Wakimoto realization of the elliptic algebra $U_{q,p}(hat{sl_N})$

We construct a free field realization of the elliptic quantum algebra $U_{q,p}(\hat{sl_N})$ for arbitrary level $k \neq 0,-N$. We study Drinfeld current and the screening current associated with $U_{q,p}(\hat{sl_N})$ for arbitrary level $k$. In the limit $p \to 0$ this realization becomes $q$-Wakimoto realization for $U_q(\hat{sl_N})$.

💡 Research Summary

The paper presents a systematic construction of a free‑field (bosonic) realization of the elliptic quantum algebra U₍q,p₎(ĥslₙ) for arbitrary level k with the restriction k ≠ 0, −N. Starting from a set of bosons a_i,n, b_{i,j,n}, c_{i,j,n} and their zero‑mode counterparts, the author defines their commutation relations in terms of the Cartan matrix of slₙ and the level k. Two elliptic parameters p = q^{2r} and p⁎ = q^{2r⁎} (with r⁎ = r − k) are introduced, together with the theta functions Θ_p(z) and Θ_{p⁎}(z) that encode the elliptic deformation.

The known q‑Wakimoto realization of U_q(ĥslₙ) is reviewed, where Drinfeld currents E_i^{±}(z), ψ_i^{±}(z) and Cartan generators h_i are expressed through the bosons and auxiliary fields γ, β. Screening currents S_i(z) are also recalled, showing their near‑commutativity with the Drinfeld currents.

The core of the work is the definition of dressing operators U_i(z) and U_i⁎(z). These are built from exponentials of the bosons B_{i,j}^{±}(z) and A_i(z), arranged in a product that reflects the structure of the underlying elliptic quantum group. Appendix A details a systematic algorithm for constructing these operators. By multiplying the q‑currents with the dressing operators, elliptic Drinfeld currents are defined: e_i(z) = U_i⁎(z) E_i⁺(z), f_i(z) = E_i⁻(z) U_i(z), Ψ_i^{±}(z) = U_i⁎(q^{±k/2}z) ψ_i^{±}(z) U_i(q^{∓k/2}z).

These elliptic currents satisfy a set of exchange relations involving the theta functions Θ_p and Θ_{p⁎} (equations (4.15)–(4.23)). The relations reproduce the standard q‑commutation structure when p→0, while for generic p they display the characteristic quasi‑Hopf features of elliptic algebras, such as non‑trivial theta‑function factors and modified Serre relations.

To obtain a full representation of U₍q,p₎(ĥslₙ), a Heisenberg algebra H generated by P_i and Q_i is tensored with the bosonic Fock space. The final currents E_i(z) = e_i(z) e^{2Q_i} z^{-P_i−1/r⁎}, F_i(z) = f_i(z) z^{h_i+P_i−1/r}, H_i^{±}(z) = Ψ_i^{±}(z) e^{±2Q_i} …, are shown to obey elliptic analogues of the Drinfeld relations, theta‑weighted Serre relations (4.28)–(4.35), and appropriate commutation rules with the screening currents.

The paper concludes that the construction provides a natural elliptic extension of the Wakimoto free‑field realization, reduces to the known q‑Wakimoto form when p→0, and opens the way for applications such as the construction of level‑k integrals of motion, elliptic vertex operators, and connections to solvable lattice models. Future directions include exploring non‑standard levels, detailed analysis of screening charges, and physical models where the elliptic deformation plays a crucial role.