Eta Products, BPS States and K3 Surfaces Yang-Hui He1 & John McKay2 1 Department of Mathematics, City University, London, EC1V 0HB, UK and School of Physics, NanKai University, Tianjin, 300071, P.R. China and Merton College, University of Oxford, OX14JD, UK
[email protected] 2 Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Quebec, H3G 1M8, Canada
[email protected] Abstract Inspired by the multiplicative nature of the Ramanujan modular discriminant, ∆, we consider physical realizations of certain multiplicative products over the Dedekind eta-function in two parallel directions: the generating function of BPS states in certain heterotic orbifolds and elliptic K3 surfaces associated to congruence subgroups of the modular group. We show that they are, after string duality to type II, the same K3 arXiv:1308.5233v3 [hep-th] 8 Jan 2014 surfaces admitting Nikulin automorphisms. In due course, we will present identities arising from q-expansions as well as relations to the sporadic Mathieu group M24. 1 Contents 1 Introduction and Motivation 3 1.1 Nomenclature . .5 2 Eta Products and Partition Functions 7 2.1 Bosonic String Oscillators . .7 2.2 Eta Products . .9 2.3 Partition Functions and K3 Surfaces . 10 2.4 Counting 1/2-BPS States . 11 3 K3 Surfaces and Congruence Groups 13 3.1 Extremal K3 Surfaces . 13 3.2 Modular Subgroups and Coset Graphs . 14 3.3 Summary . 17 3.4 Beyond Extremality . 20 3.5 Elliptic Curves . 21 4 Monsieur Mathieu 24 5 A Plethystic Outlook 28 6 Conclusions and Prospects 30 A Further Salient Features of Eta 33 2 A.1 Modularity .