E-strings and N = 4 topological Yang-Mills theories

Abstract

We study certain properties of six-dimensional tensionless E-strings (arising from zero size E 8 instantons). In particular we show that n E-strings form a bound string which carries an E 8level- n current algebra as well as a left-over conformal system with c = 12 n−4− (248 n/ n+30), whose characters can be computed. Moreover we show that the characters of the n-string bound state are captured by N = 4 U( n) topological Yang-Mills theory on 1 2 K3 . This relation not only illuminates certain aspects of E-strings but can also be used to shed light on the properties of N = 4 topological Yang-Mills theories on manifolds with b 2 + = 1. In particular the E-string partition functions, which can be computed using local mirror symmetry on a Calabi-Yau threefold, give the Euler characteristics of the Yang-Mills instanton moduli space on 1 2 K3 . Moreover, the partition functions are determined by a gap condition combined with a simple recurrence relation which has its origins in a holomorphic anomaly that has been conjectured to exist for N = 4 topological Yang-Mills on manifolds with b 2 + = 1 and is also related to the holomorphic anomaly for higher genus topological strings on Calabi-Yau threefolds.