  We propose a many-body index that extends Fredholm index theory to many-body
systems. The index is defined for any charge-conserving system with a
topologically ordered $p$-dimensional ground state sector. The index is
fractional with the denominator given by $p$. In particular, this yields a new
short proof of the quantization of the Hall conductance and of
Lieb-Schulz-Mattis theorem. In the case that the index is non-integer, the
argument provides an explicit construction of Wilson loop operators exhibiting
a non-trivial braiding and that can be used to create fractionally charged
Abelian anyons.
