This talk will explore topological invariants of susy
gauge theories, with some emphasis on index-like
quantities and the notion of holonomy saddles.
We start with 1d refined Witten index computations
where the twisted partition functions typically show
rational, rather than integral, behavior. We will explain
how this oddity is a blessing in disguise and propose
a universal tool for extracting the truely enumerative
Witten indices. In part, this finally put to the rest
a two-decade-old bound state problems which had
originated from the M-theory hypothesis.
Along the way, we resolve an old and critical conflict
between Kac+Smilga and Staudacher/Pestun, circa
1999~2002, whereby the notion of H-saddles emerges
and plays a crucial role. More importantly, H-saddles
prove to be universal features of supersymmetric
gauge theories when the spacetime include a small
circle: H-saddles are explored further for d=4, N=1
theories, with much ramifications on some recent
claims on Cardy exponents of their partition functions.