Stark Units Proven Algebraic via Quantum Dilogarithms and Fusion Categories
Radchenko and Wheeler prove that Stark–Shintani ray class invariants for real quadratic fields—special values of Faddeev's modular quantum dilogarithm—are algebraic numbers, a long-sought instance of Hilbert's 12th problem-style explicit class field theory. Their method shows these special values satisfy an overdetermined polynomial system tied to Andersen–Kashaev quantum dilogarithms, which lets them invoke Ocneanu rigidity via a categorification of Izumi fusion rings; as byproducts they construct new irrational near-group fusion categories and prove conjectured quadratic relations for Stark units linked to Zauner's SIC-POVM conjecture. Twitter commentary, notably a thread from Steve Flammia, called the result 'amazing,' framing it within the broader program of using transcendental functions to explicitly construct algebraic numbers and abelian extensions, and highlighting the surprising bridge it builds to SIC-POVM/quantum-information conjectures.
Discussion: 2 tweets from 2 authors · @__alpoge__, @S_Flammia