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Algebra used in certain conformal field theories
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 ν λμ describe fusion of primary fields.
Verlinde formula
In terms of the
modular S-matrix , the fusion coefficients are given by
[1]
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}
.
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
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., 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.