a unique realization of the boundary homomorphism.

3. This SUSY is physically extremely interesting since irreducible polynomials of degree n> 1 have interpretation as bound states. Therefore bound states, which are the basic problem of perturbative quantum field theory, would have purely number theoretic meaning. As a matter fact, infinite rationals reducing to real units in real sense represent zero energy states in zero energy ontology and it is natural to assign Galois group hierarchies also to the poles of this rational function.

Summarizing, the infinite prime - irreducible polynomial - braid - quantum state connection suggests very deep connections between number theory, algebraic geometry, topological quantum field theories, and super-symmetric quantum field theories. The article Motives and Infinite Primes gives a more detailed discussion.

Defining integration in p-adic context is one of the basic challenges of quantum TGD in which real and various p-adic physics ought to be unified to a larger theory by realizing what I have called number theoretical Universality. Grothendieck's motivic comology can be seen as a program for the realization of integration of forms making sense also in p-adic context. In the following I shall discuss some aspects of the problem in TGD framework. The discussion of course fails to satisfy all standards of mathematical rigor but it relies of extremely deep and general physical principles and my conviction is that good physics is the best guideline for developing good mathematics.

Number theoretic universality, residue integrals, and symplectic symmetry

A key challenge in the realization of the number theoretic universality is the definition of p-adic definite integral. In twistor approach integration reduces to the calculation of multiple residue integrals over closed varieties. These could exist also for p-adic number fields. Even more general integrals identifiable as integrals of forms can be defined in terms of motivic cohomology.

Yangian symmetry (see this and this) is the symmetry behind the successes of twistor Grassmannian approach and has a very natural realization in zero energy ontology (see this). Also the basic prerequisites for twistorialization are satisfied. Even more, it is possible to have massive states as bound states of massless ones and one can circumvent the IR difficulties of massless gauge theories. Even UV divergences are tamed since virtual particles consist of massless wormhole throats without bound state condition on masses. Space-like momentum exchanges correspond to pairs of throats with opposite sign of energy.

Algebraic universality could be realized if the calculation of the scattering amplitudes reduces to multiple residue integrals just as in twistor Grassmannian approach. This is because also p-adic integrals could be defined as residue integrals. For rational functions with rational coefficients field the outcome would be an algebraic number apart from power of 2π, which in p-adic framework is a nuisance unless it is possible to get rid of it by a proper normalization or unless one can accepts the infinite-dimensional transcendental extension defined by 2π. It could also happen that physical predictions do not contain the power of 2π.

Motivic cohomology defines much more general approach allowing to calculate analogs of integrals of forms over closed varieties for arbitrary number fields. In motivic integration - to be discussed below - the basic idea is to replace integrals as real numbers with elements of so called scissor group whose elements are geometric objects. In the recent case one could consider the possibility that (2π)ⁿ is interpreted as torus (S¹)ⁿ regarded as an element of scissor group which is free group formed by formal sums of varieties modulo certain natural relations meaning.

Motivic cohomology allows to realize integrals of forms over cycles also in p-adic context. Symplectic transformations are transformation leaving areas invariant. Symplectic form and its exterior powers define natural volume measures as elements of cohomology and p-adic variant of integrals over closed and even surfaces with boundary might make sense. In TGD framework symplectic transformations indeed define a fundamental symmetry and quantum fluctuating degrees of freedom reduce to a symplectic group assignable to δ M⁴_+/-× CP₂ in well-defined sense (see this). One might hope that they could allow to define scissor group with very simple canonical representatives- perhaps even polygons- so that integrals could be defined purely algebraically using elementary area (volume) formulas and allowing continuation to real and p-adic number fields. The basic argument could be that varieties with rational symplectic volumes form a dense set of all varieties involved.

How to define the p-adic variant for the exponent of Kähler action?

The exponent of Kähler function defined by the Kähler action (integral of Maxwell action for induced Kähler form) is central for quantum at least in the real sector of WCW. The question is whether this exponent could have p-adic counterpart and if so, how it should be defined.

