Modular properties of 6d (DELL) systems

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 6 d SU( N ) theory with two independent modular parameters τ and τ , the modular anomaly equation changes, because the modular transform of τ is accompanied by an ( N -dependent!) shift of τ and vice versa. This is a new peculiarity of double-elliptic systems, which deserves further investigation.