Verlinde algebra

In mathematics, a Verlinde algebra is a finite-dimensional associative algebra introduced by Erik Verlinde (1988). It is defined to have basis of elements φ_λ corresponding to primary fields of a rational two-dimensional conformal field theory, whose structure constants N^ν
_λμ describe fusion of primary fields.

In the context of modular tensor categories, there is also a Verlinde algebra. It is defined to have a basis of elements ${\displaystyle [A]}$ corresponding to isomorphism classes of simple obejcts and whose structure constants ${\displaystyle N_{C}^{A,B}}$ describe the fusion of simple objects.

In terms of the modular S-matrix for modular tensor categories, the Verlinde formula is stated as follows.¹ Given any simple objects ${\displaystyle A,B,C\in {\mathcal {C}}}$ in a modular tensor category, the Verlinde formula relates the fusion coefficient ${\displaystyle N_{C}^{A,B}}$ in terms of a sum of products of ${\displaystyle S}$-matrix entries and entries of the inverse of the ${\displaystyle S}$-matrix, normalized by quantum dimensions.

In terms of the modular S-matrix for conformal field theory, Verlinde formula expresses the fusion coefficients as²

${\displaystyle N_{\lambda \mu }^{\nu }=\sum _{\sigma }{\frac {S_{\lambda \sigma }S_{\mu \sigma }S_{\sigma \nu }^{*}}{S_{0\sigma }}}}$

where ${\displaystyle S^{*}}$ is the component-wise complex conjugate of ${\displaystyle S}$.

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 ${\displaystyle S_{0\sigma }}$ is equal to the quantum dimension of ${\displaystyle \sigma }$.

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 Freed, Hopkins & Teleman (2001) showed that the Verlinde algebra can be identified with twisted equivariant K-theory of G.