Quasi-Exactly Solvable Models and W-Algebras

TL;DR

This paper explores the link between quasi-exactly solvable models and W-algebras, introducing a new method to construct multi-dimensional solvable models using Lie algebra generators.

Contribution

It presents a novel general method for building multi-dimensional quasi-exactly solvable models based on Lie algebra and W-algebra generators, expanding the class of solvable Hamiltonians.

Findings

Certain Lie algebras produce models with hermitian Hamiltonians.

The method applies to specific algebras like B(n), C(n+1), D(2n+2), A(1), G(2), F(4), E(7), E(8).

The applicability to other algebras remains uncertain.

Abstract

The relationship between the quasi-exactly solvable problems and W-algebras is revealed. This relationship enabled one to formulate a new general method for building multi-dimensional and multi-channel exactly and quasi-exactly solvable models with hermitian hamiltonians. The method is based on the use of multi-parameter spectral differential equations constructable from generators of finite-dimensional representations of simple Lie algebras and from generators of the associated W-algebras. It is shown that algebras B(n), C(n+1), D(2n+2) with n>1 and also algebras A(1), G(2), F(4), E(7) and E(8) always lead to models with hermitian hamiltonians. The situation with the remaining algebras A(n), D(2n+3) with n>1 and E(6) is still unclear.