I’ve had several people ask me about this paper, so background 🧵 Radchenko–Wheeler, “Real quadratic fields and finite quantum dilogarithms I” (arxiv.org/abs/2609.21892). This result is amazing. They prove that certain special values of a transcendental function are actually algebraic. The goal of “explicit class field theory” is to construct algebraic numbers (numbers that are roots of polynomials with coefficients in ℤ), and ultimately abelian extensions of number fields, by evaluating transcendental functions at special points. It traces back to Hilbert's 12th problem. Ex 1: For ℚ, you have z = exp(2πix) at rational x. Clearly algebraic since z^n=1 if x=k/n. Ex 2: For imaginary quadratic fields, you have "complex multiplication". Take τ in the upper half-plane and Λ = ℤ + τℤ. Then ℂ/Λ is an elliptic curve. At imaginary quadratic (“CM”) values of τ, the transcendental modular function j(τ) (j-invariant) is an algebraic integer. Hilbert called CM "not only the most beautiful part of mathematics but also of the whole of science." If τ is real, this breaks completely because Λ = ℤ + τℤ is dense in ℝ (despite still being rank 2 as a ℤ-module). So ℂ/(ℤ+τℤ) is not a nice complex torus anymore and not an elliptic curve. This is one reason Hilbert’s 12th problem gets nasty immediately after ℚ and imaginary quadratic fields and is still almost totally open. (... in the Archimedean case. The p-adic case was proven by @samit_dasgupta and Kakde.) Stark and Shintani predicted that certain analytic quantities attached to real quadratic fields ("Stark units") should nonetheless be algebraic and generate abelian extensions. These can be written in terms of special values of Faddeev’s modular quantum dilogarithm, or a function Appleby, Kopp and I called "ש" (Shin), in homage to Shintani. Quantum dilogarithms are analytic objects, but are expected to satisfy special polynomial identities at certain special points. In our paper on Zauner's conjecture (arXiv:2501.03970), we gave a very explicit quadratic identity that they should satisfy ("the twisted convolution conjecture"), which, together with some more detailed conjectures of Stark, would prove Zauner in every dimension. Radchenko and Wheeler prove that the quantum dilogarithm indeed satisfies polynomial identities at certain special points. They also prove explicitly the twisted convolution conjecture for the case of the principal ideal (the admissible pair (d, ⟨1,1-d,1⟩) in the language of arXiv:2501.03970).  They claim the general case as well, though they omit the proof. Very interestingly, they prove the algebraicity by showing that the quantum dilogarithm maps to a fusion category, and since fusion categories obey "Ocneanu rigidity" they must be algebraic. Rigidity says, roughly, there are only finitely many equivalence classes with fixed fusion data and no continuous deformations. Because the structure constants solve polynomial equations over an algebraic field, that forces them into ℚ̅. Specifically, they say they "categorify Izumi fusion rings", but I don't know what that means and I look forward to understanding this better. Although they have proven algebraicity, they have not actually proven that their algebraic numbers are the "correct" ones, in the sense that "correct" Stark units should live in a specific field extension and have specific transformation properties under the action of the Galois group. This is left open. The consequences for Zauner's conjecture are: It's still completely open for any infinite family! To prove it, you would at least need to prove that there is a Galois automorphism in Gal(ℚ̅/ℚ) that takes the quantum dilogarithm special values to points on the unit circle. You might also need to prove some properties of the Galois group action; I will need to read this to understand it better. Congrats to the authors on a wonderful result!