### Piljin Yi

*Korea Institute for Advanced Study*

#### "Indices, Twisted Partition Functions, H-Saddles, and Gauge Theories on Compact Spacetimes"

In recent years, varieties of index-like quantities have been computed by exact path integrals, a.k.a. the localization. However, the interpretation of the results, which should be really called the twisted partition functions rather than the indices, requires even more care.

As a prototype, we start with supersymmetric Yang-Mills quantum mechanics. Exploiting the problematic rational structure of the twisted partition functions, we propose a universal tool for extracting the truly enumerative and integral index for theories with noncompact Coulombic directions, such as quiver theories. For pure Yang-Mills, this solves an old problem, dating back to 1990's, and along the way also resolves a critical conflict between Kac/Smilga and Staudacher/Pestun, circa 1999~2002.

The latter brings us to the new notion of H-saddles which proves to be a universal feature of twisted partition functions on a compact spacetime in any dimensions. We close with its ramifications on some recent claims on Casimir energies and Cardy exponents in even dimensions.