  In this note we prove an identity that equates the elliptic genus partition
function of a supersymmetric sigma model on the N-fold symmetric product
$M^N/S_N$ of a manifold M to the partition function of a second quantized
string theory on the space $M \times S^1$. The generating function of these
elliptic genera is shown to be (almost) an automorphic form for O(3,2,Z). In
the context of D-brane dynamics, this result gives a precise computation of the
free energy of a gas of D-strings inside a higher-dimensional brane.
