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Dynamical OPE coefficients as a TOE

Towards the universal equations for quantum gravity in all forms

In the 1960s, before string theory was really born, people studied the bootstrap and the S-matrix theory. The basic idea – going back to Werner Heisenberg (but driven by younger folks such as Geoffrey Chew who died in April 2019) – was that the consistency was enough to determine the S-matrix. In such a consistency-determined quantum theory, there would be no clear difference between elementary and composite fields and everything would just fit together.

Veneziano wrote his amplitude in 1968 and a few years later, it became clear that strings explained that amplitude – and the amplitude could have been created in a "constructive" way, just like QCD completed at roughly the same time which was "constructively" made of quark and gluon fields (although most of the smartest people had believed the strong force not to have any underlying "elementary particles" underneath throughout much of the 1960s). A new wave of constructive theories – colorful and stringy generalizations of the gauge theories – prevailed and downgraded bootstrap to a quasi-philosophical semi-dead fantasy.

On top of that, the constructive theory – string theory – has led to progress that made it clear that it has a large number of vacua so the complete uniqueness of the dynamics was an incorrect wishful thinking.

Still, all these vacua of string/M-theory are connected and the theory that unifies them is unique. Since the 1990s, we have understood its perturbative aspects much more than before, uncovered limited nonperturbative definitions for some superselection sectors, but it's true that the perturbative limit of string theory is the most thoroughly understood portion of quantum gravity that we have.

Various string vacua are being constructed in very different ways. We start with type II-style \(N=1\) world sheet dynamics, or with the heterotic \(N=(1,0)\) dynamics, add various GSO projections and corresponding twisted and antiperiodic sectors. Extra orientifold and orbifold decorations may be added, along with D-branes and fluxes. And the hidden dimensions may be treated as some Ricci-flat manifolds – with fluxes etc. – but also as more abstract CFTs (conformal field theories) resembling minimal models such as the Ising model.

The diversity looks too large. It seems that while the belief in the single underlying theory is totally justified by the network of dualities and topological phase transitions etc., the unity isn't explicitly visible. Isn't there some way to show that all these vacua of string/M-theory are solutions to the same conditions?

It's a difficult task (and there is no theorem guaranteeing that the solution exists at all) because the individual constructions are so different – even qualitatively different. The different vacua – or non-perturbative generalizations of world sheet CFTs – have to be solutions to some conditions or "equations". But how can so heavily different "stories" be "solutions" to the same "equations"? Only relatively recently, I decided that to make progress in this plan, one has to "shut up and calculate".

And to calculate, to have a chance of equations and their solutions, one needs to convert the "stories" to "quantitative objects". We need to "quantify the rules defining theories and orbifolds etc.". To do so, we need to write a more general Ansatz that interpolates between all the different theories and vacua that we have in string/M-theory i.e. quantum gravity.

What is the Ansatz that is enough for a full world sheet CFT? Well, it's necessary and almost sufficient to define the spectrum of all local operators and their OPEs (operator product expansions). The latter contain most of the information and are encoded in OPE coefficients \(C_{12}^3\) – closely related to three-point structure constants \(C_{123}\) and \(B_3\), see e.g. these pages for some reminder about the crossing symmetry and the conformal bootstrap.

What you should appreciate is that coefficients such as \(C_{12}^3\) encode most of the information about the world sheet CFT – e.g. a perturbative string vacuum – but they are treated as completely static numbers that define the theory at the very beginning. You can't touch them afterwards; they can't change. It became clear to me that it's exactly this static condition that is incompatible with the desire to unify the string vacua as solutions to the same equations.

To have a chance for a unifying formulation of a theory of everything (TOE), we apparently need to treat all these coefficients such as \(C_{12}^3\) as dynamical ones. "Dynamical" means "the dependence on time". Which time? I think that in the case of \(C_{12}^3\), we need the dependence on the spacetime's time \(x^0\) (and its spatial partners \(x^i\), if you wish), not the world sheet time \(\tau\), because the values of all these coefficients \(C_{12}^3\) carry the "equivalent information" as any more direct specification of the point on the configuration space of the effective spacetime QFT (a point in the configuration space of mainly scalar fields in the spacetime, if you wish).

Most of the degrees of freedom in \(C_{12}^3\) are non-dynamical ones. There is a lot of freedom hidden in the ability to linearly mix the operators into each other; and on top of that, all these coefficients are apparently obliged to obey the crossing symmetry constraints. But there should still exist a quantization of states on the configuration space of these \(C_{12}^3\) coefficients and the resulting space should be equivalent to a configuration space of fields in the target spacetime.

Since the 1995 papers by Kutasov and Martinec, I've been amazed by the observation that "whatever super-general conditions of quantum gravity hold in the spacetime, they must also apply in the world sheet, and vice versa". So I think that there are analogous operators to the local operators in the world sheet CFT but in the spacetime – labeling the "creation operators for all black hole microstates" – and their counterpart of \(C_{12}^3\) tells us about all the "changes of one microstate to another" that you get when a black hole devours another particle (or another black hole microstate). These probably obey some bootstrap equations as well although as far as I know, the research of those is non-existent in the literature and may only be kickstarted after the potential researchers learn about this possibility from me.

I tend to think that the remaining degrees of freedom in \(C_{12}^3\) aren't fully eliminated. They're just very heavy on the world sheet – and responsible for quantum gravity, non-local phenomena on the world sheet that allow the world sheet topology to change.

In the optimal scenario, one may write down the general list of the fields like \(C_{12}^3\) – possibly, three-point functions may fail to be enough and \(n\)-point functions for all positive \(n\) may be needed as well – and there will be some universal conditions. These should have the solutions in terms of the usual consistent, conformal, modular invariant world sheet theories with the state-operator correspondence. The rules could have a direct generalization outside the weakly coupled limit but the solutions should simplify in the weakly coupled limit. Dualities should be manifest – the theories or vacua dual to each other would explicitly correspond to the same values of the degrees of freedom such as \(C_{12}^3\) and the particular "dual descriptions" would be just different strategies to construct these solutions, usually starting with some approximations.

More generally, the term "quantum gravity" sounds a bit more general than "string/M-theory" although they're ultimately equivalent. "Quantum gravity" doesn't have any obvious strings in it to start with – and has just all the black hole microstates and their behavior. It seems clear to me that people need to understand the equivalence between "quantum gravity" and "string theory" better and to do so, they have to rewrite the rules of string theory in the language of a much larger number of degrees of freedom. The word "consistency" before "of quantum gravity" sounds too verbal and qualitative so far – we should have a much clearer translation of this abstract noun to the language of equations.

It seems to me that the number of people in the world who are really intensely thinking on foundational questions of any similar kind or importance is of order one and the ongoing anti-science campaign has the goal to reduce this estimate to a number much smaller than one.

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