Contents

# Contents

## Idea

For $G$ a suitable Lie group, the Verlinde ring is the collection of isomorphism classes of positive energy representations of the corresponding loop group, equipped with the “fusion” tensor product.

The Verlinde ring is also understood as being the ring of equivariant twisted K-theory classes on $G$ (FHT) and, essentially equivalently, of Chan-Paton gauge fields over D-branes in the WZW model (see there for further references).

## References

Due to

• Erik Verlinde, Fusion rules and modular transformations in 2D conformal field theory, Nuclear Physics B 300 (3): 360–376, (1988) doi90603-7)

We study conformal field theories with a finite number of primary fields with respect to some chiral algebra. It is shown that the fusion rules are completely determined by the behavior of the characters under the modular group. We illustrate with some examples that conversely the modular properties of the characters can be derived from the fusion rules. We propose how these results can be used to find restrictions on the values of the central charge and conformal dimensions.