Monodromy of elliptic curve convolution and $G_2$-motives of Beauville-Katz type

We study the convolution product on the derived category with bounded constructible cohomology on elliptic curves and compute its monodromy. This leads to a monodromy based proof of a result of Katz on the existence of $G_2$-motives in the middle cohomology of deformations of Beauville surfaces.