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Verlinde algebra

Algebra used in certain conformal field theories From Wikipedia, the free encyclopedia

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In mathematics, a Verlinde algebra is a finite-dimensional associative algebra introduced by Erik Verlinde (1988). It is defined to have basis of elements φλ corresponding to primary fields of a rational two-dimensional conformal field theory, whose structure constants Nν
λμ
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 corresponding to isomorphism classes of simple obejcts and whose structure constants describe the fusion of simple objects.

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Verlinde formula

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In terms of the modular S-matrix for modular tensor categories, the Verlinde formula is stated as follows.[1]Given any simple objects in a modular tensor category, the Verlinde formula relates the fusion coefficient in terms of a sum of products of -matrix entries and entries of the inverse of the -matrix, normalized by quantum dimensions.

Thumb
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[2]

where is the component-wise complex conjugate of .

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 is equal to the quantum dimension of .

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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.

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