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We propose a many-body index that extends Fredholm index theory to many-bodysystems. The index is defined for any charge-conserving system with atopologically ordered $p$-dimensional ground state sector. The index isfractional with the denominator given by $p$. In particular, this yields a newshort proof of the quantization of the Hall conductance and ofLieb-Schulz-Mattis theorem. In the case that the index is non-integer, theargument provides an explicit construction of Wilson loop operators exhibitinga non-trivial braiding and that can be used to create fractionally chargedAbelian anyons.

A many-body Fredholm index for ground state spaces and Abelian anyons