  We derive the partition function of an $\mathcal N=2$ chiral multiplet on
topologically twisted $H^2\times S^1$. The chiral multiplet is coupled to a
background vector multiplet encoding a real mass deformation. We consider an $
H^2\times S^1$ metric containing two parameters: one is the $S^1$ radius, while
the other gives a fugacity $q$ for the angular momentum on $H^2$. The
computation is carried out by means of supersymmetric localization, which
provides a finite answer written in terms of $q$-Pochammer symbols and multiple
Zeta functions. Especially, the partition function of normalizable fields
reproduces three-dimensional holomorphic blocks.
