On Diff(S^1) Covariantization Of Pseudodifferential Operator

TL;DR

This paper explores the covariant properties of pseudodifferential operators on the circle, linking their transformation behavior to Hamiltonian flows and constructing covariant forms for specific degrees.

Contribution

It establishes conditions for diff(S^1) covariance of pseudodifferential operators and constructs their covariant form using inverse covariant derivatives.

Findings

Hamiltonian flow defined by second Gelfand-Dickey bracket

Covariant form of pseudodifferential operators constructed

Existence of primary basis for W_{KP}^{(n)}

Abstract

A study of diff($S^1$) covariant properties of pseudodifferential operator of integer degree is presented. First, it is shown that the action of diff($S^1$) defines a hamiltonian flow defined by the second Gelfand-Dickey bracket if and only if the pseudodifferential operator transforms covariantly. Secondly, the covariant form of a pseudodifferential operator of degree n not equal to 0, 1, -1 is constructed by exploiting the inverse of covariant derivative. This, in particular, implies the existence of primary basis for W_{KP}^{(n)} (n not equal to 0, 1, -1).