Physical Meaning of $spin_{ \mathbb{C}}$ Structure

Question

I have been studying the $spin_{ \mathbb{C}}$ structure. Is that true that the $U(1)$ part represents the $U(1)$ gauge field in physics? If this is indeed the physical meaning of the $U(1)$ part, is there any generalization for $SU(2)$-gauge theory?

It turned out that for a spinor field interacting with gauge field, the mathematical description is called "twisted bundle", which is introduced in section 12.6.2 in Nakahara. Now I believe that the $U(1)$ part of $spin_{ \mathbb{C}}$ is not the actual ordinary $U(1)$ gauge field. In classical field theory, the spinor field should be grassmann number-valued. Are there any books explaining more details about grassmann number-valued spinor bundles?

Answer 1

A possible answer is given at the following post

https://physicsoverflow.org/33037/physical-derivation-applications-mayer-integrality-theorem

In such post the $spin_{ \mathbb{C}}$ structure is generalized to 

In such post some physical examples are presented.

Comments

Thank you.

Answer 2

The $U(1)$ part of $Spin^c$ is the electromagnetic field. The point is that for fermionic wavefunctions to be well-defined, around every contractible loop it has to be antiperiodic. Singular curves around which it's periodic instead can be removed by placing $\pi$ units of magnetic flux there.

Answer 3

Please consider the following theorem

According with such theorem, $w_{2}(X)$ is a topological obstruction or topological accident or topological wound  which does not permit the existence of spinors on $X$.  In order to heal the topological wound of $X$ a magnetic monopole  is introduced via the cohomological flux  $c$.  With such therapy, now $X$ admits the existence of spinors.·

Comments

Thank you very much for your answer. Could you also leave a comment about the grassmann number nature of classical spinors? How to construct a grassmann number-valued spinor bundle?

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Please tell me the reference textbook of the theorem.

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@New Student,  please look at

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.49.8926&rep=rep1&type=pdf

on page 15.