q-Virasoro algebra, q-conformal dimensions and and free q-superstring

Abstract The commutators of standard Virasoro generators and fields generate various representations of the centreless Virasoro algebra depending on a conformal dimension J of the field in question ( J is related to the Bargman index of SU (1, 1) generated by L m , m = 0, ±1). We introduce the notion of q -conformal dimension for various oscillator realizations of q -deformed Virasoro (super) algebras proposed earlier. We use the field theoretical approach introduced recently in which the q -Virasoro currents L α ( z ) are expressed as Schwinger-like point-split normally ordered quadratic expressions in elementary fields We extend this approach and probe the elementary fields A ( z ) (the q -superstring coordinate, momentum and fermionic field) and their powers by the 1-Virasoro generators L m α (i.e. we calculate the commutators [ L m α , A ( z )]) and show that to all of them can be assigned just the standard non-deformed conformal dimension.