Embeddings for Non-Critical Superstrings

It was previously shown that at critical central charge, $N$-extended superstrings can be embedded in $(N+1)$-extended superstrings. In other words, $(N=0,c=26)\to (N=1,c=15)\to (N=2,c=6)\to (N=3,c=0) \to (N=4,c=0) $. In this paper, we show that similar embeddings are also possible for $N$-extended superstrings at non-critical central charge. For any $x$, the embedding is $(N=0,c=26+x) \to (N=1,c=15+x) \to (N=2,c=6+x) \to (N=3,c=x) \to (N=4,c=x)$. As was conjectured by Vafa, the $(N=2,c=9) \to (N=3,c=3)$ embedding can be used to prove that $N=0$ topological strings are special vaccua of N=1 topological strings.