A note on Nahm’s conjecture in rank 2 case. (English) Zbl 1350.11071

Huang, An; Lee, Chul-Hee

Commun. Number Theory Phys. 4, No. 4, 609-622 (2010).

From the text: The investigation of torus partition functions of certain conformal field theories is one of the central topics relating physics and number theory. From physical considerations, in general, one expects those functions to have nice modular transformation properties. Y. Zhu’s article [J. Am. Math. Soc. 9, No. 1, 237–302 (1996; Zbl 0854.17034)] is a fundamental paper devoted to establish such expectations rigorously by using the theory of vertex operator algebras. In [Frontiers in number theory, physics, and geometry II. Berlin: Springer 2003, 67–132 (2007; Zbl 1193.81092)], W. Nahm considered certain rational conformal field theories with integrable perturbations, and argued that their partition functions have a canonical sum representation in terms of \(q\)-hypergeometric series. Consequently, he conjectured a partial answer to the question of when a particular \(q\)-hypergeometric series is modular. This is called Nahm’s conjecture, and is discussed in [D. Zagier, Frontiers in number theory, physics, and geometry II. Berlin: Springer 2003, 3–65 (2007; Zbl 1176.11026)].
The aim of this paper is to get a complete list of positive definite symmetric matrices with integer entries \((\begin{smallmatrix} a b\\ b d\end{smallmatrix})\) such that all complex solutions to the system of equations \[ 1-x_1 = x^a_1x^b_2,\quad 1-x_2 = x^b_1x^d_2 \] are real. This result is related to Nahm’s conjecture in rank 2 case.

MSC:

11G55	Polylogarithms and relations with \(K\)-theory
33D70	Other basic hypergeometric functions and integrals in several variables

Keywords:

positive definite symmetric matrices with integer entries; \(q\)-hypergeometric series; Nahm’s conjecture

Citations:

Zbl 0854.17034; Zbl 1193.81092; Zbl 1176.11026