In mathematics, a Verlinde algebra is a finite-dimensional associative algebra introduced by Erik Verlinde (1988). It is defined to have basis of elements φλ corresponding to primary fields of a rational two-dimensional conformal field theory, whose structure constants Nν 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 [ A ] {\displaystyle [A]} corresponding to isomorphism classes of simple obejcts and whose structure constants N C A , B {\displaystyle N_{C}^{A,B}} describe the fusion of simple objects.

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 ∈ C {\displaystyle A,B,C\in {\mathcal {C}}} in a modular tensor category, the Verlinde formula relates the fusion coefficient N C A , B {\displaystyle N_{C}^{A,B}} in terms of a sum of products of S {\displaystyle S}-matrix entries and entries of the inverse of the S {\displaystyle S}-matrix, normalized by quantum dimensions.

The Verlinde formula for modular tensor categories.

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

N λ μ ν = ∑ σ S λ σ S μ σ S σ ν ∗ S 0 σ {\displaystyle N_{\lambda \mu }^{\nu }=\sum _{\sigma }{\frac {S_{\lambda \sigma }S_{\mu \sigma }S_{\sigma \nu }^{*}}{S_{0\sigma }}}}

where S ∗ {\displaystyle S^{*}} is the component-wise complex conjugate of S {\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 S 0 σ {\displaystyle S_{0\sigma }} is equal to the quantum dimension of σ {\displaystyle \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 Freed, Hopkins & Teleman (2001) showed that the Verlinde algebra can be identified with twisted equivariant K-theory of G.

See also

Notes

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  • Bott, Raoul (1991), "On E. Verlinde's formula in the context of stable bundles", International Journal of Modern Physics A, 6 (16): 2847–2858, Bibcode:, doi:, ISSN , MR
  • Faltings, Gerd (1994), "A proof for the Verlinde formula", Journal of Algebraic Geometry, 3 (2): 347–374, ISSN , MR
  • Freed, Daniel S.; Hopkins, M.; Teleman, C. (2001), , Turkish Journal of Mathematics, 25 (1): 159–167, arXiv:, Bibcode:, ISSN , MR
  • Verlinde, Erik (1988), "Fusion rules and modular transformations in 2D conformal field theory", Nuclear Physics B, 300 (3): 360–376, Bibcode:, doi:, ISSN , MR
  • Witten, Edward (1995), "The Verlinde algebra and the cohomology of the Grassmannian", Geometry, topology, & physics, Conf. Proc. Lecture Notes Geom. Topology, IV, Int. Press, Cambridge, MA, pp. 357–422, arXiv:, Bibcode:, MR
  • with a number of references.