Eigensystem and full character formula of theW 1+? algebra withc = 1

Abstract

By using the free field realizations, we analyze the representation theory of the W 1+∞ algebra with c = 1. The eigenvectors for the Cartan subalgebra of W 1+∞ are parametrized by the Young diagrams, and explicitly written down by W 1+∞ generators.Moreover, their eigenvalues and full character formula are also obtained. hep-th/93122081

Chiral deformations of conformal field theories

“…The representation theory of this algebra has been intensely studied, see e.g. [15,16]. In §4 we will treat the c = 1 free field theory realization of W 1+∞ .…”

Determinant formulae of quasi-finite representation of W1+∞ algebra at lower levels

“…The character formula (30) for C = −1 is rigorously proved as follows. Since we have already demonstrated the spectral flow, we can restrict ourselves to the simplest case, b(w) = w. We follow the argument of [5]: The basis of the representation space are β −r 1 · · · β −r k γ −s 1 · · · γ −s k |0 with r 1 ≥ · · · ≥ r k ≥ 1 and s 1 ≥ · · · ≥ s k ≥ 0. These states are simultaneous eigenstates of W (D k ), since…”

“…It was mainly due to the fact that infinitely many states possibly appear at each energy level, reflecting the infinite number of generators. Although we have explicit representations by free fields [2,3] and we can even construct its full character formula in a special case of C = 1 [4,5], it remained a longstanding problem to classify all the possible representations.…”

Free fermions, KdV charges, generalised Gibbs ensembles and modular transforms

“…(Note that this is entirely analogous to the calculation of W 1+∞ characters in [18].) If we note that I 1 = L 0 − c/24, this can equally be written as…”