Elliptic genera of symmetric products and second quantized strings

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 × S¹. 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.