Free field constructions for the elliptic algebra ${\mathcal{A}}_{q,p}({\widehat{sl}}_2)$ and Baxter's eight-vertex model

Three examples of free field constructions for the vertex operators of the elliptic quantum group ${\mathcal{A}}_{q,p}({\widehat{sl}}_2)$ are obtained. Two of these (for $p^{1\mathrm{/}2}=\pm q^{3\mathrm{/}2},p^{1\mathrm{/}2}=-q^2$) are based on representation theories of the deformed Virasoro algebra, which correspond to the level 4 and level 2 $Z$-algebra of Lepowsky and Wilson. The third one ($p^{1\mathrm{/}2}=q^3$) is constructed over a tensor product of a bosonic and a fermionic Fock spaces. The algebraic structure at $p^{1\mathrm{/}2}=q^3$, however, is not related to the deformed Virasoro algebra. Using these free field constructions, an integral formula for the correlation functions of Baxter’s eight-vertex model is obtained. This formula shows different structure compared with the one obtained by Lashkevich and Pugai.