Confusions in interpreting Dirac strings (wormholes) in Haldane's Hilbert space picture ft. two tori joined by strings at gapless points

Question

I had a question about Haldane's wormhole interpretation (picture below). I believe he first proposed it in his paper Berry Curvature on the Fermi Surface: Anomalous Hall Effect as a Topological Fermi-Liquid Property. To my understanding, this is for 3D band structures, where a gapless point initially gives one Dirac string, which then immediately splits to two Dirac points and creates a Berry flux loop.

My question is, let's say we have a 2D model (instead of 3D) - such as gapped graphene or the Haldane model with a K and K' point - with a band touching ONLY at K'. Am I correct in saying that:

1. the Hilbert spaces of the two tori (one torus per band) have a wormhole/string there as well (since the gapless point is a source of Berry flux, where the Berry curvature is a dirac delta function)? i.e. this isn't exclusively a 3D interpretation?
2. that instead of having a pair of wormholes (Dirac point splitting), we have only one source of Berry curvature flux (i.e. only one Dirac string connecting points of each torus corresponding to the K' point)?

Or, does this picture fail in 2D?

Answer 1

I found the following slides helpful: https://physics.princeton.edu//~haldane/talks/dirac.pdf

Here are my answers:

1. Yes. From what I understood, the above is not necessarily a 3D effect. The wormholes appear only at gapless points, so we can expect only one Dirac string in my example.
2. Partly correct. We have only one wormhole from the gapless point. However, there may be other non-singular sources of Berry curvature flux, where the curvature may be peaked at the other valley. I do not know the extent to which one may consider a non-singular region a source of flux vs not.

Someone please feel free to correct me if anything above is off. Thanks.