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E-strings and \(N=4\) topological Yang-Mills theories. (English) Zbl 0951.81025
Summary: 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 \(c=12n-4-(248n/n+30)\), 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 \(\frac 12 K3\). 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 \(\frac 12 K3\). 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.

MSC:
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
32J81 Applications of compact analytic spaces to the sciences
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
58D27 Moduli problems for differential geometric structures
81T60 Supersymmetric field theories in quantum mechanics
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