Does the quantization of Kähler coupling strength reduce to the quantization of Chern-Simons coupling at partonic level?
Kähler coupling strength associated with Kähler action (Maxwell action for the induced Kähler form) is the only coupling constant parameter in quantum TGD, and its value (or values) is in principle fixed by the condition of quantum criticality since Kähler coupling strength is completely analogous to critical temperature. The quantum TGD at parton level reduces to almost topological QFT for light-like 3-surfaces. This almost TQFT involves Abelian Chern-Simons action for the induced Kähler form.
This raises the question whether the integer valued quantization of the Chern-Simons coupling k could predict the values of the Kähler coupling strength. I considered this kind of possibility already for more than 15 years ago but only the reading of the introduction of the recent paper by Witten about his new approach to 3-D quantum gravity led to the discovery of a childishly simple argument that the inverse of Kähler coupling strength could indeed be proportional to the integer valued Chern-Simons coupling k: 1/αK=4k if all factors are correct. k=26 is forced by the comparison with some physical input. Also p-adic temperature could be identified as Tp=1/k.
1. Quantization of Chern-Simons coupling strength
For Chern-Simons action the quantization of the coupling constant guaranteing so called holomorphic factorization is implied by the integer valuedness of the Chern-Simons coupling strength k. As Witten explains, this follows from the quantization of the first Chern-Simons class for closed 4-manifolds plus the requirement that the phase defined by Chern-Simons action equals to 1 for a boundaryless 4-manifold obtained by gluing together two 4-manifolds along their boundaries. As explained by Witten in his paper, one can consider also "anyonic" situation in which k has spectrum Z/n2 for n-fold covering of the gauge group and in dark matter sector one can consider this kind of quantization.
2. Formula for Kähler coupling strength
The quantization argument for k seems to generalize to the case of TGD. What is clear that this quantization should closely relate to the quantization of the Kähler coupling strength appearing in the 4-D Kähler action defining Kähler function for the world of classical worlds and conjectured to result as a Dirac determinant. The conjecture has been that gK2 has only single value. With some physical input one can make educated guesses about this value. The connection with the quantization of Chern-Simons coupling would however suggest a spectrum of values. This spectrum is easy to guess.
3. Justification for Wick rotation
- The U(1) counterpart of Chern-Simons action is obtained as the analog of the "instanton" density obtained from Maxwell action by replacing J wedge *J with J wedge J. This looks natural since for self dual J associated with CP2 extremals Maxwell action reduces to instanton density and therefore to Chern-Simons term. Also the interpretation as Chern-Simons action associated with the classical SU(3) color gauge field defined by Killing vector fields of CP2 and having Abelian holonomy is possible. Note however that instanton density is multiplied by imaginary unit in the action exponential of path integral. One should find justification for this "Wick rotation" not changing the value of coupling strength and later this kind of justification will be proposed.
- Wick rotation argument suggests the correspondence k/4π = 1/4gK2 between Chern-Simons coupling strength and the Kähler coupling strength gK appearing in 4-D Kähler action. This would give
The spectrum of 1/αK would be integer valued
The result is very nice from the point of number theoretic vision since the powers of αK appearing in perturbative expansions would be rational numbers (ironically, radiative corrections might vanish but this might happen only for these rational values of αK!).
- It is interesting to compare the prediction with the experimental constraints on the value of αK. The basic empirical input is that electroweak U(1) coupling strength reduces to Kähler coupling at electron length scale (see this). This gives αK= αU(1)(M127)≈ 104.1867, which corresponds to k=26.0467. k=26 would give αK= 104: the deviation would be only .2 per cent and one would obtain exact prediction for αU(1)(M127)! This would explain why the inverse of the fine structure constant is so near to 137 but not quite. Amusingly, k=26 is the critical space-time dimension of the bosonic string model. Also the conjectured formula for the gravitational constant in terms of αK and p-adic prime p involves all primes smaller than 26 (see this).
