There seems to be a media campaign to revive the beliefs in this thinking. As superstrings have turned out to be unable to say anything interesting about physics, some super string gurus have decided to "prove" mathematically that superstring theory is the only possible theory of quantum gravitation and blog audience is expected to believe the "proofs". Superstrings have been the only possible quantum theory of gravitation but for reasons having very little to do with science.

There is a popular article about these attempts in Quanta Magazine. The opinions of various academic authorities are asked about about this claim.

The arguments in favor of this claim - certainly not proofs in any imaginable sense of the word - represent some pieces of evidence that certain conformal field theories in 2-D Minkowski defining boundary of AdS_{3} have a mass spectrum with exponentially growing density of states characterizing strings like objects so that AdS_{3}/CFT is plausible and theories could be dual to a string theory in 3-dimensional AdS_{3}. From this it is concluded by some magic leap of lawyer logic completely incomprehensible to me that super string theory is the only possible theory of quantum gravity. An additional conclusion of Lubos is that all those who do not believe this are idiots and enemies of science.

Peter Woit points out some of the fallacies of the argument. First of all, 3-D quantum gravity is topological quantum field theory so that it cannot have anything to do with super string model and gravitation since there are no gravitons: here one could however argue that the corresponding string theory is higher dimensional and involve S^{7} factor. Secondly the dimension is wrong: it should be 4 rather than 3.

These articles are excellent examples of the sloppy thinking prevailing in recent day theoretical physic - to say nothing about the catastrophic situation in blogs! One should first of all precisely define what string theory means. There are many theories involving string like objects as basic dynamical entities. That some conformal field theory leads to density of states of form assignable to strings like objects tells absolutely nothing about the status of super strings as theory of quantum gravitation.

Below some attempts to encourage more precise thinking based by posing three questions: What does one really mean with a theory of gravity and what does one believe to be true about gravity? What does one really mean with super-conformal invariance? What does one really mean with string like objects?

** What does one mean with a theory of gravity? **

Can one just start from the quantization attempts of general relativity and require generalization of the lowest order Feynman diagrammatics in Minkowski space background? Is gravitation really what we believe it to be. There are anomalies even at the level of solar system: Pioneer and Flyby anomalies. Dark matter is still mystery and all the proposed models have failed. Galactic dark matter does not seem to be a spherical halo as has been believed.

TGD has led to rather detailed view about new quantum physics realized in astrophysical scales via the hierarchy of Planck constants and explaining various anomalies (see for instance this and this). This view about quantum gravitation is based on experimental findings and differs dramatically from the ultra-naive assumption behind super string models assuming that everything is understood down to Planck scale.

** What does one mean with super-conformal symmetry?**

When one speaks of AdS/CFT, one should not forget that it is based on one particular definition of super-conformal invariance: the conformal invariance associated with 2-D surfaces. In TGD framework 2-D surfaces are replaced by 3-D surfaces with one light-like direction so that one still has metric 2-dimensionality. This implies a gigantic extension of super-conformal symmetries and replaces AdS/CFT duality involving 10-D space-times AdS_{n} ×S^{10-n} with a duality which realizes the counterpart of ordinary holography stating that 2-D partonic surfaces and their tangent space data plus possibly string world sheets code for physics.

This of course shows that superstrings are certainly not the only game in the town: the super-conformal symmetries of super-string models are infinitesimal as compared to those of TGD. TGD also predicts space-time dimension correctly, is unique since 4-D Minkowski space M^{4} and CP_{2} are twistorially completely unique (their twistor space allows Kähler structure). And of course, the symmetries of TGD are those of standard model and TGD leads to a flow of non-trivial predictions in all scales: applications to quantum biology, neuroscience, and consciousness the most fascinating examples.

I do not want that TGD would be the only game in the town although it is very regrettable that I am still the only developer of this marvellous theory. The sociology of science is profoundly irrational.

**What does one mean with super-symmetry? **

Superstrings assume a very specific form of space-time super symmetry. The realization of space-time SUSY relies typically on the assumption of Majorana spinors leading to the non-conservation of fermion numbers. The failure of LHC to find N=1 SUSY has finally activated the attempts to build more realistic SUSY scenarios with Dirac fermions.

