Rational vs Polynomial Character of W$_n^l$-Algebras

TL;DR

This paper investigates the structure of W$_n^l$-algebras, revealing that their degenerate constraints lead to a rational basis for gauge-invariants, contrasting with the polynomial basis in non-degenerate cases.

Contribution

It introduces an algorithm to construct rational bases for degenerate constraints in W$_n^l$-algebras and shows how to convert them into polynomial bases.

Findings

Degenerate constraints lead to rational gauge-invariant bases.

A method for constructing rational bases from degenerate constraints.

Conversion techniques from rational to polynomial bases.

Abstract

The constraints proposed recently by Bershadsky to produce $W^l_n$ algebras are a mixture of first and second class constraints and are degenerate. We show that they admit a first-class subsystem from which they can be recovered by gauge-fixing, and that the non-degenerate constraints can be handled by previous methods. The degenerate constraints present a new situation in which the natural primary field basis for the gauge-invariants is rational rather than polynomial. We give an algorithm for constructing the rational basis and converting the base elements to polynomials.