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Verlinde algebra

{{Short description|Algebra used in certain conformal field theories}}

In mathematics, a Verlinde algebra is a finite-dimensional associative algebra introduced by {{harvs|txt|first=Erik|last=Verlinde|authorlink=Erik Verlinde|year=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 N{{su|p=ν|b=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 [A] corresponding to isomorphism classes of simple obejcts and whose structure constants N^{A,B}_{C} 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.{{Cite book |last1=Bakalov |first1=Bojko |title=Lectures on Tensor Categories and Modular Functors |last2=Kirillov |first2=Alexander |date=2000-11-20 |publisher=American Mathematical Society |isbn=978-0-8218-2686-7 |series=University Lecture Series |volume=21 |location=Providence, Rhode Island |language=en |doi=10.1090/ulect/021 |s2cid=52201867}}Given any simple objects A,B,C\in\mathcal{C} in a modular tensor category, the Verlinde formula relates the fusion coefficient N^{A,B}_{C} in terms of a sum of products of S-matrix entries and entries of the inverse of the S-matrix, normalized by quantum dimensions.

File:Verlinde-formula.png.]]

In terms of the modular S-matrix for conformal field theory, Verlinde formula expresses the fusion coefficients as{{Cite book |last=Blumenhagen |first=Ralph |url=https://archive.org/details/introductiontoco00blum_953 |title=Introduction to Conformal Field Theory |date=2009 |publisher=Springer |others=Plauschinn, Erik |isbn=9783642004490 |location=Dordrecht |pages=[https://archive.org/details/introductiontoco00blum_953/page/n150 143] |oclc=437345787 |url-access=limited}}

: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 {{harvtxt|Freed|Hopkins|Teleman|2001}} showed that the Verlinde algebra can be identified with twisted equivariant K-theory of G.

See also

Notes

{{Reflist}}

References

  • {{Citation | last1=Beauville | first1=Arnaud | editor1-last=Teicher | editor1-first=Mina|editor-link= Mina Teicher | title=Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993) | chapter-url=http://math1.unice.fr/~beauvill/pubs/Hirz65.pdf | publisher=Bar-Ilan Univ. | location=Ramat Gan | series=Israel Math. Conf. Proc. | mr=1360497 | year=1996 | volume=9 | chapter=Conformal blocks, fusion rules and the Verlinde formula | arxiv=alg-geom/9405001|pages=75–96}}
  • {{Citation | last1=Bott | first1=Raoul | author1-link=Raoul Bott | title=On E. Verlinde's formula in the context of stable bundles | doi=10.1142/S0217751X91001404 | mr=1117752 | year=1991 | journal=International Journal of Modern Physics A | issn=0217-751X | volume=6 | issue=16 | pages=2847–2858|bibcode = 1991IJMPA...6.2847B }}
  • {{Citation | last1=Faltings | first1=Gerd | title=A proof for the Verlinde formula | mr=1257326 | year=1994 | journal=Journal of Algebraic Geometry | issn=1056-3911 | volume=3 | issue=2 | pages=347–374}}
  • {{Citation | last1=Freed | first1=Daniel S. |first2=M. |last2=Hopkins |first3=C. |last3=Teleman| title=The Verlinde algebra is twisted equivariant K-theory | url=http://mistug.tubitak.gov.tr/bdyim/abs.php?dergi=mat&rak=0103-10 | mr=1829086 | year=2001 | journal=Turkish Journal of Mathematics | issn=1300-0098 | volume=25 | issue=1 | pages=159–167| arxiv=math/0101038 | bibcode=2001math......1038F }}
  • {{Citation | last1=Verlinde | first1=Erik | title=Fusion rules and modular transformations in 2D conformal field theory | doi=10.1016/0550-3213(88)90603-7 | mr=954762 | year=1988 | journal=Nuclear Physics B | issn=0550-3213 | volume=300 | issue=3 | pages=360–376|bibcode = 1988NuPhB.300..360V }}
  • {{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 }}
  • [https://mathoverflow.net/questions/151221/verlindes-formula MathOverflow discussion] with a number of references.

Category:Representation theory

Category:Conformal field theory