Notes on Non-Critical Superstrings in Various Dimensions

We study non-critical superstrings propagating in $d \le 6$ dimensional Minkowski space or equivalently, superstrings propagating on the two-dimensional Euclidean black hole tensored with d-dimensional Minkowski space. We point out a subtlety in the construction of supersymmetric theories in these backgrounds, and explain how this does not allow a consistent geometric interpretation in terms of fields propagating on a cigar-like spacetime. We explain the global symmetries of the various theories by using their description as the near horizon geometry of wrapped NS5-brane configurations. In the six-dimensional theory, we present a CFT description of the four-dimensional moduli space and the global O(3) symmetry. The worldsheet action invariant under this symmetry contains both the N=2 sine-Liouville interaction and the cigar metric, thereby providing an example where the two interactions are naturally present in the same worldsheet lagrangian already at the non-dynamical level.