Modular properties of 6d (DELL) systems

TL;DR

This paper explores the modular properties of 6d super-Yang-Mills theories, revealing a novel interplay between two modular parameters in double-elliptic systems that affects the modular anomaly equations.

Contribution

It introduces a new modular anomaly equation for 6d SU(N) theories with two modular parameters, highlighting their interdependent transformations.

Findings

Modular transform of $ au$ shifts $ au$ and $\hat{ au}$ in a coupled manner.

The modular anomaly equation is modified in 6d double-elliptic systems.

New dependence of the prepotential on Eisenstein series $E_2$ and modular parameters.

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 $\tau = \frac{\theta}{2\pi}+ \frac{4\pi\imath}{g^2}\longrightarrow -\frac{1}{\tau}$. 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 $\tau$ and $\hat \tau$, the modular anomaly equation changes, because the modular transform of $\tau$ is accompanied by an ($N$-dependent!) shift of $\hat\tau$ and vice versa. This is a new peculiarity of double-elliptic systems, which deserves further investigation.