It is known that the $PSU(2) = SO(3)$ and there is an associated global anomaly labeled by the second Stiefel-Whitney class $w_2$.
This second Stiefel-Whitney class $w_2$ can detect the 1+1 dimensional Haldane phase (antiferromagnet spin-1 gapped state with SO(3) symmetry, called as symmetry protected topological "SPT" state). We can say the bulk probed action is $$\int w_2(V_{SO(3))},$$ the second Stiefel-Whitney class of the associated vector bundle $V$ of $SO(3)$.
On the other hand, if we regard the 1+1 dimensional Haldane phase protected by time reversal symmetry, then we can say the bulk probed action is $$\int (w_1(TM))^2,$$ where $w_1$ is the first Stiefel-Whitney class of the spacetime tangent bundle $TM$ of the spacetime manifold $M$.
There is a relation between $w_1$ and $w_2$ in lower dimensions to relate the two bulk topological invariants. (Could you prove this?)
More generally, we may consider the general $PSU(n)$, where the corresponding nontrivial bulk topological invariant in 1+1 dimensions can be written as $$\int u_2$$ where $u_2$ may be regarded as a general characteristic class such that when $n=2$, it becomes $u_2=w_2$.
The generalized second Stiefel-Whitney class seems to relate to the 't Hooft twisted torus construction.
question:
What are the References for 't Hooft twisted torus construction?
What is this $u_2$ class? Is this a certain math characteristic class?
Given $w_i$ is the $i$th Stiefel-Whitney class, do we have a $u_n$ is the certain math characteristic class generalizing $u_2$ class?
