  We study elliptic vortices on $\mathbb{C}\times T^2$ by considering the 2d
quiver gauge theory describing their moduli spaces. The elliptic genus of these
moduli spaces is the elliptic version of vortex partition function of the 4d
theory. We focus on two examples: the first is a $\mathcal{N}=1$,
$\mathrm{U}(N)$ gauge theory with fundamental and anti-fundamental matter; the
second is a $\mathcal{N}=2$, $\mathrm{U}(N)$ gauge theory with matter in the
fundamental representation. The results are instances of 4d "holomorphic
blocks" into which partition functions on more complicated surfaces factorize.
They can also be interpreted as free-field representations of elliptic Virasoro
algebrae.
