Topological B-model and $\widehat{c}=1$ String Theory

We study the topological B-model on a deformed ${\mathbb{Z}}_2$ orbifolded conifold by investigating variation of complex structures via quantum Kodaira-Spencer theories. The fermionic/brane formulation together with systematic utilization of symmetries of the geometry gives rise to a free fermion realization of the amplitudes. We derive Ward identities which solve the perturbed free energy exactly. We also obtain the corresponding Kontsevich-like matrix model. All these confirm the recent conjecture on the connection of the theory with ${\hat c}=1$type0Astringtheorycompactifiedattheradius$R=\sqrt{\alpha’/2}$.