We solve the discrete Hirota equations (Kirillov-Reshetikhin Q-systems) for Ar, and their analogue for Dr, for the cases where the second variable ranges over either a finite set or over all integers. Until now only special solutions were known. We find all solutions for which no component vanishes, as required in the known applications. As an introduction we present the known solution where the second variable ranges over the natural numbers.
When is a q-series modular? This is an interesting open question in mathematics that has deep connections to conformal field theory. In this paper, we define a particular r-fold q-hypergeometric series fA, B, C, with data given by a matrix A, a vector B and a scalar C, all rational, and ask when fA, B, C is modular. In the past much work has been done to predict which values of A give rise to modular fA, B, C; however, there is no straightforward method for calculating corresponding values of B. We approach this problem from the point of view of conformal field theory, by considering (2n + 3, 2)-minimal models, and coset models of the form . By calculating the characters of these models and comparing them to the functions fA, B, C, we succeed in computing appropriate B-values in many cases.
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