On principal realization of modules for the affine Lie algebra $A_1 ^{(1)}$ at the critical level

Abstract

We present complete realization of irreducible $A_1 ^{(1)}$-modules at the critical level in the principal gradation. Our construction uses vertex algebraic techniques, the theory of twisted modules and representations of Lie conformal superalgebras. We also provide an alternative Z-algebra approach to this construction. All irreducible highest weight $A_1 ^{(1)}$-modules at the critical level are realized on the vector space $M_{\tfrac{1}{2} + \Bbb Z} (1) ^{\otimes 2}$ where $M_{\tfrac{1}{2} + \Bbb Z} (1) $ is the polynomial ring ${\Bbb C}[\alpha(-1/2), \alpha(-3/2), ...]$. Explicit combinatorial bases for these modules are also given.