Algebraic Structures and Eigenstates for Integrable Collective Field Theories

TL;DR

This paper investigates the algebraic structures and eigenstates in integrable collective field theories, revealing unique polynomial eigen-operators for specific potentials and constructing associated Lie algebras.

Contribution

It identifies conditions for polynomial eigen-operators in collective string field theories and constructs a Lie algebra for a particular potential, advancing understanding of integrable models.

Findings

Polynomial eigen-operators exist only for quadratic potential.

A $w_{}$-algebra isomorphic to 2d gravity vertex operators is found.

A Lie algebra of polynomial eigen-operators is constructed for the potential $v(x)=  x^2 + g/x^2$.

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