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't Hooft twisted torus construction and its relation to characteristic (e.g. Stiefel-Whitney) class

+ 3 like - 0 dislike
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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 $$\int w_2.$$ 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 $$\int (w_1)^2$$ where $w_1$ is the first Stiefel-Whitney class. There is a relation between $w_1$ and $w_2$ 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 
$$\int u_2$$
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:

1) What are the References for 't Hooft twisted torus construction?

2) What is this $u_2$ class? Is this certain math characteristic class?

3) 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?

asked Jun 9 in Theoretical Physics by wonderich (1,400 points) [ revision history ]

2 Answers

+ 4 like - 0 dislike

As you may know, all (ordinary) characteristic classes come from the cohomology of classifying spaces, eg. BSO(3) and BPSU(n). Sometimes it's easy to interpret these characteristic classes. For instance, $H^2$ always has to do with group extensions, and we have the two important group extensions $Spin(3) \to SO(3)$ corresponding to $w_2$ and $SU(n) \to PSU(n)$ corresponding to $u_2$. As explained in Milnor-Stasheff, the classifying spaces for classical Lie groups can be thought of as infinite Grassmannians, and in the end all these cocycles have to do with certain families of matrices of fixed rank. Thus one gets characterizations of, eg. $w_k$ of a rank $n$ bundle as the locus where $n-k+1$ generic sections have a linear dependence. One has an analogous characterization of the $u_k$'s for projective complex bundles with a volume form (the S in PSU(n)).

By the way, what is 't Hooft's twisted torus construction?

answered Jun 17 by Ryan Thorngren (1,605 points) [ revision history ]
+ 3 like - 0 dislike

I think that the so called Wu classes are relevant for this question.   Wu classes are polynomials in the Stiefel-Whitney classes :

$$v_{1} = w_{1}$$

$$v_{2} =  w_{2} + w_{1} ^{2}$$

At general, the total Stiefel-Whitney class $w$ is the total Steenrod square of the total Wu class $v$:

$$w = Sq (v) $$.

It is well known that $v_{1} =w_{1}$ is the obstruction to manifold orientation; and $v_{2}$  is the obstruction to having a $Pin^{−}$ structure.  The $Pin^{−}$ structure has many interesting applications to low-dimensional topology which are relevant for topological phases of matter.

More details at 

https://ncatlab.org/nlab/show/Wu+class

https://arxiv.org/pdf/1109.4461.pdf

answered Jun 12 by juancho (860 points) [ revision history ]
edited Jun 16 by juancho

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