  We propose and prove the Verlinde formula for the quantization of the Higgs
bundle moduli spaces and stacks for any simple and simply-connected group. This
generalizes the equivariant Verlinde formula for the case of $SU(n)$ proposed
previously by the second and third author. We further establish a Verlinde
formula for the quantization of parabolic Higgs bundle moduli spaces and
stacks. Finally, we prove that these dimensions form a one-parameter family of
$1+1$-dimensional TQFT, uniquely classified by the complex Verlinde algebra,
which is a one-parameter family of Frobenius algebras. We construct this
one-parameter family of Frobenius algebras as a deformation of the classical
Verlinde algebra for $G$.
