ON THE EXTENDED POINCARE POLYNOMIAL

TL;DR

This paper introduces the extended Poincare polynomial, linking the elliptic genus and orbifold structure in N=2 superconformal theories, with applications to Landau-Ginzburg models.

Contribution

It defines the extended Poincare polynomial P(t,x) for non-diagonal N=2 theories, connecting elliptic genus computations to algebraic orbifold structures.

Findings

Expresses generations and anti-generations via elliptic genus and Poincare polynomial

Shows no cancellation of Euler characteristic contributions within twisted sectors

Provides explicit formula for P(t,x) in Landau-Ginzburg orbifolds

Abstract

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 Poincare 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. For our considerations we recast the orbifolding procedure into an algebraic language using simple currents. Turning the argument around, this allows us to define the `extended Poincare 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.