  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 derive 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.
