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.
Due to
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.
See also
Domenico Fiorenza, Alessandro Valentino, $(3,2,1)$-TQFTs and Verlinde algebras (MO question, MO answer)
Wikipedia, Verlinde algebra
On twisted ad-equivariant K-theory of compact Lie groups and the identification with the Verlinde ring of positive energy representations of their loop group:
Daniel S. Freed, Michael Hopkins, Constantin Teleman,
Loop Groups and Twisted K-Theory I,
J. Topology, 4 (2011), 737-789
Loop Groups and Twisted K-Theory II,
J. Amer. Math. Soc. 26 (2013), 595-644
Loop Groups and Twisted K-Theory III,
Annals of Mathematics, Volume 174 (2011) 947-1007
Daniel S. Freed, Constantin Teleman,
Dirac families for loop groups as matrix factorizations,
Comptes Rendus Mathematique, Volume 353, Issue 5, May 2015, Pages 415-419
Review:
Last revised on March 10, 2021 at 09:42:35. See the history of this page for a list of all contributions to it.