Pure and Applied Mathematics Quarterly

Volume 19 (2023)

Number 1

Special Issue in honor of Don Zagier

Guest editors: Benedict H. Gross, Ken Ono, and Fernando Rodriguez Villegas

Electrotechnics, quantum modularity and CFT

Pages: 341 – 370

DOI: https://dx.doi.org/10.4310/PAMQ.2023.v19.n1.a13


Werner Nahm (Dublin Institute for Advanced Studies, Dublin, Ireland)


Based on older and recent work of Don Zagier and collaborators, a relation between algebraic K‑theory and modularity is investigated. It arises from the study of the behaviour of certain $q$-hypergeometric functions near the roots of unity. In typical cases, one finds intricate phenomena described by “quantum modularity”. Here it is investigated when they reduce to ordinary modular invariance. For the $q$-hypergeometric functions under study (“Nahm sums”) this leads to systems of algebraic equations. I had conjectured an equivalence between modular invariance of the $q$-hypergeometric functions and a K‑theoretic torsion property of all solutions of the algebraic equations. This rough and ready conjecture was not quite correct, though it pointed in a rewarding direction. In particular, Zagier’s recent work with Calegari and Garoufalidis proved that modular invariance implies the torsion property for a special solution of the algebraic system. Here their argument is generalised. The $q$-hypergeometric series should be understood as convolutions of Jacobi forms (as defined and explored by Eichler and Zagier). The Jacobi forms are vector valued, with components described by some finite set $\mathcal{M}$. For each element of $\mathcal{M}$ one has a specific system of algebraic equations and for all of them at least one solution must have a modified K‑theoretic torsion property. A related inversion property conjectured twenty years ago is proven. At greater depth new structures do appear. There is a relation to algebraic geometry, namely to vector bundles over tori. Particularly intriguing is a possible relation between CFT (mathematics) and CFT (physics).


CFT, Nahm sums, K3 group

2010 Mathematics Subject Classification

11F11, 81T40

Received 10 January 2022

Received revised 11 July 2022

Accepted 19 September 2022

Published 3 April 2023