Semi-Infinite Realization of Unitary Representations of the $N=2$ Algebra and Related Constructions

In the examples of the $N=2$ super-Virasoro algebra and the affine $\widehat{s\ell}(2)$ algebra, we investigate the construction of unitary representations of infinite-dimensional algebras in terms of “collective excitations” over a filled Dirac sea of fermionic or bosonic operators satisfying a generalized exclusion principle and represented by semi-infinite forms in the modes of one of the generators. We develop the methods for investigating properties of semi-infinite spaces (polynomial realization of the dual space) and for constructing the appropriate algebra action on these spaces (a filtration by subspaces similar to Demazure modules). We also consider relations of the semi-infinite realizations to the Rogers–Ramanujan-type identities, to the expression of coinvariants through meromorphic functions on products of Riemann surfaces with a prescribed behavior on multiple diagonals, and to some combinatorial facts; we also consider the relation between modular functors and fusion rules for the $N=2$ and $\widehat{s\ell}(2)$ theories.