- Note however that if k is allowed to have values in Z/n2, the strongest possible coupling strength is scaled to n2/4 if hbar is not scaled: already for n=2 the resulting perturbative expansion might fail to converge. In the scalings of hbar associated with M4 degrees of freedom hbar however scales as 1/n2 so that the spectrum of αK would remain invariant.
It is not too difficult to believe to the formula 1/αK =qk, q some rational. q=4 however requires a justification for the Wick rotation bringing the imaginary unit to Chern-Simons action exponential lacking from Kähler function exponential.
In this kind of situation one might hope that an additional symmetry might come in rescue. The guess is that number theoretic vision could justify this symmetry.
4. Is p-adicization needed and possible only in 3-D sense?
- To see what this symmetry might be consider the generalization of the Montonen-Olive duality obtained by combining theta angle and gauge coupling to single complex number via the formula
What this means in the recent case that for CP2 type vacuum extremals (see this) Kähler action and instanton term reduce by self duality to Kähler action obtained by the replacement g2 with -iτ/4π. The first duality τ→τ+1 corresponds to the periodicity of the theta angle. Second duality τ→-1/τ corresponds to the generalization of Montonen-Olive duality α→ 1/α. These dualities are definitely not symmetries of the theory in the recent case.
- Despite the failure of dualities, it is interesting to write the formula for τ in the case of Chern-Simons theory assuming gK2=π/k with k>0 holding true for Kac-Moody representations. What one obtains is
The allowed values of τ are integer spaced along a line whose direction angle corresponds to the phase exp(i2π/n), n=4. The transformations τ→ τ+ 4(1-i) generate a dynamical symmetry and as Lorentz transformations define a subgroup of the group E2 leaving invariant light-like momentum (this brings in mind quantum criticality!). One should understand why this line is so special. One should understand why this line is so special.
- This formula conforms with the number theoretic vision suggesting that the allowed values of τ belong to an integer spaced lattice. Indeed, if one requires that the phase angles are proportional to vectors with rational components then only phase angles associated with orthogonal triangles with short sides having integer valued lengths m and n are possible. The additional condition that the phase angles correspond to roots of unity! This leaves only m=n and m=-n>0 into consideration so that one would have τ= n(1-i) from k>0.
- Notice that theta angle is a multiple of 8kπ so that a trivial strong CP breaking results and no QCD axion is needed (this of one takes seriously the equivalence of Kähler action to the classical color YM action).
The action of CP2 type extremal is given as S=π/8αK= kπ/2. Therefore the exponent of Kähler action appearing in the vacuum functional would be exp(kπ) known to be a transcendental number (Gelfond's constant). Also its powers are transcendental. If one wants to p-adicize also in 4-D sense, this raises a problem.
Before considering this problem, consider first the 4-D p-adicization more generally.
If one remains optimistic and wants to p-adicize also in 4-D sense, the transcendental value of the vacuum functional for CP2 type vacuum extremals poses a problem (not the only one since the p-adic norm of the exponent of Kähler action can become completely unpredictable).
- The definition of Kähler action and Kähler function in p-adic case can be obtained only by algebraic continuation from the real case since no satisfactory definition of p-adic definite integral exists. These difficulties are even more serious at the level of configuration space unless algebraic continuation allows to reduce everything to real context. If TGD is integrable theory in the sense that functional integral over 3-surfaces reduces to calculable functional integrals around the maxima of Kähler function, one might dream of achieving the algebraic continuation of real formulas. Note however that for lightlike 3-surface the restriction to a category of algebraic surfaces essential for the re-interpretation of real equations of 3-surface as p-adic equations. It is far from clear whether also preferred extremals of Kähler action have this property.
