nLab Verlinde ring

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.

Related concepts

• fusion ring

• twisted ad-equivariant K-theory

• twisted ad-equivariant Tate K-theory

References

General

Due to

• Erik Verlinde, Fusion rules and modular transformations in 2D conformal field theory, Nuclear Physics B 300 (3): 360–376, (1988) (doi:10.1016/0550-3213(88)90603-7)

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.

See also

• Domenico Fiorenza, Alessandro Valentino, $(3,2,1)$-TQFTs and Verlinde algebras (MO question, MO answer)

• Wikipedia, Verlinde algebra

Relation to twisted ad-equivariant K-theory

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, The Verlinde algebra is twisted equivariant K-theory, Turk. J. Math 25 (2001) 159-167 (arXiv:math/0101038)

• Daniel S. Freed, Michael Hopkins, Constantin Teleman,

  1. Loop Groups and Twisted K-Theory I,

     J. Topology, 4 (2011), 737-789

     (arXiv:0711.1906, doi:10.1112/jtopol/jtr019)

  2. Loop Groups and Twisted K-Theory II,

     J. Amer. Math. Soc. 26 (2013), 595-644

     (arXiv:0511232, doi:10.1090/S0894-0347-2013-00761-4)

  3. Loop Groups and Twisted K-Theory III,

     Annals of Mathematics, Volume 174 (2011) 947-1007

     (arXiv:math/0312155, doi:10.4007/annals.2011.174.2.5)

• 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

  (arxiv/1409.6051, doi:10.1016/j.crma.2015.02.011)

Review:

• Valentin Zakharevich, K-Theoretic Computation of the Verlinde Ring, thesis 2018 (hdl:2152/67663, pdf)