Verlinde algebra

Short description: Algebra used in certain conformal field theories

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 Nνλμ 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 $[A]$ corresponding to isomorphism classes of simple obejcts and whose structure constants $N_C^{A,B}$ 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 $A,B,C\in {𝒞}$ in a modular tensor category, the Verlinde formula relates the fusion coefficient $N_C^{A,B}$ in terms of a sum of products of $S$-matrix entries and entries of the inverse of the $S$-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

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

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

See also

• Fusion rules

References

• Beauville, Arnaud (1996), "Conformal blocks, fusion rules and the Verlinde formula", in Teicher, Mina, 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, http://math1.unice.fr/~beauvill/pubs/Hirz65.pdf 
• Bott, Raoul (1991), "On E. Verlinde's formula in the context of stable bundles", International Journal of Modern Physics A 6 (16): 2847–2858, doi:10.1142/S0217751X91001404, ISSN 0217-751X, Bibcode: 1991IJMPA...6.2847B 
• Faltings, Gerd (1994), "A proof for the Verlinde formula", Journal of Algebraic Geometry 3 (2): 347–374, ISSN 1056-3911 
• Freed, Daniel S.; Hopkins, M.; Teleman, C. (2001), "The Verlinde algebra is twisted equivariant K-theory", Turkish Journal of Mathematics 25 (1): 159–167, ISSN 1300-0098, Bibcode: 2001math......1038F, http://mistug.tubitak.gov.tr/bdyim/abs.php?dergi=mat&rak=0103-10 
• Verlinde, Erik (1988), "Fusion rules and modular transformations in 2D conformal field theory", Nuclear Physics B 300 (3): 360–376, doi:10.1016/0550-3213(88)90603-7, ISSN 0550-3213, Bibcode: 1988NuPhB.300..360V 
• 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, Bibcode: 1993hep.th...12104W 
• MathOverflow discussion with a number of references.