  I review the multi-dimensional generalizations of the Virasoro algebra, i.e.
the non-central Lie algebra extensions of the algebra vect(N) of general vector
fields in N dimensions, and its Fock representations. Being the Noether
symmetry of background independent theories such as N-dimensional general
relativity, this algebra is expected to be relevant to the quantization of
gravity. To this end, more complicated modules which depend on dynamics in the
form of Euler-Lagrange equations are described. These modules can apparently
only be interpreted as quantum fields if spacetime has four dimensions and both
bosons and fermions are present.
