Verlinde algebra

In mathematics, a Verlinde algebra is a finite-dimensional associative algebra introduced by Erik Verlinde (1988), with a 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 terms of the modular S-matrix, the fusion coefficients are given by^[1]

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

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

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.

• Fusion rules

• Beauville, Arnaud (1996), "Conformal blocks, fusion rules and the Verlinde formula" (PDF), in Teicher, Mina (ed.), Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993), Israel Math. Conf. Proc., vol. 9, Ramat Gan: Bar-Ilan Univ., pp. 75–96, arXiv:alg-geom/9405001, MR 1360497
• 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:1991IJMPA...6.2847B, doi:10.1142/S0217751X91001404, ISSN 0217-751X, MR 1117752
• Faltings, Gerd (1994), "A proof for the Verlinde formula", Journal of Algebraic Geometry, 3 (2): 347–374, ISSN 1056-3911, MR 1257326
• Freed, Daniel S.; Hopkins, M.; Teleman, C. (2001), "The Verlinde algebra is twisted equivariant K-theory", Turkish Journal of Mathematics, 25 (1): 159–167, arXiv:math/0101038, Bibcode:2001math......1038F, ISSN 1300-0098, MR 1829086
• Verlinde, Erik (1988), "Fusion rules and modular transformations in 2D conformal field theory", Nuclear Physics B, 300 (3): 360–376, Bibcode:1988NuPhB.300..360V, doi:10.1016/0550-3213(88)90603-7, ISSN 0550-3213, MR 0954762
• 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:hep-th/9312104, Bibcode:1993hep.th...12104W, MR 1358625
• MathOverflow discussion with a number of references.