On sums of tensor and fusion multiplicities

TL;DR

This paper proves invariance properties of tensor and fusion multiplicities in Lie algebra representations, revealing symmetries related to the modular S matrix and exploring their implications in conformal and topological quantum theories.

Contribution

It provides rigorous proofs of multiplicity invariance under conjugation and explores their significance in conformal field theories and related areas.

Findings

Sum of tensor multiplicities is invariant under conjugation.

Vanishing of the sum of S-matrix elements for complex and quaternionic representations.

Illustrations in boundary conformal field theories and topological quantum field theories.

Abstract

The total multiplicity in the decomposition into irreducibles of the tensor product i x j of two irreducible representations of a simple Lie algebra is invariant under conjugation of one of them sum_k N_{i j}^{k}= sum_k N_{ibar j}^{k}. This also applies to the fusion multiplicities of affine algebras in conformal WZW theories. In that context, the statement is equivalent to a property of the modular S matrix, Sigma(k)= sum_j S_{j k}=0 if k is a complex representation. Curiously, this vanishing of Sigma(k) also holds when k is a quaternionic representation. We provide proofs of all these statements. These proofs rely on a case-by-case analysis, maybe overlooking some hidden symmetry principle. We also give various illustrations of these properties in the contexts of boundary conformal field theories, integrable quantum field theories and topological field theories.