  The moduli space $M$ of semi-stable rank 2 bundles with trivial determinant
over a complex curve carries involutions naturally associated to 2-torsion
points on the Jacobian of the curve. For every lift of a 2-torsion point to a
4-torsion point, we define a lift of the involution to the determinant line
bundle $\L$. We obtain an explicit presentation of the group generated by these
lifts in terms of the order 4 Weil pairing. This is related to the triple
intersections of the components of the fixed point sets in $M$, which we also
determine completely using the order 4 Weil pairing. The lifted involutions act
on the spaces of holomorphic sections of powers of $\L$, whose dimensions are
given by the Verlinde formula. We compute the characters of these vector spaces
as representations of the group generated by our lifts, and we obtain an
explicit isomorphism (as group representations) with the
combinatorial-topological TQFT-vector spaces of [BHMV]. As an application, we
describe a `brick decomposition', with explicit dimension formulas, of the
Verlinde vector spaces. We also obtain similar results in the twisted (i.e.,
degree one) case.
