  In this work, we first solve complex Morse flow equations for the simplest
case of a bosonic harmonic oscillator to discuss localization in the context of
Picard-Lefschetz theory. We briefly touch on the exact non-BPS solutions of the
bosonized supersymmetric quantum mechanics on algebraic geometric grounds and
report that their complex phases can be accessed through the cohomology of WKB
1-form of the underlying singular spectral curve subject to necessary
cohomological corrections for non-zero genus. Motivated by Picard-Lefschetz
theory, we write down a general formula for the index of $\mathcal{N} = 4$
quantum mechanics with background $R$-symmetry gauge fields. We conjecture that
certain symmetries of the refined Witten index and singularities of the moduli
space may be used to determine the correct intersection coefficients. A few
examples, where this conjecture holds, are shown in both linear and closed
quivers with rank-one quiver gauge groups. The $R$-anomaly removal along the
"Morsified" relative homology cycles also called "Lefschetz thimbles" is shown
to lead to the appearance of Stokes lines. We show that the Fayet-Iliopoulos
(FI) parameters appear in the intersection coefficients for the relative
homology of the quiver quantum mechanics resulting from dimensional reduction
of $2d$ $\mathcal{N}=(2,2)$ gauge theory on a circle and explicitly calculate
integrals along the Lefschetz thimbles in $\mathcal{N}=4$ $\mathbb{CP}^{k-1}$
model. The Stokes jumping of coefficients and its relation to wall crossing
phenomena is briefly discussed. We also find that the notion of "on-the-wall"
index is related to the invariant Lefschetz thimbles under Stokes phenomena. An
implication of the Lefschetz thimbles in constructing knots from quiver quantum
mechanics is indicated.