- Is 4-D p-adicization the really needed? The extension of light-like partonic 3-surfaces to 4-D space-time surfaces brings in classical dynamical variables necessary for quantum measurement theory. p-Adic physics defines correlates for cognition and intentionality. One can argue that these are not quantum measured in the conventional sense so that 4-D p-adic space-time sheets would not be needed at all. The p-adic variant for the exponent of Chern-Simons action can make sense using a finite-D algebraic extension defined by q=exp(i2π/n) and restricting the allowed lightlike partonic 3-surfaces so that the exponent of Chern-Simons form belongs to this extension of p-adic numbers. This restriction is very natural from the point of view of dark matter hierarchy involving extensions of p-adics by quantum phase q.
5. Is the p-adic temperature proportional to the Kähler coupling strength?
- One can also consider extending p-adic numbers by introducing exp(π) and its powers and possibly also π. This would make the extension of p-adics infinite-dimensional which does not conform with the basic ideas about cognition. Note that ep is not p-adic transcendental so that extension of p-adics by powers e is finite-dimensional and if p-adics are first extended by powers of π then further extension by exp(π) is p-dimensional.
- A more tricky manner to overcome the problem posed by the CP2 extremals is to notice CP2 type extremals are necessarily deformed and contain a hole corresponding to the lightlike 3-surface or several of them. This would reduce the value of Kähler action and one could argue that the allowed p-adic deformations are such that the exponent of Kähler action is a p-adic number in a finite extension of p-adics. This option does not look promising.
Kähler coupling strength would have the same spectrum as p-adic temperature Tp apart from a multiplicative factor. The identification Tp=1/k is indeed very natural since also gK2 is a temperature like parameter. The simplest guess is
Also gauge couplings strengths are expected to be proportional to gK2 and thus to 1/k apart from a factor characterizing p-adic coupling constant evolution. That all basic parameters of theory would have simple expressions in terms of k would be very nice from the point of view quantum classical correspondence.
If U(1) coupling constant strength at electron length scales equals αK=1/104, this would give 1/Tp≈ 1/26. This means that photon, graviton, and gluons would be massless in an excellent approximation for say p=M89, which characterizes electroweak gauge bosons receiving their masses from their coupling to Higgs boson. For fermions one has Tp=1 so that fermionic lightlike wormhole throats would correspond to the strongest possible coupling strength αK=1/4 whereas gauge bosons identified as pairs of light-like wormhole throats associated with wormhole contacts would correspond to αK=1/104. Perhaps Tp=1/26 is the highest p-adic temperature at which gauge boson wormhole contacts are stable against splitting to fermion-antifermion pair. Fermions and possible exotic bosons created by bosonic generators of super-canonical algebra would correspond to single wormhole throat and could also naturally correspond to the maximal value of p-adic temperature since there is nothing to which they can decay.
A fascinating problem is whether k=26 defines internally consistent conformal field theory and is there something very special in it. Also the thermal stability argument for gauge bosons should be checked.
What could go wrong with this picture? The different value for the fermionic and bosonic αK makes sense only if the 4-D space-time sheets associated with fermions and bosons can be regarded as disjoint space-time regions. Gauge bosons correspond to wormhole contacts connecting (deformed pieces of CP2 type extremal) positive and negative energy space-time sheets whereas fermions would correspond to deformed CP2 type extremal glued to single space-time sheet having either positive or negative energy. These space-time sheets should make contact only in interaction vertices of the generalized Feynman diagrams, where partonic 3-surfaces are glued together along their ends. If this gluing together occurs only in these vertices, fermionic and bosonic space-time sheets are disjoint. For stringy diagrams this picture would fail.
To sum up, the resulting overall vision seems to be internally consistent and is consistent with generalized Feynman graphics, predicts exactly the spectrum of αK, allows to identify the inverse of p-adic temperature with k, allows to understand the differences between fermionic and bosonic massivation, and reduces Wick rotation to a number theoretic symmetry. One might hope that the additional objections (to be found sooner or later!) could allow to develop a more detailed picture.
For more details see the chapter An Overview About Quantum TGD.