Abstract :
We introduce a free field realization of the central extension of the Lie algebra image of difference operators on the circle in terms of the fermionic image system. This realization admits a nontrivial Jordan block structure. We also review the free field realization of image algebra, and point out some relations between its generators of weight zero and the local integrals of motion of Bazhanov, Lukyanov and Zamolodchikov. Finally we compute the finitized characters, and the continuum characters of the Local Integrals of Motion, and find out and interesting analogy with the generating functions for the counting of branched covers of elliptic curves.
