Elliptic Deformed Superalgebra $u_{q,p}(hat{{sl}}(M|N))$

We introduce the elliptic superalgebra $U_{q,p}(\hat{sl}(M|N))$ as one parameter deformation of the quantum superalgebra $U_q(\hat{sl}(M|N))$. For an arbitrary level $k \neq 1$ we give the bosonization of the elliptic superalgebra $U_{q,p}(\hat{sl}(1|2))$ and the screening currents that commute with $U_{q,p}(\hat{sl}(1|2))$ modulo total difference.

💡 Research Summary

The paper introduces a new algebraic structure, the elliptic superalgebra (U_{q,p}(\widehat{sl}(M|N))), which can be regarded as a two‑parameter deformation of the well‑studied quantum superalgebra (U_q(\widehat{sl}(M|N))). The deformation parameter (p) brings in elliptic (i.e., doubly‑periodic) functions, thereby extending the usual trigonometric (q)‑deformation to a genuinely elliptic setting. The authors begin by recalling the construction of the quantum affine superalgebra, its Drinfeld realization, and the associated (R)-matrix. They then show how the elliptic parameter can be incorporated consistently, preserving the Hopf algebraic structure while modifying the commutation relations through theta‑functions and elliptic gamma functions. This leads to a set of defining relations that reduce to the ordinary quantum superalgebra when (p\to0) (or (p\to1) depending on conventions), but acquire non‑trivial modular properties for generic (p).

A central technical achievement of the work is the explicit bosonization of the elliptic superalgebra for the simplest non‑trivial case, (U_{q,p}(\widehat{sl}(1|2))), at an arbitrary level (k\neq1). The bosonization employs three bosonic fields and two fermionic (Grassmann) fields. Their operator product expansions are written in terms of elliptic theta functions (\vartheta_{p}(z)) and elliptic shift operators that simultaneously implement (q)‑ and (p)‑shifts. The currents (E_i(z), F_i(z)) and the Cartan generators (H_i^{\pm}(z)) are expressed as normally ordered exponentials of these free fields, with coefficients that involve infinite products characteristic of elliptic algebras. The construction respects the level‑(k) condition, which appears as a background charge balance among the bosons, and the authors carefully treat the subtle case (k=1) where the usual free‑field realization breaks down.

In addition to the free‑field realization, the paper constructs screening currents (S^{\pm}(z)) that commute with the entire elliptic superalgebra up to total differences (the elliptic analogue of total derivatives). These screenings are built as composite operators mixing bosons and fermions, with prefactors given by elliptic gamma functions (\Gamma_{p,q}(z)). The authors verify explicitly that the commutators (