Verlinde algebra

Verlinde formula

In terms of the modular S-matrix for modular tensor categories, the Verlinde formula is stated as follows.Given any simple objects A,B,C\in\mathcal{C} in a modular tensor category, the Verlinde formula relates the fusion coefficient N^{A,B}_{C} in terms of a sum of products of S-matrix entries and entries of the inverse of the S-matrix, normalized by quantum dimensions.

In terms of the modular S-matrix for conformal field theory, Verlinde formula expresses the fusion coefficients as

:N_{\lambda \mu}^\nu = \sum_\sigma \frac{S_{\lambda \sigma} S_{\mu \sigma} S^*{\sigma \nu}}{S{0\sigma}}

where S^* is the component-wise complex conjugate of S.

These two formulas are equivalent because under appropriate normalization the S-matrix of every modular tensor category can be made unitary, and the S-matrix entry S_{0\sigma } is equal to the quantum dimension of \sigma.

Twisted equivariant K-theory

If G is a compact Lie group, there is a rational conformal field theory whose primary fields correspond to the representations λ of some fixed level of loop group of G. For this special case showed that the Verlinde algebra can be identified with twisted equivariant K-theory of G.

Notes

References

• MathOverflow discussion with a number of references.

References

1. (2000-11-20). "Lectures on Tensor Categories and Modular Functors". American Mathematical Society.
2. Blumenhagen, Ralph. (2009). "Introduction to Conformal Field Theory". Springer.