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 $τ= \fracθ{2π}+ \frac{4π\imath}{g^2}\longrightarrow -\frac{1}τ$. The low-energy Seiberg-Witten prepotential ${\cal F}(a)$, however, is not explicitly invariant, because the flat moduli also change $a \longrightarrow a_D = \partial{\cal F}/\partial a$. 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 {\it two} independent modular parameters $τ$ and $\hat τ$, the modular anomaly equation changes, because the modular transform of $τ$ is accompanied by an ($N$-dependent!) shift of $\hatτ$ and vice versa. This is a new peculiarity of double-elliptic systems, which deserves further investigation.