E-Strings and N=4 Topological Yang-Mills Theories

TL;DR

This paper explores the properties of six-dimensional tensionless E-strings and their relation to N=4 topological Yang-Mills theories, revealing new connections between string bound states, current algebras, and topological invariants.

Contribution

It establishes a novel link between E-string bound states and N=4 U(n) topological Yang-Mills theory on half K3 surfaces, including explicit character computations and partition function relations.

Findings

E-strings form bound states with specific current algebra and conformal properties.

Characters of bound states are captured by N=4 U(n) topological Yang-Mills theory.

Partition functions relate to Euler characteristics of instanton moduli spaces and exhibit a holomorphic anomaly.

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 $E_8$ level $n$ current algebra as well as a left-over conformal system with $c=12n-4-{248n \over n+30}$, 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 $\half K3$. 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 three-fold, give the Euler characteristics of the Yang-Mills instanton moduli space on $\half K3$. 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.