Infinite Distances and the Axion Weak Gravity ConjectureThe first thing I needed to clarify was "what is the exact form of the 'axion weak gravity conjecture'" that they are using. There must surely be a standalone paper that formulates this variation of our conjecture. And oops, the relevant paper was [4] AMNV. I have already heard the M-name somewhere.

Yes, of course I knew the main point we wrote about the "axion weak gravity conjecture". That point – discussed in a paper by Banks, Dine, Fox, and Gorbatov (and in some lore I could have had heard from Tom many years earlier, unless I told him) – had largely stimulated the research into the "normal" weak gravity conjecture itself.

The conjecture says that the decay constant of an axion shouldn't be too high – in fact, its product with the action of the relevant instanton is smaller than one in Planck units. This is a generalization of the "normal" weak gravity conjecture because the instanton is a lower-dimensional generalization of the charged massive point-like particles (higher-dimensional ones exist as well) and its action is a generalization of the mass/tension of the objects.

Our claim implies (the previous formulation by Banks et al.) that either the decay constant or the instanton action or both have to be small. And this condition has a nice implication: quantum gravity doesn't want to allow you to emulate flat potentials too closely, unless they're exactly flat, so the axion "wants" to be visible either because its decay constant is low or because the instanton corrections to its potential are sufficiently wiggly.

This is one of the particular insights that indicates that string theory's predictivity always remains nonzero – string theory doesn't want you to approximate the effective field theory of one vacuum by another vacuum too closely.

In the older Banks et al. formulation, the "axion weak gravity conjecture" was considered as a bad news because it indicated that some natural attempts to construct natural inflation were actually forbidden in quantum gravity.

Fine, now the two Dutchpersons look at a sufficiently wide and rich class of string compactifications to test the "axion weak gravity conjecture" – at type IIA string theory vacua on Calabi-Yau compactifications. Note that type IIB has the "point-like in spacetime" instanton, the D(-1)-instanton, and similarly all the other odd ones. The Dutch paper looks at type IIA so they need to look at the even D-brane instantons.

OK, the "generic" Calabi-Yau has everything of order one. To make the decay constants and instanton actions parameterically large or small, so that you may study whether some inequalities are parameterically obeyed or violated, they need to study extreme shapes of Calabi-Yaus. They look at extreme corners of the complex structure moduli space. The analysis of these "extreme directions" is somewhat analogous to my and Banks' dualities vs singularities.

And indeed, for every extreme direction in the Calabi-Yau complex structure moduli space, they find a tower of the D2-brane instantons that is predicted by the "axion weak gravity conjecture" – with the parameterically correct actions. That's quite a nice test of the conjecture. Curiously enough, to argue that the instantons exist, they need to use another swampland conjecture, the "swampland distance conjecture". Because the weak gravity conjectures should be counted as "swampland conjectures", they use one swampland conjecture to complete the partial proof of another one. I guess that a "swampland skeptic" could remain skeptical and call the proof circular.

*OK, Vengaboys are Dutch, too.*

At any rate, the "axion weak gravity conjecture" has passed a test (at least assuming that other conjectures hold) and it looks like a nontrivial test because the limits in the space of shapes of a Calabi-Yau aren't quite simple. The authors of the weak gravity conjectures arguably weren't idiots, it seems once again. The situation is really provoking because the weak gravity conjectures may be motivated and formulated rather easily and have a "philosophical beauty, naturalness, and coherence" which are very important in theoretical physics.

On the other hand, the proofs are partial, context-dependent, and very technical.

Cannot there be a universal proof of the "weak gravity conjecture(s)" that really unifies and clarifies all the partial proofs and that is as straightforward as the proof of Heisenberg's\[

\Delta x \cdot \Delta p \geq \frac{\hbar}{2}

\] or the generalized uncertainty principle inequalities? And don't these weak gravity conjectures have some direct far-reaching philosophical consequences for quantum gravity – much like the uncertainty principle basically implies that probabilities must be predicted relatively to an observer and from complex amplitudes?

Well, let me give you another, more detailed hint what you need to do to make a breakthrough analogous to the quantum mechanical one. In quantum mechanics, you first needed to realize that \(x,p\) from the inequality should be replaced with Hermitian operators. Here, we are talking about the values of parameters in

*effective actions*of quantum gravity. So these parameters that enter the WGC-like conjectures must correspond to

*some objects*, let's call them

*prdelators*because they're like operators but probably not quite, constructed within the full theory of quantum gravity or string/M-theory (which is more abstract than just an effective field theory). Your main task is to figure out what a "prdelator" is and why it has the property analogous to noncommutativity that is responsible for the swampland inequalities. And Czech readers must be warned that their partial understanding could be illusory.

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