Algebraic Structures and Eigenstates for Integrable Collective Field Theories

Abstract

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 $U(s\ell (2)).$

27 pages

References in corpus (3)

• Algebraic Structures and Differential Geometry in 2D String Theory
• Interaction of Discrete States in Two-Dimensional String Theory
• String field actions from W-infinity

Cited by in corpus (17)

• Collective fields, Calogero-Sutherland model and generalized matrix models
• Excited States of Calogero-Sutherland Model and Singular Vectors of the $W_N$ Algebra
• Classical R-matrix structure for the Calogero model
• A Deformed Matrix Model and the Black Hole Background in Two-Dimensional String Theory
• w_{\infty} Algebras, Conformal Mechanics, and Black Holes
• The Exact $S$-matrix of the Deformed $c=1$ Matrix Model
• Quantum and Classical Aspects of Deformed $c=1$ Strings.
• Collective field theory of the matrix-vector model
• States and quantum effects in the collective field theory of a deformed matrix model
• Quantum toroidal algebras and solvable structures in gauge/string theory
• The $W_{1+\infty }$ effective theory of the Calogero- Sutherland model and Luttinger systems.
• Topological B-Model, Matrix Models, $\hat{c}=1$ Strings and Quiver Gauge Theories
• Interacting Theory of Collective and Topological Fields in 2 Dimensions
• Observable algebra for the rational and trigonometric Euler Calogero Moser models
• The Collective Field Theory of a Singular Supersymmetric Matrix Model
• Integrable extensions of the rational and trigonometric $A_N$ Calogero Moser potentials
• Higher Spin Gravity in $Ad{S}_3$ and Folds on Fermi Surface