Modular properties of 6d (DELL) systems

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 [InlineMediaObject not available: see fulltext.]. The low-energy Seiberg-Witten prepotential ℱ(a), however, is not explicitly invariant, because the flat moduli also change a − → aD = ∂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 6d 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.