Generally, a Verlinde formula gives the dimension of a space of states of Chern-Simons theory for a given gauge group . 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).
Mukai moduli space book
Dowker, On Verlinde’s formula for the dimensions of vector bundles on moduli spaces, iopscience