In the real context the replacement of the exponent with power of p changes nothing but in the p-adic context the interpretation is affected in a dramatic manner. Physical intuition provided by p-adic thermodynamics (see this) suggests that the exponent of Kähler function is analogous to Bolzmann weight replaced in the p-adic context with non-negative power of p in order to achieve convergence of the series defining the partition function not possible for the exponent function in p-adic context.

1. The quantization of Kähler function as K= rlog(m/n), where r is integer, m>n is divisible by a positive power of p and n is indivisible by a power of p, implies that the exponent of Kähler function is of form (m/n)^r and therefore exists also p-adically. This would guarantee the p-adic existence of the vacuum functional for any prime dividing m and for a given prime p would select a restricted set of p-adic space-time sheets (or partonic 2-surfaces) in the intersection of real and p-adic worlds. It would be possible to assign several p-adic primes to a given space-time sheet (or partonic 2-surface). In elementary particle physics a possible interpretation is that elementary particle can correspond to several p-adic mass scales differing by a power of two (see this). One could also consider a more general quantization of Kähler action as sum K=K₁+K₂ where K₁=rlog(m/n) and K₂=n, with n divisible by p since exp(n) exists in this case and one has exp(K)= (m/n)^r × exp(n). Also transcendental extensions of p-adic numbers involving n+p-2 powers of e^1/n can be considered.

2. The natural continuation to p-adic sector would be the replacement of integer coefficient r with a p-adic integer. For p-adic integers not reducing to finite integers the p-adic norm of the vacuum functional would however vanish and their contribution to the transition amplitude vanish unless the number of these space-time sheets increases with an exponential rate making the net contribution proportional to a finite positive power of p. This situation would correspond to a critical situation analogous to that encounted in string models as the temperature approaches Hagedorn temperature and the number states with given energy increases as fast as the Boltzmann weight. Hagedorn temperature is essentially due to the extended nature of particles identified as strings. Therefore this kind of non-perturbative situation might be encountered also now.

3. Rational numbers m/n with n not divisible by p are also infinite as real integers. They are somewhat problematic. Does it make sense to speak about algebraic extensions of p-adic numbers generated by p^1/n and giving n-1 fractional powers of p in the extension or does this extension reduce to something equivalent with the original p-adic number field when one redefines the p-adic norm as |x|_p → |vert x|^1/n? Physically this kind of extension could have a well defined meaning. If this does not make sense, it seems that one must treat p-adic rationals as infinite real integers so that the exponent would vanish p-adically.

4. If one wants that Kähler action exists p-adically a transcendental extension of rational numbers allowing all powers of log(p) and log(k), where k<p is primitive p-1:th root of unity in G(p). A weaker condition would be an extension to a ring with containing only log(p) and log(k) but not their powers. That only single k<p is needed is clear from the identity log(k^r)=rlog(k), from primitive root property, and from the possibility to expand log(k^r+pn), where n is p-adic integer, to powers series with respect to p. If the exponent of Kähler function is the quantity coding for physics and naturally required to be ordinary p-adic number, one could allow log(p) and log(k) to exists only in symbolic sense or in the extension of p-adic numbers to a ring with minimal dimension.

   Remark: One can get rid of the extension by log(p) and log(k) if one accepts the definition of p-adic logarithm as log(x)=log(p^-kx/x₀) for x=p^k(x₀+ py), |y|_p<1. To me this definition looks somewhat artificial since this function is not strictly speaking the inverse of exponent function but it might have a deeper justification.

5. What happens in the real sector? The quantization of Kähler action cannot take place for all real surfaces since a discrete value set for Kähler function would mean that WCW metric is not defined. Hence the most natural interpretation is that the quantization takes place only in the intersection of real and p-adic worlds, that is for surfaces which are algebraic surfaces in some sense. What this actually means is not quite clear. Are partonic 2-surfaces and their tangent space data algebraic in some preferred coordinates? Can one find a universal identification for the preferred coordinates- say as subset of imbedding space coordinates selected by isometries?

