Schubert Calculus and Threshold Polynomials of Affine Fusion

TL;DR

This paper integrates the concept of threshold levels in affine fusion with Schubert calculus, introducing threshold polynomials that unify calculations across all levels and relate to q-deformed coefficients.

Contribution

It presents a novel modification of the Pieri rule within Schubert calculus to incorporate threshold levels, leading to the definition of threshold polynomials for affine fusion.

Findings

Threshold polynomials generalize fusion coefficients across levels.

Modified Pieri rule simplifies calculations for all levels.

Connections established between threshold polynomials and q-deformed coefficients.

Abstract

We show how the threshold level of affine fusion, the fusion of Wess-Zumino-Witten (WZW) conformal field theories, fits into the Schubert calculus introduced by Gepner. The Pieri rule can be modified in a simple way to include the threshold level, so that calculations may be done for all (non-negative integer) levels at once. With the usual Giambelli formula, the modified Pieri formula deforms the tensor product coefficients (and the fusion coefficients) into what we call threshold polynomials. We compare them with the q-deformed tensor product coefficients and fusion coefficients that are related to q-deformed weight multiplicities. We also discuss the meaning of the threshold level in the context of paths on graphs.