In TGD framework a different kind of SUSY indeed emerges as a dynamical symmetry and is unavoidably broken at the level of quantum state. The fermionic oscillator operators at partonic surfaces labelled by fermion number, helicity, weak quantum numbers, and what might be called classical 8-momentum generate the SUSY. There is a hierarchy of SUSY breakings. Right-handed neutrino generates N=2 SUSY with smallest breaking. The other H-spinor components generate N=4 SUSYs for both quarks and leptons combining to form N=8 SUSY of N=8 supergravity. These states can still have relatively low mass since statistics allows same wave function at the partonic 2-surface for the spinor modes involved. For higher modes there is "wiggling" in CP_{2} scale and masses are measured using CP_{2} mass as a natural unit. Hence a theory resembling N=4 SUSY should be a good approximation to TGD. By adding conformal weight one obtains super-conformal algebra.

** What does one mean with strings?**

First of all, I hate neither strings nor physics. I hate only attempts to build hegemonies based on "The only game in the town" thinking: intelligence and arrogance are mutually exclusive!;-). I just want want to tell that superstrings were the first and - sad to say - wrong guess and a much more feasible option emerged 7 years befor the first super string revolution (1984).

* Strings in TGD*

Consider first the notion of string in TGD. TGD predicts 4-D string like objects with 2-D string world sheet as projection to dominate the cosmology before transition to a radiation dominated phase by the counterpart of inflationary period. String like objects which I call magnetic flux tubes dominate physics of later times in all scales.

Also genuine 2-D string world sheets emerge also from TGD from a very general argument forcing the induced spinor fields to 2-D string world sheets: the electromagnetic charge for the modes of induced spinor fields must be well-defined and this requires that classical W boson fields vanish implying restriction to string world-sheets.

Recently a stronger variant of this condition has emerged. The octonionic variant of Dirac operator allows to define generalization of the basic formulas for 4-D twistors. The octonionic Dirac operator in its algebraic version must be equivalent with the ordinary one: associativity condition requires that all induced weak gauge fields must vanish at string world sheets so that they must have 1-D CP_{2} projection and can appear only in regions with Minkowskian signature of the induced metric.

Furthermore, the gamma matrices associated with string world sheets are most naturally induced gamma matrices and SUSY requires that world sheet area appear besides Kähler action in Minkowskian regions. Gravitational constant would appear at the level of basic definition of theory and quantum criticality condition would fix its ratio to CP_{2} length squared. The view about the relationship between inertial and gravitational four-momentum becomes more precise. This suggests that string world sheets do somewhat more than just "emerge" in TGD framework.

* Twistor strings and their TGD counterparts*

Strings could also mean twistor strings of Witten, which he proposed as a manner to understand the scattering amplitudes of super-symmetric Yang-Mills theories (SYMs). This construction generalizes in a natural manner to TGD by replacing Witten's string world sheets (2-D surfaces) with unions of partonic 2-surfaces and string world sheets assumed to be surface of space-time surface which is base space of its twistor space having imbedding as a 6-D surface in the production of twistor spaces of M^{4} and CP_{2}.

I have been just working with the twistorial construction of the generalization of S-matrix in TGD framework inspired by several breakthroughs in the mathematical understanding of TGD.

- TGD as a physics based of 4-D space-time surface in 8-D imbedding space can be lifted to that for their 6-D twistor spaces representable as 6-surfaces in 12-D Cartesian product of twistor spaces of M
^{4}and CP_{2}(see this).

- Extended conformal invariance can be realized as conformal gauge condition and classically these conditions state that all conformal charges of space-time surfaces vanish. This realized the notion of preferred extremal precisely (see this). As a matter fact, one has infinite hierarchy of conformal symmetry breakings defined by the vanishing conditions for the sub-algebras of conformal algebras with conformal weights coming as multiples of integer n. This gives rise to the hierarchy of Planck constants and dark matters.

- Preferred extremal property fixes the space-time regions to a very high degree and there are arguments suggesting that in Euclidian regions one as quaternion-Kähler manifolds having twistor spaces which are so called Fano spaces (see this).

- The generalization of ordinary complex analyticity to quaternion analyticity seems to make sense after all and extends the 3-D light-like conformal symmetry to its quaternionic analog in the interior of 4-D space-time surfaces (see this).

