The tau functions of the constrained CKP hierarchy

The CKP hierarchy is one important sub-hierarchy of the KP hierarchy, which is quite special due to its tau function. Here we construct the tau functions for the constrained CKP hierarchy $(L^k{)}_{<0}={\sum }_{i=1}^m(q_{1,i}{\mathrm{\partial }}^{-1}{q}_{2,i}-(-1{)}^k{q}_{2,i}{\mathrm{\partial }}^{-1}{q}_{1,i})$ with $k$ being odd or even positive integer by using the CKP Darboux transformations.