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
A two-dimensional conformal field theory is a quantum field theory on a Euclidean two-dimensional space, that is invariant under local conformal transformations.
In contrast to other types of conformal field theories, two-dimensional conformal fie ...
, 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
In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural gene ...
, 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
In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, ...
of ''G''.
See also
*
Fusion rules
In mathematics and theoretical physics, fusion rules are rules that determine the exact decomposition of the tensor product of two representations of a group into a direct sum of irreducible representations. The term is often used in the context of ...
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