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We propose a way -- universal wave function overlap -- to extract universaltopological data from generic ground states of gapped systems in anydimensions. Those extracted topological data should fully characterize thetopological orders with gapped or gapless boundary. For non-chiral topologicalorders in 2+1D, this universal topological data consist of two matrices, $S$and $T$, which generate a projective representation of $SL(2,\mathbb Z)$ on thedegenerate ground state Hilbert space on a torus. For topological orders withgapped boundary in higher dimensions, this data constitutes a projectiverepresentation of the mapping class group $MCG(M^d)$ of closed spatial manifold$M^d$. For a set of simple models and perturbations in two dimensions, we showthat these quantities are protected to all orders in perturbation theory.

Universal Wave Function Overlap and Universal Topological Data from Generic Gapped Ground States