On the extended Poincaré polynomial

We show that the numbers of generations and anti-generations of a (2,2) string compactification with diagonal internal theory can be expressed in terms of certain specifications of the elliptic genus of the untwisted internal theory which can be computed from the Poincaré polynomial. To establish this result we show that there are no cancellations of positive and negative contributions to the Euler characteristic within a fixed twisted sector. Turning the argument around, this allows us to define the ‘extended Poincaré polynomial’ P(t,x), which encodes information on the orbits of the spinor current under fusion, for non-diagonal N = 2 superconformal field theories. As an application, we derive an explicit formula for P(t,x) for general Landau-Ginzburg orbifolds.