Modular properties of 6d (DELL) systems

Abstract

Abstract If super-Yang-Mills theory possesses the exact conformal invariance, there is an additional modular invariance under the change of the complex bare charge . The low-energy Seiberg-Witten prepotential ℱ(a), however, is not explicitly invariant, because the flat moduli also change a − → a D = ∂ℱ/∂a. In result, the prepotential is not a modular form and depends also on the anomalous Eisenstein series E 2. This dependence is usually described by the universal MNW modular anomaly equation. We demonstrate that, in the 6d SU(N) theory with two independent modular parameters τ and τ^ $$ \widehat{\tau} $$, the modular anomaly equation changes, because the modular transform of τ is accompanied by an (N -dependent!) shift of τ^ $$ \widehat{\tau} $$ and vice versa. This is a new peculiarity of double-elliptic systems, which deserves further investigation.

Keywords

• Anomalies in Field and String Theories
• Integrable Hierarchies
• Supersymmetric Gauge Theory
• Supersymmetry and Duality