Elliptic Genera of Symmetric Products and Second Quantized Strings

TL;DR

This paper proves an identity linking the elliptic genus of symmetric product manifolds to second quantized string theory, revealing automorphic properties and enabling precise D-brane free energy calculations.

Contribution

It establishes a new identity connecting symmetric product elliptic genera with second quantized strings, and demonstrates their automorphic nature.

Findings

Elliptic genus of symmetric products equals second quantized string partition function.

Generating function is an automorphic form for O(3,2,Z).

Enables exact computation of D-string free energy in brane systems.

Abstract

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.