Hilbert schemes and symmetric products: a dictionary

Given a closed complex manifold $X$ of even dimension, we develop a systematic (vertex) algebraic approach to study the rational orbifold cohomology rings $\orbsym$ of the symmetric products. We present constructions and establish results on the rings $\orbsym$ including two sets of ring generators, universality and stability, as well as connections with vertex operators and W algebras. These are independent of but parallel to the main results on the cohomology rings of the Hilbert schemes of points on surfaces as developed in our earlier works joint with W.-P. Li. We introduce a deformation of the orbifold cup product and explain how it is reflected in terms of modification of vertex operators in the symmetric product case. As a corollary, we obtain a new proof of the isomorphism between the rational cohomology ring of Hilbert schemes and the ring $\orbsym$ (after some modification of signs), when X is a projective surface with a numerically trivial canonical class; we show that no sign modification is needed if both cohomology rings use complex coefficients.