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 potentialv(x)=x² in the collective field theory. They form aw-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 potentialv(x)=x²+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 subalgebra ofU(sl(2)).Work supported in part by the Department of Energy under contract DE-AC02-76ER03130-Task AWork supported by Brown University Exchange Program P.I. 135