# Interpretation of inequivalent spin structures (on the circle)

I was wondering about the physical interpretation of inequivalent spin structures on a given configuration space. For simplicity, I'd be satisified by only discussing the case of the circle.

There are two non-isomorphic $\mathrm{Spin}(1)$-principal bundles on the circle, they are simply given by two-sheeted coverings of the circle; one is connected, the other is not.

They give rise to two (non-isomorphic) spinor bundles, whose sections (i.e., the spinors) are:

(connected case): $2\pi$-anti-periodic maps of the real line with values in $\mathbb{C}^n$ (anti-periodic means $f(x+2\pi)=-f(x)$));

(non-connected case): $2\pi$-periodic maps of the real line (i.e., maps of the circle) with values in $\mathbb{C}^n$.

Above, $n$ depends on the spin quantum number of the particle, e.g., $n=2$ for an electron. Edit: after some thinking, I'm no longer sure about the meaning of $n$; anyhow we can safely assume $n=1$ for the discussion.

Question: in this case, how do we interpret these two classes of spinors? Can it be that only one is "physical"? Depending on the situation at hand, can one propose physical considerations in favor of one or the other class?

It might be relevant (?) that the connected case arises considering the unique spin structure of the disk at its boundary.

• The physics words for these inequivalent spin structures are 'the "Neveau-Schwartz" and "Ramond" sectors of the theory'. Your precise question is explained on page 40 (page 50 of the pdf) of the 2004 Clay Monograph on Dbranes. Let me know if you have any remaining questions.
– zzz
Jan 15, 2017 at 2:35
• @zzz Afer reading your link, I'm still wondering, if we have a space with more than one inequivalent spin structure then something like the Dirac lagrangian would have as many "copies" of itself as there were inequivalent structures (different spinor fields or cross sections of the different bundles)??? Aug 4, 2020 at 6:25