- Witten's twistor string theory generalizes in natural manner to TGD (see this and this) framework. 2-D objects appear as partonic 2-surfaces and string world sheets carrying induced fermion fields and both are needed.

- Also new insights about how projective sub-manifolds of twistor Grassmannians appearing as surfaces over which twistor amplitudes are obtained as residue integrals emerge. One can identify the complex moduli characterizing polynomials defining partonic 2-surfaces in the twistor space of imbedding space as complex coefficients of these polynomials and defined modulo overall complex scaling so that the moduli space

is projective sub-manifold of a projective space or more generally, Grassmannian.

One can also generalize also the notion of positive Grassmannian to complex situation. In the real case positivity follows from the condition that the integrand of the twistor integral is projectively well-defined and therefore non-negative. In complex case the positivity can be generalized to the assumption that various complex coordinates are positivity in the sense that they have values in hyperbolic space realized as upper half-plane of complex plane. Hyperbolic space defining various 2-D hyperbolic geometries. Positivity and its generalization have deep number theoretic meaning in TGD framework since they allow the algebraic continuation of the complex amplitudes to p-adic number fields (see this and this) .

## 4 comments:

"When one speaks of AdS/CFT, one should not forget that it is based on one particular definition of super-conformal invariance: the conformal invariance associated with 2-D surfaces"

Sorry, I can't understand what do you mean. Yangian structure? T-duality in the AdS sigma model ? two dimensional Ising model with CFT?

Could you please elaborate more? Thank you very much.

Thank you for asking since this is extremely important point that I have tried to get through for more that two decades!

2-D conformal invariance is due to 2-dimensionality of the basic geometric objects, Riemann surfaces. One can generalise 2-D conformal symmetry in non-trivial manner to light-like 3-surfaces (having one-light-like direction), which are metrically 2-dimensional.

Light-cone boundary with topology S^2xR_+ is the simplest example. Denote by r the radial light-like coordinate.

*There is no contribution from the light-like radial direction to the induced metric and it is just ds^2= r^2dOmega^2, where dOmega^2 is the metric of S^2. The metric is effectively 2-D and allows conformal transformations of S^2 depending parametrically on r as generalised conformal transformations. By a suitable choice of radial dependence the conformal transformations act even as isometries so that ordinary 2-D conformal group acts as isometries.

Another huge conformal symmetry emerges in M^4xCP_2. Now one can consider symplectic transformations of S^2xCP_2 depending parametrically on r. This group has structure of conformal group with r taking the role of z so that finite-D group defining Kac-Moody ext rends to infinite-D symplectic group of M^4xCP_2.

These two kinds of symmetries mean gigantic extension of the conformal symmetries of string models. These symmetries act at imbedding space level. This of course also replaces AdS/CFT correspondence with the correspondence realised in TGD framework.

A further generalisation of the conformal symmetries of light-cone boundary are to the

conformal symmetries of light-like partonic orbits

since the effective metric 2-dimensionality is enough. These conformal symmetries correspond to ordinary Kac-Moody type symmetries acting at space-time level.

Thank you for clarifying that point about conformal invariance!

So, in two dimensions, infinitesimal conformal transformations are functions which obey the Cauchy-Riemann equations.

Conformal vector fields can be considered as a natural generalization of Killing vector fields.

For example, in the case of the black hole, there are metrics which allow the existence of Killing horizons and these Killing horizons coincide with their event horizons.

In terms of the dual CFT, CFT observers cannot exchange information faster than light.

The motivation behind their new work (the meaning of "the only game in the town") seems to be that string-like theory in flat space is, broadly defined, the simplest example of a weakly coupled theory of gravity.

Thanks again for your very interesting post and your kind answer!

I want to emphasise that I have nothing against string models. Strings in 4-D space-time are part of TGD and it is now rather clear that gravitational constant appears as fundamental constant besides Kahler coupling strength and CP_2 radius defining universal scale. The ratio hbarG/R^2 is dictated by quantum criticality.

The especially interesting aspect is quaternion conformal symmetry generalising complex analyticity to quaternion conformal analyticity: the generalisation of Cauchy-Riemann equations is carried out decades ago but I found it only now.

Post a Comment