  We analyze exact conformal invariance of string worldsheet theory in
non-trivial backgrounds using hamiltonian framework. In the first part of this
talk we consider the example of type IIB superstrings in Ramond-Ramond pp-wave
background. In particular, we discuss the quantum definition of energy-momentum
(EM) tensor and two methods of computing Virasoro algebra. One of the methods
uses dynamical supersymmetries and indirectly establishes (partially) conformal
invariance when the background is on-shell. We discuss the problem of operator
ordering involved in the other method which attempts to compute the algebra
directly. This method is supposed to work for off-shell backgrounds and
therefore is more useful. In order to understand this method better we attempt
a background independent formulation of the problem which is discussed in the
second half of the talk. For a bosonic string moving in an arbitrary
metric-background such a framework is obtained by following DeWitt's work
(Phys.Rev.85:653-661,1952) in the context of particle quantum mechanics. In
particular, we construct certain background independent analogue of quantum
Virasoro generators and show that in spin-zero representation they satisfy the
Witt algebra with additional anomalous terms that vanish for Ricci-flat
backgrounds. We also report on a new result which states that the same algebra
holds true in arbitrary tensor representations as well.
