Algebraic Structures and Eigenstates for Integrable Collective Field Theories

Conditions for the construction of polynomial eigen–operators for the Hamiltonian of collective string field theories are explored. Such eigen–operators arise for only one monomial potential v(x) = μx2 in the collective field theory. They form a w∞– algebra isomorphic to the algebra of vertex operators in 2d gravity. Polynomial potentials of orders only strictly larger or smaller than 2 have no non–zero–energy polynomial eigen–operators. This analysis leads us to consider a particular potential v(x) = μx2 + g/x2. A Lie algebra of polynomial eigen–operators is then constructed for this potential. It is a symmetric 2–index Lie algebra, also represented as a sub– algebra of U(sl(2)). ⋆ Work supported in part by the Department of Energy under contract DE-AC02-76ER03130Task A † Work supported by Brown University Exchange Program P. I. 135 ‡ On leave of absence from LPTHE Paris 6, France