If this picture inspired by p-adic thermodynamics holds true, p-adic integration at the level of WCW would give analog of partition function with Boltzman weight replaced by a power of p reducing a sum over contributions corresponding to different powers of p with WCW integra.l over space-time sheets with this value of Kähler action defining the analog for the degeneracy of states with a given value of energy. The integral over space-time sheets corresponding to fixed value of Kähler action should allow definition in terms of a symplectic form defined in the p-adic variant of WCW. In finite-dimensional case one could worry about odd dimension of this sub-manifold but in infinite-dimensional case this need not be a problem. Kähler function could defines one particular zero mode of WCW Kähler metric possessing an infinite number of zero modes.

One should also give a meaning to the p-adic integral of Kähler action over space-time surface assumed to be quantized as multiples of log(m/n).

1. The key observation is that Kähler action for preferred extrememals reduces to 3-D Chern-Simons form by the weak form of electric-magnetic duality. Therefore the reduction to cohomology takes place and the existing p-adic cohomology gives excellent hopes about the existence of the p-adic variant of Kähler action. Therefore the reduction of TGD to almost topological QFT would be an essential aspect of number theoretical universality.

2. This integral should have a clear meaning also in the intersection of real and p-adic world. Why the integrals in the intersection would be quantized as multiple of log(m/n), m/n divisible by a positive power of p? Could log(m/n) relate to the integral of ∫₁^p dx/x, which brings in mind ∮ dz/z in residue calculus. Could the integration range [1,m/n] be analogous to the integration range [0,2π]. Both multiples of 2π and logarithms of rationals indeed emerge from definite integrals of rational functions with rational coefficients and allowing rational valued limits and in both cases 1/z is the rational function responsible for this.

3. log(m/n) would play a role similar to 2π in the approach based on motivic integration where integral has geometric objects as its values. In the case of 2π the value would be circle. In the case of log(m/n) the value could be the arc between the points r=m/n>1 and r=1 with r identified the radial coordinate of light-cone boundary with conformally invariant length measures dr/r. One can also consider the idea that log(m/n) is the hyperbolic angle analogous to 2π so that these two integrals could correspond to hyper-complex and complex residue calculus respectively.

4. TGD as almost topological QFT means that for preferred extremals the Kähler action reduces to 3-D Chern-Simons action, which is indeed 3-form as cohomology interpretation requires, and one could consider the possibility that the integration giving log(m/n) factor to Kähler action is associated with the integral of Chern-Simons action density in time direction along light-like 3-surface and that the integral over the transversal degrees of freedom could be reduced to the flux of the induced CP₂ Kähler form. The logarithmic quantization of the effective distance between the braid end points the in metric defined by modified gamma matrices has been proposed earlier.

Since p-adic objects do not possess boundaries, one could argue that only the integrals over closed varieties make sense. Hence the basic premise of cohomology would fail when one has p-adic integral over braid strand since it does not represent closed curve. The question is whether one could identify the end points of braid in some sense so that one would have a closed curve effectively or alternatively relative cohomology. Periodic boundary conditions is certainly one prerequisite for this kind of identification.

1. In one of the many cohomologies known as quantum cohomology one indeed assumes that the intersection of varieties is fuzzy in the sense that two surfaces for points are connected by a curve of certain kind known as pseudo-holomorphic curve can be said to intersect at these points.

2. In the construction of the solutions of the modified Dirac equation one assumes periodic boundary conditions so that in physical sense these points are identified (see this). This assumption actually reduces the locus of solutions of the modified Dirac equation to a union of braids at light-like 3-surfaces so that finite measurement resolution for which discretization defines space-time correlates becomes an inherent property of the dynamics. The coordinate varying along the braid strands is light-like so that the distance in the induced metric vanishes between its end points (unlike the distance in the effective metric defined by the modified gamma matrices): therefore also in metric sense the end points represent intersection point. Also the effective 2-dimensionality means are effectively one and same point.

