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 "Image missing". 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 $$ \widehat{\tau} $$ τ ^ , the modular anomaly equation changes, because the modular transform of τ is accompanied by an (N -dependent!) shift of $$ \widehat{\tau} $$ τ ^ and vice versa. This is a new peculiarity of double-elliptic systems, which deserves further investigation.