Today, I single out the first hep-th paper:

- "M" is parameterized by expectation values of dynamical scalar fields, i.e. there are no adjustable non-dynamical continuous parameters in string theory. That's a well-known piece of string theory folklore.
- for any point "P0" of the moduli space "M", there exist points "P" in "M" that are arbitrarily far from "P0"; the metric is measured by the kinetic terms for the scalar fields defined as fields in the large dimensions. In other words, the diameter of "M" is infinite regardless of the way how you look at "M". Note that this is true despite the fact that the volume of the moduli space is finite whenever its dimension exceeds one, according to the first swampland paper.
- in the previous picture, as you take points "P" that are very far from "P0", by distance "D", you will find a whole infinite tower of states whose mass is of order "exp(-A.D)" where "A" is a coefficient (exponent). We have an exponential in the formula because in the most obvious example, the distances are measured by the differences of the dilaton, and the powers of the coupling constant are exponentials of this dilaton. The existence of new light states may imply a breakdown of effective field theory, following particular rules.
- the scalar curvature of the moduli space "M" near these points at infinity - where "P" lived - is never positive; the space resembles a saddle in these regions. The only way how the inequality may be saturated is that you deal with a one-dimensional moduli space.
- every one-cycle in the moduli space "M" is contractible to a vanishing distance. This is morally true because otherwise the winding number around the moduli space, in the context of a compactification on a circle, would have a global symmetry (counting the winding number) and global symmetries don't exist in quantum gravity.

## snail feedback (1) :

Dear Lubos,

thank you very much for inimitable delight!

"the set "M" of all consistent quantum gravity backgrounds (which is assumed to be the set of backgrounds of the fully "completed" string theory)"

The set "M" include subset "G" of all quantum gravity backgrounds AND uncountable bigger subset "C" of all continuous degeneracy of nongeometrical backgrounds, which are U-dual itself.

But distinguishing between "G" and "C" is unnatural becouse of quantum gravity backgrounds are quantum combination of the whole set "M" (which is superposition of the U-dual manifold).

"Your task is to verify these conjectures or disprove them, and if you don't disprove them, you should eventually prove them."

Solid evidence for these 5 conjectures imposes, in itself, U-duality:

0 g = uncountable many g

which we can represent by irreps. of the U-dual modular group:

Coherent state of K-theoretic condensates of hte multi-flux warped octonionic black hole throat

which root lattice = Z = continuous degeneracy of vacua = octonionic matrix-valued Hagedorn tachyon = 0.

Salute, Planckeon

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