3. The effective metric 2-dimensionality of the light-like 2-surfaces implies the counterpart of conformal invariance with the light-like coordinate varying along braid strands so that it might make sense to say that braid strands are pseudo-holomorphic curves. Note also that the end points of a braid along light-like 3-surface are not causally independent: this is why M-matrix in zero energy ontology is non-trivial. Maybe the causal dependence together with periodic boundary conditions, light-likeness, and pseudo-holomorphy could imply a variant of quantum cohomology and justify the p-adic integration over the braid strands.

Motivic integration

While doing web searches related to motivic cohomology I encountered also the notion of motivic measure proposed first by Kontsevich. Motivic integration is a purely algebraic procedure in the sense that assigns to the symbol defining the variety for which one wants to calculate measure. The measure is not real valued but takes values in so called scissor group, which is a free group with group operation defined by a formal sum of varieties subject to relations. Motivic measure is number theoretical universal in the sense that it is independent of number field but can be given a value in particular number field via a homomorphism of motivic group to the number field with respect to sum operation.

Some examples are in order.

1. A simple example about scissor group is scissor group consisting operations needed in the algorithm transforming plane polygon to a rectangle with unit edge. Polygon is triangulated; triangles are transformed to rectangle using scissors; long rectangles are folded in one half; rectangles are rescaled to give an unit edge (say in horizontal direction); finally the resulting rectangles with unit edge are stacked over each other so that the height of the stack gives the area of the polygon. Polygons which can be transformed to each other using the basic area preserving building bricks of this algorithm are said to be congruent.

   The basic object is the free abelian group of polygons subject to two relations analogous to second homology group. If P is polygon which can be cut to two polygons P₁ and P₂ one has [P]=[P₁]+[P₂]. If P and P' are congruent polygons, one has [P]=[P']. For plane polygons the scissor group turns out to be the group of real numbers and the area of polygon is the area of the resulting rectangle. The value of the integral is obtained by mapping the element of scissor group to a real number by group homomorphism.

2. One can also consider symplectic transformations leaving areas invariant as allowed congruences besides the slicing to pieces as congruences appearing as parts of the algorithm leading to a standard representation. In this framework polygons would be replaced by a much larger space of varieties so that the outcome of the integral is variety and integration means finding a simple representative for this variety using the relations of the scissor group. One might hope that a symplectic transformations singular at the vertices of polygon combined with with scissor transformations could reduce arbitrary area bounded by a curve into polygon.

3. One can identify also for discrete sets the analog of scissor group. In this case the integral could be simply the number of points. Even more abstractly: one can consider algebraic formulas defining algebraic varieties and define scissor operations defining scissor congruences and scissor group as sums of the formulas modulo scissor relations. This would obviously abstract the analytic calculation algorithm for integral. Integration would mean that transformation of the formula to a formula stating the outcome of the integral. Free group for formulas with disjunction of formulas is the additive operation(see this). Congruence must correspond to equivalence of some kind. For finite fields it could be bijection between solutions of the formulas. The outcome of the integration is the scissor group element associated with the formula defining the variety.

4. For residue integrals the free group would be generated as formal sums of even-dimensional complex integration contours. Two contours would be equivalent if they can be deformed to each other without going through poles. The standard form of variety consists of arbitrary small circles surrounding the poles of the integrand multiplied by the residues which are algebraic numbers for rational functions. This generalizes to rational functions with both real and p-adic coefficients if one accepts the identification of integral as a variety modulo the described equivalence so that (2π)ⁿ corresponds to torus (S¹)ⁿ. One can replace torus with 2π if one accepts an infinite-dimensional algebraic extension of p-adic numbers by powers of 2π. A weaker condition is that one allows ring containing only the positive powers of 2π.

5. The Grassmannian twistor approach for two-loop hexagon Wilson loop gives classical polylogarithms L_k(s) (see this). General polylogarithm is defined by obey the recursion formula:

   Li_s+1(z)= ∫₀^zLi_s(t)dt/t .

