We introduce a web of strongly correlated interacting 3+1D topological superconductors/ insulators
of 10 particular global symmetry groups of Cartan classes, realizable in electronic condensed matter systems, and
their generalizations. The symmetries include SU(N), SU(2), U(1), fermion parity, time reversal and
relate to each other through symmetry embeddings. We overview the lattice Hamiltonian formalism.
We complete the list of bulk symmetry-protected topological invariants
(SPT invariants/partition functions that exhibit boundary 't Hooft
anomalies) via cobordism calculations, matching their full classification.
We also present explicit 4-manifolds that detect these SPTs.
On the other hand, once we dynamically gauge part of their global symmetries, we arrive in various new phases of $SU(N)$ Yang-Mills (YM) theories,
realizable as quantum spin liquids with emergent gauge fields.
We discuss how coupling YM theories to time reversal-SPTs affects the strongly coupled theories at low energy.
For example, we point out
a possibility of having two deconfined gapless time-reversal symmetric
$SU(2)$ YM theories at $\theta=\pi$ as two distinct conformal field theories,
which although are secretly indistinguishable by gapped SPT states nor by correlators of local operators on oriented spacetimes, can be distinguished on non-orientable spacetimes or potentially by correlators of extended operators.
We develop a local cohomology theory for FI^m-modules, and show that it in many ways mimics the classical theory for multi-graded modules over a polynomial ring. In particular, we define an invariant of FI^m-modules using this local cohomology theory which closely resembles an invariant of multi-graded modules over Cox rings defined by Maclagan and Smith. It is then shown that this invariant behaves almost identically to the invariant of Maclagan and Smith.