Topological string in harmonic space and correlation functions in $S^3$ stringy cosmology

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 $$ and lowest components $\upsilon_{(0,n)}$ of $SL(2,C)$ spin $s=\frac{n}{2}$ multiplets $$, $0\leq k\leq n$ 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.