Topological Quantum Field Theories (TQFTs)
pertinent to some emergent low energy phenomena of condensed matter lattice models
in 2+1 and 3+1 dimensions are explored.
Many of our TQFTs 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 TQFTs (namely the 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 analytically calculate both the Abelian and non-Abelian braiding statistics data of anyonic particle and string excitations in these theories, where
the statistics data can one-to-one characterize the underlying topological orders of TQFTs.
Namely, we derive path integral expectation values of links formed by line and surface operators in these TQFTs.
The acquired link invariants include not only the familiar Aharonov-Bohm linking number,
but also Milnor triple linking number in 3 dimensions,
triple and quadruple linking numbers of surfaces,
and intersection number of surfaces in 4 dimensions.
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 the Rokhlin invariant.
As an example, we can use these invariants and other physical 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 $\nu \in \mathbbm{Z}_8$ classes of 2+1 dimensional gauged
$\mathbb{Z}_2$-Ising-symmetric $\mathbb{Z}_2^f$-fermionic Topological Superconductors
(realized by stacking $\nu$ layers of a pair of chiral and anti-chiral $p$-wave superconductors [$p_x+ip_y$ and $p_x-ip_y$], where boundary supports non-chiral Majorana-Weyl modes)
with continuum spin-TQFTs.