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 τ=(θ/(2π))+((4πı)/(g²))⟶−(1/τ). 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 E₂. This dependence is usually described by the universal MNW modular anomaly equation. We demonstrate that, in the 6dSU(N) theory with two independent modular parameters τ and τ̂, the modular anomaly equation changes, because the modular transform of τ is accompanied by an (N-dependent!) shift of τ̂ and vice versa. This is a new peculiarity of double-elliptic systems, which deserves further investigation.