Toda Darboux transformations and vacuum expectation values

Determinant formulas for vacuum expectation values $⟨s+k+n-m,-s\mathrm{\mid }{e}^{H(\mathbf{t})}{\beta }_m^{\ast }\cdots {\beta }_1^{\ast }{\beta }_n\cdots {\beta }_{1}g\mathrm{\mid }k⟩$ are given by using Toda Darboux transformations. Firstly notice that 2–Toda hierarchy can be viewed as the 2–component bosonizations of fermionic KP hierarchy, then two elementary Toda Darboux transformation operators $T_{+}(q)=\mathrm{\Lambda }(q)\cdot \mathrm{\Delta }\cdot q^{-1}$ and $T_{-}(r)={\mathrm{\Lambda }}^{-1}(r{)}^{-1}\cdot {\mathrm{\Delta }}^{-1}\cdot r$ are constructed from the changes of Toda (adjoint) wave functions by using 2–component boson–fermion correspondence. Based on this, the above vacuum expectation values now can be realized as the successive applications of Toda Darboux transformations. So the corresponding determinant formulas can be derived from the determinant representations of Toda Darboux transformations. Finally by similar methods, we also give the determinant formulas for $⟨n-m\mathrm{\mid }{e}^{\mathcal{H}(\mathbf{x})}{\beta }_m^{\ast }\cdots {\beta }_1^{\ast }{\beta }_n\cdots {\beta }_{1}g\mathrm{\mid }k⟩$ related with KP tau functions.