  We develop the harmonic space method for conifold and use it to study local
complex deformations of $T^{\ast}S^{3}$ preserving manifestly $SL(2,C) $
isometry. We derive the perturbative manifestly $SL(2,C) $ invariant partition
function $\mathcal{Z}_{top}$ of topological string B model on locally deformed
conifold. Generic $n$ momentum and winding modes of 2D $c=1$ non critical
theory are described by highest $% \upsilon_{(n,0)}$ and lowest components
$\upsilon_{(0,n)}$ of $SL(2,C) $ spin $s=\frac{n}{2}$ multiplets $% (\upsilon
_{(n-k,k)}) $, $0\leq k\leq n$ and are shown to be naturally captured by
harmonic monomials. Isodoublets ($n=1$) describe uncoupled units of momentum
and winding modes and are exactly realized as the $SL(2,C) $ harmonic variables
$U_{\alpha}^{+}$ and $V_{\alpha}^{-}$. We also derive a dictionary giving the
passage from Laurent (Fourier) analysis on $T^{\ast}S^{1}$ ($S^{1}$) to the
harmonic method on $T^{\ast}S^{3}$ ($S^{3}$). The manifestly $SU(2,C) $
covariant correlation functions of the $S^{3}$ quantum cosmology model of
Gukov-Saraikin-Vafa are also studied.
