nLab Verlinde formula

Idea

Generally, a Verlinde formula gives the dimension of a space of states of Chern-Simons theory for a given gauge group $G$. 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).

Related concepts

• fusion ring.

References

Original article:

• Erik Verlinde, Fusion rules and modular transformations in 2D conformal field theory, Nuclear Physics B 300 (1988) 360-376 $[$doi:10.1016/0550-3213(88)90603-7$]$

Proofs of the Verlinde formula:

• A. Tsuchiya, Kenji Ueno, 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:

• Martin Schottenloher, Mathematical Aspects of the Verlinde Formula, in: A Mathematical Introduction to Conformal Field Theory, Lecture Notes in Physics 759, Springer (2008) $[$doi:10.1007/978-3-540-68628-6, web, pdf$]$

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.

• Ralph Blumenhagen, Erik Plauschinn, §4.3 of: Introduction to Conformal Field Theory – With Applications to String Theory, Lecture Notes in Physics 779, Springer (2009) [doi:10.1007/978-3-642-00450-6]

Introduction:

• 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 of the Verlinde formula to logarithmic 2d CFTs:

• David Ridout, Simon Wood, The Verlinde formula in logarithmic CFT, J. Phys.: Conf. Ser. 597 012065 $[$arXiv:1409.0670, doi:10.1088/1742-6596/597/1/012065$]$

• Thomas Creutzig, Terry Gannon, Logarithmic conformal field theory, log-modular tensor categories and modular forms, J. Phys. A 50, 404004 (2017) (doi:10.1088/1751-8121/aa8538, arXiv:1605.04630)

• Thomas Creutzig: Resolving Verlinde’s formula of logarithmic CFT [arXiv:2411.11383]