Mahler Measure Variations, Eisenstein Series and Instanton Expansions

Abstract

This paper points at an intriguing inverse function relation with on the one hand the coefficients of the Eisenstein series in Rodriguez Villegas’ paper on “Modular Mahler Measures” and on the other hand the instanton numbers in papers on “Non-Critical Strings” by Klemm- Mayr-Vafa and Lerche-Mayr-Warner. In a companion paper Mahler measures will be related to dimer models. Thus, if it is not just a lucky coincidence in a few examples, our inverse function relation may be an incarnation of the duality between string models and dimer models proposed by Okounkov, Reshetikhin and Vafa in their paper “Quantum Calabi-Yau and Classical Crystals”. http://www.arxiv.org/abs/math.NT/0502193