  Among gauged dynamics motivated by string theory, we find many with gapless
asymptotic directions. Although the natural boundary condition for ground
states is $L^2$, one often turns on chemical potentials or supersymmetric mass
terms to regulate the infrared issues, instead, and computes the twisted
partition function. We point out how this procedure generically fails to
capture physical $L^2$ Witten index with often misleading results. We also
explore how, nevertheless, the Witten index is sometimes intricately embedded
in such twisted partition functions. For $d=1$ theories with gapless continuum
sector from gauge multiplets, such as non-primitive quivers and pure
Yang-Mills, a further subtlety exists, leading to fractional expressions. Quite
unexpectedly, however, the integral $L^2$ Witten index can be extracted
directly and easily from the twisted partition function of such theories. This
phenomenon is tied to the notion of the rational invariant that appears
naturally in the wall-crossing formulae, and offers a general mechanism of
reading off Witten index directly from the twisted partition function. Along
the way, we correct early numerical results for some of $\mathcal N=4,8,16$
pure Yang-Mills quantum mechanics, and count threshold bound states for general
gauge groups beyond $SU(N)$.
