It is known that the PSU(2)=SO(3) and there is an associated global anomaly labeled by the second Stiefel-Whitney class w2.
This second Stiefel-Whitney class w2 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 ∫w2.
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
∫(w1)2
where
w1 is the first Stiefel-Whitney class. There is a relation between
w1 and
w2 in lower dimensions to relate the two bulk topological invariant.
More generally, we may consider the general PSU(n), where the corresponding nontrivial bulk topological invariant in 1+1 dimensions can be written as
∫u2
where
u2 may be regarded as a general characteristic class such that when
n=2, it becomes
u2=w2.
The generalized second Stiefel-Whitney class seems to relate to the
't Hooft twisted torus construction.
question:
1) What are the References for 't Hooft twisted torus construction?
2) What is this u2 class? Is this certain math characteristic class?
3) Given wi is the ith Stiefel-Whitney class,
do we have a un is the certain math characteristic class generalizing u2 class?