We propose a many-body index that extends Fredholm index theory to many-body systems. The index is defined for any gapped systems with topologically ordered, possibly, degenerate ground states and requires $U(1)$-gauge symmetry. It is fractional with the denominator given by the ground state degeneracy. In the case that this fraction is not an integer, the construction of the index provides a non-trivial braiding and gives an explicit construction of unitary operators that create fractionally charged Abelian anyons.
Nguyen-Viet Dang