  We explore the path integration -- upon the contour of hermitian
(non-auxliary) field configurations -- of topologically twisted $\mathcal{N}=2$
Chern-Simons-matter theory (TTCSM) on $\mathbb{S}_2$ times a segment. In this
way, we obtain the formula for the 3D topologically twisted index, first as a
convolution of TTCSM on $\mathbb{S}_2$ times halves of $\mathbb{S}_1$, second
as TTCSM on $\mathbb{S}_2$ times $\mathbb{S}_1$ -- with a puncture --, and
third as TTCSM on $\mathbb{S}_2 \times \mathbb{S}_1$. In contradistinction to
the first two cases, in the third case, the vector multiplet auxiliary field
$D$ is constrained to be anti-hermitian.
