# Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions

Topological Quantum Field Theories (TQFTs) pertinent to some emergent low energy phenomena of condensed matter lattice models in 2+1 and 3+1D are explored. Many of our field theories are highly-interacting without free quadratic analogs. Some of our bosonic TQFTs can be regarded as the continuum field theory formulation of Dijkgraaf-Witten twisted discrete gauge theories. Other bosonic TQFTs beyond the Dijkgraaf-Witten description and all fermionic spin TQFTs are either higher-form gauge theories where particles must have strings attached, or fermionic discrete gauge theories obtained by gauging the fermionic Symmetry-Protected Topological states (SPTs). We calculate both Abelian and non-Abelian braiding statistics data of anyon particle and string excitations, where the statistics data can one-to-one characterize the underlying topological orders of TQFTs. We calculate path integral expectation values of links formed by line and surface operators in the TQFTs. The acquired link invariants include not only the Aharonov-Bohm linking number, but also Milnor triple linking number in 2+1D, triple and quadruple linking numbers of surfaces, and intersection number of surfaces in 3+1D. We also construct new spin TQFTs with the corresponding knot/link invariants of Arf(-Brown-Kervaire), Sato-Levine and others. We propose a new relation between the fermionic SPT partition function and Rokhlin invariant. We can use these invariants and other observables, including ground state degeneracy, reduced modular $\mathcal{S}^{xy}$ and $\mathcal{T}^{xy}$ matrices, and the partition function on $\mathbb{RP}^3$ manifold, to identify all $\mathbb{Z}_8$ classes of 2+1D gauged $\mathbb{Z}_2$-Ising-symmetric $\mathbb{Z}_2^f$-fermionic Topological Superconductors (TSC, realized by stacking layers of a pair of $p+ip$ and $p-ip$ SC, where boundary supports non-chiral Majorana-Weyl modes) with continuum spin-TQFTs.

**Comments:**36 pages, 8 figures, 3 tables. On 2+1D and 3+1D TQFTs / spin TQFTs. v2: Misprints corrected, clarifications and Refs added

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