In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Verlinde algebra is a finite-dimensional
associative algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
introduced by , with a basis of elements φ
λ corresponding to primary fields of a rational
two-dimensional conformal field theory, whose structure constants ''N'' describe fusion of primary fields.
Verlinde formula
In terms of the
modular S-matrix, the fusion coefficients are given by
where
is the component-wise complex conjugate of
.
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
In mathematics, a loop group is a Group (mathematics), group of Loop (topology), loops in a topological group ''G'' with multiplication defined pointwise.
Definition
In its most general form a loop group is a group of continuous mappings from a ...
of ''G''. For this special case showed that the Verlinde algebra can be identified with twisted equivariant
K-theory of ''G''.
See also
*
Fusion rules
Notes
References
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*{{Citation , last1=Witten , first1=Edward , author1-link=Edward Witten , title=Geometry, topology, & physics , arxiv=hep-th/9312104 , publisher=Int. Press, Cambridge, MA , series=Conf. Proc. Lecture Notes Geom. Topology, IV , mr=1358625 , year=1995 , chapter=The Verlinde algebra and the cohomology of the Grassmannian , pages=357–422, bibcode=1993hep.th...12104W
MathOverflow discussionwith a number of references.
Representation theory
Conformal field theory