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) = mux2 in the collective field theory. They form a w(infinity)-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) = mux2 + 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 of U(sl(2)).