nLab Verlinde formula




Generally, a Verlinde formula gives the dimension of a space of states of Chern-Simons theory for a given gauge group GG. Depending on which one of various different algebraic means to expresses these spaces is used, the Verlinde formula equivalently computes the dimension of spaces of non-abelian theta functions, the dimension of objects in a modular tensor category and so forth.

There are also Verlinde formulas in algebraic geometry (proved by Faltings) and a related one in the theory of vertex operator algebras (proved only in very special cases).


Original article:

Proofs of the Verlinde formula:

  • A. Tsuchiya, Kenji Ueno, and Y. Yamada, Conformal field theory on the universal family of stable curves with gauge symmetry In: Conformal field theory and solvable lattice models Adv. Stud. Pure Math. 16 (1989), 297–372.

  • Gerd Faltings, A proof for the Verlinde formula, J. Alg. Geom. 3 (1994) 347–374

  • Yi-Zhi Huang, Vertex operator algebras, the Verlinde conjecture and modular tensor categories, Proc. Nat. Acad. Sci. 102 (2005) 5352-5356 [[arXiv:math/0412261, doi:10.1073/pnas.0409901102]]

Review in:

where it says (p. 213):

Several mathematicians have worked on the problem of proving the Verlinde formula, starting with [[TUY89]] and coming to a certain end with [[Fal94]]. These proofs are all quite difficult to understand.


  • Shigeru Mukai, An introduction to invariants and moduli, Cambridge Univ. Press 2003

See also:

  • Dowker, On Verlinde’s formula for the dimensions of vector bundles on moduli spaces, iopscience

  • Juergen Fuchs, Christoph Schweigert, A representation theoretic approach to the WZW Verlinde formula, 1997

A generalization to logarithmic 2d CFTs has been suggested in:

Last revised on December 17, 2022 at 12:52:20. See the history of this page for a list of all contributions to it.