Algebraic Structures and Eigenstates for Integrable Collective Field Theories

Details Algebraic Structures and Eigenstates for Integrable Collective Field Theories Algebraic Structures and Eigenstates for Integrable Collective Field Theories

1992-02-19

arxiv.org/abs/hep-th/9202065v2

Commun.Math.Phys. 150 (1992) 149-166

Physics

High Energy Physics

High Energy Physics - Theory

27 pages

Scientific paper

10.1007/BF02096570

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 $v(x) = \mu x^2$ in the collective field theory. They form a $w_{\infty}$--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 $v(x)= \mu x^2 + g/x^2$. 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 $U (s\ell (2)).$

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