Second-Quantized Mirror Symmetry

We propose and give strong evidence for a duality relating Type II theories on Calabi-Yau spaces and heterotic strings on $K3 \times T^2$, both of which have $N=2$ spacetime supersymmetry. Entries in the dictionary relating the dual theories are derived from an analysis of the soliton string worldsheet in the context of $N=2$ orbifolds of dual $N=4$ compactifications of Type II and heterotic strings. In particular we construct a pairing between Type II string theory on a self-mirror Calabi-Yau space $X$ with $h^{11}= h^{21}= 11$ and a $(4,0)$ background of heterotic string theory on $K3\times T^2$. Under the duality transformation the usual first-quantized mirror symmetry of $X$ becomes a second-quantized mirror symmetry which determines nonperturbative quantum effects. This enables us to compute the exact quantum moduli space. Mirror symmetry of $X$ implies that the low-energy $N=2$ gauge theory is finite, even at enhanced symmetry points. This prediction is verified by direct computation on the heterotic side. Other branches of the moduli space, and corresponding dual pairs which are not finite $N=2$ theories, are connected to this one via black hole condensation.