   Ordinary logarithm Li₁(s) = -log(1-s) exists p-adically and generates a hierarchy containing dilogarithm, trilogarithm, and so on, which each exist p-adically for |x|_p < 1 as is easy to see. If one accepts the general definition of p-adic logariths one finds that the entire function series exists p-adically for integer values of s. An interesting question is how strong constraints p-adic existence gives to the thetwistor loop integrals and to the underlying QFT.

6. The ring having p-adic numbers as coefficients and spanned by transcendentals log(k) and log(p), where k is primitive root of unity in G(p) emerges in the proposed p-adicization of vacuum functional as exponent of Kähler action. The action for the preferred extremals reducing to 3-D Chern-Simons action for space-time surfaces in the intersection of real and p-adic worlds would be expressible p-adically as a linear combination of log(p) and log(k). log(m/n) expressible in this manner p-adically would be the symbolic outcome of p-adic integral ∫ dx/x between rational points. x could be identified as a preferred coordinate along braid strand. A possible identification for x earlier would be as the length in the effective metric defined by modified gamma matrices appearing in the modified Dirac equation (see this).

Infinite rationals and multiple residue integrals as Galois invariants and Galois groups as symmetry groups of quantum physics

In TGD framework one could consider also another kind of cohomological interpretation. The basic structures are braids at light-like 3-surfaces and space-like 3-surfaces at the ends of space-time surfaces. Braids intersects have common ends points at the partonic 2-surfaces at the light-like boundaries of a causal diamond. String world sheets define braid cobordism and in more general case 2-knot (see this)). Strong form of holography with finite measurement resolution would suggest that physics is coded by the data associated with the discrete set of points at partonic 2-surfaces. Cohomological interpretation would in turn would suggest that these points could be identified as intersections of string world sheets and partonic 2-surface defining dual descriptions of physics and would represent intersection form for string world sheets and partonic 2-surfaces.

Infinite rationals define rational functions and one can assign to them residue integrals if the variables x_n are interpreted as complex variables. These rational functions could be replaced with a hierarchy of sub-varieties defined by their poles of various dimensions. Just as the zeros allow realization as braids or braids also poles would allow a realization as braids of braids. Hence the n-fold residue integral could have a representation in terms of braids. Given level of the braid hierarchy with n levels would correspond to a level in the hierarchy of complex varieties with decreasing complex dimension.

One can assign also to the poles (zeros of polynomial in the denominator of rational function) Galois group and obtains a hierarchy of Galois groups in this manner. Also the braid representation would exists for these Galois groups and define even cohomology and homology if they do so for the zeros. The intersections of braids with of the partonic 2-surfaces would represent the poles in the preferred coordinates and various residue integrals would have representation in terms of products of complex points of partonic 2-surface in preferred coordinates. The interpretation would be in terms of quantum classical correspondence.

Galois groups transform the poles to each other and one can ask how much information they give about the residue integral. One would expect that the n-fold residue integral as a sum over residues expressible in terms of the poles is invariant under Galois group. This is the case for the simplest integrals in plane with n poles and probably quite generally. Physically the invariance under the hierarchy of Galois group would mean that Galois groups act as the symmetry group of quantum physics. This conforms with the number theoretic vision and one could justify the formula for the residue integral also as a definition motivated by the condition of Galois invariance. Of course, all symmetric functions of roots would be Galois invariants and would be expected to appear in the expressions for scattering amplitudes.

The Galois groups associated with zeros and poles of the infinite rational seem to have a clear physical significance. This can be understood in zero energy ontology if positive (negative) physical states are indeed identifiable as infinite integers and if zero energy states can be mapped to infinite rationals which as real numbers reduce to real units. The positive/negative energy part of the zero energy state would correspond to zeros/poles in this correspondence. An interesting question is how strong correlations the real unit property poses on the two Galois group hierarchies. The asymmetry between positive and negative energy states would have interpretation in terms of the thermodynamic arrow of geometric time (see this) implied by the condition that either positive or negative energy states correspond to state function reduced/prepared states with well defined particle numbers and minimum amount of entanglement.

For more details see the new chapter Infinite Primes and Motives of "TGD as Generalized Number Theory" or the article with same title.