In mathematics, a Verlinde algebra is a finite-dimensional associative algebra introduced by Erik Verlinde (1988), with a basis of elements φλ corresponding to primary fields of a rational two-dimensional conformal field theory, whose structure constants \( N_{\lambda \mu }^{\nu } \) describe fusion of primary fields.
Verlinde formula
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\( {\displaystyle S}. \)
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 and Teleman (2001) showed that the Verlinde algebra can be identified with twisted equivariant K-theory of G.
See also
Fusion rules
Notes
Blumenhagen, Ralph (2009). Introduction to Conformal Field Theory. Plauschinn, Erik. Dordrecht: Springer. pp. 143. ISBN 9783642004490. OCLC 437345787.
References
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., 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. (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.
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