Partition functions for BPS states of the non-critical \(E_8\) string (Q1381304)

There are found explicit expressions (in terms of modular forms) for the momentum partition functions \(G_n\) for the states with winding number \(n\) by considering the BPS states of the \(E_8\) non-critical string wound around one of the circles of a toroidal compactification to four dimensions. It is formulated an algorithm for generating the \(G_n\) by proving a recurrence relation being reminiscent of the holomorphic anomaly equations of Kodaira-Spencer theory.