Adjoint affine fusion and tadpoles

TL;DR

This paper provides universal formulas and lattice interpretations for adjoint affine fusion in simple Lie algebras, including explicit polynomial formulas for classical types and a method to derive off-diagonal fusion results from tensor products.

Contribution

It introduces elementary, universal formulas for adjoint affine fusion, extending the affine depth rule, and offers explicit polynomial formulas for classical Lie algebras' adjoint tadpoles.

Findings

Universal formulas for diagonal adjoint affine fusion coefficients

Lattice-polytope interpretation of fusion coefficients

Explicit polynomial formulas for classical Lie algebras' adjoint tadpoles

Abstract

We study affine fusion with the adjoint representation. For simple Lie algebras, elementary and universal formulas determine the decomposition of a tensor product of an integrable highest-weight representation with the adjoint representation. Using the (refined) affine depth rule, we prove that equally striking results apply to adjoint affine fusion. For diagonal fusion, a coefficient equals the number of nonzero Dynkin labels of the relevant affine highest weight, minus 1. A nice lattice-polytope interpretation follows, and allows the straightforward calculation of the genus-1 1-point adjoint Verlinde dimension, the adjoint affine fusion tadpole. Explicit formulas, (piecewise) polynomial in the level, are written for the adjoint tadpoles of all classical Lie algebras. We show that off-diagonal adjoint affine fusion is obtained from the corresponding tensor product by simply dropping non-dominant representations.