  In gauge theories on a spacetime equipped with a circle, the holonomy
variables, living in the Cartan torus, play special roles. With their periodic
nature properly taken into account, we find that a supersymmetric gauge theory
in $d$ dimensions tends to reduce in the small radius limit to a disjoint sum
of multiple $(d-1)$ dimensional theories at distinct holonomies, called
$H$-saddles. The phenomenon occurs regardless of the spacetime dimensions, and
here we explore such $H$-saddles for $d=4$ $\cal N=1$ theories on $T^2$ fibred
over $\Sigma_g$, in the limits of elongated $T^2$. This naturally generates
novel relationships between 4d and 3d partition functions, including ones
between 4d and 3d Witten indices, and also leads us to re-examine recent
studies of the Cardy exponents and the Casimir energies and of their purported
connections to the 4d anomalies.
