Families of N = 2 strings

In a given 4d spacetime bakcground, one can often construct not one but a family of distinct N = 2 string theories. This is due to the multiple ways an N = 2 superconformal algebra can be embedded in a given worldsheet theory. We formulate the principle of obtaining different physical theories by gauging different embeddings of the same symmetry algebra in the same “pre-theory.” We then apply it to the N = 2 strings and formulate the recipe for finding the associated parameter spaces of gauging. Flat and curved target spaces of both (4, 0) and (2, 2) signatures are considered. We broadly divide the gauging choices into two classes, denoted by α and β, and show them to be related by T-duality. The distinction between them is formulated topologically and hinges on some unique properties of 4d manifolds. We determine what their parameter spaces of gauging are under certain simplicity ansatz for generic flat spaces (R4 and its toroidal compactifications) as well as some curved spaces. We briefly discuss the spectra of D-branes for both α and β families.