I've been watching Tachikawa's talk.

Recent advances in Supersymmetry

around 38 minutes he talks about objects in \(H^3(M,\mathbb{Z}_N)\) with the property that

\(\int_C a \in \mathbb{Z}_N\)

He says due to self duality in 6D dimensions two intersecting cycles C and C' with \(C \cap C' \neq 0\) are mutually non local.

It seems that mutually non-local means \(\int_M a \wedge b \neq 0\)

But why is this true, what happens when a theory is self dual?

What happens when you add supersymmetry?

He talks about the self dual 3 form what about other fields in the theory?

Is there a fermionic(or supersymmetric) generalization to \(H^3(M,\mathbb{Z}_N)\)?

What is the recipe to construct the partition vector?

Is there any detailed exposition of the subject ? What books discuss the object \(H^3(M,\mathbb{Z}_N)\)? I am familiar with the basic ideas of co-homology for U(1) gauge groups, but I have never really studied its generalization to other gauge groups.

Please excuse me if my questions are naive or too simple.

Edit: I have attached the related slides